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 2020-2021 school year due to school closures, see our 7th Grade Scope and Sequence Recommended Adjustments.

Subscribe to Fishtank Plus to unlock access to additional resources for this unit, including:

Unit Launch
Expanded Assessment Package
Problem Sets for Each Lesson
Student Handout Editor
Google Classroom Integration
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 Unit Assessment

Take 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 "Unit Summary."
Notice the progression of concepts through the unit using “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Guide to Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Read Progressions for the Common Core Standards in Mathematics, The Number System, 6-8 for the relevant standards within the Number System domain.

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.

Unit Materials, Representations and Tools

Activities to Increase Fluency with Integer Computation
Number line
Game piece or token (optional)
Calculators
Graph paper
Standard deck of playing cards (optional)

Vocabulary

opposite

rational number

distributive property

terminating decimal

absolute value

additive inverse

commutative property

associative property

multiplicative inverse

repeating decimal

Related Teacher Tools:

7th Grade Vocabulary Glossary

Lesson Map

Topic A: Adding and Subtracting Rational Numbers

1

7.NS.A.1

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

2

7.NS.A.1

Compare and order rational numbers. Write and interpret inequalities to describe the order of rational numbers.

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.

5

7.NS.A.1.B

7.NS.A.1.D

Determine efficient ways to add rational numbers with and without the number line.

6

7.NS.A.1.B

7.NS.A.1.D

Efficiently add and reason about sums of rational numbers.

7

7.NS.A.1.C

7.NS.A.1.D

Understand subtraction as addition of the opposite value (or additive inverse).

8

7.NS.A.1.C

Find and represent the distance between two rational numbers as the absolute value of their difference.

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.

11

7.NS.A.1

7.NS.A.1.A

7.NS.A.1.B

7.NS.A.1.C

7.NS.A.1.D

Add and subtract rational numbers using a variety of strategies.

Topic B: Multiplying and Dividing Rational Numbers

12

7.NS.A.2.A

7.NS.A.2.C

Determine the rules for multiplying signed numbers.

13

7.NS.A.2.A

7.NS.A.2.C

Multiply signed rational numbers and interpret products in real-world contexts.

14

7.NS.A.2.B

7.NS.A.2.C

Determine the rules for dividing signed numbers.

15

7.NS.A.2.B

7.NS.A.2.C

Divide signed rational numbers and interpret quotients in real-world contexts.

16

7.NS.A.2.D

Convert rational numbers to decimals using long division and equivalent fractions.

17

7.NS.A.2

7.NS.A.2.C

Multiply and divide with rational numbers using properties of operations.

Topic C: Using all Four Operations with Rational 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.

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

The Number System

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.

Foundational Standards

Number and Operations in Base Ten

4.NBT.B.5

Number and Operations in Base Ten

4.NBT.B.5 — 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.

Number and Operations—Fractions

4.NF.B.3

Number and Operations—Fractions

4.NF.B.3 — Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
5.NF.A.1

Number and Operations—Fractions

5.NF.A.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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.A.2

Number and Operations—Fractions

5.NF.A.2 — 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. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
5.NF.B.3

Number and Operations—Fractions

5.NF.B.3 — 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. 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?

Operations and Algebraic Thinking

1.OA.B.3

Operations and Algebraic Thinking

1.OA.B.3 — Apply properties of operations as strategies to add and subtract. Students need not use formal terms for these properties. 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.) If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.)
3.OA.B.5

Operations and Algebraic Thinking

3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: 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.) Example: 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.)
3.OA.B.6

Operations and Algebraic Thinking

3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

The Number System

6.NS.A.1

The Number System

6.NS.A.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. 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?
6.NS.B.2

The Number System

6.NS.B.2 — Fluently divide multi-digit numbers using the standard algorithm.
6.NS.B.3

The Number System

6.NS.B.3 — Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.C.5

The Number System

6.NS.C.5 — 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.
6.NS.C.6

The Number System

6.NS.C.6 — 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.
6.NS.C.7

The Number System

6.NS.C.7 — Understand ordering and absolute value of rational numbers.
6.NS.C.7.C

The Number System

6.NS.C.7.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. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
6.NS.C.8

The Number System

6.NS.C.8 — 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.

Future Standards

High School — Number and Quantity

N.RN.A.1

High School — Number and Quantity

N.RN.A.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. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.
N.RN.A.2

High School — Number and Quantity

N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N.RN.B.3

High School — Number and Quantity

N.RN.B.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.

The Number System

8.NS.A.1

The Number System

8.NS.A.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.
8.NS.A.2

The Number System

8.NS.A.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., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

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.

Operations with Rational Numbers

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Unit Materials, Representations and Tools

Vocabulary

Related Teacher Tools:

Lesson Map

Topic A: Adding and Subtracting Rational Numbers

1

2

3

4

5

6

7

8

9

10

11

Topic B: Multiplying and Dividing Rational Numbers

12

13

14

15

16

17

Topic C: Using all Four Operations with Rational Numbers

18

Common Core Standards

Core Standards

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

Foundational Standards

Number and Operations in Base Ten

Number and Operations in Base Ten

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Operations and Algebraic Thinking

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

Future Standards

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

The Number System

The Number System

The Number System

Standards for Mathematical Practice

Unit 1

Proportional Relationships