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Fourth Grade Mathematics
Incorporated Throughout the Year
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reason of others.
Model with mathematics
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Grading Period 4
Measurement and Data
Represent and interpret data.
3.
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.
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.
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.
Columbus City Schools
2014-2015
What Can I Do At Home?
Grade 4
Fourth Grading Period
During the second grading period, in math,
your child will be expected to:
Here’s what you can do to help your child
master these skills:
Measurement and Data
♦ 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.
♦ Recognize angles as geometric shapes that formed
wherever two rays share a common endpoint and
understand concepts of angle measurements.
♦ Use two straws, toothpicks, chopsticks, etc. to
form intersecting lines. Have your child
identify the types of angles formed (right,
acute, obtuse, etc.)
♦ Use a pair of straws and a pair of toothpicks to
form identical angles. Ask your child which is
the bigger angle. Discuss that the angle size is
not affected by the length of the rays.
♦ Ask your child to use two straws, toothpicks,
etc. to make “benchmark angles” (90·, 45·,
180·, etc.).
♦ Go on an angle hunt in your neighborhood.
Help your child to see angles in a variety of
situations.
♦ Understand that each ray determines a
direction and the angle size measures the
change from one direction to another.
♦ Measure angles in whole-number degrees using a
protractor. Sketch angles of specified measure.
♦ Recognize angle measure as additive; decompose
an angle measure 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.
Geometry
♦ Draw points, lines, line segments, rays, angles
(right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional
figures.
♦ Classify two-dimensional figures based on the
parallel or perpendicular lines or specified angle
size.
♦ Recognize a line of symmetry for a twodimensional 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.
Columbus City Schools
♦ Draw three lines on a piece of paper. Ask your
child to draw a second line with each of yours so
that one grouping makes a set of parallel lines,
another makes a set of perpendicular lines and
finally the third makes a set of intersecting lines.
♦ Play “I Spy a 2-D Shape” with your child. Ask
them to use precise vocabulary when they give
their clues. For example, “I spy a shape with
four right angles, four vertices and two sets of
parallel sides.”
♦ Give your child various household objects (e.g.,
cans, boxes, pencil, drinking glass, etc.). Ask
him/her to sort the objects into different categories
and explain the sort to you. Ask for the objects to
be resorted another way.
♦ Find different objects in your home that have right
angles. Then, find other objects that have either
larger or smaller angles. Ask your child to sort the
objects into different groups according to the size
of the angle: right angles, acute angles, or obtuse
angles. Also, have your child tell you about the
measures of the angles (acute is less than 90
degrees, obtuse is more than 90 degrees, right is
equal to 90 degrees) and estimate the measure of
each angle.
2012-2013
What Can I Do At Home?
Grade 4
Fourth Grading Period Cont.
Here’s what you can do to help your child
master these skills:
♦ Find items around your home or neighborhood
that represent angles (corners), skew (objects
hanging from a mobile), parallel (train tracks),
and perpendicular (90° intersections).
♦ Compare similar angles using different objects
in your house and identifying that the angle is
the same measure but the sides or rays that
create it can all be different.
♦ Find any object in your home, and determine if
it is symmetrical (when folded in half, both
halves are mirror images of each other). If the
object is symmetrical, identify the lines of
symmetry.
♦ Draw a variety of polygons for your child. Ask
them to guess how many lines of symmetry each
polygon has. Let your child cut out each
polygon to fold the shape and check for
symmetry.
♦ Search “line plots” on the internet. Discuss the
kind of data that is displayed in a line plot.
Look at examples and ask your child questions
about the line plots you see.
♦ Help your child to measure small objects in your
home to the nearest 1/8 of an inch. Work
together to create a line plot that displays this
information. Ask your child questions like
♦ “How many objects measured 1/8 inch? What
would be the total lengths of all of the objects if
we laid them end to end?”
Columbus City Schools
2012-2013
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 4
Common Core Domain
Time Range: 20 Days
Measurement and Data
Common Core Standards
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 ndegrees.
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.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and
analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects
and the use of them to solve problems involving symmetry.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
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Columbus City Schools 2013-2014
Mathematics Model Curriculum
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
Instructional Strategies
Angles are geometric shapes composed of two rays that are infinite in length. Students can 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 larger. Ask questions about the types of angles
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Columbus City Schools 2013-2014
Mathematics Model Curriculum
created. Responses may be in terms of the relationship to right angles. Introduce angles as acute (less than
the measure of a right angle) 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.
Students can also create an angle explorer (two strips of cardboard attached with a brass fastener) to learn
about angles.
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.
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.
Instructional Resources/Tools from ODE Model Curriculum
Cardboard cut in strips to make an angle explorer
Brass fasteners
Protractor
Angle ruler
Straws
Transparencies
Angle explorers
Ohio Resource Center
Sir Cumference and the Great Knight of Angleland: 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
From the National Council of Teachers of Mathematics, Figure This: What’s My Angle? math Challenge # 10 Grade 4 Angle Measurement
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Mathematics Model Curriculum
Students can estimate the measures of the angles between their fingers when they spread out their hand.
From PBS Teachers: 3rd Grade Measuring Game, Identify acute, obtuse and right angles in this online interactive
game
From PBS Teachers: Star Gazing, Determine the correct angle at which to place a telescope in order to see as many
stars as possible in this online interactive game.
Misconceptions/Challenges
As you teach the lessons, identify the misconceptions/challenges that students have with the concepts being taught.
from ODE Model Curriculum
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.
Please read the Teacher Introductions, included in this document, for further understanding.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Equal
Groups
Arrays, 4
Area, 5
Compare
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
Number of Groups Unknown (“How
many groups?” Division
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Unknown Product
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
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Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
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students used math tools to solve the problem, they should include a picture to represent how the
tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
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whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Angles
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Measurement and Data domain is helpful when assessing student
work and student thinking. This information could be used to guide classroom discussions,
understand student misconceptions and provide differentiation opportunities. The focus of
instruction is the Common Core Mathematics Standards.
Angles
Two rays that share the same endpoint or vertex form an angle. The rays are the sides of the
angle. The angle is named with a letter (the name of its vertex) or with three letters (three points
on the angle, with the vertex in the middle).
The interior of an
The exterior of
an angle is the
B
Angle Names:
angle is the area
A
C
B,
ABC,
CBA
between the two
rays. the size of an angle is degrees. A degree can be defined as
area
of used to describe
The
unitoutside
of measure
a wedge shaped unit. All angles are measured in terms of a circle. One complete rotation is
the two
rays.
360º.
If both
rays of an angle were placed together, the measure of that angle would be 0º.
When one ray of the angle is rotated a quarter turn from 0º, it is a 90º angle. If the rotation
continues a quarter of the way each turn, the angles modeled would be 180º, 270º, and 360º
angles.
0º
90º
180º
270º
360º
beginning
quarter turn
half turn
quarter turn
(three-quarters)
full turn
Angles are classified and named by their angle measures.
Right Angle
exactly 90º
Grade 4 Angle Measurement
A right angle measures exactly 90º.
Students can use the corner of a sheet of
paper to determine an angle
classification by placing its corner onto
the vertex and lining up one edge of the
paper with one ray of the angle. If the
corner fits into the angle perfectly so
both rays are touching the edge of the
paper, it is a right angle.
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Obtuse Angle
greater than 90º
and less than 180º
An obtuse angle measures greater than
90º but less than 180º. An obtuse angle
can be identified when one edge of the
paper is lined on one ray of an angle and
the other ray can be seen outside the
other edge of the paper.
Acute Angle
less than 90º
An acute angle measures less than 90º.
An acute angle can be identified when
one edge of the paper is lined up on one
ray and the other ray lies under the
paper.
Straight Angle
exactly 180º
A straight angle measures exactly 180º.
A straight angle can also be called a
straight line. A straight angle can be
identified if one edge of the paper can
line up with both rays of the angle.
A reflex angle measures greater than
180 and less than 360 . A reflex angle
can look like an acute or obtuse angle,
depending on its position. Using a pie
as a model, remove one slice. The
missing portion could look like an acute
or obtuse angle depending on the size of
the slice.
The measure of an angle is determined by the degree or rotation of an angle side, or more simply
stated, how far one side is turned from the other side. Stated differently, an angle’s measure is
the measure of the spread of its rays. Some students may have the misconception that the length
of the rays determines the measure of an angle. Hands-on experiences with measuring angles
will help clear up any misunderstandings.
Both angles are the same
measure, although the length of
their rays is distinctly different.
Reflex Angle
greater than 180 and less than 360
Students will be expected to use benchmark angles (e.g., 45 , 90 , 120 ) to estimate the measure
of angles and use a protractor to measure and draw angles. Before students can effectively use a
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protractor to determine angle measures, they must have a conceptual foundation of the relative
size of angles and their relationships to one another. Many protractors are confusing for
students, as two sets of numbers, an inner ring and an outer ring, appear on a protractor, and
students must decide which set of numbers to use. When students rely on their conceptual
knowledge and the benchmark angles to measure, they build a deeper understanding of angle
sizes.
In addition to estimating angle measures based on the benchmark angles, students will use a
protractor to determine angle measures. To measure an angle using a protractor, place the arrow
of the center point of the protractor located on or along the bottom edge on the vertex of the
angle. Align one of the angle’s rays with the 0 mark. Find the place where the other ray crosses
the protractor and determine which row of numbers to use. Estimating the angle’s measure first
will eliminate confusion about which ring of numbers to use.
Students will also use physical models to determine the sum of the interior angles of triangles
and quadrilaterals. An interior angle can be defined as an angle on the inside of a shape.
Determining the sum of the interior angles can easily be illustrated with a physical model. A
triangle has three interior angles. To determine their sum, tear the triangle into three parts so that
each angle is in a separate part, and line up the angle rays. When the three interior angles are put
together, they measure 180 , or a straight line.
To determine the sum of the interior angles of a quadrilateral, the same exploration can be used.
A quadrilateral has four interior angles. To determine their sum, tear the quadrilateral into four
parts so that each vertex is in one part and line up the angle rays. When the four interior angles
are put together, they measure 360 , or the measure of a circle. This idea applies to any
quadrilateral.
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Another way to determine the number of degrees in a quadrilateral is to remember that every
quadrilateral can be divided into two triangles. If the measure of the sum of the angles in a
triangle is 180°, then the measure of the sum of the angles in a quadrilateral must be 360°
(180° + 180°).
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PRACTICE ASSESSMENT ITEMS
Label the vertex, rays, interior, and exterior of the angle below.
Justin is building a ramp for his skateboard from the arrow on the ground to the
front of the top step. Draw the ramp, label the angle formed by the ramp and the
ground with a letter, and give the measure of the angle formed.
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PRACTICE ASSESSMENT ITEM
Answers/Rubrics
Label the vertex, rays, interior, and exterior of the angle below.
interior
Answer:
ray
ray
vertex
exterior
A 2-point response must have all parts labeled correctly as shown.
A 1-point response has one or two incorrect or missing labels.
A 0-point response shows no mathematical understanding of the question.
Justin is building a ramp for his skateboard from the arrow on the ground to the front of the top
step. Draw the ramp, label the angle formed by the ramp and the ground with a letter, and give
the measure of the angle formed.
A
Answer: A = 20
Accept answers that are within +/- 5 of 20 . The students can label the angle with any letter(s). If
three letters are used there should be one letter on each ray and one letter on the vertex. If one
letter is used it should be on the vertex or in the interior of the angle near the vertex. The response
may also show an arc to indicate the angle, but this does not need to be included for full credit.
A 2-point response includes a ramp, a labeled angle (e.g.,
measure for the angle within 5 degrees of 20 .
A or
BAC), and an appropriate
A 1-point response correctly completes two of the three tasks required or draws a ramp to a point
other than the front of the top step and properly labels and measures that angle.
A 0-point response completes only one of the three tasks required (e.g., only draws the ramp) or
shows no understanding of this task.
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PRACTICE ASSESSMENT ITEMS
In the box, explain how to use a protractor to measure A. Then use a protractor
to find the measure of A. Draw the obtuse angle that when placed with A
would form a straight angle. Label that angle and give its measure.
A=
A
At the school carnival Rey went in a straight line from the Fish Bowl Toss to the
Dunking Booth. He then decided that he was thirsty and headed in a straight line
to the Lemonade Stand. Draw the path that Rey took with his three stops at the
carnival. Label the angle formed with a letter(s), use a protractor to measure the
angle, and tell what type of angle is formed by Rey's path.
Carnival Map
Dunking Booth
Beanbag Toss
Fish Bowl Toss
Hula Hoop Contest
Pie Eating Contest
Lemonade Stand
French Fry Stand
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
In the box, explain how to use a protractor to measure A. Then use a protractor
to find the measure of A. Draw the obtuse angle that when placed with A
would form a straight angle. Label that angle and give its measure.
Answer:
A
B
B=
A=
145°
35°
A 4-point response includes a correct measure for angle A + /- 5°, an obtuse angle that
forms a straight angle with a label and the correct measure and a complete explanation of
how to use a protractor to measure an angle.
A 3-point response includes a complete explanation with a minor error in the angle
measures (e.g., the sum of the two angles is not 180°) or no label on the second angle.
A 2-point response includes correct angle measures with a weak or no explanation or a
complete explanation with incorrect angle measures.
A 1-point response includes one correct angle measure with a weak explanation.
At the school carnival Rey went in a straight line from the Fish Bowl Toss to the Dunking Booth. He then
decided
thatresponse
he was thirsty
andno
headed
in a straightunderstanding
line to the Lemonade
Stand.
A 0-point
shows
mathematical
of this
task.Draw the path that Rey
took with his three stops at the carnival. Label the angle formed with a letter(s), use a protractor to
measure the angle, and tell what type of angle is formed by Rey's path.
Carnival Map
Answer: The angle is
115° and is an obtuse
angle. Letters used to
label the angle will
vary.
A
B
Dunking Booth
Fish Bowl Toss
A
Beanbag Toss
Hula Hoop Contest
Pie Eating Contest
C
Lemonade Stand
French Fry Stand
A 2-point response includes a correctly drawn angle that has been labeled, measured correctly
within 5 of 115 , and classified as obtuse.
A 1-point response includes correct information for an angle that is drawn to different booths than
the ones specified in the problem or includes the correct angle measure for the given path but
neglects to address one of the other parts of the problem (i.e., angle is not labeled or angle is not
classified as obtuse).
A 0-point response shows no understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Select the correct angle measure and name for
ABC.
C
B
A
A. 135; acute
B. 135 ; acute
C. 135 ; obtuse
D. 135; obtuse
Use a protractor to measure the interior angles of Figure A and Figure B. Which
answer shows the correct sum for the interior angles of each figure?
Figure A
Figure B
A. Figure A = 270 Figure B = 180
B. Figure A = 180 Figure B = 360
C. Figure A = 360 Figure B = 90
D. Figure A = 360 Figure B = 180
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Select the correct angle measure and name for
ABC.
C
B
A
A. 135; acute
B. 135 ; acute
C. 135 ; obtuse
D. 135; obtuse
Answer: C
Use a protractor to measure the interior angles of Figure A and Figure B. Which
answer shows the correct sum for the interior angles of each figure?
A. Figure A = 270 Figure B = 180
B. Figure A = 180 Figure B = 360
C. Figure A = 360 Figure B = 90
D. Figure A = 360 Figure B = 180
Answer: D
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PRACTICE ASSESSMENT ITEMS
Find the measure of
F in the quadrilateral below.
159
F
47
103
A. 43
B. 51
C. 61
D. 73
.
Diagonal BD bisects parallelogram ABCD. Without using a protractor, determine
the measure of the angles below. Explain how you found the measure of the
angles.
A
B
30°
D
D
Grade 4 Angle Measurement
30°
B
120°
A
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Find the measurement of
F in the quadrilateral below.
159
F
47
103
A. 43
B. 51
C. 61
D. 73
Answer: B
Diagonal BD bisects parallelogram ABCD. Without using a protractor determine the measure of the
angles below. Explain how you found the measure of the angles.
A
B
30°
Answer:
D
60°
B 60°
A 120°
30°
D
120°
C
A 4-point response includes correct angle measures and a complete explanation (e.g., A
quadrilateral has a total of 360°. The triangle that is labeled has 180° and is exactly half of
the parallelogram. Adding together two smaller 30° angles to get 60° gives the measure of
ADC and CBA. 120° ( BCD) + 60° ( ADC) + 60° ( CBA) = 240°. 360° - 240° = 120°
which is the measure of A.)
A 3-point response includes correct angle measures with an explanation that has minor
omissions.
A 2-point response includes correct angle measures with a weak or no explanation.
A 1-point response gives one correct measure with no explanation.
A 0-point response shows no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Without using a protractor, use the following drawing to find the measure of
BAD. BAC measures 90°. Explain your answer.
B
75°
D
A
C
Amy has drawn a quadrilateral with interior angles that measure 63 , 78 , and
100 . She forgot to record the measure of the final interior angle. Explain how to
determine the measure of that angle. Write the measure in the angle.
100
78
63
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Without using a protractor, use the following drawing to find the measure of
measures 90°. Explain your answer.
BAD.
BAC
Answer: The sum of the interior angles in a triangle is 180 . If you add
ACD which is 90° (the box on the angles indicates
B
90°) and ADC which is 75° the total is 165°.
75°
180° - 165° = 15° so DAC would be 15°.
BAC is 90°
so 90° - 15° (measure of DAC) = 75° which is the
measure of BAD.
A
A 4-point response includes a correct angle measure for BAD with a complete
explanation.
D
C
A 3-point response includes a complete explanation with a minor error in computation that
leads to an incorrect angle measure.
A 2-point response includes a correct angle measure with a weak or no explanation.
A 1-point response shows major errors in reasoning.
A 0-point response shows no mathematical understanding of this task.
Amy has drawn a quadrilateral with interior angles that measure 63 , 78 , and
100 . She forgot to record the measure of the final interior angle. Explain how to
determine the measure of that angle. Write the measure in the angle.
100
78
63
Answer: The missing measure is 119°. The sum of the interior angles in a quadrilateral is
360 . If you add up the three angles that you know and subtract that sum from 360, the
difference will be the number of degrees in the fourth interior angle.
A 2-point response includes a correct angle measure with a complete explanation.
A 1-point response includes a correct angle measure with a weak or no explanation or a
complete explanation with an incorrect angle measure due to a computational error.
A 0-point response shows no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
1) Label the interior and exterior of this triangle. Then, using letters, label the
vertices.
2) The triangle above is an equilateral triangle. Without using a protractor, label
the measure of each of its angles. Explain how you were able to determine the
measures without a protractor.
3) In this isosceles triangle, would the sum of the angles be different than in the
equilateral triangle? Explain your answer.
4) Using a protractor, measure and label each angle of the isosceles triangle.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
B
60°
Exterior
Interior
A
60°
60°
C
1) Label the interior and exterior of this triangle. Then, using letters, label the
vertices.
2) This is an equilateral triangle. Without using a protractor, label the measure of
each of its angles. Explain how you were able to determine the measures
without a protractor.
Label 60 for each of the three angles. The sum of the three interior angles of all
triangles is 180 . Knowing that this is an equilateral triangle means all of the angles
are equal, so each angle must be 60 (180 ÷ 3 = 60°).
3) In this isosceles triangle, would the sum of the angles be different than in the equilateral
triangle? Explain your answer.
No, every triangle has a sum of 180°. The angles will not all measure 60°, but the sum
of all three angles will still be 180°.
B
20°
A
C
4) Using a protractor, measure and label each angle of the isosceles triangle.
Answer:
Grade 4 Angle Measurement
BAC = 75
ACB = 75
ABC = 30
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: angle, circle, circular arc, points, end point, rays, endpoints, degree, intersect,
one-degree angle, protractor, n degree, straight, obtuse, acute, right, vertex, ray, degree,
equation, variable, difference, additive, decompose, measure
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
Find the value for Angle X. Label it as acute, obtuse, or right. Explain how you got
your answer.
X
165°
A water sprinkler rotates one-degree each interval. If the sprinkler rotated threefourths of the way around the circle before it broke, how many one-degree turns did
the sprinkler make?
Stanley is in an extremely slow revolving door. It makes 30 one-degree turns every 4
minutes. At that rate, how long will it take Stanley to complete the full circle and get
back to where he started?
Members of the Ohio State Marching Band were practicing on the field. At the end
of a song, they perform an “about face” and rotate until they are facing the opposite
end of the field. How many one-degree turns are in an “about face”?
A toy chest is broken. When new, the lid opened 90. Now it only opens 37.
How much further does the lid need to open to be like new? Using your protractor,
make a diagram showing how the toy chest opens.
Use your protractor to draw and measure an acute angle. Switch angles with a
partner. What is the difference in degrees between the two angles?
A flagpole was damaged in a wind storm. On Friday, it was leaning 18°from its
original position. On Saturday, it fell another 29°. How many degrees from the
ground is the pole now standing?
Ralphie was looking at a clock and noticed the hands made an angle of about 120.
Classify the angle as acute, obtuse, or right. What time could it have been?
Stevie used his protractor to measure angle HIJ. What measurement did he get for
angle HIJ ? He went to lunch and when he returned, his protractor was gone.
Explain to Stevie how he can find the measurement of angle JIK without his
protractor. Find the measurement of angle JIK.
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J
K
H
I
If the two rays are perpendicular, what is the value of h?
41°
h 35°
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 7
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may or may not use all of the tasks. There are four
different types of tasks. Scaffolding tasks build up to the constructing tasks which develop a
deep understanding of the concept. Next, there are practice tasks and finally performance
tasks which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Using the straight edge of a protractor, students draw an assortment of different-sized angles.
Students need to label their angles as follows: A, B, etc. Students trade papers with a
partner and decide whether the angles are acute, obtuse, or right. Give each student an
“Angle Measurement Lab Sheet” (included in this Curriculum Guide). Record the angle
name and the type of each angle on the lab sheet. Then, students record their estimates of
the measure of each angle. Finally, students use a protractor to measure each angle and
record the actual measurement. Students need to get into the habit of estimating angle
measures before actually using the protractor to measure. On most protractors, there are two
sets of numbers. If a student knows that an angle is acute, he or she will know which number
to use. For example, the student sees the numbers 60° and 120° at the same point on the
protractor. If the student has decided that the angle is acute, prior to measuring it, the angle
must measure 60°
4. Using a straight edge, draw an angle on the overhead. As a group, estimate the size of the
angle in relation to 90°. Give each student a protractor and explain that it is a tool used to
measure angles. Ask if anyone knows how to use the protractor to measure an angle. Allow
students to come to the overhead to show how they would measure the angle. The steps for
using a protractor are included below:
Find the center point on the straight edge of the protractor.
Place the center point over the vertex, or point, of the angle you wish to measure.
Use the benchmark angles to estimate the measure of the angle (45 , 90 , 120 , and
180 ). This will help students determine which set of numbers to use on the protractor.
Line up the zero on the straight edge of the protractor with one of the sides of the angle
and look to see if the 0 is in the bottom set of numbers or the top set of numbers.
Determine which set of numbers will be used on the protractor.
Make sure the second ray of the angle is pointed towards the curved side of the
protractor.
Find the point where the second side of the angle intersects the curved edge of the
protractor.
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6.
7.
8.
9.
Read the number that is written on the protractor at the point of intersection. This is the
measure of the angle in degrees. Make sure that the answer is reasonable. If the angle is
acute, then the measure should be less than 90°. If the angle is obtuse, then the measure
should be more than 90°. If your answer is not reasonable, check to make sure you have
used the correct set of numbers on the protractor.
Next, ask a student to label the vertex, interior, exterior, and rays on the angle. Demonstrate
with several other angles how to use the protractor and how to label the different parts of an
angle. Tell students to use the straight edge of their protractor to draw an angle less than
180 in their math journals or on a plain piece of paper. Instruct students to measure their
angle, record the measurement on their worksheet, and then label each part of their angle.
Have students find a partner, switch papers, measure their partner’s angle, check the labeling,
and then draw a new angle for their partner to measure and label.
Distribute “Angle Measure” (included in this Curriculum Guide) to all students and have
them complete it. Discuss the labels and angle measures as a class, focusing on any
differences. Tell students they need to be within 5 of the correct angle measure.
Provide students with copies of “Circle Geoboard Dot Paper” (included in this Curriculum
Guide). Instruct students to draw one angle on each of the geoboards and estimate the
measure of the angle. They should include a label (e.g., A, B, C, etc.) and the estimated
measure of the angle they drew in degrees inside each angle. (Colored pencils can be used to
help students easily see their angles. The label and the number of degrees for each angle
could be written in the same color as the rays of the angle.) Make sure that students include
benchmark measurements of a 90 angle, a 180 angle, and a 270 angle somewhere on the
geoboards. Ask students to record and explain on another sheet of paper how they estimated
the measure of each angle. For example, a student might draw a 45 angle on the geoboard
and label it A. On their other paper they would state that angle A is a 45 angle because they
drew a line down the middle of a 90 angle, cutting it in half. Finally, students could trade
Circle Geoboard Dot Paper with a partner and have the partner actually measure the angles to
see if the estimates are reasonable.
Demonstrate on the overhead how to draw an angle of a given measure. Start by drawing a
straight line with a dot at one end that will be one of the rays of the angle. The dot will be
the vertex. Place the protractor so that the middle point is on the dot and the line goes toward
0. Find the measure of the angle you want to make and put a tick mark above it. Take the
protractor away and use a straight edge to connect the dot on the line at the vertex with the
tick mark, creating the angle. The line that is drawn is the other ray of the angle. Model
drawing several other angles of different sizes. Give each student a large piece of paper and
a protractor. Have students fold the paper so that it is divided into eight sections. Put up
eight different angle measures on the overhead or chalkboard. Students label the top of each
section of the paper with one of the measures. Using a protractor, students create the given
angles. Once the angles are drawn, label the interior, exterior, vertex, and rays of each angle.
In each section, also write whether the angle is acute, obtuse, straight, or right.
Have students divide a sheet of paper into six rectangles. Using the straight edge of a
protractor, students draw a different-sized angle in each rectangle. Students trade papers
with a partner and decide whether the angles on the paper are acute, right, or obtuse and then
estimate the measure of each angle. Using a protractor, students then find the exact measure
of each angle. If students get into a habit of estimating angles first and then using a
protractor to measure the angle, they will have a better understanding of what the angle
measures look like. For example: If an angle is first estimated as an acute angle, when the
protractor is placed on the vertex of the angle students can look at the two possible numbers
and make a decision. If students see the numbers 60° and 120 where the ray crosses the
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protractor and they know from their estimate that this angle is acute, then they can determine
the correct measure of the angle is 60 rather than 120.
10. Have students draw and label five different angles on the “Angles Comparison Worksheet”
(included in this Curriculum Guide). Be sure that the angles include acute, right, and obtuse
examples. Exchange papers with a partner and have them estimate the measure of each angle
using benchmark measures of 45°, 90°, 120°, and 180°. Return the papers to the original
student. Measure each angle using a protractor and determine the amount of difference
between the estimate and actual measure. Who was closest? What are you using to help you
make accurate estimates? What are other strategies you could use to estimate more
effectively?
11. Students need to understand that the length of the rays (angle sides) do not influence the size
or measure of the angle. Have students find the same size angle in different-sized objects in
the classroom and/or have them draw angles and then reduce or double the length of the
angle sides. Have students measure the angle again to determine its size.
12. Divide the class into six groups. Distribute “The Angler Project Cards” (included in this
Curriculum Guide). There should be one card at each station/desk. Each student will need a
pencil, a protractor, and paper. Assign each group to a beginning table station. Students are
given a set number of minutes (the time given may vary from class to class based on the
readiness level of the students) to read the question and complete the task. Instruct students
to draw any straight lines they need using the straight edge on their protractor. They should
not draw any lines freehand. At the end of the allotted time, ring a bell and have students
rotate stations. When each group has been to all six stations, have them create a poster with
the papers from their group and title it The Angler. Make sure each group discusses the
completed tasks from their group and self correct any errors before putting the papers on the
poster. The cards can also be distributed to small groups, and groups can choose four of the
six cards to complete.
13. Distribute the “Maddening Measurements” sheet (included in this Curriculum Guide) to each
student. Students estimate the angle measures and then find the actual measures using a
protractor. Ask students to find pairs of angles that are the same measure. Does the length of
the ray have an effect on the measure of the angle? Have students write in their math
journals or on a sheet of paper about their discoveries when measuring angles with different
ray lengths.
14. Give each student a copy of “Sum of the Angles in a Triangle" (included in this Curriculum
Guide). Ask students to measure the three angles of triangle ABC and put the measure of
each angle in the appropriate box located in the angle. Tell students not to use the black dots
near the vertices to measure the angles; they should use the vertex itself. The black dots are
for the other part of the activity. They should then find the sum of the three angles of that
triangle and record it in the center of the triangle. Students then use a ruler or the edge of
their protractor to draw a straight line on a blank sheet of paper. Have students use their
protractor to measure the straight line and determine that it measures 180 . Students cut out
triangle ABC and tear it into three pieces so that each piece contains one of the angles (see
the teacher introduction for a model). Rearrange the angles so that the black dots in the
vertices of the angles are toward the straight angle. Glue all the angles from the triangle on
the line so that the angle rays are touching each other. Instruct students to complete the same
process for triangles LMN and XYZ. Challenge students to draw a triangle that they think
this procedure will not work for. Have them cut the triangle out and give it to you. You can
quickly tear off the angles and arrange them on the overhead to show that they equal 180 .
Repeat this process with as many of the student-generated triangles as necessary until the
class is convinced that the sum of the angles of every triangle is 180 . Discuss: What
conclusions can you draw based on what we just did? What do you predict will happen if we
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repeated the same experiment with quadrilaterals? Can you figure out another way to test the
conclusion?
15. Give each student a copy of “Congruent Triangles” (included in this Curriculum Guide) that
has been copied on colored paper, scissors, glue, a protractor, and a blank sheet of paper.
Have students cut out the triangles. Ask them to find a way to arrange each pair of congruent
triangles so that they form a quadrilateral, and glue the quadrilateral on the blank paper. Ask
students what they know about the sum of the angles in one triangle. Since there are 180° in
one triangle and each quadrilateral is made up of two triangles, how many degrees are there
in a quadrilateral? (360°) Have students measure the angles of their quadrilateral to verify
that they add up to 360°. Give students a copy of “Making Triangles” (included in this
Curriculum Guide). Once students have completed the activity, challenge students to come
to the overhead and draw a quadrilateral that cannot be divided into two triangles. They will
soon discover that every quadrilateral can be divided into two triangles, each of which
contains 180°. This proves that every quadrilateral has a total of 360°.
16. Give each student a copy of “Degrees in a Quadrilateral” (included in this Curriculum
Guide). Ask students to measure the angles in each quadrilateral with a protractor and put
the measure of each angle in the appropriate box located in the angle. Cut out the
quadrilaterals one at a time and tear them into four parts so that each part contains an angle.
Draw a point on a blank sheet of paper. Arrange the angles so that the vertices all touch the
point creating 360°. Draw a point for the next quadrilateral and then cut it out and repeat the
process from above until all three of the quadrilaterals have been completed. Students will
see that the four angles of a quadrilateral will always equal 360 .
17. Divide students into pairs. Distribute a protractor and the “Polygon Shapes Record Sheet”
(included in this Curriculum Guide) to each pair of students. Direct students to count the
sides of each regular polygon (all sides and all angles of the polygon are the same) and
measure each angle. Record the name of the polygon, the number of sides, the measure of
each angle, and the sum of all angles on the table. Discuss any patterns that relate the
number of sides a polygon has to the measure of its angles and to its angle sum. Students
will see the measure of each interior angle increases with the number of sides. They will also
see that the sum of all of the interior angles increases by 180º with each additional side that is
added to a polygon. Have students write facts that they can conclude from the chart into their
math journals.
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 440-443
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 98
Practice Master pp. 98
Problem Solving Master pp. 98
Reteaching Master pp. 98
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections:
Sir Cumference and the great Knight of Angleland by Cindy Neuschwander
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Problem Solving Questions
Find the value for Angle X. Label it as acute, obtuse, or right. Explain how you got
your answer.
X
165°
A water sprinkler rotates one-degree each interval. If the sprinkler rotated three-fourths
of the way around the circle before it broke, how many one-degree turns did the
sprinkler make?
Stanley is in an extremely slow revolving door. It makes 30 one-degree turns every 4
minutes. At that rate, how long will it take Stanley to complete the full circle and get
back to where he started?
Members of the Ohio State Marching Band were practicing on the field. At the end of a
song, they perform an “about face” and rotate until they are facing the opposite end of
the field. How many one-degree turns are in an “about face”?
A toy chest is broken. When new, the lid opened 90. Now it only opens 37. How
much further does the lid need to open to be like new? Using your protractor, make a
diagram showing how the toy chest opens.
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Problem Solving Questions
Use your protractor to draw and measure an acute angle. Switch angles with a partner.
What is the difference in degrees between the two angles?
A flagpole was damaged in a wind storm. On Friday, it was leaning 18 from its
original position. On Saturday, it fell another 29 How many degrees from the ground
is the pole now standing?
Ralphie was looking at a clock and noticed the hands made an angle of about 120.
Classify the angle as acute, obtuse, or right. What time could it have been?
Stevie used his protractor to measure angle HIJ. What measurement did he get for angle
HIJ? He went to lunch and when he returned, his protractor was gone. Explain to
Stevie how he can find the measurement of Angle JIK without his protractor.
Find the measurement of angle JIK.
J
K
H
I
If the two rays are perpendicular, what is the value of h?
41°
Grade 4 Angle Measurement
h 35°
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Problem Solving Questions Answers
Find the value for Angle X. Explain how you got your answer.
Answer Key
Answer: Angle X = 105 degrees = obtuse.
90 + 165 = 255 degrees
360 – 255 = 105 degrees
X=
105°
165°
A water sprinkler rotates one-degree each interval. If the sprinkler rotated three-fourths of the way
around the circle before it broke, how many one-degree turns did the sprinkler make?
Answer: The sprinkler made 270 one-degree turns.
¾ of 360 = 270.
Stanley is in an extremely slow revolving door. It makes 30 one-degree turns every 4 minutes. At that
rate, how long will it take Stanley to complete the full circle and get back to where he started?
Answer: It will take Stanley 48 minutes to complete the full circle.
There are 12 30-degree sections in 360 degrees.
12 x 4 min. = 48 min.
Members of the Ohio State Marching Band were practicing on the field. At the end of a song, they
perform an “about face” and rotate until they are facing the opposite end of the field. How many onedegree turns are in an “about face”?
Answer: There are 180 one-degree turns in an “about face.”
If they face the opposite direction, then the band members
have turned a half circle, or ½ of 360°.
A toy chest is broken. When new, the lid opened 90°. Now it only opens 37°. How much further
does the lid need to open to be like new? Using your protractor, make a diagram showing how the toy
chest opens.
Answer: The lid needs to open another 53° to reach 90°.
90° 37° = 53°
37
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Problem Solving Questions Answers
Use your protractor to draw and measure an acute angle. Switch angles with a partner. What is the
difference in degrees between the two angles?
Answers will vary. Students may calculate the degree difference by subtracting the
measurements or by using the protractor to see the difference.
A flagpole was damaged in a wind storm. On Friday, it was leaning 18°from its original position. On
Saturday, it fell another 29°. How many degrees from the ground is the pole now standing?
Answer: The pole is now at a 43°angle from the ground.
18° + 29 °= 47°total slant
90°- 47°= 43°
Ralphie was looking at a clock and noticed the hands made an angle of about 120°. Classify the angle
as acute, obtuse, or right. What time could it have been?
Answers will vary. It is an obtuse angle. A 120°angle is a span of 20 minutes on the clock.
Possible answers: 4:00, 12:20, 8:00, 3:35 etc.
Stevie used his protractor to measure angle HIJ. What measurement did he get for angle HIJ? He went
to lunch and when he returned, his protractor was gone. Explain to Stevie how he can find the
measurement of angle JIK without his protractor. Find the measurement of angle JIK.
Answer: Angle JIK = 45°. Since the two angle degrees added together
must equal 180°,and Stevie had already found Angle HIJ to be 135°,
then he knows the second angle must be 45°. 135° + 45° = 180°
J
K
H
I
If the two rays are perpendicular, what is the value of h?
41
h 35
Answer: h = 14.
41 + h + 35
Grade 4 Angle Measurement
= 90
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Angle Measurement Lab Sheet
Angles drawn by ____________________________
Angles measured by _________________________
Angle Name ( A )
Type of Angle
(acute, obtuse, right)
Estimate of Angle
Size
Actual Measure
of Angle
Attach the sheet of paper where the angles were drawn.
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Angle Measure
Name
Directions: Use three letters to label each angle and measure each angle.
1.
2.
3.
4.
Directions: Write the words vertex, interior, exterior, and ray on each angle to
indicate their location.
5.
6.
7.
8.
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Angle Measure
Answer Key
Directions: Use three letters to label each angle and measure each angle.
Letters used for labeling the angles will vary.
1.
2.
30
M
A
L
60
B
C
N
3.
4.
Q
F
G
112
65
E
R
S
Directions: Write the words vertex, interior, exterior, and ray on each angle to indicate their
location.
Ray
5.
6.
Ray
Exterior
Ray
Exterior
Interior
Ray
Interior
Exterior
Ray
vertex
vertex
8.
7.
Ray
Ray
Interior
Ray
Interior
vertex
Exterior
vertex
Grade 4 Angle Measurement
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Circle Geoboard Dot Paper
Name
Grade 4 Angle Measurement
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Angles Comparison Worksheet
Name
Angle #1
Estimate
Angle #2
Actual Measure
Estimate
Actual Measure
Angle #3
Estimate
Angle #4
Estimate
Actual Measure
Angle #5
Actual Measure
Grade 4 Angle Measurement
Estimate
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Actual Measure
Columbus City Schools 2013-2014
The Angler
Project Cards
Draw a triangle where all
three angles are acute. Give
the measure of each angle.
Now, try to draw a triangle
where all three angles are
obtuse. What do you notice?
Use your protractor to measure
angle A. Give the measure of
angle A and tell if it is an acute,
an obtuse, or a right angle.
Explain how you know.
A
Draw a triangle that contains a
45 angle. Then, draw a
polygon that is not a triangle
that also contains a 45 angle.
How are the polygons similar?
Different?
Draw a polygon that contains
at least two acute angles.
Then, draw a polygon that
contains an obtuse angle.
What strategy did you use to
draw the polygon?
Draw one acute, one obtuse,
and one right angle. Give the
measure of each angle. Then,
draw a different acute, obtuse,
and right angle and give their
measures.
Draw two angles. Make one a
55 angle and the other a 125
angle. What would happen if
you combined the angles?
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Maddening Measurements
Name
A
B
estimate
estimate
actual
actual
D
C
estimate
estimate
actual
actual
Grade 4 Angle Measurement
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Maddening Measurements
page 2
F
E
estimate
estimate
actual
actual
H
G
estimate
estimate
actual
actual
List the pairs of angles that are congruent.
Explain the relationship between the measure of an angle and the length of the
rays.
Grade 4 Angle Measurement
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Maddening Measurements
Answer Key
Name
A
B
estimate
actual
estimate
30°
90°
actual
D
C
estimate
actual
estimate
45°
Grade 4 Angle Measurement
actual
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30°
Columbus City Schools 2013-2014
Maddening Measurements
page 2
Answer Key
F
E
estimate
estimate
45°
actual
120°
actual
H
G
estimate
estimate
90°
actual
actual
List the pairs of angles that are congruent.
90°,
C and
E are 45°,
F and
A and
D are 30°,
120°
B and
G are
H are 120°
Explain the relationship between the measure of an angle and the length of the
rays.
There is no relationship between the angle measure and the length of the rays. The angle
measure is determined by the rotation of the rays not the length.
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Sum of the Angles in a Triangle
Name
B
A
C
N
L
M
Y
Z
X
Grade 4 Angle Measurement
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Sum of the Angles in a Triangle
Answer Key
B
98
41
41
A
C
N
64
90
L
Z
26
M
18
90
Y
72
X
Grade 4 Angle Measurement
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Congruent Triangles
Grade 4 Angle Measurement
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Making Triangles
Name
Draw one straight line that connects any two opposite vertices of each quadrilateral
below. A diagonal line must divide the quadrilateral into two triangles.
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Degrees in a Quadrilateral
Name
A
B
D
C
E
F
G
H
Q
R
T
Grade 4 Angle Measurement
S
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Degrees in a Quadrilateral
Answer Key
A
90
D
90
90
B
90
C
E
108
108
G
H
Q
48
132
48
R
132
T
Grade 4 Angle Measurement
F
72
72
S
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Polygon Shapes Record Sheet
Name
Polygon
Number of Sides
Grade 4 Angle Measurement
Measure of
Interior Angle
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Angle Sum
Columbus City Schools 2013-2014
Polygon Shapes Record Sheet
Answer Key
Polygon
Number of Sides
Measure of
Interior Angle
Angle Sum
Triangle
3
60
180
Square
4
90
360
Pentagon
5
108
540
Hexagon
6
120
720
Grade 4 Angle Measurement
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Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 4
Common Core Domain
Time Range: 20 Days
Geometry
Common Core Standards
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.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and
analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects
and the use of them to solve problems involving symmetry.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Grade 4 Lines and Angles
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Mathematics Model Curriculum
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
Angles
Students can and should make geometric distinctions about angles without measuring or mentioning
degrees. Angles should be classified in comparison to right angles, such as larger than, smaller than or the
same size as a right angle.
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Mathematics Model Curriculum
Students can use the corner of a sheet of paper as a benchmark for a right angle. They can use a right angle
to determine relationships of other angles.
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 in sorting activities. This can be
done informally by folding paper, tracing, creating designs with tiles or investigating reflections in mirrors.
With the use of a dynamic geometric program, students can easily construct points, lines and geometric
figures. They can also draw lines perpendicular or parallel to other line segments.
Two-dimensional shapes
Two-dimensional shapes are classified based on relationships by 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 and perpendicular lines are used to draw rectangles. Repeated experiences in
comparing and contrasting shapes enable students to gain a deeper understanding about shapes and their
properties.
Informal understanding of the characteristics of triangles is developed through angle 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.
Instructional Resources/Tools from ODE Model Curriculum
Mirrors
Geoboards
GeoGebra (a free software for learning and teaching); http://www.geogebra.com.
Misconceptions/Challenges
The following are some common misconceptions for this topic. As you teach the lessons, identify the misconceptions/challenges that
students have with the concepts being taught.
Common Misconceptions from ODE Model Curriculum
Students believe a wide angle with short sides may seem smaller than a narrow angle with long sides. Students can
compare two angles by tracing one and 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 measure of the angle does
not change.
Please read the Teacher Introductions, included in this document, for further understanding.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
Grade 4 Lines and Angles
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Equal
Groups
Arrays, 4
Area, 5
Compare
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
Number of Groups Unknown (“How
many groups?” Division
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Unknown Product
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
Grade 4 Lines and Angles
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Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
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students used math tools to solve the problem, they should include a picture to represent how the
tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
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whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Teacher Introduction
Geometry
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Geometry domain is helpful when assessing student work and student
thinking. This information could be used to guide classroom discussions, understand student
misconceptions and provide differentiation opportunities. The focus of instruction is the
Common Core Mathematics Standards.
Geometry surrounds us in our daily lives. Everywhere we look we are bombarded by geometric
images. Geometry is the study of the properties and relationships among points, lines, angles,
surfaces, and solids. The study of geometry builds interest in mathematics for many types of
learners. Geometry appeals to artistic and spatial learners, as well as engaging all students.
Sound geometry instruction provides opportunities for hands-on and exploratory experiences
with shapes in as many different forms as possible. Multiple and varied geometric experiences
will develop spatial reasoning and problem-solving skills.
Students build upon the foundation built in the primary grades and begin to use a standard
geometric vocabulary to solve and analyze problems. Students expand their geometric
knowledge to include angles, coordinate systems, and geometric models. Students identify,
describe, and model points, planes, and intersecting, parallel, and perpendicular lines and line
segments. In addition, students will be expected to describe, classify, compare, and model twoand three-dimensional objects using their attributes. Students will also identify and define
triangles based on angle measures and side lengths.
Points, Lines, and Planes
An understanding of points, lines, and planes is the foundation to understanding and
communicating geometric ideas. They help us draw pictures, make maps, and build objects.
Points are places in space that can be described by a location. The intersection of two lines
happens at a point. A point has no length or width. A line is a straight path of points that has no
endpoints. It goes on forever in both directions which is indicated by the arrows at either end of
the line. Lines are named by marking two points along the line with letters, although a single
letter can also be used to name them. A line segment is a part of a line that is measured by two
endpoints. The line segment includes all the points between those endpoints. A ray is a part of a
line that includes one endpoint and all the points on the line that go in one direction from that
endpoint. Lines, line segments, and rays have length but no width. An example of a point, a
line, a line segment, and a ray with the symbols representing each are included in the following
chart.
Name
Example
Say
Symbol
point
M
point M
point M
b
line
Grade 4 Lines and Angles
J
K
line J K, line
K J, or line b
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JK
KJ b
Columbus City Schools 2013-2014
Name
line
segment
Example
A
ray
Say
C
L
M
Symbol
line segment
AC
ray LM
AC
CA
LM
Planes are flat surfaces that are infinitely wide and long without depth. Two-dimensional figures
are referred to as plane figures because they have length and width, but no depth. Lines, line
segments, and rays are referred to as one-dimensional figures because they have length but no
width or depth. There are an infinite number of points and lines in a plane. Planes can intersect
each other or be parallel to each other. A line is formed when planes intersect each other.
Types of Lines
Students will also be expected to describe and identify different types of lines and line segments
in the environment. Recognition of the different types of lines is also embedded within the
discussions of the characteristics of two- and three-dimensional shapes. The following table
discusses characteristics of different lines.
Type of Line
Description
Parallel
Parallel lines never touch one another because
they are always the same distance apart.
Intersecting
Intersecting lines cross over one another.
Perpendicular lines are intersecting lines, but
not all intersecting lines are perpendicular.
Perpendicular
Perpendicular lines form a right angle (90°)
where they meet, or intersect. A corner of a
page can fit in a right angle.
Skew
A
B
C
D
Grade 4 Lines and Angles
Skew lines are lines that are neither parallel
nor intersecting because they do not lie in the
same plane. In the mobile shown at the left,
AB and CD are not parallel and they will
never intersect because they are not in the
same plane. Another example is a bookcase.
The top of a bookcase, the bottom of a
bookcase, and the shelves of a bookcase are all
in different planes, therefore they will never
intersect.
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Two-Dimensional Figures
Students will describe, classify, compare, and model two- and three-dimensional objects using
their attributes. Students have to develop a common mathematical vocabulary to describe
characteristics and properties of different figures.
Figure
Circle
Square
Triangle
Characteristics and/or Properties
A circle is a plane figure with all points the same
distance from the center. It has infinite lines of
symmetry. A circle is not a polygon because it is
not formed by line segments.
A square is a quadrilateral (four-sided figure) with
four right angles and four equal sides. A square can
also be classified as a parallelogram, a rectangle,
and/or a rhombus.
A triangle is a three-sided polygon with three
angles. Triangles can have different names based
on the length of sides and/or the size of the angles.
The number of lines of symmetry depends on the
type of triangle.
Hexagon
A hexagon is a six-sided polygon with six angles.
A regular hexagon has six lines of symmetry.
Trapezoid
Parallelogram
Rectangle
Rhombus
Grade 4 Lines and Angles
A trapezoid is a quadrilateral (four-sided figure)
with exactly one pair of parallel lines. A trapezoid
may or may not have one line of symmetry
depending upon its shape.
A parallelogram is a quadrilateral (four-sided
figure) that has two pairs of sides that are parallel
and the same length (congruent). Unless the
parallelogram is a rhombus or a square (all four
sides the same) there are no lines of symmetry.
A rectangle is a parallelogram with four right
angles. A rectangle has two pairs of sides that are
parallel and congruent, but not all four sides are the
same length. A rectangle has two lines of
symmetry.
A rhombus is a parallelogram with all four sides
equal in length. A rhombus with four right angles is
called a square. All other rhombi do not have right
angles. A rhombus has two lines of symmetry.
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Figure
Characteristics and/or Properties
Ellipse
An ellipse is an oval. An ellipse has two lines of
symmetry.
Pentagon
A pentagon is a five-sided figure. A regular
pentagon has five lines of symmetry.
Triangles
Students will extend their understanding of triangles to identify and define triangles based on
angle measures and side lengths. Students must have experiences with different types of
triangles in order to be able to develop the notion that lines of symmetry, congruence, and
similarity can vary between different types of triangles.
Triangles can be defined and classified by angle measures.
Triangles Classified by Angle Measures
Type of Triangle
Equiangular
Right
Characteristics
An equiangular triangle has three angles of
the same measure (60 ). If a triangle is
equiangular, it is also equilateral.
A right triangle has one right angle (90 ).
A right triangle has one line of symmetry if
it is also an isosceles triangle,
Acute
An acute triangle has no angles measuring
90 or greater.
Obtuse
An obtuse triangle has one obtuse angle.
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Triangles can also be defined and classified by side lengths.
Triangles Classified By Side Lengths
Type of Triangle
Equilateral
Characteristics
An equilateral triangle is a triangle that has
three sides of equal length. Equilateral
triangles are also equiangular (all angles have
the same measure). Equilateral triangles have
three lines of symmetry. All equilateral
triangles are similar to each other because an
equilateral triangle is a regular figure.
Isosceles
An isosceles triangle is a triangle that has two
sides of equal length. An isosceles triangle has
one line of symmetry.
Scalene
A scalene triangle is a triangle that has three
sides of different lengths. It has no lines of
symmetry.
All triangles can be classified and described by two names using the angle measures and the side
lengths. However, equilateral triangles can only be named as equiangular because the angles
must all be equal. An equilateral triangle cannot be named as a right or an obtuse triangle, as
triangles cannot be made with three right angles or three angles greater than 90°.
Example: Describe the triangle below using angle measures and side lengths.
Answer: The triangle can be described as isosceles and right. Two of the sides are congruent
(isosceles triangle). One angle is a right angle (right triangle).
Example: Describe the triangle below using angle measures and side lengths.
Answer: The triangle can be described as an obtuse scalene triangle. It has one obtuse angle
(obtuse triangle) and all three sides are different lengths (scalene triangle).
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Quadrilaterals
A quadrilateral is a four-sided figure. Students will explore different kinds of quadrilaterals and
identify similarities and differences among them, recognizing that some quadrilaterals can be
named in more than one way based on their characteristics.
The table below lists different quadrilaterals and their characteristics.
Type of Quadrilateral
Characteristics
rectangle
A rectangle has two pairs of congruent parallel
sides and four right angles. A rectangle is also
a parallelogram.
square
A square has four congruent sides and four
right angles. A square is also a rectangle, a
parallelogram, and a rhombus.
parallelogram
A parallelogram has two pairs of congruent
parallel sides. Opposite angles are congruent,
but they are not right angles.
rhombus
A rhombus has four congruent sides. Opposite
angles are congruent, but they are not right
angles. A rhombus is also a parallelogram.
trapezoid
A trapezoid is a quadrilateral with exactly one
pair of parallel sides.
Some quadrilaterals may be named and/or described with different names based on their
characteristics. For example, a square can also be named as a rectangle, a parallelogram, and a
rhombus. A rhombus and rectangle are parallelograms. In addition, some quadrilaterals may
share certain characteristics with other quadrilaterals, but can never be described as others. For
example, a rhombus can also be a square, but it can never be a trapezoid, as trapezoids have
exactly one pair of parallel sides. Since a trapezoid has only one pair of parallel lines, the only
other name that can be used to classify it is quadrilateral. There are many other four sided
figures that do not have parallel sides and therefore are not given special names. They are just
classified as quadrilaterals. The figures below are examples.
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PRACTICE ASSESSMENT ITEMS
How are these two shapes alike?

A. They both have four right angles.

B. They both have perpendicular lines.

C. They both have two pairs of parallel sides.

D. They both have four angles that are congruent.
Which pattern block piece has right angles?
orange square
yellow hexagon

A. orange square

B. yellow hexagon

C. blue rhombus

D. green triangle
Grade 4 Lines and Angles
blue rhombus
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green triangle
Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
How are these two shapes alike?

A. They both have four right angles.

B. They both have perpendicular lines.

C. They both have two pairs of parallel sides.

D. They both have four angles that are congruent.
Answer: C
Which pattern block piece has right angles?
orange square
yellow hexagon

A. orange square

B. yellow hexagon

C. blue rhombus

D. green triangle
blue rhombus
green triangle
Answer: A
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PRACTICE ASSESSMENT ITEMS
Draw a polygon. Label the interior with Point A and Point C. Label the exterior
with Point B.
Compare the sides and angles of the two quadrilaterals below. Describe two ways
that these quadrilaterals are different.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Draw a polygon. Label the interior with Point A and Point C. Label the exterior
with Point B.
A 2-point response includes a polygon and points drawn so that Point A and Point C are
inside the polygon and Point B is outside the polygon. For example:
A
B
C
A 1-point response includes a drawing of a polygon and points, but the points do not meet
the requirements in the problem or includes a drawing of a shape that is a closed figure,
but not a polygon, with points drawn so that Point A and Point C are inside the figure and
point B is outside the figure.
A 0-point response shows no mathematical understanding of the task.
Compare the sides and angles of the two quadrilaterals below. Describe two ways
that these quadrilaterals are different.
Answer: Differences given by students will vary. Accept reasonable differences that are
related to the sides and the angles of these quadrilaterals.
A 2-point response includes two differences between the figures that are related to the
angles and the sides (e.g., the rectangle has four 90 angles and the trapezoid has two angles
that are less than 90 and two angles that are greater than 90 ; the rectangle has both sets
of opposite sides parallel and the trapezoid has only one set of opposite sides parallel, etc.).
A 1-point response includes only one difference between the figures or includes two or
more ways that the quadrilaterals are the same.
A 0-point response indicates no understanding of this task.
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PRACTICE ASSESSMENT ITEMS
Draw a figure that has two pairs of parallel lines, four congruent sides, four
vertices, and four right angles.
What is the name of this figure?
Underline all the characteristics that tell how the figures are alike.
1. They are all made up of line segments.
2. They all contain right angles.
3. They have at least two sides equal in length.
4. They all have parallel lines.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Draw a figure that has two pairs of parallel lines, four congruent sides, four
vertices, and four right angles.
What is the name of this figure?
A 2-point response draws a square and identifies the polygon as a square.
A 1-point response either draws a square or states that it would be a square.
A 0-point response shows no mathematical understanding of the task.
Underline all the characteristics that tell how the figures are alike.
4. They are all made up of line segments.
5. They all contain right angles.
6. They have at least two sides equal in length.
4. They all have parallel lines.
Answer:
1. They are all made up of line segments.
2. They all contain right angles.
3. They have at least two sides equal in length.
4. They all have parallel lines.
A 2-point response correctly selects choices 1 and 3.
A 1-point response selects one of the correct answers.
A 0-point response indicates no understanding of the task.
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PRACTICE ASSESSMENT ITEMS
Describe how Figure A is more like Figure B than Figure C?
A
B
C
Which letter contains a set of parallel lines?
A, T, H, or K
 A. the letter A
 B. the letter T
 C. the letter H
 D. the letter K
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Describe how Figure A is more like Figure B than Figure C?
A
B
C
Answer: Figure A and Figure B are similar because they have sides that are perpendicular,
all angles are right angles, and opposite sides are parallel.
A 2-point response includes a complete description of how Figure A and Figure B are more
alike than Figure C.
A 1-point response has a weak description of how the figures are similar.
A 0-point response shows no mathematical understanding of this task.
Which letter contains a set of parallel lines?
A, T, H, or K
 A. the letter A
 B. the letter T
 C. the letter H
 D. the letter K
Answer: C
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PRACTICE ASSESSMENT ITEMS
Which lines are perpendicular lines?
A
B
C
D
 A. Lines A and C
 B. Lines B and A
 C. Lines C and D
 D. There are no perpendicular lines.
Use the Venn diagram to sort the following figures.
Parallel
Grade 4 Lines and Angles
Perpendicular
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
Which lines are perpendicular lines?
A
B
C
D
 A. Lines A and C
 B. Lines B and A
 C. Lines C and D
 D. There are no perpendicular lines.
Answer: A
Use the Venn diagram to sort the following figures.
Answer:
Parallel
Perpendicular
A 2-point response correctly sorts all the shapes.
A 1-point response makes an error in sorting one of the shapes by the given attributes.
A 0-point response has no mathematical understanding of this task.
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PRACTICE ASSESSMENT ITEMS
The map below shows some of the streets in German Village.
City Park Ave.
High Street
Whittier Street
Stewart Avenue
Reinhard Avenue
Schiller Park
Bike Path
Name a pair of streets that are intersecting.
__________________________________ and _______________________________
Name a pair of streets that are parallel.
_________________________________ and _________________________________
Name a pair of streets that are perpendicular.
_________________________________ and __________________________________
On the map above label a right angle with an R, an acute angle with an A, and an
obtuse angle with an O.
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
The map below shows some of the streets in German Village.
City Park Ave.
High Street
Whittier Street
Reinhard Avenue
Stewart Avenue
Schiller Park
Bike Path
Name a pair of streets that are intersecting.
__________________________________ and _______________________________
Name a pair of streets that are parallel.
_________________________________ and _________________________________
Name a pair of streets that are perpendicular.
_________________________________ and __________________________________
On the map above label a right angle with an R, an acute angle with an A, and an obtuse angle
with an O.
A 4-point response includes correct answers for all parts of the problem.
A 3-point response includes an error in one part of the response.
A 2-point response does not address all parts of the response (e.g., names all of the streets
correctly but does not label the angles).
A 1-point response includes correct answers for at least two parts of the problem.
A 0-point response includes a correct answer for one part of the problem or shows no
mathematical understanding of the task.
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: : polygon, point, line, plane, two-dimensional, intersecting, parallel,
perpendicular, line segments, ray, quadrilateral, angle, equiangular, right, acute, obtuse,
isosceles, equilateral, scalene, interior, exterior, congruent, symmetry, similar
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
Sort the polygons into a Venn Diagram. Label the right side of your Venn “At least one
set of parallel lines.” Label the left side of your Venn “At least one right angle.” Explain
why you sorted the polygons as you did.
The city is planning to extend a new road through Frieda’s neighborhood. If the new
road extension will intersect Home Avenue at a right angle, which road could they be
extending?
Home Ave.
Safe St.
Cross Rd.
Main St.
Broad St.
Dante was sketching the following polygons. He was having difficulty completing one of
them. Which polygon do you think Dante had trouble sketching and why?
1) an isosceles right triangle
2) a rectangle that is not a parallelogram
3) a parallelogram with exactly one right angle
On the clock face, draw hands to show 1:45. Draw a second hand in your clock that will
divide the 45 minute section into one acute and one obtuse angle. Label your angles.
How many of each type of angle are now on the entire clock?
Amber drew a polygon with one pair of parallel sides and 2 right angles. What shape
best describes the polygon that Amber drew? Draw the polygon.
Draw a map of what Tune Town could look like.
A) Piano, Trumpet, and Horn Streets are all parallel.
B) Treble Lane is perpendicular to Piano, and ends at Horn St.
C) Drum Rd. is intersects Horn, but not at a right angle.
Greta drew a horizontal line through the center of the figure below to show a line of
symmetry. Clark drew a vertical line through the center to show a line of symmetry.
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Which student has done it correctly? Why?
Draw all the lines of symmetry through each of the regular polygons below. What
pattern do you see? How many lines of symmetry do you think a regular polygon with 9
sides will have?
2.
3.
4.
5.
6.
I cut a shape along a line of symmetry. I noticed my new shape had its own line of
symmetry. What could my original shape have looked like?
CLYDE says that his name, when spelled with capital letters, has a total of 2 lines of
symmetry. Is he right? List the lines of symmetry you can find in the name CLYDE.
https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 6
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may or may not use all of the tasks. There are four
different types of tasks. Scaffolding tasks build up to the constructing tasks which develop a
deep understanding of the concept. Next, there are practice tasks and finally performance
tasks which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
Give each pair of students a geoboard, rubber bands, and a blank index card. Tell students to
make three or four different triangles on their geoboard. Students use the index card as a
“right angle tester”. Show students how to use one corner of the card to determine if their
triangles have right angles (90°), acute angles (less than 90°), and/or obtuse angles (greater
than 90° but less than 180°). Instruct students to draw sketches of their triangles in their
math journal and identify each angle as right, acute, or obtuse. Ask students the following
questions: Is it possible for a triangle to have two right angles? Can all of the angles be
obtuse? Can all of the angles be acute? Does the size of the triangle affect the size of the
angles? (i.e., does a “big” triangle have obtuse angles, while a “small” triangle has acute
angles?) Have a student that has an example of a triangle with a right angle come up and
draw it on the overhead. Discuss its characteristics (a right triangle has one angle that is a
right angle). Continue with a triangle with an obtuse angle and discuss its characteristics (an
obtuse triangle has one obtuse angle and two acute angles) and the characteristics of a
triangle with all acute angles (an equilateral triangle has all acute angles with the same
measure of 60 ).
Distribute “Identify the Angles” (included in this Curriculum Guide). Using the shapes
found on the lab sheet, students will decide if each shape has right angles, acute angles,
and/or obtuse angles, and how many of each type of angle.
Give each pair of students a bag that has one of each pattern block shape in it (i.e., a yellow
hexagon, a red trapezoid, a green triangle, an orange square, a blue rhombus, and a tan
rhombus). Students take the shapes out of the bag and put them on the desk in front of them.
Give students clues about the characteristics of the shapes. Students hold up the shape they
think is being described. Find the shape that has two pairs of parallel lines, four equal sides,
and four right angles (orange square). Find the shape that has three equal sides and three
equal angles (green triangle). Find the shape that has one pair of parallel lines and a pair of
sides that are equal in length (red trapezoid). Find the shape that has two pairs of parallel
lines and four sides equal in length, but no right angles (blue or tan rhombus). Find the shape
that has three pairs of parallel lines and all sides are equal (yellow hexagon).
Distribute “Dot Paper” (included in the Grids and Graphics section of this Curriculum Guide)
and “Isometric Dot Paper” (included in this Curriculum Guide). Instruct students to draw a
right triangle and an obtuse triangle on each sheet. Review the characteristics of each
triangle and quickly check each student’s triangles. Next instruct students to draw an
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equilateral triangle (a triangle that has three equal sides). Which sheet of paper will you have
to use to make an equal sided triangle? (Dot Paper-Isometric) Explain that you cannot use
Dot Paper to make an equal sided triangle because one side will always be longer than the
other two sides. Ask students to look at all of the angles. What do they notice about the
three angles? (they are all acute, the angles all have measures of 60 ) Discuss that this
triangle also has another special name; equiangular. Explain to students that there is another
triangle that is like the equiangular triangle because it has three acute angles but the three
angles do not have to be equal. This is called an acute triangle. Distribute Dot Paper and
have students draw examples of acute triangles.
7. Explain to students that they are going to help sort some triangles into three groups. Prior to
this activity, create a chart that is divided into three columns. At the top of the columns draw
a picture of an isosceles triangle, a scalene triangle, and an equilateral triangle. Also cut out
the “Triangle Cards” (included in this Curriculum Guide). Distribute the cards to students.
Have students come up one at a time to categorize their triangle under a column. If a student
makes a mistake, continue categorizing triangles. At the end, ask students if they think all
the triangles are in the correct columns. If students want to move triangles, they need to
provide their reasoning. After all triangles are correctly placed, make a list of the
characteristics of each group of triangles. Lead students into writing a description for each
triangle column. Have students write the description and draw the triangles in their math
journals. Explain to students that all three of these triangles have special names: isosceles
(two sides are the same length), scalene (no side is the same length), and equilateral (all sides
are the same length). Have students label the triangles in their journals.
8. Give each pair of students a supply of toothpicks and miniature marshmallows (leave the
bag open for a few hours so they are not so soft). Tell students to make specific twodimensional shapes (triangle, square, rectangle, pentagon, etc.) by sticking the ends of the
toothpicks into the marshmallows. Discuss the characteristics of each shape (number of sides
and vertices, length of sides, etc.).
9. Distribute “Shape Match” (included in this Curriculum Guide) to pairs of students. Students
cut apart the cards and make two groups, cards with shapes and cards with characteristics.
Have students shuffle each group of cards so the order is mixed up. Students work in pairs or
small groups to match a characteristic (property) card with the appropriate shape card.
10. Read A Cloak For The Dreamer by Aileen Friedman. Divide students into pairs. Give each
pair of students pattern blocks and a piece of drawing paper. Pairs create as many new
shapes as possible that are a combination of the pattern block shapes. The blocks are traced
on the drawing paper to show the new shape and then the name and characteristics of the new
shape are written under that tracing. Use an Ellison® machine or cut out paper shapes using
the “Shape Templates” (included in this Curriculum Guide) so that each student has two
copies of the following shapes: a square, a rectangle, a trapezoid, an equilateral triangle, a
rhombus, and a circle. Divide three pieces of chart paper in half to create six columns. Glue
a different shape at the top of each column and draw a line under the shapes. Divide the rest
of the column in half by drawing a line so that halves and fourths of shapes can be shown on
the chart (See sample).
Square
Circle
1
2
1
4
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Display the charts at the front of the class. Discuss congruence and symmetry and model
these concepts for the class using one or more of the paper shapes. Explain that each figure
is going to be folded into halves along a line of symmetry and also into fourths along another
line of symmetry if possible. The new shapes made by folding will then be compared and
contrasted with the original shape from which they were created. Using one shape ask the
students to fold it in half along a line of symmetry. Ask for a volunteer to come to the front
of the room and compare the new shape with the original one. Write a couple of words to
describe the new shape and glue or tape it in the appropriate place on the chart. Ask if
anyone else folded the shape in half a different way. Continue asking students to compare
new shapes and glue them on the chart until all of the possible new shapes have been shared.
Next, ask students to take the other copy of the same shape and see if it can be divided into
fourths. Have students share their new shapes and glue them to the chart. Write statements
on the chart comparing and contrasting the shapes created. Repeat the same process until all
six of the shapes have been discussed. Cut many of the six shapes above from different
colors of construction paper (an Ellison® machine works well to cut the shapes). If
construction paper shapes are not possible, this activity could be completed using pattern
blocks and attribute blocks. Place the shapes on a table where they are accessible to the
entire class. Divide students into groups of three or four. Ask each group to sketch a design
for a cloak using any combination of the six shapes used to create the charts. Have one
member of the group go to the table and select the construction paper shapes needed, based
on the group plan, to create the cloak. Give each group a piece of white drawing paper and
have them glue the construction paper shapes on the paper to create their final cloak.
Ask students to discuss the characteristics of the figures that they included in their cloak and
to describe any patterns that they used. Explain to the class that when figures can cover an
area with no gaps and no overlaps it is called a tessellation. Display the cloaks.
11. Divide the class into six groups with four to five students in each group. Cut out and make
six sets of “Where Is That Figure?” (included in this Curriculum Guide). Students match the
characteristic card with the correct shape or line card and name card. Each shape or line card
should be matched with one characteristic card and the corresponding name card. Students
may need to readjust cards that have already matched so that all cards will have a match at
the end of the activity.
12. Two different sets of “Shape Cards” (included in this Curriculum Guide), one with letters on
the shapes and another with numbers are provided. Both sets can be used together for a
Daily Sign In activity or this activity can be done twice with different sets of cards. The sets
can also be used together to find congruent and similar shapes between the two sets of cards.
Make a transparency of one sheet of the shape cards. Review the terms interior and exterior.
Put a point outside of the shape and ask the students whether it is on the interior or the
exterior of the shape. Then put a point on the interior of the shape and ask the same question.
Finally, put a point on the line that forms the perimeter of the shape and ask whether that
point is on the interior or the exterior (it is neither because it lies on the line that forms the
perimeter of the shape). Ask a volunteer to come to the front and point to all of the interior
angles of the first shape. Show the students that the shape also has exterior angles. Using an
index card determine whether the interior angles are acute, obtuse, or right. Divide students
into groups of four. Give each group a set of “Shape Cards” (included in this Curriculum
Guide) and each student a ruler, a pair of scissors, and an index card (to use as a right angle
benchmark). Ask the group to work together to cut out the shape cards. Students then
equally divide the shapes among the group. Each student first looks at all of the interior
angles of the six shapes they have been given using the corner of the index card as the
benchmark for a right angle. Students write A(acute) in angles that are less than a right
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angle, R(right) in right angles, and O(obtuse) in angles that are more than right angles but
less than straight angles. See sample:
A
A
TO
A
13. Students then measure the sides of the shape to determine if they are equal or different and
look to see if opposite sides are parallel. Finally, students determine how many lines of
symmetry the shape has and draw them on the shape. Distribute the “Shape Sorting Record
Sheet” (included in this Curriculum Guide) to each student and have students work as a
group to divide the shapes into the given categories doing one sort at a time. Once the group
agrees that the sort is correct, each student records the letters of the shapes under the
appropriate column headings on his or her record sheet. If a shape fits in both given
categories, then the letter is placed only under the Both column. Letters of shapes that do not
fit in any category of the sort are listed on the Other Shapes line under the table. As a class,
discuss the placement of the shapes and any differences that groups have on their record
sheets. Distribute the “Venn Diagram” graphic (included in this Curriculum Guide) and have
students complete the sort using the Venn diagram either before putting the letters on the
record sheet or to check the completed record sheets. Large circles can also be created for
the Venn diagram by connecting several pipe cleaners to create the circles or by drawing an
empty Venn diagram template on chart paper.
14. Review the terms parallel, perpendicular, and intersecting. Give each student a geoboard and
two rubber bands. Orally give students clues from “Geoboard Shapes” (included in this
Curriculum Guide) that tell them what shapes to create on the geoboard. After each set of
clues is given, have students hold up their geoboards. Ask students to look around the room
and verify that all of the shapes they see have been made following the clues. If students
have different shapes, ask them justify that their figure followed the given set of clues.
15. Give each student a copy of the “Letters” page and the “Venn Diagram” (included in this
Curriculum Guide). Students sort the letters into the categories given on the Venn diagram,
those letters that have only parallel lines, only intersecting lines, both intersecting and
parallel lines, and those that have neither. Before students begin, remind them that lines and
line segments are straight and not curved. This should help them correctly place all of the
letters with curved lines outside both of the circles. Share the sort as a group and have
students justify their placement of letters.
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RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. 438-439, 440-443,
444- 447, 456-457
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. 99, 102
Practice Master pp. 97, 99, 102
Problem Solving Master pp. 97, 99, 102
Reteaching Master pp. 97, 99, 102
INTERDISCIPLINARY CONNECTIONS
1. Literature Connections:
A Cloak for the Dreamer by Aileen Friedman
The Greedy Triangle by Marilyn Burns
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Problem Solving Questions
Sort the polygons into a Venn Diagram. Label the right side of your Venn “At least one set of parallel
lines.” Label the left side of your Venn “At least one right angle.” Explain why you sorted the
polygons as you did.
The city is planning to extend a new road through Frieda’s neighborhood. If the new road extension
will intersect Home Avenue at a right angle, which road could they be extending?
Home Ave.
Safe St.
Cross Rd.
Main St.
Broad St.
Dante was sketching the following polygons. He was having difficulty completing one
of them. Which polygon do you think Dante had trouble sketching and why?
1) an isosceles right triangle
2) a rectangle that is not a parallelogram
3) a parallelogram with exactly one right angle
On the clock face, draw hands to show 1:45. Draw a second hand in your clock that will divide the 45
minute section into one acute and one obtuse angle. Label your angles. How many of each type of
angle are now on the entire clock?
Amber drew a polygon with one pair of parallel sides and 2 right angles. What shape
best describes the polygon that Amber drew? Draw the polygon.
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Problem Solving Questions
Draw a map of what Tune Town could
look like.
Answer
Key
A) Piano, Trumpet, and Horn Streets are all parallel.
B) Treble Lane is perpendicular to Piano, and ends at Horn St.
C) Drum Rd. is intersects Horn, but not at a right angle.
Greta drew a horizontal line through the center of the figure below to show a line of
symmetry. Clark drew a vertical line through the center to show a line of symmetry.
Which student has done it correctly? Why?
Draw all the lines of symmetry through each of the regular polygons below. What pattern do you see?
How many lines of symmetry do you think a regular polygon with 9 sides will have?
I cut a shape along a line of symmetry. I noticed my new shape had its own line of
symmetry. What could my original shape have looked like?
CLYDE says that his name, when spelled with capital letters, has a total of 2 lines of
symmetry. Is he right? List the lines of symmetry you can find in the name CLYDE.
Grade 4 Lines and Angles
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Problem Solving Questions Answers
Sort
theall
polygons
a Venn Diagram.
thethe
right
side polygons
of your Venn
“AtWhat
least one
set of
Draw
the linesinto
of symmetry
through Label
each of
regular
below.
pattern
do parallel
lines.”
Label
the
left
side
of
your
Venn
“At
least
one
right
angle.”
Explain
why
you
sorted
the
you see? How many lines of symmetry do you think a regular polygon with 9 sides will have?
polygons as you did.
At least one At least one set of
right angle parallel lines
Answer:
The city is planning to extend a new road through Frieda’s neighborhood. If the new road extension
will intersect Home Avenue at a right angle, which road could they be extending?
Home Ave.
Safe St.
Main St.
Cross Rd
Broad St.
Answer: They will extend Broad St because it is perpendicular to Home.
Dante was sketching the following polygons. He was having difficulty completing one of them.
Which polygon do you think Dante had trouble sketching and why?
1) an isosceles right triangle
2) a rectangle that is not a parallelogram
3) a parallelogram with exactly one right angle
Answer: Dante could not sketch a rectangle that is not a parallelogram. Because they have two
sets of parallel lines, all rectangles are parallelograms.
On the clock face, draw hands to show 1:45. Draw a second hand in your clock that will divide the 45
minute section into one acute and one obtuse angle. Label your angles. How many of each type of
each angle are now on the entire clock?
Answers will vary. 2 obtuse and 1 acute angle
obtuse
are now on the clock. Possible answer:
acute
obtuse
Amber drew a polygon with one pair of parallel sides and 2 right angles. What shape best describes
the polygon that Amber drew? Draw the polygon.
Answers will vary. The shape that best describes the polygon is a trapezoid.
(Discuss with students the difference and similarity of this trapezoid and
the trapezoid students are more familiar with that is in the pattern block set.
Grade 4 Lines and Angles
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)
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Draw a map of what Tune Town could look like.
A) Piano, Trumpet, and Horn Streets are all parallel.
B) Treble Lane is perpendicular to Piano, and ends at Horn St.
C) Drum Rd. is intersects Horn, but not at a right angle.
Answers will vary, but maps should demonstrate that students understand the terms parallel,
perpendicular, intersect, and right angle.
Greta drew a horizontal line through the center of the figure below to show a line of symmetry. Clark
drew a vertical line through the center to show a line of symmetry. Which student has done it
correctly? Why?
Answer: Clark has drawn the line of symmetry correctly because with a vertical line through
the center, each side is a mirror image.
Draw all the lines of symmetry through each of the regular polygons below. What pattern do you see?
How many lines of symmetry do you think a regular polygon with 9 sides will have?
Answer: The number of lines of symmetry equals the number of sides in a regular polygon with
an odd number of sides. Each line of symmetry extends from one vertex to the midpoint of the
opposite side. A regular polygon with nine sides will have 9 lines of symmetry.
I cut a shape along a line of symmetry. I noticed my new shape had its own line of symmetry. What
could my original shape have looked like?
Answers will vary. Circles, rectangles, squares will work. Hearts, trapezoids, etc. will not.
CLYDE says that his name, when spelled with capital letters, has a total of 3 lines of symmetry. Is he
right? List the lines of symmetry you can find in the name CLYDE.
Answer: Clyde’s name has four lines of symmetry: 1) horizontally through the C
2) vertically through the Y 3) horizontally through the D 4) horizontally through the E
Grade 4 Lines and Angles
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Identify the Angles
Name
Look at each of the shapes below. Fill in the number of each type of angle that is
on the inside (interior) of each shape.
Shape
Grade 4 Lines and Angles
Right
Angles
Acute
Angles
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Obtuse
Angles
Columbus City Schools 2013-2014
Identify the Angles
Answer Key
Name
Look at each of the shapes below. Fill in the number of each type of angle that is
on the inside (interior) of each shape.
Shape
Right
Angles
Acute
Angles
1
2
Obtuse
Angles
6
5
2
2
2
2
1
1
4
2
Grade 4 Lines and Angles
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Isometric Dot Paper
Grade 4 Lines and Angles
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Triangle Cards
Grade 4 Lines and Angles
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Shape Match
Grade 4 Lines and Angles
I am a
quadrilateral
that has two sets
of parallel line
segments, four
right angles,
and two sets of
congruent sides.
I am a
quadrilateral
that has two sets
of parallel line
segments, four
right angles,
and four
congruent sides.
I am a
quadrilateral
that has one set
of parallel line
segments.
I am a polygon
that has three
sides and one
right angle.
I am a polygon
that has three
sets of parallel
line segments
and six sides.
I am a
quadrilateral that
has two sets of
parallel line
segments and my
opposite angles
are equal, but not
equal to 90 .
I am a polygon
that has three
sides, two of
which are equal
in length.
I am a polygon
that has three
equal sides.
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Shape Template
Grade 4 Lines and Angles
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Shape Template
Grade 4 Lines and Angles
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Shape Template
Grade 4 Lines and Angles
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Where Is That Figure?
triangle
A polygon with
three sides
quadrilateral
A polygon with
four sides
parallelogram
A quadrilateral
with opposite
sides that are
equal in length
and parallel
square
A parallelogram
with four equal
sides and four
equal angles
Grade 4 Lines and Angles
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Columbus City Schools 2013-2014
Where Is That Figure?
rectangle
A parallelogram
with four right
angles and
opposite sides of
equal lengths
rhombus
A parallelogram
with four equal
sides and no
right angles
trapezoid
A quadrilateral
with only one pair
of parallel sides
pentagon
A polygon with
five sides
Grade 4 Lines and Angles
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Columbus City Schools 2013-2014
Where Is That Figure?
hexagon
A polygon with
six sides
perpendicular
lines
Lines or line
segments that
intersect to
form right
angles
intersecting
lines
Lines or line
segments that
meet or cross
each other
parallel lines
Lines or line
segments that
are in the same
plane but never
intersect
Grade 4 Lines and Angles
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Columbus City Schools 2013-2014
Where Is That Figure?
circle
A set of points
in a plane that
are the same
distance from a
given point
called the center
line segment
Part of a line
between two
points called
endpoints
ray
Part of a line
that has one
endpoint
Grade 4 Lines and Angles
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Shape Cards
A
B
C
D
F
E
G
Grade 4 Lines and Angles
H
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Shape Cards
I
J
L
K
N
M
P
O
Grade 4 Lines and Angles
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Shape Cards
Q
R
S
T
V
U
X
W
Grade 4 Lines and Angles
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Shape Cards
1
2
3
4
5
6
8
7
Grade 4 Lines and Angles
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Shape Cards
9
10
12
11
14
13
16
15
Grade 4 Lines and Angles
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Shape Cards
18
17
20
19
21
22
23
Grade 4 Lines and Angles
24
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Columbus City Schools 2013-2014
Shape Sorting Record Sheet
Name
Sort the shapes according to the given categories. Look only at the interior angles
of each figure. Record your sort by putting the letter of each shape under the
column where it belongs. If a shape belongs in both of the first two columns, then
list it only in the Both column. Put the shapes back in one pile and then do the
next sort.
Sort 1
All sides the same length
At least one right angle
Both
Other Shapes
All opposite sides
parallel
Sort 2
At least one angle larger
than a right angle
Both
Other Shapes
At least two lines of
symmetry
Sort 3
At least one angle larger
than a right angle
Both
Other Shapes
Grade 4 Lines and Angles
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All opposite sides equal
Sort 4
At least one line of
symmetry
Both
Other Shapes
Sort 5
All opposite angles are
equal
No right angles
Both
Other Shapes
At least one set of
perpendicular lines
Sort 6
At least one angle
smaller than a right
angle
Both
Other Shapes
Grade 4 Lines and Angles
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Columbus City Schools 2013-2014
Shape Sorting Record Sheet
Answer Key
Name
Sort the shapes according to the given categories. Look only at the interior angles
of each figure. Record your sort by putting the letter of each shape under the
column where it belongs. If a shape belongs in both of the first two columns, then
list it only in the Both column. Put the shapes back in one pile and then do the
next sort.
Sort 1
All sides the same length
At least one right angle
Both
C, D, G, J, P, Q, T
B, E, M, N, S, V
A
Other Shapes
F, H, I, K, L, O, R, U, W, X
All opposite sides
parallel
A, B, N
Other Shapes
At least two lines of
symmetry
A, B, D, J, N
Grade 4 Lines and Angles
Sort 2
At least one angle larger
than a right angle
H, O, Q, R, S, U, V, W,
X
Both
C, F, G, I, P, T
D, E, J, K, L, M
Sort 3
At least one angle larger
than a right angle
F, H, I, O, R, S, U, V, W,
X
Page 57 of 63
Both
C, G, P, Q, T
Columbus City Schools 2013-2014
Other Shapes
E, K, L, M
All opposite sides equal
F, I
Other Shapes
Sort 4
At least one line of
symmetry
Both
D, E, H, J, K, M, Q, R, V
A, B, C, G, N, P, T
L, O, S, U, W, X
Sort 5
All opposite angles are equal
No right angles
Both
A, B, N
D, H, J, K, L, O, Q, R,
U, W, X
C, F, G, I, P, T
Other Shapes
At least one set of
perpendicular lines
A, B, N
Other Shapes
Grade 4 Lines and Angles
E, M, S, V
Sort 6
At least one angle
smaller than a right
angle
Both
C, D, F, H, I, J, K, L, O,
R, U, W, X
E, M, S, V
G, P, Q, T
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Venn Diagram
Grade 4 Lines and Angles
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Geoboard Shapes
(Sample Clues)
1) Make a shape that has two sets of parallel lines, four equal sides and four right
angles. (square)
2) Make a shape that has four right angles and the opposite sides are equal and
parallel, but not all four sides are equal. (rectangle)
3) Create one set of parallel lines.
4) Create a pair of intersecting lines.
5) Create a pair of perpendicular lines.
6) Create a shape that has three sides that would intersect if the sides were all
extended. (triangle)
7) Make a shape that has four sides and only one pair of parallel lines. (trapezoid)
8) Make a shape that has four equal sides, no right angles, two pairs of parallel
lines, and the opposite angles are equal. (rhombus)
9) Make a shape that has two pairs of parallel lines, no right angles, opposite sides
are equal but not all sides are the same length, and opposite angles are equal.
(parallelogram)
Grade 4 Lines and Angles
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Letters
A B C D E F
G H
I J K L
M N O P Q R
S T U V W X
Y Z
Grade 4 Lines and Angles
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Parallel
Intersecting
Venn Diagram
Name
Grade 4 Lines and Angles
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Grade 4 Lines and Angles
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Y
K
A
X
V
T
L
Intersecting
`
Columbus City Schools 2013-2014
W
Z
N
M
H
F
E
Parallel
Venn Diagram
Answer Key
O B
R P D C
S I
U
Q G
J
Mathematics Model Curriculum
Grade Level: Fourth Grade
Grading Period: 4
Common Core Domain
Time Range: 5 Days
Measurement and Data
Common Core Standards
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.
The description from the Common Core Standards Critical Area of Focus for Grade 4 says:
Students develop understanding of fraction equivalence and operations with fractions. They recognize that two
different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing
equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of
fractions and the meaning of multiplication to multiply a fraction by a whole number.
Content Elaborations
This section will address the depth of the standards that are being taught.
from ODE Model Curriculum
Ohio has chosen to support shared interpretation of the standards by linking the work of multistate partnerships as
the Mathematics Content Elaborations. Further clarification of the standards can be found through these reliable
organizations and their links:
Achieve the Core Modules, Resources
Hunt Institute Video examples
Institute for Mathematics and Education Learning Progressions Narratives
Illustrative Mathematics Sample tasks
National Council of Supervisors of Mathematics (NCSM) Resources, Lessons, Items
National Council of Teacher of Mathematics (NCTM) Resources, Lessons, Items
Partnership for Assessment of Readiness for College and Careers (PARCC) Resources, Items
Expectations for Learning (Tasks and Assessments)
Classroom tasks and assessments should be constructed in a way that challenges students to think and apply the information they are
learning. Students should be able to summarize, analyze, organize, evaluate, predict, design, create, innovate, etc.
Expectations for Learning (in development) from ODE Model Curriculum
Ohio has selected PARCC as the contractor for the development of the Next Generation Assessments for
Mathematics. PARCC is responsible for the development of the framework, blueprints, items, rubrics, and scoring
for the assessments. Further information can be found at Partnership for Assessment of Readiness for College and
Grade 4 Data
Page 1 of 26
Columbus City Schools 2013-2014
Mathematics Model Curriculum
Careers (PARCC). Specific information is located at these links:
Model Content Framework
Item Specifications/Evidence Tables
Sample Items
Calculator Usage
Accommodations
Reference Sheets
Sample assessment questions are included in this document.
The following website has problem of the month problems and tasks that can be used to assess students and help
guide your lessons.
http://www.noycefdn.org/resources.php
http://illustrativemathematics.org
http://nces.ed.gov/nationsreportcard/itmrlsx/search.aspx
http://nrich.maths.org
At the end of this topic period students will demonstrate their understanding by….
Some examples include:
Constructed Response
Performance Tasks
Portfolios
***A student’s nine week grade should be based on multiple academic criteria (classroom assignments and teacher made
assessments).
Instructional Strategies
Columbus Curriculum Guide strategies for this topic are included in this document.
Also utilize model curriculum, Ohiorc.org, textbooks, InfoOhio, etc. as resources to provide instructional
strategies.
Websites
http://illuminations.nctm.org
http://illustrativemathematics.org
http://insidemathematics.org
http://www.lessonresearch.net/lessonplans1.html
Instructional Strategies from ODE Model Curriculum
Data has been measured and represented on line plots in units of whole numbers, halves or 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 involving addition or subtraction of
fractions.
Have students create line plots with fractions of a unit ( 12 , 14 , 18 ) and plot data showing multiple data points
for each fraction.
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Mathematics Model Curriculum
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 count the fractional numbers as they
would with whole-number counting, but using the fraction name.
3
8
“What is the total number of inches for insects measuring
inches?” Students can use skip counting with
fraction names to find the total, such as, “three-eighths, six-eighths, nine-eighths. 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
to represent the problem and solution such as,
1
8
1
8
+
inch or
1
8
+
5
8
5
8
=
inches?” Have students write number sentences
7
8
inches.
Use visual fraction strips and fraction bars to represent problems to solve problems involving addition and
subtraction of fractions.
Instructional Resources/Tools from ODE Model Curriculum
Fraction bars or strips
Cuisenaire Rods
Number lines
Misconceptions/Challenges
As you teach the lessons, identify the misconceptions/challenges that students have with the concepts being taught.
Students use whole-number names when counting fractional parts on a number line. The fraction name should be
used instead. For example, if two-fourths is represented on the line plot three times, then there would be sixfourths.
Please read the Teacher Introductions, included in this document, for further understanding.
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Teacher Introduction
Problem Solving
The Common Core State Standards for Mathematical Practices focus on a mastery of
mathematical thinking. Developing mathematical thinking through problem solving empowers
teachers to learn about their students’ mathematical thinking. Students progressing through the
Common Core curriculum have been learning intuitively, concretely, and abstractly while
solving problems. This progression has allowed students to understand the relationships of
numbers which are significantly different than the rote practice of memorizing facts. Procedures
are powerful tools to have when solving problems, however if students only memorize the
procedures, then they never develop an understanding of the relationships among numbers.
Students need to develop fluency. However, teaching these relationships first, will allow
students an opportunity to have a deeper understanding of mathematics.
These practices are student behaviors and include:
1.
Make sense of problems and persevere in solving them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Teaching the mathematical practices to build a mathematical community in your classroom is
one of the major reforms in mathematics. The role of the teacher is to facilitate student thinking.
These practices are not taught in isolation, but instead are connected to and woven throughout
students’ work with the standards. Using open-ended problem solving in your classroom can
teach to all of these practices.
A problem-based approach to learning focuses on teaching for understanding. In a classroom
with a problem-based approach, teaching of content is done THROUGH problem solving.
Important math concepts and skills are embedded in the problems. Small group and whole class
discussions give students opportunities to make connections between the explicit math skills and
concepts from the standards. Open-ended problem solving helps students develop new strategies
to solve problems that make sense to them. Misconceptions should be addressed by teachers and
students while they discuss their strategies and solutions.
When you begin using open-ended problem solving, you may want to choose problems from the
Common Core Glossary, Table 2 (below) (multiplication and division situations) to pose to your
students. Included are descriptions and examples of multiplication and division word problem
structures. It is helpful to understand the type of structure that makes up a word problem. As a
teacher, you can create word problems following the structures and also have students follow the
structures to create word problems. This will deepen their understanding and give them
important clues about ways they can solve a problem.
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Table 2 includes word problem structures/situations for multiplication and division from
www.corestandards.org
Equal
Groups
Arrays, 4
Area, 5
Compare
3×6=?
There are 3 bags with 6 plums in
each bag. How many plums are
there in all?
Measurement example: You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
Measurement example: You have
18 inches of string, which you will
cut into 3 equal pieces. How long
will each piece of string be?
There are 3 rows of apples with 6
apples in each row. How many
apples are there?
If 18 apples are arranged into 3
equal rows, how many apples will
be in each row?
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example: What is the area of a
3 cm by 6 cm rectangle?
Area example: A rectangle has an
area of 18 square centimeters. If
one side is 3 cm long, how long is
a side next to it?
A red hat costs $18 and that is 3
times as much as a blue hat costs.
How much does a blue hat cost?
Area example: A rectangle has an area
of 18 square centimeters. If one side is
6 cm long, how long is a side next to it?
Measurement example: A rubber
band is stretched to be 18 cm long
and that is 3 times as long as it was
at first. How long was the rubber
band at first?
a × ? = p, and p ÷ a = ?
Measurement example: A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now as
it was at first?
? × b = p, and p ÷ b = ?
A blue hat costs $6. A red hat costs
3 times as much as the blue hat.
How much does the red hat cost?
Measurement example: A rubber
band is 6 cm long. How long will
the rubber band be when it is
stretched to be 3 times as long?
General
Number of Groups Unknown (“How
many groups?” Division
Group Size Unknown
(“How many in each group?”
Division)
3 × ? = 18, and 18 ÷ 3 = ?
If 18 plums are shared equally into
3 bags, then how many plums will
be in each bag?
Unknown Product
a×b=?
? × 6 = 18, and 18 ÷ 6 = ?
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example: You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
A red hat costs $18 and a blue hat costs
$6. How many times as much does the
red hat cost as the blue hat?
The problem structures become more difficult as you move right and down through the table (i.e.
an “Unknown Product-Equal Groups” problem is the easiest, while a “Number of Groups
Unknown-Compare” problem is the most difficult). Discuss with students the problem
structure/situation, what they know (e.g., groups and group size), and what they are solving for
(e.g., product). The Common Core State Standards require students to solve each type of
problem in the table throughout the school year. Included in this guide are many sample
problems that could be used with your students.
There are three categories of word problem structures/situations for multiplication and division:
Unknown Product, Group Size Unknown and Number of Groups Unknown. There are three
main ideas that the word structures include; equal groups or equal sized units of measure,
arrays/areas and comparisons. All problem structures/situations can be represented using
symbols and equations.
Unknown Product: (a × b = ?)
In this structure/situation you are given the number of groups and the size of each group. You
are trying to determine the total items in all the groups.
Grandma has 4 piles of her famous oatmeal cookies. There are 3 oatmeal cookies in
each pile. How many cookies does Grandma have?
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Group Size Unknown: (a × ? = p and p ÷ a = ?)
In this structure/situation you know how many equal groups and the total amount of items. You
are trying to determine the size in each group. This is a partition situation.
Grandma gives 12 cookies to 4 grandchildren and each grandchild is given the same
number of cookies. How many cookies will each grandchild get?
Number of Groups Unknown: (? × b = p and p ÷ b = ?)
In this structure/situation you know the size in each group and the total amount of items is
known. You are trying to determine the number of groups. This is a measurement situation.
Grandma gave 12 cookies to some of her grandchildren. She gave each grandchild 3
cookies. How many grandchildren got cookies?
Students should also be engaged in multi-step problems and logic problems (Reason abstractly
and quantitatively). They should be looking for patterns and thinking critically about problem
situations (Look for and make use of structure and Look for and express regularity in repeated
reasoning). Problems should be relevant to students and make a real-world connection whenever
possible. The problems should require students to use 21st Century skills, including critical
thinking, creativity/innovation, communication and collaboration (Model with mathematics).
Technology will enhance the problem solving experience.
Problem solving may look different from grade level to grade level, room to room and problem
to problem. However, all open-ended problem solving has three main components. In each
session, the teacher poses a problem, gives students the freedom to solve the problem (using
math tools, drawing a picture, acting it out, etc.), and record their thinking. Finally, the students
share their thinking and strategies (Construct viable arguments and critique the reasoning of
others).
Component 1- Pose the problem
Problems posed during the focus lesson lead to a discussion that focuses on a concept being
taught. Once an open-ended problem is posed, students should solve the problem independently
and/or in small groups. As the year progresses, some of the problems posed during the Math
APSS part of the One-Hour Block of Math will take multiple days to solve (Make sense of
problems and persevere in solving them). Students may solve the problem over one or more
class sessions. The sharing and questioning may take place during a different class session.
Component 2- Solve the problem
During open-ended problems, students need to record their thinking. Students can record their
thinking either formally using an ongoing math notebook or informally using white boards or
thinking paper. Formal math notebooks allow you, parents, and students to see growth as the
year progresses. Math notebooks also give students the opportunity to refer back to previous
strategies when solving new problems. Teachers should provide opportunities for students to
revise their solutions and explanations as other students share their thinking. Students’ written
explanations should include their “work”. This could be equations, numbers, pictures, etc. If
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students used math tools to solve the problem, they should include a picture to represent how the
tools were used. Students’ writing should include an explanation of their strategy as well as
justification or proof that their answer is reasonable and correct.
Component 3- Share solutions, strategies and thinking
After students have solved the problem, gather the class together as a whole to share students’
thinking. Ask one student or group to share their method of solving the problem while the rest of
the class listens. Early in the school year, the teacher models for students how to ask clarifying
questions and questions that require the student(s) presenting to justify the use of their strategy.
As the year progresses, students should ask the majority of questions during the sharing of
strategies and solutions. Possible questions the students could ask include “Why did you solve
the problem that way?” or “How do you know your answer is correct?” or “Why should I use
your strategy the next time I solve a similar problem?” When that student or group is finished,
ask another student in the class to explain in his/her own words what they think the student did to
solve the problem. Ask students if the problem could be solved in a different way and encourage
them to share their solution. You could also “randomly” pick a student or group who used a
strategy you want the class to understand. Calling on students who used a good strategy but did
not arrive at the correct answer also leads to rich mathematical discussions and gives students the
opportunity to “critique the reasoning of others”. This highlights that the answer is not the most
important part of open-ended problem solving and that it’s alright to take risks and make
mistakes. At the end of each session, the mathematical thinking should be made explicit for
students so they fully understand the strategies and solutions of the problem. Misconceptions
should be addressed.
As students share their strategies, you may want to give the strategies names and post them in the
room. When you give the strategy a name (i.e., “What Thomas did is called the ‘Guess and
Check’ strategy.”) it helps students to write and discuss their work more precisely. Add to the list
as strategies come up during student sharing rather than starting the year with a whole list posted.
This keeps students from assuming the strategies on a pre-printed list are the ONLY strategies
that can be used. Possible strategies students may use include:
Act It Out
Make a Table
Find a Pattern
Guess and Check
Make a List
Draw a Picture
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When you use open-ended problem solving in your mathematics instruction, students should
have access to a variety of problem solving “math tools” to use as they find solutions to
problems. Several types of math tools can be combined into one container that is placed on the
table so that students have a choice as they solve each problem, or math tools can be located in a
part of the classroom where students have easy access to them. Remember that math tools, such
as place value blocks or color tiles, do not teach a concept, but are used to represent a concept.
Therefore, students may select math tools to represent an idea or relationship for which that tool
is not typically used (e.g., ten bears may be used to represent a group of 10 rather than selecting a
rod or ten individual cubes). Some examples of math tools may include: one inch color tiles,
centimeter cubes, Unifix® or snap cubes, two-color counters, and frog, bear or other type of
animal counters. Students should also have access to a hundred chart and a number line.
Problem solving can occur in a variety of settings in classrooms. Students may sit in pairs or
small student groups using math tools to solve a problem while recording their thinking on
whiteboards. Students could solve a problem using role playing or SMART Board manipulatives
to act it out. Occasionally whole group thinking with the teacher modeling how to record
strategies can be useful. These whole group recordings can be kept in a class problem solving
book. As students have more experience solving problems they should become more refined in
their use of tools.
The first several weeks of open-ended problem solving can be a daunting and overwhelming
experience. A routine needs to be established so that students understand the expectations during
problem solving time. Initially, the sessions can seem loud and disorganized while students
become accustomed to the math tools and the problem solving process. Students become more
familiar with the routine as the year progresses, so don’t abandon the process if it doesn’t run
smoothly the first few times you try. The more regularly you engage students in open-ended
problem solving, the more organized the sessions become. The benefits and rewards of using a
problem-based approach to teaching and learning mathematics far outweigh the initial confusion
of this approach.
Using the open-ended problem solving approach helps our students to grow as problem solvers
and critical thinkers. Engaging your students in this process frequently will prepare them for
success as 21st Century learners.
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Line Plots
Teacher Introduction
The purpose of the Teacher Introductions is to build content knowledge for teachers. Having
depth of knowledge for the Measurement and Data domain is helpful when assessing student
work and student thinking. This information could be used to guide classroom discussions,
understand student misconceptions and provide differentiation opportunities. The focus of
instruction is the Common Core Mathematics Standards.
Line plots are used to show the frequency of data on a number line. Line plots make it easy to
see how data is grouped. A line plot is made of a number line with the appropriate number of
marks above each number to represent the data. A line plot needs a title so the reader knows
what the data represents. An advantage using a line plot is that it will include every piece of
data. It is similar to a bar graph in that in that there is a bar for every possible value. Line plots
share factual information about the data that is represented and allows opportunities to make
inferences that are not directly stated on the graph.
Line plots provide many opportunities to ask questions similar to questions that can be asked
about bar graphs.
Identify the number of ants that had the longest length?
How do you think the graph would look different if we measured the length of a group of
spiders? (Possible answers: The spiders might be longer in length.)
What’s the difference in length between the longest and shortest ant.
Use the data to create a horizontal bar graph.
What is the combined length of all the ants you measured? Show your work.
0
8
X
X
X
X
X
1
8
2
8
3
8
X
X
X
X
X
X
X
X
X
X
X
X
X
4
8
5
8
6
8
X
X
X
X
X
X
7
8
8
8
Length of Ants
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
The local paper recorded the snowfall for the month of December. Make a line plot, using
intervals of one-eighth. Write an equation representing the difference between the highest
amount of snowfall and the lowest amount of snowfall.
Day
Inches of Snowfall
December 10
1
8
December 15
5
8
December 16
8
8
December 23
11
December 24
12
December 29
5
8
8
8
.
A track coach recorded how far students were able in run in 11 minutes. How many students ran
at least 1.5 miles?
0
4
1
4
X
X
X
2
4
3
4
4
4
X
X
X
X
X
X
X
X
5
4
6
4
7
4
8
4
Distance in Miles
 A. 1 student
 B. 4 students
 C. 8 students
 D. 11 students
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PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
The local paper recorded the snowfall for the month of December. Make a line plot, using
intervals of one-eighth. Write an equation representing the difference between the highest
amount of snowfall and the lowest amount of snowfall.
Day
Inches of Snowfall
December 10
1
8
December 15
5
8
December 16
8
8
December 23
11
December 24
12
December 29
5
8
8
8
X
X
X
0
8
1
8
2
8
3
8
4
8
5
8
6
8
7
8
X
X
X
8
8
9
8
10
8
Inches of Snowfall
Answer:
10
8
1
8
- =
9
1
or 1
8
8
A 2-point response shows a correct line plot including titles, labels, and intervals.
A 1-point response shows a line plot which may be missing a title or labels or intervals.
A 0-point response shows no mathematical understanding of the problem.
A track coach recorded how far students were able in run in 11 minutes. How many students ran
at least 1.5 miles?
A 2-point response includes a correct line plot with a title.
A 1-point response includes a line plot
X
withX some errors
X
in the
X
X
X
data,
X
scale,
X or
X
X
title.
X
1
3
4 shows
5 no mathematical
6
7
 A. response
1 student
A 0-point
includes 0a line plot
with2 many errors
or
4
4
4
4
4
4
4
4
understanding of this task.
Distance in Miles
 B. 4 students
8
4
 C. 8 students
 D. 11 students
Answer: C
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
The students in Mrs. Weaver’s classroom measured objects on a table in the
classroom. On the line plot below is the measured data that was collected. If you
put all the objects together end to end what would be the total length of all objects?
X
X
X
X
4
8
5
8
6
8
7
8
8
8
9
8
X
X
X
X
X
X
X
X
X
10
8
11
8
12
8
13
8
Length of Objects in Inches
Recetta recorded in a table and line plot the length of glass beads she has in her craft box.
Recetta made a mistake. Identify the error.
Number of Beads
Length in
Inches
3
1
4
2
3
8
3
1
2
2
7
8
4
8
8
5
6
8
X
X
X
X
X
X
X
X
1
8
2
8
3
8
4
8
5
8
X
X
X
X
X
X
X
X
X
X
X
6
8
7
8
8
8
Bead Length in Inches
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Columbus City Schools 2013-2014
PRACTICE ASSESSMENT ITEMS
Answers/Rubrics
The students in Mrs. Weaver’s classroom measured objects on a table in the
classroom. On the line plot below is the measured data that was collected. If you
put all the objects together end to end what would be the total length of all objects?
X
X
X
X
4
8
5
8
6
8
7
8
8
8
9
8
X
X
X
X
X
X
X
X
X
10
8
11
8
12
8
13
8
Length of Objects in Inches
Answer: 15
7
8
=
4
8
6
8
8
8
8
8
10
8
10
8
10
8
11
8
11
8
11
8
12
8
13
8
13
8
127
8
15
7
8
or
127
8
Recetta recorded in a table and line plot the length of glass beads she has in her craft box.
Recetta made a mistake. Identify the error.
Bead Lengths
Number of beads Length in
Inches
3
1
4
2
3
8
3
1
2
2
7
8
4
8
8
5
6
8
X
1
8
X
X
X
4
8
X
X
2
8
3
8
5
8
X
X
X
X
X
6
8
X
X
X
X
X
X
7
8
8
8
Length in Inches
Answer: The line plot has the incorrect value data for 82 . The table has 3 data values and
the line has 2 data values.
A 2-point response identifies the correct error.
A 1-point response identifies the correct length but gives the wrong data value.
A 0-point response shows no mathematical understanding of the problem.
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TEACHING STRATEGIES/ACTIVITIES
Vocabulary: line plot, fractions, addition, subtraction data, analyze, compare, evaluate,
graphic organizer, symbols, title, labels, key, symbolic representation, survey, scale,
predict, frequency, interval
1. The word problems below can be presented to students in a variety of ways. Some options
include: using the questions to create a choice-board, Math-O board, one problem per class
session, one problem each evening for homework, partner work or as an assessment question.
The strength of problem solving lies in the rich discussion afterward. As the school year
progresses students should be able to justify their own thinking as well as the thinking of
others. This can be done through comparing strategies, arguing another student’s solution
strategy or summarizing another student’s sharing.
Amanda grew bean sprouts in science class. After a week, she measured the length of
each plant and recorded her data in the table. Record her data on the line plot, label
the line plot, and explain why you placed the data where you did.
0
8
1
8
Number of plants
Height in inches
3
1
4
2
1
8
3
1
2
1
1
2
8
3
8
4
8
5
8
6
8
7
8
8
8
Employees of the zoo were tracking the amount of gallons of milk produced by the
pygmy goats. They recorded the data in the table. Transfer their data to the line plot
and label the line plot. How many total gallons of milk did the goats produce?
# of Goats
0
4
1
4
2
4
Gallons of milk
1
1
4
2
4
4
2
5
4
2
7
4
3
4
4
4
5
4
6
4
7
4
8
4
Zoey has used a line plot to record the length, in inches, of the insects in her
collection. How many of her insects are over one inch long? If Zoey placed her
longest insects end to end, what would the total length be?
Grade 4 Data
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Columbus City Schools 2013-2014
X
X
0
8
X
X
1
8
2
8
3
8
4
8
5
8
6
8
X
X
X
X
X
X
X
X
X
7
8
8
8
9
8
10
8
Wilma kept track of the number of cups of water each of her five dogs drank in a day.
Make a line plot of her data. How much water did the five dogs drink?
Roxie……….1 cup
Sweetums…..2 ½ cups
Ralphie……..2 cups
Mr. Snarls….4 cups
Frito………..2 ½ cups
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
How many one-eighths are shown on the line plot? What is their total? How many
five-eighths are shown? What is their total?
X
X
X
X
X
1
8
2
8
0
8
3
8
X
X
X
X
X
X
X
X
X
X
X
X
X
4
8
5
8
6
8
X
X
X
X
X
X
7
8
8
8
Using the line plot data, add the number of ½ inch items and the number of ¾ inch
items together.
0
8
X
X
X
1
8
2
8
3
8
X
X
X
X
X
X
X
4
8
5
8
6
8
X
7
8
8
8
If all the longest items on the line graph were added together, how much longer
would they be than all the shortest items?
0
4
Grade 4 Data
1
4
X
X
X
2
4
3
4
4
4
Page 15 of 26
X
X
X
X
X
X
X
X
5
4
6
4
7
4
8
4
Columbus City Schools 2013-2014
Jesse recorded the rainfall in his neighborhood over a two-week period. Make a line
plot, using intervals of one-eighth. Write two different equations that represent the
difference between the highest amount of rainfall and the lowest.
Day
Inches of
Rainfall
June 12
11
4
June 13
June 18
1
June 19
1
4
June 26
3
8
June 28
3
8
5
8
Customers at Good Grief Tattoo Parlor can choose from a
1
8
inch,
2
8
inch, or
4
8
inch
tattoo. The Parlor had 5 customers on Thursday. Make a line plot that displays the
sizes of tattoos they could have sold. Use your line plot to calculate how many inches
of tattoo they inked that day.
Customers at Crazy 8 Tattoos have a choice of getting a
1
4
inch,
3
8
inch, or
1
2
inch
tattoo. The shop inked a total of 3 inches of tattoos on Wednesday. Make a line plot
that displays the sizes and numbers of tattoos they could have sold.
2. https://www.georgiastandards.org/Common-Core/Pages/Math-K-5.aspx
Go to the above web page, open grade 4 on the right side of the page. Open “Unit 7
Framework Student Edition.” These tasks are from the Georgia Department of Education.
The tasks build on one another. You may or may not use all of the tasks. There are four
different types of tasks. Scaffolding tasks build up to the constructing tasks which develop a
deep understanding of the concept. Next, there are practice tasks and finally performance
tasks which are a summative assessment for the unit. Select the tasks that best fit with your
lessons and the standards being taught at that time.
3. Divide students into groups of four. Each group will be responsible for working together to
come up with a list of survey questions that could be asked of the class. Explain to the class
that each question posed must have a purpose. These purposes include comparing and
contrasting sets of data, making predictions from sets of data, summarizing sets of data, and
determining preferences from sets of data.
For example:
o Do the number of fractional parts of an hour or more than an hour it takes
students to complete their homework differ between 4th and 5th grade
students?
o What is the typical foot measure of students in our class?
o Which foot length was the most popular among girls or boys in the 4th grade?
Have groups complete their lists. Record the group responses by writing the questions and
the purpose for each question on chart paper. Discuss which survey questions would enable
students to gather obtainable classroom data. Next, have each group choose one of their
survey questions, and create a plan for collecting the data. Groups write their question and
plan on chart paper. Share all charts. Hang up charts to use the next day. On day two have
Grade 4 Data
Page 16 of 26
Columbus City Schools 2013-2014
students gather in their same groups. They use their survey question and plan to collect data
from their class members. Give students 10 minutes to gather data from each other. After
students have recorded the data, have one person from each group share the method they
used to record the data. Discuss the methods for organizing data. Model organizing data by
taking a class survey to find the favorite colors of the class. Organize the information into a
table with tallies and also create a frequency table. Give groups the opportunity to display
their question and data using one of these methods on chart paper. Have each group
represent their data using a line plot.
4. Pose the question: How many hours do you watch television on a school night? Give the
choices of 1 hour, 1 1 , 1 1 , 1 3 , 2, 2 1 , 2 1 , 2 3 , and 3. Set up the choices so they are the
4
2
4
4
2
4
categories across the bottom of what will become a line plot. Have students use sticky notes
to represent their responses. Have the students use the collected data to create a line plot.
Make sure to emphasize correct labeling, including a title. Next, have students get back into
their groups from the previous lesson and use the data from the survey they conducted to
create a group line plot. Have each group share their completed line plot and discuss the
process they used to choose their topic, define their purpose, collect their data, and create the
line plot.
5. Divide the students in groups of four and assign each group a letter (A-F, assign two groups
the same letter if necessary). Give each student a “Survey Tally Sheet” (included in this
Curriculum Guide). Combine two groups of four together to collect data from each other. If
students have more responses than there are spaces on the chart, then have them create a tally
chart for the other numbers on the back of the paper. Each group is responsible for collecting
data for only the table that has the same letter as their group letter. Keep changing the groups
that are together until every group has had an opportunity to collect data from every other
group in the class. Each group is then responsible for creating a line plot of the data that they
collected. Make sure students give their line plot a title and have the appropriate number of
X’s above each number to represent their data. Have one member from each group present
the group’s line plot to the rest of the class.
Grade 4 Data
Page 17 of 26
Columbus City Schools 2013-2014
RESOURCES
Textbook: Mathematics, Scott Foresman-Addison Wesley (2004): pp. None
Supplemental: Mathematics, Scott Foresman-Addison Wesley (2004):
Enrichment Master pp. None
Practice Master pp. None
Problem Solving Master pp. None
Reteaching Master pp. None
Grade 4 Data
Page 18 of 26
Columbus City Schools 2013-2014
Problem Solving Questions
Amanda grew bean sprouts in science class. After a week, she measured the length of each plant and
recorded her data in the table. Record her data on the line plot, label the line plot, and explain why
you placed the data where you did.
Number of plants Height in inches
0
8
1
8
0
8
2
8
1
8
3
1
4
2
1
8
3
1
2
1
1
3
8
2
8
4
8
3
8
5
8
4
8
6
8
5
8
7
8
6
8
8
8
7
8
8
8
Employees of the zoo were tracking the amount of gallons of milk produced by the pygmy goats.
They recorded the data in the table. Transfer their data to the line plot and label the line plot. How
many total gallons of milk did the goats produce?
# of Goats
Gallons of milk
Insert number line with the following intervals marked:
0/4 ¼ 2/4 ¾ 4/4 5/4 6/4 7/4 8/4
0
4
0
4
1
4
1
4
2
4
2
4
3
4
3
4
4
4
4
4
5
4
5
4
6
4
6
4
7
4
1
1
4
2
4
4
2
5
4
2
7
4
8
4
7
4
8
4
Zoey has used a line plot to record the length, in inches, of the insects in her collection. How many of
her insects are over one inch long? If Zoey placed her longest insects end to end, what would the total
length be?
X
1
8
X
X
X
2
8
3
8
4
8
5
8
6
8
X
X
X
7
8
X
X
X
X
X
X
8
8
9
8
10
8
Length of Insects in Inches
Grade 4 Data
Page 19 of 26
Columbus City Schools 2013-2014
Problem Solving Questions
Wilma kept track of the number of cups of water each of her five dogs drank in a day. Make a line
plot of her data. How much water did the five dogs drink?
Roxie……….1 cup
Sweetums…..2 ½ cups
Ralphie……..2 cups
Mr. Snarls….4 cups
Frito………..2 ½ cups
1
2
2
2
3
2
4
2
5
2
6
2
7
2
8
2
How many one-eighths are shown on the line plot? What is their total? How many five-eighths are
shown? What is their total?
0
8
X
X
X
X
X
1
8
2
8
3
8
X
X
X
X
X
X
X
X
X
X
X
X
X
4
8
5
8
6
8
X
X
X
X
X
X
7
8
8
8
Using the line plot data, add the number of ½ inch items and the number of ¾ inch items together.
0
8
Grade 4 Data
X
X
X
1
8
2
8
3
8
X
X
X
X
X
X
X
4
8
5
8
6
8
Page 20 of 26
X
7
8
8
8
Columbus City Schools 2013-2014
Problem Solving Questions
If all the longest items on the line graph were added together, how much longer would they be than all
the shortest items?
0
4
1
4
X
X
X
2
4
3
4
4
4
X
X
X
X
X
X
X
X
5
4
6
4
7
4
8
4
Jesse recorded the rainfall in his neighborhood over a two-week period. Make a line plot, using intervals of
one-eighth. Write two different equations that represent the difference between the highest amount of rainfall
and the lowest.
Day
June 12
Inches of Rainfall
June 13
June 18
1
June 19
1
4
June 26
3
8
June 28
3
8
1
1
4
5
8
Customers at Good Grief Tattoo Parlor can choose from a
1
8
inch,
2
8
inch, or
4
8
inch tattoo. The
Parlor had 5 customers on Thursday. Make a line plot that displays the sizes of tattoos they could have
sold. Use your line plot to calculate how many inches of tattoo they inked that day.
Customers at Crazy 8 Tattoos have a choice of getting a
1
4
inch,
3
8
inch, or
1
2
inch tattoo. The shop
inked a total of 3 inches of tattoos on Wednesday. Make a line plot that displays the sizes and
numbers of tattoos they could have sold.
Grade 4 Data
Page 21 of 26
Columbus City Schools 2013-2014
Problem Solving QuestionsAnswers
Amanda grew bean sprouts in science class. After a week, she measured the length of each plant and
recorded her data in the table. Record her data on the line plot, label the line plot, and explain why
you placed the data where you did.
Number of plants Height in inches
1
3
4
0
80
8
XX X
XX X
XX X
XX X
XX X
1
81
8
2
82
8
2
1
8
3
1
2
1
1
XX X
XX X
XX X
3
83
8
XXX
4
84
8
5
85
8
6
86
8
7
87
8
8
88
8
Heightof
ofPlants
Plantsin
inInches
Inches
Height
Answer: Line plot should be marked as above. Students should explain that they converted
fractions in the table to equivalent eighths in order to plot the points in the line plot.
1
1
= 2
= 4 1= 8
4
8
2
8
8
Employees of the zoo were tracking the amount of gallons of milk produced by the pygmy goats.
They recorded the data in the table. Transfer their data to the line plot. How many total gallons of
milk did the goats produce?
# of Goats
Gallons of milk
1
1
4
2
4
4
2
5
4
2
7
4
X
0
4
1
4
2
4
3
4
X
X
X
X
4
4
5
4
X
X
6
4
7
4
8
4
Gallons of Milk Produced By Pygmy Goats
Answer: Line plot should be marked as above. The goats produced
1
+ 44 + 44 + 54 + 54 + 74 + 74 = 33
or 8 14 gallons of milk.
4
4
Grade 4 Data
Page 22 of 26
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Zoey has used a line plot to record the length, in inches, of the insects in her collection. How many of
her insects are over one inch long? If Zoey placed her longest insects end to end, what would the total
X
X
length be?
X
X
X
X
X
X
X
1
8
2
8
3
8
4
8
5
8
6
8
X
X
X
X
7
8
8
8
9
8
10
8
Length of Insects in Inches
Length of Insects in Inches
Answer: Five insects are over an inch long. The longest insects end to end total
6
8
10
8
inches.
10
8
+
+
10
8
=
30
8
30
8
inches, or 3
=36.
8
Wilma kept track of the number of cups of water each of her five dogs drank in a day. Make a line
plot of her data and add a label. How much water did the five dogs drink?
Roxie……….1 cup
Sweetums…..2 ½ cups
Ralphie……..2 cups
Mr. Snarls….4 cups
Frito………..2 ½ cups
X
1
2
2
2
X
X
X
4
2
5
2
3
2
X
6
2
7
2
8
2
Water Dogs Drank in Cups
Answer: Wilma’s dogs drank a total of
2
2
+
4
2
+
5
2
+
5
2
+
8
2
=
24
2
24
2
, or 12 cups of water. Length of Insects in Inches
= 12 cups
How many one-eighths are shown on the line plot? What is their total? How many five-eighths are
shown? What is their total?
X
0
8
X
X
X
X
X
1
8
2
8
3
8
X
X
X
X
X
X
X
X
X
X
X
X
4
8
5
8
6
8
Answer: There are 4 one-eighths on the line plot. They total
line plot. They total 20 eighths:
5
8
+
5
8
+
5
8
+
5
8
=
20
8
4
8
X
X
X
X
X
X
7
8
8
8
. There are 4 five-eighths on the
. Students can skip count by fives to arrive
at the answer.
Grade 4 Data
Page 23 of 26
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Using the line plot data, add the number of
0
8
4
8
Answer:
+
4
8
+
6
8
+
6
8
X
X
X
1
8
2
8
=
20
8
= 2
1
2
X
X
X
X
X
X
X
4
8
5
8
6
8
3
8
4
8
or 2
3
4
inch items and the number of
inch items together.
X
7
8
8
8
1
2
If all the longest items on the line graph were added together, how much longer would they be than
X
all the shortest items?
XX
X
0
0
4
4
1
1
4
4
X
XX
X
X
X
2
2
4
4
3
3
4
4
4
4
4
4
XX
XX
X
X
X
XX
XX
X
5
5
4
4
6
6
4
4
7
7
4
4
8
8
4
4
Answer: The longest items would be a total of 17/4, or 4 ¼ units longer than the shortest items.
2
7
21
+ 2 = 4
+ 7 + 7 = 21
– 4 = 174 = 4 1
4
4
4
4
4
4
4
4
4
4
Jesse recorded the rainfall in his neighborhood over a two-week period. Make a line plot, using
intervals of one-eighth. Write two different equations that represent the difference between the
highest amount of rainfall and the lowest.
Day
Inches of Rainfall
June 12
1 1
4
0
8
1
8
June 13
June 18
1
June 19
1
4
June 26
3
8
June 28
3
8
X
X
X
2
8
3
8
5
8
X
X
4
8
5
8
6
8
7
8
X
8
8
9
8
10
8
Inches of Rainfall
Answer:
The difference between the highest and lowest rainfalls is 1 inch.
Grade 4 Data
Page 24 of 26
10
8
–
2
8
=
8
8
= 1 or 1
1
4
-
1
4
=1
Columbus City Schools 2013-2014
Problem Solving Questions Answers
Customers at Good Grief Tattoo Parlor can choose from a
1
8
inch,
2
8
inch, or
4
8
inch tattoo. The
Parlor had 5 customers on Thursday. Make a line plot that displays the sizes of tattoos they could
have sold. Use your line plot to calculate how many inches of tattoo they inked that day.
Answers will vary. Encourage students to create line plots that show at least one of each size of
tattoos.
Customers at Crazy 8 Tattoos have a choice of getting a
1
4
inch,
3
8
inch, or
1
2
inch tattoo. The shop
inked a total of 3 inches of tattoos on Wednesday. Make a line plot that displays the sizes and
numbers of tattoos they could have sold.
Answers will vary. Students should convert the tattoos to eighths before plotting on a line plot.
Encourage students to include at least one of each size of tattoos.
Grade 4 Data
Page 25 of 26
Columbus City Schools 2013-2014
Survey Tally Sheet
Name
Group A
Group B
Length of
index finger
Number of
Students
Group C
Circumference
of head
Number of
Students
Group D
Height
Number of
Students
Width of palm
Number of
Students
Group E
Grade 4 Data
Length of foot
Group F
Length of arm
span
Page 26 of 26
Number of
Students
Number of
Students
Columbus City Schools 2013-2014