The Number System
Unit Guide
Math 7/8
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions
to add, subtract, multiply, and divide rational numbers (7.NS.1-7.NS.3).
 Know that there are numbers that are not rational, and approximate
them by rational numbers (8.NS.1).
Edited 6/26/14
Renton School District
Domain: The Number System (7.NS.1-3 and 8.NS.1)
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
 Know that there are numbers that are not rational, and approximate them by rational numbers
Standard 7.NS.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a
horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0.
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 realworld contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
Standard 7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of
operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers.
Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a
rational number. If p and q are integers then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world
contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually
repeats.
2
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Standard 7.NS.3
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).
Standard 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.
3
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Relevant Math Practices and Student Actions
MP1 Make sense of problems and persevere in solving them.
 Perseveres when solving problems.
 Understands what the problem asks and the relationship among the
problem’s parts.
 Uses multiple strategies and representations.
 Checks process, answers and ask “Does this make sense?”
 Explains why a solution is reasonable.
MP 2 Reason abstractly and quantitatively.
 Interprets problems in context.
 Uses representations to make meaning of problems.
 Translates a problem from situation to equation.
 Creates a situation for a given equation.
 Understands the meaning of quantities and units.
 Flexibly uses properties of operations and place value.
MP 3 Construct viable arguments and critique the reasoning of
others.
 Makes conjectures and evaluates their accuracy.
 Justifies conclusions with mathematical evidence and responds to
arguments of others.
 Asks clarifying and probing questions to improve argument.
MP 5 Use appropriate tools strategically.
 Selects tools strategically for visualizing, exploring, comparing,
predicting, and solving problems.
MP 6 Attend to precision.
 Communicates mathematical thinking accurately both orally and in
writing.
 Understands the meaning of mathematical symbols and vocabulary
and uses them appropriately.
 Labels consistently and accurately on graphs and diagrams.
 Calculates accurately and efficiently.
MP 7 Look for and make use of structure.
 Looks for, identifies, develops and generalizes patterns and
relationships.
 Makes connections to prior mathematical knowledge to solve new
problems.
MP 8 Look for and express regularity in repeated reasoning.
 Notices repeated calculations and looks for general methods and
shortcuts to solve a problem.
 Identifies patterns to develop algorithm, formula, or calculation.
MP 4 Model with mathematics.
 Identifies quantities necessary to solve a problem and uses
representations to map their relationships.
 Checks to see if an answer makes sense within the context of a
situation and improves model when necessary.
4
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
SBAC Required Evidence (Claim 1)
7.NS.1-3
 Student interprets rational number values on a number line, including modeling addition and subtraction expressions (SBAC Claim 1 Grade 7,
Target B required evidence #1).
 Student applies properties of operations as strategies to add and subtract rational numbers (SBAC Claim 1 Grade 7, Target B required evidence #2).
 Student applies properties of operations as strategies to multiply and divide rational numbers (SBAC Claim 1 Grade 7, Target B required evidence
#3).
 Student converts from a fractional form of rational numbers to a decimal form of rational numbers (SBAC Claim 1 Grade 7, Target B required
evidence #4).
 Student solves real-world and mathematical problems involving the four operations with rational numbers (SBAC Claim 1 Grade 7, Target B
required evidence #5).
A calculator is not an allowable tool for this cluster of standards. For more detail on the assessment of the standards, read the Claim 1 SBAC
Item Specifications-Target B. Students will also be assessed through Claim 2 (Problem Solving), Claim 3 (Communicating Reasoning) and Claim 4
(Modeling and Data Analysis) assessment items on the Smarter Balanced summative assessment.
8.NS.1
 The student classifies real numbers as rational or irrational (SBAC Claim 1 Grade 8, Target A required evidence #1).
 The student converts a repeating decimal into a fraction (SBAC Claim 1 Grade 8, Target A required evidence #2).
A calculator is not an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of
standards, read the Grade 8 Claim 1 Target A Item Specifications.
5
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Vocabulary
Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical
language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter
Balanced assessment items.
Absolute value
Addend
Additive inverse
Associative Property
Commutative Property
Comparison
Complex Fraction
Difference
Distance
Distributive Property
Factor
Finite decimal
Infinite decimal
Integer
Integer Chip
Irrational Number
Negative number
Number line
Opposite p + (-p) = 0
Order of operations
Place Value
Positive number
Product
Property of identity
Quotient
Rational numbers
Repeating decimal
Signed number
Sum
Terminating decimal
Variables
6
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Domain: The Number System (7.NS.1-3 and 8.NS.1)
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
 Know that there are numbers that are not rational, and approximate them by rational numbers
Standard 7.NS.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a
horizontal or vertical number line diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents
are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or
negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational
numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
See Grade 7 Flip Book pages 13-15 for explanations and examples of this standard.
Learning Objectives

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
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


Understand p + q as a number located | | units from p on a number line in either direction depending on sign of q (Grade 7 Claim 1 Target B
SBAC ALD Level 3*).
Understand subtraction is the same as adding the additive inverse (Grade 7 Claim 1 Target B SBAC ALD Level 3*).
Solve addition and subtraction mathematical problems using rational numbers (Grade 7 Claim 1 Target B SBAC ALD Level 3*).
Describe situations in which opposite quantities combine to make zero.
Determine the absolute value of a given rational number and graph on number line.
Model addition and subtraction expressions on a number line.
Apply properties of operations to add and subtract rational numbers.
7
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
SBAC Required Evidence (Claim 1)
Connections to Prior Learning
 Student interprets rational number values on a number line, including
modeling addition and subtraction expressions (required evidence
#1).
 Student applies properties of operations as strategies to add and
subtract rational numbers (required evidence #2).
Standard 5.NF.1
Add and subtract fractions with unlike denominators (including mixed
numbers) by replacing given fractions with equivalent fractions in such a
way as to produce an equivalent sum or difference of fractions with like
denominators.
Calculators are not an allowable tool for this cluster of standards.
See Claim 1 SBAC Item Specifications-Target B for more detail.
Standard 6.NS.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
Students will also be assessed through Claim 2 (Problem Solving),
Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data
Analysis) assessment items on the Smarter Balanced summative
assessment.
Standard 6.NS.5
Understand that positive and negative numbers are used together to
describe quantities having opposite directions or values; use positive and
negative numbers to represent quantities in real-world contexts, explaining
the meaning of 0 in each situation.
Standard 6.NS.6a-c
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.
Connections to Curriculum Resources
Accentuate the Negative Investigations 1 and 2 partially cover this
standard. Engage NY Grade 7 module 2 lessons 4, 7, 8 and 9 will help
cover the remaining content missing from the CMP2 unit. Some of the
additional resources may also help support missing content.
Standard 6.NS.7a-d
Understand ordering and absolute value of rational numbers.
Additional Resources/Technology Resources
Engage NY Grade 7 Module 2 Lessons 4,7,8 and 9
Grade 6-8 Number Systems Progression of Learning document
Sign your name integer addition activity (NC DPI)
Operations with rational numbers tasks (GA DOE)
Add and Subtract integers on number line video (HCPSS)
Illustrative Mathematics Grade 7 tasks
Khan Academy Common Core Student Practice for 7.NS.1
8
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Domain: The Number System (7.NS.1-3 and 8.NS.1)
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
 Know that there are numbers that are not rational, and approximate them by rational numbers
Standard 7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of
operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers.
Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a
rational number. If p and q are integers then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world
contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually
repeats.
See Grade 7 Flip Book pages 16-18 for explanations and examples of this standard.
Learning Objectives
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

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Understand (negative)(negative) equals a positive (Grade 7 Claim 1 Target B SBAC ALD Level 3*).
Understand and use the rules for multiplying and dividing signed numbers (Grade 7 Claim 1 Target B SBAC ALD Level 3*).
Solve mathematical problems using multiplication and division of signed and unsigned rational numbers (Grade 7 Claim 1 Target B SBAC ALD
Level 3*).
Convert from a fraction to a decimal with denominators that are a factor of a power of 10 to decimals (Grade 7 Claim 1 Target B SBAC ALD
Level 3*).
Understand you can divide rational numbers provided the divisor is not 0.
Interpret products and quotient of rational numbers in real-world contexts.
Apply properties of operations to multiply and divide rational numbers.
9
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
SBAC Required Evidence (Claim 1)
 Student applies properties of operations as strategies to multiply and
divide rational numbers (required evidence #3).
 Student converts from a fractional form of rational numbers to a
decimal form of rational numbers (required evidence #4).
Calculators are not an allowable tool for this cluster of standards.
See Claim 1 SBAC Item Specifications-Target B for more detail.
Students will also be assessed through Claim 2 (Problem Solving),
Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data
Analysis) assessment items on the Smarter Balanced summative
assessment.
Connections to Curriculum Resources
Accentuate the Negative Investigations 3 and 4 partially cover this
standard. Engage NY Grade 7 module 2 lessons 13, 14 and 15 will
help cover the remaining content missing from the CMP2 unit. Some
of the additional resources may also help support missing content.
Connections to Prior Learning
Standard 5.NF.4
Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction..
Standard 6.NS.1
Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction
models and equations to represent the problem.
Standard 6.NS.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
Additional Resources/Technology Resources
Engage NY Grade 7 Module 2 Lessons 13-15
Grade 6-8 Number Systems Progression of Learning document
Operations with rational numbers tasks (GA DOE)
Multiplying integers on a number line video (HCPSS)
Multiplying integers using integer counters video (HCPSS)
Dividing integers using a number line video- (student video)
Illustrative Mathematics Grade 7 tasks
Khan Academy Common Core Student Practice for 7.NS.2
10
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Domain: The Number System (7.NS.1-3 and 8.NS.1)
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
 Know that there are numbers that are not rational, and approximate them by rational numbers
Standard 7.NS.3
Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the
rules for manipulating fractions using complex fractions for division).
See Grade 7 Flip Book pages 19-21 for explanations and examples of this standard.
Learning Objectives

Solve real-world problems involving rational numbers with addition, multiplication, subtraction, and divisions using at least integers and proper
fractions (SBAC ALD Level 4*).
SBAC Required Evidence (Claim 1)
 Student solves real-world and mathematical problems involving the
four operations with rational numbers (required evidence #5).
Calculators are not an allowable tool for this cluster of standards.
See Claim 1 SBAC Item Specifications-Target B for more detail.
Students will also be assessed through Claim 2 (Problem Solving),
Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data
Analysis) assessment items on the Smarter Balanced summative
assessment.
Connections to Prior Learning
Standard 5.NF.1
Add and subtract fractions with unlike denominators (including mixed
numbers) by replacing given fractions with equivalent fractions in such a
way as to produce an equivalent sum or difference of fractions with like
denominators.
Standard 5.NF.4
Apply and extend previous understandings of multiplication to multiply a
fraction or whole number by a fraction..
Standard 6.NS.1
Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction
models and equations to represent the problem.
Standard 6.NS.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
11
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Connections to Curriculum Resources
Accentuate the Negative Investigations 1-4 partially cover this standard.
Engage NY Grade 7 module 2 lessons 16 will help cover the remaining
content missing from the CMP2 unit. Some of the additional resources
may also help support missing content.
Additional Resources/Technology Resources
Engage NY Grade 7 Module 2 Lessons 16
Grade 6-8 Number Systems Progression of Learning document
Operations with rational numbers tasks (GA DOE)
Using Positive and Negative Numbers in Context task (MAP)
Rational Numbers module – Grade 7 (engageny.org)
Illustrative Mathematics Grade 7 tasks
Khan Academy Common Core Student Practice for 7.NS.3
12
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Domain: The Number System (7.NS.1-3 and 8.NS.1)
Big Idea (Cluster):
 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
 Know that there are numbers that are not rational, and approximate them by rational numbers
Standard 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.
See Grade 8 Common Core Flip Book pages 4-6 for explanations and examples of this standard.
Learning Objectives



Convert between fractions and repeating decimals (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 3).
Classify a number as rational or irrational based on its decimal expansion.
Know that real numbers that are not rational are irrational.
SBAC Required Evidence (Claim 1)
 The student classifies real numbers as rational or irrational (required
evidence #1).
 The student converts a repeating decimal into a fraction (required
evidence #2).
Connections to Prior Learning
7.NS.2 a-d Apply and extend previous understandings of multiplication
and division and of fractions to multiply and divide rational numbers.
A calculator is not an allowable tool for this cluster of standards on
an assessment. For more information on the assessment of this set of
standards, read the Grade 8 Claim 1 Target A Item Specifications.
Connections to Curricular Materials
CMP2 does not have a unit that supports the 8.NS.1-2 cluster of
standards. The Grade 8 curriculum team has suggested supplemental
resources in the pacing guide below and within the additional resources
section. These resources have been marked with an asterisk (*).
Additional Resources/Technology Resources
Grade 6-8 Number Systems Progression of Learning document
Repeating Decimals additional task (MAP)*
Terminating and Repeating Decimals lesson*
Classifying Numbers PPT (SlideShare.doc)*
Classifying Numbers II PPT (NMS)*
Real Number Race task (NCDPI)*
Roots and Classifying Real Numbers student task (NMS)
Exponents and Equations module (georgiastandards.org)
13
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Explanations and Examples
Students should be able to describe, distinguish and classify rational and irrational numbers. Students can use a graphic
organizer to show the relationship between the subsets of the real number system. Any number that can be expressed
as a fraction is a rational number. Students recognize that the decimal equivalent of a fraction will either terminate or
repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding
builds on work in 7th grade when students used long division to distinguish between repeating and terminating
decimals. Students should also have an understanding of irrational values such as, pi, e, and square roots of nonperfect squares, and how to describe them. Students should understand the decimal expansion of an irrational number
neither terminates nor repeats.
Students will convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning. Students may investigate repeating
patterns for denominators of 9, 99, and 11.
One method of converting a repeating decimal to a fraction algebraically is shown below.
Change ̅ to a fraction. Set x = ̅ Then, multiply both sides
of the equation by 10 so the repeating value becomes the whole
number. The new equation will be 10x = ̅ . Subtract both
equations and solve for x. If the repeating pattern has more than
one repeating decimal value, multiply the original equation by
the power of 10 for the number of repeating decimal places.
14
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
The Engage NY Grade 7 Module 2 is being used to supplement this cluster of standards where CMP2 curricular materials do not provide resources
aligned to Common Core Standards 7.NS.1-3. Please be selective in the resources within the Engage NY modules you use for your students in order
to meet proficiency or higher on this standard. The Engage NY is fairly absent of the Standards for Mathematical Practice which should be equally
weighted with content on a daily basis in classroom instruction. For the time-being, teachers will need to use the Engage NY resources as a
foundation to access these standards. But, teachers are highly encouraged to improve the recommended tasks to be less direct instruction based,
have students engage in rich mathematical talk and provide opportunities for students to work collaboratively
Math 7/8 Number Systems Unit Pacing Guide using Accentuate the Negative and supplemental lessons
Accentuate the Negative and
CCSS-M Aligned Lessons
AN 1.1
AN 1.2
AN 1.3
AN 1.4
Alignment Notes
Have students model scenarios on number lines (horizontal) as well as number sentences.
Begin to introduce opposites in relationship to the number line. Connect the meaning of
opposites as the same distance from zero in opposite directions. Students should begin
making connections to absolute value as well.
This investigation uses vertical number lines which students need to be equally fluent with
as horizontal number lines. Inequalities could be used during this equation which would be
review of 6.EE.7a. Students should order signed and unsigned rational numbers which is an
extension from integer ordering in Grade 6. Begin to explore number line and meaning of
absolute value. Problem F4 is the main summary of opposites for this investigation. Take
the opportunity to revisit property of identity as opposites are added.
Have students connect the number sentence to the number line, vertical and horizontal. Use
number line patterns (add positive, go right, subtract positive, go left, etc.) for students to
begin understanding how number lines will help develop understanding of the patterns for
addition and subtraction of signed numbers.
Investigation emphasizes finding missing values using integer counters. Students should
verify solutions on a number line. Extend number line usage to reinforce sums and
differences with basic rational numbers (1/2 + -3/4).
CCSS-M
Standards
7.NS.1a, 8.NS.1
6.NS.5,
6.NS.6c,
7.NS.1a
6.NS.6c,
7.NS.1b
7.NS.3
7.NS.1b
7.NS.1c
7.NS.3
15
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
AN 2.1
Engage NY Number System
module Lesson 4: Efficiently
Adding Integers and Other Rational
Numbers
AN 2.2
AN 2.3
Modify AN 2.4
Engage NY Number System
module Lesson 7:
Engage NY Number System
module Lesson 8 and 9
Use only number lines to support student understanding of adding integers. Use integer
counters if needed as a kinesthetic model for students. Integer counters only have the
ability to model integers, not all rational numbers. Reinforce commutative property as a
strategy for modeling addition of rational numbers. Continue to identify opposites and
their sum being zero on number line or using chips. Revisit number line patterns from
elementary school and identify how they apply to negative values (add positive, right, add
a negative, left, etc.) Problem C allows for students to write a contextual situation for a
given equation. Have students predict sum, whether positive or negative, to reinforce
understanding of addition patterns. As students are learning the patterns/rules for adding
rational numbers, notate algebraically and numerically.
Lesson 4 of the Engage NY Number System module will support the further development
of addition of integers using number lines, addition of rational numbers and contextual
rational numbers. Be careful not to give students questions in this lesson that discuss
absolute value since this concept will not be explored until Investigation 2.1.
During the investigation, focus on number line models, absolute value, absolute value on
number line, additive inverse (add the opposite) and using commutative property to solve.
For Problem A5, check strategies with number line. For problem B, have students model
their solution on a number line. Continually ask students to predict difference, whether
positive or negative, before solving. As students are learning the patterns/rules for
subtracting rational numbers, notate algebraically and numerically.
Have students predict whether the sum or differences will be positive or negative. Have
students together justify reasoning before solving or using calculator to verify.
Modify 2.4 to use fact families with rational numbers beyond integers. Important to
reinforce finding missing values, additive inverse, sum and difference.
Lesson 7 of the Engage NY Number System module will support the further development
of addition and subtraction of rational numbers along with real world connections to
additional and subtraction of rational numbers.
Lesson 8 and 9 of the Engage NY Number System module will support the use of the
properties of operations to solve addition and subtraction of rational problems. Students
will use commutative and associative properties to rewrite numerical expressions in
equivalent but different forms in order to solve more efficiently.
7.NS.1b
7.NS.1d
7.NS.1b
7.NS.3
7.NS.1c
7.NS.1d
7.NS.1c
7.NS.1d
7.NS.1d
7.NS.1c
7.NS.3
7.NS.1d
7.NS.3
16
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
AN 3.1
As you introduce the investigation, on page 42, have students focus in on the expression –
(-8) = 8, opposite of -8 is 8. As students write number sentences for problem A1-5, verify
solutions on number line as students run left or right with a given rate. In order to
continually connect to the properties of operations, discuss whether multiplication is
commutative or associative and how that information might be helpful when finding the
patterns. Take the opportunity to discuss the multiple meanings of the negative sign,
negative, opposite, subtract, take away, difference, etc.
AN 3.2
When students are developing understanding of patterns of rational multiplication, notate
algebraically and numerically. You may need to extend this lesson for students to have
more opportunities to practice the patterns of rational multiplication with integers and
rational numbers. Investigation 3 ACE questions 1, 26 and Skill worksheets, and
Additional Practice pages will provide students an opportunity to practice multiplication
of rational numbers mathematically and with contextual situations.
AN 3.3
As students are dividing signed rational numbers, include use of complex fractions and
reinforce fraction bar as a symbol for division. Introduce notations of division where sign
could be on numerator, denominator, or outside fraction. Continue to use rational values
beyond integers. Students need to understand that integers can be divided, provided the
divisor in not zero. Students also need to understand that every quotient of integers is a
rational number.
Engage NY Number System
Lesson 15 of the Engage NY Number System module will support students recognizing
module Lesson 15: Multiplication
how the rules of multiplying and dividing integers apply to rational numbers. Students
and Division of Rational Numbers
will also solve for products and quotients using real-world contexts.
Converting between fraction and
Lessons 13 and 14 in the Rational Numbers Number Systems module for Grade 7 focus
decimal – terminating and repeating on converting between fractions and decimals with a result of looking at patterns for
terminating and repeating decimals. Students will practice with both positive and
negative fractions. Students should convert positive and negative common fractions (e.g.
2/9, -5/6, 7/12, -28/99, 3/11, etc.) and fractions with denominators that are a factor of a
power of 10 to decimals (-5/32, 7/8, 113/125, etc.). You may need to develop additional
practice for your students.
Converting decimal to fraction
The Repeating Decimals task from MAP will cover converting a repeating decimal back
to a fraction. Students will need instruction on the algorithm for converting a repeating
decimal to a fraction. Converting a repeating decimal to a fraction (rational number) will
be a launch into to teaching the Real Number System with rational and irrational numbers.
7.NS.2a
7.NS.2c
7.NS.2a
7.NS.3 (if
using ACE
1 and 26)
7.NS.2b
7.NS 3
7.NS.2b
7.NS.3
7.NS.2d
8.NS.1
17
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Classifying numbers in the Real
Number System as rational,
irrational, integer, and whole
number
Students will need instruction on the meaning of the rational and irrational numbers
within the Real Number System and how they are classified. The Classifying Numbers II
PowerPoint may help to supplement this content. The Real Number Race task from
NCDPI will provide an opportunity for students to make sense of classifying and
identifying numbers as rational, irrational, and a real number.
Students will be asked to apply properties of operations as strategies to add, subtract,
multiply, and divide. These properties include order of operations along with the
commutative and associative properties which is lightly covered in the unit.
8.NS.1
4.2
Distributive property is introduced to students beginning in Grade 3 with Common Core
standards. The application of the skill in Grade 7 is with rational numbers, primarily
signed integers.
District-developed common
assessment
For understanding the content required to be assessed, see the rubrics at the end of the
unit guide which is based on the Grade 7 SBAC Claim 1 Item Specification for Target B
and Grade 8 SBAC Claim 1 Item Specifications for Target A . Based on the SBAC item
specifications for these targets, a calculator is an allowable tool during the assessment.
6.NS.4
6.EE.3
7.NS.1d
7.NS.2c
7.NS.3
7.NS.1,
7.NS.2,
7.NS.3
8.NS.1
4.1
6.EE.1
7.NS.1d
7.NS.2c
*SBAC ALD Level 3 or Level 4 means Smarter Balanced Achievement Level Descriptors for entering into a Level 3 or Level 4.
Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math/compacted-math-7-8-resources .
The following resources were used to create this curriculum guide: Grade 7 SBAC Claim 1 Item Specification for Target B and Grade 8 SBAC
Claim 1 Item Specifications for Target A and Grade 6-8 Number Systems Progression document.
18
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Standard 7.NS.1 a-d
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.
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems
 Use precise and relevant
communication to justify
mathematical thinking
 Connects knowledge to
other learning targets and/or
advance problem sets.




7.NS.1
3
Solve mathematical problems 
using addition and subtraction
using signed and unsigned

rational numbers.
Model addition and

subtraction of rational
numbers on a number line.
Understand subtraction is the
same as adding the additive
inverse.
Understand p + q as a number
located | | units from p on a
number line in either direction
depending on sign of q.
2
Identify the absolute value

of a rational number.
Understand opposites
combine to make 0.
Solve mathematical
problems using addition or
subtraction using signed and

unsigned rational numbers.

1
With help, minimal
success solving
mathematical problems
using addition or
subtraction using signed
and unsigned rational
numbers.
Add and subtract nonnegative rational numbers.
Add and subtract rational
numbers with number line
or manipulative.
For example, explain how
subtraction of rational numbers
is the same as addition of the
additive inverse of the rational
numbers.
For more detail on the assessment of the standard, read the SBAC Claim 1 Grade 7 Item Specifications-Target B. The rubric was constructed from
the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. These rubrics should be further developed at the PLC level as learning
continues through the transition to Common Core State Standards and the embedded Standards for Mathematical Practice. Students will also be assessed on these
standards through Claim 2 (Problem Solving), Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data Analysis). These documents should also be
consulted by teachers when designing lessons and assessments.
19
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Standard 7.NS.2 a-d
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (Computations
with rational numbers extend the rules for manipulating fractions to complex fractions).
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems
 Use precise and relevant
communication to justify
mathematical thinking
 Connects knowledge to
other learning targets and/or
advance problem sets.
For example, solve an advanced
equation with negative and
positive rational numbers using
order of operations.




7.NS.2
3
Solve mathematical problems 
using multiplication and
divisions of signed and

unsigned rational numbers.
Convert from a fractional
form of rational numbers to a 
decimal form of rational
numbers.
Demonstrate understanding of
the rules for multiplication
and division of signed
numbers.
Understanding of zero as a
divisor (undefined) in rational
division.
2
Convert between familiar
fractions and decimals.
Multiply or divide signed
and unsigned rational
numbers.
Demonstrate partial
understanding of the rules
for multiplication and
division of signed numbers.




1
With help, minimal
success converting
between familiar fractions
and decimals.
With help, minimal
success multiplying or
dividing signed and
unsigned rational
numbers.
Multiply and divide nonnegative rational numbers.
Multiply and divide
rational numbers with
number line or
manipulative.
For more detail on the assessment of the standard, read the SBAC Claim 1 Grade 7 Item Specifications-Target B. The rubric was constructed from
the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. These rubrics should be further developed at the PLC level as learning
continues through the transition to Common Core State Standards and the embedded Standards for Mathematical Practice. Students will also be assessed on these
standards through Claim 2 (Problem Solving), Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data Analysis). These documents should also be
consulted by teachers when designing lessons and assessments.
20
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Standard 7.NS.3
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).
7.NS.3
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems
 Use precise and relevant
communication to justify
mathematical thinking
 Connects knowledge to
other learning targets and/or
advance problem sets.
3

Solve real-world problems
involving rational numbers with
addition, multiplication,
subtraction, and division with at
least integers and proper
fractions.


2
Add or subtracts rational
numbers (integers or proper
fractions or decimals) in
real-world problems.
Multiplies or divides
rational numbers in realworld problems.


1
With help, minimal
success adding or
subtracting rational
numbers (integers or
proper fractions or
decimals) in real-world
problems.
Add or subtract integers in
real-world problems with
errors or misconceptions.
For example, solves a realworld problem with an
unfamiliar context using
addition, subtraction,
multiplication and/or division.
For more detail on the assessment of the standard, read the SBAC Claim 1 Grade 7 Item Specifications-Target B. The rubric was constructed from
the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. These rubrics should be further developed at the PLC level as learning
continues through the transition to Common Core State Standards and the embedded Standards for Mathematical Practice. Students will also be assessed on these
standards through Claim 2 (Problem Solving), Claim 3 (Communicating Reasoning) and Claim 4 (Modeling and Data Analysis). These documents should also be
consulted by teachers when designing lessons and assessments.
21
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)
Standard 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.
4
In addition to being proficient
on the standard, student can
demonstrate one or more of the
following:
 Applies understanding of
standard to unfamiliar
situations and/or to solve
complex problems.
 Use precise and relevant
communication to justify
mathematical thinking.
 Connects knowledge to
other learning targets and/or
advance problem sets.



8.NS.1
3
Classify and define real
numbers as rational or
irrational.
Convert rational number to
terminating and repeating
decimals.
Convert repeating decimal
into a fraction.



2
Convert rational number to
terminating or repeating
decimals.
Classify real numbers as
rational or irrational.
Convert simple repeating
decimal into a fraction.



1
With help, student has
minimal success with
classify real numbers as
rational or irrational.
With help, student has
partial success with
conversion process.
Convert a simple
repeating decimal into a
fraction with omissions
and/or misconceptions.
For example, provide a clear
explanation of the process of
converting a repeating decimal
to a fraction.
For more detail on the assessment of the standard, read the SBAC Claim 1 Grade 8 Item Specifications-Target A. The rubric was constructed from
the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. These rubrics should be further developed at the PLC level as learning
continues through the transition to Common Core State Standards and the embedded Standards for Mathematical Practice.
22
This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (7.NS.1-3, 8.NS.1)