Lesson 10 | Operations with Rational Numbers | 7th Grade Mathematics

Objective

Add and subtract rational numbers efficiently using properties of operations.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Foundational Standards

The foundational standards covered in this lesson

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.
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.)

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Rewrite problems with subtraction as addition problems in order to use the commutative property.
Rewrite mixed numbers as a sum of the whole number and the fraction in order to find efficient ways to solve problems.
Analyze and discuss different approaches to adding and subtracting rational numbers.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

As students become more efficient and less reliant on using the number line to find sums and differences, they also learn how using properties of operations can be valuable in understanding the problem as a whole and increasing efficiency even more (MP.1).
Continue building fluency with mental math and other activities.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Represent the following using a single rational number.

$${-2{2\over5}+3{1\over4}-{3\over5}}$$

How many different ways to approach this problem can you think of?

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic A > Lesson 8 — Exercise 1

Grade 7 Mathematics > Module 2 > Topic A > Lesson 8 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Jamal is completing a math problem and represents the expression $${-5{5\over7}+8-3{2\over7}}$$ with a single rational number as shown in the steps below. Explain what Jamal did in each step and check his work along the way.

$${=-5{5\over7}+8+ \left (-3{2\over7} \right)}$$

$${=-5{5\over7} +\left (-3{2\over7} \right) + 8}$$

$${=-5 + \left( -{5\over7} \right) +(-3)+ \left (-{2\over7} \right) +8}$$

$${=-5 + \left( -{5\over7} \right) + \left (-{2\over7} \right) + (-3) +8}$$

$${= -5 + (-1) + (-3) + 8}$$

$${= -6 + (-3) + 8}$$

$${=(-9)+8}$$

$${=-1}$$

Guiding Questions

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Problem 3

Which expression is equivalent to $${-14.6-(-5.75)-8.4}$$? Use it to find a single rational number to represent the expression.

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Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Find a single integer that represents the expression. Explain the strategy you use.

$${-14+8-3+20+(-6)-(-2)+5}$$

Student Response

An example response to the Target Task at the level of detail expected of the students.

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Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Math Mistakes Adding and Subtracting Negative Numbers
EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic A > Lesson 8 — Problem Set
EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic A > Lesson 9 — Problem Set
Open Middle Absolute Value 2. — Challenge

Lesson 9

Lesson 11

Lesson Map

Topic A: Adding and Subtracting Rational Numbers

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

7.NS.A.1

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

7.NS.A.1

Describe situations in which opposite quantities combine to make zero.

7.NS.A.1.A

Model the addition of integers using a number line.

7.NS.A.1.B 7.NS.A.1.D

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

7.NS.A.1.B 7.NS.A.1.D

Efficiently add and reason about sums of rational numbers.

7.NS.A.1.B 7.NS.A.1.D

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

7.NS.A.1.C 7.NS.A.1.D

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

7.NS.A.1.C

Subtract rational numbers with and without the number line.

7.NS.A.1.C 7.NS.A.1.D

Add and subtract rational numbers efficiently using properties of operations.

7.NS.A.1.D

Add and subtract rational numbers using a variety of strategies.

7.NS.A.1 7.NS.A.1.A 7.NS.A.1.B 7.NS.A.1.C 7.NS.A.1.D

Topic B: Multiplying and Dividing Rational Numbers

Determine the rules for multiplying signed numbers.

7.NS.A.2.A 7.NS.A.2.C

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

7.NS.A.2.A 7.NS.A.2.C

Determine the rules for dividing signed numbers.

7.NS.A.2.B 7.NS.A.2.C

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

7.NS.A.2.B 7.NS.A.2.C

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

7.NS.A.2.D

Multiply and divide with rational numbers using properties of operations.

7.NS.A.2 7.NS.A.2.C

Topic C: Using all Four Operations with Rational Numbers

Solve problems with rational numbers and all four operations.

7.NS.A.3

Operations with Rational Numbers

Objective

Common Core Standards

Core Standards

The Number System

Foundational Standards

Number and Operations—Fractions

Operations and Algebraic Thinking

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

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