EM2_G7-8_M1_Teach_23A_881593_Updated 08.23 by Great Minds

7–8

A Story of Ratios®

Proportions and Linearity TEACH ▸ Module 1 ▸ Rational and Irrational Numbers

Teacher Edition: Grade 7–8, Module 1, Cover What does this painting have to do with math? The French neoimpressionist Paul Signac worked with painter Georges Seurat to create the artistic style of pointillism, in which a painting is made from small dots. Signac’s Vue de Constantinople, La Corne d’Or Matin shows the Golden Horn, a busy waterway in Istanbul, Turkey. How many dots do you think were used to make this pointillist painting? How would you make an educated guess? On the cover Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907 Paul Signac, French, 1863–1935 Oil on canvas Private collection Paul Signac (1863–1935), Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907. Oil on canvas. Private collection. Photo credit: Peter Horree/Alamy Stock Photo

Teacher Edition: Grade 7-8, Module 1, Copyright

Great Minds® is the creator of Eureka Math®, Wit & Wisdom®, Alexandria Plan™, and PhD Science®. Published by Great Minds PBC. greatminds.org © 2023 Great Minds PBC. All rights reserved. No part of this work may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including photocopying or information storage and retrieval systems—without written permission from the copyright holder. Where expressly indicated, teachers may copy pages solely for use by students in their classrooms. Printed in the USA A-Print 1 2 3 4 5 6 7 8 9 10 XXX 27 26 25 24 23 ISBN 979-8-88588-159-3

Teacher Edition: Grade 7–8, Module 1, Title

A Story of Ratios®

Proportions and Linearity ▸ 7–8 TEACH Module

1 2 3 4 5 6

Rational and Irrational Numbers

One- and Two-Variable Equations

Two-Dimensional Geometry

Graphs of Linear Equations and Systems of Linear Equations

Functions and Three-Dimensional Geometry

Probability and Statistics

Teacher Edition: Grade 7–8, Module 1, Module Overview

Before This Module

Overview

Grade 5 Module 1

Rational and Irrational Numbers

In grade 5, students use whole-number exponents to express powers of 10 and to explain patterns when multiplying and dividing by powers of 10.

Topic A

Grade 6 Modules 2 and 3 In grade 6 module 2, students are introduced to division of non–unit fractions and compute fluently with decimals. In grade 6 module 3, students extend their previous understanding of the number system to include integers and rational numbers. They learn that an integer is a number that can be presented either as a whole number or as the opposite of a whole number, and they use absolute value to determine the distance between a number and 0 on a number line. They understand a rational number to be a number that can be written as a fraction or the opposite of that fraction.

Add and Subtract Rational Numbers To begin the topic, students use directed line segments to model integer addition expressions on a number line. Then they apply decomposition, properties of operations, and patterns to evaluate addition expressions involving rational numbers. Students use their understanding of rational number addition to create and solve KAKOOMA® puzzles. Students use unknown addend equations to represent subtraction expressions on a number line, and they recognize that subtracting a number has the same result as adding the opposite of that number. They write subtraction expressions as equivalent addition expressions and use their understanding of rational number addition to evaluate subtraction expressions.

8 – (–10) –10 + ____ = 8

Grade 6 Module 4 In grade 6 module 4, students write and evaluate numerical expressions involving whole-number exponents and positive rational number bases.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Topic B Multiply and Divide Rational Numbers Students identify patterns and use properties of operations to make sense of rational number multiplication. They relate exponential expressions involving positive or negative bases to repeated multiplication. Students use unknown factor equations to make

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connections between multiplication and division and then evaluate division expressions that include rational numbers in fraction and decimal forms. At the end of the topic, students write rational numbers as either terminating decimals or repeating decimals.

p = 12 q=3

p q

−p q

12 =4 3

− 12 = −4 3

___

____

p ___ −q

− (__ q)

−p −q

12 ___ = −4

12 − (__ ) = −4

− 12 =4 −3

−3

___ ____

Topic C Properties of Exponents and Scientific Notation Students explore very large and very small positive numbers and relate them to real-world measurements and contexts. To preview scientific notation, students write large and small positive numbers as a single digit times a power of 10 and operate with large positive numbers by writing repeated factors of 10. By examining repeated factors, they make conjectures and learn the properties and definitions of exponents. Students extend their work with powers of 10 to expressions with integer exponents. They apply the properties and definitions of exponents to simplify exponential expressions. Students are then introduced to scientific notation, a × 10n. Students apply the properties and definitions of exponents to efficiently operate with numbers in scientific notation to solve real-world problems, including by choosing appropriate units of measurement. The topic closes with an open-ended modeling task in which students approximate the number of brushstrokes in a piece of fine art.

1.1 × 107 − 2 × 106 = (1.1 × 107) − (2 × 10−1 × 107) = (1.1 × 107) − (0.2 × 107) = (1.1 − 0.2) × 107 = 0.9 × 107 = (9 × 10−1) × 107 = 9 × (10−1 × 107) = 9 × 106 © Great Minds PBC

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Topic D

After This Module

Rational and Irrational Numbers Students apply their understanding of squares and cubes to reason intuitively about square roots and cube roots. They explore and apply the Pythagorean theorem to find the length of the hypotenuse c of right triangles. Students use the Pythagorean theorem to solve problems involving values of c 2 that are not perfect squares, which motivates the need for square root notation. Then students approximate values of square roots by using number sense to determine which whole-number interval includes the value. Through a digital interactive, students refine the interval to consecutive tenths, hundredths, and thousandths to increase the precision of their approximation. Once students approximate numbers that are not rational, they extend their understanding of the number system to include irrational numbers and real numbers. With this understanding, students classify real numbers as rational or irrational, order irrational numbers, and solve equations of the form x 2 = p and x 3 = p in which the solutions are irrational numbers.

7–8 Module 3 In module 3, students prove the Pythagorean theorem and its converse by using their understanding of rigid motions and congruence. They also apply the Pythagorean theorem to determine side lengths of a right triangle and to determine the distance between two points in the coordinate plane.

7–8 Module 5 In module 5, students apply the Pythagorean theorem to solve problems involving three-dimensional objects.

42 + 92 = c 2

High School: Algebra

16 + 81 = c 2

Students’ knowledge of the properties

97 = c 2

and definitions of exponents and root

—

√97 = c

— The length of the hypotenuse is √97 units, which is between 9 units and 10 units.

—

notation supports their ability to write √ab

—

as a 2 ⋅ b 2 or √a ⋅ √b . Students apply this __1

__1

—

understanding to write irrational numbers in other forms, such as writing √12 as 2 √3 .

Contents Rational and Irrational Numbers Why. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Achievement Descriptors: Overview . . . . . . . . . . . . . . . . . . . . 12 Topic A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Add and Subtract Rational Numbers

Topic B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Multiply and Divide Rational Numbers Lesson 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Multiplying Integers and Rational Numbers • Use repeated addition and the properties of operations to determine the product of a negative number and a positive number.

Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

• Informally verify that the product of two negative numbers is a positive number.

Adding Integers and Rational Numbers • Recognize that opposite integers sum to 0.

Lesson 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

• Use number lines and strategies to add rational numbers.

Lesson 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 KAKOOMA® with Rational Numbers • Use estimation and the properties of operations to add rational numbers. • Add rational numbers to solve and create puzzles.

Lesson 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Finding Distances to Find Differences • Show that the distance between two integers on a number line is the absolute value of their difference. • Evaluate integer subtraction expressions by finding the unknown addends in related addition equations.

Lesson 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Subtracting Integers

Exponential Expressions and Relating Multiplication to Division • Evaluate exponential expressions that include rational numbers. • Write division expressions as unknown factor equations to determine the value of the quotients.

Lesson 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Dividing Integers and Rational Numbers • Write rational numbers as quotients of integers. • Divide rational numbers given in different forms.

Lesson 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Decimal Expansions of Rational Numbers • Determine whether the decimal form of a rational number is a terminating decimal or a repeating decimal by analyzing the factors of the denominator. • Write rational numbers as either terminating decimals or repeating decimals.

• Express subtraction of an integer as addition of its opposite. • Subtract integers by using equivalent addition expressions.

Topic C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Properties of Exponents and Scientific Notation

Lesson 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Lesson 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Subtracting Rational Numbers • Evaluate expressions involving subtraction of rational numbers. • Subtract rational numbers by using equivalent addition expressions.

Large and Small Positive Numbers • Approximate very large and very small positive numbers and write them as a single digit times a power of 10 or as a single digit times a unit fraction with a denominator written as a power of 10. • Compare large and small positive numbers by using times as much as language.

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Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Products of Exponential Expressions with Positive Whole-Number Exponents

• Apply the product of powers with like bases property to write equivalent expressions given an expression of the form x m ⋅ x n.

Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 More Properties of Exponents

Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Rational and Irrational Numbers Lesson 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Solving Equations with Squares and Cubes • Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers.

• Apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers.

Lesson 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

• Describe the Pythagorean theorem and the conditions required to use it. • Apply the Pythagorean theorem to determine the length of a hypotenuse.

Making Sense of Integer Exponents

The Pythagorean Theorem

• Confirm that the definition of the exponent of 0 upholds the properties of exponents. • Apply the definition of a negative exponent to write equivalent expressions.

Lesson 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

Lesson 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

• Use square root notation to express lengths that are not rational and place them on a number line. • Approximate the value of square roots by using whole-number benchmarks.

Writing Very Large and Very Small Numbers in Scientific Notation

Using the Pythagorean Theorem

• Write numbers given in standard form in scientific notation. • Order numbers written in scientific notation.

Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

• Approximate values of square roots and cube roots.

Operations with Numbers Written in Scientific Notation • Interpret numbers in scientific notation displayed on digital devices. • Operate with numbers written in standard form and in scientific notation.

Lesson 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Applications with Numbers Written in Scientific Notation • Choose appropriate units of measurement and convert units of measurement with numbers written in standard form and in scientific notation. • Operate with numbers written in scientific notation in real-world situations.

Approximating Values of Roots

Lesson 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 Rational and Irrational Numbers • Classify real numbers as rational or irrational by their decimal form. • Compare and order rational and irrational numbers.

Lesson 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 Revisiting Equations with Squares and Cubes • Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers.

Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Get to the Point • Model a situation by operating with numbers in scientific notation.

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Resources

Fluency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Sample Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

Achievement Descriptors: Proficiency Indicators . . . . . . . . . . . . . . . 510 Terminology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 Math Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

Works Cited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556

Teacher Edition: Grade 7–8, Module 1, Why

Why Rational and Irrational Numbers How does this module integrate the learning of grades 7 and 8 of Eureka Math2? This module combines lessons from grade 7 module 2 about operations with rational numbers with lessons from grade 8 module 1 about scientific notation, properties of exponents, and irrational numbers. The design of this learning sequence facilitates important connections for students. These include relating the definition of a rational number to the definition of an irrational number and applying computation with integers to the properties of exponents, including when operating with numbers in scientific notation. Introducing these concepts together at the beginning of the accelerated course allows students to work with all types of real numbers in later modules, such as when they solve equations in module 2 and solve problems involving geometric figures in modules 3 and 5. Learning about rational and irrational numbers in module 1 also facilitates connections in later modules. For example, now that students can identify irrational numbers, they can recognize π as an irrational number when they learn to determine the area and circumference of a circle in module 3.

−0.75 − (−0.25)

I notice that this module does not emphasize integer rules. Do students ever learn the rules? Students begin using and reasoning with properties of operations in the primary grades. In this module, students extend this knowledge to discover that the properties of addition, subtraction, multiplication, and division also apply to integers and rational numbers.

–2

–1

Students formulate their own strategies based on properties and patterns, and they are encouraged to generalize those strategies into “rules.” Providing a set of rules to follow before conceptual understanding can hinder students from making sense of the problem or applying the properties of operations. For this reason, instead of displaying and discussing “rules,” encourage students to record their strategies throughout the module. 8

0 −0.25 +

= −0.75

−0.25 + (−0.5) = −0.75

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Why are exponential expressions written as 105+2 and 35 ⋅ 4?

In lesson 11, the expression 105 ⋅ 102 is intentionally shown as the equivalent expression 105+2 to emphasize the use of the product of powers with like bases property. Writing 105+2 has more instructional value than writing 107. When students are ready, ask them to express the sum of the exponents as a single integer. In lesson 12, the expression (34)5 is intentionally shown as the equivalent expression 35 ⋅ 4 instead of 320 to emphasize the use of the power of a power property. Students learn that (34)5 is 5 factors of 34, which is 5 factors of 4 factors of 3, which is 5 ⋅ 4 factors of 3. Students may wish to write expressions of the form (x m) n as x m ⋅ n rather than as x n ⋅ m. In the Properties and Definitions of Exponents graphic organizer and in future lessons, (x m) n will be written as x m ⋅ n for readability. When students are ready, ask them to express the product of the exponents as a single integer.

x n = __ x m = x m−n not included x n and ___ Why are _ n n

(y )

Properties of Exponents Description

Property

Example

Product of powers with like bases

x m ∙ x n = x m+n

105 ∙ 102 = 105+2 = 107

Power of a power

(x m)n = x m ⋅ n

(52)6 = 52 ⋅ 6 = 512

Power of a product

(xy)n = x n y n

(7 ∙ 6)2 = 72 ⋅ 62

Definitions of Exponents Description

Definition

Base with an exponent of 0

x0 = 1

in the Properties and Definitions of Exponents xn The power of a quotient, _x n = __ n , can be viewed as an

(y ) y extension of the definition of a power. So the power of a

Base with a negative exponent

quotient is not included in the Properties and Definitions of Exponents graphic organizer.

100 = 1

x is nonzero 1 x −n = __ n

graphic organizer?

Example

1 1 10−4 = ___ =_ 4

1 = xn x −n

1 ____ = 104 = 10,000

x is nonzero

___

x is nonzero

10,000

10−4

x m−n, is an application of the definition of a The quotient of powers with like bases, ___ n =x m

base with a negative exponent and the product of powers with like bases property. xm = x m ⋅ x −n = x m+(−n) = x m−n xn

___

So the quotient of powers is not included in the Properties and Definitions of Exponents graphic organizer. © Great Minds PBC

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Why does the Pythagorean theorem appear in two modules? Module 1 is an introduction to the Pythagorean theorem. Students solve for an unknown hypotenuse length c when given the leg lengths of a right triangle. In this module, the Pythagorean theorem motivates the need for using square roots to solve equations in the form x 2 = p. Students solve problems involving values of c 2 that are not perfect squares and estimate a range of possible values for c before transitioning to using square root notation. In module 3, the Pythagorean theorem supports learning about rigid motions, congruence, and similarity. Students apply rigid motions to prove the Pythagorean theorem and its converse. Later in the module, they use their understanding of similar triangles to prove the Pythagorean theorem and to find unknown leg and hypotenuse lengths of similar right triangles.

8 3

c a 2 + b2 = c 2 32 + 82 = c 2 9 + 64 = c 2

73 = c 2 — √73 = c

—

The length of the hypotenuse is √73 units.

Teacher Edition: Grade 7–8, Module 1, Achievement Descriptors: Overview

Achievement Descriptors: Overview Rational and Irrational Numbers Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module. Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year. ADs and their proficiency indicators support teachers with interpreting student work on

7–8.Mod1.AD1 Evaluate sums, differences, products, and

quotients of two rational numbers.

7.NS.A.2.c

7–8.Mod1.AD2 Solve real-world and mathematical problems

involving the four operations with rational numbers.

quotients of rational numbers by describing real-world contexts. 7.NS.A.1.a 7.NS.A.1.b 7.NS.A.1.c 7.NS.A.2.a

• data from other lesson-embedded formative assessments,

7.NS.A.2.b

• Exit Tickets,

7.NS.A.2.c

• Topic Quizzes, and

This module contains the 18 ADs listed.

7.NS.A.3

7–8.Mod1.AD3 Interpret sums, differences, products, and

• informal classroom observations,

• Module Assessments.

7.NS.A.1.d

7.NS.A.3

7–8.Mod1.AD4 Show on a number line that the distance

between two rational numbers is the same as the absolute value of their difference, and apply this principle in real-world contexts. 7.NS.A.1.c 7.NS.A.3

7–8.Mod1.AD5 Determine the sign of a product by looking at

the signs of its factors.

7.NS.A.2.a

7–8.Mod1.AD6 Write the fraction form of a rational number in

its decimal form by using long division.

7.NS.A.2.d

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7–8.Mod1.AD7 Determine whether numbers are rational or

7–8.Mod1.AD17 Interpret scientific notation that is generated

irrational.

by technology.

7.NS.A.2.d 8.NS.A.1

7–8.Mod1.AD8 Use rational approximations of irrational

numbers to compare the size of irrational numbers.

8.NS.A.2

7–8.Mod1.AD9 Locate irrational numbers approximately on a

number line.

8.NS.A.2

7–8.Mod1.AD10 Apply the properties of integer exponents to

generate equivalent numerical expressions.

8.EE.A.1

7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p,

where p is a positive rational number.

8.EE.A.2

7–8.Mod1.AD12 Evaluate square roots of small perfect squares

and cube roots of small perfect cubes.

8.EE.A.2

7–8.Mod1.AD13 Approximate and write very large and very

small numbers in scientific notation.

8.EE.A.3

7–8.Mod1.AD14 Express how many times as much one number

is than another number when both are written in scientific notation. 8.EE.A.3 7–8.Mod1.AD15 Operate with numbers written in standard

form and scientific notation, including problems where both decimal and scientific notation are used. 8.EE.A.4 7–8.Mod1.AD16 Operate with numbers written in scientific

notation to solve real-world problems and choose units of appropriate size for measurements of very large or very small quantities. 8.EE.A.4

8.EE.A.4

7–8.Mod1.AD18 Apply the Pythagorean theorem to determine

the unknown rational or irrational length of a hypotenuse for a right triangle in mathematical problems. 8.G.B.7 The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance. An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource. ADs have the following parts: • AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for course 7–8 module 1 is coded as 7–8.Mod1.AD1. • AD Language: The language is crafted from standards and concisely describes what will be assessed. • AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category. • Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.

Partially Proficient

7–8 ▸ M1

Proficient

Highly Proficient

Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

Evaluate square roots of rational numbers related to perfect squares and cube roots of rational numbers related to perfect cubes.

—

Evaluate.

AD Code: Grade.Mod#.AD#

—__9

EUREKA MATH2

√25 AD Language

√16

Evaluate.

—

3 √ 64

—

3 √ 0.27

7–8.Mod1.AD13 Approximate and write very large and very small numbers in scientific notation.

Related Standard

RELATED CCSSM

Partially Proficient

Approximate very large or very small numbers in scientific notation.

Compare very large and very small numbers written in scientific notation.

Write each number in scientific notation.

Round 6,190,000 to the nearest million. Then write that value in scientific notation.

Consider the numbers 4.11 × 10−6 and 4.11 × 10−5. Which number is smaller? Explain how you know.

2,500,000

Highly Proficient

Write very large or very small numbers in scientific notation.

0.00008

518

Proficient

AD Indicators

Round 0.000 0489 to the nearest hundred thousandth. Then write that value in scientific notation.

Teacher Edition: Grade 7–8, Module 1, Topic A

Topic A Add and Subtract Rational Numbers In topic A, students explore addition and subtraction of rational numbers. Students first use negative numbers in grade 6 when they identify opposites, define absolute value, compare rational numbers, and use positive and negative numbers to represent quantities in context. In this course, students begin to operate with rational numbers.

–7 + 10

Students begin the topic by modeling opposite actions by using a number line and defining the term additive inverse. While using a number line to model addition expressions with integers, students identify patterns that allow them to predict the signs of sums. Students use decomposition and properties of operations to evaluate addition expressions with integers and then with rational numbers.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10

–10 – 8 8 + ____ = –10

In lesson 2, students analyze a set of rational –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 numbers and realize that estimation is beneficial in determining the sign and an approximate value of a sum of rational numbers. Students choose a strategy to find the sum of rational numbers and continue to practice these skills by solving a KAKOOMA® puzzle. Working backward, students then create a KAKOOMA® puzzle. In lessons 3 and 4, students subtract integers and solve real-world subtraction problems by using a variety of strategies. They first use unknown addend equations, a familiar subtraction strategy from earlier grades, to make sense of integer subtraction and to evaluate subtraction expressions. By identifying patterns in the length and direction of directed line segments that represent unknown addends, students understand that the distance between two numbers on a number line is the absolute value of their difference. Then, by exploring patterns in sequences of subtraction expressions, students formalize the concept that every subtraction expression can be written as an equivalent addition expression. 16

9 10

−1_ − (−3 1_) = −1_ + 3 _1 4

= −1_ + 3 + 1_ 4

= −_1 + 3 + 1_ + 1_ 4

= (−_1 + 1_) + 3 + 1_ 4

= 0 + 3 + _1 = 3 1_

EUREKA MATH2

7–8 ▸ M1 ▸ TA

In the last lesson of the topic, students extend their understanding of integer subtraction to subtract rational numbers. At first, they model subtraction expressions by using a number line. Then they incorporate other familiar strategies, such as decomposition, to evaluate expressions and note that these strategies also work for rational numbers. In topic B, students use their understanding of addition and subtraction to develop strategies to multiply and divide rational numbers.

Progression of Lessons Lesson 1

Adding Integers and Rational Numbers

Lesson 2

KAKOOMA® with Rational Numbers

Lesson 3

Finding Distances to Find Differences

Lesson 4

Subtracting Integers

Lesson 5

Subtracting Rational Numbers

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 1 LESSON 1

Adding Integers and Rational Numbers Recognize that opposite integers sum to 0. Use number lines and strategies to add rational numbers.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1 Name

EXIT TICKET

Date

3. Consider the following expression.

−62 + 23 a. Use the number line to model the expression.

For problems 1 and 2, use the number line to model the expression and determine the sum. 1. −6 + 6

23 62

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

–62

9 10

–39

0 b. Determine the sum.

−62 + 23 = −39 + (−23) + 23 = −39 + (−23 + 23) = −39 + 0 = −39 2. 3 + (−7)

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

−4

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Lesson at a Glance Students begin this lesson by using a number line to model opposite actions and by relating opposite actions to additive inverses. They predict whether the sum of two integers is a positive number or a negative number and then evaluate the addition expression by using a number line to model it or by using decomposition. Students also observe how the sums of integers relate to the difference of the absolute values of the addends. Students then use strategies to predict whether the sum of two rational numbers is a positive number or a negative number. When considering how to add negative rational numbers, students discuss how to correctly decompose rational numbers into integer parts and non-integer parts. This lesson defines the term additive inverse.

Key Questions • Can we predict whether the sum of two rational numbers is a positive number or a negative number? How? • What strategies can we use to add integers and rational numbers?

Achievement Descriptors 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD3 Interpret sums, differences, products, and quotients of rational numbers

by describing real-world contexts. (7.NS.A.1.a; 7.NS.A.1.b; 7.NS.A.1.c; 7.NS.A.2.a; 7.NS.A.2.b; 7.NS.A.2.c; 7.NS.A.3)

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Computer or device*

Learn 30 min • Additive Inverses • Adding Integers • Adding Rational Numbers

Land 10 min

• Projection device* • Teach book* • Blank paper (3 sheets) • Marker • Tape

Students • Dry-erase marker* • Learn book* • Number Lines removable • Pencil* • Personal whiteboard* • Personal whiteboard eraser*

Lesson Preparation • Use the blank paper to prepare three signs with the following labels: Strongly Agree, Strongly Disagree, and Undecided. Post the signs around the classroom. *These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Fluency Add Fractions and Decimals Students add fractions and decimals to prepare for adding rational numbers. Directions: Add. 1.

7 + 4 10 10

__ __

1 1 __

3 _ + 2 _1

7_ + 4_

12 4_

2.3 + 3.4

5.7

6.6 + 5.7

12.3

8.4 + 34.24

42.64

3 4

5 8

7 8

Teacher Note Fluency activities are short sets of sequenced practice problems that students work on in the first 3–5 minutes of class. Administer a fluency activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response. Directions for administration can be found in the Fluency resource.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Launch

Students identify actions that are opposites and relate opposite actions to integers. Facilitate a Whiteboard Exchange to have students identify opposite numbers. When I say a number, write its opposite on your whiteboard and hold it up for me to see. Present the following sequence of numbers. Present one number at a time and check students’ answers after each number. Movement through these numbers should be quick, as students will ideally need only a few seconds per number.

Teacher Note This lesson combines and expands on the learning activities from Eureka Math2 grade 7 module 2 topic A. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

5, 10, _1 , 0, −4, −6 1_ 2

Pose the next question to have students recall the meaning of absolute value. Consider the numbers that were given and the numbers you wrote. How can you relate those numbers to what you recall about absolute value? Opposites are numbers that have the same distance from 0 on the number line but are on different sides of 0, which means they have the same absolute value. The numbers 4 and −4 have the same absolute value. What are some examples of opposite actions in real-world situations? Earning money and spending money Temperature dropping and temperature rising

Teacher Note The dialogue shown provides suggested questions and sample responses. To maximize every student’s participation, facilitate discussion by using tools and strategies that encourage student-to-student discourse. For example, make flexible use of the Talking Tool, turn and talk, think–pair–share, and the Always Sometimes Never routine.

An elevator going up and going down Invite students to stand. Use the following prompts to guide them through an activity that emphasizes opposite actions. Have students complete the actions as they are said. The location where you are standing now is your starting place. Now, move 4 steps to your right. Move 4 steps to your left.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

How does your current location compare to your starting place?

Language Support

It is the same. I am back to my starting place. Why did you end back at your starting place? The actions are opposites. How can you describe your starting place and actions mathematically? My starting place can be represented by 0. I can describe the action of moving right 4 steps by adding 4. I can describe the action of moving left 4 steps by subtracting 4. Invite students to turn and talk about the following question. Can we describe these actions by using only addition and not subtraction? Today, we will consider actions that are opposites and those that are not opposites to add integers and rational numbers.

The terms opposites, integer, and rational number are familiar from grade 6. To support these terms, consider having students recall the definitions and record an example of each of these terms in their classwork as the terms are used during the lesson facilitation. • Opposites are numbers that are the same distance from 0 on the number line but are on different sides of 0. Opposites have the same absolute value. • Integers are whole numbers and their opposites. • A rational number is a number that can be represented as a fraction or as the opposite of a fraction.

Learn Language Support

Additive Inverses Students identify additive inverses on a number line and observe that they sum to 0.

Consider using strategic, flexible grouping throughout the module.

Display the model that represents the situation from Launch.

• Pair students who have different levels of mathematical proficiency. • Pair students who have different levels of English language proficiency. • Join pairs to form small groups of four.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Introduce the Take a Stand routine to the class. Draw students’ attention to the signs posted around the classroom labeled Strongly Agree, Strongly Disagree, and Undecided.

As applicable, complement any of these groupings by pairing students who speak the same native language.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Present the statement and invite students to stand beside the sign that best describes their thinking. The model correctly represents the actions we made when beginning at a starting place, moving 4 steps to the right, and then moving 4 steps to the left. When all students are standing near a sign, allow 2 minutes for groups to discuss the reasons why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. As a class, reflect on the model and what it represents.

Teacher Note In module 3, students use transparencies as they explore rigid motions. Consider demonstrating how to use a transparency to represent the opposite actions in the situation. The following directions are written for a horizontal number line; however, this demonstration can also be done with a vertical number line.

What do you notice about the model?

• Place transparency over the number line.

I notice that the model has a number line and arrows.

• Draw a directed line segment on the transparency from 0 to 4 to represent the first addend.

I notice that the model starts at 0, goes 4 units to the right, and then goes 4 units to the left. I notice that one arrow begins in the opposite direction from where the other arrow ends. We call these arrows directed line segments. They begin at a number, point in a specific direction, and end at another number. Present the statement and invite students to stand beside the sign that best describes their thinking. Repeat the Take a Stand routine.

• Because the second addend is negative, lift and rotate the transparency so the directed line segment points left and goes from 4 to 0. • The sum of 4 and −4 is where the directed line segment ends, 0.

Earlier we stated that we can describe the action of moving left 4 steps by subtracting 4. However, we can also use the expression 4 + (−4) to represent the situation. As a class, reflect on using the expression 4 + (−4) to represent the situation. Why can we represent moving to the right from the starting place with the number 4? When we move right from 0 on the number line, the numbers are positive, and we moved 4 steps to the right. Why can we represent moving left 4 steps with adding the number −4? We are not starting at 0 anymore. We can consider 4 the new starting point on the number line. When we move left 4 units from the starting point, we are going in the opposite direction. The distance is still 4, but in the opposite direction. 24

EUREKA MATH2

What do you notice about the absolute values of 4 and −4? They are the same. They are both 4. We can say that −4 is the additive inverse of 4. We can also say that 4 is the additive inverse of −4. What do you think the term additive inverse means? I think additive inverses are opposites. −4 and 4 are opposites. The additive inverse of a number is a number such that the sum of the two numbers is 0. The additive inverse of a number x is the opposite of x because x + (−x) = 0. Why are 4 and −4 additive inverses? They are opposites, and they sum to 0. What part of the model shows us that the sum is 0? The second directed line segment ends at 0, so that is the sum. What addition sentence represents the situation?

4 + (−4) = 0

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Teacher Note Eureka Math2 uses the number line to show addition and subtraction of rational numbers by showing directed line segments. Directed line segments represent a length. Lengths are not negative numbers, so the numbers above the direct line segments should always be positive. The direction of the directed line segment indicates whether the addend is positive or negative. The use of the phrase directed line segment is intentional. It is imprecise to use the term ray or arrow to refer to these directed line segments. Use this as an opportunity to model precise mathematical terminology with students.

What number sentence represents the following situation: We begin at a starting point, move 4 steps to the left, and then move 4 steps to the right?

−4 + 4 = 0 What do you notice about the addends and the sums for this addition sentence and the previous addition sentence? The addends are 4 and −4 in both sentences, just in opposite order. The sums are both 0. We know that the commutative property of addition applies to whole numbers. It also applies to integers, so additive inverses in any order sum to 0.

Differentiation: Support Consider having students provide other examples of additive inverses. If needed, provide real-world situations to support students’ understanding of additive inverses and why they sum to 0. For example, an elevator that goes up 5 floors and then goes down 5 floors may be represented by the additive inverses 5 and −5. Students should notice that 5 + (−5) = 0 and that in this situation, the elevator returns to where it starts from, or 0.

7–8 ▸ M1 ▸ TA ▸ Lesson 1

EUREKA MATH2

Adding Integers Students predict whether the sum of two integers is a positive number or a negative number and add integers by using the number line and different strategies. Direct students to problem 1. Use the following sequence of questions to guide students on how to use the number line to model the addition expression. Have students follow along and record the work for problem 1 in their books. We can model any addition expression with directed line segments, as we did with the expression 4 + (−4). For problem 1, what number should we start with when modeling the expression? Why? We should start with 0 because that is the number where we begin counting. Which direction should the first directed line segment point? Why? The first directed line segment should point to the right because the first addend is positive. How long should the first directed line segment be? Why? The first directed line segment should be 3 units long because 3 is the first addend. Where should the second directed line segment start? Why? The second directed line segment should start at 3 because we are adding on to that number. Which direction should the second directed line segment point? Why? The second directed line segment should point to the right because the second addend is positive. How long should the second directed line segment be? Why? The second directed line segment should be 5 units long because 5 is the second addend.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

For problems 1 and 2, use the number line to model the addition expression. 1. 3 + 5

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Direct students to problem 2. Have students turn and talk about how this addition expression is different from the first and what approach they would take to model the expression. Have students share their ideas with the class. Consider repeating the sequence of questions from problem 1 to reinforce skills for using a number line to model addition. 2. −3 + (−5)

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Use the following prompts to discuss how the model is used to determine the sum. Does the model help us determine the sum? How? Yes. The second directed line segment ends 8 units to the left of 0 at −8, so the sum is −8. What do you notice about the models for problems 1 and 2? I notice that when both addends are positive, both directed line segments point to the right of 0. So the sum is a positive number.

Teacher Note Students may use the vertical, horizontal, or blank number lines given in the Number Lines removable throughout the topic to model expressions.

I notice that when both addends are negative, both directed line segments point to the left of 0. So the sum is a negative number. Do you think this will always be the case?

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Have students insert either the vertical number line or horizontal number line removable into their whiteboards to try to show a case when both addends are positive and the sum is a negative number. Allow students to productively struggle to formulate an instance when the addends are both positive and the sum is a negative number. Consider repeating with an instance when the addends are both negative and the sum is a positive number. What can you confirm about the sum when both addends are positive and when both addends are negative? When both addends are positive, the sum is a positive number. When both addends are negative, the sum is a negative number. Display the two expressions.

3+5

−3 + (−5)

What situations might the expressions represent? What do the sums represent? The expression 3 + 5 can represent an elevator going up 3 floors and then going up another 5 floors. The sum represents that the elevator went up 8 floors total. The expression −3 + (−5) could represent the temperature dropping 3 degrees and then dropping another 5 degrees. The sum represents the temperature dropping 8 degrees total. Introduce the concept of adding integers with different signs by having students think–pair–share about the following question. Encourage students to use their whiteboards as needed. In problem 3, Abdul earns $6 and then spends $2. What addition expression represents the situation? How can we use a number line to represent that?

6 + (−2) I think the first directed line segment would start at 0, point to the right, and end at 6. The second directed line segment would start at 6, point to the left, and have a length of 2 units.

Promoting the Standards for Mathematical Practice Students reason quantitatively and abstractly (MP2) as they discuss situations that can be represented by the given addition expressions. Ask the following questions to promote MP2: • What real-world situations are modeled by the expression? • How does this expression represent the situation? • Does your answer make sense in this situation?

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

How much money does Abdul have after these actions? How does the model show this? Abdul has $4 because he spent $2 of his $6. The second directed line segment ends at 4. Why doesn’t the second directed line segment end at −4? He earns more money than he spends, so he has a positive amount of money. Have students complete problem 3. For problems 3 and 4, write the addition expression that represents the situation. Then use the number line to model the addition expression and determine the sum. 3. Abdul earns $6. He then spends $2.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 6 + (−2) 4 4. The temperature rises 2°F from 0°F. Then the temperature drops 6°F.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 2 + (−6) −4

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Direct students to problem 4 and read the prompt aloud. What is similar about the situation in this problem and the situation in problem 3? Both situations have actions where one action can be represented by a positive number and one action can be represented by a negative number. Have students write an addition expression for the situation and use the number line to model it. Then have students compare their addition expression and model with a partner. What is similar and what is different about the model in problem 4 and the model in problem 3? In both models, the first directed line segment starts at 0. The first directed line segment in both models goes to the right because the first addend is positive. The second directed line segment in both models goes to the left because the second addend is negative. The second directed line segment in the model for problem 3 ends at a positive number, but the second directed line segment in the model for problem 4 ends at a negative number. Have students think–pair–share about the following question. Circulate and monitor student reasoning. Why is the sum in problem 4 a negative number? The sum is a negative number because the first directed line segment goes to the right, but the length of the second directed line segment is longer than the first, and it goes in the opposite direction.

Language Support Consider directing students to use the Share Your Thinking section of their Talking Tool as they explain to their partner why the sum in problem 4 is a negative number.

The sum is a negative number because −6 has a greater absolute value than 2. The sum is a negative number because when we start the second directed line segment at 2 and go 6 units to the left, the second directed line segment ends to the left of 0. Do you think there is a way to predict when sums will be positive numbers and when sums will be negative numbers when one addend is positive and one addend is negative? How? I think it depends on the greater absolute value of the addends. If the positive addend has the greater absolute value, then the sum is a positive number. If the negative addend has the greater absolute value, then the sum is a negative number.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 1

Direct students to problem 5. Have students work with a partner to determine the absolute value of each addend in the expressions. Then have them use the number line to model the expressions and determine the sum. 5. Consider the expressions. Expression

Absolute Values

5 + ( −9 )

|5| = 5 |−9| = 9

Model

−4 –10 –8 –6 –4 –2

9 + (−5)

8 10

4 0

8 10

|−5| = 5 |9| = 9

4 –10 –8 –6 –4 –2

−9 + 5

|9| = 9 |−5| = 5 –10 –8 –6 –4 –2

−5 + 9

Sum

8 10

|−9| = 9 |5| = 5

−4 –10 –8 –6 –4 –2

8 10

a. Determine the absolute values of the addends in each expression. b. Use the number line to model each expression and determine the sum.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

c. What do you notice about the distance on the number line between the first addend and the sum in each expression? I notice that in each expression the distance on the number line between the first addend and the sum is the same as the absolute value of the second addend. d. How can absolute value be used to help determine the sign of the sum? Write a conjecture. When the addend with the greater absolute value is positive, the sum is a positive number. When the addend with the greater absolute value is negative, the sum is a negative number. When most students are finished, facilitate a class discussion by using the following prompts. What do you notice about the signs of the addends in each expression? I notice that one addend is positive and one addend is negative in each expression. What do you notice about the absolute values of the addends in each expression? I notice that in each expression the absolute values are the same, 5 and 9. What do you notice about the absolute value of each sum? Every sum has the same absolute value, 4. What do you notice about the distance on the number line between the first addend and the sum in each expression? In each expression, the distance on the number line between the sum and the first addend is the same as the absolute value of the second addend. Highlight each addend that has the greater absolute value in each expression. What do you notice about the numbers you highlighted and the sums? Use this relationship to write a conjecture about how we can use absolute value to help us determine the sign of the sum.

Language Support The word sign has many mathematical and nonmathematical meanings. Facilitate a class discussion about the multiple meanings. Consider showing a picture for each of these meanings: • Road sign • Sign language • Equal sign • Inequality sign • Negative sign • The sign of a number: positive or negative

They have the same sign. When the addend with the greater absolute value is positive, the sum is a positive number. When the addend with the greater absolute value is negative, the sum is a negative number.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Direct students to problem 6. Have students complete the Sign of the Sum column and then compare their answers with a partner. Once the partners agree, have students independently complete the Model column for the expression 12 + (−10) by creating their own model. Circulate and note the approaches to the models students draw. Have students turn and talk about the following question. I see most models have a number line with tick marks drawn for each number. I bet that took a lot of time. Do you think we need a tick mark for every number in every model? Direct students’ attention to the expression −30 + 24. The magnitude of each number in this expression is pretty large. We have already predicted that the sum will be negative. It would take a while to draw all the tick marks we need to model this expression. Can you think of a way to model this expression in a simpler way, without drawing all the tick marks?

Differentiation: Support If students need more practice determining whether the sum is a positive number or a negative number, display the following expressions in rapid succession. Have students give a thumbs-up if they think the sum is a positive number and a thumbs-down if they think the sum is a negative number. • 14 + 11 • −11 + 14 • −14 + 11 • −14 + (−11)

We can start with a blank number line and model the first addend with a directed line segment. Then considering the distance of the first directed line segment is 30, we can model the second addend in the expression with a shorter directed line segment in the opposite direction because the absolute value of the second addend is less than the first addend. Display the model for the expression.

24 30

–30

–6

How does this model help us determine the sum? It helps us determine that the sum is a negative number because we can see that the sum is a number to the left of 0 on the number line. It helps us determine that the sum is −6 because the second directed line segment ends

6 units to the left of 0. © Great Minds PBC

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

We stated earlier that we can represent opposites by using subtraction. Let’s look at Shawn’s work to explore another strategy to help us determine the sum and relate addition to subtraction. Present Shawn’s work and explain that the work shown is an example of how the expression −30 + 24 is evaluated. Shawn’s work:

−30 + 24 The addends have opposite signs.

30 − 24 = 6 The negative addend in the expression has the greater absolute value, so the sum is −6. Did Shawn get the right answer? Why? Yes. In the model, the second directed line segment ends 6 units to the left of 0. What do you think Shawn’s strategy is? I think Shawn’s strategy is to determine the difference between the absolute values of the numbers when the signs of the addends are different. Because the first addend is negative and has the greater absolute value, the sum is a negative number. Use the following prompts to demonstrate how to use decomposition to find 12 + (−10). Emphasize the use of properties of operations. We can decompose 12 many ways. We can decompose 12 as 10 and 2 because 10 and −10 are additive inverses that sum to 0. We can use the commutative property of addition to change the order of the addends. We can use the associative property of addition to group the additive inverses. Have students use decomposition to find −30 + 24.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

6. Complete the table. Expression

Sign of the Sum

Model

Decomposition

12 + (−10) = 10 + 2 + (−10) = 2 + 10 + (−10) 12 + (−10)

= 2 + (10 + (−10))

Positive –6 –4 –2 0 2 4 6 8 10 12 14

24 −30 + 24

=2 −30 + 24 = −6 + (−24) + 24

Negative

=2+0

= −6 + (−24 + 24) = −6 + 0

–30

–6

= −6

Have students think–pair–share about the sign of the sum in problem 7. After sharing their thoughts with the class, have students complete the problem. Students may choose their strategy. Circulate as students work and identify different strategies. 7. Determine the sum.

− 40 + 25 + (−5) −40 + 25 + (−5) = −40 + 20 + 5 + (−5) = −40 + 20 + (5 + (−5)) = −20 + 0 = −20

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Once most students are finished, display the identified strategies and compare them. Students may have used any of the following strategies. Shawn’s work:

−40 + 25 + (−5) The first two addends have opposite signs.

−40 + 25 40 − 25 = 15 Of the first two addends in the expression, the negative addend has the greater absolute value. So the sum of the first two addends is −15.

−40 + 25 + (−5) = −15 + (−5) = −20 Number line:

5 25 40

–40

–20

–15

Decomposition:

−40 + 25 + (−5) = −15 + (−25) + 25 + (−5) = −15 + (−25 + 25) + (−5) = −15 + 0 + (−5) = −15 + (−5) = −20

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Adding Rational Numbers Students predict whether the sum of two rational numbers is a positive number or a negative number and add rational numbers. Direct students to problem 8. Do you think we can predict the sign of the sum of an expression such as 2.5 + (−3.4)? Why? Yes. I don’t think anything is different between this expression and expressions with integers, except this expression has decimals. So I think we can still predict that the sign of the sum of 2.5 + (−3.4) is negative because −3.4 has the greater absolute value. Use the following prompts to discuss how to decompose negative rational numbers.

UDL: Representation Consider presenting the information in another format. For example, draw a number bond or use a number line to model the decomposition of −3.4.

Let’s say that Sara decomposes −3.4 as −3 and −0.4. Dylan decomposes −3.4 as −3 and 0.4. Who is correct? Why?

–3.4

Sara is correct because she decomposed −3.4 into two negative numbers.

−3.4 can be decomposed into the integer part, −3, and the non-integer part −0.4.

–3

Have student complete problems 8–11 with a partner. Circulate to confirm answers. Consider asking the following questions when needed. • Will the sum be a positive number or a negative number?

0.4

–0.4 3

• How can you decompose 9.8 so that you have an additive inverse? • How can you decompose the addends in problem 10 into integer parts and non-integer parts?

–3.4 –3

• What happens when you find the sum of two negative numbers?

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

For problems 8–11, determine the sum.

Teacher Note

8. 2.4 + (−3.4)

2.4 + (−3.4) = 2.4 + (−2.4) + (−1) = (2.4 + (−2.4)) + (−1) = 0 + (−1) 3 1 9. −5 _ + (−3 _) 8 8

= −1

This activity in lesson 1 is intended to show students how the reasoning and strategies they use to add integers also apply to rational numbers. Students gain more practice in adding different types of positive and negative rational numbers in lesson 2.

−5 _ + (−3 1_) = −5 + (− _) + (−3) + (− 1_) 3 8

3 8

= (−5 + (−3)) + (− _ + (− _1)) 3 8

= −8 + (− 4_) = −8 4_

10. −3.4 + 9.8

9.8 has the greater absolute value. 9.8 − 3.4 = 6.4 The sum is a positive number because the addend with the greater absolute value is positive. The sum is 6.4.

4 1 11. −3 _ + 2 _ 7 7

−3 _4 + 2 _1 = −1 _ + (−2 _1) + 2 _1 7

3 7

= −1 _ + (−2 _1 + 2 _1)

= −1 _ + 0

3 7

= −1 _ 3 7

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

When most students are finished, facilitate a class discussion by using the following prompts. When adding rational numbers, do you prefer using a model or another strategy? Why? I prefer to start by visualizing the model in my head to help me predict whether the sum is a positive number or a negative number. However, the model doesn’t always provide the answer, so I have to rely on using other strategies. Did you choose to use additive inverses to determine the sum of the expression in problem 9? Why? No. Additive inverses are opposites. Both addends in the expression in problem 9 have the same sign, so I decomposed the addends into their integer parts and non-integer parts.

Land Debrief 5 min Objectives: Recognize that opposite integers sum to 0. Use number lines and strategies to add rational numbers. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. Can we predict whether the sum of two rational numbers is a positive number or a negative number? How? When the addends are positive, the sum is a positive number. When the addends are negative, the sum is a negative number. When one addend is positive and one addend is negative, the sum has the sign of the addend with the greater absolute value. How can we use a number line to represent addition expressions? On a number line, we can represent the first addend with a directed line segment that starts at 0 and points to the right if the addend is positive or to the left if the addend is negative. The first directed line segment is drawn directly above the number line. From the endpoint of the first directed line segment, we represent the second addend © Great Minds PBC

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

by drawing a directed line segment to the left if the addend is negative or to the right if the addend is positive. What strategies can we use to add integers and rational numbers? We can add integers and rational numbers by modeling them on a number line. We can use decomposition to make additive inverses, which sum to 0, or to make a simpler problem by decomposing numbers into integer parts and non-integer parts. We can relate addition to subtraction by subtracting the absolute values of the addends. The sum has the sign of the addend with the greater absolute value.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Recap

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1

RECAP

Name

Date

2. Last month, Ethan withdrew $53 from his savings account. This month, he withdrew another $40. a. Write an addition expression to represent the changes in the balance of Ethan’s savings account.

Adding Integers and Rational Numbers

−53 + (−40)

In this lesson, we •

Terminology

identified additive inverses and used a number line to model them.

Withdrawing money means taking money out of an account, so the amounts Ethan withdrew are represented by negative numbers in the expression.

The additive inverse of a

•

used a number line to model addition expressions.

•

predicted the sign of the sum of addition expressions.

•

used strategies to determine sums of rational numbers.

number is a number such that the sum of the two numbers is 0. The additive inverse

b. Use the number line to model your addition expression.

of a number x is the opposite of x because x + (−x) = 0.

Use a blank number line to draw a model that represents the addition

40 53

Examples

expression. Mark the location of 0 and have

1. Use the number line to model the expression and determine the sum.

8 + (−13)

Start where the first directed line segment ends: 8. Because the second

–93

Start at 0. Because the first addend is 8, draw the directed line segment to the right and make it 8 units long.

addend is −13, draw the directed

–53

the lengths of the directed line segments relate to the absolute value of the numbers they represent.

line segment to the left and make it 13 units long.

For problems 3–5, predict whether the sum of the expression is a positive number, 0, or a negative number. Explain your reasoning.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

3. −15.3 + (−5.4)

9 10

The sum is a negative number because both addends are negative.

When the addends have the same sign, the sum also has that sign.

8 + (−13) = −5 The endpoint of the second directed line segment is the value of the sum. In this case, the sum is −5.

3 1 4. 11 _ + (−5 _) 7 7

When two addends

The sum is a positive number because the absolute value of 11 _ 7 1 is greater than the absolute value of −5 _ . 7

RECAP

have opposite signs, the sum has the same sign as the addend with the greater absolute value.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

7_ + − 7_ 8 ( 8)

When the addends are opposite values, they sum to 0.

The sum is 0 because _ and − _ are additive inverses. 7 8

7 8

For problems 6 and 7, determine the sum. 6. −2 7_ + 1 1_ 8 8

7. −5.2 + (−4.5)

−2 7_ + 1 1_ = −1 6_ + (−1 _1) + 1 1_ 8

= −1 86_ + (−1 81_ + 1 8_1) = −1 6_8 + 0

−5.2 + (−4.5) = −5 + (−0.2) + (−4) + (−0.5) = (−5 + (−4)) + (−0.2 + (−0.5)) = −9 + (−0.7) = −9.7

= −1 6_8

The two addends have opposite signs, so using decomposition and additive inverses is an effective strategy. To find the additive inverse of 1 _1 , 8

The two addends have the same sign, so decomposing the addends into integer and non-integer parts is an effective strategy to determine the sum.

decompose −2 _ into −1 _ and −1 _1˜ . 7 8

6 8

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 1 Name

PRACTICE Date

2. On Monday, the temperature in Chicago was 0°F at 1:00 a.m. The temperature decreased 7°F. Then the temperature increased 10°F. a. Write an addition expression to represent the changes in temperature.

1. Sara gains 10 points.

−7 + 10

a. Describe an opposite action.

b. Use the number line to represent the addition expression you created in part (a).

Sara loses 10 points. b. Write an addition expression that represents both actions.

10 + (−10)

–10 –9

c. Use the number line to model the addition expression you created in part (b). Then determine the sum.

–8

–7

–6

–5

–4

–3

–2

–1

c. What is the temperature after these temperature changes?

3°F 10 9 8 7 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10

3. What is the additive inverse of −4.3? Explain how you know. The additive inverse is 4.3 because −4.3 + 4.3 = 0. 4. Predict whether the sum of each expression is a positive number, zero, or a negative number. Expression

Positive

−13 + 14

Zero

Negative

6 + (−17)

−2.4 + (−0.7)

14 + (− 14) 9 9

−25.4 + 6.2 6 1_ + (−3 2_) 3

X X

0 © Great Minds PBC

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

8. Consider the following addition expression.

For problems 5–7, use the number line to model the expression and determine the sum.

−73 + 18

5. 3 + (−6)

a. Use the number line to model the expression.

18 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

−3

–73

–55

b. Determine the sum.

−55

6. −6 + (−3)

For problems 9–12, determine the sum.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9. −60 + (−15) + 35

9 10

10. −4.6 + (−2.7)

−40

−7.3

−9

1 1 11. 4 _ + (−3 _) 2 2

7. −6 + 3

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

5 1 12. −8 _ + 12 _ 9 9 4 4_ 9

9 10 Remember

−3

For problems 13–16, add. 13.

P R ACT I C E

_ _

1 4 + 3 3 5 3

P R ACT I C E

14.

_ _

5 2 + 8 8 7 8

15.

_ _

1 7 + 9 9 8 9

16.

__ __

1 + 14 10 10 15 10

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 1

17. If 4 people share 9 cups of popcorn equally, how many cups of popcorn does each person get? Each person gets 2 _ cups of popcorn. 1 4

18. Use each location to plot its corresponding point on the number line. Location

Point

the opposite of 3

−1.5

the opposite of −1.5

3 4

3 the opposite of _ 4

B ˜5

˜4

˜3

C ˜2

F ˜1

E 0

D 1

A 2

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 2 LESSON 2

KAKOOMA® with Rational Numbers Use estimation and the properties of operations to add rational numbers. Add rational numbers to solve and create puzzles.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2 Name

Date

EXIT TICKET

3. Noor uses decomposition to find −3 _ + 7 _ . Her work is shown. How does she use properties 9 9 of operations in her work? 4

3 5 1. −2 + (− ) 2 8 Estimation: −2 + (−3) = −5

Sum:

−3 4_ + 7 5_ = −3 + (− 4_) + 7 + 5_

For problems 1 and 2, estimate, and then find the sum.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

= −3 + 7 + (− 4_) + 5_ 9

= (−3 + 7) + (− 4_ + 5_) = 4 + 1_

−2 3_ + (− 5_) = −2 + (− 3_) + (− 5_) 8

= 4 1_

__ = −2 + (− 3_ + (− 20 ))

__ = −2 + (− 23 )

Noor uses the commutative property of addition to rearrange the integer parts and the non-integer parts. She then uses the associative property of addition to group the integer parts together and the non-integer parts together.

= −2 + (−2 7_) = −4 7_

2. 7.29 + (−12.74) Estimation: 7 + (−13) = −6 Sum:

7.29 + (−12.74) = 7.29 + (−7.29) + (−5.45) = 0 + (−5.45) = −5.45

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Lesson at a Glance Students analyze a set of four rational numbers to determine which number is the sum of two other numbers. When finding the sum, they realize the benefits of using estimation, decomposition, and properties of operations. With a partner, students choose a strategy to find the sums of addition expressions. Next, they revisit the four rational numbers from the beginning of the lesson and notice that they solved one portion of a KAKOOMA® puzzle. Students then finish the rest of the puzzle. Once students understand how to solve the puzzle, they are asked to create a KAKOOMA® puzzle of their own. This lesson introduces the word refer.

Key Questions • How can we estimate the sum of rational numbers? • What strategies can we use to create a KAKOOMA® puzzle with sums of rational numbers?

Achievement Descriptor 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c)

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Index cards

Learn 30 min

Students

• Estimate and Evaluate

• None

• Solving a KAKOOMA®

Lesson Preparation

• Creating a KAKOOMA

Land 10 min

• Write the numbers −4 _, 6 _ , 2 _ , 1 3

5 6

1 2

and −3 on separate index cards. Create one set of index cards for each group of four students.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Fluency Add Integers Students add integers to prepare for adding rational numbers by using decomposition. Directions: Add the integers. 1.

−3 + 5

−7 + 4

−3

3 + (−5)

−2

7 + (−4)

−3 + (−5)

−8

−7 + (−4)

−11

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Launch

Students reason about the sums of rational numbers. Divide students into groups of four. Distribute a set of numbered index cards to each group. Give each group member one of the following number cards: −4 _ , 6 _ , 2 _ , or −3. Direct 1 3

groups to identify the number that is the sum of two other numbers.

5 6

1 2

Then facilitate a brief discussion to highlight the benefits of using estimation when finding a sum of rational numbers. If students do not offer any of the sample responses, present them as your own.

Teacher Note This lesson combines and expands on the learning activities from Eureka Math2 grade 7 module 2 lessons 4 and 6. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

What strategies did your group use to determine the answer? We know that the sum of two positive numbers is positive and the sum of two negative numbers is negative, but there were only two numbers with the same sign. We realized that we were looking for the sum of a positive number and a negative number. We rounded each number to the nearest integer and added integers.

1 1 We knew the answer wasn’t −4 _ because all the remaining numbers are greater than −4 _ , 3 3 1 so the sum of any two of the remaining numbers is also greater than −4 _ . Similarly, 3 6 6_5 was not the answer because two of the remaining numbers are negative and the 5 remaining positive number is less than 6 _ , so the sum of any two of the remaining 6 5 5 1 numbers is less than 6 _. Because neither −4 _ nor 6 _ could be the sum, we knew that 3

Teacher Note Students will return to their group of four when they complete the Creating a KAKOOMA® segment. They will also reuse their numbered index cards. Encourage students to place their cards in a location that will be easy to access later.

at least one of them was one of the addends.

How can we use estimation to help determine which two numbers have a sum of one of the remaining numbers? We can round the numbers to integers and then find the sum of the integers to estimate the sum of the rational numbers. Have students think–pair–share about the following question.

_ _ _ Can you use decomposition to confirm that −4 + 6 = 2 ? How? 1 3

5 6

1 2

Yes. I can decompose 6 _ into 4 _ and 2 _ and then add the additive inverses −4 _ and 4 _ .

The result is 2 _ . 50

1 2

5 6

2 6

3 6

1 3

2 6

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 2

Yes. I can decompose 6  _ into 6 and _  and then add the integer parts of the expression 5 6

5 6

and the non-integer parts of the expression. The result is 2  _ . 1 2

Record students’ reasoning as they share their thinking with the class. Sample student work is shown. Decomposition by Using Additive Inverses

−4 1_  + 6 _  = −4 2_  + 6 _  5 6

5 6

Decomposition into Integer and Non-Integer Parts

UDL: Representation

−4 1_  + 6 _ = −4 + ( − _1 )+ 6 + _  

As students share their work, highlight where they have applied properties of operations. Emphasize each occurrence with the same color, or label, to activate prior learning.

5 6

= −4 2_  + 4 2_  + 2 _  3 6

= 0 + 2 _  = 2 _ 

= −4 + 6 + ( − 1_ ) + _   5 6

=( −4 2_ + 4 2_ ) + 2 _  6

5 6

= (−4 + 6) + ( − 2_ + _  )

3 6

= 2 + _  

3 6

= 2 _ 

3 6

5 6

3 6

__ 3

3 = 2 +  __ 3 = 2  __

−4 _1 and 4 _2 is 0.

1 = 2  __

I used the additive inverse when I decomposed 6 _ into 4 _ + 2 _ because the sum of 3

3 6

1 2 I used the associative property of addition when I grouped −4 _ and 4 _ . 3

I used the commutative property of addition when I rearranged the expression so that the integer parts were together and the non-integer parts were together. Today, we will estimate and use properties of operations to add rational numbers. Then we will apply what we know to solve and create puzzles.

__ __

What is one property of operations you used, and how did you use it? 2 6

5 1 +  __ = −4 + 6 + −  __

= (−4 + 6) + ( −   2 +   5 )

5 6

( 3)

= 2 1_ 

−4   1  + 6   5 = −4 + ( −   1 ) + 6 +   5

3 6

= 2 1_ 

Decomposition into Integer and Non-Integer Parts

Commutative property of addition: Rearrange the addends so that the integer parts are together and the non-integer parts are together. Associative property of addition: Add the integer parts and add the non-integer parts.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Learn Estimate and Evaluate Students estimate and find the sum of rational numbers by applying properties of operations. Display problem 1. What can we determine about the sign of the sum? Explain. The sign of the sum is negative. The sum of two negative numbers is a negative number. The sign of the sum is negative. On a number line, the directed line segments that represent −3.45 and −1.52 both point to the left of 0. So the second directed line segment will end at a number to the left of 0. How can we estimate this sum? We can round the addends and add them. Because −3 + (−2) = −5, the sum of −3.45 and −1.52 is probably close to −5.

Differentiation: Support If students need more support with determining the sign of the sum, encourage them to use an imprecise number line and directed line segments.

What strategy can we use to determine the sum? We can decompose the addends into integer and non-integer parts. Why is decomposing rational numbers into integer parts and non-integer parts helpful? We can use the commutative property of addition to rearrange the expression so that the integer parts are together and the non-integer parts are together. Then we can apply the associative property of addition to add integer parts with integer parts and non-integer parts with non-integer parts. This strategy makes adding rational numbers more efficient.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Pair students and have them complete problem 1. For problems 1–5, estimate, and then find the sum. 1. −3.45 + (−1.52) Estimation: −3 + (−2) = −5 Sum:

−3 + (−0.45) + (−1) + (−0.52) = −3 + (−1) + (−0.45) + (−0.52) = (−3 + (−1)) + (−0.45 + (−0.52)) = −4 + (−0.97) = −4.97 Have pairs compare answers with another pair. Then use the following prompts to guide student thinking. If we were evaluating 3.45 + (−1.52), would our approach be different? Explain. Yes. The addends 3.45 and −1.52 have different signs, so we cannot just add all the integer parts and non-integer parts. Instead, we can decompose 3.45 into 1.93 + 1.52, so we can use additive inverses. Have pairs complete problems 2–6. Students may choose any strategy to evaluate the expressions, as shown in the sample student work.

UDL: Action & Expression As students evaluate each expression, prompt them to show and explain their work. Showing all steps reduces the likelihood of errors and allows for easier identification of errors if they occur. When students are given the opportunity to think about their thinking, they can identify their own errors and consider what steps they might do differently, or the same way, the next time.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

5 1 2. −13 _ + 8 _ 4 8

1 1 3. −6 _ + (−1 _) 2 3

Estimation: −13 + 9 = −4

Estimation: −7 + (−1) = −8

Sum:

Sum: The addends have the same sign.

−13 1_ + 8 _ = −13 + (− 1_) + 8 + _ 4

5 8

6 1_ + 1 1_ = 6 _ + 1 2_

5 8

= −13 + 8 + (− 2_) + _ 5 8

= (−13 + 8) + (− 2_ + _) 8

= −5 + _

5 8

3 8

= −4 _ + (− _) + _ 5 8

3 8

5 8

3 8

3 6

= 7_

5 6

The addends are both negative, so the sum is a negative number. The sum 5 is −7 _ . 6

3 8

= −4 _ + (− _ + _)

= −4 _ + 0

3 8

5 8

= −4 _ 5 8

4. −9.35 + 9.75

5. 7.63 + (−10.42)

Estimation: −9 + 10 = 1

Estimation: 8 + (−10) = −2

Sum:

−9.35 + 9.75 = −9.35 + 9.35 + 0.4 = (−9.35 + 9.35) + 0.4 = 0 + 0.4 = 0.4

The addends have opposite signs.

10.42 − 7.63 = 2.79 The negative addend has the greater absolute value, so the sum is negative. The sum is −2.79.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

6. Vic has $52.13 in his checking account. At midnight, a debit of $3.06 and a debit of $18.24 will post to his account. a. Write an addition expression to represent the situation. Then estimate the sum.

52.13 + (−3.06) + (−18.24) Estimation: 52 + (−3) + (−18) = 52 + (−21) = 31 b. What is the value of the expression in part (a)?

52.13 + (−3.06) + (−18.24) = 52 + 0.13 + (−3) + (−0.06) + (−18) + (−0.24) = 52 + (−3) + (−18) + 0.13 + (−0.06) + (−0.24) = 52 + (−3 + (−18)) + 0.13 + (−0.06 + (−0.24)) = 52 + (−21) + 0.13 + (−0.30) = (52 + (−21)) + (0.13 + (−0.30))

Language Support Problem 6 refers to a context that may be unfamiliar to some students. Consider facilitating a class discussion to preview the terminology of checking accounts, debits, withdrawals, credits, and deposits by facilitating a class discussion. Ask students what they know about bank accounts. Emphasize the following ideas and terminology: • Debits and withdrawals result in deducting money from an account. Some examples of debits include purchases, bills, and fees. • Credits and deposits result in adding money to an account. Some examples of credits are refunds from returned purchases, paychecks, and earned interest.

= 31 + (−0.17) = 30.83 + 0.17 + (−0.17) = 30.83 + 0 = 30.83 c. What is the amount of money in Vic’s checking account after the debits post? Vic has $30.83 in his checking account after the debits post. Have pairs compare answers with another pair. Then use the following prompt to debrief and clarify misunderstandings. Did you find any of the problems more complicated than the others? Explain how you used properties of operations to help you find the sum. Yes. In problem 2, we used decomposition, but when we added the integer parts

separately from the non−integer parts, we had sums of −5 and _ . Then we decomposed 3 8

−5 into −4 _5 + (− _3) so that we could use additive inverses. Because − _3 + _3 = 0, the sum 8 3 8

8 5 8

of −5 and _ is −4 _ .

Teacher Note Highlight various strategies students use when finding each sum: • Using decomposition to create additive inverses • Using decomposition to group integer parts together and non-integer parts together • Consider the sign of each number to determine whether to add the absolute values of the numbers and keep the signs of the addends or to subtract the absolute values of the addends and keep the sign of the addend with the greatest absolute value.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Yes. In problem 6, the sum of three numbers was more complicated than the sum of two numbers. After we used decomposition and added the integer parts separately from the non-integer parts, we had sums of 31 and −0.17. We decomposed 31 into 30.83 + 0.17 so that we could use additive inverses. Because 0.17 + (−0.17) = 0, the sum of 31 and −0.17 is 30.83.

Solving a KAKOOMA® Students add rational numbers to solve a puzzle. Introduce the KAKOOMA® puzzle and read aloud the directions for problem 7. Direct students to the top square of the puzzle. The numbers in the top piece of the puzzle are the same numbers that we analyzed at the beginning of class. Which number is the solution to the top piece of the puzzle? Explain. The number 2 _ is the solution to the top piece of the puzzle because −4 _ + 6 _ = 2 _ . 1 2

1 3

5 6

1 2

Have students write 2 _ in the top square of the final puzzle as indicated by the arrows. 2 Then have pairs work to solve problem 7. 1

7. In each four-number square, find the one number that is the sum of two other given numbers. Transfer all four sums to the puzzle below to create one final puzzle, and then solve.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

1st Sum

–4–31

–3

2–12

6–56

–3.2 0.4

5.1 6.8

–0.8 3.6

–2.1 –7.2

–34

––14

Find the number that is the sum of 2 others. a

–1

––12

2–12

Final Answer c

0.4 d

–2.1

0.4

––14

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

After pairs complete problem 7, have them compare their answers and strategies with another pair. Then facilitate a class discussion. Did you try every combination of addends to determine which sums you needed for the puzzle? If not, what did you do instead? No, we did not try every sum. In the square on the right side, we knew that the sum of two negative numbers is negative. Because −2.1 and −7.2 are the only two negative numbers in the square, there was no reason to find their sum because there is not a third negative number available for the possible sum. Similarly, we did not find the sum of 5.1 and 6.8 because the sum is positive, but there is not another positive number available for the possible sum. Was the final puzzle easier or more difficult for you to solve than the four original squares? Why? The final puzzle was more difficult for us to solve than the original squares because some of the numbers are written in decimal form while other numbers are written in fraction form. The final puzzle was more difficult for us to solve than the original squares because all of our answers from the original squares had to be correct for us to be able to answer the final puzzle. The final answer is 0.4. Which two numbers in the final puzzle have a sum of 0.4?

2 2_1 and −2.1

How did you find the sum of 2 and −2.1? 2

We know that 2 _ can be written as 2.5. We found the sum by decomposing 2.5 into 2 2.1 and 0.4, so we could use additive inverses. The sum of 2.1 and −2.1 is 0, and 0 + 0.4 = 0.4. 1

Do you think you can create this type of puzzle? How? Yes. I think I can create this type of puzzle by working backward and listing four numbers where two of the numbers add up to one of the others.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Creating a KAKOOMA® Students work backward with rational numbers to create a new KAKOOMA® puzzle. Direct students to problem 8. Read the first two sentences of the prompt to the class. Facilitate a class discussion focusing on students’ prior knowledge of the word refer. What do you think the word refer means in the prompt? I think it means to look at the KAKOOMA® puzzle that we solved in problem 7. In the prompt, refer does suggest that we look back at the KAKOOMA® puzzle that we solved in problem 7. What other words do you think of when you hear the word refer? I think of examine, review, analyze, or use. Finish reviewing the remaining directions with the class. Then have students transition back into their groups of four. Ask students to turn their original numbered index cards facedown on the desk, have a student in each group shuffle the four cards, and then have another student choose one of the cards. The number that the student chooses must be the solution to the KAKOOMA® puzzle created by the group. Direct students to write this number in the Final Answer square on their blank KAKOOMA® puzzle. Give students 8 minutes to work on this task. After each group of students has created a puzzle, lead a debrief.

Language Support This is the first use of the word refer in the curriculum. As students work, encourage them to actively use the synonyms to replace the word refer to make sense of the problems. Some students may need to be directed to consistently use one synonym, such as use.

Differentiation: Challenge Direct groups to exchange their puzzle with another group. Have each group solve the new puzzle. Encourage groups to critique the puzzle they are solving by making sure that each four-number square has only one solution and that the final puzzle has only one solution. Have students leave feedback about the puzzle on the back of the puzzle paper and return it to its creators. If the feedback indicates that the puzzle has multiple solutions, ask the creators of the puzzle to modify their puzzle so it has only one solution.

7–8 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

Describe your strategies for creating the puzzle. Did those strategies work? We started with one of the outer squares and found four numbers where only one number was the sum of two others. When we tried this on each of the four squares, we had a hard time making sure that the four numbers gave us the right mix of numbers so that the final puzzle would have only one solution that matched our group’s number. We worked backward. We started with the solution and determined two numbers that have a sum equal to the solution. Then we thought of one more number to include in the list. We checked each of the numbers by adding each number to the other three numbers in the square to make sure only one number was the sum of two others. If we had two numbers that could have been the solution to the puzzle, we changed numbers within the list so there would only be one solution. 8. Create your own KAKOOMA® puzzle by using the blank puzzle shown. Refer to problem 7, and make sure that your KAKOOMA® puzzle follows this structure. Use rational numbers between −10 and 10. Each four-number square must include at least two non-integer rational numbers and at least one number less than 0. The same number cannot be used more than once in a square.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Promoting the Standards for Mathematical Practice

1st Sum

As students add rational numbers to create a KAKOOMA® puzzle by finding entry points, monitoring their own progress, and questioning whether the values chosen for each square make sense, they are making sense of problems and persevering in solving them (MP1). Ask the following questions to promote MP1: • What are some things you can try to start solving the problem? • What facts do you need to determine the four numbers within one of the square sections of the puzzle? • Does starting with the outer squares work? Can you try something else?

Find the number that is the sum of 2 others.

b Final Answer

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Land Debrief 5 min Objectives: Use estimation and the properties of operations to add rational numbers. Add rational numbers to solve and create puzzles. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. How can we estimate the sum of rational numbers and how is it helpful? We can estimate the sum of rational numbers by rounding the rational numbers to integers and then finding the sum of the integers. Estimating the sum helps us to know the sign of the actual sum and whether the actual sum is reasonable. What properties of operations do we use while using decomposition? Give an example. We use the commutative and associative properties of addition. For example, when

we add −2 _ + 3 _ , we can use the commutative property to rearrange the integer parts 1 4

3 4

and the non-integer parts. Then we can use the associative property to group the integer parts together and the non-integer parts together. What strategies can we use to create a KAKOOMA® puzzle with sums of rational numbers? When creating a puzzle, we can start with the answer and find two rational numbers that have a sum equal to the answer. Then we can pick another number to complete the set of four numbers, making sure that no other pair of numbers has a sum equal to another number in the set. By continuing to work backward, we can pick numbers that have a sum equal to the number we need in each of the four-number square pieces.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem. 62

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

2. −15.6 + (−3.25) Estimation: −16 + (−3) = −19 Sum:

KAKOOMA® with Rational Numbers

−15.6 + (−3.25) = −15 + (−0.6) + (−3) + (−0.25)

In this lesson, we

= −15 + (−3) + (−0.6) + (−0.25)

•

estimated the sums of addition expressions.

•

used decomposition and properties of operations to add rational numbers.

•

evaluated addition expressions to solve and create puzzles.

= (−15 + (−3)) + (−0.6 + (−0.25)) = −18 + (−0.85) = −18.85

Examples 1. −5 2_ + 2 1_ 3 3 Estimation: −6 + 2 = −4

Estimate the sum by rounding each addend to an integer. Then add the integers.

Sum:

3. −21.4 + 18.45

−5 2_ + 2 1_ = −3 1_ + (−2 1_) + 2 1_ 3

Estimation: −21 + 18 = −3

= −3 1_3 + (−2 1_3 + 2 1_3)

Sum:

= −3 _13 + 0

strategy is to decompose one of the addends and use additive inverses.

Apply the commutative property of addition to rearrange the integer parts and the non-integer parts.

Apply the associative property of addition to group the integer parts together and the non-integer parts together.

For problems 1–3, estimate, and then find the sum.

The addends have opposite signs, so an effective

The addends have the same sign, so decomposing each addend into its integer part and non-integer part is an effective strategy.

= −3 1_3

The addends have opposite signs, so finding the difference between the absolute values of the two addends is an effective strategy.

The addends have opposite signs.

21.4 − 18.45 = 2.95

The additive inverse of

The negative addend has the greater absolute value, so the sum is negative. The sum is −2.95.

2 _1 is −2 _1 . Decompose 3

the rational number −5 _2 3

into −3 _1 and −2 _1 to use 3

additive inverses.

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

4. In the five-number pentagon, find the one number that is the sum of two other numbers. It is not necessary to find the sum of every

–3–41 –1–43

pair of numbers. Because 2 and 1 _1 are 2

the only two positive numbers and the sum of two positive numbers is a positive number, their sum is not one of the

–3

remaining numbers.

1–21 © Greg Tang

−3 1_ + 1 1_ = −3 + (−1_) + 1 + 1_ 4

= (−3 + 1) + (− 1_4 + 2_4) = −2 + 1_4

= −1 3_4 + (− 1_4) + 1_4 = −1 3_4

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 2

PRACTICE

Name

Date

2 1 1. Jonas uses decomposition to find −7 + 5 . His work is shown. How does Jonas use properties 3 3

For problems 3–11, estimate, and then find the sum. 3 4 3. 4 _ + (−3 _) 5 5

Estimation: 5 + (−4) = 1 Sum: 1 _

of operations in his work?

1 1 4. −20 _ + (−2 _) 3 4

Estimation: −20 + (−2) = −22 Sum: −22 __ 7 12

1 5

−7 2_ + 5 1_ = −7 + (− 2_) + 5 + 1_ 3

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

= −7 + 5 + (− 2_3) + 1_3

= (−7 + 5) + (− 2_3 + 1_3) = −2 + (− 1_3)

= −2 1_3

5. −13.7 + (−2.05)

Jonas uses the commutative property of addition to rearrange the integer parts and the non-integer parts. He then uses the associative property of addition to group the integer parts together and the non-integer parts together.

Estimation: −14 + (−2) = −16 Sum: −15.75

2 1 6. −10 _ + 4 _ 3 6

Estimation: −11 + 4 = −7 Sum: −6 _ 3 6

2. Ava uses decomposition to find 8.14 + (−8.34). Her work is shown. How does Ava use properties of operations in her work?

8.14 + (−8.34) = 8.14 + (−8.14) + (−0.2)

3 1 7. −5 _ + 1 _ 4 8

= (8.14 + (−8.14)) + (−0.2)

Estimation: −5 + 2 = −3

= 0 + (−0.2)

3 Sum: −3 _

= −0.2

Ava uses additive inverses and then uses the associative property of addition to group the additive inverses together.

P R ACT I C E

8. 0.75 + (−2.6) Estimation: 1 + (−3) = −2 Sum: −1.85

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

9. 5.15 + (−3.78)

7–8 ▸ M1 ▸ TA ▸ Lesson 2

9 4 10. −5 _ + 2 __ 5 10

Estimation: 5 + (−4) = 1 Sum: 1.37

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

13. Ethan has $147.89 in his savings account. On the last day of the month, the bank posted a monthly service charge of $5.00 and interest earned of $0.49.

Estimation: −6 + 3 = −3 Sum: −2 __

a. Write an addition expression to represent the situation. Then estimate the sum.

9 10

147.89 + (−5.00) + 0.49 Estimation: 148 + (−5) + 0 = 143

1 2 11. 10 _ + (−8 _) 5 6

Estimation: 10 + (−8) = 2

b. What is the value of the expression in part (a)?

Sum: 1 __ 23 30

143.38

12. Yu Yan has $35.27 in her checking account. At midnight, debits of $12.82 and $4.93 will post to her account.

c. What is the amount of money in Ethan’s savings account after the transactions post? Ethan has $143.38 in his account after the transactions post.

a. Write an addition expression to represent the situation. Then estimate the sum.

35.27 + (−12.82) + (−4.93) Estimation: 35 + (−13) + (−5) = 17 14. Shawn evaluated the following expression but got an incorrect answer. Find and describe any errors Shawn made in his work. 5 3 5 3 + −5 __ + (−5) + __ 8 __ = 8 + __ 11 ( 11) 11 11

b. What is the value of the expression in part (a)?

17.52

5 3 + __ = (8 + (−5)) + (__ 11 11) 8 = 3 + __ 11

8 = 3 __ 11

c. What is the amount of money in Yu Yan’s checking account after the debits post? Yu Yan has $17.52 in her checking account after the debits post.

Shawn decomposed −5 __ as −5 + __ , but −5 __ should be decomposed as −5 + (− __). 3 11

P R ACT I C E

3 11

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

15. In each five-number pentagon, find the one number that is the sum of two other numbers. Use all five sums to create one final puzzle and solve.

–2 5 –1 3 –7

1 1– 5

–3 5

1.3

4 –2– 3 –– 5 5

2.6 –2.4

–1–54

1.4

For problems 16–19, add.

–3.8

0.4

17.

_5 + − 2_ 2 ( 2) _3 2

18.

_1 + − 7_ 9 ( 9) − _6 9

19.

1 __ __ + − 14

10 ( 10) 13 − __ 10

20. Lily walks 60 feet in 10 seconds. Maya walks 25 feet in 5 seconds. Who walks at a faster rate? Explain how you know.

–0.7 –0.6

–3

Remember 16. − 1_ + 5_ 3 3 4_ 3

2 –– 5

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 2

1.1

Lily walks at a faster rate. Lily walks at a rate of 6 feet per second. Maya walks at a rate of

5 feet per second. So Lily walks at a faster rate.

21. Which equation correctly models the following statement?

−30 is 30 units from 0 on the number line.

6.4

A. (−30) = 30 B. −30 = 30 C. |30| = −30

b a

–3 e

3 –2– 5

D. |−30| = 30

Final Answer

2 –– 5

0.4

3 –2– 5

2.6

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 3 LESSON 3

Finding Distances to Find Differences Show that the distance between two integers on a number line is the absolute value of their difference. Evaluate integer subtraction expressions by finding the unknown addends in related addition equations.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

EXIT TICKET

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3 Name

Date

1. Consider the expression 7 − 12. a. Draw a number line and plot the integers in the expression.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10 11 12

b. What is the distance on the number line between the integers in the expression?

5 units

Lesson at a Glance In this lesson, students explore different ways of thinking about subtraction to understand how to subtract integers. They realize that some subtraction problems involving integers are impossible to interpret as “taking away” a number, so students look for other ways of understanding subtraction. Students practice using number lines and writing unknown addend equations to find unknown differences. By observing a number line, students also notice patterns in the distance between two numbers and the difference of those two numbers.

c. Write the expression as an unknown addend equation.

12 +

Key Questions

d. What is the unknown addend?

• What are some ways we can model a subtraction problem?

−5

• What are some strategies we can use to subtract integers?

2. Consider the expression −4 − (−10). a. Draw a number line and plot the integers in the expression.

–10

–9

–8

–7

–6

–5

–4

–3

Achievement Descriptors –2

–1

7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients

of two rational numbers. (7.NS.A.1.d; 7.NS.A.2.c)

b. What is the distance on the number line between the integers in the expression?

6 units

7–8.Mod1.AD4 Show on a number line that the distance between two

rational numbers is the same as the absolute value of their difference, and apply this principle in real-world contexts. (7.NS.A.1.c; 7.NS.A.3)

c. Write the expression as an unknown addend equation.

−10 +

= −4

d. What is the unknown addend?

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Relating the Distance

• None

• Finding the Unknown

Lesson Preparation

• Reasoning About Integer Subtraction

• None

Land 10 min

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Fluency Use a Number Line to Add Students write an addition expression for each model to prepare for subtracting integers. Directions: Write the addition expression represented by each model. 1.

8 + (−4)

–10 –8 –6 –4 –2 0

8 10

6 + (−6)

–10 –8 –6 –4 –2 0

8 10

−4 + (−2)

–10 –8 –6 –4 –2 0

8 10

−4 + 2

–10 –8 –6 –4 –2 0

8 10

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Students observe patterns in number sentences. Present the subtraction expression 10 − 8. How can you interpret the expression 10 − 8?

10 take away 8 10 minus 8 The difference of 10 and 8 Take 8 from 10

Language Support As needed throughout this lesson and topic, have students recall that the term difference has multiple meanings. In general, this lesson uses difference to indicate the result of a subtraction expression.

Facilitate a brief discussion to prompt further student thinking about different ways to conceptualize subtraction. How far apart are 10 and 8 on a number line?

Teacher Note

2 units What number can we add to 8 to get to 10?

2 How much is left when we take 8 away from 10?

2 What do you notice about all these answers? The result is always the same. Depending on the numbers in the situation, we can reason about subtraction by thinking about how far apart two numbers are on a number line, by determining an unknown addend, or by thinking about what remains after taking away.

Some students might offer a comparison interpretation for 10 − 8. They could interpret 10 − 8 as “10 is how much greater than 8?” This interpretation is accurate in this case, but interpreting 8 − 10 the same way could introduce confusion. Following suit, students may interpret 8 − 10 as “8 is how much greater than 10?” However, that does not produce an accurate response. Should the conversation arise, encourage students to make conjectures as to why this interpretation does not translate to all subtraction expressions.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Present the table showing a series of numbers being subtracted from 10.

10 − 8 = 2 10 − 9 = 1 10 − 10 = 0 10 − 11 = 10 − 12 = Invite students to think–pair–share about the following question. Can you determine the unknown values in the last two rows of the table? How? Yes. I can follow the pattern to determine that 10 − 11 = −1 and 10 − 12 = −2. For problem 1, invite students to write a conjecture about how to subtract integers by using the pattern they see in the table. Challenge them to explain their conjecture. 1. Write a conjecture about how to subtract integers by using the pattern you see in the displayed table. Explain how the pattern you see supports your conjecture. Sample: I can visualize subtraction on a number line. For example, if I start at 10 on a number line and take away 10, then I end up at 0. By taking 11 away from 10, I go one unit past 0 to the left and end up at −1. By taking 12 away from 10, I go two units past 0 to the left and end up at −2. I notice that when I subtract a larger positive number from a smaller positive number, the difference is a negative number.

Differentiation: Challenge Challenge students by presenting them with other examples of integer subtraction, such as 10 − (−23), −10 − 23, and −10 − (−23), and having them think about how their conjecture might apply to those expressions.

When students are finished, invite them to share their conjectures with the class. Challenge students’ thinking by asking them how to find the unknown value to a problem such as 10 − 23, where the answer is not readily apparent from the pattern in the table.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Have students turn and talk about the problem and how they might approach finding a solution. Do not answer 10 − 23 =

yet to pique student interest in the next segment. UDL: Representation

Today, we will explore how to evaluate any integer subtraction expression.

Learn Relating the Distance Students calculate the distance between integers on a number line and make observations about how the distance relates to a subtraction expression. Display the blank number line and the expression 10 − 8 from problem 2. Model how to find the distance between 10 and 8 by plotting points on the number line.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Have students recall that distance is a measurement, so it is always represented as a nonnegative number. Have students record the distance, 2, in the Distance Apart on the Number Line column. Then have them work in pairs and use the number line to complete the column for problems 3–9. As students work, circulate and listen for possible misconceptions about which integers are being subtracted and how to find the distance between them.

Model how to use precise language when reading subtraction expressions. Additionally, consider using a visual cue, such as color-coding or highlighting, to emphasize the two integers in a subtraction expression. For example, use a visual cue to show that the two integers in 8 − 10 are 8 and 10, while the two integers in 8 − (−10) are 8 and −10. Display a chart like the one shown for students to reference.

p−q

Values of p and q

10 − 8

p = 10, q = 8

8 − 10

p = 8, q = 10

−8 − 10

p = −8, q = 10

−8 − (−10)

p = −8, q = −10

Differentiation: Support If needed, encourage students to use their personal whiteboards with the Number Lines removable from lesson 1 to plot the points and find the distance for each problem.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

For problems 2–9, use the number line to find the distance between the two integers in the subtraction expression. Record your answer in the column labeled Distance Apart on the Number Line.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

Subtraction Expression

Distance Apart on the Number Line (units)

10 − 8

8 − 10

10 − (−8)

8 − (−10)

−10 − 8

−8 − 10

−8 − (−10)

−10 − (−8)

Unknown Addend Equation

9 10

Unknown Addend

Select a few students to share their answers. Address any misconceptions that arise. Facilitate a class discussion about patterns that students notice in the expressions and the distances between the integers in the expressions. Use the following questions as a guide.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

Are any of the subtraction expressions the same? No. Are any of the distances between the two integers in the expressions the same? Yes. Look at the subtraction expressions 10 − 8 and 8 − 10. The integers in the expressions are the same distance apart on the number line. What do you notice about these expressions? They have the same integers, 8 and 10, but they are being subtracted in a different order. We know that in an addition expression, addends can be added in any order and the sum is the same. So 8 + 10 has the same sum as 10 + 8. Can numbers be subtracted in any order? Explain. No. We can’t subtract numbers in any order. If we have 10 objects, we can take 8 away. But if we have 8 objects, we don’t have enough to take 10 away. Look at the subtraction expressions in the table with integers the same distance apart on the number line. Do they also have the same integers being subtracted but in a different order? Give some examples. No. The expressions −10 − 8 and −8 − 10 both represent 18 units on the number line, but the integers in the expressions are not the same. The integers in the first expression are −10 and 8, and the integers in the second expression are −8 and 10. In which expressions do the two integers have the same sign?

10 − 8, 8 − 10, −8 − (−10), and −10 − (−8) What is similar about the expressions in which the two integers have the same sign? What is different? The expressions 10 − 8 and 8 − 10 both have the integers 8 and 10, which are 2 units apart on a number line. The expressions −8 − (−10) and −10 − (−8) both have the integers −8 and −10, which are 2 units apart on a number line. The expressions are different because the integers are subtracted in a different order. In which expressions do the two integers have different signs?

10 − (−8), 8 − (−10), −10 − 8, and −8 − 10 © Great Minds PBC

7–8 ▸ M1 ▸ TA ▸ Lesson 3

EUREKA MATH2

What is similar about the expressions in which the two integers have different signs? What is different? Each expression has one positive number and one negative number. The expressions are different because the integers are subtracted in a different order. The expressions have either the integers −8 and 10 or −10 and 8. The integers in each expression are 18 units apart on a number line. At this point, encourage student observations, questions, and conjectures about the relationship between distance and the value of the subtraction expression. It is not yet necessary for students to make any formal observations about subtracting integers.

Finding the Unknown Students write integer subtraction expressions as unknown addend equations and find the unknown addends. Direct student attention back to problem 2. Use the following prompts to help students recall how to write a subtraction expression as its related unknown addend equation. We can think of the expression 10 − 8 as asking the question, What number do I need to add to 8 to make 10? How can we represent the subtraction expression 10 − 8 as an unknown addend equation?

= 10

The number that we add to 8 to make 10 is called the unknown addend. What is the unknown addend in this equation?

2 What is the value of 10 − 8?

2 What do you notice about the unknown addend and the difference of the two integers? They’re the same number.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

Direct students to complete the Unknown Addend Equation and Unknown Addend columns for problem 2. Display the number line that shows the directed line segment from 0 to 8.

We can also use the number line to help us identify the unknown addend. We will start with a problem we already know the answer to so we can focus on how to use the number line with more complex problems. The number line shows a directed line segment to represent 8, the first number in the unknown addend equation. In problem 2, we are looking for the number we add to 8 to make 10. What do we need to draw on our model to get from 8 to 10?

Differentiation: Support When students first learn to subtract, they relate subtraction problems to known addition facts by writing a subtraction expression as the related unknown addend equation. The number bond is a useful tool to help students visualize and comprehend how subtraction relates to addition. To help students make this connection for integer subtraction expressions, model how to draw a number bond diagram while asking, “What do I add to 10 to get 8?”

Draw a directed line segment that starts at 8, points to the right, and is 2 units long to get from 8 to 10. Using the same number line, draw a directed line segment from 8 to 10, or display the following number line.

10 0

The unknown addend, 2, that we add to 8 to get 10 is represented by the directed line segment that is 2 units in length and points to the right. Examine problem 3, which is 8 − 10. What is the unknown addend equation related to this subtraction expression?

10 +

–2

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Direct students to complete the Unknown Addend Equation column for problem 3. Display the number line with the directed line segment from 0 to 10.

The image shows a directed line segment that represents 10. We are looking for the number we add to 10 to make 8. What do we need to draw on our model? Draw a directed line segment that starts at 10, points to the left, and is 2 units long to get from 10 to 8. Using the same number line, draw a directed line segment from 10 to 8, or display the following number line. Promoting the Standards for Mathematical Practice

What is the unknown addend?

−2 What is the value of 8 − 10 ?

−2 Direct students to complete the Unknown Addend column for problem 3. How are the model and unknown addend in problem 2 different from the model and unknown addend in problem 3? In problem 2, the unknown addend is a positive number, and the directed line segment that represents it goes to the right and has a length of 2 units. In problem 3, the unknown addend is a negative number, and the directed line segment that represents it goes to the left and has a length of 2 units.

When students use the relationship between addition and subtraction to write integer subtraction expressions as unknown addend equations and relate the unknown addend to the number line, they are making use of structure (MP7). Ask the following questions to promote MP7: • How are addition and subtraction related? How can that help you solve integer subtraction problems? • How can what you know about addition help you with subtraction? • How is this problem similar to problems that you have used a number line and a model to solve?

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

Have students work with their partners to complete the Unknown Addend Equation and Unknown Addend columns for problems 4–9. For problems 2–9, write the related unknown addend equation. Then use the number line to find the unknown addend. Record your answers in the columns labeled Unknown Addend Equation and Unknown Addend.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

Subtraction Expression

Distance Apart on the Number Line (units)

10 − 8

= 10

8 − 10

10 +

−2

10 − (−8)

−8 +

= 10

8 − (−10)

−10 +

−10 − 8

= −10

−18

−8 − 10

10 +

= −8

−18

−8 − (−10)

−10 +

= −8

−10 − (−8)

−8 +

= −10

−2

Unknown Addend Equation

Unknown Addend

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Display the models for the unknown addend equations in problems 4–9. 4.

10 – (–8) –8 + ____ = 10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 5.

9 10

–10 – 8 8 + ____ = –10

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

8 – (–10) –10 + ____ = 8

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 6.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

–8 – 10 10 + ____ = –8

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 8.

9 10

–8 – (–10) –10 + ____ = –8

–10

–9

–8

–7

–6

–5

–4

–3

–2

–1

–3

–2

–1

–10 – (–8) –8 + ____ = –10

–10

–9

–8

–7

–6

–5

–4

7–8 ▸ M1 ▸ TA ▸ Lesson 3

EUREKA MATH2

Select a few students to share their unknown addend equations and answers. Facilitate a brief discussion. What is the relationship between the unknown addend and the value of the subtraction expression? Every subtraction expression can be written as an unknown addend equation, and the unknown addend is the difference of the two integers. How is the distance between two integers on a number line related to their unknown addend? Give an example. The distance between two integers on a number line is the absolute value of the difference of the two integers. For example, the distance between 8 and 10 is 2 because 10 − 8 = 2 and 8 − 10 = −2. How can we use unknown addends and directed line segments to determine whether a difference is a positive number or a negative number? To determine whether a difference is a positive number or a negative number, we need to know which direction the directed line segment is pointing to reach the sum of the unknown addend equation. When the directed line segment points to the right, as in 8+ = 10, the unknown addend, or difference, is positive. When the directed line segment points to the left, as in 10 + = 8, the unknown addend, or difference, is a negative number. Invite students to think–pair–share about the following questions. Circulate and listen for students who begin to understand the meaning and structure of integer subtraction. Ask them to share their responses with the class. Look at problems 2–9 in the table. What patterns do you notice? What do you know so far about subtracting integers? The distance between the two integers on the number line is always the absolute value of their difference. To determine whether the difference is a positive or negative number, I need to look at the order of the integers in the subtraction expression. When the first number in a subtraction expression is greater than the second number, the difference is a positive number. I know this because when I draw a model to represent the unknown addend equation, I draw a directed line segment that points to the right to reach the sum.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

When the first number in a subtraction expression is less than the second number, like in 8 − 10, the difference is a negative number. I know this because when I draw a model to represent the unknown addend equation, I draw a directed line segment that points to the left to reach the sum.

Reasoning About Integer Subtraction Students reason about integer subtraction. Introduce the Critique a Flawed Response routine and present problem 10.

Differentiation: Challenge If students need additional challenge, consider asking them to fill in the blanks to make true number sentences.

8−

(−4)

= 12

−10 −

= −17

Give students 1 minute to identify the error and explain it in part (a). Then invite students to share what they noticed. Call attention to the following error if students do not share it: • The work shows subtracting 45 from 32 instead of subtracting −45 from 32. Give students 1 minute to correctly evaluate the expression for part (b) based on their own understanding. Circulate to ensure that students correct the flawed response and get the answer 77. If you notice students with different answers, invite a few students to share their work with the class until there is consensus about the correct response. 10. Consider the following sample work.

32 − (−45) = 32 − 45 = −13 a. Explain an error in the sample work. Sample: The work shows that 32 − (−45) is equal to 32 − 45, but those subtraction expressions are not the same. The subtraction expression 32 − (−45) has the related unknown addend equation −45 + = 32, and the unknown addend must be a positive number. The subtraction expression 32 − 45 has the related unknown addend equation 45 + = 32, and the unknown addend must be a negative number.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

b. Correctly evaluate the expression.

32 − (−45) −45 +

= 32

−45 + 77 = 32 77 Invite students to complete problem 11 with a partner. 11. Sea level is represented by 0 feet. A gannet bird is at an elevation of 28 feet. It dives straight down to an elevation of −19 feet to get a fish. How many feet does the gannet dive? Draw a model to represent the distance that the gannet dives.

–19

Teacher Note Students are familiar with both a horizontal and a vertical number line from prior grades. Throughout this topic, they have opportunities to choose how to use a number line to model situations. Consider displaying different examples of student work for problem 11 to emphasize that a variety of models can accurately represent the situation. For example, some students may use a horizontal number line or may use directed line segments to represent the distance that the gannet dives as an unknown addend.

–19

28 47

The gannet dives 47 feet. When students are finished, invite them to share their answers and reasoning for problem 11.

–19

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

What subtraction expression represents this situation? How do you know? The expression 28 − (−19) represents this situation because we need an expression to represent the difference between 28 and −19, or the distance between those two integers on a number line. How does a model help you determine the distance the bird dives? The model helps me visualize that the difference of 28 and −19 is the distance between the two integers on a number line. Ask students to revisit problem 1 and refine their conjecture about subtracting integers based on patterns they noticed in this lesson.

Land Debrief 5 min Objectives: Show that the distance between two integers on a number line is the absolute value of their difference. Evaluate integer subtraction expressions by finding the unknown addends in related addition equations. Initiate a class discussion by asking the following questions. Encourage students to restate or add on to their classmates’ responses. Earlier, I posed the problem 10 − 23. What do you know now that helps you determine the difference? I can think about this problem as an unknown addend equation and ask the question, What number do I add to 23 to get 10? Because 10 is less than 23, I know the unknown addend is a negative number. I also know that by using a number line, I can model an unknown addend equation with directed line segments. For this problem, the first

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

directed line segment goes to the right from 0 to 23, and the second line segment goes to the left from 23 to 10. The unknown addend is represented by the direction and the length of the second line segment, which is 13 units. So the difference is −13. How is subtraction involving integers different from subtraction you have seen previously? How is it similar? It is possible for a subtraction problem involving integers to take away a negative amount. We can’t always physically take away an amount to figure out the difference when we subtract integers. However, subtraction problems involving integers and involving whole numbers can both be thought of as unknown addend equations. Subtraction problems can also be represented on a number line. What are some ways we can model a subtraction problem involving integers? We can plot the two integers in the subtraction problem on a number line, find the distance between them, and write the problem as an unknown addend equation. For example, to evaluate 8 − 10, we can look for the number that can be added to 10 to get 8, which is −2. The directed line segment from 10 to 8 on the number line points to the left and is 2 units long. What are some strategies we can use to subtract integers? We can look for patterns when observing how subtraction problems differ, like we did at the beginning of the lesson with integers being subtracted from 10, such as 10 − 9, 10 − 10, and 10 − 11. We can write a subtraction expression as the related unknown addend equation and use a number line or integer addition to find the unknown addend, which is the difference.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

2. Consider the expression −4 − 6. a. Write the expression as an unknown addend equation.

Finding Distances to Find Differences

= −4 Any subtraction expression can be written as an unknown addend equation. The value of the subtraction expression −4 − 6 is −10,

In this lesson, we •

examined subtraction methods and their constraints when subtracting integers.

•

found the distance between the two numbers in a subtraction expression by plotting them on a number line.

which is also the unknown addend that is added to 6 to get a sum of −4.

•

wrote subtraction expressions as unknown addend equations and used a number line to model the equations.

•

found unknown addends by using a number line and drawing directed line segments.

To determine the unknown addend, ask the question, What can be added to 6 to make −4?

b. What must be added to 6 to get a sum of −4?

−10

Examples 1. Use the number line to plot points that represent A and B. State the distance between them. Then determine the number that should be added to A to get a sum of B.

Distance (units)

Number Line

B 6

−4

–10 –8 –6 –4 –2

Because addition and subtraction are related, the unknown addend is also the difference of the subtraction expression.

Number Added to A to Get a Sum of B

6 + (−10) = −4 −4 − 6 =

−10

A 0

−10

To find the number to add to A to get B, it might be helpful to draw a directed line segment from A to B.

Because the directed line segment from 6 to −4 goes to the left and has a length of 10 units, it represents −10.

The number −10 can be added to A to get a sum of B.

B –10 –8 –6 –4 –2

A 0

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 3

PRACTICE

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

7. Consider the expression −3 − (−2). a. Write the expression as an unknown addend equation.

−2 +

For problems 1–6, use the number line to plot points that represent A and B. State the distance between them. Then determine what must be added to A to get a sum of B.

−2

Number Line

–10

–8

–6

–4

–2

−2

Distance (units)

Number Added to A to Get a Sum of B

−9

–6

–4

–2

c. What is the value of the subtraction expression −3 − (−2)? Explain how you know. The value is −1. The difference of −3 and −2 is the same as the unknown addend that is added to −2 to get a sum of −3. 8. Consider the expression −2 − 9. a. Write the expression as an unknown addend equation.

9 –8

−1

7 –10

b. What must be added to −2 to get a sum of −3?

A 0

= −3

= −2

b. What must be added to 9 to get a sum of −2?

−11

c. What is the value of the subtraction expression −2 − 9? Explain how you know. A

−7 −2

5 –10

–8

–6

–4

–2

−7

9. Consider the expression 4 − (−10).

9 –8

–6

–4

–2

–6

–4

–2

b. What must be added to −10 to get a sum of 4?

c. What is the value of the subtraction expression 4 − (−10)? Explain how you know. The value is 14. The difference of 4 and −10 is the same as the unknown addend that is added to −10 to get a sum of 4.

9 –8

−5

−7 –10

–6

–4

–2

−10 +

5 –8

a. Write the expression as an unknown addend equation.

−2 −7 –10

2 –10

The value is −11. The difference of −2 and 9 is the same as the unknown addend that is added to 9 to get a sum of −2.

−9

P R ACT I C E

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 3

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

10. In Baltimore, the temperature is 35°F. In Milwaukee, the temperature is −7°F. How much warmer in degrees Fahrenheit is it in Baltimore than in Milwaukee?

17. Complete each comparison by using <, >, or =. a.

It is 42°F warmer in Baltimore than in Milwaukee.

Remember For problems 11–14, add. 11.

_1 + 5_ 3 6 _7 6

12.

_5 + 2_ 2 5

13.

29 __ 10

_4 + 2_ 9 3

14.

10 __ 9

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 3

7 _5 + __ 6 30

_3 8

5 __

2 b. −8_ 3

−3_4

c. 5.43

|−6.2|

d. 1.7

|−4.32|

| | > |2_|

|−3.21|

1 −4_8

4 5

|3.21|

32 __ 30

15. Use the number line to model the expression. Then determine the sum.

−5 + 8

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

9 10

3 16. Determine the sum.

−5.3 + (−4.5) −9.8

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 4 LESSON 4

Subtracting Integers Express subtraction of an integer as addition of its opposite. Subtract integers by using equivalent addition expressions.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4 Name

Date

EXIT TICKET

Write each expression as an equivalent addition expression. Then find the sum. 1. −4 − (−20)

−4 + 20 −4 + 20 = 16 2. 7 − (−36)

7 + 36

Lesson at a Glance In this lesson, students consider how subtracting a negative value from a positive value is different from subtracting a negative value from a negative value. Students observe patterns to support their understanding that subtraction can be written in an equivalent form, as addition of the opposite. Then they complete a card sort involving different representations of subtraction of integers, including real-world situations.

Key Questions

7 + 36 = 43

• How can a subtraction expression be written as an equivalent addition expression?

3. −44 − 9

−44 + (−9) −44 + (−9) = −53

• Does writing a subtraction expression as its related addition expression always result in an equivalent answer? How do you know?

4. 37 − 62

37 + (−62)

Achievement Descriptors

37 + (−62) = −25

7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients

of two rational numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD3 Interpret sums, differences, products, and quotients

of rational numbers by describing real-world contexts. (7.NS.A.1.a; 7.NS.A.1.b; 7.NS.A.1.c; 7.NS.A.2.a; 7.NS.A.2.b; 7.NS.A.2.c; 7.NS.A.3)

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 4

Agenda

Materials

Fluency

Teacher

Launch 10 min

• None

Learn 25 min

Students

• Subtracting Negative Values

• Equivalence Card Match cards

• Equivalent Expressions

Lesson Preparation

Land 10 min

• Copy and cut out the Equivalence Card Match cards (in the teacher edition). Prepare one card set for each group of four students.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Fluency Add Integers Students add integers to prepare for expressing subtraction of an integer in an equivalent form, as addition of its opposite. Directions: Add. 1.

−4 + (−5)

−9

7. −12 + (−15)

−27

−4 + 5

8. 12 + (−15)

−3

4 + (−5)

−1

9. −12 + 15

−6 + (−8)

−14

10. −34 + (−45)

−79

6 + (−8)

−2

11. −34 + 45

−6 + 8

12. 34 + (−45)

−11

Teacher Note Instead of this lesson’s Fluency, consider administering the Addition of Integers Sprint, which can be found in Eureka Math2 grade 7 module 2.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 4

Launch

Students compare addition and subtraction expressions. Introduce the Which One Doesn’t Belong? routine. Display the four representations and invite students to study them.

B A

5−3 –10 –8 –6 –4 –2

−3 + 5

−3 − (−5)

8 10

Give students 1 minute to find a category into which three of the items belong, but a fourth item does not. When time is up, invite students to explain their chosen categories and to justify why one item does not fit. Which one doesn’t belong? Why? Representation A does not belong because it shows two positive integers. Representation B does not belong because it uses a number line, and the other representations are expressions. Representation C does not belong because it shows adding a positive integer. Representation D does not belong because it shows subtracting a negative integer. © Great Minds PBC

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Highlight responses that emphasize reasoning about the value of the expression and unknown addends. Ask questions that invite students to use precise language, make connections, and ask questions of their own. Sample questions: • What do all four representations have in common? • How do you know that the value of the expression in representation A is 2? • What expression is represented by the model in representation B? • How do you know that the value of the expression in representation C is 2? • How is the expression in representation D different from other representations you have seen before? • In what other ways can you represent the expression −3 + 5? • How can you write the expression 5 − 3 as an unknown addend equation? • Now that you know how to add integers, do you think you can write the expression 5 − 3 as an integer addition expression with the same result? How? Display the Subtracting Integers interactive.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 Invite a student to represent the expression −5 − 3 by adjusting the model. Facilitate a class discussion by asking the following questions. What is the value of −5 − 3? How do you know?

Teacher Note Students know how to use a number line to represent −5 as a directed line segment that starts at 0, points to the left, and has a length of 5 units. They also know that subtracting 3 can be represented by a directed line segment that points to the left 3 units. If needed, prompt students to recall this knowledge when inviting a student up to model −5 − 3 with the interactive.

The value is −8. The first directed line segment starts at 0 and ends at −5, and the second directed line segment starts at −5 and ends at −8.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 4

The model represents the true subtraction sentence −5 − 3 = −8. Consider the related addition sentence. What integer must we add to −5 to get a sum of −8? How do you know? We must add −3 because a directed line segment pointing to the left with a length of 3 units represents −3. What equivalent addition sentence can we write?

−5 + (−3) = −8 Repeat this sequence with several different expressions that involve subtracting a positive integer, such as the following:

4−9 −1 − 7 −5 − 5 5−3 For each expression, invite a student to adjust the model in the interactive to represent different integer subtraction problems. Then show the related addition sentence and the result. After several examples, continue the discussion. What do you notice about the relationship between subtraction and addition? I notice that we can write a subtraction sentence as an equivalent addition sentence. I notice that subtracting a positive integer, such as 3, is the same as adding the opposite of that integer, −3.

UDL: Engagement The interactives in this lesson promote engagement by providing immediate, formative feedback as students use number lines to model subtraction.

By writing a subtraction sentence as an equivalent addition sentence, we are adding the opposite of the integer we were subtracting. Describe the similarities between subtracting a positive integer and adding its opposite. Subtracting a positive integer has the same result as adding the opposite of that integer. The subtraction of a positive integer can be written as the addition of that integer’s opposite. Subtracting a positive integer and adding a negative integer both require a directed line segment that points to the left.

7–8 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

When using a number line to model subtracting a postive integer, we start from the end of the first directed line segment and draw a second directed line segment that points to the left. When we add a negative integer, we also draw a directed line segment that points to the left. When subtracting a positive integer, we can write subtraction expressions as equivalent addition expressions. For now, accept any student responses to the following question. This idea will be revisited later in the lesson. Do you think that it is always possible to write a subtraction expression as an equivalent addition expression? Why? Yes. I think it is always possible because we can draw models to represent subtraction expressions that are the same types of models we drew in earlier lessons when adding negative integers. I think it is possible to write a subtraction expression as an addition expression when subtracting a positive integer. I am not sure that we can write a subtraction expression as addition when subtracting a negative integer. Have students turn and talk about the following questions. It is not expected that students will give a precise answer at this time. Some students may have difficulty making sense of what it means to subtract a negative integer. What do you think it means to subtract a negative integer from a positive integer? For example, what do you think is the value of 7 − (−4)? What do you think the model that represents the expression 7 − (−4) would look like? Today, we will explore what it means to subtract a negative integer.

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 4

Learn

Subtracting Negative Values Students observe patterns, number lines, and unknown addend equations to discover that subtracting a negative value is the same as adding the opposite positive value. Display the Subtracting Negative Values interactive.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7–2=5 This model represents the expression 7 − 2. How would this model change to represent 7 − 1? The second directed line segment would still start at 7 and point to the left, but it would be 1 unit long. Reveal the new number line and number sentence.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7–2=5 7–1=6

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

How would this model change to represent 7 − 0? The model wouldn’t have a second directed line segment because we can’t have a directed line segment with a length of 0 units. Reveal the new number line and number sentence.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7–2=5 7–1=6 7–0=7 Display 7 − (−1) =

Then ask the following questions. Encourage all students to share, but do not acknowledge whether the student responses are correct. What do you think the model that represents the expression 7 − (−1) would look like? Explain. The first directed line segment would start at 0 and go to the right 7 units. I think the second directed line segment would start at 7, have a length of 1 unit, and point to the right. What do you think is the value of the expression 7 − (−1)? Explain. I notice the pattern is that we are subtracting 1 less each time. And each time, the difference gets larger by 1. So I think the value of 7 − (−1) is 8. How can we use an unknown addend equation to determine the value of 7 − (−1)? The unknown addend equation that represents 7 − (−1) is −1 +

−1 to 7, we need to add 8, so 7 − (−1) = 8.

= 7. To get from

UDL: Representation Consider presenting the subtraction expressions and their related unknown addition equations in another format to emphasize the relationship between them. For example, display the subtraction expression 7 − (−1) and its related unknown addition equation −1 + =7 and have students record them during this discussion.

We can continue with the pattern to see whether we are correct.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Reveal the new number line and number sentence.

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7–2=5 7–1=6 7–0=7 7 – (–1) = 8 Invite students to share whether their predictions about the value of the expression 7 − (−1) were correct, and if not, why. Then ask students to revisit their predictions from the end of Launch about the expression 7 − (−4). Reveal the new number line and number sentence. Continue doing this until the model for the expression 7 − (−4) is displayed.

Promoting the Standards for Mathematical Practice When students notice repeating patterns in integer subtraction expressions reinforced by a number line, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8:

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7–2 =5 7–1 =6 7–0 =7 7 – (–1) = 8 7 – (–2) = 9 7 – (–3) = 10 7 – (–4) = 11

• What patterns do you notice when we change the number we are subtracting? • What is the same about the model for the subtraction expression and the model for the addition expression?

Invite students to turn and talk about whether their predictions were correct.

7–8 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

How can we use an unknown addend equation to check that 7 − (−4) = 11? We can find the value of 7 − (−4) by thinking about what number we add to −4 to get 7. Because −4 + 11 = 7, we know that 7 − (−4) = 11. What is another expression that can be represented by this model? How do you know? Another expression that can be represented by this model is 7 + 4. For the expression 7 + 4, the first directed line segment starts at 0, points to the right, and is 7 units in length. The second directed line segment starts at 7, points to the right, and is 4 units in length. We know that when we are subtracting a positive integer, subtraction expressions can be written as equivalent addition expressions. Based on the models with the number line, do you think we can also write subtraction expressions as equivalent addition expressions when subtracting a negative integer? Why? Yes. When we looked at the pattern on a number line, we found that the model for subtracting a negative integer can also be used to model adding a positive integer. Have students work in pairs on problems 1–6. Circulate as students work, and ask the following questions: • How can you write an equivalent addition expression? • How can you write an equivalent subtraction expression? • How do you know whether the value of the expression is positive or negative? • What strategies can you use to determine the value of the expression?

100

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

For problems 1–6, complete the table. An example is provided.

Teacher Note

Subtraction Expression

Equivalent Addition Expression

Sign of the Value of the Expressions

Value of the Expressions

Example

−5 − 3

−5 + (−3)

Negative

−8

2−8

2 + (−8)

Negative

−6

−1 − 6

−1 + (−6)

Negative

−7

−4 − 5

−4 + (−5)

Negative

−9

3 − (−7)

3+7

Positive

−9 − (−4)

−9 + 4

Negative

−5

−9 − (−13)

−9 + 13

Positive

Encourage students to use unknown addend equations to check their work for problems 1–6.

Differentiation: Support Encourage students who need additional support for problems 1–6 to use a number line to model the given expression. Students may use the Number Lines removable from lesson 1.

When students are finished, ask the following questions. How can we tell whether the value of a subtraction expression is positive or negative? Give an example. We can write the expression as an equivalent addition expression and determine which integer has the greater absolute value to know whether the value of the expression is positive or negative. For example, −5 − (−3) is equivalent to −5 + 3. Because −5 has the greater absolute value and is negative, the value of the expression is negative.

101

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Do you find it is easier to add integers or subtract integers? Why? I find it is easier to add integers because I am more efficient with addition than with subtraction. I find it is easier to add integers because with subtraction, I might have an additional step of writing the subtraction expression as an equivalent addition expression. Is subtracting an integer always equivalent to adding the opposite of that integer? Give an example. Yes. Subtracting 4 is equivalent to adding −4, and subtracting −4 is equivalent to adding 4. Can we write any subtraction expression as an equivalent addition expression? Explain. Yes. Subtracting an integer is equivalent to adding the opposite of that integer. To facilitate a conversation about a common student error, display the following student work for determining the value of −23 − (−3). Invite students to turn and talk about the student’s mistake.

−23 − (−3) = −23 − 3 = −23 + (−3) = −26 Circulate and listen to student conversations to ensure that students recognize that subtracting −3 is not equivalent to subtracting 3.

Equivalent Expressions Students identify equivalent representations of subtraction expressions. Arrange students into groups of four. Distribute one Equivalence Card Match card set to each group. Direct students to set aside the two cards that ask them to write a situation. Have them sort the remaining cards into categories that represent the same information written in different ways.

102

Language Support The card sort activity includes the context of American football. The term American football is used because the sport called “soccer” in the US is called “football” elsewhere. Consider previewing the context of American football by facilitating a class discussion. Ask students about the basic rules of the game. Emphasize the following ideas: • A team makes a play when they try to move the ball along the field toward the end zone. • A team can gain or lose yards in a play. • A team has four consecutive plays to gain at least ten yards from their starting location on the first play. Consider projecting pictures or showing a short video of American football.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Once students have correctly matched their cards, have them write situations for the expressions 2 − 5 and −7 − (−10) on the cards they set aside. As students work in their groups, circulate and be prepared to help students think of appropriate contexts. Select two or three contexts to share with the class. Possible examples include the following: • 2 − 5: The temperature in Chicago is 2°F. The temperature in Ithaca, NY, is 5°F cooler. • −7 − (−10): A bank account balance is −$7. The bank cancels a previous charge of $10.

Land Debrief 5 min Objectives: Express subtraction of an integer as addition of its opposite. Subtract integers by using equivalent addition expressions. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. We have discussed that a negative sign can be understood to mean the opposite of whatever follows it. How would a model of the expression seven minus the opposite of three be different from the expression seven minus three? The model showing the expression seven minus three would have a directed line segment that points to the left and is 3 units long. The model showing the expression seven minus the opposite of three would show a directed line segment pointing in the opposite direction. Taking away the opposite of 3 is the same as adding 3. Adding 3 is represented by a directed line segment that points to the right and is 3 units in length. Using the values from problem 6, describe how to change a subtraction expression to an equivalent addition expression. We know that subtracting an integer is equivalent to adding the opposite of that integer. In the expression −9 − (−13), instead of subtracting −13, we can add the opposite of −13, which is 13. The resulting expression is −9 + 13. © Great Minds PBC

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Does writing a subtraction expression as its related addition expression always result in an equivalent answer? Explain. Yes. We can use unknown addend equations and number lines with models to show this. I am not able to come up with a situation where it doesn’t result in an equivalent answer, so I believe it always will.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

104

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4

RECAP

Name

Date

Subtracting Integers In this lesson, we •

represented subtraction expressions by using a number line and observed patterns to understand the relationship between subtraction and addition of integers.

•

determined the difference when subtracting a negative integer.

•

evaluated subtraction expressions by writing equivalent addition expressions.

Example Complete the table. Write an equivalent addition expression and find the sum. Then write an unknown addend equation and find the unknown addend. Subtraction Expression

16 − 24

−21 − (−6)

Once a subtraction expression is written as an equivalent addition expression, use the integer addition strategies to find the sum.

Subtraction expressions can be written as equivalent addition expressions. Change the subtraction sign to the addition sign and change the second integer to its opposite. In this case, 16 − 24 is written as 16 + (−24).

Equivalent Addition Expression and Sum

16 − 24 = 16 + (−24)

24 +

= 16

24 + (−8) = 16

= −8 −21 − (−6) = −21 + 6 = −15

Unknown Addend Equation and Unknown Addend

−6 +

= −21

−6 + (−15) = −21

When using unknown addend equations to check your work, ask this question: What number can I add to the second integer in the original subtraction expression to get the first integer? In this case ask, What number can I add to −6 to get −21?

105

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 4

PRACTICE

Name

Date

For problems 3–10, write an equivalent addition expression and find the sum. Then write an unknown addend equation and find the unknown addend.

1. Use the expression −2 − 5 to complete parts (a)–(d).

Subtraction Expression

a. Use the number line to model the expression.

–10

–8

–6

–4

–2

12 − (−6)

34 − (−19)

−18 − 4

−45 − 46

−9 − (−9)

25 − 29

−12 − (−5)

10.

−21 − (−30)

b. Write the subtraction expression as an equivalent addition expression.

−2 + (−5) c. Determine the sum of the expression from part (b).

−7 d. Evaluate −2 − 5.

−7

2. Which expressions have the same value as −16 − 29? Choose all that apply. A. 16 + (−29) B. −16 + (−29) C. −16 − (−29) D. 16 − 29

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Equivalent Addition Expression and Sum

12 − (−6) = 12 + 6

Unknown Addend Equation and Unknown Addend

−6 +

= 18 34 − (−19) = 34 + 19

−19 +

= 53 −18 − 4 = −18 + (−4) = −22 −45 − 46 = −45 + (−46) = −91 −9 − (−9) = −9 + 9

= −4 −12 − (−5) = −12 + 5 = −7 −21 − (−30) = −21 + 30 =9

= 34

−19 + 53 = 34 4+

= −18

4 + (−22) = −18 46 +

= −45

46 + (−91) = −45 −9 +

=0 25 − 29 = 25 + (−29)

= 12

−6 + 18 = 12

= −9 −9 + 0 = −9

29 +

= 25

29 + (−4) = 25 = −12

−5 +

−5 + (−7) = −12 −30 +

= −21

−30 + 9 = −21

E. (0 − 16) + (0 − 29)

106

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4

Remember For problems 11–14, add. 11.

_2 + − 5_ 3 ( 6)

− 1_

5 3 12. − _ + _ 2 5

_8 + − 2_ 9 ( 3)

13.

__ − 19

_2 9

15. Estimate, and then find the sum. Estimation: −4 + (−2) = −6 5 Sum: −5_ 8

5 4 14. − _ + __ 6 15

__ − 17 30

−37_ + (− 7_) 8

16. Which expressions have a value of 105? Choose all that apply. A. 10 × 5 B. 10 + 5 C. 10,000 D. 100,000

E. 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10

F. 10 + 10 + 10 + 10 + 10

P R ACT I C E

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Equivalence Card Match

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 4

Write a situation.

2−5

7 − 10

−5 − 2

−10 − 7

108

2 + (−5)

7 + (−10)

−5 + (−2)

−10 + (−7)

An American football team gains 7 yards on a play. On the next play, the team loses 10 yards.

A diver descends 5 meters below the surface of the water. A sea star catches his eye, and he swims down 2 more meters to have a better look.

The temperature is −10°F today and will be 7°F lower tomorrow.

This page may be reproduced for classroom use only.

–18 –16 –14 –12 –10 –8

–6

–4

–2

–18 –16 –14 –12 –10 –8

–6

–4

–2

–18 –16 –14 –12 –10 –8

–6

–4

–2

–18 –16 –14 –12 –10 –8

–6

–4

–2

EUREKA MATH2 7–8 ▸ M1 ▸ TA ▸ Lesson 4 ▸ Equivalence Card Match

Write a situation.

−7 − (−10)

−2 − (−5)

−10 − (−7)

−7 + 10

−2 + 5

−10 + 7

Kabir loses 2 points in a card game. Then he draws a card that reverses an earlier loss of 5 points.

Eve borrows $10 from her uncle. Her uncle is away for the weekend, and she feeds his parrot. So her uncle takes away $7 of her debt.

–18 –16 –14 –12 –10 –8

–6

–4

–2

–18 –16 –14 –12 –10 –8

–6

–4

–2

–18 –16 –14 –12 –10 –8

–6

–4

–2

This page may be reproduced for classroom use only.

109

Teacher Edition: Grade 7–8, Module 1, Topic A, Lesson 5 LESSON 5

Subtracting Rational Numbers Evaluate expressions involving subtraction of rational numbers. Subtract rational numbers by using equivalent addition expressions.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5 Name

Date

EXIT TICKET

Evaluate each expression. 1. − 4.7 − (− 3.7)

− 4.7 − (− 3.7) = − 4.7 + 3.7 = − 4 + (− 0.7) + 3 + 0.7 = (− 4 + 3) + (− 0.7 + 0.7) = −1 + 0 = −1 2. − 3_ − 4 1_ 4 8

− 3_ − 4 1_ = − 3_ + (− 4 1_) 4

= − 3_ + (− 4) + (− 1_) 4

= − 6_ + (− 4) + (− 1_)

= − 4 + (− 7_) = − 4 _7 8 3. 3 + (− 0.2) − 15.25

To begin the lesson, students are presented with a number line and a model. They determine that many subtraction expressions can be represented by the model, including expressions that involve fractions and decimals. Students then work with a partner to evaluate subtraction expressions involving rational numbers by using number lines and unknown addend equations. Students confirm that writing a subtraction expression as an equivalent addition expression by adding the opposite—a strategy developed earlier in the topic—also applies to subtraction expressions involving rational numbers. Students use decomposition and apply properties of operations to make simpler problems when evaluating subtraction expressions that involve rational numbers.

Key Questions

= − 4 + (− 6_ + (− 1_)) 8

Lesson at a Glance

• How is subtracting non-integer rational numbers similar to subtracting integers?

• What strategies are helpful when evaluating subtraction expressions that involve rational numbers? What makes them helpful?

3 + (− 0.2) − 15.25 = 3 + (− 0.2) + (− 15.25) = 3 + (− 0.2) + (− 15) + (− 0.25) = (3 + (− 15)) + (− 0.2 + (− 0.25)) = − 12 + (− 0.45) = − 12.45

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Achievement Descriptors 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD2 Solve real-world and mathematical problems involving the four operations

with rational numbers. (7.NS.A.3) 7–8.Mod1.AD3 Interpret sums, differences, products, and quotients of rational numbers

by describing real-world contexts. (7.NS.A.1.a; 7.NS.A.1.b; 7.NS.A.1.c; 7.NS.A.2.a; 7.NS.A.2.b; 7.NS.A.2.c; 7.NS.A.3)

Agenda

Materials

Fluency

Teacher

Launch 10 min

• None

Learn 25 min

Students

• Subtracting Rational Numbers

• None

• Decomposing to Subtract

Lesson Preparation

• How Much?

• None

Land 10 min

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7–8 ▸ M1 ▸ TA ▸ Lesson 5

Fluency Add and Subtract Integers Students add and subtract integers to prepare for subtracting rational numbers.

Instead of this lesson’s Fluency, consider administering the Addition and Subtraction of Integers Sprint. Directions for administration can be found in the Fluency resource.

Directions: Add or subtract. 1.

5+6

Teacher Note

7. 15 + 16

EUREKA MATH2

7–8 ▸ M1 ▸ Sprint ▸ Addition and Subtraction of Integers

Number Correct:

Add or subtract the integers.

−5 + (−6) 5 + (−6) 5−6

−11 −1 −1

8. −15 + 16

9. 15 + (−16)

10. 15 − 16

1 −1 −1

1+6

23.

−30 + (−25)

−1 + (−6)

24.

30 + (−25)

2+6

25.

−30 + 25

−2 + (−6)

26.

45 + (−32)

−8 + (−6)

−8 + (−12)

28.

−6 − 9

4−9

29.

−11 − 9

−5 − (−6)

11. −15 − 16

−31

112

−5 − 6

−11

12. −15 − (−16)

−9 − 11

4 − 80

31.

−9 − (−11)

4 − 100

32.

−20 − (−20)

11.

4 − 200

33.

−30 − (−75)

12.

10 + (−9)

34.

−3 + (−2) + (−1)

13.

10 + (−10)

35.

−7 + 8 + (−3)

14.

−12 + 10

36.

−7 + 8 + (−7)

15.

−14 + 10

37.

2 + (−15) + 5

16.

−17 + 10

38.

−2 + (−15) + 5

4 − (−3)

39.

−3 − (−3)

18.

4 − (−7)

40.

−3 − (−4)

19.

4 − (−27)

41.

−4 − (−3)

20.

4 − (−127)

42.

−14 − (−3)

21.

20 − (−8)

43.

−23 − (−8)

22.

8 − (−20)

44.

−8 − 23 − 5

350

30.

45 + 32

10.

17.

4 − 10

27.

This page is inte

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Students write subtraction expressions involving rational numbers to represent a given model. Pair students and ensure each pair has one whiteboard and marker to complete the following activity. Display the model with two directed line segments.

Teacher Note This lesson combines and expands on the learning activities from Eureka Math2 grade 7 module 2 lessons 10 and 11. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

0 With your partner, write a subtraction expression that can be represented by the model. Be prepared to explain why the model represents your expression. Then evaluate your subtraction expression. When most pairs have finished, have them show their whiteboards. Consider inviting a few pairs to share their subtraction expression and reasoning. Expect students to reason that the model shows a positive number minus a larger positive number, such as 2 − 6, based on the approximate length of the directed line segments. Students may reason that the number being subtracted appears to be about 3 times larger than the first number, but that level of specificity is not essential at this point. Display the model with the unlabeled tick marks on the number line.

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EUREKA MATH2

Invite students to think–pair–share about the following question. Does the model still accurately represent the subtraction expression you wrote? How do you know? No. The model does not accurately represent our subtraction expression. We wrote the expression 2 − 6, which evaluates to −4. If the bottom directed line segment has a length of 2, then each tick mark represents an interval of 0.5 units. If each tick mark represents an interval of 0.5 units, then the top directed line segment represents subtracting 6.5, not 6. Continue the class discussion with the following prompts. Suppose each tick mark represents an interval of 1 unit. What subtraction expression does the model represent?

4 − 13 Suppose each tick mark represents an interval that is not 1 unit. Quickly write and evaluate as many subtraction expressions as you can that can be represented by the model. Be prepared to explain why the model represents each expression. Give students 1 minute to write possible subtraction expressions, and then select a few pairs to share one of their subtraction expressions and reasoning. Expect students to reason about the length of each directed line segment by using the tick marks and the interval that each tick mark must represent. Student responses may include expressions such as 40 − 130 and 8 − 26. Then invite pairs to complete the following task. Suppose each tick mark on the number line represents an interval that is a fraction or decimal less than 1 unit. Working with your partner, decide which interval length you will use. Draw the model on your whiteboard and label the tick marks on the number line. Then write the subtraction expression that is represented by your model. Use the model to evaluate the expression.

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EUREKA MATH2

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Invite several pairs to share the interval length they chose, their labeled model, and the subtraction expression that is represented by their model. Two sample student responses are shown.

0.4 − 1.3

–0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2 –0.1

0.1

0.2

0.3

0.4

1 3

2 3

1 1_ − 4 1_ 3

–3

–2 3 –2 3

–2

–1 3 –1 3

–1

–3

Invite students to think–pair–share about the following question. Compare the subtraction expressions you wrote earlier, when the tick marks were unlabeled, to the subtraction expressions you wrote when the tick marks represented intervals of less than 1 unit. How are the expressions alike? How are they different? Both expressions are represented by the model that does not have labeled tick marks. Both expressions have a positive number minus a greater positive number, and both expressions have a negative difference. The expressions are different because one subtraction expression uses only integer numbers, and the other subtraction expression uses non-integer numbers.

Teacher Note To ensure a variety of answers, consider revising the last task by assigning each student pair a different interval length to use for their number line.

Today, we will subtract non-integer rational numbers.

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Learn

EUREKA MATH2

Subtracting Rational Numbers Students explore subtraction of fractions and decimals and confirm that the subtraction strategies they have developed apply to rational numbers. Direct students to problem 1. We observed that we can use number lines with different interval lengths to model and evaluate subtraction expressions involving rational numbers, just as we did with integers. Have students complete problem 1 with a partner. Circulate as students work, and listen for students’ reasoning. Provide support as needed. • If students appear to have difficulty working with the fractions in this problem, consider encouraging them to first evaluate 3 − 1 and then to apply that thinking to the subtraction expression with fractions. • If students appear to have difficulty working with the decimals in this problem, consider encouraging them to think about the numbers without the decimal points and then to apply that thinking to the subtraction expression with decimals. For example, instead of 0.75 − (−0.25), direct students to think of 75 − (−25).

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

1. Use the given number lines to model each subtraction expression. Then evaluate the expression. Write and solve an unknown addend equation to check your work. Subtraction Expression

Model

Difference

_ _

2 4

–1

− 4_ 4

–1

−0.5

−0.75 − (−0.25)

–1

3 4

_1 + _2 = 3_

− 0.25 +

= 0.75

− 0.25 + 1 = 0.75

− _ − 1_

=_ 4

0.75 − (−0.25)

1 + 4

3 1 − 4 4

3 4

Unknown Addend Equation

1 + 4

= −_

3 4

3 1 + −4 = − 4 ( 4) 4

− 0.25 +

= − 0.75

−0.25 + (− 0.5) = − 0.75

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Once most students have finished, engage the class in a discussion by asking the following question. How is subtracting non-integer rational numbers similar to subtracting integers? When subtracting non-integer rational numbers, we use the same strategies we use when subtracting integers. We can use number lines to model the subtraction expressions or use unknown addend equations. We can also write subtraction of a non-integer rational number as adding the opposite of that number, just as we do with integers.

Decomposing to Subtract Students decompose numbers and use properties of operations to make simpler problems when subtracting rational numbers. Direct students to problem 2.

UDL: Representation

Have students think–pair–share about the following question. What do you notice about this subtraction expression?

Consider using color coding and annotating as the problem is explained.

Permit students to share anything they notice about the expression. They may notice that both numbers in the expression include a non-integer part and that both fractional parts are thirds. If it is not mentioned, point out that when this subtraction expression is written as an equivalent addition expression, there is a negative addend and a positive addend.

• The decomposing step can be highlighted. • The application of properties of operations can be annotated.

Have students write the subtraction expression as its equivalent addition expression.

• The application of additive inverses can be annotated.

Guide students to use decomposition and properties of operations to find the value of this subtraction expression. Encourage students to provide multiple strategies. Consider recording different strategies for students to refer to. How can we use decomposition to evaluate this expression?

Both mixed numbers can be decomposed. We can decompose − 4 _ into − 4 + (−_2) and

decompose 2 _1 into 2 + _1 . 3

2 3

We can decompose − 4 _2 into − 2 _1 and − 2 _1 . Then we can group − 2 _1 with the addend 2 _1 3

to sum to 0 because they are additive inverses. 118

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

This expression can be evaluated with either of those strategies. Have the class decide which strategy they prefer to use. The sample dialogue is based on decomposing − 4 _2 into 3 integer and non-integer parts. What can we do to make this expression simpler to evaluate?

We can further decompose −_2 into −_1 + (−_1). We can rearrange the numbers so that _1 3

and −_1 are together. They are additive inverses and sum to 0. Then we can find the sum 3

of the remaining numbers. While students are describing how to write a simpler expression, ask them to describe the properties of operations they are using. Follow the suggested dialogue if needed. When we rearrange the expression, what property are we using? We are using the commutative property of addition.

When we group −_ and _1 together, what property are we using? 1 3

We are using the associative property of addition. Why do we want to group −_1 and _1 together? 3

We want to group them together because they are additive inverses and sum to 0.

Differentiation: Support Some students may need support in making sense of the expressions in problems 2–4 and knowing where to begin. Have students use estimation and number lines. Direct students to round the numbers in the expression to the nearest integer and then to visualize or model the expression by using a number line with directed line segments. For example, students may write the expression −5 − (−2), or −5 + 2, to estimate the value of the expression in problem 2.

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Allow students to finish evaluating problem 2 with a partner. Once they have finished, ask them to complete problems 3 and 4 independently. For problems 2–4, evaluate the expression. 2 1 2. − 4 _ − (− 2 _) 3 3

− 4 _2 − (− 2 1_) = − 4 2_ + 2 1_ 3

= − 4 + (− _2) + 2 + _1 3

= − 4 + (− 1_) + (− _1) + 2 + 1_ 3

= (− 4 + 2) + (− 1_) + (− 1_ + _1) 3

= − 2 + (− _1) + 0

= − 2 1_

1 1 3. − _ − (− 3 _) 2 4

− 1_ − (− 3 1_) = − 1_ + 3 1_ 4

= − _1 + 3 + _1 4

= − _1 + 3 + 2_ = − 1_ + 3 + 1_ + 1_ 4

= (− _1 + _1) + 3 + _1 4

= 0 + 3 + 1_ = 3 _1

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

5 1 2 4. 5 _ − (− 2 _) − 3 _ 3 3 6

5 1_ − (− 2 _) − 3 _2 = 5 1_ + 2 _ + (− 3 _2) 3

5 6

= 5 + 1_ + 2 + _ + (− 3) + (− 2_) 3

5 6

= 5 + 2_ + 2 + _ + (− 3) + (− 4_)

= (5 + 2 + (− 3)) + (2_ + _ + (− 4_)) =4+_ = 4_

5 6

3 6

When most students have finished, lead a class debrief of the properties students used when writing a simpler expression to evaluate. Direct students to problem 5. Have students write the expression as an equivalent addition expression. Facilitate a class discussion by asking the following questions. How can you use decomposition here? We can decompose −3.4 into −3 + (−0.4) and −2.1 into −2 + (−0.1). What can we do to make this expression simpler to evaluate? Once both numbers are decomposed, we can rearrange the expression and group −0.4 and −0.1 together to find a sum of −0.5. Then we can add the remaining integers. Have students finish problem 5 with a partner and then complete problems 6–8 independently. Encourage pairs to write the problem as an equivalent addition expression, use decomposition, and apply properties of operations to evaluate the expression.

Promoting the Standards for Mathematical Practice When students use decomposition strategies and properties of operations to make simpler problems to subtract rational numbers, they are looking for and using structure (MP7). Ask the following questions to promote MP7: • How can what you know about decomposition help you evaluate the expression? • How is the expression similar to addition problems you have evaluated before? • What is another way you could arrange the terms that would help you evaluate the expression?

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

For problems 5–8, evaluate the expression. 5. −3.4 − 2.1

−3.4 − 2.1 = −3.4 + (−2.1) = −3 + (−0.4) + (−2) + (−0.1) = (−3 + (−2)) + (−0.4 + (−0.1)) = −5 + (−0.5) = −5.5 6. 0.4 − 1.4

0.4 − 1.4 = 0.4 + (−1.4) = 0.4 + (−1) + (−0.4) = (0.4 + (−0.4)) + (−1) = 0 + (−1) = −1 7. −2.7 − 1.35 − (−6.25)

−2.7 − 1.35 − (−6.25) = −2.7 + (−1.35) + 6.25 = −2 + (−0.7) + (−1) + (−0.35) + 6 + 0.25 = (−2 + (−1) + 6) + (−0.7 + (−0.35)) + 0.25 = 3 + (−1.05) + 0.25 = (3 + (−1)) + (−0.05 + 0.25) = 2 + 0.2 = 2.2

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

8. 1.85 − 6.45 + 4.3

1.85 − 6.45 + 4.3 = 1.85 + (−6.45) + 4.3 = 1.85 + (−1.85) + (−4.6) + 4.3 = (1.85 + (−1.85)) + (−0.3) + (−4.3 + 4.3) = 0 + (−0.3) + 0 = −0.3 When most students have finished, lead a class debrief of the properties students used when writing a simpler expression to evaluate.

How Much? Students write and evaluate a subtraction expression to represent a real-world situation. Direct students to complete problem 9 individually. Then have students compare answers with a partner. 9. Liam owes his grandmother $5.49. His grandmother tells Liam that she will remove $2.50 from his debt after he cleans the bathroom. a. Write a subtraction expression that represents this situation.

−5.49 − (−2.50) b. Evaluate the expression from part (a). Explain what the result means in this situation.

−5.49 − (−2.50) = −5.49 + 2.50 = −5 + (−0.49) + 2 + 0.5 = −5 + 2 + (−0.49) + 0.5 = −3 + 0.01 = −2.99 Liam still owes his grandmother $2.99 after she removes $2.50 from his debt.

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EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Land Debrief 5 min Objectives: Evaluate expressions involving subtraction of rational numbers. Subtract rational numbers by using equivalent addition expressions. Initiate a class discussion by asking the questions. Encourage students to revoice or build on their classmates’ responses. How is subtracting non-integer rational numbers similar to subtracting integers? When subtracting non-integer rational numbers, we use the same strategies we use when subtracting integers. We can use a number line, unknown addend equations, decomposition, and properties of operations. What strategies are helpful when evaluating subtraction expressions that involve rational numbers? What makes these strategies helpful? Use one of the expressions we evaluated as an example. Decomposing expressions and applying the properties of operations are helpful strategies. Decomposing allows us to separate the integer and non-integer parts of mixed numbers. Applying the properties of operations allows us to rearrange numbers and group them together in ways that make evaluating the expression simpler. In problem 2, we used decomposition and properties of operations to make a simpler problem. When we grouped the integer parts and non-integer parts together, we found a pair of additive inverses, which made the expression simpler to evaluate.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

124

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

3. − 4 3_ − 1_ − (− 2 1_) 2 8 4

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

RECAP

Name

Date

EUREKA MATH2

− 4 3_ − _1 − (− 2 1_) = − 4 3_ + (− 1_) + 2 1_ 8

Subtracting Rational Numbers

In this lesson, we

= − 4 + (− 3_) + (− 1_) + 2 + 1_ 8

= − 4 + (− 3_) + (− 2_) + 2 + 4_

•

used number lines and unknown addend equations to confirm that a subtraction expression involving rational numbers can be written as an equivalent addition expression.

•

used decomposition and properties of operations to make subtraction expressions simpler to evaluate.

= − 2 + (− 5_ + 4_)

•

evaluated subtraction expressions involving rational numbers.

•

wrote and evaluated a subtraction expression to represent a real-world situation.

= − 2 + (− 1_)

For problems 1–4, evaluate the expression. 1 − 6 3_ 1. − 5 __ 5 10

− 3.4 − 0.06 − (− 6.45) = − 3.4 + (− 0.06) + 6.45 = − 3 + (− 0.4) + (− 0.06) + 6 + 0.45 = (− 3 + 6) + (− 0.4 + (− 0.06)) + 0.45 = 3 + (− 0.46) + 0.45 = 3 + (− 0.45 + (− 0.01)) + 0.45 = 3 + (− 0.45 + 0.45) + (− 0.01)

6 1 + (− 6) + (− __ = − 5 + (− __ 10 ) 10 )

= 3 + 0 + (− 0.01)

6 1 + (− __ = (− 5 + (− 6)) + (− __ ))

7 = − 11 + (− __ ) 7 = − 11 __

= 2.99

One strategy that is useful in this case is to decompose − 0.46 into − 0.45 + (−0.01). That way, an additive inverse with 0.45 is created. Use the associative property of addition to group the

2. − 7.43 − (− 4.31)

− 7.43 − (− 4.31) = − 7.43 + 4.31 = − 7 + (− 0.43) + 4 + 0.31 A decimal can be decomposed and written as an integer part and a non-integer part.

= (− 7 + 4) + (− 0.43 + 0.31) = − 3 + (− 0.12)

When using the associative property, consider grouping negative addends together.

4. − 3.4 − 0.06 − (−6.45)

1 _3 = − 5 + (− __ ) + (− 6) + (− ) 10

= − 2 1_

Each mixed number can be decomposed into an integer part and a non-integer part.

1 1 − 6 3_ = − 5 __ + (− 6 3_) − 5 __ 10

Write the subtraction expression as an equivalent addition expression first.

Examples

= (− 4 + 2) + (− 3_ + (− 2_)) + 4_

additive inverses together.

Use the commutative and associative properties of addition to rearrange the expression and to group the integers and non-integer parts together before evaluating.

= − 3.12 73

RECAP

125

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

5. Vic owes his brother $4.28 for a snack and $7.43 for a shirt. a. Write a subtraction expression that represents this situation.

− 4.28 − 7.43 Vic owes his brother $4.28, so that can be represented by − 4.28.

b. Evaluate the expression from part (a). Explain what the result means in this situation.

− 4.28 − 7.43 = − 4.28 + (− 7.43) = − 4 + (− 0.28) + (− 7) + (− 0.43) = (− 4 + (− 7)) + (− 0.28 + (− 0.43)) = − 11 + (− 0.71) = − 11.71 Vic owes his brother $11.71.

126

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

Student Edition: Grade 7–8, Module 1, Topic A, Lesson 5

PRACTICE

Name

Date

7–8 ▸ M1 ▸ TA ▸ Lesson 5

EUREKA MATH2

7. − 0.03 − (−2.8) − 11.97

− 9.2

For problems 1–10, evaluate the expression. 1. − 3 1_ − 3_ 5 5

− 3 4_ 5

2. 7.2 − (−6.1)

13.3 3 8. − 5 __ − 1_ − − 1 1_) 10 5 ( 2

−4

3. − 2 1_ − (− 3_) 4 8

− 1 3_ 8

4. − 7.94 − 10 9. 3 1_ − 10 3_ + 4 1_ 2 4 8

− 17.94

− 2 5_ 8

5. − 8.27 − (− 2.17)

− 6.1

1 6. 4 1_ − (− 1 __ 5 10 )

10. − 13.42 + 7.56 − 1.2

− 7.06

3 5 __ 10

P R ACT I C E

127

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

EUREKA MATH2

7–8 ▸ M1 ▸ TA ▸ Lesson 5

17. Consider the expression − 8 − (−6).

11. Identify and correct the error in the sample work shown.

− 1.23 − (− 5.26) = − 1.23 + 5.26

a. Draw a number line and plot the integers in the expression.

= − 1 + 0.23 + 5 + 0.26 = (−1 + 5) + (0.23 + 0.26)

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1

= 4 + 0.49 = 4.49

9 10

b. What is the distance on the number line between the integers in the expression?

The error is that − 1.23 is not decomposed into − 1 and − 0.23. The following work is correct.

2 units

− 1.23 − (− 5.26) = − 1.23 + 5.26

c. Write the expression as an unknown addend equation.

= − 1 + (− 0.23) + 5 + 0.26 = (− 1 + 5) + (− 0.23 + 0.26)

−6 +

= 4 + 0.03 = 4.03

= −8

d. What is the unknown addend?

−2 12. Maya goes deep-sea diving. She first dives 18.28 meters below sea level. Then she dives down another 4.71 meters.

18. Complete each statement.

a. Write a subtraction expression that represents this situation. a. The opposite of − 6 is

− 18.28 − 4.71 b. Evaluate the expression from part (a). Explain what the result means in this situation. Maya is − 22.99 meters below sea level after her second dive.

c. The opposite of 0 is d. The value of − (− _5 ) is

Remember

13. − 5_ + 3_ 3 5

5 + − 3_ 14. _ 3 ( 5)

8 + − 1_ 15. _ 9 ( 6)

4 16. − 5_ + __ 8 20

__ − 16

16 __ 15

13 __ 18

__ − 34

128

b. The opposite of the opposite of − 9 is

− 22.99

For problems 13–16, add.

−9

_5 6

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic B

Topic B Multiply and Divide Rational Numbers This topic explores multiplying and dividing rational numbers. From previous grade levels, students are familiar with strategies for multiplying and dividing positive rational numbers. In this topic, students learn how to evaluate multiplication and division expressions involving negative rational numbers and how to write rational numbers in fraction form as decimals. In lesson 6, students determine the sign of the product of two integers or two rational numbers. They model a real-world situation by using a number line, repeated addition, and multiplication expressions. When the first factor of a multiplication expression is positive and the second factor is negative, students use prior knowledge of multiplication to find the product by reasoning about one factor as the number of groups and the other factor as the size of the groups. By analyzing patterns and applying properties of operations, students determine that the product of a negative number and a positive number is a negative number and that the product of two negative numbers is a positive number. Students also use decomposition and the distributive property to evaluate multiplication expressions with rational numbers. Students then build on their understanding of exponents from grade 6 to evaluate exponential expressions with bases that are negative, rational numbers. They predict the sign of the product of a multiplication expression involving more than two factors and predict the sign of the value of an exponential expression. Students relate multiplication to division when they use unknown factor equations to determine the quotient of two integers. As they did with multiplication, students first determine whether a quotient is a positive

2(−2) = −4 1(−2) = −2 0(−2) = 0 −1(−2) = 2 −2(−2) = 4 −3(−2) = 6 −32 = −(3 ⋅ 3) = −9

(−3)2 = (−3)(−3) =9

number or a negative number by looking at the signs of the dividend and the divisor. Students formally define what a rational number is, and they recognize that rational numbers can be written in different but equivalent ways. For any integers p and q where

q ≠ 0,    

130

___ ___  

EUREKA MATH2

7–8 ▸ M1 ▸ TB

numbers can be decomposed and manipulated, and look at how the relationship between multiplication and division is used to evaluate expressions. Then students apply their understanding of rational number division to solve multi-step and real-world problems. In lesson 9, students write rational numbers as decimals by determining an equivalent decimal fraction or by using long division. They discover that any number that can be written as a decimal fraction can also be written as a terminating decimal. They then use their understanding of the long division algorithm to write the decimal form of a rational number and realize that sometimes the decimal form is a repeating decimal. Students formally conclude that all rational numbers can be written as either terminating decimals or repeating decimals by analyzing the prime factorization of the denominator to determine whether the decimal repeats or terminates. For repeating decimals, students use bar notation to precisely write the decimal form of the number. Students apply these skills related to multiplying and dividing rational numbers when they work with integer exponents in topic C and when they solve equations and inequalities in module 2.

0.0 8 3 3 3 1 2 1.0 0 0 0 0 – 96 40 –36 40 –36 40 –36 4

Progression of Lessons Lesson 6

Multiplying Integers and Rational Numbers

Lesson 7

Exponential Expressions and Relating Multiplication to Division

Lesson 8

Dividing Integers and Rational Numbers

Lesson 9

Decimal Expansions of Rational Numbers

131

Teacher Edition: Grade 7–8, Module 1, Topic B, Lesson 6 LESSON 6

Multiplying Integers and Rational Numbers Use repeated addition and the properties of operations to determine the product of a negative number and a positive number. Informally verify that the product of two negative numbers is a positive number.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6 Name

Date

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

For problems 2 and 3, determine the product. 2. −4(5)

−20

1. Ethan is scuba diving. He descends 6 feet from the water’s surface and rests. Then he descends another 6 feet and rests. Finally, he descends 6 more feet and examines a fish. a. Use a number line to model the situation.

3. − 1_(−1 1_) 2 4

− 1_(−1 1_) = − 1_(−1 + (− 1_)) 4

= (− 1_)(−1) + (− 1_)(− 1_) 4

= 1_ + 1_

= 2_ + 1_

u12

= 3_

u18

b. Write an addition expression to represent this situation.

−6 + (−6) + (−6) c. Write a multiplication expression to represent this situation.

3(−6) d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

−18 The product −18 means Ethan examines the fish at a depth of 18 feet below the water’s surface.

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Lesson at a Glance This lesson begins with students analyzing multiplication expressions with integers and predicting the products. Students apply prior knowledge of the relationship between repeated addition and multiplication to evaluate multiplication expressions where the first factor is a positive number and the second factor is a negative number, including an expression they write to represent a real-life situation. Through identifying patterns and applying properties of operations, students evaluate other multiplication expressions involving negative numbers. They identify the product of a negative number and a positive number as a negative number. In an informal, teacher-guided proof, students verify the product of two negative numbers is a positive number. Students use decomposition and the distributive property to evaluate multiplication expressions involving rational numbers. This lesson defines the zero product property.

Key Questions • Can we determine the product of a negative number and a positive number? How? • Can we determine the product of two negative numbers? How?

Achievement Descriptors 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD3 Interpret sums, differences, products, and quotients of rational numbers

by describing real-world contexts. (7.NS.A.1.a; 7.NS.A.1.b; 7.NS.A.1.c; 7.NS.A.2.a; 7.NS.A.2.b; 7.NS.A.2.c; 7.NS.A.3)

7–8.Mod1.AD5 Determine the sign of a product by looking at the signs of its factors. (7.NS.A.2.a)

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7–8 ▸ M1 ▸ TB ▸ Lesson 6

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Blank paper (3 sheets)

Learn 30 min • A Positive Number Times a Negative Number • A Negative Number Times a Positive Number • A Negative Number Times a Negative Number • Multiplication and Decomposition

Land 10 min

134

• Marker • Tape

Students • None

Lesson Preparation • Use the paper to prepare three signs with the following labels: −6, 6, and Neither. Post the signs in three different areas of the classroom.

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

Fluency Multiplication Expressions Students write and evaluate expressions to prepare for multiplying integers. Directions: Write an equivalent addition expression, multiplication expression, or product. problem number

Addition Expression

Multiplication Expression

Product

2+2+2

3(2)

3+3+3

3(3)

6+6+6

3(6)

5+5+5+5

4(5)

7+7+7+7+7

5(7)

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Launch

Students predict the products of multiplication expressions with integer factors. Display the expression 3(2). Have students think–pair–share about the following question. Encourage a wide range of interpretations and representations. What are different ways that you can interpret or represent this expression? Use your whiteboard to list and draw your ideas. Sample:

Teacher Note This lesson combines and expands on the learning activities from Eureka Math2 grade 7 module 2 lessons 13–15. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

3 groups of 2 2+2+2 Students may draw the following types of representations.

Introduce the Take a Stand routine to the class. Draw students’ attention to the signs posted around the classroom: −6, 6, Neither. Display the expression 3(−2). Then invite students to stand beside the sign they think shows the product of the expression.

136

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

When all students are standing near a sign, allow 1 minute for groups to discuss the reasons why they chose to stand at that sign. Then ask each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. Repeat the routine for each of the following expressions:

−3(2) −3(−2) Have students return to their seats. As a class, reflect on what the product means in terms of the factors. Today, we will explore different ways to determine products of integers and rational numbers.

Teacher Note In prior grades, students use the phrases number of groups and size of each group to interpret multiplication and division. To relate students’ prior conceptual knowledge to multiplication and division of integers, these phrases continue to be used in this course, enabling students to interpret these operations with familiarity. Although the size of a group cannot technically be negative, refer to the value of a group containing a negative number as the size of the group to continue the coherent phrasing from earlier grades.

Learn A Positive Number Times a Negative Number Students write expressions to represent a situation, to relate repeated addition to multiplication, and to determine the product of a positive number and a negative number. Have students complete problem 1 with a partner. Circulate and look for students who may have modeled the situation incorrectly or who may have written −2(3) for part (c). Ask those students the following questions. • Which direction should the directed line segments go? How long should they be? • If we think of the first factor as the number of groups, which number should the first factor be? • What is the size of each group?

137

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

1. An American football team has 3 consecutive plays where they lose 2 yards during each play. a. Use a number line to model the situation.

Consider displaying student work for problem 1(a) that uses a vertical number line and that uses a horizontal number line to emphasize that either model can accurately represent the situation.

2 2 2

–6

–4

–2

Teacher Note

b. Write an addition expression to represent this situation.

−2 + (−2) + (−2) c. Write a multiplication expression to represent this situation.

3(−2) d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

−6

2 n4

The product −6 means that the football team loses a total of 6 yards. When most students are finished, facilitate a class discussion by using the following prompts. How can we write a repeated addition expression as a multiplication expression?

2 n6

With repeated addition, the size of each group is the same. So one factor is the number of groups, and the other factor is the size of each group. How does the relationship between repeated addition and multiplication help us make sense of and evaluate an expression where the first factor is a positive number and the second factor is a negative number? We can think of the first factor as the number of groups and the second factor as the size of each group. For example, we can think of 3(−2) as 3 groups of −2, or −2 + (−2) + (−2), which evaluates to −6. 138

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

How can we make sense of the expression 2(−8)? We can think of the expression 2(−8) as 2 groups of −8. What does the product 2(−8) represent in the football situation? The product represents the total number of yards that are lost. If the team loses 8 yards for 2 consecutive plays, they lose a total of 16 yards. The product of 2 and −8 is −16. What do you notice about the product of a positive number and a negative number? The product of a positive number and a negative number appears to always be a negative number.

How can we make sense of  (−8)? What is the product? 2

We can think of the expression _1  (−8) as _1  of −8, which is −4. 2

The product is −4. The expression _1  (−8) is a positive number times a negative number, 2

so it should have a negative product. Have students complete problems 2–5 with a partner. Circulate to confirm answers or correct miscalculations. For problems 2–5, determine the product. 2. 2(−11)

−22 3. 3(−10)

−30 1 4. 3 _ (−1) 2 −3 1_ 2 1_ 5. (−4.5) 2

Differentiation: Support If students need support with determining the product of integers, have them represent the multiplication expression by using repeated addition. For example, the expression 2(−11) can be represented as −11 + (−11), which sums to −22.

−2.25

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

A Negative Number Times a Positive Number Students use patterns and the commutative property to determine the product of a negative number and a positive number. Direct students to problem 6. Let’s explore how to make sense of a negative number times a positive number. See if you can identify any patterns to help you complete the table in problem 6. Have students complete the table with a partner. Circulate and listen for the patterns that students notice to make true number sentences. 6. Complete the number sentences to make them true.

2(2) = 4 1(2) = 2 0(2) = 0

UDL: Representation Consider annotating the table to emphasize the patterns that students notice. For example, draw downward arrows and + (−1) to show that the first factor in each row is 1 less than the previous number sentence’s first factor. Highlight the second factor, 2, to show that it is the same in each row. Draw downward arrows in a different color and + (−2) to show that the product in each row is 2 less than the product in the previous row.

−1(2) = −2 −2(2) = −4

+(–1)

−2(2) = −6

+(–1) +(–1)

When most students are finished, facilitate a class discussion by asking the following questions. What patterns did you notice when completing the table? The first factor is 1 less than the first factor of the previous row. The second factor is always 2. The product is 2 less than the product of the previous row. What do you think −1(2) evaluates to? Why?

+(–1) +(–1)

2(2) = 4 1(2) = 2 0(2) = 0 –1(2) = –2 –2(2) = –4 –3(2) = –6

+(–2) +(–2) +(–2) +(–2) +(–2)

I think −1(2) evaluates to −2 because −2 is 2 less than 0.

140

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

What about −2(2)? What do you think it evaluates to? I think it evaluates to −4 because −4 is 2 less than −2. What do you think −3(2) evaluates to? Why? I think it evaluates to −6 because −6 is 2 less than −4. We want to be able to use the properties of operations with positive and negative numbers. If we apply the commutative property of multiplication, what expression is −3(2) equivalent to?

2(−3) What is the product of 2 and −3?

−6 By using the commutative property of multiplication, what is the product of −3 and 2? How does that compare to the product you found while using patterns? The product is −6. When I used patterns, I thought the product would be −6. How does the commutative property of multiplication help us evaluate multiplication expressions where the first factor is negative and the second factor is positive? By using the commutative property of multiplication, we can rearrange the factors, so the positive number is the first factor. Then we can think of the multiplication expression as the number of groups times the size of each group.

How can we make sense of −    (20)? What is the product? 2

We can use the commutative property of multiplication to write the expression as

20( −  _21), which means 20 groups of −  _21 . The product is −10.

Have students complete problems 7–9 with a partner. Circulate to confirm answers or correct miscalculations.

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

For problems 7–9, use the commutative property to determine the product. 7. −0.25(10)

−0.25(10) = 10(−0.25) 3 8. − _ (7) 4

= −2.5 − _ (7) = 7(− _) 3 4

9 1 9. − _ _

2(3)

__ = − 21

3 4

− _(1_) = _1 (− _) 9 2 3

= − _

9 2

9 6

When most students are finished, have them compare the signs of the factors in each expression to the sign of the product in problems 7–9. How can we determine the product when one factor is negative and one factor is positive? We can multiply the factors as if they were both positive numbers and then find the opposite of that product. We can multiply the absolute values of the factors and then find the opposite of that product. A positive number times a negative number is a negative number. A negative number times a positive number is a negative number.

142

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

A Negative Number Times a Negative Number Students use patterns and properties of operations to determine the product of two negative numbers. Have the class think–pair–share about the following question. Do not confirm or deny any responses. Do you think the product of two negative numbers is positive or negative? I think the product is positive because the opposite of the product of a positive number and a negative number is positive. I think the product is negative because all products involving a negative factor so far have a negative product. Let’s look at another set of patterns to explore the product of two negative numbers. Have students complete problems 10 and 11 independently. Consider having students compare their answers with a partner after they complete each table. For problems 10 and 11, complete the number sentences to make them true. 10.

2(−2) = −4 1(−2) = −2 0(−2) = 0 −1(−2) = 2

Promoting the Standards for Mathematical Practice When students use patterns to make true number sentences and make conjectures about the product of two negative numbers, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8:

−2(−2) = 4 −3(−2) = 6

• What patterns do you notice when you create true number sentences? • Will this pattern always be true?

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

11.

2(−10) = −20 1(−10) = −10 0(−10) = 0 −1(−10) = 10 −2(−10) = 20 −3(−10) = 30

Facilitate a class discussion by using the following prompts. What do you notice about the number sentences that have 0 as the product? When 0 is the product, one of the factors is 0.

Language Support Consider having students recall multiplication expressions with a positive factor and a factor of 0. Have them state the product. Emphasize that 0 groups of any size result in a product of 0, and any number of groups with a size of 0 also result in a product of 0. After defining the zero product property, consider displaying the following equation and emphasizing that at least one of the two factors must be 0.

⋅

That is called the zero product property. It states that if the product of two numbers is zero, then at least one of the numbers is zero. This means that when , at least one of the factors is zero. What do you notice about the product when both factors are negative numbers? If both factors are negative numbers, then the product is a positive number. Do you think the product is always a positive number when both factors are negative numbers? Will there ever be a time when the product is a negative number? I do not know for sure. I would want to try some other numbers to have a better feel for it. Let’s use the properties of operations to determine whether the product of two negative factors is always positive.

144

Teacher Note If students suggest that 0 is a negative number and that 0 times a negative number produces a negative product, remind them that 0 is not a positive or negative number because it is its own opposite.

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

Direct students’ attention to problem 12 and work through the problem as a class. Use the prompts to guide students’ thinking for each part of the problem: • Conjecture: In problem 10, you used patterns to determine that the product of −3 and −2 is 6. We will use properties to see whether that is true. • Line 1: What number makes this number sentence true? (−3) (

)=0

• Line 2: Why is there a −2 here? What number do we add to −2 to get a sum of 0? • Line 3: If we distribute −3, what goes in the first and the second blanks? • Line 4: What number goes in the blank, and how do you know? • Line 5: What number must go in the blank to make a true number sentence?

Differentiation: Support If students become overwhelmed with the lines of work in problem 12, consider covering the additional rows. Only have the rows that are completed and the row they are working on visible. Another option is to highlight the changes from one row to the next. This can aid in seeing how the problem evolves.

• Conclusion: What conclusion can we draw about −3 times −2? Was our conjecture true or false? 12. Use properties to fill in the blanks to determine whether the conjecture is true or false. Conjecture: (−3)(−2) = 6 Justification: Line 1: (−3)(

)=0

Zero product property

Line 2: (−3)(−2 + (

)) = 0

Line 3: (−3)(−2) + (

−3

)(

Line 4: (−3)(−2) + (

−6

)=0

Line 5:

+ (−6) = 0

Additive inverse

)=0

Distributive property

Teacher Note Students will learn the formal definition of proof in module 3 when they generate and explain a proof of the Pythagorean theorem. Problem 12 provides an informal, teacher-led introduction on how to use a written, logical argument to show that a mathematical statement is true. Do not expect students to use the term proof.

Product of a positive number and a negative number Additive inverse

Conclusion:

(−3)(−2) = 6 The conjecture is true. How does the product of −3 and −2 in this problem compare to the product of −3 and −2 in problem 10? Compare the two strategies. In both problems, the product is 6. Problem 10 is based on patterns, but this problem uses properties of operations. © Great Minds PBC

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7–8 ▸ M1 ▸ TB ▸ Lesson 6

If we repeat the lines of work from problem 12 but use −5 times −10, do you think the result will be 50 or −50? Why?

Differentiation: Challenge

The work will be the same, but the numbers will be different. The product will be 50.

What if we repeat the lines of work from problem 12 but use −  

think the result will be or −   ?

The product will be _ . 3 4

−1  _1? Do you 2

Challenge students by asking them to evaluate multiplication expressions with more than two factors.

−4(−2)(−3)

What conclusion can we make about the product of two negative numbers? The product of two negative numbers is a positive number.

5(−4)(−6)

1  (7)(−20) −  __ 2

9(−0.02)(20)

Multiplication and Decomposition Students apply decomposition and the distributive property to evaluate a multiplication expression.

Students evaluate multiplication expressions with more than two factors in the next lesson.

Direct students’ attention to problem 13. Guide students through decomposing the expression by using the following question. How can we use decomposition to make this expression easier to evaluate? We can decompose −2  _ 1 2

−2 and − _ 21

  , and add the products. Then we can multiply −4 by −2, multiply −4 by − _ 1 2

Have students work with a partner to complete problems 13 and 14. As students work, circulate and provide feedback about their use of decomposition and the distributive property.

Providing timely feedback sustains students’ effort and persistence. Focus your praise on student effort and strategies; emphasize working hard.

For problems 13 and 14, evaluate the expression.

13. −4( − 2 1_) 2

UDL: Engagement

−4(−2 1_) = −4( −2 + (− 1_)) 2 2

= −4(−2) + (−4)(− 1_) 2

=8+2 = 10 146

EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 6

14. −3(1.5)

−3(1.5) = −3(1 + 0.5) = −3(1) + (−3)(0.5) = −3 + (−1.5) = −4.5 If time allows, invite a few students to share their work on problems 13 and 14 with the class.

Differentiation: Challenge Challenge students by having them create as many true number sentences as they can for the following problem.

(

)⋅(

1 ) = −3  __ 4

Land Debrief 5 min Objectives: Use repeated addition and the properties of operations to determine the product of a negative number and a positive number. Informally verify that the product of two negative numbers is a positive number. Initiate a class discussion by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Explain why some multiplication expressions can be written as repeated addition expressions and how this relates to multiplying integers. The first factor in a multiplication expression can tell us how many groups, and the second factor can tell us the size of each group. To write as repeated addition, we use the size as the addend and add it as many times as needed according to the number of groups. For a multiplication expression with a positive number times a negative number, we can think of the expression as the positive number of groups of the negative number, which has a negative product.

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Can we determine the product of a negative number and a positive number? How? Yes. We can multiply the absolute values of the factors and find the opposite of that product. We discovered that the product of a negative number and a positive number is a negative number. Can we determine the product of two negative numbers? How? Yes. We can multiply the absolute values of the factors. We discovered that the product of two negative numbers is a positive number.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

148

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

b. Write an addition expression to represent this situation.

−10 + (−10) + (−10)

Because Lily descends 10 feet each time, the addends are −10.

Multiplying Integers and Rational Numbers In this lesson, we •

related repeated addition to multiplication to make sense of a multiplication expression where the first factor is positive and the second factor is negative.

•

analyzed patterns in tables to determine products.

•

used properties of operations to determine products of rational numbers.

•

applied decomposition and the distributive property to evaluate multiplication expressions.

Terminology

c. Write a multiplication expression to represent this situation.

The zero product property states that if the product of two numbers is zero, then at least one of the numbers is zero.

3(−10)

⋅ = 0, at least one of the factors is zero.

This multiplication expression describes an addition expression that has 3 groups of −10.

This means that when

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

−30 The product −30 means that Lily is 30 feet below the water’s surface when she reaches the reef.

Examples 1. Lily is scuba diving. She descends 10 feet from the water’s surface and rests. Then she descends another 10 feet and rests. Finally, she descends 10 more feet to reach a reef.

For problems 2–5, determine the product.

a. Use a number line to model the situation. The directed line segments point down because each of them represents descending

0 10

2. 4(−6)

3. (−4)(−6)

−24

10 feet. There are 3 of them because Lily descends 3 separate times.

−10

The absolute value of 4 times the absolute value of −6 is 24. The product of a positive

The product of two negative numbers is a positive number.

number and a negative number is a

negative number, so 4(−6) = −24.

−20 10 −30

The end of the last directed line segment represents Lily’s depth below the water’s surface when she reaches the reef.

RECAP

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7–8 ▸ M1 ▸ TB ▸ Lesson 6

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

4. (−0.43)(6)

(−0.43)(6) = (−0.4 + (−0.03))(6) = (−0.4)(6) + (−0.03)(6) = −2.4 + (−0.18) = −2.58 −0.43 can be decomposed into −0.4 and −0.03.

1 5. (−5 _)(−20) 4

After decomposing, use the distributive property.

(−5 4)(−20) = (−5 + (− 4))(−20) 1_

= (−5)(−20) + (− 1_4)(−20)

= 100 + 5

−5 _1 can be decomposed 4

into −5 and − _1 . 4

= 105

150

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 6

PRACTICE

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

For problems 2 and 3, write the expression as repeated addition. Then evaluate the expression. 2. 4(2)

3. 4(−2)

4(2) = 2 + 2 + 2 + 2

1. The temperature at 6:00 p.m. is −4°F. By 7:00 p.m., the temperature has dropped 4°F. By 8:00 p.m., the temperature has dropped another 4°F.

4(−2) = −2 + (−2) + (−2) + (−2)

= −8

a. Use a number line to model the situation.

For problems 4–19, determine the product.

4. 3(−6)

5. −3(−6)

−18

6. −5(−4)

7. −4(5)

−20

8. −4(−5)

9. (−6)(−3)

−4 4 −8 4 −12

b. Write an addition expression to represent this situation.

10. 6(−3)

11. −6(3)

−4 + (−4) + (−4) c. Write a multiplication expression to represent this situation.

3(−4)

−18

d. Evaluate the expression from part (c) and explain the meaning of the product in this situation.

−18

−12 The product −12 means the temperature at 8:00 p.m. is −12°F.

P R ACT I C E

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

12. − 3_ ⋅ 16 4

13.

−12

_1 (−9) 2 −4 1_ 2

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 6

For problems 24 and 25, write the expression as an equivalent addition expression and then evaluate. 24. 12 − (−46)

12 − (−46) = 12 + 46 = 58

14. −0.4 ⋅ 2

15. (−0.38)(−5)

−0.8

1.9

25. −14 − 49

−14 − 49 = −14 + (−49) = −63 16.

_2 − 4_ 3( 5) 8 − __ 15

1 17. − _ − _5 2( 8) 5 __ 16

26. Evaluate the expression.

−5 3_ + 2 1_ − 1_ 1 18. (−2 _)(−12) 3

−3 1_

19. −0.2(−1.4)

0.28

Remember For problems 20–23, subtract. 20.

_6 − 2_ 7 7 _4 7

152

21.

_4 − _2 9 9 _2 9

22.

_8 − 5_ 3 3 1

23.

12 __ __ − 2 14 14 10 __ 14

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic B, Lesson 7 LESSON 7

Exponential Expressions and Relating Multiplication to Division Evaluate exponential expressions that include rational numbers. Write division expressions as unknown factor equations to determine the value of the quotients.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7 Name

Date

EXIT TICKET

For problems 1 and 2, evaluate the expression. 3 3 1. (− _) 4

(− 4) = (− 4)(− 4)(− 4) 3 3

9 3_ = __ (− ) 16

__ = − 27

Lesson at a Glance In this lesson, students apply prior knowledge of multiplying rational numbers to evaluate exponential expressions with negative numbers. Students identify and follow a pattern to predict whether the product or the value of an exponential expression is positive or negative. As a class, students share errors they identified in sample work of evaluating an exponential expression. Students then divide integers by writing unknown factor equations and recognize that the patterns for multiplying rational numbers apply to division.

Key Questions

2 2 2. −( _) 3

_ _

2 −( 2) = −(2)(2) 3 3 3

• What details are important to think about when determining whether the value of an exponential expression is positive or negative?

= −(4_) 9

= − 4_ 9

• How are division and multiplication of integers related? For problems 3 and 4, write the unknown factor equation related to the expression and then determine the unknown factor.

Achievement Descriptors

3. −20 ÷ 4

4⋅

7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients

= −20

−5

of two rational numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD5 Determine the sign of a product by looking at the

4. −45 ÷ (−5)

−5 ⋅

signs of its factors. (7.NS.A.2.a)

= −45

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7–8 ▸ M1 ▸ TB ▸ Lesson 7

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Math Chat

• None

• Correcting the Errors

Lesson Preparation

• Relating Multiplication and Division

• None

Land 10 min

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Fluency Exponents Students evaluate exponential expressions to prepare for evaluating exponential expressions with rational numbers. Directions: Evaluate. 1.

3⋅3 2

9 9

2⋅2⋅2

Instead of this lesson’s Fluency, consider administering the Squares Sprint. Directions for administration can be found in the Fluency resource.

10 ⋅ 10

21.

1 2 (2)

3⋅3

22.

(2)

23.

(2)

24.

(4)

25.

(4)

26.

(4)

27.

(8)

28.

(8)

29.

(8)

10.

30.

(10)

11.

31.

32 + 3

12.

0.42

32.

32 + 6

13.

33.

32 + 9

14.

0.62

34.

32 + 32

15.

35.

72 + 3

16.

0.82

36.

72 + 9

17.

112

37.

72 + 32 72 − 3

1. 2.

5. 6. 7. 8.

156

52 2

25 36 64

10.

11.

12.

1,000

103 3

(0.1)

__1 3

(10)

0.001 1 ____ 1,000

Number Correct:

Evaluate the expression.

EUREKA MATH2

7–8 ▸ M1 ▸ Sprint ▸ Squares

Teacher Note

102

5⋅5 52

7⋅7

_ 2 3

_ 2 5

_ 2 1

_ 2 3

_ 2 5

_ 2 1

_ 2 3

_ 2 5

18.

1.12

38.

19.

122

39.

72 − 9

20.

1.22

40.

72 − 32

362

This page is int

_1 2

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Students predict whether the values of multiplication expressions and exponential expressions are positive or negative. Display each of the following expressions in rapid succession. Have students give a thumbs-up if they believe the product is a positive number and give a thumbs-down if they believe the product is a negative number. • 5(−6) • −3(4) • −8(−2) • −3(2)(5) • −3(−2)(5) • −3(−2)(−5) • 425(−1)(−12.4)

5 • −3,428(−2.451)(− _) 3

Then display all the previous expressions. Do we need to evaluate the expression to know whether the product is a positive number or a negative number? No, we can look at the number of negative factors in each expression. When there is one negative factor, the product is a negative number. When there are two negative factors, the product is a positive number. When there are three negative factors, the product is a negative number.

Differentiation: Support If students need support identifying whether the product is a positive number or a negative number, move them forward by having them analyze the number of negative factors and determine the product of each of the following expressions. • 1(3)(5)(2) • 1(3)(5)(−2) • 1(3)(−5)(−2) • 1(−3)(−5)(−2) • −1(−3)(−5)(−2)

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Have students turn and talk about whether they think this pattern continues. Have students continue to signal whether the product is a positive number or a negative number for the following expressions. • (−1)(−1)(−1) • (−1)(−1)(−1)(−1) • (−1)(−1)(−1)(−1)(−1) • (−1)(−1)(−1)(−1)(−1)(−1) Have students think–pair–share about the following question. How does the number of negative factors in a multiplication expression relate to whether the product is a positive number or a negative number? If there is an even number of negative factors, the product is a positive number. If there is an odd number of negative factors, the product is a negative number. Continue the activity with the following expressions. • (32)(−5) • (−3)2(5) • (−3)2(−5)2 • (−3)(53) • (33)(−5)2

Teacher Note Students are introduced to exponents in grade 5 as they multiply and divide by powers of 10. Then in grade 6, students evaluate exponential expressions with whole-number exponents and positive rational number bases.

• (−3)3(−5)3 How did you determine whether the product was a positive number or a negative number? I used the same reasoning as with the multiplication expressions. I thought about the exponential factor as repeated multiplication and determined how many factors in the equivalent multiplication expression are negative. Exponential expressions can be written as equivalent multiplication expressions. The base is the factor and the exponent represents the number of times the base is a factor. If the equivalent multiplication expression has an even number of negative factors, the product is a positive number. If the equivalent multiplication expression has an odd number of negative factors, the product is a negative number. 158

Differentiation: Challenge Challenge students to generate two expressions that evaluate to a positive number and two expressions that evaluate to a negative number.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Today, we will calculate products of rational numbers, evaluate exponential expressions, and use what we know about rational number multiplication to divide rational numbers.

Learn Math Chat Students multiply rational numbers and evaluate exponential expressions that include negative numbers. Direct students’ attention to the Math Chat segment in their book. Give students 2 minutes to begin working individually on problems 1–20. Pause the class. Have students determine which problem they would like to discuss. Take a quick poll of the class to choose one problem. Use the Math Chat routine to engage students in mathematical discourse about that problem. Give students 1 minute of silent think time to pick a strategy to evaluate the expression. Have students give a silent signal to indicate they are finished. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies. Then facilitate a class discussion. Invite students to share their thinking with the class, and record their reasoning. Evaluate the expression by using the discussed reasoning. Have students work on the remaining problems individually. Encourage students to compare their answers with a partner.

Language Support For problems 1–20, some students may work faster than others. The Math Chat routine holds valuable conversation, so do not rush through it. The problems can be finished later. Consider conducting a second Math Chat routine a few minutes after students have returned to working on the problems.

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Circulate as students work and provide support as needed. Consider using the following prompts: • What is the sign of the product of a positive number and a negative number? • What is the sign of the product of two negative numbers? • What is the base in this expression? • What is the exponent? For problems 1–20, evaluate the expression. 2. (−3)2

1. (−3)(−3)

(−3)(−3) = 9

(−3)2 = (−3)(−3) =9

3. −(3)2

4. −(32)

= −(9)

−(32) = −(3 ⋅ 3)

= −9

−(3)2 = −(3)(3)

= −(9)

1 1 6. (_)(_) 3 3

5. −(−3)2

_ _1

(3)(3) = 9

−(−3)2 = −(−3)(−3)

= −(9)

= −9 1 2 7. (_) 3

_1 2

_1 _1

(3) = (3)(3)

= 1_

1 1 8. (− _)(− _) 3 3

_ _ _ (− 3)(− 3) = 9 1

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EUREKA MATH2

1 2 9. (− _) 3

7–8 ▸ M1 ▸ TB ▸ Lesson 7

_ _ _ (− 3) = (− 3)(− 3) 1 2

1 1 10. (− _)(_) 3 3

_ _ _ (− 3)(3) = − 9 1

1 11. −(_) 3

Teacher Note 4 In problem 12, the product of − __ and _5_ is 5

written as − _4_ for ease of calculation.

=1 9

−(1_) = −(_1)(_1) 3 3 3

4 5 3 12. (− _)(_)(_) 5 7 8

Consider discussing this strategy as a class.

_ _ _ __ _ (− 5)(7)(8) = − 35(8) 4

= −(1_) = − 1_

20 3

= − 4_(_) 3 7 8

__ = − 12

14. (−0.5)2

13. (−1.1)(−0.4)

(−1.1)(−0.4) = (−1 + (−0.1))(−0.4)

(−0.5)2 = (−0.5)(−0.5)

= (−1)(−0.4) + (−0.1)(−0.4)

= 0.25

= 0.4 + 0.04 = 0.44 15. (−1.5)2

16. (1.5)(−1.5)2

(−1.5)2 = (−1.5)(−1.5)

(1.5)(−1.5)2 = (1.5)((−1.5)(−1.5))

= (−1 + (−0.5))(−1.5)

= (1.5)(2.25)

= (−1)(−1.5) + (−0.5)(−1.5)

= 3.375

= 1.5 + 0.75 = 2.25

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

1 3 18. (− _) 3

17. (−3)3

_ _ _ _ (− 3) = (− 3)(− 3)(− 3) 1 3

(−3)3 = (−3)(−3)(−3) = 9(−3)

= 1_(− 1_) 9

= −27

1 = − __

1 3 19. −(_) 3

1 3 20. −(− _) 3

−(1_) = −(1_)(1_)(1_) 3 3 3 3

−(− 1_) = −(− _1)(− 1_)(− 1_) 3 3 3 3

= −(1_)(1_) 9

1 = −(__ )

= −(1_)(− _1)

1 = −(− __ )

1 = − __

1 = __

Call the class back together. Display the expressions for problems 2–4.

Teacher Note

Problem 2

Problem 3

Problem 4

(−3)2

−(3)2

−(32)

If students chose one of these problems in the Math Chat routine, adapt the discussion to build on the prior conversation.

What do you notice about the expressions for problems 2–4? Each expression has an exponent of 2. Each expression has a negative sign. Each expression has the digit 3. The expression in problem 2 has −3 inside the parentheses and the exponent outside the parentheses.

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The expression in problem 3 has 3 inside the parentheses and the negative sign and the exponent outside the parentheses. The expression in problem 4 has 32 inside the parentheses and the negative sign outside the parentheses. Why does the expression in problem 2 evaluate to a positive number? The base of the expression is −3 because everything in the parentheses is raised to the second power. The equivalent multiplication expression is (−3)(−3). The product of two negative numbers is a positive number. Display the following work and explain it as needed to emphasize this learning.

(−3)2 = (−3)(−3) =9 The expression in problem 2 has a base of −3 and the exponent is 2. This exponential expression can be written as (−3)(−3), which evaluates to 9. How is the expression in problem 3 different from the expression in problem 2? The expression in problem 3 has the negative sign outside the parentheses instead of inside the parentheses. With the negative sign outside the parentheses, the base is 3 and not −3. What does the expression in problem 3 evaluate to? Why? It evaluates to −9 because 32 = 9, and the opposite of 9 is −9. Why does the expression in problem 4 evaluate to a negative number? For the expression in problem 4, I know I need to evaluate 32 first because it is inside the parentheses, and according to the order of operations, we evaluate operations inside parentheses first. Then I need to find the opposite of the result, because the negative sign is outside the parentheses. The opposite of 32 is −9. When we evaluate expressions, should we evaluate the exponent first or multiply first? Evaluating the exponent needs to come before multiplication.

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

We evaluate exponents before multiplication, and we evaluate operations inside parentheses before any other operation. In the expressions −(3)2 and −(32), we need to first square a number and then find the opposite. We can think of finding the opposite of a number as multiplying the number by −1. In each of these expressions, after squaring 3, we find the opposite of the result. Display the following work to emphasize this learning.

Problem 3

Problem 4

−(3)2 = −(3)(3) = −(9)

−(32) = −(3 ⋅ 3)

= −9

= −(9)

Then display the expression −32. Have students turn and talk about the following questions: • How is this expression different from the expressions in problems 2–4? • Do you think this expression evaluates to −9 or 9? Why? • How can you use the order of operations to help you evaluate this expression? Circulate and listen to student discussions. Anticipate productive struggle. Select a few students to share their reasoning. Then display the following work.

−32 = −1 ⋅ 32

Have students think–pair–share about the following question. We can think of this expression as −1 times 32. With this idea in mind, what do you think the expression evaluates to? Why? I think that it evaluates to −9. Because I can think of the negative sign as a factor of −1, I know that I need to first evaluate 32, because it has an exponent, and then multiply the result by −1. 164

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Display the following work as needed to confirm student thinking.

−32 = −1 ⋅ 32 = −1 ⋅ 9 = −9

Confirm the answers to problems 5–20. Have students make corrections, if needed.

Differentiation: Challenge Challenge students by asking them to determine whether (− _2_) 3

1,622

2 and (− __ ) 3

1,621

evaluate to positive numbers or negative numbers and to explain their reasoning.

Correcting the Errors Students identify and correct the errors in sample work. Introduce the Critique a Flawed Response routine and present problem 21. Give students 1 minute to identify errors in the work and explain one of the errors for part (a). Then invite students to share what they noticed. Call attention to the following errors if students do not share them. For lines 3–6, prior errors also still appear. • In line 2, the product of −1.5 and −1.5 is shown as −2.25 instead of 2.25. • In line 3, −1.5 is decomposed as −1 + 0.5 but should be −1 + (−0.5).

Promoting the Standards for Mathematical Practice When students analyze the given work for evaluating the exponential expression with a negative base, they are critiquing the reasoning of others (MP3).

• In line 4, −2.25, not 2.25, needs to be distributed to 0.5.

Ask the following questions to promote MP3:

• In line 5, the product of −2.25 and −1 is shown as −2.25 instead of 2.25.

• What parts of this work do you question? Why?

• In line 6, the sum of −2.25 and 1.125 is shown as 1.125 instead of −1.125. Give students 2 minutes to correctly evaluate the expression for part (b) based on their own understanding. Circulate to ensure students are correcting the flawed response and getting the answer −3.375. If you notice students with different answers, invite a few of them to share their work with the class until there is consensus about the correct response.

• How can you change this work to make it accurate?

Teacher Note The sample work in problem 21 has many errors. The errors are purposely included to initiate rich discussion about mistakes that commonly occur when evaluating an expression.

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21. Use the given sample work to answer parts (a) and (b).

(−1.5)3 = (−1.5)(−1.5)(−1.5) = −2.25(−1.5) = −2.25(−1 + 0.5) = −2.25(−1) + 2.25(0.5) = −2.25 + 1.125 = 1.125 a. Explain an error in the sample work. Multiplying two negative factors results in a positive product, so the second line should be 2.25(−1.5). In the third line, the number −1.5 should be decomposed into −1 and −0.5. To distribute −2.25, both addends in the parentheses need to be multiplied by −2.25. So in the fourth line, the correct way to distribute is −2.25(−1) + (−2.25)(0.5). Multiplying two negative factors results in a positive product. So based on previous work, the fifth line should be 2.25 + 1.125. The addend with the greater absolute value is −2.25, so the sum should be −1.125. b. Correctly evaluate the expression (−1.5)3.

(−1.5)3 = (−1.5)(−1.5)(−1.5) = 2.25(−1.5) = 2.25(−1 + (−0.5)) = 2.25(−1) + 2.25(−0.5) = −2.25 + (−1.125) = −3.375

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Relating Multiplication and Division Students write division expressions as unknown factor equations and determine the quotient. Guide students to think–pair–share about the following question. How are multiplication and division related? Use a specific example to support your answer. Sample:

We can use unknown factor equations to determine quotients. For 18 ÷ 9, we can use 9⋅ = 18 to help us determine the quotient, which is 2.

Teacher Note In grades 5 and 6, students relate division to multiplying by a reciprocal. If students offer this strategy, allow them to explain how it relates to the learning of this lesson. Promote further learning by inviting students to study this topic’s lessons more deeply by using that prior knowledge. Encourage students to frequently verify that their strategy is valid.

Every division expression with a nonzero divisor can be written as an equivalent multiplication expression. For example, 18 ÷ 9 is equivalent to 18(_). 1 9

Display the table from problem 22. Use the following prompts to help students recall the unknown factor equation.

4⋅

What is the unknown factor equation related to 8 ÷ 4? What does it mean?

4 times what number is equal to 8? What is the unknown factor?

UDL: Representation Consider activating prior knowledge by reviewing the terms dividend, divisor, quotient, factor, and product with students. Post the following equations and label each number with the correct term for students to refer to throughout the lesson.

5 ∙ 3 = 15

15 ÷ 5 = 3 Also post the following equations for students to refer to when determining how to write a division problem as a related unknown factor equation.

p÷q= q∙

Use a consistent sentence frame to prompt student thinking throughout this lesson: “q times what number is equal to p?”

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7–8 ▸ M1 ▸ TB ▸ Lesson 7

Demonstrate how to complete the second and third columns by writing the unknown factor equation 4 ⋅ = 8, the unknown factor, and the division number sentence 8 ÷ 4 = 2. Then introduce fraction form. We know that division can be represented as a related fraction. What fraction represents the quotient 8 ÷ 4?

8 fourths

Demonstrate how to complete the fourth column by writing _ = 2. Encourage students 8 4

to use what they already know about integer multiplication, along with what they know about how division can be represented as an unknown factor equation, to discover how to divide integers. The unknown factor is 2. What is 2 in the division expression?

2 is the quotient. When you determine the unknown factor, you are also determining the quotient of the related division expression. Direct students to work with a partner to complete the second row by using the problem in the previous row as a model. When most students have finished, present the solutions, and briefly discuss the first two rows. What do you notice about the signs of the dividend, divisor, and quotient in the first two problems? How are they related to what you know about integer multiplication? The dividend and divisor have the same sign, and both quotients are positive numbers. It follows the same rule as integer multiplication: When two factors have the same sign, the product is a positive number. Have students work with their partners to complete the last two rows.

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

22. Complete the table.

Division Expression

Teacher Note Unknown Factor Equation

8÷4

4⋅

−8 ÷ (−4)

−4 ⋅

4⋅2=8 =8

−4 ⋅ 2 = −8 = −8

−8 ÷ 4

4⋅

8 ÷ (−4)

−4 ⋅

4 ⋅ (−2) = −8

= −8

−4 ⋅ (−2) = 8

Division Expression Evaluated

Fraction Form Evaluated

8÷4=2

8 =2 4

−8 ÷ (−4) = 2

−8 =2 −4

−8 ÷ 4 = −2

8 ÷ (−4) = −2

___

−8 = −2 4

8 ___ = −2 −4

When most students have finished, present the solutions and facilitate a brief discussion. What do you notice about the dividend, divisor, and quotient in the last two rows? How are they related to what you know about integer multiplication?

When displaying the unknown factor equation in the table in problem 22, use a blank or a ? to indicate the unknown factor. Avoid using a variable, as it would need to be defined within the question prompt. It is important to model for students the mathematical practice of always defining a variable when it is presented and to avoid using variables that are not defined. In the fourth column, when displaying the quotient in fraction form and then evaluating 8 it, such as __ = 2, write the fraction as 4

shown instead of an equivalent fraction. The relationship between division problems and numbers in fraction form is examined more closely later in the lesson, and it is an important part of students’ understanding of integer division and rational numbers. The relationship is intentionally previewed in this activity.

When the dividend and divisor have different signs, the quotient is a negative number. It follows the same rule as integer multiplication. Look at the two division expressions in the last two rows that are written in fraction form. What is similar about

−8 8 ___ ___ and ? What is different? 4

−4

They both have one negative number and one positive number, but the negative sign is in a different place in each expression. In the first expression, the numerator is a negative number. In the second expression, the denominator is a negative number. The expressions are similar because they have the same quotient, −2.

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Display the expression −(_). Invite students to turn and talk about the following question. 8 4

In the table, which division expressions written in fraction form do you think are 8 equivalent to −(_) ? Why? 4 8 8 Have students recall that they can think of the expression −(_) as being the opposite of _ . 4

Tell students that they will work more with the idea of the opposite of a number and with

fraction form in the next lesson. Direct students to complete problem 23 individually by writing what they understand about integer division so far. 23. What is the process for dividing integers? We divide integers the same way we divide whole numbers except we need to determine the sign of the quotient. When the signs of the dividend and divisor are the same, their quotient is a positive number. When the signs of the dividend and divisor are different, their quotient is a negative number. When most students have finished, ask them to share their process with a partner. Select one or two students to read their processes to the class. Emphasize responses that demonstrate precise language.

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Land Debrief 5 min Objectives: Evaluate exponential expressions that include rational numbers. Write division expressions as unknown factor equations to determine the value of the quotients. Initiate a class discussion by using the following prompts. Encourage students to add to their classmates’ responses. What details are important to think about when determining whether the value of an exponential expression is positive or negative without evaluating the expression? We need to know how many negative factors are in the equivalent multiplication expression. An even number of negative factors results in a positive number. An odd number of negative factors results in a negative number. Determining the opposite of a number is like multiplying by −1, which is considered a negative factor. How are division and multiplication of integers related? We can find the quotient of a division expression by writing the related unknown factor equation. The rules for the signs of quotients in integer division problems are the same as the rules for the signs of products in integer multiplication problems. When the dividend and divisor have the same sign, either both are positive numbers or both are negative numbers, the quotient is a positive number. When the dividend and divisor have different signs, the quotient is a negative number.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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7–8 ▸ M1 ▸ TB ▸ Lesson 7

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

For problems 4–7, evaluate the expression. 2 4 4. (− _) 3

4 factors of − _2 .

predicted whether the value of a multiplication or exponential expression would be positive or negative.

•

evaluated exponential expressions that included negative numbers.

•

wrote unknown factor equations to divide integers.

= 4_(− 2_)(− 2_)

In this lesson, we •

(− 3) = (− 3)(− 3)(− 3)(− 3) 2 4

Exponential Expressions and Relating Multiplication to Division

The exponent is 4, so there are

8 2_ = − __ (− ) 27

__ = 16 81

Examples

The exponent is 3, so there are

5. (−0.4) 3 3

(−0.4) = (−0.4)(−0.4)(−0.4)

For problems 1–3, determine whether the value of the expression is positive or negative. Explain your answer.

3 factors of −0.4 .

= 0.16(−0.4) = −0.064

1. (−3)(0.5)(−9) The value of the expression is positive because there is an even number of negative factors. 6. −0.3 2

The value of the expression is negative because there is an odd number of negative factors.

3. −(−4) 3 The value of the expression is positive because there is an even number of negative factors.

−0.3 2 = −1 ⋅ 0.3 2

To determine the opposite of a number, multiply that number by −1. Follow the order

= −1 ⋅ 0.09

The base is −2, and the exponent is 3. So there are 3 factors of −2.

2. (−2) 3

of operations by evaluating exponents before multiplication. So determine the value of 0.32

= −0.09

(−2)(−2)(−2)

before taking the opposite.

The base is −4, and the exponent is 3. So there are 3 factors of −4. The negative sign outside the

1 2 7. −(− _) 5

parentheses can be thought of as multiplying by −1, so that makes 4 negative factors.

−(− 1_) = −(− 1_)(− 1_) 5

1 = −(__ )

1 = − __ 25

The negative sign outside the parentheses represents taking the opposite of (− _1) . 5

172

107

108

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

For problems 8 and 9, write the unknown factor equation related to the expression and then determine the unknown factor. 8. −80 ÷ 4

4⋅

= −80

−20

The quotient of −80 ÷ 4 is the number that is multiplied by 4 to get a product of −80. The unknown factor is shown as a blank in the equation.

9. −60 ÷ (−15)

−15 ⋅

= −60

RECAP

109

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 7 Name

PRACTICE Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

6. −(−7) 5 The value of the expression is positive because there is an even number of negative factors.

For problems 1–7, determine whether the value of the expression is positive or negative. Explain your answer. 1 1. (−4)(− _)(−0.3) 2

1 7. −(_) 3

The value of the expression is negative because there is an odd number of negative factors.

1 2. (−4)(− _)(0.3) 2

For problems 8–18, evaluate.

The value of the expression is positive because there is an even number of negative factors.

1 3. (−4)(− _)(−0.3)(−1) 2

The value of the expression is positive because there is an even number of negative factors.

1 1 8. (− _)(− _) 4 4 __1 16

3 3 9. (− _)(_) 4 4 9 − __ 16

10. (−0.2)(−1.5)

11. (−6) 2

0.3

4. (−5) 2

12. −6 2

The value of the expression is positive because there is an even number of negative factors.

−36

5. (−7) 5

174

111

3 2 13. (− _) 4

__9 16

3 2 14. −(− _) 4

15. 1.1(−1.1) 2

1.331

9 − __

The value of the expression is negative because there is an odd number of negative factors.

112

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 7

_1 4 17. (− 2) __1 16

16. (−6) 3

−216

7–8 ▸ M1 ▸ TB ▸ Lesson 7

EUREKA MATH2

For problems 27 and 28, evaluate. 27. −6.8 − (−2.9)

−3.9

1 18. −(__ 10) 1 − _____ 10,000 4

7 1 28. − _ − 3 __ 8 12 __ −3 23 24

For problems 19–22, write the unknown factor equation related to the expression and then determine the unknown factor. 20. −90 ÷ 3

19. 20 ÷ 5

5⋅

3⋅

= 20

22. −77 ÷ (−11)

21. 54 ÷ (−9)

−9 ⋅

= −90

−30

−11 ⋅

= 54

−6

= −77

Remember For problems 23–26, subtract. 23.

_ _ 6 1 − 5 2 7 10

24.

_ _ 8 2 − 9 5 22 45

25.

_ _ 3 6 − 2 7 9 14

26.

_ _ 8 8 − 3 5 16 15

P R ACT I C E

113

114

P R ACT I C E

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Teacher Edition: Grade 7–8, Module 1, Topic B, Lesson 8 LESSON 8

Dividing Integers and Rational Numbers Write rational numbers as quotients of integers. Divide rational numbers given in different forms.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8 Name

Date

EXIT TICKET

1. Which numbers shown are equivalent to − _5 ? Circle all that apply. − ___5 7

− ___5 −7

− (5_) 7

_7 5

5_ 7

5 ___

− 7_

−5 − (___ )

and explain patterns about equivalent forms of rational numbers −3 ___ of why −_3 = −(_3) = ___ = 3 . Students evaluate a variety of division 5

−5

expressions involving different forms of rational numbers. They discuss efficient strategies, consider how numbers can be manipulated, and

−7

look at how the relationship between multiplication and division is used to evaluate expressions. This lesson defines the term rational

For problems 2 and 3, divide. 2. − 2_ ÷ (− 0.6) 5

number by expanding on the grade 6 description. 6 − 2_ ÷ (− 0.6) = − 2_ ÷ (− __ ) 5

Key Questions

__ = − 2_(− 10 ) 5

__ = 20

• In what ways can we write a quotient of two given integers? How do we know these forms are all equivalent?

1_ 2 3. ___ − _3 5

In this lesson, students work with partners to make observations expressed as quotients of integers. This leads to the discovery

−7

Lesson at a Glance

• What strategies can we use to divide rational numbers given in different forms?

_1 ÷ − 3_ = 1_ − 5_ 2 ( 5) 2( 3) = − 5_ 6

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Achievement Descriptors 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD2 Solve real-world and mathematical problems involving the four operations

with rational numbers. (7.NS.A.3) 7–8.Mod1.AD3 Interpret sums, differences, products, and quotients of rational numbers

by describing real-world contexts. (7.NS.A.1.a; 7.NS.A.1.b; 7.NS.A.1.c; 7.NS.A.2.a; 7.NS.A.2.b; 7.NS.A.2.c; 7.NS.A.3)

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Patterns in Integer Division

• Highlighters (2 different colors per student pair)

• Division Expressions • Applications of Division

Land 10 min

Lesson Preparation • None

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Fluency Multiply Rational Numbers Students multiply rational numbers to prepare for dividing integers.

Teacher Note

Directions: Multiply.

178

−5 ⋅ 8

−40

_ (− 2)(−8)

−3(−10)

10(− _1) 5

−2

6(−4)

−24

−16(− _1)

−7(−2)

10.

3 (−20) 4

−15

−9 ⋅ 5

−45

11.

− _1 (27)

−3

−8(−8)

12.

_ _ (− 4)(− 2)

Instead of this lesson’s Fluency, consider administering the Multiplication of Integers Sprint, which can be found in Eureka Math2 grade 7 module 2.

3 8

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Students compare quotients of integers. Display the five expressions. Expression A

Expression B

Expression C

Expression D

Expression E

3 4

3 −4

−3 −4

−_

−(_)

−3 _

What do the expressions have in common? Each expression has either −3 or 3 as its numerator and −4 or 4 as its denominator. Each expression has at least one negative sign. What are some differences in the expressions? One expression has parentheses. The negative signs are in different places. Ask students to work in pairs to conjecture about what decimal each expression evaluates to. Then invite students to share their thinking, and ask questions such as the following: • Can you use what you learned in the previous lesson about unknown factor equations to make sense of the value of each expression? • Do you think all the expressions evaluate to the same number? Why?

Language Support As pairs discuss each expression, encourage them to use the Agree or Disagree section of the Talking Tool.

Today, we will divide integers and rational numbers.

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Learn Patterns in Integer Division Students explain patterns in the signs of integer division expressions. Present the table in problem 1. Direct students to work with a partner to complete the table one row at a time. For each cell, they should write the expression with the values of p and q substituted in as the dividend and divisor and then evaluate the expression. If needed, consider modeling how to complete the first row of the table. Circulate as students work and observe to ensure they are completing the table accurately.

Teacher Note Students may think that an expression such as

−p ___ must have a negative value because q

the expression shows a negative sign. Complete the table in problem 1 by using opposite language to clarify the value of

1. Complete the table by calculating each quotient with the values of p and q given in that row. p −q

−(q_)

−p −q

12 = −4 −3

12 −(_ ) = −4

−12 _ =4

−(−18) −18 ______ = −9 _____ = −9

−(_) = −9

−(−18) ______ =9

10 =2 −(−5)

−(_) = 2 10 −5

−10 = −2 −(−5)

−3 _ = 0.75

−(_) = 0.75

−(−3) _____ = −0.75

p and q given

p q

−p q

p = 12 q=3

12 _ =4

−12 _ = −4

p = −18 q = −2

−18 _ =9

p = 10 q = −5

10 = −2 −5

p = −3 q=4

−(−3) −3 _ = −0.75 _____ = 0.75

−2

−10 _ =2 −5

−(−2)

_____

−4

−18 −2

−3 4

−3

−(−2)

_____

the expression versus the sign shown in the expression.

Differentiation: Challenge Students see more complex examples of division of rational numbers later in this lesson. However, if students are prepared, add an additional challenge to problem 1 by replacing the last few rows with non-integer rational number values for p and q. In the discussion after completing the problem, have students note that the patterns in integer division hold true for division with all rational numbers.

−4

Present the solutions for each row. Then have students turn and talk about any patterns they observe in the quotients within the table.

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After students discuss the patterns they notice, direct them to use two highlighters of different colors to color-code equivalent quotients in each row. For example, in the first row, the first quotient is 4. Highlight every quotient of 4 in that row with one color. The next quotient in that row is −4. Highlight every quotient of −4 in that row with a second color. Repeat the process with the quotients in each row of the table. When students finish highlighting, facilitate a discussion about the table.

UDL: Representation Color-coding the equivalent quotients in each row of problem 1 emphasizes the relationship between the structures of the expressions and the resulting signs of the quotients.

What patterns do you notice? In each row, the first and last columns have the same quotient. The middle three columns all have the same quotient, which is the opposite of the first and last columns. How do the expressions with p and q relate to the quotients in the table?

___ _ ___ For any quotient _ q , the expressions q , − q , and −q all represent the opposite of that p

−p

quotient. Changing the sign of either p or q changes the sign of the quotient. The

___ seems to be equivalent to _ . quotient − q q −p

Display the following two statements to refer to in the rest of the discussion. −p _ p p _ = =− _ q

−q

_ _

(q)

p −p q = −q

Do the equivalent expressions remain equivalent even when the numbers you substitute are negative? Explain. Yes. The pattern continues no matter what the signs of the numbers we substitute for

p and q are. Whether one is a negative number, both are negative numbers, or both

_ ___ are positive numbers, the expressions ___ q , −q , and −(q ) are all equivalent, and the −p

___ expressions _ q and −q are equivalent. p

−p

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What do you notice about the quotients in the last row? How are they different from the quotients in the previous rows? How are they similar? The quotients in the last row are different from the quotients in the previous rows because they are decimals. However, they still follow the same pattern. The

_ ___ _ ___ expressions ___ q , −q , and −(q ) are all equivalent, and the expressions q and −q are −p

−p

equivalent.

Display the expressions from Launch. Expression A

Expression B

Expression C

Expression D

Expression E

3 4

3 −4

−3 −4

−_

−(_)

−3 _

Have students turn and talk about whether their conjectures were correct. Ask them to determine which expressions are equivalent and why. Circulate and listen for any remaining misconceptions about quotients of integers. −3 Display the expression −___ . −4

What is the decimal value of this expression? How do you know? The value is −0.75 because −3 ÷ (−4) = 0.75 and the negative sign before the fraction indicates the opposite of 0.75. Then use the following sequence of questions to help students understand why the quotient of any number and 0 is undefined. What is the quotient of __ q if the value of p is 0? How can we confirm the quotient p

by using the related unknown factor equation?

The quotient is 0 because 0 divided by any nonzero number is 0. For example, in the division expression 0 ÷ 2, the related unknown factor equation would be 2 ⋅ = 0. Because 2 ⋅ 0 = 0, the unknown factor is 0, which confirms that the quotient is 0.

Pose the following question.

What is the quotient of __ q if the value of q is 0? How can we confirm the quotient p

by using the related unknown factor equation?

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If students have a valid response, skip the next sequence of questions and transition to the formal definition of a rational number. If students respond with “0” or are unable to explain why the quotient is undefined, use the following questions to develop their understanding. Let’s think about __ q concretely. Let p = − 5, and let q = 0. What is the related unknown p

0⋅

factor equation?

= −5

What number can we put in the blank to make a true number sentence? There isn’t a number that when multiplied by 0 is any other value than 0, so there isn’t a number we can put in the blank to make a true number sentence. Because there is no number that makes the number sentence true, we say that the

quotient of __ q is undefined when q = 0. We can revise our two statements from earlier p

to say that they are true for any integer values of p and q if q ≠ 0.

Summarize this learning for students and use this opportunity to present the definition of a rational number. Since grade 6, we have been working with rational numbers, which we described as whole numbers, fractions, decimals, and their opposites. Throughout this segment, we examined numbers in the form __ q , where p and q are integers. In every case p

we examined, as long as q was not 0, we were able to find a quotient that was be written in the form __ q , where p and q are integers and q ≠ 0.

a rational number. The formal definition of a rational number is any number that can p

Language Support To reinforce the meaning of the term rational number, consider having students label problem 1 with the title Examples of Rational Numbers.

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Division Expressions Students estimate quotients and divide rational numbers. Display the four expressions. Expression A

Expression B

Expression C

Expression D

1.96 ÷ 7

4.5 ÷ 0.375

32 4 ÷ 10 100

9 18 ÷ 100 10

_ _

UDL: Representation

_ _

If needed, activate prior knowledge during

Without evaluating the expressions, can you tell whether each quotient is greater than 1 or less than 1? How? Yes. I can look at the size of the dividend and the size of the divisor. For example, in expression A, the dividend, 1.96, is much smaller than the divisor, 7, so the quotient is less than 1. Invite students to turn and talk about which quotients are greater than 1 and which are less than 1.

this discussion by showing number bonds of division. Use one example where the divisor is larger than the dividend and one example where the divisor is smaller than the dividend. multiplication expressions, such as 4 ⋅ 100 , Or present students with the related

__ ___ 10

and have them estimate those products first.

Display the next set of four expressions and ask the following questions. Expression A

Expression B

Expression C

Expression D

−1.96 ÷ (−7)

−4.5 ÷ 0.375

32 4 ÷ − 10 ( 100)

18 100

−_ ÷ _ 9 10

Which quotients are less than − 1? Between − 1 and 0? Between 0 and 1? Greater than 1? Have students work with a partner to answer the questions. Select pairs to share their responses and reasoning. Anticipate any of the following sample student responses. The quotient in expression A is between 0 and 1. Because −1.96 is close to −2, the

−2 , or _ 2. quotient is about ___ −7

The quotient in expression B is less than −1. Because 0.375 is close to 0.4 and −4.5 is close to −4.4, an estimate for the quotient is −11.

40 , dividing ___ 40 4 is equivalent to ___ The quotient in expression C is less than −1. Because __ 32 by − ___ is equivalent to dividing 40 by −32.

100

9 is equivalent to ___ 90 , The quotient in expression D is between −1 and 0. Because __

18 90 is equivalent to dividing −18 by 90. dividing − ___ by ___ 100

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EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 8

Direct students to problems 2–7. What ideas and strategies do we need to keep in mind as we determine the quotients? We need to keep in mind that we can use rounding to estimate the quotient and to determine whether our answer is reasonable. We need to keep in mind how to write equivalent fractions and how to make a common unit if needed. We need to remember how to use the long division algorithm to divide decimals. We need to remember the idea that dividing by a fraction is equivalent to multiplying by its reciprocal. We also need to remember the patterns of integer multiplication and division and to pay attention to whether the quotient is a negative number or a positive number. How is estimation helpful when finding the quotients of numbers? Estimation helps us know whether our quotients are reasonable. For example, if an estimate is less than −1 and our quotient is a positive number, then we know we made an error. Have students complete problems 2–7 with a partner. Circulate as students work, encouraging students to use estimation strategies to determine whether their quotients are reasonable.

( 4)

3. (−0.9) ÷ (−0.4)

___

1.96 4. −7

−8

2.25

−0.28

4.5 5. − ______   −0.375

9 18 6. − ___     ÷ __     100 10

3 7. −__   ÷ 0.8 10

− __   2

3 − _

When most pairs have finished, select a few students to share their answers and thinking with the class. As students share, listen for opportunities to identify the multiplicative inverses. Use the following discussion questions as a guide. What is the relationship between −_   and −_  ? The fractions are reciprocals.

3 4

The fractions are multiplicative inverses. © Great Minds PBC

Accept any equivalent form of an answer.

For example, 2   1 is an acceptable quotient 4

in problem 3, and − 0.2is an acceptable quotient in problem 6. Use different forms

For problems 2–7, divide. 3 2. 6 ÷ − _

Teacher Note

4 3

of numbers as an opportunity to discuss the efficiency of different strategies that students might have used to divide.

Teacher Note In grade 5, students understand that they can interpret division of the numerator by

the denominator as  a , where a and b are whole numbers and b ≠ 0. After students b

learn to divide integers, they can extend their form   where p and q are integers and q ≠ 0.

understanding of fractions as numbers of the

__p q

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Have students recall that a number and its reciprocal have a product of 1, so reciprocals are also multiplicative inverses. How do you know that 6 ÷ (− _) is equivalent to 6(− _4) ? 3 4

Dividing by a number is equivalent to multiplying by its multiplicative inverse. For at least one problem, have students who used two different solution pathways share their work with the class. Then invite them to compare their work. Consider using the following discussion questions.

Language Support Students are likely more comfortable using the term reciprocals from grade 6. Students learn the meaning of multiplicative inverse in grade 6 when solving equations. Encourage students to start using the term multiplicative inverses to refer to these types of numbers.

What strategy did you use to evaluate problem 7? Why did you choose that strategy? We chose to write both numbers as fractions because we noticed that 0.8 goes to the 8 and conveniently has the same units as the tenths place, so it is equivalent to __ 10

3 3 dividend, − __ . The problem − __ ÷ 0.8 is equivalent to −3 ÷ 8, so the quotient is − _3 . 10

3 We chose to write both numbers as decimals because we can write the dividend, − __ ,

as −0.3. Then we used the division algorithm to divide 3 by 8. The quotient, −0.375,

is a negative number because the dividend is a negative number and the divisor is a positive number. When is it more efficient to write both numbers in fraction form, and when is it more efficient to write both numbers in decimal form? If a fraction has a denominator that is a power of 10, it may be more efficient to write the number in decimal form. If a decimal can quickly be converted to a fraction, such as 0.5 or 0.01, it may be more efficient to write the number in fraction form. Ask students to complete problems 8 and 9 with a partner. Circulate as students work, encouraging students to use estimation strategies to determine whether their answers are reasonable. In addition, encourage students to apply their understanding of the order of operations to evaluate each expression.

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EUREKA MATH2 7–8 ▸ M1 ▸ TB ▸ Lesson 8

For problems 8 and 9, evaluate the expression. 2 8. −_ (−12 − (−9)) 3

9. −4 + ______   12   −1 − 1_ 2

−2_ (−12 − (−9)) = −2_ (−12 + 9) 3

−4 + ______   12   = −4 + 12 ÷ ( −1 1_ ) 1_ 2 −1 −

= −2_ (−3)

= −4 + (−8) = −12

Invite students to share their solutions and strategies, highlighting proper use of the order of operations. Display the complex fraction.

_ 3_   2

   __ 5   7

Ask students to think–pair–share about the following question. What do you think this expression means? Why?

I think it means to divide _  2  by _  3  because _  2  is the numerator and _  3  is the denominator. 5

Division expressions that involve a fraction divided by a fraction are sometimes called complex fractions.

Then ask students to turn and talk about how to divide _  2  by _  3  . 5

Have students complete problems 10–12 with a partner. Circulate as students work, asking the following questions: • What strategies can you use to divide two fractions? • Should the quotient be a positive number or a negative number? How do you know?

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

For problems 10–12, divide. 3_ 2_4_

− 4 10. ____ 25

_2 1_

3 −_ 3_ 24 4 __ ____ 24 __ = −4 ÷ 25 25

= −_(__) 3 25 4 24

= −__ 75 96

_ 3 −_ 1

− 2 12. ____

3 11. − __ 2_ __ − 3 = −(_2 ÷ _1) 1_ 3 6 6

= −(_2 ⋅ _) 6 3 1

12 = −(_ ) 3

1_ 3 −_

− 3 2 ____ = − _1 ÷ − _ 5

( 5)

= − _1(− _)

5 3

5 6

= −4 When students are finished, invite them to share solutions and strategies with a partner if time permits.

Applications of Division Students explore equivalent division expressions and real-world situations that involve division. Display the question about equivalent expressions. Give students 1 minute to turn and talk with a partner about the answers. Which expressions are equivalent to −20 ÷ _2 ? Choose all that apply. 5 2 A. _ ÷ (−20) 5 B. C.

2 (−20) 5 −20 ⋅ 5 ______

D. −10 ⋅ 5

188

−20 ÷ _

__ (− 2 )(5)

5 2

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Present each answer choice, asking students to show a thumbs-up if they believe that the expression is equivalent to the original expression or a thumbs-down if they believe it is not equivalent. Monitor student responses for accuracy, and reveal the correct expressions after noting all student responses. As needed, facilitate a brief discussion about why the expressions are equivalent, and review division of fractions to address any student misconceptions. Refer to the following rationales and suggested interventions for incorrect answers. A. Incorrect. The student may have applied the commutative property to a division expression. Demonstrate, by evaluating the expression or a simpler division expression, that division is not commutative. B. Incorrect. The student may have incorrectly recalled how to divide fractions. Review the concept of multiplying by the reciprocal and show by evaluating that multiplying the original divisor by the dividend does not produce an equivalent answer. C. Correct. D. Correct. E. Incorrect. The student may have incorrectly recalled how to divide fractions. Review the concept of multiplying by the reciprocal and show by evaluating that dividing by the reciprocal does not produce an equivalent answer. F.

Correct.

Display the question containing four situations. Direct students to turn and talk with the same partner about the answer to the question.

When students contextualize the division

Which situation can be represented by the expression shown?

expression − 20 ÷ 2 , they are reasoning

−20 ÷ _2

A. A desert is located 20 meters below sea level. An insect flies _2 meters above the 5 ground. What is the elevation of the insect? B. A person has a debt of $20 before paying some of the money they owe. What is the debt balance if only _2 of the original $20 debt remains? 5

C. The temperature is −20°F. The temperature drops _2 °F. What is the new temperature? 5 D. A fish at the water’s surface swims to an elevation of −20 feet in _2 seconds. At what rate in feet per second does the elevation of the fish change?

Promoting the Standards for Mathematical Practice

quantitatively and abstractly (MP2). Ask the following questions to promote MP2: • What does the division symbol in the expression tell you about the situation? • How does this expression represent that situation? • Does your answer make sense?

189

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Ask students to use a hand signal, such as showing one finger for A, two fingers for B, etc., to indicate the response they selected. Monitor student responses for accuracy and address any misconceptions. Reveal the correct answer, D, after noting all student responses. How do you know that situation D is the correct response?

We know that the fish’s elevation changes by −20 feet in _2 seconds. To determine how 5

much the elevation changes in 1 second, we need to divide −20 by _2 . 5

Ask students to complete problems 13 and 14 with a partner. Circulate as students work, and ask any of the following questions:

Differentiation: Support If students need additional support determining that situation D is the correct response, consider using an annotated double number line or a table to demonstrate the reasoning involved.

Time Elevation (seconds) (feet)

• What is the change in temperature? • What does a negative change in temperature mean? • How can you determine average rate of change? • What is the change in elevation of the dolphin? How do you know? • How can you determine how long it takes the dolphin to reach the desired elevation? What kind of diagram can you draw to help you? 13. At 9 p.m., the temperature is 4°F. At 6 a.m. the next morning, the temperature is −12°F. Assume the temperature decreases at a constant rate. What is the approximate rate at which the temperature changes in degrees per hour?

÷2 5

2 5

-20

÷2 5

Differentiation: Challenge

−12 − 4 = −16 The temperature changes by −16°F. The time changes by 9 hours.

−16 ÷ 9 ≈ −1.78

If students need additional challenge, ask them to write mathematical expressions for situations A, B, and C.

The temperature changes by about −1.78°F per hour.

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14. A dolphin is underwater at an elevation of −10 feet. The dolphin swims down at a rate of −24 feet per minute. How long does the dolphin take to descend to an elevation of −286 feet?

−286 − (−10) = −276 The dolphin’s elevation changes by −276 feet.

−276 ÷ (−24) = 11.5 The dolphin takes 11.5 minutes to descend to an elevation of −286 feet.

Differentiation: Support If students need additional support with problems 13 and 14, encourage them to draw diagrams to reason about the contexts. For example, a student might draw a number line for problem 13 to visualize how the temperature changes during the 9 hours.

When students are finished, confirm answers.

Land Debrief 5 min Objectives: Write rational numbers as quotients of integers. Divide rational numbers given in different forms. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. In what ways can we write a quotient of two given integers? How do we know they are all equivalent? When dividing two integers with opposite signs where the divisor is not 0, the quotient is a negative number no matter which number has the negative sign or whether it is −3 ___ written in front of the expression. For example, ___ = 3 = −(_3) . 4

−4

When dividing two integers with the same sign, the quotient is a positive number no matter whether both integers are positive numbers or both are negative numbers. −3 For example, _3 = ___ . 4

−4

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

How did we use equivalent expressions to evaluate division expressions?

When dividing fractions, we wrote a division expression, such as 6 ÷ (− _3), as an 4

equivalent multiplication expression, 6(− _4), by using the multiplicative inverse. 3

What strategies can we use to divide rational numbers given in different forms? We can write both numbers in decimal form or write both numbers in fraction form.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

192

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

9 5. − 4 __ ÷ 1.4 10 9 ÷ 1.4 = − 4.9 ÷ 1.4 − 4 __ 10

Dividing Integers and Rational Numbers In this lesson, we •

___ __ ways: ___ q = − q = − (q ) if q ≠ 0. −p

wrote unknown factor equations to make sense of division expressions containing 0.

•

evaluated division expressions that contained rational numbers in fraction form and decimal form.

•

solved real-world problems by dividing rational numbers.

− (−5) 5_ − a _____ ___ = =

−_2 is the denominator, so divide _4 by −_2 . 3

__ = − (− 12 ) 14

7. A fish swims at a constant rate from sea level to an elevation of − 52 feet in 6 _1 seconds. Write 2 and evaluate a division expression to show the change in elevation of the fish in feet per second.

− 52 ÷ 6 1_ = − 52 ÷ 6.5 2

= −8 The elevation of the fish changes by − 8 feet per second.

To find the rate of the fish’s change in elevation, divide the change in elevation in feet by the number of seconds the fish swims. The rate is negative because the elevation is decreasing.

Think about the related unknown factor equation:

− 15 times what number equals 0? The unknown factor is 0.

− 15

When 0 is the divisor, the quotient is undefined.

3. − 90 ÷ 0

Think about the related unknown factor equation:

Undefined

0 times what number equals − 90? There is no such number, so the quotient is undefined.

− 5 ÷ − 1 1_ = − 5 ÷ − 5_ 4

= − 5(− 4_) 5

__ = 12

0 divided by any nonzero number has a quotient of 0.

0 ____ =0

− (4_ ÷ (− 2_)) = − (_4(− 3_))

0 ÷ (− 15)

4. − 5 ÷ − 1 1_ 4

−

opposite of − 5,” which is 5.

For problems 2–6, divide. 2.

7 _ 4 In the expression − __ _2 , is the numerator and

A negative sign means taking the opposite. The expression − (−5) can be read as “the

− a for a = − 5 and b = 4. 1. Evaluate ___

q ≠ 0.

where p and q are integers and

•

_ 4

7 6. − ___ − 2_ 3

A rational number is any number p that can be written in the form _ q,

Examples

= − 3.5 Terminology

observed that the quotient of any two integers p and q can be written equivalently in different

9 Write − 4 __ in decimal form, − 4.9, and evaluate 10 the expression by using decimal division.

Dividing by −_5 has the same result 4 as multiplying by its reciprocal, −_4 . 5

121

122

RECAP

193

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 8 Name

PRACTICE Date

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

For problems 3–10, divide. 3. − 6 ÷ 1 1_ 3

4. 8 ÷ (−0.25)

− 32

18 − __ 4

1. Which expressions are equivalent to − 10 ÷ 5? Choose all that apply. − 10 A. ____ −5 − 10 B. ____ 5 10 C. ___ −5

__ D. 10 5

5. − 3_ ÷ 0 8 Undefined

__ E. − 10 5

6. 0 ÷ (−1.8)

2. Evaluate each expression for a = − 3 and b = 6. a a. _ b

7. − 16.7 ÷ (−0.4)

− _3 6

41.75

−a b. ___ b 3_ 6

9. − 3_ ÷ (−0.15) 5

− 1 1_

____2 − 5_

18 __ 10

10. 2 1_ ÷ (−0.375) 4

−6

a ___ −b _3 6

194

123

124

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 8

11. Evaluate.

− 2_ (− 16 + 10 ÷ (− 2)) 3

14 12. At 5 p.m., the temperature is 12°F. At 5 a.m. the next morning, the temperature is − 9°F. Assume the temperature decreases at a constant rate. At what rate does the temperature change in degrees per hour? The temperature changes by − 1.75°F per hour.

Remember For problems 13–16, subtract. 13.

7 − 1_ __ 12

14.

3 __ 12

9 − 2_ __ 10

5 __ 10

13 − 3_ 15. __ 16 4

11 − 3_ 16. __ 4 8

1 __

_5 8

For problems 17 and 18, determine the product. 17. − 8(7)

− 56 1 18. − 3_(− 1 _ 3) 4

19. Evaluate the expression.

13 + (8 − 5)3 ÷ 3 + 6 28

P R ACT I C E

125

195

Teacher Edition: Grade 7–8, Module 1, Topic B, Lesson 9 LESSON 9

Decimal Expansions of Rational Numbers Determine whether the decimal form of a rational number is a terminating decimal or a repeating decimal by analyzing the factors of the denominator. Write rational numbers as either terminating decimals or repeating decimals.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9 Name

Date

EXIT TICKET

Write the number as a decimal. Use bar notation where appropriate. 15 1. __ 6

2.5 4 2. − __ 25

−0.16 1 3. _ 3

0.¯ 3 4. − 2_ 9

In this lesson, students watch a video of a real-world situation involving division that does not result in a terminating decimal. Students write rational numbers in fraction form as decimals by determining an equivalent decimal fraction and by using long division. Students observe that some rational numbers can be written as decimal fractions and some cannot. Students determine that when 1 is the only common factor of the numerator and denominator of a rational number and the denominator can be written with only prime factors of 2, 5, or both, the decimal terminates; otherwise, it repeats. Students learn the notation for representing repeating decimals. This lesson defines the terms terminating decimal, repeating decimal, and bar notation.

Key Questions

−0.¯ 2

__ 5. − 10 11

• Is the prime factorization of the denominator of a rational number helpful when writing the number as a decimal? Why?

−0.¯ 90 6.

Lesson at a Glance

7 __

• How can we use what we know about long division to write a rational number in decimal form?

0.4¯ 6

• Can we predict when the decimal form of a rational number repeats? How?

129

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Achievement Descriptors 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational

numbers. (7.NS.A.1.d; 7.NS.A.2.c) 7–8.Mod1.AD6 Write the fraction form of a rational number in its decimal form by using

long division. (7.NS.A.2.d) 7–8.Mod1.AD7 Determine whether numbers are rational or irrational. (7.NS.A.2.d; 8.NS.A.1)

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Chart paper (1 sheet)

Learn 30 min

• Marker

• Nora’s Method vs. Long Division

Students

• Repeating Decimals

• None

• Terminating or Repeating?

Lesson Preparation

Land 10 min

• None

197

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Fluency Convert Fractions to Decimals and Decimals to Fractions Students convert fractions to decimals and decimals to fractions to prepare for writing a rational number as a decimal. Directions: Convert the fractions to decimals and the decimals to fractions. 1.

__1

0.1

0.7

7 10

__3

0.3

0.9

9 10

__8

0.8

1.6

16 10

___

0.11

10.

0.27

27 ___

___

0.05

11.

0.87

87 ___

____

0.014

12.

0.056

56 ____

11 100

5 100

14 1,000

198

__ __ __

100

1,000

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Teacher Note

Students analyze a real-world context involving division that results in a decimal that does not terminate. Play the Split the Check video and ask the following questions. What do you notice? What do you wonder? I notice that the bill is $20.

This lesson combines and expands on the learning activities from Eureka Math2 grade 7 module 2 lessons 19 and 20. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

I notice that the friends ask the waiter to split the check evenly. I notice that the friends don’t pay the same amount of money. I wonder how $20 can be evenly split between 3 people. I wonder why the friends don’t pay the same amount of money. How can you represent the amount of money that each friend should pay? 20 dollars. Each friend should pay __ 3

Today, we will write rational numbers in decimal form.

UDL: Representation Presenting the split the check situation in a video format supports students in understanding the problem context by removing barriers associated with written and spoken language.

Learn Nora’s Method vs. Long Division Students write rational numbers as terminating decimals by using decimal fractions and long division. Facilitate a Whiteboard Exchange to review writing a rational number as a decimal fraction and as an equivalent decimal. For each rational number, write the number as a decimal fraction and then as an equivalent decimal on your whiteboard. When you are done, hold it up for me to see.

Teacher Note Any fraction with a denominator that is a power of 10 is a decimal fraction. The term decimal fraction is first introduced in grade 4.

199

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Display each rational number one at a time. As students show their work, provide quick, individual feedback as needed.

Rational Number

Decimal Fraction

Decimal

− __

−0.6

__4

6 10

__2

0.2

− __

12 − ___

−0.12

__1

100

Not possible

6 10

3 25

Which numbers can be written as decimal fractions?

6 __ , 4 , and − __ can all be written as decimal fractions. The numbers − __ 3 25

10 20

Why can we write each of those numbers as a decimal fraction? We can write each of those numbers as a decimal fraction because we can write each number as an equivalent fraction with a denominator that is a power of 10. 1 Why can we not write __ as a decimal fraction? 12

We cannot get a power of 10 by multiplying 12 by a whole number. Have students complete problem 1 with a partner. Circulate and use questions such as the following to advance student thinking. • What do you notice in the second line of Nora’s work? • Why do you think Nora placed 2’s in the denominator? • What do you notice about the denominator in the final line of Nora’s work?

200

Language Support The term prime factorization is familiar to students from grade 6. However, because this is its first use in this course, consider reviewing this term. Break it down and discuss its parts. What does it mean to factor a number? What is a prime number? What does the combination of these terms mean? Consider reviewing the following example if further clarification is needed.

60 = 2 · 2 · 3 · 5

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

1. Nora says that she can use the prime factorization of the denominator to determine how to write a decimal fraction. Her work is shown. Explain Nora’s thinking. 2 __2 = ____ 25

5⋅5

2⋅2⋅2 = ________ 5⋅5⋅2⋅2

2⋅2⋅2 = _________ (5 ⋅ 2)(5 ⋅ 2)

2⋅2⋅2 = ______ 10 ⋅ 10

= ___ 8 100

I think Nora noticed that she can multiply each 5 in the denominator by 2 to get 10’s. Then to get an equivalent fraction, she multiplied by the same number of 2’s in the numerator. Display problem 1. Call the class back together and have students share their responses. Point to the displayed problem while students are sharing to emphasize their reasoning. Listen for them to include each of the following concepts in their explanation: • The denominator has two factors of 5. • The denominator of a decimal fraction must be a power of 10. • Each factor of 5 in the denominator must be multiplied by a factor of 2 to create a factor of 10.

Differentiation: Support If students need support with determining whether all powers of 10 will have the same number of factors of 2 and 5, consider having them find the prime factorization of several powers of 10 to determine whether there is a pattern. For example, ask them to factor 10, 100, and 1,000.

10 = 2 · 5

100 = 10 · 10 = 2 · 5 · 2 · 5

1,000 = 10 · 10 · 10 = 2 · 5 · 2 · 5 · 2 · 5

Ask students how having a denominator that is a power of 10 helps them in writing a rational number as a decimal. If the denominator of a rational number is a power of 10, then the decimal form has units of tenths, hundredths, or thousandths. The power of 10 in the denominator helps with determining the place value of the digits in the numerator.

• To write an equivalent expression, the numerator must be multiplied by the same factors as the denominator. 2 Why do you think Nora wanted to write __ as a decimal fraction? 25

When a number is written as a decimal fraction, its denominator is a power of 10, so the number can be written as a decimal by considering place value.

201

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Have students work with a partner on problems 2 and 3. Circulate and provide support when needed by asking students any of the following questions: • How many factors of 2 are in the prime factorization of the denominator? How many factors of 5 are needed so that the denominator is a power of 10? • When you realize that you need to multiply the denominator by three factors of 5, what must you do to write an equivalent fraction?

3 • How can you write __ as an equivalent fraction? Can you write the equivalent fraction 60

as a decimal fraction? For problems 2 and 3, use Nora’s method to write the number as a decimal fraction. Then write the decimal fraction as a decimal. 3 2. − _ 8

− _ = − ______ 3 8

3 2⋅2⋅2

3⋅5⋅5⋅5 2⋅2⋅2⋅5⋅5⋅5

= − ____________

3 3. __ 60

__ ________

3 3 = 60 2 ⋅ 2 ⋅ 5 ⋅ 3 1 = ______ 2⋅2⋅5

1⋅5 2⋅2⋅5⋅5

= − _____________

= ________

= − ________

= _________

= − ____

= _____

= −0.375

= ___

375 (2 ⋅ 5)(2 ⋅ 5)(2 ⋅ 5) 375 10 ⋅ 10 ⋅ 10 375 1,000

5 (2 ⋅ 5)(2 ⋅ 5) 5 10 ⋅ 10

5 100

= 0.05 Engage students in a class discussion by asking the following questions. How can we form a power of 10 in the denominator of a rational number? We can count the number of factors of 2 and 5 in the prime factorization of the denominator. Then we need to keep multiplying by the number that appears fewer times until we have the same number of 2’s and 5’s. 202

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

What makes problem 3 different from problems 1 and 2? There is a factor of 3 in the denominator.

Do you agree with the statement that __ can’t be written as a decimal fraction 3 60

because the denominator includes a factor of 3? Why?

3 1 , and __ 1 can be written as a decimal I do not agree. The number __ is equivalent to __ 20

fraction by using Nora’s method.

3 If needed, ask the following question to emphasize why __ can be written as a 60 decimal fraction. 1 Why are we able to write __ as __ ? 3 60

3 The numerator and denominator of __ have a common factor of 3, so we can divide the 60

numerator and denominator by 3 to write an equivalent fraction. To clearly identify a rational number as one that can be written as a decimal fraction, divide the numerator and denominator by any common factors first. When a rational number is written in fraction form so that the numerator and denominator have only a common factor of 1, we can determine that the rational number can be written as a decimal fraction if the denominator has only prime factors of 2, 5, or both. What operation does a fraction represent? A fraction represents division. Because a fraction represents division, what other method can we use to write a fraction as a decimal?

Differentiation: Support

We can use long division to write a fraction as an equivalent decimal. How do we divide when the dividend is less than the divisor? We can write a decimal point after the dividend and write 0’s after the decimal point. Have students work with a partner on problems 4 and 5.

This is the first time that students use long division in the course. If students need more support, consider completing problem 4 as a class to guide them through the process.

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

For problems 4 and 5, use the long division algorithm to verify that the decimal forms found in problems 2 and 3 are correct. 3 4. − _ 8

3 5. __ 60

0.3 7 5 8 3.0 0 0 –2 4 60 –56 40 –40 0

−0.375

0.0 5 6 0 3.0 0 – 0 300 –300 0

0.05

Call the class back together to discuss the use of long division.

In problems 2 and 4, we wrote − _ as a decimal by using two different strategies: 3 8

decimal fractions and long division. Which strategy do you prefer and why?

I prefer writing − _3 as a decimal fraction because I quickly wrote − _3 as an equivalent 8

fraction with a power of 10 in the denominator. Then I used place value to write the decimal fraction as a decimal. When completing long division, how do we know when we’ve arrived at the answer? How do we know when to stop? When we have a remainder of 0, we can stop dividing. Once we have a remainder of 0, how can we be sure that there will not be any other nonzero digits in this decimal? Once the remainder is 0, there is nothing left to divide. In mathematics, we say that numbers that can be written as decimal fractions have a decimal form that terminates in 0. From now on, we will call these numbers terminating decimals. A terminating decimal is a decimal that can be written with a finite number of nonzero digits. 204

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Use chart paper to create an anchor chart to illustrate this new terminology. Write the term terminating decimal on the chart. Then ask students to provide examples of terminating decimals. Record them on the chart. The term repeating decimal is defined later in this lesson. If possible, leave space on the anchor chart to add this term and several examples.

Repeating Decimals Students write decimal expansions of rational numbers by using long division and bar notation.

Language Support To reinforce the meaning of the term terminating decimal, consider having students discuss the meaning of the base word terminate. Terminate means to “bring to an end,” “to finish,” or “to complete.” So a terminating decimal is a decimal form that terminates, or ends, in 0.

1 At the beginning of the lesson, we said that we could not write __ as a decimal 12

1 fraction. Explain what happens when we try to apply Nora’s method to write __ as a 12

decimal fraction.

1 as ______ 1 . Because the denominator has With Nora’s method, we would first write __ 12

2⋅2⋅3

a factor of 3 and the only common factor of the numerator and denominator is 1, we cannot write the denominator as a factor of 10.

1 What other method could we use to write the fraction __ as a decimal?

We could use long division to divide 1 by 12.

1 and set up the related long division algorithm. Display the number __ 12

Have students work with a partner to complete the long division on their whiteboards. Then have a student pair model the process for the class.

0.0 8 3 3 3 1 2 1.0 0 0 0 0 – 96 40 –36 40 –36 40 –36 4 © Great Minds PBC

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Previously, we said that when the remainder is 0, we can stop doing the long division. How did you know when to stop doing long division in this problem? We stopped doing long division when we noticed that we kept getting a remainder of 4, which led to another digit of 3 in the quotient. This pattern will never end. When we see remainders repeating, the decimal form of the number repeats as well. We call this a repeating decimal. A repeating decimal is a decimal in which, after a certain digit, all remaining digits consist of a block of one or more digits that repeats indefinitely. Every rational number can be expressed by either a terminating decimal or a repeating decimal.

1 = 0.083333…, and write Add the term repeating decimal to the anchor chart. Then display __ 12 it as an example below the term.

There are three dots after the last 3. What do you think the dots mean? I think the dots mean that the decimal keeps going and every digit after 8 is a 3. The set of three dots following 3 is called an ellipsis. An ellipsis indicates that something has been left out or omitted. Because we can’t write a never-ending list of 3’s, we use the ellipsis to indicate that the pattern continues but the remaining digits of 3 are omitted. In mathematics, we use bar notation for repeating decimals. Bar notation indicates which block of the decimal repeats. The bar is placed over the shortest block ¯ is a compact way of repeating digits after the decimal point. For example, 0.083 to write the repeating decimal expansion 0.08333... . The bar is not above any other part of the decimal.

1 by using bar notation. Add the Model for students how to write the decimal expansion of __ 12

following example below the word repeating decimal on the anchor chart. To ensure that students are recognizing the notation, point to the bar as you write the decimal.

0.08333… = 0.083̄

206

Language Support To reinforce the meaning of the ellipsis symbol in mathematics, have students recall where they have seen the symbol used before and explain what it means in that situation. For example, students may see the ellipsis symbol used when a set of numbers is listed. • Whole numbers: 0, 1, 2, 3, 4, 5, … • Integers: …, −3, −2, −1, 0, 1, 2, 3, …

Teacher Note Throughout the curriculum, bar notation is the term used in reference to marking repeating decimals with a bar symbol. The formal term for this symbol is vinculum. Use the formal term at your discretion.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Provide the following examples of numbers written correctly and incorrectly in bar notation. Ask students to give a thumbs-up or thumbs-down to indicate whether the notation is correct. If the notation is incorrect, ask students what the correct notation should be.

0.44444… = 0.4̄

Correct.

0.454545… = 0.45̄

Incorrect. It should be 0.¯ 45.

4.4444… = 4̄

Incorrect. It should be 4.4̄.

This allows for quick formative assessment of student understanding and emphasizes the importance of the notation in understanding what repeats. Have students work with a partner to complete problems 6–8.

Promoting the Standards for Mathematical Practice When students determine where to draw the bar for bar notation by considering which block of digits repeats in a repeating decimal, they are attending to precision (MP6). Ask the following questions to promote MP6: • How can we write a repeating decimal by using bar notation? • What details are important to think about in this problem? • What does the bar mean in this context?

For problems 6–8, use the long division algorithm to find the decimal form of the following rational numbers. Use bar notation when necessary. 2 6. _ 3

0.666… = 0.6̄

5 7. − _ 6

− 0.8333… = −0.83̄

6 8. __ 11

0.545454… = 0.¯ 54

Call the class back together to verify answers and clarify misunderstandings about bar notation. Then ask students for more examples of repeating decimals to add to the anchor chart.

Terminating or Repeating? Students determine whether rational numbers are terminating decimals or repeating decimals. Display the following examples of rational numbers one at a time. Ask students to show the red side of their whiteboard if the rational number is a repeating decimal and to show the white side of their whiteboard if the rational number is a terminating decimal. Consider using the phrase “red for repeating” to help students remember what side to show. © Great Minds PBC

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1 3

Repeating decimal

− __ 9 16

Terminating decimal

__3

Terminating decimal

__5

Repeating decimal

− __ 30 12

Terminating decimal

− __ 10 9

Repeating decimal

8 − 22

Repeating decimal

20 3

Repeating decimal

25 30

Debrief by asking the following questions. How did you decide whether the rational number is a terminating decimal or repeating decimal? When the denominator has no prime factors other than 2 or 5, the decimal terminates; otherwise, it repeats. Why is − __ a terminating decimal when its denominator has a factor of 3? 30 12

30 The numerator and denominator of − __ have a common factor of 3. When I divide the

Teacher Note Consider demonstrating how calculators display a repeating decimal. Project your calculator and enter 2 ÷ 3. The calculator shows the quotient ending with the digit 7, which is different from the repeating decimal found in problem 6. Consider discussing this with students by asking the following questions. • Why do you think that is? The calculator screen can only show a certain number of digits. The calculator is probably programmed to round the last digit it can display, and it is rounding up the last digit to 7. • How do you think the calculator would display − 5 as a decimal? Why?

__ 6

I think the calculator would display

− 5 the same way we did because it 6

has a repeating digit of 3, which would not round up. If time allows, have students use a calculator to check their answers.

numerator and denominator by 3, the denominator of the equivalent fraction only has prime factors of 2.

Where have we seen _2_0 in this lesson? 3

We watched a video at the beginning of the lesson about splitting a $20 check between 20 3 friends, so each owed __ dollars. 3

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Using what you have learned about repeating decimals and terminating decimals, can you explain why the friends in the video received bills for different amounts?

20 20 Each friend owes __ dollars. The decimal expansion of __ is a repeating decimal because 3

the denominator has a prime factor other than 2 or 5. Dollars are rounded to the hundredths place, so 6.6̄ is approximately 6.67. If each friend pays only $6.66, then

Teacher Note If time permits, consider replaying the Split the Check video.

the total is $6.66 + $6.66 + $6.66, or $19.98. If each friend pays $6.67, then the total is

$6.67 + $6.67 + $6.67, or $20.01. Instead, the waiter split the check so that one person pays $6.66 and the other two pay $6.67. This way, the total is $6.66 + $6.67 + $6.67, or $20.

Land Debrief 5 min Objectives: Determine whether the decimal form of a rational number is a terminating decimal or a repeating decimal by analyzing the factors of the denominator. Write rational numbers as either terminating decimals or repeating decimals. Initiate a class discussion by using the following prompts. Encourage students to revoice or build on their classmates’ responses. What method did Nora use to write a rational number as a decimal fraction? Why is that method helpful when writing a rational number as a decimal fraction? Nora used prime factorization to write the denominator of the rational number as a product of prime factors. When the denominator only has factors of 2, 5, or both, the rational number can be written as a decimal fraction because 2 and 5 are the factors of 10, and the denominator of a decimal fraction is a power of 10.

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How can we use what we know about long division to write a rational number in decimal form? Because a rational number in fraction form represents division, we can use long division to write the number in decimal form. The numerator is the dividend, and the denominator is the divisor. Can we predict whether the decimal form of a rational number repeats? How? Yes. If the numerator and denominator have only a common factor of 1 and the denominator has only prime factors of 2, 5, or both, the decimal terminates. If the numerator and denominator have only a common factor of 1 and the denominator has any prime factors other than 2 or 5, the decimal repeats.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

210

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9 Name

RECAP Date

Decimal Expansions of Rational Numbers In this lesson, we

Terminology

wrote rational numbers given in fraction form as decimals by first writing them as decimal fractions.

•

wrote rational numbers given in fraction form as decimals by using long division.

A repeating decimal is a decimal in which, after a certain digit, all remaining digits consist of a block of one or more digits that repeats indefinitely.

determined that a rational number can be written as a terminating decimal when the denominator has only prime factors of 2, 5, or both.

A common notation for a repeating decimal expansion is bar notation. The bar is placed over the shortest block of repeating digits after the decimal point. For example, 3.1¯ 25 is a compact way to write

•

then the number can be written as a decimal fraction. This means it can be written as a terminating decimal.

A terminating decimal is a decimal that can be written with a finite number of nonzero digits.

represented repeating decimals by using bar notation.

14 14 2. Factor the denominator of − __ by using prime factorization. Write − __ as a decimal fraction and 80 80 then as a decimal. 7 __ = − __ Divide the numerator − 14 40 80 Factor the denominator into prime factors. If the prime factors are only 2, 5, or both,

•

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

2⋅2⋅2⋅5

7 = − ________

7⋅5⋅5 = − ____________ 2⋅2⋅2⋅5⋅5⋅5

7⋅5⋅5 = − _____________

and the denominator by any common factors so that their only common factor is 1.

(2 ⋅ 5)(2 ⋅ 5)(2 ⋅ 5) 10 ⋅ 10 ⋅ 10

175 = − ________

175 = − ____ 1,000

= − 0.175 Because decimal fractions have a denominator that is a power of 10, decimal fractions always

have an equal number of factors of 2 and 5 in

the repeating decimal expansion 3.12525252525… .

the denominator. To write the number as a decimal fraction, multiply by additional factors of either 2 or 5

Examples

to produce a power of 10 in the denominator.

12 1. Write − __ as a decimal fraction. Then write it as a decimal. 25

A decimal fraction has a denominator that is a power of 10.

12 ⋅ 4 __ = − ____ − 12 25

25 ⋅ 4

48 = − ___ 100

= − 0.48

Create an equivalent fraction by multiplying both the numerator and the denominator by the same number.

Multiplying the denominator by 4 produces a power of 10.

To write decimal fractions as decimals, use the power of 10 in the denominator to determine the place value.

131

132

RECAP

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EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

For problems 3 and 4, write the number in decimal form. Use bar notation where appropriate. 3.

3 __ 11

3 __ = 3 ÷ 11 11

0.2 7 2 7 11 3.0 0 0 0 –2 2 80 –77 30 –22 80 –77 3

To write fractions as decimals by using the long division algorithm, interpret the fraction as division.

__ 4. − 17 15 __ = − (17 ÷ 15) − 17 15

1.1 3 3 15 1 7.0 0 0 – 15 20 –1 5 50 –45 50 –45 5

Because the remainders begin to repeat, the quotients are repeating decimals.

0.¯ 27

The block of digits 2 and

7 repeats, so the bar is over both 2 and 7.

3 is the only digit that repeats, so the bar is only over 3.

− 1.1¯ 3

The quotient is a negative number 17 because −__ is interpreted as 15

the opposite of the quotient 17 that __ produces. 15

212

RECAP

133

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Student Edition: Grade 7–8, Module 1, Topic B, Lesson 9

PRACTICE

Name

Date

For problems 8–13, indicate whether the decimal form of the number terminates or repeats. Number

For problems 1 and 2, write the decimal in fraction form. 1. 0.58

2. − 9.76

58 ___ 100

___ − 976 100

For problems 3–5, write the number as a decimal fraction. Then write it as a decimal. 7 3. − __ 25

28 − ___ ; − 0.28 100

27 5. − __ 60

31 4. __ 20

45 − ___ ; − 0.45

155 ; 1.55 ___ 100

100

5 _5 = ______ 8 2⋅2⋅2

625 ; 0.625 ____ 1,000

__ 7. − 21 40

_5 7

− 8_ 3

10.

5 − __

11.

15 __ 8

12.

40 __ 75

13.

__ − 27

X X

9 15. 3 __ 40

0.¯ 4

2⋅2⋅2⋅5

3.225

19 16. __ 15

1.2¯ 6

525 − ____ ; − 0.525 1,000

− 0.5625

9 17. − __ 16

Repeats

14. _4 9

21 __ = − ________ − 21 40

Terminates

For problems 14–20, use long division to write the number in decimal form. Use bar notation where appropriate.

For problems 6 and 7, factor the denominator by using prime factorization. Write the number as a decimal fraction and then as a decimal. 6. _5 8

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

135

136

P R ACT I C E

43 18. __ 60

0.71¯ 6

__ 19. − 58 99 − 0.¯ 58

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7–8 ▸ M1 ▸ TB ▸ Lesson 9

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

7 20. − __ 24

EUREKA MATH2

7–8 ▸ M1 ▸ TB ▸ Lesson 9

Remember For problems 24–27, subtract.

− 0.291¯ 6

9 24. − __ − 1_ 20 2

7 − 3_ 25. − __ 16 4

__ − 19

21. Henry says that every fraction with a denominator of 9 is a repeating decimal because the only factors of the divisor 9 are 3, not 2 or 5. Do you agree with Henry? Explain why. I disagree with Henry. Some fractions with a denominator of 9 are repeating decimals, but when the numerator also has two factors of 3, then the number is a terminating decimal. For example, 18 is a fraction with a denominator of 9, but it is a terminating decimal, 2. __

− 9_ 8

3 28. (− 5_) 6

___ − 125 216

22. Together, 6 friends buy a $20 gift for their other friend. How do the 6 friends share the cost of the gift evenly?

To split the_cost of the gift, $20, evenly among 6 friends, each friend has to pay $3.3 because 20 ÷ 6 = 3.3. However, dollars are usually rounded to the hundredths place.

9 29. − (__ 10)

81 − ___ 100

One way to split the cost of the gift is to have 4 friends pay $3.33 and 2 friends pay $3.34 because 3.33 + 3.33 + 3.33 + 3.33 + 3.34 + 3.34 = 20. 23. Is the decimal form of − _9 a terminating decimal or a repeating decimal? Explain how you know. 6

The decimal form of − _9 is a terminating decimal. 6

The numerator and the denominator share a common factor of 3. If I divide the numerator

30. Which expressions are equivalent to _2 (− 4)? Choose all that apply. 3 A. _8 3 B. − 8_ 3

and the denominator by 3, I can write − _9 as − _3 . Because the equivalent expression − _3 has 2

__ − 39

27. − 7_ − 1_ 8 4

For problems 28 and 29, evaluate the expression.

__ − 19

26. − 6_ − 3_ 5 4

8 C. − __ 12

only a factor of 2 in the denominator, the decimal form of the rational number is a terminating decimal.

D. − 2_ (4) 3 E. 1_ (−8) 3 F. _4 (−2) 3

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P R ACT I C E

137

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P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic C

Topic C Properties of Exponents and Scientific Notation Topic C begins with thought-provoking contexts that spark students’ interest and create a need to further their understanding of exponents. Throughout the topic, students learn and apply the properties and definitions of exponents and use their place value understanding and knowledge of powers of 10 from grade 5 to write and operate with numbers in scientific notation. Students begin the topic by exploring very large and very small positive numbers by relating them to real-world objects and by writing the numbers in equivalent forms. To preview scientific notation, students write large positive numbers as a single digit times a power of 10. They write small positive numbers as a single digit times a unit fraction with the denominator written as a power of 10 in exponential form. By applying their understanding of place value and powers of 10, students approximate large positive numbers and solve how many times as much as problems. To begin their exploration of the properties and definitions of exponents, students examine the products of powers of 10 to make a conjecture about the relationship between the exponents of the factors and the exponent of the product. Students then extend their conjecture for powers of 10 to include like bases and positive whole-number exponents, arriving at the product of powers with like bases property of exponents: x m · x n = x m+n. In lesson 12, a digital lesson, and lesson 13, this property of exponents is used to develop additional properties involving powers of powers and powers of products and the definitions of an exponent of 0 and negative exponents. Students apply the properties and 216

Properties of Exponents Description

Property

Example

Product of powers with like bases

x m · x n = x m+n

105 · 102 = 105+2 = 107

Power of a power

(x m)n = x m · n

(52)6 = 52 · 6 = 512

Power of a product

(xy)n = x n y n

(7 · 6)2 = 72 · 62

Definitions of Exponents Description

Definition

Base with an exponent of 0

x0 = 1

x is nonzero 1 x −n = __ n x

Base with a negative exponent

x is nonzero 1 ___ = xn

x−n

x is nonzero

Example

100 = 1 1 10−4 = ___ 4 10

1 ____ = 104

10−4

EUREKA MATH2 7–8 ▸ M1 ▸ TC

definitions of exponents to write equivalent expressions. These properties and definitions are recorded on a graphic organizer that students use as a reference throughout the topic. In lesson 14, students are introduced to scientific notation and determine whether numbers are written in the form a × 10n. They then write very large and very small numbers that are in standard form in scientific notation and write very large and very small numbers that are in scientific notation in standard form. Students order numbers written in scientific notation by using each number’s order of magnitude and first factor.

First Factor

a × 10 n

Order of Magnitude

Students use properties of operations, properties of exponents, and definitions of exponents in lesson 15 to operate with numbers written in scientific notation, including raising a number written in scientific notation to an exponent. To check their work, students input the expressions into their calculator and interpret the display of the number in scientific notation. In a digital lesson, students operate with numbers written in scientific notation to solve real-world problems. They also choose appropriate units of measurement for a given situation and convert between units. Students end this topic by exploring a piece of art and a painting technique called pointillism. They complete an open-ended activity that applies their understanding of scientific notation and properties of exponents. In topic D, students continue to work with exponents, namely squares and cubes. Students apply their ability to recognize numbers written in equivalent forms when they encounter square root notation and cube root notation.

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Progression of Lessons Lesson 10 Large and Small Positive Numbers Lesson 11 Products of Exponential Expressions with Positive Whole-Number Exponents Lesson 12 More Properties of Exponents Lesson 13 Making Sense of Integer Exponents Lesson 14 Writing Very Large and Very Small Numbers in Scientific Notation Lesson 15 Operations with Numbers Written in Scientific Notation Lesson 16 Applications with Numbers Written in Scientific Notation Lesson 17 Get to the Point

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Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 10 LESSON 10

Large and Small Positive Numbers Approximate very large and very small positive numbers and write them as a single digit times a power of 10 or as a single digit times a unit fraction with a denominator written as a power of 10. Compare large and small positive numbers by using times as much as language.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10 Name

Date

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

b. The number of years since bacterial life appeared on Earth is about how many times as much as the number of years since mammals appeared?

200,000,000 = 2 × 108

1. Consider the number 0.000 0285.

4 × 109 =

⋅ 2 × 108

4______ × 109 __________________________________ = 4 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 2 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 2 × 108 10 _ 10 _ 10 _ 10 4 _ _ × 10 × _ × 10 × _ × 10 × _ × 10 = × 10 × _ 2 10 10 10 10 10 10 10 10

a. Approximate the number by rounding to the nearest hundred thousandth.

0.000 0285 ≈ 0.00003

= 2 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 10 = 2 × 10 = 20 The number of years since bacterial life appeared on Earth is about 20 times as much as the number of years since mammals appeared.

b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 3 0.00003 = _ 100,000

1 =3×_ 100,000

1 = 3 × ___ 5 10

2. Bacterial life appeared on Earth about 3.6 billion years ago. Mammals appeared about 200,000,000 years ago. a. Write when bacterial life approximately appeared on Earth as a single digit times a power of 10 in exponential form.

3.6 billion ≈ 4 billion

= 4,000,000,000 = 4 × 109

Bacterial life appeared on Earth approximately 4 × 109 years ago.

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EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Lesson at a Glance In this lesson, students lay the foundation for scientific notation by applying their understanding of the unit form of a number and powers of 10. Students develop ideas of just how large or how small positive numbers really are by relating the sizes of known objects to powers of 10 through a teacher interactive. Then students approximate very large and very small positive quantities by writing numbers as a single digit times a power of 10 in exponential form or as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. Students then participate in a stations activity to answer real-world how many times as much as questions. This lesson formally defines the term approximate.

Key Questions • Why do we write some very large or very small positive numbers as a single digit times a power of 10 in exponential form? • What strategies are helpful when solving how many times as much as problems? • What forms of numbers are useful when solving how many times as much as problems with large numbers? Why?

Achievement Descriptors 7–8.Mod1.AD13 Approximate and write very large and very small numbers in scientific

notation. (8.EE.A.3) 7–8.Mod1.AD14 Express how many times as much one number is than another number

when both are written in scientific notation. (8.EE.A.3)

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Agenda

Materials

Fluency

Teacher

Launch 10 min

• None

Learn 25 min

Students

• Writing Very Large and Very Small Positive Numbers

• Times As Much As Stations Cards

• Approximating Very Large and Very Small Positive Numbers

• Copy and cut four sets of Times As Much As Stations Cards (in the teacher edition). Set up four stations. Place one set of cards at each station.

• Times As Much As

Land 10 min

222

Lesson Preparation

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Fluency Write Numbers by Using Powers of 10 Students write numbers as a power of 10 in exponential form to prepare for writing large positive numbers as a single digit times a power of 10 in exponential form. Directions: Write each number as a power of 10 in unit form and in exponential form. Unit Form

Exponential Form

100

1 hundred

102

1,000

1 thousand

103

100,000

1 hundred thousand

105

1,000,000

1 million

106

10,000,000

1 ten million

107

1,000,000,000

1 billion

109

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7–8 ▸ M1 ▸ TC ▸ Lesson 10

Launch

Students relate very large and very small positive numbers to the size of real-world objects.

UDL: Engagement

Open and display the Large and Small Positive Numbers teacher interactive for the class. Progress through each power of 10 in the demonstration by starting with an input of 100 and moving upward to 1027. The table shows a subset of the objects found in the demonstration and the corresponding inputs. Each input represents the distance, height, length, or width of the item in meters approximated to the next highest power of 10. The input does not represent the actual distance, height, length, or width of the item.

The teacher demonstration promotes relevance by connecting to possible student interests and familiar contexts. Share additional information from the Facilitation column for select objects that spark student interest.

Input (meters)

102

Facilitation

El Castillo

El Castillo is the most recognizable ancient Mayan step pyramid in Chichén Itzá on Mexico’s Yucatán peninsula.

103

Knock Nevis

This is the largest ship ever built, but it is not in use anymore.

104

Mount Everest

The peak of Mount Everest is the highest point on Earth.

Earth’s moon

Our moon is bigger than Pluto. Pluto was considered a planet in our solar system until 2006, when it was downgraded to a dwarf planet.

107

224

Visible Object

Teacher Note Information provided for the objects from the demonstration is based on data collected in 2020.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Input (meters)

Visible Object

Facilitation

1011

Polaris

This is the name of the North Star. Earth’s geographic North Pole points directly at Polaris.

1016

Light-year

This is the distance that light travels through space in one Earth year.

Milky Way galaxy

Our solar system is within this galaxy. However, our solar system is still very far from the center of this big disk.

Observable universe

The universe is a very mysterious place. We continue to make new discoveries as our technology advances.

1021

1027

Debrief the activity to this point by asking students what they notice and wonder. Then create a need for students to want a more efficient way to write very large positive numbers. There are 1 billion trillion stars in the known universe. That is a 1 followed by 21 zeros. How long will it take us to write out that number? Write the number 1,000,000,000,000,000,000,000 for the class to see and ask students to keep track of the time. There has to be a faster way to write out this number. But first, we will examine some very small objects.

Teacher Note 1 billion trillion is also referred to as 1 sextillion.

In the teacher interactive, jump back to 100 in the demonstration. From there, progress

1 . down through inputs to ____ 13 10

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7–8 ▸ M1 ▸ TC ▸ Lesson 10

Input (meters) 1 ___

Facilitation

T. namibiensis

This is the largest known bacteria.

10 4

Newly hatched tardigrade

Also called a water bear, this is the smallest known animal.

1 ___

Spider silk

Spider silk is very thin, but it is strong for its size.

10 3 1 ___

10 5 1 ___

10 6 1 ___

This is the largest known virus. Its Megavirus

1 of the width is still only about ____ 1,000

length of the largest bacteria. Semiconductor

This is used in current smartphones.

Water molecule

Water is made of two parts hydrogen and one part oxygen. This molecule is very simple, but it is important to life as we know it.

1 ___

X-ray

This is where the shortest wavelength for x-rays can be seen. X-ray wavelengths measure from 0.01 to 10 nanometers.

1 ___

Uranium-235 nucleus

10 8 1 ___

10 9

10 11

10 12

226

Visible Object

This is the type of uranium that is used in nuclear power plants.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Lead a class discussion by asking students what they notice and wonder. Then create a need for students to want a more efficient way to write very small positive numbers. The width of a proton is 833 quintillionths of a meter. That is a decimal written with 15 zeros after the decimal point followed by 833. How long does it take us to write out that decimal? Write the number 0.000 000 000 000 000 833 for the class to see and ask students to keep track of the time.

Language Support Consider supporting the word proton by asking students what they know about atoms and protons. Show a drawing of an atom and explain that atoms are the building blocks for all matter. One of the basic particles in an atom is a proton.

Today, we will learn to represent both very large and very small positive numbers in a concise and efficient way. Teacher Note

Learn

Spaces are included in very small positive numbers, such as 0.000 000 000 000 000 833, to assist in readability. It assists students in finding the place values of each digit.

Writing Very Large and Very Small Positive Numbers Students write very large positive numbers as a single digit times a power of 10 in exponential form. Students write very small positive numbers as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. Display problem 1. Tell students that the measurements shown in problem 1 are close approximations rather than measurements that have been rounded to the next highest power of 10. Guide the class through the Eiffel Tower row. Use the following prompts to guide students’ thinking. After they respond to each prompt, complete that part of the table and ask students to do the same.

Teacher Note Students will learn the formal definition of scientific notation in lesson 14.

The Eiffel Tower, one of the objects from the demonstration, has a height of approximately 300 meters. First, we write 300 in unit form. What kind of units do we use to write a number in unit form? We use place value units to write a number in unit form.

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How can we write 300 in unit form?

3 hundreds The number 3 hundreds can be written as a single digit times what number?

100 How can we write 100 as a power of 10 in exponential form?

102 How can we write 300 as a single digit times a power of 10 expressed in exponential form?

3 × 102 Have students complete problem 1 for Mount Everest and Venus. 1. Complete the table. The table shows an approximate measurement of objects seen in the demonstration. Approximate Measurement (meters) Objects

Standard Form

Unit Form

Single Digit Times a Power of 10 (expanded form)

Single Digit Times a Power of 10 (exponential form)

Eiffel Tower (height)

300

3 hundreds

3 × 100

3 × 102

Mount Everest (height)

9,000

9 thousands

9 × 1,000

9 × 103

1 × 10,000,000

1 × 107

Venus (width)

228

10,000,000 1 ten million

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

When most students have completed the table, ask for volunteers to complete the displayed table. Use the following prompt to help students recognize that each row shows the same number written in a different form. What do you notice about the values we wrote in the Mount Everest row? I notice that each value we wrote in the Mount Everest row represents the same number, but each column shows a different form of the number. Some forms include words, products, or exponents. All the numbers in the Mount Everest row are equivalent to 9,000, which is the approximate height of Mount Everest in meters. Display problem 2 and guide the class through the Grape row. Use the following prompts to guide students’ thinking. After they respond to each prompt, complete that part of the table and ask students to do the same. Now, we will look at different representations of small positive numbers. What is the unit form for the approximate width in meters of a grape?

3 hundredths How do we write 3 hundredths as a fraction?

3 ___

100

3 as a single digit times a unit fraction? Recall that unit fractions How do we write ___ 100

have a numerator of 1. 1 3 × ___ 100 How do we write the unit fraction with a denominator written as a power of 10 in exponential form? 1 ___ 10 2 How do we write 0.03 as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form? 1 3 × ___ 10 2

Teacher Note Later in this topic, students learn the meaning of a negative exponent and will then be expected to express small positive numbers by using a power of 10 in exponential form.

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Have students complete problem 2 for the approximate length of a grain of rice and the approximate width of a human hair. 2. Complete the table. The table shows an approximate measurement of objects seen in the demonstration. Approximate Measurement (meters)

Objects

Standard Form

Unit Form

Fraction

Single Digit Times a Unit Fraction (expanded form)

Grape (width)

0.03

3 hundredths

3 ___

1 3 × ___ 100

1 3 × ___ 2

Grain of Rice (length)

0.006

6 thousandths

6 ____

1 6 × ____ 1,000

1 6 × ___ 3

Human Hair (width)

0.00008

8 hundred thousandths

8 ______

1 8 × ______

1 8 × ___ 5

100

1,000

100,000

Single Digit Times a Unit Fraction (exponential form)

When most students have completed the table, ask for volunteers to complete the displayed table. Use the following prompts to lead a class discussion. Do you find it helpful to first write the number in unit form? Why? Yes. Writing the number in unit form helps us know which power of 10 to use for the denominator when writing the number as a fraction.

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Do you find it helpful to write the number as a fraction? Why? Yes. Writing the number as a fraction helps us know to use the value of the numerator for the single digit when writing the number as a single digit times a unit fraction. What is similar about how we write the approximate measurements of the Eiffel Tower and a grape in the last column of the tables? They are similar because the approximate measurements of the Eiffel Tower and a grape both have a 102. What is different about how we write the approximate measurements of the Eiffel Tower and a grape in the last column of the tables? They are different because 102 is in the denominator for the approximate measurement 1 . For the approximate measurement of the of a grape, so 3 is multiplied by ___ 2

Language Support

Eiffel Tower, 3 is multiplied by 102.

Because the unit forms of the numbers are written in hundreds and hundredths, there is a 102 in both numbers. A positive single digit times a power of 10 in exponential form, such as 3 × 102, represents a large positive number. A positive single digit times a unit fraction with a denominator written as a power 1 of 10 in exponential form, such as 3 × ___ , represents a small positive number. 2 10

Approximating Very Large and Very Small Positive Numbers

This is the first time the word approximate is used as a verb in the curriculum. Consider highlighting the use of the word approximate in different parts of speech. Lead a class discussion to clarify the different forms of approximate in these three examples. Emphasize the differences in pronunciation and the suffix changes, as applicable.

Part of Speech

Example

Students approximate very large and very small positive numbers. Students write these approximations as a single digit times a power of 10 in exponential form or as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

Adjective

The approximate height of Mount Everest is 9 thousand meters.

Direct students to problem 3. Read the problem aloud as students follow along. Use the following prompts to define approximate.

Verb

Approximate the height of Mount Everest in meters.

Noun

An approximation of Mount Everest’s height is 9 thousand meters.

What are you asked to do in part (a)? In part (a), I am asked to approximate the length of Rhode Island to the nearest ten thousand meters.

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What do you think the word approximate means? I think approximate means to find a value that is close to a given value. When we approximate, we represent the quantity by using a similar number that is found by rounding to a place value. What is another scenario where we might approximate values? Accept all reasonable answers, such as approximating the world population, the distance between Earth and other planets, or a person’s age in days. In problems 1 and 2, the tables are titled Approximate Measurement. What do you think the meaning of approximate is in this situation? I think it means that the values in the table are not the actual measurements. They may be estimates or values close to the actual measurements. In the tables for problems 1 and 2, the word approximate is used as an adjective. It indicates that the values in the tables are not exact measurements. Have students complete problems 3 and 4. Use prompts such as the following to guide students’ thinking: • Is 77,249 closer to 70,000 or 80,000? • Because we want to write the approximation as a single digit times a power of 10 in exponential form, which single digit will we use? • How do you write 80,000 as a single digit times a power of 10 in exponential form? Explain your strategy. 3. The length of Rhode Island, from the northernmost point to the southernmost point, is 77,249 meters. a. Approximate the length of Rhode Island by rounding to the nearest ten thousand meters.

80,000 meters

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b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

80,000 = 8 × 10,000 = 8 × 104 8 × 104 meters 4. The width of a water molecule is 0.000 000 000 28 meters. a. Approximate the width of a water molecule by rounding to the nearest ten billionth of a meter.

0.000 000 0003 meters b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 3 1 1 0.000 000 0003 = ___________ = 3 × ___________ = 3 × ___ 10,000,000,000 10,000,000,000 1010 1 ___ 3 × 10 meters 10 When most students are finished, confirm answers as a class.

Times As Much As Students write and solve equations with unknown factors to compare two quantities. Ask students to read problem 5. Then ask the following questions. What is 9 billion written as a single digit times a power of 10 in exponential form?

9 × 109 What is 3,000 written as a single digit times a power of 10 in exponential form?

3 × 103 How can we represent the question with an unknown factor equation?

9 × 109 =

⋅ 3 × 103

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Demonstrate how to use the unknown factor equation to determine the answer to problem 5. Ask students to record the work.

9 × 109 =

⋅ 3 × 103

______ __________________________________

9 × 10 9 9 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 3 × 10 × 10 × 10 3 × 10 3

= _ × __ × __ × __ × 10 × 10 × 10 × 10 × 10 × 10 9 3

10 10

Teacher Note Writing the factors of 10 helps make the division clearer by allowing students to pair factors of 10 in the numerator and denominator to create quotients of 1.

= 3 × 1 × 1 × 1 × 10 × 10 × 10 × 10 × 10 × 10 = 3 × 106 9 billion is 3 × 106 times as much as 3,000. 5. 9 billion is how many times as much as 3,000?

9 × 109 =

⋅ 3 × 103

______ __________________________________

9 × 10 9 9 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 3 3 × 10 × 10 × 10 3 × 10

= _ × __ × __ × __ × 10 × 10 × 10 × 10 × 10 × 10 9 3

10 10

UDL: Engagement

= 3 × 1 × 1 × 1 × 10 × 10 × 10 × 10 × 10 × 10 = 3 × 106 9 billion is 3 × 106 times as much as 3,000. Arrange students in pairs and have them complete the Times As Much As stations. They may complete the stations in any order. Every pair should finish at least two of the four stations in the remaining class time before the debrief. Direct students to record their work and answers under the appropriate headings in their books.

234

Support students’ effort and persistence by providing timely and specific feedback. Recognize and encourage students’ flexible thinking about numbers during the partner activity by highlighting a variety of solution paths. For example, most of the sample student work shows division with powers of 10, but station 1 uses place value thinking.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Circulate and use prompts such as the following to focus students’ thinking: • Why does the number

round to

• Explain how you determined the unknown factor equation for this problem. Record your work for each of the stations under the appropriate heading. Station 1

Station 2

UDL: Engagement Including a variety of real-world problems with varying degrees of difficulty provides an opportunity for student choice. Allowing students to select problems they find interesting and challenging puts students in charge of their learning and promotes relevance.

Promoting the Standards for Mathematical Practice When students determine approximately how many times as much one quantity is than another quantity, they are reasoning abstractly and quantitatively (MP2). Ask the following questions to promote MP2:

Station 3

Station 4

• How do the units involved in each quantity help you think about this problem? • What does writing each quantity as a single digit times a power of 10 tell you about this context? • How does writing a missing factor equation represent this context?

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After all pairs have completed at least two stations, call the class back together to debrief. Consider discussing that these are approximations, meaning that finding an approximation for each value and then calculating with that value contributes to some errors in the final answer. However, let students know that generating approximations is still a reasonable method to use when needing to calculate with very large numbers.

Land

Differentiation: Challenge If students need additional challenge, consider asking the following question. If the width of a red blood cell is 7.5 × 10−3 millimeters and the width of a fat cell is 1.5 × 10−1 millimeters, how many times as much is the width of the fat cell? The width of the fat cell is 20 times as much as the width of the red blood cell.

Debrief 5 min Objectives: Approximate very large and very small positive numbers and write them as a single digit times a power of 10 or as a single digit times a unit fraction with a denominator written as a power of 10. Compare large and small positive numbers by using times as much as language. Facilitate a class discussion by using the following prompts. Encourage students to restate or build upon one another’s responses. Why do we write some very large or very small positive numbers as a single digit times a power of 10 in exponential form? Writing some very small or very large positive numbers as a single digit times a power of 10 in exponential form is faster than writing many zeros between the decimal point and the single digit. What strategies are helpful when solving how many times as much as problems? Explain. A helpful strategy is to use an unknown factor equation. We can write an equation in the form Product = Factor · Factor to solve these problems. We could be given the product and one of the factors. To solve, we need to divide the product by the given factor to find the unknown factor. We could be given both factors. To solve, we need to multiply both factors to find the product. 236

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What forms of numbers are useful when solving how many times as much as problems with large numbers? Why? Numbers written as a single digit times a power of 10 in exponential form are useful because we can multiply or divide by grouping the single digits and grouping the 10’s.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

2. Complete the table. The table shows the typical speed in miles per hour of a sea star. Typical Speed (miles per hour)

Large and Small Positive Numbers In this lesson, we •

explored very large and very small positive numbers by relating them to the sizes of real-world objects.

•

analyzed equivalent forms of large and small positive numbers.

•

approximated very large and very small positive numbers.

•

wrote unknown factor equations to answer how many times as much as questions.

Object

Standard Form

Unit Form

Fraction

Single Digit Times a Unit Fraction (expanded form)

Single Digit Times a Unit Fraction (exponential form)

Sea Star

0.01

1 hundredth

1 100

1 1 × ___

1 1 × ___ 2

100

A unit fraction has a numerator of 1.

Each number in the Sea Star row is an equivalent form of 0.01.

Examples 1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the Eiffel Tower.

3. The land area of France is 212,954 square miles. The land area of Canada is 3,854,083 square miles. The land area of Canada is about how many times as large as the land area of France?

Approximate Number of Stacked Pennies

Object

Standard Form

Unit Form

Single Digit Times a Power of 10 (expanded form)

Eiffel Tower

200,000

2 hundred thousand

2 × 100,000

Use place value units when writing numbers in unit form.

The expanded form of 2 hundred thousand is 2 × 100,000, where the power of 10 is written

in standard form.

212,954 ≈ 200,000

Single Digit Times a Power of 10 (exponential form)

2 × 10

The exponential form of

1 hundred thousand is 105, so write 2 hundred thousand as 2 × 105.

Approximate the value of each land area. Then write the approximation

= 4 × 106

as a single digit times a power of 10 in exponential form.

3,854,083 ≈ 4,000,000 To decide how to write the unknown factor equation, determine

= 2 × 105

which quantity is being multiplied. To determine the unknown factor, divide 4 × 106 by 2 × 105.

4 × 106 =

⋅ 2 × 105

4______ × 106 4_______________________ = × 10 × 10 × 10 × 10 × 10 × 10 2 × 10 × 10 × 10 × 10 × 10 2 × 105 10 __ 10 __ 10 × 10 × __ × 10 × __ × 10 = _4 × __ 2 10 10 10 10 10

= 2 × 1 × 1 × 1 × 1 × 1 × 10 = 2 × 10

Write the factors of 10 in the numerator and in the denominator. Then pair the factors of 10 to create quotients 10 , which is 1. of __ 10

= 20 The land area of Canada is about 20 times as large as the land area of France.

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147

148

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 10 Name

PRACTICE Date

7–8 ▸ M1 ▸ TC ▸ Lesson 10

EUREKA MATH2

3. There are 907,200,000 milligrams in 1 ton. a. Approximate the number of milligrams in 1 ton by rounding to the nearest hundred million milligrams.

1. Complete the table. The table shows the approximate number of stacked pennies needed to reach the height of the given objects.

900,000,000 milligrams b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

Approximate Number of Stacked Pennies

9 × 108 milligrams

Objects

Standard Form

Unit Form

Single Digit Times a Power of 10 (expanded form)

Single Digit Times a Power of 10 (exponential form)

Mount Everest

6,000,000

6 million

6 × 1,000,000

6 × 106

Empire State Building

300,000

3 hundred thousand

3 × 100,000

3 × 105

4. The smallest insect on the planet is a type of parasitic wasp that measures 0.000 139 meters. a. Approximate the length of the insect by rounding to the nearest ten thousandth of a meter.

0.0001 meter b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form. 1 1 × ___ meter 4 10

5. 800,000 is how many times as much as 2,000?

400 2. Complete the table. The table shows the typical speed of the given animals. Typical Speed (miles per hour) Objects

Standard Form

Unit Form

Galapagos Tortoise

0.2

2 tenths

Sloth

0.07

7 hundredths

6. 60,000,000 is how many times as much as 30,000?

2,000

Fraction

Single Digit Times a Unit Fraction (expanded form)

Single Digit Times a Unit Fraction (exponential form)

2 __

1 2 × __ 10

1 2 × ___ 1

7 ___ 100

1 7 × ___

100

7. 6 × 105 is 2 × 103 times as much as what number?

300

102

149

150

P R ACT I C E

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7–8 ▸ M1 ▸ TC ▸ Lesson 10

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

For problems 8 and 9, use the values in the table to answer the questions.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10

Remember For problems 11–14, subtract.

Location

Russia

United States

Brazil

Belize

Slovenia

Jamaica

Approximate Land Area (square miles)

7 × 106

4 × 106

3 × 106

9 × 103

8 × 103

4 × 103

11.

( 5)

8 − − 4_ __ 15

20 __ 15

8. Based on land area, about how many islands the size of Jamaica does it take to equal the size of Russia?

12.

( 5)

7 − − 2_ __ 10

11 __ 10

3 13. − 5_ − (− _ 4) 6

4 14. − 7_ − (− _ 5 6)

1 − __

__ − 22

For problems 15 and 16, divide. 15. − 3_ ÷ (−0.5) 5

It takes about 1,750 islands the size of Jamaica to equal the size of Russia.

30 __ 25

1 as large as the land area of the United States? 9. Which country’s land area is about ___ 500

1 as large as the land area of the United States. The land area of Slovenia is about ___ 500

16.

− 1_

___8 4_ 5

5 − __ 32

10. The Atlantic Ocean contains about 310,410,900 cubic kilometers of water. Lake Superior, which is the largest lake in the United States, contains about 12,000 cubic kilometers of water. Approximately how many Lake Superiors would it take to fill the Atlantic Ocean? It would take about 30,000 Lake Superiors to fill the Atlantic Ocean.

17. Find the value of the expression shown.

− 1_ ÷ 2_ ⋅ (− 1 2_) 4

21 __ 40

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P R ACT I C E

151

152

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10 ▸ Times As Much As Stations Cards

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 10

Station 1 In the fall of 2019, approximately 50,800,000 students attended school in the United States in prekindergarten through grade 12. About 11,683,000 of those students attended middle school. The overall number of students in prekindergarten through grade 12 was about how many times as much as the number of students in middle school?

This page may be reproduced for classroom use only.

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Station 2 In 2019, the total outstanding consumer debt in the United States, including mortgages, auto loans, credit cards, and student loans, was about $3.9 trillion. That same year, the published national debt of the United States was $22,460,468,000,000. The published United States national debt was about how many times as much as the total United States consumer debt?

242

This page may be reproduced for classroom use only.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 10 ▸ Times As Much As Stations Cards

Station 3 As of July 1, 2018, the US Census Bureau estimated the population of New York City at 8,398,748 people and the population of New York State at 19,542,209 people. The population of New York State was about how many times as much as the population of New York City?

This page may be reproduced for classroom use only.

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Station 4 The total global carbon dioxide emissions for 2018 were about 33.1 billion tons. That same year, the carbon dioxide emissions from natural gas in the United States were about 1.629 billion tons. Approximately what fraction of the total global carbon dioxide emissions was from the use of natural gas in the United States?

244

This page may be reproduced for classroom use only.

5 1

5 ten million 1 ten million

= __________

⋅ 1 × 107

2 10 10 10 10 10 10 10 10 10 10 10 4 10 10 10 10 10 10 10 10 10 10 10 2_ = × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 10 4 2 _ = × 10 4 20 = __ 4

10 10

= _ × __ × __ × __ × __ × __ × __ × __ × __ × __ × __ × __ × __ × 10

2 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 4 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

⋅ 4 × 1012 = ________________________________________________

2 × 1013 =

22,460,468,000,000 ≈ 20,000,000,000,000 = 2 × 1013

= 4 × 1012

In 2019, the published United States national debt was about 5 times as much as the total United States consumer debt.

4 × 10

2______ × 1013

≈ 4,000,000,000,000

3.9 trillion = 3,900,000,000,000

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 10

The published United States national debt was about how many times as much as the total United States consumer debt?

In 2019, the total outstanding consumer debt in the United States, including mortgages, auto loans, credit cards, and student loans, was about $3.9 trillion. That same year, the published national debt of the United States was $22,460,468,000,000.

Station 2

In the fall of 2019, the overall number of students in prekindergarten through grade 12 was about 5 times as much as the number of students in middle school.

1 × 10

5______ × 107

5 × 107 =

11,683,000 ≈ 10,000,000 = 1 × 107

= 5 × 107

50,800,000 ≈ 50,000,000

The overall number of students in prekindergarten through grade 12 was about how many times as much as the number of students in middle school?

In the fall of 2019, approximately 50,800,000 students attended school in the United States in prekindergarten through grade 12. About 11,683,000 of those students attended middle school.

Station 1

Times As Much As Stations Answer Key

EUREKA MATH2 7–8 ▸ M1 ▸ TC ▸ Lesson 10 ▸ Times As Much As Stations Answer Key

This page may be reproduced for classroom use only.

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246 This page may be reproduced for classroom use only.

= 2.5

2 10 10 10 10 10 10 8 10 10 10 10 10 10 2 = _ × 1 × 1 × 1 × 1 × 1 × 1 × 10 8 2 _ = × 10 8 20 __ = 8

= _ × __ × __ × __ × __ × __ × __ × 10

2 × 10 × 10 × 10 × 10 × 10 × 10 × 10 8 × 10 × 10 × 10 × 10 × 10 × 10

= ___________________________

⋅ 8 × 106

In 2018, about

3 × 1010

10 10

2 3 2_ __ 1 = × 3 10 2 = __ 30 1 = __ 15

1 10

= _ × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × __

2 3

10 10

1 10

= _ × __ × __ × __ × __ × __ × __ × __ × __ × __ × __

3 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10

__1 of the total global carbon dioxide emissions was from the use of natural gas in the United States. 15

⋅ 3 × 1010 2 × 109 2 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 ______ = ______________________________________

2 × 109 =

= 2 × 109

1.629 billion = 1,629,000,000 ≈ 2,000,000,000

= 3 × 1010

33.1 billion = 33,100,000,000 ≈ 30,000,000,000

Approximately what fraction of the total global carbon dioxide emissions was from the use of natural gas in the United States?

The total global carbon dioxide emissions for 2018 were about 33.1 billion tons. That same year, the carbon dioxide emissions from natural gas in the United States were about 1.629 billion tons.

Station 4

In 2018, the population of New York State was about 2.5 times as much as the population of New York City.

8 × 10

2______ × 107

2 × 107 =

= 2 × 107

19,542,209 ≈ 20,000,000

= 8 × 106

8,398,748 ≈ 8,000,000

The population of New York State was about how many times as much as the population of New York City?

As of July 1, 2018, the US Census Bureau estimated the population of New York City at 8,398,748 people and the population of New York State at 19,542,209 people.

Station 3

7–8 ▸ M1 ▸ TC ▸ Lesson 10 ▸ Times As Much As Stations Answer Key EUREKA MATH2

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 11 LESSON 11

Products of Exponential Expressions with Positive Whole-Number Exponents Apply the product of powers with like bases property to write equivalent expressions given an expression of the form x m ⋅ x n.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11 Name

Date

EXIT TICKET

For problems 1–4, apply the product of powers with like bases property to write an equivalent expression. 1. 53 ⋅ 56

2. (− 5)3 (− 5)8

(−5)11

3 4 3_ 3. (_ 5 ) (5 )

This lesson introduces students to the exponent property associated with products of powers. Students use their knowledge of powers to explore expressions of the form x m ⋅ x n for any number x and positive whole numbers m and n. Through practice with concrete examples, they make a conjecture about an equivalent expression to x m ⋅ x n and use repeated multiplication to show that x m ⋅ x n = x m+n. With this new property associated with the product of powers with like bases, students write equivalent expressions that include coefficients and multiple bases.

Key Questions

_3 5 (5 )

• What conditions are necessary to apply the product of powers with like bases property?

4. 5a ⋅ 5b

• Why does x m ⋅ x n = x m+n?

5a+b

5. Explain what the expression 43 ⋅ 46 represents and how it can be written as a single power.

Achievement Descriptor

The expression represents 3 factors of 4 multiplied by 6 factors of 4. Because there are 9 factors of 4, I can write the expression as 49.

Lesson at a Glance

7–8.Mod1.AD10 Apply the properties of integer exponents to generate

equivalent numerical expressions. (8.EE.A.1)

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Agenda

Materials

Fluency

Teacher

Launch 5 min

• Chart paper (1 sheet)

Learn 30 min

• Markers (3 different colors)

• Multiplying Powers with Like Bases

Students

• Applying a Property of Exponents

• Properties and Definitions of Exponents graphic organizer

• Multiplying Powers with Unlike Bases

Land 10 min

Lesson Preparation • Use the chart paper to create an anchor chart with the same tables as the Properties and Definitions of Exponents graphic organizer to display to the class. The anchor chart will be filled in throughout lessons 11–13.

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Fluency Write Expressions by Using Exponents Students write products in exponential notation to prepare for writing equivalent expressions of the form x m . x n. Directions: Write each product as a single base raised to an exponent. 1.

8.8.8.8.8.8.8.8.8.8

810

(−4)(−4)(−4)

(− 4)3

g.g.g.g

2.3 . 2.3 . 2.3 . 2.3 . 2.3 . 2.3

2.36

1 1 1 1 1 1 1 1 . . . . . . . 4 4 4 4 4 4 4 4

_ _ _ _ _ _ _ _

(4 )

250

_1 8

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TC ▸ Lesson 11

Students multiply two powers with like bases. Have students attempt to complete problem 1 without a calculator. Circulate as students work and observe whether they try to write out all the 10’s by hand. 1. Multiply. Write the product as a power of 10 in exponential form.

1050 . 1020 1050 . 1020 = 1070 After about one minute of productive struggle time, engage in a brief discussion by using the following questions. What strategy did you decide to use for problem 1? Why? Did it work? I wrote out the 10’s because the exponents indicate repeated multiplication. It didn’t work because there were too many 10’s to write out. Students who try to write out 10’s by hand and then multiply may express frustration about the problem, while a few students may recognize that the expression represents using 10 as a factor 70 times. Allow for varying opinions to be shared with the class. Today, we will find a more efficient way to write the product of powers with like bases.

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Learn Multiplying Powers with Like Bases Students look for patterns when multiplying powers with like bases. Direct students to problem 2. How is problem 2 similar to problem 1? Problems 1 and 2 are similar because they are both products of powers of 10, and they both have exponents involving 5 and 2. How are they different? They are different because the exponents in problem 1 are 50 and 20, and the exponents in problem 2 are 5 and 2. Have students complete problem 2 independently. 2. Multiply. Write the product as a power of 10 in exponential form.

105 . 102 105 . 102 = 10 . 10 . 10 . 10 . 10 . 10 . 10 = 107 Engage in a discussion by posing the following questions. Did you prefer to complete problem 1 or 2? Why? Allow for a variety of responses. Students who tried to write out all the 10’s in problem 1 will likely indicate that they prefer completing problem 2 because smaller exponents are used, while students who found a shortcut in problem 1 may not see a difference in complexity.

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Often, we want to find ways to make calculations more efficient. Can we always write out all the factors of a power individually? Explain. I suppose we could try, but it will not necessarily be easy or efficient. It would take a long time in problem 1 to write out 50 tens and 20 tens. How do the exponents in the factors in problem 2 relate to the exponent in the product? The exponent in the product, 7, is the sum of the exponents in the factors, 5 and 2. We can explore why this works. Display problem 2. Consider using color-coding to help students relate the equivalent expressions in each step.

105 . 102 How do we expand the power 105 as a product of 10’s?

105 = 10 . 10 . 10 . 10 . 10

How do we expand the power 102 as a product of 10’s?

UDL: Representation Color-coding can serve as a visual cue to help students distinguish between the factors of 10 that result from expanding each of the powers, 105 and 102.

102 = 10 . 10 Now write 105 . 102 as a product of 10’s.

105 . 102 = 10 . 10 . 10 . 10 . 10 . 10 . 10 5 times

2 times

How can we determine how many factors of 10 we have? We can count how many 10’s there are. We can add 5 and 2 because we have 5 factors of 10 and 2 factors of 10.

105 . 102 = 10 . 10 . 10 . 10 . 10 . 10 . 10 (5 + 2) times 105 . 102 = 105+2

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

We can write the product of 105 . 102 as 105+2, which equals 107. Why do you think the exponent is 7? There are 7 factors of 10. Is it important that both powers in a product have the same base? Why? Yes. It is important that both powers in the product have the same base. If the bases are different, then we do not have the same factor repeating. For example, in a product like 105 . 42, there are 5 factors of 10 and 2 factors of 4. When we write 105 and 102 as a product of 10’s, we can see that we have 5 + 2 factors of 10. So we can write 105 . 102 as 105+2. To write a product as a single power by adding exponents, the powers in the product must have the same base. We can use variables to make a general property for any base. Display the following expression. Use color-coding to help students relate the equivalent expressions in each step.

xm . xn

What do you think it means for x to have an exponent of m? What does m represent?

Teacher Note If students have difficulty with a variable base and variable exponents, consider displaying x 4 . x 7.

I think m represents the number of x’s that are being multiplied together. Can m be any number? I’m not sure. I think it can only be a positive integer. How many factors of x are represented by x m? There are m factors of x. How many factors of x are represented by x n? There are n factors of x. Label the next steps by using braces. Express the expansion of factors of x by using ellipses.

x = x ... x . x ... x m times n times n

Promoting the Standards for Mathematical Practice When students expand products of powers such as 10m . 10n and 10m+n and notice the pattern x m . x n = x m+n, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8: • What patterns did you notice when you expanded the powers? • Will this pattern always work?

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When we multiply x m by x n, how many factors of x do we have?

UDL: Action & Expression

We have m + n factors of x. Display this work to the class.

Students continue to use the Properties and Definitions of Exponents graphic organizer to formalize properties and definitions involving exponents in lessons 12 and 13. Display the tables as an anchor chart that is gradually completed throughout the topic.

x m . x n = x ... x . x ... x = x m+n (m + n) times Review with students the general property that results from this work. If x is any number and m and n are positive whole numbers, then x m . x n = x m+n. For us to add the exponents, the factors must have like bases. Instruct students to add this property to the first blank row in their Properties and Definitions of Exponents graphic organizer. Do the same on chart paper for the class Properties and Definitions of Exponents anchor chart. Consider using color-coding for the Property and Example columns.

Properties of Exponents Description

Property

Example

Product of powers with like bases

x m . x n = x m+n

105 . 102 = 105+2 = 107

Encourage students to refer to their graphic organizers or the class anchor chart regularly throughout the topic. They can use it for problems within the lesson and for Practice problems.

Teacher Note The property names are provided to help students distinguish each property, but it is not critical that students know the property name. It is most important that students know how to apply the property properly.

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Definitions of Exponents Description

Definition

Example

Ask students to complete problem 3 independently and to show their work by using the product of powers with like bases property. Circulate to confirm answers. 3. Multiply. Write the product as a power of 10 in exponential form.

1057 . 10342 1057 . 10342 = 1057+342 = 10399

Applying a Property of Exponents Students apply the product of powers with like bases property to write equivalent expressions with like bases. Direct students to preview problems 4–7 and ask them to share what they notice about the problems. Then have students work through problems 4–7 with a partner.

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For problems 4–15, apply the product of powers with like bases property to write an equivalent expression. 4. 53 . 54

5. (− 5)6 . (− 5)10 (− 5)16

57 2 7 2 5 6. (− _) . (− _) 3 3 2_ 12

(− 3 )

7. y 8 . y 11

y 19

Confirm the answers for problems 4–7. Consider sharing student work that shows different possible answers, including work that shows the exponent of each power as an addition expression, such as y 8+11, and work that shows the exponent of each power as a sum, such as y 19. Then have students complete problems 8–18 independently. Encourage students who write the exponent of each power as an addition expression to write the exponent as a sum. 8. 5.823 . 5.88

9. a 23 . a 8

5.831

a 31

10. 7210 . 72

11. f 10 . f

7211

f 11

Students may overlook the power of 1 for problems 10 and 11. For these students, ask the following questions: • For 7210 . 72, by how many additional factors do you multiply 7210 ?

78 5 22 12. (_) . 5_ (9) 9

78 b 22 13. (_) . b_ (2) 2

14. (− 3)9 . (− 3)6

15. 2 x 6 . 3 x 5

5_ 100 (9 )

(− 3)15

Teacher Note

• How are the additional factors accounted for in your answer?

_b 100 (2 )

2x6 . 3x5 = 2 . x6 . 3 . x5 = 2 . 3 . x6 . x5 = 6 . x 6+5 = 6 x 11

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16. Solve for x.

74 . 7x = 712 8 17. Write three exponential expressions that are equivalent to 916. Sample:

915 . 9, 910 . 96, 914 . 92 18. Which expressions have a value of − 16? Choose all that apply. A. − 1 . 24 B. − 24

Differentiation: Support If students need additional support completing problem 18, encourage them to write each exponential expression as a product of factors. For example, − 1 . 24 = − 1 . 2 . 2 . 2 . 2.

C. (− 2)4 D. − (24) E. − (− 2)4 F. (− 2)2 . (2)2 Confirm answers as a class. Invite students to share their answers and reasoning for problem 18. If needed, display the following expressions and ask the questions to clarify student thinking for problem 18.

− 520 (− 5)20 Is the value of the expression − 520 negative or positive? How do you know? The value of the expression is negative because the expression is equivalent to − 1 multiplied by twenty 5’s, and − 1 multiplied by any positive number has a negative value. Is the value of the expression (−5)20 negative or positive? How do you know? The value of the expression is positive because the expression is equivalent to twenty

− 5’s multiplied together, and an even number of negative factors produces a positive product.

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Multiplying Powers with Unlike Bases Students apply the product of powers with like bases property to write equivalent expressions with unlike bases. Introduce the Which One Doesn’t Belong? routine. Present four expressions and invite students to study them.

23 . 53

106

22 . 52 . 24 . 54

2.2.2.2.2.2.5.5.5.5.5.5

Give students 1–2 minutes to find a category in which three of the expressions belong, but a fourth expression does not. When time is up, invite students to explain their chosen categories and to justify why one expression does not fit. Celebrate all valid reasonings. Which expression does not belong? Why? The expression 23 . 53 does not belong because it is not equivalent to the other expressions. The expression 106 does not belong because it has one base and one exponent. The expression 22 . 52 . 24 . 54 does not belong because it has multiple powers with the same base. The expression 2 . 2 . 2 . 2 . 2 . 2 . 5 . 5 . 5 . 5 . 5 . 5 does not belong because there are no exponents. Highlight responses that emphasize the product of powers with like bases property. Ask the following questions that invite students to use precise language, make connections, and ask questions of their own. How is the expression 23 . 53 in this activity different from the expressions in problems 8–14? The expression 23 . 53 has two different bases. For the expressions in problems 8–14, there are two bases, but they are the same number. Display the expression 26 . 56. What do you notice about this expression? I notice that the bases are different, but the exponents are the same. © Great Minds PBC

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How do we write this expression as a product of factors without any exponents?

2.2.2.2.2.2.5.5.5.5.5.5 We have 6 factors of 2 and 6 factors of 5. Invite students to think–pair–share about the following questions. Do you think this expression is equivalent to 106? Why? Yes. We can use the commutative property of multiplication to rearrange the factors, and then we’ll have 6 groups of 2 . 5, and the expression (2 . 5)6 is equivalent to 106. Demonstrate how to use the commutative and associative properties of multiplication to show that 26 . 56 = 106.

26 . 56 = 2 . 2 . 2 . 2 . 2 . 2 . 5 . 5 . 5 . 5 . 5 . 5 =2.5.2.5.2.5.2.5.2.5.2.5 = (2 . 5)(2 . 5)(2 . 5)(2 . 5)(2 . 5)(2 . 5) = 10 . 10 . 10 . 10 . 10 . 10 = 106 Display the expression 22 . 52 . 24 . 54. Is this expression equivalent to 106? How do you know? Yes. We can use the commutative property of multiplication to rearrange the factors of 2 and 5. Then we can use the associative property of multiplication to group the factors of 2 and 5 into 6 factors of 10. How does the expression 22 . 52 . 24 . 54 relate to the product of powers with like bases property? The property x m . x n = x m+n applies only to products of powers with like bases. When the expression has powers with two different bases, we apply the property to powers with like bases separately.

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Instruct students to complete problem 19 independently. For problems 19–21, apply the product of powers with like bases property to write an equivalent expression. 19. 93 . 94 . 42 . 4

97 . 43 Problem 19 provides a rich opportunity to draw out possible errors when multiplying powers, such as adding the bases and adding the exponents or multiplying unlike bases. Use the following prompts to clarify student thinking. Display the incorrect answer 133+4+2+1. Suppose Logan gets 133+4+2+1 as his answer for problem 19. Is Logan correct? Why? Logan is incorrect. He added the bases 9 and 4 and added all the exponents. The property only applies to powers with the same base, and we add only the exponents, not the bases. So Logan should have given the answer 93+4 . 42+1, or 97 . 43. Display the incorrect answer 363+4+2+1. Suppose Lily gets 363+4+2+1 as her answer for problem 19. Is Lily correct? Why? Lily is incorrect. She multiplied the bases 9 and 4 and added all the exponents. The property only applies to powers with the same base, and we keep that base in the product. So Lily also should have given the answer 93+4 . 42+1, or 97 . 43. Confirm the correct answer. Emphasize to students that we can only apply the property to powers with like bases. When there are different bases, we apply the property to powers with like bases separately. Ask students to complete problems 20 and 21 in pairs. When pairs have finished, invite them to check their work with a neighboring pair. 20. (−2)2 . 195 . 197 . (− 2)3

(−2)5 . 1912

21. 4 . 104 . 105

4 . 109

Differentiation: Challenge If students need additional challenge, consider asking the following question. Is 218 . 524 equivalent to 1018 . 56 ? How do you know?

218 . 524 = 218 . 518 . 56 = 1018 . 56.

Yes. The expressions are equivalent because

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Land Debrief 5 min Objective: Apply the product of powers with like bases property to write equivalent expressions given an expression of the form x m . x n. Facilitate a class discussion by using the following prompts. Encourage students to restate or build upon one another’s responses. What conditions are necessary to apply the product of powers with like bases property? To apply the product of powers with like bases property, powers must be multiplied and have the same base. Why does x m . x n = x m+n? Give an example. Because x m has m factors of x and x n has n factors of x, the product x m . x n has m + n factors of x. Therefore, x m . x n can be written as x m+n. For example, 45 . 43 can be written as 45+3 because the product 45 . 43 has 5 + 3, or 8, factors of 4. To preview upcoming learning, facilitate the following discussion. Display the expression 30 . 35. Invite students to turn and talk about what the product should be according to the product of powers with like bases property. Then invite students to make conjectures about what the value of 30 is and why. Do not confirm or deny any responses.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2 7–8 ▸ M1 ▸ TC ▸ Lesson 11

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

2. (− 1_)(− 1_) 6 6

1_ 1_ 1_ (− 6)(− 6) = (− 6)

1+3

Products of Exponential Expressions with Positive Whole-Number Exponents

= (− 1_) 6

In this lesson, we •

discovered a pattern when multiplying powers with like bases.

•

learned the product of powers with like bases property.

•

applied a property of exponents to write equivalent expressions.

3. 74 ⋅ 83 ⋅ 74 ⋅ 84

74 ⋅ 83 ⋅ 74 ⋅ 84 = 74 ⋅ 74 ⋅ 83 ⋅ 84

Product of Powers with Like Bases Property x

x is any number

x =x when

= 74+4 ⋅ 83+4 = 78 ⋅ 87

m+n

Negative bases and fractional bases have parentheses to avoid confusion.

The first factor represents 1 factor of

(− 6 ), so it has an exponent of 1.

Because there are two different bases, 7 and 8, use the commutative property of multiplication to rearrange the powers with like bases together. Then apply the property of exponents to powers with like bases separately.

m and n are positive whole numbers

Examples Apply a property of exponents to write an equivalent expression. 1. 94 ⋅ 92

94 ⋅ 92 = 94+2 = 96

94 . 92 = 9 . 9 . 9 . 9 . 9 . 9 4 times 2 times 4

9 is 4 factors of 9. 92 is 2 factors of 9. So 94 . 92 is 4 + 2 factors of 9, which can be

written as 94+2.

161

162

RECAP

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 11

PRACTICE

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

11. Which expressions are equivalent to 53 . 55? Choose all that apply. A. 53+5

B. 53 . 5

For problems 1–10, apply the product of powers with like bases property to write an equivalent expression. 1. 106 . 105

2. 24 . 23 . 22

3. (−4)8 (−4)2

4. (− y)8 (− y)10 (− y)6

1011

(−4)10

D. 515

E. 54 . 54

F. 258

(− y)24

12. Which expressions are equivalent to 912? Choose all that apply. A. 36 . 36

B. 99 . 93

5. 32 . 3

1 1 __ 6. (__ 10 ) (10 )

_1 5 _1 _1 5 _1 7. (a ) (a ) (a ) (a )

8. 1010 . 108 . (−2)7 . (−2)9

_1 (a )

9. d 3 . c . c . d 3

c 2 . d6

C. 58

C. 94 . 93

1 31 __ (10)

D. 9 . 912

E. 96 . 96 F. 96 . 92

1018 . (−2)16

13. Sara states that when two powers with the same base are multiplied, the exponents are multiplied together. She uses an example to support this claim.

42 . 42 = 42 . 2 = 44

1 1_ 1_ 10. (1_) (_ 4 ) (2 ) (4 ) 2 28

_1 54 1_ 54 (2) (4 )

Fill in the boxes to create an equation that shows that Sara’s claim is incorrect. Sample:

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163

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P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

For problems 20–23, solve for b.

14. Write the area of the rectangle as a single base raised to an exponent.

105 ft

20. 3b = 311 . 33

21. 7b . 74 = 712

22. 4 . 4b = 46

23. 65 . 63 = 62b

102 ft

107 square feet

For problems 24–26, fill in the boxes with digits 1–6 to make the equation true. Each digit can be used only once.

For problems 15–18, indicate whether the value of the expression is a positive number or a negative number. 15. (− 3)2 Positive number

17. (− 5)133 Negative number

24.

16. (− 4)3 Negative number

x2 . 4x

= 8 x6

)(3x) =

26. (x 4 y 2)(xy) = x

25. (2 x

18. (− 6)4,592 Positive number

19. The product (− 1)3 . (− 1)n is a negative number. Which of the following values of n are possible? Choose all that apply. A. 2

Remember For problems 27–30, multiply. 1.6 27. _ 3

B. 5 C. 7

28. 6(− 2_) 3

29. − 10(2_) 5

−4

4 . 14 30. _ 7

D. 8 E. 10

P R ACT I C E

165

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P R ACT I C E

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 11

For problems 31–34, write each number as a decimal. Use bar notation where appropriate. 36 31. − ____ 1,000

9 32. − __ 25

− 0.036

33. − 7_ 9

7 − 0.¯

− 0.36

34.

5 __ 11

45 0.¯

Estimate, and then find the quotient. 35. 85.92 ÷ 12 Estimation: 84 ÷ 12 = 7 Quotient: 7.16

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P R ACT I C E

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Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 12 LESSON 12

More Properties of Exponents Apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12 Name

Date

EXIT TICKET

Apply the properties of exponents to write an equivalent expression. Assume y is nonzero. 1. (9 3 ) 6

9 18

2. (2 x) 4

Lesson at a Glance In this digital lesson, students encounter and apply properties of exponents by making use of a mustard spot context. Students engage with raising powers to powers by considering the repetition of a base. They examine raising products to powers by determining whether two expressions are equivalent. Finally, they make a conjecture about an equivalent expression for a quotient raised to a power. This lesson introduces the term verify. Use the digital platform to prepare for and facilitate this lesson. Students will also interact with lesson content and activities via the digital platform.

2 4x 4

3. (3__ y)

4 2

38 __ y2

Key Question

• How can we write equivalent expressions for (x m) n, (x y) n, and (_y ) ? x n

Achievement Descriptor 7–8.Mod1.AD10 Apply the properties of integer exponents

to generate equivalent numerical expressions. (8.EE.A.1)

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Agenda

Materials

Fluency

Teacher

Launch 5 min

• Markers (3 different colors)

Learn 30 min

• Properties and Definitions of Exponents anchor chart (from lesson 11)

• Raising Powers to Powers • Raising Products to Powers • Raising Quotients to Powers

Land 10 min

Students • Computers or devices (1 per student pair) • Properties and Definitions of Exponents graphic organizer (from lesson 11)

Lesson Preparation • None

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Fluency Expand Powers as a Product of the Bases Students expand powers as products of the bases to prepare for generating x n expressions of the form (x m) n, (xy) n, and ( __y ) . Directions: Expand each power as a product of the bases. For example, expand 3 4 as 3 . 3 . 3 . 3. 1.

10 6

10 . 10 . 10 . 10 . 10 . 10

2.2.2.2.2

a.a.a.a

(−3) 7

(−3)(−3)(−3)(−3)(−3)(−3)(−3)

(5 )

_ 3

(5 )(5 )(5 )

(− 7 )

270

_3 2

_2 _2 _2 _3

(− 7 )(− 7 )

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Students write equivalent forms of exponential expressions.

Students begin the lesson by considering examples of similar expressions that can be represented tediously as repeated multiplication of a power or succinctly as a power of a power. Students use the product of powers with like bases property of exponents, x m . x n = x m+n, to represent similarly structured expressions. Students write expressions equivalent to the following expressions:

34 . 34 34 . 34 . 34 34 . 34 . 34 . 34 . 34 . 34 . 34 . 34 . 34 What patterns do you notice? When the product has 2 factors of 3 4, we can write the product as 3 4+4. When the product has 3 factors of 3 4, we can write the product as 3 4+4+4. So if a product has n factors of 3 4, we can write the product as a power with 3 as the base and a sum of n 4’s as the exponent. Students consider alternate ways to write equivalent expressions.

34 . 34 . 34 . 34 . 34 . 34 . 34 . 34 . 34 3 4+4+4+4+4+4+4+4+4 Can you find or think of a more efficient way to write either expression? The first expression can be written as (3 4) 9. The second expression can be written as 3 9 . 4. Students watch a video about a student, Vic, and a mustard spot.

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Learn Raising Powers to Powers Students consider expressions of the form (x m)n. Students transition to thinking more abstractly about exponents and how exponents influence the structure of expressions. Specifically, students see that any expression raised to the fifth power can still be written as a product with 5 factors of that expression. They then generalize to see that this is true regardless of the base and the idea can be extended to other exponents.

(34)5 = ( ) ( ) ( )( )( ) Students explore the case in which 3 4 is under the mustard spot. How is (3 4)5 similar to the mustard spot expression? These expressions are similar because both are raised to an exponent of 5, which represents 5 factors of the term in the parentheses.

UDL: Representation Digital activities align to the UDL principle of Representation by including the following elements: • Scaffolds that connect new information to prior knowledge. Students connect new uses of exponents, such as raising powers to powers, raising products to powers, and raising quotients to powers, to prior exponent skills and knowledge, such as products of powers with like bases. • Opportunities for generalizing learning to new situations. Students generalize exponent properties to new, unfamiliar situations.

Explain the meaning of (3 4)5.

The expression (3 4) 5 means that we multiply 5 factors of 3 4. Students use the mustard spot pattern from earlier in the lesson to write their new expression in a more succinct equivalent form. What expression is equivalent to (3 4)5 ? Explain your reasoning.

(3 4 ) 5 is equivalent to 3 5 . 4. I can think of the power inside the parentheses as 4 factors of 3. The exponent outside the parentheses tells me I have 5 factors of 3 4. This means I have 5 factors of 4 factors of 3, which is 5 . 4, or 20, factors of 3.

Students discuss the pattern for these expressions and add the property (x m ) n = x m . n to the next blank row in their Properties and Definitions of Exponents graphic

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organizer from lesson 11. Add the new property to the Properties and Definitions of Exponents anchor chart for this topic and add an example, such as (5 2) 6 = 5 2 . 6.

Properties of Exponents Description

Property

Example

Product of powers with like bases

x m . x n = x m+n

10 5 . 10 2 = 10 5+2 = 10 7

Power of a power

(x m) n = x m . n

(5 2) 6 = 5 2 . 6 = 5 12

Teacher Note At this stage, have students leave the exponent in the form of a product to help make the use of the property more visible. Students may wish to write expressions of the form (x m) n as x m . n rather than as x n . m. Consider highlighting the fact that both responses are correct. In the Properties and Definitions of Exponents graphic organizer and future lessons, these expressions will be written as x m . n for readability.

Definitions of Exponents Description

Definition

Example

Raising Products to Powers Students consider expressions of the form (xy)n. Students examine two problems that require using exponents from Vic’s work that are partially covered in mustard. They explore the case in which 4 . 3 is the covered expression.

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Language Support

7. (

6. ( =4.3.4.3.4.3.4.3.4.3.4.3.4.3

=4.4.4.4.4.4.4.3.3.3.3.3.3.3

This is the first use of the term verify in Eureka Math2™. Consider previewing the meaning of the term before students see it in print. Facilitate a class discussion focusing on students’ prior experience with checking their work or making sure something is true.

Students consider two different perspectives on expressions that could be equivalent.

(4x) 3 = (4x)(4x)(4x) (4x) 3 = 4 3 x 3 Are the expressions (4 x)3, (4 x)(4 x)(4x), and 4 3x 3 equivalent? The expressions contain the same number of factors of 4 and of x. Therefore, the expressions are equivalent. After they have identified a new exponent pattern, students verify that the expressions are equivalent. When we have an idea for what is correct, but we need to make sure it is right, we verify it. Have you verified that (4x)3 = 4 3x 3 ? How do you know?

Yes, because the first line of work is (4x) 3 and the following lines lead to 4 3x 3 after I commute the terms in the second to last line. Students add the property (x y) n = x n y n to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 11. Add the new property to the Properties and Definitions of Exponents anchor chart for this topic and briefly discuss an example, such as (7 . 6) 2 = 7 2 . 6 2.

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Properties of Exponents Description

Property

Example

Product of powers with like bases

x m . x n = x m+n

10 5 . 10 2 = 10 5+2 = 10 7

Power of a power

(x m) n = x m . n

(5 2) 6 = 5 2 . 6 = 5 12

Power of a product

(x y) n = x n y n

(7 . 6) 2 = 7 2 . 6 2 Teacher Note

Definitions of Exponents Description

xn __ n ___ , can be n

The power of a quotient, ( xy ) =

Definition

Example

viewed as an extension of the definition of a power, so it is not included in the Properties and Definitions of Exponents graphic organizer.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12 Name

LESSON Date

More Properties of Exponents Raising Powers to Powers

Raising Quotients to Powers

Students guess and verify equivalent expressions of the form (__yx ) .

Raising Products to Powers

Students use their understanding of exponents and their properties to write an equivalent expression for (_y ) . After students examine other equivalent x n

expressions with similar structure, such as (_y ) , they verify their equivalent expression x 3

for (_yx) , for any nonzero number y.

Raising Quotients to Powers

_ __ Display (_y ) = __ n . Briefly discuss an example, such as ( ) = 6 . x n

xn y

3 6 4

36 4

169

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Land Debrief 5 min Objective: Apply properties of exponents, including raising powers to powers, raising products to powers, and raising quotients to powers. Facilitate a class discussion by asking the following prompts. How can we write an equivalent expression for (x 2)4 ?

We can write (x 2) 4 as x 4 . 2 because we can think of the power inside the parentheses as 2 factors of x, and the exponent outside the parentheses tells us we have 4 factors of x 2. This means we have 4 factors of 2 factors of x, which is 4 . 2, or 8, factors of x. How can we write an equivalent expression for (5x)4 ?

We can write (5x) 4 as 5 4x 4. Because (5x) 4 has 4 factors of 5x, we can rearrange the expression so all the 5’s are together and all the x’s are together. Then we have 4 factors of 5 multiplied by 4 factors of x, or 5 4 x 4. How can we write an equivalent expression for (_x) ? 5

The expression (_ ) represents 4 factors of _ . We can also write this as __4 , which x 4 5

x 5

x4 5

is the same as a fraction with 4 factors of x in the numerator and 4 factors of 5 in the denominator. So we can write (_ ) as __4 . x 4 5

x4 5

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Apply the properties of exponents to write an equivalent expression. 1. (w 2 ) 3

2. (3r 7 ) 4

More Properties of Exponents

3 4 r 28

3 times 2 3

(w ) = w

In this lesson, we

4 times

= w2+2+2

•

established two new properties of exponents.

•

used the properties of exponents to write equivalent expressions.

The entire term 3r 7 is raised to the fourth power.

(3r7)4 = 3r7 . 3r7 . 3r7 . 3r7

3.2

=w 2 3

(w ) is 3 factors of w 2 . This is 3 factors of 2 factors of w,

Power of a Power Property

which is 3 . 2 factors of w.

(x m)n = x m . n when

m and n are positive whole numbers

x is any number

3. ( 9__ ) 10 2

4. 23 . 53

103

912 ___

106

Power of a Product Property n

(xy) = x y when

x and y are any numbers

Both the numerator, 9 2 , and

Both factors have a power of 3, so there are

to the sixth power.

commutative and associative properties of multiplication, this expression can be written as 3 factors of (2 . 5), which is 10 3.

the denominator, 10, are raised

n n

92 6 92 92 92 92 92 92 ( 10) = 10 . 10 . 10 . 10 . 10 . 10

___ ___ ___ ___ ___ ___

3 factors of 2 and 3 factors of 5. By using the

23 . 53 = (2 . 2 . 2) . (5 . 5 . 5)

= (2 . 5) . (2 . 5) . (2 . 5)

n is a positive whole number

= (2 . 5)3

= 103

173

174

RECAP

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 12

PRACTICE

Name

Date

11. Jonas says (5n 2 ) 3 is equivalent to 5n 2 . 3. Do you agree? Explain your reasoning.

I disagree with Jonas because the expression 5n 2 is being raised to the third power. That means there will be a product of 3 factors of 5’s and 3 factors of n 2 ’s.

For problems 1–10, apply the properties of exponents to write an equivalent expression. Assume m is nonzero. 1. (3 4 ) 5

2. (a 2 ) 3

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EUREKA MATH2

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(5n 2) 3 = (5n 2) (5n 2) (5n 2) = 53n6

In problems 12–16, fill in the boxes with values that make the equation true. Assume s is nonzero. 12. (a 3 b 4 )

7 8 9

3. (6 b )

4. (8c)

6 63 b 72

8 9c 9

6. ( fg h )

d 11 g 11

f 4 g 8 h 12

38 6 7. ( __6 ) 4

1 13 8. (___2) 10

48 3___ 436

113 ___

14. ( 3

× 10 5 ) 2 = 9 × 10

10 26

10. ((p 2 ) 4 ) 6

n15 ___

p 48

278

3 4

n3 5 9. (____7) m m 35

= a 6b

r r 10 ____ 13. (____ ) = s3 6 s 5

5. (dg)

15. 34 . ( 2 )4 = 64

175

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P R ACT I C E

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16.

7–8 ▸ M1 ▸ TC ▸ Lesson 12

3x 20. The side length of a square measures ___ meters. What is the area of the square? y5 Assume y is nonzero. 4

_______ = 1_ 6 (2 )

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2 x8 3____ square meters y 10

Remember 17. Which expressions are equal to 8 24 ? Choose all that apply.

For problems 21–24, multiply. 21. − 1_ . 5 3

A. (8 20 ) 4 B. (8 12 ) 2

C. 8 8 . 8 3

22. 6(2_) 5

− 5_

12 __ 5

D. 8 14 . 8 10

23. − 10(− 2_) 3 20 __ 3

24. − 4_ . 8 7

__ − 32 7

25. There are 0.000 000 001 102 293 tons in 1 milligram.

E. 8 8 + 8 16

a. Approximate the number of tons in 1 milligram by rounding to the nearest billionth of a ton.

0.000 000 001 ton b. Write your answer from part (a) as a single digit times a unit fraction with a denominator written as a power of 10 in exponential form.

18. Which expressions are equal to x 12 y 8 z 4 ? Choose all that apply.

1 1 × ___ ton 9

A. (x 8 y 6 z) 4 3 2

B. (x y z)

26. The world population is expected to reach about 9.7 billion people in the year 2050.

C. (x 6 y 4 z 2 ) 2

a. Approximate the expected world population in 2050 by rounding to the nearest billion people.

D. (x 3 y 4 z 2 )(x 4 y 2 z 2 )

10 billion people

E. (x 6 y 5 z)(x 6 y 3 z 3 )

b. Write your answer from part (a) as a single digit times a power of 10 in exponential form.

1 × 1010 people c. The population of the United States is expected to reach about 400 million people in the year 2050. The expected world population in 2050 is about how many times as much as the expected United States population in 2050?

19. The edge of a cube measures 4n 2 inches. What is the volume of the cube?

4 3 n 6 cubic inches

The expected world population in 2050 is about 25 times as much as the expected United States population in 2050.

P R ACT I C E

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P R ACT I C E

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 12

27. Consider the rational number − _5 . 9

a. Is the decimal form of − _5 a terminating decimal? Explain how you know. 9

No. The decimal value of − _5 is not a terminating decimal. A fraction that corresponds with 9

a terminating decimal must have a denominator that can be written with only prime factors of 2, 5, or both. The denominator of − _5 has a prime factor of 3. 9

b. Write − _5 in decimal form.

− 0.¯ 5

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P R ACT I C E

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Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 13 LESSON 13

Making Sense of Integer Exponents Confirm that the definition of the exponent of 0 upholds the properties of exponents. Apply the definition of a negative exponent to write equivalent expressions.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13 Name

Date

EXIT TICKET

For problems 1 and 2, apply the definition of the exponent of 0 to write an equivalent expression. 2 0 1. (_ 3)

In this lesson, students begin by making an educated guess about

x m · x n = x m+n to determine that x 0 = 1 for all nonzero values of x and the value of a power with an exponent of 0. They use the property

apply this definition to evaluate expressions. Students also define

2. 7 ⋅ x

Lesson at a Glance

the meaning of negative exponents and relate negative exponents to multiplicative inverses. Then students analyze different strategies

x to write equivalent expressions for quotients of the form ___ n where x m

3. Use the definition of a negative exponent to write 5−7 with a positive exponent. __1 57

is nonzero.

8 . The table shows their work. 4. Kabir and Yu Yan each write an equivalent expression for __ 5

Key Questions

Kabir’s Work 4 5 89 8_____ __ = ⋅8

5 = 8 5 ⋅ 84 8

• What value does x 0 have to be to uphold the properties of exponents? Why?

Yu Yan’s Work 89 __ = 89 ⋅ 8−5 85

• What does x−n represent?

= 89+(−5)

• Can we use the properties and definitions of exponents to write

= 84

x an equivalent expression for a quotient of the form ___ where x is xn nonzero? How? m

Explain how Kabir and Yu Yan each use the properties and definitions of exponents to get the final expression. 8 is 1, so he writes his final expression as 84. to write 89 as 85 ⋅ 84 . He notices __ 5

Kabir notices he can express 89 as 85+4 . He uses the product of powers with like bases property 5

8 as 89 ⋅ 8−5. Then she uses the Yu Yan uses the definition of a negative exponent to write __ 5 8

product of powers with like bases property to write 89 ⋅ 8−5 as 89+(−5). She writes her final 8

Achievement Descriptor 7–8.Mod1.AD10 Apply the properties of integer exponents

expression as 84.

to generate equivalent numerical expressions. (8.EE.A.1)

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Agenda

Materials

Fluency

Teacher

Launch 5 min

• Blank paper (4 sheets)

Learn 30 min • Defining the Exponent of 0 • Integer Exponents • Quotients of Powers

Land 10 min

• Marker • Properties and Definitions of Exponents anchor chart (from lesson 11) • Tape

Students • Properties and Definitions of Exponents graphic organizer (from lesson 11)

Lesson Preparation • None

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Fluency Find the Multiplicative Inverse Students find the multiplicative inverse to prepare for defining negative exponents. Directions: Write the number that completes the equation. 1. 2. 3. 4. 5.

284

2·

10 ·

100 · 102 · 104 ·

1 2

1 10

1 100

=1 =1

___ 1 ___ 102

1 ___ 104

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Students make a prediction about the value of a power with an exponent of 0. Draw students’ attention to the Properties and Definitions of Exponents anchor chart. Our previous two lessons focused on these three properties of exponents. We defined these properties to work with positive whole-number exponents. What are some types of numbers we haven’t yet worked with as exponents? We haven’t worked with negative integers or 0 as exponents. Display the expression 100. Think for a moment about what the value of 100 might be. Give students time to think about a possible value. Call on students to share their predictions with the class. Anticipate student responses such as 10, 0, and 1. Write the four most common predictions on separate sheets of blank paper to create four signs. Once all predictions have been recorded, post the signs around the room. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs labeled with their predictions posted in the classroom. Invite students to stand beside the sign that they think shows the correct value of 100. When all students are standing near a sign, allow groups to discuss the reasons why they chose that sign. Then call on each group to share the reasons for their selection. Invite students who change their minds during the discussion to join a different group. Have students return to their seats. Do not reveal that 100 is equal to 1. Students continue their exploration in the next segment. Until this point, we have been working with only positive integer exponents. Today, we will define the exponent of 0 and negative integer exponents.

Teacher Note At the end of lesson 11, students think about what the product of 30 and 35 should be according to the product of powers with like bases property. They then conjecture about what the value of 30 is. Consider having students recall their conjectures before the Take a Stand activity.

Language Support During the Take a Stand routine, direct students to use the Agree or Disagree section of the Talking Tool as they discuss the reasons why they chose a specific sign.

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Learn Defining the Exponent of 0 Students test their predictions about the value of a power with an exponent of 0. Once students return to their seats, engage them in a discussion about the exponent of 0 by using the following prompts. Can we think about 100 in the same way that we think about 103? Explain.

No. We can think of 103 as 3 factors of 10, or 10 ∙ 10 ∙ 10, but we don’t know how to think about 0 factors of 10.

When we define a power with an exponent of 0, we want to make sure the definition works with our existing properties of exponents. That way, we don’t need to learn a new set of properties. At the beginning of class, you made predictions about the value of 100. Let’s test some predictions to see whether they uphold the properties of exponents. Could 100 = 10? Let’s determine whether the property x m · x n = x m+n holds if 100 = 10.

Using the property x m · x n = x m+n, what is 100 ⋅ 103? Why?

100 ∙ 103 = 103 because 0 + 3 = 3.

Let’s test the idea that 100 is equivalent to 10 by substituting 10 for 100. Display the following statement: If 100 = 10, then 100 ∙ 103 = 10 ∙ 103 = 10 ∙ 10 ∙ 10 ∙ 10 = 104. What do you notice? The property does not hold if 100 = 10 because 103 does not equal 104. So 100 is not equal to 10. Invite students to think–pair–share about the following questions. Could 100 = 0? Explain.

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No. By the product of powers with like bases property, we know that 100 ∙ 103 = 103. If 100 = 0, then 100 ∙ 103 equals 0 ∙ 103, which is 0. But we know that 103 does not equal 0. So 100 does not equal 0. Could 100 = 1? Explain.

Yes. By the product of powers with like bases property, we know that 100 ∙ 103 = 103. If 100 = 1, then 100 ∙ 103 equals 1 ∙ 103, which is 103. We just found that the product of powers with like bases property holds when we define 100 as 1. Let’s see whether our definition works for any nonzero base. Display x 0 ∙ x n. Use the following statements to show that the definition x 0 = 1 upholds the property x m ∙ x n = x m+n for any nonzero x. By the product of powers with like bases property, x 0 · x n = x 0+n = x n.

If x = 1, then the property holds because x · x = 1 · x = x . 0

So the definition of a base with an exponent of 0 is x 0 = 1 for any nonzero x.

Instruct students to add this definition to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 11. Add the new definition to the Properties and Definitions of Exponents anchor chart for this topic.

Properties of Exponents Description

Property

Power of a power

(x m)n = x m ⋅ n

105 ⋅ 102 = 105+2 = 107

Power of a product

(xy)n = x ny n

(7 ⋅ 6)2 = 72 ⋅ 62

The definition x 0 = 1 applies for any nonzero x. Students may wonder what happens when x is equal to 0. The number 00 is considered indeterminate, which means its value cannot be determined. To help students see why, show them the following patterns and ask them what they think the value of 00 should be based on the patterns. Power

Value

Power

Value

Example

x m ⋅ x n = x m+n

Product of powers with like bases

UDL: Representation

(52)6 = 52 ⋅ 6 = 512

Students will likely answer that the value should be 1 based on the first table and 0 based on the second table. This shows students that two valid patterns can result in two possible values for 00, which is why its value cannot be determined.

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Definitions of Exponents Description

Definition

Base with an exponent of 0

x0 = 1

Example 10 0 = 1

x is nonzero

Invite students to complete problem 1 with a partner. Circulate as students work, and ask them to explain their choices. 1. Circle all expressions that have a value of 1. Assume x is nonzero.

(−6)0

0 ⋅ 60

60 __ 6

_1 0

(6)

10 __

6x0

(6 x)0

When students are finished, consider inviting them to share their choices and reasoning with the class.

Integer Exponents Students define negative exponents. Engage the class in a discussion about negative exponents by using the following prompts. We know that the properties of exponents work for exponents that are positive whole numbers or 0. Let’s see whether the properties work for negative exponents.

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Display the expression 104 ⋅ 10−4.

If we apply the product of powers with like bases property, what should the value of 104 ⋅ 10−4 be? Why?

The value of 104 ⋅ 10−4 should be 1. By using the product of powers with like bases property, 104 ⋅ 10−4 should equal 104+(−4), which is 100, or 1.

Display the equation 104 ⋅ 10–4 = 1.

Are 104 and 10–4 multiplicative inverses? How do you know?

Teacher Note Clarify that 10−4 is not equivalent to −104. To demonstrate this, ask students to evaluate 104 ⋅ (−104). They will find that 104 ⋅ (−104) is −108, which is not 1.

104 ⋅ (−104) = 104 ⋅ (−1) ⋅ 104 = −1 ⋅ (104 ⋅ 104) = −1 ∙ 108

Yes. Their product equals 1.

Display the unknown factor equation 104 ⋅

= −108

= 1.

Think about reciprocals. What is another way to write the multiplicative inverse of 104? 1 . Another way to write the multiplicative inverse of 104 is ___ 4

1 = 1. Display the equation 104 ⋅ ___ 4

1 If ___ and 10−4 are both multiplicative inverses of 104, what does that tell us about 104 −4

1 and ___ ? 104

1 . It tells us that we can write 10−4 as ___ 4 10

If we want the properties of exponents to be true for negative exponents, then 10−4 1 must be equivalent to ___ . 104

1 Display the expression 10−4 = ___ . 4 10

Let’s make a general rule for a base that has a negative exponent.

Display the expression x 4 ⋅ x−4.

Consider a nonzero base x. If we want the properties of exponents to work for negative exponents, then what should the value of x 4 ⋅ x−4 be? Why?

The value of x ⋅ x should be 1. By using the product of powers with like bases property, x 4 ⋅ x−4 should equal x 4+(−4), which is x 0, or 1. 4

−4

Language Support Students work with reciprocals in grade 6 when dividing fractions. To activate this prior knowledge, consider providing students with examples, such as the following:

__ 5 3 is __ 2. • The reciprocal of __ • The reciprocal of 5 is 1 . 2

Also consider providing students with the following sentence frame: • The reciprocal of

Offer more examples and allow students to work with a partner to appropriately complete the sentence frame.

Teacher Note If students question why the base must be nonzero, have them write 0−4 as an expression with a positive exponent. The

expression 1 has a 0 in the denominator, 04

which is undefined. © Great Minds PBC

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Because the product is 1, what is the relationship between x 4 and x−4?

Teacher Note

The expressions are multiplicative inverses. How can we write x−4 with a positive exponent? Explain. −4

We can write x multiplicative inverses of x 4.

upholds the property x m ⋅ x n = x m+n,

of a negative exponent supports the other properties in the following ways.

• Power of a power property: (x m)n = x m ⋅ n

Then display x n ⋅ x−n. 4

(104)−1 = 104(−1) (10,000)−1 = 10−4

Now consider an expression with integer exponents n and −n. What is the relationship between x n and x−n?

1 = ___ 1 _____

How can we write x−n with a positive exponent? x

We can define a negative exponent for any nonzero x and any integer n as x

−n

1 = __ .

x 1 is equivalent to __ , which is the multiplicative inverse, or reciprocal, of x n.

Why must x be nonzero?

The value of x cannot be 0 because x is in the denominator and we cannot divide by 0. Invite students to think–pair–share about the following question.

1 Can we write ___ with a positive exponent? How? x−n 1 1 as x n because the expression ___ 1 is equivalent to 1 ÷ __ Yes. We can write ___ n , which x−n

104

10,000

• Power of a product property: (x y)m = x m y m

1 We can write x−n as __ n.

equals 1 ⋅ x n, or x n.

10,000

1 = _____ 1 _____

The expressions are multiplicative inverses.

So x

104

consider showing them that the definition

1 The expressions x and __ are equivalent because they are both multiplicative x4 1 inverses of x 4. So we can write x−4 with a positive exponent as __ . x4 1 Display x −4 = __ . −4

−n

___

After students determine that 10−4 = 1

1 because both x−4 and __ 1 are with a positive exponent as __

x−n

(5 ⋅ 2)−4 = 5−4 ⋅ 2−4

(10)−4 = 5−4 ⋅ 2−4 1 = __ 1 ⋅ __ 1 ___ 104

10,000

625

10,000

1 1 = ___ 1 ⋅ __ _____ 16

1 = _____ 1 _____

Consider adding examples that include negative exponents to the Properties and Definitions of Exponents graphic organizer.

Have students add the definition of a base with a negative exponent to the next blank row in their Properties and Definitions of Exponents graphic organizer from lesson 11. Add the new definition to the Properties and Definitions of Exponents anchor chart for this topic.

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Properties of Exponents Description

Property

Example

Product of powers with like bases

x m ∙ x n = x m+n

105 ∙ 102 = 105+2 = 107

Power of a power

(x m)n = x m ⋅ n

(52)6 = 52 ⋅ 6 = 512

Power of a product

(x y)n = x n y n

(7 ∙ 6)2 = 72 ∙ 62

Promoting the Standards for Mathematical Practice Students attend to precision when they write an equivalent expression for an expression that has a negative exponent (MP6). Ask the following questions to promote MP6: • How can we write the division expression 1 ÷ 10−4 with a positive exponent? • How are you using the definition of a negative exponent when you write an equivalent expression for 1 ?

____ −4

Definitions of Exponents Description

Definition

Base with an exponent of 0

x0 = 1 x is nonzero 1 x−n = __ n

Base with a negative exponent

Example 0

10 = 1

1 10−4 = ___ 4

1 = xn x−n

1 ____ = 104

x is nonzero

___

x is nonzero

Teacher Note In future grades, students explore rational exponents.

10−4

Now, our first three properties work for exponents that are positive integers, zero, and negative integers. In other words, they work for all integer exponents.

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Have students complete problems 2–5 with a partner. Circulate as students work, and refer them to the definition of a negative exponent if needed. For problems 2–5, use the definition of a negative exponent to write an equivalent expression with a positive exponent. Assume x is nonzero. 2. 10−7 1 ___

3. (−5)−9 1 ____

(−5)9

4. x−8

1 5. ____ 10−2

__1

1 ____ = 1 ÷ 10−2

UDL: Representation To emphasize the difference between negative exponents and negative bases, have students highlight the base in one color and the exponent in another color in each expression. Help students recall that a negative base means the repeated factor in a related expression is negative. A negative exponent represents the multiplicative inverse. Consider posting a visual with color-coding and labeling for students to refer to throughout the topic.

10−2

1 = 1 ÷ ___ 2 10

Differentiation: Support

102 1

If students need support with problem 5,

point out that a fraction can be represented

= 1 ⋅ ___ = 10 Confirm answers when students are finished.

Quotients of Powers Students evaluate quotients of powers by using the properties and definitions of exponents. Direct students to problem 6. Have them study So-hee’s work before engaging in a class discussion.

____ as

as division. So they can think of 1 1 . 1 ÷ 10 , or 1 ÷ ___ −2

10−2

Differentiation: Challenge If students complete problems 2–5 early or need additional challenge, consider asking them to write an equivalent expression with positive exponents for the following expression.

____

3x −6 y −5

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10 6. So-hee applies the properties of exponents to write an equivalent expression for ___ . Her 107 work is shown. 3

103 _______ 103 ___ =

107

103 ⋅ 104

1 = ___3 ⋅ ___ 4 103

1 = ___ 4

Use the Five Framing Questions routine to invite students to analyze So-hee’s strategy. Encourage students to add to their classmates’ responses.

Notice and Wonder I notice that So-hee wrote 107 as 103 · 104. What do you notice about this work?

I notice that she wrote the fraction so the numerator and denominator each have a factor of 103.

___ is 1. I notice that 10 3 3

I notice that the difference between the exponents of the denominator and the numerator is the exponent of the answer. I wonder why she wrote 107 as 103 · 104.

From your observations, what do you wonder? 10 is 1. I wonder why she skipped the step that shows ___ 3 3

I wonder why she didn’t use a negative exponent instead and write 10−7 because 107 is in the denominator.

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Organize What steps did So-hee take? How do you know? So-hee noticed 103 in the numerator and decided to write the denominator as a product 10 is 1. When she wrote the denominator where one factor is 103. She did this because ___ 3 3

by using a factor of 103, she had to include a factor of 104 because 103 · 104 is 103+4, or 107. 10

1 1 because 1 ⋅ ___ 1 . She got the answer of ___ is ___ 4 4 4

So-hee could have written the denominator as 105 ⋅ 102. Why didn’t she? 10

So-hee didn’t write the denominator as 105 · 102 because she wanted to make a fraction that is equal to 1 by using the factor of 103 in the numerator. So she wrote the denominator as 103 · 104.

Have students complete problem 6 independently. Circulate as students work. If needed, suggest that they write the fraction as a whole number times a unit fraction. Identify students to share their work with the class. Apply the properties and definitions of exponents to verify So-hee’s answer. 103 1 ___ = 103 ⋅ ___

107

= 103 ⋅ 10−7

= 103+(−7) = 10−4 1 = ___ 4 10

When students are finished, invite identified students to share their work. Advance the discussion to focus on the definition of a negative exponent by encouraging student thinking that makes connections between multiplicative inverses and negative exponents.

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Reveal Let’s focus on the definition of a negative exponent. Where do you see that in your work?

1 as 10−7 and 10−4 My work shows the definition of a negative exponent when I wrote ___ 7 1 . as ___ 4

Distill What difference does using the definition of a negative exponent make when 10 evaluating ___ ? Explain.

Teacher Note

It makes a difference in the work, but we still get the same answer. When So-hee

Writing a quotient of powers with like

107

103 in her work, she gets the same answer as if she had applied the definition evaluates ___

of a negative exponent to write the expression as 103 · 10−3. 103

10 By using the definition of a negative exponent, we can write ___ as 103 · 10−7. Then 7 3

we can apply the product of powers with like bases property. We can use the definition 1 . of a negative exponent to write the answer of 10−4 with a positive exponent as ___ 4

___m

bases, x n , as x m−n is an application of the x

definition of a negative exponent and the product of powers with like bases property.

x m = x m ⋅ x−n = x m+(−n) = x m−n ___ n x

So it is not included in the Properties and Definitions of Exponents graphic organizer.

Know How is using the properties and definitions of exponents helpful for writing an equivalent expression when compared with writing out all the factors of 10 as we did in previous lessons? We can write equivalent expressions much faster by using the properties and definitions of exponents than by writing out all the factors of 10. Have students complete problems 7 and 8 with a partner. Circulate as students work, and advance their thinking by asking the following questions: • In a quotient of powers with like bases, if the exponent is greater in the denominator, will the answer have a positive exponent in the denominator? Why? • Can we subtract the exponents in a quotient of powers with like bases? Why?

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For problems 7 and 8, apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume d is nonzero. 813 7. ___ 821

813 ___ = 813 ⋅ 8−21 21

= 813+(−21) = 8−8 1 = __ 88

7d 4 8. ____7 14d

7d 4 7 d4 ____ = __ ⋅ __

14d 7

14 d 7

= __ ⋅ d 4 ⋅ d −7 7 14

= _1 ⋅ d 4+(−7) 2

= _1 ⋅ d −3 2

1 = ___ 3

Differentiation: Support A common error is to apply the properties and definitions of exponents to the coefficient of the expressions. For extra support, consider having students complete the following problems individually before completing problem 8. •

__7 14

4 • d 7 d d4 • 7 ⋅ 14 d7

__ __

Confirm answers when students are finished.

Differentiation: Challenge If students need additional challenge, consider asking them to show that x 0 = 1

___m

upholds x n = x m−n. x

___m

We know that x m = x m−m, so x m = x 0. We also

___m

know that x m = 1 because the numerator x

and denominator are the same number.

___m

So x 0 = 1 upholds x n = x m−n. x

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7–8 ▸ M1 ▸ TC ▸ Lesson 13

Land Debrief 5 min Objectives: Confirm that the definition of the exponent of 0 upholds the properties of exponents. Apply the definition of a negative exponent to write equivalent expressions. Facilitate a class discussion by asking the following questions. Encourage students to restate or build upon one another’s responses. What value does x 0 have to be to uphold the properties of exponents? Why?

The value of x 0 for any nonzero x has to be 1 to uphold the properties of exponents. For example, by the product of powers with like bases property, x 0 ⋅ x m = x 0+m, or x m. If x 0 = 1, then x 0 ⋅ x m = x m. What does x−n represent?

1 . The power x−n represents the multiplicative inverse of x n, which is __ n x

Can we use the properties and definitions of exponents to write an equivalent x expression for __ ? How? 3

x as x 3 ⋅ x −8, which Yes. We can use the properties and definitions of exponents to write __ 8 x 1 −5 is x , or __5 . x8

x as _____ x 1. Yes. We can use the properties of exponents to write __ , which is __ 5 8 3 5 3

Exit Ticket 5 min

x ⋅x

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

1 2. Without using the definition of a negative exponent, how can we show 10−5 = ___ ? 5 10

By using the product of powers with like bases property and the definition of the exponent of 0, we know 105 ⋅ 10−5 = 105+(−5) = 100 = 1. 1 We also know 105 ⋅ ___ = 1. 5

Making Sense of Integer Exponents

1 are both multiplicative inverses This means 10−5 and ___ 5

In this lesson, we •

used the product of powers with like bases property to determine that x 0 = 1.

•

related negative exponents to multiplicative inverses.

•

learned the definition of a negative exponent.

•

applied the definition of a negative exponent to write equivalent expressions.

1 . of 105. So 10−5 = ___ 5

Two factors that have a product of 1 are multiplicative inverses.

Definition of the Exponent of 0

For problems 3 and 4, use the definition of a negative exponent to write an equivalent expression with a positive exponent.

x0 = 1 x is nonzero

3. 17−4 1 ___

17 4

Definition of a Negative Exponent x –n = 1n

1 ___

A fraction represents division.

−3

1 ___ = 1 ÷ 5−3 5−3

1 = 1 ÷ __ 3 5

= 1 ⋅ 5__

when

= 53

x is nonzero

n is an integer

Examples 1. Write an equivalent expression for 5 c 0. Assume c is nonzero.

5c 0 = 5 ⋅ c 0 =5⋅1 =5

298

The exponent of 0 applies only to the base c, not to the entire expression 5c.

185

186

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

5. Apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume r is nonzero. 5 2 r5 2 ___ ____ = __ · r

16 r 11

2 = __ ⋅ r 5 ⋅ r −11 16

= 1_ ⋅ r 5+(−11) 8

= 1_ ⋅ r −6 8

1 = ___ 6 8r

To use the product of powers with like bases r as r 5 ⋅ r −11. Then use the property, write ___ 11 5

definition of a negative exponent to write the answer with a positive exponent.

2 r5 ____

16 r 11

5 2 r5 = 2 ⋅ r 16 r 11 16 r 11

____ _ ___

2 ___ =_ ⋅ 5r 6 5

16 r ⋅ r

r __ = 1_ ⋅ __ ⋅ 16 5 5

8 r

= 1 ⋅ 16 8 r

_ __

1 = ___ 6 8r

r , To make a quotient of __ 5

or 1, write r

as r ⋅ r . 5

RECAP

187

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 13 Name

PRACTICE Date

For problems 1–15, apply the properties and definitions of exponents to write an equivalent expression with only positive exponents. Assume all variables are nonzero. 1. 20

3 0 4. (− __ f 10 )

7. x −10 1 ___ x 10

2. 110 ⋅ 113

113

5. (−4)2 (− 4)0

(− 4)2

8. ( h0 k)8

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

For problems 16–19, apply the properties and definitions of exponents to determine the value of t.

16. 80 ⋅ 82 = 8t

0 3 t 11 17. (3_) (_ = (3_) 8) 8 8

18. 25 ⋅ 20 = t

19. (− 0.25)0 (− 0.25)t = 1

3. 10−5 1 ___ 105

2 0 2_ 0 6. (_ 9 ) (9 )

For problems 20 and 21, write a number in the box to make the equation true. 20. (− 2)0 (− 2)

9. 6 ⋅ 20 ⋅ 63 ⋅ 60 ⋅ 2 ⋅ 22

= (− 2) 5

1 1_ 0 _____ 21. (_ = 1 4 )(4 )

64 ⋅ 23

22. Choose the expression that has a value of 9,804.

10. (− 6) −3 1 ____ (−6)3

13.

1 ___

f −3

___

8 11. 105 10

103

12. 6 ⋅ 6 −5

B. 9 × 104 + 8 × 103 + 4 × 101

__1

C. 9 × 104 + 8 × 103 + 4 × 100

D. 9 × 103 + 8 × 102 + 4 × 100

3 14. __ 36

8t 15. ___ 24t 9

3t7

__1

A. 9 × 103 + 8 × 102 + 4 × 101

23. What is the value of 6 × 104 + 3 × 102 + 2 × 100?

A. 632 B. 6,032

1 ___

C. 6,320 D. 60,302 E. 60,300

300

189

190

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 13

24. Maya says 10−5 is equivalent to (− 10)5. Do you agree? Explain your reasoning. 100,000

4 2 8 31. (2_) (_ 7) 7

(− 10)(− 10)(− 10)(− 10)(− 10), or − 100,000. So the expressions 10−5 and (− 10)5 are not equivalent.

_2 12

(7 )

1 ? 25. Which expression does not have a value of __ 16

411 A. ___ 413 4 B. __ 42 C. 4−2

32. 9a ⋅ 9b

2 D. __ 29

9a+b

26. Order the values from least to greatest.

(−1)5 5

(− 1) , 0.1,

EUREKA MATH2

For problems 31 and 32, apply the product of powers with like bases property to write an equivalent expression.

1 , or ______ 1 I disagree with Maya. The value of 10−5 is ___ , but the value of (− 10) 5 is 5 10

7–8 ▸ M1 ▸ TC ▸ Lesson 13

5−1 5−1,

100,

10 0

1 ___

0.1

4−1

2 , ___ 1 , 3−6 ⋅ 38 __

3−6 ⋅ 38

2__6

33. Explain what the expression 53 ⋅ 57 represents and how it can be written as a single power.

The expression represents the product of 3 factors of 5 and 7 factors of 5. That is a total of 10 factors of 5, which can be written as 510.

4−1

Remember For problems 27–30, multiply. 6 ⋅ 5_ 27. _ 5 2

__ 28. − 3_ ⋅ 16 4 6 −2

__ −__6 29. (− 10 3 )( 20 ) 1

__ 30. − 4_ ⋅ 21 7 2

17 in decimal form. 34. Write __

−6

1.0625

P R ACT I C E

191

192

P R ACT I C E

301

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 14 LESSON 14

Writing Very Large and Very Small Numbers in Scientific Notation Write numbers given in standard form in scientific notation. Order numbers written in scientific notation.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14 Name

Date

EXIT TICKET

For problems 1 and 2, write the number in scientific notation. 1. 60,630,000

6.063 × 107

2. 0.0051

5.1 × 10−3

For problems 3 and 4, write the number in standard form.

Lesson at a Glance This lesson opens with a historical context that addresses the need for using scientific notation. Students learn the definition of scientific notation and leverage their prior knowledge of place value, powers of 10, and the definition of a negative exponent to write numbers in scientific notation. In a partner activity that involves moving around the room, students determine whether their number written in scientific notation is less than or greater than a given number. Then students determine their number’s place in an ordered set. During this activity, students briefly experience negative numbers written in scientific notation and their place in the order of a set of numbers. This lesson formally defines the terms scientific notation and order of magnitude.

3. 3 × 10−5

0.00003

Key Questions • Can you tell when a number is written in scientific notation? How?

4. 3.4 × 1010

• What details are important to think about when ordering positive numbers written in scientific notation?

34,000,000,000

Achievement Descriptor

5. Order the numbers from least to greatest.

6.0 × 108, 9.2 × 10−15, 5.1 × 109, 8.4 × 109, 7.4 × 10−10, 6.8 × 10−15

7–8.Mod1.AD13 Approximate and write very large and very small

6.8 × 10−15, 9.2 × 10−15, 7.4 × 10−10, 6.0 × 108, 5.1 × 109, 8.4 × 109

numbers in scientific notation. (8.EE.A.3)

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7–8 ▸ M1 ▸ TC ▸ Lesson 14

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Blank paper (2 sheets)

Learn 30 min • Another Way to Represent Numbers

• Marker • Tape

• Small Positive Numbers

Students

• Ordering Numbers in Scientific Notation

• Scientific Notation Ordering Cards (1 card per student pair)

Land 10 min

Lesson Preparation • Use the paper to prepare two signs with the following labels: Less Than and Greater Than. Post the Less Than sign at the front, left corner of the room and post the Greater Than sign at the front, right corner of the room. • Copy and cut one set of Scientific Notation Ordering Cards (in the teacher edition). Separate the four cards that are shaded to use for display.

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Fluency Order Numbers Students order numbers from least to greatest to prepare for ordering numbers written in scientific notation. Directions: Place the following numbers in order from least to greatest. 1.

4.5, 45, 0.45, 54, 0.54, 5.4

0.3, 0.03, 0.333, 3.003, 0.003, 0.33

− 0.022, 0.202, − 0.22, 0.022, − 2.2, 2.02

304

0.45, 0.54, 4.5, 5.4, 45, 54 0.003, 0.03, 0.3, 0.33, 0.333, 3.003 − 2.2, − 0.22, − 0.022, 0.022, 0.202, 2.02

EUREKA MATH2 7–8 ▸ M1 ▸ TC ▸ Lesson 14

Launch

Students identify different representations of numbers through the history of Archimedes. Direct students to the historical information on Archimedes. Give students about 2 minutes to read silently and complete problems 1 and 2. Archimedes was a Greek mathematician whose ideas were ahead of his time. He lived in Sicily during the third century BCE and was the first to develop fundamental concepts in geometry, calculus, and physics. Fascinated by very large numbers and living on the coast of the Ionian Sea, Archimedes set out to determine how many grains of sand are needed to fill the known universe.

Teacher Note In addition to this activity, consider incorporating the Math Past material provided in the Math Past resource.

Language Support To offer additional support, consider the following strategies: • Read the passage aloud to a small group of students. • Pair students and have partners take turns reading aloud. • Provide highlighters for students to visually emphasize key ideas. • Summarize the text as a class before assigning problems 1 and 2.

Teacher Note The known universe for Archimedes is not the known universe today. The Math Past materials provide a description of the differences.

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7–8 ▸ M1 ▸ TC ▸ Lesson 14

Mathematicians used Ionic Greek notation in the third century BCE. So instead of numbers, they used symbols and letters from the Greek alphabet. name

symbol

value

name

symbol

value

name

symbol

value

alpha

iota

rho

100

beta

kappa

sigma

200

gamma

lambda

tau

300

delta

upsilon

400

epsilon

phi

500

digamma

chi

600

zeta

omicron

psi

700

eta

omega

800

theta

koppa

sampi

900

1. In this system, χνδ represents the number 654. What does σπε represent?

285 2. What does ψοζ represent?

777

306

EUREKA MATH2

Review the answers to problems 1 and 2, and then discuss the following questions as a class. What is challenging about using Ionic Greek notation? The ancient Greeks used many more characters to represent numbers. We just use

10 digits, 0 through 9.

We can’t express very large numbers by using Ionic Greek notation. Display the value 3 × 106. Can we write this number in standard form? How? Yes. 3,000,000 Can we write this number in Ionic Greek notation? Have students work with a partner to try to write the number in Ionic Greek notation. Give students time to engage in a productive struggle. This number is impossible to write in Ionic Greek notation given the limited scope of that number system. Were you and your partner able to write the number 3,000,000 in Ionic Greek notation? Explain.

7–8 ▸ M1 ▸ TC ▸ Lesson 14

UDL: Engagement To foster student interest in mathematical history, consider reading the story of the death of Archimedes to the class. Because the exact details of his death are not confirmed, this story is one of many theories about how Archimedes died during the Second Punic War. Archimedes’s native city of Syracuse, Italy, was captured by Roman forces in the year 212 BCE. While the city was under siege, Archimedes was drawing a diagram of circles in the sand.

Archimedes’s Twin Circles

We were not able to write the number in Ionic Greek notation because the system cannot represent such a large number. What is the largest number we can write by using the Greek symbols in the table? The largest number we can write by using the Greek symbols is 999. Archimedes wanted to know how many grains of sand are needed to fill the known universe. Is the number 999 useful to estimate the number of grains of sand needed to fill the universe? Explain. No. It takes far more than 999 grains of sand to fill the universe. Consider engaging students in additional discussion about the number of grains of sand needed to fill the universe by asking the following questions: • Would finding the number of grains of sand needed to fill the known universe be an interesting task in Archimedes’s day? • How many grains of sand do you think are needed to fill the universe?

As he contemplated his work, the 75-year-old mathematician was approached by a Roman soldier demanding that Archimedes meet immediately with the general of the Roman Army. Archimedes refused, insisting that he would not go until he finished his math problem. This infuriated the Roman soldier, and he pulled out his sword and killed Archimedes on the spot. Archimedes’s last words are thought to be “Do not disturb my circles.” His work would go unknown until the year 530 CE, 742 years after his death.

• Is there a number that is too high? What is a number that is too low?

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Archimedes had to develop a new way to write very large numbers. He called this new notation a myriad, represented by the capital letter M. He used the symbol M for the number 10,000. With his new notation, Archimedes was able to use myriads to estimate the number of grains of sand needed to fill the known universe. He arrived at the value of 1 followed by 63 zeros. How would we represent that value as a single digit times a power of 10?

1 × 1063 Which is more efficient to write: 1 followed by 63 zeros, or 1 × 1063? It is more efficient to write 1 × 1063 because it would take much longer to write 1 followed by 63 zeros. Archimedes developed the myriad when he recognized the limitations of Ionic Greek notation. Similarly, scientists developed scientific notation, which is a condensed way of writing large and small numbers in our current number system. In this lesson, we will learn how to write numbers in scientific notation.

Learn

Language Support

Another Way to Represent Numbers Students identify and write large numbers in scientific notation. Introduce students to the term scientific notation by displaying the following number with the general expression written directly underneath.

6.02 × 103 a × 10 n Examine the structure of the two expressions. What do you notice? I notice the structure is very similar. Both expressions have a power of 10. The power of 10 is multiplied by a number in the first expression and a variable in the second expression. 308

Consider placing new terms alongside the expression a × 10 n to illustrate the significance of the new definitions.

First Factor

a × 10 n

Order of Magnitude

To solidify the concept, provide examples of numbers written in scientific notation and ask students to identify the first factor and the order of magnitude of each number.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. Numbers written in scientific notation are in the form a × 10 n. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10. The exponent n, which we call the order of magnitude, is the exponent on the power of 10 for a number written in scientific notation. Display the following number.

6.02 × 103 What is the first factor and the order of magnitude of this number? The first factor is 6.02, and the order of magnitude is 3. How does scientific notation relate to how we wrote very large numbers in previous lessons? We wrote large numbers as a single digit times a power of 10. With scientific notation, the first factor does not have to be a single digit, but the absolute value of the first factor needs to be at least 1 but less than 10. Have students work together in pairs to record these definitions in problems 3 and 4. 3. Fill in the blanks to complete the statements. A number is written in scientific notation when it is represented as a number a multiplied power by a of 10 . The general expression that represents a number written in scientific notation is a × 10n . The absolute value of a must be at

least 1

but

less

than

4. Identify the first factor and the order of magnitude of the expression 8.86 × 106. The first factor is 8.86. The order of magnitude is 6. Begin a short discussion to prepare students for problem 5. Display the number − 7.1 × 103 and ask the class the following questions.

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What is the first factor of the expression? The first factor of the expression is − 7.1. What is the absolute value of the first factor? The absolute value of − 7.1 is 7.1. Is the number −7.1 × 103 written in scientific notation? How do you know? The number − 7.1 × 103 is written in scientific notation. It is written in the form a × 10 n, and the absolute value of the first factor is at least 1 but less than 10. What is the order of magnitude? The order of magnitude is 3. As we have seen from the number −7.1 × 10 3, numbers written in scientific notation do not always need to be positive. A negative number can also be written in scientific notation because it can also be written in the form a × 10 n where a represents the first factor with an absolute value of at least 1 but less than 10 and n represents the order of magnitude. Have students complete problem 5 individually. Encourage students who finish early to share their choices with a partner. 5. Indicate whether each number is written in scientific notation. Number

Yes

0.38 × 102 4.8 × 103

X X

19.04 × 107

310

Encourage students to identify the first factor and the order of magnitude of each number in problem 5. Highlight the corresponding values by using different colors to show students the parallel structure between the examples and the general form of a number written in scientific notation, a × 10 n.

− 6.75 × 106

1 × 109

10 × 1010

UDL: Representation

EUREKA MATH2

Review students’ reasoning by using the following prompt. Identify one number from the list that is written in scientific notation. Explain how you know. Sample: The number 4.8 × 10 3 is written in scientific notation because 4.8 represents the first factor, which is multiplied by a power of 10. The absolute value of 4.8 is at least 1 but less than 10.

7–8 ▸ M1 ▸ TC ▸ Lesson 14

Language Support Have students use the Agree or Disagree section of the Talking Tool as they discuss whether the numbers in problem 5 are written in scientific notation.

Have pairs of students identify as partner A and partner B, and share the following instructions with the class: • Partner A identifies a number from the list that is not written in scientific notation and explains their reasoning. • After partner A finishes explaining, partner B has a chance to agree or disagree and explain reasoning as needed. • When the partners agree on a number that is not written in scientific notation, partner B then writes the number in scientific notation on their personal whiteboard. • Once the partners agree that the number is written in scientific notation, the partners switch roles. Identify one number from problem 5 that is not written in scientific notation. Explain how you know. Sample: The number 19.04 × 107 is not written in scientific notation because the first factor has an absolute value of 19.04, which is greater than 10. What was your process for writing the number in scientific notation? I wrote the number in standard form and then wrote the number in scientific notation. I wrote the first factor to have an absolute value of at least 1 but less than 10 and thought about how that affects the power of 10.

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Have students complete problems 6–11 with a partner. Circulate to confirm answers or correct miscalculations. For problems 6–11, complete the table. Number in Standard Form

Number in Scientific Notation

400,000

4 × 105

20,000

2 × 104

− 27,000

− 2.7 × 104

2,350,000

2.35 × 106

10.

− 525,000

− 5.25 × 105

11.

2,050,000

2.05 × 106

Small Positive Numbers Students recognize and write small positive numbers in scientific notation. Direct students to the table in problem 12 and ask students to recall the activity they did with very large and very small numbers in lesson 10. Then ask them the following questions. We completed this table in a previous lesson when discussing small numbers. Now we are able to write these numbers in scientific notation. What is the definition of scientific notation? A number is written in scientific notation when it is represented as a number a, which has an absolute value of at least 1 but less than 10, multiplied by a power of 10.

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1 Look at the row with the width of a grape. Is ___ a power of 10? Why? 102

No. There is a power of 10 in the denominator, but the whole expression is not a power of 10. 1 as _1_ , which is a power of __ 1 , not a power of 10. No. I can write ___ ( ) 2 10

10 1

Can we write ___ as a power of 10? How?

102 Yes. I can write 12 as 10−2 by using the definition of a negative exponent. 10

___

Direct students to complete the table with a partner by using the definition of a negative exponent. 12. Complete the table. Approximate Measurement (meters) Single Single Digit Digit Times Times a Unit Scientific Fraction a Unit Fraction Notation Fraction (exponential (expanded form) form)

Objects

Standard Form

Unit Form

Grape (width)

0.03

3 hundredths

3 _

1 3×_

1 3 × ___ 2

3 × 10−2

Grain of Rice (length)

0.006

6 thousandths

6 _

1 6×_

1 6 × ___ 3

6 × 10−3

Human Hair (width)

0.00008

8 hundred

1 8×_

1 8 × ___ 5

8 × 10−5

thousandths

100

1,000

8 100,000

100

1,000

100,000

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Call the class back together and confirm answers. Use the following prompts to engage students in a discussion about writing numbers in scientific notation or standard form. Look more closely at the number 6 × 10−3. What do you notice about the power of 10 in scientific notation and the unit of the number in unit form? I notice that the power of 10 is 10−3 and the unit of the number is thousandths. How do we know 10−3 represents thousandths?

1 1 1 = ____ Because 10−3 = ___ and because ___ , we know 10−3 represents thousandths. 3 3 10

1,000

What is the order of magnitude? What does it tell us about the number 6 × 10−3 when written in standard form? The order of magnitude is − 3, and it tells us that 6 is in the thousandths place when written in standard form. We can use the power of 10 to determine the highest place value of the number written in standard form. For example, the order of magnitude of 8 × 10−5 is −5, which tells us that the digit 8 in the first factor is in the hundred-thousandths place when written in standard form. Have students work with a partner on problems 13–18. For problems 13–18, complete the table.

314

Number in Standard Form

Number in Scientific Notation

13.

0.0007

7 × 10−4

14.

0.0002

2 × 10−4

15.

0.00023

2.3 × 10−4

16.

0.000 0023

2.3 × 10−6

17.

0.000 0062

6.2 × 10−6

18.

0.000 030 18

3.018 × 10−5 © Great Minds PBC

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

Display the completed table and allow students to check their answers. Then use the following prompts to facilitate a discussion about number representations. In problem 16, why is the order of magnitude −6 if the number written in standard form has 7 digits after the decimal point? The order of magnitude of − 6 tells us that the first nonzero digit of our number is in the 10−6 place value when the number is written in standard form. It does not tell us the place value of the last digit of the number in standard form. In problem 16, the order of magnitude of −6 indicates that the first nonzero digit of the number is found 6 places after the decimal point. So 2 in 0.000 0023 is in the sixth place after the decimal point. Looking only at the numbers in the table written in scientific notation, how can we tell that these numbers are small positive values? The powers of 10 have negative exponents, so the orders of magnitude are negative. That tells us the first nonzero digit of each number is found after the decimal point. Because the first factors are positive, the numbers are between 0 and 1.

Teacher Note Students might ask whether the first digit of a number with a positive order of magnitude n is found n places before the decimal point. Consider providing students with the following example.

2 × 103 = 2 × 1,000 = 2,000

All the numbers in problems 13–18 have a negative order of magnitude. Why are none of the numbers negative when written in standard form? All the numbers have a positive first factor, so the numbers are positive. The order of magnitude just helps us find the highest place value of the number written in standard form. The order of magnitude does not affect the sign of the number. If the number in problem 13 is −0.0007 instead, how would the scientific notation change? The first factor would be negative, but the order of magnitude would stay the same. We would write − 0.0007 as − 7 × 10−4 in scientific notation. The power of 10 in scientific notation tells us the highest place value of a number written in standard form. If the first factor is positive, the number is positive. If the first factor is negative, the number is negative. However, scientific notation is generally used to represent measurements, so most of the scientific notation you encounter will represent positive numbers instead of negative numbers.

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Ordering Numbers in Scientific Notation Students order numbers given in scientific notation from least to greatest. Distribute one white Scientific Notation Ordering card to each pair of students. The gray cards are used for display. Look at the number on your card. Discuss with your partner whether the number is less than 1 or greater than 1. While partners discuss, tape the 3 × 102 card at the front of the room, halfway between the Less Than and Greater Than signs. Have students determine whether the number on their card is less than or greater than

3 × 102, and then stand by the appropriate sign.

In your group, justify why your number is less than or greater than the number I have displayed. As students explain, listen for the terms first factor and order of magnitude. If you do not hear these terms, encourage students to use them as a way to explain their number’s location. If students determine that a number is in the wrong group, have the students assigned to that number move to the other group and explain why their number belongs there. When students have finished explaining, display the cards with the numbers 7 × 10−2 and 6 × 104. Discuss how to order these two numbers with 3 × 102 by using their orders of magnitude. Tape these numbers appropriately between the Less Than sign and 3 × 102 and between 3 × 102 and the Greater Than sign. Ask student pairs to reorganize themselves in relation to 7 × 10−2, 3 × 102, and 6 × 104. Because there are now four groups, students may need more support understanding where to stand. If needed, point out the four groups to students as less than 7 × 10−2, between 7 × 10−2 and 3 × 102, between 3 × 102 and 6 × 104, and greater than 6 × 104. When pairs are settled in their new groups, direct them to discuss why their number is in that location. Again, if any students realize they belong in another group, have them move to and participate with that group.

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Promoting the Standards for Mathematical Practice When students explain why a number written in scientific notation is less than or greater than a second number written in scientific notation and discuss their choices, they are constructing viable arguments and critiquing the reasoning of others (MP3). Ask the following questions to promote MP3: • Is your sign choice of Less Than (or Greater Than) a guess, or do you know for sure? • What questions can you ask another pair of students to make sure you understand their reasoning?

UDL: Representation Consider using a horizontal number line to model the values of the numbers, smaller numbers to the left and larger numbers to the right. Including the numbers 0 and 1 on the number line can further support learning about small positive numbers. Alternatively, use a vertical number line with larger numbers near the top.

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7–8 ▸ M1 ▸ TC ▸ Lesson 14

When students finish discussing, ask them to organize themselves in their current group from least to greatest number. Invite each student pair to tape their number in the correct location starting at one of the given numbers, 7 × 10−2, 3 × 102, or 6 × 104. Then have them explain to the class why their number is in that location. Again, encourage students to use the terms first factor and order of magnitude to explain their number’s location.

2.05 × 10−3, 2.1 × 10−3, 4.6 × 10−2, 7 × 10−2, 7.1 × 10−2, 3.6, 4.5 × 10, 3 × 102, 3.2 × 102, 2.5 × 103, 6.1 × 103, 6 × 104, 6.1 × 104, 7.02 × 104, 6.08 × 105 Have students return to their seats. Direct them to problem 19, which includes some of the numbers used in the activity. Have students complete the problem with their partner. 19. Order the given numbers from least to greatest.

3.6 7 × 10−2 3.2 × 102 7.02 × 104 6 × 104 3 × 102 4.6 × 10−2 4.6 × 10−2, 7 × 10−2, 3.6, 3 × 102, 3.2 × 102, 6 × 104, 7.02 × 104 When students have finished ordering the numbers from least to greatest, engage students in a class discussion by using the following prompts. What was your strategy to order the numbers from least to greatest? We organized the numbers from least to greatest based on their orders of magnitude. If two numbers had the same order of magnitude, then the number with the larger first factor was the larger number.

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Display the last gray card with the number − 2 × 103. Use the following prompts to engage in a class discussion. Is −2 × 103 written in scientific notation? Explain how you know. Yes. The absolute value of − 2 is at least 1 but less than 10, and it is multiplied by a power of 10. What does the first factor of −2 tell us? The − 2 tells us that the number written in standard form is negative. If −2 × 103 was also in the set of numbers in problem 19, where would it be in the ordering activity? Explain.

Differentiation: Challenge If students need additional challenge, create another card with −2 × 10−3. Have students use their knowledge of order of magnitude to explain why −2 × 103 is less than −2 × 10−3.

It would be less than 4.6 × 10−2 because negative numbers are less than positive numbers. Could −2 × 103 represent a distance? Why? No. Distance is always positive. Because the first factor is negative, the number is negative and cannot represent a distance.

Land Debrief 5 min Objectives: Write numbers given in standard form in scientific notation. Order numbers written in scientific notation. Facilitate a class discussion by using the following prompts. Encourage students to restate or build upon one another’s responses. Can you tell when a number is written in scientific notation? How? Yes. Numbers written in scientific notation are in the form a × 10n. We call a the first factor. The absolute value of the first factor must be at least 1 but less than 10. The exponent n represents the number’s order of magnitude. 318

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When is it useful to write a number in scientific notation? It is useful to write a number in scientific notation when it is a very large or very small number with many zeros. Is there a time when writing a number in scientific notation is not useful? If so, give an example. Yes. I do not think it is useful to write 4 or 645,741 in scientific notation. Writing these numbers in scientific notation would take longer than writing them in standard form. Can a number written in scientific notation have a negative value? Explain. Yes. A number written in scientific notation can have a negative value if the first factor is negative. What details are important to think about when ordering positive numbers written in scientific notation? The power of 10 indicates the highest place value of the number written in standard form. Larger orders of magnitude mean larger powers of 10, larger place values, and therefore, larger positive numbers. If the orders of magnitude are the same, the value of the number is dependent on the first factor; then, the number with the larger first factor is the larger number.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Students may benefit from a list of numbers to explain their process. Consider displaying the numbers 7.2 × 1015, 7.2 × 10−12, 3.1 × 10−12, and 7.25 × 1015.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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Recap

EUREKA MATH2

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14 Name

7–8 ▸ M1 ▸ TC ▸ Lesson 14

2. Complete the table.

RECAP Date

Writing Very Large and Very Small Numbers in Scientific Notation In this lesson, we •

defined scientific notation.

•

identified examples and nonexamples of numbers written in scientific notation.

•

wrote numbers in scientific notation and in standard form.

•

ordered numbers written in scientific notation.

Examples

Terminology •

A number is written in scientific notation when it is represented as a number a multiplied

by a power of 10. Numbers written in scientific notation are in the form a × 10n. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10. •

The order of magnitude n is the exponent

on the power of 10 for a number written in scientific notation.

1. Circle all the numbers written in scientific notation.

9.82 × 10

15 million

0.6 × 1011

− 6 × 105

4.2 × 10−3

4,200,000

320

Number in Standard Form

Number in Scientific Notation

0.0005

5 × 10−4

90,000

9 × 104

0.000 0007

7 × 10−7

8,000

8 × 103

0.000 002 09

2.09 × 10−6

0.000 000 032

3.2 × 10−8 The first nonzero digit is 5. The place value of the digit 5

The first nonzero digit is 3. The place value of the digit 3 is represented

Numbers written in scientific notation must be in the form a × 10n.

is represented by 10−4, which 1 1 . is equal to ___ , or _____ 4

by 10−8. The first factor is 3.2, which is at least 1 but less than 10.

10,000

The power of 10, 10−6, shows the place value of the first nonzero digit only, which is 2. The digits 0 and 9 are

The absolute value of a, the first factor, must be at least 1 but less than 10.

EUREKA MATH2

written after the decimal.

201

202

RECAP

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

3. The table shows the approximate weights of animals in pounds. Order the animals from heaviest to lightest. Animal

Approximate Weight (pounds)

African Elephant

9.5 × 103

Blue Whale

3.5 × 105

Colossal Squid

4.4 × 102

Giraffe

2.2 × 103

Grizzly Bear

8 × 102

Hippopotamus

6 × 103

Zebra

8.8 × 102

Blue whale, African elephant, hippopotamus, giraffe, zebra, grizzly bear, colossal squid

The blue whale is the heaviest because its weight has the greatest order of magnitude, which is 5.

The orders of magnitude are equal for the weights of the African elephant, the hippopotamus, and the giraffe. In this case, use the first factor of each weight to determine the correct order.

9.5 > 6 > 2.2

RECAP

203

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 14 Name

PRACTICE Date

1. Circle all the numbers written in scientific notation. 7

− 9.99 × 10

5.5 trillion

6.758 × 10

10 × 10

1,570,000,000

− 0.28 × 102

8 × 10

48.6 × 10−5

−6

2. Match each number written in standard form with its corresponding number written in scientific notation.

8,940

8,940,000

8.904 × 105

8.94 × 103

8,094

8.94 × 104

890,400

8.094 × 103

89,400

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

For problems 3–14, complete the table. Number in Standard Form

Number in Scientific Notation

3,000

3 × 103

0.003

3 × 10−3

2,000,000

2 × 106

575,000

5.75 × 105

0.0006

6 × 10−4

450,000

4.5 × 105

0.00045

4.5 × 10−4

10.

0.000 007

7 × 10−6

11.

0.000 0092

9.2 × 10−6

12.

0.00601

6.01 × 10−3

13.

5,505

5.505 × 103

14.

− 8,095,000

− 8.095 × 106

15. In 2014, the United States discarded a total of 5.08 × 109 pounds of trash. Write this number in standard form.

8.94 × 106

5,080,000,000 16. Sara believes that the number 3 × 10 is not written in scientific notation. Do you agree or disagree? Explain. I disagree. 3 × 10 is written in scientific notation because it is in the form a × 10n, where n is 1 and the absolute value of a is at least 1 but less than 10.

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205

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P R ACT I C E

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EUREKA MATH2

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17. In 2021, the richest person in the world had a net worth totaling about $177 billion. Write this number in scientific notation.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 14

Remember For problems 20–23, multiply.

1.77 × 1011

7 20. − _2(__ 3 10)

18. The mass of a neutron is about 1.67493 × 10−27 kg. The mass of a proton is about 1.67262 × 10−27 kg. Explain which is heavier.

__ − 14 30

21. − 2_(7_) 5 9

__ − 14 45

22. − 5_(− _1) 5 6

7 23. _2(− __ 10) 9

__5

__ − 14

Because the numbers have the same order of magnitude, the first factor determines which is heavier. A neutron is heavier because 1.67493 is greater than 1.67262. For problems 24–26, apply the properties of exponents to write an equivalent expression. Assume y is nonzero.

19. Before 2006, Pluto was considered to be one of the planets in our solar system. Many people thought it should be classified as a dwarf planet instead. The table lists planets in our solar system, the dwarf planet Pluto, and their approximate width in meters. Planet

Approximate Width (meters)

Mercury

4.88 × 106

Venus

1.21 × 107

Earth

1.28 × 10

Mars

6.79 × 106

Jupiter

1.43 × 10

Saturn

1.2 × 108

Uranus

5.11 × 107

Neptune

4.95 × 107

Pluto

2.4 × 106

24. (84)3

812

25. (3 x 3)2

32 x 6

26.

(y)

42 4 __

4__8 y4

List the planets, including Pluto, from least to greatest based on their width. Pluto, Mercury, Mars, Venus, Earth, Neptune, Uranus, Saturn, Jupiter © Great Minds PBC

P R ACT I C E

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208

P R ACT I C E

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27. Choose all the true number sentences. A. 82 = 8 ⋅ 8

B. 52 = 5 ⋅ 2 C. 42 = 8

D. 4 ⋅ 4 ⋅ 4 = 34

E. 7 ⋅ 7 ⋅ 7 = 73

28. Choose one false number sentence from problem 27. Explain why it is false. Sample:

In the number sentence for choice B, 52 means to multiply two 5’s. 52 is equal to 5 ⋅ 5, not 5 ⋅ 2.

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P R ACT I C E

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6.08 × 105

3.2 × 102 6.1 × 103

2.05 × 10−3

−2 × 103 7 × 10−2

7.1 × 10−2

6 × 104 3 × 102 © Great Minds PBC

4.5 × 10

7–8 ▸ M1 ▸ TC ▸ Lesson 14 ▸ Scientific Notation Ordering Cards

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 14

This page may be reproduced for classroom use only.

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326

3.6

4.6 × 10−2

2.5 × 103

7.02 × 104

2.1 × 10−3

6.1 × 104

7–8 ▸ M1 ▸ TC ▸ Lesson 14 ▸ Scientific Notation Ordering Cards

This page may be reproduced for classroom use only.

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 15 LESSON 15

Operations with Numbers Written in Scientific Notation Interpret numbers displayed in scientific notation on digital devices. Operate with numbers written in standard form and in scientific notation.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 15

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15 Name

Date

EXIT TICKET

1. After a series of calculations, a calculator screen displays this result.

4.399e14 Write the displayed value in scientific notation.

4.399 × 1014

Lesson at a Glance Students discover how numbers written in scientific notation are displayed on a digital device. They learn how to input numbers written in scientific notation into devices so that they can check their answers after making calculations. In groups and through class discussion, students explore how to operate with numbers written in scientific notation by using the properties of operations and the properties and definitions of exponents.

Key Questions

For problems 2–4, evaluate the expression. Write the answer in scientific notation. 2. 1.3 × 10−4 + 2.1 × 10−4

• What details are important to consider when we operate with numbers written in scientific notation?

1.3 × 10−4 + 2.1 × 10−4 = (1.3 + 2.1) × 10−4 −4

= 3.4 × 10

• Are the properties and definitions of exponents helpful when operating with numbers written in scientific notation? How? 3. 4.7 × 105 − 4.1 × 105

Achievement Descriptors

4.7 × 105 − 4.1 × 105 = (4.7 − 4.1) × 105 = 0.6 × 105

7–8.Mod1.AD14 Express how many times as much one number is than

= (6 × 10−1) × 105 = 6 × (10−1 × 105)

another number when both are written in scientific notation. (8.EE.A.3)

= 6 × 104

7–8.Mod1.AD15 Operate with numbers written in standard form

4. (3 × 10−4)3

and scientific notation, including problems where both decimal and scientific notation are used. (8.EE.A.4)

(3 × 10−4)3 = 33 × (10−4)3 = 27 × 10−12 = (2.7 × 10) × 10−12 = 2.7 × (10 × 10−12)

7–8.Mod1.AD17 Interpret scientific notation that is generated

= 2.7 × 10−11

by technology. (8.EE.A.4) 217

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7–8 ▸ M1 ▸ TC ▸ Lesson 15

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Scientific calculator

Learn 30 min

Students

• Adding and Subtracting

• Scientific calculator

• Multiplying and Dividing

Lesson Preparation

• Power to a Power

• Prepare a display of the scientific calculator so keystrokes and answers can be shown to the class.

Land 10 min

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Fluency Write in Scientific Notation Students write numbers in scientific notation to prepare for operating with numbers written in scientific notation.

Instead of this lesson’s Fluency, consider administering the Scientific Notation with Positive and Negative Exponents Sprint. Directions for administration can be found in the Fluency resource.

Directions: Write each number in scientific notation. 1.

56,000

5.6 × 104

0.00056

5.6 × 10−4

Teacher Note

EUREKA MATH2

7–8 ▸ M1 ▸ Sprint ▸ Scientific Notation with Positive and Negative Exponents

Number Correct:

Write the number in standard form.

3. 4. 5. 6.

−0.50006 560 −5,000.6 0.00506

−5.0006 × 10−1 5.6 × 102 3

−5.0006 × 10 5.06 × 10−3

3 × 10

19.

7.3 × 104

3 × 102

20.

7.03 × 104

4.1 × 10

21.

7.032 × 104

4.1 × 103

22.

7.0324 × 104 7.03242 × 104

5 × 10−1

23.

5 × 10−2

24.

80 × 10−1

5 × 10−4

25.

8,000 × 10−2

8.4 × 10−1

26.

800 × 10−3

8.4 × 10−3

27.

80,000 × 10−4

10.

5.25 × 103

28.

0.9 × 10

11.

5.25 × 104

29.

0.09 × 102

12.

2.367 × 102

30.

0.009 × 102

13.

2.367 × 103

31.

0.009 × 103

14.

6.25 × 10−1

32.

6,000 × 10−4

15.

6.25 × 10−2

33.

6,000,000 × 10−5

16.

2.358 × 10−1

34.

6,000,000 × 10−6

17.

2.358 × 10−2

35.

60 × 10−7

18.

2.358 × 10−3

36.

6,000 × 10−8

358

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EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TC ▸ Lesson 15

Students become familiar with reading and entering numbers displayed in scientific notation on digital devices. Distribute a calculator to each student. Display a calculator so the class can see the keys pushed to enter data. Direct students to problem 1 and read the prompt aloud. What do you notice? What do you wonder? I notice there is an “e” and a “+10” on Liam’s screen. I wonder why there is a “+” on the answer screen. I wonder what the “e” stands for. I wonder why the calculator displays the answer like this.

Teacher Note Digital devices display numbers in scientific notation in many ways. If possible, have students use the calculator or other digital device they will use throughout the year, including on assessments, to become familiar with its input and output characteristics.

How can we multiply 200,000 and 450,000 without using a calculator? We can multiply 200,000 and 450,000 by using the multiplication algorithm. We can multiply 45 by 2 and then use place value reasoning to determine the product of 200,000 and 450,000. Allow students to find the product by using the mentioned strategies. What is the product of 200,000 and 450,000?

90,000,000,000 Input 90,000,000,000 into your calculator. What do you notice? My screen does not hold that many digits. It can’t display them all at once. Compare 90,000,000,000 to the product 9e+10 that is shown on Liam’s calculator. What do you notice? I notice they both have 9 at the beginning. I notice the product 9e+10 has a 10 and there 10 zeros in 90,000,000,000. What is 90,000,000,000 written in scientific notation?

9 × 1010

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Have students complete problem 1 with a partner. 1. Liam enters 200,000 × 450,000 into his calculator. The screen shows the following display.

9e+10 a. Write 200,000 in scientific notation.

2 × 105 b. Write 450,000 in scientific notation.

4.5 × 105 Call the class back together for a discussion. What is 200,000 written in scientific notation?

2 × 105

What is 450,000 written in scientific notation?

4.5 × 105

Display the equation (2 × 105)(4.5 × 105) = 9 × 1010. Compare the two factors written in scientific notation with the product written in scientific notation. What do you notice? I notice if I multiply 2 and 4.5, the product is 9. If I multiply 105 and 105, the product is 1010. I notice I can rearrange the numbers so that the first factors are multiplied to get 9 and the powers of 10 are multiplied to get 1010. Now, let’s look at how Liam’s screen displays the product. What do you think the “e” represents? The “e” represents that the answer is in scientific notation. What do you think the “+10” represents? The “+10” represents that the order of magnitude is positive 10.

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Teacher Note Clarify that the “e” does not indicate an error and that the “+10” does not indicate the base of 10 in scientific notation. If students are not convinced that the “+10” represents the order of magnitude, ask them to calculate 2,000,000 ⋅ 4,500,000 on their devices and determine the interpretation of the “+12.”

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 15

The calculator displays numbers in scientific notation by having the first factor followed by an “e” and the order of magnitude. Why do you think the calculator displays the number in this way? Calculators use scientific notation to show numbers with a lot of digits, just like we do. Calculators have a limited space to display an answer. If a number is very large or very small, a calculator must display the answer in a different way.

Teacher Note

Let’s input (2 × 105)(4.5 × 105) into our calculators.

Depending on the type of calculator they use, students may notice that their screen displays large numbers differently from Liam’s screen. Have students compare how their calculator displays the product to how Liam’s calculator displays the product.

Use the displayed calculator to guide students as they input (2 × 105)(4.5 × 105). Today, we will use the properties of operations and the properties and definitions of exponents to operate with numbers written in scientific notation. We will check our calculations with a calculator.

Learn Adding and Subtracting Students add and subtract numbers written in scientific notation and write the results in scientific notation. Place students in groups of three and direct them to problem 2. Have them work in groups on parts (a)–(c). After each part, have students check their answer with a calculator. Guide students to operate with numbers written in scientific notation instead of in standard form. Circulate as students work, and provide support by asking the following questions as needed. • What operation can you use to answer this question? • How might the units of these numbers help you think about how to add (or subtract) them? • How might you apply the distributive property to add (or subtract) these numbers? © Great Minds PBC

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• How might you apply any of the properties of operations to add (or subtract) these numbers? • Is there a property or definition of exponents that is helpful here? Why? • How can you write the result in scientific notation? 2. The table shows the number of views for the following online videos. Video

Number of Views

Singing cat

55,000,000

Science experiment

1.1 × 107

Dancing baby

2 million

Talking parrot

5.1 × 107

a. How many total views do the singing cat video and the science experiment video receive? Write the answer in scientific notation.

55,000,000 = 5.5 × 107 5.5 × 107 + 1.1 × 107 = (5.5 + 1.1) × 107 = 6.6 × 107 The singing cat video and the science experiment video receive a total of 6.6 × 107 views. b. How many more views does the singing cat video receive than the science experiment video? Write the answer in scientific notation.

5.5 × 107 − 1.1 × 107 = (5.5 − 1.1) × 107 = 4.4 × 107 The singing cat video receives 4.4 × 107 more views than the science experiment video.

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Teacher Note Encourage students to use their Properties and Definitions of Exponents graphic organizer to help them identify properties and definitions as they evaluate the expressions.

Differentiation: Support If students need more support with part (a), have them find the sum of the following problems to recall unit reasoning and the distributive property. • 3 fifths + 2 fifths + 4 fifths

9 fifths • 3m + 2m + 4m

9m • 3 × 100 + 2 × 100 + 4 × 100

9 × 100

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 15

c. How many more views does the singing cat video receive than the talking parrot video? Write the answer in scientific notation.

5.5 × 107 − 5.1 × 107 = (5.5 − 5.1) × 107 = 0.4 × 107 = (4 × 10−1) × 107 = 4 × (10−1 × 107) = 4 × 106 The singing cat video receives 4 × 106 more views than the talking parrot video. When most students are finished, facilitate a class discussion by asking the following questions. If all students wrote the answer to part (c) as 4 × 106, consider mentioning 0.4 × 107 as a possible answer. What was your answer for part (c)?

4 × 106 0.4 × 107 Are the answers 4 × 106 and 0.4 × 107 equivalent? Why do our answers look different? Yes. After we subtract, our answer is 0.4 × 107, which is not written in scientific notation because the absolute value of 0.4 is not at least 1. When we write 0.4 × 107 in scientific notation, we get 4 × 106, which looks different but is equivalent. How can we write the answer in scientific notation? We can write the answer in standard form and then write it in scientific notation. We can write 0.4 × 107 as 4,000,000 and then as 4 × 106.

We can use the associative property of multiplication and the property x m ⋅ x n = x m+n to write the answer in scientific notation. We can write 0.4 as 4 × 10−1 and then multiply 10−1 and 107 to get 106, which gives us an answer of 4 × 106.

Promoting the Standards for Mathematical Practice When students perform operations with numbers written in scientific notation and make sure their answer is in scientific notation, they are attending to precision (MP6). Ask the following questions to promote MP6: • How are you using the properties and definitions of exponents in your work? • What details are important to think about when adding or subtracting numbers written in scientific notation? • How can we write the answer by using scientific notation?

If needed, have students return to part (c) to write their answers in scientific notation.

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Have students read part (d) and determine which operation to use. Then have them work in their groups to write an addition expression involving numbers written in scientific notation. How is the expression in part (d) different from the previous addition and subtraction expressions? In the previous expressions, the numbers had a common power of 10 when written in scientific notation. In part (d), one addend has 107 and the other addend has 106. Have students complete part (d) in their groups. Circulate as students work. Identify groups that use 106 as the common power of 10 and groups that use 107. Ask the following questions to provide support. • Can we still use the distributive property? What do we need to do first? • If we write the numbers with a common power of 10, how does one of the first factors change? • Is the answer written in scientific notation? d. How many total views do the talking parrot video and the dancing baby video receive? Write the answer in scientific notation.

2 million = 2 × 106 5.1 × 107 + 2 × 106 = (5.1 × 107) + (2 × 10 − 1 × 107) = (5.1 × 107) + (0.2 × 107) = (5.1 + 0.2) × 107 = 5.3 × 107 The talking parrot video and the dancing baby video receive a total of 5.3 × 107 views.

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Invite an identified group that used 106 as the common power of 10 to explain their work. Write the work as students explain. Consider using color-coding or highlighting to emphasize the relationships between the numbers. If no group used 106 as the common power of 10, use the following work. 6

2 million = 2 × 10

5.1 × 107 + 2 × 106 = (5.1 × 101 × 106) + (2 × 106) = (51 × 106) + (2 × 106) = (51 + 2) × 106 = 53 × 106

UDL: Representation Color-coding can serve as a visual cue when operating with numbers written in scientific notation. It helps students distinguish between the addends and see how the lines of work relate to one another. Another way to represent the relationships between the numbers is by using annotations such as circling or underlining.

= (5.3 × 10) × 106 = 5.3 × (10 × 106) = 5.3 × 107 The talking parrot video and the dancing baby video receive a total of 5.3 × 107 views.

Then invite an identified group that used 107 as the common power of 10 to explain their work. Write the work as students explain. Consider using color-coding or highlighting if needed. If no group used 107 as the common power of 10, use the provided sample response in part (d). When the powers of 10 are different, we can choose either power of 10 to be our common power of 10. Why was there less work when using 107 as the common power of 10 instead of 106? When the common power of 10 was 106, the sum was not written in scientific notation. When the common power of 10 was 107, the sum was written in scientific notation.

Have students complete part (e) in their groups. Circulate and use the bulleted list of questions from part (d) to provide support.

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e. How many more views does the science experiment video receive than the dancing baby video? Write the answer in scientific notation.

1.1 × 107 − 2 × 106 = (1.1 × 107) − (2 × 10−1 × 107) = (1.1 × 107) − (0.2 × 107) = (1.1 − 0.2) × 107 = 0.9 × 107 = (9 × 10−1) × 107 = 9 × (10−1 × 107) = 9 × 106 The science experiment video receives 9 × 106 more views than the dancing baby video. When most students have finished, confirm their responses. What common power of 10 did your group use?

106

Differentiation: Challenge Challenge students to evaluate the expression in part (e) with a different common power of 10 than what they originally used.

107 Which common power of 10 produced a difference that was in scientific notation?

Differentiation: Challenge

Is it wrong to use 107 as the common power of 10? Explain.

Challenge students to predict which common power of 10—the power 10−2, 10−5, or neither— produces a difference that is in scientific notation for the following expression.

106

No. We can use either power of 10 as the common power of 10, but using 107 involves more work to write our answer in scientific notation. Use the following prompts to discuss how students used the properties of operations and the property x m ⋅ x n = x m+n to operate with numbers written in scientific notation. What properties of operations did you use to complete problem 2? Explain.

6.7 × 10−2 − 8 × 10−5 Then have students evaluate the expression to test their prediction.

When I added or subtracted numbers with a common power of 10, I used the distributive property to write the expression as a power of 10 times a sum or difference.

When the common power of 10 is 10−2, the difference is in scientific notation.

When the result of an operation was not written in scientific notation, I used the associative property of multiplication to group the powers of 10.

6.692 × 10−2

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What properties and definitions of exponents did you use to complete problem 2? Explain.

When the result of an operation was not written in scientific notation, I used the property x m ⋅ x n = x m+n to multiply the powers of 10.

When I wrote numbers to have a common power of 10, I used the property x m ⋅ x n = x m+n to multiply the powers of 10. When I wrote a decimal as a number in scientific notation with a negative exponent, 1 −n I used the definition __ n=x . x

Multiplying and Dividing Students multiply and divide numbers written in scientific notation. Direct students to problem 3. Display the map of the continental United States and point to Colorado. The state of Colorado is approximately what shape? Colorado looks almost like a rectangle. Read the directions for parts (a) and (b) to the class. What is an advantage and a disadvantage of finding the approximate area instead of the exact area?

Teacher Note Consider having students continue to check each answer with a calculator. This provides students with immediate feedback and allows them to practice entering and reading numbers displayed in scientific notation on digital devices.

An advantage of finding the approximate area is that the calculations are simpler. A disadvantage is that the answer is not precise. When might the approximate area be used instead of the exact area? The approximate area would be used when exact measurements are unknown. The approximate area would be used when we want a general idea of how large the area is. Have students turn and talk about the following question. We used the properties of operations and the properties and definitions of exponents to add and subtract numbers written in scientific notation. Do you think those properties still apply when we multiply numbers written in scientific notation?

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Have students complete parts (a) and (b) with a partner. Guide students to operate with numbers written in scientific notation instead of in standard form. 3. The length of Colorado is about 611,551 meters. The width is about 450,616 meters.

Colorado

a. Approximate the length and width of Colorado by rounding to the nearest hundred thousand meters. Then write the length and width in scientific notation. The length of Colorado is approximately 600,000 meters, which is 6 × 105 meters. The width of Colorado is approximately 500,000 meters, which is 5 × 105 meters. b. Approximate the area of Colorado. Write the answer in scientific notation.

(6 × 105)(5 × 105) = (6)(5) × (105)(105) = 30 × 1010 = (3 × 10) × 1010 = 3 × (10 × 1010) = 3 × 1011 The approximate area of Colorado is 3 × 1011 square meters.

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Call the class back together and confirm their responses. What properties of operations did you apply in part (b)? Explain. I used the commutative property of multiplication and the associative property of multiplication to multiply the first factors and to multiply the powers of 10. I used the associative property of multiplication to group the powers of 10 as I wrote the result in scientific notation. I used the property x m ⋅ x n = x m+n to multiply the powers of 10.

What property of exponents did you apply in part (b)? Explain. Read part (c) to the class. Then have students turn and talk about the following question. What operation can we use to determine how many times as large the area of Colorado is than the area of Denver? How do you know? Have students complete part (c) with a partner. Circulate as students work and provide support by asking the following questions as needed. • How can we use an unknown factor equation to solve this problem? • What does ____ mean? 8 1011 10

• Is the result written in scientific notation?

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c. The area of Denver, the capital of Colorado, is approximately 4 × 108 square meters. The area of Colorado is about how many times as large as the area of Denver? Write the answer in scientific notation.

3 × 1011 =

⋅ (4 × 108)

1011 ______ = 3_ × ____

3 × 1011 4 × 108

(4)

( 108 )

Differentiation: Challenge Consider having students choose a different city and state and answer the following question: The area of the state is about how many times as large as the area of the city?

= 0.75 × 103 = (7.5 × 10−1) × 103 = 7.5 × (10−1 × 103) = 7.5 × 102 The area of Colorado is about 7.5 × 102 times as large as the area of Denver. Call the class back together for a discussion. What properties and definitions of exponents did you apply in part (c)? Explain.

I used the definition of a negative exponent to understand ____ as 1011 ⋅ 10−8. Then I used 8

the property x m ⋅ x n = x m+n to multiply the powers of 10.

1011 10

What is another way we can write the answer for part (c)? We can say the area of Colorado is 750 times as large as the area of Denver because 7.5 × 102 is equivalent to 750.

Display the following work for another problem and facilitate a discussion about the error. The length of a blue whale is about 2.8 × 10 meters. The length of a tube of lip balm is about 6.7 × 10−2 meters. A blue whale is about how many times as long as a tube of lip balm?

(2.8 × 10)(6.7 × 10−2) = (2.8)(6.7) × (10)(10−2) = 18.76 × 10−1 = 1.876 A blue whale is about 1.876 times as long as a tube of lip balm. 342

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Have students think–pair–share about the following question. Jonas’s work to solve this problem is shown. Do you agree with his work? Explain. No. Jonas multiplied the length of a blue whale by the length of a tube of lip balm. He should have divided the length of a blue whale by the length of a tube of lip balm.

Power to a Power Students use properties of exponents to calculate squares and cubes of numbers written in scientific notation. Direct students to problem 4. How does this problem compare with previous problems? This problem is similar to previous problems because it involves a number written in scientific notation. This problem is different from previous problems because it has an expression raised to the second power. In the previous problems, we added, subtracted, multiplied, and divided expressions. Have students work in pairs to complete problems 4 and 5. Allow students to use a calculator to evaluate 8.852. Circulate to confirm answers or correct miscalculations.

Teacher Note When using certain calculators, students do not need to enter parentheses around the number in scientific notation before raising it to an exponent if they use the scientific notation button. However, it is important for students to understand when to use parentheses to prevent computational errors.

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For problems 4 and 5, use the properties of exponents and the properties of operations to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator. 4. (8.85 × 103)2

(8.85 × 103)2 = 8.852 × (103)2 = 78.3225 × 106 = (7.83225 × 10) × 106 = 7.83225 × (10 × 106) = 7.83225 × 107 5. (2 × 10−9)3

(2 × 10−9)3 = 23 × (10−9)3 = 8 × 10−27 Facilitate a class discussion by asking the following questions. What properties of exponents did you apply to evaluate these expressions? Explain. I used the property (xy)n = x n y n to raise the first factor to an exponent and the power of 10 to the same exponent.

I used the property (x m)n = x m ⋅ n to evaluate the power of 10 raised to an exponent. I used the property x m ⋅ x n = x m+n to multiply the powers of 10.

What property of operations did you apply to evaluate these expressions? Explain.

I used the associative property of multiplication to group the powers of 10 as I wrote the product in scientific notation.

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Land Debrief 5 min Objectives: Interpret numbers displayed in scientific notation on digital devices. Operate with numbers written in standard form and in scientific notation. Facilitate a class discussion by asking the following questions. Encourage students to restate or build upon one another’s responses. How is a number in scientific notation displayed on a digital device? Most digital devices display the first factor followed by an “e” and the order of magnitude. What details are important to consider when we operate with numbers written in scientific notation? When we add or subtract numbers written in scientific notation, we need to think about whether the numbers have the same unit, or a common power of 10, which we need to use the distributive property. If the numbers that we are adding or subtracting do not have a common power of 10, we need to choose a common power of 10 and write both numbers using it. If the first factor of a result does not have an absolute value of at least 1 but less than 10, we have to consider how to use powers of 10 to correctly write the result in scientific notation. Are the properties and definitions of exponents helpful when operating with numbers written in scientific notation? How? Yes. The properties and definitions of exponents are helpful in different ways. When the result is not written in scientific notation, we can write the first factor to have an absolute value of at least 1 and less than 10. Then we can use the property x m ⋅ x n = x m+n to multiply the powers of 10.

For addition and subtraction problems, we can use the property x m ⋅ x n = x m+n to get common powers of 10 so that we can use the distributive property.

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For division problems, we can use the definition x−n = __n to understand the quotient 1 x

of the powers of 10 as a multiplication expression. Then we can multiply by using the property x m ⋅ x n = x m+n.

For problems involving a power of a number in scientific notation, we can use the property (xy)n = x n y n to raise the first factor to an exponent and the power of 10 to the same exponent. Then we can use the property (x m)n = x m ⋅ n to evaluate the power of 10 raised to an exponent.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support. For the Practice problems, students should have access to a scientific calculator or similar digital device.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 15

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 15

3. (9 × 10 −7)(3.4 × 10 2)

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15

RECAP

Name

Date

EUREKA MATH2

(9 × 10−7)(3.4 × 102) = (9)(3.4) × (10−7)(102) = 30.6 × 10−5

Operations with Numbers Written in Scientific Notation

= (3.06 × 10) × 10−5 = 3.06 × (10 × 10−5)

In this lesson, we •

interpreted numbers displayed in scientific notation on digital devices.

•

used the properties and definitions of exponents and the properties of operations to efficiently operate with numbers written in scientific notation.

•

wrote sums, differences, products, and quotients in scientific notation.

Use the commutative and associative properties of multiplication to rewrite the original expression. Then multiply the factors in each pair.

= 3.06 × 10−4

4. (1.6 × 10−8)3

Examples

(1.6 × 10−8)3 = 1.63 × (10−8)3 = 4.096 × 10−24

1. A calculator displays 3.45e−4. Write this number in scientific notation and in standard form.

3.45 × 10−4 = 0.000 345

For problems 2–4, evaluate the expression. Write the answer in scientific notation.

= 10.8 × 10−9 The addends have a common power −9

of 10, 10 . So the distributive property can be applied.

−9

= (1.08 × 10) × 10

= 1.08 × (10 × 10−9) −8

= 1.08 × 10

(x m)n = x m ⋅ n to evaluate (1.6 × 10−8)3.

5. Use the table of approximate animal weights to complete each part.

2. 2.7 × 10−9 + 8.1 × 10−9

2.7 × 10−9 + 8.1 × 10−9 = (2.7 + 8.1) × 10−9

Use the properties (xy)n = x n y n and

The number 10.8 × 10−9 is not written in scientific notation because the absolute value of the first factor is greater than 10. Use the associative property of multiplication and the property

x m ⋅ x n = x m+n to write 10.8 × 10−9 in scientific

Animal

Approximate Weight (pounds)

Aphid

4.4 × 10−7

Emperor scorpion

6.6 × 10−2

Gray tree frog

1.6 × 10−3

Termite

3.3 × 10−6

notation.

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a. About how many more pounds does a gray tree frog weigh than an aphid?

(1.6 × 10−3) − (4.4 × 10−7) = (1.6 × 10−3) − (4.4 × 10−4 × 10−3) To determine how many more pounds the gray tree frog

= (1.6 × 10−3) − (0.00044 × 10−3) = (1.6 − 0.00044) × 10−3 = 1.59956 × 10−3

weighs than the aphid, find the difference of their weights.

Write 10−7 as 10−4 × 10−3. Then apply the distributive property.

Another strategy is to use 10−7 as the common power of 10.

1.6 × 10−3 can be written as 1.6 × 104 × 10−7.

A gray tree frog weighs about 1.59956 × 10−3 pounds more than an aphid.

b. An emperor scorpion is about how many times as heavy as a termite?

6.6 × 10−2 = Divide the weight of the scorpion by the weight of the termite.

________

___

⋅ (3.3 × 10−6)

____

−2 6.6 × 10 = 6.6 × 10 3.3 × 10−6 (3.3) (10−6) −2

= 2 × 104 = 20,000 An emperor scorpion is about 20,000 times as heavy as a termite.

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

Student Edition: Grade 7–8, Module 1, Topic C, Lesson 15 Name

PRACTICE Date

9. 5 × 105 + 3 × 104

5.3 × 105

10. (2.2 × 104)(5.4 × 107)

1.188 × 1012

For problems 1–4, write the answer displayed on the calculator screen in scientific notation and in standard form. 1.

5 . 678e+17

5.678 × 10

0.0000023*0.00005 1.15˜ -10 1.15 × 10−10

567,800,000,000,000,000

0.000 000 000 115

11.

_______

5.6 × 105 2 × 10−3

2.8 × 108

1 .386471e9 1.386471 × 109 1,386,471,000

12. (2 × 10−3)4

1.6 × 10−11

(1 .3 9 * 10 9 ) 1. 9321 * 10 18

1.9321 × 1018

13. 9.6 × 1010 − 3 × 1010 + 2.2 × 1010

1,932,100,000,000,000,000

8.8 × 1010

For problems 5–14, evaluate the expression. Write the answer in scientific notation. 5. 4 × 103 + 3 × 103

7 × 103

6. 9 × 10−8 − 2.7 × 10−8

6.3 × 10−8 14. 9.8 × 105 + 7.4 × 105 + 8.9 × 105

2.61 × 106 7. (9 × 1010)(1.1 × 103) 13

9.9 × 10

8. 6 × 10−6 + 8 × 10−6

1.4 × 10−5

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d. About how many more pounds does a zebra weigh than a wood mouse?

15. Use the table of approximate animal weights to complete each part. Check each answer with a calculator.

Animal

Approximate Weight (pounds)

Giraffe

2.2 × 103

Housefly

2.6 × 10−5

Hummingbird

8.8 × 10−3

Monarch butterfly

1.1 × 10−3

Wood mouse

4.4 × 10−2

Zebra

8.8 × 102

A zebra weighs about 879.956 pounds more than a wood mouse.

e. Which animal or insect is about 8 times as heavy as a monarch butterfly? A hummingbird is about 8 times as heavy as a monarch butterfly.

A zebra is about how many times as heavy as a hummingbird? A zebra is about 1 × 105 times as heavy as a hummingbird.

Remember

a. About how many more pounds does a hummingbird weigh than a monarch butterfly? A hummingbird weighs about 7.7 × 10−3 pounds more than a monarch butterfly.

For problems 16–19, divide. 1 16. −3 ÷ 1_ 17. _ ÷ 3 4 4 __1 −12 12

1 18. 6 ÷ (− _) 3

−18

19. − 1_ ÷ (−3) 6 __1 18

b. About how many more pounds does a wood mouse weigh than a monarch butterfly? A wood mouse weighs about 4.29 × 10−2 pounds more than a monarch butterfly.

c. A wood mouse is about how many times as heavy as a monarch butterfly? A wood mouse is about 40 times as heavy as a monarch butterfly.

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20. Eve and Lily each write an equivalent expression for __3 . The table shows their work. x7 x

Eve’s Work

Lily’s Work

3 4 x 7 x_____ __ = ⋅x

x7 = x 7 ⋅ x −3 x3

= x 7+(−3)

= x4

= x4 Explain how Eve and Lily each use the properties and definitions of exponents to get the final expression. Eve uses the definition of a negative exponent to write __3 as x 7 ⋅ x −3. Then she uses the x7 x

product of powers with like bases property to write x ⋅ x −3 as x 7+(−3). She writes her final 7

expression as x 4.

Lily notices she can express x 7 as x 3+4. She uses the product of powers with like bases property x3 ⋅ x 4 x

to write __3 as _____ . She notices __3 is 1, so she writes her final expression as x 4. 3 x7 x

x3 x

21. Which expression is equivalent to 5 6 ⋅ 54? A. 2524 B. 1010 C. 524 D. 510

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Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 16 LESSON 16

Applications with Numbers Written in Scientific Notation Choose appropriate units of measurement and convert units of measurement with numbers written in standard form and in scientific notation. Operate with numbers written in scientific notation in real-world situations.

EUREKA MATH2

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Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16 Name

Date

EXIT TICKET

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 16

c. The entire series has 15 episodes for each of the 4 seasons. What is the approximate number of minutes per episode? There are 60 seconds in a minute.

A popular television series is added to your favorite online streaming service. It will take 1.902 × 105 seconds to watch the entire series.

_________

____

1.902 × 105 = (1.902) × 105 60 60

= 0.0317 × 105

a. Choose a more appropriate unit of measurement to describe the amount of time needed to watch the entire series. Explain why you chose that unit.

= (3.17 × 10−2) × 105 = 3.17 × (10−2 × 105)

Sample:

= 3.17 × 103

A more appropriate unit of measurement is days because it uses a larger unit for time, which makes the measurement simpler to interpret.

The entire series is 3.17 × 103 minutes long. There are 4 seasons in the series and 15 episodes per season.

4(15) = 60 There are 60 episodes in the series.

3.17 × 103 ________ ____ × 103 = 3.17 60

b. Convert the unit from seconds to the unit of measurement you chose in part (a). Round your answer to the nearest tenth of a unit.

( 60 )

≈ 0.05283 × 103

= 52.83 There are approximately 52.83 minutes per episode.

Sample: There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute.

24(60)(60) = 86,400 = 8.64 × 104 4

There are 8.64 × 10 seconds in a day.

_________

____

___

5 1.902 × 105 = (1.902) × 104 8.64 (10 ) 8.64 × 104

≈ 0.22 × 10

= 2.2

It will take approximately 2.2 days to watch the entire series.

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Lesson at a Glance In this digital lesson, students operate with numbers written in standard form and in scientific notation by engaging with digitally enhanced problems. Students compare the operations and strategies they use to solve the problems and realize that there are various paths to a correct response. Faced with problems involving a variety of units of measurement, students discuss which units of measurement might be most appropriate for the given situation. Then they compare quantities by converting them to the same unit. Scaffolds involving less complex mathematical operations prepare students to persevere with more challenging questions. Use the digital platform to prepare for and facilitate this lesson. Students will also interact with lesson content and activities via the digital platform.

Key Questions • Why is it important to choose an appropriate unit of measurement for a given situation? • What tools or strategies can help you solve real-world problems that use numbers written in scientific notation?

Achievement Descriptors 7–8.Mod1.AD14 Express how many times as much one number is than another number

when both are written in scientific notation. (8.EE.A.3) 7–8.Mod1.AD15 Operate with numbers written in standard form and scientific notation,

including problems where both decimal and scientific notation are used. (8.EE.A.4) 7–8.Mod1.AD16 Operate with numbers written in scientific notation to solve real-world

problems and choose units of appropriate size for measurements of very large or very small quantities. (8.EE.A.4)

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Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Operating with Numbers Written in Scientific Notation

• Scientific calculator

• Comparing and Converting Units

Lesson Preparation

• How Many Times as Large and How Much Larger

• None

• Computers or devices (1 per student pair)

Land 10 min

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Fluency Convert to the Equivalent Unit of Measurement Students convert to equivalent units of measurement to prepare for choosing appropriate units of measurement. Directions: Convert to the equivalent unit of measurement. 1.

6 minutes is

seconds.

360

18 months is

years.

1.5

7 feet is

4 kilograms is

2 centimeters is

4 pounds is

Teacher Note Consider giving students a conversion chart to highlight the relationships in this Fluency.

inches. grams. meters. ounces.

4,000 0.02 64

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Launch

Students estimate the number of breaths it will take to inflate a balloon to a specific volume and determine what information they need to find the exact number. Students begin the lesson with a simulation of inflating a balloon one breath at a time. They realize that it will take a very long time to fill the balloon to the desired volume. Students ultimately use the desired volume of the balloon and the volume of a breath to calculate the number of breaths it will take to fill the balloon to the desired volume.

Teacher Note The contexts for the problems in this lesson are intentionally unconventional. These contexts are meant to capture students’ imaginations while the students engage in meaningful mathematics.

What do you notice? What do you wonder? I notice that the balloon does not inflate much with each breath. I wonder how many breaths it takes to fill the balloon to the desired volume.

UDL: Engagement

I wonder how long it takes to fill the balloon to the desired volume. Students use scientific notation in an interactive and responsive context to help build a foundation for transitioning to scientific notation contexts that are more challenging to predict. Using what they learn from inflating the balloon, students estimate how many breaths it would take to fill a balloon with the same volume as the moon. Given that the volume of the moon is about 2.1968 × 1019 cubic meters, students apply their understanding of scientific notation to determine the number of breaths it would take to fill a balloon with the same volume as the moon. How is finding the number of breaths for the first balloon different from finding the number of breaths for the moon-size balloon? With the first balloon, I am able to use the standard form of the number of breaths. Trying to use standard form for the number of breaths for the moon-size balloon is much more difficult because the number is so large. When I used scientific notation, it made the numbers simpler for me to write.

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Digital activities align to the UDL principle of Engagement by including the following elements: • Engaging and interesting topics. Students engage in intriguing contexts that pique their interest, such as discovering the size of their life-breath balloon. • Various levels of challenge. Students respond to questions with different levels of precision when they calculate the volume of air that a pump produces in cubic meters per second and cubic kilometers per minute. • Immediate formative feedback. Students use digital tools that display the consequences of their answers, such as seeing the size of their balloon compared to real-world objects.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 16

Learn Operating with Numbers Written in Scientific Notation

Students operate with numbers written in standard form and in scientific notation. Students notice that filling the first balloon and the moon-size balloon by using one breath at a time is not efficient. An air pump is then provided to students to help them speed up the process. They continue trying to inflate the balloons and determine the volume of air they are able to pump. While inflating the balloons, students work with a variety of units, including cubic meters per second, cubic meters per minute, and cubic kilometers per minute. Given examples of basic unit conversions, students reason how to convert from one unit to another. When converting the volume of air you are able to pump, how was converting cubic meters per minute to cubic kilometers per minute different from converting cubic meters per second to cubic meters per minute? When converting from cubic meters per minute to cubic kilometers per minute, I only needed to adjust the order of magnitude of my number written in scientific notation. It was more work when converting from cubic meters per second to cubic meters per minute because I had to multiply the value by 60.

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Comparing and Converting Units Students convert units of measurement to compare quantities written in scientific notation. Students choose which automated pump to use to inflate the balloon based on rates given with different units, such as cubic meters per hour and cubic centimeters per second. The class’s choices are displayed in a graph, and students discuss which units are appropriate to use in certain contexts. Students are then challenged to compare rates that are given with different units and determine which automated pump fills the balloon the fastest. They discuss a variety of solution paths and find that it is necessary to convert values to a common unit when comparing. Common units can vary from student to student. Did you have to choose the same units as your classmates? Explain. No. I chose cubic meters per hour and converted all four rates to that unit. My classmate chose cubic centimeters per second and converted all four rates to that unit. When all four rates are converted to the same unit, we can just compare the numerical part of the rate.

Promoting the Standards for Mathematical Practice When students convert and compare rates written in scientific notation to understand the units of measurement, they are making sense of problems and persevering in solving them (MP1). Ask the following questions to promote MP1: • What information or facts do you need to solve this problem? • What are some strategies you can try to solve the problem? • How can you make the problem simpler?

Students further explore the context of balloons and breaths and consider a life-breath balloon that is filled with every breath they have taken throughout their lives. The volume of this balloon drives the need for using scientific notation while continuing to build on a context made familiar throughout the lesson. An interactive allows students to visualize their life-breath balloon in contrast with real-world objects.

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How Many Times as Large and How Much Larger Students make additive and multiplicative comparisons involving numbers written in scientific notation.

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Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16 Name

LESSON Date

Applications with Numbers Written in Scientific Notation

Having had an opportunity to work with numbers written in scientific notation with the balloon context, students move to more abstract examples. They choose objects to compare, both from an additive and multiplicative perspective, to the life-breath balloon.

Operating with Numbers in Scientific Notation

Comparing and Converting Units

How did you determine whether the problem asked for a multiplicative comparison or an additive comparison? The first problem asked how many times as large one object is than another object, so I knew it was asking for a multiplicative comparison. The second problem asked how much larger one object is than another object, so I knew it was asking for an additive comparison.

How Many Times as Large and How Much Larger

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Land Debrief 5 min Objectives: Choose appropriate units of measurement and convert units of measurement with numbers written in standard form and in scientific notation. Operate with numbers written in scientific notation in real-world situations. Facilitate a class discussion by using the following questions. Encourage students to restate and build on one another’s responses. What did you find interesting about today’s lesson? I found it interesting to think about filling a balloon the size of the moon as well as a balloon with all the breaths I have taken in my life.

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Why is it important to choose an appropriate unit of measurement for a given situation? It is important to choose an appropriate unit of measurement that makes sense in a given situation. The numerical part of the measurement can help us determine whether the unit of measurement is appropriate. For example, if the numerical part is really large, we might switch to a larger unit of measurement so that the number is smaller, and we can quickly interpret it. What tools or strategies can help you solve real-world problems that use numbers written in scientific notation? I can use properties and definitions of exponents to solve problems. I sometimes use technology to help calculate my answer. For today’s problems, it was also helpful to use the digital tools to visualize my answers. What is important to think about when comparing two quantities? It is important to compare quantities that have the same unit of measurement. For example, it is appropriate to compare a quantity in cubic meters per hour with other quantities in cubic meters per hour. If the unit of measurement for a given quantity is different from that of another quantity, we must convert them so that the two quantities have the same unit of measurement.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

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Recap

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Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16

RECAP

Name

Date

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c. The longest distance around Earth is about 4 × 104 kilometers. Using your answer from part (b), about how many weeks would it take for a hawk to travel that distance? Assume the hawk travels at a constant rate. (1 mile ≈ 1.6 kilometers)

240 ⋅ 1.6 = 384

Applications with Numbers Written in Scientific Notation

A hawk travels about 384 kilometers per day.

In this lesson, we •

determined an appropriate unit of measurement for a given situation.

•

converted units of measurement to more appropriate units of measurement.

•

operated with numbers written in scientific notation.

4 4______ × 104 ________ = 4 × 10

384

Convert the number of miles traveled in 1 day

3.84 × 102

10 4 ___4 = (____ )× 2 3.84

to kilometers.

(10 )

≈ 1.04 × 102

Examples

Divide the longest distance around Earth by the distance traveled by the hawk per day.

= 104

It would take about 104 days. There are 7 days in 1 week.

1. A hawk can travel about 1.552 × 107 inches per day.

104 ___ ≈ 14.86

a. Choose a more appropriate unit of measurement to describe the distance a hawk travels per day. Explain why you chose that unit.

It would take about 15 weeks for a hawk to travel the longest distance around Earth.

A more appropriate unit of measurement is miles per day because it uses a larger unit for the distance, which makes the measurement simpler to interpret.

Round 14.86 to the nearest whole number to determine the approximate number of weeks.

2. Water flows from a shower head at a rate of 7.57 × 103 cubic centimeters per minute.

b. Convert the unit from inches per day to the unit of measurement you chose in part (a).

a. Determine the rate that water flows from the shower head in cubic meters per minute.

1 foot = 12 inches

There are 106 cubic centimeters in 1 cubic meter.

1 mile = 5,280 feet

7.57 × 103 ________ ___3 = 7.57 × 10

_________ _________ 1.552 × 107 1.552 × 107 = 63,360 12(5,280)

106

−3

= 7.57 × 10

= ________4 1.552 × 107

There are 12(5,280), or 63,360, inches in 1 mile. Divide 1.552 × 107 inches by

63,360 inches to find how many miles

the hawk can travel each day.

6.336 × 10

Water flows from the shower head at a rate of 7.57 × 10−3 cubic meters per minute.

107 ____ × ___ = (1.552 ) 4 6.336

≈ 0.24 × 10

(106)

(10 )

Visualize a cube with an edge length of 1 meter, which is 100 centimeters.

100 cm 1m

100 cm

The volume of the cube can be expressed as 1 cubic meter or 1,000,000 cubic

= (2.4 × 10−1) × 10 3

centimeters.

= 2.4 × (10−1 × 103)

1 cubic meter = 106 cubic centimeters

= 2.4 × 102 2

A hawk can travel about 2.4 × 10 , or 240, miles per day.

233

234

RECAP

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b. Determine the rate that water flows from the shower head in cubic meters per second. There are 60 seconds in 1 minute.

7.57 × 10−3 _________ ____ × 10−3 = 7.57 60

( 60 )

≈ 0.126 × 10−3

= (1.26 × 10−1) × 10−3 = 1.26 × (10−1 × 10−3) = 1.26 × 10−4 Water flows from the shower head at a rate of about 1.26 × 10−4 cubic meters per second.

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

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Student Edition: Grade 7–8, Module 1, Topic C, Lesson 16 Name

PRACTICE Date

1. Use the following information to fill in the blanks. On Earth, there are about 1 quadrillion (1,000,000,000,000,000) ants and about 8 billion people. a. On Earth, there are about

3. The table shows the distances of several stars from Earth measured in light-years. A light-year is the distance that light travels in 1 year. For example, the distance from the closest star, Proxima Centauri, to Earth is 4.243 light-years because it takes light 4.243 years to reach Earth from Proxima Centauri. Light travels at a speed of approximately 9.46 × 1015 meters per year.

more ants than people.

Star Name

9.99992 × 1014

b. On Earth, there are about

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times as many ants as people.

1.25 × 105

Distance from Earth (light-years)

Alpha Canis Majoris (Sirius)

8.6

Alpha Canis Minoris (Procyon)

11.41

Alpha Lyrae (Vega)

25.3

Delta Eridani (Rana)

29.5

Delta Pavonis

19.92

Zeta Tucanae

28.03

a. Determine which stars given in the table are between 1.3 × 1017 and 2.5 × 1017 meters from Earth. Alpha Lyrae (Vega) and Delta Pavonis are between 1.3 × 1017 and 2.5 × 1017 meters from Earth. 2. It takes Earth 8.61624 × 104 seconds to complete a single rotation on its axis. Choose a more appropriate unit of measurement to report this data. Explain why you chose that unit.

b. Approximately how many kilometers per day does light travel? Light travels approximately 2.59 × 1010 kilometers per day.

Sample: I would use hours to report this data because 8.61624 × 104 seconds is 23.934 hours, which is simpler to interpret.

c. Choose a more appropriate unit of measurement to report the speed in which light travels. Explain. Sample: I would report the speed in kilometers per day because light travels very quickly and a day is a smaller amount of time than a year.

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P R ACT I C E

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9. Write 870,000,000 in scientific notation.

4. Maya bought a fish tank that has a volume of 175 liters. She read a fun fact that it would take 7.43 × 1018 fish tanks of water at that tank size to fill all the oceans in the world.

8.7 × 108

a. Use the fun fact to determine the total volume of water in the world’s oceans in liters. Write your answer in scientific notation. The total volume of the water in the world’s oceans is 1.30025 × 1021 liters.

10. Write 5.8 × 10−3 in standard form.

0.0058

b. Given that 1 liter is 1 × 10−12 cubic kilometers, find the total volume of water in the world’s oceans in cubic kilometers. Write your answer in scientific notation.

11. Order the numbers from least to greatest.

7.2 × 105, 8.1 × 10−3, 6.1 × 108, 4.7 × 108, 9.3 × 10−12, 5.6 × 10−8

The total volume of water in the world’s oceans is 1.30025 × 10 cubic kilometers.

9.3 × 10−12, 5.6 × 10−8, 8.1 × 10−3, 7.2 × 105, 4.7 × 108, 6.1 × 108

12. Which expressions are equal to 520? Choose all that apply.

c. Use your answer from part (b) to find the total volume of water in the world’s oceans in cubic centimeters.

A. (515)5

The total volume of water in the world’s oceans is 1.30025 × 10 cubic centimeters.

B. (510)2

C. 510 ⋅ 510 D. 54 ⋅ 55

E. 512 + 58

d. The Atlantic Ocean has a volume of about 323,600,000 cubic kilometers. You bought a fish tank that holds 75 more liters than Maya’s tank. About how many tanks like yours would it take to fill the Atlantic Ocean? Write your answer in scientific notation. It would take about 1.2944 × 1018 tanks like mine to fill the Atlantic Ocean.

Remember For problems 5–8, divide. 5 5. − _ ÷ 1_ 6 6

−5

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_ _ 7 1 ÷ 8 8

7. − 3_ ÷ (− 1_) 2 2

__7 ÷ − __3

10 ( 10) −7 3

P R ACT I C E

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240

P R ACT I C E

Teacher Edition: Grade 7–8, Module 1, Topic C, Lesson 17 LESSON 17

Get to the Point Model a situation by operating with numbers in scientific notation.

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Student Edition: Grade 7–8, Module 1, Topic C, Lesson 17 Name

Date

EXIT TICKET

Reflect on the lesson.

Lesson at a Glance In this open-ended modeling lesson, students learn about an impressionist painting technique and then examine a piece of art composed of small, distinct dots. Students estimate how many brushstrokes the painting has and compare their estimates with their initial guesses by operating with very large and very small numbers. Using the appraised value of the painting and the Read–Represent–Solve–Summarize routine, students estimate the value of each brushstroke. Then groups of students present their solution strategies to the class.

Key Question • When is it useful to write numbers in scientific notation before operating with them? Why?

Achievement Descriptors 7–8.Mod1.AD14 Express how many times as much one number is than

another number when both are written in scientific notation. (8.EE.A.3) 7–8.Mod1.AD15 Operate with numbers written in standard form

and scientific notation, including problems where both decimal and scientific notation are used. (8.EE.A.4) 7–8.Mod1.AD16 Operate with numbers written in scientific notation

to solve real-world problems and choose units of appropriate size for measurements of very large or very small quantities. (8.EE.A.4) © Great Minds PBC

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Agenda

Materials

Fluency

Teacher

Launch 5 min

8 of each of the following:

Learn 30 min • Point by Point • How Large? • Expensive Dots

Land 10 min

• Highlighters • Markers • Paper (lined, grid, and blank) • Rulers, inch and metric • Scientific calculators • Transparency film

Students • Modeling in A Story of Ratios

Lesson Preparation • Arrange materials in a designated area of the classroom.

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Fluency Operate with Numbers Written in Scientific Notation Students operate with numbers written in scientific notation to prepare for modeling a situation with numbers written in scientific notation. Directions: Evaluate. Write your answer in scientific notation. (2 × 105)(4 × 10−6)

8 × 10−1

______

2 × 105

(1 × 10−9) + (7 × 10−9)

8 × 10−9

(8 × 106) − (2.5 × 106)

5.5 × 106

(3 × 10−4)2

9 × 10−8

(4 × 103)(5 × 108)

2 × 1012

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6 × 107

3 × 102

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Launch

7–8 ▸ M1 ▸ TC ▸ Lesson 17

Students engage with the artistic technique of pointillism and generate a question about a piece of art. Display the image and pose the following questions. What do you notice? What do you wonder? I notice a bunch of different-color patches of paint. I notice that the image looks like it is textured. I wonder what the image is supposed to be. I wonder how many patches of paint that is.

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Display the image and pose the following questions. What do you notice? What do you wonder? I notice that the painting is blurry. I notice that it looks like the same style as the first image. I wonder if the two images are related. I wonder how many brushstrokes it took to create this painting.

Have students look at the front cover of their book. Introduce the painting and the painting technique used to create it. The images are of the painting La Corne d’Or. Matin by Paul Signac. The painting technique that Signac used is called pointillism. As we can see in the first image, which is a zoomed-in portion of the painting, pointillism is a technique that involves using dots of distinct colors to form images that can be seen as the painting is viewed from farther back. Each dot of color is a brushstroke.

Language Support To familiarize students with pointillism, consider displaying other examples of this type of painting technique.

Today, we will apply our understanding of exponents and scientific notation to solve problems involving this piece of art.

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Learn Point by Point Students choose tools that help them estimate the number of brushstrokes in a painting. Direct students to problem 1. Have them study the painting more closely and record their questions. 1. Study the painting La Corne d’Or. Matin by Paul Signac.

Differentiation: Challenge If students need additional challenge, consider modifying this lesson. Instead of facilitating the Point by Point and Expensive Dots segments as written, have students use the Read–Represent–Solve–Summarize routine to answer the following question. If Signac’s painting La Corne d’Or. Matin appraises for $8 million, what is the value of each brushstroke?

Record some questions that come to mind. © Great Minds PBC

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After most students have a few questions recorded, lead the class in a discussion by using the following prompts. When you look at the painting, what questions come to mind? Resist the urge to answer the questions. Instead, use this time to encourage student curiosity. Record questions, one at a time, and display them for the class. Expect students to be curious about the colors, the painting technique, or the location depicted in the painting. When a student volunteers a question, ask the class who else has that same question. Record the number of students who have the same question by placing check marks, plus signs, or other counting marks next to each question. If students do not question how many dots or brushstrokes are in the painting, offer that as your question. Find out how many students also find that question interesting. We won’t be able to address all these questions today. Let’s focus on estimating the number of brushstrokes in Signac’s painting. Direct students to record the focus question in problem 2. Encourage them to make an educated guess that answers the focus question and to write their guess in scientific notation.

Emphasize that although pointillism seems to be dots of paint, each dot is considered one brushstroke.

Promoting the Standards for Mathematical Practice

Guess:

When students apply their understanding of area to estimate the total number of brushstrokes in the painting La Corne d’Or. Matin and model the painting as a rectangle and each brushstroke as a unit of area, they are modeling with mathematics (MP4).

Sample: 2 × 106

Ask the following questions to promote MP4:

2. Focus Question: How many brushstrokes did Signac use to paint La Corne d’Or. Matin?

Divide the class into groups of four. Distribute the following tools to each group: calculator, highlighter, marker, paper (lined, grid, and blank), ruler, and transparency. Give groups time to brainstorm a plan for how to use the provided tools to help them estimate the number of brushstrokes in the painting. Circulate to monitor progress as groups discuss their plans. Expect students to generate plans similar to the following: • We can use a ruler to measure a small area of the painting and count the brushstrokes by using a highlighter within that area. Then we can multiply to estimate the number of brushstrokes in the entire painting.

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Teacher Note

• What assumptions can you make to simplify the problem of finding the number of brushstrokes in the painting? • How can you improve your method of estimation to better determine the number of brushstrokes in the painting? • What math can you write to represent this problem?

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• We can count the number of brushstrokes along the base and along the height of the painting. Then we can multiply those numbers to estimate the number of brushstrokes in the entire painting. Have students work with their groups to complete problem 3. Gather data to make an informed estimate that answers the focus question. Record your strategy and work in the space provided in problem 3. Describe which tools you used and how you used them to gather your information and make an estimate. Estimates will vary, but they should be within a range of 15,000 to 25,000 brushstrokes. 3. Strategy: Sample: We can use a ruler to measure a small area of the painting and count the brushstrokes by using a highlighter within that area. Then we can multiply to estimate the number of brushstrokes in the entire painting.

Tools Used: Sample: Ruler and highlighter

Estimate: Sample: 19,000 brushstrokes When all groups of students finish, invite them to share their solution strategies. After all groups have shared, engage students in a discussion by asking the following questions: • Are the answers you found reasonable? Are you surprised by the results? • What assumptions did you make? How did those assumptions affect your solution? • How is your solution strategy similar to another group’s strategy? How is it different? • Were your estimates close to each other?

7–8 ▸ M1 ▸ TC ▸ Lesson 17

Promoting the Standards for Mathematical Practice Students use appropriate tools strategically (MP5) when they make strategic choices about using calculators, rulers, transparencies, and different types of paper to help develop a strategy to estimate the number of brushstrokes in the painting. Ask the following questions to promote MP5: • Which tools or materials could help you solve this problem? • Why did you choose to use this tool? Did that work well?

UDL: Engagement Consider assigning group roles, such as notetaker and timekeeper. Define responsibilities for each role. Review the activity goal, directions, and group norms before groups begin. Suggest a target completion time and project a visual timer.

Language Support As students discuss their solution strategies, encourage them to use the Share Your Thinking section of the Talking Tool.

• How could you have increased the accuracy of your estimates? © Great Minds PBC

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How Large? Students compare the guess and the estimate to distinguish between how many times as large and how large. Have students complete problem 4 individually. Encourage students to give a thumbs-up to signal that they are finished. 4. Consider your answers from problems 2 and 3. a. Which is larger: your guess from problem 2 or your estimate from problem 3? Sample: My guess is larger. b. How many times as large is it as your other answer? Sample:

2______ × 106 _______ 2 × 106 = 19,000

1.9 × 104

2 = (___ )× 1.9

(104)

___ 106

≈ 1.05 × 102

My guess is about 1.05 × 102, or 105, times as large as my estimate. c. How much larger is it than your other answer? Sample:

(2 × 106) - 19,000 = (2 × 106) - (1.9 × 104) = (2 × 102 × 104) - (1.9 × 104) = (200 - 1.9) × 104 = 198.1 × 104 = (1.981 × 102) × 104 = 1.981 × (102 × 104) = 1.981 × 106 My guess is 1.981 × 106, or 1,981,000, brushstrokes larger than my estimate. 374

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When most students have completed the problem, consider using the following prompts to engage students in a discussion about the similarities and differences in parts (b) and (c): • Are your answers from parts (b) and (c) the same or different? Why? • What is the same about the questions asked in parts (b) and (c)? • What is different about the questions asked in parts (b) and (c)?

Expensive Dots Students estimate the dollar value of each brushstroke in the painting. Display the following value: 8 × 106. This number represents the painting in some way. What do you think the number 8 × 106 represents? Record the answers students share. Then give the class an additional piece of information to launch the next focus question. Signac’s painting appraises for about 8 million dollars. Introduce the problem and have students recall the problem-solving routine Read–Represent–Solve–Summarize they first encountered in grade 6. Have students remove the Modeling in A Story of Ratios page from their book and use it as a reference as they progress through the problem. Read problem 5 aloud. Engage students in the Read portion of the problem-solving routine by having them turn and talk with a partner about these two questions: • What does this problem ask me to find? • What do I know?

Language Support To support the context of appraising art, consider previewing the meaning of the term appraises by facilitating a class discussion. Ask students why certain pieces of art are considered more valuable than other pieces of art. One factor that affects the value of art is demand. When officials or experts appraise a piece of art, they are setting its price or value.

Encourage students to read the problem as many times as needed to understand what is being asked. Before using the following prompt, consider having students share their understanding with the class. Now that we have read the problem and we understand it, another productive habit is to find some way to represent the relationship we read about. We can represent problems by using a tape diagram, an equation, a graph, a table, a double number

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line, or any other model. Representing the relationship helps us move from reading the problem to solving it and summarizing our findings. Allow students to work in pairs to represent and solve the problem by selecting a model and making calculations. As students work, circulate and listen as they determine how to model the problem and use operations with numbers written in scientific notation to efficiently solve the problem. Encourage students to ask themselves the following questions after solving the problem. If their answer to either question is no, have them revise their model or create a new one. • Does my answer make sense? • Does my result answer the question? 5. What is the approximate dollar value for each brushstroke in the painting? Create a model to support your response.

Teacher Note Expensive Dots includes references to an important resource: the problem-solving routine Read–Represent–Solve–Summarize. Modeling in A Story of Ratios includes the routine Read–Represent–Solve–Summarize. This intermediate problem-solving routine links the Read–Draw–Write (RDW) method from A Story of Units to the formal mathematical modeling cycle used in A Story of Functions. EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 17 ▸ Modeling in A Story of Ratios

Student Edition: Grade 7-8, Module 1, Topic C, Lesson 17

Modeling in A Story of Ratios Read

Read the problem all the way through. Ask yourself: •

What is this problem asking me to find?

Then reread a chunk at a time. As you reread, ask yourself:

Sample:

•

What do I know?

Model the situation by using tools such as tables, graphs, diagrams, and equations.

Represent

Model:

Represent the problem by using your chosen model. Ask yourself: •

What labels do I use on the table, graph, or diagram?

•

How should I define the variables?

•

Are the known and the unknown clear in the model?

Add to or revise your model as necessary.

Number of Brush Strokes

1.9 × 10 4

Solve Solve the problem to determine whether your result answers the question. Ask yourself: •

Does my answer make sense?

•

Does my result answer the question?

If your answer to either question is no, then revise your model or create a new one. Then ask yourself these questions again using your new result.

Summarize Summarize your result and be ready to justify your reasoning.

Value (dollars) ?

8 × 10 6 © Great Minds PBC

Estimate:

8______ × 106 _______ 8 × 106 = 19,000

1.9 × 104

(104)

8 106 = ___ × ___

(1.9)

≈ 4.21 × 10

The value of each brushstroke is about 4.21 × 102, or 421, dollars.

376

245

Teacher Note Although the dollar value of a painting is not based on the number of brushstrokes, it serves as an intriguing context for students to explore.

EUREKA MATH2

When pairs finish, invite them to summarize their work and to share their estimates with the class. Then use the following prompts to engage students in a class discussion: • Did you work out the problem by using numbers written in scientific notation? Why? • Did you all calculate similar dollar values? • What could account for the variety of results in the class? • How did you use the Read–Represent–Solve–Summarize process in your work?

7–8 ▸ M1 ▸ TC ▸ Lesson 17

Language Support Provide students with sentence starters to respond to the posed questions. • I worked with numbers written in scientific notation to represent . • I wrote the number in scientific notation because .

Land

• We did/did not calculate similar values because . • The variety of results is caused by

Debrief 5 min Objective: Model a situation by operating with numbers in scientific notation. Use the following questions to help students recognize where they used operations with numbers written in scientific notation. Encourage students to add to their classmates’ responses. When is it useful to write numbers in scientific notation before operating with them? Why? It is useful to write numbers in scientific notation before operating with them when the numbers are very large. Writing numbers in scientific notation sometimes makes operations simpler because we can apply properties of exponents to solve the problem. Select from the following questions to further debrief the lesson: • What assumptions did you make about the painting, and how did that affect your results? • Where did your group need more support? • How did your group overcome obstacles? • Is there anything your group would do differently? • What was most helpful? • How could we be more precise in the estimates we made today? © Great Minds PBC

Teacher Note The Exit Ticket asks students to reflect on this lesson. Consider guiding students’ reflection with a question that was not asked during the debrief.

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Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. For students who were not present for the lesson, provide them the estimate of 19,000 brushstrokes in the painting La Corne d’Or. Matin to complete problems 4 and 5. For the Practice problems, students should have access to a scientific calculator or similar digital device.

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 17

Student Edition: Grade 7-8, Module 1, Topic C, Lesson 17 Name

PRACTICE Date

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7–8 ▸ M1 ▸ TC ▸ Lesson 17

b. What assumptions did you make in answering part (a)? Sample: I assumed all of Signac’s paintings have the same number of brushstrokes. I also assumed he never had to repaint a brushstroke.

1. What assumptions did you make when you modeled the number of brushstrokes in the painting La Corne d’Or. Matin? We assumed that each brushstroke was the same size.

c. Do you think your estimation for part (a) is reasonable? Sample: No, I do not think my estimate is reasonable. Signac likely did not use the same number of brushstrokes in each of his 500 paintings. Also, I think Signac worked on more than 500 paintings, but he may not have completed all those paintings.

2. Was writing numbers in scientific notation helpful in the lesson? If so, when? If not, why not? Yes, in problem 5 when I had to find the cost of each brushstroke, writing the number of brushstrokes and the appraised value of the painting in scientific notation was helpful.

5. You want to recreate the painting La Corne d’Or. Matin. Each brushstroke takes approximately 20 seconds to paint, which includes mixing the paint, painting the canvas, and washing your brush. 3. If you had more time to explore another question related to Paul Signac’s paintings, what question would you choose? What would be your plan to determine the answer?

a. How long does it take you to recreate the painting if you paint at a constant rate? Sample:

Student responses will vary.

Each brushstroke takes 20 seconds. There are 1.9 × 104 brushstrokes. It takes (2 × 10)(1.9 × 104), or 3.8 × 105, seconds to recreate the painting.

There are 60 seconds in 1 minute. It takes _______ , or approximately 6.3 × 103, minutes to recreate the painting.

There are 60 minutes in 1 hour. It takes _______ , or approximately 105, hours to recreate

For problems 4 and 5, use your estimations from the lesson. 4. Paul Signac completed about 500 pointillism paintings during his career as a painter.

There are 24 hours in 1 day. It takes ___ , or 4 _ , days to recreate the painting. 105 24

Sample:

(19,000)(500) = (1.9 × 104)(5 × 102)

3 8

b. What unit of measurement did you choose to report the amount of time it takes in part (a)? Explain why you chose this unit of measurement.

= (1.9)(5) × (104)(102) = 9.5 × 106

Sample:

Paul Signac painted about 9.5 × 106, or 9,500,000, brushstrokes during his entire career.

6.3 × 103 60

the painting.

a. Estimate the number of brushstrokes Paul Signac painted during his entire career.

3.8 × 105 60

I chose to report the amount of time in days because the approximate time was over 24 hours.

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250

P R ACT I C E

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EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 17

EUREKA MATH2

7–8 ▸ M1 ▸ TC ▸ Lesson 17

Remember For problems 6-9, divide. 8 __ ÷ 3 6. - __ 10 10 - _8 3

7. - 5_ ÷ 2_ 8 8 - _5 2

__ 8. - 9_ ÷ (- 10 4 4) __9 10

6 ÷ -4 5 ( 5) -6 4

For problems 10 and 11, evaluate the expression. Write the answer in scientific notation. 10. 6.7 × 106 - 6.2 × 106

5 × 105

11. (2.5 × 10-5)3

1.5625 × 10-14

For problems 12-14, apply the properties and definitions of exponents to write an equivalent expression. Assume all variables are nonzero. 12.

__ 36 3

13. (15x 2)3

153x 6 14. a2 b4 ⋅ a0 b7

a2 b11

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P R ACT I C E

251

Teacher Edition: Grade 7–8, Module 1, Topic D

Topic D Rational and Irrational Numbers In topic D, students begin to explore irrational numbers. They solve equations of the forms x 2 = p and x 3 = p where p is a rational number by first applying their grade 6 understanding of squared and cubed numbers. Then an introduction to the Pythagorean theorem motivates and supports the need for expressing numbers that are not rational. Students approximate values of square roots and cube roots, classify real numbers as rational or irrational, and solve equations of the form x 2 = p and x 3 = p in which the solutions are irrational numbers. Students begin topic D by working with perfect squares and perfect cubes. They relate these concepts to area and volume before considering squares and cubes of any rational number. Students rely on previous understanding of squares and cubes as products of equal factors to reason intuitively about square roots and cube roots before using square root and cube root notation. Students apply similar intuitive reasoning to solve equations of the forms x 2 = p and x 3 = p where p is a perfect square, a perfect cube, or a number related to a 4 perfect square or perfect cube, such as _ or 0.16. Students revisit these equations where the 9 solutions are irrational numbers in lesson 23. In lesson 19, students use digital interactives to explore the relationship among the areas of squares that can be made on the sides of a right triangle. Students make a conjecture about the relationship among the side lengths of a right triangle and test their conjecture. Then the Pythagorean theorem is introduced, and students apply it to find hypotenuse lengths that are rational numbers. 2

When using the Pythagorean theorem in lesson 20, students encounter values of c that are whole numbers but not perfect squares. This experience creates a need for square roots. Students first develop an understanding of square roots by finding two consecutive whole numbers that the value c must lie between. Square root notation is then formalized, and students use it to express hypotenuse lengths. Students create their own Spiral of Theodorus and use the lengths from the spiral to observe the placement of square roots on a number line, which lays the groundwork for approximating values of irrational numbers.

382

1 1

— √4

— √5

— √3

— √6

— √2

— √7

— — √17

— √8

— — √16

— √9

— — √10

1 1

— — — — — √13 √ 1 1 √— 12

— — √14

— — √15

1 1

EUREKA MATH2

7-8 ▸ M1 ▸ TD

In a digital lesson, students approximate the length of the hypotenuse of a right triangle by using a number line to consider which whole-number interval includes the value of the length. Then they continue to increase the precision of the interval to consecutive tenths, hundredths, and thousandths. Students learn cube root notation as they consider how to find the edge length of a cube when given only its volume. After defining the term irrational number, students classify real numbers as rational or irrational and use number sense to compare and order rational and irrational numbers. Then they revisit equations of the forms x 2 = p and x 3 = p and write the 0 1 solutions of the equations by using square root and cube root notation. Additionally, 1 students use the relationship between the 9 square and square root and the cube and cube root to generalize the number of solutions for those equations.

√2

√27

√17

√27

√120

Students revisit irrational numbers in later modules. In module 3, students solve additional problems involving the Pythagorean theorem, and they encounter π when examining the relationship between the circumference and area of a circle. Students work with π again in module 5 as they solve problems involving the volumes of spheres, cylinders, and cones.

Progression of Lessons Lesson 18 Solving Equations with Squares and Cubes Lesson 19 The Pythagorean Theorem Lesson 20 Using the Pythagorean Theorem Lesson 21 Approximating Values of Roots Lesson 22 Rational and Irrational Numbers Lesson 23 Revisiting Equations with Squares and Cubes © Great Minds PBC

383

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 18 LESSON 18

Solving Equations with Squares and Cubes Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18 Name

Date

EXIT TICKET

For problems 1–3, state whether the number is a perfect square, a perfect cube, both, or neither. Explain. 1. 121 The number 121 is a perfect square because it is the square of the integers 11 and − 11.

2. 75 The number 75 is neither because there is no integer that can be squared or cubed to get 75.

3. 64 The number 64 is both a perfect square and a perfect cube. It is a perfect square because it is the square of the integers 8 and − 8. It is a perfect cube because it is the cube of the integer 4.

The solutions are 7 and − 7.

6. r 2 = 0.81 The solutions are 0.9 and − 0.9.

In this lesson, students formalize their understanding of the terms perfect square and perfect cube and classify numbers as perfect squares, perfect cubes, both, or neither. They apply this understanding to determine which number or numbers square to a given number and which numbers cube to a given number. Students then solve equations of the forms x 2 = p and x 3 = p intuitively, without square root or cube root notation. They begin with equations for which p is a perfect square or a perfect cube, progress to equations for which p and the solutions are rational numbers, and also consider cases in which p is negative. This lesson defines perfect square and perfect cube.

Key Questions • What do you know about the number of solutions to an equation of the form x 2 = p?

For problems 4–7, solve the equation.

x 2 = 49

Lesson at a Glance

− 64 = y 3 The solution is − 4.

• What do you know about the number of solutions to an equation of the form x 3 = p?

Achievement Descriptors

1 7. k 3 = ___ 125

7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p,

The solution is 1_ . 5

where p is a positive rational number. (8.EE.A.2)

7–8.Mod1.AD12 Evaluate square roots of small perfect squares and

cube roots of small perfect cubes. (8.EE.A.2) © Great Minds PBC

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Agenda

Materials

Fluency

Teacher

Launch 10 min

• Blank paper (4 sheets)

Learn 25 min • Squares and Cubes of Fractions and Decimals

• Marker • Tape

Students

• Solving Equations of the Form x 2 = p

• Venn Diagram removable

• Solving Equations of the Form x 3 = p

Lesson Preparation

Land 10 min

• Use the paper to prepare four signs with the following labels: 2 and − 2, − 2, 2, and No Solution. Post one sign in each corner of the classroom.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Fluency Evaluate Squares and Cubes Students evaluate squares and cubes to prepare for solving equations of the forms x 2 = p and x 3 = p. Directions: Evaluate each expression.

386

1. 32

2. (− 3)2

3. 33

4. (− 3)3

− 27

1 2 5. (_) 3

1 9

1 3 6. (_) 3

_ 27

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Students categorize integers as perfect squares, perfect cubes, both, or neither. Have students think–pair–share about the following questions. Suppose a square has an area of 25 square units. What is its side length? Explain.

Its side length is 5 units. The side lengths of a square are equal. The area is 25 square units, so the side lengths must be 5 units because 5 ⋅ 5 = 25.

Suppose a cube has a volume of 125 cubic units. What is its edge length? Explain.

Its edge length is 5 units. The edge lengths of a cube are equal. The volume is 125 cubic units, so the edge length must be 5 units because 5 ⋅ 5 ⋅ 5 = 125.

Use the following prompts to introduce students to the term perfect square.

When we multiply two of the same number, we are squaring the number. When we multiply three of the same number, we are cubing the number. The number 25 is an example of a perfect square. What do you think is the definition of a perfect square? I think a perfect square is the number you get after squaring a number. A perfect square is the square of an integer. What types of numbers are integers?

Differentiation: Support If students need more support determining the side length or edge length, consider providing the following problems: Use what you know about the area of a square to complete the table. Side Length (units)

Area (square units)

Use what you know about the volume of a cube to complete the table. Edge Length (units)

Volume (cubic units)

Integers are all the positive and negative whole numbers and 0. What are some examples of perfect squares? How do you know? The number 4 is a perfect square because 22 = 4.

Teacher Note

The number 9 is a perfect square because 32 = 9. The number 16 is a perfect square because 42 = 16. How do you know that 25 is a perfect square? I know 25 is a perfect square because it is the square of the integer 5. Are there any other integers we can square to get 25?

This lesson combines and expands on the learning activities from Eureka Math2 grade 8 module 1 lessons 16 and 17. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

Yes. We can square − 5 to get 25.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

The integers 5 and − 5 can both be squared to get the perfect square 25. Have students think–pair–share about the following question. Is − 25 a perfect square? Explain. The number − 25 is not a perfect square. There is no integer that can be squared to get − 25. Direct students to remove the Venn diagram from their books. Have them use the label Perfect Squares for the left section in their Venn diagram and record the examples of perfect squares from the discussion.

UDL: Representation Consider using the Perfect Squares and Perfect Cubes interactive from grade 8 module 1 lesson 16 Launch to provide a visual representation of the specific square and cube used in the discussion.

Language Support

Sample:

Perfect Squares

Students are introduced to the terms squared and cubed in grade 6 module 4 lesson 4. To reinforce the meaning of squaring and cubing a number, consider using the following statements:

4 16 9

• We are squaring a number when we multiply a number by itself. • We are cubing a number when we multiply a number by itself and then by itself again.

388

EUREKA MATH2

Use the following prompts to introduce the term perfect cube. The number 125 is an example of a perfect cube. What do you think is the definition of a perfect cube? I think a perfect cube is the number you get after cubing a number. For a perfect cube, do you think the number that is cubed needs to be an integer? Why?

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Teacher Note Later in the lesson, students discuss whether perfect squares and perfect cubes can be negative numbers.

Yes. I think the number that is cubed needs to also be an integer like it is for perfect squares. The number that is cubed does need to be an integer. A perfect cube is the cube of an integer. What are some examples of perfect cubes? How do you know? The number 8 is a perfect cube because 23 = 8. The number 27 is a perfect cube because 33 = 27. The number 125 is a perfect cube because 53 = 125. If students did not provide 8 as a perfect cube, ask them whether it is a perfect cube before asking the following question. Have them explain their reasoning. Are there any other integers we can cube to get 8? No. There are no other integers we can cube to get 8. The integer 2 is the only integer we can cube to get the perfect cube 8. Have students think–pair–share about the following question. Is − 8 a perfect cube? Explain.

The number − 8 is a perfect cube because (− 2)3 = − 8.

Have students use the label Perfect Cubes for the right section in their Venn diagram and record the examples of perfect cubes from the discussion.

Differentiation: Support To support students with understanding why −2 is not identified as an integer that when cubed results in 8, consider reviewing multiplication of integers. Help students examine and compare the products that generate perfect squares and perfect cubes by guiding students step-by-step through the products of (−5)2 and (−2)3.

(−5)2 = (−5)(−5) = 25 (−2)3 = (−2)(−2)(−2) = (4)(−2) = −8

389

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Sample:

Perfect Squares

Perfect Cubes

125

−8

Next, have students use the label Both for the overlapping section of the Venn diagram and use the label Neither for the outside section of the Venn diagram. Display the following numbers.

−25

121

−4

Have students work in pairs to record each number in the correct part of the Venn diagram. Circulate and provide support as needed.

390

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Sample:

Perfect Squares

Perfect Cubes Both

4 16

125 1

−8

0 121 Language Support

Neither

−25

−4

After most students have finished, invite a pair of students to share with the class where they placed a number on the diagram and explain why. Repeat this process for each number. Consider having other students indicate agreement by using a thumbs-up or another nonverbal signal. For students who disagree with the placement on the diagram, ask them to share their reasoning. Depending on the responses and explanations provided by students, consider using any, or all, of the following prompts to debrief with the class or with individual pairs.

Consider providing students with the following sentence frame examples as they discuss where each number belongs on the diagram: • The number is a perfect square because it is the square of . • The number is a perfect cube because it is the cube of . • The number is both a perfect square and a perfect cube because it is the square of and the cube of . • The number is neither a perfect square nor a perfect cube because .

391

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

If the explanations for the nonzero perfect squares only mention squaring a positive integer, probe further and ask the following question: • Is there another integer you can square that results in the same perfect square? If so, what is it? If students classify 0 as neither a perfect square nor a perfect cube, ask the following questions: • What are the definitions of a perfect square and a perfect cube? • Which number has a square of 0? Which number has a cube of 0? • Is 0 an integer? If students classify − 4 as a perfect square, ask the following questions: • Is there an integer you can square to get − 4? • Is the product of two negative numbers a positive number or a negative number? After correct responses and explanations are shared, have students correct their Venn diagrams so they are accurate for future reference. Students continue to add to their Venn diagrams throughout the lesson. Today, we will use our knowledge of perfect squares and perfect cubes to solve equations.

Learn

Squares and Cubes of Fractions and Decimals Students apply what they know about perfect squares and perfect cubes to explore squares and cubes of fractions and decimals. 81 Facilitate a class discussion about the fraction ___ . 100

392

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

81 What do you notice about the number ___ ? 100

I notice that it is a fraction.

I notice both 81 and 100 are perfect squares. I notice it is a number less than 1 but greater than 0. Both 81 and 100 are perfect squares. Do you think there is a fraction that we can

81 square to get ___ ? If so, what fraction can we square? If not, explain why. 100

9 81 Yes. I think we can square __ to get ___ because 92 = 81 and 102 = 100. 10

100

Demonstrate and display for the class how this line of thinking looks symbolically. 9⋅9 9 9 9 2 81 ___ = _____ = __ __ = __ 100

10 ⋅ 10

(10)(10)

(10)

81 Is there another fraction we can square to get ___ ? If so, what fraction can 100 we square? If not, explain why. 9 81 Yes. We can square −__ to get ___ . 10

100

Differentiation: Support If students need support determining the positive fraction that when squared is 81 , 100 use the following prompts:

___

• What positive number squared is 81? • What positive number squared is 100?

9 81 ___ If needed, show calculations to demonstrate that (−__ . ) = 10

100

81 Where should ___ go in the Venn diagram? Why? 100 9 2 ___ 81 81 The number ___ should go in the Neither section. Even though ___ = __ , 81 does not 100 100 (10 ) 100 9 fit the definition of a perfect square because __ is not an integer. 10 81 Have students add ___ to the Neither section of their Venn diagram. 100

Use the following prompts to highlight similar thinking for cubes. What do you notice about the number 0.027? I notice it is a decimal. I notice 27 is part of the decimal, and 27 is a perfect cube. I notice it is a number less than 1 but greater than 0. What number cubed is 27?

393

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Invite students to turn and talk about the following question. We know that 3 cubed is 27. How can you use that fact to determine which number can be cubed to get 0.027? Circulate and listen to students discuss. If needed, prompt students to write 0.027 as a 27 decimal fraction, ____ . 1,000

How can we use fractions to determine which number cubed is 0.027?

27 The number 0.027 can be written as the decimal fraction ____ . The numbers 27 and 1,000 1,000

are perfect cubes. Because 33 = 27 and 103 = 1,000, we know the number cubed 3 is __ , or 0.3. 10

Is there another number we can cube to get 0.027? If so, what is the number? If not, explain why. No. There is no other number we can cube to get 0.027. When we cube a negative number it results in a negative number.

Differentiation: Support

Where should 0.027 go in the Venn diagram? Why?

The number 0.027 should go in the Neither section. Even though 0.027 = (0.3)3, 0.027 does not fit the definition of a perfect cube because 0.3 is not an integer.

If students need support determining that no other number can be cubed to get 0.027, use the following prompts: • What is the opposite of 0.3?

Have students add 0.027 to the Neither section of their Venn diagram. Display the following statement and have students think–pair–share about whether they agree with it.

• What is −0.3 cubed?

Yu Yan says 0.3 is the number that is squared to get 0.9. Do you agree? • If so, explain why.

Teacher Note

• If not, explain why not and describe her error. I do not agree because (0.3)2 = 0.09. I think Yu Yan identified the digit 9 as a perfect square and then thought 0.3 is squared to result in 0.9. The number 0.9 can be written

9 as __ . The numerator is a perfect square, but the denominator is not. 10

3 9 I do not agree. Yu Yan may have thought of 0.3 as __ and then squared 3 to get __ , or 0.9. 10

9 9 But she forgot to square the 10 in the denominator. So the square of 0.3 is ___ , not __ . 100

394

If students ask about what number to square to get 0.9, let them know that there is a number that can be squared to get 0.9, but it is not rational. Students are introduced to irrational numbers in lesson 22 and explore them further in high school.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Given a decimal, it can be helpful to write it as a fraction and determine whether both the numerator and denominator are perfect squares or perfect cubes to avoid Yu Yan’s error. Have students complete problems 1–4 with a partner. Circulate as students work. Ask the following questions to advance student understanding: • Is the given number a perfect square? • Is the given number related to a perfect square? • How can you write the number in fraction form? • Are both the numerator and denominator perfect squares or perfect cubes? 1. What are all the numbers that square 49 to __ ? 64

_7 and −_7 8

3. What are all the numbers that square to 1.69?

1.3 and − 1.3

2. What are all the numbers that cube 1 to −__ ?

− 1_

4. What are all the numbers that cube to 0.125?

0.5

As students finish, confirm answers and have students state what they notice. What do you notice about the answers to problems 1 through 4? I notice that in problems 1 and 3, there are two numbers that square to get a given number, but in problems 2 and 4, only one number cubes to get a given number. Display the following statements one at a time. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas.

Teacher Note

• Perfect squares are positive numbers.

If time permits, consider including the following statements in the routine:

• Perfect cubes are positive numbers. • The square of a number is greater than the number being squared. • Fractions and decimals can be perfect squares and perfect cubes. Give students 1 minute of silent think time to evaluate whether the statement is always, sometimes, or never true. © Great Minds PBC

• The cube of a number is greater than the number cubed. • The square of a negative number is greater than the number being squared.

395

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite the identified students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim. Conclude by having the class come to the following consensus: • Most perfect squares are positive numbers because a positive number squared is a positive number and a negative number squared is a positive number. The only perfect square that is not a positive number is 0. • Some perfect cubes are positive numbers, some perfect cubes are negative numbers, and one perfect cube is neither. For example, 8 is a perfect cube because 23 = 8, and − 8 is a perfect cube because (− 2)3 = − 8. The perfect cube 0 is neither a positive number nor a negative number. • Some squares of a number are greater than the number being squared, but not all. For 2 example, 25 is greater than 5, and 52 = 25; however, _1 is less than _1 , and (_1) = _1 . 4

• Fractions and decimals are not perfect squares or perfect cubes because fractions and decimals are not integers. Only integers can be perfect squares or perfect cubes. Have students return to their Venn diagram and write the opposites of the numbers they already have listed for perfect cubes. The opposites of the nonzero numbers in the Both section will be negative numbers, so ensure that students list these in the Perfect Cubes section.

396

Teacher Note In grade 6, students learn that 0 is neither a positive number nor a negative number.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Sample:

Perfect Squares

Perfect Cubes Both

4 16

125 1

−8

−27

−1

121

−125

−64

Neither

−25

−4

81 100

Teacher Note

0.027

Solving Equations of the Form x 2 = p Students solve equations of the form x 2 = p, where p is a rational number. Direct students’ attention to problem 5. The directions say to solve the equation. What does it mean to solve an equation? Solving an equation means finding a value for the variable that makes the equation true.

Consider encouraging students to add numbers in the appropriate places in their Venn diagram as they identify other examples.

Teacher Note The concept of an equation having more than one solution may surprise some students. Because students solve only linear equations before this course, some students may think that an equation can have only one solution. Consider taking a moment to acknowledge this new development with students.

397

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Have students turn and talk about why 36 is a perfect square. What value can we substitute for x to make this equation true? How do you know? We can substitute 6 for x to make this equation true. I know this because 62 = 36. So 6 is a solution to the equation. Is there another value we can substitute for x that makes the equation true? Explain. Yes. − 6 also makes the equation true because (− 6)2 = 36.

UDL: Action & Expression Consider creating an anchor chart that displays the numbers 1–20 squared and the numbers 1–10 cubed. Students may want to add some of these numbers to their Venn diagram to have a more complete list.

Two values for x make this equation true: 6 and − 6. So the solutions to the equation x 2 = 36 are 6 and − 6. Ask students to complete problems 5–12 independently or with a partner. Circulate as students work, and ask the following questions to advance student understanding:

Differentiation: Support

• Is the square of the variable a perfect square or related to a perfect square?

If students need more support with problems 5–12, consider providing the following problem:

• Are both the numerator and denominator perfect squares? For problems 5–13, solve the equation. 5. x 2 = 36

The solutions are 6 and − 6. 2

7. a = 1

The solutions are 5 and − 5. 2

The solutions are 1 and − 1. 2

9. 225 = n

The solutions are 15 and − 15.

4 11. x = __ 2

6. 25 = m2

The solutions are _2 and −_2 . 5

8. p = 169 The solutions are 13 and − 13. 10. c 2 = 0

(

) = 16 2

4 and −4 Then encourage students to write the equations for problems 5–12 in a similar manner.

The solution is 0. 12. 0.49 = r 2 The solutions are 0.7 and − 0.7.

Confirm answers as a class. Ask the following question to probe for recognition that equations of the form x 2 = p have two solutions when p is a positive number but only one solution when p is 0.

398

Determine all values that make the equation true.

Teacher Note In later lessons, students revisit solving equations of the form x 2 = p, where p is any real number.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

What patterns do you notice for the number of solutions in problems 5 through 12? I notice that most of the equations have two solutions except for the equation in problem 10. The equation in problem 10 has only one solution because 0 is the only value that makes the equation true. The square of 0 is 0. I notice that the equations in problems 5 through 9 and in problems 11 and 12 have two solutions because there is one positive value and one negative value that make the equation true. The square of a positive number is always a positive number, and the square of a negative number is also always a positive number. Next, have students complete problem 13 independently. Although some confusion or questions may arise, refrain from providing any immediate assistance. 13. w2 = − 4 No solution After giving students a minute to work, introduce the Take a Stand routine to the class. Draw students’ attention to the four signs posted in the classroom: 2 and − 2, − 2, 2, No Solution. Invite students to stand beside the sign that best describes the solution to the equation in problem 13. When all students are standing near a sign, allow one minute for groups to discuss the reasons why they chose that sign.

Teacher Note In this course, students work only within the real number system and conclude that there is no solution for equations such as w 2 = −4. Note that real numbers are defined in lesson 22. In later courses, students eventually discover that such equations do have solutions, and those solutions are part of the complex number system.

Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group. Have students return to their seats. As a class, reflect on the reasons why the equation has no solution. Watch for students who may insist that − 2 is the solution because − 22 = − 4. If this situation arises, review the order of operations to help students recognize the difference between − 22, which is the opposite of 2 squared, and (− 2)2, which is the square of − 2.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

When we square any nonzero number, the result is always a positive number because the product of two positive numbers is a positive number and the product of two negative numbers is also a positive number. So with the numbers we know, it is impossible to get − 4 from squaring a number. Because there are no numbers that make the equation w 2 = − 4 true, the equation has no solution. Have students return to problem 13 to revise their answer, if needed.

Solving Equations of the Form x 3 = p Students solve equations of the form x 3 = p, where p is a rational number. Direct students’ attention to the Solving Equations of the Form x3 = p segment. Have students turn and talk about the following question. How are these equations different from the equations we just solved? How does that affect how we solve these equations?

Teacher Note In later lessons, students revisit solving equations of the form x 3 = p, where p is any real number.

Have students complete problems 14–21 independently or with a partner. Circulate as students work. For problem 19, some students may see the negative number and think the equation has no solution. Ask students how this equation is different from the equation in problem 13. For problems 14–21, solve the equation. 14. x3 = 27

15. 8 = a3

The solution is 3. 16. g3 = 1

17. 64 = t 3

The solution is 1. 18. r 3 = 125

The solution is 4. 19. h3 = − 8

The solution is 5.

1 20. __ = a3 64

The solution is _1 . 4

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The solution is 2.

The solution is − 2. 21. b3 = 0.001 The solution is 0.1.

EUREKA MATH2

Confirm answers as a class. Use the following prompts to reinforce an understanding that equations of the form x 3 = p have only one solution. Also reflect on why equations of the form x 3 = p where p is negative have a solution. How many values make each equation in problems 14 through 21 true? How do you know? One value makes each equation true. When we cube a positive number, we get a positive number. When we cube a negative number, we get a negative number. How is the equation in problem 19 similar to the equation in problem 13? How is it different? The equations are similar because they both have a negative number. They are different because the variable in problem 13 is squared and the variable in problem 19 is cubed. Why is it that h 3 = − 8 has a solution, but w 2 = − 4 does not have a solution? We can cube − 2 to get − 8, but there are no numbers we can square to get − 4. When we cube a positive number, the result is a positive number. When we cube a negative number, the result is a negative number. When we cube 0, the result is 0. So equations of the form x 3 = p have one solution for any value of p.

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Promoting the Standards for Mathematical Practice When students notice that there are generally two integers that can be squared to get a perfect square, the exception being 0, but there is only one integer that can be cubed to get a perfect cube, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8: • Will there always be two integers you can square to get a perfect square? • Will there always be one integer you can cube to get a perfect cube?

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Land Debrief 5 min Objective: Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are rational numbers. Facilitate a class discussion by using the following prompts. Encourage students to restate or build upon one another’s responses. How can you use what you know about perfect squares to find the number or numbers 25 that square to __ ? 64

25 The numerator and denominator of __ are perfect squares. We can look at the 64

numerator and the denominator individually to determine the positive numbers we can square to get 25 and 64. Then we can write those positive numbers as the numerator

25 and the denominator of a fraction. To get __ , we square _5 . Because the square of any 64

25 positive or negative number is a positive number, we can also square −_5 to get __ . 8

What do you know about the number of solutions to an equation of the form x 2 = p? An equation of the form x 2 = p has one solution, two solutions, or no solution. When p is a positive number, the equation x 2 = p has two solutions because there are two numbers, one negative and one positive, that we can square to get p. When p is 0, the equation x 2 = p has one solution because only 02 is 0. When p is a negative number, the equation x 2 = p has no solution because if we square any number we know, we can only get a positive number or 0. What do you know about the number of solutions to an equation of the form x 3 = p? An equation of the form x 3 = p has only one solution. When p is a positive number, a negative number, or 0, the equation x 3 = p has one solution because there is only one number that we can cube to get p.

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If students focus on only one case (a positive number, a negative number, or 0) for p, probe further. Consider scaffolding by asking a series of questions for each case, including actual values for p, if necessary. For example, consider asking the following questions: • When p is a positive number such as 16, how many solutions does the equation x 2 = 16 have? Why? • When p is a positive number such as 1, how many solutions does the equation x 3 = 1 have? Why? Continue as needed for the other cases.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 18

For problems 3 and 4, determine all the numbers that cube to the given number. 3. − 27

Solving Equations with Squares and Cubes In this lesson, we • •

•

Terminology

classified numbers as perfect squares, perfect cubes, both, or neither.

For problems 1 and 2, determine all numbers that square to the given number.

2. 0.64

0.8 and − 0.8

= (9)(3)

= −27

= 27

The fraction includes the perfect cubes 8 and 125.

Solutions of x 2 = p

Examples

9 and − 9

33 = (3)(3)(3)

= (9)(−3)

125

A perfect cube is the cube of an integer.

solved equations of the form x 3 = p, where p is either a perfect cube or a rational number related to a perfect cube.

(−3)3 = (−3)(−3)(−3)

8 ___ 2 5

A perfect square is the square of an integer.

solved equations of the form x 2 = p, where p is either a perfect square or a rational number related to a perfect square.

1. 81

Only one integer has a cube of − 27.

−3

Multiplying two numbers with the same sign results in a positive number. So there are two numbers, one positive and one negative, that have the same positive square.

92 = 81 and (− 9)2 = 81

When p is a negative number:

When p is 0:

the equation has two solutions because both positive and negative numbers have positive squares.

the equation has no solution because no numbers have negative squares.

the equation has one solution because only 0 has a square of 0.

When p is a positive number:

When p is a negative number:

When p is 0:

the equation has one solution because only positive numbers have positive cubes.

the equation has one solution because only negative numbers have negative cubes.

the equation has one solution because only 0 has a cube of 0.

Solutions of x 3 = p

When the number given is not a perfect square, determine whether it is related to a perfect square. For decimals, write the number as a fraction and determine whether the numerator and the denominator are perfect squares. 8⋅8 10 ⋅ 10

When p is a positive number:

0.64 = ___ = _____ = (__)(__) = (__) 64 100

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8 10

261

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RECAP

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For problems 5–10, solve the equation.

5. x 2 = 121

The phrase solve the equation means to find the values that, when substituted for the variable, make the equation true.

6. − 8 = a3

The solutions are 11 and − 11.

25 7. __ = m2 49

The solutions are _5 and − _5 . 7

Use the perfect squares 25 and 49 to determine the solutions.

9. k2 = − 25 No solution

The solution is − 2.

8. q3 = 0.064 The solution is 0.4.

Use a fraction to make sense of the place value.

4⋅4⋅4 4 __4 __4 __4 0.064 = ____ = ________ = (__ )( )( ) = ( ) 64 1,000

10 ⋅ 10 ⋅ 10

10. 0 = n3 The solution is 0.

No number squared equals − 25 because no numbers have negative squares, so there is no solution.

RECAP

263

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Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

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Student Edition: Grade 7–8, Module 1, Topic D, Lesson 18 Name

PRACTICE Date

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7–8 ▸ M1 ▸ TD ▸ Lesson 18

12. Liam says that when he squares or cubes any number, the result is always a greater number. Do you agree with Liam? Explain why and provide an example to support your thinking. I do not agree with Liam. When I square or cube a positive fraction less than 1 or a positive decimal less than 1, I get a number that is smaller. For example, (0.2)2 is 0.04 and (0.2)3 is 0.008.

For problems 1–3, state whether the number is a perfect square, a perfect cube, both, or neither. Explain. 1. 1 The number 1 is both a perfect square and a perfect cube. It is a perfect square because it is the square of the integers 1 and − 1. It is a perfect cube because it is the cube of the integer 1.

For problems 13–16, determine all the values that make the equation true. 13. 52 =

14.

2. 25

(

)

= 144

12 and − 12

The number 25 is a perfect square because it is the square of the integers 5 and − 5. 15. 2

3. 60 The number 60 is neither because there is no integer that can be squared or cubed to get 60.

16.

(

)

= 125

For problems 4–7, determine all the numbers that square to the given number. 16 4. __ 81

_4 and − 4_ 9 9

6. 2.25

1.5 and − 1.5

9 5. __ 64

For problems 17–28, solve the equation. 17. x 2 = 100

_3 and − 3_ 8 8

10 and − 10

7. 0.49

19. 64 = r 2

0.7 and − 0.7

8 and − 8

18. m 3 = 27

20. h 3 = − 64

−4

For problems 8–11, determine all the numbers that cube to the given number. 8. 0.008

0.2

_3 4

1 10. − ___ 125

11. (2.3)3

− 1_ 5

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27 9. __ 64

__ 21. w 2 = 49 81

_7 and − 7_ 9 9

23. 0 = b 3

2.3

265

266

22. 196 = a2

14 and − 14

24. − 1 = z 2 No solution

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__ 25. c 3 = 27 64 3_ 4

26. n3 = − 1

−1

27. m3 = 0.125

28. 0.01 = g 2

0.5

0.1 and − 0.1

29. Fill in the box to create an equation that has the solutions − 13 and 13.

x 2 = 169 30. Fill in the box to create an equation that has − _1 as its only solution. 4

1 x 3 = − __ 64

Remember For problems 31–34, divide. 31. − 3_ ÷ 3_ 5 4

__ − 12

6 32. _ ÷ − 6_ 9 ( 5)

__ − 30 54

33. − 7_ ÷ (− 3_) 4 8

34.

28 __ 24

3 7_ __ ÷ 10

27 __ 70

35. A used car costs 4.5 × 103 dollars. A luxury car costs 9 × 104 dollars. The cost of the luxury car is how many times greater than the cost of the used car? The cost of the luxury car is 20 times greater than the cost of the used car. 36. Which expressions are equivalent to 1.2 × 105? Choose all that apply. A. 120,000 B. (1 × 106) + (2 × 105) C. (2 × 103)(6 × 102) 4.8 × 107 D. _______ 4 × 102

E. (2 × 105) − (8 × 104)

P R ACT I C E

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Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 19

Teacher 7–8,Module Module4,1,Topic TopicC,D,Lesson Lesson21 19 Teacher Edition: Edition: Grade Grade A1, LESSON 19

The Pythagorean Theorem Describe the Pythagorean theorem and the conditions required to use it. Apply the Pythagorean theorem to determine the length of a hypotenuse.

EUREKA MATH2

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Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19 Name

Date

EXIT TICKET

1. To which type of triangle does the Pythagorean theorem apply? The Pythagorean theorem applies only to right triangles. 2. Identify the sides of each right triangle by labeling each side as a leg or a hypotenuse.

hypotenuse

leg leg

leg

Lesson at a Glance In this lesson, students use interactives to explore the Pythagorean theorem. Students are introduced to tools used by builders to create right angles and consider what combinations of side lengths create a right triangle with a knotted rope. Students use areas of squares to develop an understanding of the relationship among side lengths in a right triangle. By applying the Pythagorean theorem, students determine the length of a hypotenuse when the square of the hypotenuse is either a perfect square or related to a perfect square. This lesson defines the Pythagorean theorem and the terms leg and hypotenuse.

Key Questions hypotenuse

leg

• When does the Pythagorean theorem apply? • What relationship does the Pythagorean theorem demonstrate?

3. Find the length of the hypotenuse c.

Achievement Descriptors

7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p,

where p is a positive rational number. (8.EE.A.2)

12 a2 + b 2 = c 2

7–8.Mod1.AD12 Evaluate square roots of small perfect squares and

122 + 92 = c 2

cube roots of small perfect cubes. (8.EE.A.2)

144 + 81 = c 2

7–8.Mod1.AD18 Apply the Pythagorean theorem to determine the

225 = c 2 15 = c

unknown rational or irrational length of a hypotenuse for a right triangle in mathematical problems. (8.G.B.7)

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Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Knotted Rope

• Highlighters (3 different colors)

• Squares and Right Triangles

Lesson Preparation

Land 10 min

• None

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Fluency Identify the Right Triangle Students identify right triangles to prepare for applying the Pythagorean theorem to right triangles. Directions: Determine whether the triangle is a right triangle. Write yes or no.

Yes

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Launch

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Students identify tools that are helpful when building a garden shed. Display the image of a garden shed. What tools could be useful to build a garden shed? Measuring tape Pencil Saw Nails Screws

Language Support To support the context, consider showing images or displaying the items as they are referred to.

Hammer Screwdriver

Display the image of a foundation.

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People often build sheds on a concrete base called a foundation. Suppose you have completed the foundation to your shed. What do you need to think about before you build the walls? I need to think about what materials I should use to make the walls. I need to think about whether the walls will be attached to the foundation. I need to think about how to attach the walls to the foundation. I need to think about how thick I want the walls to be. I need to think about how to make the walls perpendicular to the foundation so the walls are not leaning. If students do not mention that the walls should be perpendicular to the foundation, consider using any of the following prompts to guide their thinking. What should you consider when building the walls? I don’t want the walls of the shed to lean in or out so I should build them perpendicular to the foundation. What does it mean for a wall to be perpendicular to the foundation? A wall is perpendicular to the foundation when the wall and the foundation meet at a 90° angle. What tools can you use in the classroom to make right angles? I can use a corner of an index card or a protractor to make right angles. In construction, other tools are used to make right angles.

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Teacher Note Students use protractors to measure, classify, and draw angles in grade 4 module 6. Then in grade 5, students use protractors and index cards as right-angle tools to classify and construct quadrilaterals.

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Display the following images.

The triangular tool is known as a rafter square, and the L-shaped tool is known as a framing square. Each tool can be used to create right angles. Consider having students predict how these tools are used to create right angles. Today, we will explore other methods to create a right angle, and we will identify a strategy to solve problems involving right triangles.

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Learn Knotted Rope Students use an interactive to simulate forming a knotted rope into the shape of a right triangle. Display the Knotted Rope interactive.

Historians believe that builders in ancient times, such as the Egyptians, used a knotted rope to make a right triangle to ensure that a right angle was formed. What do you notice about the knots? I notice there are 12 knots, and they are evenly spaced. If students do not mention that the knots are equally spaced, ask about the spacing of the knots.

Teacher Note Consider preparing an example of the knotted rope by using a string with equally spaced tick marks to represent the knots.

UDL: Engagement To build interest, consider reading to the class from the book What’s Your Angle, Pythagoras? located in grade 8 module 1 lesson 18.

Invite a few students to use the interactive to create a right triangle with a knot at each vertex.

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Display the triangles that the students created, which may look like any of the following right triangles.

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Teacher Note In the interactive, when a right triangle is constructed with a knot at each vertex, a small square or box appears to mark the right angle.

What do you notice? I notice all the right triangles have side lengths of 3 units, 4 units, and 5 units. I notice that the sides with 3 units and 4 units make a right angle. Have students think–pair–share about the following question. Do these triangles look identical? Explain. Yes. These triangles look identical because they look like copies of one another. I am not sure if these triangles are identical. I would want to place them on top of each other to see if they line up perfectly. Overlap the triangles in the interactive. Are the triangles identical? How do you know?

Language Support To support the word identical, consider asking one of the following questions: • Do these triangles look the same? • Do these triangles look exactly alike?

Yes. The triangles are placed on top of each other, and they line up perfectly. We can call figures identical if we can line them up on top of each other so that there are no overlapping parts. Therefore, these triangles are identical. Each of these triangles is a right triangle with side lengths of 3 units, 4 units, and 5 units. Consider returning to the Knotted Rope interactive and having students attempt to make a right triangle with a side length of 1 unit and a knot at each vertex or with a side length of 2 units and a knot at each vertex. Anticipate that students may productively struggle before they realize that making a right triangle with these side lengths and a knot at each vertex is impossible.

Teacher Note In this lesson, students informally discuss the concept of identical triangles and congruence. In module 3, students explore the conditions that determine a unique triangle and use rigid motions to explore the concept of congruence.

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Do you think any right triangle we make from this rope with a knot at each vertex will have side lengths of 3 units, 4 units, and 5 units? Why? Yes. I think any right triangle we make with this rope will be a triangle with side lengths of 3 units, 4 units, and 5 units because all the triangles we created were identical. No one created a right triangle with different side lengths. Have students turn and talk about the following question. Do not confirm or deny student responses at this time. If you had a longer rope with more knots, do you think you could make a different right triangle that still had a knot at each vertex?

Squares and Right Triangles Students explore the Pythagorean theorem. Display the Right Triangle Squares interactive. What do you notice? What do you wonder? I notice the triangle is a right triangle. I notice the white square is the largest square. I notice the blue square is the second-largest square. I notice the red square is the smallest square. I notice the side length of each square is the same as a side length of the right triangle. I wonder what squares have to do with right triangles. Reveal the blue tiles and the red tiles. The blue square and red square are separated into square tiles. What do you notice? What do you wonder? I notice the blue square and the red square are separated into many square tiles. I notice the white square is not separated into square tiles. I notice that two of the triangle’s side lengths are 3 units and 4 units. I notice the area of the blue square is 16 square units, and the area of the red square is 9 square units.

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I wonder if the length of the third side of the right triangle is 5 units like in the knotted rope activity. I wonder if all the blue tiles and red tiles fit in the large white square at the bottom. Have students think–pair–share about the following question. If you could move the square tiles into the large white square at the bottom, how many square tiles do you think would fit? Why? Some students may reason that 25 square tiles can fit into the bottom square, while others may attempt to estimate with spatial reasoning. Encourage different rationales. We can move the square tiles to see if any of our predictions are correct. Use the interactive to fill the white square with tiles. What do you notice? I notice the number of blue tiles plus the number of red tiles equals the number of square tiles that fill the white square exactly. What is the length of the third side of the triangle? How do you know? The length of the third side of the triangle is 5 units because 5 square tiles fit along the third side of the triangle. The length of the third side of the triangle is 5 units. There are 25 square tiles and they fit perfectly into the white square, which means it has an area of 25 square units. So the side length must be 5 units. Return the tiles to their original squares and display only the large blue square, large red square, and white square. Have students discuss problem 1 as a class and have them write a conjecture about how the areas of the blue square and the red square relate to the area of the white square. Students will continue to refine this conjecture throughout this lesson.

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1. Make a conjecture about how the areas of the blue square and the red square relate to the area of the white square.

Blue Red

White The area of the blue square plus the area of the red square is equal to the area of the white square. Invite students to think–pair–share about the following question. Does our conjecture still hold true if the side lengths of the white square are changed and the side lengths of the blue square and the red square remain unchanged? Explain your reasoning. I don’t think so. The area of the blue square is 16 square units, and the area of the red square is 9 square units. This means the area of the white square needs to be 25 square units. A square must have side lengths of 5 units to have an area of 25 square units. I don’t think so. If the side lengths were shorter, the area of the white square would be smaller, so fewer square tiles would fit inside. If the side lengths were longer, the area of the white square would be larger, so more square tiles would fit inside.

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Display the Triangle Side Length interactive. Change the third side length and fill the white square to show that all the tiles do not fit exactly inside it if the third side length is not 5 units. Repeat the process with different side lengths. Does our conjecture hold true when the third side length is not 5 units? Explain. No. When the third side length is less than 5 units, the area of the white square is smaller, so all the tiles do not fit inside it. When the third side length is larger than 5 units, the area of the white square is larger, so there is extra room in it. Why do you think the conjecture only works when the third side length is 5 units? I think the conjecture only works when the third side length is 5 units because the triangle is a right triangle. In the triangle, what do you notice about the angle that is opposite the third side if the length of the third side is not 5 units? When the length is not 5 units, the angle is not a right angle. When the third side is longer than 5 units, the angle measure is greater than 90°. When the third side is shorter than 5 units, the angle measure is less than 90°. If we change the third side length, the angle opposite from it changes, and the triangle is no longer a right triangle. Our conjecture seems to only hold true when the triangle is a right triangle. We can look at some other right triangles to see if our conjecture holds true with them.

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Display the following triangle.

6 Our conjecture is “The area of the blue square plus the area of the red square is equal to the area of the white square.” Do you think we can test our conjecture on this triangle? Not really. Our conjecture discusses areas of squares. This image does not show squares. It only shows a triangle. Suppose we use this triangle to draw squares to make a diagram similar to the one we used for our conjecture. What would the area of each square be? How do you know? One square would have an area of 64 square units because 82 = 64. Another square would have an area of 36 square units because 62 = 36. The last square would have an area of 100 square units because 102 = 100. Our conjecture states to add the areas of the blue square and the red square. Which areas would we add together for this triangle? Explain. We would add the areas of the squares related to the two shorter side lengths because the blue square and the red square have the shorter side lengths in problem 1.

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Have students return to problem 1. As a class, restate the conjecture so that it relates to the side lengths of a right triangle. Conjecture: The area of the blue square plus the area of the red square is equal to the area of the white square. The sum of the squares of the side lengths of the two shorter sides of a right triangle is equal to the square of the longest side length of the right triangle. Have students complete problem 2 with a partner. 2. Test the conjecture with the following triangle. State whether the conjecture holds true and explain your reasoning.

Promoting the Standards for Mathematical Practice When students create and refine a conjecture about the relationship among the side lengths of a right triangle, they are attending to precision (MP6). Ask the following questions to promote MP6: • How can we describe this relationship by using squares of numbers?

• What details are important to think about when generalizing our conjecture?

Square of each shorter side length:

82 = 64 62 = 36 Square of the longest side length:

102 = 100

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Sum of the squares of the two shorter side lengths:

64 + 36 = 100 Compare the sum of the squares of the two shorter side lengths to the square of the longest side length:

100 = 100 The conjecture holds true. The sum of the squares of the side lengths of the two shorter sides of the triangle is equal to the square of the longest side length of the triangle. When most students are done, discuss when the conjecture holds true. What side lengths of right triangles does our conjecture hold true for? Our conjecture holds true for a right triangle with side lengths of 3, 4, and 5 units and a right triangle with side lengths of 6, 8, and 10 units. Have students turn and talk about the following question. Do you think this conjecture will work for other right triangles? Have students work on problem 3 with their partner. 3. Test the conjecture with the following triangle. State whether the conjecture holds true and explain your reasoning.

13 Square of each shorter side length:

52 = 25 122 = 144 Square of the longest side length:

132 = 169

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Sum of the squares of the two shorter side lengths:

25 + 144 = 169 Compare the sum of the squares of the two shorter side lengths to the square of the longest side length:

169 = 169 The conjecture holds true. The sum of the squares of the side lengths of the two shorter sides of the triangle is equal to the square of the longest side length of the triangle. When most students are finished, use the following prompts to introduce the Pythagorean theorem. Did the conjecture hold true? Did this surprise you? The conjecture held true. This did surprise me because the side lengths of this triangle were not multiples of 3, 4, and 5. The conjecture held true. This did not surprise me because the triangle was a right triangle. A philosopher and mathematician in ancient Greece named Pythagoras also identified this pattern and wrote what is known as the Pythagorean theorem to describe it. Display the Pythagorean Theorem interactive.

A leg of a right triangle is a side adjacent to the right angle. The hypotenuse of a right triangle is the side opposite the right angle.

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The Pythagorean theorem describes the relationship in a right triangle in which the sum of the squares of the leg lengths is equal to the square of the hypotenuse length. In a right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a2 + b2 = c2. Distribute 3 highlighters to each student. Have students highlight the sides of the triangle, annotate the triangle with the terms legs and hypotenuse, write the equation a2 + b2 = c2, and highlight the corresponding variables to support the new vocabulary terms. Ensure that students are not highlighting the exponents but only the variable and the side of the triangle the variable represents.

Teacher Note In grade 4, students use the term adjacent to describe angles. Then in grade 5, students use the term adjacent when discussing place value and when discussing the classification of quadrilaterals.

Hypotenuse c

Legs

b a 2 + b2 = c2

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Discuss the Pythagorean theorem. The Pythagorean theorem can be used to solve many different types of problems involving right triangles. In this lesson, we will use the Pythagorean theorem to determine the length of the hypotenuse when given the lengths of the legs of a right triangle. Throughout this year, we will see other situations in which the Pythagorean theorem is helpful. Which type of triangle does the Pythagorean theorem apply to? The Pythagorean theorem applies only to right triangles. According to the Pythagorean theorem, which variables represent the lengths of the legs of a right triangle? Which variable represents the length of the hypotenuse of a right triangle? The variables a and b represent the lengths of the legs of a right triangle. The variable c represents the length of the hypotenuse of a right triangle. How did we distinguish between the side lengths of the right triangle in our conjecture? We called the lengths of the legs the two shorter side lengths, and we called the length of the hypotenuse the longest side length. Have students complete problem 4 with their partner. Have one partner use 8 as the value of a, and have the other partner use 15 as the value of a. Circulate and advance student thinking by asking the following questions: • If the value of a is 8, what is the value of b? • What number times itself has a product of 289? • Is there another number that when multiplied by itself has a product of 289? Would that number make sense in this context?

UDL: Action & Expression If an anchor chart with the squares of 1–20 was created in lesson 18, consider displaying it to support determining the length of a hypotenuse in this lesson.

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7–8 ▸ M1 ▸ TD ▸ Lesson 19

4. Use the Pythagorean theorem to find the length of the hypotenuse c.

a2 + b2 = c2 82 + 152 = c2 64 + 225 = c2 289 = c2 17 = c

a2 + b2 = c2 152 + 82 = c2 225 + 64 = c2 289 = c2 17 = c

Teacher Note Students are introduced to the term square roots in lesson 20. If students are using a calculator to determine the value of c, they must square numbers to get the value of c2. For example, in problem 4, students could do the following calculations:

202 = 400 162 = 256 172 = 289

The length of the hypotenuse is 17 units. After partners have compared their work, engage the class in a discussion by asking the following questions. What did you notice when you compared your work and answer with that of your partner? We both determined the length of the hypotenuse is 17 units. Our work is similar. I had 82 + 152, which gave me 64 + 225. My partner had 152 + 82, which results in 225 + 64. Other than that, our work was the same. Does it matter which leg length is used for a and b? Explain. No. The variables a and b represent the leg lengths of a right triangle, but it does not matter which leg has length a and which leg has length b. Due to the commutative property of addition a2 + b2 is equivalent to b2 + a2. Why is the answer not −17 or both 17 and −17? The answer represents the length of the hypotenuse. Length is always a positive number.

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Have students complete problem 5 with their partner. Circulate to confirm answers or correct miscalculations. 5. Find the length of the hypotenuse c.

0.3

0.4 a2 + b2 = c2 (0.3)2 + (0.4)2 = c2 0.09 + 0.16 = c2 0.25 = c2 0.5 = c The length of the hypotenuse is 0.5 units. When most students are finished, emphasize the conditions needed to use the Pythagorean theorem. Why are we able to use the Pythagorean theorem for problem 5? We are able to use the Pythagorean theorem because we are given a right triangle and the length of each leg.

Differentiation: Challenge If students need additional challenge, consider having them think of other uses for the Pythagorean theorem besides determining the length of a hypotenuse.

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EUREKA MATH2

Land Debrief 5 min Objectives: Describe the Pythagorean theorem and the conditions required to use it. Apply the Pythagorean theorem to determine the length of a hypotenuse. Facilitate a class discussion by asking the following questions. Encourage students to restate or build on one another’s responses. When does the Pythagorean theorem apply? The Pythagorean theorem only applies to right triangles. We need to know both leg lengths to determine the length of the hypotenuse. What relationship does the Pythagorean theorem demonstrate? The Pythagorean theorem demonstrates a relationship among the side lengths of a right triangle. If the leg lengths are a and b and the hypotenuse length is c, then a 2 + b 2 = c 2. If students answer with only an equation, ask them what a, b, and c represent. We compared side lengths of squares with the side lengths of a right triangle. How can we use the squares and right triangle to explain the Pythagorean theorem? For a right triangle, the areas of the squares that have the same side lengths as the lengths of the triangle’s legs sum to the area of the square that has the same side length as the length of the hypotenuse. Did you use your understanding of perfect squares while using the Pythagorean theorem? Explain. Yes. I used the strategy of solving an equation of the form x 2 = p for x. That allowed me to determine the value of c. To preview upcoming learning, consider posing the following questions. Allow students to make predictions and discuss but do not confirm or deny any responses.

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7–8 ▸ M1 ▸ TD ▸ Lesson 19

Do you think c2 will always be a perfect square or related to a perfect square? Do you think the Pythagorean theorem could be helpful if we know the length of one leg of a right triangle and the length of the hypotenuse but not the length of the other leg? How?

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Examples Find the length of the hypotenuse c.

The Pythagorean Theorem In this lesson, we •

developed a conjecture about the relationship among the side lengths of a right triangle.

•

discussed the conditions required for using the Pythagorean theorem.

•

found the hypotenuse length of right triangles when the square of the hypotenuse length was a perfect square or related to a perfect square.

1. Terminology

a2 + b2 = c 2

A leg of a right triangle is a side adjacent to the right angle.

The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of a right triangle.

The Pythagorean Theorem

72 + 242 = c 2 Substitute the values of the leg lengths for a and b in either order.

49 + 576 = c 2 625 = c 2 25 = c

The hypotenuse is the side opposite the right angle.

In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse length.

a2 + b2 = c 2

The legs of a right triangle are the sides that are adjacent to the right angle.

The length of the hypotenuse is 25 units.

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277

278

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

3 2

a2 + b2 = c 2

2 + (3_) = c 2 2

Use the perfect squares 25 and 4 to determine the length of the hypotenuse.

_ _

4 + 9_ = c 2 4

__ _

16 9 + = c2 4 4

25 5 5 5 2 = ( )( ) = ( ) 2 2 2 4

25 = c2 4

5 =c 2

The length of the hypotenuse is _5 units. 2

RECAP

279

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 19

PRACTICE

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

6. So-hee and Dylan each make errors while finding the length of the hypotenuse c of the following right triangle.

1. When does the relationship a2 + b 2 = c 2 hold true? The relationship a2 + b 2 = c 2 holds true for right triangles when the leg lengths are a and b and the hypotenuse length is c.

For problems 2–5, find the length of the hypotenuse c.

So-hee’s work:

Dylan’s work:

a2 + b2 = c 2 2

0.6

0.8

a2 + b2 = c 2

92 + 122 = c 2

144 + 81 = c 2

18 + 24 = c 2

225 = c 2

42 = c 2

12 + 9 = c 40

112.5 = c The length of the hypotenuse is 112.5 units.

The length of the hypotenuse is 42 units.

1 unit a. Describe all the errors So-hee makes.

50 units

So-hee divides 225 by 2 instead of finding the number that squares to get 225.

b. Describe all the errors Dylan makes.

4 3

432

c. Find the correct length of the hypotenuse.

5 4

5_ units 3

Dylan multiplies 9 and 12 by 2 instead of squaring each number. He also says that the length of the hypotenuse is the value of c 2 instead of c.

15 units

13 units __ 4

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P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 19

Remember For problems 7–10, divide. 3 ÷ 8_ 7. _ 4 9 27 __ 32

9 2_ ÷ 8. − __ 10 3

__ − 27 20

9. − 4_ ÷ (− 8_) 7 3 12 __ 56

4 3 ÷ ___ 10. _ 4 (− 3 ) 9 − __ 16

For problems 11 and 12, evaluate. Write the answer in scientific notation. 11. (7.5 × 109) + (4.7 × 109)

1.22 × 1010

12. (6.4 × 108) ÷ 3,200

2 × 105

13. On a given day, the population of Earth was 7,597,175,534 people. Which number is the closest estimate of Earth’s population on that day? A. 7 × 109 B. 8 × 109 C. 7 × 1010 D. 8 × 1010

P R ACT I C E

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Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 20 LESSON 20

Using the Pythagorean Theorem Use square root notation to express lengths that are not rational and place them on a number line. Approximate the value of square roots by using whole-number benchmarks.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20 Name

EXIT TICKET

Date

1. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

c 7

a2 + b2 = c2 2

(5) + (7)2 = c2 25 + 49 = c2 74 = c2 —

√ 74 = c

The number 74 is between 64 and 81. So the value of c is between 8 and 9 because 82 = 64 and 92 = 81.

—

The length of the hypotenuse is √ 74 units, which is between 8 units and 9 units.

— 2 2. (√ 9 )

For problems 2 and 3, evaluate.

—

• What details are important to think about when estimating the length of the hypotenuse c when c is not a whole number?

—

4. Order the following numbers from least to greatest: 3, √ 8 , 1, √ 5 , 4, √ 2 .

— — — 1, √ 2 , √ 5 , √ 8 , 3, 4

In this lesson, students use the Pythagorean theorem to determine the length of the hypotenuse c of a right triangle and realize that c 2 is not always a perfect square or related to a perfect square, which means that the length of the hypotenuse is not always a rational number. In these cases, students consider which two consecutive whole numbers have squares that c 2 is between to determine an approximation for the length of the hypotenuse. The introduction of the square root symbol allows students to find the exact length of the hypotenuse of a right triangle when the length is not a rational number. Students continue to apply the Pythagorean theorem while engaging in a hands-on activity by recreating the Spiral of Theodorus. By placing hypotenuse lengths from the spiral on a number line, students begin to approximate decimal values of nonperfect squares. This lesson formally defines the term square root.

Key Questions

— 2 3. (√ 3 ) 3

Lesson at a Glance

• Why do we need square root notation?

—

• What do you know about the location of √x on a number line when x is not a perfect square?

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Achievement Descriptors 7–8.Mod1.AD7 Determine whether numbers are rational or irrational. (7.NS.A.2.d, 8.NS.A.1) 7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p, where p is a positive rational

number. (8.EE.A.2)

7–8.Mod1.AD12 Evaluate square roots of small perfect squares and cube roots of small

perfect cubes. (8.EE.A.2)

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Using Square Root Notation

• Spiral of Theodorus

• The Spiral of Theodorus

• Index card

• Square Roots on a Number Line

Lesson Preparation

Land 10 min

• None

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Fluency Solve Equations of the Form x 2 = p Students solve equations of the form x 2 = p to prepare for applying the Pythagorean theorem to find the unknown length of the hypotenuse. Directions: Solve each equation. 1.

x2 = 1

1 and −1

x 2 = 36

6 and −6

x 2 = 64

8 and −8

x 2 = 121

11 and −11

49 = x 2

7 and −7

4 = x2 9

2 2 and − 3 3

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EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Students find which two whole numbers the length of the hypotenuse c is between when c is not rational. Display the triangle and have students find the length of the hypotenuse.

Teacher Note This lesson combines and expands on the learning activities from Eureka Math2 grade 8 module 1 lessons 19 and 20. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

9 Note that students may seek guidance when they get to 97 = c 2 because this is their first encounter with a solution that is not rational. Use the following prompts to facilitate a discussion. What is different about finding the length of the hypotenuse for this problem compared to the problems we saw in the previous lesson? This problem is different because when I found 97 = c 2, I couldn’t think of a number that I could square to get 97. What do you know about the length of the hypotenuse? It must be a positive number because length is a positive number. It’s not a whole number because 97 is not a perfect square. I think it’s between 9 units and 10 units because 97 is between 81 and 100. Is 97 a perfect square? How do you know? No. There is no integer that can be squared to get 97. So we do not currently have a way to describe the number with a square of 97. We can use reasoning to determine a range in which the square of 97 must be located. What is the closest perfect square that is less than 97? The number 81 is the closest perfect square that is less than 97. © Great Minds PBC

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EUREKA MATH2

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What is the closest perfect square that is greater than 97? The number 100 is the closest perfect square that is greater than 97. Have students turn and talk about the following question. We just identified that the value of c 2, which is 97, is between the perfect squares 81 and 100. Can we determine two consecutive whole numbers that the value of c is between? If so, how? Yes. The value of c is between 9 and 10 because 92 = 81 and 102 = 100. We know the value of c is a positive number because it is the length of the hypotenuse and lengths are positive numbers. We can also determine that the value of c is between 9 and 10 because c 2, which is 97, is between the perfect squares 81, which is 92, and 100, which is 102. Today, we will approximate the length of the hypotenuse c when c2 is not a perfect square or related to a perfect square, and we will discover how to precisely represent a nonperfect square.

Learn Using Square Root Notation Students use square root notation to express the length of the hypotenuse c when c2 is not a perfect square or related to a perfect square.

Language Support To support the term consecutive as it relates to numbers within a mathematical context, consider displaying the words consecutive and nonconsecutive. Ask students to provide examples and nonexamples of consecutive whole numbers. Record responses next to the appropriate word.

Teacher Note At this point, students need to recognize only that the value of c is between two consecutive whole numbers. Students will delve into closer approximations of square roots in lesson 21. To prepare for this thinking, consider asking students to make a guess as to whether the value of c is closer to 9 or 10.

Display the following equation. Then use the prompts to help students understand the need for new notation.

97 = c 2 We know that the length of the hypotenuse is the positive number that equals 97 when squared. We know that it’s between 9 and 10, but we need a way to represent that number.

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—

Display the following equation.

√97 = c

The symbol in this equation is the square root symbol. We read this equation as The square root of 97 equals c. This means c is the number that, when squared, is 97.

A square root of a nonnegative number x is a number with a square that is x. The — expression √x represents the positive square root of x when x is a positive number. If — x is 0, then √0 = 0.

—

Have students complete problems 1 and 2 to record the square root symbol, √1 , and its meaning in their books.

—

1. Write the square root symbol. √1

2. Describe why the square root symbol is needed and how it is used to find the length of the hypotenuse.

9 42 + 92 = c 2 16 + 81 = c 2 97 = c 2 — √97 = c

Teacher Note This is students’ initial introduction to the definition and notation of square root. Because the square root symbol resembles the long division symbol, some students may evaluate the square root of a number by dividing by 2. Encourage students to recognize the differences in the square root symbol and the long division symbol, annotate or highlight these differences in problem 1, and look for these differences when determining the intent of a problem. Students will explore square roots as values on a number line at the end of this lesson and examine closer approximations in lesson 21.

Language Support The word root has many mathematical and nonmathematical meanings. Facilitate a class discussion about other meanings of root. Consider showing a picture for each of these meanings: • Plant root • Root vegetables • Tooth root

I need the square root symbol because 97 = c 2, and no integer squared is 97. The symbol is used when the square of the hypotenuse is not a perfect square or related to a perfect square.

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When students have completed the problems, use the following prompts to facilitate a conversation about square root notation. Encourage students to add more examples to their notes as needed.

—

We can practice reading this new notation. If √97 is the positive number that we square to get 97, how would you describe each of the following numbers?

—

Display the numbers √5 and √16 and point to them one at a time.

—

The number √5 is the positive number we square to get 5.

—

The number √16 is the positive number we square to get 16.

—

Teacher Note If students question why −4 is not also — equivalent to √16 , they may be thinking of solving the equation x 2 = p, where x could be a positive number or a negative number when p is positive. Restate that when considering — problems of the form √x , we define x to be nonnegative.

Are either of these expressions equivalent to a whole number? Explain.

Yes. √16 is equivalent to the whole number 4 because we can square 4 to get 16.

Display each of the following expressions. Have students give a thumbs-up if the number is equivalent to a whole number and give a thumbs-down if the number is not equivalent to a whole number.

—

√10

—

√1

—

√4

—

√27

Have students work with a partner to complete problems 3 and 4.

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—

√0

Teacher Note Look for students who confuse square roots with division. Because the square root symbol resembles a long division symbol, some students may mistakenly divide by 2 rather than take the square root. If students give a thumbs-down for all the expressions with odd radicands, they are likely working under that misconception.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

For problems 3 and 4, use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know. 3.

Differentiation: Challenge Consider pairing students and having one student make up a value for c2 while the other student determines which two consecutive whole numbers c is between. Then have students switch roles and repeat the activity.

5 c 6

a2 + b2 = c 2 52 + 62 = c 2 25 + 36 = c 2 61 = c 2 — √61 = c

The number 61 is between 49 and 64. So the value of c is between 7 and 8 because 72 = 49 and 82 = 64

—

The length of the hypotenuse is √61 units, which is between 7 units and 8 units.

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1 1 a2 + b2 = c 2 12 + 12 = c 2 1 + 1 = c2 2 = c2 — √2 = c

The number 2 is between 1 and 4. So the value of c is between 1 and 2 because 12 = 1 and 22 = 4.

—

The length of the hypotenuse is √2 units, which is between 1 unit and 2 units.

Discuss responses as needed. Then use the following prompt to solidify understanding about the need for square root notation. Why do we need to use square root notation to express the length of the hypotenuse in problems 3 and 4?

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20 ▸ Spiral of Theodorus

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20

We need to use square root notation because the exact length of the hypotenuse is not a whole number.

We can represent the exact length of the hypotenuse by using square root notation. Stating the length of the hypotenuse by using square root notation is more precise than approximating its length by stating the consecutive whole numbers the length is between. 0

The Spiral of Theodorus Students construct the Spiral of Theodorus to relate square roots to lengths. Have students remove the Spiral of Theodorus removable from their books. Distribute one index card to each student. 442

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Invite students to study the triangle, and then lead them through a class discussion by using the following prompts. What do you notice about the triangle?

To support the fine motor and executive function demands of the task, consider offering the following options for completing the Spiral of Theodorus:

It is the same triangle we worked with in problem 4.

—

UDL: Action & Expression

What is the length of the hypotenuse?

The length of the hypotenuse is √2 units.

—

Label the length of the hypotenuse √2 .

• Allow students to work with a partner to complete the Spiral of Theodorus.

We will create a spiral made by connecting right triangles, called the Spiral of Theodorus, starting from the triangle we already have. Each triangle has one leg with a length of 1 unit and another leg with the same length as the hypotenuse of the triangle before it.

Guide students through the creation of the next triangle. Instruct students to use their index card to measure one leg of the triangle and mark a length of 1 unit on the card. Then have students line up the corner of their index card along the hypotenuse to make a right angle.

• Provide visual and written directions that students can refer to as they create the Spiral of Theodorus. • Open and display the Spiral of Theodorus teacher interactive, pausing as desired so that students can calculate hypotenuse lengths. Then provide students with a copy of the Completed Spiral of Theodorus (in the teacher edition). Have them use an index card to transfer lengths from the spiral to the number line.

1 √2

√8

√9

√10

√11

√13

√12

√15

√14

√16

√17 1

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 20

This page may be reproduced for classroom use only.

√7

√3

7–8 ▸ M1 ▸ TD ▸ Lesson 20 ▸ Completed Spiral of Theodorus

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√6

√4

√5

1 1

1 0

EUREKA MATH2

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Then, by using the index card as a straightedge, have students draw a 1-unit segment from the mark on the card to the top vertex of the triangle.

1 1

Instruct students to mark a right angle and a length of 1 unit for the new segment. Finally, have students use the index card as a straightedge to connect the end of the new segment to the center point of the spiral to form the hypotenuse of the new triangle.

1 √2 1

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When the first drawn triangle of the spiral is complete, have students work with a partner to determine the length of the hypotenuse. Anticipate that students may seek guidance — 2 evaluating (√2 ) because this is their first encounter with squaring a number written with square root notation. Use the following prompts to facilitate a discussion. To determine the length of the hypotenuse, we must use the Pythagorean theorem. Display the following work.

a2 + b2 = c 2 2 (√—2 ) + 12 = c 2

What is different about finding the length of the hypotenuse in this triangle than in previous triangles?

UDL: Representation

problems are whole numbers.

Consider highlighting or drawing boxes

— One of the leg lengths of this triangle is √2 units, and the leg lengths in previous

Let’s consider what it means to square a number written with square root notation. — According to the definition of a square root, what is √2 ?

— — 2 Because √2 is the number with a square of 2, what is (√ 2) ?

It is the number with a square that is 2.

After students have a clear understanding about squaring numbers written with square root notation, finish calculating the length of the hypotenuse c as a class.

a2 + b2 = c 2 2 (√—2 ) + 12 = c 2

— 2

around (√2 ) and 2 in the following work to emphasize their equivalence.

a2 + b2 = c 2

(√—2)2 + 12 = c 2 2 + 1 = c2 3 = c2 —

√3 = c

2 + 1 = c2 3 = c2 — √3 = c

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—

Have students label the hypotenuse of their new triangle √3 . Then have students repeat the entire process for the next triangle and each triangle after that. If students need support to get started on the next triangle, guide them through one more triangle before they continue on their own. The goal is for each student to create their own Spiral of Theodorus, but they may check in with another student as they work. Circulate as students work to assist as needed in the construction process and to offer support in squaring square roots in the calculations. Students may use the workspace provided to do the hypotenuse length calculations.

EUREKA MATH2

Teacher Note Because students are finding the length of the hypotenuse of a triangle, only the positive solution makes sense. Students solve equations of the form of x 2 = p by using square root notation in lesson 23.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

The Spiral of Theodorus

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LESSON

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Allow students to work on their spirals for about 5–7 minutes. Students do not need — to complete the spiral all the way to the triangle with the hypotenuse length of √17 units if time does not permit.

√4

√5

√6

√3

√8

√9

1 √2

√7

Theodorus taught mathematics to Plato and Theaetetus and is known for his contribution to the development of irrational numbers. The completed spiral shown is where Theodorus stopped. The spiral could go on forever, but the 17th triangle begins to overlap with the rest of the spiral. Although there are plausible theories, the reason Theodorus stopped at — the triangle with a hypotenuse length of √17 units is unknown.

√10

1 1

√11

√12

√17

√13

√14

√15

Teacher Note

√16

To further engage students, consider displaying the following examples of spirals found in nature.

1 1

Use the following prompts to debrief the activity. What do you notice about the length of each hypotenuse as each triangle is added to the spiral? The number under the square root symbol increases by 1 each time. Consider discussing with students that some of the hypotenuse lengths can be written as whole numbers.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Square Roots on a Number Line Students use the lengths from the Spiral of Theodorus to place square roots on a number line. Direct students back to their spiral. Make sure they have their index card accessible. The spiral shows us that each square root has a numerical value because each square root represents the length of a hypotenuse. Let’s use our index card to measure each hypotenuse and then mark the square root lengths on the number line.

Have students line up the corner of their index card on the center point of the spiral and line — up the edge on the hypotenuse labeled √2 . Then, have the students mark the length of the hypotenuse on their card with a tick mark. Next, move the index card to the number line, — aligning the corner of the card at 0, and plot a point at √2 on the number line. Label the — point √2 .

1 1

q10

q17

—

q16

q11 1

q12

When students repeatedly solve for the value of c in a2 + b2 = c 2 for each subsequent triangle in the spiral and recognize that the radicand increases by 1 each time, they are looking for and expressing regularity in repeated reasoning (MP8). Ask the following questions to promote MP8: • What patterns do you notice when you solve for c from one triangle to the next? • How can this pattern help you solve for c more efficiently? • Will this pattern always work? Explain.

Promoting the Standards for Mathematical Practice

q13 1

q14 1

q15

Recall that √2 is between 1 and 2. We can use an inequality to describe this. 448

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

—

Display 1 < √2 < 2.

—

We read this inequality as “√2 is greater than 1 and less than 2.”

Teacher Note

— Yes. The number line verifies this estimate because the point for √2 is between 1 and 2. — Can we say with certainty which whole number √2 is closer to? Why? — Yes. We can say with certainty that √2 is closer to 1. We can verify this by measuring Does your number line verify this estimate? How?

—

Using the inequality 1 < √2 < 2 to describe — √2 is an informal introduction to compound inequalities. Students define and learn more about compound statements in future courses.

—

on the number line.

What might be a good estimate to the nearest tenth for the value of √2 based on your number line? How can we check our estimate? Student estimates may vary, but listen for estimates around 1.3 and 1.4. Students can check their estimates by squaring their value and seeing how close their estimate is to 2.

—

Have students use their index card to repeat the transfer process through at least √9 , or for as many lengths as completed on their spiral. If short on time, consider displaying the finished number line after students have had the opportunity to transfer at least two lengths to the number line.

v5 v6 v7 v8

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Once students complete the transfer process or are viewing a finished number line, have them choose one of the square roots that is not a whole number. Have them write an inequality to describe its location on the number line between two consecutive whole — numbers, similar to the inequality to describe √2 . Have students share different examples by displaying them and reading them aloud.

— 1 < √3 < 2 — 2 < √5 < 3 — 2 < √6 < 3 — 2 < √7 < 3 — 2 < √8 < 3 Sample:

Differentiation: Challenge For students ready for more of a challenge, ask questions about other square roots similar — to the questions posed to the class about √2 . • Is ____ closer to ____ or to ____?

— — —

—

Have students think–pair–share about the following question.

Why are the values of √5 , √6 , √7 , and √8 all between 2 and 3?

— 2

We know that (√5 ) = 5, (√6 ) = 6, (√7 ) = 7, and (√8 ) = 8. The numbers 5, 6, 7, and 8 are between two perfect squares, 4 and 9. Because 22 = 4 and 32 = 9, the numbers we square to get 5, 6, 7, and 8 must be between 2 and 3. Have students compare their number line with their partner’s number line. Students may discover that the points on their number line are in slightly different locations than their — — partner’s number line. They may also notice that the points for √4 and √9 are not exactly on 2 and 3 as expected. Such discoveries provide a basis for rich discussion. Use the following questions to engage the class in a discussion about accuracy:

• Based on your number line, what might be a good estimate for ____ to the nearest tenth? Consider extending even further to foreshadow lesson 21. • How might we come up with more precise — approximations of numbers such as √2 ?

• Why might your points be placed slightly differently than your partner’s points? • In what ways might the materials or the method we used to create the spiral lead to slight inaccuracies on our number lines? • What kind of human errors may have affected our accuracy? • How do you think you could determine a more accurate location for each number?

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7–8 ▸ M1 ▸ TD ▸ Lesson 20

Land Debrief 5 min Objectives: Use square root notation to express lengths that are not rational and place them on a number line. Approximate the value of square roots by using whole-number benchmarks. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. What details are important to think about when estimating the length of the hypotenuse c when c is not a whole number? We must think about which two perfect squares are closest to the value of c 2 and then determine that c must be between the square roots of the two closest perfect squares. Why do we need square root notation? We need square root notation to precisely represent a number whose square is not a perfect square or related to a perfect square.

—

What do you notice about the location of √x on a number line when x is not a perfect square?

When x is not a perfect square, the square root of x is located between two whole numbers on the number line.

—

How do we determine the approximate location of √2 on the number line?

— 2

We know that (√2 ) = 2, and 2 is between the perfect squares 1 and 4. Because 12 = 1 and 22 = 4, the number that is squared to get 2 must be between 1 and 2.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

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7–8 ▸ M1 ▸ TD ▸ Lesson 20

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20 Name

RECAP Date

—2

In this lesson, we used square root notation to express hypotenuse lengths that are not rational.

•

determined which two consecutive whole numbers the length of a hypotenuse is between.

•

found the squares of numbers written with square root notation.

•

created the Spiral of Theodorus to relate square roots to length measurements.

•

plotted points on a number line to represent locations of square roots.

Because √70 is the number with — 2 a square that is 70, then (√70 ) = 70.

3. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

Terminology

•

—

2. Evaluate (√70 ) .

Using the Pythagorean Theorem

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

A square root of a nonnegative number x is a number with a square that is x. The — expression √x represents the

positive square root of x when x is a positive number. If x is 0, — then √0 = 0.

— √7

—2 12 + (√7 ) = c 2 1+7=c

Examples 1. Use square root notation to express the length of the hypotenuse c. Then approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. Explain how you know.

—

(√—x ) = x for any nonnegative 2

number x.

8 = c2 —

√8 = c

The number 8 is between 4 and 9. So the value of c is between 2 and 3 because 22 = 4 and 32 = 9.

The length of the hypotenuse is √8 units, which is between 2 units and 3 units.

c 2

a + b2 = c 2 52 + 92 = c 2 25 + 81 = c 2

What number squared is 106? Use the square root symbol to represent the exact value.

106 = c 2 —

This expression means the square root of 106.

√106 = c

The number 106 is between 100 and 121. So the value of c is between 10 and 11 because 102 = 100 and 112 = 121.

—

The length of the hypotenuse is √106 units, which is between 10 units and 11 units.

452

293

294

RECAP

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

—

4. Which point represents the approximate location of √12 on a number line? 2 (√— 12 ) = 12, and 12 is between the two

perfect squares 9 and 16.

B C D

E F

Because 32 = 9 and 42 = 16, the number squared to get 12 must be between 3 and 4.

—

Point B represents the approximate location of √12 on the number line. —

—

The inequality 3 < √12 < 4 indicates that √12 is greater than 3 and less than 4.

RECAP

295

453

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 20

PRACTICE

Name

Date

For problems 1 and 2, determine which two consecutive whole numbers the length of the hypotenuse c is between. 2.

For problems 12–17, use square root notation to express the length of the hypotenuse. If the length is not a whole number, approximate the length by stating the two consecutive whole number lengths the length of the hypotenuse is between. 12. The leg lengths of a right triangle measure 1 unit and 3 units.

—

√10 units

14. The leg lengths of a right triangle measure — 1 unit and √5 units.

c 11 units and 12 units

—

√6 units

7 units and 8 units

— 4. √40

— 5. √125

— 6. √144

Yes, 4

No, 11 and 12

(√—2 ) = 2

16.

No, 6 and 7

True

—

√137 units

The length of the hypotenuse is between

(√—4 ) = 16

(√— 16 ) = 4 2

False

— 11. √81 = 32

11 units and 12 units.

False

—2

3 units

—

√85 units

The length of the hypotenuse is between 9 units and 10 units.

18. Consider a right triangle with hypotenuse length c and leg lengths a and b. Is it possible for c 2 to be a perfect square when a2 and b2 are not perfect squares? If so, give an example.

10. 11 = (√11 ) True

15. The leg lengths of a right triangle measure — — √2 units and √7 units.

Yes, 12

The length of the hypotenuse is between

10 units and 11 units.

17.

For problems 7–11, determine whether the statement is true or false. 7.

—

√116 units

The length of the hypotenuse is between 2 units and 3 units.

For problems 3–6, state whether the expression represents a whole number. If yes, state the whole number that is equal to the expression. If no, state the two consecutive whole numbers that the expression is between.

— 3. √16

13. The leg lengths of a right triangle measure 4 units and 10 units.

The length of the hypotenuse is between 3 units and 4 units.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

Sample:

—

Yes. A right triangle with leg lengths of √3 units and √6 units has a hypotenuse length that measures 3 units.

True

454

297

298

P R ACT I C E

EUREKA MATH2 7–8 ▸ M1 ▸ TD ▸ Lesson 20

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

—

19. Which point represents the approximate location of √8 on a number line? Write an inequality — to describe the location of √8 between two consecutive whole numbers on a number line.

A 0

B 2

D 3

F 5

For problems 27 and 28, solve.

1 28. k 2 = __ 81

27. x 2 = 64

G 6

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20

_1 and −_1

8 and −8

29. While working on calculations for her science homework, Eve saw the following display on her calculator.

— 2 < √8 < 3

Point C

4.1633363e -17 — B. √4 — D. √12

— A. √2 — C. √10 — E. √15

Rad

20. Which numbers are between 3 and 4? Choose all that apply.

(

)

∏

sin

cos

log

𝗑

Rad e

Rad tan

Rad √

Rad 1

Rad 2

Rad 3

Rad –

Rad Ans

Rad EXP

Rad xy

Rad 0

Rad .

Inv

Rad =

AC ÷

Rad +

Write this number in scientific notation to help Eve interpret her calculator’s display.

Remember

4.1633363 × 10−17

For problems 21–24, divide. 1 1 21. 2 ÷ 2 _ 22. −6 ÷ 2 _ 2 2

_4 5

__ − 12 5

1 23. 8 ÷ (−1 _) 4

__ − 32

1 24. −5 ÷ (−6 _) 3

15 __ 19

For problems 25 and 26, state whether the number is a perfect square, a perfect cube, both, or neither. Explain how you know. 25. 110 The number 110 is neither because there is no integer that can be squared or cubed to get 110. 26. −8 The number −8 is a perfect cube because it is the cube of the integer −2.

P R ACT I C E

299

300

P R ACT I C E

455

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 20 ▸ Completed Spiral of Theodorus

√12

√8 1

√7

√6

√9

√5

√10

√4

√11

√3

√2

√13

√14

√15

√16

√17

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 20

456

This page may be reproduced for classroom use only.

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 21 LESSON 21

Approximating Values of Roots Approximate values of square roots and cube roots.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21 Name

Date

EXIT TICKET

—

Approximate the value of √17 to the nearest tenth by using consecutive whole numbers, tenths, and hundredths. Explain your thinking.

—

The value of √17 is approximately 4.1.

—

Because 17 is between the perfect squares 16 and 25, the value of √17 is between 4 and 5. 2

Next, I consider the tenths from 4 to 5. I find that 17 is between 4.1 and 4.22, or 16.81 and 17.64.

Then, I consider the hundredths from 4.1 to 4.2. I find that 17 is between 4.122 and 4.132, or 16.9744 — and 17.0569. Both 4.12 and 4.13 round to 4.1, so √17 ≈ 4.1.

Lesson at a Glance In this digital lesson, students consider the values of square roots and cube roots and approximate the locations of these values on a number line. Students find the approximate locations of these less familiar numbers by using intervals with rational number benchmarks such as perfect squares and perfect cubes. Students then use a visual representation of the squares and cubes of numbers to determine more precise approximations. They continue with their approximations by testing smaller and smaller intervals, including to the tenths place, the hundredths place, and the thousandths place. This lesson introduces the term adjust and defines the term cube root. Use the digital platform to prepare for and facilitate this lesson. Students will also interact with lesson content and activities via the digital platform.

Key Questions • Can we approximate the value of a square root more precisely than by using whole-number benchmarks? How? • Can we approximate the value of a cube root more precisely than by using whole-number benchmarks? How?

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Achievement Descriptors 7–8.Mod1.AD8 Use rational approximations of irrational numbers to compare the size

of irrational numbers. (8.NS.A.2) 7–8.Mod1.AD9 Locate irrational numbers approximately on a number line. (8.NS.A.2) 7–8.Mod1.AD12 Evaluate square roots of small perfect squares and cube roots of small

perfect cubes. (8.EE.A.2)

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Approximating Square Roots

• Computers or devices (1 per student pair)

• Approximating Cube Roots

Lesson Preparation

Land 10 min

• None

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Fluency Plot on a Number Line Students plot square roots of perfect squares on a number line to prepare for approximating values of square roots. Directions: Plot and label the number on the number line. —

9 10

—

2 3 √4

—

√64

9 10

√9

9 10

√1

9 10

√36

9 10

√4

√64

√9

√1

√36

460

Teacher Note

—

Students may use the Number Lines 0 to 10 removable.

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TD ▸ Lesson 21

—

Students approximate the value of √8 and consider what would improve their approximation.

Given a right triangle with leg lengths of 2 units, students find the exact length of — — the hypotenuse to be √8 units. They then determine that the value of √8 is between — 2 and 3. Students plot the approximate location of √8 on a number line. After viewing — the class’s approximations, students reason that √8 is closer to 3 than 2 and consider what would improve their approximation. How did you determine which two consecutive whole numbers the — value of √8 is between?

I know that 8 is between the perfect squares 4 and 9. Because 22 = 4 and 32 = 9, I determined that the value of — √8 is between 2 and 3.

Why do you think the class approximations are grouped closer to 3 on the number line?

The number 8 is closer to 9 than it is to 4. Because 22 = 4 and 32 = 9, the — approximations for the value of √8 are closer to 3 on the number line.

461

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Learn Approximating Square Roots

—

Students approximate the values of square roots.

Students improve their approximation of the value of √8 by determining intervals to the tenths place, the hundredths place, and the thousandths place. For each

— 2

interval, they square the first number and last number of the interval and compare

these values to the value of (√8 ) to determine whether the inequality that represents the interval is a true statement.

— 2 2.82 < (√8 ) < 2.92 7.84 < 8 < 8.41 — 2.8 < √8 < 2.9

— Students use these intervals to approximate the value of √8 by rounding to the nearest — 2.8 < √8 < 2.9 — 2.82 < √8 < 2.83 — 2.828 < √8 < 2.829

whole number, tenth, and hundredth.

462

UDL: Representation Digital activities align to the UDL principle of Representation by including the following elements: • Multiple formats and modes. Students choose an interval on a number line and see the first number and last number of the interval in an inequality. This allows students to move from a pictorial representation to an abstract representation of the interval. • Scaffolds that connect new information to prior knowledge. Students connect their prior knowledge of lengths, areas, and volumes to interpret numbers with square root or cube root notation as lengths.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

As students adjust their intervals, they are also given a visual representation of the squares of these values to see how they form a smaller square and a larger square — than the one created by the length of the hypotenuse, √8 units. They use the visual — representation to help them determine whether √8 is in the selected interval. The difference in the sizes of the squares becomes less apparent, prompting students to rely on the values in their inequalities.

√8

As the intervals become smaller, can we rely on the animated squares to help — us determine the two numbers that the value of √8 is between? Explain.

No. As the intervals become smaller, the animated squares look to be almost the same sizes. We have to focus on the inequalities instead.

463

7–8 ▸ M1 ▸ TD ▸ Lesson 21

—

How does the work we have done so far help us approximate the value of √8 ?

—

We know the value of √8 is between 2.828 and 2.829.

— How can we approximate the value of √8 by rounding to the nearest hundredth? — Because the value of √8 must be greater than 2.828, we know the digit in the — thousandths place is greater than 5. So we can approximate the value of √8 by rounding up to 2.83. — Students then consider a triangle with a hypotenuse length of √32 units. They follow — a similar iterative process to approximate the value of √32 .

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21 Name

LESSON Date

Approximating Values of Roots Approximating Square Roots

Approximating Cube Roots Students approximate the values of cube roots. 3—

301

Students apply what they learned about approximating square roots to approximate the value of √18 . They select an interval, cube the first and last number of that 3— 3

interval, and compare those values to the value of ( √18 ) to determine whether the 3— 3 23 < ( √18 ) < 33

inequality that represents the interval is a true statement.

8 < 18 < 27 3— 2 < √18 < 3

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

As students adjust their intervals, they are also given a visual representation of the 3—

cubes of these values to see how they form a smaller cube or a larger cube than the one created by the edge length of the cube with edge length √18 units. Students 3—

use the visual representation to help them determine whether √18 is in the selected interval. The difference in the size of the cubes becomes less apparent, prompting students to rely on the values in their inequalities.

Each iteration of this process helps students make increasingly precise approximations. At the end of the activity, students engage in an error analysis activity that highlights how approximating a cube root differs from approximating a square root. 3—

√134

112 < 134 < 122 121 < 134 < 144

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7–8 ▸ M1 ▸ TD ▸ Lesson 21

—

√

1. Write the cube root symbol. 3

2. Explain why the cube root symbol is needed to determine the edge length of the cube shown.

Volume: 18 cubic units The volume of the cube is found by cubing the edge length, so the edge length is the 3—

cube root of the volume. I need the cube root symbol because the edge length of the cube is √18 units and no integer cubed is 18.

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EUREKA MATH2

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Land Debrief 5 min Objective: Approximate values of square roots and cube roots. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses.

—

Suppose the hypotenuse length of a right triangle is √12 units. Which two consecutive — whole numbers is the value of √12 between?

—

The value of √12 is between 3 and 4.

—

Can we approximate the value of √12 more precisely? How?

Yes. We can consider the tenths from 3 to 4. Because 12 is between 3.42 and 3.52, the — value of √12 is between 3.4 and 3.5. To get a more precise approximation, we can look at consecutive hundredths, thousandths, and so on. How is approximating the value of a square root similar to approximating the value of a cube root? How are they different? They are similar because both processes involve determining two numbers that the value of a square root or the value of a cube root is between. Both processes also involve finding smaller intervals to get a closer approximation. They are different because we consider squares of numbers when finding intervals that include the value of a square root. We consider cubes of numbers when finding intervals that include the value of a cube root. 3—

Suppose the edge length of a cube is √136 units. What strategy will help us 3—

approximate the value of √136 ? What two consecutive whole numbers is the value 3—

of √136 between?

3—

We can begin by identifying a perfect cube less than 136 and a perfect cube greater

than 136. We know 53 is 125 and 63 is 216, so the value of √136 is between √125 and √216 . 3—

This means the value of √136 is between 5 and 6.

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7–8 ▸ M1 ▸ TD ▸ Lesson 21

EUREKA MATH2

3—

Can we approximate the value of √136 more precisely? How? 3—

Yes. We can consider the tenths from 5 to 6. Because 136 is between 5.13 and 5.23, the

value of √136 is between 5.1 and 5.2. To get a more precise approximation, we can look at consecutive hundredths, thousandths, and so on.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support. For the Practice problems, consider providing students with a calculator to support their computation as they approximate square roots and cube roots.

468

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21

RECAP

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

3—

5. Approximate the value of √46 to the nearest tenth. Explain your thinking. 3—

Because 46 is between the perfect cubes 27 and 64, the value of √46 is between 3 and 4. Next, I consider the tenths from 3 to 4. I find that 46 is between 3.53 and 3.63, or 42.875 and 46.656.

Approximating Values of Roots In this lesson, we

Then, I consider the hundredths from 3.5 to 3.6. I find that 46 is between 3.583 and 3.593, 3— or 45.882712 and 46.268279. Both 3.58 and 3.59 round to 3.6, so √46 ≈ 3.6.

Terminology

•

approximated values of square roots.

•

explored and defined cube root notation and approximated the values of cube roots.

The cube root of a number x is a

number with a cube that is x. The 3— expression √x represents the cube root of x.

Examples — < √68 <

3— < √12 <

For problems 1 and 2, determine the two consecutive whole numbers each value is between.

68 is between the perfect squares 64 and 81. —

—

√64 < √68 < √81

— 8 < √68 < 9

—

12 is between the perfect cubes 8 and 27. 3— 3— 3— √8 < √12 < √27 3— 2 < √12 < 3

—

For problems 3 and 4, round each value to the nearest whole number. 3 4. √53

3. √200

200 is between the perfect squares 196 and 225.

53 is between the perfect cubes 27 and 64.

200 is between 14.12 and 14.22, or 198.81 and 201.64. Both 14.1 and 14.2 round to 14.

53 is between 3.73 and 3.83, or 50.653 and 54.872. Both 3.7 and 3.8 round to 4.

— — — √196 < √200 < √225 — 14 < √200 < 15

—

3 3 3 √ 27 < √53 < √64 3—

3 < √53 < 4

305

306

RECAP

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 21

PRACTICE

Name

Date

— < √17 <

— < √88 <

— < √6 <

5 3

3—

< √30 <

12. √57

10 4

Hundredth

7.5

7.55

10.9

10.91

4.1

4.12

14. √70

15. Find the exact length of the hypotenuse c. Then approximate the length by rounding to the nearest tenth.

—

Tenth

3—

6. √90

Whole Number

—

Root

13. √119

For problems 5–8, round each value to the nearest whole number. 5. √2

For problems 12–14, approximate the value by rounding to the nearest whole number, tenth, and hundredth.

—

For problems 1–4, determine the two consecutive whole numbers each value is between.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

3—

8. √10

7. √150

—

9. Henry states that the value of √30 is between 5.5 and 5.6. Do you agree with Henry? Why?

—

No. The value of √30 is not between 5.5 and 5.6 because 5.52 = 30.25 and 5.62 = 31.36, which are both greater than 30. 3—

10. Is the value of √25 greater than or less than 3? How do you know? 3—

3—

—

√41 units, 6.4 units

3—

The value of √25 is less than 3. Because √8 = 2 and √27 = 3, I know the value of √25 must be between 2 and 3.

Remember

— 11. Approximate the value of √23 to the nearest tenth. Explain your thinking. — Because 23 is between the perfect squares 16 and 25, the value of √23 is between 4 and 5.

For problems 16–19, divide. 3 17. −1 4_ ÷ 1_ 16. − 1_ ÷ (−1 _) 5 7 5 4 5 __

I consider the tenths from 4 to 5. I find that 23 is between 4.72 and 4.82, or 22.09 and 23.04.

470

__ − 55

Next, I consider the hundredths from 4.7 to 4.8. I find that 23 is between 4.79 and 4.80 , — or 22.9441 and 23.04. Both 4.79 and 4.80 round to 4.8, so √23 ≈ 4.8. 2

307

308

P R ACT I C E

18.

_1 ÷ −2 1_ 3 ( 3)

3 − __ 21

19. 4 1_ ÷ 1_ 6 5 125 ___ 6

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 21

20. Find the length of the hypotenuse c.

12 c

13 units

21. The total volume of fresh water on Earth is approximately 3.5 × 107 cubic kilometers. The total volume of all water on Earth is approximately 1.4 × 109 cubic kilometers. What is the approximate volume of the water on Earth that is not fresh water? Write your answer in scientific notation. The approximate volume of the water on Earth that is not fresh water is 1.365 × 109 cubic kilometers.

P R ACT I C E

309

471

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 22 LESSON 22

Rational and Irrational Numbers Classify real numbers as rational or irrational by their decimal form. Compare and order rational and irrational numbers.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22 Name

EXIT TICKET

Date

1. Indicate whether each number is rational or irrational. Number

Rational

_4 5

√81

—

Irrational

0.20406081…

2.134 × 10−3

0.¯ 3

—

3 √ 9

—

3 √ 40

— 3— I know that the value of √27 is between 5 and 6. I know that the value of √40 is between 3 and 4. — 3— So the value of √27 must be greater than the value of √40 . —

—

3. Plot and label a point at the approximate location of each value on the number line. 14 3 √24 , 4.2¯ 3, √16 , __ , √125

— —

√16

14 3

4.23 4.5

In this lesson, students classify numbers by determining whether the decimal form of each number eventually repeats. They use that distinction and their prior knowledge of rational numbers to develop a definition of irrational numbers. Through the Take a Stand routine, students determine whether different number categories include only rational numbers, only irrational numbers, or both rational and irrational numbers and justify their reasoning. A playful game of Battle Cards allows students to compare rational and irrational numbers, one pair of numbers at a time. Then students order the whole set of cards from least to greatest and plot some of the values on a number line. Students conclude the lesson by using approximation and number sense to order a set of expressions that include irrational numbers. This lesson formally defines the terms irrational number and real number.

Key Questions

2. Compare the values by using the < or > symbol. Explain your reasoning. √27

Lesson at a Glance

— —

√2 4

• What are some similarities and differences between rational and irrational numbers? • What strategies can we use to compare and order rational and irrational numbers?

—— 3—

√125

5 313

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Achievement Descriptors 7–8.Mod1.AD7 Determine whether numbers are rational or irrational. (7.NS.A.2.d, 8.NS.A.1) 7–8.Mod1.AD8 Use rational approximations of irrational numbers to compare the size

of irrational numbers. (8.NS.A.2) 7–8.Mod1.AD9 Locate irrational numbers approximately on a number line. (8.NS.A.2)

Agenda

Materials

Fluency

Teacher

Launch 5 min

• Blank paper (3 sheets)

Learn 30 min • Is It Rational? • Battle Cards • Ordering Expressions

Land 10 min

• Chart paper • Marker • Tape

Students • Battle Cards (1 set per student pair)

Lesson Preparation • Prepare three signs with the following labels: Only Rational Numbers, Only Irrational Numbers, and Both. Post the signs around the classroom. • Copy and cut out the Battle Cards (in the teacher edition). Prepare enough sets for 1 per student pair.

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EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Fluency Evaluate Roots Students evaluate square roots and cube roots to prepare for approximating irrational numbers and expressions that include roots. Directions: Evaluate the expression. 1. 2. 3. 4. 5. 6.

474

—

√49

—

√64

—

√121 3—

√64

3—

√125 3—

√−1

7 8 11 4 5 −1

EUREKA MATH2

Launch

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Students identify the next decimal digit for repeating and nonrepeating decimal forms of numbers. Have students complete problems 1–6 independently. For problems 1–6, identify the decimal digit that comes next in the decimal form of the number. If you cannot identify the next decimal digit, indicate that. 1_ 1. = 0.333333333… 3 3 3—

2. √12 = 2.2894284… 3.

___

__5 = 3.41666666…

—

5. √42 = 6.480740698… 6.

3 = 0.428571428571… 7

This lesson combines and expands on the learning activities from Eureka Math2 grade 8 module 1 lessons 22 and 23. Refer to those lessons as needed for additional questions, problems, and activities that may enhance student understanding.

Cannot identify

144 = 1.45454545… 99

4. 3

Teacher Note

Cannot identify

Bring the class together and invite students to share how they identified which decimal digit comes next in the decimal form of each number. Focus the discussion on whether the decimal form of each number eventually repeats. Guide students to realize that a block of digits eventually repeats in the decimal forms of the numbers in problems 1, 3, 4, and 6, but that does not happen in the decimal forms of the numbers in problems 2 and 5. Based on the observations you made, how would you classify the numbers in problems 1 through 6? I would classify the numbers based on whether they have a decimal form that eventually repeats. © Great Minds PBC

475

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

A rational number has a decimal form that terminates or eventually repeats. Any rational number can be written as a fraction. What could we call numbers that have a decimal form that does not terminate and does not repeat? Allow students to brainstorm, and invite several students to share their suggestions. Consider writing a list of these suggestions for the class to see, and engage the class in a discussion about the suggestions. We call these numbers irrational numbers. An irrational number is a number that is

__p

not rational and cannot be expressed as q for integer p and nonzero integer q. An

UDL: Representation Consider playing the Long Division with Repeating Decimals video to activate prior learning about terminating and repeating decimals.

Language Support

irrational number has a decimal form that neither terminates nor repeats.

If a number is written in decimal form, how can we tell whether it is rational? If the decimal form terminates or eventually repeats, the number is rational. If the decimal form does not terminate or repeat, the number is irrational.

Consider calling students’ attention to the prefix ir-, meaning not in words such as irrelevant and irregular. This will help students understand the distinction between rational and irrational.

Any number that is either rational or irrational is considered a real number. Invite students to return to problems 1–6 and to record whether each number is rational or irrational. Confirm answers as a class.

1 = 0.333333333… 3

Rational

Cannot identify

Irrational

144 ___ = 1.45454545…

Rational

3 __ = 3.41666666… 12

Rational

Cannot identify

Irrational

Rational

3—

√12 = 2.2894284…

2. 3.

—

√42 = 6.480740698…

_3 = 0.428571428571…

Teacher Note If students ask about numbers that are not real, celebrate this question and tell them that non-real numbers do exist. These numbers are called imaginary numbers and are studied in later courses.

Today, we will compare and order rational and irrational numbers. 476

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Learn Is It Rational? Students classify numbers by using the definitions of rational and irrational numbers.

UDL: Representation

Create an anchor chart on chart paper with the table shown. Rational Number

Irrational Number

Real Number

Definition

A number with a decimal form that terminates or eventually repeats

A number with a decimal form that neither terminates nor repeats

A number that is either rational or irrational

Examples

Sample:

_1 2

−4

_ 1 6

7.65656565…

0.121314151… — √2

_1 2

−4

_ 1 6

7.65656565…

1.23444444…

5.678678678…

0.25

3.4642548…

0.¯ 25 — √25

3.4642548…

Consider inviting students to create their own version of the anchor chart on a piece of paper so that they can easily refer to it as needed.

0.121314151… 0.25

0.¯ 25 — √25 — √2 477

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Rational Number Features

Any fraction Any decimal form with bar notation Any decimal form that terminates

Irrational Number

Real Number

Any number that cannot be written as a fraction

All the numbers we know

Any decimal form that does not terminate or repeat

All rational and irrational numbers

Any decimal form that eventually repeats

Invite students to help you complete the table. Begin by asking students to share their definitions of a rational number, an irrational number, and a real number. Then invite students to share examples of rational and irrational numbers and to justify their selections. Encourage students to provide numbers that are not in decimal form. For now, add examples only to the Rational Number and Irrational Number columns of the table. As students share justifications for their examples, note their reasoning about rational and irrational numbers in the Features row. Display the numbers 0.25 and 0. ¯ 25. Are these numbers rational or irrational? How do you know? Both numbers are rational. The first number is rational because the decimal form terminates. The second number is rational because it has bar notation, which means it is a repeating decimal.

478

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

If the decimal form of a number terminates or can be written with bar notation, then the number is rational.

—

Add the numbers 0.25 and 0. ¯ 25 to the table. Then display the following numbers. √2 = 1.41421356…

—

√25 = 5

—

Are these numbers rational or irrational? How do you know?

√2 is irrational because its decimal form neither terminates nor repeats. √25 is rational

—

because its decimal form terminates.

Differentiation: Support If students need additional support with bar notation, consider asking them to return to problems 1–6 and to write each rational number with bar notation.

Can we determine that √2 is irrational without seeing the decimal expansion? How?

—

Yes. There is no rational number we can square to get 2, so we know √2 must be irrational.

—

Add the numbers √2 and √25 to the table.

Display the number 0.121314151… .

Is the number 0.121314151… rational or irrational? Why? It is irrational because it does not terminate or repeat. Rational numbers have a decimal form that either terminates or repeats. The number 0.121314151… follows a pattern but not a repeating pattern, so it is irrational. Add the number 0.121314151… to the table. Invite students to share which numbers listed in the table belong in the Real Number column, highlighting that all rational and irrational numbers are real numbers. Add all examples to the Real Number column. Then have students provide features of real numbers and record their responses in the table. Once the table is complete, introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: Only Rational Numbers, Only Irrational Numbers, and Both. Present a number category and use the following prompt.

Teacher Note When given a number in decimal form that does not repeat or terminate, such as 0.121314151…, it cannot be determined for sure whether the number is rational or irrational. It is possible that the decimal form will repeat or terminate at some point. In this lesson, students make an assumption about numbers being rational or irrational based on the information provided.

479

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Determine whether this number category includes only rational numbers, only irrational numbers, or both rational and irrational numbers. Invite students to stand beside the sign that best describes their thinking. When all students are standing near a sign, allow groups to discuss the reasons why they chose that sign. Then call on each group to share reasons for their selection. Invite students who change their minds during the discussion to join a different group.

EUREKA MATH2

Promoting the Standards for Mathematical Practice When students justify their own reasoning and then listen to and analyze their peers’ reasoning in the Take a Stand routine, they are constructing viable arguments and critiquing the reasoning of others (MP3).

Repeat the process with each number category. For square roots, consider asking students to give an example of a square root that is a rational number and a square root that is an irrational number.

Ask the following questions to promote MP3:

Number categories:

• What questions can you ask your peers who chose other signs to make sure you understand the reason for their choice?

• Integers

• Is your sign choice a guess or do you know for sure? How do you know for sure?

Only rational numbers • Square roots Both • Negative numbers Both • Fractions Only rational numbers Have students return to their seats.

Battle Cards Students compare and order rational and irrational numbers. Pair students and distribute a set of Battle Cards to each pair. Instruct pairs to shuffle the cards and split them equally into two piles laid facedown, with one pile in front of each student. Have students flip over the top card in their pile. Encourage students to determine which two consecutive whole numbers the value of their card is between when the value

480

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

is not a whole number. The student who turns over the card with the greater value collects both cards and sets them aside in their own winning pile. The game ends when students have gone through all their cards once. When most students have finished playing, discuss students’ strategies for approximating an irrational number and examples of comparison statements as a class. Have students keep their winning piles separate during the discussion. Choose a card you played that has an irrational number. How did you determine which two consecutive whole numbers the value of the card is between? 3—

Sample: I played the card with √120 . Because 43 = 64 and 53 = 125, I knew that the value 3— of √120 is between 4 and 5.

Invite students to choose another card. Have them state an inequality that compares the value of this card to the value of the first card they chose. Write the inequalities for students to refer to. Samples are shown. 3—

√120 > 35 12

16 √— < 2 √__

__ 49

Direct students to arrange the cards in their winning pile from least to greatest. After most students have their winning pile in order, tell pairs to order all the cards from least to greatest.

Language Support

After most pairs have their cards in order, give students time to study the card arrangement of a neighboring pair of students. Have pairs discuss any discrepancies and revise the order of their cards if necessary.

Encourage students to use the Share Your Thinking and Agree or Disagree sections of the Talking Tool to support discussions in the remaining lesson activities.

__ — 3 — 3 — 35 3 — 3 — — 3 — — 16 11 3 0, _1 , √__ , 0.9375, 40, √2 , √8 , √10 , __ , √27 , √64 , √17 , √120 , √27 , 5 .¯ 85714, (__ )

Show the correct order, one card at a time.

3 12 9 49 Note reactions from students and ask the reason for any moments of surprise, confusion, or realization. Instruct pairs to rearrange their cards as necessary. Consider asking the following questions to debrief the activity:

• What differences did you notice between your card arrangement and the arrangement of your peers?

481

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

• Did you make any revisions to your arrangement? If so, what revisions did you make? • Which cards were the most challenging to order? • Which cards were the least challenging to order? Have students complete problem 7 in pairs.

—

3—

—

3—

—

Differentiation: Challenge

7. Plot and label the approximate location of each value on the number line.

√2

√17 , √120 , 1 , √2 , √27 , √27

_ 9

√27 3

√17

1 9

If students need additional challenge, consider asking them to plot all numbers from 11 3 the set of Battle Cards, except (__ , on the 3) number line.

√27

√120

Display the correct locations of the numbers on the number line. How is a number line helpful for ordering numbers? A number line shows how a number’s location relates to the whole numbers, which helps me to order numbers. To deepen student understanding of squares and square roots, ask the following questions. If 0 < a < b, is 0 < a2 < b2 ? How do you know? Yes. We can think of a and b as the side lengths of two squares. If one square has a longer side length than another square, its area is also larger.

—

If 0 < a < b, is 0 < √a < √b ? How do you know?

Yes. We can think of a and b as the areas of two squares. If one square has a larger area than another square, its side length is also longer.

482

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Ordering Expressions Students order expressions that include rational and irrational numbers. Use the Numbered Heads routine. Organize students into groups of three and assign each student a number, 1 through 3. Present problem 8. Give students 1 minute to answer the question as a group. Remind students that any one of them could be the spokesperson for the group, so they should be prepared to answer. After 1 minute, call a number, 1 through 3. Have the students assigned that number respond to the following prompt.

—

Share your group’s findings and explain your reasoning.

—

The expressions from least to greatest are √28 − 5, √22 , √28 , and √22 + 5. We know that

—

√28 is greater than √22 because 28 is greater than 22. √28 is also greater than √28 − 5

—

because we are subtracting 5 from √28 . √22 + 5 is greater than √22 because we are

—

adding 5 to √22 . We also know that the value of √22 is between 4 and 5 and that the

—

value of √28 is between 5 and 6. Adding 5 to √22 gives a value between 9 and 10.

—

So √22 + 5 is greater than √28 . Subtracting 5 from √28 gives a value between 0 and 1.

—

So √28 − 5 is less than √22 .

Invite students to complete problem 9 in their groups.

—

For problems 8 and 9, order the expressions from least to greatest. 8. √28 , √22 , √28 − 5, √22 + 5

—

√28 − 5, √22 , √28 , √22 + 5

—

9. √17 − 2, 2 ⋅ √17 , √17 + 2

—

√17 − 2, √17 + 2, 2 ⋅ √17

Differentiation: Challenge If students need additional challenge, — 3— consider asking them to change √28 to √28 3— in problem 8. Students can approximate √28 — to the nearest whole number to compare √22 3— and √28 . The rest of the comparison work is similar to that described for problem 8.

Confirm answers to problem 9 as a class.

483

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Land Debrief 5 min Objectives: Classify real numbers as rational or irrational by their decimal form. Compare and order rational and irrational numbers. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. What are some similarities and differences between rational and irrational numbers? Rational and irrational numbers are similar because they can be classified as real numbers. Rational and irrational numbers are different because a rational number can be expressed as a fraction, but an irrational number cannot. Also, the decimal form of a rational number is either terminating or repeating, but the decimal form of an irrational number does not terminate or repeat. What strategies can we use to compare and order rational and irrational numbers? We can approximate irrational numbers before comparing them to rational numbers. Sometimes we can compare irrational numbers just by knowing which consecutive whole numbers they are between. After approximating and comparing two numbers in this way, we can order a set of irrational and rational numbers. We can approximate irrational numbers. Then we can visualize the approximate locations of the rational and irrational numbers on a number line to help us compare and order the numbers.

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

484

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22 Name

RECAP Date

¯ 5. 3.94

identified real numbers as rational or irrational.

•

compared and ordered rational and irrational numbers.

•

plotted the approximate locations of rational and irrational numbers on a number line.

3—

is less than √65 .

An irrational number is a number that is not p rational and cannot be expressed as _ q for

integer p and nonzero integer q. An irrational number has a decimal form that neither terminates nor repeats. A real number is any number that is either rational or irrational.

For problems 1–4, identify whether the number is rational or irrational. 4_ 3 Rational

A rational number can be written as a fraction.

—

2. −0.24680153… Irrational 3—

4. −√8

3. √3

Irrational

Rational

—

3 7. √120

6. √50

3—

When the decimal form of a number neither terminates nor repeats, the number cannot be written as a fraction and is irrational.

Negative numbers can be rational or irrational.

is between 4 and 5.

62 __ 9

—

63 Because __ = 7, 9 62 __ is less than 7.

√50 is greater than 7.

—

Because √64 = 4, 3— 3— and √125 = 5, √120

Because √49 = 7,

3 8. √25 + 4

√15

3—

Examples 1.

—

3 √ 65

Because √64 = 4, √65 is greater than 4. So 3.¯ 94

Terminology

•

—

For problems 5–8, compare the values of the expressions by using the < or > symbol.

Rational and Irrational Numbers In this lesson, we

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

—

3— 2 · √25

3—

The value of √25 is between 2 and 3. Adding 4

Because √9 = 3 — — and √16 = 4, √15

3—

to √25 gives a number that is between 6 and 7. 3— Multiplying √25 by 2 gives a number that

is between 3 and 4.

is between 4 and 6.

9. Plot and label a point at the approximate location of each value on the number line. — 3— 8_ √ , 30 , 2.¯ 4, √17 3

– 2. 4

Roots can be rational or irrational.

−√3 = −1.73205… 3—

√8 = 2

√30

√17

2 8 3

3—

Because √27 = 3 3— 3— and √64 = 4, √30

5 —

Because √16 = 4 — — and √25 = 5, √17

is between 4 and 5.

is between 3 and 4.

315

316

RECAP

485

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 22 Name

PRACTICE Date

1. Write each value in the appropriate column of the table.

—

3 √ 5

−_3 2

—

−√18

0.23742374… Rational

−2_ 3

115 0

—

115

√25

—

3 √ 64

—

−5.¯ 67

Irrational

0.23742374… —

√25

—

3 √ 64

—

3 √ 5

For problems 4–9, compare the values by using the < or > symbol.

—

4. √49

— −√18

1 7.¯

√15

87 8

—

5. √20

—

3 √ 20

0.67534702…

—

3 6. √27

2. Abdul can predict the next decimal digit in the number 0.010305070… , so he concludes that the number is rational. Explain the error in Abdul’s thinking. The decimal digits in the number follow a pattern, but the pattern does not repeat. The decimal form of a rational number must either terminate or repeat. So this number is irrational.

486

—

Both Ethan and Nora are correct. Ethan is correct because √5 has a decimal form that neither terminates nor repeats. Nora is correct because all rational and irrational numbers are real numbers.

−5. ¯ 67

—

3. Ethan states that √5 is an irrational number. Nora states that √5 is a real number. Who is correct? Explain.

0.67534702…

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

317

—

8. √121

318

—

P R ACT I C E

—

3 7. √64

__ 34 7

37 __ 9

—

3 √ 125

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

For problems 10 and 11, plot and label a point at the approximate location of each value on the number line.

—

— 3 — √15

√4

— 3 — √2 7

—

17.

√11

— ——

√2 7

√2 0

— — — 4, √17 , √12 , 3, √8

19. Order the numbers from least to greatest.

— — — √8 , 3, √12 , 4, √17

— —

40 7

— 2 (√30 )

— —

3— 3— 40 √— √— 11. 4.1¯ 6, __ , 40 , 90 − 1, √120 , √3 + 6 7 √12 0

18.

25 — —

– 4.16

— 2 (√25 )

For problems 17 and 18, evaluate.

3 3 10. √27 , √27 , √11 , √20 , √15 , √4

–

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22

–

— —

√3 + 6

√4 0

√9 0 – 1

For problems 20 and 21, solve. g 20. _ = 8 6

— — — √8 , √5 + 2, √10 − 3

12. Order the expressions from least to greatest.

—

√10 − 3, √8 , √5 + 2

Remember

21. 40 = 3n

For problems 13–16, divide. 2 1 13. −1 _ ÷ 1 _ 5 2 __ −14 15

1 1 14. −2 _ ÷ (−1 _) 4 8

4 1 15. 4 _ ÷ 1 _ 5 5

40 3

1 1 16. 8 _ ÷ (−2 _) 3 4 100 −___ 27

P R ACT I C E

319

320

P R ACT I C E

487

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 22 ▸ Battle Cards

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 22

—

35 12

3 √ 120

—

3 √ 10

—

1 9

—

√ 27

√ 17

—

3 √ 64

—

3 √ 8

16 √__ 49

3 √ 27

488

—

This page may be reproduced for classroom use only.

—

√2

(3)

11 3

0.9375

5.¯ 85714

Teacher Edition: Grade 7–8, Module 1, Topic D, Lesson 23 LESSON 23

Revisiting Equations with Squares and Cubes Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23 Name

Date

EXIT TICKET

For problems 1–3, solve the equation. Identify all solutions as rational or irrational. 1. x 2 = 13

— — The solutions are √13 and −√13 .

— x = √13

— x = −√13

x 2 = 13 or

Irrational

Lesson at a Glance In this lesson, students build on their previous work in this topic to solve equations of the forms x 2 = p and x 3 = p, where the solutions are real numbers. Students also engage in the Critique a Flawed Response routine to analyze a student’s work, correct errors, and clarify the meaning of the solutions to an equation. At the end of the lesson, students apply the Pythagorean theorem to solve a problem involving a three-dimensional figure.

Key Questions 2. x 3 = 36 3—

The solution is √36 . Irrational

• Given that p is a perfect square and q is not a perfect square, how are the solutions to x 2 = p and x 2 = q similar? How are the solutions different?

x 3 = 36 3— x = √36

• Given that p is a perfect cube and q is not a perfect cube, how are the solutions to x 3 = p and x 3 = q similar? How are the solutions different?

Achievement Descriptors 8 3. x 3 = − __ 27

—

7–8.Mod1.AD7 Determine whether numbers are rational or

8 x 3 = − __

√ 27

irrational. (7.NS.A.2.d, 8.NS.A.1)

8 x = − __ 3

The solution is − _ . 2 3

7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p,

x = − 2_

where p is a positive rational number. (8.EE.A.2)

7–8.Mod1.AD12 Evaluate square roots of small perfect squares and

Rational

325

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Agenda

Materials

Fluency

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Revisiting Equations of the Form x 2 = p

• None

• Revisiting Equations of the Form x 3 = p

Lesson Preparation

• The Baseball Bat Problem

• None

Land 10 min

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Fluency Rational or Irrational Students classify square roots and cube roots as rational or irrational numbers to prepare for solving equations with rational and irrational solutions. Directions: Identify whether each number is rational or irrational. —

√3

Irrational

√18

Irrational

√121

492

—

36 √__ 49

3—

Rational

√16

Irrational

√27

Rational

3—

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Launch

Students determine the number of solutions to equations of the forms x 2 = p and x 3 = p and classify solutions as rational or irrational. Invite students to complete problem 1 with a partner. Circulate as students work, and ask the following questions: • How many solutions does that equation have? How do you know? • What are the solutions? 1. Indicate whether each equation has one solution, two solutions, or no solution. Equation

One Solution

x2 = 0

x2 = 1

Two Solutions

x 2 = −1

x 3 = −1

x3 = 8

x2 = 8

No Solution

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EUREKA MATH2

After most students have finished, bring the class together for discussion. Anticipate that students may incorrectly determine the number of solutions to the equation x 2 = 8. Welcome all considerations about this equation at this point in the lesson. Which equations have only one solution? Why? Equations of the form x 3 = p only have one solution because there is only one number we can cube to get another number. The equation x 2 = 0 only has one solution because the only number we can square to get 0 is 0. Which equations have two solutions? Why? Equations of the form x 2 = p where p is not 0 or a negative number have two solutions because there are two different numbers we can square to get the same positive number. What are the solutions to the equation x 2 = 1?

1 and −1 Invite students to turn and talk about whether the equations in problem 1 have rational or irrational solutions and why. Today, we will revisit equations involving squares and cubes. This time, we will use square root and cube root notation to provide exact solutions when the solutions are irrational.

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Learn Revisiting Equations of the Form x 2 = p Students solve equations of the form x 2 = p, where p is a rational number and the solutions are real numbers. Have students complete problem 2 in pairs. For problems 2–5, solve the equation. Identify the solutions as rational or irrational. 2. x 2 = 25

3. x 2 = 8

The solutions are 5 and −5. Rational

— — x = √8 or x = −√8 — — The solutions are √8 and −√8 . x2 = 8

UDL: Representation Reviewing how to solve the equation x 2 = 25 helps activate prior knowledge and supports making connections to more complex equations.

Irrational 49 4. __ = h2 4

—

5. t 2 = 0.01

The solutions are _ and − _ .

Rational

—

49 = h2 4

√4__49 = h or −√4__49 = h _

7 =h 2

7 2

−_ = h 7 2

7 2

t 2 = 0.01 — — t = √0.01 or t = −√0.01 t = 0.1

t = −0.1

The solutions are 0.1 and −0.1.

Rational

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Confirm the answer to problem 2 with the class. Use the following prompts to reinforce the need to include both the positive solution and the negative solution. What question can we ask ourselves to solve the equation in problem 2? We can ask the question, What numbers can we square to get 25? How many numbers can we square to get 25? Explain. We can square two numbers, 5 and −5, to get 25. The square of a positive number is a positive number, and the square of a negative number is also a positive number. What do you notice about the two solutions to the equation x 2 = 25? The two solutions are opposites of each other. When we used the Pythagorean theorem to solve problems in previous lessons, how many solutions did we have? Why? We had only one solution because we were finding the hypotenuse length of a triangle, which is always a positive number. When an equation does not represent quantities within a context, we need to consider all possible values of the variable. So equations of the form x 2 = p have two solutions for positive values of p, and those two solutions are opposites of each other. Direct students to problem 3. How is the equation in problem 3 similar to the equation in problem 2? How is it different? The equation in problem 3 is similar because it has a variable squared. It is different because 8 is not a perfect square. When we used the Pythagorean theorem to find the length of a hypotenuse, we solved equations that did not involve a perfect square. With those equations, how did we write the exact value of c ? We used a square root symbol. The square root symbol represents the positive square root. But what do we know about an equation of the form x 2 = p when it does not represent a context? The equation has two solutions: a positive number and its opposite.

496

Teacher Note In this course, students write solutions to x 2 = p — — as √p and −√p . Students transition to the — shorthand notation ±√p in later courses. Highlight that the solutions are a positive number and its opposite, and that the negative sign belongs outside of the square root symbol.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Have students complete problems 3–5 in pairs. Circulate as students work. Encourage them to consider both the positive and negative solutions to the equations, and highlight that the values are opposites. When students are finished, confirm responses as a class. If students do not notice that

—

the solutions for problem 5 are rational, allow them to debate whether the solutions are √0.01 and −√0.01 or 0.1 and −0.1.

Revisiting Equations of the Form x 3 = p Students solve equations of the form x 3 = p, where p is a rational number and the solutions are real numbers. Invite students to examine problems 6–9.

Teacher Note When p is a perfect square, some students might apply the intuitive approach and skip the step of writing the solutions with square root notation. For example, in problem 2, students may proceed directly to identifying 5 and −5 as the solutions. Allow for either approach.

How are these equations different from the equations in problems 2–5? In these equations, the variable is cubed, not squared. Have students complete problem 6 in pairs. For problems 6–9, solve the equation. Identify all solutions as rational or irrational. 6. t 3 = 27 The solution is 3. Rational

7. −25 = w 3

−25 = w 3 3— √−25 = w 3— The solution is √−25 . Irrational

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1 8. _ = w 3 8

—

√1_8 = w

1 = w3 8

The solution is _ . 1 2

1 =w 2

9. 38 = z 3

38 = z 3 3— √38 = z 3— The solution is √38 .

Irrational

Rational Confirm the response to problem 6 with the class. Then use the following prompts to reinforce that equations of the form x 3 = p have only one solution. What question can we ask ourselves to solve the equation in problem 6? We can ask, What numbers can we cube to get 27? How many numbers can we cube to get 27? Explain. We can cube only one number, 3, to get 27. The cube of a positive number is a positive number. Equations of the form x 3 = p have only one solution for any value of p. Direct students to problem 7. Engage students in a discussion about how to solve the equation. Have students follow along in their books. How is the equation in problem 7 similar to the equation in problem 6? How is it different? The equation in problem 7 is similar because it has a variable cubed. It is different because the variable cubed equals a negative number. It is also different because −25 is not a perfect cube. The equation in problem 7 does not involve a perfect cube. But we can solve the equation by applying what we know about solving equations with squares. What does the equation x 3 = −25 tell us about x ? It tells us that x is a number we can cube to get −25.

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How can we write an exact value of x? We can use a cube root symbol.

3—

The solution to the equation x 3 = −25 is √−25 . Is √−25 rational or irrational? Why? 3—

√−25 is irrational because there is no rational number we can cube to get −25.

Have students complete problems 8 and 9 in pairs. Look for students who need support understanding that there is only one solution to each equation. Confirm responses as a class. Then introduce the Critique a Flawed Response routine and present problem 10. Read the prompt aloud as students follow along. Give students 1 minute to identify the errors in Dylan’s work. Then invite students to share what they noticed. Responses may include the following:

Teacher Note To help students understand proper placement of negative signs for solutions with roots, consider highlighting the difference between the following statements:

—

• One solution to x 2 = p is −√p .

3—

• The solution to x 3 = −p is √−p .

• The equation x 3 = −64 has only one solution. • Dylan found a square root rather than a cube root. Give students 1 minute to correctly solve the equation. Circulate and ensure students determine that the only solution is −4. 10. Dylan solves the equation x 3 = −64 but makes some errors. His work is shown. Help Dylan correct his work.

x 3 = −64 3— 3— x = √−64 or x = − √−64 x = −8

x = −(−8) x=8

The solutions are −8 and 8.

x 3 = −64 3— x = √−64 x = −4

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7–8 ▸ M1 ▸ TD ▸ Lesson 23

Use the following prompts to debrief the activity. Why does the equation x 3 = −64 have only one solution? There is only one number we can cube to get −64. That number is −4. How can we verify that −4 is a solution and 4 is not a solution? We can substitute the numbers −4 and 4 into the equation and see whether they each make the equation true. Only −4 makes the equation true. What are the solutions to the equation x 2 = −64? Explain. There is no solution to the equation because the square of any number is always a positive number.

The Baseball Bat Problem Students solve a real-world problem by applying the Pythagorean theorem and solving equations of the form x 2 = p. Invite students to complete problem 11 with a partner. Circulate to support students in correctly applying the Pythagorean theorem. If students immediately indicate that the length, width, and height are too short for the bat to fit in the box, encourage them to think creatively about other ways the bat might be packed into the box.

UDL: Representation Consider offering students a hands-on experience for problem 11. For example, provide a box and a few objects of different lengths so students can explore various ways the objects can fit in the box.

Promoting the Standards for Mathematical Practice Students reason quantitatively and abstractly as they apply the Pythagorean theorem and their knowledge of irrational numbers to solve a real-world problem involving a three-dimensional figure (MP2). Ask the following questions to promote MP2: • What real-world situations are modeled by right triangles? • What does the problem ask you to do? • What does the Pythagorean theorem tell you about this situation?

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7–8 ▸ M1 ▸ TD ▸ Lesson 23

11. Lily wants to ship a baseball bat that is 32 inches long. She can buy a box that measures 20 inches by 20 inches by 20 inches. Will the baseball bat fit in the box? Explain your reasoning.

Base Diagonal

202 + 202 = c 2 400 + 400 = c 2 800 = c 2 — √800 = c

c 20 inches

Teacher Note Consider allowing students to use a calculator to determine a decimal approximation for the length of the diagonal from a top corner to the opposite bottom corner of the box. They could also approximate the length by determining the two consecutive whole numbers that the length is between.

20 inches 20 inches

Differentiation: Support

Diagonal from Top Corner to Opposite Bottom Corner

— (√800 )2 + 202 = d 2

800 + 400 = d 2 1,200 = d 2 — √1,200 = d 34.64 ≈ d

If students need support with problem 11, consider providing a picture of a cube that they can draw on to represent the box and the bat.

Differentiation: Challenge

Yes, the 32-inch bat can fit in the box. The length of the diagonal from a top corner to the opposite bottom corner of the box is about 34.64 inches.

If students need additional challenge after completing problem 11, consider offering the following extension:

When pairs finish, confirm that the bat can fit in the box. If needed, invite a pair to share their solution strategy with the class. Consider advancing the class discussion by asking any or all of the following questions:

• Part 1: Find possible dimensions of another box that can fit the bat but still has a length, width, and height that are each less than 32 inches.

• What difference does the context make when solving equations of the form x 2 = p ? Explain. • What assumptions did you and your partner make to solve the problem? • Could the width of the bat make any difference in this situation? Why? • Could additional items in the box make any difference in this situation? Why?

• Part 2: Find possible dimensions of another box that can fit the bat and uses less cardboard than the box in part 1. Do you think this box uses the least amount of cardboard? Explain.

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7–8 ▸ M1 ▸ TD ▸ Lesson 23

Land Debrief 5 min Objective: Solve equations of the forms x 2 = p and x 3 = p, where p is a rational number and the solutions are real numbers. Facilitate a class discussion by using the following prompts. Encourage students to restate or build on one another’s responses. How are the solutions to x 2 = 100 and x 2 = 10 similar? How are the solutions different? The solutions are similar because both equations have two solutions: a positive number

—

and its opposite. The solutions are different because the solutions to x 2 = 100, 10 and −10, are rational numbers. The solutions to x 2 = 10, √10 and −√10 , are irrational numbers.

How are the solutions to x 3 = 27 and x 3 = 3 similar? How are the solutions different? The solutions are similar because both equations have one positive solution. The solutions are different because the solution to x 3 = 27, 3, is a rational number. The solution to 3— x 3 = 3, √3 , is an irrational number.

Language Support As the discussion unfolds, consider providing students with key terminology and sentence frames as a scaffold. For example, post the terms positive, negative, square root, cube root, rational, and irrational, and post the following sentence frames: The solutions are similar because

The solutions are different because

How are the solutions to x 2 = 15 and x 3 = 15 similar? How are the solutions different?

The solutions are similar because they are roots. The solutions to both equations are also

— — 3— solution, √15 , which is a cube root.

irrational numbers. The solutions are different because the equation x 2 = 15 has two solutions, √15 and −√15 , which are square roots. The equation x 3 = 15 has only one

Exit Ticket 5 min Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

502

Teacher Note Assign the Practice problems for completion outside of class or use them in class if time remains after the lesson. Refer students to the Recap for support.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Recap

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23 Name

RECAP Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

3. A square has an area of 46 square centimeters. What is the side length of the square? Let s represent the side length of the square in centimeters.

s2 = 46 — s = √46

Revisiting Equations with Squares and Cubes

—

The side length of the square is √46 centimeters.

In this lesson, we •

solved equations of the forms x 2 = p and x 3 = p.

•

expressed irrational solutions by using square root and cube root notation.

The area of a square can be found by squaring the side length. The side length of a square cannot be negative, so the positive solution is the only answer to the problem.

Examples For problems 1 and 2, solve the equation. Identify all solutions as rational or irrational. 1. m2 = 35

m2 = 35 — — m = √35 or m = −√35 — — The solutions are √35 and −√35 .

4. A cube has a volume of 12 cubic inches. What is the edge length of the cube? Let x represent the edge length of the cube in inches.

There are two numbers that equal a positive number when squared: a positive number and its opposite. So the equation has two solutions.

x 3 = 12 3— x = √12 3— The edge length of the cube is √12 inches.

Irrational

2. −100 = r 3

—

3 The solution is √−100 .

Irrational

−100 = r 3 3— √ −100 = r

The volume of a cube can be found by cubing the edge length.

Notice that the variable is cubed. Although 100 is a perfect square,

100 and −100 are not perfect cubes.

327

328

RECAP

503

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

Student Edition: Grade 7–8, Module 1, Topic D, Lesson 23

PRACTICE

Name

Date

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

—

49 8. t 2 = __ 64

—

7. h2 = 8

√8 and −√8 , Irrational

_7 and − _7 , Rational 8

1. Which numbers are solutions to the equation x 3 = 27? Choose all that apply. A. 3

B. −3

3—

—

C. − √27

3 D. √27

—

E. √27

9. − 1_ = t 3 8

F. −√27

10. b2 = −100

− _1 , Rational

No solution

2. The number −1 is a solution to which equations? Choose all that apply. A. x 2 = 0

B. x 2 = 1

C. x 2 = −1

D. x 3 = 1

11. In the equations, n, m, p, and q are positive numbers. Order n, m, p, and q from least to greatest.

n2 = 15

E. x 3 = −1

m3 = 30

p3 = 25

q2 = 20

p, m, n, q

For problems 3–10, solve the equation. Identify all solutions as rational or irrational.

—

4. m3 = 15

5. 200 = j2 — — √200 and −√200 , Irrational

6. n3 = −1

√27 and −√27 , Irrational

504

—

3. y 2 = 27

3 √ 15 , Irrational

12. A cube has a volume of 58 cubic centimeters. What is the edge length of the cube?

—

−1, Rational

3 √ 58 centimeters

329

330

P R ACT I C E

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

EUREKA MATH2

7–8 ▸ M1 ▸ TD ▸ Lesson 23

19. Ava solves the equation _ x = 26 but makes an error. 2 _1 x = 26 2 1

13. A square has an area of 32 square inches. A cube has a volume of 120 cubic inches. Which is greater, the side length of the square or the edge length of the cube? Explain.

—

The side length of the square is greater. The side length of the square is √32 inches, which 3— is between 5 inches and 6 inches. The edge length of the cube is √120 inches, which is between 4 inches and 5 inches.

2(1_ x) = 1_ (26) 2

x = 13 a. Identify Ava’s error. Explain what she should have done.

Ava multiplied the left side of the equation by 2 and the right side of the equation by _ . She should have multiplied both sides of the equation by 2.

1 2

Remember For problems 14–17, evaluate. 3 2 3 2 15. (− _)(− _) 14. _ + _ 5 4 5 4

__6

23 20

16. − 5_ − 2_ 8 5 __ − 41 40

17. − 5_ ÷ 2_ 8 5 __ − 25 16

b. Show the correct work to solve the equation. 1_ x = 26 2

2(1_ x) = 2(26) 2

x = 52 The solution is 52.

—

18. Approximate the value of √21 to the nearest tenth by using consecutive whole numbers, tenths, and hundredths. Explain your thinking.

—

The value of √21 is approximately 4.6.

—

Because 21 is between the perfect squares 16 and 25, the value of √21 is between 4 and 5. 2

Next, I consider the tenths from 4 to 5. I find that 21 is between 4.5 and 4.62, or 20.25 and 21.21.

Then, I consider the hundredths from 4.5 to 4.6. I find that 21 is between 4.582 and 4.592, — or 20.9764 and 21.0681. Both 4.58 and 4.59 round to 4.6, so √21 ≈ 4.6.

P R ACT I C E

331

332

P R ACT I C E

505

Teacher Edition: Grade 7–8, Module 1, Standards

Standards Module Content Standards Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 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. 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 of |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.

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. 7.NS.A.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.

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EUREKA MATH2

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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 − (_ q ) = q = − q . Interpret quotients of rational p

−p

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. 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.1 Know that there are numbers that are not rational, and approximate them by rational numbers. 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 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., π 2). 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.

Work with radicals and integer exponents. 8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent 1 1 numerical expressions. For example, 32 × 3−5 = 3−3 = __ = __ . 3 3

8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square

Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

507

EUREKA MATH2

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—

roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

8.EE.A.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. 8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Understand and apply the Pythagorean Theorem. 8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Standards for Mathematical Practice MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

MP6

Attend to precision.

MP7

Look for and make use of structure.

MP8

Look for and express regularity in repeated reasoning

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Teacher Edition: Grade 7–8, Module 1, Achievement Descriptors: Proficiency Indicators

Achievement Descriptors: Proficiency Indicators 7–8.Mod1.AD1 Evaluate sums, differences, products, and quotients of two rational numbers. RELATED CCSSM

7.NS.A.1.d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.A.2.c Apply properties of operations as strategies to multiply and divide rational numbers.

Partially Proficient

Proficient

Evaluate sums, differences, products, and quotients of two integers.

Evaluate sums, differences, products, and quotients of two rational numbers.

Evaluate.

Highly Proficient

− 3(2 1 )

−3 + 9

6 − (− 5)

7–8.Mod1.AD2 Solve real-world and mathematical problems involving the four operations with rational numbers. RELATED CCSSM

7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers.

Partially Proficient

Proficient

Highly Proficient

Solve real-world and mathematical problems involving the four operations with rational numbers. A dolphin is underwater at an elevation of − 12 feet. The elevation of the dolphin changes at a rate of − 6 feet per second. How long does the dolphin take to descend to an elevation of − 114 feet? seconds

510

EUREKA MATH2

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7–8.Mod1.AD3 Interpret sums, differences, products, and quotients of rational numbers by describing real-world contexts. RELATED CCSSM

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.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 p −p p and q are integers, then − ( q ) = q = − 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.3 Solve real-world and mathematical problems involving the four operations with rational numbers.1

Partially Proficient

Proficient

Highly Proficient

Write expressions for sums, differences, products, and quotients of rational numbers in real-world contexts.

Write and evaluate expressions to interpret sums, differences, products, and quotients of rational numbers to describe real-world contexts.

Create real-world contexts that can be represented by products and quotients of rational numbers.

A pulley lifts an object up 3 1 feet and then lowers it

by the expression 3(− 2 ).

__ 4 3 __ down 7 feet. Write an expression to represent the

A pulley lifts an object up 3 1 feet and then lowers it 4

situation.

__ 4

down 7 3 feet. Write an expression to represent the 4

Create a real-world scenario that can be represented

__ 3

situation. Evaluate your expression and explain what the solution means in this situation.

Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

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7–8.Mod1.AD4 Show on a number line that the distance between two rational numbers is the same as the absolute value

of their difference, and apply this principle in real-world contexts. RELATED CCSSM

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.3 Solve real-world and mathematical problems involving the four operations with rational numbers.1

Partially Proficient Model whole-number subtraction on a number line. Use a number line to model the expression 9 − 12.

Proficient

Highly Proficient

Show on a number line that the distance between two rational numbers is the same as the absolute value of their difference and apply this principle in real-world contexts. The base of an in-ground swimming pool is 8 meters below ground level. The height of the pool diving platform is 5 meters above ground. Write an expression to represent the distance in meters between the base of the pool and the top of the diving platform. Draw a number line to support your reasoning.

Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

512

EUREKA MATH2

7–8 ▸ M1

7–8.Mod1.AD5 Determine the sign of a product by looking at the signs of its factors. RELATED CCSSM

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.

Partially Proficient

Proficient

Highly Proficient

Determine the sign of a product by looking at the signs of its factors. For each expression, indicate whether the value is positive or negative.

Expression

Positive

Negative

Rational

Irrational

_4 (− 2) 5 − 4_ (2) 5

(− 3.4)(2.1)(− 5.5) 5_ (− 4(2.1)) 8

Number

3 — number in its decimal form by using long division. 7–8.Mod1.AD6 Write the fraction form of a rational √ 11

7_ 8

RELATED CCSSM

—

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.

Partially Proficient

√2

Proficient

Highly Proficient

4.12 × 10−6

Write the fraction form of a rational number in its terminating decimal form by using long division.

Write − 3 in decimal form. 8

Write the_fraction form of a rational number in its 0 . 13 repeating decimal form by using long division and appropriate repeating-decimal notation. — √ 5 Write − 81 in decimal form.

__ 6

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7–8.Mod1.AD7 Determine whether numbers are rational or irrational. RELATED CCSSM

Expression Positive Negative 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. 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 4_ (− 2) expansion which repeats eventually into a rational number. that the decimal expansion repeats eventually, and convert a decimal 5

Partially Proficient Determine whether the decimal form of a rational number terminates in 0’s or eventually repeats. Determine whether the decimal form of each number terminates or repeats.

−5

_5_ 7

−9 2

− 4_ (2) 5

Proficient

Highly Proficient

(− 3.4)(2.1)(− 5.5) Determine whether numbers are rational or irrational. 5_ (− 4(2.1)) 8 Determine whether each number is rational or irrational. Number

—

3 √ 11

Rational

Irrational

7_ 8

—

√2

4.12 × 10−6 _ 0 . 13 —

√81

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7–8.Mod1.AD8 Use rational approximations of irrational numbers to compare the size of irrational numbers. 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., π 2). 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. RELATED CCSSM

Partially Proficient

Proficient

Highly Proficient

Approximate the values of irrational numbers to compare their sizes.

—

Compare the numbers by using the <, >, or = symbol. √60

28 __ 3

—

3 √ 206

_ 4 . 36

7.6 — √90

—

√35

— 3 + √3

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7–8.Mod1.AD9 Locate irrational numbers approximately on a number line. 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., π 2). 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. RELATED CCSSM

Partially Proficient Determine between which two integers the value of an irrational number lies.

—

The value of √29 lies between which two whole numbers?

Proficient

Highly Proficient

Locate irrational numbers approximately on a number line.

—

Plot √29 at its approximate location on a number line.

A. 2 and 9 B. 5 and 6 C. 6 and 7 D. 14 and 15

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7–8.Mod1.AD10 Apply the properties of integer exponents to generate equivalent numerical expressions.

__ __

RELATED CCSSM

8.EE.A.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3−5 = 3−3 = 1 = 1 . 33

Partially Proficient

Proficient

Apply the properties of integer exponents to identify equivalent numerical expressions.

Apply the properties of integer exponents to generate equivalent numerical expressions.

Which expressions are equivalent to 76? Choose all that apply.

Use the properties of exponents to write an equivalent expression for 35 ⋅ 3−9 by using only positive exponents.

__ ___ __

A. 7 75 −3 B. 7 7−9 4 C. 7 72

Highly Proficient

D. 7−3 ⋅ 73 E. 7−2 ⋅ 78

7–8.Mod1.AD11 Solve equations of the forms x 2 = p and x 3 = p, where p is a positive rational number. 8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate square — roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Partially Proficient

Proficient

Solve equations of the forms x 2 = p and x 3 = p, where the solutions are integers.

Solve equations of the forms x 2 = p and x 3 = p, where p is a positive rational number.

Solve.

x 2 = 16

x 2 = 0.81

x 3 = 64

x 2 = 23

Highly Proficient

x3 = 8

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7–8.Mod1.AD12 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. 8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p, where p is a positive rational number. Evaluate — square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. RELATED CCSSM

Partially Proficient

Proficient

Highly Proficient

Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

Evaluate square roots of rational numbers related to perfect squares and cube roots of rational numbers related to perfect cubes.

—

Evaluate.

√16

—9 √__ 25

Evaluate.

—

3 √ 64

—

3 √ 0.27

7–8.Mod1.AD13 Approximate and write very large and very small numbers in scientific notation. RELATED CCSSM

Partially Proficient

Highly Proficient

Write very large or very small numbers in scientific notation.

Approximate very large or very small numbers in scientific notation.

Compare very large and very small numbers written in scientific notation.

Write each number in scientific notation.

Round 6,190,000 to the nearest million. Then write that value in scientific notation.

Consider the numbers 4.11 × 10−6 and 4.11 × 10−5. Which number is smaller? Explain how you know.

0.00008 2,500,000

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Proficient

Round 0.000 0489 to the nearest hundred thousandth. Then write that value in scientific notation.

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7–8.Mod1.AD14 Express how many times as much one number is than another number when both are written in scientific

notation.

RELATED CCSSM

Partially Proficient

Proficient

Identify how many times as much one number is than another, when both numbers are written in scientific notation.

Express how many times as much one number is than another number when both are written in scientific notation.

Consider the numbers 5 × 109 and 5 × 104. Which statement is true?

A desktop computer costs about 5 × 102 dollars. A supercomputer costs about 1 × 108 dollars.

A. 5 × 109 is 10,000 times as large as 5 × 104.

The cost of a supercomputer is about how many times as much as the cost of a desktop computer?

B. 5 × 109 is 100,000 times as large as 5 × 104.

Highly Proficient Estimate how many times as much one number is than another by using scientific notation. During a musician’s concert tour, fans spent 13 million dollars on tickets. If 1.9 × 105 tickets were sold, about how much did the average ticket cost? Write the cost in scientific notation.

C. 5 × 109 is 1,000,000 times as large as 5 × 104. D. 5 × 109 is 10,000,000 times as large as 5 × 104.

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7–8.Mod1.AD15 Operate with numbers written in standard form and scientific notation, including problems where both

decimal and scientific notation are used. RELATED CCSSM

8.EE.A.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Partially Proficient

Proficient

Add or subtract numbers written in scientific notation with like units where the result has a single-digit first factor.

Operate with numbers written in standard form and scientific notation, including problems where both decimal and scientific notation are used.

Multiply or divide numbers written in scientific notation and evaluate exponential expressions that have numbers written in scientific notation.

Write a number in scientific notation that is equivalent to the given expression.

(6.2 × 10−6) + (9.1 × 10−6) (9.4 × 109) − (3.8 × 109) + (4.9 × 109)

(3 × 105) + (6 × 105)

(0.0009)(7.5 × 10−4)

(2 × 10−3)3

(8.4 × 105) ÷ 2,100

Highly Proficient Add or subtract numbers written in scientific notation with unlike units. Write a number in scientific notation that is equivalent to the given expression.

(3.7 × 10−2) + (4.1 × 103) (9.8 × 106) − (5.2 × 104)

(3 × 10−2)(2 × 105)

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7–8.Mod1.AD16 Operate with numbers written in scientific notation to solve real-world problems and choose units

of appropriate size for measurements of very large or very small quantities. RELATED CCSSM

Partially Proficient

Proficient

Highly Proficient

Operate with numbers written in scientific notation to solve real-world problems and choose units of appropriate size for measurements of very large or very small quantities. Jonas lives 2 × 10−1 miles from the store. It takes him about 240 seconds to walk to the store. Part A Assume Jonas walks at a constant rate. At what rate does Jonas walk in miles per second? Part B Choose another unit to express Jonas’s walking rate.

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7–8 ▸ M1

7–8.Mod1.AD17 Interpret scientific notation that is generated by technology. RELATED CCSSM

Partially Proficient

Proficient

Highly Proficient

Interpret scientific notation that is generated by technology. A calculator is used to perform several operations. The result is shown.

3. 375e+12 Write the result by using scientific notation.

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Apply the Pythagorean theorem to determine the unknown rational or irrational length of a hypotenuse for a right triangle in mathematical problems. 7–8.Mod1.AD18

RELATED CCSSM

8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Partially Proficient

Proficient

Apply the Pythagorean theorem to determine the unknown whole-number length of a hypotenuse for a right triangle in mathematical problems in two dimensions.

Apply the Pythagorean theorem to determine the unknown rational or irrational length of a hypotenuse for a right triangle in mathematical problems in two dimensions.

Find the length of the hypotenuse c.

Find the length of the hypotenuse c for each triangle.

Highly Proficient

c c

0.4

24 0.3

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Teacher Edition: Grade 7–8, Module 1, Terminology

Terminology The following terms are critical to the work of course 7–8 module 1. This resource groups terms into categories called New, Familiar, and Academic Verbs. The lessons in this module incorporate terminology with the expectation that students work toward applying it during discussions and in writing. Items in the New category are discipline-specific words that are introduced to students in this module. These items include the definition, description, or illustration as it is presented to students. At times, this resource also includes italicized language for teachers that expands on the wording used with students. Items in the Familiar category are discipline-specific words introduced in prior modules or in previous grade levels. Items in the Academic Verbs category are high-utility terms that are used across disciplines. These terms come from a list of academic verbs that the curriculum strategically introduces at this grade level.

New additive inverse The additive inverse of a number is a number such that the sum of the two numbers is 0. The additive inverse of a number x is the opposite of x because x + (− x) = 0. (Lesson 1)

bar notation A common notation for a repeating decimal expansion is bar notation. The bar is placed over the shortest block _ of repeating digits after the decimal point. For example, 3.125 is a compact way to write the repeating decimal expansion 3.12525252525… . (Lesson 9) cube root The cube root of a number x is a number with a cube that is x. The 3 — expression √x represents the cube root of x. (Lesson 21) hypotenuse The hypotenuse of a right triangle is the side opposite the right angle. It is also the longest side of a right triangle. (Lesson 19) irrational number An irrational number is a number that is not rational and cannot

be expressed as _ q for integer p and nonzero integer q. An irrational p

number has a decimal form that neither terminates nor repeats. (Lesson 22) leg A leg of a right triangle is a side adjacent to the right angle. (Lesson 19) order of magnitude The order of magnitude n is the exponent on the power of 10 for a number written in scientific notation. (Lesson 14) perfect cube A perfect cube is the cube of an integer. (Lesson 18)

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perfect square A perfect square is the square of an integer. (Lesson 18) Pythagorean theorem In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse length. (Lesson 19)

a 2 + b2 = c 2

terminating decimal A terminating decimal is a decimal that can be written with a finite number of nonzero digits. (Lesson 9) zero product property The zero product property states that if the product of two numbers is zero, then at least one of the numbers is zero. (Lesson 6)

Familiar

absolute value associative property (of multiplication and addition)

rational number p A rational number is any number that can be written in the form _ q, where p and q are integers and q ≠ 0. (Lesson 8) real number A real number is any number that is either rational or irrational. (Lesson 22) repeating decimal A repeating decimal is a decimal in which, after a certain digit, all remaining digits consist of a block of one or more digits that repeats indefinitely. (Lesson 9) scientific notation A number is written in scientific notation when it is represented as a number a multiplied by a power of 10. Numbers written in scientific notation are in the form a × 10n. The number a, which we call the first factor, is a number with an absolute value of at least 1 but less than 10. (Lesson 14) square root A square root of a nonnegative number x is a number with a square — that is x. The expression √x represents the positive square root of x — when x is a positive number. If x is 0, then √0 = 0. (Lesson 20)

base commutative property (of multiplication and addition) cubed decimal fraction distributive property equation equivalent equivalent expression exponent expression integer inverse multiplicative inverse negative

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opposites

Academic Verbs

positive

adjust

power

approximate

rational number (grade 6 description)

refer

reciprocal

verify

right triangle solution squared

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Teacher Edition: Grade 7–8, Module 1, Math Past

Math Past Archimedes: The Sand Reckoner How many grains of sand are there on the beach? How many grains of sand would fill the world? How many grains of sand would fill the visible universe? Ask your students how they would go about estimating the number of grains of sand needed to fill a container. This may spark memories of “guess the number of jelly beans in the jar” or some similar contest. Students may remember trying to count the jelly beans in one layer and then trying to count the layers. That would be pretty hard to do with sand! More than 2,000 years ago, Greek mathematician and scientist Archimedes set out to determine how many grains of sand are needed to fill the ultimate container—the entire known universe! Archimedes (287–211 BCE) lived in the Greek city-state of Syracuse on the island of Sicily. He is considered to be one of the greatest mathematicians and scientists of all time. In his book The Sand Reckoner, Archimedes determined that no more than 1 × 1063 grains of sand are needed to fill the universe. The universe of Archimedes’s time was a much smaller space than the known universe of today. People assumed that Earth

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was at the center of the universe and that the moon, the planets, and the sun orbited Earth. Archimedes’s universe was the sphere containing the sun’s apparent orbit. The distant stars were thought to lie on the surface of this sphere. However, when thinking about the size of the universe, Archimedes considered a different opinion that had been expressed by his colleague Aristarchus. Remarkably, Aristarchus had anticipated by centuries the findings of the famous astronomer Copernicus by asserting that the sun, not Earth, was at the center of the universe. Aristarchus’s calculations implied a universe that was 10,000 times as large in diameter as the universe of Archimedes. So Archimedes, wanting to err on the high side, calculated the sand needed to fill Aristarchus’s larger universe. This led to the figure 1 × 1063. Archimedes didn’t actually write 1 × 1063 for his answer. Here, we use the modern representation of an extremely large number by writing it as a power of 10 with a coefficient greater than or equal to 1 and less than 10. We call this method of writing numbers scientific notation. Archimedes knew that he needed very large numbers to answer the question about grains of sand. But there was no such notation for large numbers in the time of Archimedes. He had to invent it. Ordinary Greek numbers (written by using letters) went as high as 1 myriad, or 10,000. A myriad of myriads produced 10,0002, or 100,000,000. That was the limit for using Greek notation. Help your students see that a myriad of myriads (i.e., 100,000,000) would be 1 × 108 in modern notation. Because this number is so large, let’s use the symbol ℳ to stand for a myriad of myriads. Archimedes treated the number we are

EUREKA MATH2

calling ℳ as a counting base for a new number system. Instead of place value based on powers of 10, Archimedes’s numbers had place value based on powers of ℳ. Show your students this example of how Archimedes’s base-ℳ system would represent a large number.

2,473,680,000,100,900,043,216 = 247,368 × ℳ2 + 1,009 × ℳ + 43,216.

Have your students notice Archimedes’s “digits”—the multipliers in front of the powers of ℳ. According to Archimedes’s base-ℳ system, any number between 0 and ℳ − 1 can appear as a multiplier. This is the same idea as in our base-ten number system, where any digit from 0 to 9 (i.e., 10 − 1) can appear as a multiplier of a power of 10. But Archimedes had 100,000,000 possible multipliers!

Archimedes called the numbers from 0 up to ℳ − 1 the first order. A number of the first order multiplied by ℳ gives a number of the second order. A number of the first order multiplied by ℳ2 gives a number of the third order, and so on. Archimedes would express the large example number this way:

247,368 units of the third order, plus 1,009 units of the second order, plus 43,216 units of the first order. Let’s get back to the grains of sand. How did Archimedes relate them to the universe? He did what your students might think of doing—he tackled the problem in stages. Grains of sand are pretty small. Archimedes decided that it would take no more than 10,000 of them to make a cluster the size of a poppy seed. He figured that no more than 40 poppy seeds laid side by side equal 1 fingerbreadth, or the width of a finger. Finally, Archimedes determined that no more than 10,000 fingerbreadths 1

7–8 ▸ M1

equal a stadium (i.e., approximately 600 feet—a standard unit of length for the Greeks). Archimedes separately determined that the diameter of his universe was no larger than 10,000,000,000 stadiums in diameter, with the Aristarchus universe having a diameter 10,000 times that much. So Archimedes put it all together one step at a time. When he finally reached the end of his calculations, he declared that the number of grains of sand that fill the (Aristarchus) universe is at most

10,000,000 units of the eighth order of numbers.1

This is 10,000,000 × ℳ7, or equivalently, 1 × 1063 when written in scientific notation. Fast-forward to modern times. The universe is vastly larger than either Archimedes or Aristarchus imagined. All matter is made of particles called atoms. Today, scientists put the number of atoms in the known universe at around 1 × 1082. Suppose that all those atoms became sand. A single grain of sand has about 1 × 1019 atoms. Turning all the atoms in the universe into sand would therefore make approximately (1 × 1082) ÷ (1 × 1019) = 1 × 1063 grains. Surprise! It’s the same as Archimedes’s number. How is it possible that the same number could emerge from two vastly different

Heath, Works of Archimedes, 232.

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interpretations of the universe? A lively student discussion might generate an explanation for this apparent paradox. Some students may observe that the atoms in the known universe are highly dispersed, meaning that they are not concentrated as Archimedes assumed. In any case, it is amusing to note that the number of grains of sand that Archimedes said would be needed to fill the Aristarchus

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universe is actually all the grains of sand the real universe could possibly provide! As noted, Archimedes is regarded as one of the greatest mathematicians and scientists of all time. His book The Sand Reckoner, with its novel number system and careful calculations, shows that Archimedes deserves the honor of that title.

Teacher Edition: Grade 7–8, Module 1, Materials

Materials The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. 27

Blank paper, sheets

Personal whiteboards

Chart paper, tablet

Personal whiteboard erasers

Dry-erase markers

Projection device

Grid paper, sheets

Rulers, inch and metric

Highlighters, in sets of 3 colors

Scientific calculators

Index cards

Student computers or devices

Learn books

Teach book

Lined paper, sheets

Teacher computer or device

Markers

Transparency films

Pencils

Transparent tape, roll

Visit http://eurmath.link/materials to learn more.

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Teacher Edition: Grade 7–8, Module 1, Fluency

Fluency Fluency activities allow students to develop and practice automaticity with fundamental skills so they can devote cognitive power to solving more challenging problems. Skills are incorporated into fluency activities only after they are introduced conceptually within the module. Each lesson in A Story of Ratios begins with a Fluency segment designed to activate students’ readiness for the day’s lesson. This daily segment provides sequenced practice problems on which students can work independently, usually in the first few minutes of a class. Students can use their personal whiteboards to complete the activity, or you may distribute a printed version, available digitally. Each fluency routine is designed to take 3–5 minutes and is not part of the 45-minute lesson structure. Administer the activity as a bell ringer or adapt the activity as a teacher-led Whiteboard Exchange or choral response.

Whiteboard Exchange This routine builds fluency through repeated practice and immediate feedback. A Whiteboard Exchange maximizes participation by having every student record solutions or strategies for a sequence of problems. These written recordings allow for differentiation: Based on the answers you observe, you can make responsive, in-the-moment adjustments to the sequence of problems. Each student requires a personal whiteboard and a whiteboard marker with an eraser for this routine. 1. Display one problem in the sequence. 2. Give students time to work. Wait until nearly all students are ready.

Bell Ringer

3. Signal for students to show their whiteboards. Provide immediate and specific feedback to students one at a time. If revisions are needed, briefly return to validate the work after students make corrections.

This routine provides students with independent work time to determine the answers to a set of problems.

4. Advance to the next problem in the sequence and repeat the process.

1. Display all the problems at once. 2. Encourage students to work independently and at their own pace. 3. Read or reveal the answers.

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Choral Response

Sprints

This routine actively engages students in building familiarity with previously learned skills, strengthening the foundational knowledge essential for extending and applying math concepts. The choral response invites all students to participate while lowering the risk for students who may respond incorrectly.

Sprints are activities that develop mathematical fluency with a variety of facts and skills. A major goal of each Sprint is for students to witness their own improvement within a very short time frame. The Sprint routine is a fun, fast-paced, adrenaline-rich experience that intentionally builds energy and excitement. This rousing routine fuels students’ motivation to achieve their personal best and provides time to celebrate their successes.

1. Establish a signal for students to respond to in unison. 2. Display a problem. Ask students to raise their hands when they know the answer. 3. When nearly all hands are raised, signal for the students’ response. 4. Reveal the answer and advance to the next problem.

Count By This routine actively engages students in committing counting sequences to memory, strengthening the foundational knowledge essential for extending and applying math concepts.

Each Sprint includes two parts, A and B, that feature closely related problems. Students complete Sprint A, followed by two count by routines—one fast-paced and one slow-paced—that include a stretch or other physical movement. Then students complete Sprint B, aiming to improve their score from Sprint A. Each part is scored but not graded. Sprints can be given at any time after the content of the Sprint has been conceptually developed and practiced. The same Sprint may be administered more than once throughout a year or across grade levels. With practice, the Sprint routine takes about 10 minutes.

1. Establish one signal for counting up and counting down and another signal for stopping the count. 2. Tell students the unit to count by. Establish the starting and ending numbers between which they should count. 3. Begin the count by providing the signals. Be careful not to mouth the words as students count.

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Directions 1.

Have students read the instructions and sample problems. Frame the task by encouraging students to complete as many problems as they can—to do their personal best.

Time students for 1 minute on Sprint A. Do not expect them to finish. When time is up, have students underline the last problem they completed.

Read the answers to Sprint A quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.

Have students count their correct answers and record that number at the top of the page. This is their personal goal for Sprint B.

Celebrate students’ effort and success on Sprint A.

To increase success with Sprint B, offer students additional time to complete more problems on Sprint A, or ask sequencing questions to analyze and discuss the patterns in Sprint A.

Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count.

Remind students of their personal goal from Sprint A.

Direct students to Sprint B.

10.

Time students for 1 minute on Sprint B. When time is up, have students underline the last problem they completed.

11.

12. 13.

14.

Sample Vignette Have students read the instructions and complete the sample problems. Frame the task: • You may not finish, and that’s okay. Complete as many problems as you can—do your personal best. • On your mark, get set, think! Time students for 1 minute on Sprint A. • Stop! Underline the last problem you did. • I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer. Read the answers to Sprint A quickly and energetically. • Count the number of answers you got correct and record that number at the top of the page. This is your personal goal for Sprint B. Celebrate students’ effort and success. Provide 2 minutes to allow students to complete more problems or to analyze and discuss patterns in Sprint A by using sequencing questions. Lead students in the fast-paced and slow-paced count by routines. Include a stretch or other physical movement during the count. • Point to the number of answers you got correct on Sprint A. Remember, this is your personal goal for Sprint B. Direct students to Sprint B.

Read the answers to Sprint B quickly and energetically. Have students call out “Yes!” if they answered correctly; have them circle the answer if they answered incorrectly.

• On your mark, get set, improve!

Have students count their correct answers and record that number at the top of the page.

• Stop! Underline the last problem you did.

Have students calculate their improvement score by finding the difference between the number of correct answers in Sprint A and in Sprint B. Tell them to record the number at the top of the page. Celebrate students’ improvement from Sprint A to Sprint B.

Time students for 1 minute on Sprint B. • I’m going to read the answers quickly. As I read the answers, call out “Yes!” if you got it right. If you made a mistake, circle the answer. Read the answers to Sprint B quickly and energetically. • Count the number of answers you got correct and record that number at the top of the page. • Calculate your improvement score and record it at the top of the page. Celebrate students’ improvement.

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The table below provides implementation guidance for the Sprints recommended in this module.

Sprint Name

Administration Guidelines

Sequence Questions

Count By Routines

Addition and Subtraction of Integers

Students add or subtract integers.

What pattern do you notice in problems 7–11?

Fast-paced: Count by twenties from − 80 to 0.

What do you notice about problems 21 and 22?

0 to 63.

Apply Properties of Positive and Negative Exponents

Administer after module 1 lesson 4 or in place of module 1 lesson 5 Fluency.

Students apply the properties and definitions of exponents to write equivalent expressions with positive exponents and a single base.

Scientific Notation with Positive and Negative Exponents

Squares

Fast-paced: Count by twos from

− 30 to 10.

What do you notice about problem 18?

Slow-paced: Count down by halves from 0 to − 10 by using mixed numbers.

How can you use problem 6 to answer problems 7 and 8?

− 50 to 10.

What do you notice about problems 18–20?

0 to 1.

Students write numbers in standard form.

What pattern do you notice in problems 5–7?

0 to 120.

Administer after module 1 lesson 14 or in place of module 1 lesson 15 Fluency.

What are the similarities and differences between problems 5–7 and 16–18?

Students evaluate expressions involving the squares of numbers.

How can you use problem 13 to answer problem 14?

0 thirds to 12 thirds.

Administer before module 1 lesson 18.

How can you use problem 4 to answer problem 31?

0 thirds to 4 by using mixed numbers.

Administer after module 1 lesson 12.

Integer Multiplication and Division

How do the answers to problems 1–4 compare with the answers to problems 10–13?

Slow-paced: Count by threes from

Students multiply or divide integers. Administer after module 1 lesson 9.

Fast-paced: Count by fives from

Slow-paced: Count by tenths from

Fast-paced: Count by tens from Slow-paced: Count by sixes from

0 to 72.

Fast-paced: Count by thirds from Slow-paced: Count by thirds from

537

538 −7 8 −8 −14 −20 −5 −6 −76 −96 −196 1 0 −2 −4 −7 7 11 31 131 28 28

−1 + (−6) 2+6 −2 + (−6) −8 + (−6) −8 + (−12) 4−9 4 − 10 4 − 80 4 − 100 4 − 200 10 + (−9) 10 + (−10) −12 + 10 −14 + 10 −17 + 10 4 − (−3) 4 − (−7) 4 − (−27) 4 − (−127) 20 − (−8) 8 − (−20)

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

350

1+6

Add or subtract the integers.

7–8 ▸ M1 ▸ Sprint ▸ Addition and Subtraction of Integers

Addition and Subtraction of Integers

44.

43.

42.

41.

40.

39.

38.

37.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

26.

25.

24.

23.

−12

−2 + (−15) + 5

−8 − 23 − 5

−23 − (−8)

−14 − (−3)

−4 − (−3)

−3 − (−4)

−36

−15

−11

−1

−8

2 + (−15) + 5

−3 − (−3)

−6

−2

−6

−20

−15

−5

−55

−7 + 8 + (−7)

−7 + 8 + (−3)

−3 + (−2) + (−1)

−30 − (−75)

−20 − (−20)

−9 − (−11)

−9 − 11

−11 − 9

−6 − 9

45 + 32

45 + (−32)

−30 + 25

30 + (−25)

−30 + (−25)

Number Correct:

EUREKA MATH2

7–8 ▸ M1 EUREKA MATH2

−9 8 −8 −14 −17 −4 −17 −47 −97 −397 2 0 −4 −6 −9 12 14 34 134 30 30

−2 + (−7) 3+5 −3 + (−5) −9 + (−5) −9 + (−8) 3−7 3 − 20 3 − 50 3 − 100 3 − 400 8 + (−6) 8 + (−8) −12 + 8 −14 + 8 −17 + 8 6 − (−6) 6 − (−8) 6 − (−28) 6 − (−128) 25 − (−5) 5 − (−25)

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

352

2+7

Add or subtract the integers.

7–8 ▸ M1 ▸ Sprint ▸ Addition and Subtraction of Integers

Addition and Subtraction of Integers

44.

43.

42.

41.

40.

39.

38.

37.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

26.

25.

24.

23.

15 −15 33 97 −17 −22 −22 4 0 45 −12 0 −10 10 −15 0 2 −4 −15 −37 46

40 + (−25) −40 + 25 65 + (−32) 65 + 32 −8 − 9 −13 − 9 −9 − 13 −9 − (−13) −40 − (−40) −30 − (−75) −3 + (−4) + (−5) −7 + 9 + (−2) −7 + 6 + (−9) 23 + (−18) + 5 −2 + (−18) + 5 −7 − (−7) −7 − (−9) −7 − (−3) −18 − (−3) −43 − (−6) −4 − (−45) − (−5)

−65

−40 + (−25)

Improvement:

Number Correct:

EUREKA MATH2

EUREKA MATH2 7–8 ▸ M1

539

540 Number Correct:

EUREKA MATH2

346

18.

17.

16.

15.

14.

13.

12.

11.

10.

9−4 • 94

9−4 • 90

9−4 • 9−1

9−4 • 9−2

9−4 • 9−3

2−3 • 2−1

2−2 • 2−1

2−1 • 2−1

20 • 2−1

7−4 • 78

7−4 • 79

7−4 • 710

7−4 • 711

105 • 100

105 • 10−1

105 • 10−2

105 • 10−3

105 • 10−4 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

102 103 104 105 77 76 75 74

_ __ __ __ __ __ __ __ 1

1 94

1 95

1 96

1 97

1 24

1 23

1 22

1 2

19.

9−8 • 27−2

3−8 • 27−2

3−8 • 9−2

8 • 4−5

23 • 4−5

8 • 2−5

23 • 2−5

2−3 • 24

z −11 • z −21

z 11 • z −21

z −11 • z 21

z 11 • z 21

(−1.7)−10 • (−1.7)4

(−38)14 • (−38)−3

8.26 • 8.2−8

8.2−6 • 8.28

x 6 • x −6

32−8 • 328

1 322

___

1 314

___

1 312

___

__1

z 32

1 ___

z 10

1 ___

z 10

z 32

1 (−1.7)6

______

(−38)11

8.22

1 ___

8.22

Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.

7–8 ▸ M1 ▸ Sprint ▸ Apply Properties of Positive and Negative Exponents

Apply Properties of Positive and Negative Exponents

7–8 ▸ M1 EUREKA MATH2

Improvement:

Number Correct:

EUREKA MATH2

348

18.

17.

16.

15.

14.

13.

12.

11.

10.

9−5 • 95

9−5 • 90

9−5 • 9−1

9−5 • 9−2

9−5 • 9−3

3−3 • 3−1

3−2 • 3−1

3−1 • 3−1

30 • 3−1

7−3 • 74

7−3 • 75

7−3 • 76

7−3 • 77

1010 • 100

1010 • 10−1

1010 • 10−2

1010 • 10−3

1010 • 10−4 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

107 108 109 1010 74 73 72 7

_ __1 __1 __1 __1 __1 __1 __1 1

1 3

19.

106

4−8 • 8−2

2−8 • 8−2

2−8 • 4−2

27 • 9−5

33 • 9−5

27 • 3−5

33 • 3−5

3−3 • 34

z −12 • z −9

z −12 • z 9

z 12 • z −9

z 12 • z 9

(−1.7)−11 • (−1.7)8

(−38)12 • (−38)−5

8.24 • 8.2−6

8.2−4 • 8.26

x 7 • x −7

32−6 • 326

222

1 ___

214

1 ___

212

1 ___

__1

z 21

1 ___

__1

z 21

1 (−1.7)3

______

(−38)7

8.22

1 ___

8.22

Apply the properties and definitions of exponents to write an equivalent expression with a positive exponent and a single base. Let all variables represent nonzero numbers.

7–8 ▸ M1 ▸ Sprint ▸ Apply Properties of Positive and Negative Exponents

Apply Properties of Positive and Negative Exponents

EUREKA MATH2 7–8 ▸ M1

541

542 4•3

36. 37. 38. 39.

−18 −18 24 −24 21 −21 −2 7 2 3 −7

−3 • 7

−14 ÷ 7 −14 ÷ (−2) −14 ÷ (−7) −21 ÷ (−7) 21 ÷ (−3)

18. 19. 20. 21. 22.

354

−7 • (−3) 17.

16.

6 • (−4)

−6 • 3

15.

13.

4•6

35.

−10 ÷ (−2)

11.

14.

34.

−3

−9 ÷ 3

10.

−3 • 6

−3

9 ÷ (−3)

12.

33.

8÷4

44.

43.

42.

41.

40.

32.

31.

30.

29.

−2

−8 ÷ 4

27.

−12

26.

28.

25.

−6

24.

−6

−8 ÷ (−4)

4 • (−3)

23.

−3 • 2

3 • (−2)

−3 • (−2)

Multiply or divide the integers.

7–8 ▸ M1 ▸ Sprint ▸ Integer Multiplication and Division

Integer Multiplication and Division

−144 ÷ (−9)

−144 ÷ 9

−144 ÷ (−3)

144 ÷ (−6)

−144 ÷ 12

(−7)(−7)(5)

(6)(−7)(4)

(−6)(−7)(−2)

(5)(4)(−3)

(5)(−2)(−3)

−72 ÷ 9

−56 ÷ (−8)

56 ÷ (−7)

−48 ÷ (−8)

−42 ÷ (−6)

28 ÷ (−7)

−9 • (−8)

7 • (−8)

−7 • (−7)

−9 • (−6)

−7 • 6

8 • (−6)

Number Correct:

−16

−24

−12

245

−168

−84

−60

−8

−4

−56

−42

−48

EUREKA MATH2

7–8 ▸ M1 EUREKA MATH2

−3 • 5

36. 37. 38. 39.

6 −30 −36 36 −42 28 −28

−18 ÷ (−3)

11.

−4 14 4 5 −7

−4 • 7

−28 ÷ 7 −28 ÷ (−2) −28 ÷ (−7) −35 ÷ (−7) 35 ÷ (−5)

18. 19. 20. 21. 22.

356

−7 • (−4) 17.

16.

6 • (−7)

−6 • 6 15.

13.

6•6

35.

−5

−15 ÷ 3

10.

14.

34.

−5

15 ÷ (−3)

−5 • 6

12 ÷ 4

12.

33.

−3

−12 ÷ 4

44.

43.

42.

41.

40.

32.

31.

30.

29.

28.

27.

−16 3

26.

−12 ÷ (−4)

25.

−15

24.

−15

4 • (−4)

4•4

23.

3 • (−5)

−3 • (−5)

Multiply or divide the integers.

7–8 ▸ M1 ▸ Sprint ▸ Integer Multiplication and Division

Integer Multiplication and Division

−144 ÷ (−12)

−144 ÷ 6

−144 ÷ (−6)

144 ÷ (−3)

−144 ÷ 2

(−7)(−7)(6)

(3)(−7)(7)

(−5)(−7)(−2)

(4)(4)(−3)

(4)(−2)(−3)

72 ÷ 9

−64 ÷ (−8)

−56 ÷ (7)

−48 ÷ (−6)

−42 ÷ (−7)

28 ÷ (−7)

−9 • (8)

9 • (−8)

−8 • (−7)

−9 • (−7)

−7 • 9

8 • (−7)

Improvement:

Number Correct:

−24

−48

−72

294

−147

−70

−48

−8

−4

−72

−63

−56

EUREKA MATH2

EUREKA MATH2 7–8 ▸ M1

543

544 300 41 4,100 0.5 0.05 0.0005 0.84 0.0084 5,250 52,500 236.7 2,367 0.625 0.0625 0.2358 0.02358 0.002 358

3 × 102 4.1 × 10 4.1 × 103 5 × 10−1 5 × 10−2 5 × 10−4 8.4 × 10−1 8.4 × 10−3 5.25 × 103 5.25 × 104 2.367 × 102 2.367 × 103 6.25 × 10−1 6.25 × 10−2 2.358 × 10−1 2.358 × 10−2 2.358 × 10−3

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

358

3 × 10

Write the number in standard form.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

26.

25.

24.

23.

22.

21.

20.

19.

60 6 0.000 006 0.00006

6,000,000 × 10−5 6,000,000 × 10−6 60 × 10−7 6,000 × 10−8

0.6

6,000 × 10−4

80,000 × 10−4

0.8

800 × 10−3

0.009 × 103

8,000 × 10−2

0.9

80 × 10−1

0.009 × 102

70,324.2

7.03242 × 104

70,324

7.0324 × 104

0.09 × 102

70,320

7.032 × 104

70,300

7.03 × 104

0.9 × 10

73,000

Number Correct:

EUREKA MATH2

7.3 × 104

7–8 ▸ M1 ▸ Sprint ▸ Scientific Notation with Positive and Negative Exponents

Scientific Notation with Positive and Negative Exponents

7–8 ▸ M1 EUREKA MATH2

500 32 3,200 0.6 0.06 0.0006 0.34 0.0034 8,670 86,700 876.5 87.65 0.225 0.0225 0.2358 0.000 2358 0.000 023 58

5 × 102 3.2 × 10 3.2 × 103 6 × 10−1 6 × 10−2 6 × 10−4 3.4 × 10−1 3.4 × 10−3 8.67 × 103 8.67 × 104 8.765 × 102 8.765 × 10 2.25 × 10−1 2.25 × 10−2 2.358 × 10−1 2.358 × 10−4 2.358 × 10−5

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

360

5 × 10

Write the number in standard form.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

26.

25.

24.

23.

22.

21.

20.

19.

30 3 0.000 003 0.00003

3,000,000 × 10−5 3,000,000 × 10−6 30 × 10−7 3,000 × 10−8

0.3

3,000 × 10−4

40,000 × 10−4

0.4

400 × 10−3

0.007 × 103

4,000 × 10−2

0.7

40 × 10−1

0.007 × 102

50,624.5

5.06245 × 104

506,240

5.0624 × 105

0.07 × 102

506,200

5.062 × 105

50,600

5.06 × 104

0.7 × 10

56,000

Improvement:

Number Correct:

EUREKA MATH2

5.6 × 104

7–8 ▸ M1 ▸ Sprint ▸ Scientific Notation with Positive and Negative Exponents

Scientific Notation with Positive and Negative Exponents

EUREKA MATH2 7–8 ▸ M1

545

546 3•3

1 16 0.16 36 0.36 64 0.64 121 1.21 144 1.44

12 42 0.42 62 0.62 82 0.82 112 1.12 122 1.22

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

362

40.

39.

38.

37.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

49 49

26.

25.

25 25

24.

23.

7•7

5•5

22.

21.

100

102

10 • 10

Evaluate the expression.

7–8 ▸ M1 ▸ Sprint ▸ Squares

Squares

9 4

_ 25 4

_ _1

_9 16

25 16

(2)

_ 2

(2)

_ 2 _ 2 3 2 (4) 5 2 (4)

(8)

_ 1 _

12 15 18 18 52 58 58 46 40 40

_1 2 32 + 3 32 + 6 32 + 9 32 + 32 72 + 3 72 + 9 72 + 32 72 − 3 72 − 9 72 − 32

(10)

(8)

100

25 64

64 2

(8)

_ _ 2

_ 2

(4)

1 4

_ 2

(2)

Number Correct:

EUREKA MATH2

7–8 ▸ M1 EUREKA MATH2

2•2

100 25 0.25 49 0.49 81 0.81 121 1.21 144 1.44

102 52 0.52 72 0.72 92 0.92 112 1.12 122 1.22

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

364

40.

39.

38.

37.

36.

35.

34.

33.

32.

31.

30.

29.

28.

27.

36 36

26.

25.

16 16

24.

23.

6•6

4•4

22.

21.

1,000

103

10 • 10 • 10

Evaluate the expression.

7–8 ▸ M1 ▸ Sprint ▸ Squares

Squares

4 9

_ 16 9

_ _1

_9 25

16 25

(3)

_ 2 (3)

_ 2 (5)

_ 2 3 2 (5) 4 2 (5)

(6)

49 36

_ 9 _

20 26 32 32 68 80 80 60 48 48

(6)

_ 2 _3 2 42 + 4 42 + 10 42 + 16 42 + 42 82 + 4 82 + 16 82 + 42 82 − 4 82 − 16 82 − 42

(10)

100

_ 5

(6)

25 36

_1 _ 2

_ 2

1 9

_ 2 1

(3)

Improvement:

Number Correct:

EUREKA MATH2

EUREKA MATH2 7–8 ▸ M1

547

Teacher Edition: Grade 7–8, Module 1, Sample Solutions

Sample Solutions Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

EUREKA MATH2

7–8 ▸ M1

EUREKA MATH2

7–8 ▸ M1 ▸ Mixed Practice 1

Student Edition: Grade 7–8, Module 1, Mixed Practice 1

Mixed Practice

Name

6. Mrs. Kondo makes bags of school supplies for her students. Mrs. Kondo has 36 pencils and 48 erasers, and she wants to place all of them in the bags.

Date

a. What is the greatest number of bags Mrs. Kondo can make with each bag containing the same number of pencils and the same number of erasers?

For problems 1–3, determine the unit rate for the situation. 1. A car travels 105 miles in 3 hours. What is the unit rate associated with the rate of miles per hour?

Mrs. Kondo can make 12 bags.

35 b. How many pencils and erasers will be in each bag?

2. Maya buys 6 limes for $1.50. What is the unit rate associated with the rate of dollars per lime?

There will be 3 pencils and 4 erasers in each bag.

0.25 3. Pedro receives 10 text messages in 2 _1 hours. What is the unit rate associated with the rate of 2 text messages per hour?

7. Consider the given equation.

3 3 _4 + __ = _____ + _____ 5 10 10

4. Liam makes green paint by mixing 3 gallons of yellow paint with 2 gallons of blue paint. He likes the shade of green paint he makes and wants to make more in the same ratio. Complete the table to show the relationship between the number of gallons of yellow paint and the number of gallons of blue paint.

Yellow Paint (gallons) Blue Paint (gallons)

3 2

_3 2

_2 3

a. Fill in the boxes to make a true number sentence. 3 . Choose all that apply. b. Find the sum of _4 and __

7 A. __ 10 11 __ B. 5

8. Find the product.

4.27 ⋅ 0.3

It will take Ethan 12 hours to drive 600 miles.

548

EM2_7-801TE_sample_solution_resource.indd 548

11 C. __ 10 1 D. 1 __ 10 7 E. __ 15

5. Ethan drives 150 miles in 3 hours. If he continues to drive at a constant rate, how many hours will it take Ethan to drive 600 miles?

1.281

333

334

06/02/23 4:28 PM

EUREKA MATH2

7-8 ▸ M1

EUREKA MATH2

7-8 ▸ M1 ▸ Mixed Practice 1

9. Use the standard algorithm to find the sum.

4.073 + 8.607 + 2.46 15.14

For problems 10 and 11, divide. 10. 575.186 ÷ 5.8

11. 857.22 ÷ 47.1

18.2

99.17 12. Order the values from least to greatest.

|0.6|, - 1-4 , 5, 0, |6|, |-4|, - 7, |- 3-2|

- 7, - -1 , 0, |0.6|, - -3 , |- 4|, 5, |6| 2 4

| |

335

549

EUREKA MATH2

7–8 ▸ M1

EUREKA MATH2

7–8 ▸ M1

7–8 ▸ M1 ▸ Mixed Practice 2

EUREKA MATH2

4. The measurements of a rectangular floor are shown in the figure.

Student Edition: Grade 7–8, Module 1, Mixed Practice 2

Mixed Practice

Name

Date

a. Find the area of the floor. The area of the floor is 126 square meters.

1. Evaluate (2.1)3. A. 4.41

b. Find the perimeter of the floor.

B. 6.3

The perimeter of the floor is 54 m.

C. 8.001

21 m

D. 9.261

2. Which expressions represent twice the product of 3 and g? Choose all that apply. A. 2 + 3 + g

B. (3g) ⋅ 2

5. Find the area of the triangle.

C. 2(3 + g) D. 2 + 3g E. 2(3g)

10 cm

3. Evaluate 4x 3 + 6 − 16 when x = 2.

6 cm The area of the triangle is 30 square centimeters.

550

EM2_7-801TE_sample_solution_resource.indd 550

337

338

06/02/23 4:28 PM

EUREKA MATH2

7-8 ▸ M1

EUREKA MATH2

7-8 ▸ M1 ▸ Mixed Practice 2

6. Two vertices of a rectangle are located at (-2, - 9) and (3, - 9). The area of the rectangle is 100 square units. Name one set of possible coordinates for the other two vertices. Use the coordinate plane as needed.

EUREKA MATH2

7-8 ▸ M1 ▸ Mixed Practice 2

7. Use the figure shown to complete parts (a)-(f).

y 30

Sample:

-30

-20

-10

a. Name a line.

⟷ CE

c. Name a line segment. -10

e. Name an obtuse angle. -20

b. Name a ray.

YN ∠ENY

⟶ YC

d. Name an acute angle. f.

∠CEN

Name a pair of parallel line segments.

CY EN - and -

-30

Sample: One set of possible coordinates for the other two vertices is (-2, 11) and (3, 11).

339

340

551

EUREKA MATH2

7-8 ▸ M1

EUREKA MATH2

7-8 ▸ M1 ▸ Mixed Practice 2

For problems 8 and 9, determine whether the data distribution is approximately symmetric or skewed. Describe the shape of the data distribution. 8.

EUREKA MATH2

7-8 ▸ M1 ▸ Mixed Practice 2

Points Scored in Each Game

Patients’ Ages 12

10 9

Number of Points

8 Frequency

Approximately symmetric around 2

7 6 5 4 3 2 1

Age (years)

Skewed to the right

552

341

342

Teacher Edition: Grade 7-8, Module 1, Works Cited

Works Cited CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018. Common Core Standards Writing Team. Progressions for the Common Core State Standards in Mathematics. Tucson, AZ: Institute for Mathematics and Education, University of Arizona, 2011-2015. https://www.math.arizona .edu/~ime/progressions/. Ellis, Julie. What’s Your Angle, Pythagoras? Watertown, MA: Charlesbridge Publishing, 2004. Heath, Thomas L., trans. The Sand Reckoner of Archimedes. London: 2008. Heath, Thomas L., ed. The Works of Archimedes. New York: Dover Publications, 2002.

554

National Governors Association Center for Best Practices, Council of Chief State School Officers (NGA Center and CCSSO). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010. Zwiers, Jeffrey, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Palo Alto: Stanford University, UL/SCALE, 2017. https://ul.stanford .edu/sites/default/files/resource/2021-11/Principles%20 for%20the%20Design%20of%20Mathematics%20 Curricula-1.pdf.

Teacher Edition: Grade 7-8, Module 1, Credits

Credits Great Minds® has made every effort to obtain permission for the reprinting of all copyrighted material. If any owner of copyrighted material is not acknowledged herein, please contact Great Minds for proper acknowledgment in all future editions and reprints of this presentation. Common Core State Standards for Mathematics Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All Rights Reserved. For a complete list of credits, visit http://eurmath.link /media-credits. Cover image copyright Paul Signac, Vue de Constantinople, La Corne d’OR (Gold Coast) matin (Morning), 1907, Private Collection. Photo credit: Peter Horree/Alamy Stock Photo; page 61, KAKOOMA® Puzzles appear courtesy of Greg Tang; Page 305, Courtesy Future Perfect at Sunrise. Public domain via Wikimedia Commons; page 307, Archimedes, Greek mathematician, c. 250 BC., Science Museum, London, Great Britain. Photo credit: SSPL/ Science Museum/Art Resource, NY; page 340, EPS/ShutterStock. com; page 358, koya979/Shutterstock.com; page 358,

Nerthuz/Shutterstock.com; pages 369, 370, 371, Paul Signac, Vue de Constantinople, La Corne d’OR (Gold Coast) matin (Morning), 1907, Private Collection. Photo credit: Peter Horree/ Alamy Stock Photo; page 411, Brinja Schmidt/Shutterstock.com; page 411, digitalreflections/Shutterstock.com; page 411, Daniel J Lee/Shutterstock.com; page 413, Vadym Zaitsev/ Shutterstock.com; page 447, anthony pietrafesa/Shutterstock. com, page 447, pixelgerm/Shutterstock.com; page 447, Macrez Black/Shutterstock.com; page 528, “Cassini apparent” by James Ferguson (1710-1776), based on similar diagrams by Giovanni Cassini (1625-1712) and Dr Roger Long (1680-1770); engraved for the Encyclopaedia Britannica (1st Edition, 1771; facsimile reprint 1971), Volume 1, Fig. 2 of Plate XL facing page 449 by Andrew Bell, courtesy Wikimedia Commons. Public domain; page 529, “A simulated view of the entire observable universe” by Andrew Z. Colvin, courtesy Wikimedia Commons, is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported license (CC BY-SA 3.0), https://creativecommons.org/licenses/by-sa/3.0/. All images are the property of Great Minds.

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Teacher Edition: Grade 7-8, Module 1, Acknowledgments

Acknowledgments Amanda Aleksiak, Tiah Alphonso, Lisa Babcock, Chris Barbee, Reshma P. Bell, David Choukalas, Mary Christensen-Cooper, Jill Diniz, Kelli Ferko, Levi Fletcher, Anita Geevarghese, Krysta Gibbs, Stefanie Hassan, Travis Jones, Robin Kubasiak, Connie Laughlin, Maureen McNamara Jones, Dave Morris, Ben Orlin, Brian Petras, April Picard, Lora Podgorny, Janae Pritchett, Aly Schooley, Erika Silva, Tara Stewart, Heidi Strate, Cathy Terwilliger, Cody Waters Ana Alvarez, Lynne Askin-Roush, Trevor Barnes, Brianna Bemel, Carolyn Buck, Lisa Buckley, Adam Cardais, Christina Cooper, Kim Cotter, Lisa Crowe, Brandon Dawley, Cherry dela Victoria, Delsena Draper, Sandy Engelman, Tamara Estrada, Ubaldo Feliciano-Hernández, Soudea Forbes, Jen Forbus,

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Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Adam Green, Dennis Hamel, Cassie Hart, Sagal Hassan, Kristen Hayes, Marcela Hernandez, Abbi Hoerst, Libby Howard, Elizabeth Jacobsen, Ashley Kelley, Lisa King, Sarah Kopec, Drew Krepp, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Patricia Mickelberry, Mary-Lise Nazaire, Corinne Newbegin, Tara O’Hare, Max Oosterbaan, Tamara Otto, Christine Palmtag, Laura Parker, Katie Prince, Gilbert Rodriguez, Todd Rogers, Karen Rollhauser, Neela Roy, Gina Schenck, Amy Schoon, Aaron Shields, Leigh Sterten, Mary Sudul, Lisa Sweeney, Tracy Vigliotti, Dave White, Charmaine Whitman, Glenda Wisenburn-Burke, Howard Yaffe

Exponentially Better Knowledge2 In our tradition of supporting teachers with everything they need to build student knowledge of mathematics deeply and coherently, Eureka Math2 provides tailored collections of videos and recommendations to serve new and experienced teachers alike. Digital2 With a seamlessly integrated digital experience, Eureka Math2 includes hundreds

Module 2 One- and Two-Variable Equations Module 3 Two-Dimensional Geometry

of clever illustrations, compelling videos, and digital interactives to spark discourse and wonder in your classroom.

Module 4 Graphs of Linear Equations and Systems of Linear Equations

Accessible2 Created with all readers in mind, Eureka Math2 has been carefully designed to ensure struggling readers can access lessons, word problems, and more.

Module 5 Functions and Three-Dimensional Geometry

Joy2 Together with your students, you will fall in love with math all over again—or for the first time—with Eureka Math2. What does this painting have to do with math? The French neoimpressionist Paul Signac worked with painter Georges Seurat to create the artistic style of pointillism, in which a painting is made from small dots. Signac’s Vue de Constantinople, La Corne d’Or Matin shows the Golden Horn, a busy waterway in Istanbul, Turkey. How many dots do you think were used to make this pointillist painting? How would you make an educated guess? On the cover Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907 Paul Signac, French, 1863–1935 Oil on canvas Private collection Paul Signac (1863–1935), Vue de Constantinople, La Corne d’Or (Gold Coast) Matin (Morning), 1907. Oil on canvas. Private collection. Photo credit: Peter Horree/Alamy Stock Photo

ISBN 979-8-88588-159-3

Module 1 Rational and Irrational Numbers

798885 881593

Module 6 Probability and Statistics