# Operations with Rational Numbers

Students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers, and build fluency with evaluating numerical expressions.

## Unit Summary

In Unit 2, seventh-grade students extend the operations of addition, subtraction, multiplication, and division to include positive and negative rational numbers. Standards 7.NS.1 and 7.NS.2 represent a culmination in the extension of the four operations to all rational numbers. In this unit, students model addition and subtraction on the number line, and through repeated reasoning and application of properties of operations, they determine efficient rules for computing with rational numbers (MP.8). Students gain the ability to model a greater scope of real-world contexts to include situations involving elevation, temperature changes, debts and credits, and proportional relationships with negative rates of change (MP.4). They also develop greater fluency with evaluating numerical expressions, using the properties of operations to increase their flexibility in approach.

Starting in first grade, students learn about the commutative and associative properties of addition, and the relationship between addition and subtraction. In third grade, students extend their understanding of the properties of operations to include multiplication and the distributive property. Throughout the years, students have applied these properties and relationships between the operations to whole numbers, fractions, and decimals. In seventh grade, all of these skills and concepts come together as students now operate with all rational numbers, including negative numbers.

In several upcoming units, seventh-grade students will rely on their increased number sense and ability to compute with rational numbers, in particular in Unit 3, Numerical and Algebraic Expressions, and in Unit 4, Equations and Inequalities. By the time students enter eighth grade, students should have a strong grasp on operating with rational numbers, which will be an underlying skill in many algebraic concepts. In eighth grade, students are introduced to irrational numbers, rounding out their understanding of the real number system before learning about complex numbers in high school.

Included in the materials for this unit are some activities that aim to support and build students’ fluency with integer computations, especially mental math. See our Guide to Procedural Skill and Fluency for additional information and strategy and activity suggestions.

Pacing: 22 instructional days (18 lessons, 3 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 7th Grade Scope and Sequence Recommended Adjustments.

• Unit Launch
• Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

## Assessment

This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

?

#### Internalization of Standards via the Post-Unit Assessment

• Take the Post-Unit Assessment. Annotate for:
• Standards that each question aligns to
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that Assessment points to

#### Internalization of Trajectory of Unit

• Read and annotate the Unit Summary.
• Notice the progression of concepts through the unit using the Lesson Map.
• Essential Understandings
• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

### Essential Understandings

?

• The properties of operations for addition, subtraction, multiplication, and division hold true for rational numbers.
• In the equation ${p+q=r}$, where $p$, $q$, and $r$ are rational numbers, $\left | q \right |$ represents the distance between $p$ and $r$, and is also represented as $\left | r-p \right |$.
• The quotient or product of two negative or two positive numbers is positive.
• The quotient or product of two numbers, in which one of the numbers is negative, is negative.

### Materials

?

• Calculators (1 per student)
• Graph Paper (1 sheet per student)
• Dry erase marker (1 per student)
• Laminated number line (1 per student)
• Optional: Game piece of token (1 per student)
• Number cards (1 per student) — These are used in Anchor Problem 1 and require some preparation.
• Optional: Standard deck of playing cards (1 per student or small group)

### Vocabulary

?

opposite

rational number

distributive property

terminating decimal

absolute value

commutative property

associative property

multiplicative inverse

repeating decimal

To see all the vocabulary for this course, view our 7th Grade Vocabulary Glossary.

# 1

7.NS.A.1

Represent rational numbers on the number line. Define opposites and absolute value.

# 3

7.NS.A.1.A

Describe situations in which opposite quantities combine to make zero.

# 4

7.NS.A.1.B

7.NS.A.1.D

Model the addition of integers using a number line.

7.NS.A.1.B

7.NS.A.1.D

# 9

7.NS.A.1.C

7.NS.A.1.D

Subtract rational numbers with and without the number line.

# 10

7.NS.A.1.D

Add and subtract rational numbers efficiently using properties of operations.

# 12

7.NS.A.2.A

7.NS.A.2.C

Determine the rules for multiplying signed numbers.

# 14

7.NS.A.2.B

7.NS.A.2.C

Determine the rules for dividing signed numbers.

# 18

7.NS.A.3

Solve problems with rational numbers and all four operations.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

?

##### The Number System
• 7.NS.A.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.

• 7.NS.A.1.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.

• 7.NS.A.1.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.

• 7.NS.A.1.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.

• 7.NS.A.1.D — Apply properties of operations as strategies to add and subtract rational numbers.

• 7.NS.A.2 — Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

• 7.NS.A.2.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.

• 7.NS.A.2.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.

• 7.NS.A.2.C — Apply properties of operations as strategies to multiply and divide rational numbers.

• 7.NS.A.2.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.

• 7.NS.A.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.

?

• 4.NBT.B.5

• 4.NF.B.3

• 5.NF.A.1

• 5.NF.A.2

• 5.NF.B.3

• 1.OA.B.3

• 3.OA.B.5

• 3.OA.B.6

• 6.NS.A.1

• 6.NS.B.2

• 6.NS.B.3

• 6.NS.C.5

• 6.NS.C.6

• 6.NS.C.7

• 6.NS.C.7.C

• 6.NS.C.8

?

• N.RN.A.1

• N.RN.A.2

• N.RN.B.3

• 8.NS.A.1

• 8.NS.A.2

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.