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MATHEMATICS
COURSE OF STUDY
2013
CHAGRIN FALLS EXEMPTED VILLAGE SCHOOLS
400 East Washington Street
Chagrin Falls, Ohio 44022
THE MATHEMATICS COURSE OF STUDY
has been approved
by the
Chagrin Falls Board of Education
on
May 20, 2013
Resolution #13-032
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ACKNOWLEDGEMENTS
The development of the Mathematics Course of Study reflects the efforts ogf the Chagrin Falls Schools‘ teaching professionals. Developing and revising this
Course of Study entailed a commitment of time and cooperation of all members. Our Course of Study review process entailed researching national and state
standards, studying best practices in mathematics education, developing a scope and sequence of knowledge and skills required at each level, and writing and
revising this Course of Study. The dedication of the members of this Review Team is deeply appreciated. Special thanks are given to the following:
Julie Albrecht, Gurney Elementary
Rachel Gebler, Gurney Elementary
Ann Kehrier, Gurney Elementary
Brian Ritz, Gurney Elementary
Kelly Shanaberger, Gurney Elementary
Lisa Todaro, Gurney Elementary
Kim Tressler, Gurney Elementary
Shelly Zdolshek, Gurney Elementary
Abby Dippel, Chagrin Falls Intermediate School
Lisa Janson, Chagrin Falls Intermediate School
Polly Mitchell, Chagrin Falls Intermediate School
Sarah Read, Chagrin Falls Intermediate School
Mike Wujnovich, Chagrin Falls Intermediate School
Barb Cymanski, Math Department Chairperson, Grades 7-12, Chagrin Falls High School
David Arundel, Chagrin Falls Middle School
Geoff Brown, Chagrin Falls Middle School
Todd Thombs, Chagrin Falls Middle School
Jeff Decker, Chagrin Falls High School
Dan Kerul, Chagrin Falls High School
Josh Maas, Chagrin Falls High School
Carolyn Petite, Chagrin Falls High School
Mike Sweeney, Chagrin Falls High School
Chuck Murphy
Director of Curriculum and Instruction
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MATHEMATICS TABLE OF CONTENTS
Page
Chagrin Falls Exempted Village School District Philosophy ................................................................................................. 1
Organization of Mathematics Course of Study ....................................................................................................................... 2
Mathematics Philosophy ......................................................................................................................................................... 3
Kindergarten ........................................................................................................................................................................... 4
Grade 1.................................................................................................................................................................................. 14
Grade 2.................................................................................................................................................................................. 28
Grade 3.................................................................................................................................................................................. 40
Grade 4.................................................................................................................................................................................. 53
Grade 5.................................................................................................................................................................................. 70
Grade 6.................................................................................................................................................................................. 83
Grade 6 Advanced Applications ........................................................................................................................................... 99
Grade 7................................................................................................................................................................................ 131
Grade 7 Advanced Applications ......................................................................................................................................... 149
Grade 8 ............................................................................................................................................................................... 179
Algebra 1 Middle School .................................................................................................................................................... 201
iii
Algebra 1 High School........................................................................................................................................................ 239
Fundamentals of Algebra and Geometry ............................................................................................................................ 269
Geometry ............................................................................................................................................................................ 280
Honors Geometry ................................................................................................................................................................ 299
Topics in Mathematics ........................................................................................................................................................ 320
Algebra 2............................................................................................................................................................................. 331
Honors Algebra 2 ................................................................................................................................................................ 358
Pre-calculus ......................................................................................................................................................................... 387
Honors Pre-calculus ............................................................................................................................................................ 414
Probability and Statistics..................................................................................................................................................... 446
AP Statistics ........................................................................................................................................................................ 458
AP Calculus AB .................................................................................................................................................................. 464
AP Calculus BC .................................................................................................................................................................. 470
Evaluation ........................................................................................................................................................................... 480
iv
PHILOSOPHY OF THE CHAGRIN FALLS EXEMPTED VILLAGE SCHOOLS
Education is a lifelong process. We, the Chagrin Falls Board of Education, as the elected body being legally responsible for the public school portion of that
education, subscribe to the following philosophy:
We recognize an obligation to organize and administer the educational program in a manner compatible with a democratic society. We recognize that the unique
responsibility of the schools is to pass along a fund of knowledge; and in order to meet this goal, to provide enriched conditions which foster academic excellence
and individual growth. We recognize that all persons in our school community have rights and responsibilities that are inherent within that society. We believe
that our school system should educate toward responsibility and responsiveness.
We recognize that students differ in their physical, intellectual, emotional, and social growth and in the way they develop these aspects. We believe that the school
must play a primary role in the students' education and that educational goals grow out of the needs of individuals.
We believe that students bring together different abilities, talents, and backgrounds. Intellectual growth will flourish in an environment of trust, respect, and
teamwork. The school should strive to heighten the students' appreciation of the cultural and individual diversity within the human family.
We believe that students need the freedom to question and should be encouraged to use independent, reflective, and critical thinking. They should have the
opportunity to exercise independent judgment by making decisions about their education in addition to following standardized requirements.
We believe that students should understand their relationship to the community, the country, and to the world. Our schools should provide meaningful
opportunities for students to become familiar with, and react to, a segment of society outside of their own community.
We believe the education received in Chagrin Falls should fit the students' needs for supporting themselves in the future by preparing for further academic study
and vocational opportunities.
We believe objectives and procedures of all educational programs are dynamic rather than static and must change to meet new conditions in an ever-changing
world. We should consider changes in the educational program because of their demonstrated worth.
1
ORGANIZATION OF THE MATHEMATICS COURSE OF STUDY
The Mathematics Course of Study is based on the Common Core State Standards for mathematics and is modeled after Ohio‘s Model Curriculum. It is divided
into grade levels throughout the elementary level and courses at the secondary level. For each cluster of standards, in each grade (K-8) or Conceptual Category
(high school), the curriculum for mathematics will contain the following sections:
Content Elaborations: These sections will provide additional clarification and examples to aid in the understanding of the standards. To support shared
interpretations across states, content elaborations are being developed through multi-state partnerships organized by Common Core State Standards Organization
(CCSSO) and other national organizations.
Instructional Strategies and Resources: The instructional strategies and resource section is designed to be fluid and improving over time, through additional
research and input from the field. These sections will contain the following subsections:
• Instructional Strategies: descriptions of effective and promising strategies for engaging students in observation, exploration and problem solving targeted
to the concepts and skills in the cluster of standards.
Instructional Resources and Tools: models, manipulatives, tasks, online tools and other resources to help students learn the concepts and skills in the
standards. Many resources are drawn from the extensive collection at the Ohio Resource Center (ORC), www.ohiorc.org.
Common Misconceptions: descriptions of common misconceptions as well as strategies for overcoming them.
This is a comprehensive curriculum that will guide our mathematics teachers‘ planning, instruction, and assessment. It is critical to the success of our district that
we follow this curriculum and find ways to challenge our students even beyond it when appropriate. It is expected that teachers will differentiate when possible
and remediate when necessary.
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MATHEMATICS PHILOSOPHY
Mathematics is a group of sciences that focuses on quantities, magnitudes and forms. By using numbers and symbols students learn the highly interconnected
areas of number and operations, geometry, algebra, data / probability and measurement. Each year students study concepts within each of these areas every school
year with varying degrees of emphasis depending on the grade level and / or course. For example, the study of mathematics in the early grades deals heavily with
basic skills in number, operations and measurement. Students in middle and high school courses shift their emphases to algebra, geometry and statistics.
Success in mathematics involves understanding, computing, applying, using logic, and engaging. Students must understand and carry out mathematical procedures
fluently. Students should be engaged by mathematics to recognize the applicability of the content within our world. They must be able to formulate problems,
apply strategies to solve them, and justify solutions. The development of student understanding within the area of mathematics is crucial in preparing our students
to become effective, contributing members of the 21st century.
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MATH
KINDERGARTEN
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Counting and Cardinality
Know number names and the count sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0–20 (with 0 representing a count of no objects).
Instructional Strategies
Common Misconceptions
Provide opportunities for students to connect mathematical language and symbols to their
Some students might not see zero as a number. Ask students to write 0 and say
everyday lives. Help students see patterns, make connections and provide repeated
zero to represent the number of items left when all items have been taken away.
experiences that give students time and opportunities to develop understandings and increase
Avoid using the word none to represent this situation.
fluency. Encourage students to explain their reasoning by asking probing questions such as
―How do you know?‖
Understand that students will rote-count before they truly understand cardinality. Students
will demonstrate understanding of cardinality by counting on from a given number (0-100)
Over time, students construct meaning for a written number (symbol), and understand that it
represents a number (cardinality). Some ways to practice this connection is to use dot cards,
dominoes and number cubes to help create mental images of the numbers.
Students should study and write numbers 0 to 20 in this order: numbers 1 to 9, the number 0,
then numbers 10 to 20. Students need to know that 0 is the number items left after all items in
a set are taken away. Do not accept ―none‖ as the answer to ―How many items are left?‖ for
this situation.
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Counting and Cardinality
Count to tell the number of objects.
4. Understand the relationship between numbers and quantities; connect counting to cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with
one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the
order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
5. Count to answer ―how many?‖ questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered
configuration; given a number from 1–20, count out that many objects.
Instructional Strategies
Common Misconceptions
One of the first major concepts in a student‘s mathematical development is cardinality.
Some students might think that the count word used to tag an item is
Cardinality, knowing that the number word said tells the quantity you have and that the
permanently connected to that item. So when the item is used again for counting
number you end on when counting represents the entire amount counted. The big idea is that
and should be tagged with a different count word, the student uses the original
number means amount and, no matter how you arrange and rearrange the items, the amount is
count word. For example, a student counts four geometric figures: triangle,
the same. Until this concept is developed, counting is merely a routine procedure done when a
square, circle and rectangle with the count words: one, two, three, four. If these
number is needed. To determine if students have the cardinality rule, listen to their responses
items are rearranged as rectangle, triangle, circle and square and counted, the
when you discuss counting tasks with them. For example, ask, ―How many are here?‖ The
student says these count words: four, one, three, two.
student counts correctly and says that there are seven. Then ask, ―Are there seven?‖ Students
may count or hesitate if they have not developed cardinality. Students with cardinality may
emphasize the last count or explain that there are seven because they counted them. These
students can now use counting to find a matching set.
Frequent opportunities to use and discuss counting as a means of solving problems relevant to
kindergarteners is more beneficial than repeating the same routine day after day.
As students develop meaning for numerals, they also compare numerals to the quantities they
represent. The models that can represent numbers, such as dot cards and dominoes, become
tools for such comparisons. Students can concretely, pictorially or mentally look for
similarities and differences in the representations of numbers. They begin to ―see‖ the
relationship of one more, one less, two more and two less, thus landing on the concept that
successive numbers name quantities that are one larger. In order to encourage this idea,
children need discussion and reflection of pairs of numbers from 1 to 10. Activities that utilize
anchors of 5 and 10 are helpful in securing understanding of the relationships between
numbers. This flexibility with numbers will build students‘ ability to break numbers into parts.
Provide a variety of experiences in which students connect count words or number words to
the numerals that represent the quantities. Students will arrive at an understanding of a number
when they acquire cardinality and can connect a number with the numerals and the number
word for the quantity they all represent.
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Counting and Cardinality
Compare numbers.
6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and
counting strategies.
7. Compare two numbers between 1 and 10 presented as written numerals.
Instructional Strategies
As children develop meaning for numerals, they also compare these numerals to the
quantities represented and their number words. The modeling numbers with manipulatives
such as dot cards and five- and ten-frames become tools for such comparisons. Activities that
utilize anchors of 5 and 10 are helpful in securing understanding of the relationships between
numbers. This flexibility with numbers will greatly impact children‘s ability to break numbers
into parts.
Children demonstrate their understanding of the meaning of numbers when they can justify
why their answer represents a quantity just counted. This justification could merely be the
expression that the number said is the total because it was just counted, or a ―proof‖ by
demonstrating a one to-one match, by counting again or other similar means (concretely or
pictorially) that makes sense. An ultimate level of understanding is reached when children can
compare two numbers from 1 to10 represented as written numerals without counting.
Students need to explain their reasoning when they determine whether a number is greater
than, less than, or equal to another number. Teachers need to ask probing questions such as
―How do you know?‖ to elicit their thinking. For students, these comparisons increase in
difficulty, from greater than to less than to equal. It is easier for students to identify
differences than to find similarities.
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations,
expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a
drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a
drawing or equation.
5. Fluently add and subtract within 5.
Instructional Strategies
Common Misconceptions
Provide contextual situations for addition and subtraction that relate to the everyday lives of
Students may over-generalize the vocabulary in word problems and think that
kindergarteners. A variety of situations can be found in children‘s literature books. Students then
certain words indicate solution strategies that must be used to find an answer. They
model the addition and subtraction using a variety of representations such as drawings, sounds,
might think that the word more always means to add and the words take away or
acting out situations, verbal explanations and numerical expressions. Manipulatives, like two-color
left always means to subtract. When students use the words take away to refer to
counters, clothespins on hangers, connecting cubes and stickers can also be used for modeling these
subtraction and its symbol, teachers need to repeat students‘ ideas using the words
operations. Kindergarten students should see addition and subtraction equations written by the
minus or subtract. For example, students use addition to solve this Take from/Start
teacher. Although students might struggle at first, teachers should encourage them to try writing the
Unknown problem: Seth took the 8 stickers he no longer wanted and gave them to
equations. Students‘ writing of equations in Kindergarten is encouraged, but it is not required.
Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin
with?
Create written addition or subtraction problems with sums and differences less than or equal to 10
using the numbers 0 to 10 and Table 1 on page 88 of the Common Core State Standards (CCSS) for
Mathematics for guidance. It is important to use a problem context that is relevant to kindergarteners. If students progress from working with manipulatives to writing numerical
After the teacher reads the problem, students choose their own method to model the problem and find expressions and equations, they skip using pictorial thinking. Students will then be
a solution. Students discuss their solution strategies while the teacher represents the situation with an more likely to use finger counting and rote memorization for work with addition
equation written under the problem. The equation should be written by listing the numbers and
and subtraction. Counting forward builds to the concept of addition while counting
symbols for the unknown quantities in the order that follows the meaning of the situation. The
back leads to the concept of subtraction. However, counting is an inefficient
teacher and students should use the words equal and is the same as interchangeably.
strategy. Teachers need to provide instructional experiences so that students
progress from the concrete level, to the pictorial level, then to the abstract level
Have students decompose numbers less than or equal to 5 during a variety of experiences to promote
when learning mathematical concepts.
their fluency with sums and differences less than or equal to 5 that result from using the numbers 0 to
5. For example, ask students to use different models to decompose 5 and record their work with
drawings or equations. Next, have students decompose 6, 7, 8, 9, and 10 in a similar fashion. As they
come to understand the role and meaning of arithmetic operations in number systems, students gain
computational fluency, using efficient and accurate methods for computing.
The teacher can use back-mapping and scaffolding to teach students who show a need for more help
with counting. For instance, ask students to build a tower of 5 using 2 green and 3 blue linking cubes
while you discuss composing and decomposing 5. Have them identify and compare other ways to
make a tower of 5. Repeat the activity for towers of 7 and 9. Help students use counting as they
explore ways to compose 7 and 9.
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Work with numbers 11–19 to gain foundations for place value.
1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or
decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six,
seven, eight, or nine ones.
Instructional Strategies
Common Misconceptions
Kindergarteners need to understand the idea of a ten so they can develop the strategy of adding onto Students have difficulty with ten as a singular word that means 10 things. For
10 to add within 20 in Grade 1. Students need to construct their own base-ten ideas about quantities many students, the understanding that a group of 10 things can be replaced by a
and their symbols by connecting to counting by ones. They should use a variety of manipulatives to single object and they both represent 10 is confusing. Help students develop the
model and connect equivalent representations for the numbers 11 to19. For instance, to represent
sense of 10 by first using groupable materials then replacing the group with an
13, students can count by ones and show 13 beans. They can anchor to five and show one group of
object or representing 10. Watch for and address the issue of attaching words to
5 beans and 8 beans or anchor to ten and show one group of 10 beans and 3 beans. Students need to
materials and groups without knowing what they represent. If this
eventually see a ten as different from 10 ones.
misconception is not addressed early on it can cause additional issues when
working with numbers 11-19 and beyond.
After the students are familiar with counting up to 19 objects by ones, have them explore different
ways to group the objects that will make counting easier. Have them estimate before they count and
group. Discuss their groupings and lead students to conclude that grouping by ten is desirable. 10
ones make 1 ten makes students wonder how something that means a lot of things can be one thing.
They do not see that there are 10 single objects represented on the item for ten in pregrouped
materials, such as the rod in base-ten blocks. Students then attach words to materials and groups
without knowing what they represent. Eventually they need to see the rod as a ten that they did not
group themselves. Students need to first use groupable materials to represent numbers 11 to 19
because a group of ten such as a bundle of 10 straws or a cup of 10 beans makes more sense than a
ten in pregrouped materials.
Kindergarteners should use proportional base-ten models, where a group of ten is physically 10
times larger than the model for a one. Nonproportional models such as an abacus and money should
not be used at this grade level.
Students should impose their base-ten concepts on a model made from groupable and pregroupable
materials (see Resources/Tools). Students can transition from groupable to pregroupable materials
by leaving a group of ten intact to be reused as a pregrouped item. When using pregrouped
materials, students should reflect on the ten-to-one relationships in the materials, such as the
―tenness‖ of the rod in base-ten blocks. After many experiences with pregrouped materials, students
can use dots and a stick (one tally mark) to record singles and a ten.
Encourage students to use base-ten language to describe quantities between 11 and 19. At the
beginning, students do not need to use ones for the singles. Some of the base-ten language that is
acceptable for describing quantities such as18 includes one ten and eight, a bundle and eight, a rod
and 8 singles and ten and eight more. Write the horizontal equation 18 = 10 + 8 and connect it to
base-ten language. Encourage, but do not require, students to write equations to represent quantities.
9
MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Measurement and Data
Describe and compare measurable attributes.
1. Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
2. Directly compare two objects with a measureable attribute in common, to see which object has ―more of‖/ ―less of‖ the attribute, and describe the difference.
For example, directly compare the heights of two children and describe one child as taller/shorter.
Instructional Strategies
It is critical for students to be able to identify and describe measureable attributes of objects.
An object has different attributes that can be measured, like the height and weight of a can of
food. When students compare shapes directly, the attribute becomes the focus. For example,
when comparing the volume of two different boxes, ask students to discuss and justify their
answers to these questions: Which box will hold the most? Which box will hold least? Will
they hold the same amount? Students can decide to fill one box with dried beans then pour the
beans into the other box to determine the answers to these questions.
Have students work in pairs to compare their arm spans. As they stand back-to-back with
outstretched arms, compare the lengths of their spans, then determine who has the smallest
arm span. Ask students to explain their reasoning. Then ask students to suggest other
measureable attributes of their bodies that they could directly compare, such as their height or
the length of their feet.
Connect to other subject areas. For example, suppose that the students have been collecting
rocks for classroom observation and they wanted to know if they have collected typical or
unusual rocks. Ask students to discuss the measurable attributes of rocks. Lead them to first
comparing the weights of the rocks. Have the class chose a rock that seems to be a ―typical‖
rock. Provide the categories: Lighter Than Our Typical Rock and Heavier Than Our Typical
Rock. Students can take turns holding a different rock from the collection and directly
comparing its weight to the weight of the typical rock and placing it in the appropriate
category. Some rocks will be left over because they have about the same weight as the typical
rock. As a class, they count the number of rocks in each category and use these counts to
order the categories and discuss whether they collected ―typical‖ rocks.
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MATH
KINDERGARTEN
Domain
Measurement and Data
Cluster
Classify objects and count the number of objects in each category.
3. Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
Content
Standards
Instructional Strategies
Provide categories for students to use to sort a collection of objects. Each category can relate
to only one attribute, like Red and Not Red or Hexagon and Not Hexagon, and contain up to
10 objects. Students count how many objects are in each category and then order the
categories by the number of objects they contain.
Ask questions to initiate discussion about the attributes of shapes. Then have students sort a
collection of two-dimensional and three-dimensional shapes by their attributes. Provide
categories like Circles and Not Circles or Flat and Not Flat. Have students count the objects
in each category and order the categories by the number of objects they contain.
Have students infer the classification of objects by guessing the rule for a sort. First, the
teacher uses one attribute to sort objects into two loops or regions without labels. Then the
students determine how the objects were sorted, suggest labels for the two categories and
explain their reasoning.
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MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Geometry
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
1. Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in
front of, behind, and next to.
2. Correctly name shapes regardless of their orientations or overall size.
3. Identify shapes as two-dimensional (lying in a plane, ―flat‖) or three-dimensional (―solid‖).
Instructional Strategies
Common Misconceptions
Develop spatial sense by connecting geometric shapes to students‘ everyday lives. Initiate
Students many times use incorrect terminology when describing shapes. For
natural conversations about shapes in the environment. Have students identify and name two- example students may say a cube is a square or that a sphere is a circle. The use
and three-dimensional shapes in and outside of the classroom and describe their relative
of the two-dimensional shape that appears to be part of a three-dimensional shape
position.
to name the three-dimensional shape is a common misconception. Work with
students to help them understand that the two-dimensional shape is a part of the
Ask students to find rectangles in the classroom and describe the relative positions of the
object but it has a different name.
rectangles they see, e.g. This rectangle (a poster) is over the sphere (globe). Teachers can use
a digital camera to record these relationships.
Hide shapes around the room. Have students say where they found the shape using positional
words, e.g. I found a triangle UNDER the chair.
Have students create drawings involving shapes and positional words: Draw a window ON
the door or Draw an apple UNDER a tree. Some students may be able to follow two- or
three-step instructions to create their drawings.
Use a shape in different orientations and sizes along with non-examples of the shape so
students can learn to focus on defining attributes of the shape.
Manipulatives used for shape identification actually have three dimensions. However,
Kindergartners need to think of these shapes as two-dimensional or ―flat‖ and typical threedimensional shapes as ―solid.‖ Students will identify two-dimensional shapes that form
surfaces on three-dimensional objects. Students need to focus on noticing two and three
dimensions, not on the words two-dimensional and three-dimensional.
12
MATH
KINDERGARTEN
Domain
Cluster
Content
Standards
Geometry
Analyze, compare, create and compose shapes
4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities,
differences, parts (e.g., number of sides and vertices/―corners‖) and other attributes (e.g., having sides of equal length).
5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
6. Compose simple shapes to form larger shapes. For example, ―Can you join these two triangles with full sides touching to make a rectangle?‖
Instructional Strategies
Common Misconceptions
Use shapes collected from students to begin the investigation into basic properties and
One of the most common misconceptions in geometry is the belief that
characteristics of two- and three-dimensional shapes. Have students analyze and compare
orientation is tied to shape. A student may see the first of the figures below as a
each shape with other objects in the classroom and describe the similarities and differences
triangle, but claim to not know the name of the second.
between the shapes. Ask students to describe the shapes while the teacher records key
descriptive words in common student language. Students need to use the word flat to describe
two-dimensional shapes and the word solid to describe three-dimensional shapes.
Use the sides, faces and vertices of shapes to practice counting and reinforce the concept of
one-to-one correspondence.
The teacher and students orally describe and name the shapes found on a Shape Hunt.
Students draw a shape and build it using materials regularly kept in the classroom such as
construction paper, clay, wooden sticks or straws.
Students can use a variety of manipulatives and real-world objects to build larger shapes with
these and other smaller shapes: squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres. Kindergarteners can manipulate cardboard shapes, paper plates,
pattern blocks, tiles, canned food, and other common items.
Students need to have many experiences with shapes in different orientations.
For example, in the Just Two Triangles activity referenced above, ask students to
form larger triangles with the two triangles in different orientations.
Another misconception is confusing the name of a two-dimensional shape with a
related three-dimensional shape or the shape of its face. For example, students
might call a cube a square because the student sees the face of the cube.
Have students compose (build) a larger shape using only smaller shapes that have the same
size and shape. The sides of the smaller shapes should touch and there should be no gaps or
overlaps within the larger shape. For example, use one-inch squares to build a larger square
with no gaps or overlaps. Have students also use different shapes to form a larger shape
where the sides of the smaller shapes are touching and there are no gaps or overlaps. Ask
students to describe the larger shape and the shapes that formed it.
13
MATH
GRADE 1
14
MATH
GRADE 1
Domain
Operations and Algebraic Thinking
Cluster
Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing,
Content
with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Standards
Instructional Strategies
Instructional Resources/Tools
Provide opportunities for students to participate in shared problem-solving activities to solve
12 situations found in Table 1 of the Common Core State Standards (CCSS) for
word problems.
Mathematics
Collaborate in small groups to develop problem-solving strategies using a variety of models
such as drawings, words, and equations with symbols for the unknown numbers to find the
solutions. Additionally students need the opportunity to explain, write and reflect on their
problem-solving strategies. The situations for the addition and subtraction story problems
should involve sums and differences less than or equal to 20 using the numbers 0 to 20. They
need to align with the 12 situations found in Table 1 of the Common Core State Standards
(CCSS) for Mathematics.
Literature is a wonderful way to incorporate problem-solving in a context that young students
can understand. Many literature books that include mathematical ideas and concepts have
been written in recent years. For Grade 1, the incorporation of books that contain a problem
situation involving addition and subtraction with numbers 0 to 20 should be included in the
curriculum. Use the situations found in Table 1 of the CCSS for guidance in selecting
appropriate books. As the teacher reads the story, students use a variety of manipulatives,
drawings, or equations to model and find the solution to problems from the story
Literature books relating to concepts
Common Misconceptions
Many children misunderstand the meaning of the equal sign. The equal sign
means ―is the same as‖ but most primary students believe the equal sign tells you
that the ―answer is coming up‖ to the right of the equal sign. This misconception
is over-generalized by only seeing examples of number sentences with an
operation to the left of the equal sign and the answer on the right. First graders
need to see equations written multiple ways, for example
5 + 7 = 12 and 12 = 5 + 7.
15
MATH
GRADE 1
Domain
Operations and Algebraic Thinking
Cluster
Represent and solve problems involving addition and subtraction. (cont.)
2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with
Content
a symbol for the unknown number to represent the problem.
Standards
Instructional Strategies
Instructional Resources/Tools
Students need the opportunity of writing and solving story problems involving three addends
Common addition and subtraction situations Table 1 on page 88 in the Common
with a sum that is less than or equal to 20. For example, each student writes or draws a
Core State Standards (CCSS) for School for Mathematics illustrates 12 addition
problem in which three whole things are being combined. The students exchange their
and subtraction problem situations.
problems with other students, solving them individually and then discussing their models and
solution strategies. Now both students work together to solve each problem using a different
Literature books relating to concepts
strategy.
Common Misconceptions
A misconception that many students have is that it is valid to assume that a key
word or phrase in a problem suggests the same operation will be used every time.
For example, they might assume that the word left always means that subtraction
must be used to find a solution. Providing problems in which key words like this
are used to represent different operations is essential. For example, the use of the
word left in this problem does not indicate subtraction as a solution method: Seth
took the 8 stickers he no longer wanted and gave them to Anna. Now Seth has 11
stickers left. How many stickers did Seth have to begin with? Students need to
analyze word problems and avoid using key words to solve them.
16
MATH
GRADE 1
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Understand and apply properties of operations and the relationship between addition and subtraction.
3. Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (commutative property of
addition).To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (associative property of addition).
4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
Instructional Strategies
Instructional Resources/Tools
One focus in this cluster is for students to discover and apply the commutative and associative A variety of objects for modeling and solving addition and subtraction problems
properties as strategies for solving addition problems. Students do not need to learn the names
for these properties. It is important for students to share, discuss and compare their strategies
Ten Frame activities and games
as a class. The second focus is using the relationship between addition and subtraction as a
strategy to solve unknown-addend problems. Students naturally connect counting on to
Dominoes
solving subtraction problems. For the problem ―15 – 7 = ?‖ they think about the number they
have to add to 7 to get to 15. First graders should be working with sums and differences less
Dot Cards and Ten Frame activities at: http://tinyurl.com/ca3dz6j
than or equal to 20 using the numbers 0 to 20.
From the National Council of Teachers of Mathematics: How many left? This
Provide investigations that require students to identify and then apply a pattern or structure in lesson encourages the students to explore unknown-addend problems using the
mathematics. For example, pose a string of addition and subtraction problems involving the
set model and the game Guess How Many?
same three numbers chosen from the numbers 0 to 20, like 4 + 13 = 17 and 13 + 4 = 17.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L117
Students analyze number patterns and create conjectures or guesses. Have students choose
other combinations of three numbers and explore to see if the patterns work for all numbers 0
to 20. Students then share and discuss their reasoning. Be sure to highlight students‘ uses of
the commutative and associative properties and the relationship between addition and
Common Misconceptions
subtraction.
A common misconception is that the commutative property applies to
subtraction. After students have discovered and applied the commutative
Expand the student work to three or more addends to provide the opportunities to change the
property for addition, ask them to investigate whether this property works for
order and/or groupings to make tens. This will allow the connections between place-value
subtraction. Have students share and discuss their reasoning and guide them to
models and the properties of operations for addition to be seen. Understanding the
conclude that the commutative property does not apply to subtraction.
commutative and associative properties builds flexibility for computation and estimation, a
First graders might have informally encountered negative numbers in their lives,
key element of number sense.
so they think they can take away more than the number of items in a given set,
resulting in a negative number below zero. Provide many problems situations
Provide multiple opportunities for students to study the relationship between addition and
where students take away all objects from a set, e.g. 19 - 19 = 0 and focus on the
subtraction in a variety of ways, including games, modeling and real-world situations.
meaning of 0 objects and 0 as a number. Ask students to discuss whether they
Students need to understand that addition and subtraction are related, and that subtraction can can take away more objects than what they have.
be used to solve problems where the addend is unknown.
17
MATH
GRADE 1
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2
+ 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction
(e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent
6 + 6 + 1 = 12 + 1 = 13).
Instructional Strategies
Instructional Resources/Tools
Provide many experiences for students to construct strategies to solve the different
A variety of objects for counting
problem types illustrated in Table 1 in the Common Core State Standards on page 88.
These experiences should help students combine their procedural and conceptual
A variety of objects for modeling and solving addition and subtraction problems
understandings. Have students invent and refine their strategies for solving problems
involving sums and differences less than or equal to 20 using the numbers 0 to 20. Ask
Five and Ten Frame Activities:
them to explain and compare their strategies as a class.
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1-2.pdf
Provide multiple and varied experiences that will help students develop a strong sense of
numbers based on comprehension – not rules and procedures. Number sense is a blend of
comprehension of numbers and operations and fluency with numbers and operations.
Students gain computational fluency (using efficient and accurate methods for computing)
as they come to understand the role and meaning of arithmetic operations in number
systems.
Primary students come to understand addition and subtraction as they connect counting
and number sequence to these operations. Addition and subtraction also involve part to
whole relationships. Students‘ understanding that the whole is made up of parts is
connected to decomposing and composing numbers.
Common Misconceptions
Students ignore the need for regrouping when subtracting with numbers 0 to 20 and
think that they should always subtract a smaller number from a larger number. For
example, students solve 15 – 7 by subtracting 5 from 7 and 0 (0 tens) from 1 to get
12 as the incorrect answer. Students need to relate their understanding of placevalue concepts and grouping in tens and ones to their steps for subtraction. They
need to show these relationships for each step using mathematical drawings, tenframes or base-ten blocks so they can understand an efficient strategy for multi-digit
subtraction.
Provide numerous opportunities for students to use the counting on strategy for solving
addition and subtraction problems. For example, provide a ten frame showing 5 colored
dots in one row. Students add 3 dots of a different color to the next row and write 5 + 3.
Ask students to count on from 5 to find the total number of dots. Then have them add an
equal sign and the number eight to 5 + 3 to form the equation 5 + 3 = 8. Ask students to
verbally explain how counting on helps to add one part to another part to find a sum.
Discourage students from inventing a counting back strategy for subtraction because it is
difficult and leads to errors
18
MATH
GRADE 1
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Work with addition and subtraction equations.
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the
following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number
that makes the equation true in each of the equations 8 + ? = 11, 5 = – 3, 6 + 6 =
Instructional Strategies
Instructional Resources/Tools
Provide opportunities for students use objects of equal weight and a number balance to model A variety of objects that can be used for modeling and solving addition and
equations for sums and differences less than or equal to 20 using the numbers 0 to 20. Give
subtraction problems
students equations in a variety of forms that are true and false. Include equations that show
the identity property, commutative property of addition, and associative property of addition.
Number balances
Students need not use formal terms for these properties.
Five-frames and ten-frames
13 = 13 Identity Property
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1-2.pdf
8 + 5 = 5 + 8 Commutative Property for Addition
3 + 7 + 4 = 10 + 4 Associative Property for Addition
Double ten-frames
Ask students to determine whether the equations are true or false and to record their work
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM15.pdf
with drawings. Students then compare their answers as a class and discuss their reasoning.
Present equations recorded in a nontraditional way, like 13 = 16 – 3 and 9 + 4 = 18 – 5, then
ask, ―Is this true?‖. Have students decide if the equation is true or false. Then as a class,
students discuss their thinking that supports their answers.
Provide situations relevant to first graders for these problem types illustrated in Table 1 of the
Common Core State Standards (CCSS): Add to / Result Unknown, Take from / Start
Unknown, and Add to / Result Unknown. Demonstrate how students can use graphic
organizers such as the Math Mountain to help them think about problems.
The Math Mountain shows a sum with diagonal lines going down to connect with the two
addends, forming a triangular shape. It shows two known quantities and one unknown
quantity. Use various symbols, such as a square, to represent an unknown sum or addend in a
horizontal equation. For example, here is a Take from / Start Unknown problem situation
such as: Some markers were in a box. Matt took 3 markers to use. There are now 6 markers in
the box. How many markers were in the box before? The teacher draws a square to represent
the unknown sum and diagonal lines to the numbers 3 and 6. (Like triangle flash cards).
Common Misconceptions
Many students think that the equals sign means that an operation must be
performed on the numbers on the left and the result of this operation is written on
the right. They think that the equal sign is like an arrow that means becomes and
one number cannot be alone on the left. Students often ignore the equal sign in
equations that are written in a nontraditional way. For instance, students find the
incorrect value for the unknown in the equation 9 = Δ - 5 by thinking 9 – 5 = 4. It
is important to provide equations with a single number on the left as in 18 = 10 +
8. Showing pairs of equations such as 11 = 7 + 4 and 7 + 4 = 11 gives students
experiences with the meaning of the equal sign as is the same as and equations
with one number to the left.
Have students practice using the Math Mountain to organize their solutions to problems
involving sums and differences less than or equal to 20 with the numbers 0 to 20. Then ask
them to share their reactions to using the Math Mountain.
19
MATH
GRADE 1
Provide numerous experiences for students to compose and decompose numbers less than or
equal to 20 using a variety of manipulatives. Have them represent their work with drawings,
words, and numbers. Ask students to share their work and thinking with their classmates.
Then ask the class to identify similarities and differences in the students‘ representations.
Domain
Number and Operations in Base Ten
Cluster
Extend the counting sequence.
1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
In this grade, students build on their counting to 100 by ones and tens beginning with
Groupable models
numbers other than 1 as they learned in Kindergarten. Students can start counting at any
Linking cubes
number less than 120 and continue to 120. It is important for students to connect different
Plastic chain links
representations for the same quantity or number. Students use materials to count by ones and
tens to a build models that represent a number, then they connect this model to the number
Pregrouped models
word and its representation as a written numeral.
Base-Ten Blocks (skinnies and bits)
Students learn to use numerals to represent numbers by relating their place-value notation to
their models. They build on their experiences with numbers 0 to 20 in Kindergarten to create
models for 21 to 120 with groupable and pregroupable materials (see Resources/Tools).
Students represent the quantities shown in the models by placing numerals in labeled
hundreds, tens and ones columns. They eventually move to representing the numbers in
standard form, where the group of hundreds, tens, then singles shown in the model matches
the left-to-right order of digits in numbers.
Listen as students orally count to 120 and focus on their transitions between decades and the
century number. These transitions will be signaled by a 9 and require new rules to be used to
generate the next set of numbers. Students need to listen to their rhythm and pattern as they
orally count so they can develop a strong number word list.
Extend hundreds charts by attaching a blank hundreds charts and writing the numbers 101 to
120 in the spaces following the same pattern as in the hundreds chart. Students can use these
charts to connect the number symbols with their count words for numbers 1 to 120.
Post the number words in the classroom to help students read and write them.
20
MATH
GRADE 1
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Understand place value.
2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a ―ten.‖
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
Instructional Strategies
Instructional Resources/Tools
Essential skills for students to develop include making tens (composing) and breaking a
Groupable models
number into tens and ones (decomposing). Composing numbers by tens is foundational for
Linking cubes
representing numbers with numerals by writing the number of tens and the number of leftover
Plastic chain links
ones. Decomposing numbers by tens builds number sense and the awareness that the order of
the digits is important. Composing and decomposing numbers involves number relationships
Pregrouped materials
and promotes flexibility with mental computation.
Base-ten blocks
Five-frame and Ten-frame
The beginning concepts of place value are developed in Grade 1 with the understanding of
Place-value mat with ten-frames
ones and tens. The major concept is that putting ten ones together makes a ten and that there
Strips (ten connected squares) and squares (singles)
is a way to write that down so the same number is always understood. Students move from
counting by ones, to creating groups and ones, to tens and ones. It is essential at this grade for
students to see and use multiple representations of making tens using base-ten blocks, bundles Common Misconceptions
of tens and ones, and ten-frames. Making the connections among the representations, the
Often when students learn to use an aid (Pac Man, bird, alligator, etc.) for
numerals and the words are very important. Students need to connect these different
knowing which comparison sign (<, >, = ) to use, the students don‘t associate the
representations for the numbers 0 to 99.
real meaning and name with the sign. The use of the learning aids must be
Students need to move through a progression of representations to learn a concept. They start
with a concrete model, move to a pictorial or representational model, then an abstract model.
For example, ask students to place a handful of small objects in one region and a handful in
another region. Next have them draw a picture of the objects in each region. They can draw a
likeness of the objects or use a symbol for the objects in their drawing. Now they count the
physical objects or the objects in their drawings in each region and use numerals to represent
the two counts. They also say and write the number word. Now students can compare the two
numbers using an inequality symbol or an equal sign.
accompanied by the connection to the names: < Less Than, > Greater Than, and
= Equal To. More importantly, students need to begin to develop the
understanding of what it means for one number to be greater than another. In
Grade 1, it means that this number has more tens, or the same number of tens,
but with more ones, making it greater. Additionally, the symbols are shortcuts for
writing down this relationship. Finally, students need to begin to understand that
both inequality symbols (<, >) can create true statements about any two numbers
where one is greater/smaller than the other, (15 < 28 and 28 >15).
21
MATH
GRADE 1
|Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Use place value understanding and properties of operations to add and subtract.
4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or
drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a
written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is
necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and
strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used.
Instructional Strategies
Instructional Resources/Tools
Provide multiple and varied experiences that will help students develop a strong sense of
Groupable models
numbers based on comprehension – not rules and procedures. *Number sense is a blend of
Linking cubes
comprehension of numbers and operations and fluency with numbers and operations.
Plastic chain links
Students gain computational fluency (using efficient and accurate methods for computing) as
they come to understand the role and meaning of arithmetic operations in number systems.
Students should solve problems using concrete models and drawings to support and record
their solutions. It is important for them to share the reasoning that supports their solution
strategies with their classmates.
Students will usually move to using base-ten concepts, properties of operations, and the
relationship between addition and subtraction to invent mental and written strategies for
addition and subtraction. Help students share, explore, and record their invented strategies.
Recording the expressions and equations in the strategies horizontally encourages students to
think about the numbers and the quantities they represent. Encourage students to try the
mental and written strategies created by their classmates. Students eventually need to choose
efficient strategies to use to find accurate solutions.
Pregrouped materials
Base-Ten Blocks
Strips (ten connected squares) and squares (singles)
Five-frame and Ten-frame
Place-value mat with ten-frames
Hundreds chart and Blank hundreds chart
Students should use and connect different representations when they solve a problem. They
should start by building a concrete model to represent a problem. This will help them form a
mental picture of the model. Now students move to using pictures and drawings to represent
and solve the problem. If students skip the first step, building the concrete model, they might
use finger counting to solve the problem. Finger counting is an inefficient strategy for adding
within 100 and subtracting within multiples of 10 between10 and 90.
Have students connect a 0-99 chart or a 1-100 chart to their invented strategy for finding 10
more and 10 less than a given number. Ask them to record their strategy and explain their
reasoning.
22
MATH
GRADE 1
Domain
Cluster
Content
Standards
Measurement and Data
Measure lengths indirectly and by iterating length units.
1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.
2. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that
the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being
measured is spanned by a whole number of length units with no gaps or overlaps.
Instructional Strategies
Instructional Resources/Tools
The measure of an attribute is a count of how many units are needed to fill, cover or match the
Clothesline rope
attribute of the object being measured. Students need to understand what a unit of measure is and
Yarn
how it is used to find a measurement. They need to predict the measurement, find the
Toothpicks
measurement and then discuss the estimates, errors and the measuring process. It is important for
students to measure the same attribute of an object with differently sized units.
Straws
It is beneficial to use informal units for beginning measurement activities at all grade levels
because they allow students to focus on the attributes being measured. The numbers for the
measurements can be kept manageable by simply adjusting the size of the units. Experiences with
informal or nonstandard units promote the need for measuring with standard units.
Paper clips
Measurement units share the attribute being measured. Students need to use as many copies of the
length unit as necessary to match the length being measured. For instance, use large footprints
with the same size as length units. Place the footprints end to end, without gaps or overlaps, to
measure the length of a room to the nearest whole footprint. Use language that reflects the
approximate nature of measurement, such as the length of the room is about 19 footprints.
Students need to also measure the lengths of curves and other distances that are not straight lines.
A variety of common two- and three-dimensional objects
Students need to make their own measuring tools. For instance, they can place paper clips end to
end along a piece of cardboard, make marks at the endpoints of the clips and color in the spaces.
Students can now see that the spaces represent the unit of measure, not the marks or numbers on a
ruler. Eventually they write numbers in the center of the spaces. Encourage students not to use the
end of the ruler as a starting point. Compare and discuss two measurements of the same distance,
one found by using a ruler and one found by aligning the actual units end to end, as in a chain of
paper clips. Students should also measure lengths that are longer than a ruler.
Have students use reasoning to compare measurements indirectly, for example to order the lengths
of Objects A, B and C, examine then compare the lengths of Object A and Object B and the
lengths of Object B and Object C. The results of these two comparisons allow students to use
reasoning to determine how the length of Object A compares to the length of Object C. For
example, to order three objects by their lengths, reason that if Object A is smaller than Object B
and Object B is smaller than Object C, then Object A has to be smaller than Object C. The order
of objects by their length from smallest to largest would be Object A - Object B - Object C.
Connecting cubes
Cuisenaire rods
Strips of tagboard or cardboard
ORC # 4329 From the National Council of Teachers of Mathematics,
Illuminations: The Length of My Feet - This lesson focuses students‘
attention on the attributes of length and develops their knowledge of and
skill in using nonstandard units of measurement.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L124
ORC # 1485 From the American Association for the Advancement of
Science: Estimation and Measurement - In this lesson students will use
nonstandard units to estimate and measure distances.
http://sciencenetlinks.com/lessons/estimation-and-measurement/
Common Misconceptions
Some students may view the measurement process as a procedural counting
task. They might count the markings on a ruler rather than the spaces
between (the unit of measure). Students need numerous experiences
measuring lengths with student-made tapes or rulers with numbers in the
center of the spaces.
23
MATH
GRADE 1
Domain
Measurement and Data
Cluster
Tell and write time.
3. Tell and write time in hours and half hours using analog and digital clocks.
Content
Standards
Instructional Strategies
Students are likely to experience some difficulties learning about time. On an analog clock,
the little hand indicates approximate time to the nearest hour and the focus is on where it is
pointing. The big hand shows minutes before and after an hour and the focus is on distance
that it has gone around the clock or the distance yet to go for the hand to get back to the top. It
is easier for students to read times on digital clocks, but these do not relate times very well.
Instructional Resources/Tools
ORC # 4328 From the National Council of Teachers of Mathematics,
Illuminations: Grouchy Lessons of Time This lesson provides an introduction to
and practice with the concept of time and hours.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L126
Students need to experience a progression of activities for learning how to tell time. Begin by
using a one-handed clock to tell times in hour and half-hour intervals, then discuss what is
happening to the unseen big hand. Next use two real clocks, one with the minute hand
removed, and compare the hands on the clocks. Students can predict the position of the
missing big hand to the nearest hour or half-hour and check their prediction using the twohanded clock. They can also predict the display on a digital clock given a time on a one- or
two-handed analog clock and vice-versa.
Have students tell the time for events in their everyday lives to the nearest hour or half hour.
Make a variety of models for analog clocks. One model uses a strip of paper marked in half
hours. Connect the ends with tape to form the strip into a circle.
24
MATH
GRADE 1
Domain
Measurement and Data
Cluster
Represent and interpret data
4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each
Content
category, and how many more or less are in one category than in another.
Standards
Instructional Strategies
Instructional Resources/Tools
The measure of an attribute is a count of how many units are needed to fill, cover or match the
Clothesline rope
attribute of the object being measured. Students need to understand what a unit of measure is
Yarn
and how it is used to find a measurement. They need to predict the measurement, find the
Toothpicks
measurement and then discuss the estimates, errors and the measuring process. It is important
for students to measure the same attribute of an object with differently sized units.
Straws
It is beneficial to use informal units for beginning measurement activities at all grade levels
because they allow students to focus on the attributes being measured. The numbers for the
measurements can be kept manageable by simply adjusting the size of the units. Experiences
with informal or nonstandard units promote the need for measuring with standard units.
Paper clips
Measurement units share the attribute being measured. Students need to use as many copies of
the length unit as necessary to match the length being measured. For instance, use large
footprints with the same size as length units. Place the footprints end to end, without gaps or
overlaps, to measure the length of a room to the nearest whole footprint. Use language that
reflects the approximate nature of measurement, such as the length of the room is about 19
footprints. Students need to also measure the lengths of curves and other distances that are not
straight lines.
A variety of common two- and three-dimensional objects
Students need to make their own measuring tools. For instance, they can place paper clips end
to end along a piece of cardboard, make marks at the endpoints of the clips and color in the
spaces. Students can now see that the spaces represent the unit of measure, not the marks or
numbers on a ruler. Eventually they write numbers in the center of the spaces. Encourage
students not to use the end of the ruler as a starting point. Compare and discuss two
measurements of the same distance, one found by using a ruler and one found by aligning the
actual units end to end, as in a chain of paper clips. Students should also measure lengths that
are longer than a ruler.
Have students use reasoning to compare measurements indirectly, for example to order the
lengths of Objects A, B and C, examine then compare the lengths of Object A and Object B and
the lengths of Object B and Object C. The results of these two comparisons allow students to
use reasoning to determine how the length of Object A compares to the length of Object C. For
example, to order three objects by their lengths, reason that if Object A is smaller than Object B
and Object B is smaller than Object C, then Object A has to be smaller than Object C. The order
of objects by their length from smallest to largest would be Object A - Object B - Object C.
Connecting cubes
Cuisenaire rods
Strips of tagboard or cardboard
ORC # 4329 From the National Council of Teachers of Mathematics,
Illuminations: The Length of My Feet This lesson focuses students‘ attention
on the attributes of length and develops their knowledge of and skill in using
nonstandard units of measurement.
ORC # 1485 From the American Association for the Advancement of
Science: Estimation and Measurement In this lesson students will use
nonstandard units to estimate and measure distances.
Common Misconceptions
Some students may view the measurement process as a procedural counting
task. They might count the markings on a ruler rather than the spaces between
(the unit of measure). Students need numerous experiences measuring lengths
with student-made tapes or rulers with numbers in the center of the spaces.
25
MATH
GRADE 1
Domain
Cluster
Content
Standards
Geometry
Reason with shapes and their attributes
1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build
and draw shapes to possess defining attributes.
2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right
rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. 2.
Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right
rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of,
fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates
smaller shares.
Instructional Strategies
Instructional Resources/Tools
Students can easily form shapes on geoboards using colored rubber bands to represent the
Paper shapes
sides of a shape. Ask students to create a shape with four sides on their geoboard, and then
copy the shape on dot paper. Students can share and describe their shapes as a class while the
Pattern blocks
teacher records the different defining attributes mentioned by the students.
Color tiles
Pattern block pieces can be used to model defining attributes for shapes. Ask students to
create their own rule for sorting pattern blocks. Students take turns sharing their sorting rules
Isosceles right triangles cut from squares
with their classmates and showing examples that support their rule. The classmates then draw
a new shape that fits this same rule after it is shared.
Tangrams
Students can use a variety of manipulatives and real-world objects to build larger shapes. The
manipulatives can include paper shapes, pattern blocks, color tiles, triangles cut from squares
(isosceles right triangles), tangrams, canned food (right circular cylinders) and gift boxes
(cubes or right rectangular prisms).
Folding shapes made from paper enables students to physically feel the shape and form the
equal shares. Ask students to fold circles and rectangles first into halves and then into fourths.
They should observe and then discuss the change in the size of the parts.
Canned food (right circular cylinders)
Gift boxes (cubes and right rectangular prisms)
ORC # 1481 From the Math Forum: Introduction to fractions for primary
students - http://mathforum.org/varnelle/knum1.html http://mathforum.org/varnelle/knum2.html http://mathforum.org/varnelle/knum5.html
This four-lesson unit introduces young children to fractions. Students learn to
recognize equal parts of a whole as halves, thirds and fourths. van Hiele Puzzle
Common Misconceptions
Students may think that a square that has been rotated so that the sides form 45degree angles with the vertical diagonal is no longer a square but a diamond.
They need to have experiences with shapes in different orientations. For
26
MATH
GRADE 1
example, in the building-shapes strategy above, ask students to orient the smaller
shapes in different ways.
Some students may think that the size of the equal shares is directly related to the
number of equal shares. For example, they think that fourths are larger than
halves because there are four fourths in one whole and only two halves in one
whole. Students need to focus on the change in the size of the fractional parts as
recommended in the folding shapes strategy. The first activity in the unit
Introduction to Fractions for Primary Students (referenced above) includes a link,
Parts of a Whole, to an interactive manipulative. It allows students to divide a
circle into the number of equal parts that they choose. Students can easily see the
change in the size of the equal shares as they increase or decrease the number of
parts.
27
MATH
GRADE 2
28
MATH
GRADE 2
Domain
Operations and Algebraic Thinking
Cluster
Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking
Content
apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Standards
Instructional Strategies
Instructional Resources/Tools
Students now build on their work with one-step problems to solve two-step problems. Second Common Core State Standards for Mathematics: Common addition and
graders need to model and solve problems for all the situations shown in Table 1 on page 88
subtraction situations
in the Common Core State Standards and represent their solutions with equations. The
Table 1 on page 88 in the Common Core State Standards (CCSS) for School for
problems should involve sums and differences less than or equal to 100 using the numbers 0
Mathematics illustrates twelve addition and subtraction problem situations.
to 100. It is vital that students develop the habit of checking their answer to a problem to
determine if it makes sense for the situation and the questions being asked.
Ask students to write word problems for their classmates to solve. Start by giving students the
answer to a problem. Then tell students whether it is an addition or subtraction problem
situation. Also let them know that the sums and differences can be less than or equal to 100
using the numbers 0 to 100. For example, ask students to write an addition word problem for
their classmates to solve which requires adding four two-digit numbers with 100 as the
answer. Students then share, discuss and compare their solution strategies after they solve the
problems.
ORC # 4243 From the National Council of Teachers of Mathematics: Get the
Picture—Get the Story - In this lesson, students act as reporters at the Super
Bowl. Students study four pictures of things that they would typically find at a
football game then create problem situations that correspond to their
interpretation of each of the pictures.
Common Misconceptions
Some students end their solution to a two-step problem after they complete the
first step. They may have misunderstood the question or only focused on finding
an answer to a problem. Students need to check their work to see if their answer
makes sense in terms of the problem situation. They need ample opportunities to
solve a variety of two-step problems and develop the habit of reviewing their
solution after they think they have finished.
Many children have misconceptions about the equal sign. Students can
misunderstand the use of the equal sign even if they have proficient
computational skills. The equal sign means ―is the same as‖ but most primary
students think that the equal sign tells you that the ―answer is coming up.‖
Students might only see examples of number sentences with an operation to the
left of the equal sign and the answer on the right, so they overgeneralize from
those limited examples. They might also be predisposed to think of equality in
terms of calculating answers rather than as a relation because it is easier for
young children to carry out steps to find an answer than to identify relationships
among quantities.
Students might rely on a key word or phrase in a problem to suggest an operation
that will lead to an incorrect solution. For example, they might think that the
word left always means that subtraction must be used to find a solution. Students
29
MATH
GRADE 2
need to solve problems where key words are contrary to such thinking. For
example, the use of the word left in this problem does not indicate subtraction as
a solution method: Seth took the 8 stickers he no longer wanted and gave them to
Anna. Now Seth has 11 stickers left. How many stickers did Seth have to begin
with? It is important that students avoid using key words to solve problems.
Domain
Operations and Algebraic Thinking
Cluster
Add and subtract within 20
2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Provide many activities that will help students develop a strong understanding of number
Five-frames and ten-frames:
relationships, addition and subtraction so they can develop, share and use efficient strategies
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1-2.pdf
for mental computation. An efficient strategy is one that can be done mentally and quickly.
ORC # 4308 From the National Council of Teachers of Mathematics: Looking
Students gain computational fluency, using efficient and accurate methods for computing, as
back and moving forward:
they come to understand the role and meaning of arithmetic operations in number systems.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L43
Efficient mental processes become automatic with use.
In the game Race to Zero at the bottom of the page, students take turns rolling a
Provide activities in which students apply the commutative and associative properties to their
number cube and subtracting the number they rolled each time from 20. The first
mental strategies for sums less or equal to 20 using the numbers 0 to 20.
person to reach 0 wins the round.
Have students study how numbers are related to 5 and 10 so they can apply these
relationships to their strategies for knowing 5 + 4 or 8 + 3. Students might picture 5 + 4 on a
ten-frame to mentally see 9 as the answer. For remembering 8 + 7, students might think
―since 8 is 2 away from 10, take 2 away from 7 to make 10 + 5 = 15.‖
Provide simple word problems designed for students to invent and try a particular strategy as
they solve it. Have students explain their strategies so that their classmates can understand it.
Guide the discussion so that the focus is on the methods that are most useful. Encourage
students to try the strategies that were shared so they can eventually adopt efficient strategies
that work for them.
Make posters for student-developed mental strategies for addition and subtraction within 20.
Use names for the strategies that make sense to the students and include examples of the
strategies.
Present a particular strategy along with the specific addition and subtraction facts relevant to
the strategy. Have students use objects and drawings to explore how these facts are alike.
ORC # 4314 From the National Council of Teachers of Mathematics: Finding
fact families: http://illuminations.nctm.org/LessonDetail.aspx?ID=L57
In this lesson, the relationship of subtraction to addition is introduced with a
book and with dominoes.
Common Misconceptions
Students may over generalize the idea that answers to addition problems must be
bigger. Adding 0 to any number results in a sum that is equal to that number.
Provide word problems involving 0 and have students model them using
drawings with an empty space for 0.
Students are usually proficient when they focus on a strategy relevant to
particular facts. When these facts are mixed with others, students may revert to
counting as a strategy and ignore the efficient strategies they learned. Provide a
list of facts from two or more strategies and ask students to name a strategy that
would work for that fact. Students explain why they chose that strategy then
show how to use it.
30
MATH
GRADE 2
Domain
Operations and Algebraic Thinking
Cluster
Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to
Content
express an even number as a sum of two equal addends.
Standards
Instructional Strategies
Instructional Resources/Tools
Students need to understand that a collection of objects can be one thing (a group) and that a group
Five-frames and ten-frames:
contains a given number of objects. Investigate separating no more than 20 objects into two equal
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1groups. Find the numbers (the total number of objects in collections up to 20 members) that will
2.pdf
have some objects and no objects remaining after separating the collections into two equal groups.
Odd numbers will have some objects remaining while even numbers will not. For an even number
of objects in a collection, show the total as the sum of equal addends (repeated addition).
Domain
Operations and Algebraic Thinking
Cluster
Work with equal groups of objects to gain foundations for multiplication. (cont.)
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as
Content
a sum of equal addends.
Standards
Instructional Strategies
Instructional Resources/Tools
A rectangular array is an arrangement of objects in horizontal rows and vertical columns. Arrays
Grid paper:
can be made out of any number of objects that can be put into rows and columns. All rows contain
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM30the same number of items and all columns contain an equal number of items. Have students use
36.pdf
objects to build all the arrays possible with no more than 25 objects. Their arrays should have up to
5 rows and up to 5 columns. Ask students to draw the arrays on grid paper and write two different
Five-frames and ten-frames:
equations under the arrays: one showing the total as a sum by rows and the other showing the total
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1as a sum by columns. Both equations will show the total as a sum of equal addends.
2.pdf
The equation by rows: 20 = 5 + 5 + 5 +5
The equation by columns: 20 = 4 + 4 + 4 + 4 + 4
Tiles
Build on knowledge of composing and decomposing numbers to investigate arrays with up to 5
rows and up to 5 columns in different orientations. For example, form an array with 3 rows and 4
objects in each row. Represent the total number of objects with equations showing a sum of equal
addends two different ways: by rows, 12 = 4 + 4 + 4; by columns, 12 = 3 + 3 + 3 + 3. Rotate the
array 90° to form 4 rows with 3 objects in each row. Write two different equations to represent 12
as a sum of equal addends: by rows, 12 = 3 + 3 + 3 + 3; by columns, 12 = 4 + 4 + 4. Have students
discuss this statement and explain their reasoning: The two arrays are different and yet the same.
Linking cubes
Ask students to think of a full ten-frame showing 10 circles as an array. One view of the ten-frame
is 5 rows with 2 circles in each row. Students count by rows to 10 and write the equation 10 = 2 + 2
+ 2 + 2 + 2. Then students put two full ten-frames together end-to-end so they form 10 rows of 2
circles or 10 columns of 2 circles. They use this larger array to count by 2s up to 20 and write an
equation that shows 20 equal to the sum of ten 2s.
31
MATH
GRADE 2
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Understand place value
1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones.
Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a ―hundred.‖
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2. Count within 1000; skip-count by 5s, 10s, and 100s.
3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
Instructional Strategies
Instructional Resources/Tools
The understanding that 100 is 10 tens or 100 ones is critical to the understanding of
Base-ten blocks
place value. Using proportional models like base-ten blocks and bundles of tens along
with numerals on place-value mats provides connections between physical and
Pictures of nickels and dimes
symbolic representations of a number. These models can be used to compare two
numbers and identify the value of their digits.
Base-ten grid paper:
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM19.pdf
Model three-digit numbers using base-ten blocks in multiple ways. For example, 236
can be 236 ones, or 23 tens and 6 ones, or 2 hundreds, 3 tens and 6 ones, or 20 tens
Five-frames and Ten-frames:
and 36 ones. Use activities and games that have students match different
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1-2.pdf
representations of the same number.
Online resource for base-ten blocks:
Provide games and other situations that allow students to practice skip-counting.
http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=category_g_1_t_1.html
Students can use nickels, dimes and dollar bills to skip count by 5, 10 and 100.
Pictures of the coins and bills can be attached to models familiar to students: a nickel
Online resource for hundreds chart, use for counting by 5s and 10s:
on a five-frame with 5 dots or pennies and a dime on a ten-frame with 10 dots or
http://nlvm.usu.edu/en/nav/frames_asid_337_g_1_t_1.html?from=category_g_1_t_1.html
pennies.
Online place value number line:
On a number line, have students use a clothespin or marker to identify the number
http://nlvm.usu.edu/en/nav/frames_asid_334_g_1_t_1.html?from=category_g_1_t_1.html
that is ten more than a given number or five more than a given number.
Have students create and compare all the three-digit numbers that can be made using
Common Misconceptions
numbers from 0 to 9. For instance, using the numbers 1, 3, and 9, students will write
Some students may not move beyond thinking of the number 358 as 300 ones plus 50
the numbers 139, 193, 319, 391, 913 and 931. When students compare the numerals
ones plus 8 ones to the concept of 8 singles, 5 bundles of 10 singles or tens, and 3
in the hundreds place, they should conclude that the two numbers with 9 hundreds
bundles of 10 tens or hundreds. Use base-ten blocks to model the collecting of 10 ones
would be greater than the numbers showing 1 hundred or 3 hundreds. When two
(singles) to make a ten (a rod) or 10 tens to make a hundred (a flat). It is important that
numbers have the same digit in the hundreds place, students need to compare their
students connect a group of 10 ones with the word ten and a group of 10 tens with the
digits in the tens place to determine which number is larger.
word hundred.
32
MATH
GRADE 2
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Use place value understanding and properties of operations to add and subtract.
5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship
between addition and subtraction; relate the strategy to written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts
hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
9. Explain why addition and subtraction strategies work, using place value and the properties of operations.
Instructional Strategies
Instructional Resources/Tools
Provide many activities that will help students develop a strong understanding of number
Groupable materials
relationships, addition and subtraction so they can develop, share and use efficient strategies
Dried beans and small cups for groups of 10 beans
for mental computation. An efficient strategy is one that can be done mentally and quickly.
Students gain computational fluency, using efficient and accurate methods for computing, as
Linking cubes
they come to understand the role and meaning of arithmetic operations in number systems.
Efficient mental processes become automatic with use.
Plastic chain links
Students need to build on their flexible strategies for adding within 100 in Grade 1 to fluently
add and subtract within 100, add up to four two-digit numbers, and find sums and differences Pregrouped materials
less than or equal to 1000 using numbers 0 to 1000.
Base-ten blocks
Initially, students apply base-ten concepts and use direct modeling with physical objects or
drawings to find different ways to solve problems. They move to inventing strategies that do
not involve physical materials or counting by ones to solve problems. Student-invented
strategies likely will be based on place-value concepts, the commutative and associative
properties, and the relationship between addition and subtraction. These strategies should be
done mentally or with a written record for support.
Dried beans and beans sticks (10 dried beans glued on a craft stick – 10 sticks
can be bundled for 100)
It is vital that student-invented strategies be shared, explored, recorded and tried by others.
Recording the expressions and equations in the strategies horizontally encourages students to
think about the numbers and the quantities they represent instead of the digits. Not every
student will invent strategies, but all students can and will try strategies they have seen that
make sense to them. Different students will prefer different strategies.
Ten-frame
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1-2.pdf
Students will decompose and compose tens and hundreds when they develop their own
strategies for solving problems where regrouping is necessary. They might use the make-ten
strategy (37 + 8 = 40 + 5 = 45, add 3 to 37 then 5) or (62 - 9 = 60 – 7 = 53, take off 2 to get
60, then 7 more) because no ones are exchanged for a ten or a ten for ones.
Hundreds chart (numbers 1-100) and blank hundreds chart (add numbers 101120 and attach to hundreds chart)
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM9-1016.pdf
Strips (ten connected squares) and squares (singles)
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM1314.pdf
Place-value mat with ten-frames
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM15.pdf
33
MATH
GRADE 2
Have students analyze problems before they solve them. Present a variety of subtraction
problems within 1000. Ask students to identify the problems requiring them to decompose the
tens or hundreds to find a solution and explain their reasoning.
Common Misconceptions
Students may think that the 4 in 46 represents 4, not 40. Students need many
experiences representing two-and three-digit numbers with groupable then
pregrouped materials.
When adding two-digit numbers, some students might start with the digits in the
ones place and record the entire sum. Then they add the digits in the tens place
and record this sum. Assess students‘ understanding of a ten and provide more
experiences modeling addition with grouped and pregrouped base-ten materials.
When subtracting two-digit numbers, students might start with the digits in the
ones place and subtract the smaller digit from the larger digit. Then they move to
the tens and the hundreds places and subtract the smaller digits from the larger
digits. Assess students‘ understanding of a ten and provide more experiences
modeling subtraction with grouped and pregrouped base-ten materials.
34
MATH
GRADE 2
Domain
Cluster
Content
Standards
Measurement and Data
Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size
of the unit chosen.
3. Estimate lengths using units of inches, feet, centimeters, and meters.
4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.
Instructional Strategies
Instructional Resources/Tools
Second graders are transitioning from measuring lengths with informal or nonstandard units
Centimeter rulers and tapes
to measuring with these standard units: inches, feet, centimeters, and meters. The measure of
length is a count of how many units are needed to match the length of the object or distance
Inch rulers and tapes
being measured. Students have to understand what a length unit is and how it is used to find a
measurement. They need many experiences measuring lengths with appropriate tools so they
Yardsticks
can become very familiar with the standard units and estimate lengths. Use language that
reflects the approximate nature of measurement, such as the length of the room is about 26
Meter sticks
feet.
Have students measure the same length with different-sized units then discuss what they
noticed. Ask questions to guide the discussion so students will see the relationship between
the size of the units and measurement, i.e. the measurement made with the smaller unit is
more than the measurement made with the larger unit and vice versa.
Insist that students always estimate lengths before they measure. Estimation helps them focus
on the attribute to be measured, the length units, and the process. After they find
measurements, have students discuss the estimates, their procedures for finding the
measurements and the differences between their estimates and the measurements.
Common Misconceptions
When some students see standard rulers with numbers on the markings, they
believe that the numbers are counting the marks instead of the units or spaces
between the marks. Have students use informal or standard length units to make
their own rulers by marking each whole unit with a number in the middle. They
will see that the ruler is a representation of a row of units and focus on the
spaces.
Some students might think that they can only measure lengths with a ruler
starting at the left edge. Provide situations where the ruler does not start at zero.
For example, a ruler is broken and the first inch number that can be seen is 2. If a
pencil is measured and it is 9 inches on this ruler, the students must subtract 2
inches from the 9 inches to adjust for where the measurement started.
35
MATH
GRADE 2
Domain
Cluster
Content
Standards
Measurement and Data
Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings
of rulers) and equations with a symbol for the unknown number to represent the problem.
6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent
whole-number sums and differences within 100 on a number line diagram.
Instructional Strategies
Instructional Resources/Tools
Connect the whole-number units on rulers, yardsticks, meter sticks and measuring tapes to
Rulers
number lines showing whole-number units starting at 0. Use these measuring tools to model
different representations for whole-number sums and differences less than or equal to 100
Yardsticks
using the numbers 0 to 100.
Meter sticks
Use the meter stick to view units of ten (10 cm) and hundred (100 cm), and to skip count by
5s and 10s.
Measuring tapes
Provide one- and two-step word problems that include different lengths measurement made
with the same unit (inches, feet, centimeters, and meters). Students add and subtract within
100 to solve problems for these situations: adding to, taking from, putting together, taking
apart, and comparing, and with unknowns in all positions. Students use drawings and write
equations with a symbol for the unknown to solve the problems.
Have students represent their addition and subtraction within 100 on a number line. They can
use notebook or grid paper to make their own number lines. First they mark and label a line
on paper with whole-number units that are equally spaced and relevant to the addition or
subtraction problem. Then they show the addition or subtraction using curved lines segments
above the number line and between the numbers marked on the number line. For 49 + 5, they
start at 49 on the line and draw a curve to 50, then continue drawing curves to 54. Drawing
the curves or making the ―hops between the numbers will help students focus on a space as
the length of a unit and the sum or difference as a length.
Cash register tapes or paper strips
ORC # 3991 From the National Council of Teachers of Mathematics: Hopping
Backward to Solve Problems:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L76 - In this lesson,
students determine differences using the number line to compare lengths.
ORC # 3979 From the National Council of Teachers of Mathematics: Where
Will I Land? http://illuminations.nctm.org/LessonDetail.aspx?ID=L118 - In this
lesson, the students find differences using the number line, a continuous model
for subtraction.
36
MATH
GRADE 2
Domain
Cluster
Content
Standards
Measurement and Data
Work with time and money.
7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ￠ symbols appropriately. Example: If you have 2 dimes and 3
pennies, how many cents do you have?
Instructional Strategies
Instructional Resources/Tools
Second graders expand their work with telling time from analog and digital clocks to
Play money
the nearest hour or half-hour in Grade 1 to telling time to the nearest five minutes
Coin Box: http://illuminations.nctm.org/ActivityDetail.aspx?ID=217 - This game will
using a.m. and p.m.
help students learn how to count, collect, exchange and make change for coins.
The topic of money begins at Grade 2 and builds on the work in other clusters in this
and previous grades. Help students learn money concepts and solidify their
understanding of other topics by providing activities where students make
connections between them. For instance, link the value of a dollar bill as 100 cents to
the concept of 100 and counting within 1000. Use play money - nickels, dimes, and
dollar bills to skip count by 5s, 10s, and 100s. Reinforce place value concepts with
the values of dollar bills, dimes, and pennies.
Students use the context of money to find sums and differences less than or equal to
100 using the numbers 0 to 100. They add and subtract to solve one- and two-step
word problems involving money situations of adding to, taking from, putting
together, taking apart, and comparing, with unknowns in all positions. Students use
drawings and equations with a symbol for the unknown number to represent the
problem. The dollar sign, $, is used for labeling whole-dollar amounts without
decimals, such as $29.
Students need to learn the relationships between the values of a penny, nickel, dime,
quarter and dollar bill.
From the National Library of Virtual Manipulatives, Utah State University: Time –
Match Clocks:
http://nlvm.usu.edu/en/nav/frames_asid_317_g_1_t_4.html?from=category_g_1_t_4.html
Students manipulate a digital clock to show the time given on an analog clock. They can
also manipulate the hands on a face clock to show the time given on a digital clock.
Times are given to the nearest five minutes.
ORC # 1133 From the National Council of Teachers of Mathematics: Number Cents:
http://illuminations.nctm.org/LessonDetail.aspx?ID=U67 - In this unit, students explore
the relationship between pennies, nickels, dimes, and quarters. They count sets of mixed
coins, write story problems that involve money, and use coins to make patterns.
Common Misconceptions
Some students might confuse the hour and minutes hands. For the time of 3:45, they say
the time is 9:15. Also, some students name the numeral closest to the hands, regardless of
whether this is appropriate. For instance, for the time of 3:45 they say the time is 3:09 or
9:03. Assess students‘ understanding of the roles of the minute and hour hands and the
relationship between them. Provide opportunities for students to experience and measure
times to the nearest five minutes and the nearest hour. Have them focus on the movement
and features of the hands.
Students might overgeneralize the value of coins when they count them. They might
count them as individual objects.
Also some students think that the value of a coin is directly related to its size, so the
bigger the coin, the more it is worth. Place pictures of a nickel on the top of five-frames
that are filled with pictures of pennies. In like manner, attach pictures of dimes and
pennies to ten-frames and pictures of quarters to 5 x 5 grids filled with pennies. Have
students use these materials to determine the value of a set of coins in cents.
37
MATH
GRADE 2
Domain
Cluster
Content
Standards
Measurement and Data
Represent and interpret data.
9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show
the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put together, take-apart, and
compare problems using information presented in a bar graph.
Instructional Strategies
Instructional Resources/Tools
Line plots are useful tools for collecting data because they show the number of things along a Common Core State Standards for Mathematics: Common addition and
numeric scale. They are made by simply drawing a number line then placing an X above the
subtraction situations - Table 1 on page 88 in the Common Core State Standards
corresponding value on the line that represents each piece of data. Line plots are essentially
(CCSS) for School for Mathematics illustrates 12 addition and subtraction
bar graphs with a potential bar for each value on the number line.
problem situations.
Pose a question related to the lengths of several objects. Measure the objects to the nearest
whole inch, foot, centimeter or meter. Create a line plot with whole-number units (0, 1, 2, ...)
on the number line to represent the measurements.
At first students should create real object and picture graphs so each row or bar consists of
countable parts. These graphs show items in a category and do not have a numerical scale.
For example, a real object graph could show the students‘ shoes (one shoe per student) lined
end to end in horizontal or vertical rows by their color. Students would simply count to find
how many shoes are in each row or bar. The graphs should be limited to 2 to 4 rows or bars.
Students would then move to making horizontal or vertical bar graphs with two to four
categories and a single-unit scale. Use the information in the graphs to pose and solve simple
put together, take-apart, and compare problems illustrated in Table 1 of the Common Core
State Standards.
National Library of Virtual Manipulatives, Utah State University: Bar Chart
This manipulative can be used to make a bar chart with 1 to 20 for the vertical
axis and 1 to 12 bars on the horizontal axis. The colors for the bars are
predetermined however users can type in their own title for the graph and labels
for the bars.
Common Misconceptions
The attributes for the same kind of object can vary. This will cause equal values
in an object graph to appear unequal. For example, when making an object graph
using shoes for boys and girls, five adjacent boy shoes would likely appear
longer than five adjacent girl shoes. To standardize the objects, place the objects
on the same-sized construction paper, then make the object graph.
38
MATH
GRADE 2
Domain
Cluster
Content
Standards
Geometry
Reason with shapes and their attributes.
1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals,
pentagons, hexagons, and cubes.
2. Partition a rectangle into rows of same-size squares and count to find the total number of them.
3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe
the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
Instructional Strategies
Instructional Resources/Tools
Modeling multiplication with partitioned rectangles promotes students‘ understanding of
Grid paper :
multiplication. Tell students that they will be drawing a square on grid paper. The length of
http://www.ablongman.com/vandewalleseries/Vol_1_BLM_PDFs/BLM30each side is equal to 2 units. Ask them to guess how many 1 unit by 1 unit squares will be
36.pdf
inside this 2 unit by 2 unit square. Students now draw this square and count the 1 by 1 unit
squares inside it. They compare this number to their guess. Next, students draw a 2 unit by 3
ORC # 1481 From the Math Forum: Introduction to fractions for primary
unit rectangle and count how many 1 unit by 1 unit squares are inside. Now they choose the
students - http://mathforum.org/varnelle/knum1.html two dimensions for a rectangle, predict the number of 1 unit by 1 unit squares inside, draw the http://mathforum.org/varnelle/knum2.html rectangle, count the number of 1 unit by 1 unit squares inside and compare this number to
http://mathforum.org/varnelle/knum5.html - This four-lesson unit introduces
their guess. Students repeat this process for different-size rectangles. Finally, ask them to
young children to fractions. Students learn to recognize equal parts of a whole as
what they observed as they worked on the task.
halves, thirds and fourths.
It is vital that students understand different representations of fair shares. Provide a collection
of different-size circles and rectangles cut from paper. Ask students to fold some shapes into
halves, some into thirds, and some into fourths. They compare the locations of the folds in
their shapes as a class and discuss the different representations for the fractional parts. To fold
rectangles into thirds, ask students if they have ever seen how letters are folded to be placed
in envelopes. Have them fold the paper very carefully to make sure the three parts are the
same size. Ask them to discuss why the same process does not work to fold a circle into
thirds.
Common Misconceptions
Some students may think that a shape is changed by its orientation. They may see
a rectangle with the longer side as the base, but claim that the same rectangle
with the shorter side as the base is a different shape. This is why is it so
important to have young students handle shapes and physically feel that the
shape does not change regardless of the orientation, as illustrated below.
Students also may believe that a region model represents one out of two, three or
four fractional parts without regard to the fact that the parts have to be equal
shares, e.g., a circle divided by two equally spaced horizontal lines represents
three thirds.
39
MATH
GRADE 3
40
MATH
GRADE 3
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in
which a total number of objects can be expressed as 5 × 7.
2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8
shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of
shares or a number of groups can be expressed as 56 ÷ 8.
3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using
drawings and equations with a symbol for the unknown number to represent the problem.¹
4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number
that makes the equation true in each of the equations 8 × ? = 48, 5 =
÷ 3, 6 × 6 = ?.
Instructional Strategies
Instructional Resources/Tools
Encourage students to solve these problems in different ways to show the same idea and be
Sets of counters
able to explain their thinking verbally and in written expression. Allowing students to present
Number lines to skip count and relate to multiplication
several different strategies provides the opportunity for them to compare strategies.
Arrays
Sets of counters, number lines to skip count and relate to multiplication and arrays/area
models will aid students in solving problems involving multiplication and division. Allow
Table 2. Common multiplication and division situations (Common Core State
students to model problems using these tools. They should represent the model used as a
Standards for Mathematics 2010)
drawing or equation to find the solution.
ORC # 4345 From the National Council of Teachers of Mathematics,
This shows multiplication using grouping with 3 groups of 5 objects and can be written as 3 × Illuminations: http://illuminations.nctm.org/LessonDetail.aspx?ID=L317 -This
four-part lesson encourages students to explore models for multiplication, the
5.
inverse of multiplication, and representing multiplication facts in equation form.
Provide a variety of contexts and tasks so that students will have more opportunity to develop
ORC # 4343 From the National Council of Teachers of Mathematics,
and use thinking strategies to support and reinforce learning of basic multiplication and
Illuminations: http://illuminations.nctm.org/LessonDetail.aspx?id=U109 - In this
division facts.
four-lesson unit students explore several meanings and representation of
multiplications and learn about properties of operations for multiplication.
Have students create multiplication problem situations in which they interpret the product of
whole numbers as the total number of objects in a group and write as an expression. Also,
National Library of Virtual Manipulatives - The National Library of Virtual
have students create division-problem situations in which they interpret the quotient of whole Manipulatives contains Java applets and activities for K-12 mathematics.
numbers as the number of shares.
http://nlvm.usu.edu/en/nav/frames_asid_197_g_2_t_1.html?open=activities&fro
Students can use known multiplication facts to determine the unknown fact in a multiplication
or division problem. Have them write a multiplication or division equation and the related
multiplication or division equation. For example, to determine the unknown whole number in
ultiplication fact of 3 × 9 = 27.
They should ask themselves questions such as, ―How many 3s are in 27?‖ or ―3 times what
number is 27?‖ Have them justify their thinking with models or drawings.
m=category_g_2_t_1.html : Illustrates arithmetic operations using a number line.
http://nlvm.usu.edu/en/nav/frames_asid_180_g_2_t_1.html?open=activities&fro
m=category_g_2_t_1.html Use bars to show addition, subtraction, multiplication,
and division on a number line.
41
MATH
GRADE 3
Common Misconceptions
Students think a symbol (? or []) is always the place for the answer. This is
Students also think that 3 ÷ 15 = 5 and 15 ÷ 3 = 5 are the same equations. The
use of models is essential in helping students eliminate this understanding.
The use of a symbol to represent a number once cannot be used to represent
another number in a different problem/situation. Presenting students with
multiple situations in which they select the symbol and explain what it represents
will counter this misconception.
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Understand properties of multiplication and the relationship between multiplication and division.
5. Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property
of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)
Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.
Instructional Strategies
Instructional Resources/Tools
Students need to apply properties of operations (commutative, associative and distributive) as Ohio Resource Center # 3998, From the National Council of Teachers of
strategies to multiply and divide. Applying the concept involved is more important than
Mathematics, Illuminations: – Multiplication:
students knowing the name of the property. Understanding the commutative property of
http://illuminations.nctm.org/LessonDetail.aspx?ID=L324 - Students skip-count
multiplication is developed through the use of models as basic multiplication facts are
and examine multiplication patterns. They also explore the commutative property
learned. For example, the result of multiplying 3 x 5 (15) is the same as the result of
of multiplication.
multiplying 5 x 3 (15).
Ohio Resource Center # 10564, From the National Council of Teachers of
To find the product of three numbers, students can use what they know about the product of
Mathematics, Illuminations:- Multiplication:
two of the factors and multiply this by the third factor. For example, to multiply 5 x 7 x 2,
http://illuminations.nctm.org/LessonDetail.aspx?ID=L325 - Students use a webstudents know that 5 x 2 is 10. Then, they can use mental math to find the product of 10 x 7
based calculator to create and compare counting patterns using the constant
(70). Allow students to use their own strategies and share with the class when applying the
function feature of the calculator. Making connections between multiple
associative property of multiplication.
representations of counting patterns reinforces students understanding of this
important idea and helps them recall these patterns as multiplication facts. From
Splitting arrays can help students understand the distributive property. They can use a known
fact to learn other facts that may cause difficulty. For example, students can split a 6 x 9 array a chart, students notice that multiplication is commutative.
into 6 groups of 5 and 6 groups of 4; then, add the sums of the groups.
The 6 groups of 5 is 30 and the 6 groups of 4 is 24. Students can write 6 x 9 as 6 x 5 + 6 x 4.
Students‘ understanding of the part/whole relationships is critical in understanding the
connection between multiplication and division.
42
MATH
GRADE 3
Domain
Operations and Algebraic Thinking
Cluster
Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one
Content
knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Standards
Instructional Strategies
Instructional Resources/Tools
Students need to understand the part/whole relationships in order to understand the
Unifix cubes or cubes
connection between multiplication and division. They need to develop efficient strategies
that lead to the big ideas of multiplication and division. These big ideas include
Grid or graph paper
understanding the properties of operations, such as the commutative and associative
properties of multiplication and the distributive property. The naming of the property is
Sets of counters
not necessary at this stage of learning.
In Grade 2, students found the total number of objects using rectangular arrays, such as a
5 x 5, and wrote equations to represent the sum. This is called unitizing, and it requires
students to count groups, not just objects. They see the whole as a number of groups of a
number of objects. This strategy is a foundation for multiplication in that students should
make a connection between repeated addition and multiplication.
As students create arrays for multiplication using objects or drawing on graph paper, they
may discover that three groups of four and four groups of three yield the same results.
They should observe that the arrays stay the same, although how they are viewed
changes. Provide numerous situations for students to develop this understanding.
To develop an understanding of the distributive property, students need decompose the
whole into groups. Arrays can be used to develop this understanding. To find the product
of 3 × 9, students can decompose 9 into the sum of 4 and 5 and find 3 × (4 + 5).
The distributive property is the basis for the standard multiplication algorithm that
students can use to fluently multiply multi-digit whole numbers in Grade 5.
Once students have an understanding of multiplication using efficient strategies, they
should make the connection to division.
Using various strategies to solve different contextual problems that use the same two
one-digit whole numbers requiring multiplication allows for students to commit to
memory all products of two one-digit numbers.
43
MATH
GRADE 3
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess
the reasonableness of answers using mental computation and estimation strategies including rounding.
9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example,
observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Instructional Strategies
Instructional Resources/Tools
Students gain a full understanding of which operation to use in any given situation through
From the National Council of Teachers of Mathematics, Illuminations: Times.
contextual problems. Number skills and concepts are developed as students solve problems.
Students can also look for patterns in the table.
Problems should be presented on a regular basis as students work with numbers and
computations.
Ohio Resource Center # 3998, From the National Council of Teachers of
Mathematics, Illuminations –
Researchers and mathematics educators advise against providing ―key words‖ for students to
http://illuminations.nctm.org/LessonDetail.aspx?ID=L324 - Students skip-count
look for in problem situations because they can be misleading. Students should use various
and examine multiplication patterns. They also explore the commutative property
strategies to solve problems. Students should analyze the structure of the problem to make
of multiplication.
sense of it. They should think through the problem and the meaning of the answer before
attempting to solve it.
Ohio Resource Center # 10564, From the National Council of Teachers of
Mathematics, Illuminations –
Encourage students to represent the problem situation in a drawing or with counters or blocks. http://illuminations.nctm.org/LessonDetail.aspx?ID=L325 - Students use a webStudents should determine the reasonableness of the solution to all problems using mental
based calculator to create and compare counting patterns using the constant
computations and estimation strategies.
function feature of the calculator. Making connections between multiple
representations of counting patterns reinforces students understanding of this
Students can use base–ten blocks on centimeter grid paper to construct rectangular arrays to
important idea and helps them recall these patterns as multiplication facts. From
represent problems.
a chart, students notice that multiplication is commutative.
Students are to identify arithmetic patterns and explain them using properties of operations.
They can explore patterns by determining likenesses, differences and changes. Use patterns in
addition and multiplication tables.
44
MATH
GRADE 3
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.
1. Use place value understanding to round whole numbers to the nearest 10 or 100.
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition
and subtraction.
3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.
Instructional Strategies
Instructional Resources/Tools
Prior to implementing rules for rounding students need to have opportunities to investigate
Number lines
place value. A strong understanding of place value is essential for the developed number
sense and the subsequent work that involves rounding numbers.
100s chart
Building on previous understandings of the place value of digits in multi-digit numbers,
place value is used to round whole numbers. Dependence on learning rules can be eliminated
with strategies such as the use of a number line to determine which multiple of 10 or of100,
a number is nearest (5 or more rounds up, less than 5 rounds down). As students‘
understanding of place value increases, the strategies for rounding are valuable for
estimating, justifying and predicting the reasonableness of solutions in problem-solving.
Strategies used to add and subtract two-digit numbers are now applied to fluently add and
subtract whole numbers within 1000. These strategies should be discussed so that students
can make comparisons and move toward efficient methods.
Number sense and computational understanding is built on a firm understanding of place
value.
Understanding what each number in a multiplication expression represents is important.
Multiplication problems need to be modeled with pictures, diagrams or concrete materials to
help students understand what the factors and products represent. The effect of multiplying
numbers needs to be examined and understood.
The use of area models is important in understanding the properties of operations of
multiplication and the relationship of the factors and its product. Composing and
decomposing area models is useful in the development and understanding of the distributive
property in multiplication.
Continue to use manipulative like hundreds charts and place-value charts. Have students use
a number line or a roller coaster example to block off the numbers in different colors.
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?from=category_g_1_t
_1.html – Visualize the multiplication of two numbers as area.
ORC # 5793 From the Mathematics TEKS Toolkit, Make a hundred - Students
roll a die seven times, each time determining whether to add that number of tens
or that number of ones to make a sum as close to 100 as possible without going
over.
Common Misconceptions
The use of terms like ―round up‖ and ―round down‖ confuses many students. For
example, the number 37 would round to 40 or they say it ―rounds up‖. The digit in
the tens place is changed from 3 to 4 (rounds up). This misconception is what
causes the problem when applied to rounding down. The number 32 should be
rounded (down) to 30, but using the logic mentioned for rounding up, some
students may look at the digit in the tens place and take it to the previous number,
resulting in the incorrect value of 20. To remedy this misconception, students need
to use a number line to visualize the placement of the number and/or ask questions
such as: ―What tens are 32 between and which one is it closer to?‖ Developing the
understanding of what the answer choices are before rounding can alleviate much
of the misconception and confusion related to rounding.
For example this chart show what numbers will round to the tens place.
Rounding can be expanded by having students identify all the numbers that will round to 30
or round to 200.
45
MATH
GRADE 3
Number and Operations – Fractions
Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by
a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that
each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its
endpoint locates the number a/b on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction
model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize
that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when
the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Instructional Strategies
Instructional Resources/Tools
This is the initial experience students will have with fractions and is best done over
Region or area models
time. Students need many opportunities to discuss fractional parts using concrete
models to develop familiarity and understanding of fractions. Expectations in this
Length or measurement models
domain are limited to fractions with denominators 2, 3, 4, 6 and 8.
Grid or dot paper (draw pictures to explore fraction ideas)
Understanding that a fraction is a quantity formed by part of a whole is essential to
number sense with fractions. Fractional parts are the building blocks for all fraction
Set models
concepts. Students need to relate dividing a shape into equal parts and representing
this relationship on a number line, where the equal parts are between two whole
Geoboards
numbers. Help students plot fractions on a number line, by using the meaning of the Fraction bars or strips
fraction. For example, to plot 4/5 on a number line, there are 5 equal parts with 4
copies of the 5 equal parts.
National Library of Virtual Manipulatives http://nlvm.usu.edu/en/nav/frames_asid_104_g_2_t_1.html?from=category_g_2_t_1.html
As students counted with whole numbers, they should also count with fractions.
– Write the fraction corresponding to the highlighted portion of a shape.
Counting equal-sized parts helps students determine the number of parts it takes to
make a whole and recognize fractions that are equivalent to whole numbers.
http://nlvm.usu.edu/en/nav/frames_asid_103_g_2_t_1.html?from=category_g_2_t_1.html
– Illustrate a fraction by dividing a shape and highlighting the appropriate parts.
Students need to know how big a particular fraction is and can easily recognize
which of two fractions is larger. The fractions must refer to parts of the same whole. http://nlvm.usu.edu/en/nav/frames_asid_102_g_2_t_1.html?from=category_g_2_t_1.html
Domain
Cluster
Content
Standards
46
MATH
GRADE 3
Benchmarks such as 1/2 and 1 are also useful in comparing fractions.
– Relates parts of a whole unit to written description and fraction.
Equivalent fractions can be recognized and generated using fraction models.
Students should use different models and decide when to use a particular model.
Make transparencies to show how equivalent fractions measure up on the number
line.
From the National Council of Teachers of Mathematics, Illuminations:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L543 - Students investigate the length
model by working with relationship rods to find equivalent fractions. Students develop
skills in reasoning and problem solving as they explain how two fractions are equivalent
(the same length).
Venn diagrams are useful in helping students organize and compare fractions to
determine the relative size of the fractions, such as more than 1/2 , exactly 1/2 or
less than 1/2 . Fraction bars showing the same sized whole can also be used as
models to compare fractions. Students are to write the results of the comparisons
with the symbols >, =, or <, and justify the conclusions with a model.
Transparencies you can make to show students how equivalent fractions measure up on
the number line. http://mathforum.org/paths/fractions/seeing.equiv.html
From the National Council of Teachers of Mathematics, Illuminations:
http://illuminations.nctm.org/LessonDetail.aspx?id=U152 - In this unit, students explore
relationships among fractions through work with the length model. This early work with
fraction relationships helps students make sense of basic fraction concepts and facilitates
work with comparing and ordering fractions and working with equivalency.
Learn Fractions with Cuisenaire Rods http://teachertech.rice.edu/Participants/silha/Lessons/cuisen2.html
- http://teachertech.rice.edu/Participants/silha/Lessons/equivalent.html
Common Misconceptions
The idea that the smaller the denominator, the smaller the piece or part of the set, or the
larger the denominator, the larger the piece or part of the set, is based on the comparison
that in whole numbers, the smaller a number, the less it is, or the larger a number, the more
it is. The use of different models, such as fraction bars and number lines, allows students
to compare unit fractions to reason about their sizes.
Students think all shapes can be divided the same way. Present shapes other than circles,
squares or rectangles to prevent students from overgeneralizing that all shapes can be
divided the same way. For example, have students fold a triangle into eighths. Provide oral
directions for folding the triangle:
1. Fold the triangle into half by folding the left vertex (at the base of the triangle) over to
meet the right vertex.
2. Fold in this manner two more times.
3. Have students label each eighth using fractional notation. Then, have students count the
fractional parts in the triangle (one-eighth, two-eighths, three-eighths, and so on).
47
MATH
GRADE 3
Domain
Cluster
Content
Standards
Measurement and Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in
minutes, e.g., by representing the problem on a number line diagram.
2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide
to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement
scale) to represent the problem.
Instructional Strategies
Instructional Resources/Tools
A clock is a common instrument for measuring time. Learning to tell time has
Beakers with whole number measures
much to do with learning to read a dial-type instrument and little with time
Graduated cylinders
measurement.
Measuring cups with liter markings
Students have experience in telling and writing time from analog and digital
clocks to the hour and half hour in Grade 1 and to the nearest five minutes, using
Balance scales
a.m. and p.m. in Grade 2. Now students will tell and write time to the nearest
Pan or bucket balances
minute and measure time intervals in minutes.
Weights in grams and kilograms
Provide analog clocks that allow students to move the minute hand.
Objects to weigh
Students need experience representing time from a digital clock to an analog clock
and vice versa.
Food coloring
Provide word problems involving addition and subtraction of time intervals in
minutes. Have students represent the problem on a number line. Student should
relate using the number line with subtraction from Grade 2.
Provide opportunities for students to use appropriate tools to measure and estimate
liquid volumes in liters only and masses of objects in grams and kilograms.
Students need practice in reading the scales on measuring tools since the markings
may not always be in intervals of one. The scales may be marked in intervals of
two, five or ten.
Allow students to hold gram and kilogram weights in their hand to use as a
benchmark. Use water colored with food coloring so that the water can be seen in
a beaker.
Students should estimate volumes and masses before actually finding the
measuring. Show students a group containing the same kind of objects. Then,
show them one of the objects and tell them its weight. Fill a container with more
objects and ask students to estimate the weight of the objects.
Use similar strategies with liquid measures. Be sure that students have
Water
Ohio Resource Center # 4021 - http://www.time-for-time.com/interactive.htm - This site has
interactive clocks, games, quizzes, worksheets, and reference materials, all related to time.
Analog and digital clocks help students in grades K-3 tell time to the nearest hour, half hour,
5 minutes, and 1 minute.
Virtual Manipulative Library http://nlvm.usu.edu/en/nav/frames_asid_316_g_2_t_4.html?from=category_g_2_t_4.html Interactively set the time on a digital and analog clock.
http://nlvm.usu.edu/en/nav/frames_asid_316_g_2_t_4.html?from=category_g_2_t_4.html –
Answer questions asking you to show a given time on digital and analog clocks.
http://nlvm.usu.edu/en/nav/frames_asid_318_g_2_t_4.html – Answer questions asking you
to indicate what time it will be before or after a given time period.
http://www.pbs.org/teachers/mathline/lessonplans/esmp/ittakesten/ittakesten_procedure.shtm
- Students estimate and measure marbles to the nearest gram and squeeze water-saturated
sponges to practice measuring in milliliters.
48
MATH
GRADE 3
opportunities to pour liquids into different size containers to see how much liquid
will be in certain whole liters. Show students containers and ask, ―How many
liters do you think will fill the container?‖
Common Misconceptions
Students may read the mark on a scale that is below a designated number on the scale as if it
was the next number. For example, a mark that is one mark below 80 grams may be read as
81 grams. Students realize it is one away from 80, but do not think of it as 79 grams.
Domain
Cluster
Content
Standards
Measurement and Data
Represent and interpret data.
3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ―how many more‖ and ―how
many less‖ problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might
represent pets.
4. Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the
horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.
Instructional Strategies
Instructional Resources/Tools
Representation of a data set is extended from picture graphs and bar graphs with single-unit
From the National Council of Teachers of Mathematics, Illuminations:
scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate to
http://illuminations.nctm.org/ActivityDetail.aspx?ID=204 This is a NCTM site
multiplication and division with 100 (product is 100 or less and numbers used in division are
that contains a bar graph tool to create bar graphs.
100 or less). In picture graphs, use values for the icons in which students are having difficulty
From the National Council of Teachers of Mathematics, Illuminations:
should use known facts to determine that the three icons represents 21 people. The intervals
http://illuminations.nctm.org/LessonDetail.aspx?ID=L317 - Students listen to the
on the vertical scale in bar graphs should not exceed 100.
counting story, What Comes in 2's, 3's, & 4's, and then use counters to set up
multiple sets of equal size. They fill in a table listing the number of sets, the
Students are to draw picture graphs in which a symbol or picture represents more than one
number of objects in each set, and the total number in all. They study the table to
object. Bar graphs are drawn with intervals greater than one. Ask questions that require
find examples of the order (commutative) property. Finally, they apply the equal
students to compare quantities and use mathematical concepts and skills. Use symbols on
sets model of multiplication by creating pictographs in which each icon
picture graphs that student can easily represent half of, or know how many half of the symbol represents several data points.
represents.
From the National Council of Teachers of Mathematics, Illuminations:
Students are to measure lengths using rulers marked with halves and fourths of an inch and
http://illuminations.nctm.org/LessonDetail.aspx?ID=L536 - Students create
record the data on a line plot. The horizontal scale of the line plot is marked off in whole
pictographs and answer questions about the data set.
numbers, halves or fourths. Students can create rulers with appropriate markings and use the
ruler to create the line plots.
Common Misconceptions
Although intervals on a bar graph are not in single units, students count each
square as one. To avoid this error, have students include tick marks between each
interval. Students should begin each scale with 0. They should think of skipcounting when determining the value of a bar since the scale is not in single
units.
49
MATH
GRADE 3
Domain
Cluster
Content
Standards
Measurement and Data
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
5. Recognize area as an attribute of plane figures and understand concepts of area measurement.
a. A square with side length 1 unit, called ―a unit square,‖ is said to have ―one square unit‖ of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
7. Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and
represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to
represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems.
Instructional Strategies
Instructional Resources/Tools
Students can cover rectangular shapes with tiles and count the number of units (tiles) to begin
Square tiles
developing the idea that area is a measure of covering. Area describes the size of an object that
is two-dimensional. The formulas should not be introduced before students discover the
Rectangle Multiplication – Visualize the multiplication of two numbers as an
meaning of area.
area. This application allows student to create different size arrays and relate the
array to the multiplication problem.
The area of a rectangle can be determined by having students lay out unit squares and count
how many square units it takes to completely cover the rectangle completely without overlaps
or gaps. Students need to develop the meaning for computing the area of a rectangle. A
connection needs to be made between the number of squares it takes to cover the rectangle and Common Misconceptions
Students may confuse perimeter and area when they measure the sides of a
the dimensions of the rectangle. Ask questions such as:
rectangle and then multiply. They think the attribute they find is length, which
• What does the length of a rectangle describe about the squares covering it?
is perimeter. Pose problems situations that require students to explain whether
• What does the width of a rectangle describe about the squares covering it?
they are to find the perimeter or area.
The concept of multiplication can be related to the area of rectangles using arrays. Students
need to discover that the length of one dimension of a rectangle tells how many squares are in
each row of an array and the length of the other dimension of the rectangle tells how many
squares are in each column. Ask questions about the dimensions if students do not make these
discoveries. For example:
• How do the squares covering a rectangle compare to an array?
• How is multiplication used to count the number of objects in an array?
Students should also make the connection of the area of a rectangle to the area model used to
represent multiplication. This connection justifies the formula for the area of a rectangle.
Provide students with the area of a rectangle (i.e., 42 square inches) and have them determine
possible lengths and widths of the rectangle. Expect different lengths and widths such as, 6
inches by 7 inches or 3 inches by 14 inches.
50
MATH
GRADE 3
Domain
Measurement and Data
Cluster
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
8. Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown
Content
side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
Standards
Instructional Strategies
Instructional Resources/Tools
Students have created rectangles before when finding the area of rectangles and connecting
Square tiles
them to using arrays in the multiplication of whole numbers. To explore finding the perimeter
1-inch or 1-centimeter grid paper
of a rectangle, have students use nonstretchy string. They should measure the string and
create a rectangle before cutting it into four pieces. Then, have students use four pieces of the Non-stretchy string
nonstretchy string to make a rectangle. Two pieces of the string should be of the same length
Geoboards and rubber bands
and the other two pieces should have a different length that is the same. Students should be
able to make the connection that perimeter is the total distance around the rectangle.
From PBS Teacher – For Real: Penned In - Explore rectangles and perimeter in
real-world applications. In this video clip from Cyberchase, Harry builds a
Geoboards can be used to find the perimeter of rectangles also. Provide students with
rectangular fence with an assortment of different-size sections but forgets to add
different perimeters and have them create the rectangles on the geoboards. Have students
a gate to get out.
share their rectangles with the class. Have discussions about how different rectangles can
have the same perimeter with different side lengths.
National Library of Virtual Manipulatives - Making a 5 Peg Triangle - Use
geoboards to illustrate area, perimeter, and rational number concepts.
Students experienced measuring lengths of inches and centimeters in Grade 2. They have also
related addition to length and writing equations with a symbol for the unknown to represent a Ohio Resource Center # 11115 Spaghetti and Meatballs for All!: A Mathematical
problem.
Story - An entry on the Ohio Resource Center Mathematics Bookshelf, this book
provides students with a real-life context for investigating variation in perimeter
Once students know how to find the perimeter of a rectangle, they can find the perimeter of
while area remains constant. In the story, small tables are pushed together to
rectangular-shaped objects in their environment. They can use appropriate measuring tools to
make one large table, until too many people show up, and the large table has to
find lengths of rectangular-shaped objects in the classroom. Present problems situations
be subdivided into smaller arrangements to provide more seating. Activities and
involving perimeter, such as finding the amount of fencing needed to enclose a rectangular
extensions are suggested at the back of the book.
shaped park, or how much ribbon is needed to decorate the edges of a picture frame. Also
present problem situations in which the perimeter and two or three of the side lengths are
From the National Council of Teachers of Mathematics, Illuminations: Junior
known, requiring students to find the unknown side length.
Architects - Finding Perimeter and Area - In this lesson, students develop
strategies for finding the perimeter and area for rectangles and triangles using
Students need to know when a problem situation requires them to know that the solution
geoboards and graph paper. Students learn to appreciate how measurement is a
relates to the perimeter or the area. They should have experience with understanding area
critical component to planning their clubhouse design.
concepts when they recognize it as an attribute of plane figures. They also discovered that
when plane figures are covered without gaps by n unit squares, the area of the figure is n
Common Misconceptions
square units.
Students think that when they are presented with a drawing of a rectangle with
only two of the side lengths shown or a problem situation with only two of the
Students need to explore how measurements are affected when one attribute to be measured is
side lengths provided, these are the only dimensions they should add to find the
held constant and the other is changed. Using square tiles, students can discover that the area
perimeter. Encourage students to include the appropriate dimensions on the other
of rectangles may be the same, but the perimeter of the rectangles varies. Geoboards can also
sides of the rectangle. With problem situations, encourage students to make a
be used to explore this same concept.
drawing to represent the situation in order to find the perimeter.
51
MATH
GRADE 3
Domain
Cluster
Content
Standards
Geometry
Reason with shapes and their attributes.
1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared
attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples
of quadrilaterals that do not belong to any of these subcategories.
2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with
equal area, and describe the area of each part as 1/4 of the area of the shape.
Instructional Strategies
Instructional Resources/Tools
In earlier grades, students have experiences with informal reasoning about particular shapes
From the National Council of Teachers of Mathematics, Illuminations:
through sorting and classifying using their geometric attributes. Students have built and
Rectangles and Parallelograms - While exploring properties of rectangles and
drawn shapes given the number of faces, number of angles and number of sides.
parallelograms using dynamic software, students identify, compare, and analyze
attributes of both shapes through physical and mental manipulation.
The focus now is on identifying and describing properties of two-dimensional shapes in more
precise ways using properties that are shared rather than the appearances of individual shapes. Exploring Properties of Rectangles and Parallelograms Using Dynamic Software
These properties allow for generalizations of all shapes that fit a particular classification.
- Dynamic geometry software provides an environment in which students can
Development in focusing on the identification and description of shapes‘ properties should
explore geometric relationships and make and test conjectures. In this example,
include examples and nonexamples, as well as examples and nonexamples drawn by students properties of rectangles and parallelograms are examined. The emphasis is on
of shapes in a particular category. For example, students could start with identifying shapes
identifying what distinguishes a rectangle from a more general parallelogram
with right angles. An explanation as to why the remaining shapes do not fit this category
should be discussed. Students should determine common characteristics of the remaining
shapes.
Common Misconceptions
In Grade 2, students partitioned rectangles into two, three or four equal shares, recognizing
Students may identify a square as a ―nonrectangle‖ or a ―nonrhombus‖ based on
that the equal shares need not have the same shape. They described the shares using words
limited images they see. They do not recognize that a square is a rectangle
such as, halves, thirds, half of, a third of, etc., and described the whole as two halves, three
because it has all of the properties of a rectangle. They may list properties of
thirds or four fourths. In Grade 4, students will partition shapes into parts with equal areas
each shape separately, but not see the interrelationships between the shapes. For
(the spaces in the whole of the shape). These equal areas need to be expressed as unit
example, students do not look at the properties of a square that are characteristic
fractions of the whole shape, i.e., describe each part of a shape partitioned into four parts as
of other figures as well. Using straws to make four congruent figures have
students change the angles to see the relationships between a rhombus and a
/ of the area of the shape.
square. As students develop definitions for these shapes, relationships between
the properties will be understood.
Have students draw different shapes and see how many ways they can partition the shapes
into parts with equal area.
52
MATH
GRADE 4
53
MATH
GRADE 4
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.
Represent verbal statements of multiplicative comparisons as multiplication equations.
2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem, distinguishing multiplicative comparison from additive comparison.
3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which
remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.
Instructional Strategies
Instructional Resources/Tools
Students need experiences that allow them to connect mathematical statements and number
Table 2. Common multiplication and division situations (Common Core State
sentences or equations. This allows for an effective transition to formal algebraic concepts.
Standards for Mathematics 2010) They represent an unknown number in a word problem with a symbol. Word problems which http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
require multiplication or division are solved by using drawings and equations.
The National Assessment of Educational Progress (NAEP) Assessments Students need to solve word problems involving multiplicative comparison (product
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx .
unknown, partition unknown) using multiplication or division as shown in Table 2 of the
Common Core State Standards for Mathematics, page 89. They should use drawings or
Diverse Learners - Information and instructional strategies for gifted students,
equations with a symbol for the unknown number to represent the problem. Students need to
English Language Learners (ELL), and students with disabilities is available in
be able to distinguish whether a word problem involves multiplicative comparison or additive the Introduction to Universal Design for Learning document located on the
comparison (solved when adding and subtracting in Grades 1 and 2).
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Present multistep word problems with whole numbers and whole-number answers using the
Learning principles can be found at www.cast.org
four operations. Students should know which operations are needed to solve the problem.
Drawing pictures or using models will help students understand what the problem is asking.
They should check the reasonableness of their answer using mental computation and
Common Misconceptions
estimation strategies.
Understand that the commutative property works for addition and multiplication
(and not subtraction and division)
Examples of multistep word problems can be accessed from the released questions on the
NAEP (National Assessment of Educational Progress) Assessment at
Understand remainders (and knowing when to round up or down)
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx.
For example, a constructed response question from the 2007 Grade 4 NAEP assessment
reads, ―Five classes are going on a bus trip and each class has 21 students. If each bus holds
only 40 students, how many buses are needed for the trip?‖
54
MATH
GRADE 4
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Gain familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given
whole number is the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or
composite.
Instructional Strategies
Instructional Resources/Tools
Students need to develop an understanding of the concepts of number theory such as prime
Calculators
numbers and composite numbers. This includes the relationship of factors and multiples.
Multiplication and division are used to develop concepts of factors and multiples. Division
Counters
problems resulting in remainders are used as counter-examples of factors.
Grid papers
Review vocabulary so that students have an understanding of terms such as factor, product,
multiples, and odd and even numbers.
The Ohio Resource Center - http://ohiorc.org/
Students need to develop strategies for determining if a number is prime or composite, in
other words, if a number has a whole number factor that is not one or itself. Starting with a
number chart of 1 to 20, use multiples of prime numbers to eliminate later numbers in the
chart. For example, 2 is prime but 4, 6, 8, 10, 12,… are composite. Encourage the
development of rules that can be used to aid in the determination of composite numbers. For
example, other than 2, if a number ends in an even number (0, 2, 4, 6 and 8), it is a composite
number.
ORC # 397 From the National Council of Teachers of Mathematics,
Illuminations: The Factor Game http://illuminations.nctm.org/LessonDetail.aspx?ID=L620 - engages students in a
friendly contest in which winning strategies involve distinguishing between
numbers with many factors and numbers with few factors. Students are then
guided through an analysis of game strategies and introduced to the definitions of
prime and composite numbers.
Using area models will also enable students to analyze numbers and arrive at an
understanding of whether a number is prime or composite. Have students construct rectangles
with an area equal to a given number. They should see an association between the number of
rectangles and the given number for the area as to whether this number is a prime or
composite number.
Understanding factoring through geometry http://mathforum.org/alejandre/factor1.html - Using square unit tiles, students
work with a partner to construct all rectangles whose area is equal to a given
number. After several examples, students see that prime numbers are associated
with exactly two rectangles, whereas composite numbers are associated with
more than two rectangles.
Definitions of prime and composite numbers should not be provided, but determined after
many strategies have been used in finding all possible factors of a number.
Provide students with counters to find the factors of numbers. Have them find ways to
separate the counters into equal subsets. For example, have them find several factors of 10,
14, 25 or 32, and write multiplication expressions for the numbers.
Another way to find the factor of a number is to use arrays from square tiles or drawn on grid
papers. Have students build rectangles that have the given number of squares.
Knowing how to determine factors and multiples is the foundation for finding common
ORC # 4209, From the National Council of Teachers of Mathematics,
Illuminations, The Product Game – Classifying Numbers. http://illuminations.nctm.org/LessonDetail.aspx?ID=L274 - Students construct
Venn diagrams to show the relationships between the factors or products of two
or more numbers in the Product Game.
ORC # 1161, From the National Council of Teachers of Mathematics,
Illuminations, The Product Game. http://illuminations.nctm.org/LessonDetail.aspx?ID=U100 - In the Product
55
MATH
GRADE 4
multiples and factors in Grade 6.
Writing multiplication expressions for numbers with several factors and for numbers with a
few factors will help students in making conjectures about the numbers. Students need to look
for commonalities among the numbers.
Game, students start with factors and multiply to find the product. In The Factor
Game, students start with a number and find its factors.
ORC # 4001, From the National Council of Teachers of Mathematics,
Illuminations, Multiplication: It‘s in the Cards – More Patterns with Products.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L327
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/vlibrary.html - The National Library of Virtual
Manipulatives contains Java applets and activities for K-12 mathematics.
Sieve of Eratosthenes – relate number patterns with visual patterns. Click on the
link for Activities for directions on engaging students in finding all prime
numbers 1-100.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Learning principles can be found at www.cast.org
Specific strategies for mathematics may include:
Some students may need to start with numbers that have only one pair of factors,
then those with two pairs of factors before finding factors of numbers with
several factor pairs.
Common Misconceptions
When listing multiples of numbers, students may not list the number itself.
Emphasize that the smallest multiple is the number itself.
Some students may think that larger numbers have more factors. Having students
share all factor pairs and how they found them will clear up this misconception.
Students often confuse ―multiples‖ and ―factors‖.
56
MATH
GRADE 4
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Generate and analyze patterns.
5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example,
given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and
even numbers. Explain informally why the numbers will continue to alternate in this way.
Instructional Strategies
Instructional Resources/Tools
In order for students to be successful later in the formal study of algebra, their algebraic
From PBS Teachers: Snake Patterns –s-s-s:
thinking needs to be developed. Understanding patterns is fundamental to algebraic
http://www.pbs.org/teachers/mathline/lessonplans/atmp/snake/snake_procedure.shtm
thinking. Students have experience in identifying arithmetic patterns, especially those
- Students will use given rules to generate several stages of a pattern and will be able
included in addition and multiplication tables. Contexts familiar to students are helpful in
to predict the outcome for any stage.
developing students‘ algebraic thinking.
From the National Council of Teachers of Mathematics, Illuminations: Patterns that
Grow – Growing Patterns. Students should generate numerical or geometric patterns that follow a given rule. They
http://illuminations.nctm.org/LessonDetail.aspx?ID=L304 - Students use numbers to
should look for relationships in the patterns and be able to describe and make
make growing patterns. They create, analyze, and describe growing patterns and
generalizations.
then record them. They also analyze a special growing pattern called Pascal‘s
triangle.
As students generate numeric patterns for rules, they should be able to ―undo‖ the pattern
to determine if the rule works with all of the numbers generated. For example, given the
From the National Council of Teachers of Mathematics, Illuminations: Patterns that
rule, ―Add 4‖ starting with the number 1, the pattern 1, 5, 9, 13, 17, is generated. In
Grow – Exploring Other Number Patterns.
analyzing the pattern, students need to determine how to get from one term to the next
http://illuminations.nctm.org/LessonDetail.aspx?ID=L304 - Students analyze
term. Teachers can ask students, ―How is a number in the sequence related to the one that
numeric patterns, including Fibonacci numbers. They also describe numeric patterns
came before it?‖, and ―If they started at the end of the pattern, will this relationship be the
and then record them in table form.
same?‖ Students can use this type of questioning in analyzing numbers patterns to
determine the rule.
From the National Council of Teachers of Mathematics, Illuminations: Patterns that
Students should also determine if there are other relationships in the patterns. In the
numeric pattern generated above, students should observe that the numbers are all odd
numbers.
Provide patterns that involve shapes so that students can determine the rule for the pattern.
Students may state that the rule is to multiply the previous number of squares by 3.
Grow – Looking Back and Moving Forward.
http://illuminations.nctm.org/LessonDetail.aspx?ID=L306 - In this final lesson of the
unit, students use logical thinking to create, identify, extend, and translate patterns.
They make patterns with numbers and shapes and explore patterns in a variety of
mathematical contexts.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in the
Introduction to Universal Design for Learning document located on the Revised
Academic Content Standards and Model Curriculum Development Web page.
Additional strategies and resources based on the Universal Design for Learning
principles can be found at www.cast.org .
57
MATH
GRADE 4
Domain
Cluster
Content
Standards
Number and Operations Base Ten
Generalize place value understanding for multi-digit whole numbers.
1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that
700 ÷ 70 = 10 by applying concepts of place value and division.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings
of the digits in each place, using >, =, and < symbols to record the results of comparisons.
3. Use place value understanding to round multi-digit whole numbers to any place.
Instructional Strategies
Instructional Resources/Tools
Provide multiple opportunities in the classroom setting and use real-world context for
Place value boxes
students to read and write multi-digit whole numbers.
Place value flip charts
Students need to have opportunities to compare numbers with the same number of
digits, e.g., compare 453, 698 and 215; numbers that have the same number in the
Number cards
leading digit position, e.g., compare 45, 495 and 41,223; and numbers that have
different numbers of digits and different leading digits, e.g., compare 312, 95, 5245 and
Diverse Learners - Information and instructional strategies for gifted students, English
10,002.
Language Learners (ELL), and students with disabilities is available in the
Introduction to Universal Design for Learning document located on the Revised
Students also need to create numbers that meet specific criteria. For example, provide
Academic Content Standards and Model Curriculum Development Web page.
students with cards numbered 0 through 9. Ask students to select 4 to 6 cards; then,
Additional strategies and resources based on the Universal Design for Learning
using all the cards make the largest number possible with the cards, the smallest number principles can be found at www.cast.org.
possible and the closest number to 5000 that is greater than 5000 or less than 5000.
In Grade 4, rounding is not new, and students need to build on the Grade 3 skill of
rounding to the nearest 10 or 100 to include larger numbers and place value. What is
Common Misconceptions
new for Grade 4 is rounding to digits other than the leading digit, e.g., round 23,960 to
There are several misconceptions students may have about writing numerals from
the nearest hundred. This requires greater sophistication than rounding to the nearest ten verbal descriptions. Numbers like one thousand do not cause a problem; however a
thousand because the digit in the hundreds place represents 900 and when rounded it
number like one thousand two causes problems for students. Many students will
becomes 1000, not just zero.
understand the 1000 and the 2 but then instead of placing the 2 in the ones place,
students will write the numbers as they hear them, 10002 (ten thousand two). There are
Students should also begin to develop some rules for rounding, building off the basic
multiple strategies that can be used to assist with this concept, including place-value
strategy of; ―Is 48 closer to 40 or 50?‖ Since 48 is only 2 away from 50 and 8 away
boxes and vertical-addition method.
from 40, 48 would round to 50. Now students need to generalize the rule for much
larger numbers and rounding to values that are not the leading digit.
Students often assume that the first digit of a multi-digit number indicates the
"greatness" of a number. The assumption is made that 954 is greater than 1002 because
students are focusing on the first digit instead of the number as a whole.
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MATH
GRADE 4
Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic
4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and
the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of
operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or
area models.
Instructional Strategies
Instructional Resources/Tools
A crucial theme in multi-digit arithmetic is encouraging students to develop strategies
Place-value mats
that they understand, can explain, and can think about, rather than merely follow a
sequence of directions that they don't understand.
bound place value flip books (so that the digit in a certain place can be switched)
It is important for students to have seen and used a variety of strategies and materials to
broaden and deepen their understanding of place value before they are required to use
standard algorithms. The goal is for them to understand all the steps in the algorithm,
and they should be able to explain the meaning of each digit. For example, a 1 can
represent one, ten, one hundred, and so on. For multi-digit addition and subtraction in
Grade 4, the goal is also fluency, which means students must be able to carry out the
calculations efficiently and accurately.
base ten blocks
Start with a student‘s understanding of a certain strategy, and then make intentional,
clear-cut connections for the student to the standard algorithm. This allows the student
to gain understanding of the algorithm rather than just memorize certain steps to follow.
Sometimes students benefit from 'being the teacher' to an imaginary student who is
having difficulties applying standard algorithms in addition and subtraction situations.
To promote understanding, use examples of student work that have been done
incorrectly and ask students to provide feedback about the student work.
Make literature connections by using the resource book, Read Any Good Math
Lately?, to identify books related to certain math topics. Books can provide a 'hook' for
learning, to activate background knowledge, and to build student interest.
It is very important for some students to talk through their understanding of connections
between different strategies and standard addition and subtractions algorithms. Give
students many opportunities to talk with classmates about how they could explain
standard algorithms. Think-Pair-Share is a good protocol for all students.
When asking students to gain understanding about multiplying larger numbers, provide
frequent opportunities to engage in mental math exercises. When doing mental math, it
is difficult to even attempt to use a strategy that one does not fully understand. Also, it
is a natural tendency to use numbers that are 'friendly' (multiples of 10) when doing
tens frames
hundreds flats
Smartboard
Diverse Learners - Information and instructional strategies for gifted students, English
Language Learners (ELL), and students with disabilities is available in the
Introduction to Universal Design for Learning document located on the Revised
Academic Content Standards and Model Curriculum Development Web page.
Additional strategies and resources based on the Universal Design for Learning
principles can be found at www.cast.org.
Common Misconceptions
Often students mix up when to 'carry' and when to 'borrow'. Also students often do not
notice the need of borrowing and just take the smaller digit from the larger one.
Emphasize place value and the meaning of each of the digits.
59
MATH
GRADE 4
mental math, and this promotes its understanding.
As students developed an understanding of multiplying a whole number up to four
digits by a one-digit whole number, and multiplying two two-digit numbers through
various strategies, they should do the same when finding whole-number quotients and
remainders. By relating division to multiplication and repeated subtraction, students can
find partial quotients. An explanation of partial quotients can be viewed at
http://www.teachertube.com, search for Outline of partial quotients. This strategy will
help them understand the division algorithm.
Students will have a better understanding of multiplication or division when problems
are presented in context.
Students should be able to illustrate and explain multiplication and division calculations
by using equations, rectangular arrays and the properties of operations. These strategies
were used in Grade 3 as students developed an understanding of multiplication.
To give students an opportunity to communicate their understandings of various
strategies, organize them into small groups and ask each group to create a poster to
explain a particular strategy and then present it to the class.
Vocabulary is important. Students should have an understanding of terms such as, sum,
difference, fewer, more, less, ones, tens, hundreds, thousands, digit, whole numbers,
product, factors and multiples.
60
MATH
GRADE 4
Number and Operations – Fractions
Extend understanding of fractions equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts
differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a
benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of
comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Instructional Strategies
Instructional Resources/Tools
Students‘ initial experience with fractions began in Grade 3. They used models such as
Pattern blocks
number lines to locate unit fractions, and fraction bars or strips, area or length models, and
Venn diagrams to recognize and generate equivalent fractions and make comparisons of
Fraction bars or strips
fractions.
Diverse Learners - Information and instructional strategies for gifted students,
Students extend their understanding of unit fractions to compare two fractions with different
English Language Learners (ELL), and students with disabilities is available in
numerators and different denominators.
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
Students should use models to compare two fractions with different denominators by creating page. Additional strategies and resources based on the Universal Design for
common denominators or numerators. The models should be the same (both fractions shown
Learning principles can be found at www.cast.org.
using fraction bars or both fractions using circular models) so that the models represent the
same whole. The models should be represented in drawings. Students should also use
benchmark fractions such as 12 to compare two fractions. The result of the comparisons
Common Misconceptions
should be recorded using <, >, and = symbols.
Students think that when generating equivalent fractions they need to multiply or
divide either the numerator or denominator, such as, changing 1/2 to sixths. They
would multiply the denominator by 3 to get 1/6, instead of multiplying the
numerator by 3 also. Their focus is only on the multiple of the denominator, not
the whole fraction.
Domain
Cluster
Content
Standards
Students need to use a fraction in the form of one such as 3/3 so that the
numerator and denominator do not contain the original numerator or
denominator.
61
MATH
GRADE 4
Domain
Cluster
Content
Standards
Number and Operations - Fractions
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of
operations and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual
fraction models and equations to represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion
by the equation 5/4 = 5 × (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction
model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the
problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast
beef will be needed? Between what two whole numbers does your answer lie?
Instructional Strategies
Instructional Resources/Tools
In Grade 3, students added unit fractions with the same denominator. Now, they begin to
Fraction tiles/bars
represent a fraction by decomposing the fraction as the sum of unit fraction and justify with a
fraction model.
Circular fraction models
Students also represented whole numbers as fractions. They use this knowledge to add and
subtract mixed numbers with like denominators using properties of number and appropriate
fraction models. It is important to stress that whichever model is used, it should be the same
for the same whole. For example, a circular model and a rectangular model should not be
used in the same problem.
Understanding of multiplication of whole numbers is extended to multiplying a fraction by a
whole number. Allow students to use fraction models and drawing to show their
understanding.
Present word problems involving multiplication of a fraction by a whole number. Have
students solve the problems using visual models and write equations to represent the
problems.
Rulers with markings of 1/2 , 1/4 and 1/8
Number lines
Students think that it does not matter which model to use when finding the sum
or difference of fractions. They may represent one fraction with a rectangle and
the other fraction with a circle. They need to know that the models need to
represent the same whole.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Learning principles can be found at www.cast.org.
62
MATH
GRADE 4
Common Misconceptions
Students think that it does not matter which model to use when finding the sum
or difference of fractions. They may represent one fraction with a rectangle and
the other fraction with a circle. They need to know that the models need to
represent the same whole.
Domain
Cluster
Content
Standards
Number and Operations - Fractions
Understand decimal notations for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective
denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a
number line diagram.
7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same
whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Instructional Strategies
Instructional Resources/Tools
The place value system developed for whole numbers extends to fractional parts represented
Length or area models
as decimals. This is a connection to the metric system. Decimals are another way to write
10 x 10 square on a grid
fractions. The place-value system developed for whole numbers extends to decimals. The
concept of one whole used in fractions is extended to models of decimals.
Decimal place-value mats
Students can use base-ten blocks to represent decimals. A 10 x 10 block can be assigned the
value of one whole to allow other blocks to represent tenths and hundredths. They can show a
decimal representation from the base-ten blocks by shading on a 10 x 10 grid.
Students need to make connections between fractions and decimals. They should be able to
write decimals for fractions with denominators of 10 or 100. Have students say the fraction
with denominators of 10 and 100 aloud. For example 410 would ―four tenths‖ or 27/100
would be ―twenty-seven hundredths.‖ Also, have students represent decimals in word form
with digits and the decimal place value, such as 410 would be 4 tenths.
Students should be able to express decimals to the hundredths as the sum of two decimals or
fractions. This is based on understanding of decimal place value. For example 0.32 would be
the sum of 3 tenths and 2 hundredths. Using this understanding, students can write 0.32 as the
sum of two fractions�3/10 + 2/100�.
Students‘ understanding of decimals to hundredths is important in preparation for performing
operations with decimals to hundredths in Grade 5.
Base-ten blocks
Number lines
From the National Council of Teachers of Mathematics, Illuminations: A Meter
of Candy – In this series of three hands-on activities, students develop and
reinforce their understanding of hundredths as fractions, decimals and
percentages. Students explore with candy pieces as they physically make and
connect a set and linear model (meter) to produce area models (grids and pie
graphs). At this time, students are not to do percents. The relationships among
fractions, decimals and percents are developed in Grade 6.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
63
MATH
GRADE 4
Learning principles can be found at www.cast.org.
In decimal numbers, the value of each place is 10 times the value of the place to its immediate
right. Students need an understanding of decimal notations before they try to do conversions
in the metric system. Understanding of the decimal place value system is important prior to
the generalization of moving the decimal point when performing operations involving
decimals.
Common Misconceptions
Students treat decimals as whole numbers when making comparison of two
decimals. They think the longer the number, the greater the value. For example,
they think that .03 is greater than 0.3.
Students extend fraction equivalence from Grade 3 with denominators of 2, 3 4, 6 and 8 to
fractions with a denominator of 10. Provide fraction models of tenths and hundredths so that
students can express a fraction with a denominator of 10 as an equivalent fraction with a
denominator of 100.
Students extend fraction equivalence from Grade 3 with denominators of 2, 3 4, 6 and 8 to
fractions with a denominator of 10. Provide fraction models of tenths and hundredths so that
students can express a fraction with a denominator of 10 as an equivalent fraction with a
denominator of 100.
When comparing two decimals, remind students that as in comparing two fractions, the
decimals need to refer to the same whole. Allow students to use visual models to compare
two decimals. They can shade in a representation of each decimal on a 10 x 10 grid. The 10 x
10 grid is defined as one whole. The decimal must relate to the whole.
Flexibility with converting fractions to decimals and decimals to fractions provides efficiency
in solving problems involving all four operations in later grades.
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MATH
GRADE 4
Domain
Cluster
Content
Standards
Measurement and Data
Solve problems involving measurement and conversion of measurement from a larger unit to a smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of
measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know
that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1,
12), (2, 24), (3, 36), ...
2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems
involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent
measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
3. Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Instructional Strategies
Instructional Resources/Tools
In order for students to have a better understanding of the relationships between units, they
Yardsticks(meter sticks) and rulers (marked with customary and metric units)
need to use measuring devices in class. The number of units needs to relate to the size of the
unit. They need to discover that there are 12 inches in 1 foot and 3 feet in 1 yard. Allow
Teaspoons and tablespoons
students to use rulers and yardsticks to discover these relationships among these units of
measurements. Using 12-inch rulers and yardstick, students can see that three of the 12-inch
Graduated measuring cups (marked with customary and metric units)
rulers, which is the same as 3 feet since each ruler is 1 foot in length, are equivalent to one
yardstick. Have students record the relationships in a two column table or t-charts. A similar
Diverse Learners - Information and instructional strategies for gifted students,
strategy can be used with rulers marked with centimeters and a meter stick to discover the
English Language Learners (ELL), and students with disabilities is available in
relationships between centimeters and meters.
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
Present word problems as a source of students‘ understanding of the relationships among
page. Additional strategies and resources based on the Universal Design for
inches, feet and yards.
Learning principles can be found at www.cast.org .
Students are to solve word problems involving distances, intervals of time, liquid volumes,
masses of objects, and money, including problems involving simple fractions or decimals,
and problems that require expressing measurements given in a larger unit in terms of a
smaller unit.
Present problems that involve multiplication of a fraction by a whole number (denominators
are 2, 3, 4, 5 6, 8, 10, 12 and 100). Problems involving addition and subtraction of fractions
should have the same denominators. Allow students to use strategies learned with these
concepts.
Common Misconceptions
Students believe that larger units will give the larger measure. Students should be
given multiple opportunities to measure the same object with different measuring
units. For example, have the students measure the length of a room with one-inch
tiles, with one-foot rulers, and with yard sticks. Students should notice that it
takes fewer yard sticks to measure the room than rulers or tiles.
Students used models to find area and perimeter in Grade 3. They need to relate discoveries
from the use of models to develop an understanding of the area and perimeter formulas to
solve real-world and mathematical problems.
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MATH
GRADE 4
Domain
Cluster
Content
Standards
Measurement and Data
Represent and interpret data.
4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by
using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens
in an insect collection.
Instructional Strategies
Instructional Resources/Tools
Data has been measured and represented on line plots in units of whole numbers, halves or
Fraction bars or strips
quarters. Students have also represented fractions on number lines. Now students are using
line plots to display measurement data in fraction units and using the data to solve problems
Diverse Learners - Information and instructional strategies for gifted students,
involving addition or subtraction of fractions.
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Have students create line plots with fractions of a unit ½ ¼ 1/8 and plot data showing
Revised Academic Content Standards and Model Curriculum Development Web
multiple data points for each fraction.
page. Additional strategies and resources based on the Universal Design for
Learning principles can be found at www.cast.org
Pose questions that students may answer, such as
• ―How many one-eighths are shown on the line plot?‖ Expect ―two one-eighths‖ as the
answer. Then ask, ―What is the total of these two one-eighths?‖ Encourage students to
Common Misconceptions
count the fractional numbers as they would with whole-number counting, but using the
Students use whole-number names when counting fractional parts on a number
fraction name.
line. The fraction name should be used instead. For example, if two-fourths is
• ―What is the total number of inches for insects measuring 38 inches?‖ Students can use skip represented on the line plot three times, then there would be six-fourths.
counting with fraction names to find the total, such as, ―three-eighths, six-eighths, nineeighths. The last fraction names the total. Students should notice that the denominator did
not change when they were saying the fraction name. have them make a statement about
the result of adding fractions with the same denominator.
• ―What is the total number of insects measuring 18 inch or 58 inches?‖ Have students write
number sentences to represent the problem and solution such as, 18 + 18 + 58 = 78 inches.
Use visual fraction strips and fraction bars to represent problems to solve problems involving
addition and subtraction of fractions.
66
MATH
GRADE 4
Domain
Cluster
Content
Standards
Measurement and Data
Geometric measurement: understand concepts of angle and measure angles.
5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between
the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a ―one-degree angle,‖ and can be used to measure
angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle
measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using
an equation with a symbol for the unknown angle measure.
Instructional Strategies
Instructional Resources/Tools
Angles are geometric shapes composed of two rays that are infinite in length. Students can
Cardboard cut in strips to make an angle explorer
understand this concept by using two rulers held together near the ends. The rulers can
represent the rays of an angle. As one ruler is rotated, the size of the angle is seen to get
Brass fasteners
larger. Ask questions about the types of angles created. Responses may be in terms of the
relationship to right angles. Introduce angles as acute (less than the measure of a right angle)
Protractor
and obtuse (greater than the measure of a right angle). Have students draw representations of
each type of angle. They also need to be able to identify angles in two-dimensional figures.
Angle ruler
Students can also create an angle explorer (two strips of cardboard attached with a brass
fastener) to learn about angles.
Straws
Transparencies
They can use the angle explorer to get a feel of the relative size of angles as they rotate the
cardboard strips around.
Students can compare angles to determine whether an angle is acute or obtuse. This will
allow them to have a benchmark reference for what an angle measure should be when using a
tool such as a protractor or an angle ruler.
Provide students with four pieces of straw, two pieces of the same length to make one angle
and another two pieces of the same length to make an angle with longer rays.
Another way to compare angles is to place one angle over the other angle. Provide students
with a transparency to compare two angles to help them conceptualize the spread of the rays
of an angle. Students can make this comparison by tracing one angle and placing it over
another angle. The side lengths of the angles to be compared need to be different.
Angle explorers
Ohio Resource Center
Sir Cumference and the Great Knight of Angleland:
http://www.ohiorc.org/for/math/bookshelf/detail.aspx?id=49 In this story, young
Radius, son of Sir Cumference and Lady Di of Ameter, undertakes a quest, the
successful completion of which will earn him his knighthood. With the help of a
family heirloom that functions much like a protractor, he is able to locate the
elusive King Lell and restore him to the throne of Angleland. In gratitude, King
Lell bestows knighthood on Sir Radius. This book is an entry on the Ohio
Resource Center Mathematics Bookshelf
http://ohiorc.org/for/math/bookshelf/default.aspx
From the National Council of Teachers of Mathematics, Figure This: What‘s My
67
MATH
GRADE 4
Students are ready to use a tool to measure angles once they understand the difference
between an acute angle and an obtuse angle. Angles are measured in degrees. There is a
relationship between the number of degrees in an angle and circle which has a measure of 360
degrees. Students are to use a protractor to measure angles in whole-number degrees. They
can determine if the measure of the angle is reasonable based on the relationship of the angle
to a right angle. They also make sketches of angles of specified measure.
Angle?
http://www.figurethis.org/challenges/c10/challenge.htm math Challenge # 10 Students can estimate the measures of the angles between their fingers when they
spread out their hand.
From PBS Teachers: 3rd Grade Measuring Game,
http://www.pbs.org/teachers/connect/resources/5488/preview/ Identify acute,
obtuse and right angles in this online interactive game
From PBS Teachers: Star Gazing,
http://pbskids.org/cyberchase/games/anglemeasurement/anglemeasurement.html/
Determine the correct angle at which to place a telescope in order to see as many
stars as possible in this online interactive game.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Learning principles can be found at www.cast.org
Common Misconceptions
Students are confused as to which number to use when determining the measure
of an angle using a protractor because most protractors have a double set of
numbers. Students should decide first if the angle appears to be an angle that is
less than the measure of a right angle (90°) or greater than the measure of a right
angle (90°). If the angle appears to be less than 90°, it is an acute angle and its
measure ranges from 0° to 89°. If the angle appears to be an angle that is greater
than 90°, it is an obtuse angle and its measures range from 91° to 179°. Ask
questions about the appearance of the angle to help students in deciding which
number to use.
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MATH
GRADE 4
Domain
Cluster
Content
Standards
Geometry
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size.
Recognize right triangles as a category, and identify right triangles.
3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts.
Identify line-symmetric figures and draw lines of symmetry.
Instructional Strategies
Instructional Resources/Tools
Angles - Students can use the corner of a sheet of paper as a benchmark for a right angle.
Mirrors
They can use a right angle to determine relationships of other angles.
Geoboards
Symmetry - When introducing line of symmetry, provide examples of geometric shapes with
and without lines of symmetry. Shapes can be classified by the existence of lines of symmetry GeoGebra (a free software for learning and teaching); http://www.geogebra.com.
in sorting activities. This can be done informally by folding paper, tracing, creating designs
with tiles or investigating reflections in mirrors.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
With the use of a dynamic geometric program, students can easily construct points, lines and
the Introduction to Universal Design for Learning document located on the
geometric figures. They can also draw lines perpendicular or parallel to other line segments.
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Two-dimensional shapes - Two-dimensional shapes are classified based on relationships by
Learning principles can be found at www.cast.org.
the angles and sides. Students can determine if the sides are parallel or perpendicular, and
classify accordingly. Characteristics of rectangles (including squares) are used to develop the
concept of parallel and perpendicular lines. The characteristics and understanding of parallel
Common Misconceptions
and perpendicular lines are used to draw rectangles. Repeated experiences in comparing and
Students believe a wide angle with short sides may seem smaller than a narrow
contrasting shapes enable students to gain a deeper understanding about shapes and their
angle with long sides. Students can compare two angles by tracing one and
properties.
placing it over the other. Students will then realize that the length of the sides
does not determine whether one angle is larger or smaller than another angle. The
Informal understanding of the characteristics of triangles is developed through angle
measure of the angle does not change.
measures and side length relationships. Triangles are named according to their angle
measures (right, acute or obtuse) and side lengths (scalene, isosceles or equilateral). These
characteristics are used to draw triangles.
69
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70
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GRADE 5
Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
2. Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the
calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7).Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to
calculate the indicated sum or product.
Instructional Strategies
Instructional Resources/Tools
Students should be given ample opportunities to explore numerical expressions with mixed
The Ohio Resource Center
operations. This is the foundation for evaluating numerical and algebraic expressions that will ORC # 11463 From the National Council of Teachers of Mathematics,
include whole-number exponents in Grade 6.
Illuminations: Order of Operations Bingo. Instead of calling numbers to play
Bingo, you call (and write) numerical expressions to be evaluated for the
There are conventions (rules) determined by mathematicians that must be learned with no
numbers on the Bingo cards. The operations in this lesson are addition,
conceptual basis. For example, multiplication and division are always done before addition
subtraction, multiplication, and division; the numbers are all single-digit whole
and subtraction.
numbers.
Begin with expressions that have two operations without any grouping symbols
(multiplication or division combined with addition or subtraction) before introducing
expressions with multiple operations. Using the same digits, with the operations in a different
order, have students evaluate the expressions and discuss why the value of the expression is
different. For example, have students evaluate 5 × 3 + 6 and 5 + 3 × 6. Discuss the rules that
must be followed. Have students insert parentheses around the multiplication or division part
in an expression. A discussion should focus on the similarities and differences in the
problems and the results. This leads to students being able to solve problem situations which
require that they know the order in which operations should take place.
After students have evaluated expressions without grouping symbols, present problems with
one grouping symbol, beginning with parentheses, then in combination with brackets and/or
braces.
Have students write numerical expressions in words without calculating the value. This is the
foundation for writing algebraic expressions. Then, have students write numerical expressions
from phrases without calculating them.
Diverse Learners - Information and instructional strategies for gifted students,
English Language Learners (ELL), and students with disabilities is available in
the Introduction to Universal Design for Learning document located on the
Revised Academic Content Standards and Model Curriculum Development Web
page. Additional strategies and resources based on the Universal Design for
Learning principles can be found at www.cast.org .
Connections: This cluster is connected to the Grade 5 Critical Area of Focus #2,
Extending division to 2-digit divisors, integrating decimal fractions into the place
value system and developing understanding of operations with decimals to
hundredths, and developing fluency with whole number and decimal operations.
More information about this critical area of focus can be found by clicking here.
Common Misconceptions
Solving multiplication before division. Same with addition and subtraction.
What they should do is solve in order as they occur from left to right.
Solve the entire equation from left to right ignoring order of operations
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Domain
Cluster
Content
Standards
Operations and Algebraic Thinking
Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of
corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number
0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the
corresponding terms in the other sequence. Explain informally why this is so.
Instructional Strategies
Instructional Resources/Tools
Students should have experienced generating and analyzing numerical patterns using a given
Grid paper
rule in Grade 4.
Given two rules with an apparent relationship, students should be able to identify the
relationship between the resulting sequences of the terms in one sequence to the
corresponding terms in the other sequence. For example, starting with 0, multiply by 4 and
starting with 0, multiply by 8 and generate each sequence of numbers (0, 4, 8, 12, 16, …) and
(0, 8, 16, 24, 32,…). Students should see that the terms in the second sequence are double the
terms in the first sequence, or that the terms in the first sequence are half the terms in the
second sequence.
Common Misconceptions
Students reverse the points when plotting them on a coordinate plane. They count
up first on the y-axis and then count over on the x-axis. The location of every
point in the plane has a specific place. Have students plot points where the
numbers are reversed such as (4, 5) and (5, 4). Begin with students providing a
verbal description of how to plot each point. Then, have them follow the verbal
description and plot each point.
Have students form ordered pairs and graph them on a coordinate plane. Patterns can be also
discerned in graphs.
Graphing ordered pairs on a coordinate plane is introduced to students in the Geometry
domain where students solve real-world and mathematical problems. For the purpose of this
cluster, only use the first quadrant of the coordinate plane, which contains positive numbers
only. Provide coordinate grids for the students, but also have them make coordinate grids. In
Grade 6, students will position pairs of integers on a coordinate plane.
The graph of both sequences of numbers is a visual representation that will show the
relationship between the two sequences of numbers.
Encourage students to represent the sequences in T-charts so that they can see a connection
between the graph and the sequences.
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Domain
Cluster
Content
Standards
Number and Operations in Base Ten
Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents
in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal
point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
3. Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 ×
(1/10) + 9 × (1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
4. Use place value understanding to round decimals to any place.
Instructional Strategies
Instructional Resources/Tools
In Grade 5, the concept of place value is extended to include decimal values to thousandths.
Base Block Decimals, Student use a Ten Frames to demonstrate decimal
The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and relationships.
decimal numbers. For example, students need to continue to represent, write and state the
value of numbers including decimal numbers. For students who are not able to read, write and
represent multi-digit numbers, working with decimals will be challenging.
Common Misconceptions
A common misconception that students have when trying to extend their
Money is a good medium to compare decimals. Present contextual situations that require the
understanding of whole number place value to decimal place value is that as you
comparison of the cost of two items to determine the lower or higher priced item. Students
move to the left of the decimal point, the number increases in value. Reinforcing
should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a the concept of powers of ten is essential for addressing this issue.
given value. Help students make connections between the number of each type of coin and
the value of each coin, and the expanded form of the number. Build on the understanding that A second misconception that is directly related to comparing whole numbers is
it always takes ten of the number to the right to make the number to the left.
the idea that the longer the number the greater the number. With whole numbers,
a 5-digit number is always greater that a 1-, 2-, 3-, or 4-digit number. However,
Number cards, number cubes, spinners and other manipulatives can be used to generate
with decimals a number with one decimal place may be greater than a number
decimal numbers. For example, have students roll three number cubes, then create the largest
with two or three decimal places. For example, 0.5 is greater than 0.12, 0.009 or
and small number to the thousandths place. Ask students to represent the number with
0.499. One method for comparing decimals it to make all numbers have the same
numerals and words.
number of digits to the right of the decimal point by adding zeros to the number,
such as 0.500, 0.120, 0.009 and 0.499. A second method is to use a place-value
chart to place the numerals for comparison.
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Number and Operations – Base Ten
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of
operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or
area models.
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Instructional Strategies
Instructional Resources/Tools
Because students have used various models and strategies to solve problems involving
Decimal place-value chart
multiplication with whole numbers, they should be able to transition to using standard
algorithms effectively. With guidance from the teacher, they should understand the
From the National Library of Virtual Manipulatives: Base Blocks Decimals–
connection between the standard algorithm and their strategies.
Add and subtract decimal values using base blocks. (Note: make sure the Base
equals 10).
Connections between the algorithm for multiplying multi-digit whole numbers and strategies
such as partial products or lattice multiplication are necessary for students‘ understanding.
You can multiply by listing all the partial products. For example:
Common Misconceptions
Students might compute the sum or difference of decimals by lining up the right234
hand digits as they would whole number. For example, in computing the sum of
×8
15.34 + 12.9, students will write the problem in this manner:
32
Multiply the ones (8 × 4 ones = 32 ones)
Domain
Cluster
Content
Standards
240 Multiply the tens (8 × 3 tens = 24 tens or 240
1600 Multiply the hundreds (8 × 2 hundreds = 16 hundreds or 1600)
1872 Add the partial products
The multiplication can also be done without listing the partial products by multiplying the
value of each digit from one factor by the value of each digit from the other factor.
Understanding of place value is vital in using the standard algorithm.
In using the standard algorithm for multiplication, when multiplying the ones, 32 ones is 3
tens and 2 ones. The 2 is written in the ones place. When multiplying the tens, the 24 tens is 2
hundreds and 4 tens. But, the 3 tens from the 32 ones need to be added to these 4 tens, for 7
tens. Multiplying the hundreds, the 16 hundreds is 1 thousand and 6 hundreds. But, the 2
hundreds from the 24 tens need to be added to these 6 hundreds, for 8 hundreds.
15.34
+ 12.9
16.63
To help students add and subtract decimals correctly, have them first estimate the
sum or difference. Providing students with a decimal-place value chart will
enable them to place the digits in the proper place.
234
×8
1872
As students developed efficient strategies to do whole number operations, they should also
develop efficient strategies with decimal operations.
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Students should learn to estimate decimal computations before they compute with pencil and
paper. The focus on estimation should be on the meaning of the numbers and the operations,
not on how many decimal places are involved. For example, to estimate the product of 32.84
× 4.6, the estimate would be more than 120, closer to 150.
Domain
Cluster
Content
Standards
Number and Operations - Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to
produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (a
2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using
visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the
reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. d + bc)/bd.)
Instructional Strategies
Instructional Resources/Tools
To add or subtract fractions with unlike denominators, students use their understanding of
From the National Library of Virtual Manipulatives - Fraction Bars – Learn
equivalent fractions to create fractions with the same denominators. Start with problems that
about fractions using fraction bars.
require the changing of one of the fractions and progress to changing both fractions. Allow
students to add and subtract fractions using different strategies such as number lines, area
From the National Library of Virtual Manipulatives - Fractions - Adding –
models, fraction bars or strips. Have students share their strategies and discuss commonalities Illustrates what it means to find a common denominator and combine.
in them.
From the National Library of Virtual Manipulatives - Number Line Bars – Use
Students need to develop the understanding that when adding or subtracting fractions, the
bars to show addition, subtraction, multiplication, and division on a number line.
fractions must refer to the same whole. Any models used must refer to the same whole.
Students may find that a circular model might not be the best model when adding or
subtracting fractions.
Common Misconceptions
Students often mix models when adding, subtracting or comparing fractions.
As with solving word problems with whole number operations, regularly present word
Students will use a circle for thirds and a rectangle for fourths when comparing
problems involving addition or subtraction of fractions. The concept of adding or subtracting
fractions with thirds and fourths. Remind students that the representations need
fractions with unlike denominators will develop through solving problems. Mental
to be from the same whole models with the same shape and size.
computations and estimation strategies should be used to determine the reasonableness of
answers. Students need to prove or disprove whether an answer provided for a problem is
reasonable.
Estimation is about getting useful answers, it is not about getting the right answer. It is
important for students to learn which strategy to use for estimation. Students need to think
about what might be a close answer.
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Domain
Cluster
Content
Standards
Number and Operations - Fractions
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to
answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as
the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of
size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole
numbers does your answer lie?
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For
example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In
general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is
the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as
rectangular areas.
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by
whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the
given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the
problem.
7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a
visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 =
1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual
fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using
visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of
chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Instructional Strategies
Instructional Resources/Tools
Connect the meaning of multiplication and division of fractions with whole-number
The National Library of Virtual Manipulatives: contains Java applets and
multiplication and division. Consider area models of multiplication and both sharing and
activities for K-12 mathematics.
measuring models for division.
Fractions - Rectangle Multiplication – students can visualize and practice
Ask questions such as, ―What does 2 × 3 mean?‖ and ―What does 12 ’ 3 mean?‖ Then,
multiplying fractions using an area representation.
follow with questions for multiplication with fractions, such as, ―What does 34 ×13 mean?‖
―What does 34 × 7 mean?‖ (7 sets of 34 ) and What does 7 × 34 mean?‖ (34 of a set of 7)
Number Line Bars – Fractions: students can divide fractions using number line
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MATH
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bars.
The meaning of 4 ÷ 12 (how many 12 are in 4) and 12 ÷ 4(how many groups of 4 are in 12 )
also should be illustrated with models or drawings like:
34 7 groups of 34 or 214
Encourage students to use models or drawings to multiply or divide with fractions. Begin
with students modeling multiplication and division with whole numbers. Have them explain
how they used the model or drawing to arrive at the solution.
Models to consider when multiplying or dividing fractions include, but are not limited to: area
models using rectangles or squares, fraction strips/bars and sets of counters.
Use calculators or models to explain what happens to the result of multiplying a whole
number by a fraction (3 × 12 , 4 × 12 , 5 × 12 …and 4 ×12 , 4 × 13 , 4 × 14 ,…) and when
multiplying a fraction by a number greater than 1.
Use calculators or models to explain what happens to the result when dividing a unit fraction
by a non-zero whole number (18 ’ 4, 18 ’ 8, 18 ’ 16,…) and what happens to the result when
dividing a whole number by a unit fraction (4 ’ 14 , 8 ’ 14 12 ’ 14 ,…).
ORC # 5812 Divide and Conquer - Students can better understand the algorithm
for dividing fractions if they analyze division through a sequence of problems
starting with division of whole numbers, followed by division of a whole number
by a unit fraction, division of a whole number by a non-unit fraction, and finally
division of a fraction by a fraction (addressed in Grade 6).
Common Misconceptions
Students may believe that multiplication always results in a larger number. Using
models when multiplying with fractions will enable students to see that the
results will be smaller.
Additionally, students may believe that division always results in a smaller
number. Using models when dividing with fractions will enable students to see
that the results will be larger.
Present problem situations and have students use models and equations to solve the problem.
It is important for students to develop understanding of multiplication and division of
fractions through contextual situations.
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GRADE 5
Domain
Measurement and Data
Cluster
Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in
Content
solving multi-step, real world problems.
Standards
Instructional Strategies
Instructional Resources/Tools
Students should gain ease in converting units of measures in equivalent forms within the
Yardsticks(meter sticks) and rulers (marked with customary and metric units)
same system. To convert from one unit to another unit, the relationship between the units
must be known. In order for students to have a better understanding of the relationships
Teaspoons and tablespoons
between units, they need to use measuring tools in class. The number of units must relate to
the size of the unit. For example, students have discovered that there are 12 inches in 1 foot
Graduated measuring cups (marked with customary and metric units)
and 3 feet in 1 yard. This understanding is needed to convert inches to yards. Using 12-inch
rulers and yardsticks, students can see that three of the 12-inch rulers are equivalent to one
From the National Council of Teachers of Mathematics, Illuminations: yardstick (3 × 12 inches = 36 inches; 36 inches = 1 yard). Using this knowledge, students can Discovering Gallon Man. Students experiment with units of liquid measure used
decide whether to multiply or divide when making conversions.
in the customary system of measurement. They practice making volume
conversions in the customary system.
Once students have an understanding of the relationships between units and how to do
conversions, they are ready to solve multi-step problems that require conversions within the
From the National Council of Teachers of Mathematics, Illuminations: – Do You
same system. Allow students to discuss methods used in solving the problems. Begin with
Measure Up? Students learn the basics of the metric system. They identify which
problems that allow for renaming the units to represent the solution before using problems
units of measurement are used to measure specific objects, and they learn to
that require renaming to find the solution.
convert between units within the same system.
Common Misconceptions
When solving problems that require renaming units, students use their
knowledge of renaming the numbers as with whole numbers. Students need to
pay attention to the unit of measurement which dictates the renaming and the
number to use. The same procedures used in renaming whole numbers should not
be taught when solving problems involving measurement conversions. For
example, when subtracting 5 inches from 2 feet, students may take one foot from
the 2 feet and use it as 10 inches. Since there were no inches with the 2 feet, they
put 1 with 0 inches and make it 10 inches.
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MATH
GRADE 5
Domain
Cluster
Content
Standards
Measurement and Data
Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems
involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker
would contain if the total amount in all the beakers were redistributed equally.
Instructional Strategies
Instructional Resources/Tools
Using a line plot to solve problems involving operations with unit fractions now includes
From the National Council of Teachers of Mathematics, Illuminations: Fractions
multiplication and division. Revisit using a number line to solve multiplication and division
in Every Day Life - This activity enables students to apply their knowledge about
problems with whole numbers. In addition to knowing how to use a number line to solve
fractions to a real-life situation. It also provides a good way for teachers to assess
problems, students also need to know which operation to use to solve problems.
students' working knowledge of fraction multiplication and division. Students
should have prior knowledge of adding, subtracting, multiplying, and dividing
Use the tables for common addition and subtraction, and multiplication and division
fractions before participating in this activity. This will help students to think
situations (Table 1 and Table 2 in the Common Core State Standards for Mathematics) as a
about how they use fractions in their lives, sometimes without even realizing it.
guide to the types of problems students need to solve without specifying the type of problem.
The basic idea behind this activity is to use a recipe and alter it to serve larger or
Allow students to share methods used to solve the problems. Also have students create
smaller portions.
problems to show their understanding of the meaning of each operation.
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MATH
GRADE 5
Domain
Cluster
Content
Standards
Measurement and Data
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
3. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
a. A cube with side length 1 unit, called a ―unit cube,‖ is said to have ―one cubic unit‖ of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as
would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number
products as volumes, e.g., to represent the associative property of multiplication.
b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the
context of solving real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the
non-overlapping parts, applying this technique to solve real world problems.
Instructional Strategies
Instructional Resources/Tools
Volume refers to the amount of space that an object takes up and is measured in cubic units
Cubes
such as cubic inches or cubic centimeters.
Students need to experience finding the volume of rectangular prisms by counting unit cubes, Rulers (marked in standard or metric units)
in metric and standard units of measure, before the formula is presented. Provide multiple
opportunities for students to develop the formula for the volume of a rectangular prism with
activities similar to the one described below.
Give students one block (a I- or 2- cubic centimeter or cubic-inch cube), a ruler with the
appropriate measure based on the type of cube, and a small rectangular box. Ask students to
determine the number of cubes needed to fill the box. Have students share their strategies
with the class using words, drawings or numbers. Allow them to confirm the volume of the
box by filling the box with cubes of the same size.
Grid paper
http://illuminations.nctm.org/ActivityDetail.aspx?ID=6 - Determining the
Volume of a Box by Filling It with Cubes, Rows of Cubes, or Layers of Cubes
By stacking geometric solids with cubic units in layers, students can begin understanding the
concept of how addition plays a part in finding volume. This will lead to an understanding of
the formula for the volume of a right rectangular prism, b x h, where b is the area of the base.
A right rectangular prism has three pairs of parallel faces that are all rectangles.
Have students build a prism in layers. Then, have students determine the number of cubes in
the bottom layer and share their strategies. Students should use multiplication based on their
knowledge of arrays and its use in multiplying two whole numbers.
Ask what strategies can be used to determine the volume of the prism based on the number of
cubes in the bottom layer. Expect responses such as ―adding the same number of cubes in
each layer as were on the bottom layer‖ or multiply the number of cubes in one layer times
the number of layers.
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MATH
GRADE 5
Domain
Cluster
Content
Standards
Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with
the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number
indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis,
with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in
the context of the situation.
Instructional Strategies
Instructional Resources/Tools
Students need to understand the underlying structure of the coordinate system and see how
Grid/graph paper
axes make it possible to locate points anywhere on a coordinate plane. This is the first time
students are working with coordinate planes, and only in the first quadrant. It is important that From the National Council of Teachers of Mathematics, Illuminations: Finding
students create the coordinate grid themselves. This can be related to two number lines and
Your Way Around - Students explore two-dimensional space via an activity in
reliance on previous experiences with moving along a number line.
which they navigate the coordinate plane.
Multiple experiences with plotting points are needed. Provide points plotted on a grid and
have students name and write the ordered pair. Have students describe how to get to the
location. Encourage students to articulate directions as they plot points.
Present real-world and mathematical problems and have students graph points in the first
quadrant of the coordinate plane. Gathering and graphing data is a valuable experience for
students. It helps them to develop an understanding of coordinates and what the overall graph
represents. Students also need to analyze the graph by interpreting the coordinate values in
the context of the situation.
From the National Council of Teachers of Mathematics, Illuminations: Describe
the Way – In this lesson, students will review plotting points and labeling axes.
Students generate a set of random points all located in the first quadrant.
Common Misconceptions
When playing games with coordinates or looking at maps, students may think the
order in plotting a coordinate point is not important. Have students plot points so
that the position of the coordinates is switched. For example, have students plot
(3, 4) and (4, 3) and discuss the order used to plot the points. Have students
create directions for others to follow so that they become aware of the
importance of direction and distance.
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MATH
GRADE 5
Domain
Cluster
Content
Standards
Geometry
Classify two-dimensional figures into categories based on their properties.
3. Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles
have four right angles and squares are rectangles, so all squares have four right angles.
4. Classify two-dimensional figures in a hierarchy based on properties.
Instructional Strategies
Instructional Resources/Tools
This cluster builds from Grade 3 when students described, analyzed and compared properties
Rectangles and Parallelograms: Students use dynamic software to examine the
of two-dimensional shapes. They compared and classified shapes by their sides and angles,
properties of rectangles and parallelograms, and identify what distinguishes a
and connected these with definitions of shapes. In Grade 4 students built, drew and analyzed
rectangle from a more general parallelogram. Using spatial relationships, they
two-dimensional shapes to deepen their understanding of the properties of two-dimensional
will examine the properties of two-and three-dimensional shapes.
shapes. They looked at the presence or absence of parallel and perpendicular lines or the
presence or absence of angles of a specified size to classify two-dimensional shapes. Now,
http://illuminations.nctm.org/LessonDetail.aspx?ID=L270 - In this lesson,
students classify two-dimensional shapes in a hierarchy based on properties. Details learned
students classify polygons according to more than one property at a time. In the
in earlier grades need to be used in the descriptions of the attributes of shapes. The more ways context of a game, students move from a simple description of shapes to an
that students can classify and discriminate shapes, the better they can understand them. The
analysis of how properties are related.
shapes are not limited to quadrilaterals.
Students can use graphic organizers such as flow charts or T-charts to compare and contrast
the attributes of geometric figures. Have students create a T-chart with a shape on each side.
Have them list attributes of the shapes, such as number of side, number of angles, types of
lines, etc. they need to determine what‘s alike or different about the two shapes to get a larger
classification for the shapes.
Common Misconceptions
Students think that when describing geometric shapes and placing them in
subcategories, the last category is the only classification that can be used.
Pose questions such as, ―Why is a square always a rectangle?‖ and ―Why is a rectangle not
always a square?‖
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MATH
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Domain
Cluster
Content
Standards
Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to
beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received
nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For
example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15
hamburgers, which is a rate of $5 per hamburger.”
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double
number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values
on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate,
how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole,
given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Instructional Strategies
Instructional Resources/Tools
Proportional reasoning is a process that requires instruction and practice. It does not develop over
100 grids (10 x 10) for modeling percents
time on its own. Grade 6 is the first of several years in which students develop this multiplicative
thinking. Examples with ratio and proportion must involve measurements, prices and geometric
Ratio tables – to use for proportional reasoning
contexts, as well as rates of miles per hour or portions per person within contexts that are relevant to
sixth graders. Experience with proportional and non-proportional relationships, comparing and
Bar Models – for example, 4 red bars to 6 blue bars as a visual
predicting ratios, and relating unit rates to previously learned unit fractions will facilitate the
representation of a ratio and then expand the number of bars to show
development of proportional reasoning. Although algorithms provide efficient means for finding
other equivalent ratios.
solutions, the cross-product algorithm commonly used for solving proportions will not aid in the
development of proportional reasoning. Delaying the introduction of rules and algorithms will
Something Fishy - ORC #257: Students will estimate the size of a large
encourage thinking about multiplicative situations instead of indiscriminately applying rules.
population by applying the concepts of ratio and proportion through the
capture-recapture statistical procedure.
Students develop the understanding that ratio is a comparison of two numbers or quantities. Ratios
that are written as part-to-whole are comparing a specific part to the whole. Fractions and percents are How Many Noses Are in Your Arm? - ORC # 130: Students will apply
examples of part-to-whole ratios. Fractions are written as the part being identified compared to the
the concept of ratio and proportion to determine the length of the Statue
whole amount. A percent is the part identified compared to the whole (100). Provide students with
of Liberty‘s torch-bearing arm.
multiple examples of ratios, fractions and percents of this type. For example, the number of girls in
the class (12) to the number of students in the class (28) is the ratio 12 to 28.
If You Hopped Like a Frog - This book introduces the concepts of ratio
and proportion by comparing what humans would be able to do if they
Percents are often taught in relationship to learning fractions and decimals. This cluster indicates that
had the capabilities of different animals.
percents are to be taught as a special type of rate. Provide students with opportunities to find percents
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in the same ways they would solve rates and proportions.
Part-to-part ratios are used to compare two parts. For example, the number of girls in the class (12)
compared to the number of boys in the class (16) is the ratio the ratio 12 to 16. This form of ratios is
often used to compare the event that can happen to the event that cannot happen.
Rates, a relationship between two units of measure, can be written as ratios, such as miles per hour,
ounces per gallon and students per bus. For example, 3 cans of pudding cost $2.48 at Store A and 6
cans of the same pudding costs $4.50 at Store B. Which store has the better buy on these cans of
pudding?
Various strategies could be used to solve this problem:
• A student can determine the unit cost of 1 can of pudding at each store and compare.
• A student can determine the cost of 6 cans of pudding at Store A by doubling $2.48.
• A student can determine the cost of 3 cans of pudding at Store B by taking ½ of $4.50.
Using ratio tables develops the concept of proportion. By comparing equivalent ratios, the concept of
proportional thinking is developed and many problems can be easily solved.
Brain Pop, Compass Learning, You Tube, Math Made Easy, United
Streaming, PowerPoint, Teacher generated worksheets, Buckle Down,
Math Textbook, Textbook Compatible Worksheets, Manipulatives
Common Misconceptions
Fractions and ratios may represent different comparisons. Fractions
always express a part-to-whole comparison, but ratios can express a partto-whole comparison or a part-to-part comparison.
Even though ratios and fractions express a part-to-whole comparison, the
addition of ratios and the addition of fractions are distinctly different
procedures. When adding ratios, the parts are added, the wholes are
added and then the total part is compared to the total whole. For example,
(2 out of 3 parts) + (4 out of 5 parts) is equal to six parts out of 8 total
parts (6 out of 8) if the parts are equal. When dealing with fractions, the
procedure for addition is based on a common denominator: (23) + (45) =
(1015) + (1215) which is equal to (2215). Therefore, the addition process
for ratios and for fractions is distinctly different.
Often there is a misunderstanding that a percent is always a natural
number less than or equal to 100. Provide examples of percent amounts
that are greater than 100%, and percent amounts that are less 1%.
Students should also solve real-life problems involving measurement units that need to be converted.
Representing these measurement conversions with models such as ratio tables, t-charts or double
number line diagrams will help students internalize the size relationships between same system
measurements and relate the process of converting to the solution of a ratio.
Multiplicative reasoning is used when finding the missing element in a proportion. For example, use
2 cups of syrup to 5 cups of water to make fruit punch. If 6 cups of syrup are used to make punch,
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how many cups of water are needed?
Other ways to illustrate ratios that will help students see the relationships follow. Begin written
representation of ratios with the words ―out of‖ or ―to‖ before using the symbolic notation of the
colon and then the fraction bar; for example, 3 out of 7, 3 to 5, 6:7 and then 4/5.
Use skip counting as a technique to determine if ratios are equal.
Labeling units helps students organize the quantities when writing proportions.
3 eggs, 2 cups of flour = z eggs, 8 cups of flour
Using hue/color intensity is a visual way to examine ratios of part-to-part. Students can compare the
intensity of the color green and relate that to the ratio of colors used. For example, have students mix
green paint into white paint in the following ratios: 1 part green to 5 parts white, 2 parts green to 3
parts white, and 3 parts green to 7 parts white. Compare the green color intensity with their ratios.
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Domain
Cluster
Content
Standards
The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and
equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the
relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a
rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Instructional Strategies
Instructional Resources/Tools
Computation with fractions is best understood when it builds upon the familiar
Models for Multiplying and Dividing Fractions - This teacher resource gives
understandings of whole numbers and is paired with visual representations. Solve a simpler
shows how the area model can be used in multiplication and division of fractions.
problem with whole numbers, and then use the same steps to solve a fraction divided by a
There is also a section on the relationship to decimals.
fraction. Looking at the problem through the lens of ―How many groups?‖ or ―How many in
each group?‖ helps visualize what is being sought.
From the National Library of Virtual Manipulatives: Fractions - Rectangle
Multiplication - Use this virtual manipulative to graphically demonstrate,
explore, and practice multiplying fractions.
Common Misconceptions
Students may believe that dividing by is the same as dividing in half. Dividing
by half means to find how many s there are in a quantity, whereas, dividing in
half means to take a quantity and split it into two equal parts. Thus 7 divided by
= 14 and 7 divided in half equals 3 .
Teaching ―invert and multiply‖ without developing an understanding of why it works first
leads to confusion as to when to apply the shortcut.
Learning how to compute fraction division problems is one part, being able to relate the
problems to real-world situations is important. Providing opportunities to create stories for
fraction problems or writing equations for situations is needed.
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GRADE 6
Domain
Cluster
Content
Standards
The Number System
Compute fluency with multi-digit numbers and find common factors and multiples.
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to
12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no
common factor. For example, express 36 + 8 as 4 (9 + 2).
Instructional Strategies
Instructional Resources/Tools
As students study whole numbers in the elementary grades, a foundation is laid in the
Greatest Common Factor Lesson: This lesson is a resource for teachers or for
conceptual understanding of each operation. Discovering and applying multiple strategies for
students after participating in lessons exploring GCF.
computing creates connections which evolve into the proficient use of standard algorithms.
Fluency with an algorithm denotes an ability that is efficient, accurate, appropriate and
flexible. Division was introduced in Grade 3 conceptually, as the inverse of multiplication. In Common Misconceptions
Grade 4, division continues using place-value strategies, properties of operations, the
Students have difficulty deciphering from grouping base numbers with higher or
relationship with multiplication, area models, and rectangular arrays to solve problems with
lower exponents when looking for Greatest Common Factor and Least Common
one digit divisors. In Grade 6, fluency with the algorithms for division and all operations with Multiples.
decimals is developed.
Fluency is something that develops over time; practice should be given over the course of the
year as students solve problems related to other mathematical studies. Opportunities to
determine when to use paper pencil algorithms, mental math or a computing tool is also a
necessary skill and should be provided in problem solving situations.
Greatest common factor and least common multiple are usually taught as a means of
combining fractions with unlike denominators. This cluster builds upon the previous learning
of the multiplicative structure of whole numbers, as well as prime and composite numbers in
Grade 4. Although the process is the same, the point is to become aware of the relationships
between numbers and their multiples. For example, consider answering the question: ―If two
numbers are multiples of four, will the sum of the two numbers also be a multiple of four?‖
Being able to see and write the relationships between numbers will be beneficial as further
algebraic understandings are developed. Another focus is to be able to see how the GCF is
useful in expressing the numbers using the distributive property, (36 + 24) = 12(3+2), where
12 is the GCF of 36 and 24. This concept will be extended in Expressions and Equations as
work progresses from understanding the number system and solving equations to simplifying
and solving algebraic equations in Grade 7.
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Domain
Cluster
Content
Standards
The Number System
Apply and extend previous understandings of numbers to the system of rational numbers. (cont.)
5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below
zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent
points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a
number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only
by signs, the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational
numbers on a coordinate plane.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a
statement that –3 is located to the right of –7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3ºC > –7ºC to express the fact that –3ºC
is warmer than –7ºC.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or
negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a
debt greater than 30 dollars.
8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to
find distances between points with the same first coordinate or the same second coordinate.
Instructional Strategies
The purpose of this cluster is to begin study of the existence of negative numbers, their
relationship to positive numbers, and the meaning and uses of absolute value. Starting with
examples of having/owing and above/below zero sets the stage for understanding that there is
a mathematical way to describe opposites. Students should already be familiar with the
counting numbers (positive whole numbers and zero), as well as with fractions and decimals
(also positive). They are now ready to understand that all numbers have an opposite. These
special numbers can be shown on vertical or horizontal number lines, which then can be used
to solve simple problems. Demonstration of understanding of positives and negatives
involves translating among words, numbers and models: given the words ―7 degrees below
zero,‖ showing it on a thermometer and writing -7; given -4 on a number line, writing a reallife example and mathematically -4. Number lines also give the opportunity to model absolute
value as the distance from zero.
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Simple comparisons can be made and order determined. Order can also be established and
written mathematically: -3° C > -5° C or -5° C < -3° C. Finally, absolute values should be
used to relate contextual problems to their meanings and solutions.
Using number lines to model negative numbers, prove the distance between opposites, and
understand the meaning of absolute value easily transfers to the creation and usage of fourquadrant coordinate grids. Points can now be plotted in all four quadrants of a coordinate
grid. Differences between numbers can be found by counting the distance between numbers
on the grid. Actual computation with negatives and positives is handled in Grade 7.
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Domain
Cluster
Content
Standards
Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5”
as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a
single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic
operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order
(Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.
3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the
equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties
of operations to y + y + y to produce the equivalent expression 3y.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For
example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Instructional Strategies
Instructional Resources/Tools
The skills of reading, writing and evaluating expressions are essential for future work with
From the National Library of Virtual Manipulatives: Online algebra tiles that can
expressions and equations, and are a Critical Area of Focus for Grade 6. In earlier grades,
be used to represent expressions and equations.
students added grouping symbols ( ) to reduce ambiguity when solving equations. Now the
focus is on using ( ) to denote terms in an expression or equation. Students should now focus
Online game Late Delivery - In this game, the student helps the mail carrier
on what terms are to be solved first rather than invoking the PEMDAS rule. Likewise, the
deliver five letters to houses with numbers such as 3(a + 2).
division symbol (3 ÷ 5) was used and should now be replaced with a fraction bar (3/5). Less
confusion will occur as students write algebraic expressions and equations if x represents only
Common Misconceptions
preferred.
Many of the misconceptions when dealing with expressions stem from the
misunderstanding/reading of the expression. For example, knowing the
Provide opportunities for students to write expressions for numerical and real-world
operations that are being referenced with notation like, x³, 4x, 3(x + 2y) is critical.
situations. Write multiple statements that represent a given algebraic expression. For
The fact that x³ means x x x, means x times x times x, not 3x or 3 times x; 4x
example, the expression x – 10 could be written as ―ten less than a number,‖ ―a number minus means 4 times x or x+x+x+x, not forty-something. When evaluating 4x when x =
ten,‖ ―the temperature fell ten degrees,‖‘, ―I scored ten fewer points than my brother,‖ etc.
7, substitution does not result in the expression meaning 47. Use of the ―x‖
Students should also read an algebraic expression and write a statement.
notation as both the variable and the operation of multiplication can complicate
this understanding.
Through modeling, encourage students to use proper mathematical vocabulary when
discussing terms, factors, coefficients, etc.
Provide opportunities for students to write equivalent expressions, both numerically and with
variables. For example, given the expression x + x + x + x + 4•2, students could write 2x + 2x
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+ 8 or some other equivalent expression. Make the connection to the simplest form of this
expression as 4x + 8. Because this is a foundational year for building the bridge between the
concrete concepts of arithmetic and the abstract thinking of algebra, using hands-on materials
(such as algebra tiles, counters, unifix cubes, "Hands on Algebra") to help students translate
between concrete numerical representations and abstract symbolic representations is critical.
Provide expressions and formulas to students, along with values for the variables so students
can evaluate the expression. Evaluate expressions using the order of operations with and
without parentheses. Include whole-number exponents, fractions, decimals, etc. Provide a
model that shows step-by-step thinking when simplifying an expression. This demonstrates
how two lines of work maintain equivalent algebraic expressions and establishes the need to
have a way to review and justify thinking.
Provide a variety of expressions and problem situations for students to practice and deepen
their skills. Start with simple expressions to evaluate and move to more complex expressions.
Likewise start with simple whole numbers and move to fractions and decimal numbers. The
use of negatives and positives should mirror the level of introduction in Grade 6 The Number
System; students are developing the concept and not generalizing operation rules.
The use of technology can assist in the exploration of the meaning of expressions. Many
calculators will allow you to store a value for a variable and then use the variable in
expressions. This enables the student to discover how the calculator deals with expressions
like X², 5x, xy, and 2(x + 5).
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Domain
Cluster
Content
Standards
Expressions and Equations
Reason about and solve one-variable equations and inequalities.
5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality
true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an
unknown number, or, depending on the purpose at hand, any number in a specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all
nonnegative rational numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the
form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Instructional Strategies
The skill of solving an equation must be developed conceptually before it is developed
procedurally. This means that students should be thinking about what numbers could possibly
be a solution to the equation before solving the equation. For example, in the equation x + 21
= 32 students know that 21 + 9 = 30 therefore the solution must be 2 more than 9 or 11, so x =
11.
Provide multiple situations in which students must determine if a single value is required as a
solution, or if the situation allows for multiple solutions. This creates the need for both types
of equations (single solution for the situation) and inequalities (multiple solutions for the
situation). Solutions to equations should not require using the rules for operations with
negative numbers since the conceptual understanding of negatives and positives is being
introduced in Grade 6. When working with inequalities, provide situations in which the
solution is not limited to the set of positive whole numbers but includes rational numbers. This
is a good way to practice fractional numbers and introduce negative numbers. Students need to
be aware that numbers less than zero could be part of a solution set for a situation. As an
extension to this concept, certain situations may require a solution between two numbers. For
example, a problem situation may have a solution that requires more than 10 but not greater
than 25. Therefore, the exploration with students as to what this would look like both on a
number line and symbolically is a reasonable extension.
The process of translating between mathematical phrases and symbolic notation will also
assist students in the writing of equations/inequalities for a situation. This process should go
both ways; Students should be able to write a mathematical phrase for an equation.
Additionally, the writing of equations from a situation or story does not come naturally for
many students. A strategy for assisting with this is to give students an equation and ask them
to come up with the situation/story that the equation could be referencing.
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Domain
Cluster
Content
Standards
Expressions and Equations
Represent and analyze quantitative relationships between dependent and independent variables.
9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent
and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and
graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Instructional Strategies
Instructional Resources/Tools
The goal is to help students connect the pieces together. This can be done by having students
Use graphic organizers as tools for connecting various representations.
use multiple representations for the mathematical relationship. Students need to be able to
Pedal Power – NCTM illuminations lesson on translating a graph to a story.
translate freely among the story, words (mathematical phrases), models, tables, graphs and
equations. They also need to be able to start with any of the representations and develop the
others.
Common Misconceptions
Students may misunderstand what the graph represents in context. For example,
Provide multiple situations for the student to analyze and determine what unknown is
that moving up or down on a graph does not necessarily mean that a person is
dependent on the other components. For example, how far I travel is dependent on the time
moving up or down.
and rate that I am traveling.
Throughout the expressions and equations domain in Grade 6, students need to have an
understanding of how the expressions or equations relate to situations presented, as well as
the process of solving them.
The use of technology, including computer apps, CBLs, and other hand-held technology
allows the collection of real-time data or the use of actual data to create tables and charts. It is
valuable for students to realize that although real-world data often is not linear, a line
sometimes can model the data.
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Domain
Cluster
Content
Standards
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume.
1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other
shapes; apply these techniques in the context of solving real-world and mathematical problems.
2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show
that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of
right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first
coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these
techniques in the context of solving real-world and mathematical problems.
Instructional Strategies
Instructional Resources/Tools
It is very important for students to continue to physically manipulate materials and make
Squares that can be joined together used to develop possible nets for a cube.
connections to the symbolic and more abstract aspects of geometry. Exploring possible nets
should be done by taking apart (unfolding) three-dimensional objects. This process is also
Use floor plans as a real world situation for finding the area of composite shapes.
foundational for the study of surface area of prisms. Building upon the understanding that a
net is the two-dimensional representation of the object, students can apply the concept of area Online dot paper:
to find surface area. The surface area of a prism is the sum of the areas for each face.
http://illuminations.nctm.org/lessons/DotPaper.pdf#search=%22dot paper%22
Multiple strategies can be used to aid in the skill of determining the area of simple twodimensional composite shapes. A beginning strategy should be to use rectangles and
triangles, building upon shapes for which they can already determine area to create composite
shapes. This process will reinforce the concept that composite shapes are created by joining
together other shapes, and that the total area of the two-dimensional composite shape is the
sum of the areas of all the parts.
A follow-up strategy is to place a composite shape on grid or dot paper. This aids in the
decomposition of a shape into its foundational parts. Once the composite shape is
decomposed, the area of each part can be determined and the sum of the area of each part is
the total area.
ORC # 5279, #5280, #5281, lessons on area:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L580
Common Misconceptions
Students may believe that the orientation of a figure changes the figure. In Grade
6, some students still struggle with recognizing common figures in different
orientations. For example, a square rotated 45° is no longer seen as a square and
instead is called a diamond.
This impacts students‘ ability to decompose composite figures and to
appropriately apply formulas for area. Providing multiple orientations of objects
within classroom examples and work is essential for students to overcome this
misconception.
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Domain
Cluster
Content
Standards
Statistics and Probability
Solve real-world and mathematical problems involving area, surface area, and volume. (cont.)
1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old
am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its
values vary with a single number.
Instructional Strategies
Instructional Resources/Tools
Grade 6 is the introduction to the formal study of statistics for students. Students need multiple
From the National Council of Teachers of Mathematics, Illuminations:
opportunities to look at data to determine and word statistical questions. Data should be
Numerical and Categorical Data - In this unit of three lessons, students
analyzed from many sources, such as organized lists, box-plots, bar graphs and stem-and-leaf
formulate and refine questions, and collect, display and analyze data.
plots. This will help students begin to understand that responses to a statistical question will
ORC # 391, 392, 393
vary, and that this variability is described in terms of spread and overall shape. At the same
time, students should begin to relate their informal knowledge of mean, mode and median to
Data Analysis and Probability Virtual Manipulatives Grades 6-8
understand that data can also be described by single numbers. The single value for each of the
#5048 Students can use the appropriate applet from this page of virtual
measures of center (mean, median or mode) and measures of spread (range, interquartile range,
manipulatives to create graphical displays of the data set. This provides an
mean absolute deviation) is used to summarize the data. Given measures of center for a set of
important visual display of the data without requiring students to spend time
data, students should use the value to describe the data in words. The important purpose of the
hand-drawing the display. Classroom time can then be spent discussing the
number is not the value itself, but the interpretation it provides for the variation of the data.
patterns and variability of the data.
Interpreting different measures of center for the same data develops the understanding of how
each measure sheds a different light on the data. The use of a similarity and difference matrix to
compare mean, median, mode and range may facilitate understanding the distinctions of
Common Misconceptions
purpose between and among the measures of center and spread.
Students may believe all graphical displays are symmetrical. Exposing
students to graphs of various shapes will show this to be false.
Include activities that require students to match graphs and explanations, or measures of center
and explanations prior to interpreting graphs based upon the computation measures of center or
The value of a measure of center describes the data, rather than a value used to
spread. The determination of the measures of center and the process for developing graphical
interpret and describe the data.
representation is the focus of the cluster ―Summarize and describe distributions‖ in the Statistics
and Probability domain for Grade 6. Classroom instruction should integrate the two clusters.
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Domain
Cluster
Content
Standards
Statistics and Probability
Summarize and describe distributions
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing
any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
Instructional Strategies
Instructional Resources/Tools
This cluster builds on the understandings developed in the Grade 6 cluster ―Develop
Graphing calculators may also be used for creating lists and displaying the data.
understanding of statistical variability.‖ Students have analyzed data displayed in various
ways to see how data can be described in terms of variability. Additionally, in Grades 3-5
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
students have created scaled picture and bar graphs, as well as line plots. Now students learn
report. American Statistical Association
to organize data in appropriate representations such as box plots (box-and-whisker plots), dot
plots, and stem-and-leaf plots. Students need to display the same data using different
Ohio Resource Center:
representations. By comparing the different graphs of the same data, students develop
Hollywood Box Office #10112. This rich problem focuses on measures of center
understanding of the benefits of each type of representation.
and graphical displays.
Further interpretation of the variability comes from the range and center-of-measure numbers.
Prior to learning the computation procedures for finding mean and median, students will
benefit from concrete experiences.
To find the median visually and kinesthetically, students should reorder the data in ascending
or descending order, then place a finger on each end of the data and continue to move toward
the center by the same increments until the fingers touch. This number is the median.
The concept of mean (concept of fair shares) can be demonstrated visually and kinesthetically
by using stacks of linking cubes. The blocks are redistributed among the towers so that all
towers have the same number of blocks. Students should not only determine the range and
centers of measure, but also use these numbers to describe the variation of the data collected
from the statistical question asked. The data should be described in terms of its shape, center,
spread (range) and interquartile range or mean absolute deviation (the absolute value of each
data point from the mean of the data set). Providing activities that require students to sketch a
representation based upon given measures of center and spread and a context will help create
connections between the measures and real-life situations.
Continue to have students connect contextual situations to data to describe the data set in
Wet Heads #275. In this lesson, students create stem-and-leaf plots and back-toback stem-and-leaf plots to display data collected from an investigative activity.
Stella‘s Stumpers Basketball Team Weight #13966. This problem situation uses
the mean to determine a missing data element.
Learning Conductor Lessons - Use the interactive applets in these standardsbased lessons to improve understanding of mathematical concepts. Scroll down
to the statistics section for your specific need.
From the National Council of Teachers of Mathematics, Illuminations: Height of
Students in our Class. This lesson has students creating box-and-whisker plots
with an extension of finding measures of center and creating a stem-and-leaf
plot.
National Library of Virtual Manipulatives. Students can use the appropriate
applet from this page of virtual manipulatives to create graphical displays of the
data set. This provides an important visual display of the data without the
tediousness of the student hand drawing the display.
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words prior to computation. Therefore, determining the measures of spread and measures of
center mathematically need to follow the development of the conceptual understanding.
Students should experience data which reveals both different and identical values for each of
the measures. Students need opportunities to explore how changing a part of the data may
change the measures of center and measure of spread. Also, by discussing their findings,
students will solidify understanding of the meanings of the measures of center and measures
of variability, what each of the measures do and do not tell about a set of data, all leading to a
better understanding of their usage.
Using graphing calculators to explore bos plots (box-and-whisker plots) removes the time
intensity from their creation and permits more time to be spent on the meaning. It is important
to use the interquartile range in box plots when describing the variation of the data. The mean
absolute deviation describes the distance each point is from the mean of that data set. Patterns
in the graphical displays should be observed, as should any outliers in the data set. Students
should identify the attributes of the data and know the appropriate use of the attributes when
describing the data. Pairing contextual situations with data and its box-and-whisker plot is
essential.
Common Misconceptions
Students often use words to help them recall how to determine the measures of
center. However, student‘s lack of understanding of what the measures of center
actually represent tends to confuse them. Median is the number in the middle, but
that middle number can only be determined after the data entries are arranged in
ascending or descending order. Mode is remembered as the ―most,‖ and often
students think this means the largest value, not the ―most frequent‖ entry in the
set. Vocabulary is important in mathematics, but conceptual understanding is
equally as important. Usually the mean, mode, or median have different values,
but sometimes those values are the same.
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Domain
Cluster
Content
Standards
UNIT 1 Grade 7, The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a
horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are
oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show
that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the
number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations,
particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational
numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number.
If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations with rational numbers.
6.NS.1-4
1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and
equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the
relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a
rectangular strip of land with length 3/4 mi and area 1/2 square mi?
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to
12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no
common factor. For example, express 36 + 8 as 4 (9 + 2).
Instructional Strategies
Instructional Resources/Tools
This cluster builds upon the understandings of rational numbers in Grade 6:
Two-color counters
quantities can be shown using + or – as having opposite directions or values,
Calculators
points on a number line show distance and direction,
opposite signs of numbers indicate locations on opposite sides of 0 on the number line,
From the National Library of Virtual Manipulatives:
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the opposite of an opposite is the number itself,
the absolute value of a rational number is its distance from 0 on the number line,
the absolute value is the magnitude for a positive or negative quantity, and
locating and comparing locations on a coordinate grid by using negative and positive
numbers.
Circle 3 – A puzzle involving adding positive real numbers to sum to
three.
Circle 21 – A puzzle involving adding positive and negative integers to
sum to 21.
Learning now moves to exploring and ultimately formalizing rules for operations (addition,
subtraction, multiplication and division) with integers.
Using both contextual and numerical problems, students should explore what happens when
negatives and positives are combined. Number lines present a visual image for students to
explore and record addition and subtraction results. Two-color counters or colored chips can be
used as a physical and kinesthetic model for adding and subtracting integers. With one color
designated to represent positives and a second color for negatives, addition/subtraction can be
represented by placing the appropriate numbers of chips for the addends and their signs on a
board. Using the notion of opposites, the board is simplified by removing pairs of opposite
colored chips. The answer is the total of the remaining chips with the sign representing the
appropriate color. Repeated opportunities over time will allow students to compare the results of
adding and subtracting pairs of numbers, leading to the generalization of the rules. Fractional
rational numbers and whole numbers should be used in computations and explorations. Students
should be able to give contextual examples of integer operations, write and solve equations for
real-world problems and explain how the properties of operations apply. Real-world situations
could include: profit/loss, money, weight, sea level, debit/credit, football yardage, etc.
Using what students already know about positive and negative whole numbers and
multiplication with its relationship to division, students should generalize rules for multiplying
and dividing rational numbers. Multiply or divide the same as for positive numbers, then
designate the sign according to the number of negative factors. Students should analyze and
solve problems leading to the generalization of the rules for operations with integers.
For example, beginning with known facts, students predict the answers for related facts, keeping
in mind that the properties of operations apply (See Tables 1, 2 and 3 below).
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Table 1
4 x 4 = 16
4 x 3 = 12
4x2=8
4x1=4
4x0=0
4 x -1 =
4x-2=
4x-3=
4x-4=
Table 2
4 x 4 = 16
4 x 3 = 12
4x2=8
4x1=4
4x0=0
-4 x 1 =
-4 x 2 =
-4 x 3 =
-4 x 4 =
Table 3
-4 x -4 = 16
-4 x -3 = 12
-4 x -2 = 8
-4 x -1 = 4
-4 x 0 = 0
-1 x - 4 =
-2 x - 4 =
-3 x - 4 =
-4 x - 4 =
Using the language of ―the opposite of‖ helps some students understand the multiplication of
negatively signed numbers ( -4 x -4 = 16, the opposite of 4 groups of -4). Discussion about the
tables should address the patterns in the products, the role of the signs in the products and
commutativity of multiplication. Then students should be asked to answer these questions and
prove their responses.
Is it always true that multiplying a negative factor by a positive factor results in a negative
product?
Does a positive factor times a positive factor always result in a positive product?
What is the sign of the product of two negative factors?
When three factors are multiplied, how is the sign of the product determined?
How is the numerical value of the product of any two numbers found?
Students can use number lines with arrows and hops, groups of colored chips or logic to explain
their reasoning. When using number lines, establishing which factor will represent the length,
number and direction of the hops will facilitate understanding. Through discussion,
generalization of the rules for multiplying integers would result.
Division of integers is best understood by relating division to multiplication and applying the
rules. In time, students will transfer the rules to division situations. (Note: In 2b, this algebraic
language (–(p/q) = (–p)/q = p/(–q)) is written for the teacher‘s information, not as an
expectation for students.) Ultimately, students should solve other mathematical and real-world
problems requiring the application of these rules with fractions and decimals.
In Grade 7 the awareness of rational and irrational numbers is initiated by observing the result
of changing fractions to decimals. Students should be provided with families of fractions, such
as, sevenths, ninths, thirds, etc. to convert to decimals using long division. The equivalents can
be grouped and named (terminating or repeating). Students should begin to see why these
patterns occur. Knowing the formal vocabulary rational and irrational is not expected.
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Domain
Cluster
Content
Standards
UNIT 1 Grade 8, The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show
that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and
estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and
1.5, and explain how to continue on to get better approximations.
6.NS.5-7
5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below
zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent
points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a
number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ
only by signs, the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational
numbers on a coordinate plane.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a
statement that –3 is located to the right of –7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3ºC > –7ºC to express the fact that –3ºC
is warmer than –7ºC.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or
negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents
a debt greater than 30 dollars.
Instructional Strategies
Instructional Resources/Tools
The distinction between rational and irrational numbers is an abstract distinction, originally
Graphing calculators
based on ideal assumptions of perfect construction and measurement. In the real world,
however, all measurements and constructions are approximate. Nonetheless, it is possible to
Dynamic geometry software
see the distinction between rational and irrational numbers in their decimal representations.
A rational number is of the form a/b, where a and b are both integers, and b is not 0. In the
elementary grades, students learned processes that can be used to locate any rational number
on the number line: Divide the interval from 0 to 1 into b equal parts; then, beginning at 0,
Common Misconceptions
Some students are surprised that the decimal representation of pi does not repeat.
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count out a of those parts. The surprising fact, now, is that there are numbers on the number
line that cannot be expressed as a/b, with a and b both integers, and these are called irrational
numbers.
Students construct a right isosceles
triangle with legs of 1 unit. Using
the Pythagorean theorem, they
determine that the length of the
hypotenuse is . In the figure
below, they can rotate the
hypotenuse back to the original
number line to show that indeed
is a number on the number line.
Some students believe that if only we keep looking at digits farther and farther to
the right, eventually a pattern will emerge.
A few irrational numbers are given special names (pi and e), and much attention
is given to sqrt (2). Because we name so few irrational numbers, students
sometimes conclude that irrational numbers are unusual and rare. In fact,
irrational numbers are much more plentiful than rational numbers, in the sense
that they are ―denser‖ in the real line.
1
0
1
2
In the elementary grades, students become familiar with decimal fractions, most often with
decimal representations that terminate a few digits to the right of the decimal point. For
example, to find the exact decimal representation of 2/7, students might use their calculator to
find 2/7 = 0.2857142857…and they might guess that the digits 285714 repeat. To show that
the digits do repeat, students in Grade 7 actually carry out the long division and recognize that
the remainders repeat in a predictable pattern—a pattern that creates the repetition in the
decimal representation (see 7.NS.2.d).
Thinking about long division generally, ask students what will happen if the remainder is 0 at
some step. They can reason that the long division is complete, and the decimal representation
terminates. If the reminder is never 0, in contrast, then the remainders will repeat in a cyclical
pattern because at each step with a given remainder, the process for finding the next
remainder is always the same. Thus, the digits in the decimal representation also repeat.
When dividing by 7, there are 6 possible nonzero remainders, and students can see that the
decimal repeats with a pattern of at most 6 digits. In general, when finding the decimal
representation of m/n, students can reason that the repeating portion of decimal will have at
most n-1 digits. The important point here is that students can see that the pattern will repeat,
so they can imagine the process continuing without actually carrying it out.
Conversely, given a repeating decimal, students learn strategies for converting the decimal to
a fraction. One approach is to notice that rational numbers with denominators of 9 repeat a
single digit. With a denominator of 99, two digits repeat; with a denominator of 999, three
digits repeat, and so on. For example,
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13/99 = 0.13131313…
74/99 = 0.74747474…
237/999 = 0.237237237…
485/999 = 0.485485485…
From this pattern, students can go in the other direction, conjecturing, for example, that the
repeating decimal 0.285714285714… = 285714/999999. And then they can verify that this
fraction is equivalent to 2/7.
Once students understand that (1) every rational number has a decimal representation that
either terminates or repeats, and (2) every terminating or repeating decimal is a rational
number, they can reason that on the number line, irrational numbers (i.e., those that are not
rational) must have decimal representations that neither terminate nor repeat. And although
students at this grade do not need to be able to prove that
is irrational, they need to know
that
is irrational (see 8.EE.2), which means that its decimal representation neither
terminates nor repeats. Nonetheless, they can approximate
without using the square root
key on the calculator. Students can create tables like those below to approximate
to one,
two, and then three places to the right of the decimal point:
x
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
x2
1.00
1.21
1.44
1.69
1.96
2.25
2.56
2.89
3.24
3.61
4.00
x
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
x2
1.9600
1.9881
2.0164
2.0449
2.0736
2.1025
2.1316
2.1609
2.1904
2.2201
2.2500
x
1.410
1.411
1.412
1.413
1.414
1.415
1.416
1.417
1.418
1.419
1.420
x2
1.988100
1.990921
1.993744
1.996569
1.999396
2.002225
2.005056
2.007889
2.010724
2.013561
2.016400
From knowing that 12 = 1 and 22 = 4, or from the picture above, students can reason that there
is a number between 1 and 2 whose square is 2. In the first table above, students can see that
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between 1.4 and 1.5, there is a number whose square is 2. Then in the second table, they
locate that number between 1.41 and 1.42. And in the third table they can locate
between
1.414 and 1.415. Students can develop more efficient methods for this work. For example,
from the picture above, they might have begun the first table with 1.4. And once they see that
1.422 > 2, they do not need generate the rest of the data in the second table.
Use set diagrams to show the relationships among real, rational, irrational numbers, integers,
and counting numbers. The diagram should show that the all real numbers (numbers on the
number line) are either rational or irrational.
Given two distinct numbers, it is possible to find both a rational and an irrational number
between them.
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Domain
Cluster
Content
Standards
UNIT 1 Grade 8 Expressions and Equations
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 3–5 = 3–3 = 1/33 = 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square
roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times
as much one is than the other. For example, estimate the population of the United States as 3 108 and the population of the world as 7 109, and determine
that the world population is more than 20 times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific
notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading).
Interpret scientific notation that has been generated by technology.
6.EE.1-9
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5
– y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a
single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic
operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order
of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.
3. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the
equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of
operations to y + y + y to produce the equivalent expression 3y.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For
example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality
true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an
unknown number, or, depending on the purpose at hand, any number in a specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative
rational numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the
form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of
as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and
independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph
ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
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Instructional Strategies
Although students begin using whole-number exponents in Grades 5 and 6, it is in Grade 8
when students are first expected to know and use the properties of exponents and to extend
the meaning beyond counting-number exponents. It is no accident that these expectations are
simultaneous, because it is the properties of counting-number exponents that provide the
rationale for the properties of integer exponents. In other words, students should not be told
these properties but rather should derive them through experience and reason.
Instructional Resources/Tools
Square tiles and cubes to develop understanding of squared and cubed numbers
For counting-number exponents (and for nonzero bases), the following properties follow
directly from the meaning of exponents.
Place value charts to connect the digit value to the exponent (negative and
positive)
1. anam = an+m
2. (an)m = anm
3. anbn = (ab)n
Students should have experience simplifying numerical expressions with exponents so that
these properties become natural and obvious. For example,
Calculators to verify and explore patterns
Webquests using data mined from sites like the U.S. Census Bureau, scientific
data (planetary distances)
Powers of 10 online video.
Common Misconceptions
Students may mix up the product of powers property and the power of a power
2
3
5
6
property. Is x x equivalent to x or x ? Students may make the
relationship that in scientific notation, when a number contains one nonzero digit
and a positive exponent, that the number of zeros equals the exponent. This
pattern may incorrectly be applied to scientific notation values with negative
values or with more than one nonzero digit.
If students reason about these examples with a sense of generality about the numbers, they
begin to articulate the properties. For example, ―I see that 3 twos is being multiplied by 5
twos, and the results is 8 twos being multiplied together, where the 8 is the sum of 3 and 5,
the number of twos in each of the original factors. That would work for a base other than two
(as long as the bases are the same).‖
Note: When talking about the meaning of an exponential expression, it is easy to say
(incorrectly) that ―35 means 3 multiplied by itself 5 times.‖ But by writing out the meaning,
, students can see that there are only 4 multiplications. So a better
description is ―35 means 5 3s multiplied together.‖
Students also need to realize that these simple descriptions work only for counting-number
exponents. When extending the meaning of exponents to include 0 and negative exponents,
these descriptions are limiting: Is it sensible to say ―30 means 0 3s multiplied together‖ or that
―32 means -2 3s multiplied together‖?
The motivation for the meanings of 0 and negative exponents is the following principle: The
properties of counting-number exponents should continue to work for integer exponents.
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For example, Property 1 can be used to reasonn
what 30 should be. Consider the following
expression and simplification:
. This computation shows that the when 30 is
multiplied by 35, the result (following Property
1) should be 35, which implies that 30 must be 1.
Because this reasoning holds for any base other
than 0, we can reason that a0 = 1 for any nonzero
number a.
To make a judgment about the meaning of 3-4,
the approach is similar:
. This computation shows that 3-4 should
be the reciprocal of 34, because their product is 1.
And again, this reasoning holds for any nonzero
base. Thus, we can reason that a−n = 1/an.
Putting all of these results together, we now have
the properties of integer exponents, shown in the
above chart. For mathematical completeness, one
might prove that properties 1-3 continue to hold
for integer exponents, but that is not necessary at
this point.
A supplemental strategy for developing meaning
for integer exponents is to make use of patterns,
as shown in the chart to the right.
The meanings of 0 and negative-integer
exponents can be further explored in a placevalue chart:
Properties of Integer Exponents
For any nonzero real numbers a and b
and integers n and m:
1. anam = an+m
2. (an)m = anm
3. anbn = (ab)n
4. a0 = 1
5. a−n = 1/an
Patterns in Exponents
54
53
625
125
52
51
50
5-1
5-2
5-3
25
5
1
1/5
1/25
1/125
As the exponent decreases by 1, the
value of the expression is divided by
5, which is the base. Continue that
pattern to 0 and negative exponents.
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thousands
hundreds
tens
ones
tenths
hundredths
thousandths
MATH
GRADE 6 ADVANCED APPLICATIONS
103
3
102
2
101
4
100
7
10-1
5
10-2
6
10-3
8
.
Thus, integer exponents support writing any decimal in expanded form like the following:
.
Expanded form and the connection to place value is important for helping students make
sense of scientific notation, which allows very large and very small numbers to be written
concisely, enabling easy comparison. To develop familiarity, go back and forth between
standard notation and scientific notation for numbers near, for example, 1012 or 10-9.
Compare numbers, where one is given in scientific notation and the other is given in standard
notation. Real-world problems can help students compare quantities and make sense about
their relationship.
Provide practical opportunities for students to flexibly move between forms of squared and
cubed numbers. For example, If
symbolically and verbally.
32 9
then
9
3 . This flexibility should be experienced
Opportunities for conceptually understanding irrational numbers should be developed. One
way is for students to draw a square that is one unit by one unit and find the diagonal using
the Pythagorean Theorem. The diagonal drawn has an irrational length of √2. Other irrational
lengths can be found using the same strategy by finding diagonal lengths of rectangles with
various side lengths.
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Domain
Cluster
Content
Standards
UNIT 2, Grade 7 Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For
example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and
observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p,
the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r)
where r is the unit rate.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and
commissions, fees, percent increase and decrease, percent error.
6.RP.1-3
*1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks
in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three
votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example,
“This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is
a rate of $5 per hamburger.”
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double
number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on
the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how
many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole,
given a part and the percent.
*d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Instructional Strategies
Building from the development of rate and unit concepts in Grade 6, applications now need to
focus on solving unit-rate problems with more sophisticated numbers: fractions per fractions.
Instructional Resources/Tools
Play money - act out a problem with play money
Advertisements in newspapers
Proportional relationships are further developed through the analysis of graphs, tables,
equations and diagrams. Ratio tables serve a valuable purpose in the solution of proportional
Unlimited manipulatives or tools (don‘t restrict the tools to one or two, give
students many options)
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problems. This is the time to push for a deep understanding of what a representation of a
proportional relationship looks like and what the characteristics are: a straight line through the
origin on a graph, a ―rule‖ that applies for all ordered pairs, an equivalent ratio or an
expression that describes the situation, etc. This is not the time for students to learn to cross
multiply to solve problems.
Because percents have been introduced as rates in Grade 6, the work with percents should
continue to follow the thinking involved with rates and proportions. Solutions to problems
can be found by using the same strategies for solving rates, such as looking for equivalent
ratios or based upon understandings of decimals. Previously, percents have focused on ―out of
100‖; now percents above 100 are encountered.
Providing opportunities to solve problems based within contexts that are relevant to seventh
graders will connect meaning to rates, ratios and proportions. Examples include: researching
newspaper ads and constructing their own question(s), keeping a log of prices (particularly
sales) and determining savings by purchasing items on sale, timing students as they walk a
lap on the track and figuring their rates, creating open-ended problem scenarios with and
without numbers to give students the opportunity to demonstrate conceptual
understanding, inviting students to create a similar problem to a given problem and explain
their reasoning.
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Domain
Cluster
Content
Standards
Unit 2, Grade 7 Expressions and Equations
Use properties to generate equivalent expressions
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For
example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Instructional Strategies
Instructional Resources/Tools
Have students build on their understanding of order of operations and use the properties of
Online Algebra Tiles - Visualize multiplying and factoring algebraic expressions
operations to rewrite equivalent numerical expressions that were developed in Grade 6.
using tiles.
Students continue to use properties that were initially used with whole numbers and now
develop the understanding that properties hold for integers, rational and real numbers.
Common Misconceptions
Provide opportunities to build upon this experience of writing expressions using variables to
As students begin to build and work with expressions containing more than two
represent situations and use the properties of operations to generate equivalent expressions.
operations, students tend to set aside the order of operations. For example having
These expressions may look different and use different numbers, but the values of the
a student simplify an expression like 8 + 4(2x - 5) + 3x can bring to light several
expressions are the same.
misconceptions. Do the students immediately add the 8 and 4 before distributing
the 4? Do they only multiply the 4 and the 2x and not distribute the 4 to both
Provide opportunities for students to experience expressions for amounts of increase and
terms in the parenthesis? Do they collect all like terms
decrease. In Standard 2, the expression is rewritten and the variable has a different
8 + 4 – 5, and 2x + 3x?
coefficient. In context, the coefficient aids in the understanding of the situation. Another
example is this situation which represents a 10% decrease: b - 0.10b = 1.00b - 0.10b which
Each of these show gaps in students‘ understanding of how to simplify numerical
equals 0.90b or 90% of the amount.
expressions with multiple operations.
One method that students can use to become convinced that expressions are equivalent is by
substituting a numerical value for the variable and evaluating the expression. For example 5(3
+ 2x) is equal to:
53 + 52x
Let x = 6 and substitute 6 for x in both equations.
5(3 + 26)
53 + 526
5(3 + 12)
15 + 60
5(15)
75
75
Provide opportunities for students to use and understand the properties of operations. These
include: the commutative, associative, identity, and inverse properties of addition and of
multiplication, and the zero property of multiplication. Another method students can use to
become convinced that expressions are equivalent is to justify each step of simplification of
an expression with an operation property.
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Domain
Cluster
Content
Standards
Unit 2, Grade 7 Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solve multi-step, real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess
the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will
make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a
door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of
these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For
example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of
the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you
want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Instructional Strategies
To assist students‘ assessment of the reasonableness of answers, especially problem situations
involving fractional or decimal numbers, use whole-number approximations for the
computation and then compare to the actual computation. Connections between performing
the inverse operation and undoing the operations are appropriate here. It is appropriate to
expect students to show the steps in their work. Students should be able to explain their
thinking using the correct terminology for the properties and operations.
Instructional Resources/Tools
Solving for a Variable This activity for students uses a pan balance to model
solving equations for a variable.
Solving an Inequality This activity for students illustrates the solution to
inequalities modeled on a number line.
Continue to build on students‘ understanding and application of writing and solving one-step
equations from a problem situation to multi-step problem situations. This is also the context
for students to practice using rational numbers including: integers, and positive and negative
fractions and decimals. As students analyze a situation, they need to identify what operation
should be completed first, then the values for that computation. Each set of the needed
operation and values is determined in order. Finally an equation matching the order of
operations is written. For example, Bonnie goes out to eat and buys a meal that costs $12.50
that includes a tax of $.75. She only wants to leave a tip based on the cost of the food. In this
situation, students need to realize that the tax must be subtracted from the total cost before
being multiplied by the percent of tip and then added back to obtain the final cost. C = (12.50
- .75)(1 + T) + .75 = 11.75(1 +T) + .75 where C = cost and T = tip.
Provide multiple opportunities for students to work with multi-step problem situations that
have multiple solutions and therefore can be represented by an inequality. Students need to be
aware that values can satisfy an inequality but not be appropriate for the situation, therefore
limiting the solutions for that particular problem.
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Domain
Cluster
Content
Standards
Unit 2, Grade 8 Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in
different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the
equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on extending the understanding of ratios and proportions. Unit rates have Carnegie Math™
been explored in Grade 6 as the comparison of two different quantities with the second unit a
unit of one, (unit rate). In seventh grade unit rates were expanded to complex fractions and
Graphing calculators
percents through solving multistep problems such as: discounts, interest, taxes, tips, and
percent of increase or decrease. Proportional relationships were applied in scale drawings, and SMART™ technology with software emulator
students should have developed an informal understanding that the steepness of the graph is
the slope or unit rate. Now unit rates are addressed formally in graphical representations,
National Library of Virtual Manipulatives (NLVM)©,
algebraic equations, and geometry through similar triangles.
The National Council of Teachers of Mathematics, Illuminations
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose
Annenberg™ video tutorials, www.nsdl.org to access applets
of illustrating how the rates of two objects can be represented, analyzed and described in
different ways: graphically and algebraically. Emphasize the creation of representative graphs Texas Instruments® website (www.ticares.com)
and the meaning of various points. Then compare the same information when represented in
an equation.
By using coordinate grids and various sets of three similar triangles, students can prove that
the slopes of the corresponding sides are equal, thus making the unit rate of change equal.
After proving with multiple sets of triangles, students can be led to generalize the slope to y =
mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
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Domain
Cluster
Content
Standards
Unit 2, Grade 8 Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the
case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a
and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students applied the properties of operations to generate equivalent expressions,
SMART Board‘s new tools for solving equations
and identified when two expressions are equivalent. This cluster extends understanding to the
process of solving equations and to their solutions, building on the fact that solutions maintain Graphing calculators
equality, and that equations may have only one solution, many solutions, or no solution at all.
Equations with many solutions may be as simple as 3x = 3x, 3x + 5 = x + 2 + x + x + 3, or 6x Index cards with equations/graphs for matching and sorting
+ 4x = x(6 + 4), where both sides of the equation are equivalent once each side is simplified.
Supply and Demand This activity focuses on having students create and solve a
Table 3 on page 90 of CCSS generalizes the properties of operations and serves as a reminder system of linear equations in a real-world setting. By solving the system, students
for teachers of what these properties are. Eighth graders should be able to describe these
will find the equilibrium point for supply and demand. Students should be
relationships with real numbers and justify their reasoning using words and not necessarily
familiar with finding linear equations from two points or slope and y-intercept.
with the algebraic language of Table 3. In other words, students should be able to state that
This lesson was adapted from the October 1991 edition of Mathematics Teacher.
3(-5) = (-5)3 because multiplication is commutative and it can be performed in any order (it is
commutative), or that 9(8) + 9(2) = 9(8 + 2) because the distributive property allows us to
distribute multiplication over addition, or determine products and add them. Grade 8 is the
Common Misconceptions
beginning of using the generalized properties of operations, but this is not something on
Students think that only the letters x and y can be used for variables.
which students should be assessed.
Students think that you always need a variable = a constant as a solution
Pairing contextual situations with equation solving allows students to connect mathematical
analysis with real-life events. Students should experience analyzing and representing
The variable is always on the left side of the equation.
contextual situations with equations, identify whether there is one, none, or many solutions,
and then solve to prove conjectures about the solutions. Through multiple opportunities to
Equations are not always in the splote intercept form, y=mx+b.
analyze and solve equations, students should be able to estimate the number of solutions and
possible values(s) of solutions prior to solving. Rich problems, such as computing the number Students confuse one-variable and two-variable equations.
of tiles needed to put a border around a rectangular space or solving proportional problems as
in doubling recipes, help ground the abstract symbolism to life.
Experiences should move through the stages of concrete, conceptual and algebraic/abstract.
Utilize experiences with the pan balance model as a visual tool for maintaining equality
(balance) first with simple numbers, then with pictures symbolizing relationships, and finally
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with rational numbers allows understanding to develop as the complexity of the problems
increases. Equation-solving in Grade 8 should involve multistep problems that require the use
of the distributive property, collecting like terms, and variables on both sides of the equation.
This cluster builds on the informal understanding of slope from graphing unit rates in Grade 6
and graphing proportional relationships in Grade 7 with a stronger, more formal
understanding of slope. It extends solving equations to understanding solving systems of
equations, or a set of two or more linear equations that contain one or both of the same two
variables. Once again the focus is on a solution to the system. Most student experiences
should be with numerical and graphical representations of solutions. Beginning work should
involve systems of equations with solutions that are ordered pairs of integers, making it easier
to locate the point of intersection, simplify the computation and hone in on finding a solution.
More complex systems can be investigated and solve by using graphing technology.
System-solving in Grade 8 should include estimating solutions graphically, solving using
substitution, and solving using elimination. Students again should gain experience by
developing conceptual skills using models that develop into abstract skills of formal solving
of equations. Provide opportunities for students to change forms of equations (from a given
form to slope-intercept form) in order to compare equations.
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Domain
Cluster
Content
Standards
Unit 3, Grade 7 Statistics and Probability
Use random sampling to draw inferences about a population
1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from
a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and
support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling
words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
6.SP.1-2
1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old
am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its
values vary with a single number.
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any
overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe data. Students
Guidelines for Assessment and instruction in Statistics Education (GAISE)
continue to use this knowledge in Grade 7 as they use random samples to make predictions
Report, American Statistical Association
about an entire population and judge the possible discrepancies of the predictions. Providing
Ohio Resource Center:
opportunities for students to use real-life situations from science and social studies shows the
Mathline Something Fishy #257: Students estimate the size of a large population
purpose for using random sampling to make inferences about a population.
by applying the concepts of ratio and proportion through the capture-recapture
statistical procedure.
Make available to students the tools needed to develop the skills and understandings required
to produce a representative sample of the general population. One key element of a
Random Sampling and Estimation # 8347: In this session, students estimate
representative sample is understanding that a random sampling guarantees that each element
population quantities from a random sample.
of the population has an equal opportunity to be selected in the sample. Have students
Bias in Sampling #11062: This content resource addresses statistics topics that
compare the random sample to population, asking questions like ―Are all the elements of the
teachers may be uncomfortable teaching due to limited exposure to statistical
entire population represented in the sample?‖ and ―Are the elements represented
content and vocabulary. This resource focuses a four-component statistical
proportionally?‖ Students can then continue the process of analysis by determining the
problem-solving process and the meaning of variation and bias in statistics and to
measures of center and variability to make inferences about the general population based on
investigate how data vary.
the analysis.
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Provide students with random samples from a population, including the statistical measures.
Ask students guiding questions to help them make inferences from the sample.
From the National Council of Teachers of Mathematics, Illuminations - Capture
Recapture: In this lesson, students
Common Misconceptions
Students may believe - One random sample is not representative of the entire
population. Many samples must be taken in order to make an inference that is
valid. By comparing the results of one random sample with the results of
multiple random samples, students can correct this misconception.
Domain
Cluster
Content
Standards
Unit 3, Grade 7 Statistics and Probability
Draw informal comparative inferences about two populations
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by
expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height
of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two
distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
For example, decide whether the words in a chapter of a 7th-grade science book are generally longer than the words in a chapter of a 4th-grade science
book.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe sets of data. In the
Ohio Resource Center:
cluster ―Use random sampling to draw inferences about a population‖ of Statistics and
Baseball Stats ORC #1494: In this lesson students explore and compare data sets
Probability in Grade 7, students learn to draw inferences about one population from a random and statistics in baseball.
sampling of that population. Students continue using these skills to draw informal
comparative inferences about two populations.
Representation of Data—Cholera and War ORC #9740: The object of this
activity is to study excellent examples of the presentation of data. Students
Provide opportunities for students to deal with small populations, determining measures of
analyze (1) a map of cholera cases plotted against the location of water wells in
center and variability for each population. Then have students compare those measures and
London in 1854 and (2) a map of Napoleon's march on Moscow in 1812-1813 to
make inferences. The use of graphical representations of the same data (Grade 6) provides
see what inferences they can draw from the data displays.
another method for making comparisons. Students begin to develop understanding of the
benefits of each method by analyzing data with both methods.
Representation of Data—The U. S. Census ORC # 9741: The object of this
When students study large populations, random sampling is used as a basis for the population activity is to study an excellent example of the presentation of data. Students
inference. This build on the skill developed in the Grade 7 cluster ―Use random sampling to
analyze an illustration of the 1930 U.S. census compared to the 1960 census to
draw inferences about a population‖ of Statistics and Probability. Measures of center and
see what inferences they can draw from the data displays.
variability are used to make inferences on each of the general populations. Then the students
have make comparisons for the two populations based on those inferences.
This is a great opportunity to have students examine how different inferences can be made
based on the same two sets of data. Have students investigate how advertising agencies uses
data to persuade customers to use their products. Additionally, provide students with two
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populations and have them use the data to persuade both sides of an argument.
Domain
Cluster
Content
Standards
Grade 7 Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a
probability near 1 indicates a likely event.
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and
predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled
roughly 200 times, but probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not
good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For
example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the
spinning penny appear to be equally likely based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the
compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday
language (e.g., ―rolling double sixes‖), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the
answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Instructional Strategies
Grade 7 is the introduction to the formal study of probability. Through multiple experiences,
students begin to understand the probability of chance (simple and compound), develop and
use sample spaces, compare experimental and theoretical probabilities, develop and use
graphical organizers, and use information from simulations for predictions.
Help students understand the probability of chance is using the benchmarks of probability: 0,
1 and ½. Provide students with situations that have clearly defined probability of never
happening as zero, always happening as 1 or equally likely to happen as to not happen as 1/2.
Then advance to situations in which the probability is somewhere between any two of these
benchmark values. This builds to the concept of expressing the probability as a number
Instructional Resources/Tools
From the National Council of Teachers of Mathematics, Illuminations:
Boxing Up: In this lesson students explore the relationship between theoretical
and experimental probabilities.
Capture-Recapture: In this lesson students estimate the size of a total population
by taking samples and using proportions to estimate the entire population.
Ohio Resource Center:
Probability Basics ORC #24 Probability Basics This is a 7+ minute video that
explores theoretical and experimental probability with tree diagrams and the
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between 0 and 1. Use this understaind to build the understanding that the closer the
probability is to 0, the more likely it will not happen, and the closer to 1, the more likely it
will happen. Students learn to make predictions about the relative frequency of an event by
using simulations to collect, record, organize and analyze data. Students also develop the
understanding that the more the simulation for an event is repeated, the closer the
experimental probability approaches the theoretical probability.
Have students develop probability models to be used to find the probability of events. Provide
students with models of equal outcomes and models of not equal outcomes are developed to
be used in determining the probabilities of events.
Students should begin to expand the knowledge and understanding of the probability of
simple events, to find the probabilities of compound events by creating organized lists, tables
and tree diagrams. This helps students create a visual representation of the data; i.e., a sample
space of the compound event. From each sample space, students determine the probability or
fraction of each possible outcome. Students continue to build on the use of simulations for
simple probabilities and now expand the simulation of compound probability.
fundamental counting principle.
Probability Using Dice ORC #9737 This activity explores the probabilities of
rolling various sums with two dice. Extensions of the problem and a complete
discussion of the underlying mathematical ideas are included.
How to Fix and Unfair Game ORC #9718 This activity explores a fair game and
―How to Fix an Unfair Game.‖
Remove One ORC # 253 A game is analyzed and the concepts of probability and
sample space are discussed. In addition to the lesson plan, the site includes ideas
for teacher discussion, extensions of the lesson, additional resources (including a
video of the lesson procedures) and a discussion of the mathematical content.
Dart Throwing ORC #10131 The object of this activity is to study an excellent
example of the presentation of data. Students analyze an illustration of the 1930
U.S. census compared to the 1960 census to see what inferences they can draw
from the data displays.
Providing opportunities for students to match situations and sample spaces assists students in
visualizing the sample spaces for situations.
Students often struggle making organized lists or trees for a situation in order to determine the
theoretical probability. Having students start with simpler situations that have fewer elements
enables them to have successful experiences with organizing lists and trees diagrams. Ask
guiding questions to help students create methods for creating organized lists and trees for
situations with more elements.
Common Misconceptions
Students often expect the theoretical and experimental probabilities of the same
data to match. By providing multiple opportunities for students to experience
simulations of situations in order to find and compare the experimental
probability to the theoretical probability, students discover that rarely are those
probabilities the same.
Students often see skills of creating organized lists, tree diagrams, etc. as the end product.
Provide students with experiences that require the use of these graphic organizers to
determine the theoretical probabilities. Have them practice making the connections between
the process of creating lists, tree diagrams, etc. and the interpretation of those models.
Students often expect that simulations will result in all of the possibilities. All
possibilities may occur in a simulation, but not necessarily. Theoretical
probability does use all possibilities. Note examples in simulations when some
possibilities are not shown.
Additionally, students often struggle when converting forms of probability from fractions to
percents and vice versa. To help students with the discussion of probability, don‘t allow the
symbol manipulation/conversions to detract from the conversations. By having students use
technology such as a graphing calculator or computer software to simulate a situation and
graph the results, the focus is on the interpretation of the data. Students then make predictions
about the general population based on these probabilities.
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Domain
Cluster
Content
Standards
Grade 7 Geometry
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale
drawing at a different scale.
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures
of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular
pyramids.
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on the importance of visualization in the understanding of Geometry. Being able to visualize and then
Straws, clay, angle rulers, protractors, rulers,
represent geometric figures on paper is essential to solving geometric problems.
grid paper
Scale drawings of geometric figures connect understandings of proportionality to geometry and lead to future work in similarity
and congruence. As an introduction to scale drawings in geometry, students should be given the opportunity to explore scale
factor as the number of time you multiple the measure of one object to obtain the measure of a similar object. It is important that
students first experience this concept concretely progressing to abstract contextual situations. Pattern blocks (not the hexagon)
provide a convenient means of developing the foundation of scale. Choosing one of the pattern blocks as an original shape,
students can then create the next-size shape using only those same-shaped blocks. Questions about the relationship of the original
block to the created shape should be asked and recorded. A sample of a recording sheet is shown.
Shape
Square
Triangle
Rhombus
Original
Side
Length
1 unit
1 unit
1 unit
Created
Side
Length
Scale
Relationship of
Created to Original
This can be repeated for multiple iterations of each shape by comparing each side length to the original‘s side length. An
extension would be for students to compare the later iterations to the previous. Students should also be expected to use side
lengths equal to fractional and decimal parts. In other words, if the original side can be stated to represent 2.5 inches, what would
be the new lengths and what would be the scale?
Shape
Square
Parallelogram
Trapezoid
Original
Side
Length
2.5 inches
3.25 cms
(Actual
measurements)
Created
Side
Length
Road Maps - convert to actual miles
Dynamic computer software - Geometer's
SketchPad. This cluster lends itself to using
dynamic software. Students sometimes can
manipulate the software more quickly than
do the work manually. However, being able
to use a protractor and a straight edge are
desirable skills.
Common Misconceptions
Student may have misconceptions about
correctly setting up proportions
How to read a ruler
Doubling side measures does not double
perimeter
Scale
Length 1
Length 2
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Provide opportunities for students to use scale drawings of geometric figures with a given scale that requires them to draw and
label the dimensions of the new shape. Initially, measurements should be in whole numbers, progressing to measurements
expressed with rational numbers. This will challenge students to apply their understanding of fractions and decimals.
After students have explored multiple iterations with a couple of shapes, ask them to choose and replicate a shape with given
scales to find the new side lengths, as well as both the perimeters and areas. Starting with simple shapes and whole-number side
lengths allows all students access to discover and understand the relationships. An interesting discovery is the relationship of the
scale of the side lengths to the scale of the respective perimeters (same scale) and areas (scale squared). A sample recording sheet
is shown.
Shape
Rectangle
Triangle
Side
Length
2 x 3 in.
1.5 inches
Scale
2
2
Original
Perimeter
10 inches
4.5 inches
Scaled
Perimeter
20 inches
9 inches
Perimeter
Scale
2
2
Original
Area
6 sq. in.
2.25 sq. in.
Scaled
Area
24 sq in.
9 sq in.
Area
Scale
4
4
Students should move on to drawing scaled figures on grid paper with proper figure labels, scale and dimensions. Provide word
problems that require finding missing side lengths, perimeters or areas. For example, if a 4 by 4.5 cm rectangle is enlarged by a
scale of 3, what will be the new perimeter? What is the new area? or If the scale is 6, what will the new side length look like? or
Suppose the area of one triangle is 16 sq units and the scale factor between this triangle and a new triangle is 2.5. What is the area
of the new triangle?
Reading scales on maps and determining the actual distance (length) is an appropriate contextual situation.
Constructions facilitate understanding of geometry. Provide opportunities for students to physically construct triangles with
straws, sticks, or geometry apps prior to using rulers and protractors to discover and justify the side and angle conditions that will
form triangles. Explorations should involve giving students: three side measures, three angle measures, two side measures and an
included angle measure, and two angles and an included side measure to determine if a unique triangle, no triangle or an infinite
set of triangles results. Through discussion of their exploration results, students should conclude that triangles cannot be formed
by any three arbitrary side or angle measures. They may realize that for a triangle to result the sum of any two side lengths must
be greater than the third side length, or the sum of the three angles must equal 180 degrees. Students should be able to transfer
from these explorations to reviewing measures of three side lengths or three angle measures and determining if they are from a
triangle justifying their conclusions with both sketches and reasoning.
This cluster is related to the following Grade 7 cluster ―Solve real-life and mathematical problems involving angle measure, area,
surface area, and volume.‖ Further construction work can be replicated with quadrilaterals, determining the angle sum, noticing
the variety of polygons that can be created with the same side lengths but different angle measures, and ultimately generalizing a
method for finding the angle sums for regular polygons and the measures of individual angles. For example, subdividing a
polygon into triangles using a vertex (N-2)180° or subdividing a polygons into triangles using an interior point 180°N - 360°
where N = the number of sides in the polygon. An extension would be to realize that the two equations are equal.
Slicing three-dimensional figures helps develop three-dimensional visualization skills. Students should have the opportunity to
physically create some of the three-dimensional figures, slice them in different ways, and describe in pictures and words what has
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been found. For example, use clay to form a cube, then pull string through it in different angles and record the shape of the slices
found. Challenges can also be given: ―See how many different two-dimensional figures can be found by slicing a cube‖ or ―What
three-dimensional figure can produce a hexagon slice?‖ This can be repeated with other three-dimensional figures using a chart to
record and sketch the figure, slices and resulting two-dimensional figures.
Domain
Cluster
Content
Standards
Grade 7 Geometry
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference
and area of a circle.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an
unknown angle in a figure.
6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
6.G.1, 2, 4
*1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other
shapes; apply these techniques in the context of solving real-world and mathematical problems.
2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and
show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes
of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these
techniques in the context of solving real-world and mathematical problems.
Instructional Strategies
This is the students‘ initial work with circles. Knowing that a circle is created by connecting
all the points equidistant from a point (center) is essential to understanding the relationships
between radius, diameter, circumference, pi and area. Students can observe this by folding a
paper plate several times, finding the center at the intersection, then measuring the lengths
between the center and several points on the circle, the radius. Measuring the folds through
the center, or diameters leads to the realization that a diameter is two times a radius. Given
multiple-size circles, students should then explore the relationship between the radius and the
length measure of the circle (circumference) finding an approximation of pi and ultimately
deriving a formula for circumference. String or yarn laid over the circle and compared to a
ruler is an adequate estimate of the circumference. This same process can be followed in
finding the relationship between the diameter and the area of a circle by using grid paper to
estimate the area.
Another visual for understanding the area of a circle can be modeled by cutting up a paper
plate into 16 pieces along diameters and reshaping the pieces into a parallelogram. In figuring
Instructional Resources/Tools
Circular objects of several different sizes
String or yarn
Tape measures, rulers
Grid paper
Paper plates
NCTM Illuminations:
Square Circles: This lesson features two creative twists on the standard lesson of
having students measure several circles to discover that the ratio of the
circumference to the diameter seems always to be a little more than 3. This
lesson starts with squares, so students can first identify a simpler constant ratio
(4) of perimeter to length of a side before moving to the more difficult case of the
circle. The second idea is to measure with a variety of units, so students can more
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area of a circle, the squaring of the radius can also be explained by showing a circle inside a
square. Again, the formula is derived and then learned. After explorations, students should
then solve problems, set in relevant contexts, using the formulas for area and circumference.
In previous grades, students have studied angles by type according to size: acute, obtuse and
right, and their role as an attribute in polygons. Now angles are considered based upon the
special relationships that exist among them: supplementary, complementary, vertical and
adjacent angles. Provide students the opportunities to explore these relationships first through
measuring and finding the patterns among the angles of intersecting lines or within polygons,
then utilize the relationships to write and solve equations for multi-step problems.
Real-world and mathematical multi-step problems that require finding area, perimeter,
volume, surface area of figures composed of triangles, quadrilaterals, polygons, cubes and
right prisms should reflect situations relevant to seventh graders. The computations should
make use of formulas and involve whole numbers, fractions, decimals, ratios and various
units of measure with same system conversions.
readily see that the ratio of the measurements remains constant, not only across
different sizes of figures, but even for the same figure with different
measurements. From these measurements, students will discover the constant
ratio of 1:4 for all squares and the ratio of approximately 1:3.14 for all circles.
Apple Pi: Using estimation and measurement skills, students will determine the
ratio of circumference to diameter and explore the meaning of π. Students will
discover the circumference and area formulas based on their investigations.
Circle Tool: With this three-part online applet, students can
explore with graphic and numeric displays how the circumference and area of a
circle compare to its radius and diameter. Students can collect data points by
dragging the radius to various lengths and clicking the "Add to Table" button to
record the data in the table.
Geometry of Circles: Using a MIRATM geometry tool, students determine the
relationships between radius, diameter, circumference and area of a circle.
Ohio Resource Center:
Circles and Their Areas: Given that units of area are squares, how can we find
the area of a circle or other curved region? Imagine a waffle-like grid inside a
circle and a larger grid containing the circle. The area of the circle lies between
the area of the inside grid and the area of the outside grid..
Exploring c/d = π: Students measure circular objects to collect data to
investigate the relationship between the circumference of a circle and its
diameter. They find that, regardless of the size of the object or the size of the
measuring unit, it always takes a little more than three times the length of the
diameter to measure the circumference.
Parallel Lines: Students use Geometer's Sketchpad® to explore relationships
among the angles formed when parallel lines are cut by a transversal. The
software is integral to the lesson, and step-by-step instructions are provided.
Common Misconceptions
Students may believe: Pi is an exact number rather than understanding that 3.14
is just an approximation of pi.
Many students are confused when dealing with circumference (linear
measurement) and area. This confusion is about an attribute that is measured
using linear units (surrounding) vs. an attribute that is measured using area units
(covering).
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Domain
Cluster
Content
Standards
Grade 8 Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and
translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections,
translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three
angles appears to form a line, and give an argument in terms of transversals why this is so.
Instructional Strategies
Instructional Resources/Tools
A major focus in Grade 8 is to use knowledge of angles and distance to analyze two- and threePattern blocks or shape sets
dimensional figures and space in order to solve problems. This cluster interweaves the
Mirrors - Miras
relationships of symmetry, transformations, and angle relationships to form understandings of
similarity and congruence. Inductive and deductive reasoning are utilized as students forge into
Geometry software like Geometer's Sketchpad, Cabri Jr or GeoGebra
the world of proofs. Informal arguments are justifications based on known facts and logical
reasoning. Students should be able to appropriately label figures, angles, lines, line segments,
Graphing calculators
congruent parts, and images (primes or double primes). Students are expected to use logical
thinking, expressed in words using correct terminology. They are NOT expected to use
Grid paper
theorems, axioms, postulates or a formal format of proof as in two-column proofs.
Patty paper
Transformational geometry is about the effects of rigid motions, rotations, reflections and
translations on figures. Initial work should be presented in such a way that students understand
From the National Library of Virtual Manipulatives:
the concept of each type of transformation and the effects that each transformation has on an
Congruent Triangles – Build similar triangles by combining sides and angles.
object before working within the coordinate system. For example, when reflecting over a line,
Geoboard - Coordinate – Rectangular geoboard with x and y coordinates.
each vertex is the same distance from the line as its corresponding vertex. This is easier to
visualize when not using regular figures. Time should be allowed for students to cut out and
Transformations - Composition – Explore the effect of applying a composition
trace the figures for each step in a series of transformations. Discussion should include the
of translation, rotation, and reflection transformations to objects.
description of the relationship between the original figure and its image(s) in regards to their
corresponding parts (length of sides and measure of angles) and the description of the
Transformations - Dilation – Dynamically interact with and see the result of a
movement, including the attributes of transformations (line of symmetry, distance to be moved,
dilation transformation.
center of rotation, angle of rotation and the amount of dilation).The case of distance –
preserving transformation leads to the idea of congruence.
Transformations - Reflection – Dynamically interact with and see the result of
a reflection transformation.
It is these distance-preserving transformations that lead to the idea of congruence.
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Transformations - Rotation – Dynamically interact with and see the result of a
rotation transformation.
Work in the coordinate plane should involve the movement of various polygons by addition,
subtraction and multiplied changes of the coordinates. For example, add 3 to x, subtract 4 from
y, combinations of changes to x and y, multiply coordinates by 2 then by . Students should
observe and discuss such questions as ‗What happens to the polygon?‘ and ‗What does making
the change to all vertices do?‘. Understandings should include generalizations about the
changes that maintain size or maintain shape, as well as the changes that create distortions of
the polygon (dilations). Example dilations should be analyzed by students to discover the
movement from the origin and the subsequent change of edge lengths of the figures. Students
should be asked to describe the transformations required to go from an original figure to a
transformed figure (image). Provide opportunities for students to discuss the procedure used,
whether different procedures can obtain the same results, and if there is a more efficient
procedure to obtain the same results. Students need to learn to describe transformations with
both words and numbers.
Transformations - Translation – Dynamically interact with and see the result of
a translation transformation.
Common Misconceptions
Students often confuse situations that require adding with multiplicative
situations in regard to scale factor. Providing experiences with geometric
figures and coordinate grids may help students visualize the difference.
Through understanding symmetry and congruence, conclusions can be made about the
relationships of line segments and angles with figures. Students should relate rigid motions to
the concept of symmetry and to use them to prove congruence or similarity of two figures.
Problem situations should require students to use this knowledge to solve for missing measures
or to prove relationships. It is an expectation to be able to describe rigid motions with
coordinates.
Provide opportunities for students to physically manipulate figures to discover properties of
similar and congruent figures, for example, the corresponding angles of similar figures are
equal. Additionally use drawings of parallel lines cut by a transversal to investigate the
relationship among the angles. For example, what information can be obtained by cutting
between the two intersections and sliding one onto the other?
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In Grade 7, students develop an understanding of the special relationships of angles and their
measures (complementary, supplementary, adjacent, vertical). Now, the focus is on learning the
about the sum of the angles of a triangle and using it to, find the measures of angles formed by
transversals (especially with parallel lines), or to find the measures of exterior angles of
triangles and to informally prove congruence.
By using three copies of the same triangle labeled and placed so that the three different angles
form a straight line, students can:
explore the relationships of the angles,
learn the types of angles (interior, exterior, alternate interior, alternate exterior,
corresponding, same side interior, same side exterior), and
explore the parallel lines, triangles and parallelograms formed.
Further examples can be explored to verify these relationships and demonstrate their relevance
in real life.
1 2 3
17 B A
16 C
15 14
4 5 6
C B 7
A B C
13
12 11
A 8
10 9
Investigations should also lead to the Angle-Angle criterion for similar triangles. For instance,
pairs of students create two different triangles with one given angle measurement, then repeat
with two given angle measurements and finally with three given angle measurements. Students
observe and describe the relationship of the resulting triangles. As a class, conjectures lead to
the generalization of the Angle-Angle criterion.
Students should solve mathematical and real-life problems involving understandings from this
cluster. Investigation, discussion, justification of their thinking, and application of their learning
will assist in the more formal learning of geometry in high school.
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Domain
Grade 8 Geometry
Cluster
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve-real world and mathematical problems.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Begin by recalling the formula, and its meaning, for the volume of a right rectangular prism:
Ohio Resource Center:
V = l ×w ×h. Then ask students to consider how this might be used to make a conjecture
#8421 ―The Cylinder Problem - Students build a family of cylinders, all from the
about the volume formula for a cylinder:
same-sized paper, and discover the relationship between the dimensions of the
paper and the resulting cylinders. They order the cylinders by their volumes and
draw a conclusion about the relationship between a cylinder's dimensions and its
volume.
h
h
l
w
Most students can be readily led to the understanding that the volume of a right rectangular
prism can be thought of as the area of a ―base‖ times the height, and so because the area of
the base of a cylinder is π r2 the volume of a cylinder is Vc = π r2h.
To motivate the formula for the volume of a cone, use cylinders and cones with the same base
and height. Fill the cone with rice or water and pour into the cylinder. Students will
discover/experience that 3 cones full are needed to fill the cylinder. This non-mathematical
derivation of the formula for the volume of a cone, V = 1/3 π r2h, will help most students
remember the formula.
In a drawing of a cone inside a cylinder, students might see that that the triangular crosssection of a cone is 1/2 the rectangular cross-section of the cylinder. Ask them to reason why
the volume (three dimensions) turns out to be less than 1/2 the volume of the cylinder. It turns
out to be 1/3.
National Library of Virtual Manipulatives :
―How High‖ is an applet that can be used to take an inquiry approach to the
formula for volume of a cylinder or cone.
NCTM:
Finding Surface Area and Volume
Blue Cube, 27 Little Cubes (Stella Stunner)
Volume of a Spheres and Cones (Rich Problem)
Common Misconceptions
A common misconception among middle grade students is that ―volume‖ is a
―number‖ that results from ―substituting‖ other numbers into a formula. For these
students there is no recognition that ―volume‖ is a measure – related to the
amount of space occupied. If a teacher discovers that students do not have an
understanding of volume as a measure of space, it is important to provide
opportunities for hands on experiences where students ―fill‖ three dimensional
objects. Begin with righ- rectangular prisms and fill them with cubes will help
students understand why the units for volume are cubed. See Cubes
http://illuminations.nctm.org/ActivityDetail.aspx?ID=6
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For the volume of a sphere, it may help to have students visualize a hemisphere ―inside‖ a
cylinder with the same height and ―base.‖ The radius of the circular base of the cylinder is
also the radius of the sphere and the hemisphere. The area of the ―base‖ of the cylinder and
the area of the section created by the division of the sphere into a hemisphere is π r2. The
height of the cylinder is also r so the volume of the cylinder is π r3. Students can see that the
volume of the hemisphere is less than the volume of the cylinder and more than half the
volume of the cylinder. Illustrating this with concrete materials and rice or water will help
students see the relative difference in the volumes. At this point, students can reasonably
accept that the volume of the hemisphere of radius r is 2/3 π r3 and therefore volume of a
sphere with radius r is twice that or 4/3 π r3. There are several websites with explanations for
students who wish to pursue the reasons in more detail. (Note that in the pictures above, the
hemisphere and the cone together fill the cylinder.)
Students should experience many types of real-world applications using these formulas. They
should be expected to explain and justify their solutions.
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GRADE 7
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GRADE 7
Domain
Ratios and Proportional Relationships
Cluster
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example,
Content
if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.
Standards
Instructional Strategies
Instructional Resources/Tools
Building from the development of rate and unit concepts in Grade 6, applications now need to Lesson 4-1 in Holt McDougal Mathematics 7
focus on solving unit-rate problems with more sophisticated numbers: fractions per fractions.
―Unfolding Fraction Multiplication‖, Mathematics Teaching in the Middle
School, December 2011/January 2012, pp. 289-294.
―Estimation‘s Role in Calculations with Fractions,‖ Mathematics Teaching in the
Middle School, September 2011, pp. 96-102‖
―Navigating Through Measurement 6-8,‖ Chapter 1 Accuracy and Estimation,
pp. 11-25.
Supplement to add rates involving fractions.
Incorporate materials on division of fractions. Reinforce the notion that division
is the reciprocal operation of multiplication.
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MATH
GRADE 7
Domain
Cluster
Content
Standards
Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems. (cont.)
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and
observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r)
where r is the unit rate.
Instructional Strategies
Instructional Resources/Tools
Proportional relationships are further developed through the analysis of graphs, tables,
Mathematics Teaching in the Middle School Resources:
equations and diagrams. Ratio tables serve a valuable purpose in the solution of proportional
1. ―Canine Conjectures: Using Data for Proportional Reasoning,‖ August
problems. This is the time to push for a deep understanding of what a representation of a
2011, pp. 26-32.
proportional relationship looks like and what the characteristics are: a straight line through the
2. ―Proportional Reasoning with a Pyramid,‖ May 2011, pp. 545-549.
origin on a graph, a ―rule‖ that applies for all ordered pairs, an equivalent ratio or an
3. ―Getting into Gear,‖ October 2011, pp. 160-165.
expression that describes the situation, etc. This is not the time for students to learn to cross
4. ―Multiple Ways to Solve Proportions,‖ April 2011, pp. 483-489.
multiply to solve problems.
Holt McDougal Mathematics 7 Activity on pages 204-205.
Because percents have been introduced as rates in Grade 6, the work with percents should
continue to follow the thinking involved with rates and proportions. Solutions to problems
Holt McDougal Mathematics 7 Lessons 4-2 through 4-4.
can be found by using the same strategies for solving rates, such as looking for equivalent
ratios or based upon understandings of decimals. Previously, percents have focused on ―out of ―Navigating Through Measurement, 6-8,‖ Chapter 3, Proportionality, pp. 45-65.
100‖; now percents above 100 are encountered.
Strong Connection to Similar Figures CC7.G
Common Misconceptions
Students need to be aware that ―cross products‖ do not apply to fraction
multiplication. ―Cross products‖ refer to the fact that the product of the means
and extremes in a proportion must be equal.
133
MATH
GRADE 7
Domain
Ratios and Proportional Relationships
Cluster
Analyze proportional relationships and use them to solve real-world and mathematical problems. (cont.)
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and
Content
commissions, fees, percent increase and decrease, percent error.
Standards
Instructional Strategies
Instructional Resources/Tools
Because percents have been introduced as rates in Grade 6, the work with percents should
Holt McDougal Mathematics 7, Lessons 6-4 through 6-6.
continue to follow the thinking involved with rates and proportions. Solutions to problems
can be found by using the same strategies for solving rates, such as looking for equivalent
Percent Error will need to be supplemented as it is not included in the text.
ratios or based upon understandings of decimals. Previously, percents have focused on ―out of
100‖; now percents above 100 are encountered.
Providing opportunities to solve problems based within contexts that are relevant to seventh
graders will connect meaning to rates, ratios and proportions. Examples include: researching
newspaper ads and constructing their own question(s), keeping a log of prices (particularly
sales) and determining savings by purchasing items on sale, timing students as they walk a
lap on the track and figuring their rates, creating open-ended problem scenarios with and
without numbers to give students the opportunity to demonstrate conceptual understanding,
inviting students to create a similar problem to a given problem and explain their reasoning.
134
MATH
GRADE 7
Domain
Cluster
Content
Standards
The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a
horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are
oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show
that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the
number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
Instructional Strategies
Instructional Resources/Tools
This cluster builds upon the understandings of rational numbers in Grade 6:
Holt McDougal Mathematics 7, Lessons 1-1 and Lessons 2-1 through 2-3,
• quantities can be shown using + or – as having opposite directions or values,
and 3-1 & 3-5.
• points on a number line show distance and direction,
• opposite signs of numbers indicate locations on opposite sides of 0 on the number line,
From the National Library of Virtual Manipulative :
• the opposite of an opposite is the number itself,
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
• the absolute value of a rational number is its distance from 0 on the number line,
Circle 21 – A puzzle involving adding positive and negative integers to sum
• the absolute value is the magnitude for a positive or negative quantity, and
to 21.
• locating and comparing locations on a coordinate grid by using negative and positive numbers.
Learning now moves to exploring and ultimately formalizing rules for operations (addition,
subtraction, multiplication and division) with integers.
Using both contextual and numerical problems, students should explore what happens when
negatives and positives are combined. Number lines present a visual image for students to
explore and record addition and subtraction results. Two-color counters or colored chips can be
used as a physical and kinesthetic model for adding and subtracting integers. With one color
designated to represent positives and a second color for negatives, addition/subtraction can be
represented by placing the appropriate numbers of chips for the addends and their signs on a
board. Using the notion of opposites, the board is simplified by removing pairs of opposite
colored chips. The answer is the total of the remaining chips with the sign representing the
appropriate color. Repeated opportunities over time will allow students to compare the results of
adding and subtracting pairs of numbers, leading to the generalization of the rules.
Fractional rational numbers and whole numbers should be used in computations and
explorations. Students should be able to give contextual examples of integer operations, write
and solve equations for real-world problems and explain how the properties of operations apply.
Real-world situations could include: profit/loss, money, weight, sea level, debit/credit, football
yardage, etc.
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MATH
GRADE 7
Domain
Cluster
Content
Standards
The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (cont.)
2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations,
particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational
numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If
p and q are integers, then – (p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Instructional Strategies
Instructional Resources/Tools
Using what students already know about positive and negative whole numbers and
Holt McDougal Mathematics 7, Lessons 2-4, 2-6, 2-7 and 3-2, 3-3, 3-6, 3-7
multiplication with its relationship to division, students should generalize rules for multiplying (Chapter 2 focuses on integers, Chapter 3 on rational numbers)
and dividing rational numbers. Multiply or divide the same as for positive numbers, then
designate the sign according to the number of negative factors. Students should analyze and
Helpful Link:
solve problems leading to the generalization of the rules for operations with integers.
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
Using the language of ―the opposite of‖ helps some students understand the multiplication of
negatively signed numbers ( -4 x -4 = 16, the opposite of 4 groups of -4). Discussion about the
tables should address the patterns in the products, the role of the signs in the products and
commutativity of multiplication. Then students should be asked to answer these questions and
prove their responses.
• Is it always true that multiplying a negative factor by a positive factor results in a negative
product?
• Does a positive factor times a positive factor always result in a positive product?
• What is the sign of the product of two negative factors?
• When three factors are multiplied, how is the sign of the product determined?
• How is the numerical value of the product of any two numbers found?
Common Misconceptions
Students quite often confuse the rules for multiplication and addition. It is
important to move beyond the rules to understand that multiplication is repeated
addition where one of the factors indicates the number of addends or how many
―times‖ the addend is repeated.
Students can use number lines with arrows and hops, groups of colored chips or logic to
explain their reasoning. When using number lines, establishing which factor will represent the
length, number and direction of the hops will facilitate understanding. Through discussion,
generalization of the rules for multiplying integers would result.
Division of integers is best understood by relating division to multiplication and applying the
rules. In time, students will transfer the rules to division situations. (Note: In 2b, this algebraic
language (–(p/q) = (–p)/q = p/(–q)) is written for the teacher‘s information, not as an
expectation for student).
Ultimately, students should solve other mathematical and real-world problems requiring the
application of these rules with fractions and decimals.
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MATH
GRADE 7
Domain
The Number System
Cluster
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (cont.)
3. Solve real-world and mathematical problems involving the four operations with rational numbers.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
In Grade 7 the awareness of rational and irrational numbers is initiated by observing the result ―What is the Difference: Using Contextualized Problems,‖ Mathematics
of changing fractions to decimals. Students should be provided with families of fractions,
Teaching in the Middle School, April 2012, pp. 473-478.
such as, sevenths, ninths, thirds, etc. to convert to decimals using long division. The
―Shifting Computational Focus,‖ Mathematics Teaching in the Middle School,
equivalents can be grouped and named (terminating or repeating). Students should begin to
November 2010, pp. 217-223.
see why these patterns occur. Knowing the formal vocabulary rational and irrational is not
expected.
Holt McDougal Mathematics 7, Lessons 2-6, 2-7.
Domain
Expressions and Equations
Cluster
Use properties to generate equivalent expressions
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Have students build on their understanding of order of operations and use the properties of
Holt McDougal Mathematics 7, Lessons 1-3, 1-5, and Lab on p. 31, Lab p.466,
operations to rewrite equivalent numerical expressions that were developed in Grade 6.
Lesson 11-1.
Students continue to use properties that were initially used with whole numbers and now
Navigating through Algebra, 6-8, Chapter 4, ―Using Algebraic Symbols,‖ pp. 60develop the understanding that properties hold for integers, rational and real numbers.
61.
Provide opportunities to build upon this experience of writing expressions using variables to
Online Algebra Tiles - Visualize multiplying and factoring algebraic expressions
represent situations and use the properties of operations to generate equivalent expressions.
using tiles.
These expressions may look different and use different numbers, but the values of the
expressions are the same.
Common Misconceptions
One method that students can use to become convinced that expressions are equivalent is by
As students begin to build and work with expressions containing more than two
substituting a numerical value for the variable and evaluating the expression. For example 5(3 operations, students tend to set aside the order of operations. For example having
a student simplify an expression like 8 + 4(2x - 5) + 3x can bring to light several
misconceptions. Do the students immediately add the 8 and 4 before distributing
Provide opportunities for students to use and understand the properties of operations. These
the 4? Do they only multiply the 4 and the 2x and not distribute the 4 to both
include: the commutative, associative, identity, and inverse properties of addition and of
terms in the parenthesis? Do they collect all like terms 8 + 4 – 5, and 2x + 3x?
multiplication, and the zero property of multiplication. Another method students can use to
Each of these show gaps in students‘ understanding of how to simplify numerical
become convinced that expressions are equivalent is to justify each step of simplification of
expressions with multiple operations.
an expression with an operation property.
137
MATH
GRADE 7
Domain
Expressions and Equations
Cluster
Use properties to generate equivalent expressions. (cont.)
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For
Content
example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Standards
Instructional Strategies
Instructional Resources/Tools
Provide opportunities for students to experience expressions for amounts of increase and
Holt McDougal Mathematics 7, Lessons 1-4, 6-3 (connections to lessons 6-4
decrease. In Standard 2, the expression is rewritten and the variable has a different
through 6-6 via percent increase and decrease).
coefficient. In context, the coefficient aids in the understanding of the situation. Another
example is this situation which represents a 10% decrease: b - 0.10b = 1.00b - 0.10b which
equals 0.90b or 90% of the amount.
Domain
Cluster
Content
Standards
Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solve multi-step, real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess
the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will
make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door
that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Instructional Strategies
Instructional Resources/Tools
To assist students‘ assessment of the reasonableness of answers, especially problem situations Holt McDougal Mathematics 7, Lessons 6-1, 6-2
involving fractional or decimal numbers, use whole-number approximations for the
computation and then compare to the actual computation. Connections between performing
Navigating through Algebra, Chapter 4, ―Using Algebraic Expressions,‖ pp. 59the inverse operation and undoing the operations are appropriate here. It is appropriate to
72.
expect students to show the steps in their work. Students should be able to explain their
thinking using the correct terminology for the properties and operations.
Helpful Links:
Provide multiple opportunities for students to work with multi-step problem situations that
have multiple solutions and therefore can be represented by an inequality. Students need to be
aware that values can satisfy an inequality but not be appropriate for the situation, therefore
limiting the solutions for that particular problem.
These links will require you to create a free account with the ORC:
Solving for a Variable This activity for students uses a pan balance to model
solving equations for a variable.
Solving an Inequality This activity for students illustrates the solution to
inequalities modeled on a number line.
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MATH
GRADE 7
Domain
Cluster
Content
Standards
Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (cont.)
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning
about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these
forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the
perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of
the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want
your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Instructional Strategies
Instructional Resources/Tools
Continue to build on students‘ understanding and application of writing and solving one-step
Holt McDougal Mathematics 7, Lessons 11-2 through 11-7.
equations from a problem situation to multi-step problem situations. This is also the context
for students to practice using rational numbers including: integers, and positive and negative
fractions and decimals. As students analyze a situation, they need to identify what operation
should be completed first, then the values for that computation. Each set of the needed
operation and values is determined in order. Finally an equation matching the order of
operations is written. For example, Bonnie goes out to eat and buys a meal that costs $12.50
that includes a tax of $.75. She only wants to leave a tip based on the cost of the food. In this
situation, students need to realize that the tax must be subtracted from the total cost before
being multiplied by the percent of tip and then added back to obtain the final cost. C = (12.50
- .75)(1 + T) + .75 = 11.75(1 +T) + .75 where C = cost and T = tip.
139
MATH
GRADE 7
Domain
Geometry
Cluster
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale
Content
drawing at a different scale.
Standards
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on the importance of visualization in the understanding of Geometry. Being able to
Holt McDougal Mathematics 7, Lessons 4-5, 4-6, Lab p. 176, Lab p.
visualize and then represent geometric figures on paper is essential to solving geometric problems.
178,
Scale drawings of geometric figures connect understandings of proportionality to geometry and lead to future
work in similarity and congruence. As an introduction to scale drawings in geometry, students should be given
the opportunity to explore scale factor as the number of time you multiple the measure of one object to obtain
the measure of a similar object. It is important that students first experience this concept concretely progressing
to abstract contextual situations. Pattern blocks (not the hexagon) provide a convenient means of developing the
foundation of scale. Choosing one of the pattern blocks as an original shape, students can then create the nextsize shape using only those same-shaped blocks. Questions about the relationship of the original block to the
created shape should be asked and recorded. A sample of a recording sheet is shown.
NCTM Addenda Series: Measurement in the Middle Grades, Activities
20A and 20B, pp. 56-57.
Road Maps - convert to actual miles
Common Misconceptions
Correctly setting up proportions
How to read a ruler
This can be repeated for multiple iterations of each shape by comparing each side length to the original‘s side
length. An extension would be for students to compare the later iterations to the previous. Students should also
be expected to use side lengths equal to fractional and decimal parts. In other words, if the original side can be
stated to represent 2.5 inches, what would be the new lengths and what would be the scale?
Provide opportunities for students to use scale drawings of geometric figures with a given scale that requires
them to draw and label the dimensions of the new shape. Initially, measurements should be in whole numbers,
progressing to measurements expressed with rational numbers. This will challenge students to apply their
understanding of fractions and decimals.
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MATH
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After students have explored multiple iterations with a couple of shapes, ask them to choose and replicate a
shape with given scales to find the new side lengths, as well as both the perimeters and areas. Starting with
simple shapes and whole-number side lengths allows all students access to discover and understand the
relationships. An interesting discovery is the relationship of the scale of the side lengths to the scale of the
respective perimeters (same scale) and areas (scale squared). A sample recording sheet is shown.
Students should move on to drawing scaled figures on grid paper with proper figure labels, scale and
dimensions. Provide word problems that require finding missing side lengths, perimeters or areas. For example,
if a 4 by 4.5 cm rectangle is enlarged by a scale of 3, what will be the new perimeter? What is the new area? or
If the scale is 6, what will the new side length look like? or Suppose the area of one triangle is 16 sq units and
the scale factor between this triangle and a new triangle is 2.5. What is the area of the new triangle?
Reading scales on maps and determining the actual distance (length) is an appropriate contextual situation.
Domain
Geometry
Cluster
Draw, construct, and describe geometrical figures and describe the relationships between them. (cont.)
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures
Content
of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Standards
Instructional Strategies
Instructional Resources/Tools
Constructions facilitate understanding of geometry. Provide opportunities for students to
Straws, clay, angle rulers, protractors, rulers, grid paper
physically construct triangles with straws, sticks, or geometry apps prior to using rulers and
protractors to discover and justify the side and angle conditions that will form triangles.
Dynamic computer software - Geometer's SketchPad. This cluster lends itself to
Explorations should involve giving students: three side measures, three angle measures, two
using dynamic software. Students sometimes can manipulate the software more
side measures and an included angle measure, and two angles and an included side measure to quickly than do the work manually. However, being able to use a protractor and a
determine if a unique triangle, no triangle or an infinite set of triangles results. Through
straight edge are desirable skills.
discussion of their exploration results, students should conclude that triangles cannot be
formed by any three arbitrary side or angle measures. They may realize that for a triangle to
Holt McDougal Mathematics 7, Lab p. 334, Lessons 8-5, Lab p. 341, Lab p. 342,
result the sum of any two side lengths must be greater than the third side length, or the sum of
the three angles must equal 180 degrees. Students should be able to transfer from these
Congruent Triangles Manipulative
explorations to reviewing measures of three side lengths or three angle measures and
141
MATH
GRADE 7
determining if they are from a triangle justifying their conclusions with both sketches and
reasoning.
This cluster is related to the following Grade 7 cluster ―Solve real-life and mathematical
problems involving angle measure, area, surface area, and volume.‖ Further construction
work can be replicated with quadrilaterals, determining the angle sum, noticing the variety of
polygons that can be created with the same side lengths but different angle measures, and
ultimately generalizing a method for finding the angle sums for regular polygons and the
measures of individual angles. For example, subdividing a polygon into triangles using a
vertex (N-2)180° or subdividing a polygons into triangles using an interior point 180°N 360° where N = the number of sides in the polygon. An extension would be to realize that the
two equations are equal.
NCTM Addenda Series: Geometry in the Middle Grades:
1. Activity 9: Polygon Angle Sum, p. 44.
2. Activities 1-2c: Explorations with Three-Dimensional Figures, pp. 1112.
NCTM Addenda Series: Measurement in the Middle Grades, Activity 22A, pp.
57-58.
Navigating through Geometry, Grades 5-8, ―Constructing Three Dimensional
Figures,‖ pp. 71-72.
Slicing three-dimensional figures helps develop three-dimensional visualization skills.
Students should have the opportunity to physically create some of the three-dimensional
figures, slice them in different ways, and describe in pictures and words what has been found.
For example, use clay to form a cube, then pull string through it in different angles and record
the shape of the slices found. Challenges can also be given: ―See how many different twodimensional figures can be found by slicing a cube‖ or ―What three-dimensional figure can
produce a hexagon slice?‖ This can be repeated with other three-dimensional figures using a
chart to record and sketch the figure, slices and resulting two-dimensional figures.
Domain
Geometry
Cluster
Draw, construct, and describe geometrical figures and describe the relationships between them. (cont.)
3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular
Content
pyramids.
Standards
Instructional Strategies
Instructional Resources/Tools
Slicing three-dimensional figures helps develop three-dimensional visualization skills.
Holt McDougal Mathematics 7, Extensions p. 378,
Students should have the opportunity to physically create some of the three-dimensional
figures, slice them in different ways, and describe in pictures and words what has been found. Navigating through Geometry, Grades 5-8, ―Cross Sections of ThreeFor example, use clay to form a cube, then pull string through it in different angles and record Dimensional Shapes,‖ pp. 67-68.
the shape of the slices found. Challenges can also be given: ―See how many different twodimensional figures can be found by slicing a cube‖ or ―What three-dimensional figure can
produce a hexagon slice?‖ This can be repeated with other three-dimensional figures using a
chart to record and sketch the figure, slices and resulting two-dimensional figures.
142
MATH
GRADE 7
Domain
Geometry
Cluster
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference
Content
and area of a circle.
Standards
Instructional Strategies
Instructional Resources/Tools
This is the students‘ initial work with circles. Knowing that a circle is created by connecting
Holt McDougal Mathematics 7, Lessons 9-1, 9-2, Lab p. 358
all the points equidistant from a point (center) is essential to understanding the relationships
NCTM Addenda Series, ―Measurement in the Middle Grades,‖ Activity 12, pp.
between radius, diameter, circumference, pi and area. Students can observe this by folding a
32-33.
paper plate several times, finding the center at the intersection, then measuring the lengths
between the center and several points on the circle, the radius. Measuring the folds through
Navigating through Measurement in Grades 6-8, ―Going in Circles,‖ pp. 37-38.
the center, or diameters leads to the realization that a diameter is two times a radius. Given
multiple-size circles, students should then explore the relationship between the radius and the Helpful Links:
length measure of the circle (circumference) finding an approximation of pi and ultimately
Square Circles
deriving a formula for circumference. String or yarn laid over the circle and compared to a
Apple Pi
ruler is an adequate estimate of the circumference. This same process can be followed in
Circle Tool
finding the relationship between the diameter and the area of a circle by using grid paper to
Geometry of Circles
estimate the area.
Circles and Their Areas
Another visual for understanding the area of a circle can be modeled by cutting up a paper
plate into 16 pieces along diameters and reshaping the pieces into a parallelogram. In figuring
area of a circle, the squaring of the radius can also be explained by showing a circle inside a
square. Again, the formula is derived and then learned. After explorations, students should
then solve problems, set in relevant contexts, using the formulas for area and circumference.
C/D = Pi
Circular objects of several different sizes
String or yarn
Tape measures, rulers
Grid paper
Paper plates
Common Misconceptions
Students may believe pi is an exact number rather than understanding that 3.14 is
just an approximation of pi.
Many students are confused when dealing with circumference (linear
measurement) and area. This confusion is about an attribute that is measured
using linear units (surrounding) vs. an attribute that is measured using area units
(covering).
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MATH
GRADE 7
Domain
Geometry
Cluster
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (cont.)
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an
Content
unknown angle in a figure.
Standards
Instructional Strategies
Instructional Resources/Tools
In previous grades, students have studied angles by type according to size: acute, obtuse and
Holt McDougal Mathematics 7, Lessons 8-1 through 8-4, Lab p. 310, Lab p. 316,
right, and their role as an attribute in polygons. Now angles are considered based upon the
Lab p. 322, Focus on Problem Solving p. 329.
special relationships that exist among them: supplementary, complementary, vertical and
adjacent angles. Provide students the opportunities to explore these relationships first through Helpful Link:
measuring and finding the patterns among the angles of intersecting lines or within polygons,
Geometer‘s Sketchpad Investigation: Parallel Lines and Transversals
then utilize the relationships to write and solve equations for multi-step problems.
Domain
Geometry
Cluster
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (cont.)
6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,
Content
quadrilaterals, polygons, cubes, and right prisms.
Standards
Instructional Strategies
Instructional Resources/Tools
Real-world and mathematical multi-step problems that require finding area, perimeter,
Holt McDougal Mathematics 7, Lessons 9-3 through 9-6, Lab p. 380, Lab p. 386,
volume, surface area of figures composed of triangles, quadrilaterals, polygons, cubes and
Real World Connections p. 393.
right prisms should reflect situations relevant to seventh graders. The computations should
make use of formulas and involve whole numbers, fractions, decimals, ratios and various
Navigating through Measurement in Grades 5-8, ― To the Surface and Beyond,‖
units of measure with same system conversions.
pp. 39-42.
NCTM Addenda Series, ―Measurement in the Middle Grades,‖ Activity 22b &
C, pp. 58-59.
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Domain
Cluster
Content
Standards
Statistics and Probability
Use random sampling to draw inferences about a population
1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population
from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples
and support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling
words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might
be.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe data. Students
Holt McDougal Mathematics 7, Lessons7-3, Lab p. 290, Focus on Problem
continue to use this knowledge in Grade 7 as they use random samples to make predictions
Solving p. 291.
about an entire population and judge the possible discrepancies of the predictions. Providing
opportunities for students to use real-life situations from science and social studies shows the
Navigating through Data Analysis in Grades 5-8, ―A Matter of Opinion,‖ pp. 58purpose for using random sampling to make inferences about a population.
59.
Make available to students the tools needed to develop the skills and understandings required
to produce a representative sample of the general population. One key element of a
representative sample is understanding that a random sampling guarantees that each element
of the population has an equal opportunity to be selected in the sample. Have students
compare the random sample to population, asking questions like ―Are all the elements of the
entire population represented in the sample?‖ and ―Are the elements represented
proportionally?‖ Students can then continue the process of analysis by determining the
measures of center and variability to make inferences about the general population based on
the analysis.
Provide students with random samples from a population, including the statistical measures.
Ask students guiding questions to help them make inferences from the sample.
NCTM Addenda Series: Dealing with Data and Chance, ―Surveys, Statistics and
Students: an Interdisciplinary Unit,‖ pp. 5-7.
Navigating through Data Analysis in Grades 5-8, Chapter 2 ― Comparing Data
Sets with Equal Numbers Elements,‖ pp. 31-48.
Common Misconceptions
Students may believe:
One random sample is not representative of the entire population. Many samples
must be taken in order to make an inference that is valid. By comparing the
results of one random sample with the results of multiple random samples,
students can correct this misconception.
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Domain
Cluster
Content
Standards
Statistics and Probability
Draw informal comparative inferences about two populations
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by
expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean
height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two
distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two
populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a
fourth-grade science book.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe sets of data. In the
Holt McDougal Mathematics 7, Lessons 7-1, 7-2, Lab p. 284, Lab p. 292, Real
cluster ―Use random sampling to draw inferences about a population‖ of Statistics and
World Connections p. 295.
Probability in Grade 7, students learn to draw inferences about one population from a random
sampling of that population. Students continue using these skills to draw informal
NCTM Addenda Series: Dealing with Data and Chance, Chapter 3 ―Focus on
comparative inferences about two populations.
Problem Solving,‖ pp. 27-28.
Provide opportunities for students to deal with small populations, determining measures of
center and variability for each population. Then have students compare those measures and
make inferences. The use of graphical representations of the same data (Grade 6) provides
another method for making comparisons. Students begin to develop understanding of the
benefits of each method by analyzing data with both methods.
When students study large populations, random sampling is used as a basis for the population
inference. This build on the skill developed in the Grade 7 cluster ―Use random sampling to
draw inferences about a population‖ of Statistics and Probability. Measures of center and
variability are used to make inferences on each of the general populations. Then the students
have make comparisons for the two populations based on those inferences.
This is a great opportunity to have students examine how different inferences can be made
based on the same two sets of data. Have students investigate how advertising agencies uses
data to persuade customers to use their products. Additionally, provide students with two
populations and have them use the data to persuade both sides of an argument.
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Domain
Cluster
Content
Standards
Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a
probability near 1 indicates a likely event.
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and
predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled
roughly 200 times, but probably not exactly 200 times.
Instructional Strategies
Instructional Resources/Tools
Grade 7 is the introduction to the formal study of probability. Through multiple experiences,
Holt McDougal Mathematics 7, Lessons 10-1, 10-2, Lab p. 422
students begin to understand the probability of chance (simple and compound), develop and
use sample spaces, compare experimental and theoretical probabilities, develop and use
Helpful Links:
graphical organizers, and use information from simulations for predictions.
Capture, RE-Capture
Random Sampling and Estimation
Help students understand the probability of chance is using the benchmarks of probability: 0,
Something Fishy
1 and ½. Provide students with situations that have clearly defined probability of never
happening as zero, always happening as 1 or equally likely to happen as to not happen as 1/2.
Then advance to situations in which the probability is somewhere between any two of these
Common Misconceptions
benchmark values. This builds to the concept of expressing the probability as a number
One random sample is not representative of the entire population. Many samples
between 0 and 1. Use this understand to build the understanding that the closer the probability must be taken in order to make an inference that is valid. By comparing the
is to 0, the more likely it will not happen, and the closer to 1, the more likely it will happen.
results of one random sample with the results of multiple random samples,
Students learn to make predictions about the relative frequency of an event by using
students can correct this misconception.
simulations to collect, record, organize and analyze data. Students also develop the
understanding that the more the simulation for an event is repeated, the closer the
experimental probability approaches the theoretical probability.
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Domain
Cluster
Content
Standards
Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models. (cont.)
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not
good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example,
if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning
penny appear to be equally likely based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound
event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday
language (e.g., ―rolling double sixes‖), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer
to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Instructional Strategies
Instructional Resources/Tools
Providing opportunities for students to match situations and sample spaces assists students in
Holt McDougal Mathematics 7, Lessons 10-3 through 10-9, Lab p. 422, Lab
visualizing the sample spaces for situations.
p. 424, Lab p. 430, Focus on Problem Solving p. 433
Students often struggle making organized lists or trees for a situation in order to determine the
theoretical probability. Having students start with simpler situations that have fewer elements
enables them to have successful experiences with organizing lists and trees diagrams. Ask guiding
questions to help students create methods for creating organized lists and trees for situations with
more elements.
Students often see skills of creating organized lists, tree diagrams, etc. as the end product. Provide
students with experiences that require the use of these graphic organizers to determine the
theoretical probabilities. Have them practice making the connections between the process of
creating lists, tree diagrams, etc. and the interpretation of those models.
Navigating through Probability in Grades 6-8 offers a wealth of activities for
probabilities.
Helpful Links:
Baseball Stats
Cholera and War,
Boxing Up
Dice Probabilities
Fixing an Unfair Game
Additionally, students often struggle when converting forms of probability from fractions to
percents and vice versa. To help students with the discussion of probability, don‘t allow the
symbol manipulation/conversions to detract from the conversations. By having students use
technology such as a graphing calculator or computer software to simulate a situation and graph
the results, the focus is on the interpretation of the data. Students then make predictions about the
general population based on these probabilities.
Link to ODE Common Core and Model Curriculum
National Council of Teachers of Mathematics
Ohio Resource Center
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Domain
Cluster
Content
Standards
UNIT 1 Grade 7, The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
7. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a
horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are
oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show
that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the
number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
8. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of
operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret
products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number.
If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
9. Solve real-world and mathematical problems involving the four operations with rational numbers.
Instructional Strategies
Instructional Resources/Tools
This cluster builds upon the understandings of rational numbers in Grade 6:
Two-color counters
quantities can be shown using + or – as having opposite directions or values,
Calculators
points on a number line show distance and direction,
opposite signs of numbers indicate locations on opposite sides of 0 on the number line,
From the National Library of Virtual Manipulatives :
the opposite of an opposite is the number itself,
Circle 3 – A puzzle involving adding positive real numbers to sum to three.
the absolute value of a rational number is its distance from 0 on the number line,
Circle 21 – A puzzle involving adding positive and negative integers to sum
the absolute value is the magnitude for a positive or negative quantity, and
to 21.
locating and comparing locations on a coordinate grid by using negative and positive
numbers.
Learning now moves to exploring and ultimately formalizing rules for operations (addition,
subtraction, multiplication and division) with integers.
Using both contextual and numerical problems, students should explore what happens when
negatives and positives are combined. Number lines present a visual image for students to
explore and record addition and subtraction results. Two-color counters or colored chips can
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be used as a physical and kinesthetic model for adding and subtracting integers. With one
color designated to represent positives and a second color for negatives, addition/subtraction
can be represented by placing the appropriate numbers of chips for the addends and their
signs on a board. Using the notion of opposites, the board is simplified by removing pairs of
opposite colored chips. The answer is the total of the remaining chips with the sign
representing the appropriate color. Repeated opportunities over time will allow students to
compare the results of adding and subtracting pairs of numbers, leading to the generalization
of the rules. Fractional rational numbers and whole numbers should be used in computations
and explorations. Students should be able to give contextual examples of integer operations,
write and solve equations for real-world problems and explain how the properties of
operations apply. Real-world situations could include: profit/loss, money, weight, sea level,
debit/credit, football yardage, etc.
Using what students already know about positive and negative whole numbers and
multiplication with its relationship to division, students should generalize rules for
multiplying and dividing rational numbers. Multiply or divide the same as for positive
numbers, then designate the sign according to the number of negative factors. Students should
analyze and solve problems leading to the generalization of the rules for operations with
integers.
For example, beginning with known facts, students predict the answers for related facts,
keeping in mind that the properties of operations apply (See Tables 1, 2 and 3 below).
Table 1
4 x 4 = 16
4 x 3 = 12
4x2=8
4x1=4
4x0=0
4 x -1 =
4x-2=
4x-3=
4x-4=
Table 2
4 x 4 = 16
4 x 3 = 12
4x2=8
4x1=4
4x0=0
-4 x 1 =
-4 x 2 =
-4 x 3 =
-4 x 4 =
Table 3
-4 x -4 = 16
-4 x -3 = 12
-4 x -2 = 8
-4 x -1 = 4
-4 x 0 = 0
-1 x - 4 =
-2 x - 4 =
-3 x - 4 =
-4 x - 4 =
Using the language of ―the opposite of‖ helps some students understand the multiplication of
negatively signed numbers ( -4 x -4 = 16, the opposite of 4 groups of -4). Discussion about
the tables should address the patterns in the products, the role of the signs in the products and
commutativity of multiplication. Then students should be asked to answer these questions and
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prove their responses.
Is it always true that multiplying a negative factor by a positive factor results in a
negative product?
Does a positive factor times a positive factor always result in a positive product?
What is the sign of the product of two negative factors?
When three factors are multiplied, how is the sign of the product determined?
How is the numerical value of the product of any two numbers found?
Students can use number lines with arrows and hops, groups of colored chips or logic to
explain their reasoning. When using number lines, establishing which factor will represent the
length, number and direction of the hops will facilitate understanding. Through discussion,
generalization of the rules for multiplying integers would result.
Division of integers is best understood by relating division to multiplication and applying the
rules. In time, students will transfer the rules to division situations. (Note: In 2b, this algebraic
language (–(p/q) = (–p)/q = p/(–q)) is written for the teacher‘s information, not as an
expectation for students.)
Ultimately, students should solve other mathematical and real-world problems requiring the
application of these rules with fractions and decimals.
In Grade 7 the awareness of rational and irrational numbers is initiated by observing the result
of changing fractions to decimals. Students should be provided with families of fractions,
such as, sevenths, ninths, thirds, etc. to convert to decimals using long division. The
equivalents can be grouped and named (terminating or repeating). Students should begin to
see why these patterns occur. Knowing the formal vocabulary rational and irrational is not
expected.
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Domain
Cluster
Content
Standards
UNIT 1 Grade 8, The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
3. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show
that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
4. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and
estimate the value of expressions (e.g., 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and
1.5, and explain how to continue on to get better approximations.
Instructional Strategies
Instructional Resources/Tools
The distinction between rational and irrational numbers is an abstract distinction, originally
Graphing calculators
based on ideal assumptions of perfect construction and measurement. In the real world,
however, all measurements and constructions are approximate. Nonetheless, it is possible to
Dynamic geometry software
see the distinction between rational and irrational numbers in their decimal representations.
A rational number is of the form a/b, where a and b are both integers, and b is not 0. In the
elementary grades, students learned processes that can be used to locate any rational number
on the number line: Divide the interval from 0 to 1 into b equal parts; then, beginning at 0,
count out a of those parts. The surprising fact, now, is that there are numbers on the number
line that cannot be expressed as a/b, with a and b both integers, and these are called irrational
numbers.
Students construct a right isosceles
triangle with legs of 1 unit. Using
the Pythagorean theorem, they
determine that the length of the
hypotenuse is . In the figure
below, they can rotate the
hypotenuse back to the original
number line to show that indeed
is a number on the number line.
1
0
1
Common Misconceptions
Some students are surprised that the decimal representation of pi does not repeat.
Some students believe that if only we keep looking at digits farther and farther to
the right, eventually a pattern will emerge.
A few irrational numbers are given special names (pi and e), and much attention
is given to sort (2). Because we name so few irrational numbers, students
sometimes conclude that irrational numbers are unusual and rare. In fact,
irrational numbers are much more plentiful than rational numbers, in the sense
that they are ―denser‖ in the real line.
2
In the elementary grades, students become familiar with decimal fractions, most often with
decimal representations that terminate a few digits to the right of the decimal point. For
example, to find the exact decimal representation of 2/7, students might use their calculator to
find 2/7 = 0.2857142857…and they might guess that the digits 285714 repeat. To show that
the digits do repeat, students in Grade 7 actually carry out the long division and recognize that
the remainders repeat in a predictable pattern—a pattern that creates the repetition in the
decimal representation (see 7.NS.2.d).
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Thinking about long division generally, ask students what will happen if the remainder is 0 at
some step. They can reason that the long division is complete, and the decimal representation
terminates. If the reminder is never 0, in contrast, then the remainders will repeat in a cyclical
pattern because at each step with a given remainder, the process for finding the next
remainder is always the same. Thus, the digits in the decimal representation also repeat.
When dividing by 7, there are 6 possible nonzero remainders, and students can see that the
decimal repeats with a pattern of at most 6 digits. In general, when finding the decimal
representation of m/n, students can reason that the repeating portion of decimal will have at
most n-1 digits. The important point here is that students can see that the pattern will repeat,
so they can imagine the process continuing without actually carrying it out.
Conversely, given a repeating decimal, students learn strategies for converting the decimal to
a fraction. One approach is to notice that rational numbers with denominators of 9 repeat a
single digit. With a denominator of 99, two digits repeat; with a denominator of 999, three
digits repeat, and so on. For example,
13/99 = 0.13131313…
74/99 = 0.74747474…
237/999 = 0.237237237…
485/999 = 0.485485485…
From this pattern, students can go in the other direction, conjecturing, for example, that the
repeating decimal 0.285714285714… = 285714/999999. And then they can verify that this
fraction is equivalent to 2/7.
Once students understand that (1) every rational number has a decimal representation that
either terminates or repeats, and (2) every terminating or repeating decimal is a rational
number, they can reason that on the number line, irrational numbers (i.e., those that are not
rational) must have decimal representations that neither terminate nor repeat. And although
students at this grade do not need to be able to prove that
is irrational, they need to know
that
is irrational (see 8.EE.2), which means that its decimal representation neither
terminates nor repeats. Nonetheless, they can approximate
without using the square root
key on the calculator. Students can create tables like those below to approximate
to one,
two, and then three places to the right of the decimal point:
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x
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
x2
1.00
1.21
1.44
1.69
1.96
2.25
2.56
2.89
3.24
3.61
4.00
x
1.40
1.41
1.42
1.43
1.44
1.45
1.46
1.47
1.48
1.49
1.50
x2
1.9600
1.9881
2.0164
2.0449
2.0736
2.1025
2.1316
2.1609
2.1904
2.2201
2.2500
x
1.410
1.411
1.412
1.413
1.414
1.415
1.416
1.417
1.418
1.419
1.420
x2
1.988100
1.990921
1.993744
1.996569
1.999396
2.002225
2.005056
2.007889
2.010724
2.013561
2.016400
From knowing that 12 = 1 and 22 = 4, or from the picture above, students can reason that there
is a number between 1 and 2 whose square is 2. In the first table above, students can see that
between 1.4 and 1.5, there is a number whose square is 2. Then in the second table, they
locate that number between 1.41 and 1.42. And in the third table they can locate
between
1.414 and 1.415. Students can develop more efficient methods for this work. For example,
from the picture above, they might have begun the first table with 1.4. And once they see that
1.422 > 2, they do not need generate the rest of the data in the second table.
Use set diagrams to show the relationships among real, rational, irrational numbers, integers,
and counting numbers. The diagram should show that the all real numbers (numbers on the
number line) are either rational or irrational.
Given two distinct numbers, it is possible to find both a rational and an irrational number
between them.
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Domain
Cluster
Content
Standards
UNIT 1 Grade 8 Expressions and Equations
Work with radicals and integer exponents.
8. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 3–5 = 3–3 = 1/33 = 1/27.
9. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate
square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
10. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many
times as much one is than the other. For example, estimate the population of the United States as 3 108 and the population of the world as 7 109, and
determine that the world population is more than 20 times larger.
11. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific
notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading).
Interpret scientific notation that has been generated by technology.
Instructional Strategies
Instructional Resources/Tools
Although students begin using whole-number exponents in Grades 5 and 6, it is in Grade 8
Square tiles and cubes to develop understanding of squared and cubed numbers
when students are first expected to know and use the properties of exponents and to extend the
meaning beyond counting-number exponents. It is no accident that these expectations are
Calculators to verify and explore patterns
simultaneous, because it is the properties of counting-number exponents that provide the
rationale for the properties of integer exponents. In other words, students should not be told
Webquests using data mined from sites like the U.S. Census Bureau, scientific
these properties but rather should derive them through experience and reason.
data (planetary distances)
For counting-number exponents (and for nonzero bases), the following properties follow
directly from the meaning of exponents.
1. anam = an+m
2. (an)m = anm
3. anbn = (ab)n
Students should have experience simplifying numerical expressions with exponents so that
these properties become natural and obvious. For example:
Place value charts to connect the digit value to the exponent (negative and
positive)
Powers of 10 online video.
Common Misconceptions
Students may mix up the product of powers property and the power of a power
2
If students reason about these examples with a sense of generality about the numbers, they
begin to articulate the properties. For example, ―I see that 3 twos is being multiplied by 5 twos,
and the results is 8 twos being multiplied together, where the 8 is the sum of 3 and 5, the
number of twos in each of the original factors. That would work for a base other than two (as
long as the bases are the same).‖
3
5
6
property. Is x x equivalent to x or x ? Students may make the
relationship that in scientific notation, when a number contains one nonzero
digit and a positive exponent, that the number of zeros equals the exponent.
This pattern may incorrectly be applied to scientific notation values with
negative values or with more than one nonzero digit.
Note: When talking about the meaning of an exponential expression, it is easy to say
(incorrectly) that ―35 means 3 multiplied by itself 5 times.‖ But by writing out the meaning,
, students can see that there are only 4 multiplications. So a better
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description is ―35 means 5 3s multiplied
together.‖
Students also need to realize that these simple
descriptions work only for counting-number
exponents. When extending the meaning of
exponents to include 0 and negative exponents,
these descriptions are limiting: Is it sensible to
say ―30 means 0 3s multiplied together‖ or that
―3-2 means -2 3s multiplied together‖?
The motivation for the meanings of 0 and
negative exponents is the following principle:
The properties of counting-number exponents
should continue to work for integer exponents.
For example, Property 1 can be used to reason
what 30 should be. Consider the following
expression and simplification:
is multiplied by 35, the result (following Property
1) should be 35, which implies that 30 must be 1.
Because this reasoning holds for any base other
than 0, we can reason that a0 = 1 for any nonzero
number a.
To make a judgment about the meaning of 3-4,
the approach is similar:
. This computation shows that 3-4 should
be the reciprocal of 34, because their product is 1.
And again, this reasoning holds for any nonzero
base. Thus, we can reason that a−n = 1/an.
Putting all of these results together, we now have
the properties of integer exponents, shown in the
above chart. For mathematical completeness, one
might prove that properties 1-3 continue to hold
for integer exponents, but that is not necessary at
this point.
A supplemental strategy for developing meaning
for integer exponents is to make use of patterns,
as shown in the chart to the right.
Properties of Integer
Exponents
For any nonzero real numbers
a and b and integers n and m:
1. anam = an+m
2. (an)m = anm
3. anbn = (ab)n
4. a0 = 1
5. a−n = 1/an
. This computation shows that the when 30
Patterns in Exponents
54
53
52
51
50
5-1
5-2
5-3
625
125
25
5
1
1/5
1/25
1/125
As the exponent decreases by 1, the
value of the expression is divided by
5, which is the base. Continue that
pattern to 0 and negative exponents.
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thousands
hundreds
tens
ones
tenths
hundredths
thousandths
The meanings of 0 and negative-integer exponents can be further explored in a place-value
chart:
103
3
102
2
101
4
100
7
10-1
5
10-2
6
10-3
8
.
Thus, integer exponents support writing any decimal in expanded form like the following:
.
Expanded form and the connection to place value is important for helping students make sense
of scientific notation, which allows very large and very small numbers to be written concisely,
enabling easy comparison. To develop familiarity, go back and forth between standard notation
and scientific notation for numbers near, for example, 10 12 or 10-9. Compare numbers, where
one is given in scientific notation and the other is given in standard notation. Real-world
problems can help students compare quantities and make sense about their relationship.
Provide practical opportunities for students to flexibly move between forms of squared and
cubed numbers. For example, If
symbolically and verbally.
32 9
then
9
3 . This flexibility should be experienced
Opportunities for conceptually understanding irrational numbers should be developed. One
way is for students to draw a square that is one unit by one unit and find the diagonal using the
Pythagorean Theorem. The diagonal drawn has an irrational length of √2. Other irrational
lengths can be found using the same strategy by finding diagonal lengths of rectangles with
various side lengths.
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Domain
Cluster
Content
Standards
UNIT 2, Grade 7 Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For
example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and
observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p,
the relationship between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1,
r) where r is the unit rate.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and
commissions, fees, percent increase and decrease, percent error.
Instructional Strategies
Instructional Resources/Tools
Building from the development of rate and unit concepts in Grade 6, applications now need to
Play money - act out a problem with play money
focus on solving unit-rate problems with more sophisticated numbers: fractions per fractions.
Advertisements in newspapers
Proportional relationships are further developed through the analysis of graphs, tables,
equations and diagrams. Ratio tables serve a valuable purpose in the solution of proportional
Unlimited manipulatives or tools (don‘t restrict the tools to one or two, give
problems. This is the time to push for a deep understanding of what a representation of a
students many options)
proportional relationship looks like and what the characteristics are: a straight line through the
origin on a graph, a ―rule‖ that applies for all ordered pairs, an equivalent ratio or an
expression that describes the situation, etc. This is not the time for students to learn to cross
multiply to solve problems.
Because percents have been introduced as rates in Grade 6, the work with percents should
continue to follow the thinking involved with rates and proportions. Solutions to problems can
be found by using the same strategies for solving rates, such as looking for equivalent ratios or
based upon understandings of decimals. Previously, percents have focused on ―out of 100‖;
now percents above 100 are encountered.
Providing opportunities to solve problems based within contexts that are relevant to seventh
graders will connect meaning to rates, ratios and proportions. Examples include: researching
newspaper ads and constructing their own question(s), keeping a log of prices (particularly
sales) and determining savings by purchasing items on sale, timing students as they walk a lap
on the track and figuring their rates, creating open-ended problem scenarios with and without
numbers to give students the opportunity to demonstrate conceptual understanding, inviting
students to create a similar problem to a given problem and explain their reasoning.
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Domain
Cluster
Content
Standards
Unit 2, Grade 7 Expressions and Equations
Use properties to generate equivalent expressions
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For
example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
Instructional Strategies
Instructional Resources/Tools
Have students build on their understanding of order of operations and use the properties of
Online Algebra Tiles - Visualize multiplying and factoring algebraic expressions
operations to rewrite equivalent numerical expressions that were developed in Grade 6.
using tiles.
Students continue to use properties that were initially used with whole numbers and now
develop the understanding that properties hold for integers, rational and real numbers.
Common Misconceptions
Provide opportunities to build upon this experience of writing expressions using variables to
As students begin to build and work with expressions containing more than two
represent situations and use the properties of operations to generate equivalent expressions.
operations, students tend to set aside the order of operations. For example having
These expressions may look different and use different numbers, but the values of the
a student simplify an expression like 8 + 4(2x - 5) + 3x can bring to light several
expressions are the same.
misconceptions. Do the students immediately add the 8 and 4 before distributing
the 4? Do they only multiply the 4 and the 2x and not distribute the 4 to both
Provide opportunities for students to experience expressions for amounts of increase and
terms in the parenthesis? Do they collect all like terms
decrease. In Standard 2, the expression is rewritten and the variable has a different
coefficient. In context, the coefficient aids in the understanding of the situation. Another
8 + 4 – 5, and 2x + 3x? Each of these show gaps in students‘ understanding of
example is this situation which represents a 10% decrease: b - 0.10b = 1.00b - 0.10b which
how to simplify numerical expressions with multiple operations.
equals 0.90b or 90% of the amount.
One method that students can use to become convinced that expressions are equivalent is by
substituting a numerical value for the variable and evaluating the expression. For example 5(3
+ 2x) is equal to
53 + 52x
Let x = 6 and substitute 6 for x in both equations.
5(3 + 26)
5(3 + 12)
5(15)
75
53 + 526
15 + 60
75
Provide opportunities for students to use and understand the properties of operations. These
include: the commutative, associative, identity, and inverse properties of addition and of
multiplication, and the zero property of multiplication. Another method students can use to
become convinced that expressions are equivalent is to justify each step of simplification of
an expression with an operation property.
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Domain
Cluster
Content
Standards
Unit 2, Grade 7 Expressions and Equations
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
3. Solve multi-step, real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and
decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess
the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will
make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a
door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by
reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of
these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For
example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of
the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you
want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
Instructional Strategies
Instructional Resources/Tools
To assist students‘ assessment of the reasonableness of answers, especially problem situations Solving for a Variable This activity for students uses a pan balance to model
involving fractional or decimal numbers, use whole-number approximations for the
solving equations for a variable.
computation and then compare to the actual computation. Connections between performing
the inverse operation and undoing the operations are appropriate here. It is appropriate to
Solving an Inequality This activity for students illustrates the solution to
expect students to show the steps in their work. Students should be able to explain their
inequalities modeled on a number line.
thinking using the correct terminology for the properties and operations.
Continue to build on students‘ understanding and application of writing and solving one-step
equations from a problem situation to multi-step problem situations. This is also the context
for students to practice using rational numbers including: integers, and positive and negative
fractions and decimals. As students analyze a situation, they need to identify what operation
should be completed first, then the values for that computation. Each set of the needed
operation and values is determined in order. Finally an equation matching the order of
operations is written. For example, Bonnie goes out to eat and buys a meal that costs $12.50
that includes a tax of $.75. She only wants to leave a tip based on the cost of the food. In this
situation, students need to realize that the tax must be subtracted from the total cost before
being multiplied by the percent of tip and then added back to obtain the final cost. C = (12.50
- .75)(1 + T) + .75 = 11.75(1 +T) + .75 where C = cost and T = tip.
Provide multiple opportunities for students to work with multi-step problem situations that
have multiple solutions and therefore can be represented by an inequality. Students need to be
aware that values can satisfy an inequality but not be appropriate for the situation, therefore
limiting the solutions for that particular problem.
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Domain
Cluster
Content
Standards
Unit 2, Grade 8 Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different
ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the
equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on extending the understanding of ratios and proportions. Unit rates have Carnegie Math™
been explored in Grade 6 as the comparison of two different quantities with the second unit a
unit of one, (unit rate). In seventh grade unit rates were expanded to complex fractions and
Graphing calculators
percents through solving multistep problems such as: discounts, interest, taxes, tips, and
percent of increase or decrease. Proportional relationships were applied in scale drawings, and SMART™ technology with software emulator
students should have developed an informal understanding that the steepness of the graph is
the slope or unit rate. Now unit rates are addressed formally in graphical representations,
National Library of Virtual Manipulatives (NLVM)©,
algebraic equations, and geometry through similar triangles.
The National Council of Teachers of Mathematics, Illuminations
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose
Annenberg™ video tutorials, www.nsdl.org to access applets
of illustrating how the rates of two objects can be represented, analyzed and described in
different ways: graphically and algebraically. Emphasize the creation of representative graphs Texas Instruments® website (www.ticares.com)
and the meaning of various points. Then compare the same information when represented in
an equation.
By using coordinate grids and various sets of three similar triangles, students can prove that
the slopes of the corresponding sides are equal, thus making the unit rate of change equal.
After proving with multiple sets of triangles, students can be led to generalize the slope to y =
mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
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Domain
Cluster
Content
Standards
Unit 2, Grade 8 Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the
case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a
and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students applied the properties of operations to generate equivalent expressions,
SMART Board‘s new tools for solving equations
and identified when two expressions are equivalent. This cluster extends understanding to the
process of solving equations and to their solutions, building on the fact that solutions maintain Graphing calculators
equality, and that equations may have only one solution, many solutions, or no solution at all.
Equations with many solutions may be as simple as 3x = 3x, 3x + 5 = x + 2 + x + x + 3, or 6x
Index cards with equations/graphs for matching and sorting
+ 4x = x(6 + 4), where both sides of the equation are equivalent once each side is simplified.
Supply and Demand This activity focuses on having students create and solve a
Table 3 on page 90 of CCSS generalizes the properties of operations and serves as a reminder system of linear equations in a real-world setting. By solving the system, students
for teachers of what these properties are. Eighth graders should be able to describe these
will find the equilibrium point for supply and demand. Students should be
relationships with real numbers and justify their reasoning using words and not necessarily
familiar with finding linear equations from two points or slope and y-intercept.
with the algebraic language of Table 3. In other words, students should be able to state that
This lesson was adapted from the October 1991 edition of Mathematics Teacher.
3(-5) = (-5)3 because multiplication is commutative and it can be performed in any order (it is
commutative), or that 9(8) + 9(2) = 9(8 + 2) because the distributive property allows us to
distribute multiplication over addition, or determine products and add them. Grade 8 is the
Common Misconceptions
beginning of using the generalized properties of operations, but this is not something on
Students think that only the letters x and y can be used for variables.
which students should be assessed.
Students think that you always need a variable = a constant as a solution.
Pairing contextual situations with equation solving allows students to connect mathematical
analysis with real-life events. Students should experience analyzing and representing
The variable is always on the left side of the equation.
contextual situations with equations, identify whether there is one, none, or many solutions,
and then solve to prove conjectures about the solutions. Through multiple opportunities to
Equations are not always in the slope intercept form, y=mx+b
analyze and solve equations, students should be able to estimate the number of solutions and
possible values(s) of solutions prior to solving. Rich problems, such as computing the number Students confuse one-variable and two-variable equations.
of tiles needed to put a border around a rectangular space or solving proportional problems as
in doubling recipes, help ground the abstract symbolism to life.
Experiences should move through the stages of concrete, conceptual and algebraic/abstract.
Utilize experiences with the pan balance model as a visual tool for maintaining equality
(balance) first with simple numbers, then with pictures symbolizing relationships, and finally
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with rational numbers allows understanding to develop as the complexity of the problems
increases. Equation-solving in Grade 8 should involve multistep problems that require the use
of the distributive property, collecting like terms, and variables on both sides of the equation.
This cluster builds on the informal understanding of slope from graphing unit rates in Grade 6
and graphing proportional relationships in Grade 7 with a stronger, more formal
understanding of slope. It extends solving equations to understanding solving systems of
equations, or a set of two or more linear equations that contain one or both of the same two
variables. Once again the focus is on a solution to the system. Most student experiences
should be with numerical and graphical representations of solutions. Beginning work should
involve systems of equations with solutions that are ordered pairs of integers, making it easier
to locate the point of intersection, simplify the computation and hone in on finding a solution.
More complex systems can be investigated and solve by using graphing technology.
System-solving in Grade 8 should include estimating solutions graphically, solving using
substitution, and solving using elimination. Students again should gain experience by
developing conceptual skills using models that develop into abstract skills of formal solving
of equations. Provide opportunities for students to change forms of equations (from a given
form to slope-intercept form) in order to compare equations.
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Domain
Cluster
Content
Standards
Unit 3, Grade 7 Statistics and Probability
Use random sampling to draw inferences about a population
1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population
from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and
support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling
words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe data. Students
Guidelines for Assessment and instruction in Statistics Education (GAISE)
continue to use this knowledge in Grade 7 as they use random samples to make predictions
Report, American Statistical Association
about an entire population and judge the possible discrepancies of the predictions. Providing
opportunities for students to use real-life situations from science and social studies shows the
Ohio Resource Center:
purpose for using random sampling to make inferences about a population.
Mathline Something Fishy #257: Students estimate the size of a large population
by applying the concepts of ratio and proportion through the capture-recapture
Make available to students the tools needed to develop the skills and understandings required
statistical procedure.
to produce a representative sample of the general population. One key element of a
representative sample is understanding that a random sampling guarantees that each element
Random Sampling and Estimation # 8347: In this session, students estimate
of the population has an equal opportunity to be selected in the sample. Have students
population quantities from a random sample.
compare the random sample to population, asking questions like ―Are all the elements of the
entire population represented in the sample?‖ and ―Are the elements represented
Bias in Sampling #11062: This content resource addresses statistics topics that
proportionally?‖ Students can then continue the process of analysis by determining the
teachers may be uncomfortable teaching due to limited exposure to statistical
measures of center and variability to make inferences about the general population based on
content and vocabulary. This resource focuses a four-component statistical
the analysis.
problem-solving process and the meaning of variation and bias in statistics and to
investigate how data vary.
Provide students with random samples from a population, including the statistical measures.
Ask students guiding questions to help them make inferences from the sample.
From the National Council of Teachers of Mathematics, Illuminations - Capture
Recapture: In this lesson, students
Common Misconceptions
Students may believe: One random sample is not representative of the entire
population. Many samples must be taken in order to make an inference that is
valid. By comparing the results of one random sample with the results of
multiple random samples, students can correct this misconception.
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Domain
Cluster
Content
Standards
Unit 3, Grade 7 Statistics and Probability
Draw informal comparative inferences about two populations
3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by
expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean
height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two
distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two
populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a
fourth-grade science book.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students used measures of center and variability to describe sets of data. In the
Ohio Resource Center:
cluster ―Use random sampling to draw inferences about a population‖ of Statistics and
Baseball Stats ORC #1494: In this lesson students explore and compare data sets
Probability in Grade 7, students learn to draw inferences about one population from a random and statistics in baseball.
sampling of that population. Students continue using these skills to draw informal
comparative inferences about two populations.
Representation of Data—Cholera and War ORC #9740: The object of this
activity is to study excellent examples of the presentation of data. Students
Provide opportunities for students to deal with small populations, determining measures of
analyze (1) a map of cholera cases plotted against the location of water wells in
center and variability for each population. Then have students compare those measures and
London in 1854 and (2) a map of Napoleon's march on Moscow in 1812-1813 to
make inferences. The use of graphical representations of the same data (Grade 6) provides
see what inferences they can draw from the data displays.
another method for making comparisons. Students begin to develop understanding of the
benefits of each method by analyzing data with both methods.
Representation of Data—The U. S. Census ORC # 9741: The object of this
activity is to study an excellent example of the presentation of data. Students
When students study large populations, random sampling is used as a basis for the population analyze an illustration of the 1930 U.S. census compared to the 1960 census to
inference. This build on the skill developed in the Grade 7 cluster ―Use random sampling to
see what inferences they can draw from the data displays.
draw inferences about a population‖ of Statistics and Probability. Measures of center and
variability are used to make inferences on each of the general populations. Then the students
have make comparisons for the two populations based on those inferences.
This is a great opportunity to have students examine how different inferences can be made
based on the same two sets of data. Have students investigate how advertising agencies uses
data to persuade customers to use their products. Additionally, provide students with two
populations and have them use the data to persuade both sides of an argument.
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Domain
Cluster
Content
Standards
Grade 7 Statistics and Probability
Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate
greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a
probability near 1 indicates a likely event.
6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and
predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled
roughly 200 times, but probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not
good, explain possible sources of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For
example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the
approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning
penny appear to be equally likely based on the observed frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the
compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday
language (e.g., ―rolling double sixes‖), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the
answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
Instructional Strategies
Instructional Resources/Tools
Grade 7 is the introduction to the formal study of probability. Through multiple experiences,
From the National Council of Teachers of Mathematics, Illuminations:
students begin to understand the probability of chance (simple and compound), develop and
Boxing Up: In this lesson students explore the relationship between theoretical
use sample spaces, compare experimental and theoretical probabilities, develop and use
and experimental probabilities.
graphical organizers, and use information from simulations for predictions.
Capture-Recapture: In this lesson students estimate the size of a total population
Help students understand the probability of chance is using the benchmarks of probability: 0,
by taking samples and using proportions to estimate the entire population.
1 and ½. Provide students with situations that have clearly defined probability of never
happening as zero, always happening as 1 or equally likely to happen as to not happen as 1/2. Ohio Resource Center:
Then advance to situations in which the probability is somewhere between any two of these
Probability Basics ORC #24 Probability Basics This is a 7+ minute video that
benchmark values. This builds to the concept of expressing the probability as a number
explores theoretical and experimental probability with tree diagrams and the
between 0 and 1. Use this understand to build the understanding that the closer the probability fundamental counting principle.
is to 0, the more likely it will not happen, and the closer to 1, the more likely it will happen.
Students learn to make predictions about the relative frequency of an event by using
Probability Using Dice ORC #9737 This activity explores the probabilities of
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simulations to collect, record, organize and analyze data. Students also develop the
understanding that the more the simulation for an event is repeated, the closer the
experimental probability approaches the theoretical probability.
Have students develop probability models to be used to find the probability of events. Provide
students with models of equal outcomes and models of not equal outcomes are developed to
be used in determining the probabilities of events.
Students should begin to expand the knowledge and understanding of the probability of
simple events, to find the probabilities of compound events by creating organized lists, tables
and tree diagrams. This helps students create a visual representation of the data; i.e., a sample
space of the compound event. From each sample space, students determine the probability or
fraction of each possible outcome. Students continue to build on the use of simulations for
simple probabilities and now expand the simulation of compound probability.
rolling various sums with two dice. Extensions of the problem and a complete
discussion of the underlying mathematical ideas are included.
How to Fix and Unfair Game ORC #9718 This activity explores a fair game and
―How to Fix an Unfair Game.‖
Remove One ORC # 253 A game is analyzed and the concepts of probability and
sample space are discussed. In addition to the lesson plan, the site includes ideas
for teacher discussion, extensions of the lesson, additional resources (including a
video of the lesson procedures) and a discussion of the mathematical content.
Dart Throwing ORC #10131 The object of this activity is to study an excellent
example of the presentation of data. Students analyze an illustration of the 1930
U.S. census compared to the 1960 census to see what inferences they can draw
from the data displays.
Providing opportunities for students to match situations and sample spaces assists students in
visualizing the sample spaces for situations.
Students often struggle making organized lists or trees for a situation in order to determine the
theoretical probability. Having students start with simpler situations that have fewer elements
enables them to have successful experiences with organizing lists and trees diagrams. Ask
guiding questions to help students create methods for creating organized lists and trees for
situations with more elements.
Common Misconceptions
Students often expect the theoretical and experimental probabilities of the same
data to match. By providing multiple opportunities for students to experience
simulations of situations in order to find and compare the experimental
probability to the theoretical probability, students discover that rarely are those
probabilities the same.
Students often see skills of creating organized lists, tree diagrams, etc. as the end product.
Provide students with experiences that require the use of these graphic organizers to
determine the theoretical probabilities. Have them practice making the connections between
the process of creating lists, tree diagrams, etc. and the interpretation of those models.
Students often expect that simulations will result in all of the possibilities. All
possibilities may occur in a simulation, but not necessarily. Theoretical
probability does use all possibilities. Note examples in simulations when some
possibilities are not shown.
Additionally, students often struggle when converting forms of probability from fractions to
percents and vice versa. To help students with the discussion of probability, don‘t allow the
symbol manipulation/conversions to detract from the conversations. By having students use
technology such as a graphing calculator or computer software to simulate a situation and
graph the results, the focus is on the interpretation of the data. Students then make predictions
about the general population based on these probabilities.
Domain
Grade 7 Geometry
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Cluster
Content
Standards
Draw, construct, and describe geometrical figures and describe the relationships between them.
1. Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale
drawing at a different scale.
2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three
measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right
rectangular pyramids.
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on the importance of visualization in the understanding of Geometry. Being able to
Straws, clay, angle rulers, protractors, rulers, grid paper
visualize and then represent geometric figures on paper is essential to solving geometric problems.
Road Maps - convert to actual miles
Scale drawings of geometric figures connect understandings of proportionality to geometry and lead to future
work in similarity and congruence. As an introduction to scale drawings in geometry, students should be given Dynamic computer software - Geometer's SketchPad. This
the opportunity to explore scale factor as the number of time you multiple the measure of one object to obtain
cluster lends itself to using dynamic software. Students
the measure of a similar object. It is important that students first experience this concept concretely
sometimes can manipulate the software more quickly than do
progressing to abstract contextual situations. Pattern blocks (not the hexagon) provide a convenient means of
the work manually. However, being able to use a protractor and
developing the foundation of scale. Choosing one of the pattern blocks as an original shape, students can then
a straight edge are desirable skills.
create the next-size shape using only those same-shaped blocks. Questions about the relationship of the
original block to the created shape should be asked and recorded. A sample of a recording sheet is shown.
Common Misconceptions
Student may have misconceptions about:
Scale
Shape
Original Side Length Created Side Length
Relationship of Created to Original
Correctly setting up proportions
Square
1 unit
How to read a ruler
Triangle
1 unit
Doubling side measures does not double perimeter
Rhombus
1 unit
This can be repeated for multiple iterations of each shape by comparing each side length to the original‘s side
length. An extension would be for students to compare the later iterations to the previous. Students should also
be expected to use side lengths equal to fractional and decimal parts. In other words, if the original side can be
stated to represent 2.5 inches, what would be the new lengths and what would be the scale?
Shape
Square
Parallelogram
Trapezoid
Original Side Length
2.5 inches
3.25 cms
(Actual measurements)
Created Side Length
Scale
Length 1
Length 2
Provide opportunities for students to use scale drawings of geometric figures with a given scale that requires
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them to draw and label the dimensions of the new shape. Initially, measurements should be in whole numbers,
progressing to measurements expressed with rational numbers. This will challenge students to apply their
understanding of fractions and decimals.
After students have explored multiple iterations with a couple of shapes, ask them to choose and replicate a
shape with given scales to find the new side lengths, as well as both the perimeters and areas. Starting with
simple shapes and whole-number side lengths allows all students access to discover and understand the
relationships. An interesting discovery is the relationship of the scale of the side lengths to the scale of the
respective perimeters (same scale) and areas (scale squared). A sample recording sheet is shown.
Shape
Rectangle
Triangle
Side
Length
2 x 3 in.
1.5 inches
Scale
2
2
Original
Perimeter
10 inches
4.5 inches
Scaled
Perimeter
20 inches
9 inches
Perimeter
Scale
2
2
Original
Area
6 sq. in.
2.25 sq. in.
Scaled
Area
24 sq in.
9 sq in.
Area
Scale
4
4
Students should move on to drawing scaled figures on grid paper with proper figure labels, scale and
dimensions. Provide word problems that require finding missing side lengths, perimeters or areas. For
example, if a 4 by 4.5 cm rectangle is enlarged by a scale of 3, what will be the new perimeter? What is the
new area? or If the scale is 6, what will the new side length look like? or Suppose the area of one triangle is
16 sq units and the scale factor between this triangle and a new triangle is 2.5. What is the area of the new
triangle?
Reading scales on maps and determining the actual distance (length) is an appropriate contextual situation.
Constructions facilitate understanding of geometry. Provide opportunities for students to physically construct
triangles with straws, sticks, or geometry apps prior to using rulers and protractors to discover and justify the
side and angle conditions that will form triangles. Explorations should involve giving students: three side
measures, three angle measures, two side measures and an included angle measure, and two angles and an
included side measure to determine if a unique triangle, no triangle or an infinite set of triangles results.
Through discussion of their exploration results, students should conclude that triangles cannot be formed by
any three arbitrary side or angle measures. They may realize that for a triangle to result the sum of any two
side lengths must be greater than the third side length, or the sum of the three angles must equal 180 degrees.
Students should be able to transfer from these explorations to reviewing measures of three side lengths or three
angle measures and determining if they are from a triangle justifying their conclusions with both sketches and
reasoning.
This cluster is related to the following Grade 7 cluster ―Solve real-life and mathematical problems involving
angle measure, area, surface area, and volume.‖ Further construction work can be replicated with
quadrilaterals, determining the angle sum, noticing the variety of polygons that can be created with the same
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side lengths but different angle measures, and ultimately generalizing a method for finding the angle sums for
regular polygons and the measures of individual angles. For example, subdividing a polygon into triangles
using a vertex (N-2)180° or subdividing a polygons into triangles using an interior point 180°N - 360° where
N = the number of sides in the polygon. An extension would be to realize that the two equations are equal.
Slicing three-dimensional figures helps develop three-dimensional visualization skills. Students should have
the opportunity to physically create some of the three-dimensional figures, slice them in different ways, and
describe in pictures and words what has been found. For example, use clay to form a cube, then pull string
through it in different angles and record the shape of the slices found. Challenges can also be given: ―See how
many different two-dimensional figures can be found by slicing a cube‖ or ―What three-dimensional figure can
produce a hexagon slice?‖ This can be repeated with other three-dimensional figures using a chart to record
and sketch the figure, slices and resulting two-dimensional figures.
Domain
Grade 7 Geometry
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Cluster
Content
Standards
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
4. Know the formulas for the area and circumference of a circle and solve problems; give an informal derivation of the relationship between the circumference
and area of a circle.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and use them to solve simple equations for an
unknown angle in a figure.
6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,
quadrilaterals, polygons, cubes, and right prisms.
Instructional Strategies
Instructional Resources/Tools
This is the students‘ initial work with circles. Knowing that a circle is created by connecting
Circular objects of several different sizes
all the points equidistant from a point (center) is essential to understanding the relationships
between radius, diameter, circumference, pi and area. Students can observe this by folding a
String or yarn
paper plate several times, finding the center at the intersection, then measuring the lengths
between the center and several points on the circle, the radius. Measuring the folds through
Tape measures, rulers
the center, or diameters leads to the realization that a diameter is two times a radius. Given
multiple-size circles, students should then explore the relationship between the radius and the Grid paper
length measure of the circle (circumference) finding an approximation of pi and ultimately
deriving a formula for circumference. String or yarn laid over the circle and compared to a
Paper plates
ruler is an adequate estimate of the circumference. This same process can be followed in
finding the relationship between the diameter and the area of a circle by using grid paper to
NCTM Illuminations:
estimate the area.
Square Circles: This lesson features two creative twists on the standard lesson of
having students measure several circles to discover that the ratio of the
Another visual for understanding the area of a circle can be modeled by cutting up a paper
circumference to the diameter seems always to be a little more than 3. This
plate into 16 pieces along diameters and reshaping the pieces into a parallelogram. In figuring lesson starts with squares, so students can first identify a simpler constant ratio
area of a circle, the squaring of the radius can also be explained by showing a circle inside a
(4) of perimeter to length of a side before moving to the more difficult case of the
square. Again, the formula is derived and then learned. After explorations, students should
circle. The second idea is to measure with a variety of units, so students can more
then solve problems, set in relevant contexts, using the formulas for area and circumference.
readily see that the ratio of the measurements remains constant, not only across
different sizes of figures, but even for the same figure with different
In previous grades, students have studied angles by type according to size: acute, obtuse and
measurements. From these measurements, students will discover the constant
right, and their role as an attribute in polygons. Now angles are considered based upon the
ratio of 1:4 for all squares and the ratio of approximately 1:3.14 for all circles.
special relationships that exist among them: supplementary, complementary, vertical and
adjacent angles. Provide students the opportunities to explore these relationships first through
Apple Pi: Using estimation and measurement skills, students will determine the
measuring and finding the patterns among the angles of intersecting lines or within polygons,
ratio of circumference to diameter and explore the meaning of π. Students will
then utilize the relationships to write and solve equations for multi-step problems.
discover the circumference and area formulas based on their investigations.
Real-world and mathematical multi-step problems that require finding area, perimeter,
volume, surface area of figures composed of triangles, quadrilaterals, polygons, cubes and
Circle Tool: With this three-part online applet, students can
right prisms should reflect situations relevant to seventh graders. The computations should
explore with graphic and numeric displays how the circumference and area of a
make use of formulas and involve whole numbers, fractions, decimals, ratios and various
circle compare to its radius and diameter. Students can collect data points by
units of measure with same system conversions.
dragging the radius to various lengths and clicking the "Add to Table" button to
record the data in the table.
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Geometry of Circles: Using a MIRATM geometry tool, students determine the
relationships between radius, diameter, circumference and area of a circle.
Ohio Resource Center
Circles and Their Areas: Given that units of area are squares, how can we find
the area of a circle or other curved region? Imagine a waffle-like grid inside a
circle and a larger grid containing the circle. The area of the circle lies between
the area of the inside grid and the area of the outside grid..
Exploring c/d = π: Students measure circular objects to collect data to
investigate the relationship between the circumference of a circle and its
diameter. They find that, regardless of the size of the object or the size of the
measuring unit, it always takes a little more than three times the length of the
diameter to measure the circumference.
Parallel Lines: Students use Geometer's Sketchpad® to explore relationships
among the angles formed when parallel lines are cut by a transversal. The
software is integral to the lesson, and step-by-step instructions are provided.
Common Misconceptions
Students may believe pi is an exact number rather than understanding that 3.14 is
just an approximation of pi.
Many students are confused when dealing with circumference (linear
measurement) and area. This confusion is about an attribute that is measured
using linear units (surrounding) vs. an attribute that is measured using area units
(covering).
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Domain
Cluster
Content
Standards
Grade 8 Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
6. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
7. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and
translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
9. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections,
translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
10..Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles
appears to form a line, and give an argument in terms of transversals why this is so.
Instructional Strategies
A major focus in Grade 8 is to use knowledge of angles and distance to analyze two- and threedimensional figures and space in order to solve problems. This cluster interweaves the
relationships of symmetry, transformations, and angle relationships to form understandings of
similarity and congruence. Inductive and deductive reasoning are utilized as students forge into
the world of proofs. Informal arguments are justifications based on known facts and logical
reasoning. Students should be able to appropriately label figures, angles, lines, line segments,
congruent parts, and images (primes or double primes). Students are expected to use logical
thinking, expressed in words using correct terminology. They are NOT expected to use theorems,
axioms, postulates or a formal format of proof as in two-column proofs.
Instructional Resources/Tools
Pattern blocks or shape sets
Transformational geometry is about the effects of rigid motions, rotations, reflections and
translations on figures. Initial work should be presented in such a way that students understand
the concept of each type of transformation and the effects that each transformation has on an
object before working within the coordinate system. For example, when reflecting over a line,
each vertex is the same distance from the line as its corresponding vertex. This is easier to
visualize when not using regular figures. Time should be allowed for students to cut out and trace
the figures for each step in a series of transformations. Discussion should include the description
of the relationship between the original figure and its image(s) in regards to their corresponding
parts (length of sides and measure of angles) and the description of the movement, including the
attributes of transformations (line of symmetry, distance to be moved, center of rotation, angle of
rotation and the amount of dilation).The case of distance – preserving transformation leads to the
idea of congruence.
Patty paper
Mirrors - Miras
Geometry software like Geometer's Sketchpad, Cabri Jr or GeoGebra
Graphing calculators
Grid paper
From the National Library of Virtual Manipulatives:
Congruent Triangles – Build similar triangles by combining sides and angles.
Geoboard - Coordinate – Rectangular geoboard with x and y coordinates.
Transformations - Composition – Explore the effect of applying a
composition of translation, rotation, and reflection transformations to objects.
Transformations - Dilation – Dynamically interact with and see the result of a
dilation transformation.
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It is these distance-preserving transformations that lead to the idea of congruence.
Transformations - Reflection – Dynamically interact with and see the result of
a reflection transformation.
Transformations - Rotation – Dynamically interact with and see the result of a
rotation transformation.
Transformations - Translation – Dynamically interact with and see the result
of a translation transformation.
Work in the coordinate plane should involve the movement of various polygons by addition,
subtraction and multiplied changes of the coordinates. For example, add 3 to x, subtract 4 from y,
combinations of changes to x and y, multiply coordinates by 2 then by . Students should observe
and discuss such questions as ‗What happens to the polygon?‘ and ‗What does making the
change to all vertices do?‘. Understandings should include generalizations about the changes
that maintain size or maintain shape, as well as the changes that create distortions of the polygon
(dilations). Example dilations should be analyzed by students to discover the movement from the
origin and the subsequent change of edge lengths of the figures. Students should be asked to
describe the transformations required to go from an original figure to a transformed figure
(image). Provide opportunities for students to discuss the procedure used, whether different
procedures can obtain the same results, and if there is a more efficient procedure to obtain the
same results. Students need to learn to describe transformations with both words and numbers.
Common Misconceptions
Students often confuse situations that require adding with multiplicative
situations in regard to scale factor. Providing experiences with geometric
figures and coordinate grids may help students visualize the difference.
Through understanding symmetry and congruence, conclusions can be made about the
relationships of line segments and angles with figures. Students should relate rigid motions to the
concept of symmetry and to use them to prove congruence or similarity of two figures. Problem
situations should require students to use this knowledge to solve for missing measures or to
prove relationships. It is an expectation to be able to describe rigid motions with coordinates.
Provide opportunities for students to physically manipulate figures to discover properties of
similar and congruent figures, for example, the corresponding angles of similar figures are equal.
Additionally use drawings of parallel lines cut by a transversal to investigate the relationship
among the angles. For example, what information can be obtained by cutting between the two
intersections and sliding one onto the other?
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In Grade 7, students develop an understanding of the special relationships of angles and their
measures (complementary, supplementary, adjacent, vertical). Now, the focus is on learning the
about the sum of the angles of a triangle and using it to, find the measures of angles formed by
transversals (especially with parallel lines), or to find the measures of exterior angles of triangles
and to informally prove congruence.
By using three copies of the same triangle labeled and placed so that the three different angles
form a straight line, students can:
explore the relationships of the angles,
learn the types of angles (interior, exterior, alternate interior, alternate exterior,
corresponding, same side interior, same side exterior), and
explore the parallel lines, triangles and parallelograms formed.
Further examples can be explored to verify these relationships and demonstrate their relevance in
real life.
1 2 3
17 B A
16 C
15 14
4 5 6
C B 7
A B C
13
12 11
A 8
10 9
Investigations should also lead to the Angle-Angle criterion for similar triangles. For instance,
pairs of students create two different triangles with one given angle measurement, then repeat
with two given angle measurements and finally with three given angle measurements. Students
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observe and describe the relationship of the resulting triangles. As a class, conjectures lead to the
generalization of the Angle-Angle criterion.
Students should solve mathematical and real-life problems involving understandings from this
cluster. Investigation, discussion, justification of their thinking, and application of their learning
will assist in the more formal learning of geometry in high school.
Domain
Grade 8 Geometry
Cluster
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve-real world and mathematical problems.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Begin by recalling the formula, and its meaning, for the volume of a right rectangular prism:
Ohio Resource Center:
V = l ×w ×h. Then ask students to consider how this might be used to make a conjecture
#8421 ―The Cylinder Problem - Students build a family of cylinders, all from the
about the volume formula for a cylinder:
same-sized paper, and discover the relationship between the dimensions of the
paper and the resulting cylinders. They order the cylinders by their volumes and
draw a conclusion about the relationship between a cylinder's dimensions and its
volume.
h
h
l
w
Most students can be readily led to the understanding that the volume of a right rectangular
prism can be thought of as the area of a ―base‖ times the height, and so because the area of
the base of a cylinder is π r2 the volume of a cylinder is Vc = π r2h.
To motivate the formula for the volume of a cone, use cylinders and cones with the same base
and height. Fill the cone with rice or water and pour into the cylinder. Students will
discover/experience that 3 cones full are needed to fill the cylinder. This non-mathematical
derivation of the formula for the volume of a cone, V = 1/3 π r 2h, will help most students
remember the formula.
In a drawing of a cone inside a cylinder, students might see that that the triangular crosssection of a cone is 1/2 the rectangular cross-section of the cylinder. Ask them to reason why
National Library of Virtual Manipulatives :
―How High‖ is an applet that can be used to take an inquiry approach to the
formula for volume of a cylinder or cone.
NCTM:
Finding Surface Area and Volume
Blue Cube, 27 Little Cubes (Stella Stunner)
Volume of a Spheres and Cones (Rich Problem)
Common Misconceptions
A common misconception among middle grade students is that ―volume‖ is a
―number‖ that results from ―substituting‖ other numbers into a formula. For these
students there is no recognition that ―volume‖ is a measure – related to the
amount of space occupied. If a teacher discovers that students do not have an
understanding of volume as a measure of space, it is important to provide
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the volume (three dimensions) turns out to be less than 1/2 the volume of the cylinder. It turns
out to be 1/3.
opportunities for hands on experiences where students ―fill‖ three dimensional
objects. Begin with right- rectangular prisms and fill them with cubes will help
students understand why the units for volume are cubed. See Cubes
http://illuminations.nctm.org/ActivityDetail.aspx?ID=6
For the volume of a sphere, it may help to have students visualize a hemisphere ―inside‖ a
cylinder with the same height and ―base.‖ The radius of the circular base of the cylinder is
also the radius of the sphere and the hemisphere. The area of the ―base‖ of the cylinder and
the area of the section created by the division of the sphere into a hemisphere is π r2. The
height of the cylinder is also r so the volume of the cylinder is π r 3. Students can see that the
volume of the hemisphere is less than the volume of the cylinder and more than half the
volume of the cylinder. Illustrating this with concrete materials and rice or water will help
students see the relative difference in the volumes. At this point, students can reasonably
accept that the volume of the hemisphere of radius r is 2/3 π r 3 and therefore volume of a
sphere with radius r is twice that or 4/3 π r3. There are several websites with explanations for
students who wish to pursue the reasons in more detail. (Note that in the pictures above, the
hemisphere and the cone together fill the cylinder.)
Students should experience many types of real-world applications using these formulas. They
should be expected to explain and justify their solutions.
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Domain
The Number System
Cluster
Know that there are numbers that are not rational, and approximate them by rational numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show
Content
that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Standards
Instructional Strategies
Instructional Resources/Tools
In the elementary grades, students become familiar with decimal fractions, most often with
Graphing Calculator
decimal representations that terminate a few digits to the right of the decimal point. For
example, to find the exact decimal representation of 2/7, students might use their calculator to Dynamic Geometry Software (Geogebra)
find 2/7 = 0.2857142857…and they might guess that the digits 285714 repeat. To show that
the digits do repeat, students in Grade 7 actually carry out the long division and recognize that 1-1 and 3-7 – Holt McDougal Common Core 8
the remainders repeat in a predictable pattern—a pattern that creates the repetition in the
decimal representation (see 7.NS.2.d).
Common Misconceptions
Thinking about long division generally, ask students what will happen if the remainder is 0 at Some students are surprised that the decimal representation of pi does not repeat.
some step. They can reason that the long division is complete, and the decimal representation Some students believe that if only we keep looking at digits farther and farther to
terminates. If the reminder is never 0, in contrast, then the remainders will repeat in a cyclical the right, eventually a pattern will emerge.
pattern because at each step with a given remainder, the process for finding the next
remainder is always the same. Thus, the digits in the decimal representation also repeat.
A few irrational numbers are given special names (pi and e), and much attention
When dividing by 7, there are 6 possible nonzero remainders, and students can see that the
is given to sqrt(2). Because we name so few irrational numbers, students
decimal repeats with a pattern of at most 6 digits. In general, when finding the decimal
sometimes conclude that irrational numbers are unusual and rare. In fact,
representation of m/n, students can reason that the repeating portion of decimal will have at
irrational numbers are much more plentiful than rational numbers, in the sense
most n-1 digits. The important point here is that students can see that the pattern will repeat,
that they are ―denser‖ in the real line.
so they can imagine the process continuing without actually carrying it out.
Conversely, given a repeating decimal, students learn strategies for converting the decimal to
a fraction. One approach is to notice that rational numbers with denominators of 9 repeat a
single digit. With a denominator of 99, two digits repeat; with a denominator of 999, three
digits repeat, and so on.
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GRADE 8
Domain
Cluster
Content
Standards
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers. (cont.)
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and
estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and
1.5, and explain how to continue on to get better approximations.
Instructional Strategies
Instructional Resources/Tools
Students construct a right isosceles triangle with legs of 1 unit. Using the Pythagorean
Graphing Calculator
theorem, they determine that the length of the hypotenuse is √2. In the figure below, they can
rotate the hypotenuse back to the original number line to show that indeed √2 is a number on
Dynamic Geometry Software (Geogebra)
the number line.
Lessons 3-6, 3-8 in Holt McDougal Common Core Grade 8
Once students understand that (1) every rational number has a decimal representation that
either terminates or repeats, and (2) every terminating or repeating decimal is a rational
Lab Explore Right Triangles pg. 131 in Holt McDougal Common Core Grade 8
number, they can reason that on the number line, irrational numbers (i.e., those that are not
rational) must have decimal representations that neither terminate nor repeat. And although
students at this grade do not need to be able to prove that √2 is irrational, they need to know
that √2 is irrational (see 8.EE.2), which means that its decimal representation neither
terminates nor repeats. Nonetheless, they can approximate √2 without using the square root
key on the calculator. Students can create tables to approximate √2.
From knowing that
= 1 and
= 4, or from the picture above, students can reason that
there is a number between 1 and 2 whose square is 2. In the first table above, students can see
that between 1.4 and 1.5, there is a number whose square is 2. Then in the second table, they
locate that number between 1.41 and 1.42. And in the third table they can locate √2 between
1.414 and 1.415. Students can develop more efficient methods for this work. For example,
from the picture above, they might have begun the first table with 1.4. And once they see that
1.422 > 2, they do not need generate the rest of the data in the second table.
Use set diagrams to show the relationships among real, rational, irrational numbers, integers,
and counting numbers. The diagram should show that the all real numbers (numbers on the
number line) are either rational or irrational.
Given two distinct numbers, it is possible to find both a rational and an irrational number
between them.
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Domain
Cluster
Content
Standards
Expressions and Equations
Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3‾5 = 3‾3 = 1/33 = 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x3 = p, where p is a positive rational number. Evaluate square
roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many
times as much one is than the other. For example, estimate the population of the United States as 3 × 10 8 and the population of the world as 7 × 10 9, and
determine that the world population is more than 20 times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific
notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading).
Interpret scientific notation that has been generated by technology.
Instructional Strategies
Instructional Resources/Tools
Although students begin using whole-number exponents in Grades 5 and 6, it is in Grade 8 when
Square tiles and cubes to develop understanding of squared and cubed
students are first expected to know and use the properties of exponents and to extend the meaning
numbers
beyond counting-number exponents. It is no accident that these expectations are simultaneous,
Calculators to verify and explore patterns
because it is the properties of counting-number exponents that provide the rationale for the
properties of integer exponents. In other words, students should not be told these properties but
Webquests using data mined from sites like the U.S. Census Bureau,
rather should derive them through experience and reason.
scientific data (planetary distances)
For counting-number exponents (and for nonzero bases), the following properties follow directly
from the meaning of exponents.
Place value charts to connect the digit value to the exponent (negative and
positive)
Powers of 10 online video.
Lessons 3-2, 3-3, 3-4, 3-5, 3-7 in Holt McDougal Grade 8
Students should have experience simplifying numerical expressions with exponents so that these
properties become natural and obvious. For example,
Explore Cube Roots Lab pg. 120 in Holt McDougal Grade 8
Common Misconceptions
Students may mix up the product of powers property and the power of a
power property.
If students reason about these examples with a sense of generality about the numbers, they begin
to articulate the properties. For example, ―I see that 3 twos is being multiplied by 5 twos, and the
results is 8 twos being multiplied together, where the 8 is the sum of 3 and 5, the number of twos
Students may make the relationship that in scientific notation, when a
number contains one nonzero digit and a positive exponent, that the number
of zeros equals the exponent. This pattern may incorrectly be applied to
scientific notation values with negative values or with more than one
nonzero digit.
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in each of the original factors. That would work for a base other than two (as long as the bases are
the same).‖ Note: When talking about the meaning of an exponential expression, it is easy to say
(incorrectly) that ―35 means 3 multiplied by itself 5 times.‖ But by writing out the meaning,
35=3∙3∙3∙3∙3, students can see that there are only 4 multiplications. So a better description is ―35
means 5 3s multiplied together.‖
Students also need to realize that these simple descriptions work only for counting-number
exponents. When extending the meaning of exponents to include 0 and negative exponents, these
descriptions are limiting: Is it sensible to say ―30 means 0 3s multiplied together‖ or that ―3-2
means -2 3s multiplied together‖?
The motivation for the meanings of 0 and negative exponents is the following principle: The
properties of counting-number exponents should continue to work for integer exponents.
For example, Property 1 can be used to reason what 30 should be. Consider the following
expression and simplification: 30∙35=30+5=35. This computation shows that the when 30 is
multiplied by 35, the result (following Property 1) should be 35, which implies that 30 must be 1.
Because this reasoning holds for any base other than 0, we can reason that a0 = 1 for any nonzero
number a.
To make a judgment about the meaning of 3-4, the approach is similar: 3−4∙34=3−4+4=30=1. This
computation shows that 3-4 should be the reciprocal of 34, because their product is 1. And again,
this reasoning holds for any nonzero base. Thus, we can reason that a−n = 1/an.
Putting all of these results together, we now have the properties of integer exponents, shown in the
above chart. For mathematical completeness, one might prove that properties 1-3 continue to hold
for integer exponents, but that is not necessary at this point.
A supplemental strategy for developing meaning for integer exponents is to make use of patterns,
as shown in the chart below.
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The meanings of 0 and negative-integer exponents can be further explored in a place-value chart:
Thus, integer exponents support writing any decimal in expanded form like the following:
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Expanded form and the connection to place value is important for helping students make sense of
scientific notation, which allows very large and very small numbers to be written concisely,
enabling easy comparison. To develop familiarity, go back and forth between standard notation
and scientific notation for numbers near, for example, 1012 or 10-9. Compare numbers, where one
is given in scientific notation and the other is given in standard notation. Real-world problems can
help students compare quantities and make sense about their relationship.
Provide practical opportunities for students to flexibly move between forms of squared and cubed
numbers. For example, If then. This flexibility should be experienced symbolically and verbally.
32=9 - 9=3
Opportunities for conceptually understanding irrational numbers should be developed. One way is
for students to draw a square that is one unit by one unit and find the diagonal using the
Pythagorean Theorem. The diagonal drawn has an irrational length of √2. Other irrational lengths
can be found using the same strategy by finding diagonal lengths of rectangles with various side
lengths.
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Domain
Cluster
Content
Standards
Expressions and Equations
Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different
ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the
equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on extending the understanding of ratios and proportions. Unit rates have Graphing calculators
been explored in Grade 6 as the comparison of two different quantities with the second unit a
unit of one, (unit rate). In seventh grade unit rates were expanded to complex fractions and
National Library of Virtual Manipulatives (NLVM)©,
percents through solving multistep problems such as: discounts, interest, taxes, tips, and
percent of increase or decrease. Proportional relationships were applied in scale drawings, and The National Council of Teachers of Mathematics, Illuminations
students should have developed an informal understanding that the steepness of the graph is
the slope or unit rate. Now unit rates are addressed formally in graphical representations,
Annenberg™ video tutorials, http://nsdl.org/ to access applets
algebraic equations, and geometry through similar triangles.
Texas Instruments® website (www.ticares.com)
Distance time problems are notorious in mathematics. In this cluster, they serve the purpose
of illustrating how the rates of two objects can be represented, analyzed and described in
Lessons 8-2, 8-3, 8-5, 9-4 in Holt McDougal Grade 8
different ways: graphically and algebraically. Emphasize the creation of representative graphs
and the meaning of various points. Then compare the same information when represented in
Explore Slope Lab pg. 343 in Holt McDougal Grade 8
an equation.
By using coordinate grids and various sets of three similar triangles, students can prove that
the slopes of the corresponding sides are equal, thus making the unit rate of change equal.
After proving with multiple sets of triangles, students can be led to generalize the slope to y =
mx for a line through the origin and y = mx + b for a line through the vertical axis at b.
Common Misconceptions
Students may confuse slope and y-intercept in the formula
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Domain
Cluster
Content
Standards
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the
case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a
and b are different numbers).
b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive
property and collecting like terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points,
determine whether the line through the first pair of points intersects the line through the second pair.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students applied the properties of operations to generate equivalent expressions, and SMART Board‘s new tools for solving equations
identified when two expressions are equivalent. This cluster extends understanding to the
Graphing calculators
process of solving equations and to their solutions, building on the fact that solutions maintain
equality, and that equations may have only one solution, many solutions, or no solution at all.
Index cards with equations/graphs for matching and sorting
Equations with many solutions may be as simple as 3x = 3x, 3x + 5 = x + 2 + x + x + 3, or 6x +
Supply and Demand This activity focuses on having students create and solve
4x = x(6 + 4), where both sides of the equation are equivalent once each side is simplified.
a system of linear equations in a real-world setting. By solving the system,
students will find the equilibrium point for supply and demand.
Table 3 on page 90 of CCSS generalizes the properties of operations and serves as a reminder
for teachers of what these properties are. Eighth graders should be able to describe these
Lessons 1-5, 1-6, 7-1, 7-3, 7-4, 8-6 in Holt McDougal Grade 8
relationships with real numbers and justify their reasoning using words and not necessarily with
the algebraic language of Table 3. In other words, students should be able to state that 3(-5) = (- Modeling Equations with Variables on Both Sides Lab pg. 308 in Holt
5)3 because multiplication is commutative and it can be performed in any order (it is
McDougal Grade 8
commutative), or that 9(8) + 9(2) = 9(8 + 2) because the distributive property allows us to
distribute multiplication over addition, or determine products and add them. Grade 8 is the
beginning of using the generalized properties of operations, but this is not something on which
Common Misconceptions
students should be assessed.
Students think that only the letters x and y can be used for variables.
Pairing contextual situations with equation solving allows students to connect mathematical
analysis with real-life events. Students should experience analyzing and representing contextual
situations with equations, identify whether there is one, none, or many solutions, and then solve
to prove conjectures about the solutions. Through multiple opportunities to analyze and solve
equations, students should be able to estimate the number of solutions and possible values(s) of
Students think that you always need a variable = a constant as a solution.
The variable is always on the left side of the equation.
Equations are not always in the slope intercept form, y=mx+b
Students confuse one-variable and two-variable equations.
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solutions prior to solving. Rich problems, such as computing the number of tiles needed to put a
border around a rectangular space or solving proportional problems as in doubling recipes, help
ground the abstract symbolism to life.
Experiences should move through the stages of concrete, conceptual and algebraic/abstract.
Utilize experiences with the pan balance model as a visual tool for maintaining equality
(balance) first with simple numbers, then with pictures symbolizing relationships, and finally
with rational numbers allows understanding to develop as the complexity of the problems
increases. Equation-solving in Grade 8 should involve multistep problems that require the use of
the distributive property, collecting like terms, and variables on both sides of the equation.
This cluster builds on the informal understanding of slope from graphing unit rates in Grade 6
and graphing proportional relationships in Grade 7 with a stronger, more formal understanding
of slope. It extends solving equations to understanding solving systems of equations, or a set of
two or more linear equations that contain one or both of the same two variables. Once again the
focus is on a solution to the system. Most student experiences should be with numerical and
graphical representations of solutions. Beginning work should involve systems of equations
with solutions that are ordered pairs of integers, making it easier to locate the point of
intersection, simplify the computation and hone in on finding a solution. More complex systems
can be investigated and solve by using graphing technology.
Contextual situations relevant to eighth graders will add meaning to the solution to a system of
equations. Students should explore many problems for which they must write and graph pairs of
equations leading to the generalization that finding one point of intersection is the single
solution to the system of equations. Provide opportunities for students to connect the solutions
to an equation of a line, or solution to a system of equations, by graphing, using a table and
writing an equation. Students should receive opportunities to compare equations and systems of
equations, investigate using graphing calculators or graphing utilities, explain differences
verbally and in writing, and use models such as equation balances.
Problems such as, ―Determine the number of movies downloaded in a month that would make
the costs for two sites the same, when Site A charges $6 per month and $1.25 for each movie
and Site B charges $2 for each movie and no monthly fee.‖
Students write the equations letting y = the total charge and x = the number of movies.
Site A: y = 1.25x + 6 Site B: y = 2x
Students graph the solutions for each of the equations by finding ordered pairs that are solutions
and representing them in a t-chart. Discussion should encompass the realization that the
intersection is an ordered pair that satisfies both equations. And finally students should relate
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the solution to the context of the problem, commenting on the practicality of their solution.
Problems should be structured so that students also experience equations that represent parallel
lines and equations that are equivalent. This will help them to begin to understand the
relationships between different pairs of equations: When the slope of the two lines is the same,
the equations are either different equations representing the same line (thus resulting in many
solutions), or the equations are different equations representing two not intersecting, parallel,
lines that do not have common solutions.
System-solving in Grade 8 should include estimating solutions graphically, solving using
substitution, and solving using elimination. Students again should gain experience by
developing conceptual skills using models that develop into abstract skills of formal solving of
equations. Provide opportunities for students to change forms of equations (from a given form
to slope-intercept form) in order to compare equations.
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Domain
Cluster
Content
Standards
Functions
Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and
the corresponding output.
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has
the greater rate of change.
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the
function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are
not on a straight line.
Instructional Strategies
Instructional Resources/Tools
In grade 6, students plotted points in all four quadrants of the coordinate plane. They also
Graphing calculators
represented and analyzed quantitative relationships between dependent and independent
variables. In Grade 7, students decided whether two quantities are in a proportional
Graphing software (including dynamic geometry software)
relationship. In Grade 8, students begin to call relationships functions when each input is
assigned to exactly one output. Also, in Grade 8, students learn that proportional relationships Lessons 2-4, 8-3, 8-5, 9-3, 9-4 in Holt McDougal Grade 8
are part of a broader group of linear functions, and they are able to identify whether a
relationship is linear. Nonlinear functions are included for comparison. Later, in high school,
students use function notation and are able to identify types of nonlinear functions.
Common Misconceptions
Some students will mistakenly think of a straight line as horizontal or vertical
To determine whether a relationship is a function, students should be expected to reason from only.
a context, a graph, or a table, after first being clear which quantity is considered the input and
which is the output. When a relationship is not a function, students should produce a
Some students will mix up x- and y-axes on the coordinate plane, or mix up the
counterexample: an ―input value‖ with at least two ―output values.‖ If the relationship is a
ordered pairs. When emphasizing that the first value is plotted on the horizontal
function, the students should explain how they verified that for each input there was exactly
axes (usually x, with positive to the right) and the second is the vertical axis
one output. The ―vertical line test‖ should be avoided because (1) it is too easy to apply
(usually called y, with positive up), point out that this is merely a convention: It
without thinking, (2) students do not need an efficient strategy at this point, and (3) it creates
could have been otherwise, but it is very useful for people to agree on a standard
misconceptions for later mathematics, when it is useful to think of functions more broadly,
customary practice.
such as whether x might be a function of y.
―Function machine‖ pictures are useful for helping students imagine input and output values,
with a rule inside the machine by which the output value is determined from the input.
Notice that the standards explicitly call for exploring functions numerically, graphically,
verbally, and algebraically (symbolically, with letters). This is sometimes called the ―rule of
four.‖ For fluency and flexibility in thinking, students need experiences translating among
these. In Grade 8, the focus, of course, is on linear functions, and students begin to recognize
a linear function from its form y = mx + b. Students also need experiences with nonlinear
functions, including functions given by graphs, tables, or verbal descriptions but for which
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there is no formula for the rule, such as a girl‘s height as a function of her age.
In the elementary grades, students explore number and shape patterns (sequences), and they
use rules for finding the next term in the sequence. At this point, students describe sequences
both by rules relating one term to the next and also by rules for finding the nth term directly.
(In high school, students will call these recursive and explicit formulas.) Students express
rules in both words and in symbols. Instruction should focus on additive and multiplicative
sequences as well as sequences of square and cubic numbers, considered as areas and
volumes of cubes, respectively.
When plotting points and drawing graphs, students should develop the habit of determining,
based upon the context, whether it is reasonable to ―connect the dots‖ on the graph. In some
contexts, the inputs are discrete, and connecting the dots can be misleading. For example, if a
function is used to model the height of a stack of n paper cups, it does not make sense to have
2.3 cups, and thus there will be no ordered pairs between n = 2 and n = 3.
Provide multiple opportunities to examine the graphs of linear functions and use graphing
calculators or computer software to analyze or compare at least two functions at the same
time. Illustrate with a slope triangle where the run is "1" that slope is the "unit rate of
change." Compare this in order to compare two different situations and identify which is
increasing/decreasing as a faster rate.
Students can compute the area and perimeter of different-size squares and identify that one
relationship is linear while the other is not by either looking at a table of value or a graph in
which the side length is the independent variable (input) and the area or perimeter is the
dependent variable (output).
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Domain
Cluster
Content
Standards
Functions
Use functions to model relationships between quantities.
4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description
of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear
function in terms of the situation it models, and in terms of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Instructional Strategies
Instructional Resources/Tools
In Grade 8, students focus on linear equations and functions. Nonlinear functions are used for Graphing calculators
comparison.
Graphing software for computers, including dynamic geometry software
Students will need many opportunities and examples to figure out the meaning of y = mx + b.
What does m mean? What does b mean? They should be able to ―see‖ m and b in graphs,
Data-collecting technology, such as motion sensors, thermometers, CBL‘s, etc.
tables, and formulas or equations, and they need to be able to interpret those values in
contexts. For example, if a function is used to model the height of a stack of n paper cups,
Graphing applets online
then the rate of change, m, which is the slope of the graph, is the height of the ―lip‖ of the
cup: the amount each cup sticks above the lower cup in the stack. The ―initial value‖ in this
Lessons 2-5, 5-8, 8-1, 8-3, 8-4, 9-3 in Holt McDougal Grade 8
case is not valid in the context because 0 cups would not have a height, and yet a height of 0
would not fit the equation. Nonetheless, the value of b can be interpreted in the context as the Finding Volume of Prisms and Cylinders Lab pg. 266
height of the ―base‖ of the cup: the height of the whole cup minus its lip.
Use graphing calculators and web resources to explore linear and non-linear functions.
Provide context as much as possible to build understanding of slope and y-intercept in a
Common Misconceptions
graph, especially for those patterns that do not start with an initial value of 0.
Students often confuse a recursive rule with an explicit formula for a function.
For example, after identifying that a linear function shows an increase of 2 in the
Give students opportunities to gather their own data or graphs in contexts they understand.
values of the output for every change of 1 in the input, some students will
Students need to measure, collect data, graph data, and look for patterns, then generalize and
represent the equation as y = x + 2 instead of realizing that this means y = 2x + b.
symbolically represent the patterns. They also need opportunities to draw graphs
When tables are constructed with increasing consecutive integers for input
(qualitatively, based upon experience) representing real-life situations with which they are
values, then the distinction between the recursive and explicit formulas is about
familiar. Probe student thinking by asking them to determine which input values make sense
whether you are reasoning vertically or horizontally in the table. Both types of
in the problem situations.
reasoning—and both types of formulas—are important for developing
proficiency with functions.
Provide students with a function in symbolic form and ask them to create a problem situation
in words to match the function. Given a graph, have students create a scenario that would fit
When input values are not increasing consecutive integers (e.g., when the input
the graph. Ask students to sort a set of "cards" to match a graphs, tables, equations, and
values are decreasing, when some integers are skipped, or when some input
problem situations. Have students explain their reasoning to each other.
values are not integers), some students have more difficulty identifying the
pattern and calculating the slope. It is important that all students have experience
From a variety of representations of functions, students should be able to classify and
with such tables, so as to be sure that they do not overgeneralize from the easier
describe the function as linear or non-linear, increasing or decreasing. Provide opportunities
examples.
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for students to share their ideas with other students and create their own examples for their
classmates.
Use the slope of the graph and similar triangle arguments to call attention to not just the
change in x or y, but also to the rate of change, which is a ratio of the two.
Emphasize key vocabulary. Students should be able to explain what key words mean: e.g.,
model, interpret, initial value, functional relationship, qualitative, linear, non-linear. Use a
―word wall‖ to help reinforce vocabulary.
Some students may not pay attention to the scale on a graph, assuming that the
scale units are always ―one.‖ When making axes for a graph, some students may
not using equal intervals to create the scale.
Some students may infer a cause and effect between independent and dependent
variables, but this is often not the case.
Some students graph incorrectly because they don‘t understand that x usually
represents the independent variable and y represents the dependent variable.
Emphasize that this is a convention that makes it easier to communicate.
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Domain
Cluster
Content
Standards
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and
translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections,
translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three
angles appears to form a line, and give an argument in terms of transversals why this is so.
Instructional Strategies
Instructional Resources/Tools
A major focus in Grade 8 is to use knowledge of angles and distance to analyze two- and
Pattern blocks or shape sets
three-dimensional figures and space in order to solve problems. This cluster interweaves the
relationships of symmetry, transformations, and angle relationships to form understandings of Geometry software like Geometer's Sketchpad, Cabri Jr or GeoGebra
similarity and congruence. Inductive and deductive reasoning are utilized as students forge
into the world of proofs. Informal arguments are justifications based on known facts and
Graphing calculators
logical reasoning. Students should be able to appropriately label figures, angles, lines, line
segments, congruent parts, and images (primes or double primes). Students are expected to
Patty paper
use logical thinking, expressed in words using correct terminology. They are NOT expected
to use theorems, axioms, postulates or a formal format of proof as in two-column proofs.
From the National Library of Virtual Manipulatives:
Congruent Triangles – Build similar triangles by combining sides and angles.
Transformational geometry is about the effects of rigid motions, rotations, reflections and
translations on figures. Initial work should be presented in such a way that students
Geoboard - Coordinate – Rectangular geoboard with x and y coordinates.
understand the concept of each type of transformation and the effects that each transformation
has on an object before working within the coordinate system. For example, when reflecting
Transformations - Composition – Explore the effect of applying a composition of
over a line, each vertex is the same distance from the line as its corresponding vertex. This is
translation, rotation, and reflection transformations to objects.
easier to visualize when not using regular figures. Time should be allowed for students to cut
out and trace the figures for each step in a series of transformations. Discussion should
Transformations - Dilation – Dynamically interact with and see the result of a
include the description of the relationship between the original figure and its image(s) in
dilation transformation.
regards to their corresponding parts (length of sides and measure of angles) and the
description of the movement, including the attributes of transformations (line of symmetry,
Transformations - Reflection – Dynamically interact with and see the result of a
distance to be moved, center of rotation, angle of rotation and the amount of dilation).The
reflection transformation.
case of distance – preserving transformation leads to the idea of congruence.
Transformations - Rotation – Dynamically interact with and see the result of a
It is these distance-preserving transformations that lead to the idea of congruence.
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Work in the coordinate plane should involve the movement of various polygons by addition,
subtraction and multiplied changes of the coordinates. For example, add 3 to x, subtract 4
from y, combinations of changes to x and y, multiply coordinates by 2 then by 12. Students
should observe and discuss such questions as ‗What happens to the polygon?‘ and ‗What does
making the change to all vertices do?‘. Understandings should include generalizations about
the changes that maintain size or maintain shape, as well as the changes that create distortions
of the polygon (dilations). Example dilations should be analyzed by students to discover the
movement from the origin and the subsequent change of edge lengths of the figures. Students
should be asked to describe the transformations required to go from an original figure to a
transformed figure (image). Provide opportunities for students to discuss the procedure used,
whether different procedures can obtain the same results, and if there is a more efficient
procedure to obtain the same results. Students need to learn to describe transformations with
both words and numbers.
Through understanding symmetry and congruence, conclusions can be made about the
relationships of line segments and angles with figures. Students should relate rigid motions to
the concept of symmetry and to use them to prove congruence or similarity of two figures.
Problem situations should require students to use this knowledge to solve for missing
measures or to prove relationships. It is an expectation to be able to describe rigid motions
with coordinates.
Provide opportunities for students to physically manipulate figures to discover properties of
similar and congruent figures, for example, the corresponding angles of similar figures are
equal. Additionally use drawings of parallel lines cut by a transversal to investigate the
relationship among the angles. For example, what information can be obtained by cutting
between the two intersections and sliding one onto the other?
rotation transformation.
Transformations - Translation – Dynamically interact with and see the result of a
translation transformation.
Lessons 4-3, 4-4, 5-2, 5-3, 5-6, 5-7, 5-8
Explore Similarity Lab pg. 168
Exterior Angles of Polygons Lab pg. 212
Explore Dilation Lab pg. 174
Explore Congruence Lab pg. 220
Combine Transformations pg. 237
Common Misconceptions
Students often confuse situations that require adding with multiplicative
situations in regard to scale factor. Providing experiences with geometric figures
and coordinate grids may help students visualize the difference.
In Grade 7, students develop an understanding of the special relationships of angles and their
measures (complementary, supplementary, adjacent, vertical). Now, the focus is on learning
the about the sum of the angles of a triangle and using it to, find the measures of angles
formed by transversals (especially with parallel lines), or to find the measures of exterior
angles of triangles and to informally prove congruence.
By using three copies of the same triangle labeled and placed so that the three different angles
form a straight line, students can:
• explore the relationships of the angles,
• learn the types of angles (interior, exterior, alternate interior, alternate exterior,
corresponding, same side interior, same side exterior), and
• explore the parallel lines, triangles and parallelograms formed.
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Further examples can be explored to verify these relationships and demonstrate their
relevance in real life.
Investigations should also lead to the Angle-Angle criterion for similar triangles. For
instance, pairs of students create two different triangles with one given angle measurement,
then repeat with two given angle measurements and finally with three given angle
measurements. Students observe and describe the relationship of the resulting triangles. As a
class, conjectures lead to the generalization of the Angle-Angle criterion.
Students should solve mathematical and real-life problems involving understandings from
this cluster. Investigation, discussion, justification of their thinking, and application of their
learning will assist in the more formal learning of geometry in high school.
Domain
Cluster
Content
Standards
Geometry
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Instructional Strategies
Instructional Resources/Tools
Previous understanding of triangles, such as the sum of two side measures is greater than the
From the National Library of Virtual Manipulatives :
third side measure, angles sum, and area of squares, is furthered by the introduction of unique Pythagorean Theorem – Solve two puzzles that illustrate the proof of the
qualities of right triangles. Students should be given the opportunity to explore right triangles
Pythagorean Theorem.
to determine the relationships between the measures of the legs and the measure of the
hypotenuse. Experiences should involve using grid paper to draw right triangles from given
Right Triangle Solver – Practice using the Pythagorean theorem and the
measures and representing and computing the areas of the squares on each side. Data should
definitions of the trigonometric functions to solve for unknown sides and angles
be recorded in a chart such as the one below, allowing for students to conjecture about the
of a right triangle.
relationship among the areas within each triangle.
Triangle
Measure of
Leg 1
Measure of
Leg 2
Area of
Square on
Leg 1
Area of
Square on
Leg 2
Area of
Square on
Hypotenuse
Common Misconceptions
Students may confuse which side to use as the hypotenuse.
1
Students should then test out their conjectures, then explain and discuss their findings.
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Finally, the Pythagorean Theorem should be introduced and explained as the pattern they
have explored. Time should be spent analyzing several proofs of the Pythagorean Theorem to
develop a beginning sense of the process of deductive reasoning, the significance of a
theorem, and the purpose of a proof. Students should be able to justify a simple proof of the
Pythagorean Theorem or its converse.
Previously, students have discovered that not every combination of side lengths will create a
triangle. Now they need situations that explore using the Pythagorean Theorem to test
whether or not side lengths represent right triangles. (Recording could include Side length a,
Side length b, Sum of a2 + b2, c2, a2 + b2 = c2, Right triangle? Through these opportunities,
students should realize that there are Pythagorean (triangular) triples such as (3, 4, 5), (5, 12,
13), (7, 24, 25), (9, 40, 41) that always create right triangles, and that their multiples also
form right triangles. Students should see how similar triangles can be used to find additional
triples. Students should be able to explain why a triangle is or is not a right triangle using the
Pythagorean Theorem.
The Pythagorean Thereom should be applied to finding the lengths of segments on a
coordinate grid, especially those segments that do not follow the vertical or horizontal lines,
as a means of discussing the determination of distances between points. Contextual situations,
created by both the students and the teacher, that apply the Pythagorean theorem and its
converse should be provided. For example, apply the concept of similarity to determine the
height of a tree using the ratio between the student's height and the length of the student's
shadow. From that, determine the distance from the tip of the tree to the end of its shadow
and verify by comparing to the computed distance from the top of the student's head to the
end of the student's shadow, using the ratio calculated previously. Challenge students to
identify additional ways that the Pythagorean Theorem is or can be used in real world
situations or mathematical problems, such as finding the height of something that is difficult
to physically measure, or the diagonal of a prism.
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Domain
Geometry
Cluster
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve-real world and mathematical problems.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Begin by recalling the formula, and its meaning, for the volume of a right rectangular prism:
Ohio Resource Center :
V = l ×w ×h. Then ask students to consider how this might be used to make a conjecture
#8421 ―The Cylinder Problem - Students build a family of cylinders, all from the
about the volume formula for a cylinder:
same-sized paper, and discover the relationship between the dimensions of the
paper and the resulting cylinders. They order the cylinders by their volumes and
draw a conclusion about the relationship between a cylinder's dimensions and its
volume.
h
h
l
w
Most students can be readily led to the understanding that the volume of a right rectangular
prism can be thought of as the area of a ―base‖ times the height, and so because the area of
the base of a cylinder is π r2 the volume of a cylinder is Vc = π r2h.
To motivate the formula for the volume of a cone, use cylinders and cones with the same base
and height. Fill the cone with rice or water and pour into the cylinder. Students will
discover/experience that 3 cones full are needed to fill the cylinder. This non-mathematical
derivation of the formula for the volume of a cone, V = 1/3 π r2h, will help most students
remember the formula.
In a drawing of a cone inside a cylinder, students might see that that the triangular crosssection of a cone is 1/2 the rectangular cross-section of the cylinder. Ask them to reason why
the volume (three dimensions) turns out to be less than 1/2 the volume of the cylinder. It turns
out to be 1/3.
National Library of Virtual Manipulatives :
―How High‖ is an applet that can be used to take an inquiry approach to the
formula for volume of a cylinder or cone.
NCTM:
Finding Surface Area and Volume
Blue Cube, 27 Little Cubes (Stella Stunner)
Volume of a Spheres and Cones (Rich Problem)
Common Misconceptions
A common misconception among middle grade students is that ―volume‖ is a
―number‖ that results from ―substituting‖ other numbers into a formula. For these
students there is no recognition that ―volume‖ is a measure – related to the
amount of space occupied. If a teacher discovers that students do not have an
understanding of volume as a measure of space, it is important to provide
opportunities for hands on experiences where students ―fill‖ three dimensional
objects. Begin with right rectangular prisms and fill them with cubes will help
students understand why the units for volume are cubed. See Cubes
http://illuminations.nctm.org/ActivityDetail.aspx?ID=6
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For the volume of a sphere, it may help to have students visualize a hemisphere ―inside‖ a
cylinder with the same height and ―base.‖ The radius of the circular base of the cylinder is
also the radius of the sphere and the hemisphere. The area of the ―base‖ of the cylinder and
the area of the section created by the division of the sphere into a hemisphere is π r2. The
height of the cylinder is also r so the volume of the cylinder is π r3. Students can see that the
volume of the hemisphere is less than the volume of the cylinder and more than half the
volume of the cylinder. Illustrating this with concrete materials and rice or water will help
students see the relative difference in the volumes. At this point, students can reasonably
accept that the volume of the hemisphere of radius r is 2/3 π r3 and therefore volume of a
sphere with radius r is twice that or 4/3 π r3. There are several websites with explanations for
students who wish to pursue the reasons in more detail. (Note that in the pictures above, the
hemisphere and the cone together fill the cylinder.)
Students should experience many types of real-world applications using these formulas. They
should be expected to explain and justify their solutions.
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Domain
Cluster
Content
Standards
Statistics and Probability
Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as
clustering, outliers, positive or negative association, linear association, and nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear
model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm
in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for
rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a
curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Instructional Strategies
Instructional Resources/Tools
Building on the study of statistics using univariate data in Grades 6 and 7, students This lesson - Glued to the Tube or Hooked to the Books? - provides step-by-step
are now ready to study bivariate data. Students will extend their descriptions and
instructions for using the graphing calculator to construct a scatter plot of class data and a
understanding of variation to the graphical displays of bivariate data.
line of best fit.
Scatter plots are the most common form of displaying bivariate data in Grade 8.
Provide scatter plots and have students practice informally finding the line of best
fit. Students should create and interpret scatter plots, focusing on outliers, positive
or negative association, linearity or curvature. By changing the data slightly,
students can have a rich discussion about the effects of the change on the graph.
Have students use a graphing calculator to determine a linear regression and
discuss how this relates to the graph. Students should informally draw a line of
best fit for a scatter plot and informally measure the strength of fit. Discussion
should include ―What does it mean to be above the line, below the line?‖
The study of the line of best fit ties directly to the algebraic study of slope and
intercept. Students should interpret the slope and intercept of the line of best fit in
the context of the data. Then students can make predictions based on the line of
best fit.
From the National Council of Teachers of Mathematics, Illuminations: Impact of a Superstar
- This lesson uses technology tools to plot data, identify lines of best fit, and detect outliers.
Then, students compare the lines of best fit when one element is removed from a data set,
and interpret the results.
From the National Council of Teachers of Mathematics, Illuminations: Exploring Linear
Data - In this lesson, students construct scatter plots of bivariate data, interpret individual
data points, make conclusions about trends in data, especially linear relationships, and
estimate and write equation of lines of best fit.
The Ohio Resource Center –the tutorial video Lines of Fit shows how to determine a line of
best fit for a set of data.
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report,
American Statistical Association
Common Misconceptions
Students may believe: Bivariate data is only displayed in scatter plots. 8.SP.4 in this cluster
provides the opportunity to display bivariate, categorical data in a table.
In general, students think there is only one correct answer in mathematics. Students may
mistakenly think their lines of best fit for the same set of data will be exactly the same.
Because students are informally drawing lines of best fit, the lines will vary slightly. To
obtain the exact line of best fit, students would use technology to find the line of regression.
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High School Conceptual Category: Number and Quantity
Domain
Quantities
Cluster
Reason quantitatively and use units to solve problems
1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and
Content
interpret the scale and the origin in graphs and data displays.
Standards
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement reporting quantities.
Instructional Strategies
Instructional Resources/Tools
In real-world situations, answers are usually represented by numbers associated with units.
NCTM. Focus in High School Mathematics (Reasoning and Sense Making)
Units involve measurement and often require a conversion. Measurement involves both
precision and accuracy. Estimation and approximation often precede more exact computations. Mathematical Sciences Education Board. High School Mathematics at Work
Students need to develop sound mathematical reasoning skills and forms of argument to make
reasonable judgments about their solutions. They should be able to decide whether a problem
calls for an estimate, for an approximation, or for an exact answer. To accomplish this goal,
teachers should provide students with a broad range of contextual problems that offer
opportunities for performing operations with quantities involving units. These problems should
be connected to science, engineering, economics, finance, medicine, etc.
Some contextual problems may require an understanding of derived measurements and
capability in unit analysis. Keeping track of derived units during computations and making
reasonable estimates and rational conclusions about accuracy and the precision of the answers
help in the problem-solving process.
For example, while driving in the United Kingdom (UK), a U.S. tourist puts 60 liters of
gasoline in his car. The gasoline cost is £1.28 per liter The exchange rate is £ 0.62978 for each
$1.00. The price for a gallon of a gasoline in the United States is $3.05. The driver wants to
compare the costs for the same amount and the same type of gasoline when he/she pays in UK
pounds. Making reasonable estimates should be encouraged prior to solving this problem.
Since the current exchange rate has inflated the UK pound at almost twice the U.S. dollar, the
driver will pay more for less gasoline.
NCTM. Principles and Standards for School Mathematics
Joint Committee of the MAA and NCTM. A Sourcebook of Applications of
School Mathematics
Common Misconceptions
Students may not realize the importance of the units‘ conversions in conjunction
with the computation when solving problems involving measurements.
Since today‘s calculating devices often display 8 to 10 decimal places, students
frequently express answers to a much greater degree of precision than the
required.
By dividing $3.05 by 3.79L (the number of liters in one gallon), students can see that 80.47
cents per liter of gasoline in US is less expensive than £1.28 or $ 2.03 per liter of the same type
of gasoline in the UK when paid in U.S. dollars. The cost of 60 liters of gasoline in UK is ( )
In order to compute the cost of the same quantity of gasoline in the United States in UK
currency, it is necessary to convert between both monetary systems and units of volume. Based
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on UK pounds, the cost of 60 liters of gasoline in the U.S. is).
The computation shows that the gasoline is less expensive in the United States and how an
analysis can be helpful in keeping track of unit conversations. Students should be able to
correctly identify the degree of precision of the answers which should not be far greater than
the actual accuracy of the measurements.
Graphical representations serve as visual models for understanding phenomena that take place
in our daily surroundings. The use of different kinds of graphical representations along with
their units, labels and titles demonstrate the level of students‘ understanding and foster the
ability to reason, prove, self-check, examine relationships and establish the validity of
arguments. Students need to be able to identify misleading graphs by choosing correct units
and scales to create a correct representation of a situation or to make a correct conclusion from
it.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
Content
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Standards
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a
factor not depending on P.
Instructional Strategies
Instructional Resources/Tools
Extending beyond simplifying an expression, this cluster addresses interpretation of the
Hands-on materials, such as algebra tiles, can be used to establish a visual
components in an algebraic expression. A student should recognize that in the expression 2x + understanding of algebraic expressions and the meaning of terms, factors and
1, ―2‖ is the coefficient, ―2‖ and ―x‖ are factors, and ―1‖ is a constant, as well as ―2x‖ and ―1‖
coefficients.
being terms of the binomial expression. Development and proper use of mathematical
language is an important building block for future content.
From the National Library of Virtual Manipulatives - Algebra Tiles – Visualize
multiplying and factoring algebraic expressions using tiles.
Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression
0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one
Common Misconceptions
minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
Students may believe that the use of algebraic expressions is merely the abstract
number of cell phone minutes used in the month. Similar real-world examples, such as tax
manipulation of symbols. Use of real-world context examples to demonstrate
rates, can also be used to explore the meaning of expressions.
the meaning of the parts of algebraic expressions is needed to counter this
misconception.
Factoring by grouping is another example of how students might analyze the structure of an
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the ―x – 5‖ is
Students may also believe that an expression cannot be factored because it does
common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should
not fit into a form they recognize. They need help with reorganizing the terms
become comfortable with rewriting expressions in a variety of ways until a structure emerges.
until structures become evident.
Have students create their own expressions that meet specific criteria (e.g., number of terms
factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten
in different forms. Additionally, pair/group students to share their expressions and rewrite one
another‘s expressions.
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High School Conceptual Category: Algebra
Domain
Creating Equations
Cluster
Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple
Content
rational and exponential functions
Standards
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a
modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to
highlight resistance R.
Instructional Strategies
Instructional Resources/Tools
Provide examples of real-world problems that can be modeled by writing an equation or
Graphing calculators
inequality. Begin with simple equations and inequalities and build up to more complex
equations in two or more variables that may involve quadratic, exponential or rational functions. Computer software that generate graphs of functions
Discuss the importance of using appropriate labels and scales on the axes when representing
functions with graphs.
Examine real-world graphs in terms of constraints that are necessary to balance a mathematical
model with the real-world context. For example, a student writing an equation to model the
maximum area when the perimeter of a rectangle is 12 inches should recognize that y = x(6 – x)
only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side
of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6.
Students can discuss the difference between the parabola that models the problem and the
portion of the parabola that applies to the context.
Explore examples illustrating when it is useful to rewrite a formula by solving for one of the
A = 1 h(b + b )
1 2 )
2
variables in the formula. For example, the formula for the area of a trapezoid (
can be solved for h if the area and lengths of the bases are known but the height needs to be
calculated. This strategy of selecting a different representation has many applications in science
and business when using formulas.
Examples of real-world situations that lend themselves to writing equations
that model the contexts
Common Misconceptions
Students may believe that equations of linear, quadratic and other functions are
abstract and exist only ―in a math book,‖ without seeing the usefulness of these
functions as modeling real-world phenomena.
Additionally, they believe that the labels and scales on a graph are not
important and can be assumed by a reader, and that it is always necessary to
use the entire graph of a function when solving a problem that uses that
function as its model.
Provide examples of real-world problems that can be solved by writing an equation, and have
students explore the graphs of the equations on a graphing calculator to determine which parts
of the graph are relevant to the problem context.
Use a graphing calculator to demonstrate how dramatically the shape of a curve can change
when the scale of the graph is altered for one or both variables.
Give students formulas, such as area and volume (or from science or business), and have
students solve the equations for each of the different variables in the formula.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Understand solving equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the
Content
original equation has a solution. Construct a viable argument to justify a solution method.
Standards
Instructional Strategies
Instructional Resources/Tools
Graphing Calculators
Challenge students to justify each step of solving an equation. Transforming 2x - 5 = 7 to
2x = 12 is possible because 5 = 5, so adding the same quantity to both sides of an equation
makes the resulting equation true as well. Each step of solving an equation can be defended,
Common Misconceptions
much like providing evidence for steps of a geometric proof.
Students may believe that solving an equation such as 3x + 1 = 7 involves ―only
Provide examples for how the same equation might be solved in a variety of ways as long as
removing the 1,‖ failing to realize that the equation 1 = 1 is being subtracted to
equivalent quantities are added or subtracted to both sides of the equation, the order of steps
produce the next step.
taken will not matter.
Additionally, students may believe that all solutions to radical and rational
3n + 2 = n - 10
3n + 2 = n - 10
3n + 2 = n - 10
equations are viable, without recognizing that there are times when extraneous
2 = -2
+ 10 = +10
-n
= -n
solutions are generated and have to be eliminated.
3n
= n – 12
OR
3n + 12 = n
OR
2n + 2 = -10
-n
= -n
-3n
= -3n
-2=–2
2n
= -12
12 = -2n
2n = -12
n = -6
n= -6
n = -6
Connect the idea of adding two equations together as a means of justifying steps of solving a
simple equation to the process of solving a system of equations. A system consisting of two
linear functions such as 2x + 3y = 8 and x - 3y = 1 can be solved by adding the equations
together, and can be justified by exactly the same reason that solving the equation 2x - 4 = 5
can begin by adding the equation 4 = 4.
Begin with simple, one-step equations and require students to write out a justification for each
step used to solve the equation.
It is very important that students are able to reason how and why extraneous solutions arise.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve equations and inequalities in one variable
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
There are two major reasons for discussing the topic of inequalities along with equations:
Graphing utilities to explore the effects of changes in parameters of equations on
First, there are analogies between solving equations and inequalities that help students
their graphs.
understand them both. Second, the applications that lead to equations almost always lead in
the same way to inequalities.
Tables, graphs and equations of real-world applications.
In grades 6-8, students solve and graph linear equations and inequalities. Graphing
experience with inequalities is limited to graphing on a number line diagram. Despite this
work, some students will still need more practice to be proficient. It may be beneficial to
remind students of the most common solving techniques, such as converting fractions from
one form to another, removing parentheses in the sentences, or multiplying both sides of an
equation or inequality by the common denominator of the fractions. Students must be aware
of what it means to check an inequality‘s solution. The substitution of the end points of the
solution set in the original inequality should give equality regardless of the presence or the
absence of an equal sign in the original sentence. The substitution of any value from the rest
of the solution set should give a correct inequality.
Careful selection of examples and exercises is needed to provide students with meaningful
review and to introduce other important concepts, such as the use of properties and
applications of solving linear equations and inequalities. Stress the idea that the application of
properties is also appropriate when working with equations or inequalities that include more
than one variable, fractions and decimals. Regardless of the type of numbers or variables in
the equation or inequality, students have to examine the validity of each step in the solution
process.
Solving equations for the specified letter with coefficients represented by letters (e.g.,
when solving for ) is similar to solving an equation with one variable. Provide
students with an opportunity to abstract from particular numbers and apply the same kind of
manipulations to formulas as they did to equations. One of the purposes of doing abstraction
is to learn how to evaluate the formulas in easier ways and use the techniques across
mathematics and science.
http://ohiorc.org/for/math/
http://illuminations.nctm.org/
Common Misconceptions
Some students may believe that for equations containing fractions only on one
side, it requires ―clearing fractions‖ (the use of multiplication) only on that side
of the equation. To address this misconception, start by demonstrating the
solution methods for equations similar to x + x + x + 46 = x and stress that
the Multiplication Property of Equality is applied to both sides, which are
multiplied by 60.
Students may confuse the rule of changing a sign of an inequality when
multiplying or dividing by a negative number with changing the sign of an
inequality when one or two sides of the inequality become negative (for ex., 3x >
-15 or x < - 5).
Some students may believe that subscripts can be combined as b1 + b2 = b3 and
the sum of different variables d and D is 2D (d +D = 2D).
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Draw students‘ attention to equations containing variables with subscripts. The same
variables with different subscripts (e.g., x1 and x2 ) should be viewed as different variables
that cannot be combined as like terms. A variable with a variable subscript, such as an, must
be treated as a single variable – the nth term, where variables a and n have different meaning.
High School Conceptual Category: Number and Quantity
Domain
The Real Number System
Cluster
Extend the properties of exponents to rational exponents
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a
Content
notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
Standards
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Instructional Strategies
Instructional Resources/Tools
The goal is to show that a fractional exponent can be expressed as a radical or a root. For
Graphing calculator
example, an exponent of 1/3 is equivalent to a cube root; an exponent of ¼ is equivalent to a
fourth root.
Computer algebra systems
, for whole number exponents (e.g. (7 2) 3 = 76.
Review the power rule,
1/2 2
1
Compare examples, such as (7 ) = 7 = 7 and
The Ohio Resource Center
, to help students establish a
connection between radicals and rational exponents:
and, in general,
The National Council of Teachers of Mathematics, Illuminations
.
Provide opportunities for students to explore the equality of the values using calculators, such
as 71/2 and
.
Offer sufficient examples and exercises to prompt the definition of fractional exponents, and
give students practice in converting expressions between radical and exponential forms.
When n is a positive integer, generalize the meaning of
and then to
,
where n and m are integers and n is greater than or equal to 2. When m is a negative integer,
the result is the reciprocal of the root
Common Misconceptions
Students sometimes misunderstand the meaning of exponential operations, the
way powers and roots relate to one another, and the order in which they should
be performed. Attention to the base is very important. Consider examples:
and
. The position of a negative sign of a term with a rational
exponent can mean that the rational exponent should be either applied first to the
base, 81, and then the opposite of the result is taken
, or the rational
.
Stress the two rules of rational exponents: 1) the numerator of the exponent is the base‘s
power and 2) the denominator of the exponent is the order of the root. When evaluating
expressions involving rational exponents, it is often helpful to break an exponent into its parts
– a power and a root – and then decide if it is easier to perform the root operation or the
exponential operation first.
exponent should be applied to a negative term
. The answer of
will be not real if the denominator of the exponent is even. If the root is odd, the
answer will be a negative number.
Students should be able to make use of estimation when incorrectly using
multiplication instead of exponentiation.
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Model the use of precise mathematics vocabulary (e.g., base, exponent, radical, root, cube
root, square root etc.).
The rules for integer exponents are applicable to rational exponents as well; however, the
operations can be slightly more complicated because of the fractions. When multiplying
exponents, powers are added
. When dividing exponents, powers are
subtracted
multiplied
Students may believe that the fractional exponent in the expression
means
the same as a factor in multiplication expression,
and multiply the base
by the exponent.
. When raising an exponent to an exponent, powers are
.
High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve systems of equations
8. Analyze and solve pairs of simultaneous linear equations.
Content
c. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of
Standards
intersection satisfy both equations simultaneously.
d. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
e. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points,
determine whether the line through the first pair of points intersects the line through the second pair.
Instructional Strategies
Instructional Resources/Tools
Contextual situations relevant to eighth graders will add meaning to the solution to a system of
SMART Board‘s new tools for solving equations
equations. Students should explore many problems for which they must write and graph pairs of
equations leading to the generalization that finding one point of intersection is the single
Graphing calculators
solution to the system of equations. Provide opportunities for students to connect the solutions
to an equation of a line, or solution to a system of equations, by graphing, using a table and
Index cards with equations/graphs for matching and sorting
writing an equation. Students should receive opportunities to compare equations and systems of
equations, investigate using graphing calculators or graphing utilities, explain differences
Supply and Demand This activity focuses on having students create and solve
verbally and in writing, and use models such as equation balances.
a system of linear equations in a real-world setting. By solving the system,
students will find the equilibrium point for supply and demand. Students
Problems such as, ―Determine the number of movies downloaded in a month that would make
should be familiar with finding linear equations from two points or slope and
the costs for two sites the same, when Site A charges $6 per month and $1.25 for each movie
y-intercept. This lesson was adapted from the October 1991 edition of
and Site B charges $2 for each movie and no monthly fee.‖
Mathematics Teacher.
Students write the equations letting y = the total charge and x = the number of movies.
Site A: y = 1.25x + 6
Site B: y = 2x
Students graph the solutions for each of the equations by finding ordered pairs that are solutions
and representing them in a t-chart. Discussion should encompass the realization that the
Common Misconceptions
Students think that only the letters x and y can be used for variables.
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intersection is an ordered pair that satisfies both equations. And finally students should relate
the solution to the context of the problem, commenting on the practicality of their solution.
Problems should be structured so that students also experience equations that represent parallel
lines and equations that are equivalent. This will help them to begin to understand the
relationships between different pairs of equations: When the slope of the two lines is the same,
the equations are either different equations representing the same line (thus resulting in many
solutions), or the equations are different equations representing two not intersecting, parallel,
lines that do not have common solutions.
Students think that you always need a variable = a constant as a solution.
The variable is always on the left side of the equation.
Equations are not always in the slope intercept form, y=mx+b
Students confuse one-variable and two-variable equations.
System-solving in Grade 8 should include estimating solutions graphically, solving using
substitution, and solving using elimination. Students again should gain experience by
developing conceptual skills using models that develop into abstract skills of formal solving of
equations. Provide opportunities for students to change forms of equations (from a given form
to slope-intercept form) in order to compare equations.
High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve systems of equations
5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system
Content
with the same solutions.
Standards
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Instructional Strategies
Instructional Resources/Tools
The focus of this standard is to provide mathematics justification for the addition
Graph paper
(elimination) and substitution methods of solving systems of equations that transform a given
system of two equations into a simpler equivalent system that has the same solutions as the
Graphing calculators
original.
Computer graphing tools to operate with matrices and to find determinants of
The Addition and Multiplication Properties of Equality allow finding solutions to certain
higher order matrices.
systems of equations. In general, any linear combination, m(Ax + By) + n(Cx + Dy) = mE
+nF, of two linear equations
Dynamic geometry software
Ax + By = E and
Cx + Dy = F
From the National Council of Teachers of Mathematics, Illuminations: Supply
intersecting in a single point contains that point. The multipliers m and n can be chosen so
and Demand - This activity focuses on having students create and solve a system
that the resulting combination has only an x-term or only a y-term in it. That is, the
of linear equations in a real-world setting. By solving a system of two equations
combination will be a horizontal or vertical line containing the point of intersection.
in two unknowns, students will find the equilibrium point for supply and demand.
In the proof of a system of two equations in two variables, where one equation is replaced by
the sum of that equation and a multiple of the other equation, produces a system that has the
same solutions, let point (x1, y1) be a solution of both equations:
http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/mak
ing_connections.pdf - Students use graphing calculators to solve systems of
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Ax1 + By1 = E (true)
Cx1 + Dy1 = F (true)
Replace the equation Ax + By = E with Ax + By + k(Cx + Dy) on its left side and with E +
kF on its right side.
The new equation is Ax + By + k(Cx + Dy) = E + kF.
Show that the ordered pair of numbers (x1, y1) is a solution of this equation by replacing (x1,
y1) in the left side of this equation and verifying that the right side really equals E + kF:
Ax1 + By1 + k(Cx1 + Dy1) = E + kF (true)
Systems of equations are classified into two groups, consistent or inconsistent, depending on
whether or not solutions exist. The solution set of a system of equations is the intersection of
the solution sets for the individual equations. Stress the benefit of making the appropriate
selection of a method for solving systems (graphing vs. addition vs. substitution). This
depends on the type of equations and combination of coefficients for corresponding variables,
without giving a preference to either method.
linear equations in two ways. They first solve the systems by graphing the
equations and finding the point of intersection. Next, they will solve systems of
equations by writing related matrices and finding the solution by using inverse
matrices.
From the National Council of Teachers of Mathematics, Illuminations:
Movement with Functions - In this lesson, students use remote-controlled cars to
create a system of equations.
Common Misconceptions
Most mistakes that students make are careless rather than conceptual. Teachers
should encourage students to learn a certain format for solving systems of
equations and check the answers by substituting into all equations in the system.
The graphing method can be the first step in solving systems of equations. .A set of points
representing solutions of each equation is found by graphing these equations. Even though the
graphing method is limited in finding exact solutions and often yields approximate values, the
use of it helps to discover whether solutions exist and, if so, how many are there
Prior to solving systems of equations graphically, students should revisit ―families of
functions‖ to review techniques for graphing different classes of functions. Alert students to
the fact that if one equation in the system can be obtained by multiplying both sides of
another equation by a nonzero constant, the system is called consistent, the equations in the
system are called dependent and the system has an infinite number of solutions that produces
coinciding graphs. Provide students opportunities to practice linear vs. non-linear systems;
consistent vs. inconsistent systems; pure computational vs. real-world contextual problems
(e.g., chemistry and physics applications encountered in science classes). A rich variety of
examples can lead to discussions of the relationships between coefficients and consistency
that can be extended to graphing and later to determinants and matrices.
The next step is to turn to algebraic methods, elimination or substitution, to allow students to
find exact solutions. For any method, stress the importance of having a well organized format
for writing solutions.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Represent and solve equations and inequalities graphically
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be
Content
a line).
Standards
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x);
find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution
set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Instructional Strategies
Instructional Resources/Tools
Beginning with simple, real-world examples, help students to recognize a graph as a set of
Examples of real-world situations that involve linear functions and two-variable
solutions to an equation. For example, if the equation y = 6x + 5 represents the amount of
linear inequalities
money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do the work), then
every point on the line represents an amount of money paid, given the amount of time worked. Graphing calculators or computer software that generate graphs and tables for
solving equations
Explore visual ways to solve an equation such as 2x + 3 = x – 7 by graphing the functions y =
2x + 3 and y = x – 7. Students should recognize that the intersection point of the lines is at (10, -17). They should be able to verbalize that the intersection point means that when x = -10 is Common Misconceptions
substituted into both sides of the equation, each side simplifies to a value of -17. Therefore, -10 Students may believe that the graph of a function is simply a line or curve
is the solution to the equation. This same approach can be used whether the functions in the
―connecting the dots,‖ without recognizing that the graph represents all
original equation are linear, nonlinear or both.
solutions to the equation.
Using technology, have students graph a function and use the trace function to move the cursor
along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the
calculator, emphasizing that every point on the curve represents a solution to the equation.
Begin with simple linear equations and how to solve them using the graphs and tables on a
graphing calculator. Then, advance students to nonlinear situations so they can see that even
complex equations that might involve quadratics, absolute value, or rational functions can be
solved fairly easily using this same strategy. While a standard graphing calculator does not
simply solve an equation for the user, it can be used as a tool to approximate solutions.
Use the table function on a graphing calculator to solve equations. For example, to solve the
equation x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine
that they intersect when x = 4 and when x = -3 by examining the table to find where the yvalues are the same.
Students may also believe that graphing linear and other functions is an isolated
skill, not realizing that multiple graphs can be drawn to solve equations
involving those functions.
Additionally, students may believe that two-variable inequalities have no
application in the real world. Teachers can consider business related problems
(e.g., linear programming applications) to engage students in discussions of how
the inequalities are derived and how the feasible set includes all the points that
satisfy the conditions stated in the inequalities.
Investigate real-world examples of two-dimensional inequalities. For example, students might
explore what the graph would look like for money earned when a person earns at least $6 per
hour. (The graph for a person earning exactly $6/hour would be a linear function, while the
graph for a person earning at least $6/hour would be a half-plane including the line and all
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points above it.) Applications such as linear programming can help students to recognize how
businesses use constraints to maximize profit while minimizing the use of resources. These
situations often involve the use of systems of two variable inequalities.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Understand the concept of a function and use function notation
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and
Content
the corresponding output.
Standards
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has
the greater rate of change.
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the
function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are
not on a straight line.
Instructional Strategies
Instructional Resources/Tools
In grade 6, students plotted points in all four quadrants of the coordinate plane. They also
Graphing calculators
represented and analyzed quantitative relationships between dependent and independent
variables. In Grade 7, students decided whether two quantities are in a proportional
Graphing software (including dynamic geometry software)
relationship. In Grade 8, students begin to call relationships functions when each input is
assigned to exactly one output. Also, in Grade 8, students learn that proportional relationships
are part of a broader group of linear functions, and they are able to identify whether a
Common Misconceptions
relationship is linear. Nonlinear functions are included for comparison. Later, in high school,
Some students will mistakenly think of a straight line as horizontal or vertical
students use function notation and are able to identify types of nonlinear functions.
only.
To determine whether a relationship is a function, students should be expected to reason from
a context, a graph, or a table, after first being clear which quantity is considered the input and
which is the output. When a relationship is not a function, students should produce a
counterexample: an ―input value‖ with at least two ―output values.‖ If the relationship is a
function, the students should explain how they verified that for each input there was exactly
one output. The ―vertical line test‖ should be avoided because (1) it is too easy to apply
without thinking, (2) students do not need an efficient strategy at this point, and (3) it creates
misconceptions for later mathematics, when it is useful to think of functions more broadly,
such as whether x might be a function of y.
Some students will mix up x- and y-axes on the coordinate plane, or mix up the
ordered pairs. When emphasizing that the first value is plotted on the horizontal
axes (usually x, with positive to the right) and the second is the vertical axis
(usually called y, with positive up), point out that this is merely a convention: It
could have been otherwise, but it is very useful for people to agree on a standard
customary practice.
―Function machine‖ pictures are useful for helping students imagine input and output values,
with a rule inside the machine by which the output value is determined from the input.
Notice that the standards explicitly call for exploring functions numerically, graphically,
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verbally, and algebraically (symbolically, with letters). This is sometimes called the ―rule of
four.‖ For fluency and flexibility in thinking, students need experiences translating among
these. In Grade 8, the focus, of course, is on linear functions, and students begin to recognize
a linear function from its form y = mx + b. Students also need experiences with nonlinear
functions, including functions given by graphs, tables, or verbal descriptions but for which
there is no formula for the rule, such as a girl‘s height as a function of her age.
In the elementary grades, students explore number and shape patterns (sequences), and they
use rules for finding the next term in the sequence. At this point, students describe sequences
both by rules relating one term to the next and also by rules for finding the nth term directly.
(In high school, students will call these recursive and explicit formulas.) Students express
rules in both words and in symbols. Instruction should focus on additive and multiplicative
sequences as well as sequences of square and cubic numbers, considered as areas and
volumes of cubes, respectively.
When plotting points and drawing graphs, students should develop the habit of determining,
based upon the context, whether it is reasonable to ―connect the dots‖ on the graph. In some
contexts, the inputs are discrete, and connecting the dots can be misleading. For example, if a
function is used to model the height of a stack of n paper cups, it does not make sense to have
2.3 cups, and thus there will be no ordered pairs between n = 2 and n = 3.
Provide multiple opportunities to examine the graphs of linear functions and use graphing
calculators or computer software to analyze or compare at least two functions at the same
time. Illustrate with a slope triangle where the run is "1" that slope is the "unit rate of
change." Compare this in order to compare two different situations and identify which is
increasing/decreasing as a faster rate.
Students can compute the area and perimeter of different-size squares and identify that one
relationship is linear while the other is not by either looking at a table of value or a graph in
which the side length is the independent variable (input) and the area or perimeter is the
dependent variable (output).
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Understand the concept of a function and use function notation
1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the
Content
range. If f is a function and x is an element of its domain, the f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
Standards
equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fiboonacci sequence is
defined recursively by F(0) = F(1) = 1, f(n + 1) = f(n) + f (n – 1) for n ≥ 1.
Instructional Strategies
Instructional Resources/Tools
Provide applied contexts in which to explore functions. For example, examine the amount of
Diagrams or drawings of function machines, as well as tables and
money earned when given the number of hours worked on a job, and contrast this with a
graphs.
situation in which a single fee is paid by the ―carload‖ of people, regardless of whether 1, 2,
or more people are in the car.
Function Machine virtual manipulatives, such as available at
nlvm.usu.edu.
Use diagrams to help students visualize the idea of a function machine. Students can examine
several pairs of input and output values and try to determine a simple rule for the function.
Common Misconceptions
Rewrite sequences of numbers in tabular form, where the input represents the term number
Students may believe that all relationships having an input and an output are
(the position or index) in the sequence, and the output represents the number in the sequence.
functions, and therefore, misuse the function terminology.
Help students to understand that the word ―domain‖ implies the set of all possible input
values and that the integers are a set of numbers made up of {…-2, -1, 0, 1, 2, …}.
Distinguish between relationships that are not functions and those that are functions (e.g.,
present a table in which one of the input values results in multiple outputs to contrast with a
functional relationship). Examine graphs of functions and non-functions, recognizing that if a
vertical line passes through at least two points in the graph, then y (or the quantity on the
vertical axis) is not a function of x (or the quantity on the horizontal axis).
Students may also believe that the notation f(x) means to multiply some value f
times another value x. The notation alone can be confusing and needs careful
development. For example, f(2) means the output value of the function f when
the input value is 2.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description
Content
of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear
Standards
function in terms of the situation it models, and in terms of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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Instructional Strategies
In Grade 8, students focus on linear equations and functions. Nonlinear functions are used for
comparison.
Instructional Resources/Tools
Graphing calculators
Graphing software for computers, including dynamic geometry software
Students will need many opportunities and examples to figure out the meaning of y = mx + b.
What does m mean? What does b mean? They should be able to ―see‖ m and b in graphs,
tables, and formulas or equations, and they need to be able to interpret those values in
contexts. For example, if a function is used to model the height of a stack of n paper cups,
then the rate of change, m, which is the slope of the graph, is the height of the ―lip‖ of the
cup: the amount each cup sticks above the lower cup in the stack. The ―initial value‖ in this
case is not valid in the context because 0 cups would not have a height, and yet a height of 0
would not fit the equation. Nonetheless, the value of b can be interpreted in the context as the
height of the ―base‖ of the cup: the height of the whole cup minus its lip.
Use graphing calculators and web resources to explore linear and non-linear functions.
Provide context as much as possible to build understanding of slope and y-intercept in a
graph, especially for those patterns that do not start with an initial value of 0.
Give students opportunities to gather their own data or graphs in contexts they understand.
Students need to measure, collect data, graph data, and look for patterns, then generalize and
symbolically represent the patterns. They also need opportunities to draw graphs
(qualitatively, based upon experience) representing real-life situations with which they are
familiar. Probe student thinking by asking them to determine which input values make sense
in the problem situations.
Provide students with a function in symbolic form and ask them to create a problem situation
in words to match the function. Given a graph, have students create a scenario that would fit
the graph. Ask students to sort a set of "cards" to match a graphs, tables, equations, and
problem situations. Have students explain their reasoning to each other.
From a variety of representations of functions, students should be able to classify and
describe the function as linear or non-linear, increasing or decreasing. Provide opportunities
for students to share their ideas with other students and create their own examples for their
classmates.
Use the slope of the graph and similar triangle arguments to call attention to not just the
change in x or y, but also to the rate of change, which is a ratio of the two.
Emphasize key vocabulary. Students should be able to explain what key words mean: e.g.,
model, interpret, initial value, functional relationship, qualitative, linear, non-linear. Use a
―word wall‖ to help reinforce vocabulary.
Data-collecting technology, such as motion sensors, thermometers, CBL‘s, etc.
Graphing applets online.
Common Misconceptions
Students often confuse a recursive rule with an explicit formula for a function.
For example, after identifying that a linear function shows an increase of 2 in the
values of the output for every change of 1 in the input, some students will
represent the equation as y = x + 2 instead of realizing that this means y = 2x + b.
When tables are constructed with increasing consecutive integers for input
values, then the distinction between the recursive and explicit formulas is about
whether you are reasoning vertically or horizontally in the table. Both types of
reasoning—and both types of formulas—are important for developing
proficiency with functions.
When input values are not increasing consecutive integers (e.g., when the input
values are decreasing, when some integers are skipped, or when some input
values are not integers), some students have more difficulty identifying the
pattern and calculating the slope. It is important that all students have experience
with such tables, so as to be sure that they do not overgeneralize from the easier
examples.
Some students may not pay attention to the scale on a graph, assuming that the
scale units are always ―one.‖ When making axes for a graph, some students may
not using equal intervals to create the scale.
Some students may infer a cause and effect between independent and dependent
variables, but this is often not the case.
Some students graph incorrectly because they don‘t understand that x usually
represents the independent variable and y represents the dependent variable.
Emphasize that this is a convention that makes it easier to communicate.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs
Content
showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing,
Standards
positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change
from a graph.
Instructional Strategies
Instructional Resources/Tools
Flexibly move from examining a graph and describing its characteristics (e.g., intercepts,
Tables, graphs, and equations of real-world functional relationships.
relative maximums, etc.) to using a set of given characteristics to sketch the graph of a
function.
Graphing calculators to generate graphical, tabular, and symbolic representations
Examine a table of related quantities and identify features in the table, such as intervals on
which the function increases, decreases, or exhibits periodic behavior.
of the same function for comparison.
Recognize appropriate domains of functions in real-world settings. For example, when
determining a weekly salary based on hours worked, the hours (input) could be a rational
number, such as 25.5. However, if a function relates the number of cans of soda sold in a
machine to the money generated, the domain must consist of whole numbers.
Common Misconceptions
Students may believe that it is reasonable to input any x-value into a function, so
they will need to examine multiple situations in which there are various
limitations to the domains.
Given a table of values, such as the height of a plant over time, students can estimate the rate
of plant growth. Also, if the relationship between time and height is expressed as a linear
equation, students should explain the meaning of the slope of the line. Finally, if the
relationship is illustrated as a linear or non-linear graph, the student should select points on
the graph and use them to estimate the growth rate over a given interval.
Students may also believe that the slope of a linear function is merely a number
used to sketch the graph of the line. In reality, slopes have real-world meaning,
and the idea of a rate of change is fundamental to understanding major concepts
from geometry to calculus.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Content
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Standards
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
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Instructional Strategies
Explore various families of functions and help students to make connections in terms of
general features. For example, just as the function y = (x + 3)2 – 5 represents a translation of
the function y = x by 3 units to the left and 5 units down, the same is true for the function y =
| x + 3 | - 5 as a translation of the absolute value function y = | x |.
Discover that the factored form of a quadratic or polynomial equation can be used to
determine the zeros, which in turn can be used to identify maxima, minima and end
behaviors.
Use various representations of the same function to emphasize different characteristics of that
function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as
(0, -12). However, rewriting the function as
y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). Furthermore, completing the square
allows the equation to be written as y = (x – 2)2 – 16, which shows that the vertex (and
minimum point) of the parabola is at (2, -16).
Examine multiple real-world examples of exponential functions so that students recognize
that a base between 0 and 1 (such as an equation describing depreciation of an automobile [
x
f(x) = 15,000(0.8) represents the value of a $15,000 automobile that depreciates 20% per
year over the course of x years]) results in an exponential decay, while a base greater than 1
Instructional Resources/Tools
Graphing utilities on a calculator and/or computer can be used to demonstrate the
changes in behavior of a function as various parameters are varied.
Real-world problems, such as maximizing the area of a region bound by a fixed
perimeter fence, can help to illustrate applied uses of families of functions.
Common Misconceptions
Students may believe that each family of functions (e.g., quadratic, square root,
etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphs.
Students may also believe that skills such as factoring a trinomial or completing
the square are isolated within a unit on polynomials, and that they will come to
understand the usefulness of these skills in the context of examining
characteristics of functions.
Additionally, student may believe that the process of rewriting equations into
various forms is simply an algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
x
(such as the value of an investment over time [ f(x) = 5,000(1.07) represents the value of an
investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth.
High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a
Content
context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by
Standards
adding a constant function to a decaying exponential and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the
temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the
location of the weather balloon as a function of time.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms.
Instructional Strategies
Instructional Resources/Tools
Provide a real-world example (e.g., a table showing how far a car has driven after a given
Hands-on materials (e.g., paper folding, building progressively larger shapes
number of minutes, traveling at a uniform speed), and examine the table by looking ―down‖
using pattern blocks, etc.) can be used as a visual source to build numerical tables
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the table to describe a recursive relationship, as well as ―across‖ the table to determine an
explicit formula to find the distance traveled if the number of minutes is known.
Write out terms in a table in an expanded form to help students see what is
happening. For example, if the y-values are 2, 4, 8, 16, they could be written as
2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used
multiple times as a factor.
Focus on one representation and its related language – recursive or explicit – at a
time so that students are not confusing the formats.
Provide examples of when functions can be combined, such as determining a function
describing the monthly cost for owning two vehicles when a function for the cost of each
(given the number of miles driven) is known.
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual
models to generate sequences of numbers that can be explored and described with both
recursive and explicit formulas. Emphasize that there are times when one form to describe the
function is preferred over the other.
for examination.
Visuals available to assist students in seeing relationships are featured at the
National Library of Virtual Manipulatives as well as The National Council of
Teachers of Mathematics, Illuminations
Common Misconceptions
Students may believe that the best (or only) way to generalize a table of data is
by using a recursive formula. Students naturally tend to look ―down‖ a table to
find the pattern but need to realize that finding the 100th term requires knowing
the 99th term unless an explicit formula is developed.
Students may also believe that arithmetic and geometric sequences are the same.
Students need experiences with both types of sequences to be able to recognize
the difference and more readily develop formulas to describe them.
Additionally, advanced students who study composition of functions may
misunderstand function notation to represent multiplication (e.g., f(g(x)) means to
multiply the f and g function values)
High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k
Content
given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
Standards
functions from their graphs and algebraic expressions for them.
Instructional Strategies
Instructional Resources/Tools
Use graphing calculators or computers to explore the effects of a constant in the graph of a
Graphing calculator that can be used to explore the effects of parameter changes
function. For example, students should be able to distinguish between the graphs of y = x2, y
on a graph
= 2x2, y = x2 + 2, y = (2x)2, and y = (x + 2)2. This can be accomplished by allowing students
to work with a single parent function and examine numerous parameter changes to make
Common Misconceptions
generalizations.
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4
units to the left (due to the subtraction symbol). Examples should be explored by
Distinguish between even and odd functions by providing several examples and helping
hand and on a graphing calculator to overcome this misconception.
students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual
approaches to identifying the graphs of even and odd functions can be used as well.
Students may also believe that even and odd functions refer to the exponent of
the variable, rather than the sketch of the graph and the behavior of the function.
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High School Conceptual Category: Functions
Domain
Linear and Exponential Models
Cluster
Construct and compare linear and exponential models and solve problems
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
Content
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Standards
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output
pairs (include reading these from a table.)
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally)
as a polynomial function.
Instructional Strategies
Instructional Resources/Tools
Compare tabular representations of a variety of functions to show that linear functions have a
Examples of real-world situations that apply linear and exponential functions to
first common difference (i.e., equal differences over equal intervals), while exponential
compare their behaviors
functions do not (instead function values grow by equal factors over equal x-intervals).
Graphing calculators or computer software that generate graphs and tables of
Apply linear and exponential functions to real-world situations. For example, a person
functions. A graphing tool such as the one found at nlvm.usu.edu is one option.
earning $10 per hour experiences a constant rate of change in salary given the number of
hours worked, while the number of bacteria on a dish that doubles every hour will have equal
factors over equal intervals.
Common Misconceptions
Provide examples of arithmetic and geometric sequences in graphic, verbal, or tabular forms,
and have students generate formulas and equations that describe the patterns.
Use a graphing calculator or computer program to compare tabular and graphic
representations of exponential and polynomial functions to show how the y (output) values of
the exponential function eventually exceed those of polynomial functions.
Students may believe that all functions have a first common difference and need
to explore to realize that, for example, a quadratic function will have equal
second common differences in a table.
Students may also believe that the end behavior of all functions depends on the
situation and not the fact that exponential function values will eventually get
larger than those of any other polynomial functions.
Have students draw the graphs of exponential and other polynomial functions on a graphing
calculator or computer utility and examine the fact that the exponential curve will eventually
get higher than the polynomial function‘s graph. A simple example would be to compare the
2
x
graphs (and tables) of the functions y = x and y = 2 to find that the y values are greater for
the exponential function when x > 4.
Help students to see that solving an equation such as 2x = 300 can be accomplished by
entering y = 22 and y = 300 into a graphing calculator and finding where the graphs intersect,
by viewing the table to see where the function values are about the same, as well as by
applying a logarithmic function to both sides of the equation.
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Use technology to solve exponential equations such as 3*10x = 450. (In this case, students can
determine the approximate power of 10 that would generate a value of 150.) Students can
also take the logarithm of both sides of the equation to solve for the variable, making use of
the inverse operation to solve.
High School Conceptual Category: Functions
Domain
Linear and Exponential Models
Cluster
Interpret expressions for functions in terms of the situation they model
5. Interpret the parameters in a linear or exponential function in terms of a context.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Use real-world contexts to help students understand how the parameters of linear and
Graphing calculators or computer software that generates graphs and tables of
exponential functions depend on the context. For example, a plumber who charges $50 for a
functions.
house call and $85 per hour would be expressed as the function y = 85x + 50, and if the rate
were raised to $90 per hour, the function would become y = 90x + 50. On the other hand, an
Examples of real-world situations that apply linear and exponential functions to
equation of y = 8,000(1.04)x could model the rising population of a city with 8,000 residents
examine the effects of parameter changes.
when the annual growth rate is 4%. Students can examine what would happen to the
population over 25 years if the rate were 6% instead of 4% or the effect on the equation and
Web sites and other sources that provide raw data, such as the cost of products
function of the city‘s population were 12,000 instead of 8,000.
over time, population changes, etc.
Graphs and tables can be used to examine the behaviors of functions as parameters are
changed, including the comparison of two functions such as what would happen to a
population if it grew by 500 people per year, versus rising an average of 8% per year over the
course of 10 years.
Common Misconceptions
Students may believe that changing the slope of a linear function from ―2‖ to ―3‖
makes the graph steeper without realizing that there is a real-world context and
reason for examining the slopes of lines. Similarly, an exponential function can
appear to be abstract until applying it to a real-world situation involving
population, cost, investments, etc.
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High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on a single count or measurement variable
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Content
2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or
Standards
more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Instructional Strategies
Instructional Resources/Tools
It is helpful for students to understand that a statistical process is a problem-solving process
TI-84 and TI emulator
consisting of four steps: formulating a question that can be answered by data; designing and
Quantitative Literacy Exploring Data module
implementing a plan that collects appropriate data; analyzing the data by graphical and/or
numerical methods; and interpreting the analysis in the context of the original question.
NCTM Navigating through Data Analysis 9-12.
Opportunities should be provided for students to work through the statistical process. In
Grades 6-8, learning has focused on parts of this process. Now is a good time to investigate a
Printed media (e.g., almanacs, newspapers, professional reports)
problem of interest to the students and follow it through. The richer the question formulated,
the more interesting is the process. Teachers and students should make extensive use of
Software such as TinkerPlots and Excel
resources to perfect this very important first step. Global web resources can inspire projects.
Show World: This website offers data about the world that is up to date.
Although this domain addresses both categorical and quantitative data, there is no reference in
the Standards 1 - 4 to categorical data. Note that Standard 5 in the next cluster (Summarize,
Iearn: This website offers projects that students around the world are working on
represent, and interpret data on two categorical and quantitative variables) addresses analysis
simultaneously.
for two categorical variables on the same subject. To prepare for interpreting two categorical
variables in Standard 5, this would be a good place to discuss graphs for one categorical
Common Misconceptions
variable (bar graph, pie graph) and measure of center (mode).
Students may believe:
Have students practice their understanding of the different types of graphs for categorical and
That a bar graph and a histogram are the same. A bar graph is appropriate
numerical variables by constructing statistical posters. Note that a bar graph for categorical
when the horizontal axis has categories and the vertical axis is labeled by either
data may have frequency on the vertical (student‘s pizza preferences) or measurement on the
frequency (e.g., book titles on the horizontal and number of students who like
vertical (radish root growth over time - days).
the respective books on the vertical) or measurement of some numerical
variable (e.g., days of the week on the horizontal and median length of root
Measures of center and spread for data sets without outliers are the mean and standard
growth of radish seeds on the vertical). A histogram has units of measurement
deviation, whereas median and interquartile range are better measures for data sets with
of a numerical variable on the horizontal (e.g., ages with intervals of equal
outliers.
length).
Introduce the formula of standard deviation by reviewing the previously learned MAD (mean
That the lengths of the intervals of a boxplot (min,Q1), (Q1,Q2), (Q2,Q3),
absolute deviation). The MAD is very intuitive and gives a solid foundation for developing
(Q3,max) are related to the number of subjects in each interval. Students
the more complicated standard deviation measure.
should understand that each interval theoretically contains one-fourth of the
total number of subjects. Sketching an accompanying histogram and
Informally observing the extent to which two boxplots or two dotplots overlap begins the
constructing a live boxplot may help in alleviating this misconception.
discussion of drawing inferential conclusions. Don‘t shortcut this observation in comparing
two data sets.
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High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on two categorical and quantitative variables
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as
Content
clustering, outliers, positive or negative association, linear association, and nonlinear association.
Standards
2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear
model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm
in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for
rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have
a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Instructional Strategies
Instructional Resources/Tools
Building on the study of statistics using univariate data in Grades 6 and 7, students are now This lesson - Glued to the Tube or Hooked to the Books? - provides step-by-step
ready to study bivariate data. Students will extend their descriptions and understanding of
instructions for using the graphing calculator to construct a scatter plot of class data
variation to the graphical displays of bivariate data.
and a line of best fit.
Scatter plots are the most common form of displaying bivariate data in Grade 8. Provide
scatter plots and have students practice informally finding the line of best fit. Students
should create and interpret scatter plots, focusing on outliers, positive or negative
association, linearity or curvature. By changing the data slightly, students can have a rich
discussion about the effects of the change on the graph. Have students use a graphing
calculator to determine a linear regression and discuss how this relates to the graph.
Students should informally draw a line of best fit for a scatter plot and informally measure
the strength of fit. Discussion should include ―What does it mean to be above the line,
below the line?‖
From the National Council of Teachers of Mathematics, Illuminations: Impact of a
Superstar - This lesson uses technology tools to plot data, identify lines of best fit,
and detect outliers. Then, students compare the lines of best fit when one element is
removed from a data set, and interpret the results.
The study of the line of best fit ties directly to the algebraic study of slope and intercept.
Students should interpret the slope and intercept of the line of best fit in the context of the
data. Then students can make predictions based on the line of best fit.
The Ohio Resource Center –the tutorial video Lines of Fit shows how to determine
a line of best fit for a set of data.
From the National Council of Teachers of Mathematics, Illuminations: Exploring
Linear Data - In this lesson, students construct scatter plots of bivariate data,
interpret individual data points, make conclusions about trends in data, especially
linear relationships, and estimate and write equation of lines of best fit.
Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report,
American Statistical Association
Common Misconceptions
Students may believe bivariate data is only displayed in scatter plots. 8.SP.4 in this
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cluster provides the opportunity to display bivariate, categorical data in a table.
In general, students think there is only one correct answer in mathematics. Students
may mistakenly think their lines of best fit for the same set of data will be exactly
the same. Because students are informally drawing lines of best fit, the lines will
vary slightly. To obtain the exact line of best fit, students would use technology to
find the line of regression.
High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on two categorical and quantitative variables
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal,
Content
and conditional relative frequencies). Recognize possible associations and trends in the data.
Standards
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residents.
c. Fit a linear function for a scatter plot that suggests a linear association.
Instructional Strategies
Instructional Resources/Tools
In this cluster, the focus is that two categorical or two quantitative variables are being
TI-83/84 and TI emulator
measured on the same subject.
Quantitative Literacy Exploring Data module
In the categorical case, begin with two categories for each variable and represent them in a
two-way table with the two values of one variable defining the rows and the two values of the NCTM Navigating through Data Analysis 9-12.
other variable defining the columns. (Extending the number of rows and columns is easily
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
done once students become comfortable with the 2x2 case.) The table entries are the joint
frequencies of how many subjects displayed the respective cross-classified values. Row totals Report
and column totals constitute the marginal frequencies. Dividing joint or marginal frequencies
Software such as TinkerPlots and Excel
by the total number of subjects define relative frequencies (and percentages), respectively.
Conditional relative frequencies are determined by focusing on a specific row or column of
the table. They are particularly useful in determining any associations between the two
Common Misconceptions
variables.
Students may believe:
That a 45 degree line in the scatterplot of two numerical variables always
In the numerical or quantitative case, display the paired data in a scatterplot. Note that
indicates a slope of 1 which is the case only when the two variables have the
although the two variables in general will not have the same scale, e.g., total SAT versus
same scaling.
grade-point average, it is best to begin with variables with the same scale such as SAT Verbal
That residual plots in the quantitative case should show a pattern of some sort.
and SAT Math. Fitting functions to such data will avoid difficulties such as interpretation of
Just the opposite is the case.
slope in the linear case in which scales differ. Once students are comfortable with the same
scale case, introducing different scales situations will be less problematic.
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High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Interpret Linear Models
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Content
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
Standards
9. Distinguish between correlation and causation.
Instructional Strategies
Instructional Resources/Tools
In this cluster, the key is that two quantitative variables are being measured on the same
TI-83/84 and TI emulator
subject. The paired data should be listed and then displayed in a scatterplot. If time is one of
the variables, it usually goes on the horizontal axis. That which is being predicted goes on the Quantitative Literacy Exploring Data module
vertical; the predictor variable is on the horizontal axis.
NCTM Navigating through Data Analysis 9-12.
Note that unlike a two-dimensional graph in mathematics, the scales of a scatterplot need not
be the same, and even if they are similar (such as SAT Math and SAT Verbal), they still need
not have the same spacing. So, visual rendering of slope makes no sense in most scatterplots,
i.e., a 45 degree line on a scatterplot need not mean a slope of 1.
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
Report
Often the interpretation of the intercept (constant term) is not meaningful in the context of the
data. For example, this is the case when the zero point on the horizontal is of considerable
distance from the values of the horizontal variable, or in some case has no meaning such as
for SAT variables.
The Ohio Resource Center
To make some sense of Pearson‘s r, correlation coefficient, students should recall their
middle school experience with the Quadrant Count Ratio (QCR) as a measure of relationship
between two quantitative variables.
Common Misconceptions
Students may believe that a 45 degree line in the scatterplot of two numerical
variables always indicates a slope of 1 which is the case only when the two
variables have the same scaling. Because the scaling for many real-world
situation varies greatly students need to be give opportunity to compare graphs of
differing scale. Asking students questions like; What would this graph look like
with a different scale or using this scale? Is essential in addressing this
misconception.
Noting that a correlated relationship between two quantitative variables is not causal (unless
the variables are in an experiment) is a very important topic and a substantial amount of time
should be spent on it.
Software such as TinkerPlots and Excel
The National Council of Teachers of Mathematics, Illuminations
Students may believe that when two quantitative variables are related, i.e.,
correlated, that one causes the other to occur. Causation is not necessarily the
case. For example, at a theme park, the daily temperature and number of bottles
of water sold are demonstrably correlated, but an increase in the number of
bottles of water sold does not cause the day‘s temperature to rise or fall.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
Content
a. Interpret parts of an expression, such as terms, factors, and coefficients.
Standards
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a
factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can
be factored as (x2 - y2)(x2 + y2).
Instructional Strategies
Extending beyond simplifying an expression, this cluster addresses interpretation of the
components in an algebraic expression. A student should recognize that in the expression 2x +
1, ―2‖ is the coefficient, ―2‖ and ―x‖ are factors, and ―1‖ is a constant, as well as ―2x‖ and ―1‖
being terms of the binomial expression. Development and proper use of mathematical
language is an important building block for future content.
Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression
0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one
minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
number of cell phone minutes used in the month. Similar real-world examples, such as tax
rates, can also be used to explore the meaning of expressions.
Factoring by grouping is another example of how students might analyze the structure of an
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the ―x – 5‖ is
common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should
become comfortable with rewriting expressions in a variety of ways until a structure emerges.
Instructional Resources/Tools
Hands-on materials, such as algebra tiles, can be used to establish a visual
understanding of algebraic expressions and the meaning of terms, factors and
coefficients.
From the National Library of Virtual Manipulatives - Algebra Tiles – Visualize
multiplying and factoring algebraic expressions using tiles.
Common Misconceptions
Students may believe that the use of algebraic expressions is merely the abstract
manipulation of symbols. Use of real-world context examples to demonstrate the
meaning of the parts of algebraic expressions is needed to counter this
misconception.
Students may also believe that an expression cannot be factored because it does
not fit into a form they recognize. They need help with reorganizing the terms
until structures become evident.
Have students create their own expressions that meet specific criteria (e.g., number of terms
factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten
in different forms. Additionally, pair/group students to share their expressions and rewrite one
another‘s expressions.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Write expressions in equivalent forms to solve problems
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
Content
a. Factor a quadratic expression to reveal the zeros of the function it defines
Standards
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression1.15t can be written as (1.151/12)12t ≈
1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on linking expressions and functions, i.e., creating connections between
Graphing utilities to explore the effects of parameter changes on a graph
multiple representations of functional relations – the dependence between a quadratic
expression and a graph of the quadratic function it defines, and the dependence between
Tables, graphs and equations of real-world applications that apply quadratic
different symbolic representations of exponential functions. Teachers need to foster the idea that and exponential functions
changing the forms of expressions, such as factoring or completing the square, or transforming
expressions from one exponential form to another, are not independent algorithms that are
Computer algebra systems
learned for the sake of symbol manipulations. They are processes that are guided by goals (e.g.,
investigating properties of families of functions and solving contextual problems).
From the National Library of Virtual Manipulatives - Grapher – A tool for
graphing and exploring functions.
Factoring methods that are typically introduced in elementary algebra and the method of
completing the square reveals attributes of the graphs of quadratic functions, represented by
http://www.learner.org/workshops/algebra/workshop5/lessonplan1.html - This
quadratic equations.
website contains a lesson and a workshop that showcases ways that teachers
can help students explore mathematical properties studied in algebra. The
The solutions of quadratic equations solved by factoring are the x – intercepts of the
activities use a variety of techniques to help students understand concepts of
parabola or zeros of quadratic functions.
2
factoring quadratic trinomials.
A pair of coordinates (h, k) from the general form f(x) = a(x – h) +k represents the vertex
of the parabola, where h represents a horizontal shift and k represents a vertical shift of the
From the National Council of Teachers of Mathematics, Illuminations parabola y = x2 from its original position at the origin.
Difference of Squares - This activity uses a series of related arithmetic
A vertex (h, k) is the minimum point of the graph of the quadratic function if a › 0 and is the
maximum point of the graph of the quadratic function if a ‹ 0. Understanding an algorithm of experiences to prompt students to explore arithmetic statements leading to a
result that is the factoring pattern for the difference of two squares. A
completing the square provides a solid foundation for deriving a quadratic formula.
geometric interpretation of the familiar formula is also included.
Translating among different forms of expressions, equations and graphs helps students to
understand some key connections among arithmetic, algebra and geometry. The reverse
Common Misconceptions
thinking technique (a process that allows working backwards from the answer to the starting
Some students may believe that factoring and completing the square are
point) can be very effective. Have students derive information about a function‘s equation,
isolated techniques within a unit of quadratic equations. Teachers should help
represented in standard, factored or general form, by investigating its graph.
students to see the value of these skills in the context of solving higher degree
equations and examining different families of functions.
Offer multiple real-world examples of exponential functions. For instance, to illustrate an
exponential decay, students need to recognize that in the equation for an automobile cost C(t) =
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20,000(0.75)t , the base is 0.75 and between 0 and 1 and the value of $20,000 represents the
initial cost of an automobile that depreciates 25% per year over the course of t years.
Similarly, to illustrate exponential growth, in the equation for the value of an investment over
time A(t) = 10,000(1.03)t, where the base is 1.03 and is greater than 1; and the $10,000
represents the value of an investment when increasing in value by 3% per year for x years.
Students may think that the minimum (the vertex) of the graph of y = (x + 5)2
is shifted to the right of the minimum (the vertex) of the graph y = x2 due to the
addition sign. Students should explore examples both analytically and
graphically to overcome this misconception.
Some students may believe that the minimum of the graph of a quadratic
function always occur at the y-intercept.
Some students cannot distinguish between arithmetic and geometric sequences,
or between sequences and series. To avoid this confusion, students need to
experience both types of sequences and series.
High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Perform arithmetic operations on polynomials
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and
Content
multiplication; add, subtract, and multiply polynomials.
Standards
Instructional Strategies
Instructional Resources/Tools
The primary strategy for this cluster is to make connections between arithmetic of integers
Graphing calculators
and arithmetic of polynomials. In order to understand this standard, students need to work
toward both understanding and fluency with polynomial arithmetic. Furthermore, to talk
Graphing software, including dynamic geometry software
about their work, students will need to use correct vocabulary, such as integer, monomial,
polynomial, factor, and term.
Computer Algebra Systems
In arithmetic of polynomials, a central idea is the distributive property, because it is
fundamental not only in polynomial multiplication but also in polynomial addition and
subtraction. With the distributive property, there is little need to emphasize misleading
mnemonics, such as FOIL, which is relevant only when multiplying two binomials, and the
procedural reminder to ―collect like terms‖ as a consequence of the distributive property. For
example, when adding the polynomials 3x and 2x, the result can be explained with the
distributive property as follows: 3x + 2x = (3 + 2)x = 5x.
An important connection between the arithmetic of integers and the arithmetic of polynomials
can be seen by considering whole numbers in base ten place value to be polynomials in the
base b = 10. For two-digit whole numbers and linear binomials, this connection can be
illustrated with area models and algebra tiles. But the connections between methods of
multiplication can be generalized further. For example, compare the product 213 x 47 with
the product
:
Algebra tiles
Area models
Common Misconceptions
Some students will apply the distributive property inappropriately. Emphasize
that it is the distributive property of multiplication over addition. For example,
the distributive property can be used to rewrite
as
, because in
this product the second factor is a sum (i.e., involving addition). But in the
product
, the second factor,
, is itself a product, not a sum.
Some students will still struggle with the arithmetic of negative numbers.
Consider the expression
. On the one hand,
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. But using the distributive property,
. Because the first calculation gave 0,
the two terms on the right in the second calculation must be opposite in sign.
Thus, if we agree that
, then it must follow that
.
Note how the distributive property is in play in each of these examples: In the left-most
computation, each term in the factor
must be multiplied by each term in the other
factor,
, just like the value of each digit in 47 must be multiplied by the value
of each digit in 213, as in the middle computation, which is similar to ―partial products
methods‖ that some students may have used for multiplication in the elementary grades. The
common algorithm on the right is merely an abbreviation of the partial products method.
The new idea in this standard is called closure: A set is closed under an operation if when any
two elements are combined with that operation, the result is always another element of the
same set. In order to understand that polynomials are closed under addition, subtraction and
multiplication, students can compare these ideas with the analogous claims for integers: The
sum, difference or product of any two integers is an integer, but the quotient of two integers is
not always an integer.
Now for polynomials, students need to reason that the sum (difference or product) of any two
polynomials is indeed a polynomial. At first, restrict attention to polynomials with integer
coefficients. Later, students should consider polynomials with rational or real coefficients and
reason that such polynomials are closed under these operations.
For contrast, students need to reason that polynomials are not closed under the operation of
division: The quotient of two polynomials is not always a polynomial. For example
is not a polynomial. Of course, the quotient of two polynomials is sometimes a
polynomial. For example,
.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve equations and inequalities in one variable (cont.)
4. Solve quadratic equations in one variable.
Content
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions
Standards
derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate
to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them in a ± bi for real numbers a and b.
Instructional Strategies
Instructional Resources/Tools
Completing the square is usually introduced for several reasons‖ to find the vertex of a
Graphing utilities to explore the effects of changes in parameters of equations on
parabola whose equation has been expanded; to look at the parabola through the lenses of
their graphs
translations of a ―parent‖ parabola y = x2; and to derive a quadratic formula. Completing the
square is a very useful tool that will be used repeatedly by students in many areas of
Tables, graphs and equations of real-world applications that apply quadratic and
mathematics. Teachers should carefully balance traditional paper-pencil skills of
exponential functions
manipulating quadratic expressions and solving quadratic equations along with an analysis of
the relationship between parameters of quadratic equations and properties of their graphs.
http://ohiorc.org/for/math/
Start by inspecting equations such as x2 = 9 that has two solutions, 3 and -3. Next, progress to
equations such as (x - 7)2 = 9 by substituting x – 7 for x and solving them either by
―inspection‖ or by taking the square root on each side:
x–7=3
and x – 7 = -3
x = 10
x=4
Graph both pairs of solutions (-3 and 3, 4 and 10) on the number line and notice that 4 and 10
are 7 units to the right of – 3 and 3. So, the substitution of x – 7 for x moved the solutions 7
units to the right. Next, graph the function y = (x – 7)2 – 9, pointing out that the x-intercepts
are 4 and 10, and emphasizing that the graph is the translation of 7 units to the right and 9
units down from the position of the graph of the parent function y = x2 that passes through the
origin (0, 0). Generate more equations of the form y = a(x – h)2 + k and compare their graphs
using a graphing technology.
http://illuminations.nctm.org/
Common Misconceptions
Some students may think that rewriting equations into various forms (taking
square roots, completing the square, using quadratic formula and factoring) are
isolated techniques within a unit of quadratic equations. Teachers should help
students see the value of these skills in the context of solving higher degree
equations and examining different families of functions.
Highlight and compare different approaches to solving the same problem. Use technology to
recognize that two different expressions or equations may represent the same relationship. For
example, since x2 -10x +25 = 0 can be rewritten as (x- 5)(x –5) = 0 or (x – 5)2 = 0 or x2 = 25,
these are all representations of the same equation that has a double solution x = 5. Support it
by putting all expressions into graphing calculator. Compare their graphs and generate their
tables displaying the same output values for each expression.
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Guide students in transforming a quadratic equation in standard form, 0 = ax2 + bx + c, to the
vertex form 0 = a(x – h)2 + k by separating your examples into groups with a = 1 and a ≠ 1
and have students guess the number that needs to be added to the binomials of the type x2 +
6x, x2 – 2x, x2 +9x, x2 - x to form complete square of the binomial (x – m)2. Then generalize
the process by showing the expression (b/2)2 that has to be added to the binomial x2 + bx.
Completing the square for an expression whose x2 coefficient is not 1 can be complicated for
some students. Present multiple examples of the type 0 = 2x2 – 5x - 9 to emphasize the logic
behind every step, keeping in mind that the same process will be used to complete the square
in the proof of the quadratic formula.
Discourage students from giving a preference to a particular method of solving quadratic
equations. Students need experience in analyzing a given problem to choose an appropriate
solution method before their computations become burdensome. Point out that the Quadratic
Formula, x =
is a universal tool that can solve any quadratic equation; however,
it is not reasonable to use the Quadratic Formula when the quadratic equation is missing
either a middle term, bx, or a constant term, c. When it is missing a constant term, (e.g., 3x2 –
10x = 0) a factoring method becomes more efficient. If a middle term is missing (e.g., 2x2 –
15 = 0), a square root method is the most appropriate. Stress the benefit of memorizing the
Quadratic Formula and flexibility with a factoring strategy. Introduce the concept of
discriminants and their relationship to the number and nature of the roots of quadratic
equation.
Offer students examples of a quadratic equation, such as x2 + 9 = 0. Since the graph of the
quadratic function y = x2 + 9 is situated above the x –axis and opens up, the graph does not
have x–intercepts and therefore, the quadratic equation does not have real solutions. At this
stage introduce students to non-real solutions, such as
or
and a new
number type-imaginary unit i that equals
. Using i in place of
is a way to present
the two solutions of a quadratic equation in the complex numbers form a ± bi, if a and b are
real numbers and b ≠ 0. Have students observe that if a quadratic equation has complex
solutions, the solutions always appear in conjugate pairs, in the form a + bi and a – bi.
Particularly, for the equation x2 = - 9, a conjugate pair of solutions are 0 +3i and 0 – 3i.
Project the same logic in the examples of any quadratic equations 0 = ax2 + bx + c that have
negative discriminants. The solutions are a pair of conjugate complex numbers
b2 – 4ac is negative.
, if D =
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve systems of equations (cont.)
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of
Content
intersection between the line y = -3x and the circle x2 + y2 = 3.
Standards
Instructional Strategies
Instructional Resources/Tools
Prior to solving systems of equations graphically, students should revisit ―families of
Graph paper
functions‖ to review techniques for graphing different classes of functions. Alert students to
the fact that if one equation in the system can be obtained by multiplying both sides of
Graphing calculators
another equation by a nonzero constant, the system is called consistent, the equations in the
system are called dependent and the system has an infinite number of solutions that produces
coinciding graphs. Provide students opportunities to practice linear vs. non-linear systems;
Common Misconceptions
consistent vs. inconsistent systems; pure computational vs. real-world contextual problems
Most mistakes that students make are careless rather than conceptual. Teachers
(e.g., chemistry and physics applications encountered in science classes).
should encourage students to learn a certain format for solving systems of
equations and check the answers by substituting into all equations in the system.
High School Conceptual Category: Number and Quantity
Domain
The Real Number System
Cluster
Use properties of rational and irrational numbers
3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the
Content
product of a nonzero rational number and an irrational number is irrational.
Standards
Instructional Strategies
Instructional Resources/Tools
This cluster is an excellent opportunity to incorporate algebraic proof, both direct and
The Ohio Resource Center
indirect, in teaching properties of number systems.
The National Council of Teachers of Mathematics, Illuminations
Students should explore concrete examples that illustrate that for any two rational numbers
written in form a/b and c/d, where b and d are natural numbers and a and c are integers, the
NCTM Principles and Standards for School Mathematics
following are true:
represents a rational number, and
represents a rational number.
Continue exploring situations where the sum of a rational number and an irrational number is
irrational (e.g., a sum of rational number 2 and irrational number √3 is (2 + √3), which is an
irrational).
Common Misconceptions
Some students may believe that both terminating and repeating decimals are
rational numbers, without considering nonrepeating and nonterminating decimals
as irrational numbers.
Students may also confuse irrational numbers and complex numbers, and
therefore mix their properties. In this case, students should encounter examples
that support or contradict properties and relationships between number sets (i.e.,
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Proofs are valid ways to justify not only geometry statements also algebraic statements.
Use indirect algebraic proof to generalize the statement that the sum of a rational and
irrational number is irrational.
Assume that x is an irrational number and the sum of x and a rational number is also rational
and is represented as
irrational numbers are real numbers and complex numbers are non-real numbers.
The set of real numbers is a subset of the set of complex numbers).
By using false extensions of properties of rational numbers, some students may
assume that the sum of any two irrational numbers is also irrational. This
statement is not always true (e.g.,
, a rational
number), and therefore, cannot be considered as a property.
x+
x=
x=
represents a rational number
Since the last statement contradicts a given fact that x is an irrational number, the assumption
is wrong and a sum of a rational number and an irrational number has to be irrational.
Similarly, it can be proven that the product of a non- zero rational and an irrational number is
irrational.
Students need to see that results of the operations performed between numbers from a
particular number set does not always belong to the same set. For example, the sum of two
irrational numbers
and
is 4, which is a rational number.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
6. Explain a proof of the Pythagorean Theorem and its converse.
Content
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three-dimensions.
Standards
8. Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
Instructional Strategies
Instructional Resources/Tools
Previous understanding of triangles, such as the sum of two side measures is greater than the
From the National Library of Virtual Manipulatives third side measure, angles sum, and area of squares, is furthered by the introduction of unique Pythagorean Theorem – Solve two puzzles that illustrate the proof of the
qualities of right triangles. Students should be given the opportunity to explore right triangles Pythagorean Theorem.
to determine the relationships between the measures of the legs and the measure of the
hypotenuse. Experiences should involve using grid paper to draw right triangles from given
Right Triangle Solver – Practice using the Pythagorean theorem and the
measures and representing and computing the areas of the squares on each side. Data should
definitions of the trigonometric functions to solve for unknown sides and angles
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be recorded in a chart such as the one below, allowing for students to conjecture about the
relationship among the areas within each triangle.
Triangle
Measure of
Leg 1
Measure of
Leg 2
Area of
Square on
Leg 1
Area of
Square on
Leg 2
Area of
Square on
Hypotenuse
of a right triangle.
Common Misconceptions
Students may confuse which side to use as the hypotenuse.
1
Students should then test out their conjectures, then explain and discuss their findings.
Finally, the Pythagorean Theorem should be introduced and explained as the pattern they
have explored. Time should be spent analyzing several proofs of the Pythagorean Theorem to
develop a beginning sense of the process of deductive reasoning, the significance of a
theorem, and the purpose of a proof. Students should be able to justify a simple proof of the
Pythagorean Theorem or its converse.
Previously, students have discovered that not every combination of side lengths will create a
triangle. Now they need situations that explore using the Pythagorean Theorem to test
whether or not side lengths represent right triangles. (Recording could include Side length a,
Side length b, Sum of a2 + b2, c2, a2 + b2 = c2, Right triangle? Through these opportunities,
students should realize that there are Pythagorean (triangular) triples such as (3, 4, 5), (5, 12,
13), (7, 24, 25), (9, 40, 41) that always create right triangles, and that their multiples also
form right triangles. Students should see how similar triangles can be used to find additional
triples. Students should be able to explain why a triangle is or is not a right triangle using the
Pythagorean Theorem.
The Pythagorean Thereom should be applied to finding the lengths of segments on a
coordinate grid, especially those segments that do not follow the vertical or horizontal lines,
as a means of discussing the determination of distances between points. Contextual situations,
created by both the students and the teacher, that apply the Pythagorean theorem and its
converse should be provided. For example, apply the concept of similarity to determine the
height of a tree using the ratio between the student's height and the length of the student's
shadow. From that, determine the distance from the tip of the tree to the end of its shadow
and verify by comparing to the computed distance from the top of the student's head to the
end of the student's shadow, using the ratio calculated previously. Challenge students to
identify additional ways that the Pythagorean Theorem is or can be used in real world
situations or mathematical problems, such as finding the height of something that is difficult
to physically measure, or the diagonal of a prism.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs
Content
showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing,
Standards
positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change
from a graph.
Instructional Strategies
Instructional Resources/Tools
Flexibly move from examining a graph and describing its characteristics (e.g., intercepts,
Tables, graphs, and equations of real-world functional relationships.
relative maximums, etc.) to using a set of given characteristics to sketch the graph of a
function.
Graphing calculators to generate graphical, tabular, and symbolic representations
of the same function for comparison.
Examine a table of related quantities and identify features in the table, such as intervals on
which the function increases, decreases, or exhibits periodic behavior.
Common Misconceptions
Recognize appropriate domains of functions in real-world settings. For example, when
Students may believe that it is reasonable to input any x-value into a function, so
determining a weekly salary based on hours worked, the hours (input) could be a rational
they will need to examine multiple situations in which there are various
number, such as 25.5. However, if a function relates the number of cans of soda sold in a
limitations to the domains.
machine to the money generated, the domain must consist of whole numbers.
Given a table of values, such as the height of a plant over time, students can estimate the rate
of plant growth. Also, if the relationship between time and height is expressed as a linear
equation, students should explain the meaning of the slope of the line. Finally, if the
relationship is illustrated as a linear or non-linear graph, the student should select points on
the graph and use them to estimate the growth rate over a given interval.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Content
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Standards
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret
these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y =
(1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Instructional Strategies
Instructional Resources/Tools
Explore various families of functions and help students to make connections in terms of
Graphing utilities on a calculator and/or computer can be used to demonstrate the
general features. For example, just as the function y = (x + 3)2 – 5 represents a translation of
changes in behavior of a function as various parameters are varied.
the function y = x by 3 units to the left and 5 units down, the same is true for the function y =
| x + 3 | - 5 as a translation of the absolute value function y = | x |.
Real-world problems, such as maximizing the area of a region bound by a fixed
perimeter
Discover that the factored form of a quadratic or polynomial equation can be used to
r fence, can help to illustrate applied uses of families of functions.
determine the zeros, which in turn can be used to identify maxima, minima and end
behaviors.
Use various representations of the same function to emphasize different characteristics of that
function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as
(0, -12). However, rewriting the function as
y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). Furthermore, completing the square
allows the equation to be written as y = (x – 2)2 – 16, which shows that the vertex (and
minimum point) of the parabola is at (2, -16).
Examine multiple real-world examples of exponential functions so that students recognize
that a base between 0 and 1 (such as an equation describing depreciation of an automobile [
Common Misconceptions
Students may believe that each family of functions (e.g., quadratic, square root,
etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphs.
Students may also believe that skills such as factoring a trinomial or completing
the square are isolated within a unit on polynomials, and that they will come to
understand the usefulness of these skills in the context of examining
characteristics of functions.
x
f(x) = 15,000(0.8) represents the value of a $15,000 automobile that depreciates 20% per
year over the course of x years]) results in an exponential decay, while a base greater than 1
x
(such as the value of an investment over time [ f(x) = 5,000(1.07) represents the value of an
investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth.
Additionally, student may believe that the process of rewriting equations into
various forms is simply an algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
Content
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Standards
b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential and relate these functions to the model.
Instructional Strategies
Instructional Resources/Tools
Provide a real-world example (e.g., a table showing how far a car has driven after a given
Hands-on materials (e.g., paper folding, building progressively larger shapes
number of minutes, traveling at a uniform speed), and examine the table by looking ―down‖
using pattern blocks, etc.) can be used as a visual source to build numerical tables
the table to describe a recursive relationship, as well as ―across‖ the table to determine an
for examination.
explicit formula to find the distance traveled if the number of minutes is known.
Visuals available to assist students in seeing relationships are featured at the
Write out terms in a table in an expanded form to help students see what is
National Library of Virtual Manipulatives as well as The National Council of
happening. For example, if the y-values are 2, 4, 8, 16, they could be written as
Teachers of Mathematics, Illuminations
2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used
multiple times as a factor.
Common Misconceptions
Focus on one representation and its related language – recursive or explicit – at a
Students may believe that the best (or only) way to generalize a table of data is
time so that students are not confusing the formats.
by using a recursive formula. Students naturally tend to look ―down‖ a table to
find the pattern but need to realize that finding the 100th term requires knowing
Provide examples of when functions can be combined, such as determining a function
the 99th term unless an explicit formula is developed.
describing the monthly cost for owning two vehicles when a function for the cost of each
(given the number of miles driven) is known.
Students may also believe that arithmetic and geometric sequences are the same.
Students need experiences with both types of sequences to be able to recognize
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual
the difference and more readily develop formulas to describe them.
models to generate sequences of numbers that can be explored and described with both
recursive and explicit formulas. Emphasize that there are times when one form to describe the Additionally, advanced students who study composition of functions may
function is preferred over the other.
misunderstand function notation to represent multiplication (e.g., f(g(x)) means to
multiply the f and g function values).
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k
Content
given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
Standards
functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2x3 for x>0 or
f(x) = (x+1)/(x-1) for x ≠1.
Instructional Strategies
Instructional Resources/Tools
Use graphing calculators or computers to explore the effects of a constant in the graph of a
Graphing calculator that can be used to explore the effects of parameter changes
function. For example, students should be able to distinguish between the graphs of y = x2, y
on a graph
= 2x2, y = x2 + 2, y = (2x)2, and y = (x + 2)2. This can be accomplished by allowing students
to work with a single parent function and examine numerous parameter changes to make
generalizations.
Common Misconceptions
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4
Distinguish between even and odd functions by providing several examples and helping
units to the left (due to the subtraction symbol). Examples should be explored by
students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual
hand and on a graphing calculator to overcome this misconception.
approaches to identifying the graphs of even and odd functions can be used as well.
Students may also believe that even and odd functions refer to the exponent of
Provide examples of inverses that are not purely mathematical to introduce the idea. For
the variable, rather than the sketch of the graph and the behavior of the function.
example, given a function that names the capital of a state, f(Ohio) = Columbus. The inverse
would be to input the capital city and have the state be the output, such that f--1(Denver) =
Additionally, students may believe that all functions have inverses and need to
Colorado.
see counter examples, as well as examples in which a non-invertible function can
be made into an invertible function by restricting the domain. For example,
2
-1
f(x) = x has an inverse ( f (x) = x ) provided that the domain is restricted to x
≥ 0.
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High School Conceptual Category: Number and Quantity
Domain
Quantities
Cluster
Reason quantitatively and use units to solve problems
1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and
Content
interpret the scale and the origin in graphs and data displays.
Standards
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement reporting quantities.
Instructional Strategies
Instructional Resources/Tools
In real-world situations, answers are usually represented by numbers associated with units. Units NCTM. Focus in High School Mathematics (Reasoning and Sense Making)
involve measurement and often require a conversion. Measurement involves both precision and
accuracy. Estimation and approximation often precede more exact computations.
Mathematical Sciences Education Board. High School Mathematics at Work
Students need to develop sound mathematical reasoning skills and forms of argument to make
reasonable judgments about their solutions. They should be able to decide whether a problem
calls for an estimate, for an approximation, or for an exact answer. To accomplish this goal,
teachers should provide students with a broad range of contextual problems that offer
opportunities for performing operations with quantities involving units. These problems should
be connected to science, engineering, economics, finance, medicine, etc.
NCTM. Principles and Standards for School Mathematics
Joint Committee of the MAA and NCTM. A Sourcebook of Applications of
School Mathematics
Some contextual problems may require an understanding of derived measurements and capability
in unit analysis. Keeping track of derived units during computations and making reasonable
estimates and rational conclusions about accuracy and the precision of the answers help in the
problem-solving process.
Common Misconceptions
Students may not realize the importance of the units‘ conversions in
conjunction with the computation when solving problems involving
measurements.
For example, while driving in the United Kingdom (UK), a U.S. tourist puts 60 liters of gasoline
in his car. The gasoline cost is £1.28 per liter The exchange rate is £ 0.62978 for each $1.00. The
price for a gallon of a gasoline in the United States is $3.05. The driver wants to compare the
costs for the same amount and the same type of gasoline when he/she pays in UK pounds.
Making reasonable estimates should be encouraged prior to solving this problem. Since the
current exchange rate has inflated the UK pound at almost twice the U.S. dollar, the driver will
pay more for less gasoline.
Since today‘s calculating devices often display 8 to 10 decimal places,
students frequently express answers to a much greater degree of precision
than the required.
By dividing $3.05 by 3.79L (the number of liters in one gallon), students can see that 80.47 cents
per liter of gasoline in US is less expensive than £1.28 or $ 2.03 per liter of the same type of
gasoline in the UK when paid in U.S. dollars. The cost of 60 liters of gasoline in UK is ( )
In order to compute the cost of the same quantity of gasoline in the United States in UK
currency, it is necessary to convert between both monetary systems and units of volume. Based
on UK pounds, the cost of 60 liters of gasoline in the U.S. is ).
The computation shows that the gasoline is less expensive in the United States and how an
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analysis can be helpful in keeping track of unit conversations. Students should be able to
correctly identify the degree of precision of the answers which should not be far greater than the
actual accuracy of the measurements.
Graphical representations serve as visual models for understanding phenomena that take place in
our daily surroundings. The use of different kinds of graphical representations along with their
units, labels and titles demonstrate the level of students‘ understanding and foster the ability to
reason, prove, self-check, examine relationships and establish the validity of arguments. Students
need to be able to identify misleading graphs by choosing correct units and scales to create a
correct representation of a situation or to make a correct conclusion from it.
High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
Content
(a) Interpret parts of an expression, such as terms, factors, and coefficients.
Standards
(b) Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a
factor not depending on P.
Instructional Strategies
Instructional Resources/Tools
Extending beyond simplifying an expression, this cluster addresses interpretation of the
Hands-on materials, such as algebra tiles, can be used to establish a visual
components in an algebraic expression. A student should recognize that in the expression 2x + understanding of algebraic expressions and the meaning of terms, factors and
1, ―2‖ is the coefficient, ―2‖ and ―x‖ are factors, and ―1‖ is a constant, as well as ―2x‖ and ―1‖ coefficients.
being terms of the binomial expression. Development and proper use of mathematical
language is an important building block for future content.
From the National Library of Virtual Manipulatives - Algebra Tiles – Visualize
multiplying and factoring algebraic expressions using tiles.
Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression
0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one
Common Misconceptions
minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
Students may believe that the use of algebraic expressions is merely the abstract
number of cell phone minutes used in the month. Similar real-world examples, such as tax
manipulation of symbols. Use of real-world context examples to demonstrate the
rates, can also be used to explore the meaning of expressions.
meaning of the parts of algebraic expressions is needed to counter this
Factoring by grouping is another example of how students might analyze the structure of an
misconception.
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the ―x – 5‖ is
common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should
Students may also believe that an expression cannot be factored because it does
become comfortable with rewriting expressions in a variety of ways until a structure emerges. not fit into a form they recognize. They need help with reorganizing the terms
until structures become evident.
Have students create their own expressions that meet specific criteria (e.g., number of terms
factorable, difference of two squares, etc.) and verbalize how they can be written and
rewritten in different forms. Additionally, pair/group students to share their expressions and
rewrite one another‘s expressions.
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High School Conceptual Category: Algebra
Domain
Creating Equations
Cluster
Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple
Content
rational and exponential functions
Standards
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a
modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to
highlight resistance R.
Instructional Strategies
Instructional Resources/Tools
Provide examples of real-world problems that can be modeled by writing an equation or
Graphing calculators
inequality. Begin with simple equations and inequalities and build up to more complex
equations in two or more variables that may involve quadratic, exponential or rational functions. Computer software that generate graphs of functions
Discuss the importance of using appropriate labels and scales on the axes when representing
functions with graphs.
Examine real-world graphs in terms of constraints that are necessary to balance a mathematical
model with the real-world context. For example, a student writing an equation to model the
maximum area when the perimeter of a rectangle is 12 inches should recognize that y = x(6 – x)
only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side
of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6.
Students can discuss the difference between the parabola that models the problem and the
portion of the parabola that applies to the context.
Explore examples illustrating when it is useful to rewrite a formula by solving for one of the
A = 1 h(b + b )
1 2 )
2
variables in the formula. For example, the formula for the area of a trapezoid (
can be solved for h if the area and lengths of the bases are known but the height needs to be
calculated. This strategy of selecting a different representation has many applications in science
and business when using formulas.
Examples of real-world situations that lend themselves to writing equations
that model the contexts
Common Misconceptions
Students may believe that equations of linear, quadratic and other functions are
abstract and exist only ―in a math book,‖ without seeing the usefulness of these
functions as modeling real-world phenomena.
Additionally, they believe that the labels and scales on a graph are not
important and can be assumed by a reader, and that it is always necessary to
use the entire graph of a function when solving a problem that uses that
function as its model.
Provide examples of real-world problems that can be solved by writing an equation, and have
students explore the graphs of the equations on a graphing calculator to determine which parts
of the graph are relevant to the problem context.
Use a graphing calculator to demonstrate how dramatically the shape of a curve can change
when the scale of the graph is altered for one or both variables.
Give students formulas, such as area and volume (or from science or business), and have
students solve the equations for each of the different variables in the formula.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Understand solving equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the
Content
original equation has a solution. Construct a viable argument to justify a solution method.
Standards
Instructional Strategies
Instructional Resources/Tools
Graphing Calculators
Challenge students to justify each step of solving an equation. Transforming 2x - 5 = 7 to
2x = 12 is possible because 5 = 5, so adding the same quantity to both sides of an equation
makes the resulting equation true as well. Each step of solving an equation can be defended,
Common Misconceptions
much like providing evidence for steps of a geometric proof.
Students may believe that solving an equation such as 3x + 1 = 7 involves
―only removing the 1,‖ failing to realize that the equation 1 = 1 is being
Provide examples for how the same equation might be solved in a variety of ways as long as
subtracted to produce the next step.
equivalent quantities are added or subtracted to both sides of the equation, the order of steps
taken will not matter.
Additionally, students may believe that all solutions to radical and rational
equations are viable, without recognizing that there are times when extraneous
3n + 2 = n - 10
3n + 2 = n - 10
3n + 2 = n - 10
solutions are generated and have to be eliminated.
2 = -2
+ 10 = +10
-n
= -n
3n
= n – 12
OR
3n + 12 = n
OR
2n + 2 = -10
-n
= -n
-3n
= -3n
-2=–2
2n
= -12
12 = -2n
2n = -12
n = -6
n= -6
n = -6
Connect the idea of adding two equations together as a means of justifying steps of solving a
simple equation to the process of solving a system of equations. A system consisting of two
linear functions such as 2x + 3y = 8 and x - 3y = 1 can be solved by adding the equations
together, and can be justified by exactly the same reason that solving the equation 2x - 4 = 5
can begin by adding the equation 4 = 4.
Begin with simple, one-step equations and require students to write out a justification for each
step used to solve the equation.
It is very important that students are able to reason how and why extraneous solutions arise.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve equations and inequalities in one variable
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
There are two major reasons for discussing the topic of inequalities along with equations:
Graphing utilities to explore the effects of changes in parameters of equations on
First, there are analogies between solving equations and inequalities that help students
their graphs.
understand them both. Second, the applications that lead to equations almost always lead in
the same way to inequalities.
Tables, graphs and equations of real-world applications.
In grades 6-8, students solve and graph linear equations and inequalities. Graphing
experience with inequalities is limited to graphing on a number line diagram. Despite this
work, some students will still need more practice to be proficient. It may be beneficial to
remind students of the most common solving techniques, such as converting fractions from
one form to another, removing parentheses in the sentences, or multiplying both sides of an
equation or inequality by the common denominator of the fractions. Students must be aware
of what it means to check an inequality‘s solution. The substitution of the end points of the
solution set in the original inequality should give equality regardless of the presence or the
absence of an equal sign in the original sentence. The substitution of any value from the rest
of the solution set should give a correct inequality.
Careful selection of examples and exercises is needed to provide students with meaningful
review and to introduce other important concepts, such as the use of properties and
applications of solving linear equations and inequalities. Stress the idea that the application of
properties is also appropriate when working with equations or inequalities that include more
than one variable, fractions and decimals. Regardless of the type of numbers or variables in
the equation or inequality, students have to examine the validity of each step in the solution
process.
Solving equations for the specified letter with coefficients represented by letters (e.g.,
when solving for ) is similar to solving an equation with one variable. Provide
students with an opportunity to abstract from particular numbers and apply the same kind of
manipulations to formulas as they did to equations. One of the purposes of doing abstraction
is to learn how to evaluate the formulas in easier ways and use the techniques across
mathematics and science.
http://ohiorc.org/for/math/
http://illuminations.nctm.org/
Common Misconceptions
Some students may believe that for equations containing fractions only on one
side, it requires ―clearing fractions‖ (the use of multiplication) only on that side
of the equation. To address this misconception, start by demonstrating the
solution methods for equations similar to x + x + x + 46 = x and stress that
the Multiplication Property of Equality is applied to both sides, which are
multiplied by 60.
Students may confuse the rule of changing a sign of an inequality when
multiplying or dividing by a negative number with changing the sign of an
inequality when one or two sides of the inequality become negative (for ex., 3x >
-15 or x < - 5).
Some students may believe that subscripts can be combined as b1 + b2 = b3 and
the sum of different variables d and D is 2D (d +D = 2D).
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Draw students‘ attention to equations containing variables with subscripts. The same
variables with different subscripts (e.g., x1 and x2 ) should be viewed as different variables
that cannot be combined as like terms. A variable with a variable subscript, such as an, must
be treated as a single variable – the nth term, where variables a and n have different meaning.
High School Conceptual Category: Number and Quantity
Domain
The Real Number System
Cluster
Extend the properties of exponents to rational exponents
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a
Content
notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3
Standards
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Instructional Strategies
Instructional Resources/Tools
The goal is to show that a fractional exponent can be expressed as a radical or a root. For
Graphing calculator
example, an exponent of 1/3 is equivalent to a cube root; an exponent of ¼ is equivalent to a
fourth root.
Computer algebra systems
Review the power rule,
, for whole number exponents (e.g. (7 2) 3 = 76.
Compare examples, such as (71/2)2 = 71 = 7 and
The Ohio Resource Center
The National Council of Teachers of Mathematics, Illuminations
, to help students establish a
connection between radicals and rational exponents:
and, in general,
.
Provide opportunities for students to explore the equality of the values using calculators, such
as 71/2 and
.
Offer sufficient examples and exercises to prompt the definition of fractional exponents, and
give students practice in converting expressions between radical and exponential forms.
When n is a positive integer, generalize the meaning of
and then to
,
where n and m are integers and n is greater than or equal to 2. When m is a negative integer,
the result is the reciprocal of the root
.
Common Misconceptions
Students sometimes misunderstand the meaning of exponential operations, the
way powers and roots relate to one another, and the order in which they should
be performed. Attention to the base is very important. Consider examples:
and
. The position of a negative sign of a term with a rational
exponent can mean that the rational exponent should be either applied first to the
base, 81, and then the opposite of the result is taken
, or the rational
exponent should be applied to a negative term
. The answer of
will be not real if the denominator of the exponent is even. If the root is odd, the
answer will be a negative number.
Stress the two rules of rational exponents: 1) the numerator of the exponent is the base‘s
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power and 2) the denominator of the exponent is the order of the root. When evaluating
expressions involving rational exponents, it is often helpful to break an exponent into its parts
– a power and a root – and then decide if it is easier to perform the root operation or the
exponential operation first.
Model the use of precise mathematics vocabulary (e.g., base, exponent, radical, root, cube
root, square root etc.).
Students should be able to make use of estimation when incorrectly using
multiplication instead of exponentiation.
Students may believe that the fractional exponent in the expression
means
the same as a factor in multiplication expression,
and multiply the base
by the exponent.
The rules for integer exponents are applicable to rational exponents as well; however, the
operations can be slightly more complicated because of the fractions. When multiplying
exponents, powers are added
. When dividing exponents, powers are
subtracted
multiplied
. When raising an exponent to an exponent, powers are
High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve systems of equations
5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system
Content
with the same solutions.
Standards
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Instructional Strategies
Instructional Resources/Tools
The focus of this standard is to provide mathematics justification for the addition
Graph paper
(elimination) and substitution methods of solving systems of equations that transform a given
system of two equations into a simpler equivalent system that has the same solutions as the
Graphing calculators
original.
Computer graphing tools to operate with matrices and to find determinants of
The Addition and Multiplication Properties of Equality allow finding solutions to certain
higher order matrices.
systems of equations. In general, any linear combination, m(Ax + By) + n(Cx + Dy) = mE
+nF, of two linear equations
Dynamic geometry software
Ax + By = E and
Cx + Dy = F
intersecting in a single point contains that point. The multipliers m and n can be chosen so
that the resulting combination has only an x-term or only a y-term in it. That is, the
combination will be a horizontal or vertical line containing the point of intersection.
From the National Council of Teachers of Mathematics, Illuminations: Supply
and Demand - This activity focuses on having students create and solve a system
of linear equations in a real-world setting. By solving a system of two equations
in two unknowns, students will find the equilibrium point for supply and
demand.
In the proof of a system of two equations in two variables, where one equation is replaced by
the sum of that equation and a multiple of the other equation, produces a system that has the
http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/mak
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same solutions, let point (x1, y1) be a solution of both equations:
Ax1 + By1 = E (true)
Cx1 + Dy1 = F (true)
Replace the equation Ax + By = E with Ax + By + k(Cx + Dy) on its left side and with E +
kF on its right side.
The new equation is Ax + By + k(Cx + Dy) = E + kF.
Show that the ordered pair of numbers (x1, y1) is a solution of this equation by replacing (x1,
y1) in the left side of this equation and verifying that the right side really equals E + kF:
Ax1 + By1 + k(Cx1 + Dy1) = E + kF (true)
Systems of equations are classified into two groups, consistent or inconsistent, depending on
whether or not solutions exist. The solution set of a system of equations is the intersection of
the solution sets for the individual equations. Stress the benefit of making the appropriate
selection of a method for solving systems (graphing vs. addition vs. substitution). This
depends on the type of equations and combination of coefficients for corresponding variables,
without giving a preference to either method.
ing_connections.pdf - Students use graphing calculators to solve systems of
linear equations in two ways. They first solve the systems by graphing the
equations and finding the point of intersection. Next, they will solve systems of
equations by writing related matrices and finding the solution by using inverse
matrices.
From the National Council of Teachers of Mathematics, Illuminations:
Movement with Functions - In this lesson, students use remote-controlled cars to
create a system of equations.
Common Misconceptions
Most mistakes that students make are careless rather than conceptual. Teachers
should encourage students to learn a certain format for solving systems of
equations and check the answers by substituting into all equations in the system.
The graphing method can be the first step in solving systems of equations. .A set of points
representing solutions of each equation is found by graphing these equations. Even though the
graphing method is limited in finding exact solutions and often yields approximate values, the
use of it helps to discover whether solutions exist and, if so, how many are there
Prior to solving systems of equations graphically, students should revisit ―families of
functions‖ to review techniques for graphing different classes of functions. Alert students to
the fact that if one equation in the system can be obtained by multiplying both sides of
another equation by a nonzero constant, the system is called consistent, the equations in the
system are called dependent and the system has an infinite number of solutions that produces
coinciding graphs. Provide students opportunities to practice linear vs. non-linear systems;
consistent vs. inconsistent systems; pure computational vs. real-world contextual problems
(e.g., chemistry and physics applications encountered in science classes). A rich variety of
examples can lead to discussions of the relationships between coefficients and consistency
that can be extended to graphing and later to determinants and matrices.
The next step is to turn to algebraic methods, elimination or substitution, to allow students to
find exact solutions. For any method, stress the importance of having a well-organized format
for writing solutions.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Represent and solve equations and inequalities graphically
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be
Content
a line).
Standards
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x);
find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution
set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Instructional Strategies
Instructional Resources/Tools
Beginning with simple, real-world examples, help students to recognize a graph as a set of
Examples of real-world situations that involve linear functions and two-variable
solutions to an equation. For example, if the equation y = 6x + 5 represents the amount of
linear inequalities
money paid to a babysitter (i.e., $5 for gas to drive to the job and $6/hour to do the work),
then every point on the line represents an amount of money paid, given the amount of time
Graphing calculators or computer software that generate graphs and tables for
worked.
solving equations
Explore visual ways to solve an equation such as 2x + 3 = x – 7 by graphing the functions y =
2x + 3 and y = x – 7. Students should recognize that the intersection point of the lines is at (10, -17). They should be able to verbalize that the intersection point means that when x = -10
is substituted into both sides of the equation, each side simplifies to a value of -17. Therefore,
-10 is the solution to the equation. This same approach can be used whether the functions in
the original equation are linear, nonlinear or both.
Using technology, have students graph a function and use the trace function to move the
cursor along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of
the calculator, emphasizing that every point on the curve represents a solution to the equation.
Begin with simple linear equations and how to solve them using the graphs and tables on a
graphing calculator. Then, advance students to nonlinear situations so they can see that even
complex equations that might involve quadratics, absolute value, or rational functions can be
solved fairly easily using this same strategy. While a standard graphing calculator does not
simply solve an equation for the user, it can be used as a tool to approximate solutions.
Common Misconceptions
Students may believe that the graph of a function is simply a line or curve
―connecting the dots,‖ without recognizing that the graph represents all solutions
to the equation.
Students may also believe that graphing linear and other functions is an isolated
skill, not realizing that multiple graphs can be drawn to solve equations involving
those functions.
Additionally, students may believe that two-variable inequalities have no
application in the real world. Teachers can consider business related problems
(e.g., linear programming applications) to engage students in discussions of how
the inequalities are derived and how the feasible set includes all the points that
satisfy the conditions stated in the inequalities.
Use the table function on a graphing calculator to solve equations. For example, to solve the
equation x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine
that they intersect when x = 4 and when x = -3 by examining the table to find where the yvalues are the same.
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Investigate real-world examples of two-dimensional inequalities. For example, students might
explore what the graph would look like for money earned when a person earns at least $6 per
hour. (The graph for a person earning exactly $6/hour would be a linear function, while the
graph for a person earning at least $6/hour would be a half-plane including the line and all
points above it.) Applications such as linear programming can help students to recognize how
businesses use constraints to maximize profit while minimizing the use of resources. These
situations often involve the use of systems of two variable inequalities.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Understand the concept of a function and use function notation
1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the
Content
range. If f is a function and x is an element of its domain, the f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the
Standards
equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fiboonacci sequence is
defined recursively by F(0) = F(1) = 1, f(n + 1) = f(n) + f (n – 1) for n ≥ 1.
Instructional Strategies
Instructional Resources/Tools
Provide applied contexts in which to explore functions. For example, examine the amount of
Diagrams or drawings of function machines, as well as tables and
money earned when given the number of hours worked on a job, and contrast this with a
graphs.
situation in which a single fee is paid by the ―carload‖ of people, regardless of whether 1, 2, or
more people are in the car.
Function Machine virtual manipulatives, such as available at
nlvm.usu.edu.
Use diagrams to help students visualize the idea of a function machine. Students can examine
several pairs of input and output values and try to determine a simple rule for the function.
Rewrite sequences of numbers in tabular form, where the input represents the term number
(the position or index) in the sequence, and the output represents the number in the sequence.
Help students to understand that the word ―domain‖ implies the set of all possible input values
and that the integers are a set of numbers made up of {…-2, -1, 0, 1, 2, …}.
Distinguish between relationships that are not functions and those that are functions (e.g.,
present a table in which one of the input values results in multiple outputs to contrast with a
functional relationship). Examine graphs of functions and non-functions, recognizing that if a
vertical line passes through at least two points in the graph, then y (or the quantity on the
vertical axis) is not a function of x (or the quantity on the horizontal axis).
Common Misconceptions
Students may believe that all relationships having an input and an output are
functions, and therefore, misuse the function terminology.
Students may also believe that the notation f(x) means to multiply some value f
times another value x. The notation alone can be confusing and needs careful
development. For example, f(2) means the output value of the function f when
the input value is 2.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs
Content
showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing,
Standards
positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change
from a graph.
Instructional Strategies
Instructional Resources/Tools
Flexibly move from examining a graph and describing its characteristics (e.g., intercepts,
Tables, graphs, and equations of real-world functional relationships.
relative maximums, etc.) to using a set of given characteristics to sketch the graph of a function.
Examine a table of related quantities and identify features in the table, such as intervals on
which the function increases, decreases, or exhibits periodic behavior.
Recognize appropriate domains of functions in real-world settings. For example, when
determining a weekly salary based on hours worked, the hours (input) could be a rational
number, such as 25.5. However, if a function relates the number of cans of soda sold in a
machine to the money generated, the domain must consist of whole numbers.
Given a table of values, such as the height of a plant over time, students can estimate the rate of
plant growth. Also, if the relationship between time and height is expressed as a linear equation,
students should explain the meaning of the slope of the line. Finally, if the relationship is
illustrated as a linear or non-linear graph, the student should select points on the graph and use
them to estimate the growth rate over a given interval.
Graphing calculators to generate graphical, tabular, and symbolic
representations of the same function for comparison.
Common Misconceptions
Students may believe that it is reasonable to input any x-value into a function,
so they will need to examine multiple situations in which there are various
limitations to the domains.
Students may also believe that the slope of a linear function is merely a
number used to sketch the graph of the line. In reality, slopes have real-world
meaning, and the idea of a rate of change is fundamental to understanding
major concepts from geometry to calculus.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Content
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Standards
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Instructional Strategies
Instructional Resources/Tools
Explore various families of functions and help students to make connections in terms of
Graphing utilities on a calculator and/or computer can be used to demonstrate the
general features. For example, just as the function y = (x + 3)2 – 5 represents a translation of
changes in behavior of a function as various parameters are varied.
the function y = x by 3 units to the left and 5 units down, the same is true for the function y =
| x + 3 | - 5 as a translation of the absolute value function y = | x |.
Real-world problems, such as maximizing the area of a region bound by a fixed
perimeter fence, can help to illustrate applied uses of families of functions.
Discover that the factored form of a quadratic or polynomial equation can be used to
determine the zeros, which in turn can be used to identify maxima, minima and end
behaviors.
Common Misconceptions
Students may believe that each family of functions (e.g., quadratic, square root,
Use various representations of the same function to emphasize different characteristics of that etc.) is independent of the others, so they may not recognize commonalities
function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as
among all functions and their graphs.
(0, -12). However, rewriting the function as y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( 2, 0). Furthermore, completing the square allows the equation to be written as y = (x – 2)2 –
Students may also believe that skills such as factoring a trinomial or completing
16, which shows that the vertex (and minimum point) of the parabola is at (2, -16).
the square are isolated within a unit on polynomials, and that they will come to
understand the usefulness of these skills in the context of examining
Examine multiple real-world examples of exponential functions so that students recognize
characteristics of functions.
that a base between 0 and 1 (such as an equation describing depreciation of an automobile [
x
Additionally, student may believe that the process of rewriting equations into
f(x) = 15,000(0.8) represents the value of a $15,000 automobile that depreciates 20% per
various forms is simply an algebra symbol manipulation exercise, rather than
year over the course of x years]) results in an exponential decay, while a base greater than 1
serving a purpose of allowing different features of the function to be exhibited.
x
(such as the value of an investment over time [ f(x) = 5,000(1.07) represents the value of an
investment of $5,000 when increasing in value by 7% per year for x years]) illustrates growth.
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a
Content
context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by
Standards
adding a constant function to a decaying exponential and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the
temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the
location of the weather balloon as a function of time.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms.
Instructional Strategies
Instructional Resources/Tools
Provide a real-world example (e.g., a table showing how far a car has driven after a given
Hands-on materials (e.g., paper folding, building progressively larger shapes
number of minutes, traveling at a uniform speed), and examine the table by looking ―down‖
using pattern blocks, etc.) can be used as a visual source to build numerical tables
the table to describe a recursive relationship, as well as ―across‖ the table to determine an
for examination.
explicit formula to find the distance traveled if the number of minutes is known.
Visuals available to assist students in seeing relationships are featured at the
Write out terms in a table in an expanded form to help students see what is
National Library of Virtual Manipulatives as well as The National Council of
happening. For example, if the y-values are 2, 4, 8, 16, they could be written as
Teachers of Mathematics, Illuminations
2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used
multiple times as a factor.
Common Misconceptions
Focus on one representation and its related language – recursive or explicit – at a
Students may believe that the best (or only) way to generalize a table of data is
time so that students are not confusing the formats.
by using a recursive formula. Students naturally tend to look ―down‖ a table to
find the pattern but need to realize that finding the 100 th term requires knowing
Provide examples of when functions can be combined, such as determining a function
the 99th term unless an explicit formula is developed.
describing the monthly cost for owning two vehicles when a function for the cost of each
(given the number of miles driven) is known.
Students may also believe that arithmetic and geometric sequences are the same.
Students need experiences with both types of sequences to be able to recognize
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual
the difference and more readily develop formulas to describe them.
models to generate sequences of numbers that can be explored and described with both
recursive and explicit formulas. Emphasize that there are times when one form to describe the Additionally, advanced students who study composition of functions may
function is preferred over the other.
misunderstand function notation to represent multiplication (e.g., f(g(x)) means to
multiply the f and g function values).
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k
Content
given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
Standards
functions from their graphs and algebraic expressions for them.
Instructional Strategies
Instructional Resources/Tools
Use graphing calculators or computers to explore the effects of a constant in the graph of a
Graphing calculator that can be used to explore the effects of parameter changes
function. For example, students should be able to distinguish between the graphs of y = x2, y
on a graph
= 2x2, y = x2 + 2, y = (2x)2, and y = (x + 2)2. This can be accomplished by allowing students
to work with a single parent function and examine numerous parameter changes to make
generalizations.
Common Misconceptions
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4
Distinguish between even and odd functions by providing several examples and helping
units to the left (due to the subtraction symbol). Examples should be explored by
students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual
hand and on a graphing calculator to overcome this misconception.
approaches to identifying the graphs of even and odd functions can be used as well.
Students may also believe that even and odd functions refer to the exponent of
the variable, rather than the sketch of the graph and the behavior of the function.
High School Conceptual Category: Functions
Domain
Linear and Exponential Models
Cluster
Construct and compare linear and exponential models and solve problems
1. Distinguish between situations that can be modeled with linear functions and with exponential functions.
Content
a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Standards
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output
pairs (include reading these from a table.)
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally)
as a polynomial function.
Instructional Strategies
Instructional Resources/Tools
Compare tabular representations of a variety of functions to show that linear functions have a
Examples of real-world situations that apply linear and exponential functions to
first common difference (i.e., equal differences over equal intervals), while exponential
compare their behaviors
functions do not (instead function values grow by equal factors over equal x-intervals).
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Apply linear and exponential functions to real-world situations. For example, a person
earning $10 per hour experiences a constant rate of change in salary given the number of
hours worked, while the number of bacteria on a dish that doubles every hour will have equal
factors over equal intervals.
Graphing calculators or computer software that generate graphs and tables of
functions. A graphing tool such as the one found at nlvm.usu.edu is one option.
Provide examples of arithmetic and geometric sequences in graphic, verbal, or tabular forms,
and have students generate formulas and equations that describe the patterns.
Common Misconceptions
Students may believe that all functions have a first common difference and need
to explore to realize that, for example, a quadratic function will have equal
second common differences in a table.
Use a graphing calculator or computer program to compare tabular and graphic
representations of exponential and polynomial functions to show how the y (output) values of
the exponential function eventually exceed those of polynomial functions.
Students may also believe that the end behavior of all functions depends on the
situation and not the fact that exponential function values will eventually get
larger than those of any other polynomial functions.
Have students draw the graphs of exponential and other polynomial functions on a graphing
calculator or computer utility and examine the fact that the exponential curve will eventually
get higher than the polynomial function‘s graph. A simple example would be to compare the
2
x
graphs (and tables) of the functions y = x and y = 2 to find that the y values are greater for
the exponential function when x > 4.
Help students to see that solving an equation such as 2x = 300 can be accomplished by
entering y = 22 and y = 300 into a graphing calculator and finding where the graphs intersect,
by viewing the table to see where the function values are about the same, as well as by
applying a logarithmic function to both sides of the equation.
Use technology to solve exponential equations such as 3*10 x = 450. (In this case, students can
determine the approximate power of 10 that would generate a value of 150.) Students can
also take the logarithm of both sides of the equation to solve for the variable, making use of
the inverse operation to solve.
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High School Conceptual Category: Functions
Domain
Linear and Exponential Models
Cluster
Interpret expressions for functions in terms of the situation they model
5. Interpret the parameters in a linear or exponential function in terms of a context.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Use real-world contexts to help students understand how the parameters of linear and
Graphing calculators or computer software that generates graphs and tables of
exponential functions depend on the context. For example, a plumber who charges $50 for a
functions.
house call and $85 per hour would be expressed as the function y = 85x + 50, and if the rate
were raised to $90 per hour, the function would become y = 90x + 50. On the other hand, an
Examples of real-world situations that apply linear and exponential functions to
equation of y = 8,000(1.04)x could model the rising population of a city with 8,000 residents
examine the effects of parameter changes.
when the annual growth rate is 4%. Students can examine what would happen to the
population over 25 years if the rate were 6% instead of 4% or the effect on the equation and
Web sites and other sources that provide raw data, such as the cost of products
function of the city‘s population were 12,000 instead of 8,000.
over time, population changes, etc.
Graphs and tables can be used to examine the behaviors of functions as parameters are
changed, including the comparison of two functions such as what would happen to a
population if it grew by 500 people per year, versus rising an average of 8% per year over the
course of 10 years.
Common Misconceptions
Students may believe that changing the slope of a linear function from ―2‖ to ―3‖
makes the graph steeper without realizing that there is a real-world context and
reason for examining the slopes of lines. Similarly, an exponential function can
appear to be abstract until applying it to a real-world situation involving
population, cost, investments, etc.
High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on a single count or measurement variable
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Content
2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or
Standards
more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Instructional Strategies
Instructional Resources/Tools
It is helpful for students to understand that a statistical process is a problem-solving process
TI-84 and TI emulator
consisting of four steps: formulating a question that can be answered by data; designing and
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implementing a plan that collects appropriate data; analyzing the data by graphical and/or
numerical methods; and interpreting the analysis in the context of the original question.
Opportunities should be provided for students to work through the statistical process. In
Grades 6-8, learning has focused on parts of this process. Now is a good time to investigate a
problem of interest to the students and follow it through. The richer the question formulated,
the more interesting is the process. Teachers and students should make extensive use of
resources to perfect this very important first step. Global web resources can inspire projects.
Quantitative Literacy Exploring Data module
Although this domain addresses both categorical and quantitative data, there is no reference in
the Standards 1 - 4 to categorical data. Note that Standard 5 in the next cluster (Summarize,
represent, and interpret data on two categorical and quantitative variables) addresses analysis
for two categorical variables on the same subject. To prepare for interpreting two categorical
variables in Standard 5, this would be a good place to discuss graphs for one categorical
variable (bar graph, pie graph) and measure of center (mode).
Show World: This website offers data about the world that is up to date.
Have students practice their understanding of the different types of graphs for categorical and
numerical variables by constructing statistical posters. Note that a bar graph for categorical
data may have frequency on the vertical (student‘s pizza preferences) or measurement on the
vertical (radish root growth over time - days).
Measures of center and spread for data sets without outliers are the mean and standard
deviation, whereas median and interquartile range are better measures for data sets with
outliers.
Introduce the formula of standard deviation by reviewing the previously learned MAD (mean
absolute deviation). The MAD is very intuitive and gives a solid foundation for developing
the more complicated standard deviation measure.
Informally observing the extent to which two boxplots or two dotplots overlap begins the
discussion of drawing inferential conclusions. Don‘t shortcut this observation in comparing
two data sets.
NCTM Navigating through Data Analysis 9-12.
Printed media (e.g., almanacs, newspapers, professional reports)
Software such as TinkerPlots and Excel
Iearn:This website offers projects that students around the world are working on
simultaneously.
Common Misconceptions
Students may believe that a bar graph and a histogram are the same. A bar graph
is appropriate when the horizontal axis has categories and the vertical axis is
labeled by either frequency (e.g., book titles on the horizontal and number of
students who like the respective books on the vertical) or measurement of some
numerical variable (e.g., days of the week on the horizontal and median length of
root growth of radish seeds on the vertical). A histogram has units of
measurement of a numerical variable on the horizontal (e.g., ages with intervals
of equal length).
Student may also believe that the lengths of the intervals of a boxplot (min,Q1),
(Q1,Q2), (Q2,Q3), (Q3,max) are related to the number of subjects in each
interval. Students should understand that each interval theoretically contains onefourth of the total number of subjects. Sketching an accompanying histogram and
constructing a live boxplot may help in alleviating this misconception.
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High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on two categorical and quantitative variables
5. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal,
Content
and conditional relative frequencies). Recognize possible associations and trends in the data.
Standards
6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residents.
c. Fit a linear function for a scatter plot that suggests a linear association.
Instructional Strategies
Instructional Resources/Tools
In this cluster, the focus is that two categorical or two quantitative variables are being
TI-83/84 and TI emulator
measured on the same subject.
Quantitative Literacy Exploring Data module
In the categorical case, begin with two categories for each variable and represent them in a
two-way table with the two values of one variable defining the rows and the two values of the NCTM Navigating through Data Analysis 9-12.
other variable defining the columns. (Extending the number of rows and columns is easily
done once students become comfortable with the 2x2 case.) The table entries are the joint
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
frequencies of how many subjects displayed the respective cross-classified values. Row totals Report
and column totals constitute the marginal frequencies. Dividing joint or marginal frequencies
by the total number of subjects define relative frequencies (and percentages), respectively.
Software such as TinkerPlots and Excel
Conditional relative frequencies are determined by focusing on a specific row or column of
the table. They are particularly useful in determining any associations between the two
variables.
Common Misconceptions
Students may believe that a 45 degree line in the scatterplot of two numerical
In the numerical or quantitative case, display the paired data in a scatterplot. Note that
variables always indicates a slope of 1 which is the case only when the two
although the two variables in general will not have the same scale, e.g., total SAT versus
variables have the same scaling.
grade-point average, it is best to begin with variables with the same scale such as SAT Verbal
and SAT Math. Fitting functions to such data will avoid difficulties such as interpretation of
Students may also believe that residual plots in the quantitative case should show
slope in the linear case in which scales differ. Once students are comfortable with the same
a pattern of some sort. Just the opposite is the case.
scale case, introducing different scales situations will be less problematic.
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High School Conceptual Category: Statistics and Probability
Domain
Interpreting Categorical and Quantitative Data
Cluster
Interpret Linear Models
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Content
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
Standards
9. Distinguish between correlation and causation.
Instructional Strategies
Instructional Resources/Tools
In this cluster, the key is that two quantitative variables are being measured on the same
TI-83/84 and TI emulator
subject. The paired data should be listed and then displayed in a scatterplot. If time is one of
the variables, it usually goes on the horizontal axis. That which is being predicted goes on the Quantitative Literacy Exploring Data module
vertical; the predictor variable is on the horizontal axis.
NCTM Navigating through Data Analysis 9-12.
Note that unlike a two-dimensional graph in mathematics, the scales of a scatterplot need not
be the same, and even if they are similar (such as SAT Math and SAT Verbal), they still need
not have the same spacing. So, visual rendering of slope makes no sense in most scatterplots,
i.e., a 45 degree line on a scatterplot need not mean a slope of 1.
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
Report
Often the interpretation of the intercept (constant term) is not meaningful in the context of the
data. For example, this is the case when the zero point on the horizontal is of considerable
distance from the values of the horizontal variable, or in some case has no meaning such as
for SAT variables.
The Ohio Resource Center
To make some sense of Pearson‘s r, correlation coefficient, students should recall their
middle school experience with the Quadrant Count Ratio (QCR) as a measure of relationship
between two quantitative variables.
Common Misconceptions
Students may believe that a 45 degree line in the scatterplot of two numerical
variables always indicates a slope of 1 which is the case only when the two
variables have the same scaling. Because the scaling for many real-world
situations varies greatly students need to be give opportunity to compare graphs
of differing scale. Asking students questions like; What would this graph look
like with a different scale or using this scale? Is essential in addressing this
misconception.
Noting that a correlated relationship between two quantitative variables is not causal (unless
the variables are in an experiment) is a very important topic and a substantial amount of time
should be spent on it.
Software such as TinkerPlots and Excel
The National Council of Teachers of Mathematics, Illuminations
That when two quantitative variables are related, i.e., correlated, that one causes
the other to occur. Causation is not necessarily the case. For example, at a theme
park, the daily temperature and number of bottles of water sold are demonstrably
correlated, but an increase in the number of bottles of water sold does not cause
the day‘s temperature to rise or fall.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
Content
(a) Interpret parts of an expression, such as terms, factors, and coefficients.
Standards
(b) Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a
factor not depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that
can be factored as (x2 - y2)(x2 + y2).
Instructional Strategies
Extending beyond simplifying an expression, this cluster addresses interpretation of the
components in an algebraic expression. A student should recognize that in the expression 2x +
1, ―2‖ is the coefficient, ―2‖ and ―x‖ are factors, and ―1‖ is a constant, as well as ―2x‖ and ―1‖
being terms of the binomial expression. Development and proper use of mathematical
language is an important building block for future content.
Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression
0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one
minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the
number of cell phone minutes used in the month. Similar real-world examples, such as tax
rates, can also be used to explore the meaning of expressions.
Factoring by grouping is another example of how students might analyze the structure of an
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the ―x – 5‖ is
common to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should
become comfortable with rewriting expressions in a variety of ways until a structure emerges.
Instructional Resources/Tools
Hands-on materials, such as algebra tiles, can be used to establish a visual
understanding of algebraic expressions and the meaning of terms, factors and
coefficients.
From the National Library of Virtual Manipulatives - Algebra Tiles – Visualize
multiplying and factoring algebraic expressions using tiles.
Common Misconceptions
Students may believe that the use of algebraic expressions is merely the abstract
manipulation of symbols. Use of real-world context examples to demonstrate the
meaning of the parts of algebraic expressions is needed to counter this
misconception.
Students may also believe that an expression cannot be factored because it does
not fit into a form they recognize. They need help with reorganizing the terms
until structures become evident.
Have students create their own expressions that meet specific criteria (e.g., number of terms
factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten
in different forms. Additionally, pair/group students to share their expressions and rewrite one
another‘s expressions.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Write expressions in equivalent forms to solve problems
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
Content
a. Factor a quadratic expression to reveal the zeros of the function it defines
Standards
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression1.15t can be written as (1.151/12)12t ≈
1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%
Instructional Strategies
Instructional Resources/Tools
This cluster focuses on linking expressions and functions, i.e., creating connections between
Graphing utilities to explore the effects of parameter changes on a graph
multiple representations of functional relations – the dependence between a quadratic
expression and a graph of the quadratic function it defines, and the dependence between
Tables, graphs and equations of real-world applications that apply quadratic and
different symbolic representations of exponential functions. Teachers need to foster the idea
exponential functions
that changing the forms of expressions, such as factoring or completing the square, or
transforming expressions from one exponential form to another, are not independent
Computer algebra systems
algorithms that are learned for the sake of symbol manipulations. They are processes that are
guided by goals (e.g., investigating properties of families of functions and solving contextual
From the National Library of Virtual Manipulatives - Grapher – A tool for
problems).
graphing and exploring functions.
Factoring methods that are typically introduced in elementary algebra and the method of
completing the square reveals attributes of the graphs of quadratic functions, represented by
quadratic equations.
The solutions of quadratic equations solved by factoring are the x – intercepts of the
parabola or zeros of quadratic functions.
A pair of coordinates (h, k) from the general form f(x) = a(x – h)2 +k represents the vertex
of the parabola, where h represents a horizontal shift and k represents a vertical shift of
the parabola y = x2 from its original position at the origin.
A vertex (h, k) is the minimum point of the graph of the quadratic function if a › 0 and is
the maximum point of the graph of the quadratic function if a ‹ 0. Understanding an
algorithm of completing the square provides a solid foundation for deriving a quadratic
formula.
http://www.learner.org/workshops/algebra/workshop5/lessonplan1.html
This website contains a lesson and a workshop that showcases ways that teachers
can help students explore mathematical properties studied in algebra. The
activities use a variety of techniques to help students understand concepts of
factoring quadratic trinomials.
Translating among different forms of expressions, equations and graphs helps students to
understand some key connections among arithmetic, algebra and geometry. The reverse
thinking technique (a process that allows working backwards from the answer to the starting
point) can be very effective. Have students derive information about a function‘s equation,
represented in standard, factored or general form, by investigating its graph.
Common Misconceptions
Some students may believe that factoring and completing the square are isolated
techniques within a unit of quadratic equations. Teachers should help students to
see the value of these skills in the context of solving higher degree equations and
examining different families of functions.
From the National Council of Teachers of Mathematics, Illuminations Difference of Squares - This activity uses a series of related arithmetic
experiences to prompt students to explore arithmetic statements leading to a
result that is the factoring pattern for the difference of two squares. A geometric
interpretation of the familiar formula is also included.
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Offer multiple real-world examples of exponential functions. For instance, to illustrate an
exponential decay, students need to recognize that in the equation for an automobile cost C(t)
= 20,000(0.75)t , the base is 0.75 and between 0 and 1 and the value of $20,000 represents the
initial cost of an automobile that depreciates 25% per year over the course of t years.
Students may think that the minimum (the vertex) of the graph of y = (x + 5)2 is
shifted to the right of the minimum (the vertex) of the graph y = x2 due to the
addition sign. Students should explore examples both analytically and
graphically to overcome this misconception.
Similarly, to illustrate exponential growth, in the equation for the value of an investment over
time A(t) = 10,000(1.03)t, where the base is 1.03 and is greater than 1; and the $10,000
represents the value of an investment when increasing in value by 3% per year for x years.
Some students may believe that the minimum of the graph of a quadratic
function always occur at the y-intercept.
Some students cannot distinguish between arithmetic and geometric sequences,
or between sequences and series. To avoid this confusion, students need to
experience both types of sequences and series.
High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Perform arithmetic operations on polynomials
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and
Content
multiplication; add, subtract, and multiply polynomials.
Standards
Instructional Strategies
Instructional Resources/Tools
The primary strategy for this cluster is to make connections between arithmetic of integers
Graphing calculators
and arithmetic of polynomials. In order to understand this standard, students need to work
toward both understanding and fluency with polynomial arithmetic. Furthermore, to talk
Graphing software, including dynamic geometry software
about their work, students will need to use correct vocabulary, such as integer, monomial,
polynomial, factor, and term.
Computer Algebra Systems
In arithmetic of polynomials, a central idea is the distributive property, because it is
fundamental not only in polynomial multiplication but also in polynomial addition and
subtraction. With the distributive property, there is little need to emphasize misleading
mnemonics, such as FOIL, which is relevant only when multiplying two binomials, and the
procedural reminder to ―collect like terms‖ as a consequence of the distributive property. For
example, when adding the polynomials 3x and 2x, the result can be explained with the
distributive property as follows: 3x + 2x = (3 + 2)x = 5x.
An important connection between the arithmetic of integers and the arithmetic of polynomials
can be seen by considering whole numbers in base ten place value to be polynomials in the
base b = 10. For two-digit whole numbers and linear binomials, this connection can be
illustrated with area models and algebra tiles. But the connections between methods of
Algebra tiles
Area models
Common Misconceptions
Some students will apply the distributive property inappropriately. Emphasize
that it is the distributive property of multiplication over addition. For example,
the distributive property can be used to rewrite
as
, because in
this product the second factor is a sum (i.e., involving addition). But in the
product
, the second factor,
, is itself a product, not a sum.
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multiplication can be generalized further. For example, compare the product 213 x 47 with
the product
:
Some students will still struggle with the arithmetic of negative numbers.
Consider the expression
. On the one hand,
. But using the distributive property,
. Because the first calculation gave 0,
the two terms on the right in the second calculation must be opposite in sign.
Thus, if we agree that
, then it must follow that
.
Note how the distributive property is in play in each of these examples: In the left-most
computation, each term in the factor
must be multiplied by each term in the other
factor,
, just like the value of each digit in 47 must be multiplied by the value
of each digit in 213, as in the middle computation, which is similar to ―partial products
methods‖ that some students may have used for multiplication in the elementary grades. The
common algorithm on the right is merely an abbreviation of the partial products method.
The new idea in this standard is called closure: A set is closed under an operation if when any
two elements are combined with that operation, the result is always another element of the
same set. In order to understand that polynomials are closed under addition, subtraction and
multiplication, students can compare these ideas with the analogous claims for integers: The
sum, difference or product of any two integers is an integer, but the quotient of two integers is
not always an integer.
Now for polynomials, students need to reason that the sum (difference or product) of any two
polynomials is indeed a polynomial. At first, restrict attention to polynomials with integer
coefficients. Later, students should consider polynomials with rational or real coefficients and
reason that such polynomials are closed under these operations.
For contrast, students need to reason that polynomials are not closed under the operation of
division: The quotient of two polynomials is not always a polynomial. For example
is not a polynomial. Of course, the quotient of two polynomials is sometimes a
polynomial. For example,
.
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve equations and inequalities in one variable (cont.)
4. Solve quadratic equations in one variable.
Content
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions.
Standards
Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate
to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them in a ± bi for real numbers a and b.
Instructional Strategies
Instructional Resources/Tools
Completing the square is usually introduced for several reasons‖ to find the vertex of a
Graphing utilities to explore the effects of changes in parameters of equations on
parabola whose equation has been expanded; to look at the parabola through the lenses of
their graphs
translations of a ―parent‖ parabola y = x2; and to derive a quadratic formula. Completing the
square is a very useful tool that will be used repeatedly by students in many areas of
Tables, graphs and equations of real-world applications that apply quadratic and
mathematics. Teachers should carefully balance traditional paper-pencil skills of
exponential functions
manipulating quadratic expressions and solving quadratic equations along with an analysis of
the relationship between parameters of quadratic equations and properties of their graphs.
http://ohiorc.org/for/math/
Start by inspecting equations such as x2 = 9 that has two solutions, 3 and -3. Next, progress to
equations such as (x - 7)2 = 9 by substituting x – 7 for x and solving them either by
―inspection‖ or by taking the square root on each side:
x–7=3
x = 10
and x – 7 = -3
x=4
Graph both pairs of solutions (-3 and 3, 4 and 10) on the number line and notice that 4 and 10
are 7 units to the right of – 3 and 3. So, the substitution of x – 7 for x moved the solutions 7
units to the right. Next, graph the function y = (x – 7)2 – 9, pointing out that the x-intercepts
are 4 and 10, and emphasizing that the graph is the translation of 7 units to the right and 9
units down from the position of the graph of the parent function y = x2 that passes through the
origin (0, 0). Generate more equations of the form y = a(x – h)2 + k and compare their graphs
using a graphing technology.
http://illuminations.nctm.org/
Common Misconceptions
Some students may think that rewriting equations into various forms (taking
square roots, completing the square, using quadratic formula and factoring) are
isolated techniques within a unit of quadratic equations. Teachers should help
students see the value of these skills in the context of solving higher degree
equations and examining different families of functions.
Highlight and compare different approaches to solving the same problem. Use technology to
recognize that two different expressions or equations may represent the same relationship. For
example, since x2 -10x +25 = 0 can be rewritten as (x- 5)(x –5) = 0 or (x – 5)2 = 0 or x2 = 25,
these are all representations of the same equation that has a double solution x = 5. Support it
by putting all expressions into graphing calculator. Compare their graphs and generate their
tables displaying the same output values for each expression.
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Guide students in transforming a quadratic equation in standard form, 0 = ax2 + bx + c, to the
vertex form 0 = a(x – h)2 + k by separating your examples into groups with a = 1 and a ≠ 1
and have students guess the number that needs to be added to the binomials of the type x2 +
6x, x2 – 2x, x2 +9x, x2 - x to form complete square of the binomial (x – m)2. Then generalize
the process by showing the expression (b/2)2 that has to be added to the binomial x2 + bx.
Completing the square for an expression whose x2 coefficient is not 1 can be complicated for
some students. Present multiple examples of the type 0 = 2x2 – 5x - 9 to emphasize the logic
behind every step, keeping in mind that the same process will be used to complete the square
in the proof of the quadratic formula.
Discourage students from giving a preference to a particular method of solving quadratic
equations. Students need experience in analyzing a given problem to choose an appropriate
solution method before their computations become burdensome. Point out that the Quadratic
Formula, x =
is a universal tool that can solve any quadratic equation; however,
it is not reasonable to use the Quadratic Formula when the quadratic equation is missing
either a middle term, bx, or a constant term, c. When it is missing a constant term, (e.g., 3x2 –
10x = 0) a factoring method becomes more efficient. If a middle term is missing (e.g., 2x2 –
15 = 0), a square root method is the most appropriate. Stress the benefit of memorizing the
Quadratic Formula and flexibility with a factoring strategy. Introduce the concept of
discriminants and their relationship to the number and nature of the roots of quadratic
equation.
Offer students examples of a quadratic equation, such as x2 + 9 = 0. Since the graph of the
quadratic function y = x2 + 9 is situated above the x –axis and opens up, the graph does not
have x–intercepts and therefore, the quadratic equation does not have real solutions. At this
stage introduce students to non-real solutions, such as
or
and a new
number type-imaginary unit i that equals
. Using i in place of
is a way to present
the two solutions of a quadratic equation in the complex numbers form a ± bi, if a and b are
real numbers and b ≠ 0. Have students observe that if a quadratic equation has complex
solutions, the solutions always appear in conjugate pairs, in the form a + bi and a – bi.
Particularly, for the equation x2 = - 9, a conjugate pair of solutions are 0 +3i and
0 – 3i. Project the same logic in the examples of any quadratic equations 0 = ax2 + bx + c that
have negative discriminants. The solutions are a pair of conjugate complex numbers
D = b2 – 4ac is negative.
, if
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High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Solve systems of equations (cont.)
7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of
Content
intersection between the line y = -3x and the circle x2 + y2 = 3.
Standards
Instructional Strategies
Instructional Resources/Tools
Prior to solving systems of equations graphically, students should revisit ―families of
Graph paper
functions‖ to review techniques for graphing different classes of functions. Alert students to
the fact that if one equation in the system can be obtained by multiplying both sides of
Graphing calculators
another equation by a nonzero constant, the system is called consistent, the equations in the
system are called dependent and the system has an infinite number of solutions that produces
coinciding graphs. Provide students opportunities to practice linear vs. non-linear systems;
Common Misconceptions
consistent vs. inconsistent systems; pure computational vs. real-world contextual problems
Most mistakes that students make are careless rather than conceptual. Teachers
(e.g., chemistry and physics applications encountered in science classes).
should encourage students to learn a certain format for solving systems of
equations and check the answers by substituting into all equations in the system.
High School Conceptual Category: Number and Quantity
Domain
The Real Number System
Cluster
Use properties of rational and irrational numbers
3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the
Content
product of a nonzero rational number and an irrational number is irrational.
Standards
Instructional Strategies
Instructional Resources/Tools
This cluster is an excellent opportunity to incorporate algebraic proof, both direct and
The Ohio Resource Center
indirect, in teaching properties of number systems.
The National Council of Teachers of Mathematics, Illuminations
Students should explore concrete examples that illustrate that for any two rational numbers
written in form a/b and c/d, where b and d are natural numbers and a and c are integers, the
NCTM Principles and Standards for School Mathematics
following are true:
represents a rational number, and
represents a rational number.
Common Misconceptions
Some students may believe that both terminating and repeating decimals are
rational numbers, without considering nonrepeating and nonterminating decimals
as irrational numbers.
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Continue exploring situations where the sum of a rational number and an irrational number is
irrational (e.g., a sum of rational number 2 and irrational number √3 is (2 + √3), which is an
irrational).
Proofs are valid ways to justify not only geometry statements also algebraic statements.
Use indirect algebraic proof to generalize the statement that the sum of a rational and
irrational number is irrational.
Assume that x is an irrational number and the sum of x and a rational number is also
rational and is represented as
Students may also confuse irrational numbers and complex numbers, and
therefore mix their properties. In this case, students should encounter examples
that support or contradict properties and relationships between number sets (i.e.,
irrational numbers are real numbers and complex numbers are non-real numbers.
The set of real numbers is a subset of the set of complex numbers).
By using false extensions of properties of rational numbers, some students may
assume that the sum of any two irrational numbers is also irrational. This
statement is not always true (e.g.,
, a rational
number), and therefore, cannot be considered as a property.
x+
x=
x=
represents a rational number
Since the last statement contradicts a given fact that x is an irrational number, the assumption
is wrong and a sum of a rational number and an irrational number has to be irrational.
Similarly, it can be proven that the product of a non- zero rational and an irrational number is
irrational.
Students need to see that results of the operations performed between numbers from a
particular number set does not always belong to the same set. For example, the sum of two
irrational numbers
and
is 4, which is a rational number.
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
Content
(a) Determine an explicit expression, a recursive process, or steps for calculation from a context.
Standards
(b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a
constant function to a decaying exponential and relate these functions to the model.
Instructional Strategies
Instructional Resources/Tools
Provide a real-world example (e.g., a table showing how far a car has driven after a given
Hands-on materials (e.g., paper folding, building progressively larger shapes
number of minutes, traveling at a uniform speed), and examine the table by looking ―down‖
using pattern blocks, etc.) can be used as a visual source to build numerical tables
the table to describe a recursive relationship, as well as ―across‖ the table to determine an
for examination.
explicit formula to find the distance traveled if the number of minutes is known.
Visuals available to assist students in seeing relationships are featured at the
Write out terms in a table in an expanded form to help students see what is
National Library of Virtual Manipulatives as well as The National Council of
happening. For example, if the y-values are 2, 4, 8, 16, they could be written as
Teachers of Mathematics, Illuminations
2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that students recognize that 2 is being used
multiple times as a factor.
Common Misconceptions
Focus on one representation and its related language – recursive or explicit – at a
Students may believe that the best (or only) way to generalize a table of data is
time so that students are not confusing the formats.
by using a recursive formula. Students naturally tend to look ―down‖ a table to
find the pattern but need to realize that finding the 100 th term requires knowing
Provide examples of when functions can be combined, such as determining a function
the 99th term unless an explicit formula is developed.
describing the monthly cost for owning two vehicles when a function for the cost of each
(given the number of miles driven) is known.
Students may also believe that arithmetic and geometric sequences are the same.
Students need experiences with both types of sequences to be able to recognize
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual
the difference and more readily develop formulas to describe them.
models to generate sequences of numbers that can be explored and described with both
recursive and explicit formulas. Emphasize that there are times when one form to describe the Additionally, advanced students who study composition of functions may
function is preferred over the other.
misunderstand function notation to represent multiplication (e.g., f(g(x)) means to
multiply the f and g function values).
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k
Content
given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
Standards
functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2x3 for x>0 or
f(x) = (x+1)/(x-1) for x ≠1.
Instructional Strategies
Instructional Resources/Tools
Use graphing calculators or computers to explore the effects of a constant in the graph of a
Graphing calculator that can be used to explore the effects of parameter changes
function. For example, students should be able to distinguish between the graphs of y = x2, y
on a graph
= 2x2, y = x2 + 2, y = (2x)2, and y = (x + 2)2. This can be accomplished by allowing students
to work with a single parent function and examine numerous parameter changes to make
generalizations.
Common Misconceptions
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3 shifted 4
Distinguish between even and odd functions by providing several examples and helping
units to the left (due to the subtraction symbol). Examples should be explored by
students to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual
hand and on a graphing calculator to overcome this misconception.
approaches to identifying the graphs of even and odd functions can be used as well.
Students may also believe that even and odd functions refer to the exponent of
Provide examples of inverses that are not purely mathematical to introduce the idea. For
the variable, rather than the sketch of the graph and the behavior of the function.
example, given a function that names the capital of a state, f(Ohio) = Columbus. The inverse
would be to input the capital city and have the state be the output, such that f--1(Denver) =
Additionally, students may believe that all functions have inverses and need to
Colorado.
see counter examples, as well as examples in which a non-invertible function can
be made into an invertible function by restricting the domain. For example,
2
-1
f(x) = x has an inverse ( f (x) = x ) provided that the domain is restricted to x
≥ 0.
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FUNDAMENTALS OF ALGEBRA AND GEOMETRY
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MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use order of operations,
including use of parenthesis and
exponents to solve multi-step
problems, and verify and
interpret the results.
INDICATOR
1. Use order of operations and
properties to simplify numerical
expressions involving integers,
fractions and decimals.
REF.
NNSO
E
7.4
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will utilize ―Please Excuse My Dear Aunt Sally‖ to remember the order of operations.
Students will write properties on index cards and play around the world.
Technology/Resources: Index cards, calculators
21st Century Skills: Critical Thinking, Collaboration
Apply and explain the use of
prime factorizations, common
factors, and common multiples in
problem situations.
2. Represent and solve problem
NNSO
situations that can be modeled by G
and solved using concepts of
7.9
absolute value, exponents and
square roots (for perfect squares).
Lesson Ideas:
Students will solve the following determining which would be greater: –getting paid $5000 in
a month or getting a penny the first day and then doubling it each day for a month.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Use and analyze the steps in
standard and non-standard
algorithms for computing with
fractions, decimals and integers.
3. Explain the meaning and effect NNSO
of adding, subtracting,
H
multiplying and dividing integers; 7.5
e.g., how adding two integers can
result in a lesser value.
Lesson Ideas:
Students will learn the concept of integers by utilizing it in money, sports or other situations.
Students will use the calculator to check their answers.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
"
4. Develop and analyze
algorithms for computing with
percents and integers, and
demonstrate fluency in their use.
NNSO
H
7.8
Lesson Ideas:
Students will learn that ―of‖ means multiply and that ―is‖ means equal.
Students will apply this knowledge in real world applications solving problems with percents
and integers.
Technology/Resources: Calculator
Use a variety of strategies,
including proportional reasoning,
to estimate, compute, solve and
explain solutions to problems
involving integers, fractions,
decimals and percents.
Line 1:
Line 2:
Line 3:
5. Simplify numerical expressions NNSO
involving integers and use
I
integers to solve real-life
7.6
problems.
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will use integers to determine the stats from a sporting event.
Technology/Resources: Calculator, newspaper
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
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FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use a variety of strategies,
including proportional reasoning,
to estimate, compute, solve and
explain solutions to problems
involving integers, fractions,
decimals and percents. (cont.)
"
Use scientific notation to express
large numbers and numbers less
than one.
INDICATOR
6. Solve problems using the
appropriate form of a rational
number (fraction, decimal or
percent).
REF.
NNSO
I
7.7
Lesson Ideas:
Students will apply the appropriate form of rational numbers to different problems involving
sales tax, sale prices and tips in order to find out the correct amount to pay or tip.
Technology/Resources: Calculator
7. Represent and solve problem
NNSO
situations that can be modeled by I
and solved using concepts of
7.9
absolute value, exponents and
square roots (for perfect squares).
8. Use scientific notation to
express large numbers and small
numbers between 0 and 1.
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
NNSO
A
8.1
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will apply equation solving skills to problems that involve absolute value, exponents
and square roots.
Technology/Resources: Calculators
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students understand how scientific notation is demonstrated on the calculator and then be able
to write it out correctly.
Students will also utilize their knowledge from science class and adapt it to problems in math
class.
Technology/Resources: Calculator
Compare, order and determine
equivalent forms of real
numbers.
9. Compare, order and determine
equivalent forms for rational and
irrational numbers.
NNSO
E
9.2
21st Century Skills: Critical Thinking, Adaptability
Lesson Ideas:
Students will be able to line up the decimals and compare each digit in order to determine
which number is larger.
Technology/Resources: Calculator
Estimate, compute and solve
problems involving real numbers,
including ratio, proportion and
percent, and explain solutions.
10. Estimate, compute and solve NNSO
problems involving rational
G
numbers, including ratio,
8.6
proportion and percent, and judge
the reasonableness of solutions.
21st Century Skills: Critical Thinking
Lesson Ideas:
Students will solve problems with proportions when given real world examples.
Students will find items in the newspaper, on the internet or on the news where a proportion
might have been applied. They will be discuss and test the accuracy of application.
Technology/Resources: Internet, newspapers
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
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FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Find the square root of perfect
squares, and approximate the
square root of non-perfect
squares.
REF.
INDICATOR
11. Find the square root of perfect NNSO
squares, and approximate the
H
square root of non-perfect squares 8.7
as consecutive integers between
which the root lies; e.g., the
square root of 130 is between 11
and 12.
Select appropriate units to
measure angles, circumference,
surface area, mass and volume,
using; 1) U.S. customary units;
e.g., degrees, square feet, pounds,
and other units as appropriate; 2)
Metric units; e.g., square meters,
kilograms and other units as
appropriate.
12. Select appropriate units for
M
measuring derived measurements; A
e.g., miles per hour, revolutions
7.1
per minute.
Use problem solving techniques
and technology as needed to
solve problems involving length,
weight, perimeter, area, volume,
time and temperature.
13. Solve problems involving
proportional relationships and
scale factors; e.g., scale models
that require unit conversions
within the same measurement
system.
M
E
7.4
Use formulas to find surface area
and volume for specified threedimensional objects accurate to a
specified level of precision.
14. Use appropriate levels of
precision when calculating with
measurements.
M
B
8.3
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will estimate answers to square roots and then have them check their answers on the
calculator.
Students will utilize this knowledge by playing a game to see who can get closest to the
answer.
Technology/Resources: Calculator
21st Century Skills: Critical Skills, Collaboration
Lesson Ideas:
Students will solve real-life problems that involve measurement. They will have to determine
method and appropriate unit to solve the problem.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will use proportions to find the approximate distances on a map.
Technology/Resources: Calculator, map, Smart Board / Mimio
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
Lesson Ideas:
Students will discuss, understand and apply appropriate rounding techniques. They must
demonstrate an understanding of the appropriate number of digits to include in a specific
answer.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
272
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FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use formulas to find surface area
and volume for specified threedimensional objects accurate to a
specified level of precision.
(cont.)
Apply indirect measurement
techniques, tools and formulas,
as appropriate, to find perimeter,
circumference and area of
circles, triangles, quadrilaterals
and composite shapes and to find
volume of prisms, cylinders, and
pyramids.
INDICATOR
15. Derive formulas for surface
area and volume and justify them,
using geometric models and
common materials. For example,
find:
the surface area of a cylinder as
a function of its height and
radius; and
that the volume of a pyramid
(or cone) is one-third of the
volume of a prism (or cylinder)
with the same base area and
height.
16. Demonstrate understanding of
the concepts of perimeter,
circumference and area by using
established formula for triangles,
quadrilaterals, and circles to
determine the surface area and
volume of prisms, pyramids,
cylinders, spheres and cones.
(Note: Only volume should be
calculated for spheres and cones.)
REF.
M
B
8.4
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will create models of cylinders, cones, etc. and take them apart based on smaller
shapes such as circles and rectangles. The students will then be able to figure out the formulas
based on adding up the area for smaller figures. Students will determine the areas by
measuring their figure.
Technology/Resources: Scissors, paper, glue, geometry templates
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
M
C
8.9
Lesson Ideas:
Students will create models of cylinders, cones, etc. and take them apart based on smaller
shapes such as circles and rectangles. The students will then be able to figure out the formulas
based on adding up the area for smaller figures. Students will determine the areas by
measuring their figure.
Students will be given real-life items to find surface areas and volumes. They will determine
the costs of items. For example, they could figure out the cost of carpeting a room or mulching
a certain area with a certain depth of mulch.
Technology/Resources: Calculators, rulers, yardsticks, items around the school
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
Use proportional reasoning and
apply indirect measurement
techniques, including right
triangle trigonometry and
properties of similar triangles, to
solve problems involving
measurements and rates.
Line 1:
Line 2:
Line 3:
17. Apply proportional reasoning
to solve problems involving
indirect measurements or rates.
M
D
8.7
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
Lesson Ideas:
Students will utilize knowledge of proportions to find lengths of sides in similar triangles.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
273
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FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use proportional reasoning and
apply indirect measurement
techniques, including right
triangle trigonometry and
properties of similar triangles, to
solve problems involving
measurements and rates. (cont.)
"
Estimate and compute various
attributes, including length, angle
measure, area, surface area and
volume, to a specified level of
precision.
Use proportions to express
relationships among
corresponding parts of similar
figures.
"
INDICATOR
18. Use scale drawings and right
triangle trigonometry to solve
problems that include unknown
distances and angle measures.
REF.
M
D
9.4
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will apply ―SOHCAHTOA‖ to right triangle geometry. Sin = opp./hyp Cos =
adj./hyp. Tan = opp./adj.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
19. Solve problems involving unit M
conversion for situations
D
involving distances, areas,
9.5
volumes and rates within the
same measurement system.
20. Use conventional formulas to M
find the surface area and volume E
of prisms, pyramids and cylinders 8.10
and the volume of spheres and
cones to a specified level of
precision.
21. Use proportional reasoning to G
describe and express relationships E
between parts and attributes of
7.1
similar and congruent figures.
22. Determine and use scale
G
factors for similar figures to solve E
problems using proportional
7.6
reasoning
Lesson Ideas:
Students will be given real world examples to solve.
Students will make a table and cancel out units in order to get correct units.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will solve real world problems in which they must research the dimensions of
different prisms, pyramids, etc. to determine surface area and volume.
Technology/Resources: Internet and calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will utilize their knowledge of proportions to express relationships between parts of
similar and congruent figures.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
Lesson Ideas:
Students will utilize their knowledge of proportions to solve real world examples involving
scale factors.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
274
MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Describe and use properties of
triangles to solve problems
involving angle measures and
side lengths of right triangles.
"
Describe and apply the properties
of similar and congruent figures;
and justify conjectures involving
similarity and congruence.
INDICATOR
23. Use and demonstrate
understanding of the properties of
triangles. For example: a). Use
Pythagorean Theorem to solve
problems involving right
triangles; and b.) Use triangle
angle sum relationships to solve
problems.
24. Apply properties of congruent
or similar triangles to solve
problems involving missing
lengths and angle measures.
REF.
G
G
7.3
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will find the distance from home to 2nd on a baseball field using Pythagorean‘s
Theorem.
Technology/Resources: Calculator, internet
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
G
G
7.5
Lesson Ideas:
Students will use their knowledge of proportions to solve for missing lengths on triangles.
Students will find angle measures on different diagrams.
Technology/Resources: Calculator
25. Use proportions in several
G
forms to solve problems
B
involving similar figures (part-to- 8.3
part, part-to-whole, corresponding
sides between figures).
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
Lesson Ideas:
Students will use their knowledge of proportions to solve problems involving similar figures.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Problem Solving
Recognize and apply angle
relationships in situations
involving intersecting lines,
perpendicular lines, and parallel
lines.
26. Recognize the angles formed G
and the relationship between the C
angles when two lines intersect
8.2
and when parallel lines are cut by
a transversal.
Lesson Ideas:
Students will apply their knowledge of different angle types and play around the world using a
large diagram on the board.
Represent and model
transformations in a coordinate
plane and describe the results.
27. Draw the results of
translations, reflections, rotations
and dilations of objects in the
coordinate plane, and determine
properties that remain fixed; e.g.,
lengths of sides remain the same
under translations.
Lesson Ideas:
Students will draw pictures on graph paper to draw a picture. They will exchange with another
student to create a reflection, rotation, etc. when instructed by the teacher. They drawing will
be circulated multiple times.
Line 1:
Line 2:
Line 3:
G
F
8.5
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
21st Century Skills: Critical Thinking, Adaptability
Technology/Resources: Graph paper, Smart Board / Mimio
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
275
MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
INDICATOR
Represent, analyze and
28. Generalize patterns by
generalize a variety of patterns
describing in words how to find
and functions with tables, graphs, the next term.
words and symbolic rules.
REF.
PFA
B
7.2
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will describe different patterns of numbers and pictures. They will find the next term
based on pattern.
Students will make different patterns on index cards and pass them around to let other students
try and solve.
Technology/Resources: Calculator and index cards
Use formulas in problem-solving
situations.
29. Use formulas in problemsolving situations.
PFA
J
7.8
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
Lesson Ideas:
Students will identify different formulas that could be used in a future career. They will
explain how to use them in real world examples.
Technology/Resources: Internet and calculator
Graph linear equations and
inequalities.
30. Represent inequalities on a
PFA
number line or a coordinate plane. K
7.6
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
Lesson Ideas:
Students will learn that the Less than symbol kind of looks like an ―L‖ = ―<‖ and then be able
to graph on a number line or a coordinate plane.
Technology/Resources: Calculator
Generalize and explain patterns
31. Generalize and explain
PFA
and sequences in order to find the patterns and sequences in order to A
next term and the nth term.
find the next term and the nth
8.2
term.
Translate information from one
representation (words, table,
graph or equation) to another
representation of a relation or
function.
Line 1:
Line 2:
Line 3:
32. Relate the various
representations of a relationship;
i.e., relate a table to graph,
description and symbolic form.
PFA
C
8.1
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
21st Century Skills: Critical Thinking
Lesson Ideas:
Students will be able to describe a pattern first in words, then using a formula.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will be able to graph by making tables and by using y=mx+b.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Adaptability
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
276
MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use algebraic representations,
such as tables, graphs,
expressions, functions and
inequalities, to model and solve
problem situations.
"
Interpret data by looking for
patterns and relationships; draw
and justify conclusions, and
answer related questions.
INDICATOR
33. Use symbolic algebra
(equations and inequalities),
graphs and tables to represent
situations and solve problems.
34. Write, simplify and evaluate
algebraic expressions (including
formulas) to generalize situations
and solve problems.
PFA
D
8.8
Lesson Ideas:
Students will write and solve equations and inequalities given real world examples.
Technology/Resources: Calculator
Describe the probability of an
event using ratios, including
fractional notation.
37. Compute probabilities of
compound events; e.g., multiple
coin tosses or multiple rolls of
number cubes, using such
methods as organized lists, tree
diagrams and area models.
Lesson Ideas:
Students will simplify while they solve equations demonstrating an understanding of the Laws
of Exponents.
Technology/Resources: Calculator
35. Construct opposing arguments DAP
based on analysis of the same
B
data, using different graphical
7.4
representations.
36. Identify misuses of statistical
data in articles, advertisements,
and other media.
Line 2:
Line 3:
PFA
D
8.7
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Evaluate conjectures and
predictions based upon data
presented in tables and graphs,
and identify misuses of statistical
data and displays.
Line 1:
REF.
DAP
G
7.6
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will interpret graphs from the internet or from a newspaper.
Students will discuss different aspects of the graph and analyze it.
Technology/Resources: Internet and newspaper
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
Lesson Ideas:
Students will bring in articles, advertisements and other media to discuss statistical data.
Technology/Resources: Internet, newspaper, and calculator
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving, Creativity
DAP
I
7.7
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
Lesson Ideas:
Students will find different probabilities such as: flipping a coin, rolling a die, rolling 2 dice,
spinning a spinner, etc.
Technology/Resources: Coins, dice, spinners
21st Century Skills: Critical Thinking, Adaptability, Collaboration, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
277
MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Create, interpret and use
graphical displays and statistical
measures to describe data; e.g.,
box-and-whisker plots,
histograms, scatter plots,
measures of center and
variability.
INDICATOR
38. Use, create and interpret
scatterplots and other types of
graphs as appropriate.
REF.
DAP
A
8.1
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will take part in the Balloon Bungee Activity where they will determine how many
rubber bands it will take to drop a water balloon off the top of the bleachers without hitting the
ground.
Students will use linear regression to make a conjecture.
Technology/Resources: Balloons, yardsticks, calculators, rubber bands
21st Century Skills: Critical Thinking, Problem Solving, Organizational Skills, Adaptability,
Interpersonal Skills, Collaboration, Leadership, Resourcefulness
Evaluate different graphical
representations of the same data
to determine which is the most
appropriate representation for an
identified purpose
"
Construct convincing arguments
based on analysis of data and
interpretation of graphs.
39. Evaluate different graphical
DAP
representations of the same data
B
to determine which is the most
8.2
appropriate representation for an
identified purpose; e.g., line graph
for change over time, circle graph
for part-to-whole comparison,
scatterplot for relationship
between two variants.
Lesson Ideas:
Students will use data to make a line graph, circle graph and a scatter plot to determine which
graph they feel is the most appropriate.
40. Differentiate between discrete DAP
and continuous data and appropri- B
ate ways to represent each.
8.3
Lesson Ideas:
Students will use data to determine the appropriate graph to use to represent the data.
41. Make conjectures about
possible relationship in a
scatterplot and approximate line
of best fit.
DAP
F
8.6
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving, Organizational Skills, Adaptability,
Interpersonal Skills, Collaboration, Leadership, Resourcefulness
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving, Organizational Skills, Adaptability,
Collaboration
Lesson Ideas:
Students will take part in the Balloon Bungee Activity where they will determine how many
rubber bands it will take to drop a water balloon off the top of the bleachers without hitting the
ground.
Students will use linear regression (line of best fit) to make a conjecture.
Technology/Resources: Balloons, yardsticks, calculators, rubber bands
21st Century Skills: Critical Thinking, Problem Solving, Organizational Skills, Adaptability,
Interpersonal Skills, Collaboration, Leadership, Resourcefulness
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
278
MATH
FUNDAMENTALS OF ALGEBRA AND GEOMETRY
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use counting techniques, such as
permutations and combinations,
to determine the total number of
options and possible outcomes.
INDICATOR
42. Use counting techniques and
the Fundamental Counting
principle to determine the total
number of possible outcomes for
mathematical situations.
REF.
DAP
H
9.7
43. Write, simplify and solve
multi-step algebraic equations.
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will determine how many ways they can line up in a row. Students will figure out a
better way than just counting to determine this answer.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving, Organizational Skills, Adaptability,
Interpersonal Skills, Collaboration, Leadership, Resourcefulness
Lesson Ideas:
Throughout the course of the year students solve multiple problems integrating multi-step
algebraic equations.
Technology/Resources: Calculator
21st Century Skills: Critical Skills and Adaptability
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
279
MATH
GEOMETRY
280
MATH
GEOMETRY
High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a
Content
line, and distance around a circular arc. G.CO.1
Standards
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane
as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal
stretch). G.CO.2
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.3
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.4
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.
Specify a sequence of transformations that will carry a given figure onto another. G.CO.5
Instructional Strategies
Instructional Resources/Tools
Review vocabulary associated with transformations (e.g. point, line, segment, angle, circle,
Tracing paper (patty paper)
polygon, parallelogram, perpendicular, rotation reflection, translation).
Transparencies
Provide both individual and small-group activities, allowing adequate time for students to
explore and verify conjectures about transformations and develop precise definitions of
Graph paper
rotations, reflections and translations.
Ruler
Provide real-world examples of rigid motions (e.g. Ferris wheels for rotation; mirrors for
reflection; moving vehicles for translation).
Protractor
Use graph paper, transparencies, tracing paper or dynamic geometry software to obtain
images of a given figure under specified transformations.
Provide students with a pre-image and a final, transformed image, and ask them to describe
the steps required to generate the final image. Show examples with more than one answer
(e.g., a reflection might result in the same image as a translation).
Work backwards to determine a sequence of transformations that will carry (map) one figure
onto another of the same size and shape.
Focus attention on the attributes (e.g. distances or angle measures) of a geometric figure that
remain constant under various transformations.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Common Misconceptions
The terms ―mapping‖ and ―under‖ are used in special ways when studying
transformations. A translation is a type of transformation that moves all the
points in the object in a straight line in the same direction. Students should know
that not every transformation is a translation.
Students sometimes confuse the terms ―transformation‖ and ―translation.‖
Make the transition from transformations as physical motions to functions that take points in
the plane as inputs and give other points as outputs. The correspondence between the initial
281
MATH
GEOMETRY
and final points determines the transformation.
Analyze various figures (e.g. regular polygons, folk art designs or product logos) to determine
which rotations and reflections carry (map) the figure onto itself. These transformations are
the ―symmetries‖ of the figure.
Emphasize the importance of understanding a transformation as the correspondence between
initial and final points, rather than the physical motion.
High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use
Content
the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.6
Standards
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent. G.CO.7
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.8
Instructional Strategies
Instructional Resources/Tools
Develop the relationship between transformations and congruency. Allow adequate time and
Tracing paper (patty paper)
provide hands-on activities for students to visually and physically explore rigid motions and
Graph paper
congruence.
Ruler
Use graph paper, tracing paper or dynamic geometry software to obtain images of a given
Protractor
figure under specified rigid motions. Note that size and shape are preserved.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri® or
Use rigid motions (translations, reflections and rotations) to determine if two figures are
Geogebra®); websites with similar tools (such as the National Library of Virtual
congruent. Compare a given triangle and its image to verify that corresponding sides and
Manipulatives that features applets for exploring triangle congruence).
corresponding angles are congruent.
Graphing calculators and other handheld technology such as TI-Nspire™.
Work backwards – given two figures that have the same size and shape, find a sequence of
rigid motions that will map one onto the other.
Common Misconceptions
Build on previous learning of transformations and congruency to develop a formal criterion
Some students may believe That combinations such as SSA or AAA are also a
for proving the congruency of triangles. Construct pairs of triangles that satisfy the ASA,
congruence criterion for triangles. Provide counterexamples for this
SAS or SSS congruence criteria, and use rigid motions to verify that they satisfy the
misconception.
definition of congruent figures. Investigate rigid motions and congruence both algebraically
(using coordinates) and logically (using proofs).
Student may also believe that all transformations, including dilation, are rigid
motions. Provide counterexamples of this misconception.
282
MATH
GEOMETRY
High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles
Content
are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s
Standards
endpoints. G.CO.9
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.10
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with congruent diagonals. G.CO.11
Instructional Strategies
Instructional Resources/Tools
Classroom teachers and mathematics education researchers agree that students have a hard
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®).
time learning how to do geometric proofs. An article by Battista and Clements (1995)
(http://investigations.terc.edu/library/bookpapers/geometryand_proof.cfm) provides
Principles and Standards for School Mathematics, pp.309-318, 342-346.
information for teachers to help students who struggle learn to do proof. The most significant
implication for instructional strategies for proof is stated in their conclusion.
Niven, Ivan, ―Can Geometry Survive in the Secondary Curriculum?‖ Learning
and Teaching Geometry, K-12. 1987 Yearbook of the National Council of
―Ironically, the most effective path to engendering meaningful use of proof in secondary
Teachers of Mathematics
school geometry is to avoid formal proof for much of students‘ work. By focusing instead on
justifying ideas while helping students build the visual and empirical foundation for higher
Pythagorean Puzzle levels of geometric thought, we can lead students to appreciate the need for formal proof.
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/pyt
Only then will we be able to use it meaningfully as a mechanism for justifying ideas.‖
hagorean_puz - Though focused on the Pythagorean theorem, this site provides the
kind of hands-on experience that should be a precursor to formal proof. In this
The article and ideas from Niven (1987) offers a few suggestions about teaching proof in
self-guided investigation, students use Geometer‘s Sketchpad® to construct a right
geometry:
triangle and discover a geometric proof of the Pythagorean theorem. Students then
• Initial geometric understandings and ideas should be taught ―without excessive emphasis on test the geometric proof with acute and obtuse triangles.
rigor.‖ Develop basic geometric ideas outside an axiomatic framework, and then let the
importance of the framework (and the framework itself) emerges from the geometry.
• Geometry is visual and should be taught in ways that leverage this aspect. Sketching,
Common Misconceptions
drawing and constructing figures and relationships between geometric objects should be
Research over the last four decades suggests that student misconceptions about
central to any geometric study and certainly to proof. Battista and Clement make a
proof abound:
powerful argument that the use of dynamic geometry software can be an important tool for
• even after proving a generalization, students believe that exceptions to the
helping students understand proof.
generalization might exist;
• ―[A]void the deadly elaboration of the obvious‖ (Niven, p. 43). Often textbooks begin the
• one counterexample is not sufficient;
treatment of formal proof with ―easy‖ proofs, which appear to students to need no proof at
• the converse of a statement is true (parallel lines do not intersect, lines that do
all. After presenting many opportunities for students to ―justify‖ properties of geometric
not intersect are parallel); and
figures, formal proof activities should begin with non-obvious conjectures.
• a conjecture is true because it worked in all examples that were explored.
• Use the history of geometry and real-world applications to help students develop conceptual
understandings before they begin to use formal proof.
283
MATH
GEOMETRY
Proofs in high school geometry should not be restricted to the two-column format. Most
proofs at the college level are done in paragraph form, with the writer explaining and
defending a conjecture. In many cases, the two-column format can hinder the student from
making sense of the geometry by paying too much attention to format rather than
mathematical reasoning.
Each of these misconceptions needs to be addressed, both by the ways in which
formal proof is taught in geometry and how ideas about ―justification‖ are
developed throughout a student‘s mathematical education.
Some of the theorems listed in this cluster (e.g. the ones about alternate interior angles and
the angle sum of a triangle) are logically equivalent to the Euclidean parallel postulate, and
this should be acknowledged.
Use dynamic geometry software to allow students to make conjectures that can, in turn, be
formally proven. For example, students might notice that the base angles of an isosceles
triangle always appear to be congruent when manipulating triangles on the computer screen
and could then engage in a more formal discussion of why this occurs.
Common Core Standards Appendix A states, ―Encourage multiple ways of writing proofs,
such as in narrative paragraphs, using flow diagrams, in two-column format, and using
diagrams without words.
Students should be encouraged to focus on the validity of the underlying reasoning while
exploring a variety of formats for expressing that reasoning‖ (p. 29). Different methods of
proof will appeal to different learning styles in the classroom.
284
MATH
GEOMETRY
High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
Content
geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the
Standards
perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.12
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.13
Instructional Strategies
Instructional Resources/Tools
Students should analyze each listed construction in terms of what simpler constructions are
Compass
involved (e.g., constructing parallel lines can be done with two different constructions of
Straightedge
perpendicular lines).
String
Using congruence theorems, ask students to prove that the constructions are correct.
Origami paper
Provide meaningful problems (e.g. constructing the centroid or the incenter of a triangle) to
offer students practice in executing basic constructions.
Challenge students to perform the same construction using a compass and string. Use paper
folding to produce a reflection; use bisections to produce reflections.
Ask students to write ―how-to‖ manuals, giving verbal instructions for a particular
construction.
Offer opportunities for hands-on practice using various construction tools and methods.
Compare dynamic geometry commands to sequences of compass-and-straightedge steps.
Prove, using congruence theorems, that the constructions are correct.
Reflection tool (e.g. Mira®).
Dynamic geometry software (e.g. Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/pyt
hagorean_puzzle.pdf - In this self-guided investigation, students use Geometer‘s
Sketchpad to construct a right triangle and discover a geometric proof of the
Pythagorean Theorem. Students test the geometric proof with acute and obtuse
triangles.
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/con
current_events.pdf - This lesson enhances student knowledge of how to use
Geometer's Sketchpad to explore geometric concepts (e.g. the points of
concurrency in a triangle). It includes the construction of line segments, triangles,
circles, perpendicular bisectors of line segments, angle bisectors, altitudes of
triangles, and medians of triangles.
http://mathforum.org/alejandre/circles.html -Students will construct a number of
compass-and-straightedge designs using ideas from this site.
Common Misconceptions
Some students may believe that a construction is the same as a sketch or drawing.
Emphasize the need for precision and accuracy when doing constructions. Stress
the idea that a compass and straightedge are identical to a protractor and ruler.
Explain the difference between measurement and construction.
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
Content
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
Standards
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.1
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations
the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.2
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.3
Instructional Strategies
Instructional Resources/Tools
Allow adequate time and hands-on activities for students to explore dilations visually and
Dot paper
physically.
Graph paper
Use graph paper and rulers or dynamic geometry software to obtain images of a given figure
under dilations having specified centers and scale factors. Carefully observe the images of
Rulers
lines passing through the center of dilation and those not passing through the center,
respectively. A line segment passing through the center of dilation will simply be shortened
Protractors
or elongated but will lie on the same line, while the dilation of a line segment that does not
pass through the center will be parallel to the original segment (this is intended as a
Pantograph
clarification of Standard 1a).
Photocopy machine
Illustrate two-dimensional dilations using scale drawings and photocopies.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Measure the corresponding angles and sides of the original figure and its image to verify that
Geogebra®).
the corresponding angles are congruent and the corresponding sides are proportional (i.e.
stretched or shrunk by the same scale factor). Investigate the SAS and SSS criteria for similar Web-based applets that demonstrate dilations, such as those at the National
triangles.
Library of Virtual Manipulatives.
Use graph paper and rulers or dynamic geometry software to obtain the image of a given
figure under a combination of a dilation followed by a sequence of rigid motions (or rigid
motions followed by dilation).
Work backwards – given two similar figures that are related by dilation, determine the center
of dilation and scale factor. Given two similar figures that are related by a dilation followed
by a sequence of rigid motions, determine the parameters of the dilation and rigid motions
that will map one onto the other.
Video: Similarity by Project Mathematics! (www.projectmathematics.com)
Common Misconceptions
Some students often do not recognize that congruence is a special case of
similarity. Similarity with a scale factor equal to 1 becomes a congruency.
Students may not realize that similarities preserve shape, but not size. Angle
measures stay the same, but side lengths change by a constant scale factor.
Using the theorem that the angle sum of a triangle is 180°, verify that the AA criterion is
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equivalent to the AAA criterion.
Given two triangles for which AA holds, use rigid motions to map a vertex of one triangle
onto the corresponding vertex of the other in such a way that their corresponding sides are in
line. Then show that dilation will complete the mapping of one triangle onto the other.
Some students often do not list the vertices of similar triangles in order.
However, the order in which vertices are listed is preferred and especially
important for similar triangles so that proportional sides can be correctly
identified.
High School Conceptual Category: Geometry
Domain
Similarity, Right Triangle, and Trigonometry
Cluster
Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the
Content
Pythagorean Theorem proved using triangle similarity. G.SRT.4
Standards
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.5
Instructional Strategies
Instructional Resources/Tools
Review triangle congruence criteria and similarity criteria, if it has already been established.
Cardboard models of right triangles.
Review the angle sum theorem for triangles, the alternate interior angle theorem and its
converse, and properties of parallelograms. Visualize it using dynamic geometry software.
Using SAS and the alternate interior angle theorem, prove that a line segment joining
midpoints of two sides of a triangle is parallel to and half the length of the third side. Apply
this theorem to a line segment that cuts two sides of a triangle proportionally.
Generalize this theorem to prove that the figure formed by joining consecutive midpoints of
sides of an arbitrary quadrilateral is a parallelogram. (This result is known as the Midpoint
Quadrilateral Theorem or Varignon‘s Theorem.)
Use cardboard cutouts to illustrate that the altitude to the hypotenuse divides a right triangle
into two triangles that are similar to the original triangle. Then use AA to prove this theorem.
Then, use this result to establish the Pythagorean relationship among the sides of a right
triangle (a2 + b2 = c2) and thus obtain an algebraic proof of the Pythagorean Theorem.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®).
Video: The Theorem of Pythagoras from Project MATHEMATICS!
Websites for the Pythagorean Theorem
Jim Loy‘s Pythagorean Theorem:
Animated proofs of the Pytharorean Theorem: Pythagorean Theorem and its
Many Proofs:
Common Misconceptions
Some students may confuse the alternate interior angle theorem and its converse
as well as the Pythagorean theorem and its converse.
Prove that the altitude to the hypotenuse of a right triangle is the geometric mean of the two
segments into which its foot divides the hypotenuse.
Prove the converse of the Pythagorean Ttheorem, using the theorem itself as one step in the
proof. Some students might engage in an exploration of Pythagorean Triples (e.g., 3-4-5, 512-13, etc.), which provides an algebraic extension and an opportunity to explore patterns.
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute
Content
angles. G.SRT.6
Standards
7. Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.7
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.8
Instructional Strategies
Instructional Resources/Tools
Review vocabulary (opposite and adjacent sides, legs, hypotenuse and complementary
Cutouts of right triangles
angles) associated with right triangles.
Rulers
Make cutouts or drawings of right triangles or manipulate them on a computer screen using
dynamic geometry software and ask students to measure side lengths and compute side ratios. Protractors
Observe that when triangles satisfy the AA criterion, corresponding side ratios are equal. Side
ratios are given standard names, such as sine, cosine and tangent. Allow adequate time for
Scientific calculators
students to discover trigonometric relationships and progress from concrete to abstract
understanding of the trigonometric ratios.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®)
Show students how to use the trigonometric function keys on a calculator. Also, show how to
find the measure of an acute angle if the value of its trigonometric function is known.
Trig Trainer® instructional aids
Investigate sines and cosines of complementary angles, and guide students to discover that
they are equal to one another. Point out to students that the ―co‖ in cosine refers to the ―sine
of the complement.‖
Clinometers (can be made by the students)
Observe that, as the size of the acute angle increases, sines and tangents increase while
cosines decrease.
Trigonometric Functions
Websites for the history of mathematics:
Trigonometric Course
Stress trigonometric terminology by the history of the word ―sine‖ and the connection
between the term ―tangent‖ in trigonometry and tangents to circles.
Have students make their own diagrams showing a right triangle with labels showing the
trigonometric ratios. Although students like mnemonics such as SOHCAHTOA, these are not
a substitute for conceptual understanding. Some students may investigate the reciprocals of
sine, cosine, and tangent to discover the other three trigonometric functions.
Use the Pythagorean theorem to obtain exact trigonometric ratios for 30°, 45°, and 60°
angles.
Video: Sines and Cosines, Part II from Project MATHEMATICS!
Common Misconceptions
Some students believe that right triangles must be oriented a particular way.
Some students do not realize that opposite and adjacent sides need to be
identified with reference to a particular acute angle in a right triangle.
Some students believe that the trigonometric ratios defined in this cluster apply to
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Use cooperative learning in small groups for discovery activities and outdoor measurement
projects.
all triangles, but they are only defined for acute angles in right triangles.
Have students work on applied problems and project, such as measuring the height of the
school building or a flagpole, using clinometers and the trigonometric functions.
High School Conceptual Category: Geometry
Domain
Circles
Cluster
Understand and apply theorems about circles
1. Prove that all circles are similar. G.C.1
Content
2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed
Standards
angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. G.C.2
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.3
Instructional Strategies
Instructional Resources/Tools
Given any two circles in a plane, show that they are related by dilation. Guide students to
Ruler
discover the center and scale factor of this dilation and make a conjecture about all dilations
of circles.
Compass
Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle
sum of a triangle is 180° to show that this angle is a right angle. Using dynamic geometry,
students can grab a point on a circle and move it to see that the measure of the inscribed angle
passing through the endpoints of a diameter is always 90°. Then extend the result to any
inscribed angles. For inscribed angles, proofs can be based on the fact that the measure of an
exterior angle of a triangle equals the sum of the measures of the nonadjacent angles.
Consider cases of acute or abtuse inscribed angles.
Use properties of congruent triangles and perpendicular lines to prove theorems about
diameters, radii, chords, and tangent lines.
Use formal geometric constructions to construct perpendicular bisectors of the sides and
angle bisectors of a given triangle. Their intersections are the centers of the circumscribed and
inscribed circles, respectively.
Protractor
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Common Misconceptions
Students sometimes confuse inscribed angles and central angles.
Students may think they can tell by inspection whether a line intersects a circle in
exactly one point. It may be beneficial to formally define a tangent line as the
line perpendicular to a radius at the point where the radius intersects the circle.
Dissect an inscribed quadrilateral into triangles, and use theorems about triangles to prove
properties of these quadrilaterals and their angles.
Constructing tangents to a circle from a point outside the circle is a useful application of the
result that an angle inscribed in a semicircle is a right angle.
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High School Conceptual Category: Geometry
Domain
Circles
Cluster
Find arc lengths and areas of sectors of circles
5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as
Content
the constant of proportionality; derive the formula for the area of a sector. G.C.5
Standards
Instructional Strategies
Instructional Resources/Tools
Begin by calculating lengths of arcs that are simple fractional parts of a circle and do this for
Ruler
circles of various radii so that students discover a proportionality relationship.
Compass
Provide plenty of practice in assigning radian measures to angles that are simple fractional parts
of a straight angle.
Protractor
Stress the definition of radian by considering a central angle whose intercepted arc has its length
equal to the radius, making the constant of proportionality 1. Students who are having difficulty
understanding radians may benefit from constructing cardboard sectors whose angles are one
radian. Use a ruler and string to approximate such an angle.
String
Compute areas of sectors by first considering them as fractional parts of a circle. Then, using
proportionality, derive a formula for their area in terms of radius and central angle. Do this for
angles that are measured in both degrees and radians and note that the formula is much simpler
when the angles are measured in radians.
Video: The Story of Pi from Project MATHEMATICES!
Derive formulas that relate degrees and radians.
Introduce arc measures that are equal to the measures of the intercepted central angles in
degrees and radians.
Computer dynamic geometry software (Geometer‘s Sketchpad)
Common Misconceptions
Sectors and segments are often used interchangeably in everyday conversation.
Care should be taken to distinguish these two geometric concepts.
The formulas for converting radians to degrees and vice versa are easily
confused. Knowing that the degree measure of a given angle is always a
number larger than the radian measure can help students use the correct unit.
Emphasize appropriate use of terms, such as, angle, arc, radian, degree, and sector.
High School Conceptual Category: Geometry
Domain
Expressing Geometric Properties with Equations
Cluster
Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given
Content
by an equation. G.GPE.1
Standards
2. Derive the equation of a parabola given a focus and directrix. G.GPE.2
Instructional Strategies
Instructional Resources/Tools
Review the definition of a circle as a set of points whose distance from a fixed point is constant. Physical models of cones sliced to show cross sections that are circles, ellipses
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parabolas and hyperbolas
Review the algebraic method of completing the square and demonstrate it geometrically.
Illustrate conic sections geometrically as cross sections of a cone.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®)
Use the Pythagorean theorem to derive the distance formula. Then, use the distance formula to
derive the equation of a circle with a given center and radius, beginning with the case where the
center is the origin.
Parabolic reflectors to illustrate practical applications of parabolas
Starting with any quadratic equation in two variables (x and y) in which the coefficients of the
quadratic terms are equal, complete the squares in both x and y and obtain the equation of a
circle in standard form.
Common Misconceptions
Because new vocabulary is being introduced in this cluster, remembering the
names of the conic sections can be problematic for some students.
Given two points, find the equation of the circle passing through one of the points and having
the other as its center.
The Euclidean distance formula involves squared, subscripted variables whose
differences are added. The notation and multiplicity of steps can be a serious
stumbling block for some students.
Define a parabola as a set of points satisfying the condition that their distance from a fixed point
(focus) equals their distance from a fixed line (directrix). Start with a horizontal directrix and a
focus on the y-axis, and use the distance formula to obtain an equation of the resulting parabola
in terms of y and x2. Next use a vertical directrix and a focus on the x-axis to obtain an equation
of a parabola in terms of x and y2. Make generalizations in which the focus may be any point,
but the directrix is still either horizontal or vertical. Allow sufficient time for students to become
familiar with new vocabulary and notation.
The method of completing the square is a multi-step process that takes time to
assimilate. A geometric demonstration of completing the square can be helpful
in promoting conceptual understanding
Given y as a quadratic equation of x (or x as a quadratic function of y), complete the square to
obtain an equation of a parabola in standard form.
Identify the vertex of a parabola when its equation is in standard form and show that the vertex
is halfway between the focus and directrix.
Investigate practical applications of parabolas and paraboloids.
Information below includes additional mathematics that students should learn in order to take
advanced courses such as calculus, advanced statistics, or discrete mathematics, and goes
beyond the mathematics that all students should study in order to be college- and career-ready:
Students preparing for advanced courses may use the distance formula and relevant focusdirectrix definitions to derive equations of ellipses and hyperbolas whose major axes are either
horizontal or vertical. Encourage students to explore conic sections using dynamic geometry
software.
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High School Conceptual Category: Geometry
Domain
Expressing Geometric Properties with Equations
Cluster
Use coordinates to prove simple geometric theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the
Content
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.4
Standards
5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point). G.GPE.5
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.6
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.GPE.7
Instructional Strategies
Instructional Resources/Tools
Review the concept of slope as the rate of change of the y-coordinate with respect to the xGraph paper
coordinate for a point moving along a line, and derive the slope formula.
Scientific or graphing calculators
Use similar triangles to show that every nonvertical line has a constant slope.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®)
Review the point-slope, slope-intercept and standard forms for equations of lines.
Investigate pairs of lines that are known to be parallel or perpendicular to each other and
discover that their slopes are either equal or have a product of –1, respectively.
Pay special attention to the slope of a line and its applications in analyzing properties of lines.
Allow adequate time for students to become familiar with slopes and equations of lines and
methods of computing them.
Common Misconceptions
Students may claim that a vertical line has infinite slopes. This suggests that
infinity is a number. Since applying the slope formula to a vertical line leads to
division by zero, we say that the slope of a vertical line is undefined. Also, the
slope of a horizontal line is 0. Students often say that the slope of vertical and/or
horizontal lines is ―no slope,‖ which is incorrect.
Use slopes and the Euclidean distance formula to solve problems about figures in the
coordinate plane such as:
• Given three points, are they vertices of an isosceles, equilateral, or right triangle?
• Given four points, are they vertices of a parallelogram, a rectangle, a rhombus, or a square?
• Given the equation of a circle and a point, does the point lie outside, inside, or on the circle?
• Given the equation of a circle and a point on it, find an equation of the line tangent to the
circle at that point.
• Given a line and a point not on it, find an equation of the line through the point that is
parallel to the given line.
• Given a line and a point not on it, find an equation of the line through the point that is
perpendicular to the given line.
• Given the equations of two non-parallel lines, find their point of intersection.
• Given two points, use the distance formula to find the coordinates of the point halfway
between them. Generalize this for two arbitrary points to derive the midpoint formula.
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Use linear interpolation to generalize the midpoint formula and find the point that partitions a
line segment in any specified ratio.
Use the distance formula to find the length of each side of a polygon whose vertices are
known, and compute the perimeter of that figure.
Given the vertices of a triangle or a parallelogram, find the equation of a line containing the
altitude to a specified base and the point of intersection of the altitude and the base. Use the
distance formula to find the length of that altitude and base, and then compute the area of the
figure.
High School Conceptual Category: Geometry
Domain
Geometric Measurement and Dimension
Cluster
Explain volume formulas and use them to solve problems
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
Content
arguments, Cavalieri’s principle, and informal limit arguments. G.GMD.1
Standards
2. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.3
Instructional Strategies
Instructional Resources/Tools
Revisit formulas and observe that the circumference is a little more than three times the
Rope or string
diameter of the circle. Briefly discuss the history of this number and attempts to compute its
value. C=πd - rCπ2=
Concrete models of circles cut into sectors and cylinders, pyramids, cones and
spheres cut into slices.
Use alternative ways to derive the formula for the area of the circle. For example, A=πr2
Rope or string
Cut a cardboard circular disk into 6 congruent sectors and rearrange the pieces to form a
shape that looks like a parallelogram with two scalloped edges. Repeat the process with 12
sectors and note how the edges of the parallelogram look ―straighter.‖ Discuss what would
happen in the case as the number of sectors becomes infinitely large. Then calculate the area
Volume relationship set of plastic shapes
of a parallelogram with base ½ C and altitude r to derive the formula A=πr2
Web sites that explore volumes of solids Mathman,
Wind a piece of string or rope to form a circular disk and cut it along a radial line. Stack the
pieces to form a triangular shape with base C and altitude r. Again discuss what would
Web site on Archimedes and the volume of a sphere:
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happen if the string became thinner and thinner so that the number of pieces in the stack
became infinitely large. Then calculate the area of the triangle to derive the formula A=πr2
http://physics.weber.edu/carroll/archimedes/method1.htm
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®)
Introduce Cavalieri‘s principle using a concrete model, such as a deck of cards. Use
Cavalieri‘s principle with cross sections of cylinders, pyramids, and cones to justify their
volume formulas.
For pyramids and cones, the factor 13will need some explanation. An informal demonstration
can be done using a volume relationship set of plastic shapes that permit one to pour liquid or
The set includes three pyramids with equal bases and altitudes that will stack to form a cube.
An algebraic approach involves the formula for the sum of squares (12 + 22 +…+ n2).
After the coefficient 1/3 has been justified for the formula of the volume of the pyramid (A =
13Bh), one can argue that it must also apply to the formula of the volume of the cone by
considering a cone to be a pyraType equation here.mid that has a base with infinitely many
sides.
The formulas for volumes of cylinders, pyramids, cones and spheres can be applied to a wide
variety of problems such as finding the capacity of a pipeline; comparing the amount of food
in cans of various shapes; comparing capacities of cylindrical, conical and spherical storage
tanks; using pyramids and cones in architecture; etc. Use a combination of concrete models
and formal reasoning to develop conceptual understanding of the volume formulas.
Video: The Story of Pi from Project MATHEMATICS!
Common Misconceptions
An informal survey of students from elementary school through college showed
the number pi to be the mathematical idea about which more students were
curious than any other. There are at least three facets to this curiosity: the symbol
π itself, the number 3.14159…, and the formula for the area of a circle. All of
these facets can be addressed here, at least briefly.
Many students want to think of infinity as a number. Avoid this by talking about
a quantity that becomes larger and larger with no upper bound.
The inclusion of the coefficient 13 in the formulas for the volume of a pyramid or
cone and 43in the formula for the volume of a sphere remains a mystery for many
students. In high school, students should attain a conceptual understanding of
where these coefficient come from. Concrete demonstrations, such as pouring
water from one shape into another should be followed by more formal reasoning.
Cavalieri‘s principle can also be applied to obtain the volume of a sphere, using an argument
similar to that employed by Archimedes more than 2000 years ago. In this demonstration,
cross sections of a sphere of radius R and a cone having radius 2R and altitude 2R are
balanced against cross sections of a cylinder having radius 2R and altitude 2R. (Details are
shown on the Archimedes website listed below.)
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High School Conceptual Category: Geometry
Domain
Geometric Measurement and Dimension
Cluster
Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twoContent
dimensional objects. G.GMD.4
Standards
Instructional Strategies
Instructional Resources/Tools
Review vocabulary for names of solids (e.g., right prism, cylinder, cone, sphere, etc.).
Concrete models of solids such as cubes, pyramids, cylinders, and spheres.
Include some models that can be sliced, such as those made from Styrofoam
Slice various solids to illustrate their cross sections. For example, cross sections of a cube can
be triangles, quadrilaterals or hexagons. Rubber bands may also be stretched around a solid to Rubber bands.
show a cross section.
Cardboard cutouts of 2-D figures (e.g. rectangles, triangles, circles)
Cut a half-inch slit in the end of a drinking straw, and insert a cardboard cutout shape. Rotate
the straw and observe the three-dimensional solid of revolution generated by the twoDrinking straws
dimensional cutout.
Web sites that illustrate geometric models. Some examples are:
Java applets on some web sites can also be used to illustrate cross sections or solids of
The Geometry Junkyard
revolution.
Wolfram Mathworld
Encourage students to create three-dimensional models to be sliced and cardboard cutouts to
be rotated. Students can also make three-dimensional models out of modeling clay and slice
through them with a plastic knife.
Web sites that can be used to create solids of revolution. An example is: The
Lathe
Common Misconceptions
Some cross sections are more difficult to visualize than others. For example, it is
often easier to visualize a rectangular cross section of a cube than a hexagonal
cross section.
Generating solids of revolution involves motion and is difficult to visualize by
merely looking at drawings.
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High School Conceptual Category: Geometry
Domain
Modeling with Geometry
Cluster
Apply geometric concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.1
Content
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.2
Standards
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios). G.MG.3
Instructional Strategies
Instructional Resources/Tools
Genuine mathematical modeling typically involves more than one conceptual category. For
A Sourcebook of Applications of School Mathematics, compiled by a Joint
example, modeling a herd of wild animals may involve geometry, measurement, proportional Committee of the Mathematical Association of America and the National
reasoning, estimation, probability and statistics, functions, and algebra. It would be somewhat Council of Teachers of Mathematics (1980).
misleading to try to teach a unit with the title of ―modeling with geometry.‖ Instead, these
standards can be woven into other content clusters.
Mathematics: Modeling our World, Course 1 and Course 2, by the Consortium
for Mathematics and its Applications (COMAP), http://www.comap.com/.
A challenge for teaching modeling is finding problems that are interesting and relevant to
high school students and, at the same time, solvable with the mathematical tools at the
Geometry & its Applications (GeoMAP). An exciting National Science
students‘ disposal. The resources listed below are a beginning for addressing this difficulty.
Foundation project to introduce new discoveries and real-world applications of
geometry to high school students. Produced by COMAP.
Measurement in School Mathematics, NCTM 1976 Yearbook.
Common Misconceptions
When students ask to see ―useful‖ mathematics, what they often mean is, ―Show
me how to use this mathematical concept or skill to solve the homework
problems.‖ Mathematical modeling, on the other hand, involves solving problems
in which the path to the solution is not obvious. Geometry may be one of several
tools that can be used.
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High School Conceptual Category: Statistics and Probability
Domain
Conditional Probability and the Rules of Probability
Cluster
Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
Content
complements of other events (―or,‖―and,‖ ―not‖). S.CP.1
Standards
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent. S.CP.2
3. Understand the conditional probability of A given B as P (A and B)/P (B), and interpret independence of A and B as saying that the conditional probability
A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.3
4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a
sample space to decide if events are independent and to approximate conditional probabilities. For example, collect: data from a random sample of students
in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will
favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.4
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the
chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.5
Instructional Strategies
Instructional Resources/Tools
The Standard for Mathematical Practice, precision is important for working with conditional
The Titanic Problem is a well-known problem that revolves around
probability. Attention to the definition of an event along with the writing and use of probability
conditional probability.
function notation are important requisites for communication of that precision. For example: Let
A: Female and B: Survivor, then P(A|B) =. The use of a vertical line for the conditional ―given‖
Focus in High School Mathematics: Reasoning and Sense Making in
is not intuitive for students and they often confuse the events B|A and A|B. Moreover, they often
Statistics and Probability, NCTM.
find identifying a conditional difficult when the problem is expressed in words in which the word
―given‖ is omitted. For example, find the probability that a female is a survivor. The standard
Navigating through Probability in Grades 9-12, NCTM.
Make sense of problems and persevere in solving them also should be employed so students can
look for ways to construct conditional probability by formulating their own questions and
working through them such as is suggested in standard 4 above. Students should learn to employ
Common Misconceptions
the use of Venn diagrams as a means of finding an entry into a solution to a conditional
Students may believe that multiplying across branches of a tree diagram has
probability problem.
nothing to do with conditional probability.
It will take a lot of practice to master the vocabulary of ―or,‖ ―and,‖ ―not‖ with the mathematical
notation of union (∪), intersection ( ∩ ), and whatever notation is used for complement.
They may also believe that independence of events and mutually exclusive
events are the same thing.
The independence of two events is defined in Standard 2 using the intersection. It is far more
intuitive to introduce the independence of two events in terms of conditional probability (stated
in Standard 3), especially where calculations can be performed in two-way tables.
The Standards in this cluster deliberately do not mention the use of tree diagrams, the traditional
way to treat conditional probabilities. Instead, probabilities of conditional events are to be found
using a two-way table wherever possible. However, tree diagrams may be a helpful tool for some
students. The difficulty is realizing that the second set of branches are conditional probabilities.
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High School Conceptual Category: Statistics and Probability
Domain
Conditional Probability and the Rules of Probability
Cluster
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B‘s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.6
Content
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P (A and B), and interpret the answer in terms of the model. S.CP.7
Standards
Instructional Strategies
Instructional Resources/Tools
Identifying that a probability is conditional when the word ―given‖ is not stated can be very
NCTM Navigating through Probability 9-12.
difficult for students. For example, if a balanced tetrahedron with faces 1, 2, 3, 4 is rolled
twice, what is the probability that the sum is prime (A) of those that show a 3 on at least one
The National Council of Teachers of Mathematics, Illuminations
roll (B)? Whether what is asked for is P(A and B), P(A or B), or P(A|B) can be problematic
for students. Showing the outcomes in a Venn Diagram may be useful. The calculation to find TI-83/84 and TI emulator (for permutations and combinations calculations)
the probability that the sum is prime (A) given at least one roll shows 3 (B) is to count the
elements of B by listing them if possible, namely in this example, there are 7 paired outcomes
(31, 32, 33, 34, 13, 23, 43). Of those 7 there are 4 whose sum is prime (32, 34, 23, 43). Hence Common Misconceptions
in the long run, 4 out of 7 times of rolling a fair tetrahedron twice, the sum of the two rolls
Students may believe that the probability of A or B is always the sum of the two
will be a prime number under the condition that at least one of its rolls shows the digit 3.
events individually.
Note that if listing outcomes is not possible, then counting the outcomes may require a
computation technique involving permutations or combinations, which is a STEM topic.
In the above example, if the question asked were what is the probability that the sum of two
rolls of a fair tetrahedron is prime (A) or at least one of the rolls is a 3 (B), then what is being
asked for is P(A or B) which is denoted as P(AB) in set notation. Again, it is often useful to
appeal to a Venn Diagram in which A consists of the pairs: 11, 12, 14, 21, 23, 32, 34, 41, 43;
and B consists of 13, 23, 33, 43, 31, 32, 34. Adding P(A) and P(B) is a problem as there are
duplicates in the two events, namely 23, 32, 34, and 43. So P(A or B) is 9/16 + 7/16 – 4/16 =
12/16 or 3/4, so 3/4th of the time, the result of rolling a fair tetrahedron twice will result in the
Additionally, they may believe that the probability of A and B is the product of
the two events individually, not realizing that one of the probabilities may be
conditional.
It should be noted that the Multiplication Rule in Standard 8 is designated as STEM when it is
connected to the discussion of independence in Standard 2 of the previous S-CP cluster. The
formula P(A and B) = P(A)P(B|A) is best illustrated in a two-stage setting in which A denotes
the outcome of the first stage, and B, the second. For example, suppose a jar contains 7 red
and 3 green chips. If one draws two chips without replacement from the jar, the probability of
getting a red followed by a green is P(red on first and green on second) = P(red on
first)P(green on second given a red on first) = (7/10)(3/9) = 21/90. Demonstrated on a tree
diagram indicates that the conditional probabilities are on the second set of branches.
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High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a
Content
line, and distance around a circular arc. G.CO.1
Standards
2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane
as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal
stretch). G.CO.2
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.3
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.4
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.
Specify a sequence of transformations that will carry a given figure onto another. G.CO.5
Instructional Strategies
Instructional Resources/Tools
Review vocabulary associated with transformations (e.g. point, line, segment, angle, circle,
Tracing paper (patty paper)
polygon, parallelogram, perpendicular, rotation reflection, translation).
Transparencies
Provide both individual and small-group activities, allowing adequate time for students to
explore and verify conjectures about transformations and develop precise definitions of
Graph paper
rotations, reflections and translations.
Ruler
Provide real-world examples of rigid motions (e.g. Ferris wheels for rotation; mirrors for
reflection; moving vehicles for translation).
Protractor
Use graph paper, transparencies, tracing paper or dynamic geometry software to obtain
images of a given figure under specified transformations.
Provide students with a pre-image and a final, transformed image, and ask them to describe
the steps required to generate the final image. Show examples with more than one answer
(e.g., a reflection might result in the same image as a translation).
Work backwards to determine a sequence of transformations that will carry (map) one figure
onto another of the same size and shape.
Focus attention on the attributes (e.g. distances or angle measures) of a geometric figure that
remain constant under various transformations.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Common Misconceptions
The terms ―mapping‖ and ―under‖ are used in special ways when studying
transformations. A translation is a type of transformation that moves all the
points in the object in a straight line in the same direction. Students should know
that not every transformation is a translation.
Students sometimes confuse the terms ―transformation‖ and ―translation.‖
Make the transition from transformations as physical motions to functions that take points in
the plane as inputs and give other points as outputs. The correspondence between the initial
and final points determines the transformation.
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Analyze various figures (e.g. regular polygons, folk art designs or product logos) to determine
which rotations and reflections carry (map) the figure onto itself. These transformations are
the ―symmetries‖ of the figure.
Emphasize the importance of understanding a transformation as the correspondence between
initial and final points, rather than the physical motion.
High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use
Content
the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.6
Standards
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent. G.CO.7
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.8
Instructional Strategies
Instructional Resources/Tools
Develop the relationship between transformations and congruency. Allow adequate time and
Tracing paper (patty paper)
provide hands-on activities for students to visually and physically explore rigid motions and
Graph paper
congruence.
Ruler
Use graph paper, tracing paper or dynamic geometry software to obtain images of a given
Protractor
figure under specified rigid motions. Note that size and shape are preserved.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri® or
Use rigid motions (translations, reflections and rotations) to determine if two figures are
Geogebra®); websites with similar tools (such as the National Library of Virtual
congruent. Compare a given triangle and its image to verify that corresponding sides and
Manipulatives that features applets for exploring triangle congruence).
corresponding angles are congruent.
Graphing calculators and other handheld technology such as TI-Nspire™.
Work backwards – given two figures that have the same size and shape, find a sequence of
rigid motions that will map one onto the other.
Build on previous learning of transformations and congruency to develop a formal criterion
for proving the congruency of triangles. Construct pairs of triangles that satisfy the ASA,
SAS or SSS congruence criteria, and use rigid motions to verify that they satisfy the
definition of congruent figures. Investigate rigid motions and congruence both algebraically
(using coordinates) and logically (using proofs).
Common Misconceptions
Some students may believe that combinations such as SSA or AAA are also a
congruence criterion for triangles. Provide counterexamples for this
misconception.
They may also believe that all transformations, including dilation, are rigid
motions. Provide counterexamples of this misconception.
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High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles
Content
are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s
Standards
endpoints. G.CO.9
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G.CO.10
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect
each other, and conversely, rectangles are parallelograms with congruent diagonals. G.CO.11
Instructional Strategies
Instructional Resources/Tools
Classroom teachers and mathematics education researchers agree that students have a hard
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®).
time learning how to do geometric proofs. An article by Battista and Clements (1995)
(http://investigations.terc.edu/library/bookpapers/geometryand_proof.cfm) provides
Principles and Standards for School Mathematics, pp.309-318, 342-346.
information for teachers to help students who struggle learn to do proof. The most significant
implication for instructional strategies for proof is stated in their conclusion.
Niven, Ivan, ―Can Geometry Survive in the Secondary Curriculum?‖ Learning
and Teaching Geometry, K-12. 1987 Yearbook of the National Council of
―Ironically, the most effective path to engendering meaningful use of proof in secondary
Teachers of Mathematics
school geometry is to avoid formal proof for much of students‘ work. By focusing instead on
justifying ideas while helping students build the visual and empirical foundation for higher
Pythagorean Puzzle levels of geometric thought, we can lead students to appreciate the need for formal proof.
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/pyt
Only then will we be able to use it meaningfully as a mechanism for justifying ideas.‖
hagorean_puz - Though focused on the Pythagorean theorem, this site provides
the kind of hands-on experience that should be a precursor to formal proof. In this
The article and ideas from Niven (1987) offers a few suggestions about teaching proof in
self-guided investigation, students use Geometer‘s Sketchpad® to construct a right
geometry:
triangle and discover a geometric proof of the Pythagorean theorem. Students then
• Initial geometric understandings and ideas should be taught ―without excessive emphasis on test the geometric proof with acute and obtuse triangles.
rigor.‖ Develop basic geometric ideas outside an axiomatic framework, and then let the
importance of the framework (and the framework itself) emerges from the geometry.
• Geometry is visual and should be taught in ways that leverage this aspect. Sketching,
Common Misconceptions
drawing and constructing figures and relationships between geometric objects should be
Research over the last four decades suggests that student misconceptions about
central to any geometric study and certainly to proof. Battista and Clement make a
proof abound:
powerful argument that the use of dynamic geometry software can be an important tool for • even after proving a generalization, students believe that exceptions to the
helping students understand proof.
generalization might exist;
• ―[A]void the deadly elaboration of the obvious‖ (Niven, p. 43). Often textbooks begin the
• one counterexample is not sufficient;
treatment of formal proof with ―easy‖ proofs, which appear to students to need no proof at • the converse of a statement is true (parallel lines do not intersect, lines that do
all. After presenting many opportunities for students to ―justify‖ properties of geometric
not intersect are parallel); and
figures, formal proof activities should begin with non-obvious conjectures.
• a conjecture is true because it worked in all examples that were explored.
• Use the history of geometry and real-world applications to help students develop conceptual
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understandings before they begin to use formal proof.
Proofs in high school geometry should not be restricted to the two-column format. Most
proofs at the college level are done in paragraph form, with the writer explaining and
defending a conjecture. In many cases, the two-column format can hinder the student from
making sense of the geometry by paying too much attention to format rather than
mathematical reasoning.
Each of these misconceptions needs to be addressed, both by the ways in which
formal proof is taught in geometry and how ideas about ―justification‖ are
developed throughout a student‘s mathematical education.
Some of the theorems listed in this cluster (e.g. the ones about alternate interior angles and
the angle sum of a triangle) are logically equivalent to the Euclidean parallel postulate, and
this should be acknowledged.
Use dynamic geometry software to allow students to make conjectures that can, in turn, be
formally proven. For example, students might notice that the base angles of an isosceles
triangle always appear to be congruent when manipulating triangles on the computer screen
and could then engage in a more formal discussion of why this occurs.
Common Core Standards Appendix A states, ―Encourage multiple ways of writing proofs,
such as in narrative paragraphs, using flow diagrams, in two-column format, and using
diagrams without words.
Students should be encouraged to focus on the validity of the underlying reasoning while
exploring a variety of formats for expressing that reasoning‖ (p. 29). Different methods of
proof will appeal to different learning styles in the classroom.
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High School Conceptual Category: Geometry
Domain
Congruence
Cluster
Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
Content
geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the
Standards
perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. G.CO.12
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.CO.13
Instructional Strategies
Instructional Resources/Tools
Students should analyze each listed construction in terms of what simpler constructions are
Compass
involved (e.g., constructing parallel lines can be done with two different constructions of
Straightedge
perpendicular lines).
String
Using congruence theorems, ask students to prove that the constructions are correct.
Origami paper
Provide meaningful problems (e.g. constructing the centroid or the incenter of a triangle) to
offer students practice in executing basic constructions.
Challenge students to perform the same construction using a compass and string. Use paper
folding to produce a reflection; use bisections to produce reflections.
Ask students to write ―how-to‖ manuals, giving verbal instructions for a particular
construction.
Offer opportunities for hands-on practice using various construction tools and methods.
Compare dynamic geometry commands to sequences of compass-and-straightedge steps.
Prove, using congruence theorems, that the constructions are correct.
Reflection tool (e.g. Mira®).
Dynamic geometry software (e.g. Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/pyt
hagorean_puzzle.pdf - In this self-guided investigation, students use Geometer‘s
Sketchpad to construct a right triangle and discover a geometric proof of the
Pythagorean Theorem. Students test the geometric proof with acute and obtuse
triangles.
http://www.nsa.gov/academia/_files/collected_learning/high_school/geometry/con
current_events.pdf - This lesson enhances student knowledge of how to use
Geometer's Sketchpad to explore geometric concepts (e.g. the points of
concurrency in a triangle). It includes the construction of line segments, triangles,
circles, perpendicular bisectors of line segments, angle bisectors, altitudes of
triangles, and medians of triangles.
http://mathforum.org/alejandre/circles.html - Students will construct a number of
compass-and-straightedge designs using ideas from this site.
Common Misconceptions
Some students may believe that a construction is the same as a sketch or drawing.
Emphasize the need for precision and accuracy when doing constructions. Stress
the idea that a compass and straightedge are identical to a protractor and ruler.
Explain the difference between measurement and construction
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
Content
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
Standards
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.1
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformatio
the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.2
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.3
Instructional Strategies
Instructional Resources/Tools
Allow adequate time and hands-on activities for students to explore dilations visually and
Dot paper
physically.
Graph paper
Use graph paper and rulers or dynamic geometry software to obtain images of a given figure
Rulers
under dilations having specified centers and scale factors. Carefully observe the images of lines
passing through the center of dilation and those not passing through the center, respectively. A
Protractors
line segment passing through the center of dilation will simply be shortened or elongated but
Pantograph
will lie on the same line, while the dilation of a line segment that does not pass through the
center will be parallel to the original segment (this is intended as a clarification of Standard 1a). Photocopy machine
Illustrate two-dimensional dilations using scale drawings and photocopies.
Measure the corresponding angles and sides of the original figure and its image to verify that
the corresponding angles are congruent and the corresponding sides are proportional (i.e.
stretched or shrunk by the same scale factor). Investigate the SAS and SSS criteria for similar
triangles.
Use graph paper and rulers or dynamic geometry software to obtain the image of a given figure
under a combination of a dilation followed by a sequence of rigid motions (or rigid motions
followed by dilation).
Work backwards – given two similar figures that are related by dilation, determine the center of
dilation and scale factor. Given two similar figures that are related by a dilation followed by a
sequence of rigid motions, determine the parameters of the dilation and rigid motions that will
map one onto the other.
Using the theorem that the angle sum of a triangle is 180°, verify that the AA criterion is
equivalent to the AAA criterion.
Given two triangles for which AA holds, use rigid motions to map a vertex of one triangle onto
the corresponding vertex of the other in such a way that their corresponding sides are in line.
Then show that dilation will complete the mapping of one triangle onto the other.
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Web-based applets that demonstrate dilations, such as those at the National
Library of Virtual Manipulatives.
Video: Similarity by Project Mathematics! (www.projectmathematics.com)
Common Misconceptions
Some students often do not recognize that congruence is a special case of
similarity. Similarity with a scale factor equal to 1 becomes a congruency.
Students may not realize that similarities preserve shape, but not size. Angle
measures stay the same, but side lengths change by a constant scale factor.
Some students often do not list the vertices of similar triangles in order.
However, the order in which vertices are listed is preferred and especially
important for similar triangles so that proportional sides can be correctly
identified.
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangle, and Trigonometry
Cluster
Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the
Content
Pythagorean Theorem proved using triangle similarity. G.SRT.4
Standards
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.5
Instructional Strategies
Instructional Resources/Tools
Review triangle congruence criteria and similarity criteria, if it has already been established.
Cardboard models of right triangles.
Review the angle sum theorem for triangles, the alternate interior angle theorem and its
converse, and properties of parallelograms. Visualize it using dynamic geometry software.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Using SAS and the alternate interior angle theorem, prove that a line segment joining midpoints
of two sides of a triangle is parallel to and half the length of the third side. Apply this theorem
to a line segment that cuts two sides of a triangle proportionally.
Generalize this theorem to prove that the figure formed by joining consecutive midpoints of
sides of an arbitrary quadrilateral is a parallelogram. (This result is known as the Midpoint
Quadrilateral Theorem or Varignon‘s Theorem.)
Video: The Theorem of Pythagoras from Project MATHEMATICS!
Use cardboard cutouts to illustrate that the altitude to the hypotenuse divides a right triangle into
two triangles that are similar to the original triangle. Then use AA to prove this theorem. Then,
use this result to establish the Pythagorean relationship among the sides of a right triangle (a2 +
b2 = c2) and thus obtain an algebraic proof of the Pythagorean Theorem.
Websites for the Pythagorean Theorem
Jim Loy‘s Pythagorean Theorem:
Animated proofs of the Pytharorean Theorem: Pythagorean Theorem and its
Many Proofs:
Common Misconceptions
Some students may confuse the alternate interior angle theorem and its
converse as well as the Pythagorean theorem and its converse.
Prove that the altitude to the hypotenuse of a right triangle is the geometric mean of the two
segments into which its foot divides the hypotenuse.
Prove the converse of the Pythagorean Ttheorem, using the theorem itself as one step in the
proof. Some students might engage in an exploration of Pythagorean Triples (e.g., 3-4-5, 5-1213, etc.), which provides an algebraic extension and an opportunity to explore patterns.
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute
Content
angles. G.SRT.6
Standards
7. Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.7
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.8
Instructional Strategies
Instructional Resources/Tools
Review vocabulary (opposite and adjacent sides, legs, hypotenuse and complementary angles)
Cutouts of right triangles
associated with right triangles.
Rulers
Make cutouts or drawings of right triangles or manipulate them on a computer screen using
dynamic geometry software and ask students to measure side lengths and compute side ratios.
Protractors
Observe that when triangles satisfy the AA criterion, corresponding side ratios are equal. Side
ratios are given standard names, such as sine, cosine and tangent. Allow adequate time for
Scientific calculators
students to discover trigonometric relationships and progress from concrete to abstract
understanding of the trigonometric ratios.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®)
Show students how to use the trigonometric function keys on a calculator. Also, show how to
find the measure of an acute angle if the value of its trigonometric function is known.
Investigate sines and cosines of complementary angles, and guide students to discover that they
are equal to one another. Point out to students that the ―co‖ in cosine refers to the ―sine of the
complement.‖
Observe that, as the size of the acute angle increases, sines and tangents increase while cosines
decrease.
Stress trigonometric terminology by the history of the word ―sine‖ and the connection between
the term ―tangent‖ in trigonometry and tangents to circles.
Have students make their own diagrams showing a right triangle with labels showing the
trigonometric ratios. Although students like mnemonics such as SOHCAHTOA, these are not a
substitute for conceptual understanding. Some students may investigate the reciprocals of sine,
cosine, and tangent to discover the other three trigonometric functions.
Use the Pythagorean theorem to obtain exact trigonometric ratios for 30°, 45°, and 60° angles.
Use cooperative learning in small groups for discovery activities and outdoor measurement
projects.
Have students work on applied problems and project, such as measuring the height of the school
building or a flagpole, using clinometers and the trigonometric functions.
Trig Trainer® instructional aids
Clinometers (can be made by the students)
Websites for the history of mathematics:
Trigonometric Functions
Trigonometric Course
Video: Sines and Cosines, Part II from Project MATHEMATICS!
Common Misconceptions
Some students believe that right triangles must be oriented a particular way.
Some students do not realize that opposite and adjacent sides need to be
identified with reference to a particular acute angle in a right triangle.
Some students believe that the trigonometric ratios defined in this cluster apply
to all triangles, but they are only defined for acute angles in right triangles.
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High School Conceptual Category: Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Cluster
Apply trigonometry to general triangles
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.9
Content
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.10
Standards
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying
problems, resultant forces). G.SRT.11
Instructional Strategies
Instructional Resources/Tools
Information below includes additional mathematics that students should learn in order to take
Scientific calculator
advanced courses such as calculus, advanced statistics, or discrete mathematics, and goes
beyond the mathematics that all students should study in order to be college- and career-ready:
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®)
Extend the definitions of sine and cosine to functions of obtuse angles.
Real-world problems that involve solving a triangle by use of Pythagorean
The formula Area = ½ ab sin(C) can be derived from the definition of sine and the formula for
Theorem, right triangle trigonometry and/or the Law of Sines and the Law of
the area of a triangle. The formulas Area = ½ ac sin(B) and Area = ½ bc sin(A) are also valid.
Cosines
The Law of Sines follows directly from equating these three formulas for the area of a given
triangle.
There are several proofs of the Law of Cosines. One of the easiest to follow uses the Pythagorean
theorem.
Dynamic geometry software can be used to show how the law of cosines generalizes the
Pythagorean theorem by moving one vertex of a triangle to make the angle acute, right or obtuse.
Given a triangle in which the measures of three parts (at least one of which is a side) are known,
the Law of Sines or the Law of Cosines can be used to find the measures of the remaining three
parts. This procedure is known as ―solving the triangle‖ or ―triangulation.‖ Triangulation is an
important tool used by surveyors for determining the location of a point by measuring angles to it
from known points at either end of a fixed baseline, rather than directly measuring distances to
the point.
Common Misconceptions
Some students may think that definitions of sine, cosine and tangent using
right triangles ratios are valid for solving any triangles. In reality, these ratios
only applicable to right triangles, necessitating the use of the Laws of Sines
and Cosines.
When applying the Law of Sines, there is an ambiguous case (SSA) in which
there are two different possible solutions for the third side of the triangle.
The Laws of Sines and Cosines can be applied to other problems in geometry, such as finding the
perimeter of a parallelogram if the lengths of the diagonals and the angle at which the diagonals
intersect are known.
The Law of Cosines can be used in finding the vector sum or difference of two given vectors. A
physical application is finding the resultant of two forces.
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High School Conceptual Category: Geometry
Domain
Circles
Cluster
Understand and apply theorems about circles
4. Prove that all circles are similar. G.C.1
Content
5. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed
Standards
angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. G.C.2
6. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.3
7. (+) Construct a tangent line from a point outside a given circle to the circle. G.C.4
Instructional Strategies
Instructional Resources/Tools
Given any two circles in a plane, show that they are related by dilation. Guide students to
Ruler
discover the center and scale factor of this dilation and make a conjecture about all dilations
of circles.
Compass
Starting with the special case of an angle inscribed in a semicircle, use the fact that the angle
sum of a triangle is 180° to show that this angle is a right angle. Using dynamic geometry,
students can grab a point on a circle and move it to see that the measure of the inscribed angle
passing through the endpoints of a diameter is always 90°. Then extend the result to any
inscribed angles. For inscribed angles, proofs can be based on the fact that the measure of an
exterior angle of a triangle equals the sum of the measures of the nonadjacent angles.
Consider cases of acute or abtuse inscribed angles.
Use properties of congruent triangles and perpendicular lines to prove theorems about
diameters, radii, chords, and tangent lines.
Use formal geometric constructions to construct perpendicular bisectors of the sides and
angle bisectors of a given triangle. Their intersections are the centers of the circumscribed and
inscribed circles, respectively.
Protractor
Computer dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®).
Common Misconceptions
Students sometimes confuse inscribed angles and central angles.
Students may think they can tell by inspection whether a line intersects a circle in
exactly one point. It may be beneficial to formally define a tangent line as the
line perpendicular to a radius at the point where the radius intersects the circle.
Dissect an inscribed quadrilateral into triangles, and use theorems about triangles to prove
properties of these quadrilaterals and their angles.
Constructing tangents to a circle from a point outside the circle is a useful application of the
result that an angle inscribed in a semicircle is a right angle.
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High School Conceptual Category: Geometry
Domain
Circles
Cluster
Find arc lengths and areas of sectors of circles
5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as
Content
the constant of proportionality; derive the formula for the area of a sector. G.C.5
Standards
Instructional Strategies
Instructional Resources/Tools
Begin by calculating lengths of arcs that are simple fractional parts of a circle and do this for
Ruler
circles of various radii so that students discover a proportionality relationship.
Compass
Provide plenty of practice in assigning radian measures to angles that are simple fractional
parts of a straight angle.
Protractor
Stress the definition of radian by considering a central angle whose intercepted arc has its
length equal to the radius, making the constant of proportionality 1. Students who are having
difficulty understanding radians may benefit from constructing cardboard sectors whose
angles are one radian. Use a ruler and string to approximate such an angle.
String
Compute areas of sectors by first considering them as fractional parts of a circle. Then, using
proportionality, derive a formula for their area in terms of radius and central angle. Do this
for angles that are measured in both degrees and radians and note that the formula is much
simpler when the angles are measured in radians.
Video: The Story of Pi from Project MATHEMATICES!
Derive formulas that relate degrees and radians.
Introduce arc measures that are equal to the measures of the intercepted central angles in
degrees and radians.
Computer dynamic geometry software (Geometer‘s Sketchpad)
Common Misconceptions
Sectors and segments are often used interchangeably in everyday conversation.
Care should be taken to distinguish these two geometric concepts.
The formulas for converting radians to degrees and vice versa are easily
confused. Knowing that the degree measure of a given angle is always a number
larger than the radian measure can help students use the correct unit.
Emphasize appropriate use of terms, such as, angle, arc, radian, degree, and sector.
High School Conceptual Category: Geometry
Domain
Expressing Geometric Properties with Equations
Cluster
Translate between the geometric description and the equation for a conic section
3. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given
Content
by an equation. G.GPE.1
Standards
4. Derive the equation of a parabola given a focus and directrix. G.GPE.2
Instructional Strategies
Instructional Resources/Tools
Review the definition of a circle as a set of points whose distance from a fixed point is
Physical models of cones sliced to show cross sections that are circles, ellipses
constant.
parabolas and hyperbolas
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Review the algebraic method of completing the square and demonstrate it geometrically.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or Geogebra®)
Illustrate conic sections geometrically as cross sections of a cone.
Parabolic reflectors to illustrate practical applications of parabolas
Use the Pythagorean theorem to derive the distance formula. Then, use the distance formula
to derive the equation of a circle with a given center and radius, beginning with the case
where the center is the origin.
Starting with any quadratic equation in two variables (x and y) in which the coefficients of the
quadratic terms are equal, complete the squares in both x and y and obtain the equation of a
circle in standard form.
Given two points, find the equation of the circle passing through one of the points and having
the other as its center.
Define a parabola as a set of points satisfying the condition that their distance from a fixed
point (focus) equals their distance from a fixed line (directrix). Start with a horizontal
directrix and a focus on the y-axis, and use the distance formula to obtain an equation of the
resulting parabola in terms of y and x2. Next use a vertical directrix and a focus on the x-axis
to obtain an equation of a parabola in terms of x and y2. Make generalizations in which the
focus may be any point, but the directrix is still either horizontal or vertical. Allow sufficient
time for students to become familiar with new vocabulary and notation.
Common Misconceptions
Because new vocabulary is being introduced in this cluster, remembering the
names of the conic sections can be problematic for some students.
The Euclidean distance formula involves squared, subscripted variables whose
differences are added. The notation and multiplicity of steps can be a serious
stumbling block for some students.
The method of completing the square is a multi-step process that takes time to
assimilate. A geometric demonstration of completing the square can be helpful in
promoting conceptual understanding
Given y as a quadratic equation of x (or x as a quadratic function of y), complete the square to
obtain an equation of a parabola in standard form.
Identify the vertex of a parabola when its equation is in standard form and show that the
vertex is halfway between the focus and directrix.
Investigate practical applications of parabolas and paraboloids.
Information below includes additional mathematics that students should learn in order to
take advanced courses such as calculus, advanced statistics, or discrete mathematics, and
goes beyond the mathematics that all students should study in order to be college- and
career-ready:
Students preparing for advanced courses may use the distance formula and relevant focusdirectrix definitions to derive equations of ellipses and hyperbolas whose major axes are
either horizontal or vertical. Encourage students to explore conic sections using dynamic
geometry software.
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High School Conceptual Category: Geometry
Domain
Expressing Geometric Properties with Equations
Cluster
Use coordinates to prove simple geometric theorems algebraically
6. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the
Content
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G.GPE.4
Standards
7. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point). G.GPE.5
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G.GPE.6
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.GPE.7
Instructional Strategies
Instructional Resources/Tools
Review the concept of slope as the rate of change of the y-coordinate with respect to the xGraph paper
coordinate for a point moving along a line, and derive the slope formula.
Scientific or graphing calculators
Use similar triangles to show that every nonvertical line has a constant slope.
Review the point-slope, slope-intercept and standard forms for equations of lines.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®)
Investigate pairs of lines that are known to be parallel or perpendicular to each other and discover
that their slopes are either equal or have a product of –1, respectively.
Pay special attention to the slope of a line and its applications in analyzing properties of lines.
Allow adequate time for students to become familiar with slopes and equations of lines and
methods of computing them.
Use slopes and the Euclidean distance formula to solve problems about figures in the coordinate
plane such as:
• Given three points, are they vertices of an isosceles, equilateral, or right triangle?
• Given four points, are they vertices of a parallelogram, a rectangle, a rhombus, or a square?
• Given the equation of a circle and a point, does the point lie outside, inside, or on the circle?
• Given the equation of a circle and a point on it, find an equation of the line tangent to the circle at
that point.
• Given a line and a point not on it, find an equation of the line through the point that is parallel to
the given line.
• Given a line and a point not on it, find an equation of the line through the point that is
perpendicular to the given line.
• Given the equations of two non-parallel lines, find their point of intersection.
• Given two points, use the distance formula to find the coordinates of the point halfway between
them. Generalize this for two arbitrary points to derive the midpoint formula.
Common Misconceptions
Students may claim that a vertical line has infinite slopes. This suggests that
infinity is a number. Since applying the slope formula to a vertical line
leads to division by zero, we say that the slope of a vertical line is
undefined. Also, the slope of a horizontal line is 0. Students often say that
the slope of vertical and/or horizontal lines is ―no slope,‖ which is incorrect.
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Use linear interpolation to generalize the midpoint formula and find the point that partitions a line
segment in any specified ratio.
Use the distance formula to find the length of each side of a polygon whose vertices are known, and
compute the perimeter of that figure.
Given the vertices of a triangle or a parallelogram, find the equation of a line containing the altitude
to a specified base and the point of intersection of the altitude and the base. Use the distance
formula to find the length of that altitude and base, and then compute the area of the figure.
High School Conceptual Category: Geometry
Domain
Geometric Measurement and Dimension
Cluster
Explain volume formulas and use them to solve problems
3. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
Content
arguments, Cavalieri’s principle, and informal limit arguments. G.GMD.1
Standards
4. (+) Give an informal argument using Cavalieri‘s principle for the formulas for the volume of a sphere and other solid figures. G.GMD.2
5. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.3
Instructional Strategies
Instructional Resources/Tools
Revisit formulas and observe that the circumference is a little more than three times the diameter of Rope or string
the circle. Briefly discuss the history of this number and attempts to compute its value. C=πd rCπ2=
Concrete models of circles cut into sectors and cylinders, pyramids, cones
and spheres cut into slices.
Use alternative ways to derive the formula for the area of the circle. For example, A=πr2
Cut a cardboard circular disk into 6 congruent sectors and rearrange the pieces to form a shape that
looks like a parallelogram with two scalloped edges. Repeat the process with 12 sectors and note
how the edges of the parallelogram look ―straighter.‖ Discuss what would happen in the case as the
number of sectors becomes infinitely large. Then calculate the area of a parallelogram with base ½
C and altitude r to derive the formula. A=πr2
Rope or string
Wind a piece of string or rope to form a circular disk and cut it along a radial line. Stack the pieces
to form a triangular shape with base C and altitude r. Again discuss what would happen if the string
became thinner and thinner so that the number of pieces in the stack became infinitely large. Then
calculate the area of the triangle to derive the formula. A=πr2
Web sites that explore volumes of solids Mathman,
Introduce Cavalieri‘s principle using a concrete model, such as a deck of cards. Use Cavalieri‘s
principle with cross sections of cylinders, pyramids, and cones to justify their volume formulas.
Dynamic geometry software (Geometer‘s Sketchpad®, Cabri®, or
Geogebra®)
For pyramids and cones, the factor 13 will need some explanation. An informal demonstration can
be done using a volume relationship set of plastic shapes that permit one to pour liquid or sand
Video: The Story of Pi from Project MATHEMATICS!
Volume relationship set of plastic shapes
Web site on Archimedes and the volume of a sphere:
http://physics.weber.edu/carroll/archimedes/method1.htm
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from one shape into another. Another way to do this for pyr
includes three pyramids with equal bases and altitudes that will stack to form a cube. An algebraic
approach involves the formula for the sum of squares (12 + 22 +…+ n2 ).
After the coefficient 1/3 has been justified for the formula of the volume of the pyramid (A =
13Bh), one can argue that it must also apply to the formula of the volume of the cone by
considering a cone to be a pyraType equation here.mid that has a base with infinitely many sides.
The formulas for volumes of cylinders, pyramids, cones and spheres can be applied to a wide
variety of problems such as finding the capacity of a pipeline; comparing the amount of food in
cans of various shapes; comparing capacities of cylindrical, conical and spherical storage tanks;
using pyramids and cones in architecture; etc. Use a combination of concrete models and formal
reasoning to develop conceptual understanding of the volume formulas.
Cavalieri‘s principle can also be applied to obtain the volume of a sphere, using an argument
similar to that employed by Archimedes more than 2000 years ago. In this demonstration, cross
sections of a sphere of radius R and a cone having radius 2R and altitude 2R are balanced against
cross sections of a cylinder having radius 2R and altitude 2R. (Details are shown on the
Archimedes website listed below.)
Common Misconceptions
An informal survey of students from elementary school through college
showed the number pi to be the mathematical idea about which more
students were curious than any other. There are at least three facets to this
curiosity: the symbol π itself, the number 3.14159…, and the formula for
the area of a circle. All of these facets can be addressed here, at least
briefly.
Many students want to think of infinity as a number. Avoid this by talking
about a quantity that becomes larger and larger with no upper bound.
The inclusion of the coefficient 13 in the formulas for the volume of a
pyramid or cone and 43in the formula for the volume of a sphere remains a
mystery for many students. In high school, students should attain a
conceptual understanding of where these coefficient come from. Concrete
demonstrations, such as pouring water from one shape into another should
be followed by more formal reasoning.
High School Conceptual Category: Geometry
Domain
Geometric Measurement and Dimension
Cluster
Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twoContent
dimensional objects. G.GMD.4
Standards
Instructional Strategies
Instructional Resources/Tools
Review vocabulary for names of solids (e.g., right prism, cylinder, cone, sphere, etc.).
Concrete models of solids such as cubes, pyramids, cylinders, and spheres.
Include some models that can be sliced, such as those made from
Slice various solids to illustrate their cross sections. For example, cross sections of a cube can be
Styrofoam
triangles, quadrilaterals or hexagons. Rubber bands may also be stretched around a solid to show a
cross section.
Rubber bands.
Cut a half-inch slit in the end of a drinking straw, and insert a cardboard cutout shape. Rotate the
straw and observe the three-dimensional solid of revolution generated by the two-dimensional
cutout.
Java applets on some web sites can also be used to illustrate cross sections or solids of revolution.
Encourage students to create three-dimensional models to be sliced and cardboard cutouts to be
rotated. Students can also make three-dimensional models out of modeling clay and slice through
them with a plastic knife.
Cardboard cutouts of 2-D figures (e.g. rectangles, triangles, circles)
Drinking straws
Web sites that illustrate geometric models. Some examples are:
The Geometry Junkyard
Wolfram Mathworld
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Web sites that can be used to create solids of revolution. An example is:
The Lathe
Common Misconceptions
Some cross sections are more difficult to visualize than others. For
example, it is often easier to visualize a rectangular cross section of a cube
than a hexagonal cross section.
Generating solids of revolution involves motion and is difficult to visualize
by merely looking at drawings.
High School Conceptual Category: Geometry
Domain
Modeling with Geometry
Cluster
Apply geometric concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.1
Content
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.2
Standards
3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with
typographic grid systems based on ratios). G.MG.3
Instructional Strategies
Instructional Resources/Tools
Genuine mathematical modeling typically involves more than one conceptual category. For
A Sourcebook of Applications of School Mathematics, compiled by a Joint
example, modeling a herd of wild animals may involve geometry, measurement, proportional Committee of the Mathematical Association of America and the National
reasoning, estimation, probability and statistics, functions, and algebra. It would be somewhat Council of Teachers of Mathematics (1980).
misleading to try to teach a unit with the title of ―modeling with geometry.‖ Instead, these
Mathematics: Modeling our World, Course 1 and Course 2, by the Consortium
standards can be woven into other content clusters.
for Mathematics and its Applications (COMAP), http://www.comap.com/.
A challenge for teaching modeling is finding problems that are interesting and relevant to
high school students and, at the same time, solvable with the mathematical tools at the
students‘ disposal. The resources listed below are a beginning for addressing this difficulty.
Geometry & its Applications (GeoMAP). An exciting National Science
Foundation project to introduce new discoveries and real-world applications of
geometry to high school students. Produced by COMAP.
Measurement in School Mathematics, NCTM 1976 Yearbook.
Common Misconceptions
When students ask to see ―useful‖ mathematics, what they often mean is, ―Show
me how to use this mathematical concept or skill to solve the homework
problems.‖ Mathematical modeling, on the other hand, involves solving problems
in which the path to the solution is not obvious. Geometry may be one of several
tools that can be used.
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High School Conceptual Category: Statistics and Probability
Domain
Conditional Probability and the Rules of Probability
Cluster
Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
Content
complements of other events (―or,‖―and,‖ ―not‖). S.CP.1
Standards
2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent. S.CP.2
3. Understand the conditional probability of A given B as P (A and B)/P (B), and interpret independence of A and B as saying that the conditional probability A
given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. S.CP.3
4. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a
sample space to decide if events are independent and to approximate conditional probabilities. For example, collect: data from a random sample of students
in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will
favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. S.CP.4
5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the
chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.5
Instructional Strategies
Instructional Resources/Tools
The Standard for Mathematical Practice, precision is important for working with conditional
The Titanic Problem is a well-known problem that revolves around
probability. Attention to the definition of an event along with the writing and use of probability
conditional probability.
function notation are important requisites for communication of that precision. For example: Let
A: Female and B: Survivor, then P (A|B) =. The use of a vertical line for the conditional ―given‖
Focus in High School Mathematics: Reasoning and Sense Making in
is not intuitive for students and they often confuse the events B|A and A|B. Moreover, they often
Statistics and Probability, NCTM.
find identifying a conditional difficult when the problem is expressed in words in which the word
―given‖ is omitted. For example, find the probability that a female is a survivor. The standard
Navigating through Probability in Grades 9-12, NCTM.
Make sense of problems and persevere in solving them also should be employed so students can
look for ways to construct conditional probability by formulating their own questions and
working through them such as is suggested in standard 4 above. Students should learn to employ
Common Misconceptions
the use of Venn diagrams as a means of finding an entry into a solution to a conditional
Students may believe that multiplying across branches of a tree diagram has
probability problem. It will take a lot of practice to master the vocabulary of ―or,‖ ―and,‖ ―not‖
nothing to do with conditional probability.
with the mathematical notation of union (∪), intersection ( ∩ ), and whatever notation is used for
Students may also believe that independence of events and mutually
complement.
exclusive events are the same thing.
The independence of two events is defined in Standard 2 using the intersection. It is far more
intuitive to introduce the independence of two events in terms of conditional probability (stated
in Standard 3), especially where calculations can be performed in two-way tables.
The Standards in this cluster deliberately do not mention the use of tree diagrams, the traditional
way to treat conditional probabilities. Instead, probabilities of conditional events are to be found
using a two-way table wherever possible. However, tree diagrams may be a helpful tool for some
students. The difficulty is realizing that the second set of branches is conditional probabilities.
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High School Conceptual Category: Statistics and Probability
Domain
Conditional Probability and the Rules of Probability
Cluster
Use the rules of probability to compute probabilities of compound events in a uniform probability model
6. Find the conditional probability of A given B as the fraction of B‘s outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.6
Content
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P A and B), and interpret the answer in terms of the model. S.CP.7
Standards
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the
model. S.CP.8
9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. S.CP.9
Instructional Strategies
Instructional Resources/Tools
Identifying that a probability is conditional when the word ―given‖ is not stated can be very
NCTM Navigating through Probability 9-12.
difficult for students. For example, if a balanced tetrahedron with faces 1, 2, 3, 4 is rolled
twice, what is the probability that the sum is prime (A) of those that show a 3 on at least one
The National Council of Teachers of Mathematics, Illuminations
roll (B)? Whether what is asked for is P(A and B), P(A or B), or P(A|B) can be problematic
TI-83/84 and TI emulator (for permutations and combinations calculations)
for students. Showing the outcomes in a Venn Diagram may be useful. The calculation to find
the probability that the sum is prime (A) given at least one roll shows 3 (B) is to count the
elements of B by listing them if possible, namely in this example, there are 7 paired outcomes Common Misconceptions
(31, 32, 33, 34, 13, 23, 43). Of those 7 there are 4 whose sum is prime (32, 34, 23, 43). Hence Students may believe that the probability of A or B is always the sum of the two
in the long run, 4 out of 7 times of rolling a fair tetrahedron twice, the sum of the two rolls
events individually.
will be a prime number under the condition that at least one of its rolls shows the digit 3.
Note that if listing outcomes is not possible, then counting the outcomes may require a
They may also believe that the probability of A and B is the product of the two
computation technique involving permutations or combinations, which is a STEM topic.
events individually, not realizing that one of the probabilities may be conditional.
In the above example, if the question asked were what is the probability that the sum of two
rolls of a fair tetrahedron is prime (A) or at least one of the rolls is a 3 (B), then what is being
asked for is P(A or B) which is denoted as P(AB) in set notation. Again, it is often useful to
appeal to a Venn Diagram in which A consists of the pairs: 11, 12, 14, 21, 23, 32, 34, 41, 43;
and B consists of 13, 23, 33, 43, 31, 32, 34. Adding P(A) and P(B) is a problem as there are
duplicates in the two events, namely 23, 32, 34, and 43. So P(A or B) is 9/16 + 7/16 – 4/16 =
12/16 or 3/4, so 3/4th of the time, the result of rolling a fair tetrahedron twice will result in the
It should be noted that the Multiplication Rule in Standard 8 is designated as STEM when it is
connected to the discussion of independence in Standard 2 of the previous S-CP cluster. The
formula P(A and B) = P(A)P(B|A) is best illustrated in a two-stage setting in which A denotes
the outcome of the first stage, and B, the second. For example, suppose a jar contains 7 red
and 3 green chips. If one draws two chips without replacement from the jar, the probability of
getting a red followed by a green is P(red on first and green on second) = P(red on
first)P(green on second given a red on first) = (7/10)(3/9) = 21/90. Demonstrated on a tree
diagram indicates that the conditional probabilities are on the second set of branches.
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High School Conceptual Category: Statistics and Probability
Domain
Using Probability to Make Decisions
Cluster
Use probability to evaluate outcomes of decisions (cont.)
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S.MD.6
Content
7. (+) Analyze decisions and strategies, using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.)
Standards
S.MD.7
Instructional Strategies
Instructional Resources/Tools
NCTM Navigating through Probability 9-12.
This domain and cluster belong to STEM, and hence need not be for all students.
A game of chance is said to be fair if the expected net winnings are 0. If the expected net
winnings is negative, then the player needs to decide if the game is worth playing. For
Data Driven Mathematics module, Probability Models
example, a spinner has 18 red, 18 black and 2 green sections. Suppose, players gain a one
score point if the spinner lands on red, otherwise the players loose a one score point. The
probability the spinner lands on red is
. The probability it lands elsewhere is
. So,
Common Misconceptions
Students may believe probabilities and expected values aren‘t useful in making
the expected probability is 1×
+ (-1) ×
= - .053 score points. This means that
decisions that affect one‘s life. Students need to see that these are not merely
players should expect to lose a little over .05 of a score point every time they play the game.
textbook exercises.
Calculating an expected value enables players to decide whether or not the game is worth
playing
Expected values may be used to decide between two strategies. For example, suppose shop
owner needs to decide whether to stock product A or product B and can only stock one of
them. Profit margins for A follow the distribution (in thousands of dollars): 5,4,3,2,1 with
probabilities .1,.45,.3,.1,.05, respectfully. Those for B follow: 8,7,6,5,4,3,2,1,0 with
probabilities: .1,.15,.15,.1,.1,0,0,0,.4. The expected profit by stocking A is
5(.1)+4(.45)+3(.3)+2(.1)+1(.05) = 3.45 thousands of dollars. The expected profit by stocking
B is 8(.1)+7(.15)+6(.15)+5(.1)+4(.1)+0(.4) = 3.65 thousands of dollars. So, based on expected
values of profit margins, the better choice would be to stock product B.
Conditional probabilities are situations where the interpretation of an observation is
dependent upon or ―conditioned on‖ some other factor. For example, a blood test has been
shown to indicate the presence of a particular disease 95% of the time when the disease is
actually present. The same blood test gives a false positive result 0.5% of the time. A false
positive result suggests that even though the blood test indicates that the person has the
disease (the positive part) but subsequent, additional testing indicates the person does not
have that disease (hence positive but false or a false positive).
Suppose that one percent of the population actually has the disease. If a person‘s blood test is
positive, how likely is it that the person has the disease?
318
MATH
HONORS GEOMETRY
This scenario can be restated as the following conditional probability problem:
―What is the probability that a person actually has the disease given that (or conditioned on)
the blood test indicates the person has the disease?‖
There are two possibilities for a person to produce a positive blood test result: the
person has the disease or the person does not have the disease.
The probability that a person has the disease given a positive blood test result is 0.01 × 0.95 =
0.0095. This represents the part of the population that is expected to both provide a positive
blood test result and have the disease. The probability that a person gives a positive blood test
result but does not have the disease is 0.99 × 0.005 = 0.00495. This represents the part of the
population that is expected to give a positive blood test result but does not have the disease.
The total, or 0.01445 = 0.0095 + 0.00495, represents a part of the population that has a
positive blood test, whether it has a disease or not. The conditional probability that a person
actually has the disease when the blood test is positive is .
.
= 0.657, i.e., the
probability that a person actually has the disease is about 66% given that the blood test results
are positive.
319
MATH
TOPICS IN MATHEMATICS
320
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Estimate, compute, and solve
problems involving real
numbers, including ratio,
proportion, and percent, and
explain solutions.
Estimate, compute, and solve
problems involving scientific
notation, square roots, and
numbers with integer exponents
Use proportional reasoning and
apply indirect measurement
techniques, including right
triangle trigonometry and
properties of similar triangles, to
solve problems involving
measurement and rates
INDICATOR
1. Demonstrate fluency in
computations using real
numbers
REF.
NNSO
G
9.4
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will utilize real numbers to determine sales tax and Income.
Students will obtain the necessary data from the Newspaper Business Section.
Technology/Resources: Converting between Decimals, Fractions and Percents http://www.purplemath.com/modules/percents.htm; Calculator
2. Apply order of operations to
simplify expressions and
perform computations
involving integer exponents and
radicals.
3. Solve problems involving
unit conversion for situations
involving distances, areas,
volumes, and rates within the
same measurement system.
NNSO
I
8.3
21st Century Skills: Quantitative Literacy, Critical Thinking
Lesson Ideas:
The student will utilize Order of Operations throughout the course to solve problems.
Students will utilize Order of Operations to solve real world problems.
Technology/Resources: Order of Operations – PEMDAS http://www.purplemath.com/modules/orderops.htm;
Evaluation - http://www.purplemath.com/modules/evaluate.htm; Calculator
M
D
9.5
21st Century Skills: Quantitative Literacy, Critical Thinking
Lesson Ideas:
Super Bowl Shipment – NCTM Navigations – Students will explore weights and conversion
factors in a trucking shipment.
Technology/Resources: Calculator
21st Century Skills: Quantitative Literacy, Critical Thinking
4. Recognize and identify when
inductive and deductive
reasoning are used.
Lesson Ideas:
The student will utilize inductive and deductive reasoning when solving a variety of multi-step
problems.
21st Century Skills: Problems Solving, Critical Thinking
Use algebraic representations,
such as tables, graphs,
expressions, functions,
inequalities, to model and solve
problem situations.
5. Solve equations and formulas
for a specified variable
PFA
D
10.3
Lesson Ideas:
Students will solve distance/rate/time and motion problems for a specified variable. They will
work collaboratively to solve problems.
Technology/Resources: Calculator
21st Century Skills: Problem Solving, Collaboration, Critical Thinking
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
321
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use algebraic representations,
such as tables, graphs,
expressions, functions,
inequalities, to model and solve
problem situations. (cont.)
INDICATOR
6. Solve simple linear and
nonlinear equations and
inequalities having square roots
as coefficients and solutions.
REF.
PFA
D
10.5
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will solve triangle, rectangle, pyramid, prism, cone, and sphere problems to find area
and volume.
Technology/Resources: Calculator
21st Century Skills: Problem Solving, Critical Thinking
Solve and graph linear equations
and inequalities.
7. Use symbolic algebra
(equations and inequalities),
graphs, and tables to represent
situations and solve problems.
PFA
F
8.7
Lesson Ideas:
Students will explore Ohm‘s Law by experiment, using tables and graphs to represent the
problem. Students will work independently as well as with peers with the application of Ohm‘s
Law.
Use algebraic representations,
such as tables, graphs,
expressions, functions,
inequalities, to model and solve
problem situations.
8. Use symbolic algebra
(equations and inequalities),
graphs, and tables to represent
situations and solve problems.
PFA
D
8.7
Technology/Resources: Calculator
Solve and graph linear equations
and inequalities.
9. Solve linear equations and
inequalities graphically,
symbolically, and using
technology.
PFA
F
8.9
Lesson Ideas:
The student will solve algebraic problems described by linear equations using symbolic,
graphic, and calculator methods.
21st Century Skills: Problem Solving, Critical Thinking, Collaboration
Technology/Resources: Graphing Linear Equations http://www.purplemath.com/modules/graphlin.htm; Calculator
21st Century Skills: Problem Solving, Critical Thinking
"
Line 1:
Line 2:
Line 3:
10. Find linear equations that
represent lines that pass
through a given set of ordered
pairs, and find linear equations
that represent lines parallel or
perpendicular to a given line
through a specific point.
PFA
F
9.8
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
Lesson Ideas:
The student will explore slope and the slope-intercept form by solving algebraic problems.
Technology/Resources: Straight Line Equations http://www.purplemath.com/modules/strtlneq.htm; Graphing Calculator
21st Century Skills: Problem Solving, Critical Thinking
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
322
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Describe and interpret rates of
change from graphical and
numerical data.
INDICATOR
11. Compute and interpret
slope, midpoint and distance
given a set of ordered pairs.
REF.
PFA
J
8.13
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will solve problems to develop skills in Arithmetic Operations including Square &
Square Root and content knowledge of: Slope of Line, Midpoint & Distance of Line Segment
Technology/Resources: Slope of a Straight Line http://www.purplemath.com/modules/slope.htm; Midpoint Formula http://www.purplemath.com/modules/midpoint.htm;
Distance Formula - http://www.purplemath.com/modules/distform.htm
21st Century Skills: Problem Solving, Critical Thinking
Solve quadratic equations with
real roots by graphing, formula,
and factoring.
12. Solve quadratic equations
with real roots by factoring,
graphing, using the quadratic
formula, and with technology.
PFA
G
9.10
Lesson Ideas:
Students will explore motion problems, using various mathematical tools to solve quadratic
equations.
Technology/Resources:
Graphing Quadratic Functions - http://www.purplemath.com/modules/grphquad.htm;
Quadratic Formula - http://www.purplemath.com/modules/quadform.htm;
Graphing Calculator
21st Century Skills: Problem Solving, Critical Thinking
"
13. Solve simple quadratic
equations graphically.
PFA
G
8.12
Lesson Ideas:
The student will solve Up & Down Motion Problems using Graphical Methods.
Technology/Resources: Graphing Calculator
21st Century Skills: Problem Solving, Critical Thinking
Analyze and compare functions
and their graphs using attributes,
such as rates of change,
intercepts, and zeros.
14. Demonstrate the relationship among zeros of a function,
roots of equations, and
solutions of equations
graphically and in words.
PFA
E
9.4
Lesson Ideas:
Students will explore zeros through Motion Problems using Quadratic Equations.
Technology/Resources: Graphing Calculator
21st Century Skills: Critical Thinking
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
323
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Solve quadratic equations with
real roots by graphing, formula,
and factoring
INDICATOR
15. Solve real-world problems
that can be modeled using
linear, quadratic, exponential,
or square root functions.
REF.
PFA
G
10.10
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will solve Compound Interest problems using the Exponential Function. The students
will utilize a mock market statement to calculate and analyze the impact of compound interest.
Students will solve Motion Problems using the Quadratic.
Students will solve Simple Interest problems using the Linear Function. http://www.purplemath.com/modules/investment.htm
Students will solve Distance/Rate/Time Word Problems using Linear Function http://www.purplemath.com/modules/distance.htm
Technology/Resources: Quadratic Applications – pp 102-104 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Quadratic Problems: Projectile Motion - http://www.purplemath.com/modules/quadprob.htm
Solve systems of linear equations
involving two variables
graphically and symbolically.
"
Use algebraic representations,
such as tables, graphs,
expressions, functions,
inequalities, to model and solve
problem situations
16. Solve 2 by 2 systems of
linear equations graphically and
by simple substitution.
PFA
H
8.10
17. Solve real-world problems
that can be modeled, using
systems of linear equations and
inequalities.
PFA
H
10.11
18. Interpret the meaning of the
solution of a 2 by 2 system of
equations.
19. Use algebraic
representations and functions to
describe and generalize
geometric properties and
relationships.
PFA
H
8.11
PFA
D
10.4
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
Students will explore 2x2 Systems of Linear Equations by Problem Solving.
Technology/Resources: Calculator
21st Century Skills: Problem Solving
Lesson Ideas:
Supply & Demand – NCTM Illuminations- Students will explore Linear Systems using linear
supply and demand economic model.
Technology/Resources:
Systems of Equations Word Problems - http://www.purplemath.com/modules/systprob.htm;
Graphing Calculator
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
If the Earth is Round, How Big is it? NCTM Navigations – Students will calculate the diameter
of the earth using similar triangles.
Chip off the Old Block – NCTM Navigations – Students will explore the relationships among
surface area and volume of cubes, cylinders, & spheres.
Technology/Resources:
Area, Volume, Perimeter - http://www.purplemath.com/modules/perimetr.htm; Calculator
21st Century Skills: Critical Thinking
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
324
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
REF.
OHIO BENCHMARK
Draw and construct
representations of two-and threedimensional geometric objects
using a variety of tools, such as
straightedge, compass, and
technology.
INDICATOR
20. Given the name or number
of sides of a regular polygon,
compute the size of the central
angle and the angle formed by
two sides of the polygon.
Estimate and compute various
attributes, including length,
angle measure, area, surface
area, and volume, to a specified
level of precision
21. Find the sum of the interior
and exterior angles of regular
convex polygons with and
without measuring the angles
with a protractor.
M
E
8.8
22. Generalize patterns using
functions or relationships
(linear, quadratic, and
exponential), and freely
translate among tabular,
graphical, and symbolic
representations.
PFA
A
9.2
Use formal mathematical
language and notation to
represent ideas, to demonstrate
relationships within and among
representation systems, and to
formulate generalizations.
23. Relate graphical and
algebraic representations of
lines, simple curves and conic
sections.
G
H
12.3
Translate information from one
representation (words, table,
graph, or equation) to another
representation of a relation or a
function.
24. Generalize patterns using
functions or relationships
(linear, quadratic, exponential),
and freely translate among
tabular, graphical, and symbolic
representations.
PFA
C
9.2
Generalize and explain patterns
and sequences in order to find
the next term and the nth term.
Line 1:
Line 2:
Line 3:
G
E
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Students will calculate interior and central angles of regular polygons through problem solving
exercises.
Technology/Resources: Calculator
21st Century Skills: Problem Solving, Critical Thinking
Lesson Ideas:
Students will explore patterns and relate symbolic, graphical, and tabular representations
through linear & quadratic problem solving and the graphing calculator.
Technology/Resources: Matching Quadratic Equations, Graphs & Tables – pp 78-81 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Collecting Data & Regression Equations – pp 82-87 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.htm; Graphing Calculator
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
21st Century Skills: Critical Thinking, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
325
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Identify and classify functions as
linear or nonlinear, and contrast
their properties using tables,
graphs, or equations
"
INDICATOR
25. Identify functions as linear
or nonlinear based on
information given in a table,
graph, or equation.
26. Describe problem situations
(linear, quadratic, and
exponential) by using tabular,
graphical, and symbolic
representations.
REF.
PFA
B
8.3
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will be given functions in tabular, graph, and equation form and asked to identify
them as linear or non-linear.
Technology/Resources: Calculator
PFA
B
9.3
21st Century Skills: Critical Thinking
Lesson Ideas:
Smokey Bear – NCTM Illuminations – Students will explore math models to characterize forest
fire pre-conditions.
National Debt & Wars – NCTM Illuminations
On Top of the World – NCTM Illuminations
Technology/Resources: Graphing Calculator
Analyze and compare functions
and their graphs using attributes,
such as rates of change,
intercepts, and zeros.
"
Model and solve problem
situations involving direct and
indirect variation.
Line 1:
Line 2:
Line 3:
27. Describe the relationship
between the graph of a line and
its equation, including being
able to explain the meaning of
slope as a constant rate of
change and y-intercept in realworld problems.
PFA
E
8.6
21st Century Skills: Critical Thinking, Problem Solving, Collaboration
Lesson Ideas:
Ohm‘s Law – Students will explore the linear relationship of voltage and current with resistance
being the slope.
Distance/Rate/Time - Students will explore the linear relationship of distance and time with rate
being the slope.
Technology/Resources:
Slope & y-intercept to graph line - http://www.purplemath.com/modules/slopgrph.htm;
x and y intercepts - http://www.purplemath.com/moduels/intrcept.htm; Calculator
28. Describe and compare
characteristics of the following
families of functions: linear,
quadratic, and exponential
functions.
PFA
E
9.5
29. Differentiate and explain
types of changes in
mathematical relationships,
such as linear vs. nonlinear,
continuous vs. non continuous,
direct variation vs. indirect
variation.
PFA
I
8.14
21st Century Skills: Critical Thinking
Lesson Ideas:
The student will collaboratively create Venn Diagrams to compare and contrast Linear,
Quadratic, and Exponential Functions.
Technology/Resources: Graphing Calculator
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
Inverse Variation: Frequency/Wave Length: c = f • λ
Direct Variation: P & T in Perfect Gas Law PV = nRT
Technology/Resources: Graphing Calculator; Direct, Inverse & Joint Variations – pp 8-10 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Direction, Inverse & Joint Variation - http://www.purplemath.com/modules/variatn.htm
21st Century Skills: Critical Thinking, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
326
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Model and solve problem
situations involving direct and
indirect variation. (cont.)
"
Use counting techniques, such as
permutations and combinations,
to determine the total number of
options and possible outcomes.
"
Design an experiment to test a
theoretical probability, and
record and explain the results.
INDICATOR
30. Model and solve problems
involving direct and inverse
variation using proportional
reasoning.
31. Describe the relationship
between slope and the graph of
a direct variation and inverse
variation.
32. Calculate the number of
possible outcomes for a
situation, recognizing and
accounting for when items may
occur more than once or when
order is important.
REF.
PFA
I
9.13
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
Inverse: Frequency/Wave Length: c = f • λ
Direct: P & T in Perfect Gas Law PV = nRT
Technology/Resources: Graphing Calculator; Direct, Inverse & Joint Variations – pp 8-10 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Direction, Inverse & Joint Variation - http://www.purplemath.com/modules/variatn.htm
PFA
I
9.14
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
Inverse: Frequency/Wave Length: c = f • λ
Direct: P & T in Perfect Gas Law PV = nRT
Technology/Resources: Graphing Calculator; Direct, Inverse & Joint Variations – pp 8-10 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Direction, Inverse & Joint Variation - http://www.purplemath.com/modules/variatn.htm
DAP
H
8.10
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
The student will conduct sampling (probability experiments) and compare results to theoretical
using:
o Dice Rolling: Sample Results versus Theoretical Probability
o Coin Flipping: Samples Results versus Theoretical Probability
Technology/Resources: Calculator, Dice
33. Use counting techniques
and the Fundamental Counting
Principle to determine the total
number of possible outcomes
for mathematical situations.
DAP
H
9.7
34. Design an experiment to
test a theoretical probability
and explain how the results
may vary.
DAP
I
6.7
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
Students will examine Car Options and Telephone Numbers to explore counting techniques and
the Fundamental Counting Principle.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
The student will conduct sampling (probability experiments) and compare results to theoretical
using:
o Dice Rolling: Sample Results versus Theoretical Probability
o Coin Flipping: Samples Results versus Theoretical Probability
Technology/Resources: Calculator, Dice
21st Century Skills: Critical Thinking, Problem Solving
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
327
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Design an experiment to test a
theoretical probability, and
record and explain the results.
(cont.)
INDICATOR
35. Make predictions based on
theoretical probabilities, design
and conduct an experiment to
test the predictions, compare
actual results to predicted
results, and explain differences.
REF.
DAP
I
7.8
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will conduct sampling (probability experiments) and compare results to theoretical
using:
o Dice Rolling: Sample Results versus Theoretical Probability
o Coin Flipping: Samples Results versus Theoretical Probability
Technology/Resources: Box & Whisker Plots http://www.purplemath.com/modules/boxwhisk.htm;
Comparative Stats – Box & Whisker Plots – pp 90-92 http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Scatter Plots & Regression - http://www.purplemath.com/modules/scattreg.htm
21st Century Skills: Critical Thinking, Problem Solving
"
36. Describe, create, and
analyze a sample space and use
it to calculate probability.
DAP
I
9.8
Lesson Ideas:
The student will create a sample space using Dice and use it to calculate probability.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking
Compute probabilities of
compound events, independent
events, and simple dependent
events.
Make predictions based on
theoretical probabilities and
experimental results.
37. Identify situations involving
independent and dependent
events, and explain differences
between and common
misconceptions about
probabilities associated with
those events.
DAP
J
9.9
38. Use theoretical and
experimental probability,
including simulations or
random numbers, to estimate
probabilities and to solve
problems dealing with
uncertainty.
DAP
K
9.10
Lesson Ideas:
The student will use Venn Diagrams to explore independent and dependent events.
Technology/Resources:
Venn Diagrams - http://www.purplemath.com/modules/venndiag.htm
21st Century Skills: Critical Thinking
Lesson Ideas:
The student will conduct sampling (probability experiments) and compare results to theoretical
using:
o Dice Rolling: Sample Results versus Theoretical Probability
o Coin Flipping: Samples Results versus Theoretical Probability
Technology/Resources: Calculator, Dice
21st Century Skills: Critical Thinking, Problem Solving
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
328
MATH
TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Identify and classify functions as
linear or nonlinear, and contrast
their properties using tables,
graphs, or equations.
"
INDICATOR
39. Define function with
ordered pairs in which each
domain element is assigned
exactly one range element.
40. Define function formally
and with f(x) notation.
REF.
PFA
B
9.1
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will explore examples of ordered pairs to determine whether a relationship is a
function.
Technology/Resources: Functions versus Relations http://www.purplemath.com/modules/fcns.htm;
Domain and Range - http://www.purplemath.com/modules/fcns2.htm;
Domain and Range – pp 93-94 - http://www.doe.virginia.gov/vdoe/enhancedsands/malgebra2.pdf;
Vertical Line Test - http://www.purplemath.com/modules/fcns.htm#vlt
PFA
B
10.1
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
The student will be introduced to the Euler Notation f(x) for a function and the formal definition
of a function.
Technology/Resources: Function Notation - http://www.purplemath.com/modules/fcnnot.htm
Describe and interpret rates of
change from graphical and
numerical data.
41. Describe and compare how
changes in an equation affect
the related graphs.
PFA
J
8.15
21st Century Skills: Quantitative Literacy, Problem Solving
Lesson Ideas:
The student will explore Linear and Quadratic Functions to examine how changes in parameters
change the related graphs.
Technology/Resources: Function Transformations & Translations http://www.purplemath.com/modules/fcntrans.htm; Graphing Calculator
Use proportional reasoning and
apply indirect measurement
techniques, including right
triangle trigonometry and
properties of similar triangles, to
solve problems involving
measurement and rates
Use right triangle trigonometric
relationships to determine
lengths and angle measures.
42. Use scale drawings and
right triangle trigonometry to
solve problems that include
unknown distances and angle
measures.
M
D
9.4
43. Define the basic
trigonometric ratios in right
triangles: sine, cosine, and
tangent.
G
I
9.1
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
The student will use right angle trig and similar triangles to determine the diameter of the moon.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving
Lesson Ideas:
The student will be introduced to the basic trig ratios through examples and practice exercises.
Content: Right Angle Trig
Skills: Trig Functions on Graphing Calculator
Technology/Resources: Calculator
21st Century Skills: Quantitative Literacy, Critical Thinking
Line 1:
Line 2:
Line 3:
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
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TOPICS IN MATHEMATICS
INSTRUCTIONAL OBJECTIVES
OHIO BENCHMARK
Use right triangle trigonometric
relationships to determine
lengths and angle measures.
(cont.)
Line 1:
Line 2:
Line 3:
INDICATOR
44. Apply proportions and right
triangle trigonometric ratios to
solve problems involving
missing lengths and angle sizes
in similar figures.
REF.
G
I
9.2
(Standard) NNSO = Number, Number Sense and Operations Standard
PFA = Pattern, Functions and Algebra
Ohio Benchmark within Standard
Grade Level Indicator
CLARIFICATION
(SKILLS, METHODS, RESOURCES)
Lesson Ideas:
The student will use right angle trig and similar triangles to determine the diameter of the moon.
Technology/Resources: Calculator
21st Century Skills: Critical Thinking, Problem Solving
M = Measurement
DAP = Data Analysis & Probability
G = Geometry and Spatial Sense
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ALGEBRA 2
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ALGEBRA 2
High School Conceptual Category: Unit 1: Polynomial, Rational, and Radical Relationships
Domain
The Complex Number System
Cluster
Perform arithmetic operations with complex numbers
1. Know there is a complex number i such as i2 = -1, and every complex number has the form a +bi with a and b real.
Content
2. 2. Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Standards
Instructional Strategies
Instructional Resources/Tools
Before introducing complex numbers, revisit simpler examples demonstrating how number
Computer algebra systems
systems can be seen as ―expanding‖ from other number systems in order to solve more equations.
For example, the equation x + 5 = 3 has no solution as a whole numbers, but it has a solution x =
Graphing calculator
-2 as an integers. Similarly, although 7x = 5 has no solution in the integers, it has a solution x =
5/7 in the rational numbers. The linear equation ax + b = c, where a, b, and c are rational
http://www.learner.org/courses/learningmath/number/session2/index.html
This session focuses on exploration of the number sets that make up the real
numbers, always has a solution x in the rational numbers:
.
number system and on the concept of infinity and the importance of zero.
When moving to quadratic equations, once again some equations do not have solutions, creating
a need for larger number systems. For example, x2 – 2 = 0 has no solution in the rational
numbers. But it has solutions
in the real numbers. (The real number line augments the
rational numbers, completing the line with the irrational numbers.)
Point out that solving the equation x2 – 2 = 0 in terms of x is equivalent to finding x-intercepts of
a graph of y = x2 – 2, which crosses the x-axis at
and
. Thus, the graph illustrates
that the solutions are
.
http://www.clarku.edu/~djoyce/complex/numberi.html
This article is a short tour into the history of number i
http://www.math.hmc.edu/calculus/tutorials/complex/
Complex numbers: forms and operations
The Ohio Resource Center
The National Council of Teachers of Mathematics, Illuminations
Next, use an example of a quadratic equation with real coefficients, such as x2 + 1 = 0, which can
be written equivalently as x2 = -1. Because the square of any real number is non-negative, it
follows that x2 = -1 has no solution in the real numbers. One can see this graphically by noticing
that the graph of y = x2 + 1 does not cross the x-axis.
The ―solution‖ to this ―impasse‖ is to introduce a new number, the imaginary unit i, where i2 = 1, and to consider complex numbers of the form a +bi, where a and b are real numbers and i is
not a real number. Because i is not a real number, expressions of the form a +bi cannot be
simplified.
The existence of i, allows every quadratic equation to have two solutions of the form a + bi –
either real when b = 0, or complex when b ≠ 0. Have students observe that if a quadratic equation
(with real coefficients) has complex solutions, the solutions always appear in conjugate pairs, in
the form a + bi and a – bi. Particularly, for an equation x2 = - 9, a conjugate pair of solutions are
0 +3i and 0 – 3i.
Common Misconceptions
If irrational numbers are confused with non-real or complex numbers, remind
students about the relationships between the sets of numbers.
If an imaginary unit iis misinterpreted as -1 instead of
definition of i.
, re-establish a
Some properties of radicals that are true for real numbers are not true for
complex numbers.
In particular, for positive real numbers a and b,
but
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In order to find solutions of quadratic equations or to create quadratic equations from its
solutions, introduce students to the condition of equality of complex numbers, with addition,
subtraction and multiplication of complex numbers.
Stress the importance of the relationships between different number sets and their properties. The
complex number system possesses the same basic properties as the real number system: that
addition and multiplication are commutative and associative; the existence of additive identity
and multiplicative identity; the existence of an additive inverse for every complex number and
the existence of multiplicative inverse or reciprocal for every non-zero complex number; and the
distributive property of multiplication over the addition. An awareness of the properties
minimizes students‘ rote memorization and links the rules for manipulations with the complex
number system to the rules for manipulations with binomials with real coefficients of the form a
+ bx.
and
but
.
If those properties are getting misused, provide students with an example,
such as
that leads to a contradiction that a positive real number is equal to a
negative real number.
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High School Conceptual Category: Number and Quantity
Domain
The Complex Number System
Cluster
Use complex numbers in polynomial identities and equations
7. Solve quadratic equations with real coefficients that have complex solutions
Content
Standards
Instructional Strategies
Revisit quadratic equations with real coefficients and a negative discriminant and point out that
this type of equation has no real number solution. Emphasize that with the extension of the real
number system to complex numbers any quadratic equation has a solution. Since the process of
solving a quadratic equation may involve the use of the quadratic formula with a negative
discriminant, defining a square root of a negative number becomes critical
, where
N is a positive real number; iis the imaginary unit and i2 = -1). After the square root of a negative
number has been defined, emphasize that the quadratic formula can be used without restriction.
While solving quadratic equations using the quadratic formula, students should observe that the
quadratic equation always has a pair of solutions regardless of the value of the discriminant. If
the discriminant, b2 – 4ac, is positive, the equation has two unequal complex solutions that are
real (the imaginary parts of complex numbers are zeros). If the discriminant is zero, the equation
has a repeated real solution – a double root (two complex solutions with equal real parts and the
imaginary parts equal to zero). If the discriminant is negative, the equation has two conjugate
complex solutions that are not real.
Information below contains additional mathematics that students should learn in order to take
advanced courses such as calculus, advanced statistics or discrete mathematics and goes beyond
the mathematics that all students should study in order to be college- and career-ready:
Since the set of real numbers is a subset of the set of complex numbers, finding the complex
solutions of a polynomial function requires finding all solutions of the form (a +bi), which
include solutions that have b = 0, or real solutions. The Fundamental Theorem of Algebra states
that every complex polynomial function f(x) (a function with complex coefficients and a
complex variable x) of degree n ≥ 1 has at least one complex zero. Point out that according to the
Fundamental Theorem of Algebra, any quadratic equation with real coefficients (a subset of
complex numbers) always has two complex solutions and can be factored into a product of two
linear factors. If a pair of conjugate complex solutions of the quadratic polynomial is (a + bi) and
(a – bi), the quadratic polynomial can be restated as the product of two linear factors as f(x) = (x
– (a + bi))∙(x – (a – bi)). For example, since the quadratic equation x2 + 4 = 0 has two complex
solutions 2i and (- 2i), the polynomial x2 + 4 can be restated as (x – (0 + 2i))∙(x – (0 – 2i)) or (x
+2i)(x - 2i) and then the sum of the squares of two quantities becomes factorable over the set of
complex numbers.
Instructional Resources/Tools
Graphing calculator
Computer algebra systems
Dynamic geometric systems
Complex number plane - explanation, history and visual representations of
complex roots
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MT200601-366a - This site provides a specific instance when the textbook answer for
finding a root of a complex number differed with the answer given by the TI83. The site contains an expansion of the definition for the integral root of a
complex number to an arbitrary complex power of a complex number.
Common Misconceptions
Students may believe that a quadratic equation with the discriminant b 2 – 4ac
= 0 has only one solution. For example, x2 – 10x + 25 has a discriminant of 0
(102 - 4∙1∙25 = 0) and the only solution is x = 5. Students should refer to the
Fundamental Theorem of Algebra which states that any quadratic equation
has two solutions including those where the discriminant equals zero. These
equations have a repeated real solution or a double root.
In the cases of quadratic equations, when the use of quadratic formula is not
critical, students sometime ignore the negative solutions. For example, for the
equation x2 = 9, students may mention 3 and forget about (– 3),or mention 3i
and forget about (- 3i) for the equation x2 = - 9. If this misconception persists,
advise students to solve this type of quadratic equation either by factoring or
by the quadratic formula. It is also beneficial to remind students about the
Fundamental Theorem of Algebra that secures the existence of two complex
solutions for any quadratic equation.
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High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context. (a) Interpret parts of an expression, such as terms, factors, and coefficients. (b) Interpret
Content
complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not
Standards
depending on P.
2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that
can be factored as (x2 - y2)(x2 + y2).
Instructional Strategies
Instructional Resources/Tools
Extending beyond simplifying an expression, this cluster addresses interpretation of the
Hands-on materials, such as algebra tiles, can be used to establish a visual
components in an algebraic expression. A student should recognize that in the expression 2x + 1, understanding of algebraic expressions and the meaning of terms, factors and
―2‖ is the coefficient, ―2‖ and ―x‖ are factors, and ―1‖ is a constant, as well as ―2x‖ and ―1‖ being coefficients.
terms of the binomial expression. Development and proper use of mathematical language is an
important building block for future content.
From the National Library of Virtual Manipulatives - Algebra Tiles –
Visualize multiplying and factoring algebraic expressions using tiles.
Using real-world context examples, the nature of algebraic expressions can be explored. For
example, suppose the cost of cell phone service for a month is represented by the expression
0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute Common Misconceptions
(40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of Students may believe that the use of algebraic expressions is merely the
cell phone minutes used in the month. Similar real-world examples, such as tax rates, can also be abstract manipulation of symbols. Use of real-world context examples to
used to explore the meaning of expressions.
demonstrate the meaning of the parts of algebraic expressions is needed to
counter this misconception.
Factoring by grouping is another example of how students might analyze the structure of an
expression. To factor 3x(x – 5) + 2(x – 5), students should recognize that the ―x – 5‖ is common
Students may also believe that an expression cannot be factored because it
to both expressions being added, so it simplifies to (3x + 2)(x – 5). Students should become
does not fit into a form they recognize. They need help with reorganizing the
comfortable with rewriting expressions in a variety of ways until a structure emerges.
terms until structures become evident.
Have students create their own expressions that meet specific criteria (e.g., number of terms
factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in
different forms. Additionally, pair/group students to share their expressions and rewrite one
another‘s expressions.
High School Conceptual Category: Algebra
Domain
Seeing Structure in Expressions
Cluster
Write expressions in equivalent forms to solve problems (cont.)
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4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate
Content
mortgage payments.
Standards
Instructional Strategies
Instructional Resources/Tools
Offer multiple real-world examples of exponential functions. For instance, to illustrate an
Graphing utilities to explore the effects of parameter changes on a graph
exponential decay, students need to recognize that in the equation for an automobile cost C(t) =
Tables, graphs and equations of real-world applications that apply quadratic
20,000(0.75)t , the base is 0.75 and between 0 and 1 and the value of $20,000 represents the
and exponential functions
initial cost of an automobile that depreciates 25% per year over the course of t years.
Similarly, to illustrate exponential growth, in the equation for the value of an investment over
time A(t) = 10,000(1.03)t, where the base is 1.03 and is greater than 1; and the $10,000
represents the value of an investment when increasing in value by 3% per year for x years.
Computer algebra systems
A problem such as, ―An amount of $100 was deposited in a savings account on January 1st each
of the years 2010, 2011, 2012, and so on to 2019, with annual yield of 7%. What will be the
balance in the savings account on January 1, 2020?‖ illustrates the use of a formula for a
http://www.learner.org/workshops/algebra/workshop5/lessonplan1.html
This website contains a lesson and a workshop that showcases ways that
teachers can help students explore mathematical properties studied in algebra.
The activities use a variety of techniques to help students understand concepts
of factoring quadratic trinomials.
geometric series
when Sn represents the value of the geometric series with the first
term g, constant ration r ≠ 1, and n terms.
Before using the formula, it might be reasonable to demonstrate the way the formula is derived,
Sn = g + gr + gr2 +gr3 + …grn-1.
Multiply by r
rSn = gr + gr2 + …+ grn-1 + grn
Subtract
Factor
S – rS = g – grn
S(1 – r) = g(1 – rn)
From the National Library of Virtual Manipulatives - Grapher – A tool for
graphing and exploring functions.
From the National Council of Teachers of Mathematics, Illuminations Difference of Squares - This activity uses a series of related arithmetic
experiences to prompt students to explore arithmetic statements leading to a
result that is the factoring pattern for the difference of two squares. A
geometric interpretation of the familiar formula is also included.
The amount of the investment for January 1, 2020 can be found using: 100(1.07) 10 + 100(1.07)9
+ … + 100(1.07). If the first term of this geometric series is g = 100(1.07), the ratio is 1.07 and
the number of terms n = 10, the formula for the value of geometric series is:
Common Misconceptions
Some students may believe that factoring and completing the square are
isolated techniques within a unit of quadratic equations. Teachers should help
students to see the value of these skills in the context of solving higher degree
equations and examining different families of functions.
S10 ≈ $1478.36
Students may think that the minimum (the vertex) of the graph of y = (x + 5)2
is shifted to the right of the minimum (the vertex) of the graph y = x2 due to
the addition sign. Students should explore examples both analytically and
graphically to overcome this misconception.
Divide by (1 – r)
Some students may believe that the minimum of the graph of a quadratic
function always occur at the y-intercept.
Some students cannot distinguish between arithmetic and geometric
sequences, or between sequences and series. To avoid this confusion, students
need to experience both types of sequences and series.
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High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Perform arithmetic operations on polynomials
1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and
Content
multiplication; add, subtract, and multiply polynomials.
Standards
Instructional Strategies
Instructional Resources/Tools
The primary strategy for this cluster is to make connections between arithmetic of integers and
Graphing calculators
arithmetic of polynomials. In order to understand this standard, students need to work toward
both understanding and fluency with polynomial arithmetic. Furthermore, to talk about their
Graphing software, including dynamic geometry software
work, students will need to use correct vocabulary, such as integer, monomial, polynomial,
factor, and term.
Computer Algebra Systems
In arithmetic of polynomials, a central idea is the distributive property, because it is fundamental
not only in polynomial multiplication but also in polynomial addition and subtraction. With the
distributive property, there is little need to emphasize misleading mnemonics, such as FOIL,
which is relevant only when multiplying two binomials, and the procedural reminder to ―collect
like terms‖ as a consequence of the distributive property. For example, when adding the
polynomials 3x and 2x, the result can be explained with the distributive property as follows: 3x +
2x = (3 + 2)x = 5x.
An important connection between the arithmetic of integers and the arithmetic of polynomials
can be seen by considering whole numbers in base ten place value to be polynomials in the base
b = 10. For two-digit whole numbers and linear binomials, this connection can be illustrated with
area models and algebra tiles. But the connections between methods of multiplication can be
generalized further. For example, compare the product 213 x 47 with the product
:
Algebra tiles
Area models
Common Misconceptions
Some students will apply the distributive property inappropriately. Emphasize
that it is the distributive property of multiplication over addition. For
example, the distributive property can be used to rewrite
as
, because in this product the second factor is a sum (i.e., involving
addition). But in the product
, the second factor,
, is itself a
product, not a sum.
Some students will still struggle with the arithmetic of negative numbers.
Consider the expression
. On the one hand,
. But using the distributive property,
. Because the first calculation gave
0, the two terms on the right in the second calculation must be opposite in
sign. Thus, if we agree that
, then it must follow that
.
Note how the distributive property is in play in each of these examples: In the left-most
computation, each term in the factor
must be multiplied by each term in the other
factor,
, just like the value of each digit in 47 must be multiplied by the value of
each digit in 213, as in the middle computation, which is similar to ―partial products methods‖
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that some students may have used for multiplication in the elementary grades. The common
algorithm on the right is merely an abbreviation of the partial products method.
The new idea in this standard is called closure: A set is closed under an operation if when any
two elements are combined with that operation, the result is always another element of the same
set. In order to understand that polynomials are closed under addition, subtraction and
multiplication, students can compare these ideas with the analogous claims for integers: The
sum, difference or product of any two integers is an integer, but the quotient of two integers is
not always an integer.
Now for polynomials, students need to reason that the sum (difference or product) of any two
polynomials is indeed a polynomial. At first, restrict attention to polynomials with integer
coefficients. Later, students should consider polynomials with rational or real coefficients and
reason that such polynomials are closed under these operations.
For contrast, students need to reason that polynomials are not closed under the operation of
division: The quotient of two polynomials is not always a polynomial. For example
is not a polynomial. Of course, the quotient of two polynomials is sometimes a polynomial. For
example
.
High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Understand the relationship between zeros and factors of polynomials
2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a)
Content
is a factor of p(x).
Standards
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the
polynomial.
Instructional Strategies
Instructional Resources/Tools
As discussed for the previous cluster (Perform arithmetic operations on polynomials),
Graphing calculators
polynomials can often be factored. Even though polynomials (i.e., polynomial expressions) can
be explored as mathematical objects without consideration of functions, in school mathematics,
Graphing software, including dynamic geometry software
polynomials are usually taken to define functions. Some equations may include polynomials on
one or both sides. The importance here is in distinction between equations that have solutions,
Computer Algebra Systems
and functions that have zeros. Thus, polynomial functions have zeros. This cluster is about the
relationship between the factors of a polynomial, the zeros of the function defined by the
polynomial, and the graph of that function. The zeros of a polynomial function are the xintercepts of the graph of the function.
Through some experience with long division of polynomials by
), students get a sense that
the quotient is always a polynomial of a polynomial that is one degree less than the degree of the
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original polynomial, and that the remainder is always a constant. In other words,
. Using this equation, students reason that
. Thus, if
, then the
remainder
, the polynomial
is divisible by
and
is a factor of
.
Conversely, if
) is a factor of
, then
.
Whereas, the first standard specifically targets the relationship between factors and zeros of
polynomials, the second standard requires more general exploration of polynomial functions:
graphically, numerically, verbally and symbolically.
Through experience graphing polynomial functions in factored form, students can interpret the
Remainder Theorem in the graph of the polynomial function. Specifically, when
is a factor
of a polynomial
, then
, and therefore
is an x-intercept of the graph
. Conversely, when students notice an x-intercept near
in the graph of a polynomial
function
, then the function has a zero near
, and
is near zero. Zeros are located
approximately when reasoning from a graph. Therefore, if
is not exactly zero, then
is not a factor of
.
Students can benefit from exploring the rational root theorem, which can be used to find all of
the possible rational roots (i.e., zeros) of a polynomial with integer coefficients. When the goal is
to identify all roots of a polynomial, including irrational or complex roots, it is useful to graph
the polynomial function to determine the most likely candidates for the roots of the polynomial
that are the x-intercepts of the graph.
When at least one rational root
is identified, the original polynomial can be divided by
, so that additional roots can be sought in the quotient. Long division will suffice in simple
cases. Synthetic division is an abbreviated method that is less prone to error in complicated
cases, but Computer Algebra Systems may be helpful in such cases.
Graphs are used to understand the end-behavior of nth degree polynomial functions, to locate
roots and to infer the existence of complex roots. By using technology to explore the graphs of
many polynomial functions, and describing the shape, end behavior and number of zeros,
students can begin to make the following informal observations:
The graphs of polynomial functions are continuous.
An nth degree polynomial has at most n roots and at most n - 1 ―changes of direction‖ (i.e.,
from increasing to decreasing or vice versa).
An even-degree polynomial has the same end-behavior in both the positive and negative
directions: both heading to positive infinity, or both heading to negative infinity, depending
upon the sign of the leading coefficient.
An odd-degree polynomial has opposite end-behavior in the positive versus the negative
directions, depending upon the sign of the leading coefficient.
An odd-degree polynomial function must have at least one real root.
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High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Use polynomial identities to solve problems
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used
Content
to generate Pythagorean triples.
Standards
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with
coefficients determined for example by Pascal‘s Triangle.
Instructional Strategies
Instructional Resources/Tools
In Grade 6, students began using the properties of operations to rewrite expressions in equivalent Graphing calculators
forms. When two expressions are equivalent, an equation relating the two is called an identity
because it is true for all values of the variables. This cluster is an opportunity to highlight
Graphing software, including dynamic geometry software
polynomial identities that are commonly used in solving problems. To learn these identities,
students need experience using them to solve problems.
Computer Algebra System
Students should develop familiarity with the following special products:
2
2
2
(x + y) = x + 2xy + y
2
2
2
(x - y) = x - 2xy + y
2
2
(x + y)(x - y) = x - y
2
(x + a)(x + b) = x + (a + b)x + ab
3
3
2
2
3
(x + y) = x + 3 x y + 3xy + y
3
3
2
2
(x - y) = x - 3x y + 3xy – y
3
Students should be able to prove any of these identities. Furthermore, they should develop
sufficient fluency with the first four of these so that they can recognize expressions of the form
on either side of these identities in order to replace that expression with an equivalent expression
in the form of the other side of the identity.
With identities such as these, students can discover and explain facts about the number system.
For example, in the multiplication table, the perfect squares appear on the diagonal. Diagonally,
next to the perfect squares are ―near squares,‖ which are one less than the perfect square. Why?
Why is the sum of consecutive odd numbers beginning with 1 always a perfect square?
Numbers that can be expressed as the sum of the counting numbers from 1to n are called
triangular numbers. What do you notice about the sum of two consecutive triangular numbers?
Explain why it works.
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The sum and difference of cubes are also reasonable for students to prove. The focus of this
proof should be on generalizing the difference of cubes formula with an emphasis toward finite
geometric series.
Some information below includes additional mathematics that students should learn in order to
take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes
beyond the mathematics that all students should study in order to be college- and career-ready:
Ask students to use the vertical multiplication format (as describe in the first cluster) to write out
term-by-term multiplication to generate (x + y) 3 from the expanded form of (x + y)2. Then use
that expanded result to expand (x + y)4, use that result to expand (x + y)5, and so on. Students
should begin to see the arithmetic that generates the entries in Pascal‘s triangle.
High School Conceptual Category: Algebra
Domain
Arithmetic with Polynomials and Rational Expressions
Cluster
Rewrite rational expressions
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the
Content
degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
Standards
Instructional Strategies
Instructional Resources/Tools
This cluster is the logical extension of the earlier standards on polynomials and the connection
Graphing calculators
to the integers. Now, the arithmetic of rational functions is governed by the same rules as the
arithmetic of fractions, based first on division.
Graphing software, including dynamic geometry software
In particular, in order to write
in the form
, students need to work through the
long division described for A-APR.2-3. This is merely writing the result of the division as a
quotient and a remainder. For example, we can rewrite
in the form
. Note that the
fraction is interpreted as the division
, so that 75 is the dividend and 8 is the divisor.
The result indicates that 9 is the quotient and 3 is the remainder. Note that for division of
integers, we expect the remainder to be between 0 and the divisor, which in this case is 8. (If the
remainder were greater than or equal to 8, we could subtract another 8, and increase the quotient
by 1.)
Computer Algebra Systems
Common Misconceptions
Students with only procedural understanding of fractions are likely to cancel
terms (rather than factors of) in the numerator and denominator of a fraction.
Emphasize the structure of the rational expression: that the whole numerator is
divided by the whole denominator. In fact, the word ―cancel‖ likely promotes
this misconception. It would be more accurate to talk about dividing the
numerator and denominator by a common factor.
In order to rewrite simple rational expressions in different forms, students need to understand
that the rules governing the arithmetic of rational expressions are the same rules that govern the
arithmetic of rational numbers (i.e., fractions). To add fractions, we use a common denominator:
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as long as
. Although in simple situations, a, b, c, and d would each be whole numbers,
in fact they can be polynomials. So now suppose that
, then
And then the numerator can be simplified further:
In order to meet A-APR.6, students will need some experiences with the arithmetic of simple
rational expressions. For most students, the above example helps illustrating the similarity of
the form of the arithmetic used with rational expressions and the form of the arithmetic used
with rational numbers. As indicated by the (+) symbol, some (but not all) students will need to
develop fluency with these skills.
High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Understand solving equations as a process of reasoning and explain the reasoning (cont.)
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Graphing calculators
Investigate the solutions to equations such as
. By graphing the two
functions, y = 3 and
, students can visualize that graphs of the functions
Computer software that generates graphs for visually examining solutions to
only intersect at one point. However, subtracting x = x from the original equation yields
equations, particularly rational and radical
which when both sides are squared produces a quadratic equation that has
two roots x = 2 and x = 6. Students should recognize that there is only one solution (x = 2) and Examples of radical equations that do and do not result in the generation of
that x = 6 is generated when a quadratic equation results from squaring both sides; x = 6 is
extraneous solutions should be prepared for exploration
extraneous to the original equation. Some rational equations, such as
. result
in extraneous solutions as well.
Begin with simple, one-step equations and require students to write out a justification for each
step used to solve the equation.
Common Misconceptions
Students may believe that solving an equation such as 3x + 1 = 7 involves
―only removing the 1,‖ failing to realize that the equation 1 = 1 is being
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subtracted to produce the next step.
Ensure that students are proficient with solving simple rational and radical equations that have
no extraneous solutions before moving on to equations that result in quadratics and possible
solutions that need to be eliminated.
Provide visual examples of radical and rational equations with technology so that students can
see the solution as the intersection of two functions and further understand how extraneous
solutions do not fit the model.
Additionally, students may believe that all solutions to radical and rational
equations are viable, without recognizing that there are times when extraneous
solutions are generated and have to be eliminated.
It is very important that students are able to reason how and why extraneous solutions arise.
High School Conceptual Category: Algebra
Domain
Reasoning with Equations and Inequalities
Cluster
Represent and solve equations and inequalities graphically (cont.)
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x);
Content
find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases
Standards
where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Instructional Strategies
Instructional Resources/Tools
Using technology, have students graph a function and use the trace function to move the cursor Examples of real-world situations that involve linear functions and two-variable
along the curve. Discuss the meaning of the ordered pairs that appear at the bottom of the
linear inequalities
calculator, emphasizing that every point on the curve represents a solution to the equation.
Graphing calculators or computer software that generate graphs and tables for
Begin with simple linear equations and how to solve them using the graphs and tables on a
solving equations
graphing calculator. Then, advance students to nonlinear situations so they can see that even
complex equations that might involve quadratics, absolute value, or rational functions can be
solved fairly easily using this same strategy. While a standard graphing calculator does not
Common Misconceptions
simply solve an equation for the user, it can be used as a tool to approximate solutions.
Students may believe that the graph of a function is simply a line or curve
Use the table function on a graphing calculator to solve equations. For example, to solve the
equation x2 = x + 12, students can examine the equations y = x2 and y = x + 12 and determine
that they intersect when x = 4 and when x = -3 by examining the table to find where the yvalues are the same.
Investigate real-world examples of two-dimensional inequalities. For example, students might
explore what the graph would look like for money earned when a person earns at least $6 per
hour. (The graph for a person earning exactly $6/hour would be a linear function, while the
graph for a person earning at least $6/hour would be a half-plane including the line and all
points above it.) Applications such as linear programming can help students to recognize how
businesses use constraints to maximize profit while minimizing the use of resources. These
situations often involve the use of systems of two variable inequalities.
―connecting the dots,‖ without recognizing that the graph represents all
solutions to the equation.
Students may also believe that graphing linear and other functions is an isolated
skill, not realizing that multiple graphs can be drawn to solve equations
involving those functions.
Additionally, students may believe that two-variable inequalities have no
application in the real world. Teachers can consider business related problems
(e.g., linear programming applications) to engage students in discussions of how
the inequalities are derived and how the feasible set includes all the points that
satisfy the conditions stated in the inequalities.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Content
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Standards
Instructional Strategies
Instructional Resources/Tools
Explore various families of functions and help students to make connections in terms of
Graphing utilities on a calculator and/or computer can be used to demonstrate
general features. For example, just as the function y = (x + 3)2 – 5 represents a translation of
the changes in behavior of a function as various parameters are varied.
the function y = x by 3 units to the left and 5 units down, the same is true for the function y = | Real-world problems, such as maximizing the area of a region bound by a fixed
x + 3 | - 5 as a translation of the absolute value function y = | x |.
perimeter fence, can help to illustrate applied uses of families of functions.
Discover that the factored form of a quadratic or polynomial equation can be used to determine
the zeros, which in turn can be used to identify maxima, minima and end behaviors.
Use various representations of the same function to emphasize different characteristics of that
function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as (0,
-12). However, rewriting the function as y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0).
Furthermore, completing the square allows the equation to be written as y = (x – 2)2 – 16,
which shows that the vertex (and minimum point) of the parabola is at (2, -16).
Common Misconceptions
Students may believe that each family of functions (e.g., quadratic, square root,
etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphs.
Students may also believe that skills such as factoring a trinomial or completing
the square are isolated within a unit on polynomials, and that they will come to
understand the usefulness of these skills in the context of examining
characteristics of functions.
Additionally, student may believe that the process of rewriting equations into
various forms is simply an algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
High School Conceptual Category: UNIT 2 Trigonometric Functions
Domain
Trigonometric Functions
Cluster
Extend the domain of trigonometric functions using the unit circle
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Content
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of
Standards
angles traversed counterclockwise around the unit circle.
Instructional Strategies
Instructional Resources/Tools
Use a compass and straightedge to explore a unit circle with a fixed radius of 1. Help students Compass and straightedge to explore the unit circle and to draw sine and cosine
to recognize that the circumference of the circle is 2π, which represents the number of radians curves and describe their periodicity.
for one complete revolution around the circle. Students can determine that, for example, π/4
radians would represent a revolution of 1/8 of the circle or 45°.
Graphing calculators or computer graphing tools to determine radian measures
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Having a circle of radius 1, the cosine, for example, is simply the x-value for any ordered pair
on the circle (adjacent/hypotenuse where adjacent is the x-length and hypotenuse is 1).
Students can examine how a counterclockwise rotation determines a coordinate of a particular
point in the unit circle from which sine, cosine, and tangent can be determined.
and to find values of the sine, cosine, and tangent functions for any given x input
value.
Some information below includes additional mathematics that students should learn in order
to take advanced courses such as calculus, advanced statistics, or discrete mathematics and
goes beyond the mathematics that all students should study in order to be college- and
career-ready:
Common Misconceptions
Students may believe that there is no need for radians if one already knows how
to use degrees. Students need to be shown a rationale for how radians are unique
in terms of finding function values in trigonometry since the radius of the unit
circle is 1.
Some students can use what they know about 30-60-90 triangles and right isosceles triangles
to determine the values for sine, cosine, and tangent for π/3, π/4, and π/6. In turn, they can
determine the relationships between, for example, the sine of π/6, 7π/6, and 11π/6, as all of
these use the same reference angle and knowledge of a 30-60-90 triangle.
Students may also believe that all angles having the same reference values have
identical sine, cosine and tangent values. They will need to explore in which
quadrants these values are positive and negative.
Provide students with real-world examples of periodic functions. One good example is the
average high (or low) temperature in a city in Ohio for each of the 12 months. These values
are easily located at weather.com and can be graphed to show a periodic change that provides
a context for exploration of these functions.
Allow plenty of time for students to draw – by hand and with technology – graphs of the three
trigonometric functions to examine the curves and gain a graphical understanding of why, for
example, cos (π/2) = 0 and whether the function is even (e.g., cos(-x) = cos(x)) or odd (e.g.,
sin(-x) = -sin(x)). Similarly, students can generalize how function values repeat one another,
as illustrated by the behavior of the curves.
High School Conceptual Category: Functions
Domain
Trigonometric Functions
Cluster
Extend the domain of trigonometric functions using the unit circle (cont.)
5. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Content
Standards
Instructional Strategies
Instructional Resources/Tools
Allow students to explore real-world examples of periodic functions. Examples include
A list of real-world applications of periodic situations that can be modeled by
average high (or low) temperatures throughout the year, the height of ocean tides as they
using trigonometric functions for students to explore.
advance and recede, and the fractional part of the moon that one can see on each day of the
month. Graphing some real-world examples can allow students to express the amplitude,
Graphing calculators or computer software to generate the graphs of
frequency, and midline of each.
trigonometric functions.
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Help students to understand what the value of the sine (cosine, or tangent) means (e.g., that
the number represents the ratio of two sides of a right triangle having that angle measure).
Using graphing calculators or computer software, as well as graphing simple examples by
hand, have students graph a variety of trigonometric functions in which the amplitude,
frequency, and/or midline is changed. Students should be able to generalize about parameter
changes, such as what happens to the graph of y = cos(x) when the equation is changed to y =
3cos(x) + 5.
Some information below includes additional mathematics that students should learn in order
to take advanced courses such as calculus, advanced statistics, or discrete mathematics and
goes beyond the mathematics that all students should study in order to be college- and
career-ready:
Some students can explore the inverse trigonometric functions, recognizing that the periodic
nature of the functions depends on restricting the domain. These inverse functions can then be
used to solve real-world problems involving trigonometry with the assistance of technology.
Common Misconceptions
Students may believe that all trigonometric functions have a range of 1 to -1.
Students need to see examples of how coefficients can change the range and the
look of the graphs.
Students may also believe that restrictions to the domain of trigonometric
functions are not necessary for defining inverse functions.
Students may also believe that sin-1A = 1/sin A, thus confusing the ideas of
inverse and reciprocal functions.
Additionally, students may not understand that when sin A = 0.4, the value of A
represents an angle measure and that the function sin-1(0.4) can be used to find
the angle measure.
High School Conceptual Category: Functions
Domain
Trigonometric Functions
Cluster
Model periodic phenomena with trigonometric functions
2
2
Content
8. Prove the Pythagorean identity sin ( ) + cos ( ) = 1 and use it to calculate trigonometric ratios.
Standards
Instructional Strategies
Instructional Resources/Tools
In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse Drawings of the unit circle can be useful in showing why the Pythagorean
2
2
relationship must be true.
is always 1, the Pythagorean relationship sin ( ) + cos ( ) = 1 is always true. Students can
make a connection between the Pythagorean Theorem in geometry and the study of
trigonometry by proving this relationship. In turn, the relationship can be used to find the
cosine when the sine is known, and vice-versa. Provide a context in which students can
practice and apply skills of simplifying radicals.
Dynamic geometry software, such as Geometer‘s Sketchpad or Geogebra, can be
used to demonstrate that, regardless of the angle measure, the Pythagorean
relationship always holds in the unit circle.
Some information below includes additional mathematics that students should learn in order
to take advanced courses such as calculus, advanced statistics, or discrete mathematics and
goes beyond the mathematics that all students should study in order to be college- and
career-ready:
Common Misconceptions
Students may believe that there is no connection between the Pythagorean
Theorem and the study of trigonometry.
Some students can explore other trigonometric identities, such as the half-angle, double-
Students may also believe that there is no relationship between the sine and
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angle, and addition/subtraction formulas to extend on the Pythagorean relationship. Formulas
should be proven and then used to determine exact values when given an angle measure, to
prove identities, and to solve trigonometric equations. For example, by dividing the formula
2
2
2
2
2
cosine values for a particular angle. The fact that the sum of the squares of these
values always equals 1 provides a unique way to view trigonometry through the
lens of geometry.
sin(A + B) = sinA + sinB and need
Additionally, students may believe that
specific examples to disprove this assumption.
sin ( ) + cos ( ) = 1 by cos ( ) , a new formula is generated ( tan ( ) +1= sec ( ) )
High School Conceptual Category: UNIT 3 Modeling with Functions
Domain
Creating Equations
Cluster
Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple
Content
rational and exponential functions.
Standards
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable in a
modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to
highlight resistance R.
Instructional Strategies
Instructional Resources/Tools
Provide examples of real-world problems that can be modeled by writing an equation or
Graphing calculators
inequality. Begin with simple equations and inequalities and build up to more complex
equations in two or more variables that may involve quadratic, exponential or rational functions. Computer software that generate graphs of functions
Discuss the importance of using appropriate labels and scales on the axes when representing
functions with graphs.
Examples of real-world situations that lend themselves to writing equations
that model the contexts.
Examine real-world graphs in terms of constraints that are necessary to balance a mathematical
model with the real-world context. For example, a student writing an equation to model the
maximum area when the perimeter of a rectangle is 12 inches should recognize that y = x(6 – x)
only makes sense when 0 < x < 6. This restriction on the domain is necessary because the side
of a rectangle under these conditions cannot be less than or equal to 0, but must be less than 6.
Students can discuss the difference between the parabola that models the problem and the
portion of the parabola that applies to the context.
Common Misconceptions
Students may believe that equations of linear, quadratic and other functions are
abstract and exist only ―in a math book,‖ without seeing the usefulness of these
functions as modeling real-world phenomena.
Explore examples illustrating when it is useful to rewrite a formula by solving for one of the
variables in the formula. For example, the formula for the area of a trapezoid ( A = 1 h(b + b ) )
2
1
2
Additionally, they believe that the labels and scales on a graph are not
important and can be assumed by a reader, and that it is always necessary to
use the entire graph of a function when solving a problem that uses that
function as its model.
can be solved for h if the area and lengths of the bases are known but the height needs to be
calculated. This strategy of selecting a different representation has many applications in science
and business when using formulas.
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Provide examples of real-world problems that can be solved by writing an equation, and have
students explore the graphs of the equations on a graphing calculator to determine which parts
of the graph are relevant to the problem context.
Use a graphing calculator to demonstrate how dramatically the shape of a curve can change
when the scale of the graph is altered for one or both variables.
Give students formulas, such as area and volume (or from science or business), and have
students solve the equations for each of the different variables in the formula.
High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs
Content
showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing,
Standards
positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change
from a graph.
Instructional Strategies
Instructional Resources/Tools
Flexibly move from examining a graph and describing its characteristics (e.g., intercepts,
Tables, graphs, and equations of real-world functional relationships.
relative maximums, etc.) to using a set of given characteristics to sketch the graph of a function.
Graphing calculators to generate graphical, tabular, and symbolic
Examine a table of related quantities and identify features in the table, such as intervals on
representations of the same function for comparison.
which the function increases, decreases, or exhibits periodic behavior.
Recognize appropriate domains of functions in real-world settings. For example, when
determining a weekly salary based on hours worked, the hours (input) could be a rational
number, such as 25.5. However, if a function relates the number of cans of soda sold in a
machine to the money generated, the domain must consist of whole numbers.
Common Misconceptions
Students may believe that it is reasonable to input any x-value into a function,
so they will need to examine multiple situations in which there are various
limitations to the domains.
Given a table of values, such as the height of a plant over time, students can estimate the rate of
plant growth. Also, if the relationship between time and height is expressed as a linear equation,
students should explain the meaning of the slope of the line. Finally, if the relationship is
illustrated as a linear or non-linear graph, the student should select points on the graph and use
them to estimate the growth rate over a given interval.
Students may also believe that the slope of a linear function is merely a
number used to sketch the graph of the line. In reality, slopes have real-world
meaning, and the idea of a rate of change is fundamental to understanding
major concepts from geometry to calculus.
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High School Conceptual Category: Functions
Domain
Interpreting Functions
Cluster
Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Content
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Standards
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and
amplitude.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret
these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y =
(1.02)5, y = (0.97)5, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Instructional Strategies
Instructional Resources/Tools
Explore various families of functions and help students to make connections in terms of general Graphing utilities on a calculator and/or computer can be used to demonstrate
features. For example, just as the function y = (x + 3)2 – 5 represents a translation of the
the changes in behavior of a function as various parameters are varied.
function y = x by 3 units to the left and 5 units down, the same is true for the function y = | x +
3 | - 5 as a translation of the absolute value function y = | x |.
Real-world problems, such as maximizing the area of a region bound by a
fixed perimeter fence, can help to illustrate applied uses of families of
Discover that the factored form of a quadratic or polynomial equation can be used to determine
functions.
the zeros, which in turn can be used to identify maxima, minima and end behaviors.
Use various representations of the same function to emphasize different characteristics of that
function. For example, the y-intercept of the function y = x2 -4x – 12 is easy to recognize as (0, 12). However, rewriting the function as y = (x – 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0).
Furthermore, completing the square allows the equation to be written as y = (x – 2)2 – 16, which
shows that the vertex (and minimum point) of the parabola is at (2, -16).
Common Misconceptions
Students may believe that each family of functions (e.g., quadratic, square root,
etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphs.
Examine multiple real-world examples of exponential functions so that students recognize that a
base between 0 and 1 (such as an equation describing depreciation of an automobile [
Students may also believe that skills such as factoring a trinomial or
completing the square are isolated within a unit on polynomials, and that they
will come to understand the usefulness of these skills in the context of
examining characteristics of functions.
x
f(x) = 15,000(0.8) represents the value of a $15,000 automobile that depreciates 20% per year
over the course of x years]) results in an exponential decay, while a base greater than 1 (such as
x
the value of an investment over time [ f(x) = 5,000(1.07) represents the value of an investment
of $5,000 when increasing in value by 7% per year for x years]) illustrates growth.
Additionally, student may believe that the process of rewriting equations into
various forms is simply an algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from
Content
a context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by
Standards
adding a constant function to a decaying exponential and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the
temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the
location of the weather balloon as a function of time.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms.
Instructional Strategies
Instructional Resources/Tools
Provide a real-world example (e.g., a table showing how far a car has driven after a given
Hands-on materials (e.g., paper folding, building progressively larger shapes
number of minutes, traveling at a uniform speed), and examine the table by looking ―down‖ the
using pattern blocks, etc.) can be used as a visual source to build numerical
table to describe a recursive relationship, as well as ―across‖ the table to determine an explicit
tables for examination.
formula to find the distance traveled if the number of minutes is known.
Visuals available to assist students in seeing relationships are featured at the
Write out terms in a table in an expanded form to help students see what is happening. For
National Library of Virtual Manipulatives as well as The National Council of
example, if the y-values are 2, 4, 8, 16, they could be written as 2, 2(2), 2(2)(2), 2(2)(2)(2), etc., Teachers of Mathematics, Illuminations
so that students recognize that 2 is being used multiple times as a factor.
Focus on one representation and its related language – recursive or explicit – at a time so that
students are not confusing the formats.
Common Misconceptions
Students may believe that the best (or only) way to generalize a table of data is
Provide examples of when functions can be combined, such as determining a function
by using a recursive formula. Students naturally tend to look ―down‖ a table to
describing the monthly cost for owning two vehicles when a function for the cost of each (given find the pattern but need to realize that finding the 100 th term requires knowing
the number of miles driven) is known.
the 99th term unless an explicit formula is developed. Students may also
believe that arithmetic and geometric sequences are the same. Students need
Using visual approaches (e.g., folding a piece of paper in half multiple times), use the visual
experiences with both types of sequences to be able to recognize the difference
models to generate sequences of numbers that can be explored and described with both
and more readily develop formulas to describe them.
recursive and explicit formulas. Emphasize that there are times when one form to describe the
function is preferred over the other.
Additionally, advanced students who study composition of functions may
misunderstand function notation to represent multiplication (e.g., f(g(x)) means
to multiply the f and g function values).
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High School Conceptual Category: Functions
Domain
Building Functions
Cluster
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k
Content
given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd
Standards
functions from their graphs and algebraic expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) = 2x3 for x>0
or f(x) = (x+1)/(x-1) for x ≠1.
Instructional Strategies
Instructional Resources/Tools
Use graphing calculators or computers to explore the effects of a constant in the graph of a
Graphing calculator that can be used to explore the effects of parameter
function. For example, students should be able to distinguish between the graphs of y = x2, y =
changes on a graph
2x2, y = x2 + 2, y = (2x)2, and y = (x + 2)2. This can be accomplished by allowing students to
work with a single parent function and examine numerous parameter changes to make
generalizations.
Common Misconceptions
Students may believe that the graph of y = (x – 4)3 is the graph of y = x3
Distinguish between even and odd functions by providing several examples and helping students
shifted 4 units to the left (due to the subtraction symbol). Examples should be
to recognize that a function is even if f(-x) = f(x) and is odd if f(-x) = -f(x). Visual approaches to
explored by hand and on a graphing calculator to overcome this
identifying the graphs of even and odd functions can be used as well.
misconception.
Provide examples of inverses that are not purely mathematical to introduce the idea. For
example, given a function that names the capital of a state, f(Ohio) = Columbus. The inverse
Students may also believe that even and odd functions refer to the exponent of
would be to input the capital city and have the state be the output, such that f--1(Denver) =
the variable, rather than the sketch of the graph and the behavior of the
Colorado.
function.
Some information below includes additional mathematics that students should learn in order to
take advanced courses such as calculus, advanced statistics, or discrete mathematics and goes
beyond the mathematics that all students should study in order to be college- and career-ready:
Students should also recognize that not all functions have inverses. Again using a
nonmathematical example, a function could assign a continent to a given country‘s input, such as
g(Singapore) = Asia. However, g-1(Asia) does not have to be Singapore (e.g., it could be China).
Additionally, students may believe that all functions have inverses and need
to see counter examples, as well as examples in which a non-invertible
function can be made into an invertible function by restricting the domain.
2
-1
For example, f(x) = x has an inverse ( f (x) = x ) provided that the
domain is restricted to x ≥ 0.
Exchange the x and y values in a symbolic functional equation and solve for y to determine the
inverse function. Recognize that putting the output from the original function into the input of the
inverse results in the original input value.
Also, students need to recognize that exponential and logarithmic functions are inverses of one
another and use these functions to solve real-world problems.
Nonmathematical examples of functions and their inverses can help students to understand the
concept of an inverse and determining whether a function is invertible.
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High School Conceptual Category: Functions
Domain
Linear and Exponential Models
Cluster
Construct and compare linear and exponential models and solve problems (cont.)
4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm
Content
using technology.
Standards
Instructional Strategies
Instructional Resources/Tools
Use technology to solve exponential equations such as 3*10x = 450. (In this case, students can
Examples of real-world situations that apply linear and exponential functions to
determine the approximate power of 10 that would generate a value of 150.) Students can also compare their behaviors
take the logarithm of both sides of the equation to solve for the variable, making use of the
inverse operation to solve.
Graphing calculators or computer software that generates graphs and tables of
functions. A graphing tool such as the one found at nlvm.usu.edu is one option.
Common Misconceptions
Students may believe that all functions have a first common difference and need
to explore to realize that, for example, a quadratic function will have equal
second common differences in a table.
Students may also believe that the end behavior of all functions depends on the
situation and not the fact that exponential function values will eventually get
larger than those of any other polynomial functions.
High School Conceptual Category: UNIT 4 Inferences and Conclusions from Data
Domain
Interpreting Categorical and Quantitative Data
Cluster
Summarize, represent, and interpret data on a single count or measurement variable (cont.)
4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets
Content
for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Standards
Instructional Strategies
Instructional Resources/Tools
It is helpful for students to understand that a statistical process is a problem-solving process
TI-84 and TI emulator
consisting of four steps: formulating a question that can be answered by data; designing and
implementing a plan that collects appropriate data; analyzing the data by graphical and/or
Quantitative Literacy Exploring Data module
numerical methods; and interpreting the analysis in the context of the original question.
Opportunities should be provided for students to work through the statistical process. In Grades NCTM Navigating through Data Analysis 9-12.
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6-8, learning has focused on parts of this process. Now is a good time to investigate a problem
of interest to the students and follow it through. The richer the question formulated, the more
interesting is the process. Teachers and students should make extensive use of resources to
perfect this very important first step. Global web resources can inspire projects.
Although this domain addresses both categorical and quantitative data, there is no reference in
the Standards 1 - 4 to categorical data. Note that Standard 5 in the next cluster (Summarize,
represent, and interpret data on two categorical and quantitative variables) addresses analysis
for two categorical variables on the same subject. To prepare for interpreting two categorical
variables in Standard 5, this would be a good place to discuss graphs for one categorical
variable (bar graph, pie graph) and measure of center (mode).
Have students practice their understanding of the different types of graphs for categorical and
numerical variables by constructing statistical posters. Note that a bar graph for categorical data
may have frequency on the vertical (student‘s pizza preferences) or measurement on the vertical
(radish root growth over time - days).
Measures of center and spread for data sets without outliers are the mean and standard
deviation, whereas median and interquartile range are better measures for data sets with outliers.
Introduce the formula of standard deviation by reviewing the previously learned MAD (mean
absolute deviation). The MAD is very intuitive and gives a solid foundation for developing the
more complicated standard deviation measure.
Informally observing the extent to which two boxplots or two dotplots overlap begins the
discussion of drawing inferential conclusions. Don‘t shortcut this observation in comparing two
data sets.
As histograms for various data sets are drawn, common shapes appear. To characterize the
shapes, curves are sketched through the midpoints of the tops of the histogram‘s rectangles. Of
particular importance is a symmetric unimodal curve that has specific areas within one, two, and
three standard deviations of its mean. It is called the Normal distribution and students need to be
able to find areas (probabilities) for various events using tables or a graphing calculator.
Printed media (e.g., almanacs, newspapers, professional reports)
Software such as TinkerPlots and Excel
Show World: This website offers data about the world that is up to date.
Iearn:This website offers projects that students around the world are working
on simultaneously.
Common Misconceptions
Students may believe that a bar graph and a histogram are the same. A bar
graph is appropriate when the horizontal axis has categories and the vertical
axis is labeled by either frequency (e.g., book titles on the horizontal and
number of students who like the respective books on the vertical) or
measurement of some numerical variable (e.g., days of the week on the
horizontal and median length of root growth of radish seeds on the vertical). A
histogram has units of measurement of a numerical variable on the horizontal
(e.g., ages with intervals of equal length).
They may also believe that the lengths of the intervals of a boxplot (min,Q1),
(Q1,Q2), (Q2,Q3), (Q3,max) are related to the number of subjects in each
interval. Students should understand that each interval theoretically contains
one-fourth of the total number of subjects. Sketching an accompanying
histogram and constructing a live boxplot may help in alleviating this
misconception.
Additionally, students may believe that all bell-shaped curves are normal
distributions. For a bell-shaped curve to be Normal, there needs to be 68% of
the distribution within one standard deviation of the mean, 95% within two,
and 99.7% within three standard deviations.
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High School Conceptual Category: Statistics and Probability
Domain
Making Inferences and Justifying Conclusions
Cluster
Understand and evaluate random processes underlying statistical experiments
1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Content
2. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin
Standards
falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Instructional Strategies
Instructional Resources/Tools
Inferential statistics based on Normal probability models is a topic for Advanced Placement
TI-83/84 and TI emulator
Statistics (e.g., t-tests). The idea here is that all students understand that statistical decisions are
made about populations (parameters in particular) based on a random sample taken from the
Quantitative Literacy The Art and Techniques of Simulation module
population and the observed value of a sample statistic (note that both words start with the letter
―s‖). A population parameter (note that both words start with the letter ―p‖) is a measure of
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
some characteristic in the population such as the population proportion of American voters who Report
are in favor of some issue, or the population mean time it takes an Alka Seltzer tablet to
dissolve.
Software such as TinkerPlots and Fathom
As the statistical process is being mastered by students, it is instructive for them to investigate
questions such as ―If a coin spun five times produces five tails in a row, could one conclude that
the coin is biased toward tails?‖ One way a student might answer this is by building a model of
100 trials by experimentation or simulation of the number of times a truly fair coin produces
five tails in a row in five spins. If a truly fair coin produces five tails in five tosses 15 times out
of 100 trials, then there is no reason to doubt the fairness of the coin. If, however, getting five
tails in five spins occurred only once in 100 trials, then one could conclude that the coin is
biased toward tails (if the coin in question actually landed five tails in five spins).
Common Misconceptions
Students may believe that population parameters and sample statistics are one
in the same, e.g., that there is no difference between the population mean
which is a constant and the sample mean which is a variable.
They may also believe that making decisions is simply comparing the value of
one observation of a sample statistic to the value of a population parameter, not
realizing that a distribution of the sample statistic needs to be created
A powerful tool for developing statistical models is the use of simulations. This allows the
students to visualize the model and apply their understanding of the statistical process.
Provide opportunities for students to clearly distinguish between a population parameter which
is a constant, and a sample statistic which is a variable.
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High School Conceptual Category: Statistics and Probability
Domain
Making Inferences and Justifying Conclusions
Cluster
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
Content
4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random
Standards
sampling.
5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
6. Evaluate reports based on data.
Instructional Strategies
Instructional Resources/Tools
This cluster is designed to bring the four-step statistical process (GAISE model) to life and help TI-83/84 and TI emulator
students understand how statistical decisions are made. The mastery of this cluster is
Quantitative Literacy The Art and Techniques of Simulation module
fundamental to the goal of creating a statistically literate citizenry. Students will need to use all
of the data analysis, statistics, and probability concepts covered to date to develop a deeper
Guidelines for Assessment and Instruction in Statistics Education (GAISE)
understanding of inferential reasoning.
Report
Students learn to devise plans for collecting data through the three primary methods of data
production: surveys, observational studies, and experiments. Randomization plays various key
roles in these methods. Emphasize that randomization is not a haphazard procedure, and that it
requires careful implementation to avoid biasing the analysis. In surveys, the sample selected
from a population needs to be representative; taking a random sample is generally what is done
to satisfy this requirement. In observational studies, the sample needs to be representative of the
population as a whole to enable generalization from sample to population. The best way to
satisfy this is to use random selection in choosing the sample. In comparative experiments
between two groups, random assignment of the treatments to the subjects is essential to avoid
damaging problems when separating the effects of the treatments from the effects of some other
variable, called confounding. In many cases, it takes a lot of thought to be sure that the method
of randomization correctly produces data that will reflect that which is being analyzed. For
example, in a two-treatment randomized experiment in which it is desired to have the same
number of subjects in each treatment group, having each subject toss a coin where Heads
assigns the subject to treatment A and Tails assigned the subject to treatment B will not produce
the desired random assignment of equal-size groups.
The advantage that experiments have over surveys and observational studies is that one can
establish causality with experiments.
Standard 4 addresses estimation of the population proportion parameter and the population
mean parameter. Data need not come from just a survey to cover this topic. A margin-of-error
formula cannot be developed through simulation, but students can discover that as the sample
size is increased, the empirical distribution of the sample proportion and the sample mean tend
Making Sense of Statistical Studies
Focus in High School mathematics: Statistics and Probability
Navigating through Data Analysis in Grades 9-12
―A Sequence of Activities for Developing Statistical Concepts,‖ Franklin and
Kader, The Statistics Teacher Network Number 68 Winder 2006,
Software such as TinkerPlots and Fathom
Common Misconceptions
Students may believe that collecting data is easy; asking friends for their
opinions is fine in determining what everyone thinks.
Students may think that causal effect can be drawn in surveys and
observational studies, instead of understanding that causality is in fact a
property of experiments.
Additionally, they may believe that inference from sample to population can be
done only in experiments. They should see that inference can be done in
sampling and observational studies if data are collected through a random
process.
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toward a certain shape (the Normal distribution), and the standard error of the statistics
decreases (i.e. the variation) in the models becomes smaller. The actual formulas will need to be
stated.
Standard 5 addresses testing whether some characteristic of two paired or independent groups is
the same or different by the use of resampling techniques. Conclusions are based on the concept
of p-value. Resampling procedures can begin by hand but typically will require technology to
gather enough observations for which a p-value calculation will be meaningful.
High School Conceptual Category: Statistics and Probability
Domain
Using Probability to Make Decisions
Cluster
Use probability to evaluate outcomes of decisions (cont.)
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
Content
7. (+) Analyze decisions and strategies, using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.)
Standards
Instructional Strategies
Instructional Resources/Tools
This domain and cluster belong to STEM, and hence need not be for all students.
NCTM Navigating through Probability 9-12.
A game of chance is said to be fair if the expected net winnings are 0. If the expected net
winnings is negative, then the player needs to decide if the game is worth playing. For example,
a spinner has 18 red, 18 black and 2 green sections. Suppose, players gain a one score point if
the spinner lands on red, otherwise the players loose a one score point. The probability the
spinner lands on red is . The probability it lands elsewhere is . So, the expected probability
is 1
+ (-1)
= - .053 score points. This means that players should expect to lose a little
over .05 of a score point every time they play the game. Calculating an expected value enables
players to decide whether or not the game is worth playing
Data Driven Mathematics module, Probability Models
Common Misconceptions
Students may believe: Probabilities and expected values aren‘t useful in
making decisions that affect one‘s life. Students need to see that these are not
merely textbook exercises.
Expected values may be used to decide between two strategies. For example, suppose shop
owner needs to decide whether to stock product A or product B and can only stock one of them.
Profit margins for A follow the distribution (in thousands of dollars): 5, 4, 3, 2, 1 with
probabilities .1,.45,.3,.1,.05, respectfully. Those for B follow: 8,7,6,5,4,3,2,1,0 with
probabilities: .1,.15,.15,.1,.1,0,0,0,.4. The expected profit by stocking A is
5(.1)+4(.45)+3(.3)+2(.1)+1(.05) = 3.45 thousands of dollars. The expected profit by stocking B
is 8(.1)+7(.15)+6(.15)+5(.1)+4(.1)+0(.4) = 3.65 thousands of dollars. So, based on expected
values of profit margins, the better choice would be to stock product B.
Conditional probabilities are situations where the interpretation of an observation is dependent
upon or ―conditioned on‖ some other factor. For example, a blood test has been shown to
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indicate the presence of a particular disease 95% of the time when the disease is actually
present. The same blood test gives a false positive result 0.5% of the time. A false positive
result suggests that even though the blood test indicates that the person has the disease (the
positive part) but subsequent, additional testing indicates the person does not have that disease
(hence positive but false or a false positive).
Suppose that one percent of the population actually has the disease. If a person‘s blood test is
positive, how likely is it that the person has the disease?
This scenario can be restated as the following conditional probability problem:
―What is the probability that a person actually has the disease given that (or conditioned on) the
blood test indicates the person has the disease?‖
There are two possibilities for a person to produce a positive blood test result: the person has
the disease or the person does not have the disease.
The probability that a person has the disease given a positive blood test result is 0.01 0.95 =
0.0095. This represents the part of the population that is expected to both provide a positive
blood test result and have the disease. The probability that a person gives a positive blood test
result but does not have the disease is 0.99 0.005 = 0.00495. This represents the part of the
population that is expected to give a positive blood test result but does not have the
disease. The total, or 0.01445 = 0.0095 + 0.00495, represents a part of the population that has
a positive blood test, whether it has a disease or not. The conditional probability that a person
actually has the disease when the blood test is positive is
= 0.657, i.e., the probability that
a person actually has the disease is about 66% given that the blood test results are positive.
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High School Conceptual Category: Unit 1: Polynomial, Rational, and Radical Relationships
Domain
The Complex Number System
Cluster
Perform arithmetic operations with complex numbers
1. Know there is a complex number i such as i2 = -1, and every complex number has the form a +bi with a and b real.
Content
2. Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Standards
Instructional Strategies
Instructional Resources/Tools
Before introducing complex numbers, revisit simpler examples demonstrating how number
Computer algebra systems
systems can be seen as ―expanding‖ from other number systems in order to solve more
equations. For example, the equation x + 5 = 3 has no solution as a whole numbers, but it has a
Graphing calculator
solution x = -2 as an integers. Similarly, although 7x = 5 has no solution in the integers, it has a
solution x = 5/7 in the rational numbers. The linear equation ax + b = c, where a, b, and c are
http://www.learner.org/courses/learningmath/number/session2/index.html This session focuses on exploration of the number sets that make up the real
rational numbers, always has a solution x in the rational numbers:
.
number system and on the concept of infinity and the importance of zero.
When moving to quadratic equations, once again some equations do not have solutions, creating
a need for larger number systems. For example, x2 – 2 = 0 has no solution in the rational
numbers. But it has solutions
in the real numbers. (The real number line augments the
rational numbers, completing the line with the irrational numbers.)
Point out that solving the equation x2 – 2 = 0 in terms of x is equivalent to finding x-intercepts
of a graph of y = x2 – 2, which crosses the x-axis at
and
. Thus, the graph
illustrates that the solutions are
.
Next, use an example of a quadratic equation with real coefficients, such as x2 + 1 = 0, which
can be written equivalently as x2 = -1. Because the square of any real number is non-negative, it
follows that x2 = -1 has no solution in the real numbers. One can see this graphically by noticing
that the graph of y = x2 + 1 does not cross the x-axis.
2
The ―solution‖ to this ―impasse‖ is to introduce a new number, the imaginary unit i, where i = 1, and to consider complex numbers of the form a +bi, where a and b are real numbers and i is
not a real number. Because i is not a real number, expressions of the form a +bi cannot be
simplified.
The existence of i, allows every quadratic equation to have two solutions of the form a + bi –
either real when b = 0, or complex when b ≠ 0. Have students observe that if a quadratic
equation (with real coefficients) has complex solutions, the solutions always appear in
conjugate pairs, in the form a + bi and a – bi. Particularly, for an equation x2 = - 9, a conjugate
pair of solutions are 0 +3i and 0 – 3i.
http://www.clarku.edu/~djoyce/complex/numberi.html - This article is a short
tour into the history of number i
http://www.math.hmc.edu/calculus/tutorials/complex/ - Complex numbers:
forms and operations
The Ohio Resource Center
The National Council of Teachers of Mathematics, Illuminations
Common Misconceptions
If irrational numbers are confused with non-real or complex numbers, remind
students about the relationships between the sets of numbers.
If an imaginary unit iis misinterpreted as -1 instead of
definition of i.
, re-establish a
Some properties of radicals that are true for real numbers are not true for
complex numbers.
In particular, for positive real numbers a and b,
but
In order to find solutions of quadratic equations or to create quadratic equations from its
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solutions, introduce students to the condition of equality of complex numbers, with addition,
subtraction and multiplication of complex numbers.
Stress the importance of the relationships between different number sets and their properties.
The complex number system possesses the same basic properties as the real number system:
that addition and multiplication are commutative and associative; the existence of additive
identity and multiplicative identity; the existence of an additive inverse for every complex
number and the existence of multiplicative inverse or reciprocal for every non-zero complex
number; and the distributive property of multiplication over the addition. An awareness of the
properties minimizes students‘ rote memorization and links the rules for manipulations with the
complex number system to the rules for manipulations with binomials with real coefficients of
the form a + bx.
and
but
. If those
properties are getting misused, provide students with an example, such as
that
leads to a contradiction that a positive real number is equal to a negative real
number.
High School Conceptual Category: Number and Quantity
Domain
The Complex Number System
Cluster
Use complex numbers in polynomial identities and equations
7. Solve quadratic equations with real coefficients that have complex solutions
Content
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x 2 + 4 as (x +2i)(x - 2i)
Standards
9. (+) Know the Fundamental Theorem of Algebra; show that it is tr