EM2CA_G5_M3_Teach_26A_116968_Updated 12.23 by General

A Story of Units®

Fractions Are Numbers TEACH ▸ Multiplication and Division with Fractions

Module 3

What does this painting have to do with math? Color and music fascinated Wassily Kandinsky, an abstract painter and trained musician in piano and cello. Some of his paintings appear to be “composed” in a way that helps us see the art as a musical composition. In math, we compose and decompose numbers to help us become more familiar with the number system. When you look at a number, can you see the parts that make up the total? On the cover Thirteen Rectangles, 1930 Wassily Kandinsky, Russian, 1866–1944 Oil on cardboard Musée des Beaux-Arts, Nantes, France Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musée des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/ Art Resource, NY

Great Minds® is the creator of Eureka Math®, Wit & Wisdom®, Alexandria Plan™, and PhD Science®. Published by Great Minds PBC. greatminds.org © 2026 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 30 29 28 27 26 ISBN 979-8-88811-696-8

A Story of Units®

Fractions Are Numbers ▸ 5 TEACH Module

Module

1 2 3 4 5 6

Place Value Concepts for Multiplication and Division with Whole Numbers

Addition and Subtraction with Fractions

Multiplication and Division with Fractions

Place Value Concepts for Decimal Operations

Addition and Multiplication with Area and Volume

Foundations to Geometry in the Coordinate Plane

Before This Module

Overview

Grade 4 Module 4

Multiplication and Division with Fractions

In grade 4 module 4, students decompose a non-unit fraction as a sum of unit fractions and then write the sum as a product of a whole number times a unit fraction. They solve word problems involving multiplication of a fraction by a whole number and express answers as fractions and mixed numbers.

Topic A Multiplication of a Whole Number by a Fraction Students extend their understanding of fractions from parts of a whole (e.g., 1 third of a shape) to parts of a set or a number (e.g., 1 third of a group of 12 items). They find fractions of a set and then transition to finding a fraction of a whole number. Students learn that finding a fraction of a whole number means they are finding the product of a fraction and a whole number. They apply this learning to converting customary measurement units.

6 units = 4

1 unit = 4

5 units = 5 x 4 = 20

? 5 of 4 is 5 parts when 4 is partitioned into 6 equal parts.

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Topic B Multiplication of Fractions Students use area models and number lines to multiply fractions by unit fractions and then fractions by fractions. Throughout the topic, they reason about the value of products by considering whether the factors involved are greater than 1 or less than 1.

1 4 4 × = 5 5 25 1 5

1 5

2 5

3 5

4 5

5 5

1 4 4 x = 5 5 25

4 5

Topic C Division with a Unit Fraction and a Whole Number Students use tape diagrams and number lines to divide a whole number by a unit fraction and to divide a unit fraction by a whole number. They solve word problems that involve multiplication of fractions and division of whole numbers and fractions, and students explain the relationship between multiplication and division.

6 ÷ 1 = 24

12 ÷ 1 = 60

1 4

... ? fourths

1 ÷3=1 2 6

12 0 5

? 1 5

2 5

3 5

4 5

5 5

0 6

1 2

1 3

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After This Module Grade 5 Module 5 In module 5, students learn to find the area of rectangles with side lengths that are fractions. The context of area is also used to show how to multiply mixed numbers.

Topic D Multi-Step Problems with Fractions Students apply their previous learning about all operations with fractions to compare and evaluate expressions that contain grouping symbols. They create and solve word problems involving fractions, and they write equations with parentheses for word problems that require multiple steps to solve. Toby spends 2 of his money on movie tickets. He spends 1 of his 3

Grade 6 Module 2

remaining money on popcorn. He has $10 left. How much money

In grade 6, students expand their understanding of division involving fractions to division of fractions in general.

did Toby have to begin with?

movies

popcorn 1 × 10 2

$10

× 5 = 25

Toby had $25 to begin with.

Contents Multiplication and Division with Fractions Why. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Achievement Descriptors: Overview. . . . . . . . . . . . . . . . . . . . . 10 Topic A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Multiplication of a Whole Number by a Fraction Lesson 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Find fractions of a set with arrays.

Lesson 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Interpret fractions as division to find fractions of a set with tape diagrams and number lines.

Lesson 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Multiply fractions by unit fractions by making simpler problems.

Lesson 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Multiply fractions greater than 1 by fractions. Lesson 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Multiply fractions.

Topic C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Division with a Unit Fraction and a Whole Number Lesson 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Lesson 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Multiply a whole number by a fraction less than 1.

Divide a nonzero whole number by a unit fraction to find the number of groups.

Lesson 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Lesson 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

Multiply a whole number by a fraction.

Divide a nonzero whole number by a unit fraction to find the size of the group.

Lesson 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Convert larger customary measurement units to smaller measurement units.

Lesson 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Lesson 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Lesson 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

Convert smaller customary measurement units to larger measurement units.

Topic B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Multiplication of Fractions Lesson 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Multiply fractions less than 1 by unit fractions pictorially.

Divide a unit fraction by a nonzero whole number.

Divide by whole numbers and unit fractions.

Lesson 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Reason about the size of quotients of whole numbers and unit fractions and quotients of unit fractions and whole numbers.

Lesson 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Solve word problems involving fractions with multiplication and division.

Lesson 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Multiply fractions less than 1 pictorially. © Great Minds PBC

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Topic D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Multi-Step Problems with Fractions

Resources

Lesson 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

Achievement Descriptors: Proficiency Indicators. . . . . . . . . . . . . . . . 458

Compare and evaluate expressions with parentheses.

Lesson 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

Big Ideas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Create and solve one-step word problems involving fractions.

Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

Lesson 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

Math Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Solve multi-step word problems involving fractions and write equations with parentheses.

Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

Lesson 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Solve multi-step word problems involving fractions.

Works Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

Lesson 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

Credits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Evaluate expressions involving nested grouping symbols. (Optional)

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Why Multiplication and Division with Fractions How are the measurement and partitive interpretations for division important to this module? When students begin their work with division in grade 3, they learn to interpret the problem based on the context. For 12 ÷ 3 = 4, the quotient 4 can mean the size of each group (e.g., 4 marbles in 3 bags) or it can mean the number of groups (e.g., 4 bags each with 3 marbles). These interpretations are revisited in grade 5 module 1 when students divide multi-digit whole numbers. Knowing these interpretations is important for students because it helps them make sense of the numbers in problems and it supports students with reasoning about whether their answers are correct. As students begin to divide with fractions in this module, understanding the meaning of the numbers in the division equation supports students with making sense of the problem and relating the division equation to multiplication. Because the quotient is always less than the dividend when they divide by whole numbers greater than 1, students often need support with division by fractions less than 1 because the quotient is greater than the dividend. This module aims to anchor division with unit fractions in contexts. Therefore, students can reason about the size of the quotient and the dividend and interpret an equation by using questions such as, How many thirds are in 2? 2 is _   of what number? _   is 2 groups of what? 1 3

1 3

Please note that students are not required to categorize problems as measurement or partitive division, but rather interpret the meaning of the divisor and quotient within the problem.

Why do students learn to multiply and divide with fractions before they do so with decimals? Students have conceptually worked with fractions for longer than they have worked with decimals. Starting as early as in kindergarten, students informally model fractions © Great Minds PBC

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EUREKA MATH2 California Edition

as parts of a whole (shape) and identify equal parts. In grade 3, the concept of a fraction is formalized, and in grade 4, operations work with fractions begins. It is not until grade 4 that the fractional units of tenths and hundredths are introduced as decimals. Because tenths and hundredths are both place value units and fractional units, students use what they know about fractions to support their conceptual development with decimals. For the same reason, students learn to multiply and divide with fractions before they do so with decimals.

Why is lesson 22 considered optional? In grade 5, students are expected to use parentheses, brackets, and 4,628 × [ ((3 ÷ 5) − _ 53 )+ ( __   7 − (14 ÷ 26))] = 0 13 braces in numerical expressions and to be able to evaluate expressions that contain these grouping symbols. Lesson 22 extends students’ understanding of how to evaluate expressions with grouping symbols by presenting students with expressions that have nested grouping symbols. Students read an expression and identify expressions in grouping symbols as a number. For example, in the equation shown, they read the expression on the left side of the equal sign as 4,628 times another number. Then students read the expression within the brackets as a number plus another number. They continue to read the expression until they find an expression that can be evaluated (i.e., 3 ÷ 5) because it is not part of another number. This natural extension to grade 5 expectations provides an opportunity for students to practice the skills they recently acquired with adding, subtracting, multiplying, and dividing with fractions and multi-digit whole numbers. Students learn that these seemingly complicated expressions are just different ways to represent numbers. Some of the more complicated expressions that students work with can be easily evaluated, which may help students embrace challenging problems in the future.

Achievement Descriptors: Overview Multiplication and Division with Fractions 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 • informal classroom observations, • data from other lesson-embedded formative assessments, • Exit Tickets, • Topic Quizzes, and • Module Assessments. This module contains the fourteen ADs listed.

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5.Mod3.AD1

5.Mod3.AD2

5.Mod3.AD3

5.Mod3.AD4

Write numerical expressions that include fractions and parentheses.

Evaluate numerical expressions that include fractions and parentheses.

Translate between numerical expressions that include fractions and mathematical or contextual verbal descriptions.

Compare the effect of each number and operation on the value of a numerical expression that includes fractions.

5.OA.A.1

5.OA.A.2

5.Mod3.AD5

5.Mod3.AD6

5.Mod3.AD7

5.Mod3.AD8

Solve multi-step problems, including word problems, involving addition, subtraction, and multiplication of fractions, division of whole numbers with fractional quotients, and division with unit fractions and whole numbers.

Multiply whole numbers or fractions by fractions.

Recognize, model, and contextualize the product of a fraction and a whole number or fraction.

Compare the effects of multiplying by fractions and whole numbers.

5.NF

5.NF.B.4

5.NF.B.4.a

5.NF.B.5, 5.NF.B.5.a

5.Mod3.AD9

5.Mod3.AD10

5.Mod3.AD11

5.Mod3.AD12

Explain the effect of multiplying by a fraction less than 1, equal to 1, or greater than 1.

Solve real-world problems involving multiplication of fractions.

Model and evaluate division of unit fractions by nonzero whole numbers.

Model and evaluate division of whole numbers by unit fractions.

5.NF.B.6

5.NF.B.7.a

5.NF.B.7.b

5.NF.B.5, 5.NF.B.5.b

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5.Mod3.AD13

5.Mod3.AD14

Solve word problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions.

Convert among units within the customary measurement system to solve problems.

5.NF.B.7.c

5.MD.A.1

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 grade 5 module 1 is coded as 5.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 California Common Core State Standards that the AD addresses.

EUREKA MATH2 California Edition

AD Code: Grade.Module.AD##

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AD Language EUREKA MATH2 California Edition

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5.Mod3.AD13 Solve word problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions. RELATED CA CCSSM

5.NF.B.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share How many

__1 -cup servings are in 2 cups of raisins?

__1 lb of chocolate equally?

Related Standard

Partially Proficient

Proficient

Highly Proficient

Identify numerical expressions that can be used to solve single-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Solve single-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Solve multi-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Mr. Perez has

__1 gallon of grape juice. He pours the juice equally 2

__1 gallon of grape juice. He pours the 2

into 8 glasses. How many gallons of juice are in each glass?

juice equally into 8 glasses. How many gallons of juice

juice equally into 8 glasses. Then he takes 1 of the

Which expression can be used to solve this problem?

are in each glass?

glasses and shares it equally among his 3 children.

__1 × 8 2 __1 ÷ 8 B. 2 __1 C. 8 × 2 __1 D. 8 ÷

AD Indicators

How many gallons of juice does each of Mr. Perez’s children get?

470

Topic A Multiplication of a Whole Number by a Fraction In topic A, students use their understanding of fractions as division to find fractions of a set and to multiply whole numbers by fractions. Students begin by using arrays to find fractions of a set and then use number lines and tape diagrams to find fractions of sets. They realize that knowing the unit fraction of a set can be helpful to find other fractions of the set when the denominator and whole number of the set remain the same. For example, if students know _  1 of 10, then finding _  3 of 10 is simpler because _  3 of 10 is 3 times as much as _  1 of 10. 5

Next, students make the connection between finding a fraction of a whole and multiplying a whole number by a fraction (i.e., _  3 of 10 has the same value as _  3 × 10). Students continue 5

to use number lines and tape diagrams to find products of fractions and whole numbers. When students begin multiplying a whole number by a fraction greater than 1, they rely on tape diagrams because tape diagrams can be more efficient for finding larger products. Students discover multiplying a whole number by a fraction less than 1 results in a product less than the whole number and multiplying a whole number by a fraction greater than 1 results in a product greater than the whole number. Students use their understanding of multiplying fractions and whole numbers to convert between customary units. They first convert larger units to smaller units and then convert smaller units to larger units. In topic B, students use their knowledge of multiplying a whole number by a fraction to multiply two fractions.

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Progression of Lessons Lesson 1

Lesson 2

Lesson 3

Find fractions of a set with arrays.

Interpret fractions as division to find fractions of a set with tape diagrams and number lines.

Multiply a whole number by a fraction less than 1.

0 1 of 18 is 6

3 of 18 is 6

I can use an array to represent a whole number and then partition the array into equal groups to find a fraction of a set.

2 2 2 2 8 + + + = 5 5 5 5 5 2 of 4 is 8 . 5 5 When I find a fraction of a set, I can use number lines or tape diagrams to represent the problem. I can use what I know about finding a unit fraction of a whole number to find other fractions of a whole number.

3 units = 6 1 unit = 63 2 units = 2 × 63 = 4

I know I can use multiplication to find a fraction of a whole number. I can multiply a whole number by a fraction by using number lines and tape diagrams.

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Lesson 4

Lesson 5

Lesson 6

Multiply a whole number by a fraction.

Convert larger customary measurement units to smaller measurement units.

Convert smaller customary measurement units to larger measurement units.

1 ft = 6

1 × 1 ft = 1 × 12 in 6 6

? I can multiply whole numbers by fractions less than 1 and by fractions greater than 1 by using tape diagrams.

__ = 1 × 12 in 6

__ in = 12 6

= 2 in I can use what I know about how to multiply fractions to convert from larger measurement units to smaller measurement units. I can use tape diagrams to understand the relationship between the units. I can draw tape diagrams to solve real-world problems.

132 in =

1 132 × 1 in = 132 ×  __ ft 12

1 132 ×  __ = ______ 12

132 × 1 12

= ___ 132 12

= 11 I can multiply a fraction by a whole number to convert from smaller units to larger units. I can use tape diagrams to understand the relationship between the units.

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Read word problems involving fractions of a set. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy, which should include an array and equations, for solving a problem involving fractions of a set.

Lesson 2

In partners and in class discussion, orally explain how to find the value of an expression that involves a fraction of a whole number. Orally compare different methods, including arrays, number lines, and tape diagrams, for finding the value of a fraction of a whole number.

Lesson 3

In partners and in class discussion, orally explain how to model and solve a problem involving multiplication of a whole number by a fraction. Interpret a fraction of a set as multiplication and write or draw to record solution strategies, which could include tape diagrams, number lines, and equations, for solving problems that involve multiplication of a whole number by a fraction.

Lesson 4

Orally and in writing, explain why the product of a whole number and a fraction is sometimes greater than the whole number. Write or draw to record a solution strategy for solving a problem that involves multiplication of a whole number by a fraction. Listen to and orally describe solution strategies, which should include number lines, tape diagrams, and equations, for multiplying a whole number and a fraction. Compare and connect the solution strategies.

Lesson 5

Read word problems involving measurement conversions. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation for solving a problem that involves measurement conversions.

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Lesson 6

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Orally and in writing, describe the multiplicative relationship between a smaller measurement unit and a larger measurement unit. Read word problems and tape diagrams involving measurement conversions and write equations for solving the problems.

LESSON 1

Find fractions of a set with arrays.

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Name

Date

4 of 15. Draw lines or boxes to show your work. 1. Use the array to find _ 5

Lesson at a Glance Students partition arrays into equal groups to find a unit fraction of a set. They use what they know about the number of objects in one unit to find non-unit fractions of a set. This lesson introduces the verb demonstrate.

Key Question • Is an array helpful when you find a fraction of a set? How? _4 of 15 is 5

Achievement Descriptors

5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

3 of the books she reads are fiction. How many fiction 2. Yuna reads 20 books during the summer. _

books does Yuna read?

1 of 20 is 5. 4

5 × 3 = 15

3 of 20 is 15. 4

Yuna reads

fiction books.

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• Computer or device*

Prepare 12 centimeter cubes of one color for each student.

Learn 35 min • Partition a Set

• Projection device* • Teach book*

• Solve a Real-World Problem

Students

• Problem Set

• Centimeter cubes (12)

Land 10 min

• Dry-erase marker* • Learn book* • Pencil* • Personal whiteboard* • Personal whiteboard eraser* * These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

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Fluency

Happy Counting by Halves Students visualize a number line while counting aloud to maintain fluency with counting by halves. Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.) Let’s count by halves. The first number you say is 0 halves. Ready? Signal up, down, or stop accordingly.

0 2

1 2

2 2

3 2

4 2

5 2

6 2

7 2

6 2

5 2

4 2

5 2

Teacher Note Choose signals you are comfortable with, such as thumbs-up, thumbs-down, and an open hand. Show your signal and gesture accordingly for each count. The goal is to be clear and crisp, so students count in unison. Avoid saying the numbers with the class; instead, listen for errors and hesitations.

6 2

Continue counting by halves within  __ . Change directions occasionally, emphasizing where 10 2

students hesitate or count inaccurately.

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Whiteboard Exchange: Relate Repeated Addition to Multiplication Students write equations with unit fractions to represent a tape diagram to prepare for multiplying a whole number by a fraction. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

? 1 2

Display the tape diagram with 2 units of  _1  .

1 2

1 1 2 + = 2 2 2 1 2 2×2=2

Write a repeated addition equation to represent the tape diagram. Write the sum as a fraction. Display the addition equation.

Write a multiplication equation to represent the tape diagram. Write the product as a fraction. Display the multiplication equation. Repeat the process with the following sequence:

1 2

1 1 1 3 + + = 2 2 2 2 1 3 3×2=2

1 3

1 1 1 + + =3 3 3 3 3 1

3× 3 = 3

1 3

1 1 1 1 1 5 + + + + = 3 3 3 3 3 3 1

5× 3 = 3

1 3

1 4

1 1 1 1 1 5 + + + + = 4 4 4 4 4 4 1 5 5× 4 = 4

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5 ▸ M3 ▸ TA ▸ Lesson 1

Whiteboard Exchange: Interpret a Fraction as Division Students write a fraction as a division expression, and as a whole number when possible, to prepare for finding fractions of a set with arrays. 1 Display _ = 2

Write the fraction as a division expression. Then express the quotient as a whole number if possible. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

1 = 1 ÷ 2 2

Display the completed equation. Repeat the process with the following sequence:

6 = 6 ÷ 2 = 3 2

2 = 2 ÷ 3 3

9 = 9 ÷ 3 = 3 3

4 = 4 ÷ 5 5

5 = 5 ÷ 5 = 1 5

20 20 ÷ 5 = 4 = 5

2 = 2 ÷ 5 5

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TA ▸ Lesson 1

Materials—S: Cubes

Students use centimeter cubes to find fractional units of a set. Distribute 12 cubes to each student and ask them to take out their whiteboards. Display the problem.

Mr. Perez has 12 eggs. He uses  _1 of the eggs to make a cake. How many eggs does 3

Mr. Perez use to make the cake? Invite students to turn and talk about what they notice about the problem and what the problem asks them to find. Direct students to use the cubes to solve the problem. Circulate and observe student work. Allow students the opportunity to struggle productively. They might not find the answer. Transition to the next segment by framing the work.

Language Support Consider using strategic, flexible grouping throughout the module. • 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. As applicable, complement any of these groupings by pairing students who speak the same native language.

Today, we will use arrays to find fractional parts of sets of objects.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Learn

Partition a Set Students partition arrays to represent a fraction of a set. Direct students to problem 1(a) in their books. a.  _ of 12 is 1 3

b.  _ of 12 is 2 3

What do you notice about the array? There are 12 circles in the array. There are 4 columns of 3 circles. There are 3 rows of 4 circles. 1 3

We know _ means 1 part of the whole when it is partitioned into 3 equal parts. How can we use the array to demonstrate, or show, that? Language Support

We can partition the array into 3 equal groups. Partition the array and direct students to do the same. Now we have 3 equal groups. Let’s look at the groups. Point to one of the groups and ask the following questions. How many circles are in each group?

This segment introduces the academic verb demonstrate. Consider rephrasing the meaning by using the term in a conversational context familiar to your students. For example, you can demonstrate that you know the alphabet by saying A, B, C … .

4 1 3

Each group of 4 is 1 third of the total. So, what is _ of 12?

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Think back to Mr. Perez and his cake. How could you model with the cubes to 1 3

show _ of 12?

We could build an array with 3 rows of 4 cubes. Each row would show  _1  . 3

Could you use an array with 4 rows of 3 cubes as a model? Why? Yes, because that array also shows 12 eggs. Where would you see 3 equal groups in the array with 4 rows of 3 cubes?

The 3 columns are the 3 equal groups. Each column is 1 group, so each column is  _1  . 3

Have students record the answer. Direct students to problem 1(b). Invite them to think–pair–share about how they can use what they know about  _1  of 12 to help find  _2  of 12.

3 3 2 3 3 1 Because   of 12 is 4,  2  of 12 is 2 × 4, or 8. 3 3 2 1     is double    , so the answer is double 4, which is 8. 3 3

Because  _1  of 12 is 4,  _ of 12 is 4 + 4, or 8.

Direct students to record the answer. 3 3

If we follow this pattern, what is _ of 12? Why?

 _3  of 12 is 12 because  _3 is equal to 1 whole and the whole set is 12 circles. 3

Because  _1  of 12 is 4,  _3  of 12 is 3 × 4, or 12. 3

Direct students to problems 2 and 3.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Use the array to find the value. Draw lines to show your work. a.  _ of 18 is

c.  _ of 18 is

a.  _ of 15 is

c.  _ of 15 is

1 6

5 6

1 5

4 5

b.  _ of 18 is 3 6

Differentiation: Challenge For students who need an additional challenge, present the following problems:

b.  _ of 15 is 2 5

a. What fraction of 18 is 6? b. What fraction of 18 is 12? 2.

Invite students to work with a partner to find each value. Circulate as students work and use the following questions to advance student thinking: • Where do you see sixths in the array? Where do you see fifths? • How can you partition the array to show sixths? To show fifths? • How can you use  _1  of 18 to find  _3  of 18? 6 6 • How can you use  1  of 15 to find  4  of 15? 5 5

What fraction of 15 is 9?

Invite students to turn and talk about how they used the arrays to find the values.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Solve a Real-World Problem Students solve a real-world problem involving finding a fractional unit of a set.

Promoting the Standards for Mathematical Practice

Direct students to problem 4(a). Read the problem chorally with the class. As students recognize they can find a non-unit fraction of a set by multiplying or repeatedly adding a unit fraction of a set, they are looking for and expressing regularity in repeated reasoning (MP8).

3 4. There are 20 people on a bus.  _ of the people are adults. The rest are children. 5

a. How many adults are on the bus?

_ 3  of 20 is 12. 5

There are 12 adults on the bus.

Ask the following questions to promote MP8:

b. What fraction of the people on the bus are children? 3 2 1 − _5 = _5 _ 2 of the people on the bus are children.

• What patterns did you notice when you

__1 __3

__5

found   ,   , and   of 18?

__3

6 6

__1

• Is   of a set always 3 times   of the same 6

set? Explain.

c. How many people on the bus are children?

_ 2  of 20 is 8. 5

8 people on the bus are children.

UDL: Action & Expression

How many people are on the bus in total?

Consider providing cubes for students to use

as they model the problem.

What does the question ask us to find? It asks the number of adults who are on the bus. What fraction of the people on the bus are adults? 3_ 5 1 2

1 2

Will the answer be more than _ the people on the bus or less than _ ? How do you know? The answer will be more than  _1 the people on the bus because  _3 is more than  _1  . 2

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Will the answer be more than 20 or less than 20? How do you know? The answer will be less than 20 because 20 is the total and we are finding a part of the total. We are finding  _3  of 20. Because  _3 is less than 1, our answer will be less than 20. 5

What can we draw to help us solve the problem? We can draw an array of 20. Draw a 5 by 4 array and direct students to do the same. Into how many equal groups do we need to partition the array? Why? We need to make 5 equal groups because we are finding  _3  of 20. 5

We need to partition our array into 5 equal groups because we are finding fifths. How can we partition the array to show 5 equal groups? We can draw lines between each row because there are 5 rows and each row has the same number of circles. Draw lines between each row. How many circles are in each group?

4 What fraction of the total does each group represent? 1_ 5 1 5

So, what is _ of 20?

3 5

1 5

How can we use _ of 20 to help us find _ of 20?

Because  _1  of 20 is 4 and I need  _3  of 20, I can add 4 three times and find that 4 + 4 + 4 = 12. 5

I know each group is 4. I need to find 3 groups, so I can find 3 × 4 to get 12. Have students record the answer.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Direct students to problem 4(b). Read the problem chorally with the class. Invite them to think–pair–share about how they can solve the problem.

We know the total number of people on the bus represents one whole, or  _5  . We can 5

decompose  _5  into  _3  and  _2  . 5

We know the total number of people on the bus represents one whole, or 1. So, we can subtract  _3  from 1 and 1 −  _  =  _  . 5

3 5

2 5

Have students record the answer. Direct students to problem 4(c). Read the problem chorally with the class. How can we solve part (c)?

We know  _1  of 20 is 4, so we can find 2 × 4 to get  _2  of 20, which is 8. 5

We know there are 20 people on the bus in all, so we can subtract 12 from 20 to find the number of children on the bus because there are 12 adults on the bus. 20 − 12 = 8. Have students record the answer. Look back at problem 4(a). We drew a 5 by 4 array to help us solve the problem. Could we have drawn and partitioned a different array? Invite students to turn and talk about whether they could have drawn a different array to help them solve the problem. Draw a 2 by 10 array. Does this array show 20 circles? Yes. How can we partition this array to help us solve the problem? We can draw vertical lines after every 4 circles to partition the array into 5 equal parts. We can partition the array into 5 equal groups of 4 circles. When you find a fraction of a number, you can draw your array however you want. Just remember to think about the denominator when you decide how many groups you will partition your array into.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Partition the array and draw boxes around the groups as shown. Invite students to turn and talk about how this model can help them solve the problem. We can partition the array or draw boxes to group our circles in any way as long as the groups are equal. Direct students to problem 5. Invite them to work with a partner to solve the problem. 3 5. Julie has 16 balloons.  _ of Julie’s balloons are blue. How many of Julie’s balloons

are not blue?

Differentiation: Support

Sample:

If students need support getting started with or completing problem 5, ask questions such as the following:

• What have you tried so far?

_ 3  of 16 is 6. 16 − 6 = 10 10 of Julie’s balloons are not blue.

• What do you know about this problem?

Blue

Not blue

Circulate as students work and use the following questions to advance student thinking: • Will there be more than 16 or fewer than 16 balloons that are not blue? How do you know? • What does  _3 represent in the problem? 8

• What fraction of the balloons are not blue? • How will you draw your array? Why? • How will you partition your array? Why?

• How can finding  _1  of 16 help you find the number of balloons that are not blue?

• How many balloons does Julie have in total? Continue with the same line of questioning provided with problem 4. Students may also benefit from moving the question to the beginning of the problem. For example, present the problem as, How many of Julie’s balloons are not blue? Julie has

16 balloons.   3 of Julie’s balloons are blue. 8

When students are finished, invite them to turn and talk about how they solved the problem.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. 32

EUREKA MATH2 California Edition

Land

5 ▸ M3 ▸ TA ▸ Lesson 1

Teacher Note

Debrief 5 min

Some students may find a fraction of a set by taking a fraction of each group in the set.

__1

Objective: Find fractions of a set with arrays.

For example, to find   of 12, students may 3

Facilitate a class discussion about finding fractional parts of a set by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Yes. You can partition an array into equal groups to find one unit. Then you can use repeated addition or multiplication to find how many are in more than one unit.

sample student work.

Sana’s Way

then shade 1 out of 3 in each group to get 4. Validate this as an acceptable way to find a

Is an array helpful when you find a fraction of a set? How?

Display the expression  _3  of 12 and the

partition the set into 4 equal groups and

fraction of a set. Students learn additional methods to find a fraction of a number in subsequent lessons.

Toby’s Way

Invite students to turn and talk about whether both arrays show  _3  of 12. 4

What do you notice about Sana’s and Toby’s arrays? They both show 12 circles. They both show 9 shaded circles. Sana partitioned her array into 4 equal parts and Toby partitioned his array into 3 equal parts. Toby partitioned his array into groups of 4 and shaded  _3 of each group. 3 Do both arrays show _ of 12? Why? 4

Yes, because they both show that  _3  of 12 is 9. 3 How did Sana show _ of 12? 4

Sana made an array with 3 rows of 4 circles. Then she partitioned so she had 4 equal groups and she shaded in 3 of those 4 groups.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Invite students to think–pair–share to discuss how Toby showed  _3  of 12. 4

Toby also made an array with 3 rows of 4 circles. He partitioned into 3 equal groups. In each group, he shaded 3 out of 4 circles.

Instead of showing  _3  of 12, he showed  _3  of 4. He showed that three times because there are 3 fours in 12.

Exit Ticket 5 min

Teacher Note Students may not understand how Toby

__3

found   of 12. Allow time for that wonder to 4

linger, as the concept will be explored further in a later lesson.

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.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

5. Use the array to complete parts (a)–(e).

1 a. _ of 15 is

2 of 15 is b. _

1 of 15 is c. _

2 of 15 is d. _

4 of 15 is e. _

Use the array to find the value. Draw lines to show your work if needed. 1.

1 of 9 is a. _

3 c. _ of 9 is

3 c. _ of 18 is

1 a. _ of 20 is

4 c. _ of 20 is

1 a. _ of 18 is 3

2 b. _ of 9 is

2 b. _ of 18 is 3

_2 of 12 is

_3 of 16 is 4

1 of 24 is a. _ 6

5 c. _ of 24 is 6

4 20

3 b. _ of 24 is 6

_2 of 27 is 3

8. 4.

Find the value. Draw lines or boxes to show your work if needed. 6.

2 b. _ of 20 is

PROBLEM SET

_3 of 24 is 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

1 of 27 help you find _ 4 of 27? 10. How does knowing _ 9

Use the Read–Draw–Write process to solve the problem.

1 of the students bring their lunch from home. 14. There are 25 students in Miss Baker’s class. _

4 is 4 groups of _ 1 . So, if I know _ 1 of 27, I multiply that answer by 4 to find _ 4 of 27. I know _ 9

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 1

The rest of the students get a school lunch. How many students bring their lunch from home? 1 of the class? Explain. a. Should the answer be more than or less than _ 2

1 of the class because _ 1 is less than _ 1. The answer should be less than _ 2

Complete the statement to find the value. 3 of 14 11. _ 7

1 of 14 is Because _ 7

1 of 28 is Because _ 7

b. How many students bring their lunch from home?

_1 of 25 is 5. 5

5 students bring their lunch from home.

5 , then _ of 28 is 5 × 4 or 7

1 − 1_ = 4_

_4 of the students get a school lunch.

d. How many students get a school lunch?

7 of 32 13. _ 8

1 of 32 is Because _ 8

c. What fraction of the students get a school lunch?

5 of 28 12. _

3 , then _ of 14 is 3 × 2 or

7 , then _ of 32 is 7 × 4 or 8

_4 of 25 is 20. 5

20 students get a school lunch.

PROBLEM SET

LESSON 2

Interpret fractions as division to find fractions of a set with tape diagrams and number lines.

EUREKA MATH2 California Edition

Name

5 ▸ M3 ▸ TA ▸ Lesson 2

Date

Find each value. Show your work by using a tape diagram. 1. _1 of 5 is 5

1 unit = 5_ = 1

• How are number lines and tape diagrams helpful when you find a fraction of a set?

• How can you use what you know about fractions as division expressions to help you find fractions of a set?

5 units = 25

Achievement Descriptors

__ = 5 1 unit = 25 5

4 units = 4 × 5 = 20

32 __

3. _2 of 16 is

Students begin by using number lines to find fractions of a set and realize it is not always efficient to use a number line. As the total number of objects in a set starts to increase, students transition from using number lines to using tape diagrams to find a fraction of a set.

Key Questions

5 units = 5 5

2. _4 of 25 is

Lesson at a Glance

5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

3 units = 16

__ 1 unit = 16 3

__ = 32 __ 2 units = 2 × 16 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Find a Unit Fraction of a Whole Number by Using a Number Line

• None

• Find a Non-Unit Fraction of a Whole Number by Using a Number Line • Find a Fraction of a Whole Number by Using a Tape Diagram • Solve a Real-World Problem • Problem Set

Land 10 min

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Fluency

Happy Counting by Halves Students visualize a number line while counting aloud to maintain fluency with counting by halves and renaming fractions greater than 1 as whole or mixed numbers. Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.) Let’s count by halves. Today we will rename the fractions as whole or mixed numbers when possible. The first number you say is 0. Ready? Signal up, down, or stop accordingly. Teacher Note

1 2

Continue counting by halves within 5. Change directions occasionally, emphasizing crossing over whole numbers and where students hesitate or count inaccurately.

Listen to student responses and be mindful of errors, hesitation, and lack of full-class participation. If needed, adjust the tempo or sequence of numbers.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Whiteboard Exchange: Relate Repeated Addition to Multiplication Students write equations with non-unit fractions to represent a tape diagram to prepare for multiplying a whole number by a fraction. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the tape diagram with 2 units of  _ . 2 3

Write a repeated addition equation to represent the tape diagram. Write the sum as a fraction.

2 3

2 3 2 2 + =4 3 3 3

Display the addition equation. Write a multiplication equation to represent the tape diagram. Write the product as a fraction.

2× 3 = 3

Display the multiplication equation. Repeat the process with the following sequence: ? 2 3

2 3

3 4

? 3 4

3 4

3 5

2 6

2 2 2 + + =6 3 3 3 3

3 3 3 + + =9 4 4 4 4

3 3 3 3 12 + + + = 5 5 5 5 5

2 2 2 2 2 10 + + + + = 6 6 6 6 6 6

3× 4 = 4

4× 5 = 5

5× 6 = 6

3× 3 = 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Whiteboard Exchange: Interpret a Fraction as Division Students write a fraction as a division expression and a whole number or mixed number to prepare for finding fractions of a set with tape diagrams and number lines. 2 Display _ = 2

Write the fraction as a division expression. Then express the quotient as a whole or mixed number. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

2 = 2 ÷ 2 = 1 2

Display the completed equation. Continue the process with the following sequence:

3 1 = 3 ÷ 2 = 12 2

8 = 8 ÷ 4 = 2 4

11 3 = 11 ÷ 4 = 2 4 4

18 = 18 ÷ 6 = 3 6

35 = 35 ÷ 7 = 5 7

60 4 = 60 ÷ 7 = 8 7 7

13 1 = 13 ÷ 6 = 2 6 6

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TA ▸ Lesson 2

Students reason about what type of model is most efficient to find a fraction of a set. Open and display the Fraction of a Set digital interactive. Present the following problem.

Mr. Evans has 48 pencils.  _ of the pencils are sharpened. How many pencils are sharpened? 5 8

Invite students to turn and talk about how they could solve the problem. Students may suggest drawing an array as they did in the previous lesson. We could draw an array to represent the problem. Create an array.

Teacher Note

Once the array is showing, ask the following questions. What do you notice about the array? It took a long time to create all the circles. The circles are arranged in a 2 by 24 array.

The circles in the array are intentionally shown one by one in the digital interactive so students realize drawing an array is not always an efficient method to find fractions of a set.

Could we have created a different array? We could have created a 6 by 8 array. We could have created a 12 by 4 array. We could have created several different arrays, as long as the array shows a total of 48 circles. 5 8

How can we use the array shown to find _ of 48? We can partition the array into 8 equal groups. Create segments. What do you notice about the groups? There are 8 groups with 6 circles in each group.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Create a tape diagram.

What do you notice about the model now? There are boxes around each of the groups. The total number of circles is labeled. 5 8

How can we find _ by using this model? We can count the number of circles in 5 groups. We can multiply 6 by 5. Does this model look like another model you know? It looks like a tape diagram. Hide the circles and invite students to turn and talk about why they might use a tape diagram instead of an array to find a fraction of a whole number.

Transition to the next segment by framing the work. Today, we will use tape diagrams and number lines to find fractional parts of a set.

EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TA ▸ Lesson 2

Find a Unit Fraction of a Whole Number by Using a Number Line Students find a unit fraction of a whole number by using a number line. Direct students to problem 1 in their books. Read the problem chorally with the class. Find the value by using the number line.

1.   of 3 is 5

 _5 3

_ _ _ _

1 1 1 3 5 + 5 + 5 = 5

Why do you think there is a number line shown and not an array? We could draw an array that has 3 circles, but we cannot partition 3 circles into 5 equal groups. Not every problem can be modeled with an array. Let’s see whether we can use 1 5

a number line to help us find _ of 3. What do you notice about the number line? It shows the intervals of 1 from 0 to 3. It is partitioned into fifths. 1 5

We know _ means 1 part of the whole when the whole is partitioned into 5 equal parts. Let’s look at part of our number line—the interval from 0 to 1.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Highlight the first fifth in the interval from 0 to 1. 1 5

The number line shows _ of 1. Now 1 5

1 1 1 3 + + = 5 5 5 5

let’s highlight _ of each of the other intervals, from 1 to 2 and 2 to 3. Highlight the first fifth in the interval from 1 to 2 and in the interval from 2 to 3. 1 5

1 5

We just found _ of each interval of 1 from 0 to 3. To find _ of 3, we can compose these fifths.

UDL: Representation The digital interactive Fraction of a Whole on a Number Line supports composing the parts of each interval of 1 to find the fraction of the whole. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

Write  _  +  _  +  _  . 1 5

1 5

What is _ + _ + _ ? 5 5 5 3_ 5 Direct students to record the answer. How is this example of finding a fraction of a whole number by using a number line similar to finding a fraction of a whole number by using arrays? How is it different? They are similar because we need to partition the set or the number line into equal groups. The difference is that when we use an array, we draw and partition circles. When we use a number line, we partition each interval of 1. Invite students to turn and talk about whether finding a fraction of 1 is helpful for finding a fraction of any whole number and why.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Find a Non-Unit Fraction of a Whole Number by Using a Number Line Students find a non-unit fraction of a whole number by using a number line. Direct students to problem 2. Use the Math Chat routine to engage students in mathematical discourse. Give students 2 minutes of silent think time to find the value and to record their thinking. 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 one or two students to share their thinking. Purposely choose work that allows for rich discussion about connections between strategies. Find the value by using the number line.

2.   of 4 is 5

 _5 8

_ _ _ _ _

2 2 2 2 8 5 + 5 + 5 + 5 = 5

Then facilitate a class discussion. Invite students to share their thinking with the whole group and record their reasoning. Display the following work samples if students do not produce one or both.

1 1 1 1 4 + + + = 5 5 5 5 5 1 of 4 is 4. 5 5 4 4 8 + = 5 5 5 © Great Minds PBC

2 2 2 2 8 + + + = 5 5 5 5 5 2 of 4 is 8 . 5 5

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Invite students to think–pair–share about how the methods are similar and different. Both methods show thinking by using a number line.

One method shows finding  _  of 4 by shading 1 fifth of each interval of 1. The model 1 5

shows  _  of 4 is  _  , and  _  is double  _ , so they added  _ twice to get  _  . 1 5

4 5

2 5

1 5

4 5

8 5

One method shows finding  _  of 4 by shading 2 fifths of each interval of 1. So,  _  of 4 is  _  . 2 5

2 5

8 5

Find a Fraction of a Whole Number by Using a Tape Diagram Students find a fraction of a whole number by using a tape diagram. Direct students to problem 3 and have them turn and talk about how they can find the value. Find the value.

3.   of 35 is 5

Would it be helpful to use an array or a number line to find the value? Why? No. It would take too long to draw 35 circles for the array or to draw a number line that goes to 35. No. It is not efficient to draw an array or a number line for large numbers. Yes. It is easier for me to see how many are in each group when I draw an array. Yes, because I like using number lines. It may not be the most efficient way to find the value. But I know I can find the correct value by drawing and partitioning a number line. What other model could we use to help us find the value? We could use a tape diagram. 3 5

We want to find _ of 35. Let’s draw a tape diagram to represent the whole number.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Draw and label a tape diagram and direct students to do the same. Into how many units should we partition the tape? How do you know? We should partition the tape into 5 units because we need fifths.

Partition the tape diagram into 5 equal parts and direct students to do the same. Write 5 units = 35 next to the tape diagram. What are we trying to find?  _  of 35 3 5

Where should we draw a question mark in our model? Why?

We should draw it under 3 of the 5 units because we need to find  _  . 3 5

Write a question mark under 3 units. Thinking about how 5 units are equal to 35, how can we find the value of 1 unit?

UDL: Engagement Remind students of their work in earlier lessons when they equated a fraction to a division expression.

We can divide 35 by 5.

Write 1 unit =  __ below the equation and direct

35 5

Students may also benefit from a context to

students to do the same.

__3

reason about the value of   of 35. Consider 5

What is 35 ÷ 5?

using a context that is familiar to students.

Guide students to write = 7.

5 units = 35

Have we found the answer to the problem? How do you know? No. We found 1 unit is equal to 7, but we need to find what 3 units is equal to.

1 unit = 35 = 7

No. We found  _  of 35, but we need to find  _  of 35. 1 5

3 5

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2 3 5

How can we find _ of 35? We can find 7 + 7 + 7. We can multiply 3 and 7 because each unit is equal to 7 and we need the value of 3 units. 3 5

What is _ of 35?

Direct students to record the answer. 3 5

35 5

When we found _ of 35, could we have evaluated _   + _   + _  ? Would you have evaluated that expression to solve the problem?

We could have evaluated that expression to solve the problem because it has the same value as 7 + 7 + 7.

I would not have evaluated that expression to solve the problem because I know  __ = 7, and it is simpler to use the whole number 7.

35 5

Invite students to turn and talk about how they used a tape diagram to find the value.

Solve a Real-World Problem Students solve a real-world problem involving finding a fraction of a whole number. Direct students to problem 4. Have students read the problem and work with a partner to use the Read–Draw–Write process to solve the problem.

Language Support To support the context of the problem, build background knowledge by showing a photograph of a quilt.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

4. Blake has 19 yards of fabric. He uses    of the fabric to make a quilt. How many yards of fabric does Blake use for the quilt?

Promoting the Standards for Mathematical Practice

19 When students decide how to model and solve a real-world problem that asks for a fraction of a whole number and how to assess the reasonableness of their answers, they are modeling with mathematics (MP4).

? 19 ÷ 3 = __ 3 19

Ask the following questions to promote MP4:

= 6 _3 1

• What can you draw to help you understand this real-world problem?

Blake uses 6  _ yards of fabric for the quilt. 1 3

• How are the key ideas in this real-world problem represented in your diagram?

Circulate as students work and use the following questions to advance student thinking: • Will the answer be greater than 19 or less than 19? How do you know?

• How could you make a simpler problem to estimate an answer?

• What can you draw to help you solve the problem? • What does the whole tape represent? • How many equal parts are in your tape? Why? • What could you divide to find the number of yards needed for 1 quilt? When students have finished, gather the class and discuss the solution. 1 3

How many yards of fabric is _ of 19 yards?

 __ yards 3 19

6  _  yards 1 3

Does it make sense that the answer is less than 19? Why?

It makes sense because Blake used  _ of the fabric.  _ is less than 1, so the answer must 1 3

be less than the total number of yards of fabric.

1 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

What if the question asked you to find how many yards of fabric Blake had left? How could you solve the problem? 2  _  . If Blake uses  _ of the fabric, he has  _ left. So, we could find 6  _  + 6  _ to get 1 1 3

2 3

1 3

We could subtract 6  _  from 17 to find the number of yards of fabric left. 1 3

2 3

Invite students to turn and talk about how they can use a tape diagram to solve a real-world problem involving finding a fraction of a whole.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Interpret fractions as division to find fractions of a set with tape diagrams and number lines. Gather the class with their Problem Sets. Facilitate a class discussion about how number lines and tape diagrams can be useful when finding a fraction of a set by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Why is a number line useful for problems 1-4 in the Problem Set? The whole numbers in problems 1–4 are small numbers, so it is simple for me to draw a number line to help find the fraction of the whole number. Why is a tape diagram useful for problems 5-8 in the Problem Set? The whole numbers in problems 5–8 are larger numbers. It is not efficient to draw a number line for those numbers because we would have to start at 0, include all the whole numbers, and partition each interval of 1.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

How are number lines and tape diagrams helpful for finding a fraction of a whole number? Both number lines and tape diagrams help us see the relationship between the fraction and the whole number. The smaller the fraction, the smaller the result. The larger the fraction, the larger the result. Number lines and tape diagrams help us break the problem into simpler parts. On a number line, we can shade to show a fraction of each interval of 1 and then add the fractions. In a tape diagram, we can find the value of one unit by using division and adding the number of units we want so we can find the answer. How can you use what you know about fractions as division expressions to help you find fractions of a whole number? Where did this help you in the Problem Set? In problem 9, I know  _  of 28 means  __  , which is 4, so I could find  _  of 28 by doubling 4. 1 7

28 7

2 7

In problem 14, I know  _  of 8 is equivalent to 8 ÷ 3. I can interpret 8 ÷ 3 as  _  . 1 3

8 3

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.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ 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 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Complete the tape diagram and find the value. Write your answer as a whole number when possible. 5. _1 of 16 4

Find the value by using the number line. Write your answer as a whole number when possible. 1. _1 of 4 is _4 or 4

4 units =

1 unit = _____ =

6. _1 of 32 4

6 2. _2 of 3 is _____ or 3 3

units =

unit =

32 4

units =

unit =

24 3

? 0

7. _2 of 24 3

3. _2 of 6 is 3

12 __ 3

? 0

4. _3 of 3 is 4

_9 4

2 units = 2 ×

8. _5 of 24 6

PROBLEM SET

units =

unit =

24 6

units =

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

Use the Read–Draw–Write process to solve each problem.

Find the value. Write your answer as a whole number when possible. 9. _2 of 28 is 7

10. _3 of 48 is 8

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 2

15. A baker makes 36 cookies. _7 of the cookies have chocolate chips. How many cookies have

chocolate chips?

_7 of 36 is 28. 9

28 cookies have chocolate chips.

11. _5 of 72 is 9

12. _2 of 45 is 5

16. A band has 56 members. _3 of the members play a brass instrument. How many members play a brass instrument?

_3 of 56 is 24. 7

24 members play a brass instrument. 13. _5 of 42 is 6

14. _1 of 8 is 3

_8 3

PROBLEM SET

LESSON 3

Multiply a whole number by a fraction less than 1.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Name

Date

Multiply. Show your work. 1. 2_ × 15 = 3

2 × 15 = 2 × 15 3 3

= 10

• What does it mean to multiply a whole number by a fraction?

24 5

Students analyze the relationship between finding a fraction of a set and multiplying fractions and whole numbers. Students realize that when they find a fraction of a whole number, they are multiplying. They multiply whole numbers and fractions by using number lines, tape diagrams, and equations.

Key Questions

=2×5

2. 3_ × 8 = 5

Lesson at a Glance

• What do you notice about the product when you multiply a whole number that is not zero by a fraction less than 1?

3 ×8=3×8 5 5 = 24 5

Achievement Descriptors 5.Mod3.AD6 Multiply whole numbers or fractions by fractions. (5.NF.B.4) 5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD8 Compare the effects of multiplying by fractions and

whole numbers. (5.NF.B.5.a)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• None

Consider tearing out the Fluency Sheets in advance of the lesson.

Learn 35 min

Students

• Interpret Finding a Fraction of a Whole Number as Multiplication

• Multiply Fractions by Whole Numbers Fluency Sheets (in the student book)

• Multiply a Whole Number by a Fraction Less Than 1 • Select a Method • Problem Set

Land 10 min

5 ▸ M3 ▸ TA ▸ Lesson 3

Fluency

EUREKA MATH2 California Edition

Contemplate Then Calculate: Multiply Fractions by Whole Numbers Materials—S: Multiply Fractions by Whole Numbers Fluency Sheets

Students determine the sum or product to prepare for multiplying a whole number by a fraction. Direct students to study the problems on Fluency Sheet 1. Have students focus on the problems in just one column to start. Consider having them cover the other problems with sticky notes or blank paper in advance. Frame the task: As you study, ask yourself, What do I notice that could help me with these problems? Provide 1–2 minutes of silent think time. Some students may make notes or answer problems as part of their study. Have students turn and talk about their thinking. Listen for students who offer solution strategies or connect problems by highlighting relationships or patterns. Select a few students to share their ideas with the class.

UDL: Representation As students share their ideas, consider displaying Fluency Sheet 1 and annotating problems to reinforce strategies, relationships, and patterns described.

EUREKA MATH2 California Edition

After 1–2 minutes, have the class pause their work. Invite students to discuss what they noticed about the problems with a partner or in a small group. Circulate and listen as students talk, advancing their discussions as needed by asking questions such as the following: • What is the same about these problems? What is different? • Did you find patterns in the problems? If so, talk about them. • What strategy did you use? Facilitate a whole-class discussion by asking different groups to share their thinking. As time allows, have students continue to work on Fluency Sheet 1. Consider reading the answers quickly to provide immediate feedback. Invite students to complete Fluency Sheet 2 at another time by using what they learned from Fluency Sheet 1.

5 ▸ M3 ▸ TA ▸ Lesson 3

Teacher Note Consider selecting a milestone on Fluency Sheet 1 to help you decide when to pause the work. For example, you might pause when everyone has worked through at least problem 11. This way pairs or groups of students can discuss problems that everyone had a chance to try. Select the milestone based on the needs of your class.

Teacher Note Consider asking the following questions to discuss the patterns in Fluency Sheet 1: • What patterns do you notice in problems 1–10? • How do problems 11–13 compare to each other? 14–16? 17–19? 20–22?

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Launch

Students consider whether a fraction of a set can be found by using multiplication. Display the picture of the array and tape diagram and invite students to study the models.

What do you notice? I notice there are 35 objects in the array and 35 is the total of the tape diagram. I notice the array is divided into 5 groups and the tape diagram is partitioned into 5 parts. 1 5

Where can you find the answer to _ of 35 in each model? What is the answer? I can find the answer 7 in the array because it is the number of objects in 1 group. I can find the answer 7 in the tape diagram when I think about the value of 1 part. 1 part in the tape diagram is 35 ÷ 5, or 7. 2 5

Where can you find the answer to _ of 35 in each model? What is the answer? I can find the answer 14 in the array because it is the number of objects in 2 groups. I can find the answer 14 in the tape diagram because each part in the tape diagram is 7, so 2 parts is 14. 60

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3 3 5

Where can you find the answer to _ of 35 in each model? What is the answer? I can find the answer 21 in the array because it is the number of objects in 3 groups. I can find the answer 21in the tape diagram because each part in the tape diagram is 7, so 3 parts is 21. Each time we find a fraction of a set, or whole number, we think about how much is in each group. How can we find the number efficiently if we have a lot of groups? For example, what if we need to find 125 groups of 7? I would multiply 7 and 125. Invite students to turn and talk about whether they can multiply when they find a fraction of a set. Transition to the next segment by framing the work. Today, we will learn how to interpret a fraction of a set and multiply fractions and whole numbers.

Learn

Interpret Finding a Fraction of a Whole Number as Multiplication Students find a fraction of a whole number by multiplying the whole number by the fraction. Write _     of 4 is 1 6

Invite students to turn and talk about how they can determine _     of 4. Affirm that they might 6 find the value by using a number line or by using a tape diagram. 1

Let’s practice finding _ of 4 by using a number line. If we want to find _ of 4 by using 6 6 a number line, what should we draw first? Why? We should draw a number line that shows whole numbers from 0to 4 because we are trying to find a fraction of 4.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Draw and label a number line and direct students to do the same.

UDL: Action & Expression

Into how many equal parts should we partition each interval of 1? Why?

Consider preparing exemplars of the number

We need to partition each whole-number interval into 6 units because we want 1 to find _     of 4.

tape diagram work for the expression    × 6.

__1

line work for the expression    × 4and the 3

Partition the number line into sixths and direct students to do the same.

Invite students to find _     of 4 by using the number line.

1 6

Display a student’s work.

How did you 1 6

find _ of 4?

First, draw a number line to show the whole.

Next, partition each whole-number interval to show the fractional units.

1 6

What addition expression represents finding _ of 4 when using a number line?

_ 1 +  _1 +  _1 +  _1

Post the exemplars after the class completes the problems together. Students can refer to these exemplars when they select their own methods. In the exemplars, break down the work to show the thinking, as in the following example.

I shaded _    of each interval of 1 and then composed all the sixths. 1 6

__2

We are repeatedly adding _ when we find _ of 4. What is another way to show 6 6 repeated addition?

Then highlight the fractional unit in each whole number.

Repeated addition can be shown with multiplication. 1 6

What is _ of 4?

_ 4

1 6

What is 4 ×_ ?

4   6

Now compose to find the fraction of the whole.

1 6

Look at the answers to _ of 4 and 4 × _ . What does that make you think about _ of 4 1 and 4 ×_ ? 6

__1 __1 __1 __ __4   +   +   +  1 =   6

_ 1 of 4 and 4 × _ 1 are equivalent because they both equal _ 4 . 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

It seems that when we find a fraction of a number, we are multiplying. Let’s explore this idea further by trying another problem. 5 6

Think about _ of 4. Do you estimate the answer is greater than or less than 4? Why? I estimate the answer is less than 4 because we are finding a fraction, or part, of 4.

I estimate the answer is less than 4. Because _    is less than 1, and 1 × 4 = 4, multiplying 4 5 6

by a number less than 1 results in a product less than 4. Because finding a fraction of a number really means we are multiplying, what 5 6

multiplication expression can we write to model _ of 4?

_ 5 × 4 6

Draw a tape diagram that represents _     of 4 and direct students to do the same. 5 6

What is 6 units equal to?

6 units = 4

1 unit = 4

Write 6 units = 4.

How can we find the value of 1 unit?

5 units = 5 x 4 = 20 6

We can divide 4 by 6.

Write 1 unit = _   . 4 6

How can we find the value of 5 units? We can multiply _     by 5.

4 6 Write 5 units = 5 ×  4 . 6

4 6

What is 5 ×_ ? How do you know?

It is __    . I thought about 5 groups of _    .

20 4 6 6 20 4 4 4 4 4 4 It is    . I added     five times:     +     +     +     +    . 6 6 6 6 6 6 6

_ _ _ _ _

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

We estimated the answer would be less than 4. Is it? How do you know? Yes. __     = 3  _ , so the answer is less than 4 . 20 6

2 6

Invite students to think–pair–share about the following question. 5 6

4 6

5 6

We found _ of 4 with 5 × _ . Earlier, we said _ of 4 could be modeled by _ × 4. 5 6

4 6

Does _ ×4=5×_ ? Analyze the tape diagram to help you determine whether that statement is true.

20 _ 5 × 4 = 5 × _ 4 because both expressions equal __    . 6 6 6 5 _ 5 _ 5 _ 5 __ 20 5 4 _ _ I know    × 4 =     +     +     +     =    . In the tape diagram, 1 unit is _    , and we found the 6 6 6 6 6 6 6 5 20 value of 5 units, which is 5 ×_  64  or __    . So _    × 4 = 5 × _  64 . 6 6 5 Write _     of 4 is 5 parts when 4 is partitioned into 6 equal parts. 6

What does 5 parts represent? It represents the 5 units of the tape diagram we want to know the value of.

6 units = 4

1 unit = 4

5 units = 5 x 4 = 20

? 5 of 4 is 5 parts when 4 is partitioned into 6 equal parts.

_ 5 is also the numerator of the fraction in the expression 5 of 4.

What does 4 represent? It represents the whole tape diagram. It represents the whole number in the expression.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

What does 6 equal parts represent? It represents the number of units that 4 is partitioned into. It represents the fractional part of each unit.

6 is the denominator of the fraction in the expression. 6 is the denominator of the answer. 20 __

Gesture to   . 6

5 20 20 _   is 5 parts when 4 is partitioned into 6 equal parts. So _   is _ of 4. 6 6 6

We began this problem by thinking that when we find a fraction of a number, 5 6

5 6

4 6

we are multiplying. We also modeled _ of 4 with _ × 4but ended up with 5 × _ . Is our conclusion still correct? When we find a fraction of a number, are we multiplying? Yes, our conclusion is correct. We are multiplying but not the expression we thought we would be multiplying. 5

To find _ of 4, we know we can use the multiplication expression _ × 4. We can 6 6 interpret this expression as finding the value of 5 parts when 4 is partitioned into 6 equal parts. Invite students to turn and talk about how they know that finding a fraction of a whole number is multiplication.

Multiply a Whole Number by a Fraction Less Than 1 Students use a tape diagram to multiply a whole number by a fraction. Write _    × 6 = 2 3

. 2 3

Describe what _ × 6means.

It means we need to find _     of 6. 2 3

Is the product greater than or less than 6? How do you know? The product is less than 6 because 6 × 1 = 6 and we are multiplying 6 by a factor that is less than 1. The product is less than 6 because _     of 6 is 6 and _    is less than _    . 3 3

2 3

3 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Invite students to work with a partner to find the product. Students might show their work by using a tape diagram or a number line. When most students are finished, display the following work sample.

3 units = 6 1 unit = 36

2 units = 2 × 36 = 4

Have students turn and talk about how their work compares to the work shown. Display the tape diagram with different equations.

2 ×6 = 2 × 1 ×6 3 3

(

= 2 × ( 31 × 6(

= 2 × 63

= 2×2 =4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Invite students to think–pair–share about how this work relates to the tape diagram. Shade or gesture to the tape diagram when students share the connections they find.

_ 2  is 2 units of _ 1 , or 2 × _ 1 . 3

They regrouped the parentheses around _    × 6because they wanted to find _     of 6, like 1 3

1 3

in the tape diagram when we found the value of 1 unit. They showed 2 ×( _   × 6)to find the value of 2 units. 1 3

_ 6  represents 1 unit, so 2 × _ 6  is 2 units. 3 3 6 _ They renamed     as 2 to find 2 × 2. 3

This person showed     as 2 ×     in their equations. Why do you think they did that? 3

I think they wanted to show finding a unit fraction of 6. When we draw a tape diagram, 1 1 1 first we find _     of 6. I know _     of 6 is _    × 6. 3

Differentiation: Support

Is it helpful to first find a unit fraction of a whole number? Why? It is helpful because if we know the value of 1 unit, we can multiply by the number of groups we need. If the numerator is 2, then we can multiply the value of 1 unit by 2. Affirm that students can find a unit fraction of a whole number by using a number line, a tape diagram, or equations. 6 3

Unit fractions are familiar from previous grades. As needed, show students examples and counterexamples of unit fractions. Then ask them what they notice. Unit fractions: _   1  , _  1  , __   1   3 8 15 13 Non-unit fractions: _   2  , _  5  ,  __   3 8 15 A unit fraction is exactly 1 of a specific fractional unit.

Did you rename _ as 2 to multiply? Or did you multiply differently? I renamed _     as 2 because I know _    = 2. 6 3

6 3

I did not rename first. I thought about how 2 groups of _     is __   . I know __   is 4, so I renamed after I knew the final answer.

6 3

12 3

Acknowledge that either approach is valid. Encourage students to continue to pay attention to when they are renaming and why.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Select a Method Students choose a method to multiply a fraction and a whole number. Use the Numbered Heads routine. Organize students into groups of three and assign each student a number, 1 through 3. Present the following problem: _    × 12 = 4 5

Give students 3 minutes to find the product as a group. Remind students that any one of them could be the spokesperson for the group, so they should be prepared to answer. Call a number, 1 through 3. Have the students assigned that number share their group’s findings. We drew a tape diagram to represent 12. We needed to find fifths, so we partitioned the

12 tape diagram into 5 equal units. We knew 5 units are equal to 12, so 1 unit is equal to __    . 5 48 12 4 Because we needed to find _    , we multiplied __   × 4 to get __   . 5 5 5 4 _ We know    × 12 means 4 parts when 12 is partitioned into 5 equal parts, so we rewrote 5 48 _ 4 × 12as 4 × __  12 , which is __   . 5

As time allows, repeat the process with the following problems:

2  × 7 3

_3  × 9 4

Promoting the Standards for Mathematical Practice Students use appropriate tools strategically (MP5) when they choose from tape diagrams, number lines, and equations to find the product of a fraction and a whole number. Ask the following questions to promote MP5: • What kind of model or method would be helpful to solve this problem? • Why did you choose the model you used? Did it work well? 48 __

• Your work shows the answer is    . Does 5 that seem reasonable?

Language Support Consider supporting student responses with the Talking Tool. Invite students to use the Share Your Thinking section to explain their group’s findings.

3  × 10 5

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

EUREKA MATH2 California Edition

Land

5 ▸ M3 ▸ TA ▸ Lesson 3

Debrief 5 min

Differentiation: Challenge

Objective: Multiply a whole number by a fraction less than 1. Facilitate a class discussion about multiplying whole numbers by fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What do you notice about the product when you multiply a whole number that is not zero by a fraction less than 1? The product is less than the whole number. Why do we have to say a whole number that is not zero when we describe the product of a whole number and a fraction less than 1? Because the product of zero and another number is zero. So if the whole number is zero, then the product is equal to the whole number.

For students who need additional challenge, consider presenting the following problems during or after the Numbered Heads routine.

  2 ×

= 14

__3

= 12

__3

3 4

  × 5

What does it mean to multiply a whole number by a fraction? When you multiply a whole number by a fraction, it means you are finding a fractional part of the whole number. It means you partition the whole number into equal parts and then you count some

of the parts. For example, _    × 12means you are finding 3 parts when 12 is partitioned 3 4

into 4 equal parts.

Write _    × 18. 1 5

Is the value of this expression greater than or less than 18? How do you know?

It is less than 18. I know _    × 18 means _     of 18 and _    is less than 1, so _    × 18is less than 18. 1 5

1 5

It is less than 18. We are multiplying 18 and a fraction less than 1, so the answer is less than 18.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Invite students to think–pair–share about the following prompt. 1 5

4 5

Without multiplying, compare the products _ × 18and _ × 18. Explain your thinking.

_ 4 × 18is greater than _ 1 × 18 because _ 4 is greater than _ 1 . 5

Both products are less than 18 because 18 is multiplied by a factor less than 1. I know

_ 4 is closer to 1, so the product is closer to 18 and because _ 1 is less than _ 4 , the product 5

will be less.

_ 1  of 18 means we are taking a smaller fraction of 18 than _ 4 , so _ 1 × 18is less than _ 4 × 18. 5

Exit Ticket 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.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ 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 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3 ▸ Multiply Fractions by Whole Numbers

Contemplate Then Calculate

1 _1

23. _ + _

Fluency Sheet 1

3 5

Write the sum or product. Use a whole or mixed number when possible.

3 5

3. _1 + _1

7. _1 + _1 + _1 4

_4 _

11. _1 + _1 + _1 3

4 7

10. 4 × _1

1 _1 5

21. 6 × _1 6

22. 7 × _1 6

41. 9 × _4

1 _1 7

1 _1

42. 9 × _ 6 7

43. 7 × _ 5 8

33. _1 + _1 + _1 + _1 3

40. 8 × _4

1 _4

32. 4 × _2 1

5 5

3 4

39. 6 × _

31. _2 + _2 + _2 + _2 7

38. 7 × _

1 _4

3 5

20. _1 + _1 + _1 + _1 + _1 + _1 6

3 5

30. 3 × _

9. _1 + _1 + _1 + _1 7

19. 6 × _1

3 5

3 4

29. _ + _ + _

36. 6 × _2 37. 5 × _

28. 3 × _2

8. 3 × _1

16. 5 × _1

11 4

18. 5 × _1

27. _2 + _2 + _2 3

15. 4 × _1

3 5

1 _1 3

26. 2 × _4 1

17. _1 + _1 + _1 + _1 + _1

6. 3 × _1

3 5

1 _1

35. 3 × _2 3

5. _1 + _1 + _1 5

4. 2 × _1

14. _1 + _1 + _1 + _1

1 _1

25. _4 + _4 5

13. 4 × _1

2. 2 × _1

12. 3 × _1

1 _1

3 5

34. 4 × _1

24. 2 × _ 1. _1 + _1

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3 ▸ Multiply Fractions by Whole Numbers

44. 8 × _ 7 9

3_ 3 4

5 _1 4

6 6 _2 5

6 7_ 5 7

4_ 3 8

6 _2 9

1 1 _1 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3 ▸ Multiply Fractions by Whole Numbers

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3 ▸ Multiply Fractions by Whole Numbers

Fluency Sheet 2 Write the sum or product. Use a whole or mixed number when possible.

_1 + _1 3

3. _1 + _1 5

6. 3 × _1

7. _1 + _1 + _1 5

10. 4 × _1 5

11. _1 + _1 2

21. 5 × _1

22. 6 × _1

41. 6 × _ 5 6

1_ 3 5

33. _1 + _1 + _1 2

40. 7 × _4

1 _1

32. 4 × _2

39. 6 × _4

31. _2 + _2 + _2 + _2 5

1 _1

30. 3 × _2

1 _1

20. _1 + _1 + _1 + _1 + _1 5

38. 7 × _2

29. _2 + _2 + _2

28. 3 × _2

19. 5 × _1

9. _1 + _1 + _1 + _1

18. 4 × _1

8. 3 × _1

17. _1 + _1 + _1 + _1

27. _2 + _2 + _2

36. 4 × _2 37. 5 × _2

42. 8 × _ 6 7

43. 5 × _ 5 8

1 _1 2

1 _1

3 5

1 _1

35. 2 × _2 2

1 _1

3 5

26. 2 × _

16. 4 × _1

3 5

15. 3 × _1

5. _1 + _1 + _1

14. _1 + _1 + _1

4. 2 × _1

25. _ + _

1 _1

13. 3 × _1

34. 3 × _1

24. 2 × _2

2. 2 × _1

12. 2 × _1

1 _1

23. _2 + _2

44. 7 × _ 7 9

3 _1 3

4 _2 3

6 5_ 3 5

5 6_ 6 7

3 _1 8

5 _4 9

1 _1 5

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

Fill in the blanks. Then complete the equation to find the product.

4. 5 of 15 is 1 part when Use the number line to find the product. Then write a repeated addition sentence to check your work. Write your answer as a whole number when possible. 1.

1 ×4= 2

2 0

_ _ _ _ 1 1 1 1 + + + 2 2 2 2

3 ×4= 4

3 =

_ _ _ _ 3 3 3 3 + + + 4 4 4 4

3 =

15 ____ =

parts when

15 × _____ = 5

3 × 15 = 5

is partitioned into 5 equal parts.

1 × 15 = 1 × 5

5. 5 of 15 is

3 15

is partitioned into

equal parts.

Multiply. Write your answer as a whole number when possible.

3 × 60 = 4

2_ 8. 5 × 8 =

16 5

30 9

5 ×6= 9

7 × 64 = 8

3. Use the tape diagram to fill in the blanks. Then complete the equation to find the product. Write your answer as a whole number when possible. a.

1 × 20 4

20 1 1 × (_4 × 20) = 1 × _____ 4

3 × (_4 × 20) =

20 × _____ 4

3 × 20 4

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

EUREKA MATH2 California Edition

10.

5 × 18 = 6

5 ▸ M3 ▸ TA ▸ Lesson 3

3 × 12 = 8

11.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 3

36 8

16. Blake correctly found 4 × 36 = 27. He was surprised that his answer was less than 36 because he thought multiplication resulted in a number greater than both factors. Explain why Blake’s answer was less than 36.

Blake’s answer was less than 36 because he multiplied 36 by a fraction less than 1. This means he was finding a part, or _3 , of 36. 4

12.

3 × 12 = 4

4_ 13. 5 × 60 =

14. The measure of ∠A is 120°. The measure of ∠B is 3 of the measure of ∠A. What is the measure of ∠B?

The measure of ∠B is 80°.

15. The trampoline park sold 84 tickets. 7 of the tickets sold were for children. How many of the tickets sold were for children?

60 of the tickets sold were for children.

PROBLEM SET

LESSON 4

Multiply a whole number by a fraction.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Name

Date

Multiply. Show your work. 1. _4 × 8 = 5

32 __ 5

_4 × 8 = 4 × _8 5

×8 = 4____ 5

• When we multiply a fraction by a whole number that is not zero, why is the product sometimes greater than the whole number?

15 _7 × 15 = 7 × __ 5

=7×3

Achievement Descriptors

= 21

5.Mod3.AD7 Recognize, model, and contextualize the product of a

3. Which expression results in a product greater than 4? Explain how you know.

_3 × 4 4

fraction and a whole number or fraction. (5.NF.B.4.a)

_5 × 4 4

5.Mod3.AD8 Compare the effects of multiplying by fractions and

_5 × 4 is greater than 4 because _5 is greater than 1. When you multiply 4 by any number greater 4

Students select a method to find the product of a fraction less than 1 and a whole number. They reason about how to use what they know about a unit fraction times a whole number to find the product of a fraction greater than 1 and a whole number. Students solve real-world problems involving multiplying fractions and whole numbers.

Key Question

32 = __

2. _7 × 15 =

Lesson at a Glance

whole numbers. (5.NF.B.5.a)

than 1, the product is greater than 4.

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• None

Learn 35 min

Students

• Multiply a Whole Number by a Fraction Less Than 1

• Multiplication Expression Cards (in the student book)

Consider whether to remove Multiplication Expression Cards from the student books and cut out the cards in advance or have students prepare them during the lesson.

• Multiply a Whole Number by a Fraction Greater Than 1

• Scissors

• Solve Real-World Problems • Problem Set

Land 10 min

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Fluency

Whiteboard Exchange: Multiply Multi-Digit Whole Numbers Students multiply a two-digit number by a two-digit number to build fluency with multiplying multi-digit whole numbers by using the standard algorithm. Display 22 × 31 =

Write and complete the equation by using the standard algorithm. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

22 × 31 = 682 31 22 62 + 620 682 ×

Display the product and the recording of the standard algorithm in vertical form. Repeat the process with the following sequence:

32 × 42 = 1,344

47 × 25 = 1,175

Happy Counting by Thirds Students visualize a number line while counting aloud to maintain fluency with counting by thirds. Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Let’s count by thirds. The first number you say is 0 thirds. Ready? Signal up, down, or stop accordingly.

0 3

1 3

2 3

3 3

4 3

3 3

2 3

3 3

4 3

5 3

6 3

7 3

8 3

Continue counting by thirds within __  15 . Change directions occasionally, emphasizing where 3

students hesitate or count inaccurately.

Choral Response: Multiply a Whole Number by a Unit Fraction Students visualize an array partitioned into 2 or 3 equal groups to find a unit fraction of a set to develop fluency with multiplying a whole number by a fraction. Display the statement and the array. 1

How could you partition the array to find _ of 4? Tell 2 your partner.

1 of 4 is 2

2 .

Provide time for students to think and share with their partner. You could draw a horizontal line between the two rows. You could draw a vertical line between the two columns. Display the partitioned array. 1 2

What is _ of 4? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

2 Display the answer.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Repeat the process with the following sequence:

1 of 6 is 2

1 1 1 3 . 2 of 10 is 5 . 2 of 20 is 10 . 2 of 14 is 7 . 3 of 6 is 2 .

1 of 9 is 3

1 1 3 . 3 of 15 is 5 . 3 of 30 is 10 . 3 of 21 is 7 .

Launch

Materials—S: Multiplication Expression Cards, scissors

Students order expressions from the least value to the greatest value by reasoning about the products. EUREKA MATH2 California Edition

Direct students to remove Multiplication Expression Cards from their books and cut out the cards. Have students spread out the cards so they can see each expression. What do you notice?

5 ▸ M3 ▸ TA ▸ Lesson 4 ▸ Multiplication Expression Cards

_4 × 4 5

_5 × 4 4

_1 × 4 3

_1 × 4 2

_1 × 4 5

3_ ×4 8

Each card has a multiplication expression. Each multiplication expression has a fraction for the first factor.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Most of the fractions are less than 1. One fraction is greater than 1. Each expression has the factor 4. Invite students to order the expressions from the least value to the greatest value. Have students reason without actually multiplying. Encourage them to use the interpretation from previous lessons and think of each expression as a fractional part of a whole number. Have students compare their order with another student’s order. Then have them push their cards in their chosen order, to the side. Students revisit the order throughout the lesson and are encouraged to affirm or revise their thinking.

Differentiation: Support Consider having students complete the ordering activity in pairs. Then have them compare their order with the order of other student pairs.

Transition to the next segment by framing the work. Today, we will multiply fractions and whole numbers.

Learn

Multiply a Whole Number by a Fraction Less Than 1 Students choose a method to find the product of a fraction less than 1 and a whole number. Write _  3 × 7 = 4

and invite students to work with a partner to find the product.

As students work, circulate and ask the following questions to advance student thinking: • Is the answer greater than 7 or less than 7? Why? • Can you draw something to help you find the product? What can you draw? • How can you use what you know about _  1 × 7to help you find the product? 4

Identify two students to share their work. Purposefully choose work that allows for rich discussion about connections between methods.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Facilitate a class discussion. Invite students to share their thinking with the whole group. Ask questions that invite students to make connections and encourage them to ask questions of their own.

Consider providing sentence starters to support students as they share their work.

If students do not produce work samples such as the following, use these samples to demonstrate other ways to find _  3 × 7. 4

• I knew I needed to find a fraction of , so I drew .

Number Line (Noah’s Way) 3 4

• Because I wanted to find ,I partitioned the tape diagram into

How did you use a number line to help you find _ × 7? I knew I needed to find a fraction 0 1 2 3 4 5 6 of 7, so I drew a number 3 3 3 3 3 3 3 21 + + + + + + = 4 4 4 4 4 4 4 4 line showing 0 through 7. Because I wanted to find _  3 , I partitioned each interval of 1 into fourths. Then I found _  3 of each 4 4 3 _ whole-number interval and composed them to find    of the total. 4

_ 3  + _ 3  + _ 3  + _ 3  + _ 3  + _ 3  + _ 3  = __  21  4

Language Support

Tape Diagram (Sana’s Way) 3 4

How did you use a tape diagram to help you find _ × 7? I knew I needed to find a fraction

of 7, so I drew a tape diagram to represent 7. Because I wanted to find fourths, I partitioned the tape into 4 equal parts. I knew

4 units = 7, so 1 unit = _ 7 . Because 4 1 unit = _ 7 , that means 4 3 units = 3 × _ 7 , which equals __  21 . 4

4 units = 7 1 unit = 47

3 units = 3 × 47 = 214

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Equations (Riley’s Way)

1 3 4x7=3x 4 x7 = 3 x 47 = 21 4

3 How did you use equations to help you find _ × 7? 4

I know _  3 = 3 × _  1 . I wanted to use a unit fraction because that helps 4

me think about _  1  of 7, which is _   7 . To find _   3  of 7, I knew I could multiply 4

by 3, and 3 ×_  7  = __  21 . 4

Invite students to turn and talk and compare their work to the work samples shown. Write 3 ×_  7  =  ____   . 3×7 4

What do you notice? What do you wonder?

The first expression shows 3 groups of _  7 , which matches Riley’s and Sana’s work. 4

The second expression shows a fraction. The fractional unit in both expressions is fourths.

3×7 I wonder whether _  3  of 7 and 3 ×_  7 can both be written as  ____   .

4 4 4 3×7 3 I wonder whether     has the same value as     of 7 and 3 ×  7 . 4 4 4

____

3 4

7 4

We know _ of 7 can be found with _ × 7. And we learned to interpret _ × 7 as 3 × _ in a 7 4

previous lesson. What repeated addition expression can we write for 3 × _ ?

_ 7  + _ 7  + _ 7  4

Differentiation: Support

1 4x7=3x 4 x7 = 3 x 47 = 47 + 47 + 47

Add a third row to Riley’s work showing the repeated addition expression. 3 Evaluate _ + _ + _ and   ____   . What do you notice?

3×7 4 7 7 21 3 × 7  7  +     +     =    , and     =  21 . They have the same answer. 4 4 4 4 4 4 3×7 The expression     is equivalent to finding 3 groups of 7 fourths, 4 7 4

_ _ _ __

7 4

____ __

____

which is what the repeated addition expression shows. So the

expressions _ × 7, 3 × _ , and  ____   all have the same value. Riley 3 4

7 4

3×7 4

could have included that step in his work.

= 21 4

Consider highlighting the three expressions in the work to emphasize that they have the same value.

1 3 4x7=3x 4 x7 = 3 x 47 = 47 + 47 + 47

= 21 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

7 Gesture to 3 ×  _ =  ____   . 4

3×7 4

How does it help us to write 3 × _ as   ____   ? 7 4

3×7 4

3×7  ____   reminds us we are making 3 groups of 7 fourths, so the fractional unit will be fourths. 4

I know a fraction is the numerator divided by the denominator. 3 × 7 tells me the numerator, and the denominator is 4. If it made sense to rename the answer as a mixed number, I would just divide the numerator by 4. 8 4

Would you record it that way if we were evaluating 3 × _ ? Why? Yes, because 3 ×_  8  =  ____   .

3×8 4 4 3×8 I would because     =  24 , and I know 24 ÷ 4 = 6. 4 4 I would not because I would rather think about  8  as 2, so I would just do 3 × 2 = 6. 4

____ __

Invite students to revisit the Multiplication Expression Cards they ordered earlier. Encourage them to confirm their thinking or to make adjustments to the order based on their new learning.

Differentiation: Challenge If students are ready for a challenge, consider asking the following question:

__b ____

b When might you record a ×   c  as   a ×  ? c

Multiply a Whole Number by a Fraction Greater Than 1 Students use a tape diagram to find the product of a fraction greater than 1 and a whole number. Write the expression _  4  × 7 and draw the 4 tape diagram.

Does the tape diagram represent the 4 4

expression _ × 7? How do you know? Yes. The tape diagram represents the expression because we are finding a fraction of 7 and it is partitioned into 4 equal parts.

Write the expression _  5 × 7 and invite students to think–pair–share about what makes this 4

expression different from others they saw in Learn. The fraction is greater than 1 and the other fractions we saw are less than 1.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Does multiplying by a fraction greater than 1 change anything about how we model the problem? Why?

We could still model the problem with a tape diagram. But _  5 means we need to partition 4 7 into 4 equal parts and then take 5 of them. 5 4

To find _ of 7, we need to take 5 parts when 7 is partitioned into 4 equal parts. Invite students to turn and talk about how they can take 5 parts when there are only 4 parts and how they can modify the tape diagram to represent _ 5  of 7. 4

5 One way to model _ of 7 is to add another part to the tape diagram. 4

Add another part to the model and label the 5 parts with a question mark.

How can we use this tape diagram to find the 5 value of _ of 7? 4

We know _  1  of 7 is _  7 , and _  5  of 7 is 5 times 4

as much as that, so we can find 5 ×_  7 . 4

5 Do you think we could model _ of 7 with a number line? Why? 4

No, I do not think we could use a number line. Each part of a tape diagram can have any value. And as long as we know we have equal parts, we can make sense of the value of each part. We can also divide an interval of 1 on a number line into any number of parts, but we can’t divide it into _  5 parts because that is more than 1. 4

Let’s continue to think about using a tape diagram to model _ of 7. We could draw 4 two tapes, each with a value of 7. We can partition each tape into 4 equal parts and 5 label 5 of the parts with a question mark to represent _ of 7. 4

Draw the tape diagram.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4 5 4

How can we use this tape diagram to find the value of _ of 7?

We can see _  4 is equal to 7. Because we are finding _  5  of 7, we need to add the value 4

of one more part. _  1  of 7 is _  7 , so to find _  5  of 7, we should add 7 and _  7 . 4

Direct students to find _  5 × 7 . Encourage them to find the product by using whichever 4

method they prefer. Circulate to find students who use the following equations to record their thinking. 5 4

Let’s explore two different ways you may have found the value of _ × 7 .

UDL: Action & Expression

Display student work or use the following work samples.

5 times as much as 1 × 7

1 of 7 more than 4 of 7 4 4

5 1 × 7 = 5× × 7 4 4 7 = 5× 4 5×7 = 4 35 = 4 3 =8 4

5 4 1 ×7 = ×7 + ×7 4 4 4 7 = 7+ 4 3 =7 + 1 4 3 =8 4

As students analyze the two ways to find the answer, consider providing additional questions for students to reflect on the two methods. • Can you explain the steps this person took to find the answer? • Do you have another way to find the answer? Explain.

Both methods result in the same product. Which method do you prefer? Why?

I prefer the method on the right because I can find _  4 × 7mentally and then add one 4

more _  7 . 4

I prefer the method on the left because I would rather multiply a whole number and a fraction. 5 4

What is _ of 7?

_ 5  of 7 is 8  _3 . 4

5 83 _is greater than 7. Does it make sense that _ × 7 is greater than 7? Why? 4 4

Yes. _  4 × 7 = 7, so it makes sense that _  5 × 7is greater than 7 because _  5 is greater than _  4 . 4

EUREKA MATH2 California Edition

Yes, it makes sense because _  5 is greater than 1. When you multiply 7 by a number 4 greater than 1, you get a product greater than 7. Invite students to revisit the Multiplication Expression Cards they ordered earlier. Encourage them to confirm their thinking or to make adjustments to the order based on their new learning.

Solve Real-World Problems Students solve real-world problems involving multiplying a whole number by a fraction. Direct students to problem 1 in their books. Invite students to turn and talk about how they could solve the problem.

5 ▸ M3 ▸ TA ▸ Lesson 4

Promoting the Standards for Mathematical Practice Students reason abstractly and quantitatively (MP2) when they read and interpret real-world problems and find the answer by multiplying a whole number by a fraction greater than 1. Ask the following questions to promote MP2: • What does the problem ask you to do? • Does your answer make sense for this real-world problem?

What do you notice about this problem?

Scott spent _  3 of his money on comic books. 4

Scott spent $9 on comic books. Use the Read–Draw–Write process to solve each problem.

1. Scott spent _  3 of his money on comic books. He spent $9 on comic books. How much 4

money did Scott have before he bought the comic books?

_ 3 × 12 = 9 4

Scott had $12 before he bought the comic books. Let’s draw a tape diagram to represent the problem. We can draw a tape diagram to represent the total amount of money and label it with a question mark because that is what the question asks us to find. How can we show how much the comic books cost? We can partition the tape into 4 equal parts and label 3 of them with $9.

Teacher Note Problem 1 introduces a new complexity by asking students to find an unknown factor. Though unknown factor problems are familiar to students from previous grades, this is the first time the unknown factor problem involves a fraction. This complexity is supported through teacher guidance. Problem 2 is less complex, and students can complete the problem independently or in pairs. Consider differentiating the work by class or by student by assigning one of the two problems. If problem 1 is omitted, then consider omitting problems 15 and 18 from the Problem Set because they are also unknown factor problems.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Draw and label a tape diagram and direct students to do the same.

How is this tape diagram different from the other tape diagrams we drew today?

3 units = 9 9 1 unit = 3 = 3 4 units = 4 × 3 = 12

The whole number we are finding a fraction of is unknown. We know the fraction _  3  and 4 we know the product 9. 3 4

So we are trying to find _ of what number is equal to 9. Do we have enough information to find the value of 1 unit? Yes. We know 3 units = 9, so we can say 1 unit= _  9 , or 3. 3

How can we find the amount of money Scott had before he bought the comic books? We know 1 unit = 3, so we can multiply 3 by 4. How much money did Scott have before he bought the comic books?

$12 What is an equation that matches this story? How do you know?

The equation _  3 × 12 = 9matches the story. We know we are finding _  3 of some number 4

and _  3 of the number is 9. We found _  3  of 12 is 9, so the unknown factor is 12. 4

Direct students to problem 2.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Have students complete problem 2 independently or in pairs. Circulate to provide guidance and support as needed by asking the following questions: • What does the problem tell us? • What does the problem ask us to find? • What multiplication expression can we use to represent the problem? • Is the answer greater than 60° or less than 60°? Why? • What can we draw to help us solve the problem? 2. Tyler drew the angle shown. Kayla draws an angle with a measure that is _  8 the measure of 5 Tyler’s angle. What is the measure of Kayla’s angle?

_ 8 × 60 = 8 × __  60  5

= 8 × 12 = 96

The measure of Kayla’s angle is 96°.

60° Gather students to discuss their work. What is the measurement of Kayla’s angle? How do you know? It is 96°. I found _  8  of 60, which is 96. 5

Invite students to turn and talk about whether their answer is reasonable.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Land

Debrief 5 min Objective: Multiply a whole number by a fraction. Facilitate a class discussion about multiplying a whole number by a fraction by using the following prompts. Encourage students to restate or add to their classmates’ responses. Display the following expressions.

3   × 19 5

7   × 19 5

Which expression has a product greater than 19? How do you know?

_ 7 × 19is greater than 19. We know _ 5 × 19 = 19and we are finding more than _ 5  of 19, 5

so the answer is greater than 19.

_ 7 × 19is greater than 19 because we are multiplying 19 by a fraction greater than 1. 5

When is the product of a fraction and a whole number that is not zero greater than the whole number? Why? When the fraction is greater than 1, the product is greater than the whole number, as long as the whole number is not zero. The product is greater than the whole number because a fraction greater than 1 means you multiply the whole number by more than 1, and 1 times the whole number is itself. So more than 1 times the whole number is more than the whole number.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Invite students to revisit their Multiplication Expression Cards to make any last revisions. Then share the correct order.

1 ×4 5

1 ×4 3

3 ×4 8

1 ×4 2

4 ×4 5

5 ×4 4

Consider asking the following questions: • Which of the expressions did you know the value of immediately? Which cards did you need to think about? • If you know _  1 × 4 = _  4 , what other similar products can you find? 5

• How does knowing the product of a unit fraction and a whole number help you find the product of another fraction with the same denominator and the same whole 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.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ 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 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

Name

Date

4. _2 × 4 is 3

parts when 4 is partitioned into

_2 × 4 = 3

1. _1 × 5 4

_ _ _ _ _ 1 1 1 1 1 + + + + 4 4 4 4 4

5. _3 × 11 is

4 × _____ = _______ =

parts when 11 is partitioned into

×4

_ 8 3

equal parts.

_3 × 11 = 4

2. _2 × 4 3

× 11

11 × _____ = ________ = 4

__ 33 4

? 0

_ _ _ _ 2 2 2 2 + + + 3 3 3 3

3 =

6. _5 × 11 is

_ 8 3

parts when

3. _1 × 5 is 4

part when 5 is partitioned into

11 5 × 11 = _ 4

×5

5 × _____ = _______ = 4

__ 11 4

× 11

= ________ = 4

__ 55 4

Complete the statement. Then find the product.

_ 5 4

7. _7 × 12 is

parts when

7_ × 12 = 5

8. _9 × 10 is

parts when

10 7

9_ × 10 = 7

equal parts.

5 1×5= 4

is partitioned into

Use the tape diagram to complete the statement. Then find the product.

equal parts.

5 4

Use the number line to represent the multiplication. Then write a repeated addition sentence to find the product.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

PROBLEM SET

12 5

is partitioned into × 12

= __________ =

equal parts.

__ 84 5

_____

is partitioned into

9 × 10 = = 7

__ 90 7

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 4

EUREKA MATH2 California Edition

Multiply. 9.

5 ▸ M3 ▸ TA ▸ Lesson 4

__8

__1 × 8 =

11. _3 × 7 =

13. _7 × 8 =

56 5

12. _5 × 6 =

30 3

__ 21 5

10. _3 × 9 =

17. Mr. Evans makes 10 pints of salsa. _3 of the pints of salsa are spicy. How many pints of the salsa 4 are spicy?

_3 × 10 = 7 _1 2 4 7 _1 pints of the salsa are spicy. 2

10 × 19 = 14. __ 6

__ 190 6

18. Lisa breaks _2 of her colored pencils while she works on an art project. She breaks 20 of the 5 pencils. How many colored pencils did Lisa have when she started?

15. _2 of a number is 24. What is the number? Show your work. 3

_2 of ? is 20.

2 units = 24

24 1 unit = __ = 12

EUREKA MATH2 California Edition

Use the Read–Draw–Write process to solve each problem.

__ 27 8

5 ▸ M3 ▸ TA ▸ Lesson 4

Lisa started with 50 colored pencils.

3 units = 3 × 12 = 36 The number is 36.

11 the measure of Sana’s angle. 16. Sana drew the angle shown. Riley draws an angle that is __ 5

a. Is the measure of Riley’s angle greater than or less than the measure of Sana’s angle? How do you know? The measure of Riley’s angle is greater than the measure of 11 of the measure of Sana’s, Sana’s angle because Riley’s is __

11 is greater than 1. and __

45°

b. What is the measure of Riley’s angle? The measure of Riley’s angle is 99°.

PROBLEM SET

LESSON 5

Convert larger customary measurement units to smaller measurement units.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Name

Date

Convert each measurement. Use the tape diagrams or reference sheet if needed. 1. _5 yards = 6

2_1 2

feet 1 yard

• Can we use multiplication to convert larger measurement units to smaller measurement units? How?

5_ × 1 yard = 5_ × 3 feet 6 6 5 × 3 __ 5_ × 3 = ____ = 15 = 2_3 = 21_ 2 6 6 6 6

Students apply their prior knowledge about multiplication involving fractions to convert larger customary measurement units to smaller customary units. They estimate before converting and then assess the reasonableness of their answers. Students discover that converting from larger units to smaller units requires multiplying a fraction and a whole number.

Key Questions

1 foot

2. 3_ pounds = 4

Lesson at a Glance

• How does multiplying a whole number by a fraction relate to converting larger measurement units to smaller measurement units?

ounces

Achievement Descriptors

1 pound

5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a)

1 ounce

_3 × 1 pound = 3_ × 16 ounces 4 4 16 _3 × 16 = 3 × __ = 3 × 4 = 12 4 4

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

5.Mod3.AD14 Convert among units within the customary

measurement system to solve problems. (5.MD.A.1)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• Grade 5 Mathematics Reference Sheet (in the teacher edition)

Save Grade 5 Mathematics Reference Sheet for use in future lessons.

Learn 35 min • Multiply to Convert Units • Conversions in the Real World • Problem Set

Students • Grade 5 Mathematics Reference Sheet (in the student book)

Land 10 min

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Fluency

15 × 23 = 345

23 15 1

115 +230 345

Display the product and the recording of the standard algorithm in vertical form. Repeat the process with the following sequence:

42 × 61 = 2,562

85 × 27 = 2,295

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Happy Counting by Thirds Students visualize a number line while counting aloud to maintain fluency with counting by thirds and renaming fractions greater than 1 as whole or mixed numbers. Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.) Let’s count by thirds. Today we will rename the fractions as whole or mixed numbers when possible. The first number you say is 0. Ready? Signal up, down, or stop accordingly.

1 3

2 3

1 3

2 3

Continue counting by thirds within 5. Change directions occasionally, emphasizing crossing over whole numbers and where students hesitate or count inaccurately.

Choral Response: Multiply a Whole Number by a Unit Fraction Students visualize an array partitioned into 2, 3, or 4 equal groups to find a unit fraction of a set to develop fluency with multiplying a whole number by a fraction. Display the statement and the array. 1

How could you partition the array to find _ of 8? 2 Tell your partner.

1 of 8 is 2

4 .

Provide time for students to think and share with their partner. You could draw a horizontal line between the two rows. You could draw a vertical line down the middle so there are 4 circles on each side of the line.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Display the partitioned array. 1 2

What is _ of 8? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

4 Display the answer. Repeat the process with the following sequence:

1 of 12 is 6 . 2

1 of 18 is 9 . 2

1 of 9 is 3 . 3

1 of 12 is 4 . 3

1 of 24 is 8 . 3

1 of 8 is 2 . 4

1 of 12 is 3 . 4

1 of 20 is 5 . 4

Launch

Students compare two different measurement units. Present the following problem and use the Math Chat routine to engage students in mathematical discourse: Would you rather play basketball for  __ hours or 1,080 seconds? Why? 3 10

Give students 1 minute of silent think time to decide. 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. 98

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5 ▸ M3 ▸ TA ▸ Lesson 5

Facilitate a class discussion. Invite students to share their thinking with the whole group and then record their reasoning.

I would rather play basketball for  __ hours because I think that is less time and I don’t like 10 playing basketball. 3

I would rather play basketball for 1,080 seconds because I think it is more time. Basketball is my favorite sport, so I would want to play for as much time as possible. What do you notice about the units? There are two different units: hours and seconds. Which is the larger unit? Hours To accurately compare measurement units, we need to convert, or rename, so the measurement units are the same. Let’s convert hours to seconds. How many seconds are in 1 hour? How do you know? There are 3,600 seconds in 1 hour because there are 60 seconds in 1 minute and 60 minutes in 1 hour. 60 × 60 = 3,600 There are 3,600 seconds in 1 hour, but we want to know the number of seconds in 3 _   hours. What could we do? 10

We could find  __ of 3,600.

 __ × 3,600 10 3

3 10

Invite students to work with a partner to find  __ of 3,600. You might need to remind some 3 10

students to think about how they can make a simpler problem. They can find  __ of 3,600 by evaluating 3 ×   ____ to get 3 × 360 = 1,080. 3,600 10

3 10

How many seconds are in _   hours?

1,080 What do you notice?

 __ hours and 1,080 seconds are the same amount of time. 10 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Why did it seem like 1,080 seconds would be longer? Because 1,080 is a much greater number than  __ . 3 10

3 10

How could 1,080 seconds and _   hours represent the same amount of time? Seconds are small, so there are a lot of seconds in a number of hours. Transition to the next segment by framing the work. Today, we will use multiplication to convert larger measurement units to smaller measurement units.

Learn

Multiply to Convert Units Materials—T/S: Grade 5 Mathematics Reference Sheet

Students multiply a whole number by a fraction to convert larger measurement units to smaller measurement units. Write _ foot = 1 6

The number of feet is a fraction less than 1.

The unknown is the number of inches equal to  _ foot. 1 6

1 6

Let’s convert _ foot to inches. What is _ foot as a multiplication expression? Record  _ × 1 foot.

100

For example, write 1 ft. Say 1 foot and write 1 foot. Direct students to do the same. Repeat the process with 2 ft. Discuss the similarities and differences between the singular and plural measurement units.

Teacher Note

We have a number of feet equal to a number of inches.

1 6

Consider providing time for students to practice saying, reading, and writing the singular and plural forms of the length measurement units. Display a measurement with abbreviated units. Model saying and writing the measurement and then invite students to say and write it.

inches.

What do you notice?

1   _ × 1 foot 6

Language Support

In the first segment of Learn, tape diagrams are used to show how many smaller measurement units equal 1 of the larger measurement units. Later in the lesson, students use tape diagrams as part of the Read–Draw–Write process. Tape diagrams serve many purposes and can be used flexibly to make sense of problems, to solve them, or to do both. Consider highlighting the many uses of tape diagrams with students.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Display the tape diagram to show 1 foot decomposed into 12 equal parts.

1 foot

1 inch Invite students to think–pair–share about what they understand from the tape diagram.

1 foot = 12 inches 1 foot is longer than 1 inch. 1 inch is shorter than 1 foot. 1 Are the number of inches equal to _ foot greater than or less than 12 inches? How 6

do you know?

Teacher Note

The number of inches is less than 12 inches because  _ foot is less than 1 foot, and 6 1 foot = 12 inches.

When some students show their work, they might show units consistently in every equation.

Looking at our tape diagram, we can see 1 foot is 12 times as long as an inch. How can you tell that from the tape diagram?

  1 × 1 ft =   1 × 12 in

__ 6

12 inches are in 1 foot, so 1 foot is 12 times as long as 1 inch. The tape diagram is partitioned into 12 equal parts. 1 6

= 2 in

What do you think that means about the number of inches equal to _ foot? The number of inches is 12 times as large as  _ . 1 6

Record  _ × 1 foot =  _ × 12 inches. Invite students to turn and talk about whether 1 6

1 6

 _ × 1 foot =  _ × 12 inches is a true equation and how they know. 1 6

1 6

Other students might get to the conversion equation and then simply multiply and recontextualize at the end.

__ 6

We were finding  _ of 12. 1 6

1 6

How did you find _ of 12?

___

= 1 ×  12

in  _ foot.

1 When you found _ × 12 inches, what did you notice? 6

  1 × 1 ft =   1 × 12 in

Direct students to continue working independently until they find how many inches are 1 6

__ 6 1_____ =   × 12 in 6 12 ___ =   in

=1×2=2

  1 ft = 2 in 6

Either approach is acceptable. You might encourage one or the other, depending on your students’ needs.

I thought about 1 part when 12 is partitioned into 6 equal parts. © Great Minds PBC

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I thought about 1 ×  __ , which is 1 × 2. Then I multiplied. 1 × 2 = 2 12 6

Is 2 inches a reasonable answer? How do you know?

Yes, 2 inches is reasonable because we estimated  _ of 1 foot is less than 12 inches, and 6 our answer of 2 inches is less than 12 inches. 1

Yes, 2 is 12 times as much as  _ . The number of inches should be 12 times as much as the 6 number of feet. 1

Complete the equation  _ foot = 2 inches.

UDL: Representation

__1

Help students see   of 12 is 2 by partitioning 6

the tape diagram into sixths.

1 6

1 foot

Direct students to problem 1 in their books. Share that in this problem, a number of pounds is equal to a number of ounces. Invite students to think–pair–share about what they understand from the tape diagram.

1 inch

1 pound = 16 ounces 1 pound is greater than 1 ounce. 1 ounce is less than 1 pound. 1 pound is 16 times as much as 1 ounce. Convert. 2_ 1. lb = 3

10 _3 oz

Teacher Note

Students do not need to memorize any of the conversions in this lesson. Rather, the focus of the lesson is the application of multiplying a whole number by a fraction.

1 pound

1 ounce

  _3 lb =   _3 × 1 lb 2

As needed, tell students the equivalent units, show the equivalence by using a tape diagram, or have them use the reference sheet.

= _3 × 16 oz 2

= _____ oz 3 2 × 16

= __ oz 3 32

= 10 _3 oz 2

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2 3

How can you convert _ pounds to ounces? How do you know? You can find  _ × 16 ounces because 1 pound = 16 ounces. 2 3

Differentiation: Support

2 3

Is the number of ounces equal to _ pounds greater than or less than 16? Why?

2 The number of ounces is less than 16 because  _ pounds is less than 1 pound. 3 2 _ If you find   × 16, the product is less than 16 because we are multiplying by a number

Consider providing pictures or realia to support students’ understanding of the units used in problems.

less than 1.

Direct students to work with a partner to determine the value of  _ of 16 ounces. Students 3 may leave their answer as a fraction greater than 1 or rename it as a mixed number. Circulate and monitor student work. Allow students 2 minutes to work before asking them to share with the whole group. 2

32 3

2 3

I notice some of you have _ ounces and some of you have 10_ ounces as an answer. Are both correct? How do you know?

Yes, both answers are correct. 10   _ =   __ 32 3

2 3

If you were weighing an object on a scale, would you rather see 10_ ounces or 32 _   ounces? Why? 3

I would rather see 10  _ ounces because we do not usually represent measurements with 3 fractions greater than 1. 2

How can we tell whether these answers are reasonable? Is 10_ ounces reasonable? 3 How do you know? We can compare the answers to 16 ounces. Because we converted a fraction of a pound to ounces, the answers should be less than 16. 2 3

32 3

Is it simpler to make that comparison with 10_ or _  ? Why?

It is simpler to compare by using 10  _ because I can see 10  _ is less than 16. I could make

2 3 32 32 the comparison with   , but I would end up thinking about   as a mixed number. 3 3 2 2 Have students make sure they completed the equation   lb = 10   oz. 3 3

2 3

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Direct students to complete problem 2 with a partner by using the tape diagram or the reference sheet to determine how many fluid ounces are in 1 cup. Circulate and monitor student work. Allow students 2–3 minutes to work before you ask them to share with the whole group.

7 2. c = 4

Promoting the Standards for Mathematical Practice As students convert customary units of measurement, they are attending to precision (MP6) by making sure that they are using the correct abbreviation and conversion factor and that their answers make sense.

fl oz

1 cup

Ask the following questions to promote MP6: • What does the abbreviation fl oz mean in the equation in problem 2?

1 fluid ounce

• When you convert units of measurement, with which steps do you need to be extra careful? Why?

7 7   4 × 1 c = 4 × 8 fl oz

= 7 ×  _4 fl oz 8

• Where is it common to make mistakes when you convert units of measurement?

= 7 × 2 fl oz = 14 fl oz 7 4

How many fluid ounces equal _ cups? What multiplication expression did you use? I found that  _ cups = 14 fluid ounces when I used the expression  _ × 8 fl oz. 7 4

7 4

What is different about this example compared to previous examples? The fraction we multiplied by is greater than 1. How did that affect your product? Why?

The product is greater than 8 fluid ounces. There are 8 fluid ounces in 1 cup, and  _ cups 4 is greater than 1 cup. 7

The product is greater than 8 fluid ounces. We multiplied 8 fluid ounces by a fraction greater than 1, which means the product is greater than 8. Invite students to turn and talk about why they can use multiplication to convert a larger unit to a smaller unit. 104

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5 ▸ M3 ▸ TA ▸ Lesson 5

Conversions in the Real World Students apply their understanding of converting units to real-world situations. Direct students to problem 3. Ask them to read the problem independently. Then read the first two sentences of the problem chorally with the class. Use the Read–Draw–Write process to solve each problem.

3. Mr. Sharma spends   of a day at work. He spends the rest of the day at home. How many 8 hours does he spend at home? 5 5   _8 days =   _8 × 1 day 1 day, or 24 hours

= _8 × 24 hours 5

= 5 × __ hours 8 24

= 5 × 3 hours

at work

at home ?

= 15 hours Mr. Sharma spends 15 hours at home. What do we know?

Mr. Sharma spends  _ of a day at work and the rest of the day at home. 3 8

Can we draw something? What can we draw? We can draw a tape diagram to represent the total: 1 day. Then we can partition the tape into 8 equal parts. What labels can we add to our tape diagram based on what we know? We can label the total with 1 day. We can label 3 of the parts with at work. We can label the remaining 5 parts with at home.

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1 day, or 24 hours

Draw the tape diagram and direct students to do the same. Read the rest of the problem with the class.

at work

What does the question ask us to find? Where can we put the question mark in our model?

at home ?

We need to find how many hours Mr. Sharma spends at home. We can put the question mark under the part that represents his time at home. Write a question mark under the at home label. Direct students to do the same. What do you notice about the measurement unit in our tape diagram and the measurement unit in the question? The tape diagram represents 1 day. The question asks for hours. How can we show 1 day as hours in our model? Label 1 day, or 24 hours. What conclusions can you make from the tape diagram so far? I know all 8 parts in the tape equal a total of 1 day, which is equal to 24 hours. I know 1 part is  _ × 24 or  __ . 1 8

24 8

Direct students to work with a partner to solve the problem. How many hours does Mr. Sharma spend at home? Mr. Sharma spends 15 hours at home. Is your answer reasonable? How do you know? Our answer is reasonable because our tape diagram shows the value is less than 24 hours. It is also greater than half of 24 because 5 out of 8 parts is greater than half of the parts in the tape diagram. Direct students to problem 4. Have them complete the problem with a partner. Share that 1 mile = 1,760 yards. Circulate and monitor student work. Allow students 3–4 minutes to work before you ask them to share with the whole group. 106

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5 ▸ M3 ▸ TA ▸ Lesson 5

__3

4. Ryan is in a 2-mile race. He jogs 1   miles and walks the rest of the distance. How many 10 yards does Ryan walk? 3 7 2 − 1   __ =   __ 10 10 7 7   __ miles =  __ × 1 mile 2 miles 10 10 7 =   __ × 1,760 yards 10 1 3 miles

____

1,760 = 7 × 10 yards

= 7 × 176 yards

jogs

walks

= 1,232 yards Ryan walks 1,232 yards. Invite students to think–pair–share about the following question: Why is the number 3 of yards Ryan walked, 1,232, greater than the number of miles he jogged, 1  __ , even though 10 Ryan jogged most of the race? The number that represents the amount of the race Ryan walked is greater than the number that represents the amount he jogged because the amounts are different units.

1  __ miles converted to yards is greater than 1,232 yards. 10 3

Yards are a much smaller unit than miles, so it takes more yards than miles to represent a given distance.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Teacher Note Accept all equivalent answers in the Problem Set. Encourage students to express answers as mixed numbers when appropriate, but fractions greater than 1 are also acceptable responses.

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Land

Debrief 5 min Objective: Convert larger customary measurement units to smaller measurement units. Facilitate a class discussion about converting larger measurement units to smaller units by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What do all the measurement unit conversions today have in common? The conversions we did used units such as pounds, ounces, and yards. There were no units of meters, grams, or liters. We converted from larger measurement units to smaller measurement units. What operation did all our equations involve when we needed to convert from a larger measurement unit to a smaller measurement unit? Multiplication How can we use multiplication to convert larger measurement units to smaller measurement units? We can write multiplication expressions that show the amount of the large unit and the small unit. Then we can write an equation by using those expressions and find a fraction of a whole number by using multiplication.

Write  _ yd =  _ × 1 yd =  _ × 3 ft =  _ ft. 3 4

3 4

9 4

3 4

9 4

Teacher Note

3 4

Compare _ and _ . Which is greater? 9 3  _ is greater than  _ . 4

3 4

Does it make sense for the number of feet in _ yards to be greater than _ ? Why? Yes. It makes sense because feet are shorter than yards, so there are more of them to cover the same length. 3 4

Comparing the size of the product to the size of each factor continues in this topic and in the next. Customary unit conversion allows for a context-specific exploration of that concept.

9 4

How does the product _ × 3, or _ , compare to 3?

 _4 is less than 3. 9

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Does it make sense that the product _ × 3 is less than 3? Why?

Yes. It makes sense because  _ is less than  _ , and  _× 3 = 3. So,  _ × 3 is less than 3. 3 4

4 4

It is reasonable because we are finding a fraction of 3.

3 4

Display the following equations. Invite students to analyze them.

2 2 × 1 lb = × 16 oz 3 3

2 2 × 1 yd = × 3 ft 3 3

7 7 × 1 c = × 8 fl oz 4 4

1 1 × 1 lb = × 16 oz 4 4

1 1 × 1 gal = × 4 qt 8 8

1 1 × 1 ft = × 12 in 6 6

What do you notice about the highlighted factor in each equation? Each highlighted factor equals one of the larger measurement units. Each highlighted factor is greater than 1. Why is each highlighted factor greater than 1? What does this tell us about converting larger measurement units to smaller measurement units? Because we are converting from large to small measurement units and you need more of the smaller measurement units to equal the same amount of the larger measurement units.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Convert each measurement.

1 gallon Convert each measurement.

14. _3 gal = 4

1 yard 1. _1 yd = 3

2. _2 yd =

3. _1 yd = 4

_3 4

1 quart 16. _1 qt =

1 foot

6. _3 ft =

1 inch

5. _1 ft =

7. _7 ft =

10 _1 in 2

15. _1 gal =

17. _5 qt =

19. _3 pt =

1 _1

1 quart

1 foot 1 ft = 4. __

1 pint

1 pint 18. _1 pt = 2

1 cup

8. Tyler buys _3 yards of ribbon for a project. How many feet of ribbon does Tyler buy? 4

Tyler buys 2 _1 feet of ribbon.

20. Mrs. Chan needs _1 gallon of milk. The store only sells quarts of milk. Mrs. Chan buys 1 quart

of milk. Does she have enough milk? How do you know?

Yes, Mrs. Chan has enough milk. She needs _1 gallon of milk, which is equivalent to _4 quarts

Convert each measurement.

1 pound

1 ounce

1 lb = 16

10. _1 lb =

11. _3 lb =

12. _1 lb =

of milk. She bought 1 quart of milk, and that is more than _4 quarts of milk.

8 5 _1 3

13. A hamburger weighs _1 pound. How many ounces does the hamburger weigh? 4

The hamburger weighs 4 ounces.

110

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 5

Use the Read–Draw–Write process to solve each problem.

21. Toby buys 3 gallons of juice. He uses 2 _3 gallons of it to make punch. How many quarts of juice 8 does he have left?

3 − 2 _3 = _5 8

_5 gal = 2 _4 qt 8

Toby has 2 _4 quarts of juice left. 8

22. Julie buys a piece of wood that is 12 feet long. She cuts the wood into 8 equal pieces. How many inches long is each piece of wood? 12 _ 12 ÷ 8 = __ =3 8

_3 ft = 18 in

Each piece of wood is 18 inches long.

PROBLEM SET

111

112 This page may be reproduced for classroom use only.

1 ton = 2,000 pounds

1 mile = 1,760 yards

1 liter = 1,000 cubic centimeters

1 gallon = 4 quarts

1 quart = 2 pints

1 pint = 2 cups

1 cup = 8 fluid ounces

Right rectangular prism �� V = B × h or V = l × w × h

Volume Formula

1 pound = 16 ounces

1 mile = 5,280 feet

Conversions

Grade 5 Mathematics Reference Sheet

5 ▸ M3 ▸ TA ▸ Lesson 5 ▸ Mathematics Reference Sheet EUREKA MATH2 California Edition

LESSON 6

Convert smaller customary measurement units to larger measurement units.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Name

Date

Convert each measurement. Use the tape diagrams or reference sheet if needed. 1. 25 quarts =

6 1_ 4

1 gallon

gallons

1 quart 25 × 1 quart = 25 × 1_ gallon 4

4 = 6 1 gallons 4

• Can we use multiplication to convert smaller measurement units to larger measurement units? How? • How does what you understand about multiplication help you convert units?

1 pound

Achievement Descriptors 1 ounce

5.Mod3.AD7 Recognize, model, and contextualize the product of a

12 oz = 12 × 1 ounce

fraction and a whole number or fraction. (5.NF.B.4.a)

1 = 12 × __ pound 16

__ pounds = 12 12 pounds of almonds. Riley buys __

Students use multiplication to convert smaller customary measurement units to larger customary measurement units. Students analyze the relationship between the size of the units to generate equations that help them convert from one unit to another. They estimate before multiplying and then assess the reasonableness of their answers. Students solve real-world problems involving unit conversion.

Key Questions

__ gallons = 25

2. Riley buys 12 ounces of almonds. How many pounds of almonds does Riley buy?

Lesson at a Glance

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

5.Mod3.AD14 Convert among units within the customary

measurement system to solve problems. (5.MD.A.1)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• Paper (2 sheets)

Learn 35 min

Students

Prepare two signs on paper. Label one sign Division to Convert and label another sign Multiplication to Convert. Hang the signs in different locations in the classroom.

• Multiply to Convert Units

• None

• Conversions in the Real World • Problem Set

Land 10 min

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Fluency

31 × 68 = 2,108 ×

68 31 68 2 + 2 040 1 2,1 0 8

Display the product and the recording of the standard algorithm in vertical form. Repeat the process with the following sequence:

93 × 24 = 2,232

116

42 × 75 = 3,150

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Happy Counting by Fourths Students visualize a number line while counting aloud to maintain fluency with counting by fourths. Invite students to participate in Happy Counting. When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) When I give this signal, stop. (Demonstrate.) Let’s count by fourths. The first number you say is 2 fourths. Ready? Signal up, down, or stop accordingly.

2 4

3 4

4 4

5 4

6 4

7 4

8 4

9 4

8 4

7 4

6 4

7 4

8 4

Continue counting by fourths within  __ . Change directions occasionally, emphasizing where 4 students hesitate or count inaccurately. 16

Choral Response: Multiply a Whole Number by a Unit Fraction Students visualize an array partitioned into 3, 4, or 5 equal groups to find a unit fraction of a set to develop fluency with multiplying a whole number by a fraction. Display the statement and the array. 1

How could you partition the array to find _ of 3? Tell 3 your partner.

1 of 3 is 3

1 .

Provide time for students to think and share with their partner. You could draw vertical lines between each circle. Display the partitioned array. 1 3

What is _ of 3? Raise your hand when you know. © Great Minds PBC

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Wait until most students raise their hands, and then signal for students to respond.

1 Display the answer. Repeat the process with the following sequence:

1 of 18 is 6 . 3

1 of 27 is 9 . 3

1 of 4 is 1 . 4

1 of 16 is 4 . 4

1 of 24 is 6 . 4

1 of 5 is 1 . 5

1 of 15 is 3 . 5

1 of 25 is 5 . 5

Launch

Materials—T: Signs

Students wonder about and discuss converting smaller measurement units to larger measurement units. Write 132 in =

ft.

What do you notice? We have a number of inches equal to a number of feet. We know the number of inches, but we do not know the number of feet. Inches are smaller than feet. In other lessons, we started with the larger unit of measurement.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Invite students to think about how they would convert 132 inches to feet. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom: Division to Convert and Multiplication to Convert. Invite students to stand beside the sign that best describes their thinking about how to convert 132 inches to feet. 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. Sample responses: I would use division because we want a smaller number of feet than inches. I would use division because I know 132 ÷ 12 = 11, so 132 inches = 11 feet. I would think of multiplication facts I know that involve 12 because a foot is 12 times as long as an inch. I know 12 × 11 = 132, so 132 inches must equal 11 feet. Have students return to their seats. Do not confirm any accurate ways to convert the units. Instead, allow student discourse to remain open until the learning of the lesson affirms how they can convert from small to large units. Transition to the next segment by framing the work. Today, we will convert a smaller measurement unit to a larger measurement unit.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Learn

Multiply to Convert Units Students multiply to convert smaller measurement units to larger measurement units. 1 foot

Display the equation and the tape diagram.

132 in =

1 inch

What does the tape diagram show us about the relationship between 1 foot and 1 inch?

1 foot is longer than 1 inch. 1 foot is 12 times as long as 1 inch. 1 inch is shorter than 1 foot. Let’s continue to analyze the relationship between 1 inch and 1 foot.

1 foot

Highlight the part labeled 1 inch. How many inches are in 1 foot?

1 inch

12 inches = 1 foot

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Write 12 inches = 1 foot.

Teacher Note

What fraction of a foot is equal to 1 inch? 1 inch = __     foot 1 12

Write 1 inch = __     foot. 1 12

What do these two equations have in common? They show two equal lengths. The number of feet is less than the number of inches for the same length. The number of inches is 12 times the number of feet in each equation. The number of feet is __    the number of inches in each equation. 1 12

__1 In each equation we can see the number of feet is     the number of inches. What

Typically, the phrase times as much as is not used with a fraction less than 1. Instead, the phrase as much as is often used. Because this may be students’ first experience with this language, highlight the multiplicative relationship by including the word times. In the future, students may not say the word times when they describe a multiplicative comparison with a fraction less than 1 1 (i.e.,     as much as).

__ 12

is true about the number of feet equal to 132 inches?

The number of feet is __    times as much as the number of inches. 1 12

1 12

Because the number of feet is     times as much as the number of inches, is the number of feet greater than or less than the number of inches? Why?

The number of feet is less than the number of inches because __    is less than 1. 1 12

Why does it make sense that a smaller number of feet is the same length as a larger number of inches? It makes sense because a foot is longer than an inch, so you need fewer feet to equal the same length in inches.

Record 1 32 × 1 in = 132 × __     ft. 1 12

Invite students to think–pair–share to discuss why this is a true equation.

1 1 inch is equal to __    foot. So, the equation shows 132 times the same length. 12

1 1 inch and __    foot are the same length, so we are multiplying the same length by 132. 12

Direct students to continue working until they find the number of feet equal to 132 inches.

Language Support Provide students with sentence frames to help them articulate the relationship between the smaller unit and the larger unit. inches in 1 foot.

• There are • 1 foot is

times as long as 1 inch.

• 1 inch is equal to

foot.

• 1 inch is

times as long as 1 foot.

• There are

ounces in 1 pound.

• 1 pound is as 1 ounce.

times as heavy

• 1 ounce is equal to • 1 ounce is as 1 pound.

pound. times as heavy

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Display sample work similar to the following. Note that some students may show units in the first line of their work to make sense of the conversion, drop the units when they work numerically, and then include units in the answer statement, as shown in the following sample work. Other students may consistently include units in each line of their work. Either is acceptable as long as it is consistent. How is converting inches to feet similar to converting feet to inches? How is it different?

132 x 1 in = 132 x 12 ft 132 x 1 = 132 x 1 12

12 132 = 12 = 11

132 in = 11 ft

Converting inches to feet is different from converting feet to inches because we start with a smaller measurement unit and convert to a larger measurement unit. Converting inches to feet is different from converting feet to inches because we multiply by a fraction less than 1. When we convert feet to inches, we multiply by a number greater than 1. Converting inches to feet is similar to converting feet to inches because we use multiplication. Converting inches to feet is similar to converting feet to inches because in both cases we rename one unit as another. The length is the same. In Launch, we used Take a Stand to name the operation we would use to convert inches to feet. Which operation is correct? Why?

UDL: Representation As students work and discover, consider recording sample equations on an anchor chart for their reference.

Convert Small to Large

Convert Large to Small

132 in = 11 ft

3 ft = 36 in

132 × 1 in = 132 ×  1 ft 12

3 × 1 ft = 3 × 12 in

Both are correct. You can multiply or divide.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

You can divide 132 by 12, like we saw in our work when we had the fraction  ___ . I know 12 fractions can be written as division expressions, so we could have divided. 132

You can multiply 132 and __    because the number of feet is __    times as much as the 12 12 number of inches. 1

Affirm that students see division when they convert units involving fractions because fractions can be written as division expressions. However, encourage students to continue to practice converting smaller measurement units to larger measurement units by using multiplication. Invite students to turn and talk to compare this learning with their thinking during Launch. Did they use the operation they expected to when they converted inches to feet? What was new, surprising, or expected?

Conversions in the Real World

Promoting the Standards for Mathematical Practice When students notice that they multiply by a fraction less than 1 to convert smaller measurement units to larger measurement units, they are looking for and making use of structure (MP7). Ask the following questions to promote MP7: • How are inches and feet related? How can that help you solve the problem? • How are problem 1 and problem 2 related? How can that help you solve problem 2?

Students apply their understanding of converting measurement units to real-world situations. Play the Conversion Confusion video that shows Mr. Perez measuring the width of his window in inches and looking puzzled when the order form for new blinds asks for the measurement in feet. Why does Mr. Perez look confused? Mr. Perez measured in inches, but the order form asks for the measurement in feet. What should Mr. Perez do? He should convert the number of inches to feet to make sure he orders the right size. Let’s estimate. Will the number of feet be more than or less than 30? Why? The number of feet will be less than 30 because feet are longer than inches. Will there be a whole number of feet equal to 30 inches? Why? There will not be a whole number of feet because 2 feet = 24 inches and 3 feet = 36 inches. No, because 30 is not a multiple of 12.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

About how many feet equal 30 inches?

Differentiation: Support

About 2 feet Between 2 and 3 feet

This concrete representation helps students 1 see 2     feet as 30 inches and understand why 2 it takes fewer feet than inches to cover the same length.

Inch

0 CM 1

20 21

Inch

0 CM 1

20 21

Inch

1 2

Is 2_ feet a reasonable answer based on our estimate?

6 1 30 6 Yes,  __ is equal to 2  _ because 3 0 ÷ 12 = 2  __  and __     can 12 2 12 12 1 be renamed as _     .

1 2

30 12

Notice that this student renamed _ ft as 2_ft. Is 2_ft correct?

Help students see the relationship between inches and feet by having them use 12-inch rulers to build 30 inches. Consider covering the remaining 6 inches with a piece of tape or paper.

1 30 x 1 in = 30 x 12 ft 30 = 12 ft 1 = 2 2 ft

0 CM 1

Direct students to work with a partner to convert 30 inches to feet. Circulate to identify sample student work to share. If students do not have similar work, share the following:

Yes, because we estimated the number of feet would be less than 30. Yes, because we estimated the number of feet would be between 2 and 3. 30 12

1 2

Should Mr. Perez write _ feet or 2_ feet on the order form? Why?

 __ feet can be challenging to visualize or easily recognize, so it is better for him to write 30 12

the measurement as a mixed number.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Direct students to problem 1 in their books and read the problem chorally. 1. A restaurant has 56 ounces of tomatoes. How many pounds of tomatoes does the restaurant have?

1 pound

1 ounce

__1 56 × 1 oz = 56 ×     lb 16 56 × 1 1 5 6 ×  __  = _____   16 16 56 __   = 16 __8 3     = 16 1_ 3   = 2

The restaurant has 3    pounds of tomatoes. 2

What does the problem ask us to do? We need to convert 56 ounces to pounds. Display the tape diagram. Gesture to the part labeled 1 ounce.

1 pound

1 ounce What is an equation that represents how many ounces are in 1 pound? 16 ounces = 1 pound

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

What is an equation that represents what fraction of a pound is 1 ounce? 1 ounce = __   1   pound 16

What do these two equations have in common? They show equal weights. The number of pounds is less than the number of ounces for the same weight. The number of ounces is 16 times the number of pounds in each equation. The number of pounds is __    the number of ounces in each equation. 1 16

1 16

In each equation we can see the number of pounds is _   feet the number of ounces. What is true about the number of pounds equal to 56 ounces?

The number of pounds is __    times as much as the number of ounces. 1 16

1 16

Because the number of pounds is _   times as much as the number of ounces, is the

Some students might be tempted to skip

number of pounds greater than or less than the number of ounces? Why?

writing an equation with multiplication

The number of pounds is less than the number of ounces because __    is less than 1. 1 16

Why does it make sense that a lesser number of pounds is the same weight as a greater number of ounces? It makes sense because a pound is heavier than an ounce, so you need fewer pounds to equal the same weight in ounces. About how many pounds equal 56 ounces? How do you know?

56 ounces is about 3 pounds because I know 16 × 3 = 48. 56 ounces is between 3 and 4 pounds because 16 × 3 = 48 and 16 × 4 = 64. Direct students to write an equation to use to convert 56 ounces to pounds. What equation can help us convert 56 ounces to pounds? 56 × 1 oz = 56 × __  1   lb 16

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Teacher Note

expressions and instead use 56 ÷ 16 to convert 56 ounces to pounds. Encourage students to write the equation with multiplication expressions so they can practice using multiplication to convert units. Multiplication emphasizes the equivalence of two different measurements, such as

1 oz =   1 lb, so students can clearly see and 16

practice finding equivalent measurements. Students may discover it is simpler to avoid mistakes by using this method, and once they have written their multiplication expressions, they can always interpret the product of a whole number and a fraction as a division expression later in their work.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Direct students to work with a partner to finish converting 56 ounces to pounds. Invite students to turn and talk about whether their answer is reasonable and how they know. Then have partners complete problems 2 and 3. Circulate as students work, and provide support by asking the following questions, as needed: • Which measurement units are you converting between? • Are you starting with a larger or a smaller measurement unit? • How can you use the tape diagram to write the multiplication equation you can use to convert?

Differentiation: Support

• What was your estimate? How did you estimate?

Consider highlighting equivalent units in the equations for each problem.

• Is your answer reasonable? Why?

Problem 2: This reminds students

2. A family planning a vacation wants to rent a cabin for 35 days. The cabin can only be rented by the week. For how many weeks must the family rent the cabin?

1 day =   1 week. 7

35 days = 35 × 1 day = 35 ×

1 week

= 1 day

1 week 7

35 weeks 7

Problem 3: This reminds students

1 35 × 1 day = 35 ×  7 week × 35 1_ 1_____ 35 ×   = 7 7 35 __ = 7

1 pint =   1 gallon. 8

2 pints = 2 × 1 pint =2× =

1 gallon 8

2 gallons 8

The family must rent the cabin for 5 weeks.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

3. A recipe needs 2 pints of milk. How many gallons of milk does the recipe need?

1 gallon

Differentiation: Challenge Consider revising problem 2 so the number of days is not equal to a whole number of weeks.

__3

For example, 38 days = 5   weeks. If the family 7

1 pint

has to rent by the week, this means they need

2 pt = 2 × 1 pt

1_ = 2 ×   gal 8 2____ ×1 = gal 8

2_ The recipe needs gallons of milk. 8

2_ =   gal 8

to rent for 6 weeks. Consider revising problem 3 so it requires converting twice when using the reference sheet. For example, a recipe needs 4 cups of milk. How many gallons of milk does the recipe require? In this instance, students need to consider how many cups are in 1 pint and then how many cups are in 1 gallon.

Consider inviting students to share their solutions with the class.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

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EUREKA MATH2 California Edition

Land

5 ▸ M3 ▸ TA ▸ Lesson 6

Debrief 5 min Objective: Convert smaller customary measurement units to larger measurement units. Facilitate a class discussion about converting smaller measurement units to larger units by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display 48 oz = 48 × __     l b =  __  lb = 3 lb. 48 16

1 16

Compare 3 and 48.

3 is less than 48. Does it make sense that the measurements are equivalent given that the number 3 representing pounds is less than the number 48 representing ounces? Why? Yes. It makes sense because a pound is heavier than an ounce, so there should be fewer pounds to equal the same weight. Yes, because 1 ounce = __     pound, so 48 ounces is 4 8 × __     pound. __    is less than 1, 1 16

1 16

so 4 8 × __    is less than 48. 1 16

1 16

How does the product 48 × _   , or 3, compare to _   ? 1 3 is greater than __     . 16

1 16

Does it make sense that the product 48 × _   is greater than _   ? Why? Yes, because 48 is greater than 1, so 4 8 × __    is greater than __     . 1 16

1 16

Can we use multiplication to convert smaller measurement units to larger measurement units? How? Yes. We can use the relationship between the small and large units to write an equation and multiply to find the amount of the large units.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Display the following table. Convert from Small to Large

Convert from Large to Small

48 × 1 oz = 48 × __     lb =  __  lb = 3 lb

3 × 1 lb = 3 × 16 oz = 48 oz

48 oz = 3 lb

3 lb = 48 oz

1 16

48 16

Invite students to think–pair–share about the following question. What is different about the second factor of the second expression in these conversions? When converting from small to large units, the second factor is a fraction less than 1. When converting from large to small units, the second factor is a number greater than 1. Thinking about what you understand about multiplication and about converting units, why does this work? Give an example to support your thinking. Pounds are heavier than ounces, so you need fewer pounds to equal the same amount of weight. We can multiply by a fraction less than 1 to find the number of pounds. Ounces are lighter than pounds, so you need more ounces to equal the same amount of weight. We can multiply by a number greater than 1 to find the number of pounds.

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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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

9. Leo makes a mistake when he converts 11 inches to feet. Consider Leo’s work.

11 in = 11 x 1 in = 11 x 12 ft = 132 ft

Convert each measurement.

1 yard

a. What mistake does Leo make?

__1

Leo multiplied by 12 feet instead of multiplying by 12 foot.

1. 1 ft =

1 foot

_ 1 3

2. 9 ft =

3. 14 ft =

4 _3 2

b. How can Leo use an estimate to help him check the reasonableness of his answer? Leo can think about how 11 inches is less than 1 foot because there are 12 inches in 1 foot. That would help him see his answer of 132 feet is not reasonable.

c. Show Leo how to correctly convert 11 inches to feet.

1 foot

11 in = 11 × 1 in

= 11 × __ ft 12 1

4. 1 in =

6. 24 in =

__ ft = 12 11

1 inch

5. 6 in =

1 2

7. 18 in =

1 _2

ft 1

Convert each measurement.

1 pound

1 ounce

8. A snake is 30 inches long. How many feet long is the snake?

10. 1 oz =

1 16

11. 4 oz =

1 4

12. 48 oz =

13. 19 oz =

1 __ 16 lb

lb 3

The snake is 2 2 feet long. 14. Sasha mails a package that weighs 54 ounces. How many pounds does the package weigh? __6 The package weighs 3 pounds. 16

PROBLEM SET

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TA ▸ Lesson 6

Convert each measurement.

1 gallon

15. 3 qt =

3 4

17. 1 pt =

1 2

gal

16. 5 qt =

1 _4

gal

18. 3 pt =

1 _2

20. 9 c =

4 _2

1 quart 1 quart

1 pint 1 pint 19. 4 c =

1 cup

21. Yuna makes 15 cups of lemonade. How many pints of lemonade does she make?

Yuna makes 7 2 pints of lemonade.

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PROBLEM SET

Topic B Multiplication of Fractions In topic B, students use their understanding of multiplying a whole number by a fraction to multiply two fractions. Students begin by multiplying a fraction less than 1 by a unit fraction. They use number lines and area models to help them find the products. Students continue to use these models as they multiply non-unit fractions by fractions less than 1. They explore the relationship between the size of the product and the size of the factors. When students encounter a product involving two fractions, they use what they know about multiplying with unit fractions and fractions less than 1 to make a simpler problem. Students apply methods such as using known products and unit language to multiply. With the known product method, students express fractions as a product of a whole number and a unit fraction. Multiplying with unit fractions can be simpler than multiplying with fractions because students can identify the fractional unit of the product and then multiply

that number by a whole number. For example, to find the product _     × _     , students express _  1 4

3 5

as _   × 3. Then they find ( _     ×  _ )× 3 = __   × 3 = __   . When they use the unit language method, 1 5

1 4

1 5

1 20

3 20

students analyze multiplication expressions to see whether the fractional unit of one factor is related to the numerator of the other factor. For example, students interpret _   × _   as

_ 1 of 3 fourths and then recognize _ 1 of 3 is 1, so _ 1 of 3 fourths is 1 fourth. 3

1 3

3 4

Students continue to use these methods to multiply fractions greater than 1 by fractions. They use reasoning to draw conclusions about the size of products when they multiply two fractions and to compare multiplication expressions involving fractions without evaluating. In topic C, students use their knowledge of multiplying fractions to divide unit fractions and whole numbers.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB

Progression of Lessons Lesson 7

Lesson 8

Lesson 9

Multiply fractions less than 1 by unit fractions pictorially.

Multiply fractions less than 1 pictorially.

Multiply fractions by unit fractions by making simpler problems.

1 4

2 4

3 4

4 4

To multiply a unit fraction and a fraction less than 1, I can partition a number line or an area model to help me find the product.

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5 7

2 5 4 5 When I multiply fractions less than 1, I can partition a number line or an area model to help me find the product.

1 5 of 7 5 I can use what I know about multiplying a whole number by a fraction to help make simpler problems. I can use unit language to help find products when I notice a relationship between the denominator of one fraction and the numerator of the other fraction.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB

Lesson 10

Lesson 11

Multiply fractions greater than 1 by fractions.

Multiply fractions.

36 _4 × _9 = _1 × 1_   × (4 × 9) = __  1 × 36 = __ 5

I can use known products and unit language to multiply fractions greater than 1 by fractions. I can check my answer by using an area model.

17 3 5 17 x < x 20 5 5 20 17 Both expressions have 20 as a factor and 53 17 17 3 x 5 is less than 55 x 20 . is less than 55 , so 20

When I multiply a number by a fraction less than 1, the product is less than the number. When I multiply a number by a fraction greater than 1, the product is greater than the number. I can use similar reasoning to help me compare multiplication expressions without evaluating them.

135

5 ▸ M3 ▸ TB

EUREKA MATH2 California Edition

Orally reason about the size of the product of two fractions by comparing the size of the product to the sizes of both factors. Orally describe the process for representing the product of two fractions on a tape diagram or an area model and make connections between the models.

Lesson 8

Compare the size of a product to the sizes of both its factors and orally reason about why the product of two fractions less than 1 is always less than either factor.

Lesson 9

Orally justify the choice of a method for making a simpler multiplication problem, such as known products or unit language, by describing the relationships between the two fractions.

Lesson 10 In partners and in class discussion, orally draw conclusions about whether the product of two fractions will be greater than, equal to, or less than 1 based on the size of the factors. Lesson 11 Orally generate rules about the size of the product of an expression when a number is multiplied by a fraction less than 1, equal to 1, or greater than 1. Orally and in writing, compare expressions involving fractions without evaluating the expressions, and justify reasoning by describing the size of the fractions in each expression.

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LESSON 7

Multiply fractions less than 1 by unit fractions pictorially.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Name

Date

1. Use the number line to find the product. Then complete the equation.

1 5

2 5

3 5

4 5

5 5

1 1 1 _____ × = 3 5 15

_ _

• Are number lines and area models helpful for finding the product of two fractions? How?

_1 × 2_ = _____ 3

Students begin by wondering how to find a fraction of a fraction. They compare a word problem to an area model that represents the multiplication of a fraction less than 1 by a unit fraction. Students use analysis and their understanding of the area model to use an area model to multiply two fractions. They also practice multiplying two fractions by using a number line. Throughout the lesson, students reason about the size of the product compared to the sizes of the factors.

Key Question

2. Draw an area model to find the product. Then complete the equation.

Lesson at a Glance

Achievement Descriptors 5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD8 Compare the effects of multiplying by fractions and

1 4

whole numbers. (5.NF.B.5.a) 2 3

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Interpret a Model

• None

• Model Fraction Multiplication • Use a Number Line • Problem Set

Land 10 min

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Fluency

Happy Counting by Fourths Students visualize a number line while counting aloud to maintain fluency with counting by fourths and renaming fractions greater than 1 as whole or mixed numbers. Invite students to participate in Happy Counting. Let’s count by fourths. Today we will rename the fractions as whole or mixed numbers when possible. The first number you say is 2 fourths. Ready? Signal up, down, or stop accordingly.

Teacher Note

__2

This sequence starts at   instead of at 0 4

to give students an opportunity to extend the count sequence with mixed numbers a

2 4

3 4

2 4

3 4

little farther.

Continue counting by fourths within 4. Change directions occasionally, emphasizing crossing over whole numbers and where students hesitate or count inaccurately.

Choral Response: Equivalent Fractions Students determine an unknown numerator or denominator to build fluency with renaming a fraction with a larger unit. 1 2 Display _  =  _____   . 4

What is the unknown equivalent fraction? Raise your hand when you know.

2 1 = 4 2

Wait until most students raise their hands, and then signal for students to respond. 1_ 2 140

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Display the completed equation. Repeat the process with the following sequence:

2 1 = 6 3

1 3 = 2 6

4 2 = 6 3

1 3 = 3 9

2 6 = 3 9

Whiteboard Exchange: Multiply a Whole Number by a Fraction Students find a fraction of a whole number by using a number line to prepare for multiplying unit fractions by fractions less than 1. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the number line and statement. Draw the number line and write the statement.

Find the value by using the number line. Write the answer as a whole number when possible.

1 2 of 2 is 3 3 .

Display the shading on the number line and the completed statement. Repeat the process with the following sequence:

1 2 of 2 is or 1 . 2 2

2 6 of 3 is or 2 . 3 3

3 6 of 2 is 4 4 .

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Launch

Students use what they know about multiplying a whole number by a fraction to reason about how to multiply a fraction by a fraction. 1 2

What is _ of 2?

1 2

What equation shows _  of 2 is 1?  _ × 2 = 1 1 2

1 2

How do you know _ of 2 is 1?

2 units partitioned into 2 equal groups means that there is 1 unit in each group. 1 2

What is _ of 1?  _  1 2

1 2

What equation shows _ of 1 is _ ?  _ × 1 =  _  1 2

1 2

How do you know _ of 1 is _ ?

1 unit partitioned into 2 equal groups means there is  _12 unit in each group. 1 2

1 2

We know _ of 2 and _ of 1. Have you ever thought about _ of _ ? Do you think _ of _ 1 2

is greater than or less than _ of 1? Why?

I think  _  of  _  is less than  _  of 1 because we are finding a fraction of a smaller number. 1 2

1 2

Invite students to think–pair–share about how they might find  _  of _ . 1 2

1 2

I would draw a tape diagram that shows _ and then partition the _ in half. 1 2

1 2

I would draw a number line that shows _ and then partition the number line into 2 equal parts to find _ of _ . 1 2

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1 2

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Direct students to sketch a model they think represents _ of _ . Have students keep their 2 2 sketch because they return to it in Learn. 1

Transition to the next segment by framing the work. Today, we will find the product of a fraction multiplied by a unit fraction.

Learn

Interpret a Model Students interpret a model that represents fraction multiplication. Direct students to problem 1 in their books. Read the problem chorally. Then invite students to think–pair–share about what they notice and wonder about the problem. I notice an area model shows the garden. I notice part of the model is shaded. I notice the area model is labeled with two fractions. I notice not all the units are the same size. I wonder why the units are not all the same size. I wonder how the model shows the story. I wonder which operation this model shows.

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EUREKA MATH2 California Edition

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1. Mr. Evans plants flowers in _ of his garden. _ of the flowers are roses. What fraction of the 5 3 garden is roses? 2

Garden

Roses

1 3 2 5 Flowers

2 __    of the garden is roses. 15

Facilitate a discussion about how the model represents the problem by using the following prompts. The problem says Mr. Evans plants flowers in his garden. Which part of the model represents the garden? The whole square 2

The problem says Mr. Evans plants flowers in _ of his garden. Where do you see _ 5 5 represented in the model? The first 2 columns represent  _ . 2 5

Gesture to the first 2 columns on the model. 2 5

If the label were not there, how would we know this is _ ?

The whole square is partitioned into 5 columns, so 2 of the columns represent  _ .

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2 5

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

1 3

The problem says _ of the flowers are roses. Where is that represented in the model? How do you know?

The shaded parts represent the roses. The 2 columns are each partitioned into 3 equal parts, and 1 of the parts in each column is shaded. The problem asks us to find the fraction of the garden that is roses. The shaded part 1 3

2 5

of the model represents _ of _ . What fraction do you think they drew first? Why?

I think they drew  _ first because the story says  _ of the garden is flowers. They needed 5 5 to show the flowers before the roses because the roses are part of the flowers. 2

1 3

2 5

What expression matches finding _ of _ ? 1 2  _  ×  _  3 5

1 3

2 5

How can we use the model to find the product of _ × _ ? We can make all the parts the same size by partitioning each of the remaining fifths into 3 equal parts. Guide students to partition the remaining fifths into thirds. How many equal-size parts are in the model?

15 What fraction of the model is shaded now? 2  __  15

2 15

So, we can say _   of the garden is roses. Direct students to record the answer. What multiplication equation represents the problem? 2 1 2  _  ×  _  =  __  15 3 5

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5 ▸ M3 ▸ TB ▸ Lesson 7

What do you notice about the size of the product compared to the size of each of the factors? How do you see that in the model?  __  < _ . I can see that in the model because 1 row shows _ , and the shaded part is less 2 15

1 3

than that.

 __  <  _  . I can see that in the model because 2 columns show  _ , and the shaded part is less 2 15

2 5

than that.

Does it make sense that the product is less than  _ ? Why? 2 5

Yes, it makes sense.  _  ×  _  means we are finding _ of  _ . Because _ < 1, that means _ of _

2 1 1 2 5 3 3 5 2 2 2 is only part of , not all of it. So it makes sense that     <     . 15 5 5 1 Does it make sense that the product is less than    ? Why? 3 1 2 2 1 Yes, it makes sense. We can think of   ×   as   ×   because we can multiply in any order. 3 5 5 3 2 1 2 1 2 2   ×   means 5  of  3 . Because 5 < 1, it means 5  of  13 is only part of  13 , not all of it. 5 3

_ _

1 3

2 5

1 3

_ _

_ _ _ _

__ _

Display the following area models and have students name an expression that is represented by the model. Students do not need to evaluate the expression.

146

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5 ▸ M3 ▸ TB ▸ Lesson 7

Area Model

Expression

1 2  _  ×  _  3 3

1 3

1 2  _  of  _  3 3

2 3

1 2  _  ×  _  4 3

1 4

1 2  _  of  _  4 3

2 3

1 2  _  ×  _  2 3

1 2

1 2  _  of  _  2 3

2 3

_1 _1

Invite students to return to their sketch of     of   to confirm their sketch is accurate or to revise it. 2

147

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Model Fraction Multiplication Students model fraction multiplication by using an area model.

_1 _1

Direct students to problem 2. Have students return to their sketch of     of   . If students think 2

_1 _1 their model shows     of   , consider having those students share their sketches with the class. 2

Validate a range of ideas but guide the class toward using an area model. 1 2

1 2

Let’s show _ of _ by using an area model.

_1 _1

Invite students to work with a partner to draw an area model that shows of . Circulate 2 2 to analyze students’ models. Guide the facilitation of problem 3 according to any misconceptions or common errors that you observe. Draw an area model to find the product.

_1 _1 2.     ×     = 2 2

 _4 1

1 2 1 2 1 2

1 2

What did you notice or discover while using an area model to find _ of _ ? I noticed both factors are the same, so I partitioned the area model in half vertically, and then I partitioned it in half horizontally. I noticed the product  _  <  _ . 1 4

1 2

Direct students to problem 3.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

What is the same or different about problem 3 compared to problem 2? The factors are not the same in problem 3. Problem 3 is also a multiplication problem with two fractions as factors, so we are finding a fraction of a fraction.  __ 

_1 _3

3 20

3.     ×     = 5

1 4 3 5 1 4

3 5

What does _ × _ mean? It means  _ of  _ . 1 4

3 5

Let’s draw an area model to help us find the product. We are starting with _ and 1 4

3 5

we need to find _ of _ . How can we represent _ on an area model? We can partition the model into 5 equal parts and label 3 of them.

Teacher Note

Guide students to draw, partition, and label an area model. 1 4

3 5

Can we use the model to find _ of _ ? How?

Some students may realize they can partition the entire area model at the same time as 3 they partition   to make equal parts.

We can partition each of the 3 fifths into 4 equal parts and then

shade  _ of each fifth. 1 4

Partition the   into 4 equal parts and shade   of each fifth. Direct 5

students to do the same.

1 4

3 5 149

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Are we ready to use the model to find the product? Why? What can we do? Not yet because the model does not show each fifth partitioned into equal parts. We can use the model to find the product if we partition each of the remaining fifths into 4 equal parts so that the entire model shows equal parts. Guide students to partition the remaining fifths into 4 equal parts. How many equal parts does our model show now?

20 1 4

3 5

What is _ × _ ? How do you know?

 __  because the model shows 3 shaded parts out of 20 total parts. 3 20

1 4

3 5

3 20

Is it reasonable that _ of _ is _   ? Why?

I think it is reasonable.  _ < 1, so we are finding part of _ , not all of it.  __ <  _ , so __ makes sense.

1 4

3 20

3 5

3 20

Yes, that makes sense.  _ of _ must be less than _ , and  __ <  _ . 1 4

3 5

3 20

3 5

Direct students to record the answer.

Use a Number Line Students find the product of a unit fraction and a fraction less than 1 by using a number line. Write the expression  _  ×  _  . 1 4

1 3

Invite students to turn and talk to compare  _  ×  _ to the expressions they saw earlier:  _  ×  _  4 3 4 5 1 1 and  _  ×  _  . 2

1 4

1 3

What does the expression _ × _ mean? It means _ of _ . 1 4

1 3

We can also show a fraction of another fraction by using a number line.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Draw the number line. What does the number line show? Why?

1 3

2 3

3 3

3 The number line shows 0 to  _ , so it shows 1, like the area model shows us 1 whole. 3 3 1 1 I think it shows 0 to  _ because we are finding  _  of  _ , so it is helpful to have all 3 thirds, like 3

we showed in the area model.

1 1 Can we use the number line to represent the expression _ of _ ? How? 4 3

Teacher Note Some students may realize that they can partition the entire number line at the same 1 time as they partition   to make equal parts.

__ 3

The number line already shows  _ . We can partition the interval from 0 to  _  into 4 equal 1 3

parts. Then we can shade 1 of the parts to show  _  of  _ . 1 4

1 3

Language Support

Partition and shade the number line to show  _  of  _ . 1 4

1 3

Are we ready to use the number line to find the product? Why?

2 3

3 3

Consider posting a visual to reinforce the language used to describe multiplication expressions.

1 1 4 x 3

No, the number line does not show equal parts for each third, so it is not clear how many units are in 1.

1 1 4 of 3

No, we need to partition the remaining thirds into 4 equal parts so that the entire number line shows equal parts. Partition the rest of the thirds into four equal parts.

•

1 3

What fraction is shaded on the number line? How do you know?

It is  __ . I know because there is 1 part shaded out of 12 parts. 1 12

1 12

1 4

1 12

1 3

2 3

3 3

Find a fourth of _  1 . 3

_1 • Start with and partition it 3

into 4 equal parts to find 1 of the parts.

Is _   greater than, equal to, or less than _ ? 1 1  __   <  _  12 4

Is _   greater than, equal to, or less than _ ? 1 1  __   <  _  12 3

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7 1 12

1 4

1 3

1 12

The product _   is less than both factors. Is it reasonable that _   of _ equals _   ? Why? Yes, it is reasonable.  _  of  _  is only a part of  _ , not all of it.

1 1 1 4 3 3 1 1 Yes, it is reasonable.    < 1, so it makes sense that our answer is less than     because 3 4

we are finding a part that is less than the whole.

Yes, we are finding a fraction of another fraction.  _ < 1, so a fraction of  _  should be a 3 3 smaller number. 1

1 4

2 3

How would the product change if we were finding _   of _ ? How do you know?

We would shade another part between  _  and  _  so the product would increase. 1 3

2 3

The product would be twice as much because we would have twice as many parts shaded. The product would increase because we are starting with  _ instead of  _ . 2 3

1 3

Direct students to problem 4. Encourage them to use the number line and the area model to find the product. As students work, circulate and ask the following questions to advance students’ thinking. • What does  _  ×  _  mean? 3 4

• What fraction will you draw first? Why?

• How do you see _  on the number line? What can you draw to represent _  on the area model? 3 4

3 4 1 3 1 3 • How can you show of on the number line? How can you show of on the area model? 2 4 2 4

_ _

If students need additional support with problem 4, ask questions to guide them in identifying which number to start with. Then 3 provide students with a physical cutout of   . 4 Encourage students to fold the papers horizontally and use the creases to help them 1 3 see   of   . Then students can transfer this 2 4 onto the model in their books.

Invite students to turn and talk about how they can represent a fraction of a fraction on a number line.

1 2

Differentiation: Support

_ _

Find the product by using the area model and the number line. 3 _1 _3  _8 4.     ×     = 2 4

__ __

Promoting the Standards for Mathematical Practice When students use an area model and a number line to represent multiplying a fraction less than 1 by a unit fraction, they are attending to precision (MP6). Ask the following questions to promote MP6: • When you use a number line to show

__1 __3

1 4

2 4

3 4

4 4

  ×   , with which steps do you need to be 2

1 2 3 4

152

precise? Why? • What details are important to think about when you represent fraction multiplication by using an area model?

EUREKA MATH2 California Edition 1 2

5 ▸ M3 ▸ TB ▸ Lesson 7

3 4

What is _   × _  ?  _  3 8

UDL: Representation

Invite students to share their number lines and area models or display the following samples.

1 4

2 4

3 4

4 4

Consider creating a chart that shows the equation from problem 4 with both the area model and the number line as a reference for students when they analyze factors and compare the size of the factors to the size of the product.

1 3 3 x = 2 4 8

1 2 3 4

Unit Fraction Factor

Fraction Factor

Product

Have students think–pair–share as they compare the number line and the area model. What is the same and different about the representations? 1 3 They both show _ of _ . 2

They both show 1. The number line goes to  _ and the area model represents 1 whole. They both represent the factor _ first. 3 4

4 4

1 4

2 4

3 4

4 4

1 2 3 4

They both show _ of _ is 3 shaded parts out of 8 equal parts. 1 2

3 4

The number line shows the 3 shaded parts separated. So the number line shows 1  _ of each fourth. 2

The area model shows the shaded parts together. You can still see  _ of each fourth 2 because each column is partitioned in half, but the shaded parts are together. 1

Invite students to turn and talk about whether they prefer to use a number line or an area model to find a fraction of a fraction and why.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex. © Great Minds PBC

153

5 ▸ M3 ▸ TB ▸ Lesson 7

Land

EUREKA MATH2 California Edition

Debrief 5 min Objective: Multiply fractions less than 1 by unit fractions pictorially. Facilitate a class discussion about multiplying fractions less than 1 by unit fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display Thirteen Rectangles, 1930, by Wassily Kandinksy. This painting is called Thirteen Rectangles. The artist who painted it is named Wassily Kandinksy. Use the following questions to help students engage with the art: • What do you notice about the painting? • What do you wonder? Guide students to think about the painting in terms of their experience with multiplying a fraction less than 1 by a unit fraction. What do you notice about the shapes in the painting? They are all rectangles. Some of the rectangles appear to be squares. Some of the rectangles and squares overlap.

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5 ▸ M3 ▸ TB ▸ Lesson 7

What thoughts about multiplying fractions less than 1 by unit fractions come to mind when you look at this piece of art? I wonder what fraction of the red square is covered by the yellow square. Could we partition the red square to find how much of it is covered by the yellow square? Are number lines and area models helpful for finding the product of two fractions? How? Yes. I can partition and shade a number line or an area model to help me see both fractions and how the product is part of a part. Yes. I can see the product on a number line or an area model by counting the total parts and the shaded parts. As time allows, use the following questions to deepen students’ exploration of the art: • Some people may see a figure such as a person dancing when they look at the painting. Do the shapes seem to form a figure when you look at them all together? • What shape do you first see when you look at the painting? Where does your eye move to next?

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

Draw a model to find the product. 1 1_ 7. _ × = 2 5

__1 10

1 3_ 8. _ × = 2 5

__3 10

Use the number line to complete the multiplication equation. 1. 1_ of 1_ 2 3

1 of 1_ 2. _ 3 4

1 3

2 3

3 3

1 4

_ _ _____ 1 1 × = 2 3

2 4

3 4

1 2

4 4

1 5

_ _ _____ 1 1 × = 3 4

1 2

1 2_ 9. _ × = 3 3

3 5

1 3_ 10. _ × = 2 4

_ 2 9

_ 3 8

Complete the area model to find the product. Each diagram represents 1. 4.

_ _

1 2

_ 2 8

3 4

2 3

2 4

1 4

11. 1_ × 1_ = 4 4

12. 3_ × 1_ = 5 4

__1 16

__3 20

_ × _1 = __1

1 4

4 1 3

156

_____ × 2_ =

1 8

1 1 × = 2 4

1 2

1 3

1 × 4

1 4

_ 2 3

__2

3 5

1 4

2 3

PROBLEM SET

1 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

15. _4 of the food items on the lunch table are sandwiches. _2 of the sandwiches are peanut butter and

13. Adesh needs to find _1 of _3 . He estimates the answer is about 1 because he thinks the value is

5 4 _1 greater than _3 . Is Adesh correct? Explain. 5

No, Adesh is not correct. His answer is less than _3 because he is finding _1 , or a part, of _3 . 5

the rest are cheese. What fraction of the food items on the lunch table are cheese sandwiches?

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 7

_ _ __

1 4 × = 4 3 5 15 4 of the food items on the lunch table are cheese sandwiches. 15

Use the Read–Draw–Write process to solve each problem.

14. Jada has _3 gallons of lemonade. She pours _1 of the lemonade into a bottle. How many gallons 4

of lemonade does Jada pour into the bottle?

_1 × 3_ = __3 6

3 Jada pours __ gallons of lemonade into the bottle. 24

PROBLEM SET

157

LESSON 8

Multiply fractions less than 1 pictorially.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Name

Date

1. Show the product by using a number line. Then complete the equation.

1 6

2 6

3 6

4 6

5 6

6 6

15 3 5 _____ × = 4 6 24

_ _

Lesson at a Glance Students use number lines and area models to help them find the product of a non-unit fraction and a fraction less than 1. Students choose a method to use to help them multiply and discuss why they prefer that method. They reason about the size of the product compared to the size of the factors and determine that the product of two fractions less than 1 is less than either factor.

Key Question • When you multiply two fractions that are less than 1 , what can you conclude about the size of the product compared to its factors?

2. Show the product by using an area model. Then complete the equation. 6 2 3 _____ × = 5 4 20

_ _

Achievement Descriptors 5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD8 Compare the effects of multiplying by fractions and

whole numbers. (5.NF.B.5.a) 2 5

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

3 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Use a Number Line

• None

• Use an Area Model • Choose a Method • Reason About the Size of the Product • Problem Set

Land 10 min

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5 ▸ M3 ▸ TB ▸ Lesson 8

Fluency

Happy Counting by Fifths Students visualize a number line while counting aloud to maintain fluency with counting by fifths. Invite students to participate in Happy Counting. Let’s count by fifths. The first number you say is 0 fifths. Ready? Signal up or down accordingly.

0 5

1 5

2 5

3 5

4 5

3 5

2 5

3 5

4 5

5 5

6 5

5 5

6 5

7 5

8 5

7 5

Teacher Note The stop signal has been removed because students are now more familiar with the Happy Counting routine. Listen to student responses and be mindful of errors, hesitation, and lack of full-class participation. If needed, adjust the tempo or sequence of the numbers. Consider using the stop signal when necessary to manage the pace or accuracy of the count sequence.

Continue counting by fifths within  __ . Change directions occasionally, emphasizing where 15 5

students hesitate or count inaccurately.

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5 ▸ M3 ▸ TB ▸ Lesson 8

Choral Response: Equivalent Fractions Students determine an unknown numerator or denominator to build fluency with renaming a fraction with a larger unit. 1 3 Display _  =  _____   . 9

What is the unknown equivalent fraction? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. 1_ 3

3 1 = 9 3

Display the completed equation. Repeat the process with the following sequence:

6 2 = 9 3

1 2 = 8 4

2 4 = 3 6

6 3 = 8 4

2 4 = 5 10

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Whiteboard Exchange: Multiply a Whole Number by a Fraction Students find the product by using a number line to prepare for multiplying non-unit fractions by fractions less than 1. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the number line and equation. Draw the number line and write the equation.

Find the product by using the number line. Write the answer as a whole number when possible.

1 ×2= 2 4 4

Display the shading on the number line and the completed equation. Repeat the process with the following sequence:

1 ×4= 4 = 2 2 2

162

1 2 ×2= 4 3 3

3 ×3= 9 4 4

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TB ▸ Lesson 8

Students identify an error and provide feedback on how to correct the error. Introduce the Critique a Flawed Response routine and present the following multiplication equation and sample student work. Give students 1 minute to identify the error. Invite students to share. The student did not partition the entire area model to make equal-size parts. Because  __  =  _  and  _ < 1, it does not make sense 1 4 4 5 5 20 1 4 1 1 that     of     is    . It should be less than   . 5 5 5 5

_ _ _

The student found  _  of  _  instead of  _  of  _  . 1 5

4 4

1 5

4 5

1 4 4 × = 5 5 20 1 5 4 5

Give students 2 minutes to find the correct product based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about creating equal-size parts to find the product.

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Then facilitate a class discussion. Invite students to share their solutions with the whole group. Lead the class to consensus about the best way to correct the flawed response. We need to partition the remaining fifth in the area model into 5 equal parts. We can use a number line to show the product.

1 4 4 × = 5 5 25

1 5

4 5

1 5

2 5

3 5

1 4 4 x = 5 5 25

4 5

5 5

Transition to the next segment by framing the work. Today, we will use number lines and area models to find the product of fractions less than 1.

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EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TB ▸ Lesson 8

Use a Number Line Students find the product of two fractions less than 1 by using a number line. Display the following expressions.

Previous Lesson

This Lesson

1 1 4×3

2 3 3×4

1 3 2×4

3 2 4×5

Here are two products that are similar to those we found in a previous lesson and two products we will find in this lesson. Invite students to think–pair–share about how the expressions in this lesson are different from the expressions in a previous lesson. In the expressions from a previous lesson, the first factor is a unit fraction because it has a numerator of 1. The expressions in this lesson do not have numerators of 1. Will that change how we represent the expressions with area models or number lines? How? Yes, because we will need to shade more than 1 part of the other fraction. 2 3

3 4

What does the expression _   × _   mean? It means  _  of  _  . 2 3

3 4

Let’s draw a number line to help us find the product. What should we draw first? Why? We should draw a number line from 0 to 1 partitioned into fourths. We are finding  _  of  _  , so we need to show fourths.

2 3

3 4

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5 ▸ M3 ▸ TB ▸ Lesson 8

Draw the number line. Invite students to turn and talk about what they notice about the number line.

1 4

2 4 2 3

3 4

4 4

3 4

How can we use this number line to represent the expression _   of _  ?

The number line already shows  _ . We can partition each of the 3 fourths into 3 equal parts. 3 4

Then we can shade 2 of the parts in each of the fourths to show  _  of  _  . 3 4

2 3

1 4

2 4

The digital interactive Fraction of a Fraction on a Number Line supports students in composing the parts of each fractional interval to find the fraction of the fraction. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

Partition and shade the number line. Are we ready to use the number line to find the product? Why?

UDL: Representation

3 4

4 4

No, because the number line does not show equal parts for each fourth. No, we need to partition the other fourth into 3 equal parts so that the entire number line shows equal parts. Partition the final fourth into 3 equal parts.

1 4

What is the product? How do you know?

2 4

3 4

4 4

It is  __ . I know because there are 6 shaded parts out of 12 total parts. 6 12

2 3

3 4

6 12

Is it reasonable that _   of _   is equal to _    ? Why?

Yes, it is reasonable.  _  of  _ is only a part of  _  . I know  __  =  _  and  _  <  _  . 2 3

3 4

6 12

1 2

3 4

 _ is less than 1, so it makes sense that our answer is less than  _ because we are finding 3 4 a part that is less than the whole. 2

Direct students to problem 1 in their books. Invite students to work with a partner to use the number line to find the product. Circulate and provide support as needed.

166

EUREKA MATH2 California Edition

3 2 1. _ × _ = 4 5

5 ▸ M3 ▸ TB ▸ Lesson 8

 __ 6 20

Teacher Note

1 5

2 5

3 5

4 5

5 5

When students finish, discuss the product. Then invite students to turn and talk about how using the number line to find the product of two fractions less than 1 is the same or different from how they previously used the number line to multiply fractions.

product they found in Launch to help them

__2 __4 5 5 1 4 __ 4 __ __ __2 __4 Because   of   is   , then   of   is 5 5 5 5 25 2×4 8 4 2 ×  __ =   ____ =  __ . find the product   ×   .

Validate this observation, as it demonstrates students’ understanding of learning from a

__2

Use an Area Model

__1

previous topic (e.g.,   = 2 ×   ). Encourage

Students find the product of two fractions less than 1 by using an area model. 2 4 Display the expression _ × _ . 5 5 2 5

Students may recognize that they can use the

4 5

them to look for similar structures in other problems.

What does _   × _   mean? It means  _  of  _  . 2 5

4 5

Differentiation: Support

__2 __4

Let’s draw an area model to help us find the product. What should we draw first? Why?

Students may say they can find   of   by 5

We should draw  _ first because we want to find  _  of  _  . 4 5

2 5

simply shading 2 of the fifths. Address this

4 5

misconception by covering the bracket and

__4

Guide students to draw, partition, and label an area model.

the label   . Then ask students what shading 5

2 4 Can we use the model to find _   of _  ? How? 5 5

We can partition each of the 4 fifths into 5 equal parts and then shade  _ of each of the 4 fifths. 2 5

2 columns would represent. They should

4 5

realize that shading 2 columns would

__2

__2 __5

represent   of 1, or   of   . Reveal the bracket 5

__4

and the label   and ask students what 5

portion of the model they should shade to

__2 __4 5 5 __2 __4 show   of   , they must shade only a fraction represent   of   . They should realize that to 5

of each of the 4 fifths and not 2 entire fifths.

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Partition the  _  into 5 equal parts and shade  _ of each fifth. 4 5

2 5

Direct students to do the same.

Are we ready to use the model to find the value of 2 4 _   × _ ? Why? 5 5

2 5

No, because the model does not show equal parts. No, we need to partition the remaining fifth into 5 equal parts so the entire area model has equal parts.

4 5

Guide students to partition the remaining fifth into 5 equal parts. How many equal parts does the model show now?

25 Now, are we ready to use the model to find the value of 2 4 _   × _ ? How do you know? 5 5

Yes, the model shows a product of  __ because there are 8 25

2 5

8 shaded squares out of 25 total squares. 2 5

8 25

4 5

Is it reasonable that _   of _   is _    ? Why?

2 4 Yes, it is reasonable.  _ < 1, so we are finding a part of  _  , 5 5 4 8 8 not all of it.  __  <  _  , so  __ makes sense. 25 5 25 4 8 2 4 4 Yes, that makes sense.  _  of  _ must be less than  _  , and  __  <  _  . 5

4 5

Invite students to turn and talk about how using the area model to find the product of two fractions less than 1 is the same or different from how they previously used an area model to multiply fractions.

168

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5 ▸ M3 ▸ TB ▸ Lesson 8

Choose a Method Students select a method to use to find the product of two fractions less than 1. Direct students to problem 2. Give them 2 minutes to solve the problem. Have students show their thinking.

2. Sasha buys a bag of almonds that weighs    pound. She uses    of the bag to make trail mix. 3

How many pounds of almonds does Sasha use to make the trail mix?

  _ × _ =  __ = _ 3 4

2 3

6 12

1 Sasha uses  _ pound of almonds to make trail mix.

Differentiation: Support Consider using the following questions to support students as they make decisions about how to make sense of the problem. • What does the problem tell us? • What does the problem ask us to find?

1 2

• What multiplication expression represents the problem?

Circulate as students work. Identify one or two students to share their thinking. Purposefully choose work that allows for rich discussion about connections between methods they used in the previous problems.

• What can you draw to represent the problem?

Look for students who use different methods such as those shown below. Note that some students might keep their answer as  __ . Any equivalent answer is valid. 6 12

0 3

1 3

2 3

3 3

3 2 6 1 x = = 4 3 12 2

Sasha uses 1 pound of almonds 2

to make trail mix.

3 2 6 1 4 × 3 = 12 = 2 3 4 2 lb of almonds 3

1 pound 2 of almonds to make trail mix.

Sasha uses

169

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Invite students to share their thinking and their work with a partner. Then gather to discuss the problem as a whole group. What did you draw first? Why?

I drew  _ first because we were finding  _  of  _  . The original bag weighed  _ pound, and then 2 3

3 4

Sasha used  _ of it to make trail mix. 3 4

2 3

What multiplication expression matches the story? 3_ 2_ × 4 3 2 3

3 4

Earlier, we found _   × _  . What is different? What is the same?

When we found  _  ×  _  , we drew  _ first because we were finding  _  of  _  . This time, 2 3

3 4

2 3

we drew  _  first because we were finding  _  of  _  . 2 3

3 4

The product is the same.

2 3

1 2

3 4

I noticed some of you recorded the product as _   . Why?

I think it makes more sense to have  _ pound of something than  __ pound of something. 1 2

6 12

Affirm that all equivalent answers are valid. Tell students that sometimes they can use the context of the story to help them decide how to record their answer. Invite students to turn and talk about which method they prefer and why.

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EUREKA MATH2 California Edition

Reason About the Size of the Product Students reason about the size of the product compared to the size of the factors. Present the following statement: The product of two fractions less than 1 is less than both the factors. Use the Always Sometimes Never routine to engage students in constructing meaning and discussing their ideas.

5 ▸ M3 ▸ TB ▸ Lesson 8

Language Support Consider supporting students with the Always Sometimes Never routine with sentence frames for their reference.

Give students 1 minute of silent think time to evaluate whether the statement is always, sometimes, or never true.

The product of two fractions less than 1 is

Have students discuss their thinking with a partner. Encourage them to refer to earlier examples or to come up with new examples to try. Circulate and listen as they talk. Identify one or two students to share their thinking.

For example,

Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide reasons to support their claim. Conclude by coming to the consensus that it is always true that the product of two fractions less than 1 is less than both the factors.

Promoting the Standards for Mathematical Practice

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

(always or sometimes or never) less than both the factors. .

When students reason about the size of a product compared to the size of its factors as they multiply two fractions less than 1, they are constructing viable arguments and critiquing the reasoning of others (MP3). Ask the following questions to promote MP3: • Can you find a situation where it is not true that the product of two fractions less than 1 is less than both factors? • What questions can you ask your partner to make sure you understand their reasoning?

171

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5 ▸ M3 ▸ TB ▸ Lesson 8

Land

Debrief 5 min Objective: Multiply fractions less than 1 pictorially. Gather the class with their Problem Sets. Facilitate a class discussion about multiplying fractions less than 1 by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What is different about the number lines and area models we used today from the models we used in previous lessons? Give an example to support your thinking. We had to shade more parts of the models in this lesson than we had to in previous

lessons. Before today, to find  _  ×  _  we only shaded  _  of  _ . In this lesson, to find  _  ×  _  , 1 3

3 4

1 3

3 4

we shaded twice as many parts because  _ is twice as much as  _  . 2 3

Direct students to problem 6 in the Problem Set.

1 3

2 3

3 4

Teacher Note

What is different about this problem compared to others we have solved? Both factors are the same.

These questions can also be asked about problem 1 in the Problem Set.

3 4

Sasha said she started with the first factor of _   when she drew her model. Riley said 3 4

she started with the second factor of _ . Who is correct? Why? Sasha and Riley are both correct. Both the factors are the same, so it does not matter which factor you draw first.

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5 ▸ M3 ▸ TB ▸ Lesson 8

Write the expression  _ ×  _ . 3 4

2 5

Without finding the product, do you estimate it is greater than _   or less than _  ? How do you know?

It is less than  _  . I know  _ < 1, so we are finding part of  _ , and the product is less than  _  .

3 2 2 2 5 5 5 4 2 2 2 It is less than     . We are multiplying    by a fraction less than 1, so the answer is less than    . 5 5 5

Without finding the product, do you estimate it is greater than _   or less than _  ? How 4 4 do you know? It is less than  _  . I know  _  ×  _  =  _  ×  _  , which means  _  of  _  . Because  _ < 1, we are finding 3 4

2 5

3 4

2 5

3 4

part of  _ , so the product is less than  _  . 3 4

3 4

2 5

3 4

2 5

It is less than  _  . We are multiplying  _ by a fraction less than 1, so the answer is less than  _  . 3 4

3 4 3 3 3 2 2 The product is less than     because     <    , and we are finding     of     . 5 5 4 4 4

_ _

3 4

When you multiply two fractions that are less than 1, what can you conclude about the size of the product compared to its factors? The product of two fractions that are less than 1 is less than either of the factors.

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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5 ▸ M3 ▸ 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 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

Use the area model to find the product. Then complete the equation. Each model represents 1. 4. 2_ × 3_ = 3 4

6 __

5. 2_ × 3_ = 3 5

6 __

Use the number line to find the product. Then complete the equation. 1. 2_ × 2_ = 3 3

4 9

1 3

2 3

3 3

2 3

3 4

_ _

2. 2 × 2 = 3 4

3. 2_ × 2_ = 4 3

174

1 4

2 4

3 4

6. 3_ × 3_ = 4 4

4 4

4 __

9 __

7. 2_ × 2_ = 5 3

3 4

4 __

1 3

2 3

3 3

PROBLEM SET

2 5 3

4 __

2 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

EUREKA MATH2 California Edition

Multiply. Draw a model to find the product. 8. 3_ × 4_ = 4 5

12 20

5 ▸ M3 ▸ TB ▸ Lesson 8

9. 4_ × 2_ = 5 3

12. 3_ × 5_ = 5 6

8 __

13. 3_ × 4_ = 8 5

15 30

3 5

3 8 4 5

5 6 2 3

4 5

_ _

10. 2 × 5 = 3 6

12 40

4 5

3 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

_ _

10 18

11. 3 × 3 = 4 5

Use >, =, or < to compare the expressions. Explain how you can compare the expressions without evaluating them.

9 __

14. 4_ × 2_ 5 7

2_ 7 2 is in both expressions, but the first expression is finding a part of _ 2 , or _ 4 of _ 2, _ Explain: 7 7 5 7 2 4×_ 2. so _ is less than _

15. 3 × 5_ 8

5_ 8 5 5 5 5 _ Explain: 3 × is the same value as 3 groups of _ , so 3 × _ is greater than _ .

3 4

2 3

5 6

3 5

16. 4

3 ×4 4

3 of 4, Explain: 4 is in both expressions, but the second expression is finding a part of 4, or _

3 × 4. so 4 is greater than _

PROBLEM SET

175

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

EUREKA MATH2 California Edition

17. 6_ × 5_ 6 9

5 ▸ M3 ▸ TB ▸ Lesson 8

5 2 of the glue to make blue slime, _ 1 of the glue to make green 20. Tara has _ gallons of glue. She uses _

5 9

5 5 6 Explain: Both expressions are equal to _ because the first expression is _ multiplied by _ ,

which is equivalent to multiplying by 1.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 8

slime, and the rest of the glue to make purple slime. How many gallons of glue does Tara use

to make purple slime?

1 − 3_ = 2_ 5

1 of a candy bar. She told Noah’s sister she could have 18. Noah’s mom told him he could have _ 4

_1 of the rest of the candy bar. Noah says it is not fair because he only gets _1 , but his sister gets _1 . 3

Explain why Noah’s reasoning is not correct.

_ _ __ _

2 5 10 1 × = = 5 8 40 4 Tara uses 1 gallon of the glue to make purple slime. 4

1 of the Noah is not correct because he only compared the fractions to each other. He gets _ 4

3 1 of the remaining candy bar, or _ 1 of _ candy bar. His sister gets _ of the candy bar, which is equal 3

3 1 . They both get the same amount. to __ or _ 12

Use the Read–Draw–Write process to solve each problem.

3 2 of his allowance. Of the money he spends, he uses _ of it to buy a toy. What fraction 19. Scott spends _ 5

of his allowance does Scott spend on the toy?

_ _ __

3 2 × = 6 4 5 20

6 Scott spends __ of his allowance on the toy. 20

176

PROBLEM SET

LESSON 9

Multiply fractions by unit fractions by making simpler problems.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Name

Date

Make a simpler problem by using a known product or unit language. Then multiply. 1_ 7 1. 1_ × 5_ = 5 7 1_ of 5 sevenths is 1 seventh. 5

Lesson at a Glance

Students apply known products such as  _  ×  _  , or  __ , to find the product

1 1 1 5 7 35 1 2 1 3 of related expressions such as     ×     and     ×     by writing expressions 5 7 5 7

_ _

as a product of unit fractions and a whole number. Students use unit language to see relationships between the denominator of one fraction and the numerator of a second fraction. They consider multiplication expressions and decide whether they can make

6 2. 1_ × __ = 3 10

2 __

a simpler problem and how.

1 of 6 tenths is 2 tenths. 3

3. 1_ × 6_ = 9 7

Key Questions • Which problems can you make simpler by using unit language? How do you know?

6 __

_ _

• Which problems can you make simpler by using known products? Why?

1 1 × ×6= 1 ×6= 6 9 7 63 63

Achievement Descriptors 4. 1_ × 4_ = 8 7

4 __

5.Mod3.AD6 Multiply whole numbers or fractions by fractions. (5.NF.B.4)

_ _

1 1 × ×4= 1 ×4= 4 56 56 8 7

5.Mod3.AD8 Compare the effects of multiplying by fractions and

whole numbers. (5.NF.B.5.a) 5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b) © Great Minds PBC

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• None

Learn 35 min

Students

• Use Known Products to Multiply

• Using Known Products (in the student book)

Consider whether to remove Using Known Products from the student books and place inside personal whiteboards in advance or have students prepare them during the lesson.

• Use Unit Language to Multiply • Identify When to Make a Simpler Problem • Problem Set

Land 10 min

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Fluency

Whiteboard Exchange: Add Fractions Students make like units in an addition equation and find the sum to build fluency with adding fractions with unlike units from module 2. 1 1 Display _ + _ = 3 2

Look at the fractional units. Do the fractions have like units? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 1 + = 3 2

5 6

1 1 1×2 1×3 + = + 3 2 3×2 2×3

= 2+3 6

No.

Teacher Note Validate all correct responses that may not be displayed. For example, a student may choose to make units of twenty-fourths to

__5 __1

evaluate   +   instead of units of twelfths as 4

After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

displayed.

Rename both fractions to make the fractional units, or denominators, the same. Show your method. Display the sample method and fractions with like units. Find the sum. Display the sum. Repeat the process with the following sequence:

1 +2 = 9 2 5 10

180

2 + 2 = 14 3 4 12

5 + 1 = 17 4 6 12

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Happy Counting by Fifths Students visualize a number line while counting aloud to maintain fluency with counting by fifths and renaming fractions greater than 1 as whole or mixed numbers. Invite students to participate in Happy Counting. Let’s count by fifths. Today we will rename the fractions as whole or mixed numbers when possible. The first number you say is 0. Ready? Signal up or down accordingly.

1 5

2 5

3 5

4 5

Continue counting by fifths within 3. Change directions occasionally, emphasizing where students hesitate or count inaccurately.

181

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Choral Response: Equivalent Fractions Students determine an unknown numerator or denominator to build fluency with renaming a fraction with a larger unit. 2 1 Display  __ =  _____   . 10

What is the unknown equivalent fraction? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. 1_ 5

2 = 1 10 5

Display the completed equation. Repeat the process with the following sequence:

1 2 = 6 12

182

6 = 3 10 5

2 8 = 3 12

10 = 5 12 6

4 12 = 5 15

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TB ▸ Lesson 9

Students consider different ways to find the products of fractions with very small units. Let’s look at three different expressions and think about how we might find their products.

Write  _ ×  _ . 3 4

1 5

Consider the multiplication expression. Invite students to think–pair–share about how they could find the product. We could use a number line. We would partition the number line into fourths, and then partition each fourth into 5 equal parts. We would shade  _ of each of the 3 fourths 1 5

to get  __  . 3 20

We could draw an area model. We would partition the area model into fourths vertically and fifths horizontally, which makes twentieths. Then we would shade the part of the model that shows  _  ×  _  to see  __  .

Write (  _  ×  _ ) × 3. 1 6

1 5

3 4

3 20

1 4

Consider this multiplication expression. Would you find the product by using a number line, an area model, or in some other way? Why? I would draw a number line or area model to find the product  _  ×  _ , and then multiply 1 6

the product by 3 to get the answer.

1 4

I can visualize the product  _  ×  _ in an area model and on a number line, so I would not 1 6

1 4

draw that part. I would multiply 3 and  __ to find the product  __ because I can multiply 1 24

a whole number and a fraction mentally.

3 24

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Write  __  ×  ___  . 1 93

93 117

Consider the multiplication expression. Would you find the product by using a number line, an area model, or in some other way? Why? I might be able to draw a number line or an area model, but I think I would lose track of how many parts I make. I do not know another way to multiply fractions, but I think drawing with these units would be difficult. I cannot visualize the models or do any mental multiplication with these numbers. Drawing a model is a good way to find the product of two fractions, but for some

Teacher Note

problems drawing a model can be challenging. By the end of the lesson, you will

__1

be able to find the product   ×   93

93 ___ more efficiently than by drawing a model. 117

Transition to the next segment by framing the work. Today, we will make simpler problems by using what we know about unit fractions and unit language to find the product of two fractions.

184

This lesson asks students to consider ways to make simpler problems to find the product of fractions. Students are not asked or expected to write products in what is often referred to as simplified form.

EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TB ▸ Lesson 9

Use Known Products to Multiply Materials—S: Using Known Products

Students use known products to multiply a unit fraction by a fraction. Have students remove Using Known Products from their books and place them into their whiteboards. Direct students to problem 1 in their books and have them record the answers as you guide them through finding products by using a known product. Look at the area model in your whiteboard. What do you notice about how the area model is partitioned? It is partitioned into sevenths vertically and fifths horizontally. It is partitioned into 7 columns and 5 rows. It is partitioned to show thirty-fifths.

Use the partitioned area model to find the product   ×   . What 5 7 is the product? 1_ 1_ __ 1 5 × 7 =   35

Record the product  __ in problem 1(a) and have students do the same. 1 35

185

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5 ▸ M3 ▸ TB ▸ Lesson 9

1. Use a known product to make a simpler problem. Show your thinking. 1 1_ 1_ __ a. × =   35 5 7

1 2 1_ 2_ 1_ 1_ __ __ b. × = 5 × 7 × 2 =   35 × 2 =   35 5 7 3 1_ 3_ 1_ 1_ 1 __ __ c. × = × × 3 =   × 3 =   5 7 5 7 35 35 1_ 4_ 1 1 1 4 d. × = _ × _ × 4 =  __ × 4 =  __ 5 7 5 7 35 35 5 1_ 5_ 1 1 1 e. × = _ × _ × 5 =  __ × 5 =  __ 5 7 5 7 35 35

_1 _2

_1 _1

Look at problem 1(b). What is different about   ×   compared to the expression   ×   ? 5

The second factor is  _  instead of  _  . 2 7

1 7

_1 _1

_1 _2

Our goal is to use what we know about the product   ×   to find the product   ×   . 5

How many sevenths are in   ?

_1 _2

_1 _1

Let’s use that to help us write   ×   as a product of unit fractions that includes   ×   5

because we already know that answer.

Record  _  ×  _  . 1 5

1 7

Differentiation: Challenge For students who recognize the repeated reasoning that is used while finding these products, consider challenging them to explore whether a product of three or more fractions can be found by using similar reasoning.

How much is   compared to   ? 7

 _  is 2 times as much as  _ . 2 7

1 7

How can we represent 2 times as much? We can multiply by 2. 186

EUREKA MATH2 California Edition

Record × 2.

5 ▸ M3 ▸ TB ▸ Lesson 9

_1 _1 __1

__1

We know   ×   =   . What is 2 times as much as   ? 5 7 35 35 2  __ 35

Record =  __ × 2 =  __  . 1 35

2 35

Will we get the same answer if we use an area model? Try it and then compare the answers. Pause to let students shade the area model to see that the result is the same.

_1 _2

_1 _1

Promoting the Standards for Mathematical Practice

__1

When we wrote   ×   as   ×   × 2 , we used a known product,   , 5

to help us find the answer to a new problem. Let’s try this again to see whether it continues to work.

_1 _3

_1 _1

Look at problem 1(c). What is different about   ×   compared to the expression   ×   ? The second factor is  _  instead of  _  . 3 7

1 7

How much is   compared to   ? 7

 _  is 3 times as much as  _  . 3 7

1 7

Ask the following questions to promote MP8:

How can we represent 3 times as much? We can multiply by 3.

_1 _1

We have everything we need to show how to use the known product   ×   to find the 5

answer to this new problem. What should we write first? 1_ 1_ 5 × 7 What else do we need to write? Why?

When students repeatedly multiply two unit fractions to determine the product of a unit fraction and another fraction, they notice that the product of a unit fraction and another fraction is a multiple of the product of two unit fractions. In doing so, they are looking for and expressing regularity in repeated reasoning (MP8).

• When you multiply a unit fraction by a fraction, does anything repeat? How can that help you find the product more efficiently? • Does this pattern always work?

We need to write × 3 to show we need 3 times as much as  _  ×  _ because the problem asks us to find  _  ×  _  . 1 5

3 7

1 5

1 7

187

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Record  _  ×  _ × 3. Then have students find the product. 1 5

1 7

Will we get the same answer if we use an area model? Try it and then compare the answers. Pause to let students shade the area model to see that the result is the same. Then record =  __ × 3 =  __  . 3 35

1 35

_1 _4

Look at problem 1(d). What is the product   ×   ? Why? 5

It is  __  because  _  is 4 times as much as  _ , and I know  _  ×  _  =  __  . 4 35

4 7

1 7

1 5

1 7

1 35

Have students complete problem 1(d) with a partner. Encourage them to write the expression as they did in the two previous problems. Consider allowing students to complete problem 1(e) as well. If needed, bring the class together and repeat the process that you used previously to discuss problems 1(d) and 1(e).

_1 _1

How does using a known product of unit fractions such as   ×   help us find the

product of   and any number of sevenths?

We can use the known product and multiply it by however many sevenths we have. Invite students to turn and talk about how they would find the product  _  ×  _  . 1 5

188

6 7

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Use Unit Language to Multiply Students make a simpler problem by reasoning about factors before they multiply.

UDL: Representation

Display the following student work.

Consider making an anchor chart that lists the methods used in this lesson. Include examples for students to reference.

5 7

1 5 of 7 5

_1 _5 5 7 5 I notice she used a tape diagram to show  _  . 7 5 1 _ I notice she labeled the value of 1 part as     of  _  . 5 7 1 _ I notice the value of 1 part is     .

Yuna said she has another way to find   ×   . What do you notice about her work?

_1 _5

Yuna’s tape diagram shows how to find   of   . Let’s think about this by making 5

a simpler problem. What is   of 5? How do you know? 5

1  _  of 5 is 1. I thought about 1 part when 5 is partitioned into 5 equal parts. 5

Write  _  of 5 is 1. 1 5

 _1 of 5 is 1. What is  _1 of 5 sevenths? How you do know? 5

1 1 1  _  of 5 sevenths is 1 seventh. I know  _  of 5 is 1, and the only difference is that for  _  of 5

5 sevenths, the unit is sevenths.

Write  _  of 5 sevenths is 1 seventh. 1 5

189

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

_1 _5

_1 _1

We found   ×   by using the known product   ×   and then multiplying by 5. We 5

__5

wrote the product as   . What product do you think Yuna wrote? Why? 35

I think Yuna wrote the product  _ because she might have used unit language.

1 7 5 1 I think Yuna wrote the product     of     in 1 part of the tape diagram and the entire tape 5 7

_ _

diagram has 7 parts.

Whether Yuna used unit language or saw the answer in her tape diagram, it makes

sense that Yuna wrote the product as   . Did we get equivalent answers? Is it true 7

_1 __5 ?

that   =   7

Yes. I know 35 is 5 times as much as 7, and 5 is 5 times as much as 1, so the fractions are equivalent. The product Yuna found is equivalent to the product we found, so our answers are equivalent.

_1 __5 numerically?

How can we show   =   7

We can multiply both the numerator and denominator in  _  by 5 to get  __  . 1 7

_1 _5

5 35

Yuna’s method to find   ×   seems like another way to make a simpler problem. Let’s 5

check to see whether it works for other problems too. Have students complete problems 2 and 3 with a partner.

_1 _4

2. Fill in the blanks to find the product     ×     .

_1     of 4 is

4 1     of 4 fifths is 4

fifths.

_1 _8

3. Fill in the blanks to find the product     ×     .

    of 8 is 8

_1     of 8 ninths is 8

190

ninths.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Direct students to problem 4. How is problem 4 different from the two previous problems?

Now we need to find  _  of 10. In the two previous problems, the whole number is the same 1 5

number as the denominator of the first factor. Have students complete problem 4. 10 _1 __   .

4. Fill in the blanks to find the product     ×

_1     of 10 is

5 1     of 10 elevenths is 5

elevenths.

Direct students to study the factors in problems 2–4 and look for similarities.

UDL: Representation

What do you notice? For problems 2 and 3, the numerator of the second factor is the same number as the denominator of the first factor.

Consider posting a visual to illustrate the relationship.

For problem 4, the numerator of the second factor is a multiple of the denominator in the first factor. Write  _  ×  __  . 6 11

1 5

1 10 x 5 11

_1 __6 ? Why?

Can we use unit language to find the product   ×   5

Multiples of 5: 5, 10, 15, 20, ...

No, because the numerator of the second factor is not the same number as the denominator of the first factor, and 6 is not a multiple of 5.

__4

If we use the known product method for problem 2 as we did earlier, the product is   .

Is that equivalent to   ? How do you know?

Yes,  __  =  _ because they are equivalent fractions. 4 20

1 5

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We would also get equivalent fractions as the answers for problems 3 and 4 if we used the known product method. That means we can make a simpler problem when we notice the numerator of the second factor is the same number as or a multiple of the denominator of the first factor. Invite students to turn and talk about whether they would get the same result if they did not notice this relationship with the numerators and denominators and used a different method to find the product.

Identify When to Make a Simpler Problem Students identify when they can make a simpler problem before they find a product of two fractions. Revisit two of the problems from Launch and two new problems to support students with recognizing when they can make a simpler problem by using the following sequence. We know we can find a product of fractions in different ways. What have we explored in this lesson that allows us to make a simpler problem? We can use the known product method to make a simpler problem that involves the product of unit fractions multiplied by a whole number. We can use unit language or look for relationships between the numerator of one factor and the denominator of the other factor to see whether we can make a simpler problem. Display the expressions. A.

_ _

1 3 5 × 4

192

_ _

6 1 7 × 3

 __ ×   ___ 93 117 1

_ __

7 1 3 ×   10

EUREKA MATH2 California Edition

Invite students to think–pair–share about how they might make a simpler problem to find the product for each expression. I would use unit language to find the product for expression C because the numerator of the second factor is the same number as the denominator of the first factor. I would use the known product method with expressions A and D because there is already a unit fraction in those expressions. I would only need to find the product of the unit fractions and then multiply by a whole number.

5 ▸ M3 ▸ TB ▸ Lesson 9

Differentiation: Support If student pairs need support to determine whether they would use known products or unit language, ask them questions such as the following:

I would use unit language with expression B, but I would change the order of the factors

• Is there a relationship between the numerator of one fraction and the denominator of the other fraction?

If we use the commutative property of multiplication to change the order of the factors in expression B, we can use unit language to find the product. It is a good habit to consider expressions first to see whether you can make a simpler problem. But even if you do not notice you can make a simpler problem, you can find the product by using other methods and models.

• Which expressions would you rewrite as a product of unit fractions multiplied by a whole number?

first to make  _  ×  _  . 1 3

6 7

• Are any of these problems like problem 3? Problem 4?

Invite students to turn and talk about why it is more efficient to use unit language to find the product of expression C (the problem from Launch) than to draw a model.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

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Land

Debrief 5 min Objective: Multiply fractions by unit fractions by making simpler problems. Gather the class with their Problem Sets. Facilitate a class discussion about finding products by making simpler problems by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the summary of the problem from Learn.

1 5 × 5 7 1 1×5 5 = = 7 7 × 5 35

Yuna’s answer: 7 5

Our answer: 35

Gesture to the expression in the middle of the equation. When we saw Yuna’s way of making a simpler problem, we discussed whether Yuna’s

__5

_1 _5

answer   is equivalent to ours,   . For that problem, we were finding the product   ×   . 7 5 7 35 1 5_ 1×5 _ _ What do you notice about the numerators in   × and the numerator in   ? 5 7 7×5 The numerators are the same because 1 × 5 = (1 × 5). Gesture to the expression in the middle of the equation.

_1 _5

What do you notice about the denominators in   ×   and the denominator in _   ? 5

The denominators are the same but in a different order.

1×5 7×5

Does multiplying the denominators in a different order mean we will get a different product? Why? No. We will get the same product because the commutative property of multiplication says we can multiply in any order and get the same answer.

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Direct students to problems 6–15 in the Problem Set. What problems did you make simpler by using unit language? How did you know you could? I used unit language to make a simpler problem for problems 8 and 13. I noticed the relationship between the numerator of the second fraction and the denominator of the first fraction because they are the same number. I used unit language to make a simpler problem for problem 7. I noticed the relationship between the denominator of the first fraction, 2, and the numerator of the second fraction, 4. That same relationship is in problem 12. What problems did you make simpler by using known products? Why? I used known products to make problem 9 simpler. It seemed that drawing an area model to show  _  and  __ would take too long, so I thought about  _  ×  __ × 6 instead. 1 5

6 11

1 5

1 11

I used known products to make problem 10 simpler. I knew the unit of the product is fortieths, so I could mentally find  __ × 2to get the answer  __  . 1 40

2 40

Invite students to turn and talk about how they might use a method from this lesson to make a simpler problem to find the product  __  ×  _  . 6 15

2 3

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Use the model to complete the statements. The diagram represents 1.

_1 of 4 fifths is

1. Use the number line to find the product. Then complete the equation.

1 3

2 3

_1 × _4 = _____ 4

_ 1 6

_ _

1 1 × = 2 3

3 3

Fill in the blanks. 3. _1 of 3 is

_1 of 3 fifths is

fifth.

1 3 × = 3 5

4. _1 of 4 is

_1 of 4 fifths is

fifths.

1 4 × = 2 5

5. _1 of 9 is

_1 of 9 sevenths is

3 sevenths .

_ _

b. 1 × 2 2 3

1 3

2 3

1 1 1 2 _____ _____ × = × × 2 3 2 3

_ _

3 3

fifth.

4 5

1 1_ a. _ × 2 3

1 6

2 6

1 3

_ _

1 5

_ _

2 5

_ _

3 7

1 4_ c. _ × 2 3

1 3

2 3

3 3

4 3

196

1 2

_×4

1 9 × = 3 7

= 6 =

_ _

1 4 × = 2 3

_ 4 6

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

Use the Read–Draw–Write process to solve the problem.

Make a simpler problem by using a known product or unit language. Show your thinking. Then multiply. 1 3_ × = 6. _ 6 2

__3

_ _

_ __

__4

_ __

7. 1 × 4 = 2 10

1 1 × ×3= 1 ×3= 3 12 12 6 2

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 9

16. Kayla plays her video game for _2 hours on Sunday. She spends _1 as many hours playing her video 3

game on Monday.

a. Does Kayla spend more time or less time playing her video game on Monday than on Sunday? Explain your reasoning.

1 × 1 ×4= 1 ×4= 4 2 10 20 20

Kayla spends less time playing her video game on Monday because she plays her video game for a part, or _1 , of _2 hours.

1 7_ 8. _ × = 7 9

_ 1 9

1 __ 9. _ × 6 = 5 11

_1 of 7 ninths is 1 ninth. 7

_ __

__6

1 × 1 ×6= 1 ×6= 6 5 11 55 55

b. What fraction of an hour does Kayla spend playing her video game on Monday? 1 2_ 10. _ × = 8 5

__2

1 11. __ × 5_ = 10 3

_ _

1 1 × ×2= 1 ×2= 2 40 40 8 5

1 4_ 12. _ × = 2 6

__4

_ _

1 1 × ×4= 1 ×4= 4 2 6 12 12

1 7_ 14. _ × = 3 9

__7 27

_ _

1 1 × ×7= 1 ×7= 7 3 9 27 27

Kayla spends _1 hour playing her video game on Monday.

__1 × 1_ × 5 = __1 × 5 = __5 10

1 __ = 13. __ × 12 5 12

_ _ _ 1 2 1 × = 4 3 6

__5 30

_ 1 5

c. For how many minutes does Kayla play her video game on Monday?

1 × 60 = 10 6

__1 of 12 fifths is 1 fifth. 12

Kayla plays her video game for 10 minutes on Monday.

1 12 15. _ × __ = 7 6

12 42

_ _

1 1 × × 12 = 1 × 12 = 12 42 42 6 7

PROBLEM SET

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LESSON 10

Multiply fractions greater than 1 by fractions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

Name

Date

Make a simpler problem. Then multiply. 1.

2 _9 × __ = 2 10

18 __ 20

1 1 1_ × __ __ × 9 × 2 = __ × 18 = 18 20 20 2 10

4 × 6_ = 2. _ 9 6

9 × 5_ = __ 10

Students use known products to multiply fractions greater than 1 by fractions. They analyze the factors in the multiplication expression and the product of the factors to look for a relationship. Students conclude that sometimes multiplying a fraction greater than 1 by a fraction results in a product greater than 1, but not always.

Key Questions • Is multiplying by a fraction greater than 1 different from multiplying by fractions less than 1? How?

_4 9

• What do you notice about the numerators and denominators in a multiplication expression and the numerator and denominator in the product?

_4 × 1 = 4_ 9 9

Lesson at a Glance

45 __ 30

Achievement Descriptors 1 1 __ __ × 1_ × 9 × 5 = __ × 45 = 45 10

5.Mod3.AD6 Multiply whole numbers or fractions by fractions. (5.NF.B.4)

5.Mod3.AD8 Compare the effects of multiplying by fractions and

whole numbers. (5.NF.B.5.a) 7 × 8_ = 4. _ 6 9

56 __ 54

5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

1 _1 × 1_ × 7 × 8 = __ __ × 56 = 56 54 54 6 9

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• None

Learn 35 min

Students

• Multiply a Fraction Greater Than 1 by a Unit Fraction

• Using Known Products with Fractions Greater Than 1 (in the student book)

Consider whether to remove Using Known Products with Fractions Greater Than 1 from the student books and place inside whiteboards in advance or have students prepare them during the lesson.

• Multiply a Fraction Greater Than 1 by a Fraction • Problem Set

Land 10 min

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Fluency

Whiteboard Exchange: Write and Evaluate Expressions Students write and evaluate an expression to build fluency with two-step calculations involving whole numbers from module 1. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the statement: The sum of 3 and 7, doubled. Write an expression to represent the statement. Display the sample expression. Write the value of the expression.

The sum of 3 and 7, doubled

(3 + 7) × 2 20

Display the answer. Repeat the process with the following sequence:

The difference between 8 and 2, divided by 3

4 times as much as the sum of 3 and 5

(8 – 2) ÷ 3

4 × (3 + 5)

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Whiteboard Exchange: Subtract Fractions Students make like units in a subtraction equation and find the difference to build fluency with subtracting fractions with unlike units from module 2. 1 Display  _1  −  _   = . 2

Look at the fractional units. Do the fractions have like units? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. No. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Rename both fractions to make the fractional units, or denominators, the same. Show your method.

1–1= 2 3

1 6

1 – 1 = 1×3 – 1×2 2 3 2×3 3×2

= 36 – 26

Display the sample method and fractions with like units. Find the difference. Display the difference. Repeat the process with the following sequence:

4 – 1 = 3 5 2 10

2 – 2 = 2 3 4 12

5 – 1 = 13 4 6 12

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Launch

Students analyze a model involving a fraction greater than 1 and identify an error in the interpretation of the model. 5 Display the area models that represent the product  _1  ×  _  . 2

1 2

5 4 What do you notice? What do you wonder? I notice the labels  _1  and  _5 . 2

I notice both area models are partitioned into fourths vertically and halves horizontally. I notice there are 5 shaded parts.

I notice  _5 is a fraction greater than 1. 4

I wonder why it shows 2 wholes. I wonder what problem this model represents.

This is the model Leo used to find  _1  ×  _ . Why do you think he used two squares in his area model to find the product?

5 4

I think he used two squares because one square has only 4 fourths and he needed to represent 5 fourths. I think he used two squares because  _5 > 1. 4

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Invite students to think–pair–share about the following question. Leo says  _1  ×  _  =  __ . Do you agree? Why? 2

5 4

5 16

I see 5 shaded parts out of a total 16 parts in the model, but I do not think Leo is correct because each square shows eighths. I do not agree with Leo because each square has 8 equal parts, so his answer should be in eighths.

I do not agree with Leo because 1 shaded part is  _1 and there are 5 shaded parts, so his answer should be  _5 .

What error did Leo make? Leo added all the parts in both models to get 16 instead of thinking about how many parts he needed to make 1 whole, which is 8 parts. When we use models to multiply a fraction greater than 1, we must be careful to use the correct number of units in 1 whole and not the total number of parts we can see in the model. Transition to the next segment by framing the work. Today, we will use known products to multiply fractions greater than 1 by fractions.

Learn

Multiply a Fraction Greater Than 1 by a Unit Fraction Materials—S: Using Known Products with Fractions Greater Than 1

Students use known products to multiply a fraction greater than 1 by a unit fraction. Have students remove Using Known Products with Fractions Greater Than 1 from their books and place them into their whiteboards. Direct students to problem 1 in their books

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and have them record the answers as you guide them through finding products by using a known product. Look at the area model in your whiteboard. What do you notice? I notice two squares. Each model is partitioned into sevenths vertically and fifths horizontally. Each model has 7 columns and 5 rows. Each model shows thirty-fifths. For which values of fractions, less than 1 or greater than 1, might we need both squares to represent the problem? Fractions with values greater than 1 Look at problem 1. Are there any problems involving fractions greater than 1? Which problems? Yes. Problem 1(c) has  _8 , which is a fraction greater than 1. 7

Problem 1(a) looks like the expressions we saw in a previous lesson. How did we make a simpler problem then? We rewrote the fractions as unit fractions times a whole number. Have students complete problem 1 with a partner. As needed, have them check their answer by using the area model. Circulate as students work to ensure they show their thinking by writing a product of unit fractions and a whole number. 1. Use a known product to make a simpler problem. Show your thinking.

204

6 a.  _1  ×  _   =

6 1  _1  ×  _1  × 6 =  __   × 6 =  __  

7 b.  _1  ×  _   =

7 1  _1  ×  _1  × 7 =  __   × 7 =  __  

8 c.  _1  ×  _   =

8 1  _1  ×  _1  × 8 =  __   × 8 =  __  

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

When students finish problem 1, facilitate a discussion by asking the following questions. In problem 1(a), we multiplied a fraction with a value less than 1 by another fraction less than 1. How can you describe what we multiplied in problem 1(b)? In part (b), we multiplied a fraction with a value less than 1 and a fraction equal to 1. How can you describe what we multiplied in problem 1(c)? In part (c), we multiplied a fraction with a value less than 1 and a fraction with a value greater than 1.

Language Support Consider providing a list of terms and phrases such as the following to encourage students to participate in the class discussion by using precise language. • Factor, product, multiply

What do you notice about the values of the fractions in the answers?

• Numerator, denominator

All the answers are fractions less than 1.

• Greater than, equal to, less than

Let’s look more closely at the factors in each problem and compare them to the answers. Invite students to turn and talk about whether the factors in problem 1(a) are greater than, equal to, or less than the answer. 6 Is the product  __  greater than, equal to, or less than each of the factors  _1  and  _6 ? Is 5

that reasonable? Why?

6  is less than both  __ 6 . This is reasonable because both fractions are less than 1 1  and  _  __ 5

and when you find a fraction of another fraction less than 1, the product is less than both factors.

6  is less than both  _ 6 . This is reasonable because both fractions are less than 1 1  and  _  __ 5

and when you multiply by a fraction less than 1, the product is less than the other factor. Both factors are less than 1, so the product is less than both factors. Invite students to turn and talk about whether the factors in problem 1(b) are greater than, equal to, or less than the answer. 7 Is the product  __  greater than, equal to, or less than each of the factors  _1  and  _7 ? Is 5

that reasonable? Why?

7 7   <  _  __  . This is reasonable because  _7  was multiplied by  _1  and  _1 < 1. 35

1 7   =  _  __  . This is reasonable because  _1  ×  _7 has the same value as  _1 × 1, so it makes sense

the product is equal to that factor. © Great Minds PBC

Teacher Note

Students may notice multiplying by  7  is 7 equivalent to multiplying by 1 and wonder why they did not use this to make a simpler problem. Validate their thinking and share that the goal is to multiply fractions and to look for connections between factors and products. For this reason, making a simpler problem is not helpful.

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Invite students to turn and talk about whether the factors in problem 1(c) are greater than, equal to, or less than the answer. 8 Is the product  __  greater than, equal to, or less than each of the factors  _1  and  _8 ? Is 5

that reasonable? Why?

7 1 8   >  _  __  . This is reasonable because  _1 is being multiplied by  _8  and  _8 > 1. I know  _1  ×  _7  =  __  . 35

Because there is another seventh in  _8 , the product is greater than  _1 . 7 5 8 8 8 1 1 _ __ _ _ _     <    . This is reasonable because    is being multiplied by     and    < 1.

Invite students to turn and talk about the relationship they notice between the factors in the equation and the answer.

Multiply a Fraction Greater Than 1 by a Fraction Students use known products to multiply fractions greater than 1 by fractions. Direct students to problem 2. Have students analyze the fractions in problem 2. 2. Use a known product to make a simpler problem. Show your thinking. 6 1 12 a.  _2  ×  _   = ( _1 × 2) × ( _1 × 6) =  _1  ×  _1 × 2 × 6 =  __   × 12 =  __   5

8 1 24 b.  _3  ×  _   =  _1  ×  _1 × 3 × 8 =  __   × 24 =  __  

Students look for and make use of structure (MP7) as they decide how to multiply fractions greater than 1 by fractions by using known products. Ask the following questions to promote MP7:

9 36 1 c.  _4  ×  _   =  _1  ×  _1 × 4 × 9 =  __   × 36 =  __  

What do you notice about these problems compared to those in problem 1? None of the fractions in these problems are unit fractions. We are multiplying fifths and sevenths again.

Using the known product  _1  ×  _ helps us make a simpler problem, so let’s continue 5

Promoting the Standards for Mathematical Practice

1 7

• How can what you know about the product of unit fractions help you find the product of a fraction greater than 1 and a fraction? • What is another way you can rewrite the multiplication expression that will help you find the product?

to do that. Look at problem 2(a). If  _2  were  _1 , we would already know how to find this 5

product. What do you think we should do with  _2 ? 5

 _2  is 2 times as much as  _1 , so we can multiply  _1  by 2. 5

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Because  _2 is 2 times as much as  _1 and  _6  is 6 times as much as  _1 , let’s see whether 5

we can multiply the known product by 2 and by 6 to find the product  _2  ×  _ .

1 Record ( _ × 2) × ( _  × 6) and have students do the same. 1 5

6 7

Is this expression still equal to the original problem? How do you know? 2 Yes, it is equal because  _1 × 2 =  _   and  _1 × 6 =  _6 . 5

1 We know  _1  ×  _ and we know 2 × 6. Can we change the order of the factors? Why? 7

Yes. Because the expression is all multiplication, we can change the order of the factors and still get the same answer. We can use the commutative property of multiplication to change the order of the

factors. Let’s write an equivalent expression that shows the known product  _1  ×  _  first and then include the whole numbers.

1 7

1 _ Record =  _   ×  1 × 2 × 6. 5

It might be helpful to show more equations than are necessary with this first example to reinforce equivalence and properties of operations. Students are not expected to show this much work moving forward unless they find it helpful.

__ __

__ __ 5 7 1 1 __ __ =   × 2 ×   × 6 5 7 1 __ 1 __ = (  ×   )× (2 × 6) 5 7 1 __ =   × 12 35 12 ___ =  

  2  ×   6 = (  1 × 2) × (  1 × 6) 5

What is  _1  ×  _ ? 2 × 6? 5

Teacher Note

1 7

1   and 12  __ 35 1 Record =  __  × 12. 35

What is the product  _2  ×  _ ? 5

12    __

6 7

Do you think we will get the same answer if we use our area model? Try it and then compare the answers. Pause while students shade the area model to see that the result is the same. Look at problem 2(b). If we continue to use the known product  _1  ×  _ , what must we also multiply to find the 1 7 5 8 product  3  ×    ? 7 5

_ _

1 We need to find  _1  ×  _   and 3 × 8. 5

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1 Record  _1  ×  _  × 3 × 8. Have students find the product and then 5

Differentiation: Support

use the area model to see that the result is the same. 8 What is the product  _3  ×  _  ? 5

24    __

1 24 Complete the equation by recording =  __  × 24 =  __  . 35

9 Look at problem 2(c). How can we show our thinking to find the product  _4  ×  _  ? 5

1 We need to write  _1  ×  _  × 4 × 9. 5

To provide additional support, continue to write each factor in the original expression as a product of a unit fraction and a whole number. Use parentheses to emphasize the relationship between the factor and its decomposed form.

1 Record  _1  ×  _  × 4 × 9. Have students find the product and then 5

use the area model to see that the result is the same. 9 What is the product  _4  ×  _  ? 36    __

36 1 Complete the equation by recording =  __  × 36 =  __  . 35

How can you describe the fractions we multiplied in problem 2(a)? 2(b)? 2(c)? In part (a), we multiplied two fractions with values less than 1. In part (b), we multiplied a fraction with a value less than 1 and a fraction with a value greater than 1. In part (c), we multiplied a fraction with a value less than 1 and a fraction with a value greater than 1. What do you notice about the values of the fraction in the answers? The answers in part (a) and part (b) are fractions less than 1. The answer in part (c) is greater than 1. This is the first time in a lesson that we have multiplied two fractions and found a product greater than 1. Let’s see whether that happens again by finding products for a variety of fractions, and not just for fifths and sevenths.

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Direct students to problem 3. Have them complete problems 3(a)–(d) with a partner.

Differentiation: Challenge

3. Multiply. Show your thinking.

6 18 1 a.  _3  ×  _   =  _1 ×  _1 × 3 × 6 =  __ × 18 =  __ 4

5 45 1 1 9   ×  _ b.  __   =  _   ×  _1 × 9 × 5 =  __ × 45 =  __   4

13 26 1 1 2   ×  __ c.  __   =  _   ×  _1 × 2 × 13 =  __   × 26 =  __   5

40 4 1 1 10   ×  _ d.  __   =  _   ×  _1 × 10 × 4 =  __   × 40 =  __   3

When students finish problem 3, facilitate a discussion by asking the following questions.

Challenge students to make a simpler problem to find the product of 3 factors, such as the following expression:

__ __ __

5 6   4 ×   ×   6 8 5 Students may use unit language or write each factor as the product of a unit fraction and a whole number before they multiply. For an additional challenge, present the factors out of order to make using unit language less obvious.

Let’s look more closely at the factors in each problem and compare them to 1. Invite students to turn and talk about whether the factors and product in problem 3(a) are greater than, equal to, or less than 1. 18  greater than, equal to, or less than 1? Are the factors  _3  and  _6 and the product  __ 4

The factor  _3 is less than 1.

The factor  _6 is greater than 1. 5

18  is less than 1. The product  __ 20

Invite students to turn and talk about whether the factors and product in problem 3(b) are greater than, equal to, or less than 1. 9   and  _ 5 and the product  __ 45  greater than, equal to, or less than 1? Are the factors  __ 10

9  is less than 1. The factor  __

10 The factor  5 is greater than 1. 4 The product  45  is greater than 1. 40

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Invite students to turn and talk about whether the factors and product in problem 3(c) are greater than, equal to, or less than 1. 13 and the product  __ 26  greater than, equal to, or less than 1? 2   and  __ Are the factors  __ 5

2  is less than 1. The factor  __

11 The factor  13 is greater than 1. 5 The product  26  is less than 1. 55

Invite students to turn and talk about whether the factors and product in problem 3(d) are greater than, equal to, or less than 1. 10   and  _ 40  greater than, equal to, or less than 1? 4 and the product  __ Are the factors  __ 13 3 10 The factor    is less than 1. 13 The factor  4 is greater than 1. 3 The product  40  is greater than 1. 39

Invite students to think–pair–share about the following question. Why do you think the product of a fraction less than 1 and a fraction greater than 1 is sometimes less than 1 and sometimes greater than 1? Give an example to support your thinking. 5 I think it depends on the size of the factors. _   9 ×  _  > 1, and I noticed _   9 is almost 10

1 and  _5 > 1. But  _3  ×  _6 < 1. I noticed  _3 is also almost 1 and  _6 > 1, but the product is less 4

than 1. Because _   9 is closer to 1 than  _3 , I think whether the product is greater or less than 10

1 depends on how close a factor is to 1.

2  , it has to be multiplied by a fraction much If the fraction less than 1 is very small, like  __ 11

13 , the 2   and  __ greater than 1 to make the product greater than 1. When we multiplied  __ 13 is almost 3. product was less than 1 even though the value of  __

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Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

UDL: Action & Expression Consider reserving time after the class completes the Problem Set for students to reflect on their learning from this module. As needed, remind students that they began with finding fractions of a set by using an array. Invite students to ask themselves questions such as the following to promote self-monitoring.

Debrief 5 min Objective: Multiply fractions greater than 1 by fractions. Gather the class with their Problem Sets. Facilitate a class discussion about multiplying fractions greater than 1 by fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Direct students to problems 2–13 in the Problem Set. Earlier we analyzed factors in multiplication equations and their answers to draw a conclusion about their relationship. We said the answer depends on the factors. Has your thinking changed? Why?

• Which model or representation do I rely on the most to find products of fractions? Has it always been the same model or representation? • What is one skill I need to continue to work on? • Which new method could I try next time?

No. Sometimes the product is less than 1 and sometimes the product is greater than 1. 9  is almost 1 No. I thought the answer to problem 4 would be greater than 1 because  __

and  _6 is more than 1 and the answer was close to, but still less than, 1.

Is multiplying by a fraction greater than 1 different from multiplying by a fraction less than 1? How? When we multiply fractions less than 1, the answers are less than 1. When we multiply a fraction greater than 1 by a fraction, sometimes the answer is less than 1 and sometimes the answer is greater than 1.

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EUREKA MATH2 California Edition

Multiplying by a fraction greater than 1 is not much different from multiplying by a fraction less than 1 because we write the fractions greater than 1 as unit fractions times a whole number. When we multiply fractions less than 1, we need one square for the area model. When we multiply fractions greater than 1 we need more than one square in the area model. Direct students to study all the numerators and denominators of the factors and the products. What do you notice about the numerators of the factors and the numerators of the products? I notice the numerator of the product is the product of the numerators of the factors. What do you notice about the denominators of the factors and denominator of the products? I notice the denominator of the product is the product of the denominators of the factors. Invite students to turn and talk about whether they think what they noticed about the numerators and denominators can help them find products efficiently.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

2 × 5_ c. _ 3 4 2_ × 5_ = 3 4

1. Use the area model to find the product. Then complete the equation. 2 × 1_ a. _ 3 4

2 3

_1 4

1 __ × 10

= 12

2 3

1 1 _2 × 1_ = _____ × _____ × 3 4 3 4

_1 3

10 __ 12

5 4

2 __ 12

1 4

Make a simpler problem. Show your thinking. Then multiply. 3 × 8_ = 2. _ 4 5

2 × 3_ b. _ 3 4

1 __ _1 × 1_ × 3 × 8 = __ × 24 = 24 20 20 4 5

1 1 _2 × 3_ = _____ × _____ × 3 4 3 4 2 3

= 3 4

24 __ 20

1 __ 12

30 __ 5 × 6_ = 12 3. _ 6 2 1 1_ × 1_ × 5 × 6 = __ __ × 30 = 30 12 12 6 2

6 __

9 × 6_ = __ 11

1 × 11 __ = 5. _ 5 2

1 1 × 1_ × 9 × 6 = __ __ __ × 54 = 54 11

54 __ 55 5

PROBLEM SET

11 __ 10 11 1 _1 × _1 × 11 = __ × 11 = __ 10 10 2 5

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EUREKA MATH2 California Edition

Multiply. 4 × 8_ = 6. __ 13

32 __ 39

5 ▸ M3 ▸ TB ▸ Lesson 10

3 × 10 __ = 7. _ 8 5

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

14. Eddie runs for _3 hours on Friday. He runs _5 as much on Saturday. 5

30 __ 40

a. Does Eddie run for more time or less time on Saturday than he did on Friday? How do you know? Eddie runs for more time on Saturday than on Friday. I know because _5 is greater than 1. Running for more than 1 of _3 means he runs longer.

8 8_ 8. _ × = 9 7

10.

2 × 5_ = __ 11 3

12.

7 × 11 __ __ = 10 10

214

64 __ 63

10 __ 33

77 ___ 100

3 × 3_ = 9. _ 5 4

5 × 3_ = 11. _ 6 7

7 12 × __ 13. __ = 12 5

9 __ 20

b. Eddie says _3 × _5 is greater than 1 because 5_ is greater than 1. Explain why his reasoning 5 4 4 is incorrect. What advice would you give to Eddie?

15 __ 42

Eddie is incorrect because it is only sometimes true when one factor is a fraction and the other factor is a fraction greater than 1 that the product is also greater than 1. Eddie should think about the size of both factors when he estimates whether the product is greater than or less than 1.

84 __ 60

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 10

Use the Read–Draw–Write process to solve each problem.

15. _5 of the animals in the pet show are dogs. _3 of the dogs are poodles. What fraction of the animals 9

in the pet show are poodles?

15 _3 × _5 = __ 5 9 45 15 of the animals in the pet show are poodles. 45

16. At a summer camp, _4 of the campers choose to play a sport as their activity. Of the campers 5

who choose a sport, _2 play football and the rest play basketball. What fraction of the campers 5

play basketball?

1 − 2_ = 3_

5 5 12 _3 × _4 = __ 5 5 25 12 of the campers play basketball. 25

PROBLEM SET

215

LESSON 11

Multiply fractions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

Name

Date

Multiply.

15 __ 16

3 × 5_ = 1. _ 4 4

3 × 5 = 15 __ ____ 4×4

24 __ 36

3 × 8_ = 2. _ 4 9

• In a multiplication expression, how does the size of the factors affect the size of the product?

Compare the expressions by using > , =, or < . Explain how you can compare the expressions without evaluating them. 1 × 1_ 3. _ 2 3

_1 × 7_ 2 8 3

• How can you compare multiplication expressions without finding the products?

Achievement Descriptors

Explain: Both expressions have _1 as a factor and _1 is less than _7 . 2

Students begin by reasoning about the size of products. They use inductive reasoning to create general rules about the size of products of expressions involving multiplying numbers by fractions less than 1, greater than 1, and equal to 1. They use this understanding to compare two expressions without multiplying. Finally, students use what they know about multiplying fractions to solve real-world problems.

Key Questions

3 × 8 = 24 __ ____ 4×9

Lesson at a Glance

5.Mod3.AD7 Recognize, model, and contextualize the product of a

fraction and a whole number or fraction. (5.NF.B.4.a) 5.Mod3.AD8 Compare the effects of multiplying by fractions and

whole numbers. (5.NF.B.5.a) 5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1,

equal to 1, or greater than 1. (5.NF.B.5.b)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 10 min

• None

Learn 30 min

Students

• Greater Than, Equal To, Less Than

• None

• Compare Expressions Without Evaluating • Solve a Real-World Problem • Problem Set

Land 10 min

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Fluency

Whiteboard Exchange: Write and Evaluate Expressions Students write and evaluate an expression to build fluency with two-step calculations involving fractions from module 2. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the statement: The difference of 2 fifths and 1 fifth, multiplied by 4.

The difference of 2 fifths and 1 fifth, multiplied by 4

Write an expression to represent the statement. Display the sample expression.

( 25 – 15 ( × 4 4 5

Write the value of the expression. Display the answer. Repeat the process with the following sequence:

2 times the sum of 2 tenths and 3 tenths

) 10 10 )

218

1 less than the total of 6 and 6

2× 2 + 3

) 26 + 56 ) – 1

1 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

Whiteboard Exchange: Add or Subtract Fractions Students make like units in an addition or subtraction equation and find the sum or difference to build fluency with adding and subtracting fractions with unlike units from module 2. Display _  1  + _     = 2

3 7

1 + 3 = 13 2 7 14 1 + 3 = 1×7 + 3×2 2 7 2×7 7×2 7 + 6 = 14 14

Display the sample method and fractions with like units. Find the sum. Display the sum. Continue the process with the following sequence:

2 – 1 = 7 3 5 15

5 + 4 = 33 4 10 20

7 – 6 = 10 6 8 24

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Launch

Students analyze incomplete representations of a multiplication expression to complete an equation. Direct students to problem 1 in their books. Read the directions chorally with the class. Have students work in pairs to complete problem 1. Circulate to support students as needed but allow time for productive struggle. 1. Use the clues in part (a) to complete the equation in part (b). a. Analyze the clues and fill in the blanks. The clues represent equivalent expressions. Clue A

_ _

1 1   ×   × 4 × 5 3

Clue B

4×

×3

Clue C

Differentiation: Support For students who need support getting started, suggest they begin with the area model clue. Ask them to label the area model with braces and fractions to represent the shaded part and then return to the other clues to complete the expressions.

________  

b. Write the multiplication expression that is represented by the clues. Then find the product to complete the equation. 4

_____   × _____   = _____  

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When most students are finished, or the struggle is no longer productive, facilitate a discussion by asking any of the following questions: • With which clue did you begin? Why? • Which clue was most helpful? Least helpful? Why? • How did you know which multiplication expression the clues represented? • How did you find the product in problem 1(b)?

_2 _4

Could we write the expression     ×    in problem 1(b)? Why?

No. _  2  × _    is not shown in the area model. The area model shows _ 4  × _   . 5

4 3

2 3

It would not match the area model, but it could match clue A because we can multiply the factors in any order.

The expression _  2  × _    results in the same product but would not match all the clues given 5

in problem 1(a).

4 3

Display the following equations.

_ _ ____

4×2 4 2   ×   =   5×3 5 3

_ _ ____

2×4 2 4   ×   =   5×3 5 3

What relationship do you notice between the numerators and denominators in the multiplication expression and in the product? The numerator of the product is the product of the numerators in the factors. The denominator of the product is the product of the denominators in the factors.

_2 _4

_4 _2

If we did not have the area model as a clue in problem 1(a), then     ×     and     ×     would 3 3 5 5 both be correct for the equation in problem 1(b). Why? Both would be correct because we can multiply in any order and both expressions result 8 in the same product,  __  . 15

Transition to the next segment by framing the work. Today, we will multiply fractions and reason about the size of the product.

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Learn

Greater Than, Equal To, Less Than Students reason about the size of the product compared to the size of the factors.

Write the statement _  1 as much as _  3 . Then invite students to turn and talk about what the 5 4 statement means. What multiplication expression can we write to represent the statement? How do you know? We can write _  1  × _     because _ 1 as much as _ 3  means _ 1 times as much as _ 3 . 4

3 5

Affirm that _  1 as much as _  3  means _  1 times as much as _  3 , just as it does with whole numbers 5 5 4 4 (2 × 4 is 2 times as much as 4) or with a fraction and a whole number _   1  × 24 is _  1  times

as much as 24).

• Is it true that the product is less than  4 ? 5 How do you know?

The product is less than _  3 . Because _  1  < 1 , we are finding a part of _  3 , not all of it. 5

When students work with a partner to decide whether each product is greater than, equal to, or less than one of the factors in the multiplication expression, they construct viable arguments and critique the reasoning of others (MP3). Ask the following questions to promote MP3:

Do you estimate that the product is greater than, equal to, or less than    ? Why? 5

Promoting the Standards for Mathematical Practice

• What questions can you ask your partner to make sure you understand their reasoning?

Invite students to find the product. Is the product reasonable? How do you know?

3 Yes, the product is  __  and that is reasonable. We knew our answer would be less than _  3 , 3 _ and  __   <    . 20

3 5

Teacher Note

Write the statement _  4 as much as _  3 . 5

What multiplication expression can we write to represent this statement?

_ 4  × _ 3  4

What is different about this expression compared to the last one? Instead of finding _  1  × _    , we are finding _ 4  × _   . 4

222

3 5

In previous lessons, students practiced comparing a product to each factor to ensure the reasonableness of the product. In this lesson, students explicitly compare a product to one of the factors to help them compare expressions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

Invite students to turn and talk about whether they estimate the product is greater than, equal to, or less than _  3 . 5

Do you estimate that the product is greater than, equal to, or less than    ? Why? The product is equal to _  3  because _  4  = 1 and _  3  × 1 = _    . 5

3 5

Write the statement _  7 as much as _  3 . Then invite students to write a multiplication 5

expression to represent the statement.

Do you estimate that the product is greater than, equal to, or less than    ? How 5 do you know?

The product is greater than _  3 . I know _  7  × _     means _ 7  of _ 3 . Because _ 7  > 1, we are finding more than the whole.

Write the following expressions:

3 5

UDL: Action & Expression Support students in generalizing. Consider modeling a think-aloud or having a student who has already demonstrated understanding

_ _

3 4   ×   5 5

_ _

to think aloud while their peers listen. Select one of the expressions from the lesson to use

__ __

for the think-aloud, such as  6  ×   4 . 5

5 4   ×   5 5

_ _

6 4   ×   5 5

Invite students to work with a partner to decide whether each product is greater than, equal to, or less than _  4 . 5

When students have finished, guide them through summarizing their observations about each type of multiplication. Use the following sentence frames to help them. • When you multiply a number by a fraction less than 1, the product is . • When you multiply a number by a fraction equal to 1, the product is . • When you multiply a number by a fraction greater than 1, the product is .

Teacher Note When you multiply a number by a fraction less than 1, the product is less than that number.

__ __ __

  3 ×   4 <   4 5 5 5 When you multiply a number by a fraction equal to 1, the product is equal to that number.

__ __ __

  5 ×   4 =   4 5 5 5 When you multiply a number by a fraction greater than 1, the product is greater than that number.

__ __ __

  6 ×   4 >   4 5

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Compare Expressions Without Evaluating Students compare two multiplication expressions without finding the actual products.

Write _  1  × 15 and _  5  × 15 . Then invite students to think–pair–share about what they notice 3

about the expressions. Both expressions show a fraction times a whole number. The second factor is the same in both expressions.

Language Support

The first factor in both expressions is a fraction. The first expression has a factor less than 1 and the second expression has a factor that is a fraction greater than 1.

In the first expression, we are finding     of 15. We know     < 1 , so what do we know 3

_1 about the product of     and 15?

It is less than 15.

Consider providing sentence frames to support students in participating in the class discussion. is less than

• •

is greater than .

, so

. ,

In the second expression, we are finding     of 15. Is the product greater than 15 or less 3

than 15? How do you know?

It is greater than 15. Because _  5  > 1, the product is greater than 15. 3

How does     × 15 compare to     × 15? 3

_ 1  × 15 < _ 5  × 15 3

Write the expressions _  3  × __     and _ 1  × __    . Then invite students to turn and talk about what 4

43 50

they notice about the two expressions. How can we use what we know about multiplying fractions to compare the expressions without finding the actual products?

Differentiation: Challenge Consider challenging students to compare by looking at only the numerators. Ask why looking at only the numerators results in the same response as when looking at the size of the factors. Students should notice that the denominators of both expressions are the same, so they need to compare only the numerators to decide which expression is greater.

We can look at the size of the factors.

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In the last problem, we could compare the products by looking at the first factor in each expression. One was a fraction less than 1 and the other was a fraction greater than 1. Is that also true in these expressions? No, both of the first factors are fractions less than 1. Can we compare the products without multiplying? Why? Yes, we know _  3  > _    , so _ 3  of __  43 is greater than _  1  of __  43 . 4

1 4

Yes, _  3  × __  43  is 3 times as much as _ 1  × __  43  because _ 3  is 3 times as much as _ 1 . 4

87 12 9 __ Write the expressions __  12  ×  ___  and  __   ×    . Have students work with a partner for 1 minute 13

100

to compare the expressions without finding the actual product. As students work, circulate and observe the reasoning they use to determine which expression is greater. Support students by asking questions such as the following: • Are any of the factors greater than 1? • Do the expressions share a common factor?

9 87 • How can you use what you know about the sizes of  ___   and  __ to compare the 100 10 expressions?

As time allows, have students work with a partner to compare the following expressions and share their reasoning. • __   17  × _     and _ 5  × __     20

3 5

17 20

49 50

• _  1  × __     and _ 3  × __     Invite students to turn and talk about how they can compare multiplication expressions involving fractions without finding the actual product.

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Solve a Real-World Problem Students solve a real-world problem involving the multiplication of fractions. Direct students to problem 2. Read the problem chorally with the class. Use the Read-Draw-Write process to solve each problem.

2. Mrs. Chan has 4 gallons of paint. She uses _  1 of it to paint her bedroom. She uses _  3 of the 3

Differentiation: Challenge

remaining paint for her living room. What fraction of the paint does Mrs. Chan use for her living room?

4 gallons

Challenge students by asking them to solve the following problems: • What fraction of the paint does Mrs. Chan have left? • How many gallons of paint does Mrs. Chan use to paint her bedroom? Her living room?

bedroom

living room

• How many gallons of paint does Mrs. Chan have left?

Mrs. Chan uses _  3 of the paint for her living room. 6

What information do we know? Mrs. Chan has 4 gallons of paint.

She uses _  1 of the paint for her bedroom. 3

She uses _  3 of the remaining paint for her living room. 4

What does the problem ask us to find? It asks for the fraction of the paint that Mrs. Chan uses to paint her living room. Read the first sentence of problem 2 to the class. Can we draw something? What should we draw? Yes. We should draw a tape diagram to represent the 4 gallons of paint. Draw and label a tape diagram and direct students to do the same.

4 gallons

Read the second sentence to the class. 226

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How can we represent this on the tape diagram? We can partition the tape into 3 equal parts and label 1 of them bedroom. Partition and label the tape diagram and direct students to do the same.

4 gallons

We can partition the remaining part of the tape into 4 equal parts and label 3 of them living room.

Some students may need support in

understanding that Mrs. Chan uses  3  of the 4

Read the third sentence to the class. How can we model this on the tape diagram?

Differentiation: Support

remaining paint for her living room, and

not   3  of the total paint. 4

bedroom

The remaining part of the tape is already partitioned into 2 parts. How can we partition those parts so we have 4 equal parts? We can partition each of the 2 parts into 2 equal parts. Partition and label the tape diagram and direct students to do the same. Does the tape show the fraction of paint that Mrs. Chan uses to paint her living room? Why? No, the entire tape is not partitioned into equal parts.

4 gallons

bedroom

living room

What can we draw so the tape shows the fraction of paint that Mrs. Chan uses to paint her living room? We can partition the third that represents the bedroom into 2 equal parts so the entire tape shows equal parts. Partition the rest of the tape diagram into equal parts.

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What fraction of the paint does Mrs. Chan use to paint her living room? How do you know? She uses _  3 of it to paint her living

4 gallons

room. I know because there are 3 parts out of 6 parts that represent the living room. Direct students to record the answer.

bedroom

living room

Invite students to turn and talk about equations that represent the problem. What equation can we write to show the fraction of paint Mrs. Chan had left after she painted her bedroom? 1 −  _ =  _ 1 3

2 3

What equation can we write to show the fraction of paint Mrs. Chan used to paint her living room?

_ _ __

6 3 2 ×   =   12 4 3

Did we get the same answer as we did when we used the tape diagram? How do you know? 6 Yes, because _  3  and  __  are equivalent. 6

Invite students to turn and talk about how the equations show what they drew in the tape diagram. Direct students to problem 3.

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3. Blake has 3 feet of ribbon. He uses _  3 of the ribbon for a project. He gives his friend _  1  of 4

the remaining ribbon. What fraction of the ribbon does Blake give to his friend?

3 feet

project

1 −  _ =  _

Blake gives  _1 of the ribbon to his friend.

3 4

1 4

friend

_1 ×  1_ =  1_ 2

Invite students to work with a partner to use the Read–Draw–Write process to solve the problem. As students work, circulate and provide guidance as needed. When students are finished, discuss the solution. Then invite students to turn and talk about how they can solve real-world problems that involve the multiplication of fractions.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

229

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Land

Debrief 5 min Objective: Multiply fractions. Facilitate a class discussion about multiplying fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Write _  3  × _     and _ 5  × _   . 4

5 6

Which expression has a product greater than    ? How do you know? 6

_ 5  × _ 5  > _ 5 because we are multiplying _ 5 by a fraction greater than 1. 4

In a multiplication expression, how does the size of the factors affect the size of the product? The size of one of the factors tells you whether the product is greater than, equal to, or less than the other factor. When you multiply a number by a fraction less than 1, the product will be less than the number. When you multiply a number by a fraction greater than 1, the product will be greater than the number. How can you compare multiplication expressions without finding the products? If one of the factors in the two expressions is the same, you can see which expression has the greater second factor.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

Name

Date

10. Lacy thinks _2 × _9 > _2 .

2. _2 as much as _3 4

3. _5 as much as _3

less than _3 4

greater than _3

less than _3 4

greater than _3

less than _3

_3 4

_1 3

_1 4

_2 3

Lacy might think the product is greater than _2 because _9 is greater than 1. More than 1 of _2 3

b. Find _2 × _9 . 8

greater than _3

_3 4

18 _ 2 × 9 = __ ____ =3 3×8

c. Is your answer reasonable? How do you know?

_5 4

Yes, my answer is reasonable because _3 is greater than _2 . 3

Use >, =, or < to compare the expressions. Explain how you can compare the expressions without evaluating them.

Multiply. 4. _1 × _8 = 4 9

is greater than _2 .

_5 3

_2 4

a. Explain why Lacy might think the product is greater than _2 .

Consider the statement. Circle to show whether the product is less than or greater than _3 . Then find 4 the product. 1. _1 as much as _3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

8 __ 36

5. _2 × _3 = 3 5

__ 11. _1 × 13 2 15

6 __ 15

_4 × 13 __ 5 15

Explain: The second factor is the same in both expressions and _1 is less than _4 . 5

6. _5 × _6 = 8 4

30 __ 32

7. _5 × _4 = 6 3

20 __ 18

41 13 × __ 12. __ 9 50

18 __ __ × 41 19 50

18 13 is greater than __ Explain: The second factor is the same in both expressions and __ . 9

3= 7 ×_ __ 10 2

21 __ 20

9. _2 × _8 = 9 7

16 __ 63

67 11 × ___ 13. __ 10 100

7 __ __ × 11 10 10

67 7 11 as a factor and ___ Explain: Both expressions have __ is less than __ . 10

101

102

PROBLEM SET

100

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5 ▸ M3 ▸ TB ▸ Lesson 11

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TB ▸ Lesson 11

Use the Read–Draw–Write process to solve each problem.

14. Tyler bakes 1 pan of brownies. He gives _1 of the brownies to his family. He gives his friends 3

_5 as much of the brownies as he gives his family. What fraction of the brownies is left? 4

5 _5 × 1_ = __ 4 3 12 5 7 12 − __ __ = __ 12 12 12 1 __ 7 −_ __ = 3 12 3 12 Tyler has 3 of the brownies left. 12

15. Miss Song’s favorite brand of almonds is sold in bags that weigh _7 pounds. She buys 1 bag

of almonds at the store and she has _1 of a bag of almonds at home. What is the total weight 8

of Miss Song’s almonds?

_1 + _2 = _3

2 2 2 5 21 _3 × _7 = __ = 1 __ 16 2 8 16

5 pounds. The total weight of Miss Song’s almonds is 1 __ 16

232

PROBLEM SET

103

Topic C Division with a Unit Fraction and a Whole Number In topic C, students use their understanding of multiplication of a whole number and a fraction to think about division of a whole number and a fraction. Students interpret quotients as the number of groups or as the size of the group, depending on the context of the problem. This topic begins the work of division with fractions. Students use tape diagrams and number lines to reason about division of a nonzero whole number by a unit fraction. They first explore the measurement interpretation of division and understand the divisor as the size of each group. Students represent the dividend with a tape diagram and partition it to show how many of the fractional units can fit into the dividend. This way,

they can clearly see there are 20 fourths in 5 when they find the quotient 5 ÷ _  1 . Students then 4

explore the partitive interpretation of division and understand the divisor as the number of groups. They represent the dividend with a tape diagram and use a number line to show that the dividend is a fraction of another number. By using these representations, students see that 5 is _  1 of 20 when they find the quotient 5 ÷ _  1 . 4

Then students use tape diagrams and number lines to reason about division of a unit fraction by a nonzero whole number. To help them find the quotient, they interpret the division expression _  1 ÷ 5 as _  1 is 5 groups of what? 4

Students solve real-world problems involving division of a whole number by a unit fraction or a unit fraction by a whole number. They determine that drawing a tape diagram helps them choose an operation and solve real-world problems. Students reason about the size of the quotients and recognize that the quotient of a whole number and a unit fraction is greater than the dividend. They also see that the quotient of a unit fraction and a whole number is less than the dividend. Students use this understanding to compare division expressions without finding the actual quotients. In topic D, students use their knowledge of dividing whole numbers by unit fractions and unit fractions by whole numbers to solve multi-step problems involving fractions. © Great Minds PBC

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Progression of Lessons Lesson 12

Lesson 13

Lesson 14

Divide a nonzero whole number by a unit fraction to find the number of groups.

Divide a nonzero whole number by a unit fraction to find the size of the group.

Divide a unit fraction by a nonzero whole number.

12 ? thirds I can interpret a unit fraction divisor as the size of the group by asking, How many

are in

1 4

I can use this interpretation to find a quotient that represents the number

0 1 2 3 4 5 5 5 5 5 5 5 I can interpret a unit fraction divisor as the number of groups by asking, is

of what number?

of groups. I can use a tape diagram to

I can use this interpretation to find a

both model and divide.

quotient that represents the size of the group. I can use a tape diagram

I can interpret a whole number divisor as the number of groups when the dividend is a unit fraction by asking, is groups of what? I can use this interpretation to find a quotient that represents the size of the group. When I model with a tape diagram, I need to show the total units in 1 so I can accurately identify the quotient.

and a number line to both model and divide.

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Lesson 15

Lesson 16

Lesson 17

Divide by whole numbers and unit fractions.

Reason about the size of quotients of whole numbers and unit fractions and quotients of unit fractions and whole numbers.

Solve word problems involving fractions with multiplication and division.

1 4

? I can use the Read–Draw–Write process to solve problems involving dividing by whole numbers and unit fractions. I can reason about the size of the quotient based on the dividend and the divisor to check the reasonableness of my answer.

1 ÷ 4 > 1 ÷4 2 4 Both expressions have a divisor of 4. Because 1 > 1 , that means 1 ÷ 4 2 2 4 has the greater quotient. I can reason about the size of quotients when I divide a whole number by a unit fraction or divide a unit fraction by a whole number. I can explain this reasoning by drawing representations or writing explanations. I can use this reasoning to compare the values of expressions without evaluating.

4 yards 1 4

...

? fourths 4 yards

? fourths

4÷ 41 = 16 Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She has enough fabric to make 16 pencil pouches, but she has 24 students. By using information from a word problem, I can decide which operation to use to solve the problem. I can discuss with classmates about why someone may have chosen a different operation or a different pathway to the solution.

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Language Objectives Language objectives indicate the language and literacy skills students need to engage with the lesson objectives. Because language learning and mathematical learning are interdependent, teaching toward language objectives helps teachers to consider language needs when supporting students in reaching the lesson objectives. Lesson 12 Read word problems involving division of a whole number by a unit fraction. Listen to and orally describe the problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation for solving a problem that involves division of a whole number by a unit fraction. Lesson 13 Read word problems involving division of a whole number by a unit fraction. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation for solving a problem that involves division of a whole number by a fraction. Lesson 14 Orally interpret division of a unit fraction by a whole number as an unknown factor problem by using the sentence frame is groups of what? Using tape diagrams and real-world contexts, orally reason about the size of the quotient in problems that involve division of a unit fraction by a whole number. Lesson 15 Read word problems involving division with whole numbers and unit fractions. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation for solving a problem that involves division with whole numbers and unit fractions. Orally and in writing, justify why a solution strategy represents a word problem. In partners, listen to a critique of the solution strategy and make revisions.

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Lesson 16 Orally and in writing, reason about the size of quotients in division expressions by using an available context or a generalization about the relationship between the size of the dividend and the size of the quotient. Use sentence frames to orally summarize observations about dividing whole numbers and unit fractions. Lesson 17 Read word problems involving fraction multiplication and division. Write or draw to record solution strategies and equations. Listen to and orally describe solution strategies, including multiplying or dividing, and compare and connect strategies.

237

LESSON 12

Divide a nonzero whole number by a unit fraction to find the number of groups.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

Name

Date

whole number by any unit fraction. They interpret a division 1 3

1 yard of ribbon to make each bow. Jada buys 7 yards of ribbon to make bows. She uses _

Read–Draw–Write process to solve problems involving division of a

nonzero whole number by a unit fraction.

7 1 2

Students relate dividing 1 by a unit fraction to dividing a nonzero expression like 4 ÷  _ as, How many thirds are in 4? Students use the

Use the Read–Draw–Write process to solve the problem. How many bows can Jada make?

Lesson at a Glance

...

Key Questions

? halves

• How do tape diagrams and number lines help you understand dividing a whole number by a unit fraction to find the number of groups?

7 ÷ 1_ = 14 2

Jada can make 14 bows.

• Does division by a unit fraction result in a quotient that is greater than the dividend? Why?

Achievement Descriptors 5.Mod3.AD3 Translate between numerical expressions that include

fractions and mathematical or contextual verbal descriptions. (5.OA.A.2) 5.Mod3.AD12 Model and evaluate division of whole numbers by unit

fractions. (5.NF.B.7.b) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c) © Great Minds PBC

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 10 min

• None

Learn 30 min

Students

• Use a Number Line and a Tape Diagram to Divide

• None

• Use a Tape Diagram to Divide • Problem Set

Land 10 min

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Fluency

Choral Response: Divide Whole Numbers Students say a division expression to represent a question and then say the quotient to prepare for division of a whole number by a unit fraction. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the question. What division expression represents the question?

How many twos are in 6?

6÷2

6÷2 3

Display the division expression. What is 6 ÷ 2?

3 Display the answer. Repeat the process with the following sequence: How many threes are in 27?

12 is 4 groups of what?

25 is 5 groups of what?

How many sixes are in 42?

63 is 7 groups of what?

How many eights are in 64?

54 is 9 groups of what?

27 ÷ 3

63 ÷ 7

240

12 ÷ 4

64 ÷ 8

25 ÷ 5

42 ÷ 6

54 ÷ 9

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

Choral Response: Fractions Equal to Whole Numbers Students count by thirds or sixths on a number line and recognize fractions as whole numbers to prepare for division of a whole number by a unit fraction. Display the number line partitioned into thirds. Use the number line to count forward by thirds from 0 thirds to 15 thirds. The first number you say is 0 thirds. Ready? Display each fraction one at a time on the number line as students count. 0_ 1_ 14 15 3 , 3 , … , __ , __ 3 3 Display a point on the line at  _  . 3 3

What whole number is equivalent to   ? Raise your hand when you know. 3

Wait until most students raise their hands, and then signal for students to respond.

1 Display the answer. Continue the

0 3

1 3

2 3

3 3

4 3

5 3

6 3

7 3

8 3

9 3

10 3

11 3

12 3

13 3

14 3

15 3

process with

 _ ,  __ , and  __ . 9 15 3 3

12 3

Repeat the process with sixths.

1 0 6

1 6

2 6

3 6

4 6

5 6

3 6 6

7 6

8 6

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

Teacher Note

Count by sixths from 0 sixths to

24 sixths and then

fractions that are equivalent to whole

display points on the line at  _  ,  __  ,  _  , and  __ . Ask students to identify the equivalent whole number for each fraction.

Consider asking students to identify other

6 12 0 6 6 6

24 6

numbers located on the number line shown

__6

(e.g.,   = 2). 3

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Whiteboard Exchange: Multiply a Whole Number by a Fraction Students determine the product to prepare for relating multiplication by unit fractions to division by unit fractions beginning in lesson 13. Display  _× 6 = 1 3

. Teacher Note

Write and complete the equation.

1 ×6= 3

Validate all correct responses that may not be displayed. For example, when a student

__3

evaluates the expression   × 2, they may

__6 __3 __2 4 __1

choose to write   ,   , 1   , or 1   . 4 2

Display the product.

Repeat the process with the following sequence:

2 ×9= 3

242

1 ×4= 4

3 ×2= 4

6 4

1 × 10 = 5

4 ×3= 5

12 5

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TC ▸ Lesson 12

Students divide 1 by a unit fraction by using a tape diagram and a number line. How many eighths are in 1?

8 eighths are in 1. Direct students to show 8 eighths are in 1 by using a number line or a tape diagram. Find a sample of each representation to display.

1 0 8

1 8

2 8

3 8

4 8

5 8

6 8

7 8

8 8

Invite students to think–pair–share about how both the number line and the tape diagram show that 8 eighths are in 1.

The number line shows each eighth, counting up from  _  to  _  . There are 8 equal intervals between  _  and  _  . 0 8

8 8

0 8

8 8

The tape diagram shows 1 partitioned into 8 equal units, so there are 8 eighths in 1. Display the following tape diagram and number line.

0 8

1 8

2 8

3 8

4 8

5 8

6 8

7 8

8 8

What do you notice? What do you wonder? I notice the tape diagram is above the number line. I notice the tape diagram is not labeled 1. I wonder why it shows both the number line and tape diagram together. © Great Minds PBC

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Invite students to turn and talk as they try to write multiple equations that match what they see in the representation. Students might use multiplication, division, addition, or subtraction. Expect some students to need support to write an equation because there is no question mark or unknown shown. Allow the experience to be open and allow them to struggle productively. Write 1 ÷  _ = 8. 1 8

One equation that matches the representation is 1 ÷   = 8. Let’s explore why 8

this is true. How is 1 shown in the representation?

The tape represents a total of  _  , which is 1.

8 8

How is   shown in the representation? 8

The labels on the number line show counting by eighths, so each interval represents  _  . 1 8

How is 8 shown in the representation? There are 8 equal units in the tape diagram. The number line shows 8 eighths. How is division represented? The tape diagram and the number line are partitioned into 8 units. Based on the representation, what does the divisor represent in the equation

1 ÷  _1 = 8? 8

 _ is the size of each unit, or the size of each group. 1 8

What does the quotient represent? The number of groups Invite students to turn and talk about how they might use what they observed in this representation to help them find 2 ÷  _  . 1 8

Transition to the next segment by framing the work. Today, we will divide whole numbers by unit fractions and interpret each quotient as the number of unit fractions in each whole number. 244

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Learn

5 ▸ M3 ▸ TC ▸ Lesson 12

Teacher Note

Use a Number Line and a Tape Diagram to Divide Students use a number line and tape diagram to divide a nonzero whole number by a unit fraction.

by a unit fraction. In measurement division, the divisor represents the size of the group and the quotient represents the number of

(or units, or groups of 2) are in 6? Likewise,

__1 4

Before we find how many halves are in 2, let’s begin by finding how many halves are in 1. How can that help us? When we know how many halves are in 1, we can double it because there are 2 ones in 2. Earlier, we thought about how many eighths are in 1, so I think we should start with 1. When we know how many are in 1, we can use that to find how many are in 2.

such as 6 ÷ 2, would think, How many twos

with 3 ÷   , a student would think, How many

Write the question: How many halves are in 2?

What can we do to show how many halves are in 1?

groups. A student who uses measurement division with a whole-number expression,

The tape diagram shows a total of 2 because it is equivalent to the distance on the number line from 0 to 2.

Guide students to locate and label 1 on the number line, and then partition the tape diagram in the same location.

interpretation of division to reason about the quotient of a nonzero whole number divided

Draw the tape diagram and number line and direct students to do the same. What total does the tape diagram represent? How do you know?

In this lesson, students use the measurement

__1

fourths (or units, or groups of   ) are in 3? 4

Another way to think about a division expression is by using the partitive interpretation. In partitive division, the divisor represents the number of groups and the quotient tells the size of each group. Students use the partitive interpretation of division in an upcoming lesson. It is not important for students to identify division as measurement or partitive, but it is important they know whether they are finding the number of groups or the size of the group when they divide.

Partition the section of the tape diagram between 0 and 1 into 2 equal parts.

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Partition the tape diagram between 0 and 1 into 2 equal parts by using dotted lines while students do the same. How many halves are in 1?

2 halves are in 1.

Differentiation: Support

Some students might find it helpful to partition the fractional part by using a different color instead of using a dotted line.

Let’s continue partitioning to show how many halves are in 2. Partition the tape diagram between 1 and 2 into 2 equal parts by using dotted lines while students do the same. How many halves are in 2?

So our tape diagram shows that 4 halves, or 4 groups of   , make 2. Let’s label and 2

count on our number line to confirm that.

Starting with  _ , label the 0 2

number line as you count along with students. Does our number line confirm that 4 halves make 2? How? Yes, because we started at 0 halves, or 0, and counted up to 4 halves, or 2.

0 0 2

1 2

2 2

3 2

2 4 2

What division equation can we write to represent the tape diagram and the number line? 1 2 ÷ _2 = 4

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Does it make sense that the answer is greater than 2? Why? Yes, it makes sense. 2 halves are in 1, so there are twice as many halves in 2 as there are in 1. Yes, it makes sense. We found that there are 2 ones in 2. Halves are smaller units than ones, so it makes sense that more halves are in 2 than there are ones in 2. Invite students to turn and talk about how many eighths are in 2. Is the number of eighths in 2 greater than or less than the number of halves in 2? Why? The number of eighths in 2 is greater than the number of halves in 2. There are more eighths than halves in 1, so there are more eighths than halves in 2. I think it is greater. Eighths are smaller than halves, so more eighths can fit into 2 than halves. Is using a tape diagram and a number line helpful to find the number of unit fractions in a whole number? How? It is helpful because we can see and count the number of unit fractions in the whole number in both the number line and the tape diagram. It is helpful because we can partition the tape diagram and then label and count the number of unit fractions by using the number line. Encourage students to be thoughtful about what they draw and show when they divide a whole number by a unit fraction. Expect some students to continue showing both the number line and tape diagram to find quotients, while other students may choose to use only one of the representations.

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Use a Tape Diagram to Divide Students use tape diagrams to divide a nonzero whole number by a unit fraction. Direct students to problem 1 in their books and read the problem chorally. Use the Read-Draw-Write process to solve each problem. 1. A family makes 3 pans of brownies for a bake sale. They plan to sell gift bags that each _1 hold    of a pan of brownies. How many gift bags can the family make? 2

3 1 2

...

? halves 3

3 ÷ _2 = 6 1

The family can make 6 gift bags of brownies. Reread the first sentence. Can we draw something? What can we draw? Yes, we can draw a tape diagram to represent the 3 pans of brownies. Let’s draw only the tape diagram. Because we will not have the number line to show us the total the tape diagram represents, we will label the total represented by the tape diagram.

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Draw and label a tape diagram and direct students to do the same. Reread the second sentence. Can we draw something? What can we draw? We can draw a part and label it  _  . 1 2

What does   represent? 2

It represents the part of the pan of brownies the family will put in a gift bag to sell. Guide students in drawing a part of the tape diagram and labeling it  _  . 1 2

Reread the third sentence. What does the problem ask us to find?

3 1 2

It asks the number of gift bags of brownies the family can make. Invite students to turn and talk about where the model should be labeled with a question mark. The problem asks how many gift bags of brownies the family can make, so we need to find how many halves are in 3. Let’s label the unknown in the tape diagram. Label the unknown with a question mark and direct students to do the same. What expression matches the tape diagram? 1 3 ÷ _2 Can the family make more than 1 gift bag of brownies? Why?

3 1 2

... ? halves

Yes, because  _ of each pan makes 1 gift bag, and there is more than 1 half in 3. 1 2

The tape diagram helped us to make sense of what the problem says, identify what we are trying to find, and write an expression. Let’s use another tape diagram to help us solve the problem.

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Draw another tape diagram, label the entire tape 3, and direct students to do the same.

This tape diagram represents the 3 pans of brownies the family has. We want to find how many halves are in 3. We might not know right away how many halves are in 3. What could we do to make a simpler problem? Earlier we first found how many halves are in 1. We could partition 3 to show each pan of brownies. Then we could partition each pan of brownies into halves. Partition the tape into 3 equal parts and direct students to do the same. Point to 1 part.

In this tape diagram, according to the problem, what does 1 part represent?

Language Support

1 pan of brownies Direct students to use a dotted line to partition 1 part to show halves. How many halves are in 1?

2 Direct students to continue partitioning the tape. As they partition, ask the following questions.

How many halves are in 2?

Consider posting a visual to support students in describing what each part of the written equation represents. Add to the visual as each equation is introduced and label the dividend, divisor, and quotient. Leave the visual posted for students to reference. For example:

Dividend Divisor

Quotient

3÷

1 2

How many halves are in 3? 3 ÷

1 =6 2

6÷

1 4

How many fourths are in 6? 6 ÷

1 = 24 4

Does 6 answer the question the problem asks? Why?

4÷

1 6

How many sixths are in 4?

4÷

1 = 24 6

How many halves are in 3?

Yes, because 6 is the number of halves in 3, and it is the number of gift bags the family can make.

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Have students record the answer statement: The family can make 6 gift bags of brownies. Let’s think about what happened in this problem. We needed to find how many halves are in 3. What equation tells us how many halves are in 3? 1 3 ÷ _2 = 6 Let’s think about what each of these numbers represent. What does the dividend, 3, represent? The dividend represents the number of pans of brownies the family has.

What does the divisor,   , represent? 2

The divisor represents the number of pans of brownies that go into each gift bag. What does 6 represent? It represents the number of gift bags of brownies the family can make.

We showed that 6 halves, or 6 groups of   , make 3. 2

To help us find 3 ÷   , what question did we ask? 2

How many halves are in 3? Why did we draw a second tape diagram? We used the second tape diagram to solve the problem. The first tape showed us we

Promoting the Standards for Mathematical Practice As students analyze and use division of a nonzero whole number by a unit fraction to solve real-world problems, they are making sense of problems and persevering in solving them (MP1). Ask the following questions to promote MP1: • How can you explain this context in your own words? • What could you try to start solving the problem? • Does your answer make sense? Why?

were dividing 3 by  _  and that  _ was the size of the group. The second tape helped 1 2

1 2

us solve the problem by finding the number of halves in 1, then in 2, and then in 3.

Teacher Note

The dividend is 3 and the quotient is 6. Why is the quotient greater than the dividend? Our tape diagram represents 3 pans of brownies, and each pan is cut into 2 parts to make halves. That is why the quotient 6 is greater than the dividend 3. I know 2 halves are in 1 and 3 ones are in 3, so there must be more than 3 halves in 3. There are 2 halves in 1 and 3 ones in 3. So if we partition each 1 into halves, we will have more than 3 halves. We will have 6 halves. Because we are dividing by a unit fraction, which is less than 1, our quotient is greater than the dividend. Can you think of a whole number for which this is not true?

A context video for the word problem in problem 2 is available. It may be used to remove language or cultural barriers and provide student engagement. Before providing the problem to students, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before being asked to interpret it mathematically.

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Could the quotient of a whole number and a unit fraction ever be less than or equal to the dividend? Yes. Zero is a whole number, and zero divided by a unit fraction is zero. So the quotient of zero and a unit fraction is equal to the dividend, not greater than the dividend. The quotient of zero and a unit fraction is zero. So the quotient of zero and a unit fraction is equal to the dividend instead of greater than the dividend, as we have seen in other problems. When we make general statements about a whole number divided by a unit fraction, we must be careful to say whole numbers except zero. Direct students to problem 2. Have them read the problem and work with a partner to draw a tape diagram to represent the situation. Circulate as students work and encourage them to think about which quantity is the dividend and which quantity is the divisor.

2. Julie paints birdhouses. She uses    pint of paint for each birdhouse. How many 4

birdhouses can Julie paint with 6 pints of paint?

6 1 4

...

1 6 ÷ 4 = 24

Julie can paint 24 birdhouses.

Have students think–pair–share about how the quantities and the unknown are represented in the tape diagram. The tape represents a total of 6. That is the total number of pints of paint Julie has. 252

Students may draw a second tape diagram to solve the problem as they did previously, or they may draw a tape diagram above a number line as they saw in the beginning of Learn. Students may also draw 6 individual parts to represent the 6 pints and partition each part into fourths to find the quotient. All methods are valid ways of using a model to solve the problem.

Differentiation: Challenge

? fourths

After most students finish, display the tape diagram.

Teacher Note

6 1 4

...

Challenge students who finish early to create their own division problems for classmates to solve. Encourage students to study the problems in their classwork first and identify the quantity that represents the total, the quantity that represents the size of each group, and the question that must be answered. Tell students to include these same components in their problems and to provide work samples on a separate page that show different ways to solve the problems.

? fourths

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

The tape shows  _ , and we are finding the number of fourths in 6. 1 4

What does the word problem ask us to find? It asks how many birdhouses Julie can paint. Ask students to revise their tape diagrams as needed. What division expression represents the problem? 1 6 ÷ _4 To evaluate the expression, what do we need to find? We need to find how many fourths are in 6. Have students work with a partner to draw a model to help them solve the problem. When most students finish, invite them to discuss the quotient by asking the following questions.

UDL: Action & Expression

The quotient is 24. It represents the number of birdhouses Julie can paint.

Consider posting guiding questions that encourage students to monitor and evaluate their progress as they complete problem 2.

Does it make sense that the quotient is greater than the dividend? Why?

Plan:

What is the quotient and what does it represent in this problem?

Yes, we are finding how many fractional parts fit into 6, so it makes sense that the answer is greater than 6.

Yes, it makes sense because Julie only needs  _ pint of paint for each birdhouse. 4 fourths 1 4

are in 1, and there are 6 ones in 6. So the number of fourths in 6 is greater than 6.

• What question are we trying to answer? Monitor: • Is our answer reasonable? • Should we try something else? Evaluate: • What worked well? • What might we do differently next time?

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Direct students to problem 3 and invite them to work with a partner to solve the problem.

3. Ryan makes bags of peanuts for snacks. He has 4 pounds of peanuts. He puts    pound of 6

peanuts into each bag. How many bags can he make?

4 1 6

... ? sixths

Differentiation: Support If partners need more support, refer to previous examples and ask students why they used two tape diagrams. Encourage them to first create a tape to make sense of what the problem tells them and what they are trying to find.

4 ÷ _6 = 24 1

Ryan can make 24 bags of peanuts. As students work, circulate and ask the following questions to advance student thinking: • What can you draw to represent the problem?

Differentiation: Challenge

• Which number is the dividend? What does it represent? • Which number is the divisor? What does it represent? • What division expression represents the problem? • Is the quotient greater than or less than the dividend? How do you know? When students finish, invite them to turn and talk about how they can use a model to divide a whole number by a unit fraction when the divisor represents the size of each group and the quotient represents the number of groups.

Problem Set

Consider challenging students to explain

__1

mathematically why 6 ÷   = 4 ÷   . 4

Expect students to explain that when they partition 1 into fourths they get 4 equal parts, so partitioning 6 into fourths results in 24 equal parts. Similarly, when they partition 1 into sixths they get 6 equal parts, so partitioning 4 into sixths also results in 24 equal parts.

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

254

EUREKA MATH2 California Edition

Land

5 ▸ M3 ▸ TC ▸ Lesson 12

Debrief 5 min Objective: Divide a nonzero whole number by a unit fraction to find the number of groups. Facilitate a class discussion about dividing a whole number by a unit fraction to find the number of groups by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the expression 3 ÷  _ and the tape diagram. 1 3

What do the dividend and the divisor represent in the expression? The dividend represents the total. The divisor represents the size of each group.

? thirds

What do we need to find? We need to find how many thirds are in 3. How does the tape diagram show the quotient? The tape is partitioned into 3 units, and each unit is partitioned into thirds. It shows the quotient is 9.

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5 ▸ M3 ▸ TC ▸ Lesson 12

EUREKA MATH2 California Edition

How do tape diagrams and number lines help you understand dividing a whole number by a unit fraction to find the number of groups? I can draw a tape diagram and a number line to represent the total, then I can divide it into ones so I can first find how many groups are in 1. Then I can keep partitioning to find out how many groups are in the total. Does division by a unit fraction result in a quotient that is greater than the dividend? Why? Yes, except when the dividend is zero. When the dividend is zero, the quotient is zero. Yes, because it takes more than 1 unit fraction to make a whole. The number of unit fractions needed is the quotient and the whole number is the dividend. So when we divide a whole number by a unit fraction, the number of unit fractions we need to make the whole is greater than the whole number, except when the whole number is zero.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

Name

Date

Draw a model to find the quotient. Then complete the statements to check your work. 5. 5 ÷ 1_ = 2

halves in 1.

There are

halves in 2.

2 ÷ 1_ = 2

1 2

3 2

10 3

thirds in 1.

There are

thirds in 2.

2÷1= 3

1 3

2 3

4 3

5 3

1 make 5. groups of _ 2

thirds in 1.

There are

thirds in 3.

3 ÷ 1_ = 3

fourths in 1.

There are

fourths in 3.

3 ÷ 1_ = 4

groups of

_1 6

make

1 gallon. How many a. 1 gallon of juice is poured equally into containers. Each container holds _ 8

8 containers can be filled with juice.

1 3

4. There are

7. Solve the related problems.

containers can be filled with juice? 3. There are

0 1 2 3 4 5 6 7 8 9 10 11 12 6 6 6 6 6 6 6 6 6 6 6 6 6

1 2

Sample:

2. There are

6. 2 ÷ 1_ = 6

Sample:

Use the model to help you complete each statement and divide. 1. There are

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

1 gallon. b. 4 gallons of juice are poured equally into containers. Each container holds _

How many containers can be filled with juice?

32 containers can be filled with juice.

1 4

109

110

PROBLEM SET

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

8. The chart shows the ingredients needed to make a small pizza. Use the chart to write an expression. Then evaluate the expression and use a sentence to write your answer. Ingredient

Amount

Dough

_1 pound

Sauce

_1 of a jar

Cheese

_1 cup

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 12

Use the Read–Draw–Write process to solve each problem. 9. A family orders 6 large sandwiches. They cut each large sandwich into thirds to make smaller sandwiches. How many smaller sandwiches does the family have?

6 ÷ 1_ = 18 3

The family has 18 smaller sandwiches.

1 cup at each meal. How many meals 10. A bag of cat food holds 12 cups. Mrs. Chan feeds her cat _

a. How many pizzas can be made with 5 pounds of dough?

can Mrs. Chan feed her cat from the bag of cat food?

5 ÷ 1_

12 ÷ 1_ = 24

15 pizzas can be made with 5 pounds of dough.

Mrs. Chan can feed her cat 24 meals from the bag of cat food.

b. How many pizzas can be made with 4 jars of sauce?

4 ÷ 1_ 5

1 cup. How many times will Mr. Evans need to fill his 11. The only measuring cup Mr. Evans has is a _

20 pizzas can be made with 4 jars of sauce.

measuring cup to measure 3 cups of flour?

3 ÷ 1_ = 12 4

Mr. Evans will need to fill his measuring cup 12 times.

c. How many pizzas can be made with 4 cups of cheese?

4 ÷ 1_ 4

16 pizzas can be made with 4 cups of cheese.

258

PROBLEM SET

111

112

PROBLEM SET

LESSON 13

Divide a nonzero whole number by a unit fraction to find the size of the group.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

Name

Date

Use the Read–Draw–Write process to solve the problem.

1 of the number of slices Lacy needs for her party. Each package holds 6 slices of cheese. This is _ 5

How many slices of cheese does Lacy need for her party?

Students compare division problems in which the divisor represents the size of the group to division problems in which the divisor represents the number of groups. They interpret a division expression like 4 ÷ _  1 3

as the question, 4 is  _ of what? Students use the Read–Draw–Write 1 3

process to solve problems involving division of a nonzero whole

number by a unit fraction in which the divisor represents the number

0 5

Lesson at a Glance

1 5

2 5

3 5

4 5

of groups.

5 5

Key Questions

6 ÷ 1_ = 30 5

• How do tape diagrams and number lines help you understand dividing a whole number by a unit fraction to find the size of each group?

Lacy needs 30 slices of cheese for her party.

• Does division by a unit fraction result in a quotient that is greater than the dividend? Why?

Achievement Descriptors 5.Mod3.AD12 Model and evaluate division of whole numbers by unit

fractions. (5.NF.B.7.b) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c) © Great Minds PBC

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Interpret a Division Expression

• None

• Use a Tape Diagram and a Number Line to Divide • Problem Set

Land 10 min

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Fluency

Choral Response: Divide Whole Numbers Students say a division expression to represent a question and then say the quotient to develop fluency with dividing a whole number by a unit fraction. After asking each question, wait until most students raise their hands, and then signal for students to respond. Raise your hand when you know the answer to each question. Wait for my signal to say the answer. Display the question. What division expression represents the question?

How many twos are in 8?

8÷2

Display the division expression.

What is 8 ÷ 2?

Display the quotient. Repeat the process with the following sequence: How many threes are in 24?

16 is 4 groups of what?

40 is 5 groups of what?

How many sixes are in 36?

56 is 7 groups of what?

How many eights are in 72?

81 is 9 groups of what?

24 ÷ 3

56 ÷ 7

262

16 ÷ 4

72 ÷ 8

40 ÷ 5

36 ÷ 6

81 ÷ 9

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

Choral Response: Fractions Equal to Whole Numbers Students count by fourths or eighths on a number line and recognize fractions as whole numbers to develop fluency with dividing a whole number by a unit fraction. Display the number line partitioned into fourths. Use the number line to count forward by fourths from 0 fourths to 16 fourths. The first number you say is 0 fourths. Ready? Display each fraction one at a time on the number line as students count. 15 16 0_ 1_ 4 , 4 , … , __ , __ 4 4 Display a point

on the line at  _  . 4 4

0 4

1 4

2 4

3 4

4 4

What whole

5 4

6 4

number is

7 4

8 4

9 4

10 4

11 4

12 4

13 4

14 4

15 4

16 4

equivalent

to   ? Raise your hand when you know. 4

Wait until most students raise their hands, and then signal for students to respond.

1 Display the answer. Continue the process with  _  ,  __  , and  __  . 8 16 4 4

Repeat the process with eighths. Count

0 8

1 8

12 4

2 8

3 8

4 8

5 8

6 8

7 8

8 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

by eighths from

0 eighths to

24 eighths, and

then display points on the line at  _  ,  _  ,  __  , and  __ . Ask students to identify the equivalent whole number for each fraction. © Great Minds PBC

0 8 24 8 8 8

16 8

263

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5 ▸ M3 ▸ TC ▸ Lesson 13

Write and complete the equation.

1 ×6= 2

Display the product. Repeat the process with the following sequence:

2 ×8= 2

264

2 × 15 = 3

3 ×3= 4

9 4

2 ×4= 5

8 5

5 ×4= 6

20 6

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TC ▸ Lesson 13

Students model and solve an unknown factor problem. Write 6 is  _ of what number? 1 2

Read the question aloud. Then invite students to answer the question by drawing a tape diagram on their whiteboards. Consider having students work in pairs. Circulate as students work to identify those who model the problem with the following tape diagram. When students finish modeling and find the answer, invite an identified student to share their model and thinking with the class. If students did not draw a similar tape diagram, offer it as your own.

Does this tape diagram represent the question? How?

Yes. It shows  _ of the total labeled with 6, and the total is unknown. 1 2

How can we use the tape diagram to answer the question? We know the value of 1 unit in the tape diagram is 6, so 2 units make 12.

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We know 6 is   of 12. What equation can we write to represent the original question, 2

6 is  _1 of what number? 2

We can write 6 =  _ × 1 2

Does 6 × 2 =

. represent the original question? Why?

No, because the question asks us to find  _ of an unknown number, and  _ is not 1 2

in that equation.

1 2

No, because to find  _ of a number means to multiply  _ and a number, and this equation shows 6 × 2.

Does 6 ÷   = 2

1 2

represent the original question? Why?

I am not sure. It uses the numbers from the original question, and the unknown number is the same answer we found by using the tape diagram, but I found the answer by multiplying, and this is a division equation. Allow this uncertainty to linger. Resolution is reached in the next segment. If possible, keep the original question visible for students to reference later. Transition to the next segment by framing the work. Today, we will divide whole numbers by unit fractions and learn a new way to interpret the quotients.

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Learn

5 ▸ M3 ▸ TC ▸ Lesson 13

Interpret a Division Expression Students interpret division of a nonzero whole number by a unit fraction as finding the size of the group. 4 Display the tape diagram.

Invite students to turn and talk about a division equation and a multiplication equation that match the tape diagram.

8 halves

What division equation can we write to represent this tape diagram? Why?

Teacher Note In this lesson, students use the partitive interpretation of division to reason about the quotient of a nonzero whole number divided by a unit fraction. In partitive division, the divisor represents the

4 ÷  _ = 8 because 8 halves are in 4.

number of groups and the quotient tells the

What multiplication equation can we write to represent this tape diagram? Why?

about division with whole numbers, partitive

1 2

size of each group. When students first learn

8 ×  _ = 4because there are 8 groups of  _ , and that is equal to 4. 1 2

1 2

1 1 Write 4 ÷  _ = 8and 8 ×  _ = 4beneath the tape diagram. 2

Invite students to think–pair–share about what each number means in each equation. In the division equation, 4 is the total being divided. In the multiplication equation,

4 is the total when you have 8 groups of  _12  . 1 In the division equation,  _ is the size of each group. In the multiplication equation, 2 _ 1 is the size of each group.

division is generally more accessible (e.g., If

6 cookies are shared equally with 2 people, how many cookies does each person get?). With whole-number division, such as 6 ÷ 2, students can think partitively and ask, 6 is

2 groups of what? By using an example with

__1

a fractional divisor, such as 3 ÷   , a student

__1 can think, 3 is   of what?

In the division equation, 8 is the number of groups. In the multiplication equation, 8 is the number of groups.

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Display the tape diagram.

_1 In the other tape diagram, we could see 8,   , and 4. 2 _1 Do you see 8,   , and 4 in this tape diagram? Where? 2 8 is the whole tape diagram. 4 is one part, or  _1 of the

tape. Two halves are shown in the tape diagram.

By using 8,   , and 4, what multiplication equation can we write to represent this tape 2

diagram? Why?

 _ × 8 = 4. The tape diagram shows  _  of 8 is 4, and I know finding a fraction of a whole 1 2

number is multiplication.

1 2

Give students 1 minute to discuss with a partner the division equation they could write by

using 8,  _  , and 4 that matches the tape diagram. Expect that students may need support, 1 2

but encourage them to make connections to their earlier learning.

I know it cannot be 8 ÷  _ = 4because when you divide 8 by  _ , the quotient is greater than 8.

1 2

I want to write 8 ÷ 2 = 4, but that does not use  _  . 1 2

The other tape diagram showed 4 ÷  _ = 8, so I do not know how that could be the same for this tape diagram.

268

1 2

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5 ▸ M3 ▸ TC ▸ Lesson 13

Guide students to realize 4 ÷  _ = 8matches the tape diagram by asking the following questions. 1 2

When we learned how to find a fraction of a set, which number, 8,   , or 4, would be the 2

unknown in the tape diagram? Why?

UDL: Representation

4 would be the unknown because we would be finding  _1  of 8. 2

_1 _1 To determine   of 8, we could ask ourselves,   of 8 is what number? 2

By using the numbers 4 and   , what question might we ask ourselves if 8 was the 2

unknown in the tape diagram?

We might ask, 4 is  _ of what number? 1 2

Divisor is the size Divisor is the of the group. number of groups.

Have we answered a question like 4 is   of what number? When? 2

Yes. It is like the question earlier about how 6 is  _ of a number. 1 2

Earlier, we were not sure whether such a question could be represented by a

While students explore the different interpretations and representations of division and the connection to multiplication, consider recording on an anchor chart for them to reference.

division equation. But now we can see that the question 4 is   of what number 2

is helpful when you find the quotient 8 in the division problem 4 ÷   = 8.

Write 4 ÷  _ = 8 and  _ × 8 = 4beneath the tape diagram. 1 2

1 2

How are these equations similar to the equations we used in our first tape diagram? How are they different? Both equations use 8,  _  , and 4. 1 2

The division equation is the same. The order of the factors in the multiplication equation is different. The first equation showed 8 groups of  _  . This equation shows  _  of 8. 1 2

1 2

8 halves How many halves are in 4? 4÷ 1 =8

4 is 21 of what? 4÷ 1 =8

2 Group Size Number of Groups

2 Number of Groups Group Size

8× 1 =4

1 ×8=4 2

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Think about the multiplication equation. What do the factors   and 8 mean? 2

Because  _ is first in the equation, it represents the number of groups. So we have  _ of a 1 2

1 2

group of 8, and 8 represents the size of the whole group.

Because division and multiplication are related, those numbers have the same meaning in the division equation, just like in our first tape diagram. In a previous lesson, we thought about division as finding the number of groups. We know we can also interpret division as finding the size of the group. Let’s explore this interpretation of division by using word problems.

Invite students to turn and talk about how both tape diagrams show 4 ÷  _ = 8. 1 2

Use a Tape Diagram and a Number Line to Divide Students use a tape diagram to solve problems by dividing a nonzero whole number by a fraction to find the size of the group. Direct students to problem 1 in their books. Read the problem chorally with the class. Use the Read-Draw-Write process to solve each problem.

1. Lacy reads 12 pages of a book. This is    of the number of pages in the book. How many 5

pages are in Lacy’s book?

? 12

0 5

1 5

2 5

3 5

4 5

5 5

12 ÷ _5 = 60 1

There are 60 pages in Lacy’s book. 270

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5 ▸ M3 ▸ TC ▸ Lesson 13

Read the first sentence of the problem aloud again. Can we draw something to represent the problem so far? What? We can draw a tape to represent 12 pages.

Draw and label a tape diagram. Label 12 inside the part. Direct students to do the same. Read the next sentence aloud. What new information do we have?

12 is  _15 of the number of pages in the book.

To help us show 12 is   of the book, let’s also start a number line below the tape diagram. 5

Draw the beginning of a number line below the tape diagram. Direct students to do the same.

Can we draw anything else to represent this part of the problem? What can we draw?

0 5

We can draw 4 more parts of the tape to represent the rest of the fifths.

1 5

Because we need all 5 fifths to represent all the pages in the book, let’s continue the number line below our tape diagram. Complete the tape diagram and direct students to do the same.

We now have 5 parts. Why did we draw 5 parts? We need to represent the number of pages in the whole book.

12 represents only 1 fifth, so we need to draw 5 fifths to represent all the pages in the book.

0 5

1 5

2 5

3 5

4 5

5 5

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Read the rest of the problem and then ask the following questions. What is the unknown in the problem? How can we show that in the tape diagram? The unknown is the number of pages in Lacy’s book. We can label the whole tape with a question mark.

Label the tape and direct students to do the same. Look at our tape diagram. What conclusions can you make?

The number of pages is more than 12. Lacy has read less than half of the book. What expression can you use to help find the number of pages in Lacy’s book? Why?

0 5

1 5

2 5

3 5

4 5

5 5

I can use the expression 5 × 12 because I see 5 groups of 12. What is 5 × 12?

60 Write 5 × 12 = 60 below the tape diagram. What does 60 represent? It represents the number of pages in Lacy’s book. Does it make sense that the answer is greater than 12? Why? Yes, because 12 is part of the number of pages. Yes, it makes sense. Lacy has read part of the book, not the whole book, so the answer should be greater than 12.

272

Teacher Note The digital interactive Partitive Fractional Division helps students see that they can divide by any unit fraction and the result is a whole number. Consider allowing students to experiment with the tool individually or demonstrating the activity for the whole class.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

Direct students to record the answer. We used the expression 5 × 12 to help us solve the problem, but that expression does not represent the problem. We can also see 5 is not in the problem, although our tape

diagram helped us see 5 × 12. So let’s consider what   means. 5

_1 Where do you see   in the tape diagram? 5

It is one part of the tape.  _ of the whole tape is 12. 1 5

We can also see 12 is   of some number we did not know. We found 5 × 12 to learn that 5

the number is 60. So we can conclude 12 is   of 60. What equation represents that? 5 1_ 12 = 5 × 60

In that equation, what does   represent?

Teacher Note Most students will recognize that the tape diagram shows that they can use multiplication to solve the problem. Help students understand that the problem itself is not a multiplication problem but rather a division problem that can be solved by using multiplication.

Differentiation: Support

The number of groups

Some students might benefit from an additional connection to a simple whole-number equation, such as the following example:

What does 60 represent? The size of the group

Let’s write a division equation equivalent to 12 =   × 60. Because 60 was the 5

unknown factor, what division equation could we write in which 60 represents the answer? 1 12 ÷ _5 = 60

10 = 2 × 5

10 ÷ 2 = 5

This highlights that the unknown factor of the multiplication equation matches the quotient of the division equation.

In the equation 12 ÷   = 60, the number   still represents the number of groups, which 5

means 60 still represents the size of the group.

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Invite students to think–pair–share about how they can see those representations of

60 and  _15 in the tape diagram and number line.

We can see  _ as the number of groups because we needed to finish drawing the group 1 5

to see it. The number line shows the whole tape is 1 group because the number line goes to  _  . 5 5

We can see 60 as the size of the group because  _ of it, or 1 group, is equal to 60. 5 5

Affirm for students that when they think of the divisor as the number of groups, the quotient represents the size of the group. So students can ask, 12 is  _ of what? We know if 12 is the 1 5

size of  _ of a group, then the quotient is the size of 1 whole group. 1 5

Invite students to turn and talk about how interpreting the fractional divisor as the number of groups is different from interpreting the fractional divisor as the size of each group. Direct students to problem 2. Have students work with a partner to solve the problem.

Promoting the Standards for Mathematical Practice As students read, interpret, and solve real-world problems with the division of a nonzero whole number by a unit fraction, they are reasoning abstractly and quantitatively (MP2). Ask the following questions to promote MP2: • What does the problem ask you to do?

__1

• How does 5 ÷   represent the context in problem 2?

• Does your answer make sense in this context?

2. Tyler has 5 lemons. This is    of the number of lemons he needs to make a pitcher of 4

lemonade. How many lemons does Tyler need to make a pitcher of lemonade?

Language Support

? Consider showing a picture of a pitcher of lemonade and discussing the use of the multiple-meaning word pitcher in problem 2.

0 4

1 4

2 4

3 4

5 ÷ _4 = 20 1

Tyler needs 20 lemons to make a pitcher of lemonade.

274

4 4

Teacher Note The number line below the tape diagram reinforces that the tape diagram represents 1 whole group, so the answer is the size of the group. Students may or may not continue to draw the number line as well.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

As students work, circulate and ask the following questions to advance student thinking: • What can you draw to represent the problem? • How many parts does your tape show? Why?

Differentiation: Support

• What division equation represents the problem? Why?

If students need additional support, consider offering cubes to provide a concrete experience. Students will likely use 5 cubes to represent the lemons Tyler has and then make 3 more groups of 5 to find the total number of lemons Tyler needs.

When students finish, invite them to turn and talk about how they can use a tape diagram to divide a whole number by a unit fraction to find the size of 1 group.

After students model the problem completely, prompt them to represent the model by drawing a tape diagram that matches their thinking while they used the cubes.

• Is the number of lemons greater than or less than 5? How do you know? • What expression can you use to find the number of lemons Tyler needs to make a pitcher of lemonade? • What does 5 represent in the problem? What does  _  represent? 1 4

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

275

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Land

Debrief 5 min Objective: Divide a nonzero whole number by a unit fraction to find the size of the group. Facilitate a class discussion about dividing a nonzero whole number by a unit fraction to find the size of the group by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the tape diagram.

This tape diagram represents 2 ÷   = 3

? .

What do the dividend and the divisor represent

in the equation?

The dividend represents the size of  _ of a group. The divisor represents the number 1 3

of groups. What do we need to find?

2 is  _13 of what number?

When 2 represents  _ of a group, how much is 1 group? 1 3

How can you use the tape diagram to find the quotient? I can multiply 2 and 3 because 2 is the value of 1 unit and there are 3 units in the total.

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How do tape diagrams and number lines help you understand dividing a whole number by a unit fraction to find the size of the group? I can draw a tape diagram to represent the dividend, which is a fraction of the group, and then add on to my model to show 1 whole group. When I draw a tape diagram, I can see the number of units in the whole tape diagram and the value of each unit. The number line helps me see the whole tape is 1 group, and the fraction represents the number of groups when we divide by a fraction. Does dividing a whole number by a unit fraction result in a quotient that is greater than the dividend? Why? Yes, except when the dividend is zero. When the dividend is zero, the quotient is zero. Yes. When I divide a whole number except zero by a unit fraction to find the size of a group, I know the dividend represents a unit fraction of 1 group. The size of that group is the quotient, so it must be greater than the dividend.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

Draw a tape diagram to find the quotient. Then complete the division equation. 1 of what number? 5. 3 is _ 5

3÷1= 5

Use the model to help you complete each statement and divide. 1 of what number? 1. 2 is _ 3 2÷1= 3

1 of 3

1 3

2 3

1 of what number? 2. 2 is _

is 2.

1 4

2 4÷1= 8 2 1 of 8 2

278

3 4

4 4

1 of the distance Sana rides her bike. How far does Sana ride her bike? 7. 3 miles is _

3 ÷ 1 = 12 4 Sana rides her bike 12 miles.

16 16

2 4

1 of what number? 4. 4 is _

3 3

Use the Read–Draw–Write process to solve each problem.

1 of what number? 3. 4 is _

4 1 4

4 ÷ 1_ =

1 of 4

5÷1= 4

is 2.

1 of what number? 6. 5 is _

2 ÷ 1_ =

4 is

117

118

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

1 of the paper a school uses in a week. How many reams of paper does the 8. 4 reams of paper is _

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 13

1 of the liters of water the elephant 10. An elephant can hold 8 liters of water in its trunk. This is _

school use in a week?

drinks in one day. How many liters of water does the elephant drink in one day?

4 ÷ 1_ = 20

8 ÷ 1_ = 72

The school uses 20 reams of paper in a week.

The elephant drinks 72 liters of water in one day.

1 of his lawn in 15 minutes. If he continues at the same pace, how long does 11. Mr. Perez mows _

1 of the total levels in the game. How many levels are 9. Noah passes 12 levels in a game. This is _

in Noah’s game?

it take Mr. Perez to mow his whole lawn?

12 ÷ 1_ = 72

15 ÷ 1_ = 75

There are 72 levels in Noah’s game.

It takes Mr. Perez 75 minutes to mow his whole lawn.

PROBLEM SET

119

120

PROBLEM SET

279

LESSON 14

Divide a unit fraction by a nonzero whole number.

EUREKA MATH2 California Edition

Name

5 ▸ M3 ▸ TC ▸ Lesson 14

Date

Lesson at a Glance Students use tape diagrams and number lines to divide a unit fraction by a whole number. They interpret a division expression

Draw a model to divide.

such as  _ ÷ 4as the question,  _  is 4 groups of what? Students use the

Read–Draw–Write process to solve problems involving division of a

1÷2= 8

1 3

1 __

1 3

unit fraction by a whole number.

Sample: 1 8

Key Questions

• How do tape diagrams help you understand dividing a unit fraction by a whole number? • Why is the quotient of a unit fraction and a whole number less than the dividend?

Achievement Descriptors 5.Mod3.AD11 Model and evaluate division of unit fractions

by nonzero whole numbers. (5.NF.B.7.a) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c)

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Agenda

Materials

Lesson Preparation

Fluency 15 min

Teacher

Launch 5 min

• None

Learn 30 min

Students

• Divide a Unit Fraction by a Whole Number

• Blank Tape Diagram (in the student book)

• Consider whether to have students remove Blank Tape Diagram from the student books and place inside whiteboards in advance or have students prepare them during the lesson.

• Interpret a Division Expression

• Multiply a Whole Number by a Fraction Fluency Sheets (in the student book)

• Use a Tape Diagram to Divide

• Consider tearing out the Fluency Sheets in advance of the lesson.

• Problem Set

Land 10 min

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Fluency

Whiteboard Exchange: Partition Tape Diagrams Materials—S: Blank Tape Diagram

Students partition a tape diagram into equal units and determine the value of one unit to develop fluency with dividing a whole number by a unit fraction. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the blank tape diagram.

Label the total of the tape diagram as 2. Display the tape diagram labeled with 2. Partition the tape into 2 equal units and label 1 below the tape.

Display the partitioned tape diagram with 1 labeled. Now partition each unit into 2 equal units. Display the partitioned tape diagram. What is the value of each unit? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond. 1_ 2 Display the answer.

282

2 1 2

1 2

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

Repeat the process with the following sequence: Label the total of the tape diagram as 2.

Label the total of the tape diagram as 3.

Partition into 2 equal units and label 1.

Partition into 3 equal units and label 1.

Partition each unit into 4 equal units. Each unit is 1 . 4

Partition each unit into 2 equal units. Each unit is 1 . 2

Partition each unit into 3 equal units. Each unit is 1 . 3

Contemplate Then Calculate: Multiply a Whole Number by a Fraction Materials—S: Multiply a Whole Number by a Fraction Fluency Sheets

Students write the product to build fluency with multiplying a whole number by a fraction from topic A. Direct students to study the problems on Fluency Sheet 1. Have students focus on the problems in just one column to start. Consider having them cover the other problems with sticky notes or blank paper in advance. Frame the task: As you study, ask yourself, What do I notice that could help me with these problems? Provide 1–2 minutes of silent think time. Some students may make notes or answer problems as part of their study. Have students turn and talk about their thinking. Listen for students who offer solution strategies or connect problems by highlighting relationships or patterns. Select a few students to share their ideas with the class.

Teacher Note Consider asking the following questions to discuss the patterns in Fluency Sheet 1: • What patterns do you notice in problems 7–12? • What method or problem might you use to find the product in problem 10? Where else can this method or another problem be used in a similar way?

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5 ▸ M3 ▸ TC ▸ Lesson 14

After students share, provide 1–2 minutes for the class to work independently on Fluency Sheet 1. Direct students to work in order starting from problem 1, or from where they left off in their study, so that they experience problems rising in complexity. Use your own ideas or the ideas you heard to help you do as many problems as you can. I do not expect you to finish. After 1–2 minutes, have the class pause their work. Invite students to discuss what they noticed about the problems with a partner or in a small group. Circulate and listen as students talk, advancing their discussions as needed by asking questions such as the following: • What is the same about these problems? What is different? • Did you find patterns in the problems? If so, talk about them. • What strategy did you use? Facilitate a whole-class discussion by asking different groups to share their thinking. As time allows, have students continue to work on Fluency Sheet 1. Consider reading the answers quickly to provide immediate feedback. Invite students to complete Fluency Sheet 2 at another time by using what they learned from Fluency Sheet 1.

284

EUREKA MATH2 California Edition

UDL: Representation As students share their ideas, consider displaying Fluency Sheet 1 and annotating problems to reinforce strategies, relationships, and patterns described.

EUREKA MATH2 California Edition

Launch

5 ▸ M3 ▸ TC ▸ Lesson 14

Students reason about division by using a tape diagram.

Spotlight on Environmental Principles and Concepts

Display the word problem. Sana has some birdseed. She divides the birdseed equally among 3 bird feeders. How much birdseed is in each feeder? What information does the problem tell us? Sana has some birdseed. She divides the birdseed equally into 3 feeders. Do we have enough information to solve the problem? Why? No, we do not know how much birdseed Sana started with. Let’s say Sana started with 6 pounds of birdseed. Do we have enough information to solve the problem now?

People can influence natural systems. Scientists are becoming concerned that feeding birds from bird feeders might affect the spread of diseases, alter migration patterns, and help invasive species outcompete native ones. Have students discuss the positive and negative aspects of feeding birds. (CA EP&C Principles II, III and V)

Yes, we know the total and the number of groups, so we can find the amount in each group. How can we model the problem by using a tape diagram? We can draw and label a tape diagram. We can label the whole tape diagram as 6 and then partition the tape diagram into 3 equal parts. Draw and label a tape diagram. Then partition it into 3 equal parts.

What is the value of each part? How do you know? Each part is 2 because 6 ÷ 3 = 2. What does 2 represent? It represents the number of pounds of birdseed in each feeder.

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5 ▸ M3 ▸ TC ▸ Lesson 14

If Sana started with 3 pounds of birdseed instead of 6, how would the tape diagram change? The tape would be labeled 3 instead of 6. How many pounds of birdseed would there be in each bird feeder? Why? Each bird feeder would have 1 pound of birdseed because 3 ÷ 3 = 1. If Sana started with 1 pound of birdseed, how would the tape diagram change? We would label the tape 1 instead of 3. How many pounds of birdseed would be in each bird feeder? Why? Each bird feeder would have  _ pound of birdseed because 1 ÷ 3 =  _  . 1 3

1 3

If Sana started with   pound of birdseed, how would the tape diagram change? 2

We would label the tape  _ instead of 1. 1 2

Display the tape diagram with a total value of  _  . 1 2

What division expression represents the problem now? 1_ 2 ÷ 3

1 2

How is this division expression different from the previous expressions? The previous expressions had whole-number dividends and whole-number divisors. This expression has a unit-fraction dividend and a whole-number divisor. How is this division expression different from the division expressions in the past few lessons? Before, we had whole-number dividends and unit-fraction divisors. Now we have a unit-fraction dividend and a whole-number divisor. Invite students to turn and talk about how this difference—dividing a unit fraction by a whole number—affects the quotient in the birdseed problem. Transition to the next segment by framing the work. Today, we will divide unit fractions by whole numbers. 286

EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TC ▸ Lesson 14

Divide a Unit Fraction by a Whole Number Students use a tape diagram and a number line to model dividing a unit fraction by a whole number. Direct students to problem 1 in their books. 1.

1 2

a. Partition the tape diagram into 3 equal units. b. Write a division expression that represents the model. 1_ 2 ÷ 3 c. What is the size of one unit? 1_ 6 The tape diagram represents what total? How do you know?

The tape represents a total of  _ . I know because it starts at 0 and ends at  _ on the number line.

1 2

Partition the tape into 3 equal units and direct students to do the same. We partitioned the tape into 3 equal units. To find the value of one unit, how can we show that in our tape diagram? Label one unit with a question mark.

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5 ▸ M3 ▸ TC ▸ Lesson 14

Label the first unit with a question mark and direct students to do the same. What division expression can we write to represent the value of the unknown, or one unit? 1_ 2 ÷ 3 Direct students to record the division expression. Then invite them to turn and talk about how they can use the number line and the tape diagram to find the quotient.

_1 2 1 1  _ ÷ 3 =  _ . The number line is partitioned into sixths. It shows each unit on the tape 2 6 1 diagram has a value of  _  . What is   ÷ 3? How do you know?

UDL: Representation

Direct students to record the answer. Earlier, we thought about how much birdseed Sana would put into each bird feeder

if she started with   pound of birdseed. Now, we see   ÷ 3 answers that question.

Consider labeling the tick marks on the number line with sixths so students can see how the number line corresponds with the tape diagram. Encourage students to do the same.

Sana would put   pound of birdseed in each bird feeder. 6

Label the number line and shade the unit that represents the quotient.

What do you notice about the size of the quotient compared to the size of the dividend?

1 6

1 2

The quotient is less than the dividend. Does it make sense that the quotient is less than the dividend? Why?

Yes, it makes sense. We started with  _ and divided it into 3 equal groups, so each group has to be smaller than the total.

1 2

 _  =  _ , so when you divide 3 sixths into 3 equal groups, you get 1 sixth in each group. So, 1 2

3 6

it makes sense  _ ÷ 3 <  _  . 288

3 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

Because we are dividing by 3, which is a whole number, the quotient is less than the dividend. Can you think of a whole-number divisor for which the quotient is not less than the dividend? Zero is a whole number, and we cannot divide by zero.

1 is a whole number, but when we divide by 1, the quotient is equal to the dividend. When we divide a unit fraction by a whole number other than zero or 1, the quotient is less than the dividend.

Interpret a Division Expression Students relate division with unknown group size to multiplication by using the sentence frame is groups of what? Write 6 ÷ 3. Draw a tape diagram that represents the expression.

What does 6 represent?

6 represents the total of the tape diagram. It is the dividend in the expression.

6÷3

What does 3 represent?

3 represents the number of groups. We know that when we divide, we are finding an unknown factor. So when the divisor represents the number of groups, we can think, 6 is 3 groups of what, or 6 = 3 × .

289

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5 ▸ M3 ▸ TC ▸ Lesson 14

Write 6 is 3 groups of what? and 6 = 3 × expression.

6 is 3 groups of what? ? 6=3×

beneath the

What is the quotient? How do you know? The quotient is 2 because 6 = 3 × 2, and 6 ÷ 3 = 2.

We interpreted the expression 6 divided by 3 as 6 is 3 groups of what? We divided 6 into 3 equal groups and found 6 is 3 groups of 2. We can use this same thinking to interpret a division expression with a unit fraction divided by a whole number.

Display the expression  _ ÷ 3 along 1 2

1 ÷3 2

with the model from problem 1.

Write the sentence frame is

groups of what? Then

discuss how to interpret the expression.

1 6

Use the sentence frame to make

1 2

a statement about how we could interpret the expression   ÷ 3. 2

We could say  _  is 3 groups of what? 1 2

What is the equation that shows the quotient as an unknown factor? 1_ 2 = 3 ×

In other words, when we divide   into 3 equal groups, how much is in each group? Is that reasonable?

 _ is in each group. That is reasonable because  _ = 3 ×  _  . 1 6

1 2

1 6

When we divide, we can use the relationship between multiplication and division to better understand what the numbers mean and to make sure the answer is reasonable.

290

EUREKA MATH2 California Edition

Use a Tape Diagram to Divide Students use a tape diagram to solve problems by dividing a unit fraction by a nonzero whole number. Direct students to problem 2. Have them read the problem and work with a partner to discuss how they might draw a tape diagram to represent the problem. Circulate as students discuss and encourage them to think about which quantity is the dividend and which quantity is the divisor.

5 ▸ M3 ▸ TC ▸ Lesson 14

Promoting the Standards for Mathematical Practice As students read and interpret real-world problems and decide how to solve the problem by using what they have learned, they model with mathematics (MP4). Ask the following questions to promote MP4:

Use the Read-Draw-Write process to solve each problem.

_1 2. Mr. Perez has    gallon of water. He pours the water equally into 4 bottles. How much 4

water is in each bottle?

1 4

• What can you draw to help you understand this real-world problem? • How are the key ideas in this problem represented in your tape diagram? • How could you represent this context mathematically?

1 1 4 ÷ 4 =   16

1 There is  __ gallon of water in each bottle. 16

Differentiation: Challenge Challenge students to find the number of quarts, pints, cups, or fluid ounces of water in each bottle.

291

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5 ▸ M3 ▸ TC ▸ Lesson 14

Display the tape diagram. Have students turn and talk about how the quantities and the unknown are represented in the tape diagram. What do you notice about how the tape diagram is labeled and partitioned?

1 4

The tape diagram is labeled  _ , which represents the total number of gallons of water 1 4

Mr. Perez has.

Problem 1 has a number line under the tape diagram; problem 2 does not have a number line. The tape diagram is partitioned into 4 equal parts to represent the 4 bottles.

1 part is labeled with a question mark because we are finding how much water is in each bottle.

Direct students to draw the tape diagram. What does the problem ask? It asks how much water is in each bottle. What expression represents how much water is in each bottle? 1_ 4 ÷ 4 How can we interpret this expression?  _  is 4 groups of what? 1 4

What conclusions can you make when you look at the tape diagram? There is less than  _ gallon of water in each bottle. 1 4

Can we tell the value of the unknown by looking at this tape diagram? Why?

No, because the tape diagram only shows a total of  _ , and we need to see a total of  _  , or 1. 1 4

4 4

What can we do with the tape diagram to help us find the value of the unknown? We can draw 3 more fourths and then partition each fourth into 4 equal parts so that the tape shows the total number of parts in 1. 292

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

Draw and partition the remaining fourths and direct students to do the same.

1 4

? What is the size of each part now? How do you know?

The size of each part is  __ . I know because we are finding 1 part out of 16 total parts when the total is 1.

1 16

Direct students to record the answer. What division equation represents the problem? 1_ 1 4 ÷ 4 =  __ 16

Think about what each of those numbers mean. What does   represent? 4

It represents the number of gallons of water Mr. Perez started with. What does 4 represent? It represents the number of equal groups, or the number of bottles. What does

__1 represent? 16

It represents the size of each group, or the number of gallons of water in each bottle. What do you notice about the size of the quotient compared to the size of the dividend? The quotient is less than the dividend.

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Is it reasonable that the quotient is less than the dividend? Why? Yes, it makes sense. We are partitioning a part that is less than 1 into smaller parts. Yes, it is reasonable. We are starting with a unit fraction and partitioning it into groups, so the amount in each group is less than the unit fraction. What do you notice about the size of the quotient compared to 1? Is that reasonable? Why? The quotient is less than 1. That is reasonable because we are dividing an already small part into several more parts, so the quotient should be really small. Direct students to problem 3 and invite them to work with a partner to solve the problem. Affirm that students can use a number line and a tape diagram or just a tape diagram when they divide, depending on which method they find the most helpful.

3. Lacy and Adesh share    quart of ice cream equally. How much ice cream does each 2

person get?

1 2

1 1 2 ÷ 2 = 4

Each person gets _    quart of ice cream.

Differentiation: Support If students need more support to determine how to partition the tape diagram to represent the unknown, consider one of the following adaptations: • Encourage students to draw a number line from 0 to 1. They can use the number line to help them match and partition the tape diagram. • Show the tape diagram they began with in problem 2. Then ask students to think about what they did with the model to help them find the value of the unknown.

1 4

As students work, circulate and ask the following questions to advance student thinking: • What can you draw to represent the problem? • Does your tape diagram show the unknown? If not, how might you partition the tape diagram to see and find the value of the unknown? • What division expression represents the problem? • Which number is the dividend? What does it represent?

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• Which number is the divisor? What does it represent? • Is the quotient greater than or less than the dividend? How do you know? When students finish, invite them to turn and talk about how they can use a tape diagram to divide a unit fraction by a nonzero whole number.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Divide a unit fraction by a nonzero whole number. Facilitate a class discussion about dividing a unit fraction by a nonzero whole number by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the expression  _ ÷ 4and the tape diagram. 1 3

What do the dividend and the divisor represent in the expression? The dividend represents the total. The divisor represents the number of groups.

1 3

What do we need to find?  _ is 4 groups of what? 1 3

How can you use the tape diagram to find the quotient? I can draw 2 more thirds and partition each of those thirds into 4 equal units. The quotient is 1 unit out of the total number of units. © Great Minds PBC

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EUREKA MATH2 California Edition

How do tape diagrams help you understand dividing a unit fraction by a whole number? When I draw a tape diagram, I can see the total and partition it into equal groups to find the size of 1 unit. Tape diagrams help me see the dividend and the divisor better, so I can tell whether the quotient is reasonable. Why is the quotient of a unit fraction and a whole number, other than zero and 1, less than the unit fraction? When we divide a unit fraction by a whole number greater than 1, the quotient is less than the unit fraction because we are dividing it into smaller parts. When we partition a unit fraction into groups, the size of each group is smaller than the unit fraction. Zero and 1 are not included when we make statements about the quotient compared to the dividend. Why? We leave out zero and 1 because we cannot divide by zero, and when we divide by 1, the quotient equals the dividend.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14 ▸ Multiply a Whole Number by a Fraction

Contemplate Then Calculate 23. _ × 6

Fluency Sheet 1

1 2

Write the product. Use a whole number or mixed number when possible.

1. _ of 6 is

2. _ × 6

3. _ of 9 is

4. _ × 9

5. _ of 12 is

6. _ × 12

7. _ of 6 is

8. _ of 6 is

9. _ of 9 is

10. _ of 9 is

11. _ of 12 is

1 3 1 3 1 3 1 3 1 3 1 3 1 2 2 2 1 3 2 3 1 4

12. _ of 12 is

13. _ × 12

14. _ × 12

15. _ × 16

16. _ × 16

17. _ × 15

18. _ × 15

19. _ × 15

20. _ × 18

21. _ × 18

22. _ × 18

2 4 1 3

3 3 1 4

3 4 1 5 3 5 5 5 1 6 3 6 5 6

24. _ × 7 1 2

25. _ × 9 1 3

26. _ × 10 1 3

27. _ × 12 1 4

28. _ × 13 1 4

29. _ × 15 1 5

30. _ × 17 1 5

31. _ × 18 1 6

32. _ × 20 1 6

33. _ × 21 1 7

125

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14 ▸ Multiply a Whole Number by a Fraction

126

34. _ × 23 1 7

3_ 1 2

3 1 3

3 3_ 1 4

3 3_ 2 5

3 3_ 2 6

2 7

35. _ × 8

36. _ × 11

1 2 2 2

37. _ × 8 2 3

38. _ × 12 3 3

39. _ × 11 2 4

40. _ × 11 3 4

41. _ × 9 2 5

42. _ × 11 4 5

43. _ × 11 2 6

44. _ × 13 5 6

5_ 1 3

12 5_ 2 4

8_ 1 4

3_ 3 5

8_ 4 5

3_ 4 6

10 _ 5 6

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14 ▸ Multiply a Whole Number by a Fraction

Fluency Sheet 2 Write the product. Use a whole number or mixed number when possible.

2. _ × 4

3. _ of 6 is

4. _ × 6

5. _ of 8 is

7. _ of 4 is

8. _ of 4 is

9. _ of 6 is

10. _ of 6 is

11. _ of 8 is

1 1. of 4 is 2 1 2 1 2 1 2 1 2

1 6. × 8 2 1 2 2 2 1 3 2 3 1 4

298

23. _ × 4 1 2

13. _ × 9

25. _ × 6

14. _ × 9

26. _ × 7

15. _ × 12

27. _ × 8

16. _ × 12

28. _ × 9

29. _ × 10

18. _ × 10

30. _ × 12

19. _ × 10

31. _ × 12

20. _ × 12

32. _ × 14

21. _ × 12

33. _ × 14

22. _ × 12

2 12. of 8 is 4 1 3

3 3 1 4

3 4

1 17. × 10 5 3 5 5 5 1 6 3 6 5 6

24. _ × 5 1 2

1 3 1 3 1 4 1 4 1 5 1 5 1 6 1 6 1 7

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5 ▸ M3 ▸ TC ▸ Lesson 14 ▸ Multiply a Whole Number by a Fraction

128

34. _ × 16 1 7

2_ 1 2

2 1 3

36. _ × 10

1 2

37. _ × 7 2 3

38. _ × 11 3 3

2_ 1 4

2 2_ 2 5

2 2_ 2 6

2 7

35. _ × 6 2 2

39. _ × 9 2 4

40. _ × 10 3 4

41. _ × 8 2 5

42. _ × 9 4 5

43. _ × 10 2 6

44. _ × 11 5 6

4_ 2 3

11 4_ 2 4

7_ 2 4

3_ 1 5

7_ 1 5

3_ 2 6

9_ 1 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

EUREKA MATH2 California Edition

Name

5 ▸ M3 ▸ TC ▸ Lesson 14

Draw a model to divide.

Date

Use the model to divide. 1. 1_ ÷ 2 = 2

1 4

0 1 2

1 2

6. 1_ ÷ 3 = 5

1 4

1 __

2 3

1 5

Sample: 1 3

1 __

Sample:

1 6

1 4

5. 1_ ÷ 4 = 4

1 8

3. 1 ÷ 2 = 3

1 8

Sample:

2. 1_ ÷ 4 = 2

4. 1_ ÷ 2 = 4

EUREKA MATH2 California Edition

131

132

PROBLEM SET

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

Use the Read–Draw–Write process to solve each problem.

1 mile. She runs an equal distance each minute. How far does Tara run 7. In 2 minutes, Tara runs _

in 1 minute?

1 gallon of olive oil to make 4 equal bottles of salad dressing. How many gallons 9. Miss Song uses _ 4

of olive oil does Miss Song use for each bottle?

1÷4= 1 4 16

1 _1 ÷ 2 = __ 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 14

1 gallon of olive oil for each bottle. Miss Song uses __

1 mile in 1 minute. Tara runs __ 12

1 pound of fudge equally between her and 4 friends. How much fudge does 10. Ms. Baker shares _

8. Ryan has half of a pie. He eats an equal amount of the pie after dinner each day for 3 days. How much of the pie does Ryan eat each day?

each person get?

1÷3=1 2 6

1+4=5

1 of the pie each day. Ryan eats _

1 ÷5= 1 15 3

1 pound of fudge. Each person gets __ 15

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PROBLEM SET

133

134

PROBLEM SET

LESSON 15

Divide by whole numbers and unit fractions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

Name

Date

Use the Read–Draw–Write process to solve each problem.

1 gallon of milk equally. How much milk does each child get? 1. There are 4 children who share _ 2

1 2

Students sort division expressions by whether the quotient is greater than or less than the dividend. They use the Read–Draw–Write process to solve word problems and the Stronger, Clearer Each Time routine to practice sharing their thinking, critiquing one another’s work, and revising their own work as needed.

Key Questions

• What does it mean to check whether a solution is reasonable?

1_ ÷ 4 = 1_ 2 8

• Why is it important to check the reasonableness of a solution to a word problem?

Each child gets 1_ gallon of milk. 8

Achievement Descriptors

1 foot pieces can be cut from 3 feet of string? 2. How many _ 3

5.Mod3.AD11 Model and evaluate division of unit fractions

3 1 3

by nonzero whole numbers. (5.NF.B.7.a)

...

5.Mod3.AD12 Model and evaluate division of whole numbers by unit

? thirds

fractions. (5.NF.B.7.b)

3 ÷ _1 = 9

5.Mod3.AD13 Solve word problems involving division of unit fractions

9 pieces that are each _1 foot long can be cut from the string.

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c)

Lesson at a Glance

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 10 min

• Envelopes (12)

Learn 30 min

• Division Expressions Cards (in the teacher edition)

• Consider whether to have students remove Blank Tape Diagram from the student books and place inside whiteboards in advance or have students prepare them during the lesson.

• Which Model Matches? Why? • Problem Set

• Division Word Problems Set 1 and Set 2 (1 per student pair, in the teacher edition)

Land 10 min

Students

• Reason, Explain, and Critique

• Blank Tape Diagram (in the student book) • Envelope of Division Expressions Cards (1 per student pair)

• Print or copy Division Expressions Cards and cut out each set. Each sheet has 2 sets. Place 1 set of cards in an envelope. Prepare enough sets for 1 per student pair. • Print or copy Division Word Problems Set 1 and Set 2 and cut out each problem. Prepare enough copies to give both sets of problems to each student pair.

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Fluency

Whiteboard Exchange: Convert Customary Length Units Students convert yards to feet or feet to inches to build fluency with converting larger customary measurement units to smaller customary measurement units from topic A. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 1 yd =

ft.

1 yard is equal to how many feet? Tell your partner. Provide time for students to think and share with their partners.

3 feet

1 yd = 1 ft 3 1 yd = 3 ft 4 4

Display the completed equation, and then display

_ 1  yd =

1 yd = 3 ft

ft.

Write and complete the equation.

Display the answer and then display _ 1  yd = 4

ft.

Write and complete the equation. Display the answer. Repeat the process with the following sequence:

1 ft = 12 in 1 ft = 1 in 12 2 ft = 8 in 3 304

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5 ▸ M3 ▸ TC ▸ Lesson 15

Whiteboard Exchange: Partition Tape Diagrams Materials—S: Blank Tape Diagram

Label the total of the tape diagram as 3. Display the tape diagram labeled with 3. Partition the tape into 3 equal units and label 1 below the tape.

Display the partitioned tape diagram with 1 labeled. Now partition each unit into 2 equal units.

Display the partitioned tape diagram. What is the value of each unit? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 2

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Display the answer. Repeat the process with the following sequence: Label the total of the tape diagram as 3.

Label the total of the tape diagram as 4.

Partition into 3 equal units and label 1.

Partition into 4 equal units and label 1.

Partition each unit into 4 equal units. 1 Each unit is . 4

Partition each unit into 3 equal units. Each unit is 1 . 3

Partition each unit into 4 equal units. Each unit is 1 . 4

Launch

Materials—S: Envelope of Division Expressions Cards Distribute one envelope of Division Expressions Cards to each pair of students. Direct them to take out the cards that name the categories: Quotient Greater Than Dividend and Quotient Less Than Dividend. Then give students 2 minutes to sort the remaining cards into the appropriate category. Students should place the cards by using reasoning only, not by evaluating. 306

Quotient Greater Than Dividend

Quotient Less Than Dividend

4÷ 4

1 ÷5 2

3÷ 6

1 ÷3 5

5÷ 2

1 ÷3 6

1 ÷7 6

8÷ 3

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

While students sort the cards, circulate and consider asking the following questions: • Why did you sort this expression into this category? • How do you know the quotient is greater than or less than the dividend without evaluating? Once students finish sorting, choose one expression from each category to discuss. How did you know the expression 4 ÷_    has a quotient greater than 4? 1 4

We thought about 4 ÷_  1 as how many fourths are in 4? There are 4 fourths in 1, so there 4

must be more than 4 fourths in 4.

We thought about 4 ÷_  1 as the question, 4 is _ 1 of what number? 4 is a part of another 4

number, so that other number is greater than 4.

Differentiation: Challenge Challenge students to order the following expressions from least to greatest by using reasoning only, and ask them to explain their thinking.

__ 5 1  ÷ 3  __ 6 __  1  ÷ 7

  1  ÷ 3

How did you know the expression _  1  ÷ 5has a quotient less than _  1 ? 2 2 1 _ When you start with    and partition it into 5 groups, the size of each group is 2 smaller than _  1 . 2

Transition to the next segment by framing the work. Today, we will solve division problems and pay careful attention to the size of the quotient compared to the size of the dividend to make sure our answers are reasonable.

Students should reason  1 ÷ 7 <   1  ÷ 3 because

dividing   1  into more parts (i.e., 7) results in 6

a smaller fraction. Then students should

__ __

reason   1 ÷ 3 <   1  ÷ 3 because  1  <   1 , and both 6

are being divided by 3.

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Learn

Which Model Matches? Why? Students use the Read–Draw–Write process to model division word problems and reason about the size of the quotient. Direct students to problem 1 in their books. Have them use the Read–Draw–Write process independently to construct a tape diagram that matches the story. Use the Read-Draw-Write process to solve the problem.

1. Miss Song has _  1 of a pan of lasagna in the refrigerator. She wants to cut the lasagna into 4

equal slices so she can have it for dinner for 3 nights. How much of the pan of lasagna will she eat each night?

1 4

_1 ÷ 3 = __   1 4

Miss Song will eat __  1  of the pan of lasagna each night. 12

Display the following two tape diagrams.

1 4

3 1 4

... ? fourths

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Invite students to think–pair–share about which tape diagram represents the story and why.

The tape diagram on the left represents the story because it shows _  1 of a pan of lasagna 4 split into 3 equal parts to represent 3 nights. The tape diagram on the left represents the story because _  1 of the pan of lasagna 4 is shared, not the 3 nights.

The tape diagram on the right does not represent the story because the unknown asks for how many fourths, and we know Miss Song has only _  1 of a pan of lasagna. 4

Affirm that the tape diagram on the left represents the story. Expect that some students may have drawn the rest of the tape diagram, as they did in previous lessons, to find how many parts are in 1. Allow students to revise their tape diagrams if needed. If a student drew the tape diagram on the right to model the story, what advice would you give them for when they solve word problems in the future? I would tell them to think about what each number means and to make sure the model you draw matches that meaning. I would tell them to think about what their tape diagram shows and then estimate the answer. Then think, Does that estimate make sense based on what is happening in the story? In this story, you noticed the _  1 of a pan of lasagna is shared, not the 3 days. You also 4

thought about whether the answer made sense. Then you gave feedback to someone who misinterpreted the story. Let’s continue to use the Read–Draw–Write process to solve problems and practice giving feedback.

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Reason, Explain, and Critique Materials—S: Division Word Problems Set 1 and Set 2

Students solve division word problems and critique a partner’s work. Use the Stronger, Clearer Each Time routine to invite written solutions and discussion as students explore different division word problems. Strategically assign partners. From set 1, distribute problem A to one partner and problem B to the other, for each student pair. Direct students to use the Read–Draw–Write process to solve their assigned problem. Give students 3–4 minutes to solve the problem independently and write an explanation or justification for their thinking. Have students exchange their written explanations with their partners. Provide time for students to read silently. Then invite pairs to ask each other clarifying questions and critique one another’s responses. Circulate and listen as students discuss. Ask targeted questions to advance their thinking. • Does your tape diagram accurately model the problem? How do you know? • What does _  1 represent in the problem? What does 5 represent? 3

• Does your answer make sense based on the problem? How do you know? • Should you revise your work? How?

Promoting the Standards for Mathematical Practice As students solve real-world problems by dividing by whole numbers and unit fractions, explain their thinking or justify their reasoning for a problem, and critique the reasoning of their partner, they are constructing viable arguments and critiquing the reasoning of others (MP3). Ask the following questions to promote MP3: • Why does your reasoning make sense? Convince your partner. • What questions can you ask your partner to make sure you understand their explanation of their work?

Language Support

Both stories are about Pablo reading. The numbers are the same.

The dividend and divisor are different in each story. In problem A, the dividend is _  1  and 3 1 _ the divisor is 5. In problem B, the dividend is 5 and the divisor is    . 3

Direct students to use the Talking Tool as they explain their work and critique another’s written explanation.

Problem A has an answer less than the dividend. Problem B has an answer greater than the dividend.

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In each story, how did you know what is shared? I thought about what made sense to share based on what the question asks. In problem A,

_ 1 of a book is shared to find how much Pablo reads each day. In problem B, 5 books are 3

shared to find how many days it takes to read all of them.

In problem A, does it make sense that the answer is less than the dividend? Why?

Yes, it makes sense that the answer is less than the dividend. Pablo starts with _  1 of the 3

book, and then he reads part of that _  1 each day, so he could not have read more than _  1 . 3

In problem B, does it make sense that the answer is greater than the dividend? Why? Yes, it makes sense that the answer is greater than the dividend. Pablo has 5 books, and he reads part of a book every day, so it would take more than 5 days to read all 5 books. Invite students to turn and talk to share what they learned from solving these problems and what they plan to try when they solve their next problem. Distribute problem A and problem B from set 2 to student pairs. Students should get the same problem letter as they did in set 1 (i.e., if they did problem A in set 1, they should do problem A in set 2). Give students 3–4 minutes to solve the problem independently and write an explanation or justification for their thinking. Have students exchange written explanations. Provide time for students to read silently. Then invite pairs to ask each other clarifying questions and critique one another’s responses. Circulate and listen as students discuss. Ask targeted questions such as the following to advance their thinking: • Does your tape diagram accurately model the problem? How do you know? • What does _  1 represent in the problem? What does 2 represent?

Differentiation: Support Consider providing a copy of the solutions to students who may benefit from seeing a correct solution when they analyze another student’s work.

• Does your answer make sense based on the problem? How do you know? • Should you revise your work? How?

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Allow students a moment to make any necessary revisions, based on their discussion and their partner’s feedback. Direct students to place their work side by side with their partner’s. What is the same and different about the stories? Both stories are about Zara running. The numbers are the same. The dividend and the divisor are different. In problem A, 2 is the dividend and the divisor is _  1 . In problem B, the dividend is _  1 and the divisor is 2. 6

Problem A has an answer greater than the dividend. Problem B has an answer less than the dividend. Problem B from set 1 and problem A from set 2 each have a whole number divided by a unit fraction. What is different about the meaning of the divisor? In problem B from set 1, the divisor is the size of each group. In problem A from set 2, the divisor is the number of groups. In problem A from set 2, why does it make sense that the answer is greater than the dividend? It makes sense that the race is more than 2 miles because we know she pauses to take a break after she runs 2 miles.

UDL: Action & Expression

In problem B from set 2, why does it make sense that the answer is less than the dividend?

Consider reserving time after the class completes the Problem Set for students to reflect on their learning. Invite them to ask themselves questions such as the following to promote self-monitoring:

Zara starts with _  1 mile, and she splits that into 2 parts to decide when to take a break, 6

so she must pause to drink water before she runs _  1  mile. 6

Invite students to turn and talk to reflect on what they learned about solving problems that involve dividing by a unit fraction or a whole number.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

312

• What problem did I feel most confident in solving? Why? • What problem did I need support to solve? Why? • What is one skill I should continue to work on? • What resources are available to help me?

EUREKA MATH2 California Edition

Land

5 ▸ M3 ▸ TC ▸ Lesson 15

Debrief 5 min Objective: Divide by whole numbers and unit fractions. Gather the class with their Problem Sets. Facilitate a class discussion about dividing by whole numbers and unit fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. What does it mean to check whether a solution is reasonable? It means to think about whether the solution makes sense. Direct students to problem 3 in their Problem Sets.

Think about this answer statement: There is __  1  problem on Sasha’s math homework. 21

Why does that answer statement not make sense? It does not make sense that there is a fraction of a problem on a homework assignment. What mistake do you think the person who wrote the statement made? I think they misinterpreted the story and found _  1  ÷ 7. 3

Direct students to problem 4. What expression matches this problem?

1   ÷ 4 2

Is the quotient greater than or less than the dividend? How do you know? It is less than the dividend because we are dividing _  1 into smaller groups. 2

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Display the incorrect work.

1 2

Do you agree with this answer? Why?

1 ÷ 4= 1 2 4

No, I do not agree because the value of 1 part is _  1 , not _  1 . 8

No, I do not agree because there are

4 students on the team, and if each of them runs _  1 mile, that means the

1 4

Each student runs 1 mile. 4

race is 1 mile long. But we know the relay race is only _  1 mile long. 2

When they checked for reasonableness, the student who did this work might have thought their answer made sense because it is less than the dividend. What advice would you give this student? Maybe they should draw another tape diagram to help them solve the problem. This tape diagram does represent the story, but it does not show the number of units in 1. So they could draw a second tape diagram to actually solve the problem. How is this mistake different from the mistake we saw on problem 3? The mistake on problem 3 was that they interpreted the problem incorrectly, and their answer did not make sense. On problem 4, they did understand the story, and their answer made sense, but they made a mistake in their work. Why is it important to check the reasonableness of a solution to a word problem? We should check for reasonableness so we can determine whether we made a mistake with interpreting the story or finding the answer. We want to be sure our answers make sense because if they do not, we need to revise our work.

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. 314

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

1 3

? 1 3 1 15

Problem A

Zara competes in a race. She runs 2 miles before she pauses for a water break. 2 miles is 6 of the race. How many miles is the race?

? 2

0 6

The race is 12 miles.

Pablo has 5 books on his reading list. He reads 3 of a book every day. How many days will

1 3

... ? thirds

15 thirds 5 ÷ _3 = 15 1

319

Pablo will finish all 5 books in 15 days.

5 6

6 6

Zara runs every day for 6 mile. She splits her run into 2 equal distances so she can pause for a water break. After how many miles will Zara pause for her water break? 1 6

? 1 6

EUREKA MATH2 California Edition

Problem B

1 12 © Great Minds PBC

This page may be reproduced for classroom use only.

5 ▸ M3 ▸ TC ▸ Lesson 15 ▸ Division Word Problems Set 1 and Set 2

it take for him to read all 5 books?

4 6

2 ÷ 6_1 = 12

Problem B

3 6

2 6

_1 ÷ 5 = __1

__1 Pablo reads of the book each day.

1 6

5 ▸ M3 ▸ TC ▸ Lesson 15 ▸ Division Word Problems Set 1 and Set 2

How much of the book does Pablo read each day?

Set 2: Solutions

This page may be reproduced for classroom use only.

Pablo decides to read 3 of a book in 5 days. He reads the same amount of the book each day.

320

Problem A

EUREKA MATH2 California Edition

Set 1: Solutions

_1 ÷ 2 = __1 12 6 1 __ Zara will pause for her water break after mile. 12

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5 ▸ M3 ▸ TC ▸ Lesson 15

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

2. Tara makes 2 gallons of lemonade by using 4 of a container of powdered lemonade. How many gallons of lemonade can she make with the whole container of powdered lemonade?

a. Draw a model to represent the problem.

1. Scott pours 4 gallon of lemonade equally into 2 glasses. How much lemonade is in each glass?

Sample:

a. Draw a model to represent the problem.

Sample:

1 4

0 4

3 4

4 4

The quotient is greater than the dividend because the 2 gallons in the problem are a fraction of the total amount of lemonade that can be made.

The quotient is less than the dividend because a fraction is being divided into smaller parts.

c. Write an equation to find how many gallons of lemonade are in each glass. Then write a statement to answer the question.

c. Write an equation to find how many gallons of lemonade Tara can make with the whole container of powdered lemonade. Then write a statement to answer the question.

1_ 1 ÷ 2 = _8 4

2 ÷ _4 = 8 1

_1 gallon of lemonade is in each glass. 8

316

2 4

b. Is the quotient less than or greater than the dividend? Explain.

1 4

Tara can make 8 gallons of lemonade.

141

142

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 15

5. Toby eats 8 pound of raisins each day. He buys a 3-pound bag of raisins. How many days will

Use the Read–Draw–Write process to solve each problem.

_1 3. Sasha does 7 problems. This is of all the problems on her math homework. How many

Toby’s bag of raisins last?

3 ÷ _8 = 24 1

problems are on Sasha’s math homework? 1 7 ÷ _3 = 21

Toby’s bag of raisins will last 24 days.

There are 21 problems on Sasha’s math homework.

4. A 2 -mile relay race is run by a team of 4 students. Each student runs an equal distance. How many miles does each student run?

6. The perimeter of a square is 5 meter. What is the length of each side of the square? 1_ 1 ÷ 4 = __ 5 20

1_ 1 ÷ 4 = _8 2

__1

The length of each side of the square is 20 meter.

_1 Each student runs mile. 8

PROBLEM SET

143

144

PROBLEM SET

317

318 This page may be reproduced for classroom use only.

_1 ÷ 3 5 3 ÷ 1_6  

_1 ÷ 3 6

4 ÷ 1_4  

_1  ÷ 5 2 _1 ÷ 3 5 3 ÷ 1_6  

_1 ÷ 7 6 8 ÷ 1_3  

5 ÷ 1_2  

Quotient Greater Than Dividend

Quotient Less Than Dividend

_1 ÷ 7 6 8 ÷ 1_3  

_1 ÷ 3 6

_1 ÷ 5 2

Quotient Less Than Dividend

5 ÷ 1_2  

4 ÷ 1_4  

Quotient Greater Than Dividend

Division Expressions Cards (2 sets)

5 ▸ M3 ▸ TC ▸ Lesson 15 ▸ Division Expressions Cards EUREKA MATH2 California Edition

it take for him to read all 5 books?

1 of a book every day. How many days will Pablo has 5 books on his reading list. He reads  _

Problem B

How much of the book does Pablo read each day?

1 of a book in 5 days. He reads the same amount of the book each day. Pablo decides to read  _

Problem A

Set 1

EUREKA MATH2 California Edition 5 ▸ M3 ▸ TC ▸ Lesson 15 ▸ Division Word Problems Set 1 and Set 2

This page may be reproduced for classroom use only.

319

320 This page may be reproduced for classroom use only.

a water break. After how many miles will Zara pause for her water break?

1 mile. She splits her run into 2 equal distances so she can pause for Zara runs every day for  _

Problem B

of the race. How many miles is the race?

1 Zara competes in a race. She runs 2 miles before she pauses for a water break. 2 miles is  _

Problem A

Set 2

5 ▸ M3 ▸ TC ▸ Lesson 15 ▸ Division Word Problems Set 1 and Set 2 EUREKA MATH2 California Edition

LESSON 16

Reason about the size of quotients of whole numbers and unit fractions and quotients of unit fractions and whole numbers.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 16

Name

Date

Use >, =, or < to compare the expressions. Explain how you can compare the expressions without evaluating them. 1 1. 6 ÷ __ 12

6 ÷ 1_ 2

1 Explain: Both expressions have the same dividend, but the expression 6 ÷ __ has a smaller

1 is less than _ 1 , so it takes more twelfths to make 6. divisor. __ 12

Lesson at a Glance Students make sense of division word problems to decide which number represents the dividend and which number represents the divisor. They use inductive reasoning to generalize about the size of quotients compared with the size of dividends in expressions. Students use those generalizations, along with their prior understanding of multiplying fractions, to compare the value of division expressions and multiplication expressions without finding the actual quotients or products.

Key Question 2. 1_ ÷ 4 5

• How can you reason about the size of a quotient without actually dividing?

1_ ÷4 3

Explain: _1 is less than _1 . When _1 is divided into 4 groups, the groups are smaller than when _1 5

is divided into 4 groups.

Achievement Descriptors

5.Mod3.AD4 Compare the effect of each number and operation on the

value of a numerical expression that includes fractions. (5.OA.A.2) 5.Mod3.AD11 Model and evaluate division of unit fractions

by nonzero whole numbers. (5.NF.B.7.a) 5.Mod3.AD12 Model and evaluate division of whole numbers by unit

fractions. (5.NF.B.7.b)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Reason About the Size of the Quotient in Context

• None

• Reason About the Size of the Quotient Without Context • Compare Expressions Without Evaluating • Problem Set

Land 10 min

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Fluency

Whiteboard Exchange: Convert Customary Weight Units Students convert pounds to ounces to build fluency with converting larger customary measurement units to smaller customary measurement units from topic A. Display 1 lb =

oz.

1 pound is equal to how many ounces? Tell your partner.

1 lb = 16 oz

Provide time for students to think and share with their partners.

1 lb = 1 oz 16

16 ounces Display the completed equation, and then display

__   1   lb = 16

oz.

Write and complete the equation. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the answer. Continue the process with the following sequence:

1 lb = 4 oz 4

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3 lb = 12 oz 4

5 lb = 10 oz 8

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 16

Whiteboard Exchange: True or False Number Sentences Students decide whether a number sentence is true or false and make false number sentences true to build fluency with reasoning about products without evaluating from topic B. Display 1 × _  5  > _ 5 . 7

Is the number sentence true or false? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1× 7 >7 False 5

1×7=7

False. Write the number sentence by using the correct comparison symbol. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the corrected number sentence. Repeat the process with the following sequence:

Teacher Note

6× 5 = 5

2 ×4< 4 3 9 9

4 ×4< 4 3 9 9

5 ×6= 5 8 6 8

False 6× 5 > 5 7 7

True

False 4 ×4> 4 3 9 9

True

For the true number sentences, consider asking students to explain to their partner how they know the number sentence is true.

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5 ▸ M3 ▸ TC ▸ Lesson 16

Launch

Students use the Co-construction routine to contextualize a statement involving division. Present the following expression to the class.

1 6 ÷ _   2

Pair students and use the Co-construction routine to have partners create a real-world situation that could be represented by the expression. Give pairs 1 minute to compare the contexts they construct with other groups. Invite students to share their ideas and explain the relationship to the expression with the class. Lacy pours 6 liters of orange juice into glasses. Each glass holds _  1 liter of orange juice. 2 How many glasses does Lacy fill? Lacy has 6 liters of orange juice to bring to the party at school. That is _  1 the amount 2

of orange juice she needs. How much orange juice will Lacy bring to the party?

Differentiation: Challenge Encourage students to write two different contexts, one in which the divisor represents the number of groups and one in which the divisor represents the size of each group.

Noah has 6 cans of paint. He uses _  1 of a can of paint for each room he paints. How many rooms can Noah paint?

Noah has 6 cans of paint. That is _  1 the amount of paint he needs to paint a room. How 2

many cans will Noah use to paint the room?

Tyler makes 6 pounds of fudge. He puts the fudge into boxes. Each box holds _  1  pound 2 of fudge. How many boxes does Tyler need? Invite students to turn and talk about how they can create a real-world situation that is represented by a mathematical statement. Transition to the next segment by framing the work. Today, we will relate division expressions to word problems and reason about the size of quotients.

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Learn

5 ▸ M3 ▸ TC ▸ Lesson 16

Reason About the Size of the Quotient in Context Students choose a division expression that represents a real-world problem and reason about the size of the quotient. Display the problem.

Blake and 3 friends share _  1 pound of frozen yogurt equally. How many pounds of frozen 3

yogurt does each person get?

Invite students to construct a tape diagram that matches the story. Then display the incorrect tape diagram. Does this tape diagram match the story?

UDL: Representation Throughout the lesson, as the class discusses what the dividend and divisor represent in each expression, consider posting both expressions and labeling the total, the size of each group, and the number of groups.

No.

Teacher Note

Why do you think this student labeled the tape diagram 4? They probably labeled it 4 because they read that 4 friends share the frozen yogurt.

It might be more meaningful to find a student

What should the tape diagram be labeled? Why?

who incorrectly labeled the tape diagram

It should be labeled _  1 because that is the amount of frozen yogurt the friends share. 3

Does this student need to change anything else about their tape diagram? Why? They need to partition it into 4 parts to represent the 4 friends who share the yogurt. When we represent a division story, we need to think about which number represents the dividend and which number represents the divisor. It is a good habit to read the problem more than once to make sure your tape diagram represents the story before you try to solve the problem.

with 4, then realized their error and changed

the label to  1 . While circulating, if you notice 3

a student who makes and then corrects their error, consider asking them to share with the class how they discovered their error and what steps they took to make the correction.

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Display the following tape diagram and have students confirm that it represents the story.

1 3

What is an expression that represents the problem? How do you know?

_ 1 ÷ 4represents the problem because there is a total 3 of _  1 being divided into 4 groups. 3 _ 1 ÷ 4represents the problem because we are finding that _ 1  is 4 groups of what amount? Invite students to think–pair–share about the following question.

Without finding the actual quotient, is the quotient greater than _  1  or less than _  1 ? Why? 3

Blake and his friends have a total of _  1 pound of frozen yogurt that they share. It does not 3

make sense for them to each get more than the total amount of frozen yogurt, so the quotient is less than _  1 . 3

I know the answer is less than _  1 because we are starting with _  1 and partitioning it into 3 3 4 equal groups, so each group is less than _ 1 . 3

Language Support Consider supporting students’ responses with the Talking Tool. Invite students to use the Share Your Thinking section to explain

whether the quotient is greater than  1  or less

than   1 .

Direct students to problem 1 in their books. Circle the expression that can be used to solve the word problem.

1. How many _   1  -pound servings of shrimp can Miss Song make with 6 pounds of shrimp? 2

6 ÷ _  1  2

_ 1 ÷ 6 2

Invite students to work with a partner to construct a tape diagram that matches the story. Then have them use their tape diagrams to determine which expression represents the problem. Circulate as students work and use the following questions to advance student thinking.

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• What does 6 represent in the first expression? In the second expression? • What does _  1 represent in the first expression? In the second expression? 2

Invite students to think–pair–share about the following question. Without finding the actual quotient, is the quotient greater than 6 or less than 6? How do you know?

The quotient is greater than 6. Because each serving is _  1 pound, there are 2 servings 2

in each pound. So, Miss Song can make more than 6 servings. It is greater than 6. We are finding how many halves are in 6. There are 2 halves in 1, so the quotient is greater than 6. How do tape diagrams help you know what expression matches a story? When I draw to show the story, I must think about what I am drawing so it matches. Once I have my tape diagram, then I can see the dividend and the divisor clearly, which helps me choose an expression. Invite students to turn and talk about how they can determine which number is the dividend and which number is the divisor in a word problem involving division.

Reason About the Size of the Quotient Without Context Students reason about the size of the quotient without finding the actual quotient. Direct students to problems 2 and 3. 1 2. 8 ÷ _ 3

greater than 8

less than 8

greater than _  1 

less than _  1 

1 3. _ ÷ 6 6 6

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Invite students to work with a partner to decide whether the quotient is greater than the dividend or less than the dividend. When students finish, guide them to summarize their observations about dividing whole numbers and unit fractions. Use the following sentence frames to support students with explaining their thinking. • When dividing because

by a unit fraction, the quotient is

the dividend

• When dividing a unit fraction by because .

, the quotient is

the dividend

Expect students to complete the sentence frames in a variety of ways based on their level of understanding of whole numbers and division with fractions. For example: • When dividing 8 by a unit fraction, the quotient is greater than the dividend because it takes 3 thirds to make a whole. • When dividing whole numbers except 0 by a unit fraction, the quotient is greater than the dividend because it takes more than 1 unit fraction to make a whole. • When dividing a unit fraction by 1, the quotient is equal to the dividend because dividing by 1 does not change the value of the unit fraction. • When dividing a unit fraction by a whole number except 0 or 1, the quotient is less than the dividend because the dividend is being divided into smaller parts.

Compare Expressions Without Evaluating Students compare two expressions without finding the actual quotients or products.

Display the expressions _  1 ÷ 4 and _  1 ÷ 4and invite students to think–pair–share about what 2

they notice about the expressions.

Both expressions show a unit fraction divided by a whole number. The divisor is the same in both expressions. The dividend in both expressions is a unit fraction. 330

Promoting the Standards for Mathematical Practice As students compare the values of multiplication and division expressions by reasoning about the effects of multiplying and dividing by whole numbers or unit fractions, they are looking for and making use of structure (MP7). Ask the following questions to promote MP7:

• How are   1 ÷ 4 and   1 ÷ 4 related? How can 2

that help you compare the values of the expressions?

__ __ 2 4 1 help you compare the values of  __ ÷ 4 2 1 and  __ ÷ 4?

• How can what you know about  1 and   1

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 16

Invite students to turn and talk about how they could compare the expressions without finding the actual quotients. Is _  1 ÷ 4greater than, equal to, or less than _  1 ÷ 4? How do you know? 2

_ 1 ÷ 4 > _ 1 ÷ 4. The divisor is the same for both expressions. Because _ 1  > _ 1 , that means _ 1 ÷ 4 4

has the greater quotient.

_ 1 ÷ 4 > _ 1 ÷ 4. I know _ 1 is a greater amount being divided into 4 groups, so the amount 4 2 2 in each group is larger than when the dividend is _  1 . 4 1 1 _ _ Display the expressions 3 ÷     and    ÷ 3and invite students to think–pair–share about what 3

they notice about the expressions. Both expressions have 3 and _  1 . 3

In the first expression, 3 is the dividend and in the second expression, 3 is the divisor. In the first expression, _  1 is the divisor and in the second expression, _  1 is the dividend. 3

Can we compare the expressions without finding the actual quotients? Why? Yes. In the first expression, the quotient is greater than 3 because we are finding how many thirds are in 3. In the second expression, we are dividing _  1  into 3 equal groups, 3 so the quotient is less than _  1 . 3

Is 3 ÷_  1 greater than, equal to, or less than _ 1 ÷ 3 ? 3 1 1 3 ÷   >   ÷ 3 3 3

_ _

Display the expressions 5 ÷_  1 and 5 ÷ _ 1 and invite students to think–pair–share about what 4

they notice about the expressions.

Both expressions show a whole number divided by a unit fraction. In both expressions, 5 is the dividend.

Is 5 ÷_  1 greater than, equal to, or less than 5? How do you know? 4

The quotient is greater than 5 because we are finding how many fourths are in 5. Is 5 ÷_  1 greater than, equal to, or less than 5? How do you know? 3

The quotient is greater than 5 because we are finding how many thirds are in 5.

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We know both quotients are greater than 5, but can we tell which expression has the greater quotient without finding the actual quotient? How do you know? Fourths are smaller than thirds, so there are more fourths in 5 than thirds in 5. So, 5 ÷ _ 1 > 5 ÷ _ 1 . 4

Display the expressions _  1 × 3 and _  1 ÷ 3and give students 1 minute to work with a partner 9

to compare the expressions without finding the actual quotients. As students work, circulate and observe the reasoning they use to determine which expression is greater. Support students by asking questions such as the following:

Differentiation: Support Consider supporting students by providing a familiar context to help them make meaning of the expressions. Also consider allowing time for students to represent each expression with a tape diagram before they compare the expressions.

• What does _  1 × 3 mean? 9

• What does _  1 ÷ 3 mean?

Teacher Note

• Is _  1 × 3greater than, equal to, or less than _  1 ? How do you know? 9 9 1 1 • Is    ÷ 3greater than, equal to, or less than    ? How do you know? 9 9

As time allows, invite students to work with a partner to compare the following expressions and share their reasoning. • _  1  × _  1  and _ 1 ÷ 2 2 3 1 • 6 ÷    and 6 ÷  1  2 3 3

  1 × 3 means   1 of 3. The product is greater 9

than   1 because it is a product of  1 and a 9

number greater than 1.

  1 ÷ 3 means   1 divided into 3 groups. The 9

quotient is less than  1 because   1 is being 9

partitioned into smaller parts.

Invite students to turn and talk about how they can compare division expressions involving unit fractions and whole numbers without finding the actual quotients.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

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Land

5 ▸ M3 ▸ TC ▸ Lesson 16

Debrief 5 min Objective: Reason about the size of quotients of whole numbers and unit fractions and quotients of unit fractions and whole numbers. Facilitate a class discussion about reasoning about the size of quotients by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the following expressions.

_ 1 ÷ 7and 7 ÷ _ 1  4

Which expression has a quotient greater than 7? How do you know? The second expression because we are finding the number of fourths in 7.

The second expression because in the first expression we are dividing _  1  into 4 7 equal groups, so each group is less than _ 1 . 4

How can you reason about the size of a quotient without actually dividing? You have to think about how the numbers in an expression relate to each other. When you divide a fraction into groups, the quotient is a smaller fraction than the fraction you started with. When you divide by a fraction, there are multiple groups in each unit of 1, so the quotient is greater than the dividend.

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

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5 ▸ M3 ▸ TC ▸ Lesson 16

Name

Date

Use >, =, or < to compare the expressions. Explain how you can compare the expressions without evaluating them. 5.

Circle the expression that could be used to solve each word problem.

_1 ÷ 3 2

1 __ ÷3 10

Explain: The divisor is 3 for both expressions. Since _ is greater than _ , that means _ ÷ 3 has 10 2 2 the greater quotient.

1 6. 4 ÷ _ 5

_1 ÷ 4 5

Explain: The expression 4 ÷ _ is greater than 4 because it represents the number of fifths in 4. 1 5

The expression 5 ÷ 4 is less than 5 because 5 is being divided into 4 equal groups.

2. How many _1 -pound burgers can Mr. Evans make with 5 pounds of meat? 4

5 ÷ 1_4

_1 ÷ 3 2

1. Kayla and her 2 brothers share 2 of a pan of lasagna equally. What fraction of the pan of lasagna does each person get?

3 ÷ 1_2

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5 ▸ M3 ▸ TC ▸ Lesson 16

1_ ÷5 4

7. 4 ÷ 2

4 × 1_2

Explain: The expressions are equal because 4 ÷ 2 has the same value as 2 of 4. In each pair, circle the description in which the pieces are longer. Explain how you know.

Rope B: 2-foot rope cut into fourths

3. Rope A: 4-foot rope cut into fourths

_1 ÷ 2 6

_1 × 1_ 6 2

Explain: The expressions are equal because _ ÷ 2 has the same value as _ of _. 1 6

Explain: Both ropes are being cut into the same number of pieces. Because it is longer than Rope B, Rope A results in longer pieces.

1 9. 4 ÷ _ 3

1 2

1 6

4 ÷ 1_4

Explain: Both expressions have the same dividend. One dividend is being divided into thirds 4. Rope C: _1 -foot rope cut into 4 equal pieces

and the other is divided into fourths. The expression divided into fourths results in more groups

Rope D: 4-foot rope cut into _1 -foot pieces

because each group is smaller. So, 4 ÷ 4 is greater.

Explain: The pieces of Rope D are _1 foot in length. Rope C is _1 foot long and is cut into multiple 2

pieces, so those pieces are each shorter than 2 foot.

10.

_1 × 2 8

_1 ÷ 2 8

Explain: The expression _1 × 2 is greater because the product of _1 and 2 is greater than _1 . The 8

other expression has a quotient less than 8 because 8 is being divided into 2 groups.

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149

150

PROBLEM SET

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11. Write the expressions in order from least to greatest. Then explain how you know which expression has the least value.

_1 ÷ 5 2

5 ÷ 1_5

_1 ÷ 5 5

5 ÷ 1_2 ,

_1 ÷ 5 2

5 ÷ 1_2

5 ÷ 1_5

Explain: 5 ÷ 5 has the smallest value because it has the least dividend and it is being divided into more groups.

Consider the expression. Write a word problem that can be represented by the given expression. 1 12. 5 ÷ _ 4

Sample:

Tyler cuts a rope that is 5 yards long into pieces that are each _1 -yard long. How many pieces can 4 Tyler cut?

13.

1_ ÷4 3

Sample:

Lacy has 3 of a pan of brownies. She cuts the pan of brownies into 4 equal-size pieces. What

fraction of the pan is each piece?

PROBLEM SET

151

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LESSON 17

Solve word problems involving fractions with multiplication and division.

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5 ▸ M3 ▸ TC ▸ Lesson 17

Name

Date

Use the Read–Draw–Write process to solve each problem.

1 of the board to make a sign. How many 1. Kayla has a wooden board that is 12 feet long. She uses _ 3 feet long is the sign?

12 ? 1_ × 12 = 4 3 The sign is 4 feet long.

Lesson at a Glance Students write multiplication and division expressions and create matching word problems to represent the same tape diagram. Then they solve a word problem that does not ask for a numerical answer. Students realize there are multiple ways of solving a problem, and that they can use different equations to solve the same problem. They share and compare their solutions and discuss the features of a word problem that could lead to multiple solution pathways by using different tape diagrams and equations.

Key Questions • Are there multiple ways to represent a problem with a tape diagram? Why?

1 cup. How many servings of rice does 2. Eddie makes 5 cups of rice. Each serving of rice measures _ 2 Eddie make?

5 1 2

• Do we all have to solve a problem the same way? Why? • Why is knowing the unit of an unknown important?

Achievement Descriptors

...

5.Mod3.AD10 Solve real-world problems involving multiplication

? halves

of fractions. (5.NF.B.6)

5 ÷ 1_ = 10

2 Eddie makes 10 servings of rice.

5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 10 min

• None

Learn 30 min

Students

• Solve a Word Problem by Multiplying or Dividing

• None

• Share, Compare, and Connect • Problem Set

Land 10 min

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Fluency

Is the number sentence true or false? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

7 ×1<7 8 8

False

7 ×1=7 8 8

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7 ×4>7 8 8

5 ×9< 9 7 11 11

5 ×9< 9 5 11 11

13 5 13 × = 20 4 20

True

False 5 ×9= 9 5 11 11

False 13 5 13 × > 20 4 20

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 17

Whiteboard Exchange: Convert Customary Capacity Units Students convert gallons to quarts or pints to cups to build fluency with converting larger customary measurement units to smaller customary measurement units from topic A. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 1 gal =

qt.

1 gallon is equal to how many quarts? Tell your partner. Provide time for students to think and share with their partners.

1 gal = 4 qt 1 gal = 1 qt 4 1 4 gal = 3 qt 3

4 quarts

Teacher Note Validate all correct responses that may not be displayed. For example, when students complete the

equation   1  gal =

1 4  or 1 __ choose to write   __  . 3

qt, they may

Display the completed equation, and then display qt.

_ 1  gal = 4

Write and complete the equation.

Display the answer and then display _  1  gal = 3

qt.

Write and complete the equation. Display the answer. Repeat the process with the following sequence:

1 pt = 2 c 1 pt = 1 c 2 1 1 pt = 2 c 4

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Launch

Students explain the connection between a multiplication expression and a division expression by using a tape diagram. Display the tape diagram.

$60

Direct students to independently try to write more than one expression that matches the tape diagram. While students work, circulate to find the following two expressions. If students do not write one or both, provide the two expressions.

• _1  × 60 5 • 60 ÷ 5 Invite students to think–pair–share about how they see both expressions represented in the tape diagram. I see 60 ÷ 5 because the whole tape is 60 and it is partitioned into 5 equal parts. The unknown is 1 of the 5 parts, so 60 ÷ 5 represents that unknown value. I see _ 1 × 60because there are 5 equal parts, and 1 of the parts has a question mark. 5

1 part out of 5 parts is _ 1 . We are finding _ 1  of 60 and _ 1 × 60represents that value. 5

Do you notice anything else in the tape diagram? The total is labeled with dollars. The context of $60 can help us think about a real-life scenario that matches the tape diagram. Have students work with a partner to construct two word problems: one that can be solved with the multiplication expression and one that can be solved with the division expression. While students work, circulate to find one of each word problem type. If students do not create one or both types, provide a sample.

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• Noah went to the store and brought $60 with him. He spent _ 1 of his money. How much 5

money did he spend?

• Lacy bought 5 copies of the same book as gifts for her friends. She spent a total of $60 on the books. How much money does each book cost? What is the answer to each of these problems?

$12 Although each problem has a different context and we used different expressions to evaluate, each problem is represented by this tape diagram and has the same answer. Transition to the next segment by framing the work. Today, we will solve word problems involving fractions with multiplication and division.

Learn

Promoting the Standards for Mathematical Practice

Solve a Word Problem by Multiplying or Dividing Students select an operation to solve a word problem involving fractions. Direct students to problem 1 in their books. 1. Ms. Baker has 4 yards of fabric. It takes _  1 yard to make a pencil pouch. Does Ms. Baker 4 have enough fabric to make one pencil pouch for each of her 24 students? Explain how you know. Sample:

4 ÷ _ 1 = 16 4

Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She has enough fabric to make 16 pencil pouches, but she has 24 students. © Great Minds PBC

As students analyze a word problem that contains several pieces of information and decide whether to use multiplication or division to solve the problem, they are making sense of problems and persevering in solving them (MP1). Ask the following questions to promote MP1: • How can you explain this context in your own words? • What could you try in order to start solving the problem? • Is your solution method working? Is there something else you could try?

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Invite students to turn and talk about what they might draw to show the fabric. Allow students to work independently or with a partner to model the problem. Encourage students to self-select their tools and methods. Circulate and observe student work. Select two or three students to share in the next segment. Look for work samples that highlight different tape diagrams to model the story. Be sure to find a student who used multiplication to solve the problem and another who used division to solve the problem. Purposefully choose work that allows for rich discussion about connections between student work samples. Use the following prompts to elicit student thinking: • Tell me how your drawing matches the problem. • Tell me about your method. • Which operation did you choose to solve the problem? Why? When you speak with students, focus on eliciting student thinking as you informally assess their understanding and select students to share.

Differentiation: Support Allow students to represent the yards by using

cubes. Label each cube as  1 . Consider 4

providing different colors to show each yard.

1 4 1 4

The student work samples shown demonstrate a variety of solutions involving multiplication and division.

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Multiply a Whole Number and a Unit Fraction

? yards

1 4

...

Teacher Note The sample student work shows common responses. Look for similar work from your students and encourage authentic classroom conversations about the key concepts.

24 fourths ? yards

24 fourths 1

24 × 4 = 6

Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She would need 6 yards of fabric to make a pencil pouch for each of her students and she only has 4 yards.

If your students do not produce similar work, choose one or two pieces of their work to share, and highlight how it shows movement toward the goal of this lesson. Then select one work sample from the lesson that works best to advance student thinking. Consider presenting the work by saying, “This is how another student solved the problem. What do you think this student did?”

Divide a Whole Number by a Unit Fraction

4 yards 1 4

... ? fourths 4 yards

? fourths 1

4 ÷ 4 = 16

Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She has enough fabric to make 16 pencil pouches, but she has 24 students.

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Share, Compare, and Connect Students share and compare solutions and reason about their connections. Gather the class and invite the students you identified in the previous segment to share their solutions one at a time. As each student shares, ask questions to elicit their thinking and clarify their reasoning. Ask the class questions that invite students to make connections between the demonstrated solutions and their own work. Encourage students to ask questions of their own. Multiply a Whole Number and a Unit Fraction (Yuna’s Way) What does Yuna show as the known and unknown information in her first tape diagram? Why did she show that? The tape diagram shows one part is _ 1 because Ms. Baker 4

needs _ 1 yard for each pencil 4

Language Support Consider inviting students to use the Share your Thinking and Ask for Reasoning sections of the Talking Tool as they share their work, ask questions about their classmates’ work, and compare their work to their classmates’.

? yards 1 4

...

pouch. The bottom is labeled

24 fourths

with 24 fourths because she

? yards

wants to make a pencil pouch for each of her 24 students. The unknown is how many yards are equal to 24 fourths because Ms. Baker wants to know whether she has enough fabric to make the

24 fourths 1

24 × 4 = 6

Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She would need 6 yards of fabric to make a pencil pouch for each of her students and she only has 4 yards.

pencil pouches. What part of the story is not included in Yuna’s tape diagram? Yuna did not show the 4 yards of fabric that Ms. Baker has. Why do you think Yuna did not show the 4 yards? She focused on the 24 students and the amount of fabric Ms. Baker needs for each pencil pouch. Yuna did not need to use the 4 yards. 344

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How did Yuna solve the problem? Yuna drew 24 fourths and represented each group of 4 fourths as 1 yard. Her drawing shows 24 fourths is equal to 6 yards. Yuna found that Ms. Baker does not have enough fabric because 6 yards of fabric is greater than 4 yards of fabric. Yuna, why did you decide to solve the problem with multiplication?

I decided to multiply because I knew Ms. Baker needs _ 1 yard of fabric to make 4

1 pencil pouch for each of her 24 students. I drew all 24 fourths and saw the number of groups, 24, and the size of each group, _ 1  yard, so I multiplied to find the total 4 number of yards. Divide a Whole Number by a Unit Fraction (Toby’s Way) What does Toby show as the known and unknown information in his first tape diagram? Why did he show that?

4 yards 1 4

The tape diagram shows 4 yards because that is the number

... ? fourths 4 yards

of yards of fabric Ms. Baker has. The tape diagram also shows

one part is _ 1  because she needs 4 _ 1  yard for each pencil pouch. 4

The unknown is how many fourths are in 4 yards of fabric because Ms. Baker wants to know whether

4 yards is enough fabric to make

? fourths 1

4 ÷ 4 = 16 Ms. Baker does not have enough fabric to make a pencil pouch for each of her students. She has enough fabric to make 16 pencil pouches, but she has 24 students.

a pencil pouch for each of her students. What part of the story is not included in Toby’s tape diagram? Toby did not show the 24 students. Why do you think Toby did not show the 24 students? He focused on the 4 yards of fabric Ms. Baker has and the amount of fabric she needs to make each pencil pouch. So, he did not need to show the 24 students. © Great Minds PBC

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How did Toby solve the problem?

He divided 4 yards by _ 1 yard to find the number of pencil pouches Ms. Baker can make 4

with 4 yards of fabric. Toby found that Ms. Baker does not have enough fabric to make a pencil pouch for each student. She only has enough to make 16 pencil pouches. Toby, why did you decide to solve the problem with division?

I knew _ 1 yard of fabric is used for 1 pencil pouch. I know Ms. Baker has 4 yards of fabric. 4

I needed to find out how many fourths are in 4 which means I can divide. Invite students to turn and talk to compare Yuna’s and Toby’s work and thinking. Display the tape diagrams Yuna and Toby used to make sense of the story and their accompanying equations.

1 4

Yuna’s Way

Toby’s Way

? yards

4 yards

... 24 fourths 1

24 × 4 = 6

1 4

UDL: Action & Expression Consider reserving time for students to reflect after all work has been shared. Ask students if they heard reasoning from another student that they might use next time. Engage the class in a discussion about reasons to try a different approach. The sharing component of this segment provides students with multiple examples of annotated peer work. When students have time to reflect on their own work relative to the examples, it serves as a formative feedback opportunity.

... ? fourths 1

4 ÷ 4 = 16

The unknown in each tape diagram is different, although they each represent the same word problem. Why is that? They each used different information from the story. Yuna used the number of yards of fabric Ms. Baker needs for each pencil pouch and the number of students. Toby used the total number of yards of fabric Ms. Baker has and the number of yards of fabric she needs for each pencil pouch. Why is _ 1 used in both tape diagrams? 4

That is the number of yards of fabric that Ms. Baker needs for each pencil pouch, so it is necessary to include in both tape diagrams.

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How are the equations different? One uses multiplication and the other uses division. The answers are different. What does 6 represent in Yuna’s equation? Ms. Baker needs 6 yards of fabric to make 24 pencil pouches. What does 16 represent in Toby’s equation? Ms. Baker can make 16 pencil pouches from 4 yards of fabric. When our tape diagrams have different unknowns and our equations have different answers, it is important to pay close attention to what the unknown represents. Why? We have to know what our unknown represents so we can properly answer the question. In Yuna’s work, her unknown was the number of yards of fabric Ms. Baker needs to make the pencil pouches. In Toby’s work, his unknown was how many pencil pouches Ms. Baker could make with the fabric she has. If Yuna and Toby did not know what their unknowns represented, they might not have answered the question correctly. Why did both solutions, with different unknowns and different equations, result in the same answer statement—that Ms. Baker did not have enough fabric? The story gave a lot of information, so you could use different information to solve the problem in different ways. The question asked about whether Ms. Baker had enough fabric. So, we did not need a specific number as the answer. We only needed to find out whether there is enough fabric to make a pencil pouch for each student. To do that, we had to find how much total fabric was needed so we could compare it with how much fabric Ms. Baker has. Or we needed to know how many pencil pouches she could make with the fabric she has and compare that with how many students she has. Reinforce how this word problem allowed for different equations with different answers by revisiting the word problems students wrote earlier in the lesson.

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Earlier, we wrote multiplication and division word problems for a tape diagram with the same unknown. But for the word problem about Ms. Baker, we had tape diagrams with different unknowns. How might you know the difference between the two types when you read word problems and draw your tape diagrams? If a word problem does not ask for a specific amount, such as the Ms. Baker problem that asked whether she had enough fabric—not how much fabric—then we could probably find different ways to solve the problem. Invite students to skim the Problem Set to find a problem or problems in which they might have student work with different unknowns and to tell why.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Solve word problems involving fractions with multiplication and division. Gather the class with their Problem Sets. Facilitate a class discussion about solving word problems involving fractions with multiplication and division by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Invite students to share their work and their thinking about problem 7 in the Problem Set with a partner. Are there multiple ways to represent this problem with a tape diagram and solve it? How do you know? Yes. My partner and I drew different tape diagrams and solved the problem differently. 348

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Yes. Problem 7 asks whether all the books fit, so we do not need a specific number for the answer. That means we could probably represent and solve the problem in different ways. Invite students to share their work and their thinking about problem 5 in the Problem Set with a partner.

5 dozen

Display the student work. How did this student think about the problem?

The student found _ 1  of 5 dozen, which 5 is 1 dozen. Did you solve the problem differently? What else could you have done to solve the problem?

? 1 5 × 5= = 1 5 5

The student misses 1 dozen baseballs.

I found 5 ÷ 5 = 1 to show 5 dozen divided into 5 equal groups, which means there is 1 dozen in each group.

I knew 5 dozen baseballs equals 60 baseballs, so I found _ 1 × 60 = __  60 = 12 . The student 5 5 misses 12 baseballs, which equals 1 dozen baseballs. We could say the student misses 12 baseballs or 1 dozen baseballs. Why is it important to know the unit of an unknown? If the unknown in the problem is the number of dozens and we did not realize it, we might answer the question with 12 dozen, which is incorrect. If the unknown in the problem is the number of baseballs the student misses and we said the answer is 1 baseball, it would be incorrect because the student misses 1 dozen baseballs, not 1 baseball.

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 17

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 17

3. A chef makes 4 pizzas. They slice each pizza into eighths. How many slices of pizza are there?

4 ÷ _8 = 32 1

There are 32 slices of pizza.

Use the Read–Draw–Write process to solve each problem.

1. Jada pours 2 gallon of fruit punch equally into 4 containers. How many gallons of fruit punch are in each container?

_1 ÷ 4 = 1_ 2 8

Each container has 8 gallon of fruit punch.

4. Mr. Sharma spends 5 minutes driving to the gym. This is 4 of the time he spends driving to work.

How many minutes does Mr. Sharma spend driving to work?

2. Eddie has 45 bottles of water. He drinks 9 of them. How many bottles of water does Eddie drink?

5 ÷ _4 = 20 1

_1 × 45 = 5 9

Mr. Sharma spends 20 minutes driving to work.

Eddie drinks 5 bottles of water.

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157

158

PROBLEM SET

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5 ▸ M3 ▸ TC ▸ Lesson 17

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TC ▸ Lesson 17

7. Mr. Perez needs to put 24 books on his bookshelf. Each book is 8 inches wide. The bookshelf

5. A student misses 5 of the 5 dozen baseballs their coach pitches to them. How many baseballs do they miss?

is 22 inches wide. Will all the books fit on the shelf? Explain.

24 × _8 = 21 7

5 × 12 = 60

_1 × 60 = 12 5

Yes, all the books will fit on the bookshelf. The shelf is 22 inches wide and the total width of all the books is 21 inches.

They miss 12 baseballs.

6. Julie uses 5 of her beads to make 2 necklaces. a. What fraction of her beads does Julie use to make 1 necklace? 1 _1 ÷ 2 = __ 5 10

__1

She uses 10 of her beads to make 1 necklace.

b. If Julie has 160 beads, how many beads does she use to make 1 necklace? 1 __ × 160 = 16 10

Julie uses 16 beads to make 1 necklace.

PROBLEM SET

159

160

PROBLEM SET

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Topic D Multi-Step Problems with Fractions In topic D, students use their understanding of operations with fractions to solve multi-step problems with fractions. Students begin the topic by writing and evaluating numerical expressions involving

parentheses, such as 5 × (  _1  −   _1  ) or   _3 × (  _1  +   _2  ). They learn to interpret statements such 3

as 5 times the difference of   _1 and   _1 or   _3 of the sum of   _1 and   _2  and write them as numerical 3

expressions. Students take their experience of writing and evaluating expressions and apply it to writing and solving word problems. Once they are equipped with the knowledge of how to multiply fractions and how to divide with a whole number and a fraction, students can solve word problems that include contexts that naturally lend themselves to fractional numbers. Students use tape diagrams to model multi-step problems that require the use of addition, subtraction, multiplication, and division with fractions. Through experience and discussion, students realize that modeling a problem with a tape diagram can reveal solution pathways that require fewer steps to find the answer. The topic concludes with students evaluating expressions involving fractions and nested grouping symbols. Lesson 22 is optional but provides a natural extension to the work with expressions involving fractions and invites students to read expressions to make sense of the relationship between the numbers. In module 4, students apply their understanding of fractions when they work with decimal operations because they can think about decimal numbers as fractions with units of tenths, hundredths, and thousandths.

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Progression of Lessons Lesson 18

Lesson 19

Lesson 20

Compare and evaluate expressions with parentheses.

Create and solve one-step word problems involving fractions.

Solve multi-step word problems involving fractions and write equations with parentheses.

? 1 4 9 1 × 10 2

1 2 × 4 5

( )( ) 9 1 1 2 × + × 10 2 4 5

I can evaluate expressions with parentheses by using tape diagrams to help me make sense of the expressions. I can then use reasoning to compare the values of expressions and statements without evaluating.

354

... ? fourths

5 pizzas are cut into fourths, How many pieces of pizza are there after all the pizzas are cut? I can write one-step word problems to match a tape diagram or an expression.

60 glasses

no glasses

(

)

60 + 4 × 60 = 108 5 There are 108 students in the fifth grade. I can use the Read–Draw–Write process to solve word problems. I can use a tape diagram to find a solution method that uses fewer steps.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD

Lesson 21

Lesson 22

Solve multi-step word problems involving fractions.

Evaluate expressions involving nested grouping symbols. (Optional)

Lacy’s cat

Lacy’s dog

3 × 8 = 24

Lacy’s dog weighs 24 pounds.

24 24 − 2 21 = 21 21

Noah’s dog

Noah’s dog weighs

21 21 pounds.

3+ 1 ×3 -2 = 5×3 -2 8 4 3 8 3 = 158 - 32 = 29 24 I can read an expression that has parentheses inside of brackets or braces to help me evaluate the expression.

I can use the Read–Draw–Write process to solve multi-step word problems involving fractions. I notice when my tape diagram represents the unknown of the word problem, and when it does not, I can revise or draw another tape.

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Language Objectives Language objectives indicate the language and literacy skills students need to engage with the lesson objectives. Because language learning and mathematical learning are interdependent, teaching toward language objectives helps teachers to consider language needs when supporting students in reaching the lesson objectives. Lesson 18 Read a statement involving fractions, whole numbers, and language related to the four operations. Write and evaluate a numerical expression that represents the statement. Orally and in writing, reason about how to compare statements and expressions without evaluating them. Lesson 19 Create word problems that match tape diagrams or equations. In partners, orally describe how the word problems represent the tape diagrams or equations. Lesson 20 Read multi-step word problems involving fractions. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation with parentheses for solving a multi-step problem that involves fractions. Lesson 21 Read multi-step and comparison word problems involving fractions. Listen to and orally describe problems’ meanings and possible solution strategies. Write or draw to record a solution strategy and an equation for solving a multi-step comparison word problem that involves fractions. Lesson 22 Read expressions involving one or more sets of grouping symbols and determine how to evaluate the expressions by breaking them up into parts. With a partner and with the class, orally describe the process of evaluating an expression that involves one or more sets of grouping symbols.

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LESSON 18

Compare and evaluate expressions with parentheses.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

Name

Date

1. Write an expression to represent the statement. Then evaluate the expression. 2 7× _ _1 Expression: _ ( − ) 35 Value: ___

_7 of the difference between _2 and 1_ 3

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Lesson at a Glance Students use tape diagrams to support them when they write and use equations to find an unknown, and then they transition to writing expressions given in word form. They use parentheses to indicate which part of an expression must be evaluated first and realize the placement of parentheses impacts the order in which it is evaluated and can impact the value of the expression. Students consider expressions given numerically and as statements and compare them without evaluating.

Key Questions • Does modeling with tape diagrams help you make sense of where parentheses belong in equations and expressions? How? • Can you compare statements and expressions without evaluating them? How?

2. Use >, =, or < to compare. Explain how you can compare without evaluating.

2 × 7 ÷ 1 ) > 4 of the quotient of 1 and 7 3 ( 9 6 9 2 4 The fractions and are equivalent, but 7 ÷ 1 is greater than 1 and the quotient of 1 and 9 3 6 9

Achievement Descriptors

7 is less than 1, so the expression on the left is greater.

5.Mod3.AD2 Evaluate numerical expressions that include fractions

and parentheses. (5.OA.A.1) 5.Mod3.AD3 Translate between numerical expressions that include

fractions and mathematical or contextual verbal descriptions. (5.OA.A.2) 5.Mod3.AD4 Compare the effect of each number and operation on the

value of a numerical expression that includes fractions. (5.OA.A.2)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Write Equations to Find Unknown Values

• None

• Write and Evaluate Expressions • Compare Statements and Expressions • Problem Set

Land 10 min

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Fluency

Counting the Math Way by Tenths Students construct a number line with their fingers while counting aloud and model a composition and a decomposition to prepare for extending place value understanding to the thousandths place beginning in module 4. Let’s count the math way. Each finger represents 1 tenth. Face the students and instruct them to mirror you. Show a fist with your right hand, palm facing out. Show me your left hand. Make a fist like me. That’s 0 tenths, or 0. Now, raise your right pinkie. Show me your left pinkie. That’s 1 tenth. Student View of Your Hand

Student View of Student’s Hand

0.1

0.2

0.3

0.4

0.5

Let’s put up the very next finger. Raise your right ring finger. Students raise their left ring finger.

Teacher Note Around the world, numbers are represented on hands in many ways. Counting the math way has the mathematical advantage of progressing from left to right without interruption, just like the number line. It also demonstrates the magnitude of numbers, as students observe and feel the quantity increase as they count forward. Students, whether they’re looking at their own hands or your hands, will see a left-to-right progression. The progression from one finger to the next mimics the number line. To you, the progression will appear in reverse. Students begin to use this method in kindergarten and continue to use it through grade 5 to model place value concepts and perform operations with whole numbers and decimal fractions.

That’s 2 tenths. Put up the next finger. 3 tenths.

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Now that students understand the routine, switch to having them say the count as they show fingers. Guide students to continue counting the math way by tenths to 10 tenths, then back down to 0. Student View of Your Hands

0.6

0.7

0.8

0.9

1.0

Student View of Student’s Hands

Let’s count the math way again by tenths from 0 tenths to 10 tenths. Have students count the math way by tenths from 0 to 1.0. What larger unit can we make with 10 tenths?

1 one We can bundle 10 tenths to make 1 one. (Clasp hands together.) Ask students to model bundling 10 tenths by clasping their hands together. Model unbundling by unclasping your hands. Ask students to model unbundling 1 one by unclasping their hands. Let’s count the math way by tenths from 10 tenths to 0 tenths. Have students count the math way by tenths from 1.0 to 0.

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Whiteboard Exchange: Interpret a Fraction as Division Students write a fraction as a division expression and determine the quotient to prepare for solving multi-step word problems involving fractions beginning in lesson 20. Display __  34 = 2

How can we represent the fraction as a division expression? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

34 = 34 ÷ 2 = 17 2

34 ÷ 2 Display the answer. Divide and express the quotient as a whole or mixed number. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the quotient. Repeat the process with the following sequence:

72 = 72 ÷ 3 = 24 3

362

76 = 76 ÷ 4 = 19 4

1 81 = 81 ÷ 5 = 16 5 5

90 = 90 ÷ 6 = 15 6

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

Choral Response: Multiply Fractions Students multiply a fraction by a fraction to prepare for solving multi-step word problems involving fractions beginning in lesson 20. Display _ 1  × _ 1  =

What is the product in fraction form? Raise your hand when you know.

1 1 1 × = 2 3 6

Wait until most students raise their hands, and then signal for students to respond.

1   6

Display the product. Repeat the process with the following sequence:

1 2 2 × = 6 2 3

1 1 1 × = 9 3 3

Launch

2 1 2 × = 9 3 3

3 2 6 × = 4 3 12

3 4 12 × = 4 3 12

2 4 8 × = 4 5 20

7 4 28 × = 4 5 20

Teacher Note

Students analyze a tape diagram to prepare for writing and evaluating expressions. Write __   9   × _  1  + _ 1  × _ 2 . 10

Invite students to turn and talk about how they would evaluate this expression. Display the following tape diagram. Direct students to work with a partner to write as many statements as they can

9 1 × 10 2

1 2 × 4 5

This lesson previews the work of the topic, in which students create and solve one-step and multi-step word problems and write expressions and equations that include parentheses. Use this activity to informally assess students’ understanding of writing and evaluating expressions and also to assess their ability to describe expressions in words by using terms such as product, sum, and unknown value.

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about what they know from looking at the tape diagram. Circulate to informally assess students and to provide support. Expect that some student pairs may identify only the parts of the tape diagram, while others may find the value of the unknown by evaluating an expression. After 2 minutes, gather the class to discuss what they found. Validate a range of ideas but guide the discussion toward finding an expression that can be used to determine the unknown value. What information is shown in the tape diagram?

The tape diagram has two parts. One part is __  9   × _  1 , and the other part is _ 1  × _ 2 . 10

The sum of the two parts is the value of the unknown, __  11 .

How can we use what we see in the tape diagram to write an expression that represents the unknown value? We can write the value of the first part plus the value of the second part.

Return to the expression __  9   × _  1  + _ 1  × _ 2 . 10

Invite students to turn and talk about whether they would evaluate the expression in the same way as they discussed before they saw the tape diagram. Now that you see the tape diagram that matches the expression, would you place parentheses in the expression? Where? Yes. I would put parentheses around __  9   × _  1  and around _ 1  × _ 2 . 10

Transition to the next segment by framing the work. Today, we will write, evaluate, and compare expressions that include parentheses.

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Learn

5 ▸ M3 ▸ TD ▸ Lesson 18

Write Equations to Find Unknown Values Students write an equation that can be used to find an unknown value in a tape diagram.

Differentiation: Support

Direct students to problem 1 in their books and invite them to study the tape diagram. Write an equation that can be used to find the unknown value for each tape diagram. Then use the equation to find the value of the unknown.

9 + 12

2 x = (9 + 12) × _   3 2_ 21 ×   = 3 42 __   = 3

For groups who may need additional support writing an equation to find an unknown in problem 1, consider removing the complexity of showing the total as 9 + 12 and instead show it as the sum 21.

Promoting the Standards for Mathematical Practice

= 14 x = 14 Based on this tape diagram, what do we know? There are 3 equal-size parts.

When students analyze a tape diagram to write an equation that can be used to find the value of the unknown, they are attending to precision (MP6).

The total is 21.

Ask the following questions to promote MP6:

The sum of the 3 equal parts is 21.

• How are you using parentheses in your equation?

The unknown value is represented by x. The unknown value is 2 of the 3 parts of the tape diagram.

• What details are important to think about when you write an equation to represent the unknown value in the tape diagram?

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What do we need to find? We need to find the value of x. We need to find _  2  of 21. 3

Because the unknown value x is 2 of the 3 parts of the tape diagram, we can find the value of x by finding _  2  of 21. What is the value of the unknown? 3

The value of the unknown is 14. Guide students to write an equation to represent how they found the unknown value in problem 1. Let’s record our thinking by writing an equation. Record x =.

We found the value of x by finding _  2  of 21. Where does 21 come from? 3

It is the sum of 9 and 12. Record (9 + 12).

To show we found the sum first, let’s put parentheses around 9 + 12. What did we do next? We found _  2  of 21. 3

Record × _  2 and direct students to check that they wrote the equation in problem 1. 3 Write x =  2  × (9 + 12). 3

2 Invite students to turn and talk about whether the equation x =  _   × (9 + 12)also gives the 3 same value of x.

Write x = _  2 × 9 + 12 . Point to the equation. 3

Does this equation, x =_     × 9 + 12 , also give the same value of x? Why? 2 3

No, because 9 + 12 is the total, so it needs to be in parentheses. No, because _  2 × 9 = 6 , and 6 + 12 = 18. 3

I am not sure. It has all the same numbers and operations as the other equation.

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Without a tape diagram or context, you may think we need to multiply _  2  and 9 first 3

and then add 12. That would mean x = 18. But we found x = 14. To ensure we all find the same value of the unknown, we use parentheses to show what we need to do first. Direct students to problem 2 and invite them to study the tape diagram.

Differentiation: Challenge Challenge students by asking them to write an equation that can be used to find the value of f in this tape diagram. Then have them find the value.

1 1 − 3 4

y=5×( 1_  − 1_ ) 3

= 5 × __  1   12 5 = __     12

5 y = __     12

How is this tape diagram different from the previous one? There is a subtraction expression that shows the value of 1 part. There are 5 equal-size parts in the tape diagram. The unknown value y is the total of the 5 equal parts. To find the value of y, what would you do first? Why?

I would find the difference _  1  − _  1 because that is the value of 1 part. Then I can multiply 3 4 that value by 5. Direct students to work with a partner to find the value of y. Then facilitate a discussion about writing the equation by using the following questions. What is the value of y? How did you find it?

The value of y is __   5  . First, I found _  1  − _  1  = __   1  , and then I found 5 × __  1  . 12

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To find the value of y, we can multiply the difference of _  1  and _  1  by 5. We can represent 3

that thinking with an equation. What must we include in the equation to show we have to find the difference first?

We have to put parentheses around _  1  − _  1 to show we need to find the difference first. 4

Invite students to write an equation that can be used to find the value of y. If students already wrote an equation, encourage them to write another equation that leads to the same result. What equation did you write to find the value of y? y = ( _ 1  − _ 1 ) × 5 3

Differentiation: Support

_ _

y = 5 × (  1  −  1 ) 3 4 Why can the value of y be represented by two different equations? Both equations result in the same value for y.

The factors 5 and ( _  1  − _ 1 )are in different orders, but we can multiply in any order and 3

get the same answer.

Students may need support to write the expression for problem 3. Consider having students draw a tape diagram first, then use the tape diagram to write the expression.

1 2 + 6 3

Write and Evaluate Expressions Students write and evaluate expressions given as a statement. Direct students to problem 3. Read the statement aloud. Write an expression to represent the statement. Then evaluate the expression. 3. _  3 of the sum of _  1  and _  2  5

_ 3  × ( _ 1  + _ 2 ) = _ 3  × ( _ 1  + _ 4 ) 5

= _  3  × _ 5  5

= _  3  6

? If students need additional support, consider providing the tape diagram and asking the following questions:

__1 __2

• What does the sum   +   represent in the 6 3 tape diagram?

__3

• Where do you see   ? 5

__3

• Can you find the value of   of the tape 5 diagram? How?

Then use the following prompts. 368

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The statement tells us to find _  3 of something. What is the something? 5

The sum of _  1  and _  2 

3 3 We need to find    of a number, and that number is the sum of  1  and  2 . Can we find 5 6 3  3 of that number yet? Why? 5 No. We do not know the sum of  1  and  2 , so we cannot find  3 of it yet. 5 3 6 6

After we find the sum of _  1  and _  2 , how can we find _  3 of it? 6

We could multiply the sum by _  3 . 5

Write an expression to represent your thinking about this statement and then evaluate your expression. Circulate as students complete problem 3 and encourage them to think about where to place parentheses in their expressions. What is _  3 of the sum of _  1  and _  2 ? 5

3   6

UDL: Action & Expression Consider comparing the correct solution for problem 3 with an incorrect work sample. Present a chart that shows the correct work in sample A and the incorrect work in sample B to emphasize how the placement of parentheses affects the value of the expression. Ask students, “How would you compare the work in sample A to the work in sample B? Why is it incorrect to evaluate problem 3 the way it is shown in sample B?” Post the chart for the remainder of the topic as an example of why parentheses are used and the importance of their placement. Use color coding and annotation to highlight these features, such as in the following example:

First we need to find the sum. Then find 3 of that sum.

Validate all equivalent responses. Some students may find the sum by using eighteenths as the unit, which results in __  9  , while others may express their answer in the largest possible 18 unit, which results in _  1 . 2

Did you include parentheses in your expression to identify what you had to do first? Where did you place them? Why?

5 3 1 2 5of the sum of 6 and 3

Sample A

Yes, I put parentheses around _  1  + _  2 because I had to find the sum before I could find _  3 of it.

Direct students to problem 4. Have them silently read the statement. 4. 4 times as much as the difference of _  6  and _  1  7

4 × ( _ 6  − _ 1 ) = 4 × ( __  12  − __   7  ) 7

= 4 × __  5   14 20 = __    

Sample B

3x 1 +2 3x 1 + 4 5 6 3 =5 6 6 3 5 =5x6 3 =6 3x 1 +2 3 +2 5 6 3 = 30 3 3 20 = 30 + 30 23 = 30

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The statement tells us to find 4 times as much as another number. What is the other number? It is the difference of _  6  and _  1 . 7

Can we multiply yet? Why? We cannot multiply until we find the difference. Once we find the difference, we can multiply. Write and evaluate an expression that represents your thinking about this statement. Consider having students work in pairs to complete problem 4. What is 4 times as much as the difference of _  6  and _  1 ? 7

__  20 

Validate all equivalent responses. Some students may express their answer in the largest unit possible, which results in __  10 , while others may express their answer as a mixed number. 7

Did you include parentheses in your expression? Why?

Yes. I put parentheses around _  6  − _  1 to show I needed to find the difference before 7 2 I multiplied.

Write 5 ×_  1 + 12. 3

Invite students to think–pair–share about the following question.

Does this expression represent the statement 5 times the sum of _  1  and 12? Why? 3

No. I interpret the expression as 5 groups of _  1  plus 12, and that is not what the 3 statement says.

No. The expression shows to evaluate 5 ×_  1 first and then add 12. That does not seem 3

to match the statement because it tells us to find the sum of _  1  and 12. 3

No. The statement tells us to multiply 5 and another number, so we need to find the sum first.

What could we include in the expression to represent 5 times the sum of _ 1  and 12? Why? 3

We could put parentheses around _  1 + 12to make sure the sum of _  1  and 12 is multiplied 3

by 5. Otherwise, someone might only multiply _  1  by 5.

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Compare Statements and Expressions Students compare statements and expressions by reasoning about the size of the parts. Present the following statement and expression. Use the Math Chat routine to engage students in mathematical discourse.

(_ 1 × 12) + 2

Add 2 to the product _  1  × 16

Give students 1 minute of silent think time to determine whether the value represented by the statement is greater than, equal to, or less than the value of the expression without evaluating either one. 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.

Language Support

Then facilitate a class discussion. Invite students to share their thinking with the whole group.

The value of the statement add 2 to the product _  1 × 16is greater than the value of the 4

expression ( _  1 × 12) + 2 . I know because both have a group that shows a product added 4

to 2, and each product has _  1 as a factor. So I only had to think about the value inside the

Direct students to use the Share Your Thinking section of the Talking Tool to support students with sharing how they can compare statements and expressions without evaluating them.

parentheses. _  1  of 16 is greater than _  1  of 12. 4

Repeat the process for the following statement and expression.

2 × ( _ 8  − _ 2 )

The difference of _  8  and _  1 , doubled

Invite students to think–pair–share about how they can compare the values of the statement and the expression without evaluating them.

The value of the statement the difference of  _8  and  _1 , doubled is greater than the value 3 9 of the expression 2 ×( _ 8  − _ 2 ). I realized multiply by 2 means the same as doubled. I only 9

needed to think about the differences in each representation. Because _  2  > _  1 , I could tell 3

that the difference between _  8  and _  1 is greater than _  8  − _  2 , so the expression 2 × ( _ 8  − _ 1 ) is greater. © Great Minds PBC

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Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Compare and evaluate expressions with parentheses. Gather the class with their Problem Sets. Facilitate a class discussion about comparing and evaluating expressions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Direct students to problems 4–10 in the Problem Set. Did you draw a tape diagram to make sense of any statements or expressions? Was it helpful? Why?

Yes. In problem 6, drawing the tape diagram helped me see I needed to find _  5  of 10 first 6

and then subtract _  1 . So when I wrote the expression, I put parentheses around _  5 × 10 3

to show that the product had to be found before subtracting.

Yes. I started to draw the tape diagram for problem 8, and once I labeled the total as 5 + 9, I realized they are equal. Does modeling expressions with tape diagrams help you make sense of where parentheses belong in expressions? How? Yes. Modeling with tape diagrams helps me see which parts of the expression need parentheses so I know which part to evaluate first. Can you compare statements and expressions without evaluating them? How? Yes. We look for what is the same and what is different between the statement and expression. When one part is the same, we can focus on the value of the part that is different. 372

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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.

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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 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

Date

Name

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

Write an expression to represent each statement. Then evaluate your expression. 4. The difference between _3 and _2 , doubled 4

Write an equation that can be used to find the unknown value for each tape diagram. Then use the equation to find the value of the unknown.

6+8

4 2 − 5 3

h = _1 × (6 + 8)

__ Value: 26 28

Expression: (_3 − _2) × 2

4 h = 14 4

5. _1 of the sum of _2 and _1 5

Expression: _1 × (_2 + _1) 3 2 5

7 Value: __ 30

m = 5 × (_4 − _2)

10 m = __

6. _1 subtracted from _5 of 10 3

3. Read the expressions. Then follow the instructions for each part.

2 × (_1 + 4)

_1 (4 + 2) − 2

Expression: (_5 × 10) − _1 6 3

_1 (2 − 2) × 4

2 + (_1 + 4) 2

Value: 8

a. Circle the expression that represents 2 more than the sum of _1 and 4.

7. Half of _1 added to the product of 3 and _3 3

Expression: (_1 × _1) + (3 × _3) 56 Value: __

b. Put a box around the expression that represents 4 times as much as the difference between

2 and _1 . 2

c. Underline the expression that represents twice as much as the sum of _1 and 4. 2

d. Draw an X through the expression that represents 2 less than the sum of _1 and 4. 2

374

165

166

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

11. The line plot shows the amount of water, in gallons, that Yuna drinks each day for one week.

Use >, =, or < to compare. Explain how you can compare the expressions without evaluating. 8. _1 the sum of 5 and 9 7

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 18

Amounts of Water Yuna Drinks

(5 + 9) ÷ 7

Explain: They both add 5 and 9, and finding _1 of a number is equal to dividing by 7, so the values 7 are equal.

1 8

1 4

× × ×

× ×

3 8

1 2

5 8

3 4

7 8

Amount of Water (gallons)

9. _3 × 6 × _4 4

(

a. Write an expression that includes multiplication to represent the total amount of water Yuna drinks in one week. _3 + 3 × _1 + _5 + 2 × _3 2) 8 ( 4) 8 (

_3 of the product of 6 and _4 7

Explain: They both multiply 6 and _4 . The value of the first one is greater because _3 of a number

is greater than _3 of a number.

b. Evaluate your expression to find the total amount of water, in gallons, that Yuna drinks in one week.

Yuna drinks 4 gallons of water in one week.

10. Subtract 4 from _1 of 11 3

( 3 × 14) − 4

Explain: They both subtract 4. I know _1 of 14 is greater than _1 of 11, so the value of the second 3 3 one is greater.

PROBLEM SET

167

168

PROBLEM SET

375

LESSON 19

Create and solve one-step word problems involving fractions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Name

Date

2 × 20. Then solve the word problem. Write a word problem that can be solved by evaluating _ 5

Sample:

2 of the money he earns for walking dogs. If Scott earns $20 for walking dogs, how Scott spends _ 5

much money does he spend?

2 × 20 2_ × 20 = _____ 5 5

__ = 40 5

Lesson at a Glance Students analyze tape diagrams and describe the information the diagram shows. With a partner, students use what is known and unknown to construct a word problem that matches the tape diagram. They continue to construct word problems when they are given only equations. The experience of writing word problems from tape diagrams and equations is intended to support students when they solve word problems in general.

Key Question

=8 Scott spends $8.

• Does creating word problems of your own help you solve word problems? How?

Achievement Descriptors 5.Mod3.AD11 Model and evaluate division of unit fractions

by nonzero whole numbers. (5.NF.B.7.a) 5.Mod3.AD12 Model and evaluate division of whole numbers by unit

fractions. (5.NF.B.7.b) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 5 min

• None

Review the Math Past resource to support delivery of Learn.

Learn 35 min

Students

• Generate Contexts to Match a Tape Diagram

• None

• Generate Contexts to Match an Equation • Problem Set

Land 10 min

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Fluency

Counting the Math Way by Hundredths Students construct a number line with their fingers while counting aloud and model a composition and a decomposition to prepare for extending place value understanding to the thousandths place beginning in module 4. Let’s count the math way. Each finger represents 1 hundredth. Face the students and instruct them to mirror you. Show a fist with your right hand, palm facing out. Show me your left hand. Make a fist like me. That’s 0 hundredths, or 0. Now, raise your right pinkie. Show me your left pinkie. That’s 1 hundredth.

Student View of Your Hand

Student View of Student’s Hand

0.01

0.02

0.03

0.04

0.05

Let’s put up the very next finger. Raise your right ring finger. Students raise their left ring finger. That’s 2 hundredths. Put up the next finger. 3 hundredths.

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5 ▸ M3 ▸ TD ▸ Lesson 19

Now that students understand the routine, switch to having them say the count as they show fingers. Guide students to continue counting the math way by hundredths to 10 hundredths, then back down to 0. Student View of Your Hands

0.6

0.7

0.8

0.9

1.0

Student View of Student’s Hands

Let’s count the math way again by hundredths from 0 hundredths to 10 hundredths. Have students count the math way by hundredths from 0 to 0.1. What larger unit can we make with 10 hundredths?

1 tenth We can bundle 10 hundredths to make 1 tenth. (Clasp hands together.) Ask students to model bundling 10 hundredths by clasping their hands together. Model unbundling by unclasping your hands. Ask students to model unbundling 1 tenth by unclasping their hands. Let’s count the math way by hundredths from 10 hundredths to 0 hundredths. Have students count the math way by hundredths from 0.1 to 0.

379

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5 ▸ M3 ▸ TD ▸ Lesson 19

___

134

Display   2 =

How can we represent the fraction as a division expression? Raise your hand when you know.

134 = 134 ÷ 2 = 67 2

Wait until most students raise their hands, and then signal for students to respond.

134 ÷ 2 Display the answer. Divide and express the quotient as a whole or mixed number. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the quotient. Repeat the process with the following sequence:

528 = 528 ÷ 4 = 132 4

380

241 = 241 ÷ 5 = 48 1 5 5

805 = 805 ÷ 7 = 115 7

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Choral Response: Multiply Fractions Students multiply a fraction by a fraction to prepare for solving multi-step word problems involving fractions beginning in lesson 20.

_1 _1

Display   2  ×   4  =

What is the product in fraction form? Raise your hand when you know. Wait until most students raise their hands, and then signal for students to respond.

1 1 × = 2 4

1 8

3 4 12 × = 5 6 30

8 4 32 × = 5 6 30

1 8 

Display the product. Repeat the process with the following sequence:

1 2 2 × = 2 4 8

1 1 1 × = 4 4 16

Launch

3 1 3 × = 4 4 16

3 2 6 × = 5 3 15

3 5 15 × = 5 3 15

Students analyze a tape diagram and consider what problem it might represent. Display the tape diagram and the word problem with sentences covered with scribbles. Eddie’s little sister scribbled on his math homework with a marker. Now, Eddie does not know what problem he needs to solve, but he does have a tape diagram that can give him some clues about what the problem asks him to find.

381

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5 ▸ M3 ▸ TD ▸ Lesson 19

Have students read the first sentence of the word problem and study the tape diagram.

There are 18 tables in a restaurant. 2/6 of the tables at a restaurant have benches. The rest of the tables have chairs. How many of the restaurant’s 18 tables have benches?

What information is shown in the tape diagram that might help Eddie determine what is hidden underneath the scribble? The tape diagram shows the total is 18, and it is partitioned into 6 equal parts.

2 of the parts are labeled as k. What equations can we use to find the value of k?

k = _6  × 18 2

k = (18 ÷ 6) × 2 k = 2 × __    6 18

If students do not suggest any of these equations, offer them as your own ideas. Invite students to choose one of the equations and think–pair–share about what might be the matching word problem.

There are 18 tables in a restaurant.  6  of the tables at the restaurant are round. The rest of the tables are square. How many of the tables are round?

There are 18 tables in a restaurant. There are 6 servers who each wait on the same number of tables. How many tables do 2 servers wait on? Transition to the next segment by framing the work. Today, we will create and solve word problems involving fractions.

382

EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TD ▸ Lesson 19

Generate Contexts to Match a Tape Diagram Students write word problems to match tape diagrams. Direct students to problem 1 in their books. Write a word problem to match each model. 1.

Promoting the Standards for Mathematical Practice

Saturday

? Students reason quantitatively and abstractly (MP2) as they work in pairs and use the Co-construction routine to create a word problem to match the given tape diagram.

Sunday Sample: I play video games on Saturday for 5 times as long as on Sunday. On Saturday, I play for 60 minutes. How many minutes do I spend playing video games on the weekend?

I ride my bike for  5  as long on Sunday as I do on Saturday. I ride for  2  hour on Sunday. How many hours do I ride my bike over the weekend?

Ask the following questions to promote MP2: • How does the tape diagram represent your word problem? • What real-world situations are modeled by the tape diagram?

What do you notice about this tape diagram? There are two tapes in the diagram. The tape diagram labeled Saturday is partitioned into 5 parts.

Teacher Note

The tape diagram labeled Sunday has 1 part. The unknown value is the total of both tapes in the tape diagram. This tape diagram shows a comparison about something that happens on Saturday and Sunday. How does the tape that represents Saturday compare to the tape that represents Sunday? The tape that represents Saturday is longer than the tape that represents Sunday.

The digital interactive Comparison Tape Diagram helps students describe the multiplicative relationship with a fraction. Consider demonstrating the activity for the whole class.

383

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5 ▸ M3 ▸ TD ▸ Lesson 19

Can we express how much longer the Saturday tape is mathematically? How? The Saturday tape is 5 times as long as the Sunday tape. Can we express how much shorter the Sunday tape is mathematically? How?

The Sunday tape is  5 times as long as the Saturday tape. What does the tape diagram show that we need to find?

Differentiation: Support If students need additional support to construct a word problem, provide a context they can use, such as one of the following:

The total for Saturday and Sunday

• Time spent doing chores or homework

Can you write an expression that matches the tape diagram to find the unknown value yet? Why?

• Number of places they may visit • Number of texts they send

No, because numbers are not given for Saturday nor Sunday. Pair students and use the Co-construction routine to have partners write a problem to match the tape diagram. Invite them to be creative and thoughtful about any value they assign to either Saturday or Sunday as they work out a possible scenario. As necessary, guide their thinking by asking the following questions: • Can you think of an activity you do more on Saturday than on Sunday? • What situation would match the relationship we noticed between the two tapes?

Differentiation: Challenge Consider having students who finish early solve their problem, or have pairs exchange problems with another pair, then solve the problems.

• Instead of thinking about 5 times as long, can you think of a relationship involving 5 times

as much or 5 times as many? Or try thinking about  5 times as much or  5 times as many. Give pairs 2 minutes to compare the problem they construct with other groups. Invite pairs to share problems and explain how their problem matches the tape diagram.

384

Language Support To support students in explaining whether the problem matches the model, direct students’ attention to the Share Your Thinking portion of the Talking Tool.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Direct students to problem 2.

1 4

... ? fourths

Sample:

5 pizzas are cut into fourths. How many pieces of pizza are there after all the pizzas are cut?

Riley needs strips of ribbon that are  4  foot long. If Riley has 5 feet of ribbon, how many strips can Riley make?

Now, let’s look at a new tape diagram. What do you notice? The total is 5.

The tape diagram shows a part labeled  4  .

The unknown value is the number of fourths in 5. What does the tape diagram show us that we need to find? We need to find how many fourths are in 5. Can you write an expression that matches the tape diagram to find the unknown value yet? Why? Yes, because we have enough information in the tape diagram, including the numbers. What expression can we write to match the tape diagram? 1 5 ÷ _  4

If we evaluate the expression 5 ÷     , is the quotient greater than or less than 5? How 4 do you know? It is greater than 5 because you can fit 4 fourths in 1, so there are a lot of fourths in 5.

385

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Have partners construct a problem to match the tape diagram. As necessary, guide their thinking by asking the following questions: • Have we previously solved a problem that involved dividing a whole number by a fraction? Can we use a similar situation for this problem? • Can you think of a real-world situation in which you would share 5 of an item? Could that item be partitioned into fourths? Give pairs 2 minutes to compare the problem they construct with other groups. Invite pairs to share problems and explain how their problem matches the tape diagram. Which tape diagram did you find more helpful when you wrote a word problem, the tape diagram from problem 1 or problem 2? Why? The tape diagram in problem 1 was more helpful because there were no numbers, so I did not feel as limited.

Teacher Note Students may find the ancient Chinese approach to division with fractions interesting, as recorded in the book titled Nine Chapters on the Mathematical Art. Consider creating an extension to this lesson by referring to the resource Math Past for a more in-depth discussion of what kinds of problems were solved in ancient China and how. Also included in the resource are suggestions for how to use the content of Math Past with students.

The tape diagram in problem 2 was more helpful because we had all the numbers and could write an expression that matched. Knowing all the numbers and an expression to match made it easier for us to come up with a story that matched.

Generate Contexts to Match an Equation Students write word problems to match equations. Direct students to problem 3.

UDL: Engagement

Write a word problem to match each equation. 3. _   × 30 = x 4 5

Sample:

There are 30 books on my summer reading list. I have read  5  of the books already. How many books have I read?

Miss Song has 30 students.   5  of her students chose math as their favorite subject. How many students chose math as their favorite subject?

386

The situations presented here are only samples. Inviting the class to brainstorm real-world situations that are relevant to their lives provides an opportunity to anchor instruction in contexts that are familiar and meaningful to students.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Now, we are given an equation instead of a tape diagram. Let’s think about 4 a situation to use to write a word problem that can be solved by evaluating _     × 30. 5

Invite students to think–pair–share about situations in which they might take a fraction of a whole number.

Reading   5  of the 30 books on my list

_4 _4 Finding   5  of the 30 secret treasures in a game _4 Completing     of a 30-mile bike race Saving   5  of a $30 gift I received

Differentiation: Support If students need support to construct a word problem to match an equation, suggest they first draw a tape diagram to represent the equation. Or provide the tape diagram to students.

Now that we have some ideas for a situation to use, let’s write a word problem to match the equation. Have partners construct a problem to match the equation.

Give pairs 2 minutes to compare with other groups the problem they construct. Invite pairs to share problems and explain how their problem matches the tape diagram. Direct students to problem 4. 1_ 4. 4 ÷   = w 3 Sample:

Leo has $4, which is  3  of the cost of a new game. How much does the game cost?

A restaurant has 4 pounds of pizza dough. They use  3  pound of dough to make each pizza. How many pizzas can they make? Earlier we wrote a problem from a tape diagram that could be represented by an expression like the expression on the left side of this equation. Before we write 1 a problem for 4 ÷ _    = w, let’s think about how we can interpret the divisor. In what 3 1 ways might we interpret _    in this equation? 3 1_   3  might be the size of each group, or it could represent the number of groups.

387

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Display the partitive and measurement tape diagrams.

4 1 3

... w thirds

0 3

1 3

2 3

3 3

Both tape diagrams match the equation. Which tape represents     as the size of the 3 group? How do you know?

The tape diagram on the right represents  3  as the size of the group because I can see

the partition labeled with  3  .

How does the tape diagram on the left show     as the number of groups? 3

It shows the number of groups by partitioning the tape diagram into 3 equal parts. Each _1 part is   3  of the total, and there are 3 thirds in the total. How would the different tape diagrams affect the word problem you create? The story would be different. You could not use the same story for each tape diagram, even though the expression is the same, because the meaning of the divisor needs to match the story. Choose one of these meanings to write your word problem to match the equation. Give students 2 minutes to independently construct a word problem. Pair students and use the Five Framing Questions routine as they review each other’s word problems. Introduce the following questions for students to consider as they discuss one another’s word problem. Make the questions available for students to reference as they work. • Notice: What do you notice about your partner’s word problem? • Organize: Which meaning of the divisor did your partner use? • Reveal: How do you know what the divisor means in the problem your partner constructed? 388

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

• Distill: What difference does the meaning of the divisor make in the answer of the problem? • Know: How does analyzing a word problem help you solve word problems? Have one partner discuss the other’s work for 2 minutes, then signal for partners to switch. Circulate as they work to identify at least one student who used a partitive interpretation and at least one student who used a measurement interpretation. Facilitate a whole-class discussion. Invite students to share their partner’s word problem.

Leo has $4, which is   3  of the cost of a new game. How much does the game cost?

Teacher Note

pizza. How many pizzas can they make?

The problem about the cost of the game

A restaurant has 4 pounds of pizza dough. They use  3  pound of dough to make each Then challenge students to connect their thinking back to solving word problems. Advance student thinking with questions such as the following: • Which tape diagram matches the word problem that your partner created? How do you know? • Does knowing the meaning of the divisor help you to solve the problem? How?

uses   1 to mean the number of groups 3

(partitive division). The problem about the

pizza dough uses  1 to mean the size of each 3

group (measurement division). Consider

having students identify how the divisor  1 is used in each problem.

Present the following equations to the class.

42 × _6  = r 1

42 ÷ _6  = n 1

Does the product or the quotient have a greater value? How do you know? The value of the quotient is greater than the value of the product. For the product, we need to find a fraction of a whole number, so the answer is less than 42. For the quotient, we need to find how many sixths are in 42, so the answer is greater than 42. Now, you can choose to think about a word problem that involves multiplication or division. As you think about a word problem, be mindful about the value of the answer. Think, Would it make sense that the answer is greater or less than 42?

389

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Invite students to think–pair–share about a word problem for one of the equations.

Scott runs a race in 42 minutes. It takes him  6 as long to walk home. How long does it take Scott to walk home?

Jada has 42 ounces of juice in a pitcher. She fills each glass with  6 of the juice. How many glasses does Jada fill? Do you think creating word problems of your own can help you solve word problems? How? It can help because it gives me practice thinking about what questions match unknowns in tape diagrams or equations. I think it can help because creating a story that makes sense can help me make sense of answers I find when I solve other word problems. Encourage students to continue to think about whether creating word problems of their own helps them solve word problems as they complete the Problem Set.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Create and solve one-step word problems involving fractions. Gather the class with their Problem Sets. Facilitate a class discussion about creating and solving one-step word problems involving fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

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Are there any problems in the Problem Set that are like those we created earlier? Which problems? We wrote a problem that compared Saturday to Sunday, and that was like problem 3 because it compares the weights of a dog and a cat. We wrote a problem involving a whole number divided by a fraction. Problem 5 is similar

because to find the number of pages in Adesh’s book, we had to find 28 ÷   8 . Did creating word problems of your own help you solve word problems in the Problem Set? How? It was helpful because some of the problems I solved in the Problem Set were a lot like the problems I wrote, so I could figure out what expression to use or how the tape diagram would look. It was helpful because it gave me experience with more types of situations that might need a comparison tape diagram or might need multiplication or division. It was helpful because it was simpler than trying to write my own word problems. The story is given to us, and we can focus on drawing and making sense of it instead of trying to write something that matches and makes sense.

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.

391

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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 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

Name

Date

3 3. A dog and a cat go to the vet. The dog weighs 48 pounds. The vet says the cat weighs __ as much

as the dog. How much does the cat weigh? 16

9 2 of the vinegar spills. How many liters of vinegar spill? liters of vinegar. _ 1. A container has __

2_ __ × 9 = 3_ 3 10 5

3 48 × __ =9

Use the Read–Draw–Write process to solve each problem. 10

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

The cat weighs 9 pounds.

_3 liters of vinegar spills. 5

1 of their workday teaching piano lessons. If they teach 6 lessons that 4. A music teacher spends _ 2

1 -foot boards from a piece of wood that is 8 feet long. How many boards 2. Lisa wants to cut _

can she cut?

are each the same length, what fraction of their workday do they spend teaching one lesson?

1_ 1 ÷ 6 = __ 2 12

1 of their workday teaching one lesson. They spend __

8 ÷ 1_ = 24 3

Lisa can cut 24 boards.

392

173

174

PROBLEM SET

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

1 of a race. She has run 500 meters so far. How long 7. A long distance runner has completed __

1 of his book each night. If he reads 28 pages each night, how many pages are 5. Adesh reads _ 8

in the book?

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

is the race?

28 ÷ 1_ = 224

1 500 ÷ __ = 5,000

There are 224 pages in the book.

The race is 5,000 meters long.

7 6. Ryan completes a race in 17 minutes. Jada completes the same race in _ as much time as Ryan.

How many minutes does Jada take to complete the race?

1 as much in one week. 8. A bakery sells 6,000 pies in one week in November. They usually sell __

How many pies does the bakery usually sell in one week?

17 × 7_ = 14 7_

1 6,000 × __ = 600

7 It takes Jada 14 _ minutes to complete the race.

The bakery usually sells 600 pies in one week.

PROBLEM SET

175

176

PROBLEM SET

393

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 19

9. The line plot shows the distances, in miles, that a student runs. Distances a Student Runs

× × ×

× × × ×

× ×

1 4

1 2

3 4

× 1

Distance (miles)

a. Write an expression that includes multiplication to represent the total distance the student runs. 1_

(3 × 4 ) + (4 × 2 ) + (2 × 4 ) + 1 b. Evaluate your expression to find the total distance the student runs. 1 The student runs 5 _ miles. 4

394

PROBLEM SET

177

LESSON 20

Solve multi-step word problems involving fractions and write equations with parentheses.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Name

Date

Use the Read–Draw–Write process to solve the problem.

Mr. Sharma buys a bag of 24 strawberries. He eats 4 of them. He then freezes 3 of the

remaining strawberries. He keeps the rest of them in the fridge. How many strawberries does Mr. Sharma freeze?

24 eats

puts in fridge freezes

Lesson at a Glance Students explore multiplicative comparison relationships with a hands-on activity. Then they use the Read–Draw–Write process to solve a multiplicative comparison word problem involving fractions. Students continue to solve other word problems involving fractions and realize when one or more than one tape is necessary, depending on context. They articulate how a tape diagram simplifies solving word problems involving fractions, noting when a tape diagram helps them solve a problem in fewer steps.

Key Questions

_1 × 24 = 12

• Does a tape diagram help you solve a problem in fewer steps?

Mr. Sharma freezes 12 strawberries.

• How does a tape diagram help you write an equation with parentheses that matches a word problem?

Achievement Descriptors 5.Mod3.AD5 Solve multi-step problems, including word problems,

involving addition, subtraction, and multiplication of fractions, division of whole numbers with fractional quotients, and division with unit fractions and whole numbers. (5.NF) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c) © Great Minds PBC

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

Launch 10 min

• Envelopes (12)

• Prepare envelopes with 3 blue construction paper strips and 3 red construction paper strips inside. Prepare 1 envelope per student pair.

Learn 30 min • Solve a Comparison Problem • Solve Word Problems • Problem Set

Land 10 min

_1 _1 • Red construction paper, 4    ″ × 2″ (36)

• Blue construction paper, 4  2 ″ × 2″ (36) 2

• Comparison Tape Diagrams Solutions (1 per student pair)

Students • Comparison Tape Diagrams (1 per student pair, in the student book)

• Print or copy Comparison Tape Diagrams Solutions. Prepare 1 copy per student pair. • Consider whether to remove Comparison Tape Diagrams from the student books in advance or have students remove them during the lesson.

• Envelope of construction paper strips (1 per student pair)

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Fluency

Whiteboard Exchange: Convert Customary Length Units Students convert feet to yards or inches to feet to build fluency with converting customary measurements in a smaller unit in terms of a larger unit from topic A. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 3 ft =

yd.

3 feet is equal to how many yards? Tell your partner. Provide time for students to think and share with their partners.

1 yard Display the answer, and then display 1 ft =

3 ft =

1 ft =

1 3

5 ft = 1 2 3

yd.

Write and complete the equation. Display the answer, and then display 5 ft =

yd.

Write and complete the equation.

Teacher Note Encourage students to write their answers by using the largest possible unit and mixed numbers when possible. For example,

Display the answer. Repeat the process with the following sequence:

__5

if a student writes 5 ft =     yd, ask them to

__5

rename     as a mixed number. If a student 3

398

12 in =

1 in =

1 12

9 in =

3 4

__9

writes 9 in =     ft, ask them to rename the 12

fraction by using the largest possible

__3

unit (i.e.,    ). 4

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Whiteboard Exchange: Subtract a Fraction from a Whole Number Students determine the difference to prepare for solving multi-step word problems involving fractions. Display 2 − _     = 1 3

Write and complete the equation. Show your method.

2 − 1 = 12 3 3

3 3

Display the difference and sample student work. Repeat the process with the following sequence:

2− 2 = 5

3− 3 = 21 4

3− 4 = 22 6

4− 2 = 37 9

6− 5 = 5 7 12

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5 ▸ M3 ▸ TD ▸ Lesson 20

Launch

Materials—T: Comparison Tape Diagrams Solutions; S: Comparison Tape Diagrams, envelope of paper strips

Students represent multiplicative comparison relationships concretely. Pair students and direct them to remove one Comparisons Tape Diagrams from their books for each student pair. Distribute one envelope of prepared paper strips to each student pair. Each paper strip represents 1 unit. We will represent different relationships by using these strips. Do not cut or overlap the strips.

UDL: Representation The digital interactive Comparison Relationships helps students see the multiplicative relationships being described. Consider allowing students to experience the activity digitally, or use it to demonstrate the comparison relationships for the whole class.

Direct students to problem 1: The red strip is twice as long as the blue strip. Invite students to turn and talk about how they might show that relationship by using the paper strips. Display the Comparison Relationships digital interactive.

Differentiation: Challenge

Which strip should be longer: red or blue? Why? Red should be longer because we know the red strip is twice as long. How could I show that the red strip is twice as long as the blue strip? Why? Show one blue strip and two red strips. Red is twice as long, which means red needs to have 2 times as many units as blue. Adjust the blue and red digital paper strips to show the comparison relationship. Align the strips vertically, as students do when they draw comparison tape diagrams. Invite students to use their paper strips to show the remaining relationships described on the page. As students are ready, give them the solutions page to compare their thinking.

400

Red

Encourage students to show that the red strip

__1

is 1     times as long as the blue strip. 2

Red Blue

Blue

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Direct students to problem 4. Which tape is longer? Why?

The red strip is longer because we know the blue strip is _    times as long, so the blue strip 3 is not as long as the red strip. 1

Compare the tape diagrams in problem 4 to those in problem 3. What is the same? What is different? They both show one color with 3 units and the other color with 1 unit.

Teacher Note

In problem 3, blue is longer, and in problem 4, red is longer. In comparison situations, the relationship may be described by using a whole number—like in problem 3—and sometimes the relationship is described by using a fraction—like in problem 4. Direct students to problem 5. What did you do to show this relationship? Why?

In module 5, students expand their understanding of multiplicative relationships to include mixed numbers. For example, the

__1

length of a rectangle is 4   times as long as 2 its width.

I showed 2 blue units and 3 red units. We know blue is _    as long as red, so because the 2 3

blue strip is described in thirds, we can make the red strip have 3 equal units and the blue strip have 2 equal units. Why do you think we call these examples comparison relationships? We call them that because we know how to draw or show the units based on the relationship between the values. We know whether one is longer than the other and how much longer. When we show a comparison relationship as we did with each of these examples, why do you think there is more than one tape? Because there are two different values to compare. Invite students to turn and talk about how they might know whether a word problem can be represented by using a comparison tape diagram. Transition to the next segment by framing the work. Today, we will solve word problems involving fractions.

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Learn

Solve a Comparison Problem Students use the Read–Draw–Write process to solve a comparison problem involving fractions and then write a matching equation. Direct students to problem 1 in their books and have them read the problem independently. Use the Read-Draw-Write process to solve each problem.

1. In the fifth grade, there are    as many students who do not wear glasses as those who 5

do wear glasses. There are 60 students who wear glasses. How many students are in the fifth grade?

60 glasses

? no glasses

60 + ( _ 5 × 60) = 108 4

There are 108 students in the fifth grade. Do you think we need a comparison tape diagram to model this story? Why? I think we need a comparison tape diagram because we have two different groups of students: students who wear glasses and students who do not. I think we need a comparison tape diagram because we know there are _    as many 4 5

students who do not wear glasses, so _    helps us compare the two groups of students. 4 5

I do not think the story needs to be modeled with a comparison tape diagram because we are considering all the students in fifth grade. Let’s go back through the story, one piece at a time. 402

EUREKA MATH2 California Edition

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Reread the first sentence. What do we know? There are some students in fifth grade who wear glasses and some students who do not wear glasses. Are there more students who wear glasses or more students who do not wear glasses? How do you know? There are more students who wear glasses because we know there are _    as many 4 5

students who do not wear glasses, and _    < 1. 4 5

Can we draw something? What can we draw? We can draw two tapes: one for the students who wear glasses and one for students who do not wear glasses. Affirm that this is a comparison problem that needs to be modeled by using two tapes: one to represent students who wear glasses and one to represent students who do not wear 4 glasses. _    describes the relationship between the two tapes. 5

Which tape should be longer? Why? The tape for students who wear glasses should be longer because there are more students who wear glasses than those who do not. How many parts should be in each tape? Why? The tape that shows students who wear glasses should have 5 parts, and the tape

that shows students who do not wear glasses should have 4 parts. There are _    as many 4 5

students who do not wear glasses, and because that amount is described in fifths, we can make the tape for glasses show 5 parts and the tape for no glasses show 4 parts. I know that because earlier when we showed _    as much, one tape had 3 parts and one tape had 2 parts.

2 3

Differentiation: Support If students need support with this multiplicative comparison relationship, allow them to use cubes to help model the relationship. Have them start with a cube to represent the group of students who do wear glasses and a cube to represent the group who do not wear glasses. Ask students whether the model represents what the story says. Continue to add cubes until the group of students who wear glasses is represented by 5 cubes. Then ask, Because the group of

__4

students who do not wear glasses is    as long, 5

how many cubes do we need to show that?

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Draw two tapes: one glasses partitioned into 5 equal parts to represent students who wear no glasses glasses and one partitioned into 4 equal parts to represent students who do not wear glasses. Instruct students to do the same. Read the next piece of information: There are 60 students who wear glasses. Can we draw something? What can we draw? I would not draw anything, but I would label the tape that shows students who wear glasses with 60. Label the glasses tape 60. Read the question: How many students are in the fifth grade? Where should we put the question mark in the model?

60 glasses

We can put the no glasses question mark on the side so it shows the total of both tapes.

Label the total of the tapes with a question mark. Look at our model. What conclusions can you make? There are 60 students who wear glasses. The number of students who do not wear glasses is less than 60, but it is not much less than 60 because that tape has only 1 less part. Each part is the same size. Direct students to work with a partner to find the number of students in the fifth grade. How many students are in the fifth grade?

108 404

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5 ▸ M3 ▸ TD ▸ Lesson 20

How did you find the total number of students in the fifth grade? We saw in the first tape that 5 parts represent 60 students. So, we divided 60 by 5 to find that each part is 12. The tape that shows students who do not wear glasses has 1 less part, so I subtracted 12 from 60. There are 48 students who do not wear glasses. Then I added 60 and 48 and got 108. We saw in the first tape that 5 parts represent 60 students. So, we divided 60 by 5 to find that each part is 12. There are 9 equal-size parts in the unknown, so 9 × 12 = 108. We saw that the second tape is _    as much as the first tape, so we found _     of 60, 4 5

which is 48. Then we added 60 and 48 to get 108.

4 5

Let’s write one equation that shows the thinking you did to solve the problem. Write the equation 60 + 48 = 108. What does 60 represent? The 60 students who wear glasses What does 48 represent? The 48 students who do not wear glasses In the story, the value 60 is given to us, but we did some work to find 48. What is an expression you may have used to find 48?

60 − 12 4 × 12

Teacher Note

4 × 60 5

Affirm all accurate and relevant expressions. Have students write an equation that replaces 48 with one of the expressions. Circulate to find an example of a student who included parentheses around the expression equal to 48. Write 60 + ( _   × 60) = 108. 4 5

Why are there parentheses around   × 60? 5

We want to add 60 and 48, and _    × 60is an expression that represents 48, so we have 4 5

to show it as a group by using parentheses.

Students may use a different expression to represent 48. Ensure that they placed parentheses around the expression they chose to represent 48. Encourage students who did not use parentheses to place them now. The following are other acceptable equations:

60 + (60 − 12) = 108 60 + (4 × 12) = 108

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We had to know what to add to 60 to find the total number of students, so we needed to find _    × 60separately. We can use parentheses to show we did that before we 4 5

added to 60. Direct students to write a solution statement. Then invite students to turn and talk about whether the tape diagram made this a simpler problem to solve and how.

Solve Word Problems Students use the Read–Draw–Write process to solve a word problem involving fractions and then write a matching equation. Direct students to problem 2 and have them read the problem independently. Do you think we need a comparison tape diagram to model this story? Why? I do not think we need a comparison tape diagram for this story because we have fractions that are part of the total, and none of the fractions compare anything. No, the problem does not use comparison language like times as much or as long. Direct students to use the Read–Draw–Write process to construct a tape diagram to match the story. Circulate to find an accurate tape diagram to share. 2. Toby spends _    of his money on movie tickets. He spends _   of the remaining money on 2 5

1 3

popcorn. He has $10 left. How much money did Toby have to begin with?

tickets

popcorn

Differentiation: Challenge Some students may be ready to model and solve the problem without additional guidance. Consider allowing those students to solve the problem independently and then have them lead the class through making sense of their tape diagram and solution.

$10

(_    × 10)× 5 = 25 1 2

Toby had $25 to begin with.

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Display a student’s tape diagram. Then ask the following questions. How did you know to label 1 part as popcorn?

The story says _    of his remaining money, and I noticed there were 3 equal-size 1 3

parts left. _     of 3 is 1. 1 3

Look at the tape diagram. What conclusions can we make? Toby had more than $10 to begin with. The number of parts that represent tickets is equal to the number of parts labeled $10. That means Toby spent $10 on the tickets. Direct students to work with a partner to solve the problem and to write an equation that matches their work. How much money did Toby have at first?

$25

5 ▸ M3 ▸ TD ▸ Lesson 20

Language Support Encourage students to listen carefully to their classmates’ explanations to identify what the tape diagram helps them see and how it helps them decide what to do next. Consider providing sentence frames to support students throughout the lesson. • I can see • So, I

in the tape diagram. and

to find • Then I saw that

. , so I

How did you solve the problem? I saw that 2 parts is $10, so I found 10 ÷ 2 = 5. There are 5 equal-size parts, and 5 × 5 = 25. I saw that 2 parts is $10, and I know half of 10 is 5. There are 5 equal-size parts, and 5 × 5 = 25. Write 5 × 5 = 25. What does the first factor 5 represent?

5 equal-size parts, or 5 groups What does the second factor 5 represent? It represents the value of each part, or the size of each group. What expression can we use to represent how we determined that the size of each part is 5?

10 ÷ 2

1 × 10 2

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Direct students to write an equation that matches the story by using parentheses. Accept any accurate equation. Write 5 × ( _   × 10) = 25. 1 2

How does this equation match the tape diagram? The tape diagram shows that 5 groups of 5 is 25. We determined the size of the group

by finding _     of 10, so we put parentheses around _    × 10because that makes it clear it is 1 2

the value of each part.

1 2

Our tape diagram shows that the product of 5 and _     of 10 is 25. 1 2

When you first read this problem, did it seem challenging? Did your thoughts about that change as we solved the problem? Why? I thought it seemed challenging because there was a lot of information. There are two fractions, both with different denominators, and I did not know what I needed to do with the fractions and the $10. But once we drew the tape diagram, we only needed to find half of 10 and then multiply that by 5.

Promoting the Standards for Mathematical Practice When students use the Read–Draw–Write process and write an equation that uses parentheses to solve a problem, they are modeling with mathematics (MP4). Ask the following questions to promote MP4: • What key ideas in this problem do you need to make sure to include in your model? • How can you write this context mathematically?

When we draw a tape diagram, it sometimes takes a challenging problem and shows us a simpler pathway to a solution, whether it is a comparison problem like our first example or a part–whole problem like our second example. Direct students to problem 3. Have them work with a partner to use the Read–Draw–Write process to solve the problem and to write an equation that matches by using parentheses. Circulate as students work and ask targeted questions as needed. • Do you think this story can be modeled by using a comparison tape diagram? Why? • Can you draw something? What can you draw? • Can you find the answer in your first tape diagram? Why? What can you do? • How does your equation match?

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3. 3 gems is _   of the total gems in a video game level. The player found _   of the gems in 1 8

the level. How many gems did the player find in the level?

5 6

gems in the level

3 gems in the level

_ 5  × 3 ÷ _1 = 20 6

(

The player found 20 gems in the level. Gather students to discuss. Invite students to turn and talk with another pair of students to share their equations. Circulate to find a pair who wrote _     × (3 ÷ _    ) = 20. Write _     × (3 ÷ _    ) = 20. 5 6

1 8

5 6

1 8

How does this equation match the story?

First we determined there were 24 gems by finding 3 ÷_     . Then we found _    of 24, so we 1 8

5 6

were finding _    of 3 ÷_     . We know _    of 24 is 20, so the player found 20 gems. 5 6

1 8

5 6

I notice you drew two tape diagrams to solve this problem. Is this a comparison problem? Why did you draw two tapes? This is not a comparison problem because it does not use comparison language. There are two tapes because there were two different steps. First we had to find how many total gems in the level, but then we needed to draw a different tape diagram to show how many gems the player found.

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Did the tape diagram help you make a simpler problem? How? It did help me make a simpler problem because after I drew the first tape, I realized I could not find the answer because that tape showed how many gems were in a level, not how many gems the player found. So, I knew to draw a second tape. In my second tape, I saw that 6 parts represented 24 gems, so 1 part is 4 gems, and 4 × 5 = 20. Sometimes, our tape diagram can help us solve a problem in fewer steps, and sometimes it cannot, but the tape diagram can still help us simplify a problem involving fractions. Invite students to turn and talk about how this problem is different from the other problems and whether they found the tape diagram helpful.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Teacher Note The work shown in the Problem Set represents the fewest calculations needed after modeling the problem with a tape diagram. For example, here is the work required to solve problem 1 without using a tape diagram.

  1 × 360 = 72 5

360 − 72 = 288

Land

  1 × 288 = 72 10

__1

With a tape diagram, only   × 360 = 72 is 5

needed to solve the problem.

Debrief 5 min

360 pizzas

Objective: Solve multi-step word problems involving fractions and write equations with parentheses. Gather the class with their Problem Sets. Facilitate a class discussion about how using the Read–Draw–Write process helps solve word problems involving the multiplication or division of fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

410

mushrooms

green peppers

pineapple

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Take a moment to review your work in the Problem Set. For which problems do you think the tape diagram helped you solve the problem with fewer steps? Why? In problem 1, I realized it was 1 out of 5 that represented the number of pizzas with pineapple. So, I only had to find 360 ÷ 5. In problem 5, I realized that half of the parts represented how many pounds of turkey was used to make sandwiches. Half of 24 is 12. How does a tape diagram help you write an equation with parentheses that matches a word problem? Give an example from the Problem Set. The tape diagram helps me make sense of the math I do to find each piece of information. Once I know the steps I used to solve the problem, I can think about how that is written as an equation. When there is a part of the equation that I evaluated first, I put that in parentheses. Or if there is a part of the equation that represents a specific value from the tape diagram, I know to put that in parentheses.

UDL: Action & Expression Consider reserving time for students to self-reflect on their experience solving multiplication and division word problems involving fractions. Students may benefit from a think-aloud to model this type of reflection. • How did your drawing help you better understand the problem? • What did you feel successful with today? • Do you need more support? Explain.

In problem 3, after I drew the tape diagram, I realized that 4 parts represented $80. So, to find the value of 1 part, I found _    × 80. That gave me the value of each part. 1 4

I noticed 9 equal parts in the unknown. I put parentheses around _    × 80because that was the value of 1 part, and then I multiplied by 9.

1 4

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

This page may be reproduced for classroom use only.

R B

1 times as long as the blue strip. 2. The red strip is _ 2

R B

5 ▸ M3 ▸ TD ▸ Lesson 20 ▸ Comparison Tape Diagrams Solutions

400

1. The red strip is twice as long as the blue strip.

3. The blue strip is 3 times as long as the red strip. R B

1 times as long as the red strip. 4. The blue strip is _ 3

R B

412

R B

EUREKA MATH2 California Edition

2 times as long as the red strip. 5. The blue strip is _

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

Name

Date

Use the Read–Draw–Write process to solve each problem.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

3. Jada spent 9 of her money on a toy. She gave 5 of what was left to a charity. She has $80 left. How much money did Jada have to start? 1_ (4 × 80) × 9 = 180

Jada had $180 to start.

1. A pizza shop sells 360 pizzas. 5 of the pizzas have mushrooms. 4 of the remaining pizzas have green peppers. The rest have pineapple. How many pizzas have pineapple?

_1 × 360 = 72 5

72 pizzas have pineapple.

2. Someone has 160 beads. They give away 5 of the beads. They use the remaining beads to make

4. Every week, Mia spends 8 hours playing the piano. Oka spends 4 as much time playing the piano as Mia. For how many more hours does Mia play the piano than Oka?

8 − (_4 × 8) = 6

6 identical necklaces. How many beads are on each necklace?

_3 (5 × 160) ÷ 6 = 16

Mia plays the piano for 6 hours more than Oka.

There are 16 beads on each necklace.

185

186

PROBLEM SET

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 20

5. A chef has 24 pounds of turkey. They use 4 of the turkey to make soup and 3 of the remaining

turkey to make sandwiches. How many pounds of turkey does the chef use to make sandwiches?

_2 × 24 = 12 4

The chef uses 12 pounds of turkey to make sandwiches.

_2 _1 5 _ cones are chocolate, and the rest are vanilla. of the customers who order a vanilla cone ask for

6. An ice cream shop sells 312 ice cream cones. 4 of the cones are strawberry. 3 of the remaining 6

sprinkles on top. How many vanilla ice cream cones with sprinkles are sold?

_5 × 1_ × 312 = 65 ) 6 (4

65 vanilla ice cream cones with sprinkles are sold.

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PROBLEM SET

187

LESSON 21

Solve multi-step word problems involving fractions.

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 21

Name

Date

Use the Read–Draw–Write process to solve the problem.

Shen bought 20 pounds of ground beef. He used __1 of the beef to make tacos. He used _2 of the 4

remaining beef to make __1 -pound burgers. How many burgers did he make?

20 pounds of beef

Students use the Read–Draw–Write process to solve multi-step word problems involving fractions. They explore how modeling with a tape diagram helps them make sense of a multi-step word problem and find a pathway toward the solution as they engage in student-driven discussions.

Key Question

beef for tacos

• Does the Read–Draw–Write process help us solve multi-step word problems? How?

beef for burgers

10 1 4

Lesson at a Glance

...

Achievement Descriptors

? burgers

5.Mod3.AD5 Solve multi-step problems, including word problems,

__1 × 20 = 10 2

involving addition, subtraction, and multiplication of fractions, division of whole numbers with fractional quotients, and division with unit fractions and whole numbers. (5.NF)

10 ÷ _1 = 40 4

Shen made 40 burgers.

5.Mod3.AD10 Solve real-world problems involving multiplication

of fractions. (5.NF.B.6) 5.Mod3.AD13 Solve word problems involving division of unit fractions

by nonzero whole numbers and division of whole numbers by unit fractions. (5.NF.B.7.c)

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Agenda

Materials

Lesson Preparation

Fluency 10 min

Teacher

None

Launch 5 min

• None

Learn 35 min

Students

• Solve a Multi-Step Word Problem

• None

• Solve a Multi-Step Comparison Word Problem • Problem Set

Land 10 min

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5 ▸ M3 ▸ TD ▸ Lesson 21

Fluency

Whiteboard Exchange: Convert Customary Weight Units Students convert ounces to pounds to build fluency with converting customary measurements in a smaller unit in terms of a larger unit from topic A. Display 16 oz =

lb.

16 ounces is equal to how many pounds? Tell your partner. Provide time for students to think and share with their partners.

16 oz =

1 oz =

1 16

1 pound Display the answer, and then display 1 oz =

lb.

8 oz =

418

1 2

10 oz =

5 8

3 lb 19 oz = 1 16

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 21

Whiteboard Exchange: Subtract a Mixed Number from a Whole Number Students determine the difference to develop fluency with solving multi-step word problems involving fractions. Display 2 − 1  _  = 1 2

Write and complete the equation. Show your method.

2 − 11 =

1 2

1 1 +2 1 2

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the difference and sample student work. Repeat the process with the following sequence:

2 − 14 = 6

2 6

3 − 1 3 = 11 4

5 −22 = 25 7

7−4 4 = 2 8 12

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Launch

Students construct a tape diagram to show how a given expression matches a real-world situation. Direct students to problem 1 in their books. Have them read the problem independently. Write  _1 × (4 − 2). 4

Construct a tape diagram to match the story. 1. Sana has 4 pounds of green beans. She gives 2 pounds of them to her friend. Sana’s

family eats  _3 of the remaining green beans. How many pounds of green beans does Sana have left?

4 pounds

given to friend

family eats

Someone wrote this expression to match the story. Work with a partner to construct a tape diagram to show that the expression matches the story. Allow students 2 minutes to work collaboratively. Circulate to observe student work. Ask targeted questions such as the following as needed. • Can you draw something? What can you draw? • What can you label? • What does this part mean? • How does this connect to the expression?

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4 pounds

After students have finished, bring the class together. Display a student’s tape diagram or use the one provided if needed. Where do you see (4 − 2) in the tape diagram?

given to friend

family eats

2 of the 4 pounds of green beans were given to a friend. We can represent that with the expression (4 − 2).

Point to the expression  _1 × (4 − 2). 4

The number _  1 is not in the story, so why does the expression show multiplying by _  1 ? 4

 _3 of the 2 pounds that remain are eaten by Sana’s family. That means Sana has 4  _1  of 2 pounds of green beans left. To find that amount, we need to multiply 2 by  _1  . 4

Remember, your tape diagram can help you see an expression you might use to solve the problem. Invite students to turn and talk about how a tape diagram helps them write expressions to find unknowns in word problems. Transition to the next segment by framing the work. Today, we will solve multi-step word problems involving fractions.

421

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Learn

Solve a Multi-Step Word Problem Students solve a multi-step word problem involving fractions.

Teacher Note

Direct students to problem 2. Read the problem chorally with the class. Use the Read–Draw–Write process to solve each problem.

1 2. Tyler has 6 cups of blueberries. He uses 2  _  cups of blueberries for pancakes. He uses  _3  of 2

the remaining blueberries for muffins. Tyler puts the rest of the blueberries into bags.

He puts  _1 cup of blueberries into each bag. How many bags of blueberries can Tyler make? 4

pancakes

muffins

This lesson is not intended to introduce new arithmetic skills; rather, it provides an opportunity to highlight the similarities and differences in how students choose to perform calculations. Consider using this lesson as a formative assessment to evaluate how students approach multi-step word problems.

bags

2 1 4

...

? fourths 6 − 2 1_= _

_ _

7 2

4 7 ×   = 2 7 2

2 ÷  1_= 8 4

Tyler can make 8 bags of blueberries. 422

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5 ▸ M3 ▸ TD ▸ Lesson 21

Read the first sentence to the class. Can we draw something? What can we draw? We can draw a tape diagram to represent the 6 cups of blueberries. Draw a tape diagram and direct students to do the same.

Continue reading the problem, stopping at the word pancakes. What can we draw now?

1 We can partition a part of the tape and label it 2  _   . 2

We can partition the tape into 6 equal parts and partition 1 of the parts into halves. 1 Then we can label 2  _  parts as pancakes. 2

Let’s partition so we can see each cup of blueberries. Then partition 1 part in half 1 so we can label 2  _ . 2

Partition and label the tape diagram. Direct students to do the same. Read the third sentence. What can we draw now? Why? We can partition each of the remaining parts into halves because that gives us 7 equal parts. Then we can label 3 of those parts as muffins.

pancakes

Partition and label the tape diagram and direct students to do the same. Do we have any other information? Can we label anything else? Yes. We know Tyler puts the rest of the blueberries into bags, so we can label the rest of the tape diagram as bags.

pancakes

muffins

bags

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Guide students to label the rest of the tape diagram. Look at our tape diagram. What conclusions can you make?

2  _1 cups of blueberries are used for pancakes. 2

1 1  _  cups of blueberries are used for muffins. 2

2 cups of blueberries are put into bags. What does the problem ask us to find?

It asks how many  _1 -cup bags of blueberries Tyler can make. 4

Does our tape diagram show that? How do you know? No, this tape diagram shows how many cups of blueberries are used for pancakes and muffins and how many cups go into bags. It does not show us how many bags Tyler can make. What should we do? We should make another tape to represent how many bags Tyler can make. Now we know Tyler has 2 cups of blueberries to put into bags. We

also know Tyler puts   cup of blueberries into each bag. Can we draw 4

something to represent that? What can we draw? We can draw another tape diagram and label it as 2. Draw and label another tape diagram and direct students to do the same.

2 1 4

...

? fourths

What can we draw to represent the amount of blueberries Tyler puts in each bag?

We can partition the tape diagram to show a part labeled  _1 , but we do not know how 4 many partitions to make.

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5 ▸ M3 ▸ TD ▸ Lesson 21

Guide students to represent the unknown in the tape. Invite students to work with a partner to solve the problem. As students work, circulate and provide support as needed. When students have finished, gather the class and discuss the solution. How many bags of blueberries can Tyler make? He can make 8 bags of blueberries. Let’s look at the tape diagram again to help us write mathematical equations and expressions to show the work we did. Display the following tape diagram.

_1 After Tyler used 2   cups of blueberries for 2

pancakes, how many half-cups remained?

7 half-cups remained.

_1 _7

pancakes

So we could write 6 − 2   =   . 2

muff ins

bags

1 Write 6 − 2  _   =  _7 and direct students to do the same. 2

Invite students to turn and talk about what each number in the equation means.

Display the following tape diagram. What expression can we write to represent the blueberries Tyler put into bags? Why? 7 We can write  _4  ×  _  . The tape diagram shows  _4  7

of  _7 . We also know Tyler used  _3 of the remaining 7

blueberries, so he has  _4 of the remaining

pancakes

muff ins

bags

7 blueberries left. To find  _4  of  _7 , we can find  _4  ×  _   . 7

_4 _7 What is   ×   ? How do you know? 7

The product is 2. I know that because 4 sevenths of 7 halves equals 4 halves, and  _4 = 2.

7 Write  _4  ×  _  = 2and direct students to do the same. 7

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Does it make sense that Tyler has 2 cups of blueberries left after he makes muffins? Why? Yes, we can see in the tape that he had 2 cups of blueberries left. Display the following tape diagram. What equation can we write to represent the number of bags Tyler can make? How do you know? 1 We can write 2 ÷  _  = 8. We know Tyler has 2 cups of blueberries left, 4

and we want to know how many fourths are in 2.

1 Write 2 ÷  _  = 8and direct students to do the same.

2 1 4

...

? fourths

Invite students to turn and talk about how they can use the Read–Draw–Write process to solve multi-step word problems involving fractions.

Differentiation: Support To assist students with the Read–Draw–Write process, consider asking the following questions: • What do you know? • Can you draw something? • What can you draw?

Solve a Multi-Step Comparison Word Problem Students solve a multi-step comparison word problem involving fractions by using self-selected methods.

• Can you label anything? • Should you revise or add to your drawing? • Do you have all the information you need to solve the problem?

Direct students to problem 3. Have them work with a partner and use the Read–Draw–Write process to solve the problem. Circulate and observe student work. As you circulate, consider using the following prompts:

Teacher Note

• Tell me about your plan to solve the problem. • Tell me about how your drawing connects to the story. • What does this number represent? • Have you revised your tape diagram? Why? • Why did you use that operation? • Why did you draw two tape diagrams? • Does your answer seem reasonable? Why?

A context video for this word problem is available. It may be used to remove language or cultural barriers and provide student engagement. Before providing the problem to students, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before being asked to interpret it mathematically.

• How can you check your answer?

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Try to find student work that demonstrates a tape diagram model that fully represents the story and each of its steps. If your students do not produce similar work, choose one or two pieces of their work to share and highlight how it shows movement toward a tape diagram model that represents a multi-step problem. 3. Lacy’s cat weighs 8 pounds. Her cat weighs  _1 as much as her dog. Noah’s dog weighs 3

2  _2 pounds less than Lacy’s dog. How much does Noah’s dog weigh? 1

Lacy’s cat

Lacy’s dog

When students use a self-selected method to solve a comparison word problem involving fractions, they are modeling with mathematics (MP4). Ask the following questions to promote MP4: • What can you draw to help you understand this problem? • What key ideas in this problem do you need to include in your model?

3 × 8 = 24

• What math can you write to represent this problem?

Lacy’s dog weighs 24 pounds. Lacy’s dog

Promoting the Standards for Mathematical Practice

Noah’s dog

_1 _1 24 − 2 = 21 2 2

1 Noah’s dog weighs 2 1  _   pounds.

Language Support

When you read the word problem, where did you first stop to draw? What did you draw? Why? I stopped after the first sentence. I know Lacy’s cat weighs 8 pounds, so I drew a tape and labeled it Lacy’s cat, and then wrote 8 inside it.

Consider directing partners to the Agree or Disagree section of the Talking Tool to support them in discussing the similarities and differences in their work and the work of their classmates throughout this segment.

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Where did you stop next? Why? I stopped after the second sentence because I had information to help me draw something to show Lacy’s dog’s weight. Did you add on to your first tape or draw a second tape? Why?

I drew a second tape because the story says the cat weighs  _1 as much as the dog, and 3

that is comparison language. So I knew the tape diagram that represents the dog’s weight should be 3 times as long as the tape that represents the cat’s weight. Display a student’s tape diagram that shows the relationship between the cat’s weight and the dog’s weight or use the one provided if needed.

Lacy’s cat Lacy’s dog

Look at this tape diagram. What conclusions can you make?

Differentiation: Challenge Challenge students by asking them to find the weight of Noah’s pet chicken. Noah’s 1 chicken weighs   the weight of his dog.

__ 9

Finding the weight of the chicken requires 43 1 students to rename 2 1   as   prior to 2 2 1 multiplying by   .

__ __

__ 9

Once they multiply, students find Noah’s chicken weighs 2

__7 pounds. 18

The cat weighs 8 pounds, and that is  _1 as heavy as the dog. 3

The cat weighs 8 pounds, and the dog weighs 3 times as much. Is there anything else we can label? How do you know? We can label the tape diagram that represents the dog’s weight with 24 because 3 × 8 = 24. Where in the word problem did you stop next? Why? I stopped after the third sentence because I had information about Noah’s dog that I could draw. Did you add on to your tape diagrams or did you draw another tape? Why? I drew another tape diagram because I wanted to show a tape to represent Noah’s dog’s weight.

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EUREKA MATH2 California Edition

Display a student’s tape diagram that shows the relationship between Lacy’s dog’s weight and Noah’s dog’s weight or use the one provided if needed.

5 ▸ M3 ▸ TD ▸ Lesson 21

Lacy’s dog

Noah’s dog

Invite students to think–pair–share about what they notice about this student’s tape diagram.

They drew another tape to represent Lacy’s dog to show that her dog weighs 24 pounds.

They drew a tape to represent Noah’s dog’s weight but made it shorter than the tape that represents Lacy’s dog and labeled how much shorter. They put a question mark on the tape that represents Noah’s dog’s weight because we want to find out how much his dog weighs. Look at this tape diagram. What conclusions can you make? Lacy’s dog weighs more than Noah’s dog, but just a little more. What expression can you use to determine how much Noah’s dog weighs? 1 24 − 2 _ 2

1 Invite students to turn and talk about how they subtracted 2  _   from 24. 2

How much does Noah’s dog weigh? 1 21  _   pounds 2

Is that reasonable? How do you know?

1 It is reasonable because Noah’s dog weighs a little less than Lacy’s dog, and 2 1  _  is not 2 too much less than 24.

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Revisit a student’s complete tape diagram model that represents the entire story.

Lacy’s cat

Why did we continue to draw after we had the first two tapes?

Lacy’s dog

The first two tapes only show how much Lacy’s cat and dog weigh. The question asks how much Noah’s dog weighs, so we needed to keep drawing. How do you know when you need another tape diagram? When the first tape diagram does not show all the information, I need to make another tape.

Lacy’s dog

Noah’s dog ?

When the first tape diagram does not show the unknown of the question, I need to make another tape. Affirm that for multi-step problems, students may need to draw a second or a third tape diagram. Once the tape diagram shows how to find the unknown of the question, then they can stop drawing.

UDL: Action & Expression Consider providing time for students to reflect prior to beginning the Problem Set. Direct them to look at their work once more and then again at their classmates’ work. Ask questions such as the following: • What worked well for you? • What might you do differently next time?

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Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min Objective: Solve multi-step word problems involving fractions. Gather the class with their Problem Sets. Facilitate a class discussion about solving multi-step word problems involving fractions by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Invite students to turn and talk about one of the problems in the Problem Set. Have them discuss why they drew the tape diagrams the way they did and how they decided which operation to use to solve the problem. How did drawing a tape diagram help you make sense of one of the problems in the Problem Set? In problem 1, I drew tape diagrams for the state quarters and for the silver dollars to help me keep track of how many of each Tara had. In problem 2, I drew a tape to represent the total number of muffins and could partition it based on the information given in the problem. I used my tape to figure out how many muffins were left and drew another tape to help me find how many muffins each of the 8 students get. How did your tape diagram help you see that this problem has multiple steps? I drew a tape diagram, and after I filled it in with the information from the problem, I realized I did not see what I needed to answer the question. So I made another tape diagram with the information I knew and the information I needed to find out next.

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5 ▸ M3 ▸ TD ▸ Lesson 21

EUREKA MATH2 California Edition

Have students think–pair–share about the following question. Does the Read–Draw–Write process help us solve multi-step word problems involving fractions? How? Yes. We read, draw, and write in chunks. When we learn new information, we pause to draw and then go back to reading and draw when we learn something new. We can write expressions or equations as we realize which operations we can use to find unknown information. Yes. The model I draw helps me decide which operation to use to find an unknown value. Each time we find new information by evaluating an expression, we can compare it to our model and ask, Does that make sense based on what I see in the model?

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

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 21

Name

Date

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 21

3. The farmer cooks 6 carrots. The carrots that the farmer cooks is 7 of his total carrots.

The farmer gives 2 of the remaining carrots to a neighbor. The rest of the carrots are divided equally among his 8 pet rabbits. How many carrots does each rabbit get?

6 ÷ _71 = 42

Use the Read–Draw–Write process to solve each problem. 1. Tara has 15 state quarters and 10 silver dollars in her coin collection. After Tara’s dad gives her

_3 × 42 = 18

more silver dollars, she finds that 5 of her collection is state quarters. How many silver dollars

18 ÷ 8 = 2 _82

did Tara’s dad give her? 1 15 ÷ _5 = 75

Each rabbit gets 2 8 carrots.

15 + 10 = 25 75 − 25 = 50 Tara’s dad gave her 50 silver dollars.

4. The measure of ∠ABC is 147°, and the angle is decomposed into smaller angles as shown. The measure of ∠DBC is 7 the measure of ∠ABC. The measure of ∠EBD is 3 the measure

remaining muffins were given away. The rest of the muffins are shared equally between the 8 students working the bake sale. How many muffins does each student get? _1 × 270 = 54 5

_2 × 54 = 36

36 ÷ 8 = 4 _21

of ∠ABD. What is the measure of ∠ABE?

2. 270 muffins were made for a school bake sale. After 5 of the muffins were sold, 3 of the

Each student gets 4 2 muffins.

_6 × 147 = 126 7

_2 × 126 = 84

The measure of ∠ABE is 84°.

193

194

PROBLEM SET

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LESSON 22

Evaluate expressions involving nested grouping symbols. (Optional)

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22

Name

Date

Evaluate each expression. 1. 15 + [3 × (1 ÷ 4)]

15 + [3 × 1_] = 15 + 3_ = 15 3_ 4

Lesson at a Glance Students reason about how to interpret expressions that include nested grouping symbols. They then use that understanding to evaluate the expressions. Students start with expressions involving whole numbers and then progress to expressions involving fractions. Students conclude that the placement of parentheses and brackets determines the result of an expression.

Key Questions • Does reading an expression help us evaluate it? How? • Why is it important to pay attention to parentheses, brackets, and braces when we are evaluating expressions with nested grouping symbols? 2. [(1_ ÷ 4) × (12 3_ − 3_)] + 3_ 3 5 5 4

Achievement Descriptors

3_ 3_ 3_ 1 __ [12 × 12] + 4 = 1 + 4 = 1 4

5.Mod3.AD1 Write numerical expressions that include fractions and

parentheses. (5.OA.A.1) 5.Mod3.AD2 Evaluate numerical expressions that include fractions

and parentheses. (5.OA.A.1)

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Agenda

Materials

Lesson Preparation

Fluency 15 min

Teacher

Launch 5 min

• None

Consider tearing out the Fluency Sheets in advance of the lesson.

Learn 30 min

Students

• Interpret and Evaluate Expressions Involving Whole Numbers

• Multiply Fractions Fluency Sheets (in the student book)

• Interpret and Evaluate Expressions Involving Fractions • Use Parentheses and Brackets to Make an Equation True • Problem Set

Land 10 min

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5 ▸ M3 ▸ TD ▸ Lesson 22

Fluency

Whiteboard Exchange: Convert Customary Capacity Units Students convert quarts to gallons or cups to pints to build fluency with converting customary measurements in a smaller unit in terms of a larger unit from module 3. After each prompt for a written response, give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display 4 qt =

gal.

4 quarts is equal to how many gallons? Tell your partner. Provide time for students to think and share with their partners.

1 gallon Display the answer, and then display 1 qt =

gal.

Write and complete the equation. Display the answer, and then display 5 qt =

4 qt =

gal

1 qt =

1 4

gal

5 qt = 1 14

gal

gal.

Write and complete the equation. Display the answer. Repeat the process with the following sequence:

436

2c=

1c=

1 2

7c=

3 12

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22

Contemplate Then Calculate: Multiply Fractions Materials—S: Multiply Fractions Fluency Sheets

Students write the product to prepare for evaluating expressions involving nested grouping symbols. Direct students to study the problems on Fluency Sheet 1. Have students focus on the problems in just one column to start. Consider having them cover the other problems with sticky notes or blank paper in advance. Frame the task: As you study, ask yourself, What do I notice that could help me with these problems? Provide 1–2 minutes of silent think time. Some students may make notes or answer problems as part of their study.

Teacher Note

Have students turn and talk about their thinking. Listen for students who offer solution strategies or connect problems by highlighting relationships or patterns. Select a few students to share their ideas with the class.

Consider asking the following questions to discuss the patterns in Fluency Sheet 1:

• What pattern do you notice in problems 1–10? • How do problems 1–5 compare to problems 6–10?

Use your own ideas or the ideas you heard to help you do as many problems as you can. I do not expect you to finish. UDL: Representation As students share their ideas, consider displaying Fluency Sheet 1 and annotating problems to reinforce strategies, relationships, and patterns described.

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5 ▸ M3 ▸ TD ▸ Lesson 22

EUREKA MATH2 California Edition

Invite students to complete Fluency Sheet 2 at another time by using what they learned from Fluency Sheet 1.

Launch

Students build a complicated expression from a simple expression. Write 22 + 8 − 9. Have students evaluate the expression on their whiteboards. What is the value of the expression 22 + 8 − 9?

21 This expression is composed of the numbers 22, 8, and 9. All expressions are composed of numbers, letters that represent unknowns, or a combination of the two, but sometimes the expression is written in a way that appears more challenging. Let’s rewrite the expression so that it appears more challenging.

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5 ▸ M3 ▸ TD ▸ Lesson 22

Have students use different operations to write an expression equal to 22. Students may offer expressions such as the following: 20 + 2, 25 − 3, 2 × 11, 44 ÷ 2. Choose one of the expressions students create to replace 22 in the original expression. For example, 25 − 3. Write 25 − 3 + 8 − 9. Now, let’s write an expression that is equal to 8. What expressions could we use in place of 8?

5+3 15 − 7

1   × 16 2

16 ÷ 2 Choose one of the expressions students share to replace 8 in the original expression. Choose an expression that uses a different operation than what was used to replace 22, such as

_ 1  × 16. 2

Write 2 5−3+_  1 × 16 − 9. 2

Now let’s write an expression equal to 9. What expressions could we use in place of 9?

8+1 10 − 1 3×3 81 ÷ 9 Choose one of the expressions students share to replace 9 in the original expression. Choose an expression that uses a different operation than what was used to replace 22, such as 8 + 1. Write 2 5−3+_  1 × 16 − 8 + 1. 2

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5 ▸ M3 ▸ TD ▸ Lesson 22

What answer do you expect to get if you evaluate this expression? Why? I expect to get 21 because that was the answer to the original expression, and all we did was replace the numbers 22, 8, and 9 with expressions equal to those numbers.

Point to the expression 2 5−3+_  1 × 16 − 8 + 1. 2

We know this expression evaluates to 21 because all we did was replace the original numbers with expressions equal to 22, 8, and 9. But if someone saw only this new expression, they may evaluate it differently. What can we do to make sure they evaluate to 21?

We can put parentheses around 25 − 3, around _  1× 16, and around 8 + 1. 2 Place parentheses around 25 − 3, _  1  × 16, and 8 + 1. 2

Now we have a new expression that looks more challenging but has the same value

Teacher Note This lesson teaches students to evaluate an expression by reading the expression as distinct parts as opposed to distinct numbers.

__ 3 1 __ means “  times another number.” What is the 1 For example, the expression   × (2 × (1 ÷ 6)) 3

other number? The expression 2 × (1 ÷ 6), which means “2 times another number.” What is that other number? That number is 1 ÷ 6,

or   1 . 6

_    × 16 − (8 + 1) as the original one. When we read an expression such as (25 − 3) + 1 (2 ) as a number plus another number minus another number, it can help us make the

In grade 6, students are formally introduced to the order of operations.

numbers we need to add and subtract simpler to identify.

In later grades, students read equations such

Transition to the next segment by framing the work. Today, we will evaluate expressions that include parentheses, brackets, and braces.

440

x+3 ____ = 9 as “5 plus a number is 9” to 17 x+3 realize that  ____ = 4 because 5 + 4 = 9.

as 5 +

EUREKA MATH2 California Edition

Learn

5 ▸ M3 ▸ TD ▸ Lesson 22

Interpret and Evaluate Expressions Involving Whole Numbers Students interpret expressions involving whole numbers and nested grouping symbols and then evaluate the expressions. Write the expression 12 + [(3 × 5) − 6] and invite students to think–pair–share about what they notice and wonder. I notice that there is addition, multiplication, and subtraction in the expression. I see parentheses inside of brackets. I wonder how we could evaluate the expression. Let’s try to read the expression to help us understand how to evaluate it. Point to the parts of the expression and say the following. This expression means that I am finding the sum of two numbers, 12 and some other number. But we don’t know what that other number is yet.

Teacher Note Brackets, shown in the expression in this segment of Learn, and braces, shown in the next segment of Learn, may be new to students. Support students with interpreting this notation by explaining that brackets and braces are grouping symbols just like parentheses. When parentheses already exist in an expression, sometimes other grouping symbols such as brackets and braces are used.

Differentiation: Support

Point to [(3 × 5) − 6]. This other number is 6 subtracted from the product of 3 and 5. So this other number is the difference of some number, 3 × 5, and 6. What is 3 × 5?

15 Write = 12 + [15 − 6] next to the original expression. Now that we know what number we’re subtracting 6 from, we can find the difference of 15 and 6. What is 15 − 6?

12 + [(3 x 5) - 6]] = 12 + [15 - 6]] = 12 + 9 = 21

Students may benefit from focusing on only one part of the expression at a time. Consider providing a written copy of the expression. Use a sticky note to cover up different parts as students make sense of the expression, as shown.

12 + [(3 × 5) − 6] 12 + [

]

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5 ▸ M3 ▸ TD ▸ Lesson 22

Write = 12 + 9. Point to [(3 × 5) − 6]. Now that we know that this other number is equal to 9, we can find the sum of 12 and 9. What is 12 + 9?

21 Write = 21. Display the expression [68 − (40 ÷ (2 + 3))] ÷ 5. Invite students to turn and talk about how they could read the expression.

EUREKA MATH2 California Edition

UDL: Representation Consider posting a written version as each expression is described aloud. Leave it posted for students to refer to.

Expression

What do we need to find to evaluate this expression? We need to find the quotient of some number and 5. Do we know what the dividend is yet? If not, how could we find it? No. The dividend is the difference of 68 and some number. Do we know what number we’re subtracting from 68? If not, how could we find it? No. The number we’re subtracting is the quotient of 40 and another number. Do we know what the divisor for that division problem is yet? If not, how could we find it? No. The divisor is the sum of 2 and 3. Invite students to work with a partner to evaluate the expression.

How to Read the Expression

12 + [(3 x 5) - 6]

The sum of two

[(3 x 5) - 6]

The difference of

[68 - (40 ÷ (2 + 3))] ÷ 5

numbers, 12 and

some other number

some number, 3 x 5, and 6

The quotient of some number and 5

When students have finished, discuss the answer. Record student thinking as they answer the following questions. What part of the expression did you evaluate first? I added 2 and 3 to get 5. What did you do next? I divided 40 by 5 to get 8. What did you do after that? I subtracted 8 from 68 to get 60.

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5 ▸ M3 ▸ TD ▸ Lesson 22

What was your final step? I divided 60 by 5 to get 12. Invite students to turn and talk about how they can read and evaluate an expression involving more than one grouping symbol.

Interpret and Evaluate Expressions Involving Fractions Students interpret expressions involving fractions and nested grouping symbols and then evaluate the expressions.  1 × (7 ÷ _ Display the expression _  1 ) + 2]and invite students to turn and talk about what [ 4

is different in this expression compared to the expressions in the previous segment. What do you notice about this expression that is different from the expressions we looked at earlier? There are fractions in this expression, and the other expressions had only whole numbers. We can read expressions with fractions the same way we read expressions with whole numbers. Invite students to turn and talk about how they could read this expression. What do we need to find to evaluate this expression?

Promoting the Standards for Mathematical Practice When students interpret and evaluate expressions involving fractions and nested grouping symbols, they are attending to precision (MP6). Ask the following questions to promote MP6: • What do the parentheses mean in this expression?

We need to find the product of _  1  and another number.

• What details are important to think about in this problem?

4 1 We want to find the product of    and another factor. How can we find what the other 4

• Where might you make mistakes when evaluating an expression with more than one grouping symbol?

We need to find the product of two factors, _  1  and some other number.

factor is?

The other factor is the sum of some number and 2. How can we find our first addend? We need to divide 7 by _  1  . 2

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5 ▸ M3 ▸ TD ▸ Lesson 22

Invite students to work with a partner to evaluate the expression. When students have finished, discuss the answer. Record student thinking as they answer the following questions. What part of the expression did you evaluate first? I divided 7 by _  1  to get 14. 2

What did you do next? I found the sum of 14 and 2, which is 16. What was your final step? I found _  1  of 16, which is 4. 4

As time allows, repeat the process with the following expressions. 2_ 3 1_ • (_ [ 8 + 4  ) × 3]− 3  

• 1_  + {[2 − (1 ÷ 4)] × 4} 2

4 • ((4 ÷ 5) − _   + 2_  − (6 ÷ 9)) × 1,385 5 ) (3 [ ]

EUREKA MATH2 California Edition

UDL: Action & Expression Consider having partners take turns thinking aloud as they evaluate the expressions. Partner A begins, and as she evaluates the expression, she explains her decisions to partner B. While partner B listens, he can ask partner A questions. Students who are thinking aloud can use the Share Your Thinking section of the Talking Tool, and students who are listening can use the I Can Ask Questions and Say It Again sections. Thinking aloud prompts students to focus on their reasoning and change course, as needed, as they talk through decisions. Taking turns ensures that each partner has an opportunity to describe their reasoning. Model the process for students and prompt the use of the Talking Tool as needed.

As students evaluate the expressions, circulate and ask the following questions to advance student thinking: • What do we need to find to evaluate the expression? • Do we know what the minuend is? The other addend? The other factor? • How can you find the minuend? The other addend? The other factor? • What part of the expression do you evaluate first? Why?

Use Parentheses and Brackets to Make an Equation True Students insert parentheses and brackets to make an equation true. Display the equation 5 × 30 − 15 ÷ 3 = 25. Invite students to work with a partner to determine where to place parentheses and brackets to make the equation true. Then invite partners to share their answer and reasoning. 444

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22

Did you place parentheses and brackets in the equation? Where did you place the parentheses? Where did you place the brackets? Yes. I placed both parentheses and brackets in the equation. I placed brackets around 30 − 15 ÷ 3 and parentheses around 30 − 15. How do you know you made the equation true? I know I made the equation true because after placing the parentheses and brackets I had the expression 5 × [(30 − 15) ÷ 3] on the left side of the equal sign. That expression told me that I was finding the product of 5 and another factor. To find the other factor, I needed to evaluate [(30 − 15) ÷ 3]. To find the quotient, I knew I had to find the dividend by subtracting 15 from 30 first. That gave me 15 ÷ 3, which is equal to 5. Because 5 × 5 = 25, I knew that where I placed the parentheses and brackets made the equation true. As time allows, invite students to place parentheses and brackets to make true equations with the following equations: 1 • _ − 1 ÷ 3 × 1 = 0 3

• 25 − 8 ÷ 1 ÷ 2 = 9 Invite students to turn and talk about how they determined where the parentheses and brackets belong.

Problem Set Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

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5 ▸ M3 ▸ TD ▸ Lesson 22

Land

EUREKA MATH2 California Edition

Debrief 5 min Objective: Evaluate expressions involving nested grouping symbols. Facilitate a class discussion about evaluating expressions involving nested grouping symbols by using the following prompts. Encourage students to restate or add on to their classmates’ responses. Display the expressions [(4 × 15) − 6] ÷ 3 and 4 × [(15 − 6) ÷ 3]. Invite students to think–pair–share about what is the same and what is different about the expressions. Both expressions have the numbers 4, 15, 6, and 3, from left to right. Both expressions have the operations multiplication, subtraction, and division, from left to right. The placement of the parentheses and brackets is different for each expression. What is the first expression asking us to find? The quotient of a number and 3 What is the second expression asking us to find? The product of 4 and another number Does reading an expression help you evaluate it? How? Yes. Reading an expression helps me see an expression that looks challenging as simply a number. For example, the expression in problem 7 in the Problem Set has parentheses, brackets, and braces. But after I read and evaluate it, its value is 0, which surprised me because the expression looked so challenging. Yes. It helps me determine which parts of the expression must be evaluated first. For example, if I read the expression as a sum of two numbers, but one number is an expression with parentheses, then I know I need to evaluate that part first before I can add the two numbers.

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5 ▸ M3 ▸ TD ▸ Lesson 22

Why is it important to pay attention to parentheses, brackets, and braces when you are evaluating expressions with nested grouping symbols? It is important to pay attention to where the parentheses and brackets are so that I know how to read the expression. I need to pay attention to where the parentheses are so that I know which part of the expression to evaluate first.

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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5 ▸ M3 ▸ 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 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22 ▸ Multiply Fractions

Contemplate Then Calculate Fluency Sheet 1

10 15

24. _ × _

15 10

25. 1_ × 1_

1 9

26. 1_ × 2_

2 9

27. 2_ × 2_

4 9

28. 2_ × _

6 6

29. 2_ × 4_

8 9

30. 2_ × _

10 9

3 5

2. 1_ × 1_

1 6

3. 1_ × 1_

1 8

4. 1_ × 1_ 2

5. 1_ × 1_ 7

_ _

__1 14

6. 1_ × 1_

1 6

7. 1_ × 1_

1 9

8. 1_ × 1_ 3

9. 1_ × 1_ 3

10. 1_ × 1_ 5

11. 1_ × 2_ 5

448

1 4

__1

__1 15

__2 15

12. 2_ × 2_ 5

13. 1_ × 1_ 4

14. 1_ × 2_ 4

15. _ × 2_ 3 4

16. 1_ × 1_ 6

17. _ × 1_ 5 6

__4 15

__1 12

__2

__6 12

__1 18

__5

18. _ × 2_

10 18

19. _ × 2_

10 12

5 6 5 4

20. 1_ × 1_ 5

21. 2_ × 2_ 5

22. 2_ × _ 5

3 5

5 3

5 2

3 2

5 3

31. _ × _ 3 2

3 5

32. _ × 1_ 3 4

__1 25

23. 2_ × _

Write the product as a fraction.

1. 1_ × 1_

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22 ▸ Multiply Fractions

33. _ × 4_ 3 4

__ _

_ _ _ _

__9 10

__3 20

12 20

34. _ × _

15 20

35. _ × _

18 20

3 4

5 5 6 5

36. 1_ × _ 4

6 5

37. 1_ × 1_ 7

38. 1_ × _ 8

3 5

39. _ × 1_ 5 6

40. _ × _ 3 4

3 4

__6 20

__1

__3 40

__5 24

__9 16

41. 2_ × _

12 18

42. _ × _

18 8

3 4

6 6

6 2

43. _ × _ 7 8

7 9

44. __ × _ 7 12

9 8

__ 49 __ 72

__ 63 96

__4 25

__6 25

197

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EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22 ▸ Multiply Fractions

EUREKA MATH2 California Edition

5 ▸ M3 ▸ TD ▸ Lesson 22 ▸ Multiply Fractions

Fluency Sheet 2

1. 1_ × 1_

1 6

2. 1_ × 1_

1 8

15 20

__4 15

24. 4_ × _

20 15

3. 1_ × 1_ 2

4. 1_ × 1_ 2

5. 1_ × 1_ 9

6. 1_ × 1_ 5

7. 1_ × 1_ 7

8. 1_ × 1_ 5

9. 1_ × 1_ 5

10. 1_ × 1_ 3

11. 1_ × 2_ 3

__1 12

25. 1_ × 1_

__1 10

3 12

26. 1_ × _

__1 18

__6 12

27. _ × _

__1 18

28. _ × 4_

12 12

__1 10

__2 18

29. _ × _

15 16

__1 35

__1 15

__2

3 5

12. 2_ × 2_ 3

13. 1_ × 1_ 3

14. 1_ × _ 3

3 4

15. 2_ × _ 3

3 4

16. 1_ × 1_ 3

17. 2_ × 1_ 3

23. _ × _

Write the product as a fraction.

3 4

3 4 3 4

5 4

5 3

3 4 3 4

5 4

__ __1 16

__3 16

__9 16

__ __

18. 2_ × _

10 18

30. _ × _

18 16

19. _ × _

9 8

31. 4_ × 4_

16 18

__1 25

32. 2_ × 1_

__9

33. 2_ × 4_

3 2

5 6 3 4

20. 1_ × 1_ 5

21. _ × _ 3 5

3 5

22. _ × 4_ 3 5

3 4

6 4

__ __2 15

__8 15

34. 2_ × _

10 15

35. 2_ × _

12 15

5 5

6 5

36. 1_ × _ 3

6 5

37. 1_ × 1_ 9

38. 1_ × _ 5

3 8

39. _ × 1_ 3 4

__6 15

__1 81

__3 40

__3 24

40. 2_ × 2_

4 9

41. _ × _

24 32

42. 2_ × _

12 9

43. _ × _

48 63

44. __ × _

56 84

3 4

6 7

7 12

8 8 6 3

8 9

8 7

__ __ __ __

12 __

199

200

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5 ▸ M3 ▸ TD ▸ Lesson 22

EUREKA MATH2 California Edition

Name

5 ▸ M3 ▸ TD ▸ Lesson 22

9. [(4 ÷ 1_) ÷ 2] × 2_ 3 4

Date

= 20 −

÷ 2)

2. 9 × ((1_ + 7_) − 2_) = 9 × ( 2 6 3

=9×

10 __ 6

5. ((2_ − 5_) ÷ 4) × 36 3 9

− 2_) 3

Evaluate the expression. 3. 20 + [(2 × 6) − 4]

10. 24 − {2 × [(15 + 5) ÷ 5]}

16 __ 3

Fill in the blanks to show how to evaluate the expression. 1. 20 − ((7 + 3) ÷ 2) = 20 − (

EUREKA MATH2 California Edition

11. {[9 ÷ (4_ − (3 ÷ 5))] − 44} × 2,000 5

4. [(20 + 2) × 6] − 4

7 12. 4,628 × [((3 ÷ 5) − 3_) + (__ − (14 ÷ 26))] 5 13

2,000

128

6. [(2_ + 1_) × 2] − 2_ 3 5 5 20 __ 15

Place parentheses and brackets to make each equation true. 13. [60 − (3 × 4)] ÷ 2 = 24 7. {1_ − [2 × (1 ÷ 8)]} × 10 4

450

14.

5 × [(40 ÷ 4) − 2] = 40

8. [91 + (36 ÷ (2 + 2))] ÷ 10

201

202

PROBLEM SET

Standards Module Content Standards Write and interpret numerical expressions. 5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product _   × q as a parts of a partition of q into b equal parts; a b

equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show _   × 4 = _   , and create a story context for 8 2 3 3 8 2 4 this equation. Do the same with   ×   =   . In general,  a×  c =  ac . 3 5 15 ( b d bd )

_ _ __

5.NF.B.5 Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product

452

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5 ▸ M3

smaller than the given number; and relating the principle of fraction equivalence _   = ____   to the effect of multiplying _  by 1. a b

n×a n×b

a b

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1 a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for _   ÷ 4, and use 1 3

a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that _   ÷ 4 = __   because 1 __   × 4 = _  1 . 12

1 12

1 3

b. Interpret division of a whole number by a unit fraction, and compute

such quotients. For example, create a story context for 4 ÷ _   , and use 1 5

a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ _   = 20 because 20 × _  = 4. 1 5

1 5

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share _   lb of chocolate equally? How many _   -cup servings are in 2 cups of raisins? 1 3

1 2

Convert like measurement units within a given measurement system. 5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

tudents able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the S relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

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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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Module Achievement Descriptors and Content Standards by Lesson A Topic 1 Lesson Achievement Descriptor

Aligned CA CCSSM

5.Mod3.AD1

5.OA.A.1

5.Mod3.AD2

5.OA.A.1

5.Mod3.AD3

5.OA.A.2

5.Mod3.AD4

5.OA.A.2

5.Mod3.AD5

5.NF

5.Mod3.AD6

5.NF.B.4

5.Mod3.AD7

5.NF.B.4.a

5.Mod3.AD8

5.NF.B.5 5.NF.B.5.a

5.Mod3.AD9

5.NF.B.5 5.NF.B.5.b

5.Mod3.AD10

5.NF.B.6

5.Mod3.AD11

5.NF.B.7.a

5.Mod3.AD12

5.NF.B.7.b

5.Mod3.AD13

5.NF.B.7.c

5.Mod3.AD14

5.MD.A.1

A 1

A 2

A 3

A 4

A 5

A 6

B 7

B 8

B 9

B 10

B 11

C 12

C 13

C 14

C 15

C 16

C 17

D 18

D 19

D 20

D 21

D 22

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Standards for Mathematical Practice by Lesson A Topic 1 Lesson Aligned Practice Standard

A 1

A 2

A 3

A 4

A 5

A 6

B 7

B 8

B 9

B 10

B 11

C 12

C 13

C 14

C 15

C 16

C 17

D 18

D 19

D 20

D 21

D 22

MP1 MP2 MP3 MP4 MP5 MP6 MP7 MP8

456

Achievement Descriptors: Proficiency Indicators 5.Mod3.AD1 Write numerical expressions that include fractions and parentheses. RELATED CA CCSSM

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Partially Proficient

Proficient

Identify the effect of parentheses in numerical expressions that include fractions.

Create numerical expressions that include fractions to equal a specified value.

Which expression is equal to 10 ×     +     +     +     ?

Insert parentheses to make a true number sentence.

__4 __7 __2 __1

__ __ __

7 2 1 4 A. ( 10 ×    ) +     +     +     3 6 5 8

__ __

__ __ __ __

Highly Proficient

12 ×   2  +   1  +   5  +   6   = 24   3  3

B. 10 × (   4  +   7 ) +   2  +   1  5 3 6 8

__ __ __

C. 10 × (   4  +   7  +   2 ) +   1  5 3 6 8

__ __ __ __

7 4 2 1 D. 10 ×     +     +      +   (5 8 3 6 )

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5.Mod3.AD2 Evaluate numerical expressions that include fractions and parentheses. RELATED CA CCSSM

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Partially Proficient

Proficient

Highly Proficient

Evaluate numerical expressions that include fractions and a single set of parentheses.

Evaluate numerical expressions that include fractions and two sets of nonnested parentheses.

Evaluate.

1 __ 6 ×  __   +   2 

__ __

(3 −   5 ) × (   1  +   2 ) 6

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5.Mod3.AD3 Translate between numerical expressions that include fractions and mathematical or contextual verbal descriptions. RELATED CA CCSSM

5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

Partially Proficient

Proficient

Highly Proficient

Translate between numerical expressions that include fractions and verbal descriptions.

Write numerical expressions that include fractions to represent context-based verbal descriptions.

Create and explain contexts that can be modeled by given numerical expressions that include fractions.

Write an expression to represent the difference of five-sixths and two-sevenths.

Write an expression that can be used to solve the problem.

Write a word problem that can be solved by using the expression shown.

Blake has   3  pounds of cheese. He uses   pound

  1 × (   3  −   1 )

__1

1 to make dinner. The next day, Blake uses  __ of the 4

__ 3

__ __ 4

cheese that is left to make a sandwich. How much cheese is on Blake’s sandwich?

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5.Mod3.AD4 Compare the effect of each number and operation on the value of a numerical expression that includes fractions. RELATED CA CCSSM

Partially Proficient

Proficient

Highly Proficient

Compare the values of two numerical expressions that have at most two operations with fractions and at most one set of parentheses without evaluating.

Compare the values of two expressions that have at least two operations with fractions or multiple sets of parentheses without evaluating.

Justify comparisons of two different expressions that include fractions without evaluating.

Compare the expressions by using >, =, or <.

__ __

6×(   6 +   2 ) 7

__ __

3×(   6 +   2 ) 7

__ __

(2 + 6) × (   6 +   2 ) 7

__ __

(1 + 3) × (   6 +   2 ) 7

__6 __2

Explain why (2 + 6) × (   +   ) is greater than

__ __

(1 + 3) × (   6 +   2 ) without evaluating either expression. 7

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5.Mod3.AD5 Solve multi-step problems, including word problems, involving addition, subtraction, and multiplication

of fractions, division of whole numbers with fractional quotients, and division with unit fractions and whole numbers. RELATED CA CCSSM

5.NF Number and Operations—Fractions

Partially Proficient

Proficient

Solve multi-step problems involving addition, subtraction, and multiplication of fractions, division of whole numbers with fractional quotients, and division with unit fractions and whole numbers.

Solve multi-step word problems involving addition, subtraction, and multiplication of fractions, division of whole numbers with fractional quotients, and division of unit fractions and whole numbers.

Evaluate.

Mr. Sharma pours   pint of cranberry juice equally

__ __ __

  1 ×   6 +   1 8

__1 2

Highly Proficient

__1

into 8 glasses. Then he pours   pint of grape juice into 4

1 of the glasses. How many total pints of juice are in the glass that has grape juice?

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5.Mod3.AD6 Multiply whole numbers or fractions by fractions. RELATED CA CCSSM

5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Partially Proficient Multiply two unit fractions or a whole number by a unit fraction. Multiply.

__ __

  1 ×   1 7

Proficient

Highly Proficient

Multiply fractions by fractions and whole numbers by fractions. Multiply.

  1 × 8 ×   2 2

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EUREKA MATH2 California Edition

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5.Mod3.AD7 Recognize, model, and contextualize the product of a fraction and a whole number or fraction. RELATED CA CCSSM

__a

5.NF.B.4.a Interpret the product   × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use

__2

__8

ac __2 __4 __8 . In general,  __a × __  c =  ___ .

a visual fraction model to show   × 4 =   , and create a story context for this equation. Do the same with   ×   =   3

Partially Proficient Recognize multiple ways of representing a fraction of a set. Consider the shaded and unshaded hexagons shown.

15 (

bd )

Proficient

Highly Proficient

Create and explain models of multiplication of fractions to solve problems, including word problems.

Create a context to fit a given expression or equation involving multiplication of fractions.

Consider the expression.

Create a story to match the equation.

__ __

  1 ×   1 3

__ __ __

  1 ×   8 =   4 2

Part A Draw a model to represent the expression. Part B Explain how your model represents the product.

Which expression represents the number of shaded hexagons?

3 A.   × 20 2

2 B.   × 20 3

3 C.   × 20 5

2 D.   × 20 5

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5.Mod3.AD8 Compare the effects of multiplying by fractions and whole numbers. RELATED CA CCSSM

5.NF.B.5 Interpret multiplication as scaling (resizing), by: 5.NF.B.5.a Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Partially Proficient

Proficient

Highly Proficient

Compare the size of a product to the size of one factor without multiplying.

Generate a fraction that yields an increase or decrease in value.

Fill in the blank with >, =, or <.

Fill in the blank with any whole number that makes a true number sentence.

  8 × 3 12

_______ × 3 < 3

_______ × 3 = 3

_______ × 3 > 3 12

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5.Mod3.AD9 Explain the effect of multiplying by a fraction less than 1, equal to 1, or greater than 1. RELATED CA CCSSM

5.NF.B.5 Interpret multiplication as scaling (resizing), by: 5.NF.B.5.b Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given n×a a __a ____ to the effect of multiplying  __ by 1.

number; and relating the principle of fraction equivalence   =   b

Partially Proficient Identify correct explanations for problems involving multiplication by a fraction less than 1, equal to 1, or greater than 1. Select the correct explanation for the product shown.

  3 × 4 5

__3 __3 5 5 3 3 B.  __ is less than 1, so  __ × 4 is greater than 4 . 5 5 3 3 __ C.   is greater than 1, so  __ × 4 is less than 4 . 5 5 3 3 __ __ D.   is greater than 1, so   × 4 is greater than 4 . A.   is less than 1, so   × 4 is less than 4 .

466

n×b

Proficient

Highly Proficient

Explain why multiplying a given number by a fraction less than 1, equal to 1, or greater than 1 results in a product less than, equal to, or greater than the given number. Explain why the number sentence is true.

  3 × 4 < 4 5

EUREKA MATH2 California Edition

5 ▸ M3

5.Mod3.AD10 Solve real-world problems involving multiplication of fractions. RELATED CA CCSSM

5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Partially Proficient

Proficient

Highly Proficient

Identify a numerical expression that can be used to solve a real-world problem involving multiplication of fractions.

Solve one-step real-world problems involving multiplication of fractions.

Solve multi-step real-world problems involving multiplication of fractions.

Jada has   pounds of snack mix. She gives   of the snack mix

Jada has   pounds of snack mix. She gives   of the

to Lacy. How many pounds of snack mix does Lacy get?

snack mix to Lacy. How many pounds of snack mix

snack mix to Lacy. At lunch, Lacy eats   of the snack

__2 3

__1 2

Which expression can be used to solve this problem?

__ __ __ __ __ __ __ __

2 1 A.   +   3 2 2 1 B.   −   3 2 2 1 C.   ×   3 2 2 1 D.   ÷   3 2

__2 3

does Lacy get?

__1 2

__2 3

__1

__1 2

mix she got from Jada. How many pounds of snack mix did Lacy eat for lunch?

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EUREKA MATH2 California Edition

5 ▸ M3

5.Mod3.AD11 Model and evaluate division of unit fractions by nonzero whole numbers.

__1

RELATED CA CCSSM

5.NF.B.7.a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for   ÷ 4, and use a visual

__1

fraction model to show the quotient. Use the relationship between multiplication and division to explain that   ÷ 4 =   because   × 4 =   . 3

Partially Proficient

Proficient

Highly Proficient

Divide unit fractions by nonzero whole numbers by using provided models.

Model and evaluate division of unit fractions by nonzero whole numbers.

Contextualize division of unit fractions by nonzero whole numbers and interpret the quotient.

Use the model to find the quotient.

Draw a model to represent the expression. Then divide.

Use the expression to answer part A and part B.

  1 ÷ 6 4

  1 ÷ 6

1  __ ÷ 6

Part A Create a story context for the expression. Part B Explain what the quotient means in your story.

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5.Mod3.AD12 Model and evaluate division of whole numbers by unit fractions. RELATED CA CCSSM

__1

5.NF.B.7.b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷   , and use a visual fraction

__1 __1 model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷   = 20 because 20 ×   = 4. 5

Partially Proficient

Proficient

Highly Proficient

Divide whole numbers by unit fractions by using provided models.

Model and evaluate division of whole numbers by unit fractions.

Contextualize division of whole numbers by unit fractions and interpret the quotient.

Use the model to find the quotient.

Draw a model to represent the expression. Then divide.

Use the expression to answer part A and part B.

5 ÷   1 4

5 ÷   1

5 ÷   1 4

Part A Create a story context for the expression. Part B Explain what the quotient means in your story.

469

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5.Mod3.AD13 Solve word problems involving division of unit fractions by nonzero whole numbers and division of whole

numbers by unit fractions. RELATED CA CCSSM

5.NF.B.7.c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using

__1

visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share   lb of chocolate equally?

__1 How many   -cup servings are in 2 cups of raisins?

Partially Proficient

Proficient

Highly Proficient

Identify numerical expressions that can be used to solve single-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Solve single-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Solve multi-step word problems involving division of unit fractions by nonzero whole numbers and whole numbers by unit fractions.

Mr. Perez has   gallon of grape juice. He pours the juice equally

Mr. Perez has   gallon of grape juice. He pours the

into 8 glasses. How many gallons of juice are in each glass?

juice equally into 8 glasses. How many gallons of juice

juice equally into 8 glasses. Then he takes 1 of the

Which expression can be used to solve this problem?

are in each glass?

glasses and shares it equally among his 3 children.

__1 2

__ __

1 A.   × 8 2 1 B.   ÷ 8 2 1 C. 8 ×   2 1 D. 8 ÷   2

__1 2

How many gallons of juice does each of Mr. Perez’s children get?

__ __

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5 ▸ M3

5.Mod3.AD14 Convert among units within the customary measurement system to solve problems. RELATED CA CCSSM

5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Partially Proficient Convert among units within the customary measurement system. Convert each measurement.

2 ft =

  5 ft =

__ 12

Proficient

Highly Proficient

Convert among units within the customary measurement system to solve multi-step word problems. Miss Song has 6 pounds 4 ounces of almonds and 3 pounds 3 ounces of cashews. How many ounces of almonds and cashews does Miss Song have altogether?

471

Big Ideas This chart identifies the Big Ideas that are central to each module. For more information on the Big Ideas, see the grade-level chapters of the California Mathematics Framework.

Big Idea

Module 1

Module 2

Module 3

Module 4

Module 5

Module 6

Plotting Patterns Telling a Data Story Factors and Groups Modeling Fraction Connections Seeing Division Powers and Place Value Layers of Cubes Shapes on a Plane

472

Terminology The following terms are critical to the work of grade 5 module 3. 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

Familiar convert cup denominator gallon mixed number numerator ounce pint pound product quart quotient whole number

Module 3 does not introduce any new terms.

Academic Verb demonstrate

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Math Past Two Fraction Problems from the Nine Chapters Did the people of ancient China add, subtract, multiply, and divide with fractions? What kinds of problems did the people of ancient China solve by using fractions? Did people in ancient China use mixed numbers? The oldest recorded civilizations—Babylon and Egypt—used fractions almost 4,000 years ago. Let’s look at a more recent (but still ancient) civilization: China, of the period from about 200 BCE to about 300 CE, and its work with fractions. Chinese mathematicians of this time recorded their knowledge in a book called Jiuzhang Suanshu, or The Nine Chapters on the Mathematical Art. From this book, we know how the people of ancient China did arithmetic. The Nine Chapters on the Mathematical Art is a collection of 246 math problems. It also contains descriptions of mathematical rules and explanations of how to solve the problems. Some of the problems were created before the Qin Dynasty, which started in 221 BCE. The full collection of problems was probably finished around the year 100 BCE. It was traditional for writers to make their own notes and additions to ancient texts like these. In 263 CE, a mathematician named Liu Hui was the first to add his own explanations of the solutions and commentary to the Nine Chapters. The Nine Chapters was an important textbook for teaching mathematics in ancient China, as well as in neighboring countries.

Many of the problems in the Nine Chapters involve fractions and mixed numbers. We see from these problems how Chinese scholars added, subtracted, multiplied, and divided fractions or mixed numbers. Two of the 246 original problems are presented here. Your students may enjoy taking a journey back in time to see the types of problems that were solved in ancient China. Problem 17 of Chapter 1 asks the reader to divide a mixed number by a whole number. Now given 7 persons share 8  _1 coins. 3

Tell: how much does each person get? 1

You might find it strange that the problem asks about 8  _1 coins. 3

How did they make _ 1 of a coin? Then the problem requires those 3

coins to be divided into even smaller pieces so they can be shared among 7 people. Aren’t coins hard to split? Chinese mathematicians sometimes made up problems that could not happen in the real world. The problems were simply used to show how to do a calculation. However, to enliven the discussion, you might suggest students pretend the coins are chocolate coins or crackers instead. It may be easier to imagine splitting those types of items. Although students have only formally seen division of whole numbers and division involving unit fractions and whole numbers so far, encourage them to think about how they could use the techniques they have learned to solve this problem. A visual model

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Liu Hui. The Nine Chapters on the Mathematical Art: Companion and Commentary, 80.

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like the one shown may help students realize they can simplify the problem by finding 8 ÷ 7 and _ 1 ÷ 7 and then adding the quotients.

one-fifth is taken away. At the inner pass one-seventh is taken away. Assume the remaining cereal is 5 dou.

Tell: how much cereal is carried originally?3

1 3

The following is some background information to help your students understand the meaning of this problem. • The cereal is a grain farmed in China, most likely millet.

By using the method described, students should find the following solution. 4 (8 ÷ 7) + ( _1 ÷ 7) = 1 _1 + __   1 = 1  __ + __   1 = 1  __ 3

3 21

Let’s see if this answer matches the one found in the Nine Chapters. 4 Answer: Each gets 1  __ coins.2 21

Congratulate your students on using their reasoning and math skills to solve a problem that is about 2,000 years old! It is interesting to note that Chinese scholars of this time always wrote

• The passes are like modern toll booths. A person who wanted to pass through had to pay a tax. But the tax in China would not have been money—it would have been some of the cereal itself. • The fraction of tax charged at each pass is based on the amount of cereal the person has going into the pass: the more you carry, the more you pay. • A dou is a unit of volume that was used in ancient China. One dou would be approximately 2 liters, or a little over half a gallon. This problem is more complicated than the previous one. Engage your students by first asking them to give their best estimate of the starting amount of cereal and ask them to give their reasoning.

their answers in a particular form. They wrote fractions greater than

Then have students start thinking about ways to improve their

1 as mixed numbers and wrote fractions by using the largest possible

estimates. Suggest that they start with an area model to get

4 unit. For example, in this problem the answer is given as 1  __ coins 21

rather than __  25 coins. Ask your students if they can think of any 21

a better idea of what is happening in the problem. Have students draw a rectangle to represent the amount of cereal the person

reasons why Chinese scholars might have chosen to do this.

in the problem starts with before going through the passes. Tell

Problem 27 of Chapter 6 is more complex.

students to partition the rectangle into thirds and imagine taking

Now given a person carrying cereal through three passes. At the outer pass, one-third is taken away as tax. At the middle pass, 2

Liu Hui. The Nine Chapters on the Mathematical Art: Companion and Commentary, 80.

away _ 1 of the cereal. Ask students what fraction of the cereal 3

remains. They should recognize that _ 2 of the cereal remains. 3

Liu Hui. The Nine Chapters on the Mathematical Art: Companion and Commentary, 345.

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At the middle pass, _ 1 of the remaining cereal is taken away. 5

Ask students how they can use their area models to show this. From

Pass

Fraction of Cereal Left

Outer

2 3

Middle

4 of 2 5 3

Inner

6 of   8 7 15

their models, students will likely recognize that _ 4 of the remaining _ 2 of cereal is left. Challenge them to determine 5what fraction of the starting amount of cereal is left. By using their models to find the

product of the two fractions, they should see that __  8 of the starting 15

amount of cereal is left.

2 3

Equation

_ _

_ __

_ _ __

8 4 2 × =   5 3 15

_ __ ___

6 8 48 ×   =   7 15 105

Ask students whether they want to revise their estimates for the starting amount of cereal. Students may recognize that ___  48 is a 105 1 little less than _ . Since the amount of cereal left is 5 dou, the 2 starting amount should be a little more than 10 dou.

4 5

Discuss with students what they think happens at the inner pass. Ask them what fraction of the remaining cereal is left after _ 1 of it is taken away. Then ask students what fraction of the

Let’s see what is written in the Nine Chapters about this problem.

Lay down 5 dou; multiply by the tax numbers 3, 5, 7 successively as dividend. Take the continued product of the remainders 2, 4, 6 as divisor. Divide, giving the number of dou to be found.4

starting amount of cereal is left. It may help them to organize the information in a table and write a multiplication expression to represent the fraction of the starting amount of cereal left after each pass. Use the table to help your students determine that ___  48 of the starting amount of cereal is left after the person travels through the inner pass.

105

Explain to students that a continued product is a product with more than two numbers multiplied together. The word remainder might also cause some confusion. Explain that here, remainder means what is left and does not

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mean a remainder after division. Walk through the calculation of the answer by using the instructions from the Nine Chapters.

5 × (3 × 5 × 7) = 5 × 105 = 525

The person originally carried 10  __ dou of cereal. 15 16

Engage students in a discussion about whether their estimates 15 were reasonable. Are they surprised the result is 10 __ , or were their 16 estimates close? Encourage students to recognize that they can solve challenging problems, even those written thousands of years ago, by breaking them down into simpler problems.

2 × 4 × 6 = 48

525 ÷ 48 = ___

525 48

Ask students whether they recognize any of these numbers or can guess where they come from. Help your students write ___  525 as a mixed number. 48

525 ___ 45 15 480 45 ___ = + __ = 10 + __ = 10 __ 48

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Materials The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. 36

Blue construction paper, 4  _″ × 2″ strips

Personal whiteboards

Centimeter cubes, set of 1,000

Personal whiteboard erasers

Dry-erase markers

Projection device

Envelopes

Red construction paper, 4  _″ × 2″ strips

Learn books

Scissors

Paper, blank sheets

Teach book

Pencils

Teacher computer or device

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Works Cited Boaler, Jo, Jen Munson, and Cathy Williams. Mindset Mathematics: Visualizing and Investigating Big Ideas: Grade 3. San Francisco, CA: Jossey-Bass, 2018. California Department of Education. California Common Core State Standards: Mathematics. Sacramento, CA: California Department of Education, 2014. California Department of Education. Mathematics Framework for California Public Schools: Kindergarten Through Grade Twelve. Sacramento, CA: California Department of Education, 2023. California Department of Resources Recycling and Recovery. “California’s Environmental Principles and Concepts.” California Education and the Environment Initiative, 2021. https://www.californiaeei.org/epc/.

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://mathematicalmusings.org/. Danielson, Christopher. Which One Doesn’t Belong?: A Teacher’s Guide. Portland, ME: Stenhouse, 2016. Danielson, Christopher. Which One Doesn’t Belong?: Playing with Shapes. Watertown, MA: Charlesbridge, 2019. Empson, Susan B. and Linda Levi. Extending Children's Mathematics: Fractions and Decimals. Portsmouth, NH: Heinemann, 2011. Flynn, Mike. Beyond Answers: Exploring Mathematical Practices with Young Children. Portsmouth, NH: Stenhouse, 2017.

Carpenter, Thomas P., Megan L. Franke, and Linda Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann, 2003.

Fosnot, Catherine Twomey, and Maarten Dolk. Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Portsmouth, NH: Heinemann, 2001.

Carpenter, Thomas P., Megan L. Franke, Nicholas C. Johnson, Angela C. Turrou, and Anita A. Wager. Young Children's Mathematics: Cognitively Guided Instruction in Early Childhood Education. Portsmouth, NH: Heinemann, 2017.

Franke, Megan L., Elham Kazemi, and Angela Chan Turrou. Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom. Portsmouth, NH: Stenhouse, 2018.

CAST. Universal Design for Learning Guidelines version 2.2. Retrieved from http://udlguidelines.cast.org, 2018.

Hattie, John, Douglas Fisher, and Nancy Frey. Visible Learning for Mathematics: What Works Best to Optimize Student Learning. Thousand Oaks, CA: Corwin Mathematics, 2017.

Clements, Douglas H. and Julie Sarama. Learning and Teaching Early Math: The Learning Trajectories Approach. New York: Routledge, 2014.

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Huinker, DeAnn and Victoria Bill. Taking Action: Implementing Effective Mathematics Teaching Practices. Kindergarten –Grade 5, edited by Margaret Smith. Reston, VA: National Council of Teachers of Mathematics, 2017. Kelemanik, Grace, Amy Lucenta, Susan Janssen Creighton, and Magdalene Lampert. Routines for Reasoning: Fostering the Mathematical Practices in All Students. Portsmouth, NH: Heinemann, 2016. Liu Hui. The Nine Chapters on the Mathematical Art: Companion and Commentary, Translated and edited by Shen Kangshen, John N. Crossley, and Anthony W. C. Lun. New York: Oxford University Press, 1999. Ma, Liping. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. New York: Routledge, 2010. National Council for Teachers of Mathematics, Developing an Essential Understanding of Multiplication and Division for Teaching Mathematics in Grades 3–5. Reston, VA: National Council for Teachers of Mathematics, 2011. Parker, Thomas and Scott Baldridge. Elementary Mathematics for Teachers. Okemos, MI: Sefton-Ash, 2004. Schwartz, Randy K. “A Classic from China: The Nine Chapters.” The Right Angle 16 no. 2 (2008): 8–12.

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Shumway, Jessica F. Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3–5. Portland, ME: Stenhouse Publishing, 2018. Smith, Margaret S. and Mary K. Stein. 5 Practices for Orchestrating Productive Mathematics Discussions, 2nd ed. Reston, VA: National Council of Teachers of Mathematics, 2018. Smith, Margaret S., Victoria Bill, and Miriam Gamoran Sherin. The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your Elementary Classroom, 2nd ed. Thousand Oaks, CA: Corwin Mathematics; Reston, VA: National Council of Teachers of Mathematics, 2020. Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. New York: Pearson, 2004. Van de Walle, John A., Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3–5, 3rd ed. New York: Pearson, 2018. Zwiers, Jeff, 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. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford.edu/resource/principles-design -mathematics-curricula, 2017.

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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 module. California 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 and page 154, Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musée des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/Art Resource, NY; pages 91, 306, choi hyekyung/Shutterstock.com; page 248, Atsushi Hirao/Shutterstock.com; All other images are the property of Great Minds.

All United States currency images Courtesy the United States Mint and the National Numismatic Collection, National Museum of American History.

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Acknowledgments Kelly Alsup, Adam Baker, Agnes P. Bannigan, Beth Barnes, Christine Bell, Reshma P Bell, Erik Brandon, Joseph T. Brennan, Dawn Burns, Amanda H. Carter, Stella Chen, David Choukalas, Mary Christensen-Cooper, Nicole Conforti, Cheri DeBusk, Lauren DelFavero, Jill Diniz, Mary Drayer, Karen Eckberg, Melissa Elias, Danielle A Esposito, Janice Fan, Scott Farrar, Krysta Gibbs, January Gordon, Lisa Gradney, Torrie K. Guzzetta, Kimberly Hager, Karen Hall, Eddie Hampton, Andrea Hart, Stefanie Hassan, Tiffany Hill, Christine Hopkinson, Rachel Hylton, Travis Jones, Laura Khalil, Raena King, Jennifer Koepp Neeley, Emily Koesters, Liz Krisher, Leticia Lemus, Marie Libassi-Behr, Courtney Lowe, Sonia Mabry, Bobbe Maier, Ben McCarty, Maureen McNamara Jones, Pat Mohr, Bruce Myers, Marya Myers, Kati O’Neill, Darion Pack, Geoff Patterson, Victoria Peacock, Maximilian Peiler-Burrows, Brian Petras, April Picard, Marlene Pineda, DesLey V. Plaisance, Lora Podgorny, Janae Pritchett, Elizabeth Re, John Reynolds, Meri Robie-Craven, Deborah Schluben, Michael Short, Erika Silva, Jessica Sims, Tara Stewart, Heidi Strate, Theresa Streeter, James Tanton, Cathy Terwilliger, Saffron VanGalder, Rafael Vélez, Janel Verrilli, Jessica Vialva, Rachael Waltke, Allison Witcraft, Jackie Wolford, Jim Wright, Caroline Yang, Jill Zintsmaster

Ana Alvarez, Lynne Askin-Roush, Stephanie Bandrowsky, Mariel Bard, Rebeca Barroso, Brianna Bemel, Rebecca Blaho, Charles Blake, Carolyn Buck, Lisa Buckley, Shanice Burton, Adam Cardais, Cindy Carlone, Gina Castillo, Ming Chan, Tatyana Chapin, Christina Cooper, Kim Cotter, Gary Crespo, Lisa Crowe, David Cummings, Brandon Dawley, Cherry dela Victoria, Timothy Delaney, Delsena Draper, Erin DuRant, Sandy Engelman, Tamara Estrada, Ubaldo Feliciano-Hernández, Soudea Forbes, Liz Gabbard, Diana Ghazzawi, Lisa Giddens-White, Laurie Gonsoulin, Adam Green, Sagal Hassan, Kristen Hayes, Tim Heppner, Marcela Hernandez, Sary Hernandez, Abbi Hoerst, Elizabeth Jacobsen, Ashley Kelley, Sonia Khaleel, Lisa King, Sarah Kopec, Drew Krepp, Jenny Loomis, Stephanie Maldonado, Christina Martire, Siena Mazero, Thomas McNeely, Cindy Medici, Ivonne Mercado, Sandra Mercado, Brian Methe, Sara Miller, Mary-Lise Nazaire, Corinne Newbegin, Tara O’Hare, Max Oosterbaan, Tamara Otto, Christine Palmtag, Laura Parker, Toy Parrish, Katie Prince, Neha Priya, Jeff Robinson, Nate Robinson, Gilbert Rodriguez, Todd Rogers, Karen Rollhauser, Neela Roy, Gina Schenck, Aaron Shields, Madhu Singh, Leigh Sterten, Mary Sudul, Lisa Sweeney, Tracy Vigliotti, Bruce Vogel, Charmaine Whitman, Glenda Wisenburn-Burke, Samantha Wofford, Howard Yaffe, Dani Zamora

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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 of clever illustrations, compelling videos, and digital interactives to spark discourse and wonder in your classroom. Accessible2 Created with all readers in mind, Eureka Math2 has been carefully designed to ensure struggling readers can access lessons, word problems, and more. Joy2 Together with your students, you will fall in love with math all over again—or for the first time—with Eureka Math2.

ISBN 979-8-88811-696-8

798888 116968

Module 1 Place Value Concepts for Multiplication and Division with Whole Numbers Module 2 Addition and Subtraction with Fractions Module 3 Multiplication and Division with Fractions Module 4 Place Value Concepts for Decimal Operations Module 5 Addition and Multiplication with Area and Volume Module 6 Foundations to Geometry in the Coordinate Plane