# 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.

Math

Unit 2

## Unit Summary

In Unit 2, 7th 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 7th 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 and in Unit 4. By the time students enter 8th grade, students should have a strong grasp on operating with rational numbers, which will be an underlying skill in many algebraic concepts. In 8th 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)

Fishtank Plus for Math

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress. ## Assessment

The following assessments accompany Unit 2.

### Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

### Mid-Unit

Have students complete the Mid-Unit Assessment after lesson 11.

### Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.

## Unit Prep

### Intellectual Prep

Unit Launch

Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning. #### 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)
• Game piece or 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)

To see all the materials needed for this course, view our 7th Grade Course Material Overview.

### Vocabulary

absolute value

associative property

commutative property

distributive property

multiplicative inverse

opposite

rational number

repeating decimal

terminating decimal

To see all the vocabulary for Unit 2, view our 7th Grade Vocabulary Glossary.

## Lesson Map

Topic A: Adding and Subtracting Rational Numbers

Topic B: Multiplying and Dividing Rational Numbers

Topic C: Using all Four Operations with Rational Numbers

## Common Core Standards

Key

Major Cluster

Supporting 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.

Unit 1

Proportional Relationships

Unit 3

Numerical and Algebraic Expressions

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