Download Alabama Standards Alignment Kindergarten through Precalculus

Alabama Standards Alignment
2010 Alabama Course of Study (Common Core), produced by the State Department of Education
Kindergarten through Precalculus
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registration in the United States and other countries.
P.O. Box 8036 • Wisconsin Rapids, WI 54495-8036
Phone: (800) 338-4204 • Fax: (715) 424-4242
www.renlearn.com
Alabama Standards Alignment
Standards List with Aligned Product Skills
The Standards List with Aligned Product Skills Report is a standards-oriented document
showing the entire list of standards for the subject and grade and the product skills aligned to
those standards. This alignment report shows the breadth of standards coverage for the purpose
and focus of this product.
Note to Educator
Kindergarten
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....................................................................................................... 1
Grade 1
............................................................................................................ 6
Grade 2
............................................................................................................ 12
Grade 3
............................................................................................................ 18
Grade 4
............................................................................................................ 26
Grade 5
............................................................................................................ 37
Grade 6
............................................................................................................ 48
Grade 7
............................................................................................................ 61
Grade 8
............................................................................................................ 72
9-12 Analytic Math
Algebra 2
.......................................................................................................... 85
Algebra 2 with Trig
Algebra I
Precalculus
................................................................................................ 97
.......................................................................................................... 110
Discrete Mathematics
Geometry
................................................................................................. 82
............................................................................................... 125
........................................................................................................... 128
............................ ............................................................................ 139
© 2012 by Renaissance Learning, Inc. All rights reserved. No portion of this document may be reproduced, by any process or technique, without the
express written consent of Renaissance Learning, Inc. Any standards referenced in this document are the property of their respective copyright holders.
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P.O. Box 8036
Wisconsin Rapids, WI 54495-8036
Phone: (800) 338-4204
Fax: (715) 424-4242
www.renlearn.com
Note to Educator:
Thank you for your interest in Renaissance Learning technology. At Renaissance
Learning, we recognize the impact that standards and assessment reform have on schools.
We share the concerns of educators and administrators that students perform well and
that teachers have the resources they need to support their efforts to address standards and
assessments.
Renaissance Learning provides alignment reports to customers to show how the skills
within each product align to the skills within academic standards. The alignment report
presents all of the academic math standards for a specific state/agency with the aligned
Renaissance Learning product skills indented below each standard.
Academic standards encompass the entire set of learning and expectations that teachers
are responsible for. Renaissance Learning recognizes that teachers are the key to using
products to address the entire set of standards. Renaissance Learning products are ideally
suited to support teachers and academic standards. The Renaissance Learning alignment
report supports teachers in this role by clearly identifying specific product skills that are
aligned to the multiple skills within the standards.
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On the alignment report you will see the word “Core” next to some of the skills. Core
skills are the most critical mathematics skills for a student to learn at a grade level. They
are key building blocks in a student’s mathematics education. Students need to have
proficiency with core skills to be successful in math at their grade levels and to progress
in the grades that follow. Core skills are indicated on Renaissance Learning’s researchbased and empirically validated Core Progress math learning progression. Core Progress
Math identifies the continuum of concepts and skills needed for success in math. The
continuum begins with early numeracy and progresses to the level of mathematics
required for college and careers. All of the skills in the Core Progress Math contribute to
STAR Math Enterprise Assessment. Alignments for STAR Math Enterprise include
alignments to the Common Core State Standards (CCSS) Mathematics Standards.
In the alignment report, you will notice that most skills have a grade-level indicator at the
end of the text. In addition to these grade-level skills, the report includes skills that are
not grade specific and do not have a grade-level indicator. These skills are designated
Overall Product Skills. They are skills that all students gain through use of the product.
We hope this report answers your questions regarding the alignment of Renaissance
Learning technology and materials to standards. The complete alignment strategy
document is available. The Math document is number R39616. If you have any questions
about the alignment report, please feel free to call us at (800) 338-4204.
Sincerely,
Renaissance Learning
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Kindergarten
Kindergarten
AL K.CC - Counting and Cardinality
Know number names and the count sequence.
AL K.CC.1 - Count to 100 by ones and by tens.
Numbers and Operations
Count on or count back with numbers 1 to 20 [Early Numeracy]
Count on by ones from a number less than 100 [Grade 1]
Core Skill
Count by 5s or 10s to 100 starting from a multiple of 5 or 10,
respectively [Grade 1]
Core Skill
AL K.CC.2 - Count forward beginning from a given number within the known sequence (instead of
having to begin at 1).
Numbers and Operations
Count on or count back with numbers 1 to 20 [Early Numeracy]
Count on by ones from a number less than 100 [Grade 1]
Core Skill
AL K.CC.3 - Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20
(with 0 representing a count of no objects).
Numbers and Operations
Recognize numbers 0-20 [Early Numeracy]
Recognize that there are 0 objects [Early Numeracy]
Read a whole number to 30 [Grade 1]
Count objects to 20 [Grade 1]
Core Skill
Count to tell the number of objects.
AL K.CC.4 - Understand the relationship between numbers and quantities; connect counting to
cardinality.
AL K.CC.4.a - When counting objects, say the number names in the standard order, pairing each
object with one and only one number name and each number name with one and only one object.
Numbers and Operations
Count up to 15 objects [Early Numeracy]
Count up to 20 objects [Early Numeracy]
Count objects to 20 [Grade 1]
Core Skill
Count on by ones from a number less than 100 [Grade 1]
Core Skill
AL K.CC.4.b - Understand that the last number name said tells the number of objects counted. The
number of objects is the same regardless of their arrangement or the order in which they were
counted.
Numbers and Operations
Count objects to 20 [Grade 1]
Core Skill
AL K.CC.4.c - Understand that each successive number name refers to a quantity that is one larger.
Numbers and Operations
Count on or count back with numbers 1 to 10 [Early Numeracy]
Count on or count back with numbers 1 to 20 [Early Numeracy]
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Count on by ones from a number less than 100 [Grade 1]
Kindergarten
Core Skill
AL K.CC.5 - Count to answer "how many?" questions about as many as 20 things arranged in a line, a
rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number
from 1-20, count out that many objects.
Numbers and Operations
Count objects to 20 [Grade 1]
Core Skill
Compare numbers.
AL K.CC.6 - Identify whether the number of objects in one group is greater than, less than, or equal to
the number of objects in another group, e.g., by using matching and counting strategies.[Include
groups with up to ten objects.]
Numbers and Operations
Compare sets of up to 5 objects [Early Numeracy]
Compare sets of up to 10 objects using "more" or "fewer" [Early
Numeracy]
Identify two sets with equal numbers of up to 10 objects [Early
Numeracy]
Compare sets of objects using words [Grade 1]
AL K.CC.7 - Compare two numbers between 1 and 10 presented as written numerals.
Numbers and Operations
Compare sets of up to 10 objects using "more" or "fewer" [Early
Numeracy]
AL K.OA - Operations and Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking apart and
taking from.
AL K.OA.8 - Represent addition and subtraction with objects, fingers, mental images, drawings,
sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.[Drawings
need not show details, but should show the mathematics in the problem. (This applies wherever
drawings are mentioned in the Standards.)]
Numbers and Operations
Add numbers with a sum up to 9 using pictures [Early Numeracy]
Add numbers with a sum up to 10 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 9 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 10 using pictures [Early
Numeracy]
AL K.OA.9 - Solve addition and subtraction word problems, and add and subtract within 10, e.g., by
using objects or drawings to represent the problem.
Numbers and Operations
Determine the difference in number between two sets of objects
[Early Numeracy]
Add numbers with a sum up to 9 using pictures [Early Numeracy]
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Kindergarten
Add numbers with a sum up to 10 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 9 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 10 using pictures [Early
Numeracy]
Know basic addition facts to 10 plus 10 [Grade 1]
Algebra
Core Skill
Determine the missing addend in an addition sentence using
pictures [Early Numeracy]
AL K.OA.10 - Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by
using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and
5 = 4 + 1).
Numbers and Operations
Compose an equivalent number out of two numbers by using
pictures [Early Numeracy]
Decompose a number into two numbers by using pictures [Early
Numeracy]
Determine equivalent forms of a number, up to 10 [Grade 1]
Core Skill
Relate a picture model to a basic addition fact [Grade 1]
Determine the basic addition fact shown by a picture model
[Grade 1]
Algebra
Determine equivalent addition or subtraction expressions
involving 1-digit numbers [Grade 1]
AL K.OA.11 - For any number from 1 to 9, find the number that makes 10 when added to the given
number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
Numbers and Operations
Determine equivalent forms of a number, up to 10 [Grade 1]
Core Skill
AL K.OA.12 - Fluently add and subtract within 5.
Numbers and Operations
Add numbers with a sum up to 5 in horizontal format [Early
Numeracy]
Subtract numbers with a minuend up to 3 in horizontal format
[Early Numeracy]
Subtract numbers with a minuend up to 5 in horizontal format
[Early Numeracy]
Know basic addition facts to 10 plus 10 [Grade 1]
Core Skill
Know basic subtraction facts to 20 minus 10 [Grade 1]
Core Skill
AL K.NBT - Number and Operations in Base Ten
Work with numbers 11-19 to gain foundations for place value.
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Kindergarten
AL K.NBT.13 - Compose and decompose numbers from 11 to 19 into ten ones and some further ones,
e.g., by using objects or drawings, and record each composition or decomposition by a drawing or
equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two,
three, four, five, six, seven, eight, or nine ones.
AL K.MD - Measurement and Data
Describe and compare measurable attributes.
AL K.MD.14 - Describe measurable attributes of objects, such as length or weight. Describe several
measurable attributes of a single object.
Geometry and Measurement
Sort objects according to one attribute [Early Numeracy]
Determine the common attributes in a set of geometric shapes
[Grade 1]
Core Skill
AL K.MD.15 - Directly compare two objects with a measurable attribute in common, to see which
object has "more of"/"less of" the attribute, and describe the difference. Example: For example,
directly compare the heights of two children and describe one child as taller/shorter.
Geometry and Measurement
Compare the sizes of two objects [Early Numeracy]
Compare the weights of two objects [Early Numeracy]
Compare the volumes of two objects [Early Numeracy]
Compare objects using the vocabulary of measurement [Grade 1]
Core Skill
Classify objects and count the number of objects in each category.
AL K.MD.16 - Classify objects into given categories; count the numbers of objects in each category and
sort the categories by count.[Limit category counts to be less than or equal to 10.]
Geometry and Measurement
Sort objects according to one attribute [Early Numeracy]
Determine the group in which a geometric shape belongs [Grade
1]
Determine the rule used to sort geometric shapes [Grade 1]
Data Analysis, Statistics, and
Probability
Determine the rule used to sort objects [Grade 1]
AL K.G - Geometry
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders,
and spheres).
AL K.G.17 - Describe objects in the environment using names of shapes, and describe the relative
positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
Geometry and Measurement
Apply the vocabulary of position or direction [Grade 1]
AL K.G.18 - Correctly name shapes regardless of their orientations or overall size.
Geometry and Measurement
Identify a circle, a triangle, a square, or a rectangle [Grade 1]
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Core Skill
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Kindergarten
AL K.G.19 - Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional
("solid").
Overall Product Skills
Build comprehension of math vocabulary
Analyze, compare, create, and compose shapes.
AL K.G.20 - Analyze and compare two- and three-dimensional shapes, in different sizes and
orientations, using informal language to describe their similarities, differences, parts (e.g., number of
sides and vertices/"corners") and other attributes (e.g., having sides of equal length).
Geometry and Measurement
Determine the common attributes in a set of geometric shapes
[Grade 1]
Core Skill
Determine the group in which a geometric shape belongs [Grade
1]
Determine the rule used to sort geometric shapes [Grade 1]
AL K.G.21 - Model shapes in the world by building shapes from components (e.g., sticks and clay
balls) and drawing shapes.
AL K.G.22 - Compose simple shapes to form larger shapes. Example: For example, "Can you join
these two triangles with full sides touching to make a rectangle?".
Geometry and Measurement
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Compose a new shape out of two other shapes [Early Numeracy]
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Grade 1
Grade 1
AL 1.OA - Operations and Algebraic Thinking
Represent and solve problems involving addition and subtraction.
AL 1.OA.1 - Use addition and subtraction within 20 to solve word problems involving situations of
adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions,
e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the
problem.
Numbers and Operations
Determine the missing portion in a partially screened (hidden)
collection of up to 10 objects [Grade 1]
Core Skill
Relate a picture model to a basic addition fact [Grade 1]
Determine the basic addition fact shown by a picture model
[Grade 1]
Relate a picture model to a basic subtraction fact [Grade 1]
Algebra
WP: Use basic addition or subtraction facts to solve problems
[Grade 1]
Core Skill
Determine a missing addend in a basic addition-fact number
sentence [Grade 1]
Core Skill
Determine a missing subtrahend in a basic subtraction-fact
number sentence [Grade 1]
Core Skill
WP: Determine a missing addend or subtrahend in a basic
addition- or subtraction-fact number sentence [Grade 1]
Core Skill
WP: Determine a basic addition- or subtraction-fact number
sentence for a given situation [Grade 1]
Core Skill
AL 1.OA.2 - Solve word problems that call for addition of three whole numbers whose sum is less than
or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number
to represent the problem.
Numbers and Operations
Add three 1-digit numbers [Grade 1]
Algebra
Determine a missing addend in an addition sentence with three 1digit numbers [Grade 1]
Understand and apply properties of operations and the relationship between addition and subtraction.
AL 1.OA.3 - Apply properties of operations as strategies to add and subtract. Example: Examples: If 8
+ 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4,
the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of
addition.).[Students need not use formal terms for these properties.]
Numbers and Operations
Use turn-around facts to find the sum of two numbers [Early
Numeracy]
Add three 1-digit numbers [Grade 1]
Complete an addition and subtraction fact family [Grade 1]
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Grade 1
AL 1.OA.4 - Understand subtraction as an unknown-addend problem. Example: For example,
subtract 10 - 8 by finding the number that makes 10 when added to 8.
Algebra
Determine the missing addend in an addition sentence using
pictures [Early Numeracy]
Determine a missing addend in a basic addition-fact number
sentence [Grade 1]
Core Skill
Determine a missing subtrahend in a basic subtraction-fact
number sentence [Grade 1]
Core Skill
Add and subtract within 20.
AL 1.OA.5 - Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
Numbers and Operations
Count on or count back with numbers 1 to 20 [Early Numeracy]
Add numbers with a sum up to 20 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 20 using pictures [Early
Numeracy]
Apply the relationship between addition and counting on [Grade
1]
Apply the relationship between subtraction and counting back
[Grade 1]
AL 1.OA.6 - Add and subtract within 20, demonstrating fluency for addition and subtraction within
10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a
number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition
and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier
or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
Numbers and Operations
Add numbers with a sum up to 20 in horizontal format [Early
Numeracy]
Add numbers with a sum up to 20 in vertical format [Early
Numeracy]
Subtract numbers with a minuend up to 20 in horizontal format
[Early Numeracy]
Subtract numbers with a minuend up to 20 in vertical format
[Early Numeracy]
Apply the relationship between addition and counting on [Grade
1]
Apply the relationship between subtraction and counting back
[Grade 1]
Know basic addition facts to 10 plus 10 [Grade 1]
Core Skill
Know basic subtraction facts to 20 minus 10 [Grade 1]
Core Skill
Complete an addition and subtraction fact family [Grade 1]
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Grade 1
Apply the inverse relationship between addition facts and
subtraction facts [Grade 1]
Work with addition and subtraction equations.
AL 1.OA.7 - Understand the meaning of the equal sign, and determine if equations involving addition
and subtraction are true or false. Example: For example, which of the following equations are true and
which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
Numbers and Operations
Determine equivalent forms of a number, up to 10 [Grade 1]
Core Skill
Complete an addition and subtraction fact family [Grade 1]
Algebra
Determine equivalent addition or subtraction expressions
involving 1-digit numbers [Grade 1]
AL 1.OA.8 - Determine the unknown whole number in an addition or subtraction equation relating
three whole numbers. Example: For example, determine the unknown number that makes the
equation true in each of the equations 8 + ? = 11, 5 = __ - 3, 6 + 6 = __.
Numbers and Operations
Add numbers with a sum up to 20 using pictures [Early
Numeracy]
Subtract numbers with a minuend up to 20 using pictures [Early
Numeracy]
Add numbers with a sum up to 20 in horizontal format [Early
Numeracy]
Add numbers with a sum up to 20 in vertical format [Early
Numeracy]
Subtract numbers with a minuend up to 20 in horizontal format
[Early Numeracy]
Subtract numbers with a minuend up to 20 in vertical format
[Early Numeracy]
Algebra
Determine a missing addend in a basic addition-fact number
sentence [Grade 1]
Core Skill
Determine a missing subtrahend in a basic subtraction-fact
number sentence [Grade 1]
Core Skill
AL 1.NBT - Number and Operations in Base Ten
Extend the counting sequence.
AL 1.NBT.9 - Count to 120, starting at any number less than 120. In this range, read and write
numerals and represent a number of objects with a written numeral.
Numbers and Operations
Read a whole number to 30 [Grade 1]
Read a whole number from 31 to 100 [Grade 1]
Count objects to 20 [Grade 1]
Core Skill
Count on by ones from a number less than 100 [Grade 1]
Core Skill
Understand place value.
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Grade 1
AL 1.NBT.10 - Understand that the two digits of a two-digit number represent amounts of tens and
ones. Understand the following as special cases:
AL 1.NBT.10.a - 10 can be thought of as a bundle of ten ones - called a "ten."
Numbers and Operations
Count objects grouped in tens and ones [Grade 1]
Core Skill
Recognize a number from a model of tens and ones to 100 [Grade
1]
AL 1.NBT.10.b - The numbers from 11 to 19 are composed of a ten and one, two, three, four, five,
six, seven, eight, or nine ones.
Numbers and Operations
Determine the value of a digit in a 2-digit number [Grade 1]
Core Skill
AL 1.NBT.10.c - The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six,
seven, eight, or nine tens (and 0 ones).
Numbers and Operations
Identify the number of 10's in 10, 20, 30, 40, 50, 60, 70, 80, 90
[Grade 1]
Represent a 2-digit number as tens and ones [Grade 1]
Core Skill
Determine the 2-digit number represented as tens and ones [Grade Core Skill
1]
AL 1.NBT.11 - Compare two two-digit numbers based on meanings of the tens and ones digits,
recording the results of comparisons with the symbols >, =, and <.
Numbers and Operations
Compare sets of up to 20 objects using "more" or "fewer" [Early
Numeracy]
Compare whole numbers to 100 using words [Grade 1]
Core Skill
Use place value understanding and properties of operations to add and subtract.
AL 1.NBT.12 - Add within 100, including adding a two-digit number and a one-digit number, and
adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies
based on place value, properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in
adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to
compose a ten.
Numbers and Operations
Add multiples of 10, sums to 90 [Grade 1]
Add a 2-digit and a 1-digit number without regrouping [Grade 1]
Core Skill
Add two 2-digit numbers without regrouping [Grade 1]
Core Skill
Add two 2-digit numbers with regrouping, given a model [Grade
2]
Add a 2-digit number to a 1-digit number with regrouping [Grade
2]
AL 1.NBT.13 - Given a two-digit number, mentally find 10 more or 10 less than the number, without
having to count; explain the reasoning used.
Numbers and Operations
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Determine ten more than or ten less than a given number [Grade 1 Core Skill
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Overall Product Skills
Grade 1
Use strategies to derive solutions for problems
AL 1.NBT.14 - Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90
(positive or zero differences), using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy
to a written method and explain the reasoning used.
Numbers and Operations
Add multiples of 10, sums to 90 [Grade 1]
Subtract multiples of 10 up to 90 [Grade 1]
AL 1.MD - Measurement and Data
Measure lengths indirectly and by iterating length units.
AL 1.MD.15 - Order three objects by length; compare the lengths of two objects indirectly by using a
third object.
Geometry and Measurement
Order objects according to size [Early Numeracy]
Compare objects using the vocabulary of measurement [Grade 1]
Core Skill
Compare and order objects by attributes of height or length
[Grade 1]
Core Skill
AL 1.MD.16 - Express the length of an object as a whole number of length units, by laying multiple
copies of a shorter object (the length unit) end to end; understand that the length measurement of an
object is the number of same-size length units that span it with no gaps or overlaps.[Limit to contexts
where the object being measured is spanned by a whole number of length units with no gaps or
overlaps.]
Geometry and Measurement
Estimate length in nonstandard units [Grade 1]
Tell and write time.
AL 1.MD.17 - Tell and write time in hours and half-hours using analog and digital clocks.
Geometry and Measurement
Tell time to the hour [Grade 1]
Tell time to the half hour [Grade 1]
Core Skill
Core Skill
Represent and interpret data.
AL 1.MD.18 - Organize, represent, and interpret data with up to three categories; ask and answer
questions about the total number of data points, how many in each category, and how many more or
less are in one category than in another.
Data Analysis, Statistics, and
Probability
Read a 2-category tally chart [Grade 1]
Use a 2-category tally chart to represent groups of objects (1
symbol = 1 object) [Grade 1]
Answer a question using information from a 2-category tally chart
[Grade 1]
Use a tally chart to represent data [Grade 2]
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Grade 1
AL 1.G - Geometry
Reason with shapes and their attributes.
AL 1.G.19 - Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus
non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining
attributes.
Geometry and Measurement
Determine the common attributes in a set of geometric shapes
[Grade 1]
Core Skill
Determine the group in which a geometric shape belongs [Grade
1]
AL 1.G.20 - Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles,
and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones,
and right circular cylinders) to create a composite shape, and compose new shapes from the composite
shape.[Students do not need to learn formal names such as "right rectangular prism."]
Geometry and Measurement
Compose a new shape out of two other shapes [Early Numeracy]
Decompose a shape into two new shapes [Early Numeracy]
Decompose a plane shape composed of three or more simpler
shapes [Grade 2]
AL 1.G.21 - Partition circles and rectangles into two and four equal shares, describe the shares using
the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe
the whole as two of, or four of the shares. Understand for these examples that decomposing into more
equal shares creates smaller shares.
Numbers and Operations
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Identify a shape divided into equal parts [Grade 1]
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Grade 2
Grade 2
AL 2.OA - Operations and Algebraic Thinking
Represent and solve problems involving addition and subtraction.
AL 2.OA.1 - Use addition and subtraction within 100 to solve one- and two-step word problems
involving situations of adding to, taking from, putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.
Numbers and Operations
Determine the missing portion in a partially screened (hidden)
collection of up to 10 objects [Grade 1]
Core Skill
WP: Add or subtract numbers without regrouping [Grade 2]
Core Skill
WP: Add or subtract up to 2-digit numbers with one regrouping
[Grade 2]
Algebra
WP: Determine a missing addend or subtrahend in a basic
addition- or subtraction-fact number sentence [Grade 1]
Core Skill
WP: Determine a missing addend or a missing subtrahend
involving 2-digit numbers [Grade 2]
Add and subtract within 20.
AL 2.OA.2 - Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know
from memory all sums of two one-digit numbers.
Numbers and Operations
Apply the relationship between addition and counting on [Grade
1]
Apply the relationship between subtraction and counting back
[Grade 1]
Know basic addition facts to 10 plus 10 [Grade 1]
Core Skill
Know basic subtraction facts to 20 minus 10 [Grade 1]
Core Skill
Complete an addition and subtraction fact family [Grade 1]
Apply the inverse relationship between addition facts and
subtraction facts [Grade 1]
Algebra
Determine equivalent addition or subtraction expressions
involving 1-digit numbers [Grade 1]
Work with equal groups of objects to gain foundations for multiplication.
AL 2.OA.3 - Determine whether a group of objects (up to 20) has an odd or even number of members,
e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum
of two equal addends.
Numbers and Operations
Count by 2s to 50 starting from a multiple of 2 [Grade 1]
Core Skill
Solve problems involving the concept of odd and even numbers
[Grade 2]
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Grade 2
AL 2.OA.4 - Use addition to find the total number of objects arranged in rectangular arrays with up to
5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Numbers and Operations
Use a repeated addition sentence to represent the sum of equal
groups [Grade 2]
Represent repeated addition as multiplication [Grade 2]
Relate a multiplication sentence to an array model [Grade 2]
AL 2.NBT - Number and Operations in Base Ten
Understand place value.
AL 2.NBT.5 - Understand that the three digits of a three-digit number represent amounts of hundreds,
tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special
cases:
AL 2.NBT.5.a - 100 can be thought of as a bundle of ten tens - called a "hundred."
Numbers and Operations
Determine the 3-digit number represented as hundreds, tens, and
ones [Grade 2]
Core Skill
AL 2.NBT.5.b - The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four,
five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
Numbers and Operations
Determine the value of a digit in a 3-digit number [Grade 2]
Determine which digit is in a specified place in a 3-digit whole
number [Grade 2]
Represent a 3-digit number as hundreds, tens, and ones [Grade 2]
AL 2.NBT.6 - Count within 1000; skip-count by 5s, 10s, and 100s.
Numbers and Operations
Count on by ones from a number less than 100 [Grade 1]
Core Skill
Core Skill
Complete a skip pattern starting from a multiple of 2, 5, or 10
[Grade 2]
Core Skill
Complete a skip pattern of 2, 5, or 10 starting from any number
[Grade 2]
Core Skill
Count on by 100s from any number [Grade 2]
Core Skill
AL 2.NBT.7 - Read and write numbers to 1000 using base-ten numerals, number names, and expanded
form.
Numbers and Operations
Read a whole number to 1,000 [Grade 2]
Core Skill
Determine the word form of a whole number to 1,000 [Grade 2]
Core Skill
Model a number using hundreds, tens, and ones to 1,000 [Grade
2]
Represent a 3-digit number as hundreds, tens, and ones [Grade 2]
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Grade 2
AL 2.NBT.8 - Compare two three-digit numbers based on meanings of the hundreds, tens, and ones
digits, using >, =, and < symbols to record the results of comparisons.
Numbers and Operations
Compare whole numbers to 1,000 using words [Grade 2]
Compare whole numbers to 1,000 using the symbols <, >, and =
[Grade 2]
Core Skill
Use place value understanding and properties of operations to add and subtract.
AL 2.NBT.9 - Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
Numbers and Operations
Determine a number pair that totals 100 [Grade 2]
Core Skill
Add two 2-digit numbers with regrouping, given a model [Grade
2]
Add a 2-digit number to a 1-digit number with regrouping [Grade
2]
Add two 2-digit numbers with regrouping [Grade 2]
Core Skill
Subtract a 1- or 2-digit number from a 2-digit number with one
regrouping [Grade 2]
Core Skill
AL 2.NBT.10 - Add up to four two-digit numbers using strategies based on place value and properties
of operations.
Numbers and Operations
Add two 2-digit numbers with regrouping, given a model [Grade
2]
Add two 2-digit numbers with regrouping [Grade 2]
Core Skill
Add three 2-digit numbers with one regrouping, sum less than 100
[Grade 2]
AL 2.NBT.11 - Add and subtract within 1000, using concrete models or drawings and strategies based
on place value, properties of operations, and/or the relationship between addition and subtraction;
relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers,
one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is
necessary to compose or decompose tens or hundreds.
Numbers and Operations
Determine a number pair that totals 1,000 [Grade 2]
Add two 3-digit numbers without regrouping [Grade 2]
Add two 3-digit numbers with one regrouping [Grade 2]
Core Skill
Subtract a 3-digit number from a 3-digit number without
regrouping [Grade 2]
Subtract a 3-digit number from a 3-digit number with one
regrouping [Grade 2]
Core Skill
Apply the inverse relationship between addition and subtraction
with 2- and 3-digit numbers [Grade 2]
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Grade 2
AL 2.NBT.12 - Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100
from a given number 100-900.
Numbers and Operations
Add multiples of 10, sums to 90 [Grade 1]
Subtract multiples of 10 up to 90 [Grade 1]
AL 2.NBT.13 - Explain why addition and subtraction strategies work, using place value and the
properties of operations.
Numbers and Operations
Add two 2-digit numbers with regrouping, given a model [Grade
2]
Overall Product Skills
Develop collaboration skills by explaining mathematical
reasoning and strategies
AL 2.MD - Measurement and Data
Measure and estimate lengths in standard units.
AL 2.MD.14 - Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.
Geometry and Measurement
Select the appropriate measurement tool for length, weight,
temperature, time, or capacity [Grade 1]
Measure length in inches [Grade 2]
Core Skill
Measure length in centimeters [Grade 2]
Core Skill
AL 2.MD.15 - Measure the length of an object twice, using length units of different lengths for the two
measurements; describe how the two measurements relate to the size of the unit chosen.
AL 2.MD.16 - Estimate lengths using units of inches, feet, centimeters, and meters.
Geometry and Measurement
Estimate length in nonstandard units [Grade 1]
AL 2.MD.17 - Measure to determine how much longer one object is than another, expressing the length
difference in terms of a standard length unit.
Geometry and Measurement
Compare and order objects by attributes of height or length
[Grade 1]
Core Skill
Relate addition and subtraction to length.
AL 2.MD.18 - Use addition and subtraction within 100 to solve word problems involving lengths that
are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a
symbol for the unknown number to represent the problem.
Geometry and Measurement
Add or subtract compound units of length in inches and feet with
no conversion [Grade 3]
WP: Add or subtract compound units of length in inches and feet
with no conversion [Grade 3]
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Grade 2
AL 2.MD.19 - Represent whole numbers as lengths from 0 on a number line diagram with equally
spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and
differences within 100 on a number line diagram.
Numbers and Operations
Identify a number to 20 represented by a point on a number line
[Grade 1]
Core Skill
Locate a number to 20 on a number line [Grade 1]
Core Skill
Relate a number-line model to a basic addition fact [Grade 1]
Determine the basic addition fact shown by a number-line model
[Grade 1]
Determine the basic subtraction fact shown by a number-line
model [Grade 1]
Relate a number-line model to a basic subtraction fact [Grade 1]
Work with time and money.
AL 2.MD.20 - Tell and write time from analog and digital clocks to the nearest five minutes, using a.m.
and p.m.
Geometry and Measurement
Tell time to the half hour [Grade 1]
Core Skill
Tell time to the quarter hour [Grade 2]
Core Skill
Tell time to 5-minute intervals [Grade 2]
Core Skill
AL 2.MD.21 - Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $
and ¢ symbols appropriately. Example: Example: If you have 2 dimes and 3 pennies, how many cents
do you have?
Numbers and Operations
Translate between a dollar sign and a cent sign [Grade 2]
Core Skill
Determine cent amounts that total a dollar [Grade 2]
Core Skill
WP: Add or subtract a collection of bills or coins [Grade 2]
Represent and interpret data.
AL 2.MD.22 - Generate measurement data by measuring lengths of several objects to the nearest whole
unit, or by making repeated measurements of the same object. Show the measurements by making a
line plot, where the horizontal scale is marked off in whole-number units.
Geometry and Measurement
Measure length in inches [Grade 2]
Data Analysis, Statistics, and
Probability
Use a line plot to represent data [Grade 3]
Core Skill
AL 2.MD.23 - Draw a picture graph and a bar graph (with single-unit scale) to represent a data set
with up to four categories. Solve simple put-together, take-apart, and compare problems using
information presented in a bar graph.
Data Analysis, Statistics, and
Probability
Read a picture graph [Grade 1]
Use a picture graph to represent groups of objects (1 symbol = 1
object) [Grade 1]
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Grade 2
Read a bar graph [Grade 1]
Use a bar graph to represent groups of objects [Grade 1]
Answer a question using information from a bar graph [Grade 1]
Determine a question that can be answered from a data
representation [Grade 3]
AL 2.G - Geometry
Reason with shapes and their attributes.
AL 2.G.24 - Recognize and draw shapes having specified attributes, such as a given number of angles
or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
Geometry and Measurement
Identify a circle, a triangle, a square, or a rectangle [Grade 1]
Core Skill
Determine the common attributes in a set of geometric shapes
[Grade 1]
Core Skill
Determine the rule used to sort geometric shapes [Grade 1]
Identify a parallelogram, a trapezoid, a pentagon, a hexagon, or an
octagon [Grade 2]
Determine attributes of a triangle or a quadrilateral from a model
[Grade 3]
Relate a model of a triangle or a quadrilateral to a list of attributes
[Grade 3]
AL 2.G.25 - Partition a rectangle into rows and columns of same-size squares and count to find the
total number of them.
Geometry and Measurement
Relate area to the number of square units [Grade 2]
WP: Determine the area of a rectangular shape given a picture on
a grid [Grade 3]
AL 2.G.26 - Partition circles and rectangles into two, three, or four equal shares, describe the shares
using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three
thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.
Numbers and Operations
Identify a shape divided into equal parts [Grade 1]
Determine a pictorial model of a fraction of a whole [Grade 3]
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Core Skill
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Grade 3
Grade 3
AL 3.OA - Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
AL 3.OA.1 - Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in
5 groups of 7 objects each. Example: For example, describe a context in which a total number of
objects can be expressed as 5 × 7.
Numbers and Operations
Relate a multiplication sentence to an array model [Grade 2]
Recognize the number of equal groups using pictures [Grade 2]
Use a multiplication sentence to represent an area or an array
model [Grade 3]
Core Skill
AL 3.OA.2 - Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number
of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares
when 56 objects are partitioned into equal shares of 8 objects each. Example: For example, describe a
context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
Numbers and Operations
Recognize the number of equal groups using pictures [Grade 2]
Divide objects into equal groups using pictures [Grade 2]
Determine the number of equal groups of objects excluding the
remainder [Grade 2]
Use a division sentence to represent objects divided into equal
groups [Grade 3]
Core Skill
AL 3.OA.3 - Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem.
Numbers and Operations
Algebra
WP: Multiply a 1-digit number by 2, 5, or 10 [Grade 2]
WP: Divide objects into equal groups by sharing [Grade 2]
Core Skill
WP: Determine a multiplication or division sentence for a given
situation [Grade 3]
Core Skill
AL 3.OA.4 - Determine the unknown whole number in a multiplication or division equation relating
three whole numbers. Example: For example, determine the unknown number that makes the
equation true in each of the equations 8 × ? = 48, 5 = __ ÷ 3, 6 × 6 = ?.
Algebra
Determine the missing multiplicand in a number sentence
involving basic facts [Grade 3]
Determine the missing dividend or divisor in a number sentence
involving basic facts [Grade 3]
Understand properties of multiplication and the relationship between multiplication and division.
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Grade 3
AL 3.OA.5 - Apply properties of operations as strategies to multiply and divide. Example: Examples: If
6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2
can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of
multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 ×
2) = 40 + 16 = 56. (Distributive property.).[Students need not use formal terms for these properties.]
Numbers and Operations
Apply the distributive property to the multiplication of a 2-digit
number by a 1- or 2-digit number [Grade 4]
Apply the distributive property to multiply a multi-digit number
by a 1-digit number [Grade 4]
Core Skill
AL 3.OA.6 - Understand division as an unknown-factor problem. Example: For example, find 32 ÷ 8
by finding the number that makes 32 when multiplied by 8.
Numbers and Operations
Know basic multiplication facts to 10 x 10 [Grade 3]
Core Skill
Know basic multiplication facts for 11 and 12 [Grade 3]
Know basic division facts to 100 ÷ 10 [Grade 3]
Core Skill
Know basic division facts for 11 and 12 [Grade 3]
Complete a multiplication and division fact family [Grade 3]
Algebra
Determine the missing multiplicand in a number sentence
involving basic facts [Grade 3]
Determine the missing dividend or divisor in a number sentence
involving basic facts [Grade 3]
Multiply and divide within 100.
AL 3.OA.7 - Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of
operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Numbers and Operations
Know basic multiplication facts to 10 x 10 [Grade 3]
Core Skill
Know basic division facts to 100 ÷ 10 [Grade 3]
Core Skill
WP: Multiply using basic facts to 10 x 10 [Grade 3]
Core Skill
WP: Divide using basic facts to 100 ÷ 10 [Grade 3]
Core Skill
Complete a multiplication and division fact family [Grade 3]
Algebra
Recognize equivalent multiplication or division expressions
involving basic facts [Grade 3]
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
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Grade 3
AL 3.OA.8 - Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using
mental computation and estimation strategies including rounding.[This standard is limited to problems
posed with whole numbers and having whole-number answers; students should know how to perform
operations in the conventional order when there are no parentheses to specify a particular order
(Order of Operations).]
Numbers and Operations
WP: Solve a 2-step problem involving addition and/or subtraction
of multi-digit whole numbers [Grade 4]
AL 3.OA.9 - Identify arithmetic patterns (including patterns in the addition table or multiplication
table), and explain them using properties of operations. Example: For example, observe that 4 times a
number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Algebra
Determine an addition or subtraction number pattern given a rule
[Grade 2]
Determine the rule for an addition or subtraction number pattern
[Grade 2]
Core Skill
Determine a rule for a table of related number pairs [Grade 3]
Determine a rule that relates two variables [Grade 4]
Core Skill
AL 3.NBT - Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.[A range
of algorithms may be used.]
AL 3.NBT.10 - Use place value understanding to round whole numbers to the nearest 10 or 100.
Numbers and Operations
Round a 2- to 4-digit whole number to its greatest place [Grade 3]
AL 3.NBT.11 - Fluently add and subtract within 1000 using strategies and algorithms based on place
value, properties of operations, and/or the relationship between addition and subtraction.
Numbers and Operations
Add two 3-digit numbers without regrouping [Grade 2]
Add 2- and 3-digit numbers with no more than one regrouping
[Grade 2]
Core Skill
Add two 3-digit numbers with one regrouping [Grade 2]
Core Skill
Subtract a 3-digit number from a 3-digit number without
regrouping [Grade 2]
Subtract a 1- or 2-digit number from a 3-digit number with one
regrouping [Grade 2]
Core Skill
Subtract a 3-digit number from a 3-digit number with one
regrouping [Grade 2]
Core Skill
AL 3.NBT.12 - Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 ×
60) using strategies based on place value and properties of operations.
Numbers and Operations
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Multiply a 1-digit whole number by a multiple of 10 to 100
[Grade 3]
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Grade 3
AL 3.NF - Number and Operations-Fractions[Grade 3 expectations in this domain are limited to fractions
with denominators 2, 3, 4, 6, and 8.]
Develop understanding of fractions as numbers.
AL 3.NF.13 - Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned
into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Numbers and Operations
Identify a unit fraction as part of a whole [Grade 2]
Identify a fraction as part of a whole [Grade 2]
AL 3.NF.14 - Understand a fraction as a number on the number line; represent fractions on a number
line diagram.
AL 3.NF.14.a - Represent a fraction 1/b on a number line diagram by defining the interval from 0 to
1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that
the endpoint of the part based at 0 locates the number 1/b on the number line.
Numbers and Operations
Identify a fraction represented by a point on a number line [Grade Core Skill
3]
Locate a fraction on a number line [Grade 3]
Core Skill
AL 3.NF.14.b - Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from
0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on
the number line.
Numbers and Operations
Identify a fraction represented by a point on a number line [Grade Core Skill
3]
Locate a fraction on a number line [Grade 3]
Core Skill
AL 3.NF.15 - Explain equivalence of fractions in special cases, and compare fractions by reasoning
about their size.
AL 3.NF.15.a - Understand two fractions as equivalent (equal) if they are the same size, or the same
point on a number line.
Numbers and Operations
Identify equivalent fractions using models [Grade 3]
Core Skill
AL 3.NF.15.b - Recognize and generate simple equivalent fractions, (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain
why the fractions are equivalent, e.g., by using a visual fraction model.
Numbers and Operations
Identify equivalent fractions using models [Grade 3]
Core Skill
Simplify a fraction [Grade 4]
Determine a set of equivalent fractions [Grade 4]
AL 3.NF.15.c - Express whole numbers as fractions, and recognize fractions that are equivalent to
whole numbers. Example: Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4
and 1 at the same point of a number line diagram.
Numbers and Operations
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Relate a fraction equal to a whole to a pictorial model [Grade 2]
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Grade 3
AL 3.NF.15.d - Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to
the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
Numbers and Operations
Compare fractions using models [Grade 3]
Core Skill
Compare fractions with like denominators [Grade 3]
Compare fractions with like numerators [Grade 3]
AL 3.MD - Measurement and Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of
objects.
AL 3.MD.16 - Tell and write time to the nearest minute and measure time intervals in minutes. Solve
word problems involving addition and subtraction of time intervals in minutes, e.g., by representing
the problem on a number line diagram.
Geometry and Measurement
Tell time to the minute [Grade 3]
Core Skill
Calculate elapsed time within an hour, given two clocks, without
regrouping [Grade 3]
Calculate elapsed time within an hour, given two clocks, with
regrouping [Grade 3]
WP: Calculate elapsed time within an hour given two clocks
[Grade 3]
WP: Calculate elapsed time within an hour [Grade 3]
AL 3.MD.17 - Measure and estimate liquid volumes and masses of objects using standard units of
grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as
a beaker with a measurement scale) to represent the problem.[Excludes compound units such as cm³
and finding the geometric volume of a container. Excludes multiplicative comparison problems
(problems involving notions of "times as much"; See Appendix A, Table 2.)]
Overall Product Skills
Build comprehension of math vocabulary
Represent and interpret data.
AL 3.MD.18 - Draw a scaled picture graph and a scaled bar graph to represent a data set with several
categories. Solve one- and two-step "how many more" and "how many less" problems using
information presented in scaled bar graphs. Example: For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
Data Analysis, Statistics, and
Probability
Use a pictograph to represent data (1 symbol = more than 1
object) [Grade 2]
Core Skill
Answer a question using information from a pictograph (1 symbol Core Skill
= more than 1 object) [Grade 2]
Use a bar graph with a y-axis scale by 2s to represent data [Grade Core Skill
2]
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Answer a question using information from a bar graph with a yaxis scale by 2s [Grade 2]
Grade 3
Core Skill
Use a bar graph with a scale interval of 5 or 10 to represent data
[Grade 3]
Answer a question using information from a bar graph with a
scale interval of 5 or 10 [Grade 3]
AL 3.MD.19 - Generate measurement data by measuring lengths using rulers marked with halves and
fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in
appropriate units-whole numbers, halves, or quarters.
Geometry and Measurement
Measure length in inches [Grade 2]
Core Skill
Measure length to the nearest half inch or quarter inch [Grade 3]
Data Analysis, Statistics, and
Probability
Use a line plot to represent data [Grade 3]
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
AL 3.MD.20 - Recognize area as an attribute of plane figures and understand concepts of area
measurement.
AL 3.MD.20.a - A square with side length 1 unit, called "a unit square," is said to have "one square
unit" of area, and can be used to measure area.
Geometry and Measurement
Relate area to the number of square units [Grade 2]
AL 3.MD.20.b - A plane figure which can be covered without gaps or overlaps by n unit squares is
said to have an area of n square units.
Geometry and Measurement
Determine the area of a shape composed of rectangles given a
picture on a grid [Grade 3]
AL 3.MD.21 - Measure areas by counting unit squares (square cm, square m, square in, square ft, and
improvised units).
Geometry and Measurement
Determine the area of a shape composed of rectangles given a
picture on a grid [Grade 3]
WP: Determine the area of a rectangular shape given a picture on
a grid [Grade 3]
AL 3.MD.22 - Relate area to the operations of multiplication and addition.
AL 3.MD.22.a - Find the area of a rectangle with whole-number side lengths by tiling it, and show
that the area is the same as would be found by multiplying the side lengths.
Geometry and Measurement
4/26/2012
WP: Determine the area of a rectangular shape given a picture on
a grid [Grade 3]
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Grade 3
AL 3.MD.22.b - Multiply side lengths to find areas of rectangles with whole-number side lengths in
the context of solving real world and mathematical problems, and represent whole-number products
as rectangular areas in mathematical reasoning.
Geometry and Measurement
Determine the area of a rectangle given the length and width
[Grade 4]
Core Skill
WP: Determine the area of a rectangle [Grade 4]
Core Skill
AL 3.MD.22.c - Use tiling to show in a concrete case that the area of a rectangle with whole-number
side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive
property in mathematical reasoning.
Numbers and Operations
Apply the distributive property to the multiplication of a 2-digit
number by a 1- or 2-digit number [Grade 4]
Overall Product Skills
Build comprehension of math vocabulary
AL 3.MD.22.d - Recognize area as additive. Find areas of rectilinear figures by decomposing them
into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this
technique to solve real world problems.
Geometry and Measurement
Determine the area of a shape composed of rectangles given a
picture on a grid [Grade 3]
Estimate the area of a shape composed of rectangles given a
reference square [Grade 3]
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between
linear and area measures.
AL 3.MD.23 - Solve real world and mathematical problems involving perimeters of polygons, including
finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles
with the same perimeter and different areas or with the same area and different perimeters.
Geometry and Measurement
WP: Determine the perimeter of a rectangular shape given a grid
with a reference unit [Grade 3]
WP: Determine the perimeter of a shape given a model showing
all side lengths [Grade 3]
Determine the perimeter of a rectangle given a picture showing
length and width [Grade 4]
Core Skill
WP: Determine the perimeter of a rectangle given a picture
showing length and width [Grade 4]
Core Skill
WP: Determine the perimeter of a square or rectangle [Grade 4]
Determine the missing side length of a rectangle given a side
length and the perimeter [Grade 4]
Core Skill
AL 3.G - Geometry
Reason with shapes and their attributes.
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Grade 3
AL 3.G.24 - Understand that shapes in different categories (e.g., rhombuses, rectangles, and others)
may share attributes (e.g., having four sides), and that the shared attributes can define a larger
category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of
quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
Geometry and Measurement
Classify a triangle by its sides [Grade 4]
Classify a quadrilateral [Grade 4]
Relate a polygon to attributes or characteristics [Grade 4]
AL 3.G.25 - Partition shapes into parts with equal areas. Express the area of each part as a unit
fraction of the whole. Example: For example, partition a shape into 4 parts with equal area, and
describe the area of each part as 1/4 of the area of the shape.
Numbers and Operations
Identify a unit fraction as part of a whole [Grade 2]
Identify a fraction as part of a whole [Grade 2]
Determine a pictorial model of a fraction of a whole [Grade 3]
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Core Skill
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Grade 4
Grade 4
AL 4.OA - Operations and Algebraic Thinking
Use the four operations with whole numbers to solve problems.
AL 4.OA.1 - Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a
statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of
multiplicative comparisons as multiplication equations.
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 4.OA.2 - Multiply or divide to solve word problems involving multiplicative comparison, e.g., by
using drawings and equations with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive comparison.
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 4.OA.3 - Solve multistep word problems posed with whole numbers and having whole-number
answers using the four operations, including problems in which remainders must be interpreted.
Represent these problems using equations with a letter standing for the unknown quantity. Assess the
reasonableness of answers using mental computation and estimation strategies including rounding.
Numbers and Operations
WP: Estimate a sum or difference of two 3- or 4-digit whole
numbers using any method [Grade 3]
Core Skill
WP: Solve a 2-step problem involving addition and/or subtraction
of multi-digit whole numbers [Grade 4]
WP: Estimate a product of two whole numbers using any method
[Grade 4]
Core Skill
WP: Solve a 2-step whole number problem using more than 1
operation [Grade 4]
WP: Solve a 2-step problem involving whole numbers [Grade 5]
Core Skill
WP: Estimate a quotient using any method [Grade 5]
Core Skill
WP: Solve a multi-step problem involving whole numbers [Grade Core Skill
6]
Gain familiarity with factors and multiples.
AL 4.OA.4 - Find all factor pairs for a whole number in the range 1-100. Recognize that a whole
number is a multiple of each of its factors. Determine whether a given whole number in the range 1100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1100 is prime or composite.
Numbers and Operations
4/26/2012
Determine if a number to 50 is prime or composite [Grade 5]
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Grade 4
Determine a complete list of whole number factor pairs for a
number to 50 [Grade 5]
Determine all the factors of a whole number to 50 [Grade 5]
Determine the prime factorization of a number to 50 [Grade 5]
Determine the common factors for two whole numbers to 50
[Grade 5]
Determine the multiple(s) of a number [Grade 5]
Determine common multiples for two whole numbers [Grade 5]
Generate and analyze patterns.
AL 4.OA.5 - Generate a number or shape pattern that follows a given rule. Identify apparent features
of the pattern that were not explicit in the rule itself. Example: For example, given the rule "Add 3"
and the starting number 1, generate terms in the resulting sequence and observe that the terms appear
to alternate between odd and even numbers. Explain informally why the numbers will continue to
alternate in this way.
Numbers and Operations
Algebra
Complete a skip pattern starting from a multiple of 2, 5, or 10
[Grade 2]
Core Skill
Complete a skip pattern of 2, 5, or 10 starting from any number
[Grade 2]
Core Skill
Count on by 100s from any number [Grade 2]
Core Skill
Determine an addition or subtraction number pattern given a rule
[Grade 2]
Extend a number pattern [Grade 3]
Determine a rule for a table of related number pairs [Grade 3]
WP: Find the missing number in a table of paired values [Grade
3]
Generate a table of paired numbers based on a rule [Grade 4]
Core Skill
AL 4.NBT - Number and Operations in Base Ten[Grade 4 expectations in this domain are limited to whole
numbers less than or equal to 1,000,000.]
Generalize place value understanding for multi-digit whole numbers.
AL 4.NBT.6 - Recognize that in a multi-digit whole number, a digit in one place represents ten times
what it represents in the place to its right. Example: For example, recognize that 700 ÷ 70 = 10 by
applying concepts of place value and division.
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
AL 4.NBT.7 - Read and write multi-digit whole numbers using base-ten numerals, number names, and
expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using
>, =, and < symbols to record the results of comparisons.
Numbers and Operations
4/26/2012
Read a 4- or 5-digit whole number [Grade 3]
Core Skill
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Grade 4
Determine the word form of a 4- or 5-digit whole number [Grade
3]
Core Skill
Determine the 4-digit whole number represented in thousands,
hundreds, tens, and ones [Grade 3]
Core Skill
Represent a 4- or 5-digit whole number in expanded form [Grade
3]
Core Skill
Determine the 4- or 5-digit whole number represented in
expanded form [Grade 3]
Core Skill
Compare 4- or 5-digit whole numbers using the symbols <, >, and
= [Grade 3]
Read a 6-digit whole number [Grade 4]
Determine the word form of a 6-digit whole number [Grade 4]
Convert between proper expanded form and improper expanded
form up to a 5-digit whole number [Grade 4]
Convert between standard form and improper expanded form up
to a 5-digit whole number [Grade 4]
AL 4.NBT.8 - Use place value understanding to round multi-digit whole numbers to any place.
Numbers and Operations
Round a 4- to 6-digit whole number to a specified place [Grade 4] Core Skill
Use place value understanding and properties of operations to perform multi-digit arithmetic.
AL 4.NBT.9 - Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Numbers and Operations
Add 3- and 4-digit whole numbers with regrouping [Grade 3]
Core Skill
Subtract 3- and 4-digit whole numbers with regrouping [Grade 3]
Core Skill
Add up to 4-digit whole numbers in expanded form [Grade 4]
Add a 5-digit or greater whole number and a 3-digit or greater
whole number [Grade 4]
Add three multi-digit whole numbers [Grade 4]
Subtract a 3-digit or greater whole number from a 5-digit or
greater whole number [Grade 4]
WP: Add a 5-digit or greater whole number and a 3-digit or
greater whole number [Grade 4]
WP: Add three multi-digit whole numbers [Grade 4]
WP: Subtract a 3-digit or greater whole number from a 5-digit or
greater whole number [Grade 4]
AL 4.NBT.10 - Multiply a whole number of up to four digits by a one-digit whole number, and
multiply two two-digit numbers, using strategies based on place value and the properties of operations.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Numbers and Operations
Multiply a 2-digit whole number by a 1-digit number [Grade 3]
Apply the distributive property to the multiplication of a 2-digit
number by a 1- or 2-digit number [Grade 4]
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Grade 4
Apply the distributive property to multiply a multi-digit number
by a 1-digit number [Grade 4]
Core Skill
Multiply a 3- or 4-digit whole number by a 1-digit whole number
[Grade 4]
Core Skill
Multiply a 2-digit whole number by a 2-digit whole number
[Grade 4]
Core Skill
WP: Multiply a multi-digit whole number by a 1-digit whole
number [Grade 4]
Core Skill
WP: Multiply a 2-digit whole number by a 2-digit whole number
[Grade 4]
Core Skill
WP: Multiply a 3-digit whole number by a 2-digit whole number
[Grade 4]
AL 4.NBT.11 - Find whole-number quotients and remainders with up to four-digit dividends and onedigit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
Numbers and Operations
Divide a 2-digit whole number by a 1-digit whole number with no
remainder in the quotient [Grade 4]
Divide a 3-digit whole number by a 1-digit whole number with no
remainder in the quotient [Grade 4]
Divide a 2-digit whole number by a 1-digit whole number with a
remainder in the quotient [Grade 4]
Divide a 3-digit whole number by a 1-digit whole number with a
remainder in the quotient [Grade 4]
Core Skill
WP: Divide a 3-digit whole number by a 1-digit whole number
with no remainder in the quotient [Grade 4]
Core Skill
WP: Divide a 2-digit whole number by a 1-digit whole number
with a remainder in the quotient [Grade 4]
WP: Divide a 3-digit whole number by a 1-digit whole number
with a remainder in the quotient [Grade 4]
Core Skill
AL 4.NF - Number and Operations-Fractions[Grade 4 expectations in this domain are limited to fractions
with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.]
Extend understanding of fraction equivalence and ordering.
AL 4.NF.12 - Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual
fraction models, with attention to how the number and size of the parts differ even though the two
fractions themselves are the same size. Use this principle to recognize and generate equivalent
fractions.
Numbers and Operations
Identify equivalent fractions using models [Grade 3]
Core Skill
Determine a set of equivalent fractions [Grade 4]
Determine equivalent fractions not in simplest form [Grade 5]
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Grade 4
AL 4.NF.13 - Compare two fractions with different numerators and different denominators, e.g., by
creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.
Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the
results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual
fraction model.
Numbers and Operations
Compare fractions using models [Grade 3]
Core Skill
Compare fractions with like denominators [Grade 3]
Compare fractions with like numerators [Grade 3]
Compare fractions on a number line [Grade 4]
Determine equivalent fractions not in simplest form [Grade 5]
Core Skill
Compare fractions with unlike denominators [Grade 5]
Core Skill
Build fractions from unit fractions by applying and extending previous understandings of operations on
whole numbers.
AL 4.NF.14 - Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
AL 4.NF.14.a - Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
Numbers and Operations
Add fractions with like denominators no greater than 10 using
models [Grade 4]
Add fractions with like denominators no greater than 10 [Grade 4] Core Skill
Add fractions with like denominators no greater than 10 and
simplify the sum [Grade 4]
Subtract fractions with like denominators no greater than 10 using
models [Grade 4]
Subtract fractions with like denominators no greater than 10
[Grade 4]
Core Skill
Subtract fractions with like denominators no greater than 10 and
simplify the difference [Grade 4]
AL 4.NF.14.b - Decompose a fraction into a sum of fractions with the same denominator in more
than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a
visual fraction model. Example: Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 =
8/8 + 8/8 + 1/8.
Numbers and Operations
Identify a mixed number represented by a model [Grade 4]
Identify an improper fraction represented by a model of a mixed
number [Grade 4]
AL 4.NF.14.c - Add and subtract mixed numbers with like denominators, e.g., by replacing each
mixed number with an equivalent fraction, and/or by using properties of operations and the
relationship between addition and subtraction.
Numbers and Operations
4/26/2012
Add mixed numbers with like denominators and simplify the sum
[Grade 5]
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Grade 4
Subtract mixed numbers with like denominators and simplify the
difference [Grade 5]
WP: Add or subtract mixed numbers with like denominators and
simplify the sum or difference [Grade 5]
Core Skill
AL 4.NF.14.d - Solve word problems involving addition and subtraction of fractions referring to the
same whole and having like denominators, e.g., by using visual fraction models and equations to
represent the problem.
Numbers and Operations
Add fractions with like denominators no greater than 10 using
models [Grade 4]
Add fractions with like denominators no greater than 10 and
simplify the sum [Grade 4]
WP: Add fractions with like denominators no greater than 10 and
simplify the sum [Grade 4]
Subtract fractions with like denominators no greater than 10 using
models [Grade 4]
Subtract fractions with like denominators no greater than 10 and
simplify the difference [Grade 4]
WP: Subtract fractions with like denominators no greater than 10
and simplify the difference [Grade 4]
Add fractions with like denominators greater than 10 and simplify
the sum [Grade 5]
Subtract fractions with like denominators greater than 10 and
simplify the difference [Grade 5]
WP: Add or subtract fractions with like denominators and
simplify the sum or difference [Grade 5]
Core Skill
WP: Add or subtract mixed numbers with like denominators and
simplify the sum or difference [Grade 5]
Core Skill
AL 4.NF.15 - Apply and extend previous understandings of multiplication to multiply a fraction by a
whole number.
AL 4.NF.15.a - Understand a fraction a/b as a multiple of 1/b. Example: For example, use a visual
fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4
= 5 × (1/4).
Numbers and Operations
Multiply a whole number by a unit fraction using a model [Grade
5]
Multiply a whole number by a unit fraction [Grade 5]
AL 4.NF.15.b - Understand a multiple of a/b as a multiple of 1/b, and use this understanding to
multiply a fraction by a whole number. Example: For example, use a visual fraction model to express
3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.).
Numbers and Operations
Multiply a whole number by a unit fraction [Grade 5]
Multiply a proper fraction by a whole number using a model
[Grade 5]
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Grade 4
Multiply a proper fraction by a whole number [Grade 5]
AL 4.NF.15.c - Solve word problems involving multiplication of a fraction by a whole number, e.g.,
by using visual fraction models and equations to represent the problem. Example: For example, if
each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party,
how many pounds of roast beef will be needed? Between what two whole numbers does your answer
lie?
Numbers and Operations
WP: Multiply or divide a whole number by a unit fraction [Grade
5]
Understand decimal notation for fractions, and compare decimal fractions.
AL 4.NF.16 - Express a fraction with denominator 10 as an equivalent fraction with denominator 100,
and use this technique to add two fractions with respective denominators 10 and 100. Example: For
example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.[Students who can generate equivalent
fractions can develop strategies for adding fractions with unlike denominators in general. But addition
and subtraction with unlike denominators in general is not a requirement at this grade.]
Overall Product Skills
Use strategies to derive solutions for problems
AL 4.NF.17 - Use decimal notation for fractions with denominators 10 or 100. Example: For example,
rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
Numbers and Operations
Determine the decimal number from a pictorial model of tenths or Core Skill
hundredths [Grade 4]
Identify a pictorial model of tenths or hundredths of a decimal
number [Grade 4]
Core Skill
Identify a decimal number to tenths represented by a point on a
number line [Grade 4]
Core Skill
Locate a decimal number to tenths on a number line [Grade 4]
Core Skill
Determine the decimal number equivalent to a fraction with a
denominator of 10 or 100 [Grade 4]
Determine a fraction equivalent to a decimal, using a denominator
of 10 or 100 [Grade 4]
Determine the decimal number equivalent to a fraction model
[Grade 4]
Core Skill
Determine the fraction equivalent to a decimal number model
[Grade 4]
Core Skill
AL 4.NF.18 - Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the results of
comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Numbers and Operations
4/26/2012
Read a decimal number through the hundredths place [Grade 4]
Core Skill
Identify a decimal number to tenths represented by a point on a
number line [Grade 4]
Core Skill
Compare decimal numbers through the hundredths place [Grade
4]
Core Skill
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Order decimal numbers through the hundredths place [Grade 4]
Grade 4
Core Skill
AL 4.MD - Measurement and Data
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller
unit.
AL 4.MD.19 - Know relative sizes of measurement units within one system of units including km, m,
cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in
a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.
Example: For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as
48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),...
Geometry and Measurement
Convert between inches, feet, and yards [Grade 3]
Convert between centimeters and meters [Grade 3]
Convert hours to minutes or minutes to seconds [Grade 3]
Convert between customary units of capacity using whole
numbers [Grade 4]
Convert between customary units of weight using whole numbers
[Grade 4]
Convert between metric units of capacity using whole numbers
[Grade 4]
Convert between metric units of mass using whole numbers
[Grade 4]
Convert between millimeters and centimeters or meters using
whole numbers [Grade 4]
AL 4.MD.20 - Use the four operations to solve word problems involving distances, intervals of time,
liquid volumes, masses of objects, and money, including problems involving simple fractions or
decimals, and problems that require expressing measurements given in a larger unit in terms of a
smaller unit. Represent measurement quantities using diagrams such as number line diagrams that
feature a measurement scale.
Numbers and Operations
Add two money amounts less than $1 in decimal notation [Grade
3]
Subtract two money amounts less than $1 in decimal notation
[Grade 3]
Add or subtract cent amounts to or from whole dollar amounts
[Grade 3]
Determine money amounts that total $10 [Grade 3]
Core Skill
WP: Add or subtract money amounts less than $1 [Grade 3]
WP: Add or subtract money amounts greater than $1 [Grade 4]
WP: Solve a money problem involving 2 steps [Grade 4]
WP: Multiply a money amount by a 1-digit number [Grade 4]
WP: Divide a money amount by a 1-digit number [Grade 4]
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Grade 4
WP: Multiply a money amount by a 2-digit whole number [Grade
5]
Geometry and Measurement
Calculate elapsed time within an hour, given two clocks, with
regrouping [Grade 3]
WP: Calculate elapsed time within an hour given two clocks
[Grade 3]
WP: Add or subtract compound units of length requiring
customary unit conversion using whole numbers [Grade 4]
Calculate elapsed time exceeding an hour with regrouping [Grade Core Skill
4]
WP: Calculate elapsed time exceeding an hour without regrouping
hours [Grade 4]
WP: Calculate elapsed time exceeding an hour with regrouping
hours [Grade 4]
Core Skill
WP: Determine the end time given the start time and the elapsed
time exceeding an hour [Grade 4]
WP: Determine the start time given the end time and the elapsed
time exceeding an hour [Grade 4]
WP: Add time intervals involving hours and minutes [Grade 4]
Solve an elapsed-time problem using a calendar [Grade 4]
WP: Compare customary units of length, weight, or capacity
using fractional amounts [Grade 5]
WP: Compare metric units of length, mass, or capacity using
decimal amounts [Grade 5]
WP: Calculate elapsed time using a.m. and p.m. [Grade 5]
WP: Use a calendar to solve a problem [Grade 5]
AL 4.MD.21 - Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. Example: For example, find the width of a rectangular room given the area of the flooring
and the length, by viewing the area formula as a multiplication equation with an unknown factor.
Geometry and Measurement
Determine the area of a shape composed of rectangles given a
picture on a grid [Grade 3]
WP: Determine the area of a rectangular shape given a picture on
a grid [Grade 3]
Determine the perimeter of a rectangle given a picture showing
length and width [Grade 4]
Core Skill
WP: Determine the perimeter of a rectangle given a picture
showing length and width [Grade 4]
Core Skill
WP: Determine the perimeter of a square or rectangle [Grade 4]
Determine the missing side length of a rectangle given a side
length and the perimeter [Grade 4]
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Core Skill
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Grade 4
Determine the area of a rectangle given a picture showing the
length and width [Grade 4]
Determine the area of a rectangle given the length and width
[Grade 4]
Core Skill
WP: Determine the area of a rectangle [Grade 4]
Core Skill
Determine the missing side length of a rectangle given a side
length and the area [Grade 4]
Core Skill
Represent and interpret data.
AL 4.MD.22 - 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. Example: For example, from a line plot find and interpret the difference in length between
the longest and shortest specimens in an insect collection.
Data Analysis, Statistics, and
Probability
Read a line plot [Grade 3]
Use a line plot to represent data [Grade 3]
Answer a question using information from a line plot [Grade 3]
Geometric measurement: understand concepts of angle and measure angles.
AL 4.MD.23 - Recognize angles as geometric shapes that are formed wherever two rays share a
common endpoint, and understand concepts of angle measurement:
AL 4.MD.23.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.
Geometry and Measurement
Associate degrees with a turn on a circle [Grade 4]
AL 4.MD.23.b - An angle that turns through n one-degree angles is said to have an angle measure of
n degrees.
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
AL 4.MD.24 - Measure angles in whole-number degrees using a protractor. Sketch angles of specified
measure.
Geometry and Measurement
Measure an angle to the nearest 5 degrees [Grade 5]
Measure an angle, between two rays or in a shape, to the nearest
degree [Grade 6]
AL 4.MD.25 - Recognize angle measure as additive. When an angle is decomposed into nonoverlapping 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 and Measurement
4/26/2012
Determine the measure of a missing angle using straight and right
angle relationships [Grade 6]
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Grade 4
AL 4.G - Geometry
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
AL 4.G.26 - Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular
and parallel lines. Identify these in two-dimensional figures.
Geometry and Measurement
Identify parallel, perpendicular, and intersecting lines [Grade 3]
Classify an angle as equal to 90°, less than 90°, or greater than
90° [Grade 3]
Classify an angle in a shape as equal to 90°, less than 90°, or
greater than 90° [Grade 3]
Classify an angle given a picture [Grade 4]
Core Skill
Classify an angle given its measure [Grade 4]
Core Skill
Identify a shape that has parallel or perpendicular sides [Grade 4]
AL 4.G.27 - 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.
Geometry and Measurement
Classify a quadrilateral [Grade 4]
Classify a triangle by its sides and angles [Grade 5]
Know the properties of a triangle or a quadrilateral [Grade 6]
AL 4.G.28 - 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.
Geometry and Measurement
Identify symmetry in a 2-dimensional shape [Grade 2]
Determine lines of symmetry [Grade 4]
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Core Skill
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Grade 5
Grade 5
AL 5.OA - Operations and Algebraic Thinking
Write and interpret numerical expressions.
AL 5.OA.1 - Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions
with these symbols.
Numbers and Operations
Evaluate a numerical expression involving three operations, with
no parentheses, using order of operations [Grade 5]
Evaluate a numerical expression involving three operations, with
parentheses, using order of operations [Grade 5]
Evaluate a numerical expression of four or more operations, with
parentheses, using order of operations [Grade 6]
Algebra
Core Skill
Evaluate a numeric expression involving two operations [Grade 4]
AL 5.OA.2 - Write simple expressions that record calculations with numbers, and interpret numerical
expressions without evaluating them. Example: For example, express the calculation "add 8 and 7,
then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 +
921, without having to calculate the indicated sum or product.
Numbers and Operations
Apply the distributive property to the multiplication of a 2-digit
number by a 1- or 2-digit number [Grade 4]
Apply the distributive property to multiply a multi-digit number
by a 1-digit number [Grade 4]
Core Skill
Evaluate a numerical expression involving three operations, with
no parentheses, using order of operations [Grade 5]
Evaluate a numerical expression involving three operations, with
parentheses, using order of operations [Grade 5]
Analyze patterns and relationships.
AL 5.OA.3 - Generate two numerical patterns using two given rules. Identify apparent relationships
between corresponding terms. Form ordered pairs consisting of corresponding terms from the two
patterns, and graph the ordered pairs on a coordinate plane. Example: For example, given the rule
"Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate
terms in the resulting sequences, and observe that the terms in one sequence are twice the
corresponding terms in the other sequence. Explain informally why this is so.
Algebra
Generate a table of paired numbers based on a rule [Grade 4]
Core Skill
Extend a number pattern in a table of related pairs [Grade 4]
Use a first quadrant graph to represent the values from a table
generated in context [Grade 5]
Core Skill
Use a first quadrant graph to represent the values in an inputoutput table [Grade 6]
AL 5.NBT - Number and Operations in Base Ten
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Grade 5
Understand the place value system.
AL 5.NBT.4 - Recognize that in a multi-digit number, a digit in one place represents 10 times as much
as it represents in the place to its right and 1/10 of what it represents in the place to its left.
Numbers and Operations
Determine the value of a digit in a 6-digit whole number [Grade
4]
Determine the value of a digit in a decimal number to thousandths
[Grade 5]
Compare decimal numbers of differing places to thousandths
[Grade 5]
Core Skill
Represent a decimal number in expanded form using powers of
ten [Grade 6]
Core Skill
Determine the decimal number represented in expanded form
using powers of ten [Grade 6]
Core Skill
AL 5.NBT.5 - Explain patterns in the number of zeros of the product when multiplying a number by
powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied
or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Numbers and Operations
Divide a multi-digit whole number by 10 or 100 with no
remainder [Grade 4]
Multiply a decimal number through thousandths by 10, 100, or
1,000 [Grade 5]
Core Skill
Determine the power of ten that relates a decimal number and a
whole number [Grade 6]
Determine the power of ten that relates two decimal numbers
[Grade 6]
Divide a decimal number by 10, 100, or 1,000 [Grade 6]
Core Skill
Relate division by a whole number power of ten to multiplication
by the related decimal fraction power of ten [Grade 6]
AL 5.NBT.6 - Read, write, and compare decimals to thousandths.
AL 5.NBT.6.a - Read and write decimals to thousandths using base-ten numerals, number names,
and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
Numbers and Operations
Read a decimal number through the hundredths place [Grade 4]
Core Skill
Determine the word form of a decimal number through the
hundredths place [Grade 4]
Determine the value of a digit in a decimal number to thousandths
[Grade 5]
Determine a decimal number represented in expanded form
[Grade 5]
Represent a decimal number in expanded form [Grade 5]
Represent a decimal number in expanded form using powers of
ten [Grade 6]
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Core Skill
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Determine the decimal number represented in expanded form
using powers of ten [Grade 6]
Grade 5
Core Skill
AL 5.NBT.6.b - Compare two decimals to thousandths based on meanings of the digits in each place,
using >, =, and < symbols to record the results of comparisons.
Numbers and Operations
Compare decimal numbers through the hundredths place [Grade
4]
Core Skill
Order decimal numbers through the hundredths place [Grade 4]
Core Skill
Compare decimal numbers to thousandths represented in
expanded form [Grade 5]
Compare decimal numbers of differing places to thousandths
[Grade 5]
Core Skill
Order decimal numbers of differing places to thousandths in
ascending or descending order [Grade 5]
Core Skill
AL 5.NBT.7 - Use place value understanding to round decimals to any place.
Numbers and Operations
Round a decimal number to a specified place through hundredths
[Grade 4]
Core Skill
Round a decimal number to a specified decimal place to
thousandths [Grade 5]
Perform operations with multi-digit whole numbers and with decimals to hundredths.
AL 5.NBT.8 - Fluently multiply multi-digit whole numbers using the standard algorithm.
Numbers and Operations
Multiply a 1- or 2-digit whole number by a multiple of 10, 100, or Core Skill
1,000 [Grade 4]
Multiply a 3- or 4-digit whole number by a 1-digit whole number
[Grade 4]
Core Skill
Multiply a 2-digit whole number by a 2-digit whole number
[Grade 4]
Core Skill
Multiply a 3-digit whole number by a 2-digit whole number
[Grade 4]
WP: Multiply a multi-digit whole number by a 1-digit whole
number [Grade 4]
Core Skill
WP: Multiply a 2-digit whole number by a 2-digit whole number
[Grade 4]
Core Skill
WP: Multiply a 3-digit whole number by a 2-digit whole number
[Grade 4]
Multiply a 3- or 4-digit whole number by a 3-digit whole number
[Grade 5]
WP: Multiply a 3- or 4-digit whole number by a 3-digit whole
number [Grade 5]
Find the product of three identical factors [Grade 6]
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Grade 5
AL 5.NBT.9 - Find whole-number quotients of whole numbers with up to four-digit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by using
equations, rectangular arrays, and/or area models.
Numbers and Operations
WP: Divide a 3-digit whole number by a 1-digit whole number
with no remainder in the quotient [Grade 4]
Core Skill
Divide a multi-digit whole number by a 2-digit whole number,
with no remainder and no zeros in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with a remainder and no zeros in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with no remainder and at least one zero in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with a remainder and at least one zero in the quotient [Grade 5]
Core Skill
Divide a multi-digit whole number by a 2-digit whole number and Core Skill
express the quotient as a mixed number [Grade 5]
AL 5.NBT.10 - Add, subtract, multiply, and divide decimals to hundredths, using concrete models or
drawings and strategies based on place value, properties of operations, and/or the relationship between
addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Numbers and Operations
Add two decimal numbers through hundredths [Grade 4]
Subtract two decimal numbers through hundredths [Grade 4]
Subtract a decimal number from a whole number or a whole
number from a decimal number [Grade 5]
Multiply a money amount by a 2- or 3-digit whole number [Grade
5]
Divide a decimal number by 10, 100, or 1,000 [Grade 6]
Core Skill
Divide a 1- to 3-digit whole number by a decimal number to
tenths where the quotient is a whole number [Grade 6]
Core Skill
AL 5.NF - Number and Operations-Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
AL 5.NF.11 - Add and subtract fractions with unlike denominators (including mixed numbers) by
replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. Example: For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.
(In general, a/b + c/d = (ad + bc)/bd.).
Numbers and Operations
Add fractions with unlike denominators using a model and do not
simplify the sum [Grade 5]
Add fractions with unlike denominators and do not simplify the
sum [Grade 5]
Add fractions with unlike denominators that have factors in
common and simplify the sum [Grade 5]
4/26/2012
Core Skill
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Add fractions with unlike denominators that have no factors in
common [Grade 5]
Grade 5
Core Skill
Subtract fractions with unlike denominators using a model and do
not simplify the difference [Grade 5]
Subtract fractions with unlike denominators and do not simplify
the difference [Grade 5]
Subtract fractions with unlike denominators that have factors in
common and simplify the difference [Grade 5]
Core Skill
Subtract fractions with unlike denominators that have no factors
in common [Grade 5]
Core Skill
Add mixed numbers with unlike denominators and simplify the
sum [Grade 5]
Subtract mixed numbers with unlike denominators and simplify
the difference [Grade 5]
WP: Add or subtract fractions with unlike denominators and
simplify the sum or difference [Grade 6]
Add mixed numbers with unlike denominators or a mixed number
and a fraction with unlike denominators and simplify the sum
[Grade 6]
Subtract a mixed number from a whole number [Grade 6]
Subtract mixed numbers with unlike denominators or a mixed
number and a fraction and simplify the difference [Grade 6]
Add and subtract three unlike-denominator fractions, mixed
numbers, or fractions and mixed numbers, and simplify the
answer [Grade 6]
WP: Add or subtract mixed numbers with unlike denominators or
a mixed number and a fraction with unlike denominators and
simplify the sum or difference [Grade 6]
AL 5.NF.12 - Solve word problems involving addition and subtraction of fractions referring to the
same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations
to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally
and assess the reasonableness of answers. Example: For example, recognize an incorrect result 2/5 +
1/2 = 3/7, by observing that 3/7 < 1/2.
Numbers and Operations
WP: Add fractions with like denominators no greater than 10 and
simplify the sum [Grade 4]
WP: Subtract fractions with like denominators no greater than 10
and simplify the difference [Grade 4]
WP: Add or subtract fractions with like denominators and
simplify the sum or difference [Grade 5]
Core Skill
WP: Add or subtract fractions with unlike denominators that have Core Skill
no factors in common [Grade 5]
WP: Add or subtract mixed numbers with like denominators and
simplify the sum or difference [Grade 5]
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Core Skill
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Grade 5
WP: Add or subtract mixed numbers with unlike denominators
that have no factors in common [Grade 5]
Round a fraction to a benchmark number of 0, 1/2, or 1 [Grade 5]
Estimate a fraction sum using benchmark numbers 0, 1/2, and 1
[Grade 5]
Estimate a fraction difference using benchmark numbers 0, 1/2,
and 1 [Grade 5]
WP: Estimate a fraction sum or difference using benchmark
numbers 0, 1/2, and 1 [Grade 5]
WP: Add or subtract fractions with unlike denominators and
simplify the sum or difference [Grade 6]
WP: Add or subtract mixed numbers with unlike denominators or
a mixed number and a fraction with unlike denominators and
simplify the sum or difference [Grade 6]
Apply and extend previous understandings of multiplication and division to multiply and divide
fractions.
AL 5.NF.13 - Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve
word problems involving division of whole numbers leading to answers in the form of fractions or
mixed numbers, e.g., by using visual fraction models or equations to represent the problem. Example:
For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and
that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people
want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person
get? Between what two whole numbers does your answer lie?
Numbers and Operations
WP: Use a mixed number to represent an amount in a sharing
situation [Grade 4]
Relate a sharing situation to a fraction [Grade 5]
AL 5.NF.14 - Apply and extend previous understandings of multiplication to multiply a fraction or
whole number by a fraction.
Numbers and Operations
Multiply a whole number by a unit fraction using a model [Grade
5]
Multiply a whole number by a unit fraction [Grade 5]
Multiply a proper fraction by a whole number using a model
[Grade 5]
Multiply a proper fraction by a whole number [Grade 5]
4/26/2012
Multiply a fraction by a fraction [Grade 6]
Core Skill
Multiply a mixed number by a whole number [Grade 6]
Core Skill
Multiply a mixed number by a fraction [Grade 6]
Core Skill
Multiply a mixed number by a mixed number [Grade 6]
Core Skill
WP: Multiply or divide a fraction by a fraction [Grade 6]
Core Skill
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WP: Multiply or divide two mixed numbers or a mixed number
and a fraction [Grade 6]
Grade 5
Core Skill
AL 5.NF.14.a - Interpret the product (a/b) × q as a parts of a partition of q into b equal parts;
equivalently, as the result of a sequence of operations a × q ÷ b. Example: For example, use a visual
fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with
(2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.).
Numbers and Operations
Multiply a whole number by a unit fraction using a model [Grade
5]
Multiply a whole number by a unit fraction [Grade 5]
Multiply a proper fraction by a whole number using a model
[Grade 5]
Multiply a proper fraction by a whole number [Grade 5]
AL 5.NF.14.b - Find the area of a rectangle with fractional side lengths by tiling it with unit squares
of the appropriate unit fraction side lengths, and show that the area is the same as would be found by
multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and
represent fraction products as rectangular areas.
Geometry and Measurement
Determine the area of a shape composed of rectangles given a
picture on a grid [Grade 3]
WP: Determine the area of a rectangular shape given a picture on
a grid [Grade 3]
Determine the area of a polygon on a grid [Grade 4]
Core Skill
Estimate the area of an irregular polygon on a grid [Grade 4]
AL 5.NF.15 - Interpret multiplication as scaling (resizing), by:
AL 5.NF.15.a - Comparing the size of a product to the size of one factor on the basis of the size of the
other factor, without performing the indicated multiplication.
Numbers and Operations
Determine a complete list of whole number factor pairs for a
number to 50 [Grade 5]
Determine all the factors of a whole number to 50 [Grade 5]
AL 5.NF.15.b - Explaining why multiplying a given number by a fraction greater than 1 results in a
product greater than the given number (recognizing multiplication by whole numbers greater than 1
as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a
product smaller than the given number; and relating the principle of fraction equivalence a/b =
(n×a)/(n×b) to the effect of multiplying a/b by 1.
Numbers and Operations
Multiply a whole number by a unit fraction using a model [Grade
5]
Multiply a whole number by a unit fraction [Grade 5]
Multiply a proper fraction by a whole number using a model
[Grade 5]
Multiply a proper fraction by a whole number [Grade 5]
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Multiply a mixed number by a fraction [Grade 6]
Grade 5
Core Skill
AL 5.NF.16 - Solve real world problems involving multiplication of fractions and mixed numbers, e.g.,
by using visual fraction models or equations to represent the problem.
Numbers and Operations
WP: Multiply or divide a whole number by a unit fraction [Grade
5]
WP: Multiply or divide a fraction by a fraction [Grade 6]
Core Skill
WP: Multiply or divide two mixed numbers or a mixed number
and a fraction [Grade 6]
Core Skill
AL 5.NF.17 - Apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions.
AL 5.NF.17.a - Interpret division of a unit fraction by a non-zero whole number, and compute such
quotients. Example: For example, create a story context for (1/3) ÷ 4, and use a visual fraction model
to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷
4 = 1/12 because (1/12) × 4 = 1/3.
Numbers and Operations
Divide a unit fraction by a whole number [Grade 5]
AL 5.NF.17.b - Interpret division of a whole number by a unit fraction, and compute such quotients.
Example: For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show
the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20
because 20 × (1/5) = 4.
Numbers and Operations
Divide a whole number by a unit fraction using a model [Grade 5]
Divide a whole number by a unit fraction [Grade 5]
AL 5.NF.17.c - Solve real world problems involving division of unit fractions by non-zero whole
numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and
equations to represent the problem. Example: For example, how much chocolate will each person get
if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Numbers and Operations
WP: Multiply or divide a whole number by a unit fraction [Grade
5]
AL 5.MD - Measurement and Data
Convert like measurement units within a given measurement system.
AL 5.MD.18 - Convert among different-sized standard measurement units within a given measurement
system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world
problems.
Geometry and Measurement
Convert between customary units of length using fractional
amounts [Grade 5]
Convert between customary units of capacity using fractional
amounts [Grade 5]
Convert between customary units of weight using fractional
amounts [Grade 5]
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Grade 5
Convert between metric units of capacity using decimal amounts
[Grade 5]
Convert between metric units of mass using decimal amounts
[Grade 5]
Convert between millimeters or centimeters and meters, or meters
and kilometers using decimal amounts [Grade 5]
WP: Compare customary units of length, weight, or capacity
using fractional amounts [Grade 5]
WP: Compare metric units of length, mass, or capacity using
decimal amounts [Grade 5]
WP: Add or subtract customary measures of weight requiring unit
conversion [Grade 6]
WP: Add or subtract metric measures of mass requiring unit
conversion [Grade 6]
WP: Multiply or divide customary measures of weight requiring
unit conversion [Grade 6]
WP: Multiply or divide metric measures of mass requiring unit
conversion [Grade 6]
Represent and interpret data.
AL 5.MD.19 - Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4,
1/8). Use operations on fractions for this grade to solve problems involving information presented in
line plots. Example: For example, given different measurements of liquid in identical beakers, find the
amount of liquid each beaker would contain if the total amount in all the beakers were redistributed
equally.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to
addition.
AL 5.MD.20 - Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
AL 5.MD.20.a - A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit"
of volume, and can be used to measure volume.
Geometry and Measurement
Determine the volume of a rectangular prism given a diagram
showing unit cubes [Grade 4]
Overall Product Skills
Build comprehension of math vocabulary
AL 5.MD.20.b - A solid figure which can be packed without gaps or overlaps using n unit cubes is
said to have a volume of n cubic units.
Geometry and Measurement
Determine the volume of an object composed of rectangular
prisms by counting units [Grade 5]
Overall Product Skills
Build comprehension of math vocabulary
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Grade 5
AL 5.MD.21 - Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and
improvised units.
Geometry and Measurement
Determine the volume of a rectangular prism given a diagram
showing unit cubes [Grade 4]
Determine the volume of an object composed of rectangular
prisms by counting units [Grade 5]
AL 5.MD.22 - Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
AL 5.MD.22.a - Find the volume of a right rectangular prism with whole-number side lengths by
packing it with unit cubes, and show that the volume is the same as would be found by multiplying
the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold
whole-number products as volumes, e.g., to represent the associative property of multiplication.
Geometry and Measurement
Determine the volume of a rectangular prism given a diagram
showing unit cubes [Grade 4]
AL 5.MD.22.b - Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find
volumes of right rectangular prisms with whole-number edge lengths in the context of solving real
world and mathematical problems.
Geometry and Measurement
Determine the volume of a rectangular prism given a diagram
[Grade 5]
Core Skill
WP: Determine the volume of a rectangular prism given a
diagram [Grade 5]
Determine the volume of a rectangular prism [Grade 5]
WP: Determine the volume of a rectangular prism [Grade 5]
Core Skill
AL 5.MD.22.c - Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying
this technique to solve real world problems.
Geometry and Measurement
Determine the volume of an object composed of rectangular
prisms by counting units [Grade 5]
Determine the volume of a prism with a right triangle base [Grade
6]
AL 5.G - Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems.
AL 5.G.23 - Use a pair of perpendicular number lines, called axes, to define a coordinate system, with
the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in
the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first
number indicates how far to travel from the origin in the direction of one axis, and the second number
indicates how far to travel in the direction of the second axis, with the convention that the names of the
two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
Algebra
4/26/2012
Use a graph to determine the entries in an input-output table
[Grade 6]
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Geometry and Measurement
Grade 5
Determine a path or location on a grid using compass directions
[Grade 4]
Identify the location of an ordered pair in the first quadrant using
compass directions [Grade 4]
Locate a point by following compass directions on a grid [Grade
5]
Use compass directions to describe a path to a point on a grid
[Grade 5]
AL 5.G.24 - Represent real world and mathematical problems by graphing points in the first quadrant
of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Algebra
Use a first quadrant graph to represent the values from a table
generated in context [Grade 5]
Geometry and Measurement
Determine the location of an ordered pair in the first quadrant
[Grade 5]
Core Skill
Determine the ordered pair of a point in the first quadrant [Grade
5]
Classify two-dimensional figures into categories based on their properties.
AL 5.G.25 - Understand that attributes belonging to a category of two-dimensional figures also belong
to all subcategories of that category. Example: For example, all rectangles have four right angles and
squares are rectangles, so all squares have four right angles.
AL 5.G.26 - Classify two-dimensional figures in a hierarchy based on properties.
Geometry and Measurement
Identify a shape that has parallel or perpendicular sides [Grade 4]
Classify a triangle by its sides [Grade 4]
Relate a polygon to attributes or characteristics [Grade 4]
Classify a triangle by its sides and angles [Grade 5]
Know the properties of a triangle or a quadrilateral [Grade 6]
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Grade 6
Grade 6
AL 6.RP - Ratios and Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.
AL 6.RP.1 - Understand the concept of a ratio and use ratio language to describe a ratio relationship
between two quantities. Example: For example, "The ratio of wings to beaks in the bird house at the
zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received,
candidate C received nearly three votes.".
Numbers and Operations
WP: Determine a ratio using whole numbers less than 50 [Grade
6]
WP: Determine the ratio of two whole numbers, at least one of
which is larger than 50 [Grade 7]
AL 6.RP.2 - Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to
0, and use rate language in the context of a ratio relationship. Example: For example, "This recipe has
a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We
paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.".[Expectations for unit rates in this
grade are limited to non-complex fractions.]
Numbers and Operations
WP: Determine a ratio using whole numbers less than 50 [Grade
6]
WP: Determine a unit rate with a whole number value [Grade 6]
Core Skill
WP: Determine a unit rate [Grade 7]
Core Skill
AL 6.RP.3 - Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
AL 6.RP.3.a - Make tables of equivalent ratios relating quantities with whole-number measurements,
find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
Numbers and Operations
Determine if ratios, using whole numbers less than 50, are
equivalent [Grade 6]
Core Skill
Determine ratios equivalent to a given ratio of two whole
numbers, at least one of which is larger than 50 [Grade 7]
Algebra
WP: Extend a pattern to solve a problem [Grade 5]
AL 6.RP.3.b - Solve unit rate problems including those involving unit pricing and constant speed.
Example: For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
Numbers and Operations
WP: Determine a part given a ratio and the whole where the
whole is less than 50 [Grade 6]
Core Skill
WP: Determine a part given a ratio and another part where the
whole is less than 50 [Grade 6]
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WP: Determine the whole given a ratio and a part where the
whole is less than 50 [Grade 6]
Core Skill
WP: Determine a unit rate with a whole number value [Grade 6]
Core Skill
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Algebra
Grade 6
WP: Use a unit rate, with a whole number or whole cent value, to
solve a problem [Grade 6]
Core Skill
WP: Determine a part, given part to whole ratio and the whole,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to part ratio and the whole,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to whole ratio and a part, where
the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to part ratio and a part, where
the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine the whole, given part to whole ratio and a part,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine the whole, given part to part ratio and a part,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a unit rate [Grade 7]
Core Skill
WP: Use a unit rate to solve a problem [Grade 7]
Core Skill
WP: Solve a proportion [Grade 7]
Core Skill
AL 6.RP.3.c - Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100
times the quantity); solve problems involving finding the whole, given a part and the percent.
Numbers and Operations
Calculate a percent of a whole number where the answer is a
whole number [Grade 6]
WP: Calculate the percent of a whole number where the answer is
a whole number [Grade 6]
Determine a percent of a whole number using less than 100%
[Grade 7]
Determine a percent of a whole number using more than 100%
[Grade 7]
Determine a whole number given a part and a percentage less than
100% [Grade 7]
WP: Determine a percent of a whole number using less than 100% Core Skill
[Grade 7]
WP: Determine a whole number given a part and a percentage
[Grade 7]
Core Skill
AL 6.RP.3.d - Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
Geometry and Measurement
Convert between customary units of length using fractional
amounts [Grade 5]
Convert between customary units of capacity using fractional
amounts [Grade 5]
Convert between customary units of weight using fractional
amounts [Grade 5]
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Grade 6
Convert between metric units of capacity using decimal amounts
[Grade 5]
Convert between metric units of mass using decimal amounts
[Grade 5]
Convert between millimeters or centimeters and meters, or meters
and kilometers using decimal amounts [Grade 5]
Convert between Fahrenheit and Celsius temperature given a
formula [Grade 6]
Determine approximate conversions between metric and
customary units of length [Grade 7]
Determine approximate conversions between metric and
customary units of capacity [Grade 7]
Determine approximate conversions between metric and
customary units of weight/mass [Grade 7]
AL 6.NS - The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
AL 6.NS.4 - Interpret and compute quotients of fractions, and solve word problems involving division
of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
Example: For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show
the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9
because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if
3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt?
How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Numbers and Operations
Divide a whole number by a unit fraction using a model [Grade 5]
Divide a whole number by a unit fraction [Grade 5]
Divide a unit fraction by a whole number [Grade 5]
Divide a whole number by a fraction, with a whole number
quotient using a model [Grade 5]
Divide a whole number by a fraction, with a whole number
quotient [Grade 5]
Divide a fraction by a whole number resulting in a fractional
quotient [Grade 6]
Core Skill
Divide a fraction by a fraction [Grade 6]
Core Skill
Divide a whole number by a fraction resulting in a fractional
quotient [Grade 6]
Core Skill
Divide a mixed number by a fraction [Grade 6]
Divide a mixed number by a mixed number [Grade 6]
4/26/2012
WP: Multiply or divide a fraction by a fraction [Grade 6]
Core Skill
WP: Multiply or divide two mixed numbers or a mixed number
and a fraction [Grade 6]
Core Skill
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Grade 6
Compute fluently with multi-digit numbers and find common factors and multiples.
AL 6.NS.5 - Fluently divide multi-digit numbers using the standard algorithm.
Numbers and Operations
Divide a multi-digit whole number by multiples of 100 or 1,000
[Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with no remainder and no zeros in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with a remainder and no zeros in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with no remainder and at least one zero in the quotient [Grade 5]
Divide a multi-digit whole number by a 2-digit whole number,
with a remainder and at least one zero in the quotient [Grade 5]
Core Skill
Divide a multi-digit whole number by a 2-digit whole number and Core Skill
express the quotient as a mixed number [Grade 5]
Divide a whole number by a 1-digit whole number resulting in a
decimal quotient through thousandths [Grade 6]
Core Skill
Divide a whole number by a 2-digit whole number resulting in a
decimal quotient through thousandths [Grade 6]
Core Skill
WP: Divide a whole number by a 1- or 2-digit whole number
resulting in a decimal quotient [Grade 6]
Core Skill
AL 6.NS.6 - Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.
Numbers and Operations
Multiply a decimal number through thousandths by a whole
number [Grade 6]
Core Skill
WP: Multiply a decimal number through thousandths by a whole
number [Grade 6]
Core Skill
WP: Multiply a money expression by a decimal number [Grade 6]
Multiply decimal numbers to thousandths using basic facts [Grade
6]
Multiply decimal numbers less than one in hundredths or
thousandths [Grade 6]
Core Skill
Multiply a decimal number greater than one by a decimal number
to thousandths that has only 1 nonzero digit [Grade 6]
Multiply decimal numbers greater than one where the product has Core Skill
2 or 3 decimal places [Grade 6]
4/26/2012
WP: Multiply two decimal numbers to thousandths [Grade 6]
Core Skill
Divide a decimal number through thousandths by a 1- or 2-digit
whole number where the quotient has 2-5 decimal places [Grade
6]
Core Skill
WP: Divide a decimal number through thousandths by a 1- or 2digit whole number [Grade 6]
Core Skill
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Grade 6
Divide a whole number or a decimal number by 0.1, 0.01, or
0.001 [Grade 6]
Locate the decimal point in the quotient of a whole number, or a
decimal number, divided by a decimal number [Grade 6]
Divide a 1- to 3-digit whole number by a decimal number to
tenths where the quotient is a whole number [Grade 6]
Core Skill
Divide a 1- to 3-digit whole number by a decimal number to
tenths where the quotient is a decimal number to thousandths
[Grade 6]
Core Skill
Divide a 2- or 3-digit whole number by a decimal number to
hundredths or thousandths, rounded quotient if needed [Grade 6]
Core Skill
Divide a decimal number by a decimal number through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
WP: Divide a whole number by a decimal number through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
WP: Divide a decimal through thousandths by a decimal through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
AL 6.NS.7 - Find the greatest common factor of two whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less than or equal to 12. Use the distributive property to
express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole
numbers with no common factor. Example: For example, express 36 + 8 as 4 (9 + 2).
Numbers and Operations
Determine the common factors for two whole numbers to 50
[Grade 5]
Determine the greatest common factor of two whole numbers to
50 [Grade 5]
Determine the multiple(s) of a number [Grade 5]
Determine common multiples for two whole numbers [Grade 5]
Determine the least common multiple of two whole numbers
[Grade 5]
Determine the greatest common factor of three numbers to 100
[Grade 6]
Apply and extend previous understandings of numbers to the system of rational numbers.
AL 6.NS.8 - Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea
level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent
quantities in real-world contexts, explaining the meaning of 0 in each situation.
Numbers and Operations
Compare two negative integers or a negative integer and a
positive integer [Grade 6]
Order negative integers or a mix of positive and negative integers
[Grade 6]
Identify a positive or negative rational number represented by a
point on a number line [Grade 7]
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Grade 6
Locate a positive or negative rational number on a number line
[Grade 7]
Order rational numbers (positive and negative) [Grade 7]
AL 6.NS.9 - Understand a rational number as a point on the number line. Extend number line
diagrams and coordinate axes familiar from previous grades to represent points on the line and in the
plane with negative number coordinates.
AL 6.NS.9.a - Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on
the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(3) = 3, and that 0 is its own opposite.
Numbers and Operations
Identify or locate an integer on a number line [Grade 6]
Determine the opposite of an integer [Grade 7]
Identify a positive or negative rational number represented by a
point on a number line [Grade 7]
Locate a positive or negative rational number on a number line
[Grade 7]
AL 6.NS.9.b - Understand signs of numbers in ordered pairs as indicating locations in quadrants of
the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the
points are related by reflections across one or both axes.
Geometry and Measurement
Determine the location of an ordered pair in the first quadrant
[Grade 5]
Determine the location of an ordered pair in any quadrant [Grade
6]
AL 6.NS.9.c - Find and position integers and other rational numbers on a horizontal or vertical
number line diagram; find and position pairs of integers and other rational numbers on a coordinate
plane.
Numbers and Operations
Identify or locate an integer on a number line [Grade 6]
Identify a positive or negative rational number represented by a
point on a number line [Grade 7]
Locate a positive or negative rational number on a number line
[Grade 7]
Geometry and Measurement
Determine the location of an ordered pair in the first quadrant
[Grade 5]
Determine the ordered pair of a point in the first quadrant [Grade
5]
Determine the location of an ordered pair in any quadrant [Grade
6]
Determine the ordered pair of a point in any quadrant [Grade 6]
AL 6.NS.10 - Understand ordering and absolute value of rational numbers.
Numbers and Operations
Evaluate the absolute value of an integer [Grade 7]
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Grade 6
Compare rational numbers (positive and negative) [Grade 7]
Order rational numbers (positive and negative) [Grade 7]
AL 6.NS.10.a - Interpret statements of inequality as statements about the relative position of two
numbers on a number line diagram. Example: For example, interpret -3 > -7 as a statement that -3 is
located to the right of -7 on a number line oriented from left to right.
Algebra
Determine the graph of an inequality on a number line [Grade 7]
Determine the graph of the solution set of a 1-step linear
inequality [Grade 7]
AL 6.NS.10.b - Write, interpret, and explain statements of order for rational numbers in real-world
contexts. Example: For example, write -3 °C > -7 °C to express the fact that -3 °C is warmer than -7 °
C.
Numbers and Operations
Order numbers in decimal and fractional forms [Grade 6]
Core Skill
Compare two negative integers or a negative integer and a
positive integer [Grade 6]
Order negative integers or a mix of positive and negative integers
[Grade 6]
Order rational numbers (positive and negative) [Grade 7]
AL 6.NS.10.c - Understand the absolute value of a rational number as its distance from 0 on the
number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world
situation. Example: For example, for an account balance of -30 dollars, write |-30| = 30 to describe
the size of the debt in dollars.
Numbers and Operations
Evaluate the absolute value of an integer [Grade 7]
Identify a positive or negative rational number represented by a
point on a number line [Grade 7]
Locate a positive or negative rational number on a number line
[Grade 7]
AL 6.NS.10.d - Distinguish comparisons of absolute value from statements about order. Example:
For example, recognize that an account balance less than -30 dollars represents a debt greater than
30 dollars.
Numbers and Operations
Evaluate the absolute value of an integer [Grade 7]
AL 6.NS.11 - Solve real-world and mathematical problems by graphing points in all four quadrants of
the coordinate plane. Include use of coordinates and absolute value to find distances between points
with the same first coordinate or the same second coordinate.
Geometry and Measurement
Determine the location of a simple shape on the Cartesian plane
given the coordinates of its vertices [Grade 7]
Determine the coordinates of a missing point determined by
geometric information [Grade 7]
Determine a side length of a shape on the Cartesian plane [Grade
7]
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Grade 6
AL 6.EE - Expressions and Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
AL 6.EE.12 - Write and evaluate numerical expressions involving whole-number exponents.
Numbers and Operations
Determine the square of a whole number to 15 [Grade 6]
Determine the repeated multiplication that is represented by a
number raised to a power [Grade 7]
Determine an exponential form of a whole number [Grade 7]
Evaluate a numerical expression, with parentheses and exponents,
using order of operations [Grade 7]
Algebra
Evaluate an algebraic expression involving whole number
exponents [Grade 7]
WP: Evaluate a variable expression involving exponents [Grade
7]
AL 6.EE.13 - Write, read, and evaluate expressions in which letters stand for numbers.
AL 6.EE.13.a - Write expressions that record operations with numbers and with letters standing for
numbers. Example: For example, express the calculation "Subtract y from 5" as 5 - y.
Algebra
Use a variable expression with one operation to represent a verbal
expression [Grade 5]
Use a verbal expression to represent a variable expression with
one operation [Grade 5]
WP: Use a variable expression with one operation to represent a
situation [Grade 5]
Use a variable expression with two operations to represent a
verbal expression [Grade 6]
Use a verbal expression to represent a variable expression with
two operations [Grade 6]
WP: Use a variable expression with two operations to represent a
situation [Grade 6]
AL 6.EE.13.b - Identify parts of an expression using mathematical terms (sum, term, product, factor,
quotient, coefficient); view one or more parts of an expression as a single entity. Example: For
example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single
entity and a sum of two terms.
Overall Product Skills
Build comprehension of math vocabulary
Demonstrate and develop understanding of mathematical concepts
and ideas
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Grade 6
AL 6.EE.13.c - Evaluate expressions at specific values of their variables. Include expressions that
arise from formulas used in real-world problems. Perform arithmetic operations, including those
involving whole-number exponents, in the conventional order when there are no parentheses to
specify a particular order (Order of Operations). Example: For example, use the formulas V = s³ and
A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.
Numbers and Operations
Evaluate a numerical expression of four or more operations, with
parentheses, using order of operations [Grade 6]
Algebra
Evaluate a 1-variable expression, with two or three operations,
using whole number substitution [Grade 6]
Core Skill
Evaluate a 2-variable expression, with two or three operations,
using whole number substitution [Grade 6]
Core Skill
WP: Evaluate a 1- or 2-variable expression or formula using
whole numbers [Grade 6]
Core Skill
WP: Generate a table of paired numbers based on a variable
expression with two operations [Grade 6]
Evaluate a 1-variable expression, with two or three operations,
using integer substitution [Grade 7]
Evaluate a 2-variable expression, with two or three operations,
using integer substitution [Grade 7]
Core Skill
Evaluate an algebraic expression involving whole number
exponents [Grade 7]
WP: Evaluate a variable expression involving exponents [Grade
7]
AL 6.EE.14 - Apply the properties of operations to generate equivalent expressions. Example: For
example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent
expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent
expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression
3y.
Algebra
Determine which property of addition or multiplication justifies a
step in the simplification of an expression [Grade 6]
AL 6.EE.15 - Identify when two expressions are equivalent (i.e., when the two expressions name the
same number regardless of which value is substituted into them). Example: For example, the
expressions y + y + y and 3y are equivalent because they name the same number regardless of which
number y stands for.
Reason about and solve one-variable equations and inequalities.
AL 6.EE.16 - Understand solving an equation or inequality as a process of answering a question: which
values from a specified set, if any, make the equation or inequality true? Use substitution to determine
whether a given number in a specified set makes an equation or inequality true.
Algebra
4/26/2012
Determine some solutions to a 1-variable linear inequality [Grade
7]
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Grade 6
AL 6.EE.17 - Use variables to represent numbers and write expressions when solving a real-world or
mathematical problem; understand that a variable can represent an unknown number, or, depending
on the purpose at hand, any number in a specified set.
Algebra
WP: Use a variable expression with one operation to represent a
situation [Grade 5]
WP: Use a variable expression with two operations to represent a
situation [Grade 6]
Use a variable expression with two operations to represent a table
of paired numbers [Grade 7]
WP: Use a 2-variable expression to represent a situation [Grade 7]
AL 6.EE.18 - Solve real-world and mathematical problems by writing and solving equations of the
form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Algebra
WP: Use a 2-variable equation to represent a situation involving a Core Skill
direct proportion [Grade 6]
WP: Use a 2-variable linear equation to represent a situation
[Grade 6]
Core Skill
Solve a 1-step equation involving whole numbers [Grade 6]
Core Skill
Solve a 1-step linear equation involving integers [Grade 7]
Core Skill
Solve a 2-step linear equation involving integers [Grade 7]
Core Skill
WP: Use a 1-variable 1-step equation to represent a situation
[Grade 7]
Core Skill
AL 6.EE.19 - Write an inequality of the form x > c or x < c to represent a constraint or condition in a
real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Algebra
Determine the graph of an inequality on a number line [Grade 7]
Determine some solutions to a 1-variable linear inequality [Grade
7]
WP: Use a 1-variable linear inequality to represent a situation
[Grade 7]
Represent and analyze quantitative relationships between dependent and independent variables.
AL 6.EE.20 - Use variables to represent two quantities in a real-world problem that change in
relationship to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze the
relationship between the dependent and independent variables using graphs and tables, and relate
these to the equation. Example: For example, in a problem involving motion at constant speed, list and
graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship
between distance and time.
Algebra
4/26/2012
Determine the variable expression with one operation for a table
of paired numbers [Grade 5]
Core Skill
WP: Determine the variable expression with one operation for a
table of paired numbers [Grade 5]
Core Skill
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Grade 6
WP: Use a 2-variable equation to represent a situation involving a Core Skill
direct proportion [Grade 6]
WP: Use a 2-variable linear equation to represent a situation
[Grade 6]
Core Skill
Use a 2-variable equation to construct an input-output table
[Grade 6]
Core Skill
Use a 2-variable equation to represent a relationship expressed in
a table [Grade 6]
Core Skill
Use a variable expression with two operations to represent a table
of paired numbers [Grade 7]
Use a table to represent a linear function [Grade 7]
Determine the graph of a 1-operation linear function [Grade 7]
Core Skill
AL 6.G - Geometry
Solve real-world and mathematical problems involving area, surface area, and volume.
AL 6.G.21 - Find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes; apply these techniques in
the context of solving real-world and mathematical problems.
Geometry and Measurement
Use a formula to determine the area of a triangle [Grade 5]
Core Skill
Use a formula to determine the area of a parallelogram [Grade 5]
WP: Determine the area of a triangle [Grade 5]
Core Skill
WP: Determine the area of a square or rectangle [Grade 5]
Core Skill
Determine the area of a complex shape [Grade 6]
Core Skill
WP: Determine the perimeter or the area of a complex shape
[Grade 6]
Core Skill
Determine the area of a trapezoid [Grade 7]
AL 6.G.22 - Find the volume of a right rectangular prism with fractional edge lengths by packing it
with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as
would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h
to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
Geometry and Measurement
Determine the volume of a rectangular prism given a diagram
[Grade 5]
Core Skill
WP: Determine the volume of a rectangular prism given a
diagram [Grade 5]
Determine the volume of a rectangular prism [Grade 5]
4/26/2012
WP: Determine the volume of a rectangular prism [Grade 5]
Core Skill
Determine the volume of a rectangular or a triangular prism
[Grade 7]
Core Skill
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Grade 6
AL 6.G.23 - Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates
to find the length of a side joining points with the same first coordinate or the same second coordinate.
Apply these techniques in the context of solving real-world and mathematical problems.
Geometry and Measurement
Determine the location of a simple shape on the Cartesian plane
given the coordinates of its vertices [Grade 7]
Determine a side length of a shape on the Cartesian plane [Grade
7]
AL 6.G.24 - Represent three-dimensional figures using nets made up of rectangles and triangles, and
use the nets to find the surface area of these figures. Apply these techniques in the context of solving
real-world and mathematical problems.
Geometry and Measurement
Determine the surface area of a cube or a rectangular prism given
a net [Grade 5]
Determine the surface area of a rectangular prism [Grade 5]
Core Skill
WP: Find the surface area of a rectangular prism [Grade 5]
Core Skill
Determine the 3-dimensional shape that can be formed from a net
[Grade 5]
Determine a net of a 3-dimensional shape [Grade 5]
Determine the surface area of a 3-dimensional shape given a net
[Grade 7]
Determine the surface area of a triangular prism [Grade 7]
WP: Determine the surface area of a geometric solid [Grade 7]
Core Skill
AL 6.SP - Statistics and Probability
Develop understanding of statistical variability.
AL 6.SP.25 - Recognize a statistical question as one that anticipates variability in the data related to
the question and accounts for it in the answers. Example: For example, "How old am I?" is not a
statistical question, but "How old are the students in my school?" is a statistical question because one
anticipates variability in students' ages.
Overall Product Skills
Build comprehension of math vocabulary
AL 6.SP.26 - Understand that a set of data collected to answer a statistical question has a distribution
which can be described by its center, spread, and overall shape.
Overall Product Skills
Build comprehension of math vocabulary
AL 6.SP.27 - Recognize that a measure of center for a numerical data set summarizes all of its values
with a single number, while a measure of variation describes how its values vary with a single number.
Overall Product Skills
Build comprehension of math vocabulary
Summarize and describe distributions.
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Grade 6
AL 6.SP.28 - Display numerical data in plots on a number line, including dot plots, histograms, and
box plots.
Data Analysis, Statistics, and
Probability
Use a histogram to represent data [Grade 7]
Core Skill
Answer a question using information from a histogram [Grade 7]
Core Skill
Use a box-and-whisker plot to organize data [Grade 8]
AL 6.SP.29 - Summarize numerical data sets in relation to their context, such as by:
AL 6.SP.29.a - Reporting the number of observations.
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 6.SP.29.b - Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 6.SP.29.c - Giving quantitative measures of center (median and/or mean) and variability
(interquartile range and/or mean absolute deviation), as well as describing any overall pattern and
any striking deviations from the overall pattern with reference to the context in which the data were
gathered.
Data Analysis, Statistics, and
Probability
Determine the range from a graph [Grade 5]
Determine the mean of a set of whole number data, whole number
results [Grade 5]
Determine the median of an odd number of data values [Grade 5]
Core Skill
Determine the range of a set of whole number data [Grade 5]
Determine the mean of a set of whole number data [Grade 6]
Determine the median of a set of whole number data [Grade 6]
Determine the mean of a set of data [Grade 7]
Determine the median of a set of data [Grade 7]
AL 6.SP.29.d - Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
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Grade 7
Grade 7
AL 7.RP - Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
AL 7.RP.1 - Compute unit rates associated with ratios of fractions, including ratios of lengths, areas
and other quantities measured in like or different units. Example: For example, if a person walks 1/2
mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour,
equivalently 2 miles per hour.
Numbers and Operations
Divide a fraction by a fraction [Grade 6]
Core Skill
WP: Determine a unit rate with a whole number value [Grade 6]
Core Skill
WP: Use a unit rate, with a whole number or whole cent value, to
solve a problem [Grade 6]
Core Skill
WP: Determine a unit rate [Grade 7]
Core Skill
WP: Use a unit rate to solve a problem [Grade 7]
Core Skill
AL 7.RP.2 - Recognize and represent proportional relationships between quantities.
AL 7.RP.2.a - Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a
straight line through the origin.
Numbers and Operations
WP: Determine the ratio of two whole numbers, at least one of
which is larger than 50 [Grade 7]
Determine ratios equivalent to a given ratio of two whole
numbers, at least one of which is larger than 50 [Grade 7]
Algebra
Solve a proportion involving decimals [Grade 7]
Core Skill
WP: Solve a proportion [Grade 7]
Core Skill
AL 7.RP.2.b - Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relationships.
Numbers and Operations
WP: Determine a unit rate with a whole number value [Grade 6]
Core Skill
WP: Determine a unit rate [Grade 7]
Core Skill
AL 7.RP.2.c - Represent proportional relationships by equations. Example: For example, if total cost
t is proportional to the number n of items purchased at a constant price p, the relationship between
the total cost and the number of items can be expressed as t = pn.
Algebra
WP: Use a 2-variable equation to represent a situation involving a Core Skill
direct proportion [Grade 6]
AL 7.RP.2.d - Explain what a point (x, y) on the graph of a proportional relationship means in terms
of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Algebra
Use a graph to represent the ordered pairs in a function table
[Grade 7]
Geometry and Measurement
Determine the location of an ordered pair in any quadrant [Grade
6]
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Grade 7
Determine the ordered pair of a point in any quadrant [Grade 6]
AL 7.RP.3 - Use proportional relationships to solve multistep ratio and percent problems. Example:
Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent
increase and decrease, percent error.
Numbers and Operations
Determine a percent of a whole number using less than 100%
[Grade 7]
Determine a percent of a whole number using more than 100%
[Grade 7]
Determine the percent a whole number is of another whole
number, with a result less than 100% [Grade 7]
Determine a whole number given a part and a percentage less than
100% [Grade 7]
WP: Determine a percent of a whole number using less than 100% Core Skill
[Grade 7]
4/26/2012
WP: Determine the percent a whole number is of another whole
number, with a result less than 100% [Grade 7]
Core Skill
WP: Determine a whole number given a part and a percentage
[Grade 7]
Core Skill
WP: Determine the percent of decrease applied to a number
[Grade 7]
Core Skill
WP: Determine the percent of increase applied to a number
[Grade 7]
Core Skill
WP: Determine the result of applying a percent of decrease to a
value [Grade 7]
Core Skill
WP: Determine the result of applying a percent of increase to a
value [Grade 7]
Core Skill
WP: Answer a question involving a fraction and a percent [Grade
7]
Core Skill
WP: Determine a part, given part to whole ratio and the whole,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to part ratio and the whole,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to whole ratio and a part, where
the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a part, given part to part ratio and a part, where
the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine the whole, given part to whole ratio and a part,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine the whole, given part to part ratio and a part,
where the whole is greater than 50 [Grade 7]
Core Skill
WP: Determine a unit rate [Grade 7]
Core Skill
WP: Use a unit rate to solve a problem [Grade 7]
Core Skill
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Determine a percent of a number given a percent that is not a
whole percent [Grade 8]
Grade 7
Core Skill
Determine the percent one number is of another number [Grade 8] Core Skill
Determine a number given a part and a decimal percentage or a
percentage more than 100% [Grade 8]
Core Skill
WP: Determine a given percent of a number [Grade 8]
Core Skill
WP: Determine the percent one number is of another number
[Grade 8]
Core Skill
WP: Determine a number given a part and a decimal percentage
or a percentage more than 100% [Grade 8]
Core Skill
Solve a problem involving simple interest [Grade 8]
Solve a problem involving annually compounded interest [Grade
8]
WP: Find the result of two consecutive percentage changes
applied to a given number [Grade 8]
Algebra
WP: Solve a proportion [Grade 7]
Core Skill
Geometry and Measurement
Convert a rate from one unit to another with a change in one unit
[Grade 7]
Core Skill
Convert a rate from one unit to another with a change in both
units [Grade 7]
Core Skill
AL 7.NS - The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and
divide rational numbers.
AL 7.NS.4 - Apply and extend previous understandings of addition and subtraction to add and
subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line
diagram.
AL 7.NS.4.a - Describe situations in which opposite quantities combine to make 0. Example: For
example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 7.NS.4.b - Understand p + q as the number located a distance |q| from p, in the positive or
negative direction depending on whether q is positive or negative. Show that a number and its
opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing
real-world contexts.
Numbers and Operations
Determine the opposite of an integer [Grade 7]
Add integers using a number line [Grade 7]
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Grade 7
AL 7.NS.4.c - Understand subtraction of rational numbers as adding the additive inverse, p - q = p +
(-q). Show that the distance between two rational numbers on the number line is the absolute value of
their difference, and apply this principle in real-world contexts.
Numbers and Operations
Subtract integers using a number line [Grade 7]
AL 7.NS.4.d - Apply properties of operations as strategies to add and subtract rational numbers.
Numbers and Operations
Add integers [Grade 7]
Core Skill
Subtract integers [Grade 7]
Core Skill
Add or subtract signed fractions or mixed numbers [Grade 8]
Add or subtract signed decimals [Grade 8]
AL 7.NS.5 - Apply and extend previous understandings of multiplication and division and of fractions
to multiply and divide rational numbers.
AL 7.NS.5.a - Understand that multiplication is extended from fractions to rational numbers by
requiring that operations continue to satisfy the properties of operations, particularly the
distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed
numbers. Interpret products of rational numbers by describing real-world contexts.
Numbers and Operations
Multiply integers [Grade 7]
Core Skill
WP: Multiply or divide integers [Grade 7]
Core Skill
AL 7.NS.5.b - Understand that integers can be divided, provided that the divisor is not zero, and
every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.
Numbers and Operations
Overall Product Skills
Divide integers [Grade 7]
Core Skill
WP: Multiply or divide integers [Grade 7]
Core Skill
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 7.NS.5.c - Apply properties of operations as strategies to multiply and divide rational numbers.
Numbers and Operations
Multiply a fraction by a fraction [Grade 6]
Core Skill
Multiply a mixed number by a fraction [Grade 6]
Core Skill
Divide a fraction by a fraction [Grade 6]
Core Skill
Multiply a decimal number through thousandths by a whole
number [Grade 6]
Core Skill
Multiply decimal numbers greater than one where the product has Core Skill
2 or 3 decimal places [Grade 6]
4/26/2012
Divide a decimal number by 10, 100, or 1,000 [Grade 6]
Core Skill
Divide a decimal number by a decimal number through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
Multiply integers [Grade 7]
Core Skill
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Divide integers [Grade 7]
Grade 7
Core Skill
Multiply or divide signed fractions or mixed numbers [Grade 8]
Multiply or divide signed decimals [Grade 8]
AL 7.NS.5.d - Convert a rational number to a decimal using long division; know that the decimal
form of a rational number terminates in 0s or eventually repeats.
Numbers and Operations
Convert a mixed number to a decimal number [Grade 6]
Convert a fraction to a repeating decimal number [Grade 6]
Convert a percentage to a decimal number greater than 1 [Grade
7]
Convert a mixed number to a percentage [Grade 7]
AL 7.NS.6 - Solve real-world and mathematical problems involving the four operations with rational
numbers.[Computations with rational numbers extend the rules for manipulating fractions to complex
fractions.]
Numbers and Operations
WP: Multiply or divide a fraction by a fraction [Grade 6]
Core Skill
WP: Multiply or divide two mixed numbers or a mixed number
and a fraction [Grade 6]
Core Skill
WP: Solve a 2-step problem involving fractions [Grade 6]
Core Skill
WP: Multiply a decimal number through thousandths by a whole
number [Grade 6]
Core Skill
WP: Divide a decimal number through thousandths by a 1- or 2digit whole number [Grade 6]
Core Skill
WP: Divide a whole number by a decimal number through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
WP: Divide a decimal through thousandths by a decimal through
thousandths, rounded quotient if needed [Grade 6]
Core Skill
WP: Solve a 2-step problem involving decimals [Grade 6]
Core Skill
WP: Determine the percent of decrease applied to a number
[Grade 7]
Core Skill
WP: Determine the percent of increase applied to a number
[Grade 7]
Core Skill
WP: Determine the result of applying a percent of decrease to a
value [Grade 7]
Core Skill
WP: Answer a question involving a fraction and a percent [Grade
7]
Core Skill
WP: Answer a question involving a fraction and a decimal [Grade
7]
4/26/2012
WP: Solve a multi-step problem involving decimal numbers
[Grade 7]
Core Skill
WP: Solve a multi-step problem involving fractions or mixed
numbers [Grade 7]
Core Skill
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Grade 7
WP: Add and subtract using integers [Grade 7]
Core Skill
WP: Multiply or divide integers [Grade 7]
Core Skill
AL 7.EE - Expressions and Equations
Use properties of operations to generate equivalent expressions.
AL 7.EE.7 - Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
Algebra
Determine which property of addition or multiplication justifies a
step in the simplification of an expression [Grade 6]
Simplify an algebraic expression by combining like terms [Grade
8]
Core Skill
Use the distributive property to simplify an algebraic expression
[Grade 8]
AL 7.EE.8 - Understand that rewriting an expression in different forms in a problem context can shed
light on the problem and how the quantities in it are related. Example: For example, a + 0.05a = 1.05a
means that "increase by 5%" is the same as "multiply by 1.05.".
Algebra
WP: Evaluate a variable expression [Grade 7]
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
AL 7.EE.9 - Solve multi-step real-life and mathematical problems posed with positive and negative
rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.
Apply properties of operations to calculate with numbers in any form; convert between forms as
appropriate; and assess the reasonableness of answers using mental computation and estimation
strategies. Example: For example: If a woman making $25 an hour gets a 10% raise, she will make an
additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel
bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar
about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Numbers and Operations
WP: Solve a multi-step problem involving whole numbers [Grade Core Skill
6]
WP: Solve a multi-step problem involving decimal numbers
[Grade 7]
Core Skill
WP: Solve a multi-step problem involving fractions or mixed
numbers [Grade 7]
Core Skill
WP: Estimate the result of dividing or multiplying a whole
number by a fraction [Grade 7]
Algebra
4/26/2012
Solve a 1-step equation involving rational numbers [Grade 8]
Core Skill
Solve a 2-step equation involving rational numbers [Grade 8]
Core Skill
WP: Use a 1-variable equation with rational coefficients to
represent a situation involving two operations [Grade 8]
Core Skill
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WP: Use a 2-variable equation with rational coefficients to
represent a situation [Grade 8]
Grade 7
Core Skill
AL 7.EE.10 - Use variables to represent quantities in a real-world or mathematical problem, and
construct simple equations and inequalities to solve problems by reasoning about the quantities.
AL 7.EE.10.a - Solve word problems leading to equations of the form px + q = r and p(x + q) = r,
where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an
algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each
approach. Example: For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its
width?
Numbers and Operations
Algebra
WP: Determine the result of applying a percent of decrease to a
value [Grade 7]
Core Skill
WP: Determine the result of applying a percent of increase to a
value [Grade 7]
Core Skill
Solve a 2-step linear equation involving integers [Grade 7]
Core Skill
Solve a 2-step equation involving rational numbers [Grade 8]
Core Skill
WP: Use a 1-variable equation with rational coefficients to
represent a situation involving two operations [Grade 8]
Core Skill
WP: Use a 2-variable equation with rational coefficients to
represent a situation [Grade 8]
Core Skill
WP: Solve a problem involving a 1-variable, 2-step equation
[Grade 8]
Core Skill
AL 7.EE.10.b - Solve word problems leading to inequalities of the form px + q > r or px + q < r,
where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret
it in the context of the problem. Example: For example: As a salesperson, you are paid $50 per week
plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number
of sales you need to make, and describe the solutions.
Algebra
Determine some solutions to a 1-variable linear inequality [Grade
7]
Solve a 1-step linear inequality [Grade 7]
Determine the graph of the solution set of a 1-step linear
inequality [Grade 7]
WP: Use a 1-variable linear inequality to represent a situation
[Grade 7]
WP: Use a 2-step linear inequality in one variable to represent a
situation [Grade 8]
WP: Solve a problem involving a 2-step linear inequality in one
variable [Grade 8]
Core Skill
Determine the graph of the solutions to a 2-step linear inequality
in one variable [Grade 8]
Core Skill
AL 7.G - Geometry
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Grade 7
Draw, construct, and describe geometrical figures and describe the relationships between them.
AL 7.G.11 - Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
Geometry and Measurement
Determine a length given a scale [Grade 6]
Determine the scale for a drawing or map question [Grade 7]
WP: Solve a problem involving a map or scale drawing [Grade 7]
WP: Solve a problem involving scale [Grade 8]
AL 7.G.12 - Draw (freehand, with ruler and protractor, and with technology) geometric shapes with
given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when
the conditions determine a unique triangle, more than one triangle, or no triangle.
AL 7.G.13 - Describe the two-dimensional figures that result from slicing three-dimensional figures, as
in plane sections of right rectangular prisms and right rectangular pyramids.
Geometry and Measurement
Visualize a 2-dimensional shape [Grade 7]
Relate a 3-dimensional shape to its top and side views [Grade 7]
Solve a problem involving 3-dimensional visualization [Grade 8]
Identify a cross section of a 3-dimensional shape [Grade 8]
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
AL 7.G.14 - Know the formulas for the area and circumference of a circle and use them to solve
problems; give an informal derivation of the relationship between the circumference and area of a
circle.
Geometry and Measurement
Determine the circumference of a circle in terms of pi [Grade 7]
Solve a problem involving the circumference of a circle [Grade 7] Core Skill
Determine the area of a circle in terms of pi [Grade 7]
Determine the area of a circle using 3.14 for pi [Grade 7]
Determine the area of a circle using 22/7 for pi [Grade 7]
WP: Determine the area of a circle using 3.14 for pi [Grade 7]
Solve a problem given the area of a circle [Grade 7]
Core Skill
AL 7.G.15 - Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
Geometry and Measurement
Determine the measure of a missing angle using straight and right
angle relationships [Grade 6]
Determine the missing angle measure in a triangle given two other
angle measures [Grade 7]
Determine the missing angle measure in a quadrilateral given
three other angle measures [Grade 7]
Identify vertical, adjacent, complementary, or supplementary
angles [Grade 7]
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Grade 7
Determine the measure of a missing angle using angle
relationships [Grade 7]
Identify angle relationships formed by parallel lines cut by a
transversal [Grade 8]
Determine an angle measure formed by parallel lines cut by a
transversal [Grade 8]
AL 7.G.16 - Solve real-world and mathematical problems involving area, volume and surface area of
two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right
prisms.
Geometry and Measurement
Determine the volume of a prism with a right triangle base [Grade
6]
Determine the surface area of a 3-dimensional shape made from
cubes [Grade 6]
Determine the area of a trapezoid [Grade 7]
Determine the volume of a rectangular or a triangular prism
[Grade 7]
Core Skill
Determine the surface area of a 3-dimensional shape given a net
[Grade 7]
WP: Determine the surface area of a geometric solid [Grade 7]
Core Skill
Determine the surface area of a pyramid or a cone [Grade 8]
WP: Determine the surface area of a pyramid or a cone [Grade 8]
Solve a problem involving the surface area or the volume of a
pyramid or a cone [Grade 8]
AL 7.SP - Statistics and Probability
Use random sampling to draw inferences about a population.
AL 7.SP.17 - Understand that statistics can be used to gain information about a population by
examining a sample of the population; generalizations about a population from a sample are valid only
if the sample is representative of that population. Understand that random sampling tends to produce
representative samples and support valid inferences.
Data Analysis, Statistics, and
Probability
Determine if a sample is likely to be representative of the larger
population [Grade 8]
AL 7.SP.18 - Use data from a random sample to draw inferences about a population with an unknown
characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge
the variation in estimates or predictions. Example: For example, estimate the mean word length in a
book by randomly sampling words from the book; predict the winner of a school election based on
randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Data Analysis, Statistics, and
Probability
Use a proportion to make an estimate, related to a population,
based on a sample [Grade 7]
Core Skill
Draw informal comparative inferences about two populations.
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Grade 7
AL 7.SP.19 - Informally assess the degree of visual overlap of two numerical data distributions with
similar variabilities, measuring the difference between the centers by expressing it as a multiple of a
measure of variability. Example: For example, the mean height of players on the basketball team is 10
cm greater than the mean height of players on the soccer team, about twice the variability (mean
absolute deviation) on either team; on a dot plot, the separation between the two distributions of
heights is noticeable.
AL 7.SP.20 - Use measures of center and measures of variability for numerical data from random
samples to draw informal comparative inferences about two populations. Example: For example,
decide whether the words in a chapter of a seventh-grade science book are generally longer than the
words in a chapter of a fourth-grade science book.
Data Analysis, Statistics, and
Probability
Determine the mean of a set of data [Grade 7]
Determine the mode(s) of a set of data [Grade 7]
Determine the median of a set of data [Grade 7]
WP: Use the mean of a data set to solve a problem [Grade 7]
Answer a question using information from two box-and-whisker
plots [Grade 8]
Compare the medians, the modes, or the ranges of the data in a
double stem-and-leaf plot [Grade 8]
Investigate chance processes and develop, use, and evaluate probability models.
AL 7.SP.21 - Understand that the probability of a chance event is a number between 0 and 1 that
expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A
probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is
neither unlikely nor likely, and a probability near 1 indicates a likely event.
Overall Product Skills
Build comprehension of math vocabulary
AL 7.SP.22 - Approximate the probability of a chance event by collecting data on the chance process
that produces it and observing its long-run relative frequency, and predict the approximate relative
frequency given the probability. Example: For example, when rolling a number cube 600 times, predict
that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Data Analysis, Statistics, and
Probability
Determine an experimental probability given a list of results
[Grade 6]
Make a prediction based on an experimental probability [Grade 6]
AL 7.SP.23 - Develop a probability model and use it to find probabilities of events. Compare
probabilities from a model to observed frequencies; if the agreement is not good, explain possible
sources of the discrepancy.
AL 7.SP.23.a - Develop a uniform probability model by assigning equal probability to all outcomes,
and use the model to determine probabilities of events. Example: For example, if a student is selected
at random from a class, find the probability that Jane will be selected and the probability that a girl
will be selected.
Data Analysis, Statistics, and
Probability
4/26/2012
Determine the probability of a single event [Grade 6]
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Grade 7
Determine the probability of the complement of a single event
[Grade 6]
Make a prediction based on a theoretical probability [Grade 6]
Make a prediction involving the probability of compound events
[Grade 8]
AL 7.SP.23.b - Develop a probability model (which may not be uniform) by observing frequencies in
data generated from a chance process. Example: For example, find the approximate probability that
a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the
outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Data Analysis, Statistics, and
Probability
Determine an experimental probability given a list of results
[Grade 6]
Make a prediction based on an experimental probability [Grade 6]
Compare predictions from experimental and theoretical
probability [Grade 6]
AL 7.SP.24 - Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
AL 7.SP.24.a - Understand that, just as with simple events, the probability of a compound event is
the fraction of outcomes in the sample space for which the compound event occurs.
Data Analysis, Statistics, and
Probability
Determine the probability of an event consisting of mutually
exclusive events [Grade 8]
Make a prediction involving the probability of compound events
[Grade 8]
AL 7.SP.24.b - Represent sample spaces for compound events using methods such as organized lists,
tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"),
identify the outcomes in the sample space which compose the event.
Data Analysis, Statistics, and
Probability
Determine all possible outcomes of a compound event using a list
[Grade 5]
Determine all possible outcomes of a compound event using a tree
diagram [Grade 5]
AL 7.SP.24.c - Design and use a simulation to generate frequencies for compound events. Example:
For example, use random digits as a simulation tool to approximate the answer to the question: If
40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one
with type A blood?
Overall Product Skills
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Use strategies to derive solutions for problems
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Grade 8
Grade 8
AL 8.NS - The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
AL 8.NS.1 - Know that numbers that are not rational are called irrational. Understand informally that
every number has a decimal expansion; for rational numbers show that the decimal expansion repeats
eventually, and convert a decimal expansion which repeats eventually into a rational number.
Numbers and Operations
Convert a repeating decimal to a fraction or a mixed number
[Grade 8]
Identify rational or irrational numbers [Grade 8]
AL 8.NS.2 - Use rational approximations of irrational numbers to compare the size of irrational
numbers, locate them approximately on a number line diagram, and estimate the value of expressions
(e.g., pi²). Example: For example, by truncating the decimal expansion of the square root of 2, show
that the square root of 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on
to get better approximations.
Numbers and Operations
Identify a positive or negative rational number represented by a
point on a number line [Grade 7]
Locate a positive or negative rational number on a number line
[Grade 7]
Determine the two closest integers to a given square root [Grade
8]
Core Skill
Approximate the location of a square root on a number line
[Grade 8]
Core Skill
Determine the square root of a whole number to the nearest tenth
[Grade 8]
Compare rational numbers and/or irrational numbers in various
forms [Grade 8]
Order rational numbers and irrational numbers [Grade 8]
AL 8.EE - Expressions and Equations
Work with radicals and integer exponents.
AL 8.EE.3 - Know and apply the properties of integer exponents to generate equivalent numerical
expressions. Example: For example, 3² × (3 to the -5 power) = (3 to the -3 power) = 1/3³ = 1/27.
Numbers and Operations
Determine the exponential notation that represents a repeated
multiplication [Grade 7]
Evaluate a whole number power of a whole number [Grade 7]
Evaluate an integer raised to a whole number power [Grade 8]
Core Skill
Evaluate a zero or negative power of an integer [Grade 8]
Evaluate a numerical expression involving integer exponents
and/or integer bases [Grade 8]
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Core Skill
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Algebra
Grade 8
Evaluate an algebraic expression involving negative integer
exponents [Grade 8]
Evaluate a fraction raised to an integer power [Algebra 1]
Apply the product of powers property to a monomial numerical
expression [Algebra 1]
Apply the power of a power property to a monomial numerical
expression [Algebra 1]
Apply the quotient of powers property to monomial numerical
expressions [Algebra 1]
AL 8.EE.4 - Use square root and cube root symbols to represent solutions to equations of the form x² =
p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and
cube roots of small perfect cubes. Know that the square root of 2 is irrational.
Numbers and Operations
Determine the cube of a whole number to 15 [Grade 6]
Determine the whole number that can be squared to make a given
number [Grade 7]
Evaluate the positive square root of a perfect square [Grade 7]
Determine the square root of a perfect-square fraction or decimal
[Grade 8]
Core Skill
Determine both square roots of a perfect square [Grade 8]
Determine the two closest integers to a given square root [Grade
8]
Core Skill
Determine the square root of a whole number to the nearest tenth
[Grade 8]
Algebra
Simplify a monomial numerical expression involving the square
root of a whole number [Algebra 1]
Core Skill
AL 8.EE.5 - Use numbers expressed in the form of a single digit times an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much one is than the
other. Example: For example, estimate the population of the United States as 3 × (10 to the 8th power)
and the population of the world as 7 × (10 to the 9th power), and determine that the world population
is more than 20 times larger.
Numbers and Operations
Convert a whole number greater than 10 to scientific notation
[Grade 7]
Convert a whole number greater than 10 from scientific notation
to standard form [Grade 7]
4/26/2012
Convert a number less than 1 to scientific notation [Grade 8]
Core Skill
Convert a number less than 1 from scientific notation to standard
form [Grade 8]
Core Skill
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Grade 8
AL 8.EE.6 - Perform operations with numbers expressed in scientific notation, including problems
where both decimal and scientific notation are used. Use scientific notation and choose units of
appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year
for seafloor spreading). Interpret scientific notation that has been generated by technology.
Numbers and Operations
Convert a whole number greater than 10 to scientific notation
[Grade 7]
Convert a whole number greater than 10 from scientific notation
to standard form [Grade 7]
Convert a number less than 1 to scientific notation [Grade 8]
Core Skill
Convert a number less than 1 from scientific notation to standard
form [Grade 8]
Core Skill
Understand the connections between proportional relationships, lines, and linear equations.
AL 8.EE.7 - Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. Example: For
example, compare a distance-time graph to a distance-time equation to determine which of two moving
objects has greater speed.
Algebra
Determine the slope of a line given its graph or a graph of a line
with a given slope [Grade 8]
Core Skill
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
Determine the slope-intercept form or the standard form of a
linear equation [Algebra 1]
Core Skill
AL 8.EE.8 - Use similar triangles to explain why the slope m is the same between any two distinct
points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the
origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Analyze and solve linear equations and pairs of simultaneous linear equations.
AL 8.EE.9 - Solve linear equations in one variable.
AL 8.EE.9.a - Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by successively transforming
the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b
results (where a and b are different numbers).
Algebra
Solve a 1-variable linear equation with the variable on both sides
[Algebra 1]
Overall Product Skills
Use strategies to derive solutions for problems
Core Skill
AL 8.EE.9.b - Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting like terms.
Algebra
4/26/2012
Solve a proportion involving decimals [Grade 7]
Core Skill
Solve a 1-step linear equation involving integers [Grade 7]
Core Skill
Solve a 2-step linear equation involving integers [Grade 7]
Core Skill
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Grade 8
Use the distributive property to simplify an algebraic expression
[Grade 8]
Solve a 1-step equation involving rational numbers [Grade 8]
Core Skill
Solve a 2-step equation involving rational numbers [Grade 8]
Core Skill
Solve a 1-variable linear equation that requires simplification and
has the variable on one side [Algebra 1]
Solve a 1-variable linear equation with the variable on both sides
[Algebra 1]
Core Skill
AL 8.EE.10 - Analyze and solve pairs of simultaneous linear equations.
AL 8.EE.10.a - Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection satisfy both
equations simultaneously.
Algebra
Determine the graph of a linear equation given in slope-intercept,
point-slope, or standard form [Algebra 1]
Core Skill
Solve a system of linear equations in two variables by graphing
[Algebra 1]
AL 8.EE.10.b - Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection. Example: For example, 3x + 2y
= 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
Algebra
Solve a system of linear equations in two variables by graphing
[Algebra 1]
Solve a system of linear equations in two variables by substitution
[Algebra 1]
Solve a system of linear equations in two variables by elimination
[Algebra 1]
Determine the number of solutions to a system of linear equations
[Algebra 1]
Solve a system of linear equations in two variables using any
method [Algebra 1]
Core Skill
Determine if a given ordered pair is a solution to a system of
linear inequalities [Algebra 1]
AL 8.EE.10.c - Solve real-world and mathematical problems leading to two linear equations in two
variables. Example: For example, given coordinates for two pairs of points, determine whether the
line through the first pair of points intersects the line through the second pair.
Algebra
Determine if an ordered pair is a solution to a 2-variable linear
inequality [Algebra 1]
WP: Determine a system of linear equations that represents a
given situation [Algebra 1]
Core Skill
WP: Solve a mixture problem that can be represented by a system
of linear equations [Algebra 1]
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Grade 8
WP: Solve a motion problem that can be represented by a system
of linear equations [Algebra 1]
Solve a number problem that can be represented by a linear
system of equations [Algebra 1]
Determine if a given ordered pair is a solution to a system of
linear inequalities [Algebra 1]
AL 8.F - Functions
Define, evaluate, and compare functions.
AL 8.F.11 - Understand that a function is a rule that assigns to each input exactly one output. The
graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Algebra
Use a table to represent a linear function [Grade 7]
Use a graph to represent the ordered pairs in a function table
[Grade 7]
Determine the graph of a 1-operation linear function [Grade 7]
Core Skill
Determine the graph of a 2-operation linear function [Grade 8]
Core Skill
Determine if a relation is a function [Algebra 1]
AL 8.F.12 - Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). Example: For example, given a linear
function represented by a table of values and a linear function represented by an algebraic expression,
determine which function has the greater rate of change.
Algebra
Use a table to represent a linear function [Grade 7]
Use a graph to represent the ordered pairs in a function table
[Grade 7]
Overall Product Skills
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
AL 8.F.13 - Interpret the equation y = mx + b as defining a linear function, whose graph is a straight
line; give examples of functions that are not linear. Example: For example, the function A = s² giving
the area of a square as a function of its side length is not linear because its graph contains the points
(1,1), (2,4) and (3,9), which are not on a straight line.
Algebra
Determine the graph of a 1-operation linear function [Grade 7]
Core Skill
Determine the graph of a 2-operation linear function [Grade 8]
Core Skill
Determine if a function is linear or nonlinear [Algebra 1]
Determine whether a graph or a table represents a linear or
nonlinear function [Algebra 1]
Use functions to model relationships between quantities.
Algebra
Use a table to represent a linear function [Grade 7]
Determine if a relation is a function [Algebra 1]
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Grade 8
AL 8.F.14 - Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from two (x, y)
values, including reading these from a table or from a graph. Interpret the rate of change and initial
value of a linear function in terms of the situation it models, and in terms of its graph or a table of
values.
Algebra
Determine the table of values that represents a linear equation
with rational coefficients in two variables [Grade 8]
Core Skill
Determine a linear equation in two variables that represents a
table of values [Grade 8]
Core Skill
Determine the slope of a line given its graph or a graph of a line
with a given slope [Grade 8]
Core Skill
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
WP: Interpret the meaning of the y-intercept of a graphed line
[Grade 8]
Core Skill
WP: Determine a linear graph that can represent a situation
[Grade 8]
WP: Determine a graph that can represent a situation involving a
varying rate of change [Grade 8]
Determine the slope of a line given two points on the line
[Algebra 1]
Core Skill
Determine an equation of a line given the slope and y-intercept of Core Skill
the line [Algebra 1]
Determine an equation that represents a graphed line [Algebra 1]
Core Skill
Determine an equation for a line given the slope of the line and a
point on the line that is not the y-intercept [Algebra 1]
Core Skill
Determine an equation of a line given two points on the line
[Algebra 1]
Core Skill
AL 8.F.15 - Describe qualitatively the functional relationship between two quantities by analyzing a
graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that
exhibits the qualitative features of a function that has been described verbally.
Algebra
Determine the graph of a 2-operation linear function [Grade 8]
Core Skill
Determine the slope of a line given its graph or a graph of a line
with a given slope [Grade 8]
Core Skill
WP: Determine a graph that can represent a situation involving a
varying rate of change [Grade 8]
AL 8.G - Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
AL 8.G.16 - Verify experimentally the properties of rotations, reflections, and translations:
AL 8.G.16.a - Lines are taken to lines, and line segments to line segments of the same length.
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Grade 8
AL 8.G.16.b - Angles are taken to angles of the same measure.
AL 8.G.16.c - Parallel lines are taken to parallel lines.
AL 8.G.17 - Understand that a two-dimensional figure is congruent to another if the second can be
obtained from the first by a sequence of rotations, reflections, and translations; given two congruent
figures, describe a sequence that exhibits the congruence between them.
Geometry and Measurement
Identify corresponding parts of congruent shapes [Grade 7]
Identify congruent shapes given side and angle measures [Grade
7]
Determine a missing dimension given two congruent shapes
[Grade 7]
AL 8.G.18 - Describe the effect of dilations, translations, rotations, and reflections on two-dimensional
figures using coordinates.
Geometry and Measurement
Determine the result of a reflection, a rotation, or a translation on
the Cartesian plane [Grade 6]
Determine the transformation that generates the image of a figure
in the Cartesian plane [Grade 6]
Determine the coordinates of a translated, a rotated, or a reflected
shape on the Cartesian plane [Grade 7]
AL 8.G.19 - Understand that a two-dimensional figure is similar to another if the second can be
obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two
similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
Geometry and Measurement
Determine the coordinates of a translated, a rotated, or a reflected
shape on the Cartesian plane [Grade 7]
Relate the coordinates of a preimage or an image to a translation
described using mapping notation [Geometry]
Relate the coordinates of a preimage or an image to a dilation
centered at the origin [Geometry]
Determine the coordinates of the image of a figure after two
transformations of the same type [Geometry]
Determine the coordinates of the image of a figure after two
transformations of different types [Geometry]
AL 8.G.20 - Use informal arguments to establish facts about the angle sum and exterior angle of
triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle
criterion for similarity of triangles. Example: For example, arrange three copies of the same triangle so
that the sum of the three angles appears to form a line, and give an argument in terms of transversals
why this is so.
Geometry and Measurement
Identify angle relationships formed by parallel lines cut by a
transversal [Grade 8]
Determine an angle measure formed by parallel lines cut by a
transversal [Grade 8]
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Grade 8
Identify angle relationships formed by multiple lines and
transversals [Geometry]
Overall Product Skills
Determine the measure of an angle formed by parallel lines and
one or more transversals [Geometry]
Core Skill
Identify a triangle similarity postulate that justifies a similarity
statement [Geometry]
Core Skill
Identify similar triangles using triangle similarity postulates or
theorems [Geometry]
Core Skill
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
Understand and apply the Pythagorean Theorem.
AL 8.G.21 - Explain a proof of the Pythagorean Theorem and its converse.
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
AL 8.G.22 - Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in
real-world and mathematical problems in two and three dimensions.
Geometry and Measurement
Determine the length of the hypotenuse of a right triangle using
the Pythagorean theorem [Grade 8]
Core Skill
Determine the length of a leg of a right triangle using the
Pythagorean theorem [Grade 8]
Core Skill
WP: Use the Pythagorean theorem to find a length or a distance
[Grade 8]
Core Skill
Solve for the length of a side of a triangle using the Pythagorean
theorem [Geometry]
AL 8.G.23 - Apply the Pythagorean Theorem to find the distance between two points in a coordinate
system.
Geometry and Measurement
Determine a distance on the Cartesian plane using the
Pythagorean theorem [Grade 8]
Determine the distance between two points [Geometry]
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
AL 8.G.24 - Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
Geometry and Measurement
Determine the volume of a cylinder [Grade 7]
WP: Determine the volume of a cylinder [Grade 7]
WP: Solve a problem involving the volume of a geometric solid
[Grade 7]
Core Skill
Determine the volume of a pyramid or a cone [Grade 8]
WP: Determine the volume of a pyramid or a cone [Grade 8]
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Grade 8
Solve a problem involving the volume of a right pyramid or a
right cone [Geometry]
Core Skill
Determine the volume of a sphere or hemisphere [Geometry]
Core Skill
WP: Determine the volume of a sphere or hemisphere [Geometry]
AL 8.SP - Statistics and Probability
Investigate patterns of association in bivariate data.
AL 8.SP.25 - Construct and interpret scatter plots for bivariate measurement data to investigate
patterns of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
Data Analysis, Statistics, and
Probability
Use a scatter plot to organize data [Grade 8]
Determine if a scatter plot shows a positive relationship, a
negative relationship, or no relationship between the variables
[Grade 8]
Answer a question using information from a scatter plot [Grade 8]
AL 8.SP.26 - Know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit by judging the closeness of the data points to the line.
Data Analysis, Statistics, and
Probability
Approximate a trend line for a scatter plot [Grade 8]
AL 8.SP.27 - Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. Example: For example, in a linear model for a
biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each
day is associated with an additional 1.5 cm in mature plant height.
Algebra
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
WP: Interpret the meaning of the y-intercept of a graphed line
[Grade 8]
Core Skill
AL 8.SP.28 - Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way
table summarizing data on two categorical variables collected from the same subjects. Use relative
frequencies calculated for rows or columns to describe possible association between the two variables.
Example: For example, collect data from students in your class on whether or not they have a curfew
on school nights and whether or not they have assigned chores at home. Is there evidence that those
who have a curfew also tend to have chores?
Data Analysis, Statistics, and
Probability
Use a frequency table to represent 2 related data sets [Grade 6]
Answer a question using information from a frequency table
representing 2 related data sets [Grade 6]
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Overall Product Skills
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Grade 8
Develop understanding and ability to analyze mathematical
relationships including those between and among arithmetic
operations.
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9-12 Analytic Math
9-12 Analytic Math
AL N - Number and Quantity
AL N-VM - Vector and Matrix Quantities
Represent and model with vector quantities.
AL N-VM.1 - Recognize vector quantities as having both magnitude and direction. Represent vector
quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes
(e.g., v, v, v), including the use of eigen-values and eigen-vectors.
Algebra
Determine the component form of a vector represented on a graph
[Algebra 2]
AL N-VM.2 - Solve problems involving velocity and other quantities that can be represented by
vectors, including navigation (e.g., airplane, aerospace, oceanic).
Algebra
WP: Add or subtract vectors [Algebra 2]
AL N-VM.3 - Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand
that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Find the dot
product and the cross product of vectors.
Algebra
Add or subtract vectors on a coordinate plane [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
AL N-VM.4 - Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum, including vectors in complex vector spaces.
Algebra
Determine the magnitude of a vector [Algebra 2]
Determine the direction of a vector [Algebra 2]
Determine a unit vector in the same direction as a given vector
[Algebra 2]
WP: Add or subtract vectors [Algebra 2]
AL N-VM.5 - Understand vector subtraction v - w as v + (-w), where (-w) is the additive inverse of w,
with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction
graphically by connecting the tips in the appropriate order, and perform vector subtraction
component-wise, including vectors in complex vector spaces.
Algebra
Add or subtract vectors on a coordinate plane [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
Perform operations on matrices and use matrices in applications.
AL N-VM.6 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network, including linear programming.
Algebra
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Represent a system of linear equations as a single matrix equation
[Algebra 2]
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9-12 Analytic Math
AL N-VM.7 - Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in
a game are doubled, including rotation matrices.
Algebra
Multiply a matrix by a scalar [Algebra 2]
AL N-VM.8 - Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix
is nonzero if and only if the matrix has a multiplicative inverse. Solve matrix equations using
augmented matrices.
Algebra
Determine the determinant of a matrix [Algebra 2]
Determine the inverse of a matrix [Algebra 2]
Core Skill
AL N-VM.9 - Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of vectors, including
matrices larger than 2 × 2.
Algebra
Combine matrix operations [Algebra 2]
Multiply matrices [Algebra 2]
AL N-VM.10 - Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute
value of the determinant in terms of area. Solve matrix application problems using reduced row
echelon form.
Algebra
Determine the determinant of a matrix [Algebra 2]
AL N-CN - Complex Numbers
Use complex numbers in polynomial identities and equations.
AL N-CN.11 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials. Understand the importance of using complex numbers in graphing functions on the
Cartesian or complex plane.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
Determine a polynomial of lowest degree that has given roots or a
graph of a given shape [Algebra 2]
AL N-L - Limits
Understand limits of functions.
AL N-L.12 - Calculate the limit of a sequence, of a function, and of an infinite series.
Algebra
Find the sum of an infinite geometric series if it exists [Algebra 2]
AL A - Algebra
AL A-SSE - Seeing Structure in Expressions
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9-12 Analytic Math
AL A-SSE.13 - Use the laws of Boolean Algebra to describe true/false circuits. Simplify Boolean
expressions using the relationships between conjunction, disjunction, and negation operations.
AL A-SSE.14 - Use logic symbols to write truth tables.
AL A-APR - Arithmetic With Polynomials and Rational Functions
AL A-APR.15 - Reduce the degree of either the numerator or denominator of a rational function by
using partial fraction decomposition or partial fraction expansion.
Algebra
Decompose rational functions with quadratic denominators into
sums of rational functions with linear denominators [Algebra 2]
AL F - Functions
AL F-TF - Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle.
AL F-TF.16 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
Algebra
WP: Determine a trigonometric function that represents a
situation [Algebra 2]
Apply trigonometry to general triangles.
AL F-TF.17 - Prove the Law of Sines and the Law of Cosines and use them to solve problems.
Understand Law of Sines = 2r, where r is the radius of the circumscribed circle of the triangle. Apply
the Law of Tangents.
Algebra
Solve a problem involving the law of sines [Algebra 2]
Geometry and Measurement
Solve a problem involving the law of cosines [Algebra 2]
WP: Solve a problem involving the law of sines or the law of
cosines [Algebra 2]
AL F-TF.18 - Apply Euler's and deMoivre's formulas as links between complex numbers and
trigonometry.
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Algebra 2
Algebra 2
AL N - Number and Quantity
AL N-CN - The Complex Number System
Perform arithmetic operations with complex numbers.
AL N-CN.1 - Know there is a complex number i such that i² = -1, and every complex number has the
form a + bi with a and b real.
Algebra
Extend polynomial identities to the complex numbers [Algebra 2]
Raise i to a power [Algebra 2]
AL N-CN.2 - Use the relation i² = -1 and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers.
Algebra
Raise i to a power [Algebra 2]
Add or subtract complex numbers [Algebra 2]
Multiply complex numbers [Algebra 2]
Simplify an expression involving a complex denominator
[Algebra 2]
Core Skill
Use complex numbers in polynomial identities and equations.[(Polynomials with real coefficients.)]
AL N-CN.3 - Solve quadratic equations with real coefficients that have complex solutions.
Algebra
Solve a quadratic equation with complex solutions [Algebra 2]
Core Skill
AL N-CN.4 - Extend polynomial identities to the complex numbers. Example: For example, rewrite
x² + 4 as (x + 2i)(x - 2i).
Algebra
Extend polynomial identities to the complex numbers [Algebra 2]
AL N-CN.5 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
Determine a polynomial of lowest degree that has given roots or a
graph of a given shape [Algebra 2]
AL A - Algebra
AL A-SSE - Seeing Structure in Expressions
Interpret the structure of expressions[(Polynomial and rational.)]
AL A-SSE.6 - Interpret expressions that represent a quantity in terms of its context.
Algebra
Evaluate a numerical expression involving one or more exponents Core Skill
and multiple forms of rational numbers [Algebra 1]
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Algebra 2
AL A-SSE.6.a - Interpret parts of an expression, such as terms, factors, and coefficients.
Overall Product Skills
Build comprehension of math vocabulary
AL A-SSE.6.b - Interpret complicated expressions by viewing one or more of their parts as a single
entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not
depending on P.
Numbers and Operations
Evaluate a numerical expression involving integer exponents
and/or integer bases [Grade 8]
Core Skill
AL A-SSE.7 - Use the structure of an expression to identify ways to rewrite it. Example: For
example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a
difference of squares that can be factored as (x² - y²)(x² + y²).
Algebra
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
Write expressions in equivalent forms to solve problems
Algebra
Simplify an algebraic expression by combining like terms [Grade
8]
Core Skill
Use the distributive property to simplify an algebraic expression
[Grade 8]
Simplify a polynomial expression by combining like terms
[Algebra 1]
Simplify a monomial algebraic radical expression [Algebra 1]
Core Skill
Simplify a rational expression involving polynomial terms
[Algebra 1]
Core Skill
Simplify a numerical expression that includes fractional
exponents and/or nth roots [Algebra 2]
Decompose rational functions with quadratic denominators into
sums of rational functions with linear denominators [Algebra 2]
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
Identify an equivalent form of an exponential expression by using
the properties of exponents [Algebra 2]
Identify equivalent logarithmic expressions using the properties of Core Skill
logarithms [Algebra 2]
AL A-SSE.8 - Derive the formula for the sum of a finite geometric series (when the common ratio is
not 1), and use the formula to solve problems. Example: For example, calculate mortgage payments.
Algebra
Find the sum of a finite geometric series [Algebra 2]
Core Skill
Find the sum of a finite geometric series given in summation
notation [Algebra 2]
WP: Solve a problem that can be represented by a finite geometric Core Skill
series [Algebra 2]
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Algebra 2
AL A-APR - Arithmetic with Polynomials and Rational Expressions
Perform arithmetic operations on polynomials[(Beyond quadratic.)]
AL A-APR.9 - Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
Algebra
Multiply two monomial algebraic expressions [Grade 8]
Core Skill
Simplify a polynomial expression by combining like terms
[Algebra 1]
Add polynomial expressions [Algebra 1]
Core Skill
Subtract polynomial expressions [Algebra 1]
Core Skill
Multiply a polynomial by a monomial [Algebra 1]
Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1]
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Multiply two binomials of the form (ax +/- by)(cx +/- dy)
[Algebra 1]
Square a binomial [Algebra 1]
Multiply two nonlinear binomials [Algebra 1]
Multiply a trinomial by a binomial [Algebra 1]
Multiply polynomial expressions [Algebra 2]
Understand the relationship between zeros and factors of polynomials
AL A-APR.10 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a,
the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
Algebra
Divide a polynomial expression by a monomial [Algebra 1]
Core Skill
Divide a polynomial expression by a binomial [Algebra 1]
Determine factors of a polynomial by applying the remainder
theorem [Algebra 2]
AL A-APR.11 - Identify zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Determine the solution(s) of an equation given in factored form
[Algebra 1]
Solve a quadratic equation by factoring [Algebra 1]
Core Skill
Identify the end behavior, the intercepts, or the zeros of a
polynomial function [Algebra 2]
Core Skill
Use polynomial identities to solve problems
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Algebra 2
AL A-APR.12 - Prove polynomial identities and use them to describe numerical relationships.
Example: For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate
Pythagorean triples.
Algebra
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Factor the difference of two squares [Algebra 1]
Factor a polynomial of degree 3 or higher [Algebra 2]
WP: Solve a problem modeled by a polynomial equation [Algebra
2]
Multiply polynomial expressions [Algebra 2]
AL A-APR.13 - Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in
powers of x and y for a positive integer n, where x and y are any numbers, with coefficients
determined for example by Pascal's Triangle.[(The Binomial Theorem can be proved by
mathematical induction or by a combinatorial argument.)]
Algebra
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Rewrite rational expressions[(Linear and quadratic denominators.)]
AL A-APR.14 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q
(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the
degree of b(x), using inspection, long division, or, for the more complicated examples, a computer
algebra system.
Algebra
Divide a polynomial expression by a monomial [Algebra 1]
Core Skill
Divide a polynomial expression by a binomial [Algebra 1]
Divide polynomial expressions [Algebra 2]
AL A-APR.15 - Understand that rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.
Algebra
Multiply rational expressions [Algebra 1]
Core Skill
Divide rational expressions [Algebra 1]
Determine the LCD of two rational expressions [Algebra 1]
Add or subtract two rational expressions with like denominators
[Algebra 1]
Add or subtract two rational expressions with unlike monomial
denominators [Algebra 1]
Add or subtract two rational expressions with unlike polynomial
denominators [Algebra 1]
Core Skill
AL A-CED - Creating Equations
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Algebra 2
Create equations that describe numbers or relationships[(Equations using all available types of
expressions, including simple root functions.)]
AL A-CED.16 - Create equations and inequalities in one variable and use them to solve problems.
[Include equations arising from linear and quadratic functions, and simple rational and exponential
functions.]
Algebra
WP: Use a 1-variable linear inequality to represent a situation
[Grade 7]
WP: Use a 1-variable equation with rational coefficients to
represent a situation involving two operations [Grade 8]
Core Skill
WP: Solve a problem involving a 1-variable, 2-step equation
[Grade 8]
Core Skill
Solve a 2-step linear inequality in one variable [Grade 8]
Core Skill
WP: Use a 2-step linear inequality in one variable to represent a
situation [Grade 8]
WP: Solve a problem involving a 2-step linear inequality in one
variable [Grade 8]
Core Skill
Solve a 1-variable linear equation with the variable on both sides
[Algebra 1]
Core Skill
WP: Determine a linear equation that can be used to solve a
percent problem [Algebra 1]
WP: Solve a direct- or inverse-variation problem [Algebra 1]
Solve a 1-variable linear inequality with the variable on one side
[Algebra 1]
Solve a 1-variable linear inequality with the variable on both sides Core Skill
[Algebra 1]
WP: Solve a problem involving a rational equation [Algebra 1]
AL A-CED.17 - Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra
WP: Use a 2-variable equation with rational coefficients to
represent a situation [Grade 8]
Core Skill
WP: Determine an equation representing a direct variation or an
inverse variation [Algebra 1]
Determine the graph of a linear equation given in slope-intercept,
point-slope, or standard form [Algebra 1]
Core Skill
WP: Solve a mixture problem that can be represented by a system
of linear equations [Algebra 1]
WP: Solve a motion problem that can be represented by a system
of linear equations [Algebra 1]
Solve a number problem that can be represented by a linear
system of equations [Algebra 1]
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Algebra 2
Determine the equation of a circle from given information
[Algebra 2]
Determine the graph of a circle given the equation in standard
form [Algebra 2]
Determine the graph of a circle or an ellipse given the equation in Core Skill
general form [Algebra 2]
Determine the graph of a hyperbola given the equation in standard
form [Algebra 2]
Determine the equation of a parabola using given information
[Algebra 2]
Core Skill
Determine the equation of an ellipse using given information
[Algebra 2]
AL A-CED.18 - Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Example: For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Algebra
WP: Determine a system of linear equations that represents a
given situation [Algebra 1]
Core Skill
WP: Determine a system of linear inequalities that represents a
given situation [Algebra 1]
WP: Determine possible solutions to a problem that can be
represented by a system of linear inequalities [Algebra 1]
WP: Solve a linear programming problem [Algebra 2]
WP: Solve a problem that can be represented by a system of linear Core Skill
equations in three variables [Algebra 2]
WP: Solve a problem that can be modeled with a quadratic
inequality [Algebra 2]
AL A-CED.19 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as
in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R.
Algebra
Rewrite an equation to solve for a specified variable [Algebra 1]
Core Skill
AL A-REI - Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning[(Simple rational and
radical.)]
AL A-REI.20 - Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
Algebra
Solve a radical equation that leads to a linear equation [Algebra 1] Core Skill
Solve a radical equation that leads to a quadratic equation
[Algebra 1]
Core Skill
WP: Solve a problem involving a radical function [Algebra 1]
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Algebra 2
Determine the excluded values of a rational algebraic expression
[Algebra 1]
Solve a proportion that generates a linear or quadratic equation
[Algebra 1]
Solve a rational equation involving terms with monomial
denominators [Algebra 1]
Core Skill
Solve a rational equation involving terms with polynomial
denominators [Algebra 1]
Core Skill
Represent and solve equations and inequalities graphically[(Combine polynomial, rational, radical,
absolute value, and exponential functions.)]
AL A-REI.21 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.
Algebra
Solve a system of linear equations in two variables by graphing
[Algebra 1]
Determine the graph of a 2-variable absolute value inequality
[Algebra 2]
Relate a graph to a polynomial function given in factored form
[Algebra 2]
Core Skill
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
AL F - Functions
AL F-IF - Interpreting Functions
Interpret functions that arise in applications in terms of the context[(Emphasize selection of
appropriate models.)]
AL F-IF.22 - For a function that models a relationship between two quantities, interpret key features
of graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship.[Key features include intercepts; intervals where the function is
increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.]
Algebra
Determine the slope of a line given its graph or a graph of a line
with a given slope [Grade 8]
Core Skill
Determine the x- or y-intercept of a line given its graph [Grade 8] Core Skill
4/26/2012
WP: Answer a question using the graph of a quadratic function
[Algebra 1]
Core Skill
Identify the vertex, axis of symmetry, or direction of the graph of
a quadratic function [Algebra 2]
Core Skill
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Algebra 2
Relate changes in the values of a, b, and/or c to the graph of the
equation y = a(x - b)^2 + c [Algebra 2]
Core Skill
Identify the end behavior, asymptotes, excluded values, or
behavior near excluded values of a rational function [Algebra 2]
Core Skill
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
AL F-IF.23 - Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.
Algebra
Determine the domain or range of a function [Algebra 1]
WP: Determine a reasonable domain or range for a function in a
given situation [Algebra 1]
Core Skill
Identify the domain or range of a radical function [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
AL F-IF.24 - Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.
Algebra
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
WP: Determine a graph that can represent a situation involving a
varying rate of change [Grade 8]
Determine the slope of a line given two points on the line
[Algebra 1]
Core Skill
WP: Interpret an interest rate, rate of change, initial amount,
frequency of compounding and other parameters of an
exponential function [Algebra 2]
Core Skill
Analyze functions using different representations[(Focus on using key features to guide selection of
appropriate type of model function.)]
AL F-IF.25 - Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
AL F-IF.25.a - Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
Algebra
Determine the graph of a 2-variable absolute value equation
[Algebra 1]
Determine the graph of a radical function [Algebra 1]
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Algebra 2
Determine the graph of a piecewise-defined function [Algebra 2]
Relate a graph to a square or cube root function [Algebra 2]
AL F-IF.25.b - Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
Algebra
Relate a graph to a polynomial function given in factored form
[Algebra 2]
Core Skill
Identify the end behavior, the intercepts, or the zeros of a
polynomial function [Algebra 2]
Core Skill
Determine a polynomial function which goes through specified
points on a graph [Algebra 2]
Core Skill
AL F-IF.25.c - Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
Algebra
Determine the graph of an exponential function [Algebra 1]
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
Determine the graph of a sine, cosine or tangent function [Algebra Core Skill
2]
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
AL F-IF.26 - Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
Algebra
Determine the domain or range of a function [Algebra 1]
Evaluate a function written in function notation for a given value
[Algebra 1]
Core Skill
Convert a quadratic function between vertex and standard form of
the equation [Algebra 2]
Compare characteristics of two linear, two quadratic, or two
exponential functions represented in different ways [Algebra 2]
AL F-IF.27 - Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph
of one quadratic function and an algebraic expression for another, say which has the larger
maximum.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
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Algebra 2
Compare characteristics of two linear, two quadratic, or two
exponential functions represented in different ways [Algebra 2]
AL F-BF - Building Functions
Build a function that models a relationship between two quantities[(Include all types of functions
studied.)]
AL F-BF.28 - Combine standard function types using arithmetic operations. Example: For example,
build a function that models the temperature of a cooling body by adding a constant function to a
decaying exponential, and relate these functions to the model.
Algebra
WP: Apply function operations or compositions of functions to
represent a situation [Algebra 2]
Build new functions from existing functions[(Include simple radical, rational, and exponential
functions; emphasize common effect of each transformation across function types.)]
AL F-BF.29 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph using technology.
Algebra
Determine the equation of a function resulting from a translation
and/or scaling of a given function [Algebra 2]
Core Skill
AL F-BF.30 - Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x is not
equal to 1.
Algebra
Determine if the inverse of a function is a function [Algebra 2]
Determine the equation of the inverse of a linear, rational root, or
polynomial function [Algebra 2]
Core Skill
AL F-LE - Linear, Quadratic, and Exponential Models
Construct and compare linear, quadratic, and exponential models and solve problems[(Logarithms as
solutions for exponentials.)]
AL F-LE.31 - For exponential models, express as a logarithm the solution to ab to the ct power = d
where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Algebra
Convert between a simple exponential equation and its
corresponding logarithmic equation [Algebra 2]
Solve an exponential equation using logarithms [Algebra 2]
Core Skill
AL S - Statistics and Probability
AL S-ID - Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
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Algebra 2
AL S-ID.32 - Use the mean and standard deviation of a data set to fit it to a normal distribution and
to estimate population percentages. Recognize that there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
AL S-IC - Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
AL S-IC.33 - Understand statistics as a process for making inferences about population parameters
based on a random sample from that population.
Data Analysis, Statistics, and
Probability
Determine if a sample is likely to be representative of the larger
population [Grade 8]
AL S-IC.34 - Decide if a specified model is consistent with results from a given data-generating
process, e.g., using simulation. Example: For example, a model says a spinning coin falls heads up
with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Data Analysis, Statistics, and
Probability
Determine if a sample is likely to be representative of the larger
population [Grade 8]
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
AL S-IC.35 - Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
Data Analysis, Statistics, and
Probability
Determine if a question is likely to produce a biased survey result
[Grade 8]
AL S-IC.36 - Use data from a sample survey to estimate a population mean or proportion; develop a
margin of error through the use of simulation models for random sampling.
Data Analysis, Statistics, and
Probability
Analyze the effect that changing elements in a data set has on the
mean, the median, or the range [Grade 8]
AL S-IC.37 - Use data from a randomized experiment to compare two treatments; use simulations to
decide if differences between parameters are significant.
AL S-IC.38 - Evaluate reports based on data.
AL S-MD - Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions
AL S-MD.39 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
Data Analysis, Statistics, and
Probability
Determine the probability of three or more independent events
[Grade 8]
Determine the probability of three or more dependent events
[Grade 8]
Determine the probability of an event consisting of mutually
exclusive events [Grade 8]
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Algebra 2
Make a prediction involving the probability of compound events
[Grade 8]
AL S-MD.40 - Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Overall Product Skills
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Use strategies to derive solutions for problems
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Algebra 2 with Trig
Algebra 2 with Trig
AL N - Number and Quantity
AL N-CN - The Complex Number System
Perform arithmetic operations with complex numbers.
AL N-CN.1 - Know there is a complex number i such that i² = -1, and every complex number has the
form a + bi with a and b real.
Algebra
Raise i to a power [Algebra 2]
AL N-CN.2 - Use the relation i² = -1 and the commutative, associative, and distributive properties to
add, subtract, and multiply complex numbers.
Algebra
Raise i to a power [Algebra 2]
Add or subtract complex numbers [Algebra 2]
Multiply complex numbers [Algebra 2]
Simplify an expression involving a complex denominator
[Algebra 2]
Core Skill
Use complex numbers in polynomial identities and equations.[(Polynomials with real coefficients.)]
AL N-CN.3 - Solve quadratic equations with real coefficients that have complex solutions.
Algebra
Solve a quadratic equation with complex solutions [Algebra 2]
Core Skill
AL N-CN.4 - Extend polynomial identities to the complex numbers. Example: For example, rewrite
x² + 4 as (x + 2i)(x - 2i).
Algebra
Extend polynomial identities to the complex numbers [Algebra 2]
AL N-CN.5 - Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
Determine a polynomial of lowest degree that has given roots or a
graph of a given shape [Algebra 2]
AL A - Algebra
AL A-SSE - Seeing Structure in Expressions
Interpret the structure of expressions[(Polynomial and rational.)]
AL A-SSE.6 - Interpret expressions that represent a quantity in terms of its context.
AL A-SSE.6.a - Interpret parts of an expression, such as terms, factors, and coefficients.
Overall Product Skills
Build comprehension of math vocabulary
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Algebra 2 with Trig
AL A-SSE.6.b - Interpret complicated expressions by viewing one or more of their parts as a single
entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not
depending on P.
AL A-SSE.7 - Use the structure of an expression to identify ways to rewrite it. Example: For
example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a
difference of squares that can be factored as (x² - y²)(x² + y²).
Algebra
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
Write expressions in equivalent forms to solve problems
Algebra
Simplify an algebraic expression by combining like terms [Grade
8]
Core Skill
Use the distributive property to simplify an algebraic expression
[Grade 8]
Simplify a polynomial expression by combining like terms
[Algebra 1]
Simplify a monomial algebraic radical expression [Algebra 1]
Core Skill
Simplify a rational expression involving polynomial terms
[Algebra 1]
Core Skill
Simplify a numerical expression that includes fractional
exponents and/or nth roots [Algebra 2]
Decompose rational functions with quadratic denominators into
sums of rational functions with linear denominators [Algebra 2]
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
Identify an equivalent form of an exponential expression by using
the properties of exponents [Algebra 2]
Identify equivalent logarithmic expressions using the properties of Core Skill
logarithms [Algebra 2]
AL A-SSE.8 - Derive the formula for the sum of a finite geometric series (when the common ratio is
not 1), and use the formula to solve problems. Example: For example, calculate mortgage payments.
Algebra
Find the sum of a finite geometric series [Algebra 2]
Core Skill
Find the sum of a finite geometric series given in summation
notation [Algebra 2]
WP: Solve a problem that can be represented by a finite geometric Core Skill
series [Algebra 2]
AL A-APR - Arithmetic with Polynomials and Rational Expressions
Perform arithmetic operations on polynomials[(Beyond quadratic.)]
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Algebra 2 with Trig
AL A-APR.9 - Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
Algebra
Multiply two monomial algebraic expressions [Grade 8]
Core Skill
Simplify a polynomial expression by combining like terms
[Algebra 1]
Add polynomial expressions [Algebra 1]
Core Skill
Subtract polynomial expressions [Algebra 1]
Core Skill
Multiply a polynomial by a monomial [Algebra 1]
Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1]
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Multiply two binomials of the form (ax +/- by)(cx +/- dy)
[Algebra 1]
Square a binomial [Algebra 1]
Multiply two nonlinear binomials [Algebra 1]
Multiply a trinomial by a binomial [Algebra 1]
Multiply polynomial expressions [Algebra 2]
Understand the relationship between zeros and factors of polynomials
AL A-APR.10 - Know and apply the Remainder Theorem: For a polynomial p(x) and a number a,
the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).
Algebra
Divide a polynomial expression by a monomial [Algebra 1]
Core Skill
Divide a polynomial expression by a binomial [Algebra 1]
Determine factors of a polynomial by applying the remainder
theorem [Algebra 2]
AL A-APR.11 - Identify zeros of polynomials when suitable factorizations are available, and use the
zeros to construct a rough graph of the function defined by the polynomial.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Determine the solution(s) of an equation given in factored form
[Algebra 1]
Solve a quadratic equation by factoring [Algebra 1]
Core Skill
Identify the end behavior, the intercepts, or the zeros of a
polynomial function [Algebra 2]
Core Skill
Use polynomial identities to solve problems
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Algebra 2 with Trig
AL A-APR.12 - Prove polynomial identities and use them to describe numerical relationships.
Example: For example, the polynomial identity (x² + y²)² = (x² - y²)² + (2xy)² can be used to generate
Pythagorean triples.
Algebra
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Factor the difference of two squares [Algebra 1]
Factor a polynomial of degree 3 or higher [Algebra 2]
WP: Solve a problem modeled by a polynomial equation [Algebra
2]
Multiply polynomial expressions [Algebra 2]
AL A-APR.13 - Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in
powers of x and y for a positive integer n, where x and y are any numbers, with coefficients
determined for example by Pascal's Triangle.[The Binomial Theorem can be proved by
mathematical induction or by a combinatorial argument.]
Algebra
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Find a specified term of a binomial expression raised to a positive Core Skill
integer power [Algebra 2]
Rewrite rational expressions[(Linear and quadratic denominators.)]
AL A-APR.14 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q
(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the
degree of b(x), using inspection, long division, or, for the more complicated examples, a computer
algebra system.
Algebra
Divide a polynomial expression by a monomial [Algebra 1]
Core Skill
Divide a polynomial expression by a binomial [Algebra 1]
Divide polynomial expressions [Algebra 2]
AL A-APR.15 - Understand that rational expressions form a system analogous to the rational
numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.
Algebra
Multiply rational expressions [Algebra 1]
Core Skill
Divide rational expressions [Algebra 1]
Determine the LCD of two rational expressions [Algebra 1]
Add or subtract two rational expressions with like denominators
[Algebra 1]
Add or subtract two rational expressions with unlike monomial
denominators [Algebra 1]
Add or subtract two rational expressions with unlike polynomial
denominators [Algebra 1]
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Core Skill
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Algebra 2 with Trig
AL A-CED - Creating Equations
Create equations that describe numbers or relationships[(Equations using all available types of
expressions, including simple root functions.)]
AL A-CED.16 - Create equations and inequalities in one variable and use them to solve problems.
[Include equations arising from linear and quadratic functions, and simple rational and exponential
functions.]
Algebra
WP: Use a 1-variable 1-step equation to represent a situation
[Grade 7]
Core Skill
WP: Use a 1-variable linear inequality to represent a situation
[Grade 7]
WP: Use a 1-variable equation with rational coefficients to
represent a situation involving two operations [Grade 8]
Core Skill
WP: Solve a problem involving a 1-variable, 2-step equation
[Grade 8]
Core Skill
Solve a 2-step linear inequality in one variable [Grade 8]
Core Skill
WP: Use a 2-step linear inequality in one variable to represent a
situation [Grade 8]
WP: Solve a problem involving a 2-step linear inequality in one
variable [Grade 8]
Core Skill
Solve a 1-variable linear equation with the variable on both sides
[Algebra 1]
Core Skill
WP: Determine a linear equation that can be used to solve a
percent problem [Algebra 1]
WP: Solve a direct- or inverse-variation problem [Algebra 1]
Solve a 1-variable linear inequality with the variable on one side
[Algebra 1]
Solve a 1-variable linear inequality with the variable on both sides Core Skill
[Algebra 1]
WP: Solve a problem involving a rational equation [Algebra 1]
AL A-CED.17 - Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra
WP: Use a 2-variable equation to represent a situation involving a Core Skill
direct proportion [Grade 6]
WP: Use a 2-variable equation with rational coefficients to
represent a situation [Grade 8]
Core Skill
WP: Determine an equation representing a direct variation or an
inverse variation [Algebra 1]
Determine the graph of a linear equation given in slope-intercept,
point-slope, or standard form [Algebra 1]
Core Skill
WP: Solve a mixture problem that can be represented by a system
of linear equations [Algebra 1]
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Algebra 2 with Trig
WP: Solve a motion problem that can be represented by a system
of linear equations [Algebra 1]
Solve a number problem that can be represented by a linear
system of equations [Algebra 1]
Determine the equation of a circle from given information
[Algebra 2]
Determine the graph of a circle given the equation in standard
form [Algebra 2]
Determine the graph of a circle or an ellipse given the equation in Core Skill
general form [Algebra 2]
Determine the graph of a hyperbola given the equation in standard
form [Algebra 2]
Determine the equation of a parabola using given information
[Algebra 2]
Core Skill
Determine the equation of an ellipse using given information
[Algebra 2]
AL A-CED.18 - Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Example: For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Algebra
WP: Determine a system of linear equations that represents a
given situation [Algebra 1]
Core Skill
WP: Determine a system of linear inequalities that represents a
given situation [Algebra 1]
WP: Determine possible solutions to a problem that can be
represented by a system of linear inequalities [Algebra 1]
WP: Solve a linear programming problem [Algebra 2]
WP: Solve a problem that can be represented by a system of linear Core Skill
equations in three variables [Algebra 2]
WP: Solve a problem that can be modeled with a quadratic
inequality [Algebra 2]
AL A-CED.19 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as
in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R.
Algebra
Rewrite an equation to solve for a specified variable [Algebra 1]
Core Skill
AL A-REI - Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning[(Simple rational and
radical.)]
AL A-REI.20 - Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
Algebra
4/26/2012
Solve a radical equation that leads to a linear equation [Algebra 1] Core Skill
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Algebra 2 with Trig
Solve a radical equation that leads to a quadratic equation
[Algebra 1]
Core Skill
WP: Solve a problem involving a radical function [Algebra 1]
Determine the excluded values of a rational algebraic expression
[Algebra 1]
Solve a proportion that generates a linear or quadratic equation
[Algebra 1]
Solve a rational equation involving terms with monomial
denominators [Algebra 1]
Core Skill
Solve a rational equation involving terms with polynomial
denominators [Algebra 1]
Core Skill
Represent and solve equations and inequalities graphically[(Combine polynomial, rational, radical,
absolute value, and exponential functions.)]
AL A-REI.21 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.
Algebra
Solve a system of linear equations in two variables by graphing
[Algebra 1]
Determine the graph of a 2-variable absolute value inequality
[Algebra 2]
Relate a graph to a polynomial function given in factored form
[Algebra 2]
Core Skill
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
AL F - Functions
AL F-IF - Interpreting Functions
Interpret functions that arise in applications in terms of the context[(Emphasize selection of
appropriate models.)]
AL F-IF.22 - For a function that models a relationship between two quantities, interpret key features
of graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship.[Key features include: intercepts; intervals where the function
is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.]
Algebra
Determine the slope of a line given its graph or a graph of a line
with a given slope [Grade 8]
Core Skill
Determine the x- or y-intercept of a line given its graph [Grade 8] Core Skill
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Algebra 2 with Trig
WP: Answer a question using the graph of a quadratic function
[Algebra 1]
Core Skill
Identify the vertex, axis of symmetry, or direction of the graph of
a quadratic function [Algebra 2]
Core Skill
Relate changes in the values of a, b, and/or c to the graph of the
equation y = a(x - b)^2 + c [Algebra 2]
Core Skill
Identify the end behavior, asymptotes, excluded values, or
behavior near excluded values of a rational function [Algebra 2]
Core Skill
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
AL F-IF.23 - Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.
Algebra
Determine the domain or range of a function [Algebra 1]
WP: Determine a reasonable domain or range for a function in a
given situation [Algebra 1]
Core Skill
Identify the domain or range of a radical function [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
AL F-IF.24 - Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.
Algebra
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
WP: Determine a graph that can represent a situation involving a
varying rate of change [Grade 8]
Determine the slope of a line given two points on the line
[Algebra 1]
Core Skill
WP: Interpret an interest rate, rate of change, initial amount,
frequency of compounding and other parameters of an
exponential function [Algebra 2]
Core Skill
Analyze functions using different representations[(Focus on using key features to guide selection of
appropriate type of model function.)]
AL F-IF.25 - Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
Algebra
4/26/2012
Determine the graph of a 2-operation linear function [Grade 8]
Core Skill
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Algebra 2 with Trig
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Determine the graph of an exponential function [Algebra 1]
Determine the graph of a radical function [Algebra 1]
Determine the graph of a rational function [Algebra 1]
Determine the graph of a piecewise-defined function [Algebra 2]
Identify the vertex, axis of symmetry, or direction of the graph of
a quadratic function [Algebra 2]
Core Skill
Determine the graph of a rational function involving linear or
quadratic denominators [Algebra 2]
Core Skill
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
Determine the graph of the inverse of a function [Algebra 2]
AL F-IF.25.a - Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
Algebra
Determine the graph of a 2-variable absolute value equation
[Algebra 1]
Determine the graph of a radical function [Algebra 1]
Determine the graph of a piecewise-defined function [Algebra 2]
Relate a graph to a square or cube root function [Algebra 2]
AL F-IF.25.b - Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
Algebra
Relate a graph to a polynomial function given in factored form
[Algebra 2]
Core Skill
Identify the end behavior, the intercepts, or the zeros of a
polynomial function [Algebra 2]
Core Skill
Determine a polynomial function which goes through specified
points on a graph [Algebra 2]
Core Skill
AL F-IF.25.c - Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
Algebra
Determine the graph of an exponential function [Algebra 1]
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
Identify the domain, range, asymptotes, or intercepts of a
logarithmic function [Algebra 2]
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Algebra 2 with Trig
Determine the graph of a sine, cosine or tangent function [Algebra Core Skill
2]
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
AL F-IF.26 - Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
Algebra
Evaluate a function written in function notation for a given value
[Algebra 1]
Core Skill
Convert a quadratic function between vertex and standard form of
the equation [Algebra 2]
Compare characteristics of two linear, two quadratic, or two
exponential functions represented in different ways [Algebra 2]
AL F-IF.27 - Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph
of one quadratic function and an algebraic expression for another, say which has the larger
maximum.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
Compare characteristics of two linear, two quadratic, or two
exponential functions represented in different ways [Algebra 2]
AL F-BF - Building Functions
Build a function that models a relationship between two quantities[(Include all types of functions
studied.)]
AL F-BF.28 - Write a function that describes a relationship between two quantities.
AL F-BF.28.a - Combine standard function types using arithmetic operations. Example: For
example, build a function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
Algebra
WP: Apply function operations or compositions of functions to
represent a situation [Algebra 2]
Build new functions from existing functions[(Include simple radical, rational, and exponential
functions; emphasize common effect of each transformation across function types.)]
AL F-BF.29 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph using technology.[Include
recognizing even and odd functions from their graphs and algebraic expressions for them.]
Algebra
4/26/2012
Determine the equation of a function resulting from a translation
and/or scaling of a given function [Algebra 2]
Core Skill
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Algebra 2 with Trig
AL F-BF.30 - Solve an equation of the form f(x) = c for a simple function f that has an inverse and
write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x is not
equal to 1.
Algebra
Determine if the inverse of a function is a function [Algebra 2]
Determine the equation of the inverse of a linear, rational root, or
polynomial function [Algebra 2]
Core Skill
AL F-LE - Linear, Quadratic, and Exponential Models
Construct and compare linear, quadratic, and exponential models and solve problems[(Logarithms as
solutions for exponentials.)]
AL F-LE.31 - For exponential models, express as a logarithm the solution to ab to the ct power = d
where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Algebra
Convert between a simple exponential equation and its
corresponding logarithmic equation [Algebra 2]
Solve an exponential equation using logarithms [Algebra 2]
Core Skill
AL F-TF - Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle
AL F-TF.32 - Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
AL F-TF.33 - Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
Algebra
Determine an angle coterminal with a given angle [Algebra 2]
AL F-TF.34 - Define the six trigonometric functions using ratios of the sides of a right triangle,
coordinates on the unit circle, and the reciprocal of other functions.
Algebra
Determine the sine, cosine or tangent of an angle given in degrees
or radians [Algebra 2]
Determine the exact value of a sine, cosine or tangent function
given an angle in degrees or radians [Algebra 2]
Determine the value of an inverse sine, cosine, or tangent
expression [Algebra 2]
WP: Determine a trigonometric function that represents a
situation [Algebra 2]
WP: Solve a problem involving a sine, cosine, or tangent function Core Skill
[Algebra 2]
Geometry and Measurement
4/26/2012
Determine a sine, cosine, or tangent ratio in a right triangle
[Geometry]
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Algebra 2 with Trig
Determine a length using a sine, cosine, or tangent ratio in a right
triangle [Geometry]
Determine the measure of an angle using a sine, cosine, or tangent
ratio in a right triangle [Geometry]
WP: Determine a length in a right triangle using a sine, cosine, or
tangent ratio [Geometry]
WP: Determine the measure of an angle in a right triangle using a
sine, cosine, or tangent ratio [Geometry]
Relate a trigonometric function to the sides of a right triangle
[Algebra 2]
Model periodic phenomena with trigonometric functions
AL F-TF.35 - Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
Algebra
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
Prove and apply trigonometric identities
AL F-TF.36 - Prove the Pythagorean identity sin²(theta) + cos²(theta) = 1 and use it to find sin(theta),
cos(theta), or tan(theta) given sin(theta), cos(theta), or tan(theta) and the quadrant of the angle.
Algebra
Solve a problem involving the Pythagorean identity sin^2(theta) +
cos^2(theta) = 1 [Algebra 2]
AL S - Statistics and Probability
AL S-ID - Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
AL S-ID.37 - Use the mean and standard deviation of a data set to fit it to a normal distribution and
to estimate population percentages. Recognize that there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
AL S-IC - Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying statistical experiments
AL S-IC.38 - Understand statistics as a process for making inferences about population parameters
based on a random sample from that population.
AL S-IC.39 - Decide if a specified model is consistent with results from a given data-generating
process, e.g., using simulation. Example: For example, a model says a spinning coin falls heads up
with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
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Algebra 2 with Trig
AL S-IC.40 - Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
AL S-IC.41 - Use data from a sample survey to estimate a population mean or proportion; develop a
margin of error through the use of simulation models for random sampling.
Data Analysis, Statistics, and
Probability
Determine if a sample is likely to be representative of the larger
population [Grade 8]
AL S-IC.42 - Use data from a randomized experiment to compare two treatments; use simulations to
decide if differences between parameters are significant.
AL S-IC.43 - Evaluate reports based on data.
AL S-MD - Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions[(Include more complex situations.)]
AL S-MD.44 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
AL S-MD.45 - Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Overall Product Skills
4/26/2012
Use strategies to derive solutions for problems
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Algebra I
Algebra I
AL N - Number and Quantity
AL N-RN - The Real Number System
Extend the properties of exponents to rational exponents.
AL N-RN.1 - Explain how the definition of the meaning of rational exponents follows from extending
the properties of integer exponents to those values, allowing for a notation for radicals in terms of
rational exponents. Example: For example, we define 5 to the 1/3 power to be the cube root of 5
because we want (5 to the 1/3 power)³ = 5 to the power (1/3 times 3) to hold, so (5 to the 1/3 power)
cubed must equal 5.
Algebra
Evaluate a numerical expression involving one or more exponents Core Skill
and multiple forms of rational numbers [Algebra 1]
Apply the product of powers property to a monomial algebraic
expression [Algebra 1]
Core Skill
Apply the power of a power property to a monomial algebraic
expression [Algebra 1]
Core Skill
Apply the power of a product property to a monomial algebraic
expression [Algebra 1]
Core Skill
Apply the quotient of powers property to monomial algebraic
expressions [Algebra 1]
Core Skill
Apply the power of a quotient property to monomial algebraic
expressions [Algebra 1]
Core Skill
Simplify a numerical expression that includes fractional
exponents and/or nth roots [Algebra 2]
AL N-RN.2 - Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
Algebra
Simplify a monomial numerical expression involving the square
root of a whole number [Algebra 1]
Core Skill
Multiply monomial numerical expressions involving radicals
[Algebra 1]
Divide monomial numerical expressions involving radicals
[Algebra 1]
Simplify a monomial algebraic radical expression [Algebra 1]
Core Skill
Multiply monomial algebraic radical expressions [Algebra 1]
Divide monomial algebraic radical expressions [Algebra 1]
Simplify a numerical expression that includes fractional
exponents and/or nth roots [Algebra 2]
Use properties of rational and irrational numbers.
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Algebra I
AL N-RN.3 - Explain why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product of a nonzero rational
number and an irrational number is irrational.
Algebra
Simplify an algebraic expression by combining like terms [Grade
8]
Core Skill
Multiply two monomial algebraic expressions [Grade 8]
Core Skill
AL N-Q - Quantities
Reason quantitatively and use units to solve problems.[(Foundation for work with expressions,
equations, and functions.)]
AL N-Q.4 - Use units as a way to understand problems and to guide the solution of multi-step
problems; choose and interpret units consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.
Algebra
WP: Solve a direct- or inverse-variation problem [Algebra 1]
WP: Solve a multi-step problem involving dimensional analysis
[Algebra 2]
Geometry and Measurement
Convert a rate from one unit to another with a change in both
units [Grade 7]
Core Skill
AL N-Q.5 - Define appropriate quantities for the purpose of descriptive modeling.
AL N-Q.6 - Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Algebra
WP: Determine the degree of accuracy of a computation [Algebra
2]
AL A - Algebra
AL A-SSE - Seeing Structure in Expressions
Interpret the structure of expressions[(Linear, exponential, quadratic.)]
AL A-SSE.7 - Interpret expressions that represent a quantity in terms of its context.
AL A-SSE.7.a - Interpret parts of an expression, such as terms, factors, and coefficients.
Overall Product Skills
Build comprehension of math vocabulary
AL A-SSE.7.b - Interpret complicated expressions by viewing one or more of their parts as a single
entity. Example: For example, interpret P(1+r) to the n power as the product of P and a factor not
depending on P.
AL A-SSE.8 - Use the structure of an expression to identify ways to rewrite it. Example: For
example, see (x to the 4th power) - (y to the 4th power) as (x²)² - (y²)², thus recognizing it as a
difference of squares that can be factored as (x² - y²)(x² + y²).
Algebra
4/26/2012
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
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Write expressions in equivalent forms to solve problems[(Quadratic and exponential.)]
Algebra
Simplify an algebraic expression by combining like terms [Grade
8]
Algebra I
Core Skill
Use the distributive property to simplify an algebraic expression
[Grade 8]
Simplify a monomial algebraic radical expression [Algebra 1]
Core Skill
Simplify a numerical expression that includes fractional
exponents and/or nth roots [Algebra 2]
Simplify a monomial algebraic expression that includes fractional
exponents and/or nth roots [Algebra 2]
Identify an equivalent form of an exponential expression by using
the properties of exponents [Algebra 2]
Solve an exponential equation using the properties of exponents
[Algebra 2]
Identify equivalent logarithmic expressions using the properties of Core Skill
logarithms [Algebra 2]
AL A-SSE.9 - Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.
AL A-SSE.9.a - Factor a quadratic expression to reveal the zeros of the function it defines.
Algebra
Factor trinomials that result in factors of the form (x +/- a)(x +/b) [Algebra 1]
Factor trinomials that result in factors of the form (ax +/- b)(cx +/- Core Skill
d) [Algebra 1]
Factor trinomials that result in factors of the form (ax +/- by)(cx
+/- dy) [Algebra 1]
Factor the difference of two squares [Algebra 1]
Factor a perfect-square trinomial [Algebra 1]
Solve a quadratic equation by factoring [Algebra 1]
Core Skill
AL A-SSE.9.b - Complete the square in a quadratic expression to reveal the maximum or
minimum value of the function it defines.
Algebra
Complete the square of a quadratic 1-variable equation [Algebra
2]
AL A-SSE.9.c - Determine a quadratic equation when given its graph or roots.
Algebra
Solve a quadratic equation by factoring [Algebra 1]
Solve a quadratic equation using the quadratic formula [Algebra
1]
4/26/2012
Core Skill
Core Skill
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Algebra I
AL A-SSE.9.d - Use the properties of exponents to transform expressions for exponential functions.
Example: For example the expression 1.15 to the t power can be rewritten as ((1.15 to the 1/12
power) to the 12t power) is approximately equal to (1.012 to the 12t power) to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
Algebra
Evaluate a numerical expression involving one or more exponents Core Skill
and multiple forms of rational numbers [Algebra 1]
Apply properties of exponents to monomial algebraic expressions
[Algebra 1]
WP: Evaluate an exponential growth or an exponential decay
function [Algebra 1]
Solve a problem involving exponential growth or exponential
decay [Algebra 1]
AL A-APR - Arithmetic with Polynomials and Rational Expressions
Perform arithmetic operations on polynomials[(Linear and quadratic.)]
AL A-APR.10 - Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
Algebra
Multiply two monomial algebraic expressions [Grade 8]
Core Skill
Simplify a polynomial expression by combining like terms
[Algebra 1]
Add polynomial expressions [Algebra 1]
Core Skill
Subtract polynomial expressions [Algebra 1]
Core Skill
Multiply a polynomial by a monomial [Algebra 1]
Multiply two binomials of the form (x +/- a)(x +/- b) [Algebra 1]
Multiply two binomials of the form (ax +/- b)(cx +/- d) [Algebra
1]
Core Skill
Multiply two binomials of the form (ax +/- by)(cx +/- dy)
[Algebra 1]
Square a binomial [Algebra 1]
Multiply two nonlinear binomials [Algebra 1]
Multiply a trinomial by a binomial [Algebra 1]
Multiply polynomial expressions [Algebra 2]
AL A-CED - Creating Equations
Create equations that describe numbers or relationships[(Linear, quadratic, and exponential (integer
inputs only); for Standard 13, linear only.)]
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Algebra I
AL A-CED.11 - Create equations and inequalities in one variable and use them to solve problems.
[Include equations arising from linear and quadratic functions, and simple rational and exponential
functions.]
Algebra
WP: Use a 1-variable 1-step equation to represent a situation
[Grade 7]
Core Skill
WP: Use a 1-variable equation with rational coefficients to
represent a situation involving two operations [Grade 8]
Core Skill
WP: Use a 2-step linear inequality in one variable to represent a
situation [Grade 8]
WP: Determine a linear equation that can be used to solve a
percent problem [Algebra 1]
WP: Solve a direct- or inverse-variation problem [Algebra 1]
Solve a 1-variable linear inequality with the variable on one side
[Algebra 1]
Solve a 1-variable linear inequality with the variable on both sides Core Skill
[Algebra 1]
AL A-CED.12 - Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Algebra
WP: Use a 2-variable equation with rational coefficients to
represent a situation [Grade 8]
Core Skill
WP: Determine an equation representing a direct variation or an
inverse variation [Algebra 1]
Determine the graph of a linear equation given in slope-intercept,
point-slope, or standard form [Algebra 1]
Core Skill
WP: Solve a mixture problem that can be represented by a system
of linear equations [Algebra 1]
WP: Solve a motion problem that can be represented by a system
of linear equations [Algebra 1]
Solve a number problem that can be represented by a linear
system of equations [Algebra 1]
AL A-CED.13 - Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Example: For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Algebra
WP: Determine a system of linear equations that represents a
given situation [Algebra 1]
Core Skill
WP: Determine a system of linear inequalities that represents a
given situation [Algebra 1]
WP: Determine possible solutions to a problem that can be
represented by a system of linear inequalities [Algebra 1]
WP: Solve a linear programming problem [Algebra 2]
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Algebra I
WP: Solve a problem that can be represented by a system of linear Core Skill
equations in three variables [Algebra 2]
AL A-CED.14 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as
in solving equations. Example: For example, rearrange Ohm's law V = IR to highlight resistance R.
Algebra
Rewrite an equation to solve for a specified variable [Algebra 1]
Core Skill
AL A-REI - Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning[(Master linear; learn
as general principle.)]
AL A-REI.15 - Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
Overall Product Skills
Develop collaboration skills by explaining mathematical
reasoning and strategies
Solve equations and inequalities in one variable[(Linear inequalities; literal that are linear in the
variables being solved for; quadratics with real solutions.)]
AL A-REI.16 - Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters.
Algebra
Solve a 1-step equation involving rational numbers [Grade 8]
Core Skill
Solve a 2-step equation involving rational numbers [Grade 8]
Core Skill
Solve a 2-step linear inequality in one variable [Grade 8]
Core Skill
Solve a 1-variable linear equation with the variable on both sides
[Algebra 1]
Core Skill
Solve a 1-variable linear inequality with the variable on one side
[Algebra 1]
Solve a 1-variable linear inequality with the variable on both sides Core Skill
[Algebra 1]
Solve a 1-variable compound inequality [Algebra 1]
AL A-REI.17 - Solve quadratic equations in one variable.
AL A-REI.17.a - Use the method of completing the square to transform any quadratic equation in
x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula
from this form.
Algebra
Complete the square of a quadratic 1-variable equation [Algebra
2]
AL A-REI.17.b - Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as a ±
bi for real numbers a and b.
Algebra
4/26/2012
Solve a quadratic equation by taking the square root [Algebra 1]
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Algebra I
Solve a quadratic equation by factoring [Algebra 1]
Core Skill
Solve a quadratic equation using the quadratic formula [Algebra
1]
Core Skill
Use the discriminant to determine the number of real solutions
[Algebra 1]
WP: Use a given quadratic equation to solve a problem [Algebra
1]
Complete the square of a quadratic 1-variable equation [Algebra
2]
Solve a quadratic equation with complex solutions [Algebra 2]
Core Skill
Solve systems of equations[(Linear-linear and linear-quadratic.)]
AL A-REI.18 - Prove that, given a system of two equations in two variables, replacing one equation
by the sum of that equation and a multiple of the other produces a system with the same solutions.
Algebra
Solve a system of linear equations in two variables by substitution
[Algebra 1]
Solve a system of linear equations in two variables by elimination
[Algebra 1]
Solve a system of linear equations in two variables using any
method [Algebra 1]
Core Skill
AL A-REI.19 - Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
Algebra
Solve a system of linear equations in two variables by graphing
[Algebra 1]
Solve a system of linear equations in two variables by substitution
[Algebra 1]
Solve a system of linear equations in two variables by elimination
[Algebra 1]
Solve a system of linear equations in two variables using any
method [Algebra 1]
Core Skill
AL A-REI.20 - Solve a simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically. Example: For example, find the points of intersection
between the line y = -3x and the circle x² + y² = 3.
Algebra
Solve a system consisting of a linear equation and a nonlinear
equation in two variables [Algebra 2]
Core Skill
Represent and solve equations and inequalities graphically[(Linear and exponential; learn as general
principle.)]
AL A-REI.21 - Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line).
Algebra
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Determine the graph of a linear equation given in slope-intercept,
point-slope, or standard form [Algebra 1]
Core Skill
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Determine the graph of a given quadratic function [Algebra 1]
Algebra I
Core Skill
Determine the graph of a rational function [Algebra 1]
Relate a graph to a square or cube root function [Algebra 2]
Determine the graph of a vertically or horizontally oriented
parabola [Algebra 2]
Core Skill
AL A-REI.22 - Explain why the x-coordinates of the points where the graphs of the equations y = f(x)
and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value,
exponential, and logarithmic functions.
Algebra
Solve a system of linear equations in two variables by graphing
[Algebra 1]
Determine the graph of a 2-variable absolute value inequality
[Algebra 2]
Relate a graph to a polynomial function given in factored form
[Algebra 2]
Core Skill
Determine an exponential function that represents a graph
[Algebra 2]
Relate a logarithmic function to its graph [Algebra 2]
AL A-REI.23 - Graph the solutions to a linear inequality in two variables as a half-plane (excluding
the boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Algebra
Solve a 2-variable linear inequality for the dependent variable
[Algebra 1]
Determine the graph of a 2-variable linear inequality [Algebra 1]
Determine the graph of the solution set of a system of linear
inequalities in two variables [Algebra 1]
AL F - Functions
AL F-IF - Interpreting Functions
Understand the concept of a function and use function notation[(Learn as general principle; focus on
linear and exponential and on arithmetic and geometric sequences.)]
AL F-IF.24 - Understand that a function from one set (called the domain) to another set (called the
range) assigns to each element of the domain exactly one element of the range. If f is a function and x
is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph
of f is the graph of the equation y = f(x).
Algebra
4/26/2012
Determine the domain or range of a function [Algebra 1]
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Algebra I
AL F-IF.25 - Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
Algebra
WP: Determine a reasonable domain or range for a function in a
given situation [Algebra 1]
Core Skill
Evaluate a function written in function notation for a given value
[Algebra 1]
Core Skill
AL F-IF.26 - Recognize that sequences are functions, sometimes defined recursively, whose domain is
a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by f(0)
= f(1) = 1, f(n+1) = f(n) + f(n-1) for n greater than or equal to 1.
Algebra
Evaluate a function written in function notation for a given value
[Algebra 1]
Core Skill
Identify a possible formula for function of n, where n is a whole
number [Algebra 2]
Determine the recursive formula for an arithmetic sequence
[Algebra 2]
Interpret functions that arise in applications in terms of the context[(Linear, exponential, and
quadratic.)]
AL F-IF.27 - For a function that models a relationship between two quantities, interpret key features
of graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship.[Key features include: intercepts; intervals where the function
is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.]
Algebra
Determine the graph of an exponential function [Algebra 1]
Determine the graph of a rational function [Algebra 1]
Identify the end behavior, asymptotes, excluded values, or
behavior near excluded values of a rational function [Algebra 2]
Core Skill
AL F-IF.28 - Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. Example: For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.
Algebra
WP: Determine a reasonable domain or range for a function in a
given situation [Algebra 1]
Core Skill
AL F-IF.29 - Calculate and interpret the average rate of change of a function (presented symbolically
or as a table) over a specified interval. Estimate the rate of change from a graph.
Algebra
WP: Interpret an interest rate, rate of change, initial amount,
frequency of compounding and other parameters of an
exponential function [Algebra 2]
Core Skill
Analyze functions using different representations[(Linear, exponential, quadratic, absolute value, step,
piecewise-defined.)]
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Algebra I
AL F-IF.30 - Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
AL F-IF.30.a - Graph linear and quadratic functions and show intercepts, maxima, and minima.
Algebra
Determine the graph of a linear equation given in slope-intercept, Core Skill
point-slope, or standard form [Algebra 1]
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Solve a quadratic equation by graphing the associated quadratic
function [Algebra 1]
Determine if the graph of a relation is modeled by a linear or
quadratic function or represents exponential growth or decay
[Algebra 2]
AL F-IF.30.b - Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
Algebra
Determine the graph of a 2-variable absolute value equation
[Algebra 1]
Determine the graph of a radical function [Algebra 1]
Determine the graph of a piecewise-defined function [Algebra 2]
Relate a graph to a square or cube root function [Algebra 2]
AL F-IF.30.c - Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
Algebra
Determine the graph of an exponential function [Algebra 1]
Determine an exponential function that represents a graph
[Algebra 2]
AL F-IF.31 - Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
Algebra
Evaluate a function written in function notation for a given value
[Algebra 1]
Core Skill
AL F-IF.31.a - Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Algebra
Solve a quadratic equation by factoring [Algebra 1]
Core Skill
Complete the square of a quadratic 1-variable equation [Algebra
2]
AL F-IF.31.b - Use the properties of exponents to interpret expressions for exponential functions.
Example: For example, identify percent rate of change in functions such as y = (1.02) to the t
power, y = (0.97) to the t power, y = (1.01) to the 12t power, y = (1.2) to the t/10 power, and classify
them as representing exponential growth or decay.
Algebra
4/26/2012
Identify an equivalent form of an exponential expression by using
the properties of exponents [Algebra 2]
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Algebra I
WP: Determine an exponential function that represents a situation
such as exponential growth or decay [Algebra 2]
Solve an exponential equation using the properties of exponents
[Algebra 2]
WP: Solve an exponential growth or decay problem [Algebra 2]
Core Skill
AL F-IF.32 - Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph
of one quadratic function and an algebraic expression for another, say which has the larger
maximum.
Algebra
Compare characteristics of two linear, two quadratic, or two
exponential functions represented in different ways [Algebra 2]
AL F-BF - Building Functions
Build a function that models a relationship between two quantities[(For standards 33 and 34, linear,
exponential, and quadratic.)]
AL F-BF.33 - Write a function that describes a relationship between two quantities.
AL F-BF.33.a - Determine an explicit expression, a recursive process, or steps for calculation from
a context.
Algebra
Determine the recursive formula for an arithmetic sequence
[Algebra 2]
Determine the recursive formula for a geometric sequence
[Algebra 2]
AL F-BF.33.b - Combine standard function types using arithmetic operations. Example: For
example, build a function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
Algebra
WP: Evaluate an exponential growth or an exponential decay
function [Algebra 1]
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
WP: Determine an exponential function that represents a situation
such as exponential growth or decay [Algebra 2]
AL F-BF.34 - Write arithmetic and geometric sequences both recursively and with an explicit
formula, use them to model situations, and translate between the two forms.
Algebra
WP: Solve a problem that can be represented by an arithmetic
sequence [Algebra 2]
Find a specific term of a geometric sequence [Algebra 2]
Determine the common ratio for a geometric sequence [Algebra
2]
Determine the recursive formula for a geometric sequence
[Algebra 2]
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Algebra I
Determine the explicit formula for a geometric sequence [Algebra
2]
WP: Solve a problem that can be represented by a geometric
sequence [Algebra 2]
Core Skill
Build new functions from existing functions[(Linear, exponential, quadratic, and absolute value; for
standard 36a, linear only.)]
AL F-BF.35 - Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k)
for specific values of k (both positive and negative); find the value of k given the graphs. Experiment
with cases and illustrate an explanation of the effects on the graph using technology.[Include
recognizing even and odd functions from their graphs and algebraic expressions for them.]
Algebra
Determine the resulting change in a linear function, f(x), when
replacing x with x + b, where b is an integer [Algebra 2]
AL F-BF.36 - Find inverse functions.
AL F-BF.36.a - Solve an equation of the form f(x) = c for a simple function f that has an inverse
and write an expression for the inverse. Example: For example, f(x) =2 x³ or f(x) = (x+1)/(x-1) for x
is not equal to 1.
Algebra
Determine values of the inverse of a function using a table or a
graph [Algebra 2]
Determine if the inverse of a function is a function [Algebra 2]
Determine the equation of the inverse of a linear, rational root, or
polynomial function [Algebra 2]
Core Skill
AL F-LE - Linear, Quadratic, and Exponential Models
Construct and compare linear, quadratic, and exponential models and solve problems
AL F-LE.37 - Distinguish between situations that can be modeled with linear functions and with
exponential functions.
AL F-LE.37.a - Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
Algebra
Determine if the graph of a relation is modeled by a linear or
quadratic function or represents exponential growth or decay
[Algebra 2]
AL F-LE.37.b - Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.
Algebra
Determine if a table or an equation represents a direct variation,
an inverse variation, or neither [Algebra 1]
WP: Determine an equation representing a direct variation or an
inverse variation [Algebra 1]
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Algebra I
AL F-LE.37.c - Recognize situations in which a quantity grows or decays by a constant percent rate
per unit interval relative to another.
Algebra
Identify the asymptotes, growth or decay, or half-life of an
exponential function [Algebra 2]
WP: Determine an exponential function that represents a situation
such as exponential growth or decay [Algebra 2]
Determine if the graph of a relation is modeled by a linear or
quadratic function or represents exponential growth or decay
[Algebra 2]
AL F-LE.38 - Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs (include reading
these from a table).
Algebra
Determine the graph of a line for a given table of values [Grade 8]
WP: Determine a linear graph that can represent a situation
[Grade 8]
Determine the graph of a line using given information [Algebra 1] Core Skill
Determine the graph of an exponential function [Algebra 1]
Determine an exponential function that represents a graph
[Algebra 2]
WP: Determine an exponential function that represents a situation
such as exponential growth or decay [Algebra 2]
Determine if the graph of a relation is modeled by a linear or
quadratic function or represents exponential growth or decay
[Algebra 2]
WP: Solve a problem that can be represented by an arithmetic
sequence [Algebra 2]
WP: Solve a problem that can be represented by a geometric
sequence [Algebra 2]
Core Skill
AL F-LE.39 - Observe using graphs and tables that a quantity increasing exponentially eventually
exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Interpret expressions for functions in terms of the situation they model[(Linear and exponential of
form f(x) = b to the x power + k.)]
AL F-LE.40 - Interpret the parameters in a linear or exponential function in terms of a context.
Algebra
WP: Interpret an interest rate, rate of change, initial amount,
Core Skill
frequency of compounding and other parameters of an
exponential function [Algebra 2]
AL S - Statistics and Probability
AL S-ID - Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
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Algebra I
AL S-ID.41 - Represent data with plots on the real number line (dot plots, histograms, and box
plots).
Data Analysis, Statistics, and
Probability
Use a histogram to represent data [Grade 7]
Core Skill
Use a box-and-whisker plot to organize data [Grade 8]
AL S-ID.42 - Use statistics appropriate to the shape of the data distribution to compare center
(median, mean) and spread (interquartile range, standard deviation) of two or more different data
sets.
Data Analysis, Statistics, and
Probability
Determine the quartiles of a data set [Grade 8]
Analyze the effect that changing elements in a data set has on the
mean, the median, or the range [Grade 8]
Determine the median of the data in a frequency table or a bar
graph [Grade 8]
Determine the mean of the data in a frequency table or a bar graph
[Grade 8]
AL S-ID.43 - Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme data points (outliers).
Data Analysis, Statistics, and
Probability
Analyze the effect that changing elements in a data set has on the
mean, the median, or the range [Grade 8]
Summarize, represent, and interpret data on two categorical and quantitative variables[(Linear focus,
discuss general principle.)]
AL S-ID.44 - Summarize categorical data for two categories in two-way frequency tables. Interpret
relative frequencies in the context of the data (including joint, marginal, and conditional relative
frequencies). Recognize possible associations and trends in the data.
Data Analysis, Statistics, and
Probability
Determine the median of the data in a frequency table or a bar
graph [Grade 8]
Determine the mean of the data in a frequency table or a bar graph
[Grade 8]
AL S-ID.45 - Represent data on two quantitative variables on a scatter plot, and describe how the
variables are related.
AL S-ID.45.a - Fit a function to the data; use functions fitted to data to solve problems in the
context of the data.[Use given functions or choose a function suggested by the context. Emphasize
linear, quadratic, and exponential models.]
Data Analysis, Statistics, and
Probability
Use a scatter plot to organize data [Grade 8]
Determine if a scatter plot shows a positive relationship, a
negative relationship, or no relationship between the variables
[Grade 8]
Answer a question using information from a scatter plot [Grade 8]
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Algebra I
AL S-ID.45.b - Informally assess the fit of a function by plotting and analyzing residuals.
AL S-ID.45.c - Fit a linear function for a scatter plot that suggests a linear association.
Data Analysis, Statistics, and
Use a scatter plot to organize data [Grade 8]
Probability
Determine if a scatter plot shows a positive relationship, a
negative relationship, or no relationship between the variables
[Grade 8]
Interpret linear models
AL S-ID.46 - Interpret the slope (rate of change) and the intercept (constant term) of a linear model
in the context of the data.
Algebra
WP: Interpret the meaning of the slope of a graphed line [Grade
8]
Core Skill
WP: Interpret the meaning of the y-intercept of a graphed line
[Grade 8]
Core Skill
AL S-ID.47 - Compute (using technology) and interpret the correlation coefficient of a linear fit.
AL S-ID.48 - Distinguish between correlation and causation.
AL S-CP - Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data[(Link to data
from simulations or experiments.)]
AL S-CP.49 - Describe events as subsets of a sample space (the set of outcomes) using characteristics
(or categories) of the outcomes, or as unions, intersections, or complements of other events ("or,"
"and," "not").
Data Analysis, Statistics, and
Probability
Answer a question using information from a Venn diagram with
three circles [Grade 8]
Use a Venn diagram to organize summarized data [Grade 8]
AL S-CP.50 - Understand that two events A and B are independent if the probability of A and B
occurring together is the product of their probabilities, and use this characterization to determine if
they are independent.
Data Analysis, Statistics, and
Probability
Determine the probability of three or more independent events
[Grade 8]
Determine the probability of an event consisting of mutually
exclusive events [Grade 8]
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Discrete Mathematics
Discrete Mathematics
AL N - Number and Quantity
AL N.1 - Analyze topics from elementary number theory, including perfect numbers and prime
numbers, to determine properties of integers.
AL N.2 - Determine characteristics of sequences, including the Fibonacci sequence, the triangular
numbers, and pentagonal numbers. Example: Write a sequence of the first 10 triangular numbers and
hypothesize a formula to find the nth triangular number.
Algebra
Find a specific term of an arithmetic sequence [Algebra 2]
Determine the recursive formula for an arithmetic sequence
[Algebra 2]
Determine the explicit formula for an arithmetic sequence
[Algebra 2]
WP: Solve a problem that can be represented by an arithmetic
sequence [Algebra 2]
Find a specific term of a geometric sequence [Algebra 2]
Determine the common ratio for a geometric sequence [Algebra
2]
Determine the recursive formula for a geometric sequence
[Algebra 2]
Determine the explicit formula for a geometric sequence [Algebra
2]
Identify a given sequence as arithmetic, geometric, or neither
[Algebra 2]
WP: Solve a problem that can be represented by a geometric
sequence [Algebra 2]
Core Skill
AL N.3 - Use the recursive process and difference equations to create fractals, population growth models,
sequences, series, and compound interest models.
Algebra
Determine the recursive formula for an arithmetic sequence
[Algebra 2]
Determine the recursive formula for a geometric sequence
[Algebra 2]
AL N.4 - Convert between base ten and other bases.
AL A - Algebra
AL A.5 - Determine results of operations upon 3 × 3 and larger matrices, including matrix addition and
multiplication of a matrix by a matrix, vector, or scalar.
Algebra
Multiply a matrix by a scalar [Algebra 2]
Add or subtract matrices [Algebra 2]
Combine matrix operations [Algebra 2]
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Discrete Mathematics
Multiply matrices [Algebra 2]
AL A.6 - Analyze determinants and inverses of 2 × 2, 3 × 3, and larger matrices to determine the nature
of the solution set of the corresponding system of equations, including solving systems of equations in
three variables by echelon row reduction and matrix inverse.
Algebra
Determine the determinant of a matrix [Algebra 2]
Represent a system of linear equations as a single matrix equation
[Algebra 2]
Determine the inverse of a matrix [Algebra 2]
Core Skill
AL A.7 - Solve problems through investigation and application of existence and nonexistence of Euler
paths, Euler circuits, Hamilton paths, and Hamilton circuits. Example: Show why a 5 × 5 grid has no
Hamilton circuit.
AL A.7.a - Develop optimal solutions of application-based problems using existing and student-created
algorithms.
AL A.8 - Apply algorithms, including Kruskal's and Prim's, relating to minimum weight spanning trees,
networks, flows, and Steiner trees.
AL A.8.a - Use shortest path techniques to find optimal shipping routes.
AL A.9 - Determine a minimum project time using algorithms to schedule tasks in order, including
critical path analysis, the list-processing algorithm, and student-created algorithms.
AL G - Geometry
AL G.10 - Use vertex-coloring techniques and matching techniques to solve application-based problems.
Example: Use graph-coloring techniques to color a map of the western states of the United States so no
adjacent states are the same color, including determining the minimum number of colors needed and
why no fewer colors may be used.
AL G.11 - Solve application-based logic problems using Venn diagrams, truth tables, and matrices.
Data Analysis, Statistics, and
Answer a question using information from a Venn diagram with
Probability
three circles [Grade 8]
Use a Venn diagram to organize summarized data [Grade 8]
AL S - Statistics and Probability
AL S.12 - Use combinatorial reasoning and counting techniques to solve application-based problems.
Example: Determine the probability of a safe opening on the first attempt given the combination uses the
digits 2, 4, 6, and 8 with the order unknown. Answer: The probability of the safe opening on the first
attempt is 1/24.
Algebra
Evaluate an expression involving factorial, permutation, or
combination notation [Algebra 2]
Data Analysis, Statistics, and
Probability
Determine the number of combinations possible in a given
situation [Grade 8]
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Discrete Mathematics
AL S.13 - Analyze election data to compare election methods and voting apportionment, including
determining strength within specific groups.
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Geometry
Geometry
AL G - Geometry
AL G-CO - Congruence
Experiment with transformations in the plane
AL G-CO.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
AL G-CO.2 - Represent transformations in the plane using, e.g., transparencies and geometry
software; describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that do not
(e.g., translation versus horizontal stretch).
Geometry and Measurement
Relate the coordinates of a preimage or an image to a translation
described using mapping notation [Geometry]
Determine the coordinates of a preimage or an image given a
reflection across a horizontal line, a vertical line, the line y = x, or
the line y = -x [Geometry]
Relate the coordinates of a preimage or an image to a dilation
centered at the origin [Geometry]
Determine the angle of rotational symmetry of a figure
[Geometry]
Determine the coordinates of the image of a figure after two
transformations of the same type [Geometry]
Determine the coordinates of the image of a figure after two
transformations of different types [Geometry]
AL G-CO.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations
and reflections that carry it onto itself.
Geometry and Measurement
Determine the coordinates of a preimage or an image given a
reflection across a horizontal line, a vertical line, the line y = x, or
the line y = -x [Geometry]
Determine the angle of rotational symmetry of a figure
[Geometry]
AL G-CO.4 - Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
Geometry and Measurement
Determine the angle of rotational symmetry of a figure
[Geometry]
AL G-CO.5 - Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence
of transformations that will carry a given figure onto another.
Geometry and Measurement
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Relate the coordinates of a preimage or an image to a translation
described using mapping notation [Geometry]
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Geometry
Determine the coordinates of a preimage or an image given a
reflection across a horizontal line, a vertical line, the line y = x, or
the line y = -x [Geometry]
Determine the angle of rotational symmetry of a figure
[Geometry]
Determine the coordinates of the image of a figure after two
transformations of the same type [Geometry]
Determine the coordinates of the image of a figure after two
transformations of different types [Geometry]
Understand congruence in terms of rigid motions[(Build on rigid motions as a familiar starting point
for development of concept of geometric proof.)]
AL G-CO.6 - Use geometric descriptions of rigid motions to transform figures and to predict the
effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in
terms of rigid motions to decide if they are congruent.
Geometry and Measurement
Relate the coordinates of a preimage or an image to a translation
described using mapping notation [Geometry]
Determine the coordinates of a preimage or an image given a
reflection across a horizontal line, a vertical line, the line y = x, or
the line y = -x [Geometry]
Determine the coordinates of the image of a figure after two
transformations of the same type [Geometry]
Determine the coordinates of the image of a figure after two
transformations of different types [Geometry]
AL G-CO.7 - Use the definition of congruence in terms of rigid motions to show that two triangles
are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are
congruent.
Geometry and Measurement
Identify a triangle congruence postulate that justifies a congruence Core Skill
statement [Geometry]
Identify congruent triangles using triangle congruence postulates
or theorems [Geometry]
Overall Product Skills
Core Skill
Use strategies to derive solutions for problems
AL G-CO.8 - Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
Geometry and Measurement
Identify a triangle congruence postulate that justifies a congruence Core Skill
statement [Geometry]
Identify congruent triangles using triangle congruence postulates
or theorems [Geometry]
Core Skill
Prove geometric theorems[(Focus on validity of underlying reasoning while using variety of ways of
writing proofs.)]
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Geometry
AL G-CO.9 - Prove theorems about lines and angles.[Theorems include: vertical angles are
congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly
those equidistant from the segment's endpoints.]
Geometry and Measurement
Identify angle relationships formed by parallel lines cut by a
transversal [Grade 8]
Determine an angle measure formed by parallel lines cut by a
transversal [Grade 8]
Identify angle relationships formed by multiple lines and
transversals [Geometry]
Determine the measure of an angle formed by parallel lines and
one or more transversals [Geometry]
Core Skill
Identify parallel lines using angle relationships [Geometry]
Determine the measure of an angle in a figure involving parallel
and/or perpendicular lines [Geometry]
Solve a problem involving a point on the bisector of an angle
[Geometry]
AL G-CO.10 - Prove theorems about triangles.[Theorems include: measures of interior angles of a
triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints
of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle
meet at a point.]
Geometry and Measurement
Identify a triangle congruence postulate that justifies a congruence Core Skill
statement [Geometry]
Identify congruent triangles using triangle congruence postulates
or theorems [Geometry]
Core Skill
AL G-CO.11 - Prove theorems about parallelograms.[Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.]
Geometry and Measurement
Determine a length or an angle measure using general properties
of parallelograms [Geometry]
Make geometric constructions[(Formalize and explain processes.)]
AL G-CO.12 - Make formal geometric constructions with a variety of tools and methods (compass
and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
[Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line
parallel to a given line through a point not on the line.]
Geometry and Measurement
Determine a length using a perpendicular bisector of a chord
[Geometry]
AL G-CO.13 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a
circle.
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Geometry
AL G-SRT - Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations
AL G-SRT.14 - Verify experimentally the properties of dilations given by a center and a scale factor:
Geometry and Measurement
Determine the scale factor relating two similar polygons
[Geometry]
Relate the coordinates of a preimage or an image to a dilation
centered at the origin [Geometry]
AL G-SRT.14.a - A dilation takes a line not passing through the center of the dilation to a parallel
line, and leaves a line passing through the center unchanged.
AL G-SRT.14.b - The dilation of a line segment is longer or shorter in the ratio given by the scale
factor.
AL G-SRT.15 - Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the meaning of
similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of
all corresponding pairs of sides.
Geometry and Measurement
Determine the scale factor relating two similar polygons
[Geometry]
Identify similar triangles using triangle similarity postulates or
theorems [Geometry]
Core Skill
AL G-SRT.16 - Use the properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
Prove theorems involving similarity
AL G-SRT.17 - Prove theorems about triangles.[Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved
using triangle similarity.]
Geometry and Measurement
Determine the length of the hypotenuse of a right triangle using
the Pythagorean theorem [Grade 8]
Core Skill
Determine the length of a leg of a right triangle using the
Pythagorean theorem [Grade 8]
Core Skill
Solve for the length of a side of a triangle using the Pythagorean
theorem [Geometry]
Identify a triangle similarity postulate that justifies a similarity
statement [Geometry]
Core Skill
Identify similar triangles using triangle similarity postulates or
theorems [Geometry]
Core Skill
AL G-SRT.18 - Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
Geometry and Measurement
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Determine the length of a side or the measure of an angle in
similar triangles [Geometry]
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Geometry
Determine a length given the perimeters of similar triangles or the
lengths of corresponding interior line segments [Geometry]
Determine a length in a triangle using a midsegment [Geometry]
Determine a length using similar triangles formed by the altitude
to the hypotenuse of a right triangle [Geometry]
WP: Determine a length using similarity [Geometry]
Define trigonometric ratios and solve problems involving right triangles
Geometry and Measurement
Determine a length using the properties of a 45-45-90 degree
triangle or a 30-60-90 degree triangle [Geometry]
Core Skill
Core Skill
Solve a problem using multiple non-trigonometric right-triangle
relationships [Geometry]
WP: Determine a length using the properties of a 45-45-90 degree
triangle or a 30-60-90 degree triangle [Geometry]
AL G-SRT.19 - Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Geometry and Measurement
Determine a sine, cosine, or tangent ratio in a right triangle
[Geometry]
AL G-SRT.20 - Explain and use the relationship between the sine and cosine of complementary
angles.
Algebra
Determine the sine, cosine or tangent of an angle given in degrees
or radians [Algebra 2]
Determine the exact value of a sine, cosine or tangent function
given an angle in degrees or radians [Algebra 2]
WP: Solve a problem involving a sine, cosine, or tangent function Core Skill
[Algebra 2]
Geometry and Measurement
Determine a sine, cosine, or tangent ratio in a right triangle
[Geometry]
WP: Determine the measure of an angle in a right triangle using a
sine, cosine, or tangent ratio [Geometry]
AL G-SRT.21 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in
applied problems.
Algebra
WP: Solve a problem involving a sine, cosine, or tangent function Core Skill
[Algebra 2]
Geometry and Measurement
WP: Use the Pythagorean theorem to find a length or a distance
[Grade 8]
Core Skill
Determine a length in a complex figure using the Pythagorean
theorem [Geometry]
Core Skill
WP: Solve a problem involving a complex figure using the
Pythagorean theorem [Geometry]
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Geometry
Determine a length using a sine, cosine, or tangent ratio in a right
triangle [Geometry]
WP: Determine a length in a right triangle using a sine, cosine, or
tangent ratio [Geometry]
WP: Determine the measure of an angle in a right triangle using a
sine, cosine, or tangent ratio [Geometry]
Apply trigonometry to general triangles
AL G-SRT.22 - Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side.
Geometry and Measurement
Approximate the area of a regular polygon using trigonometry
[Geometry]
Determine the area of a triangle using the formula A = 1/2absin(c)
[Algebra 2]
AL G-SRT.23 - Prove the Laws of Sines and Cosines and use them to solve problems.
Algebra
Solve a problem involving the law of sines [Algebra 2]
Geometry and Measurement
Solve a problem involving the law of cosines [Algebra 2]
AL G-SRT.24 - Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Algebra
Solve a problem involving the law of sines [Algebra 2]
Geometry and Measurement
Solve a problem involving the law of cosines [Algebra 2]
WP: Solve a problem involving the law of sines or the law of
cosines [Algebra 2]
AL G-C - Circles
Understand and apply theorems about circles
Geometry and Measurement
Solve a problem involving intersecting chords, tangents, and/or
secants of a circle [Geometry]
Determine the radius, center, or diameter of a circle given an
equation [Geometry]
Determine the area of a segment of a circle [Geometry]
WP: Determine a length or an area involving a sector of a circle
[Geometry]
Core Skill
Determine the measure of an arc or an angle given the area of a
sector of a circle [Geometry]
Core Skill
AL G-C.25 - Prove that all circles are similar.
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Geometry
AL G-C.26 - Identify and describe relationships among inscribed angles, radii, and chords.[Include
the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter
are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the
circle.]
Geometry and Measurement
Determine the measure of an arc or a central angle using the
relationship between the arc and the central angle [Geometry]
Core Skill
AL G-C.27 - Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
AL G-C.28 - Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles[(Radian introduced only as unit of measure.)]
Geometry and Measurement
Determine the measure of an arc or a central angle using the
Core Skill
relationship between the arc and the central angle [Geometry]
Determine the area of a sector of a circle [Geometry]
Core Skill
AL G-C.29 - Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
Geometry and Measurement
Determine the measure of an arc or a central angle using the
relationship between the arc and the central angle [Geometry]
Core Skill
Determine the area of a sector of a circle [Geometry]
Core Skill
AL G-GPE - Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
AL G-GPE.30 - Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an equation.
Geometry and Measurement
Determine an equation of a circle [Geometry]
Determine the radius, center, or diameter of a circle given an
equation [Geometry]
Use coordinates to prove simple geometric theorems algebraically[(Include distance formula; relate to
Pythagorean Theorem.)]
AL G-GPE.31 - Use coordinates to prove simple geometric theorems algebraically. Example: For
example, prove or disprove that a figure defined by four given points in the coordinate plane is a
rectangle; prove or disprove that the point (1, the square root of 3) lies on the circle centered at the
origin and containing the point (0, 2).
Geometry and Measurement
Determine the midpoint of a line segment given the coordinates of Core Skill
the endpoints [Geometry]
Determine the area of a right triangle or a rectangle given the
coordinates of the vertices of the figure [Geometry]
Core Skill
Determine if lines through points with given coordinates are
parallel or perpendicular [Geometry]
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Geometry
Determine the coordinates of a point through which a line must
pass in order to be parallel or perpendicular to a given line
[Geometry]
AL G-GPE.32 - Prove the slope criteria for parallel and perpendicular lines and use them to solve
geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that
passes through a given point).
Algebra
Determine if two lines are perpendicular or parallel given the
equations of the lines [Algebra 1]
Geometry and Measurement
Determine if lines through points with given coordinates are
parallel or perpendicular [Geometry]
Determine the coordinates of a point through which a line must
pass in order to be parallel or perpendicular to a given line
[Geometry]
AL G-GPE.33 - Find the point on a directed line segment between two given points that partitions
the segment in a given ratio.
Geometry and Measurement
Determine the midpoint of a line segment given the coordinates of Core Skill
the endpoints [Geometry]
Solve a problem involving the midpoint formula [Geometry]
AL G-GPE.34 - Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.
Geometry and Measurement
Solve a problem involving the distance formula [Geometry]
Core Skill
Determine the area of a right triangle or a rectangle given the
coordinates of the vertices of the figure [Geometry]
Core Skill
Use coordinates to prove simple geometric theorems algebraically.
AL G-GPE.35 - Determine areas and perimeters of regular polygons, including inscribed or
circumscribed polygons, given the coordinates of vertices or other characteristics.
Geometry and Measurement
Solve a problem involving perimeters or areas of two different
shapes [Grade 8]
Determine the ratio of the perimeters or areas of similar shapes
[Grade 8]
Determine the area of a right triangle or a rectangle given the
coordinates of the vertices of the figure [Geometry]
Core Skill
Determine the area of a quadrilateral [Geometry]
Core Skill
Solve a problem involving regular polygons and/or circles that
have the same area or perimeter [Geometry]
AL G-GMD - Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems
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Geometry
AL G-GMD.36 - Give an informal argument for the formulas for the circumference of a circle, area
of a circle, volume of a cylinder, pyramid, and cone.[Use dissection arguments, Cavalieri's principle,
and informal limit arguments.]
AL G-GMD.37 - Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Geometry and Measurement
Solve a problem involving the surface area or the volume of a
pyramid or a cone [Grade 8]
Determine the volume of a sphere or hemisphere [Geometry]
Core Skill
WP: Determine the volume of a sphere or hemisphere [Geometry]
Determine the volume of a complex solid figure [Geometry]
WP: Solve a problem involving the volume of a complex solid
figure [Geometry]
AL G-GMD.38 - Determine the relationship between surface areas of similar figures and volumes of
similar figures.
Geometry and Measurement
Solve a problem involving the surface area of a cone or a pyramid
that has a rectangle or right triangle as a base [Geometry]
Solve a problem involving the volume of a right pyramid or a
right cone [Geometry]
Core Skill
Determine the radius, diameter, or circumference of a sphere
given its surface area [Geometry]
WP: Solve a problem involving the surface area of a sphere
[Geometry]
Determine the volume of a sphere or hemisphere [Geometry]
Core Skill
Solve a problem involving the surface areas of similar solid
figures [Geometry]
Solve a problem involving the volumes of similar solid figures
[Geometry]
Visualize relationships between two-dimensional and three-dimensional objects
AL G-GMD.39 - Identify the shapes of two-dimensional cross-sections of three-dimensional objects,
and identify three-dimensional objects generated by rotations of two-dimensional objects.
Geometry and Measurement
Identify a cross section of a 3-dimensional shape [Grade 8]
AL G-MG - Modeling with Geometry
Apply geometric concepts in modeling situations
AL G-MG.40 - Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
AL G-MG.41 - Apply concepts of density based on area and volume in modeling situations (e.g.,
persons per square mile, BTUs per cubic foot).
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Geometry
AL G-MG.42 - Apply geometric methods to solve design problems (e.g., designing an object or
structure to satisfy physical constraints or minimize cost; working with typographic grid systems
based on ratios).
AL S - Statistics and Probability
AL S-CP - Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data[(Link to data
from simulations or experiments.)]
AL S-CP.43 - Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.
Data Analysis, Statistics, and
Probability
Determine the probability of an event consisting of mutually
exclusive events [Grade 8]
AL S-CP.44 - Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to decide if
events are independent and to approximate conditional probabilities. Example: For example, collect
data from a random sample of students in your school on their favorite subject among math, science,
and English. Estimate the probability that a randomly selected student from your school will favor
science given that the student is in tenth grade. Do the same for other subjects and compare the
results.
Data Analysis, Statistics, and
Probability
Determine if a sample is likely to be representative of the larger
population [Grade 8]
AL S-CP.45 - Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations. Example: For example, compare the chance of having
lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of compound events in a uniform probability
model
AL S-CP.46 - Find the conditional probability of A given B as the fraction of B's outcomes that also
belong to A, and interpret the answer in terms of the model.
Data Analysis, Statistics, and
Probability
Determine the probability of an event consisting of mutually
exclusive events [Grade 8]
AL S-CP.47 - Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the
answer in terms of the model.
AL S-CP.48 - Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P
(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
AL S-CP.49 - Use permutations and combinations to compute probabilities of compound events and
solve problems.
Data Analysis, Statistics, and
Probability
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Determine the number of permutations possible in a given
situation [Grade 8]
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Geometry
Determine the number of combinations possible in a given
situation [Grade 8]
AL S-MD - Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions[(Introductory; apply counting rules.)]
AL S-MD.50 - Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
AL S-MD.51 - Analyze decisions and strategies using probability concepts (e.g., product testing,
medical testing, pulling a hockey goalie at the end of a game).
Overall Product Skills
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Use strategies to derive solutions for problems
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Precalculus
Precalculus
AL N - Number and Quantity
AL N-CN - The Complex Number System
Represent complex numbers and their operations on the complex plane.
AL N-CN.2 - Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms of a given
complex number represent the same number.
Algebra
Identify a complex number represented as a vector on a
coordinate plane [Algebra 2]
AL N-CN.3 - Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation. Example:
For example, (-1 + the square root of 3i)³ = 8 because (-1 + the square root of 3i) has modulus 2 and
argument 120°.
Algebra
Add or subtract complex numbers [Algebra 2]
Multiply complex numbers [Algebra 2]
AL N-CN.4 - Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
AL N-L - Limits
Understand limits of functions.
AL N-L.5 - Determine numerically, algebraically, and graphically the limits of functions at specific
values and at infinity.
AL N-L.5.a - Apply limits in problems involving convergence and divergence.
AL N-VM - Vector and Matrix Quantities
Represent and model with vector quantities.
AL N-VM.6 - Recognize vector quantities as having both magnitude and direction. Represent vector
quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes
(e.g., v, |v|, ||v||, v).
Algebra
Determine the component form of a vector represented on a graph
[Algebra 2]
AL N-VM.7 - Find the components of a vector by subtracting the coordinates of an initial point from
the coordinates of a terminal point.
Algebra
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Determine the component form of a vector represented on a graph
[Algebra 2]
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Precalculus
AL N-VM.8 - Solve problems involving velocity and other quantities that can be represented by
vectors.
Algebra
WP: Add or subtract vectors [Algebra 2]
Perform operations on vectors.
AL N-VM.9 - Add and subtract vectors.
AL N-VM.9.a - Add vectors end-to-end, component-wise, and by the parallelogram rule.
Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
Algebra
Add or subtract vectors on a coordinate plane [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
AL N-VM.9.b - Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum.
Algebra
Determine the magnitude of a vector [Algebra 2]
Determine the direction of a vector [Algebra 2]
Add or subtract vectors on a coordinate plane [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
Determine a unit vector in the same direction as a given vector
[Algebra 2]
WP: Add or subtract vectors [Algebra 2]
AL N-VM.9.c - Understand vector subtraction v - w as v + (-w), where -w is the additive inverse of
w, with the same magnitude as w and pointing in the opposite direction. Represent vector
subtraction graphically by connecting the tips in the appropriate order, and perform vector
subtraction component-wise.
Algebra
Add or subtract vectors on a coordinate plane [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
AL N-VM.10 - Multiply a vector by a scalar.
AL N-VM.10.a - Represent scalar multiplication graphically by scaling vectors and possibly
reversing their direction; perform scalar multiplication component-wise, e.g., as c(v subscript x, v
subscript y) = (cv subscript x, cv subscript y).
Algebra
Multiply a matrix by a scalar [Algebra 2]
Determine the direction of a vector [Algebra 2]
Add or subtract vectors component-wise [Algebra 2]
AL N-VM.10.b - Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the
direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c >
0) or against v (for c < 0).
Algebra
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Multiply a matrix by a scalar [Algebra 2]
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Precalculus
Determine the magnitude of a vector [Algebra 2]
Perform operations on matrices and use matrices in applications.
AL N-VM.11 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
Algebra
Represent a system of linear equations as a single matrix equation
[Algebra 2]
AL N-VM.12 - Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs
in a game are doubled.
Algebra
Multiply a matrix by a scalar [Algebra 2]
AL N-VM.13 - Add, subtract, and multiply matrices of appropriate dimensions.
Algebra
Add or subtract matrices [Algebra 2]
Multiply matrices [Algebra 2]
AL N-VM.14 - Understand that, unlike multiplication of numbers, matrix multiplication for square
matrices is not a commutative operation, but still satisfies the associative and distributive properties.
Algebra
Multiply matrices [Algebra 2]
AL N-VM.15 - Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix
is nonzero if and only if the matrix has a multiplicative inverse.
Algebra
Determine the determinant of a matrix [Algebra 2]
Determine the inverse of a matrix [Algebra 2]
Core Skill
AL N-VM.16 - Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of vectors.
Algebra
Multiply matrices [Algebra 2]
AL N-VM.17 - Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute
value of the determinant in terms of area.
Algebra
Multiply matrices [Algebra 2]
Determine the determinant of a matrix [Algebra 2]
AL A - Algebra
AL A-REI - Reasoning with Equations and Inequalities
Solve systems of equations
AL A-REI.18 - Represent a system of linear equations as a single matrix equation in a vector
variable.
Algebra
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Represent a system of linear equations as a single matrix equation
[Algebra 2]
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Precalculus
AL A-REI.19 - Find the inverse of a matrix if it exists and use it to solve systems of linear equations
(using technology for matrices of dimension 3 × 3 or greater).
Algebra
Determine the inverse of a matrix [Algebra 2]
Core Skill
AL F - Functions
AL F-CS - Conic Sections
Understand the graphs and equations of conic sections.
AL F-CS.20 - Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and
degenerate conics, from second-degree equations. Example: Graph x² - 6x + y2 - 12y + 41 = 0 or y² 4x + 2y + 5 = 0.
Algebra
Determine the graph of a given quadratic function [Algebra 1]
Core Skill
Determine a polynomial of lowest degree that has given roots or a
graph of a given shape [Algebra 2]
Determine the graph of a circle given the equation in standard
form [Algebra 2]
Relate a graph of an ellipse centered at the origin to its equation
[Algebra 2]
Core Skill
Determine the graph of a circle or an ellipse given the equation in Core Skill
general form [Algebra 2]
Determine the graph of a hyperbola given the equation in standard
form [Algebra 2]
Determine the standard form of the equation for a conic section
given its general form equation [Algebra 2]
Determine which conic section a given equation represents
[Algebra 2]
Determine the equation of a parabola using given information
[Algebra 2]
Core Skill
Determine the equation of a hyperbola using given information
[Algebra 2]
AL F-CS.20.a - Formulate equations of conic sections from their determining characteristics.
Example: Write the equation of an ellipse with center (5, -3), a horizontal major axis of length 10,
and a minor axis of length 4. Answer: (x - 5)²/25 + (y + 3)²/4 = 1.
Algebra
Determine the standard form of the equation for a conic section
given its general form equation [Algebra 2]
Determine which conic section a given equation represents
[Algebra 2]
AL F-IF - Interpreting Functions
Analyze functions using different representations[(Logarithmic and trigonometric functions.)]
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Precalculus
AL F-IF.21 - Graph rational functions, identifying zeros and asymptotes when suitable factorizations
are available, and showing end behavior.
Algebra
Determine the graph of a rational function [Algebra 1]
Identify the end behavior, asymptotes, excluded values, or
behavior near excluded values of a rational function [Algebra 2]
Core Skill
Determine the graph of a rational function involving linear or
quadratic denominators [Algebra 2]
Core Skill
AL F-BF - Building Functions
Build a function that models a relationship between two quantities
AL F-BF.22 - Compose functions. Example: For example, if T(y) is the temperature in the
atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time,
then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Algebra
Determine the composition of two functions [Algebra 2]
WP: Apply function operations or compositions of functions to
represent a situation [Algebra 2]
Build new functions from existing functions
AL F-BF.23 - Determine the inverse of a function and a relation.
Algebra
Determine values of the inverse of a function using a table or a
graph [Algebra 2]
Determine if the inverse of a function is a function [Algebra 2]
Determine the equation of the inverse of a linear, rational root, or
polynomial function [Algebra 2]
Core Skill
AL F-BF.24 - Verify by composition that one function is the inverse of another.
Algebra
Determine values of the inverse of a function using a table or a
graph [Algebra 2]
Determine the composition of two functions [Algebra 2]
Determine if the inverse of a function is a function [Algebra 2]
AL F-BF.25 - Read values of an inverse function from a graph or a table, given that the function has
an inverse.
Algebra
Determine values of the inverse of a function using a table or a
graph [Algebra 2]
AL F-BF.26 - Produce an invertible function from a non-invertible function by restricting the
domain.
AL F-BF.27 - Understand the inverse relationship between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents.
Algebra
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Convert between a simple exponential equation and its
corresponding logarithmic equation [Algebra 2]
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Precalculus
Solve an exponential equation using logarithms [Algebra 2]
Core Skill
Solve a logarithmic equation [Algebra 2]
Core Skill
Evaluate a logarithmic expression using the change of base
formula [Algebra 2]
WP: Solve a problem that can be modeled using a logarithmic
equation [Algebra 2]
WP: Solve an exponential growth or decay problem [Algebra 2]
Overall Product Skills
Core Skill
Demonstrate and develop understanding of mathematical concepts
and ideas
AL F-BF.28 - Compare effects of parameter changes on graphs of transcendental functions.
Example: Explain the relationship of the graph y = e to the (x-2) power to the graph y = e to the x
power.
AL F-TF - Trigonometric Functions
Recognize attributes of trigonometric functions and solve problems involving trigonometry.
AL F-TF.29 - Determine the amplitude, period, phase shift, domain, and range of trigonometric
functions and their inverses.
Algebra
Determine the value of an inverse sine, cosine, or tangent
expression [Algebra 2]
Determine the amplitude, frequency, period, or phase shift of a
sine or a cosine function [Algebra 2]
Determine the graph of an inverse sine, cosine, or tangent
function [Algebra 2]
AL F-TF.30 - Use the sum, difference, and half-angle identities to find the exact value of a
trigonometric function.
AL F-TF.31 - Utilize parametric equations by graphing and by converting to rectangular form.
AL F-TF.31.a - Solve application-based problems involving parametric equations.
AL F-TF.31.b - Solve applied problems that include sequences with recurrence relations.
Extend the domain of trigonometric functions using the unit circle
AL F-TF.32 - Use special triangles to determine geometrically the values of sine, cosine, tangent for
pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x,
pi+x, and 2pi-x in terms of their values for x, where x is any real number.
Algebra
Determine the exact value of a sine, cosine or tangent function
given an angle in degrees or radians [Algebra 2]
Geometry and Measurement
WP: Determine a length using the properties of a 45-45-90 degree
triangle or a 30-60-90 degree triangle [Geometry]
Determine a sine, cosine, or tangent ratio in a right triangle
[Geometry]
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Precalculus
AL F-TF.33 - Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
Algebra
Determine an angle coterminal with a given angle [Algebra 2]
Model periodic phenomena with trigonometric functions
AL F-TF.34 - Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
Algebra
Determine the graph of an inverse sine, cosine, or tangent
function [Algebra 2]
Overall Product Skills
Demonstrate and develop understanding of mathematical concepts
and ideas
AL F-TF.35 - Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context.
Algebra
WP: Determine a trigonometric function that represents a
situation [Algebra 2]
WP: Solve a problem involving a sine, cosine, or tangent function Core Skill
[Algebra 2]
Prove and apply trigonometric identities
AL F-TF.36 - Prove the addition and subtraction formulas for sine, cosine, and tangent and use them
to solve problems.
AL G - Geometry
AL G-GPE - Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
AL G-GPE.37 - Derive the equation of a parabola given a focus and directrix.
Algebra
Determine the equation of a parabola using given information
[Algebra 2]
Core Skill
AL G-GPE.38 - Derive the equations of ellipses and hyperbolas given the foci, using the fact that the
sum or difference of distances from the foci is constant.
Algebra
Determine the equation of an ellipse using given information
[Algebra 2]
Determine the equation of a hyperbola using given information
[Algebra 2]
Explain volume formulas and use them to solve problems
AL G-GPE.39 - Give an informal argument using Cavalieri's principle for the formulas for the
volume of a sphere and other solid figures.
Geometry and Measurement
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Determine the volume of an oblique prism or an oblique cylinder
[Geometry]
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Precalculus
Determine the volume of a right pyramid or a right cone
[Geometry]
Solve a problem involving the volume of a right pyramid or a
right cone [Geometry]
Core Skill
Determine the volume of an oblique pyramid or an oblique cone
[Geometry]
Determine the volume of a sphere or hemisphere [Geometry]
Core Skill
WP: Determine the volume of a sphere or hemisphere [Geometry]
WP: Solve a problem involving the volume of a complex solid
figure [Geometry]
Solve a problem involving the volumes of similar solid figures
[Geometry]
WP: Solve a problem involving two solid figures that have the
same surface area or volume [Geometry]
AL S - Statistics and Probability
AL S-MD - Using Probability to Make Decisions
Calculate expected values and use them to solve problems
AL S-MD.40 - Define a random variable for a quantity of interest by assigning a numerical value to
each event in a sample space; graph the corresponding probability distribution using the same
graphical displays as for data distributions.
AL S-MD.41 - Calculate the expected value of a random variable; interpret it as the mean of the
probability distribution.
AL S-MD.42 - Develop a probability distribution for a random variable defined for a sample space in
which theoretical probabilities can be calculated; find the expected value. Example: For example,
find the theoretical probability distribution for the number of correct answers obtained by guessing
on all five questions of a multiple-choice test where each question has four choices, and find the
expected grade under various grading schemes.
AL S-MD.43 - Develop a probability distribution for a random variable defined for a sample space in
which probabilities are assigned empirically; find the expected value. Example: For example, find a
current data distribution on the number of TV sets per household in the United States, and calculate
the expected number of sets per household. How many TV sets would you expect to find in 100
randomly selected households?
Use probability to evaluate outcomes of decisions
AL S-MD.44 - Weigh the possible outcomes of a decision by assigning probabilities to payoff values
and finding expected values.
AL S-MD.44.a - Find the expected payoff for a game of chance. Example: For example, find the
expected winnings from a state lottery ticket or a game at a fast-food restaurant.
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Precalculus
AL S-MD.44.b - Evaluate and compare strategies on the basis of expected values. Example: For
example, compare a high-deductible versus a low-deductible automobile insurance policy using
various, but reasonable, chances of having a minor or a major accident.
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