EM2 TEACH Level 4 Module 1 Lesson 1 Thin Slice

4 A Story of Units® Fractional Units

TEACH ▸ Module 1 ▸ Place Value Concepts for Addition and Subtraction

What

does

this painting have to do with math?

American abstract painter Frank Stella used a compass to make brightly colored curved shapes in this painting. Each square in this grid includes an arc that is part of a design of semicircles that look like rainbows. When Stella placed these rainbow patterns together, they formed circles. What fraction of a circle is shown in each square?

On the cover

Tahkt-I-Sulayman Variation II, 1969

Frank Stella, American, born 1936

Acrylic on canvas

Minneapolis Institute of Art, Minneapolis, MN, USA

Frank Stella (b. 1936), Tahkt-I-Sulayman Variation II, 1969, acrylic on canvas. Minneapolis Institute of Art, MN. Gift of Bruce B. Dayton/Bridgeman Images. © 2020 Frank Stella/Artists Rights Society (ARS), New York

Great Minds® is the creator of Eureka Math® , Wit & Wisdom® , Alexandria Plan™, and PhD Science®

Published by Great Minds PBC. greatminds.org

© 2021 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 B-Print

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ISBN 978-1-64497-173-4

Module

1 Place Value Concepts for Addition and Subtraction

2 Place Value Concepts for Multiplication and Division

3 Multiplication and Division of Multi-Digit Numbers

4 Foundations for Fraction Operations

5 Place Value Concepts for Decimal Fractions

6 Angle Measurements and Plane Figures

Before This Module

Overview

Grade 3 Module 1

In grade 3 module 1, students build a conceptual understanding of multiplication as a number of equal groups (e.g., 4 × 3 = 12 can be interpreted as 4 groups of 3 is 12).

Grade 3 Module 2

In grade 3 module 2, students compose and decompose metric measurement units and relate them to place value units up to 1 thousand. They use place value understanding and the vertical number line to round two- and three-digit numbers. Grade 3 students also add and subtract two- and three-digit numbers by using a variety of strategies, including the standard algorithm.

Place Value Concepts for Addition and Subtraction

Topic A

Multiplication as Multiplicative Comparison

Students identify, represent, and interpret multiplicative comparisons in patterns, tape diagrams, multiplication equations, measurements, and units of money. They describe the relationship between quantities as times as much as or use other language as applicable to a given context (e.g., times as many as, times as long as, and times as heavy as). Students use multiplication or division to find an unknown quantity in a comparison.

Topic B

Place Value and Comparison Within 1,000,000

Students name the place value units of ten thousand, hundred thousand, and million. They recognize the multiplicative relationship between place value units—the value of a digit in one place is ten times as much as the value of the same digit in the place to its right. Students write and compare numbers with up to 6 digits in standard, expanded, word, and unit forms.

56,348

50,000 + 6,000 + 300 + 40 + 8

fifty-six thousand, three hundred forty-eight

56 thousands 3 hundreds 4 tens 8 ones

Topic C Rounding Multi-Digit Whole Numbers

Students name multi-digit numbers in unit form in different ways by using smaller units (e.g., 245,000 as 24 ten thousands 5 thousands or 245 thousands), and they find 1 more or 1 less of a given unit in preparation for rounding on a vertical number line. Students round four-digit, five-digit, and six-digit numbers to the nearest thousand, ten thousand, and hundred thousand. They determine an appropriate rounding strategy to make useful estimates for a given context.

Topic D

Multi-Digit Whole Number Addition and Subtraction

Students build fluency with addition and subtraction of numbers of up to 6 digits by using the standard algorithm. They add and subtract to solve two-step and multi-step word problems. The Read–Draw–Write process is used to help students make sense of the problem and find a solution path. Throughout the topic, students round to estimate the sum or difference and check the reasonableness of their answers.

Topic E

Metric Measurement Conversion Tables

Students use multiplicative comparisons to describe the relative sizes of metric units of length (kilometers, meters, centimeters), mass (kilograms, grams), and liquid volume (liters, milliliters). They express larger units in terms of smaller units and complete conversion tables. Students add and subtract mixed unit measurements.

After This Module

Grade 5 Modules 1 and 4

In grade 5 modules 1 and 4, students extend the work of grade 4 by adding, subtracting, rounding, and comparing multi-digit numbers with digits to the thousandths place. Students recognize that the value of a digit in one place is 1 10 of what it represents in the place to its left.

Solve multiplicative comparison problems with unknowns in various positions.

Describe relationships between measurements by using multiplicative comparison.

Represent the composition of larger units of money by using multiplicative comparison.

Write

Organize, count, and represent a collection of objects.

Demonstrate that a digit represents 10 times the value of what it represents in the place to its right.

Rounding Multi-Digit Whole Numbers

Multi-Digit Whole Number Addition and Subtraction

Add by using the standard algorithm.

Solve multi-step addition word problems by using the standard algorithm.

Subtract by using the standard algorithm, decomposing larger units once.

Subtract by using the standard algorithm, decomposing larger units up to 3 times.

Subtract by using the standard algorithm, decomposing larger units multiple times.

Solve two-step word problems by using addition and subtraction.

Solve multi-step word problems by using addition and subtraction.

Why Place Value Concepts for Addition and Subtraction

Why does the place value module begin with a topic on multiplicative comparisons?

Beginning with multiplicative comparison enables students to build on their prior knowledge of multiplication from grade 3 and provides a foundation upon which students can explore the relationships between numbers and place value units. This placement also activates grade 3 knowledge of multiplication and division facts within 100 and provides students with opportunities to continue building fluency with the facts in preparation for multiplication and division in modules 2 and 3.

Students are familiar with additive comparison—relating numbers in terms of how many more or how many less. Multiplicative comparison—relating numbers as times as many—is a new way to compare numbers. Students use multiplicative comparison throughout the year to relate measurement units, whole numbers, and fractions. This important relationship between factors, where one factor tells how much larger the product is compared to the other factor, is foundational to ratios and proportional relationships in later grades. Taking time to develop this understanding across the grade 4 modules sets students up for success with interpreting multiplication as scaling in grade 5 and applying or finding a scale factor in scale drawings, dilations, and similar figures.

Figure L

Figure M

Figure N

Figure O

Figure A

Figure B

Why is the vertical number line used for rounding numbers?

The vertical number line is used to help support conceptual understanding of rounding. In grade 3, students first see the vertical number line as an extension of reading a vertical measurement scale. Using the context of temperature, students identify the tens (i.e., benchmarks) between which a temperature falls, the halfway mark between the benchmark temperatures, and the benchmark temperature the actual temperature is closer to. Students then generalize to round numbers to the nearest ten and hundred.

In grade 4, students round numbers with up to 6 digits to any place. They continue to use the vertical number line as a supportive model. Labeling the benchmark numbers and halfway tick mark in both standard form and unit form helps emphasize the unit to which a number is being rounded. This way, the place values line up vertically, helping students see the relationship between the numbers.

The pictorial support of the vertical number line when rounding is eventually removed, but the conceptual understanding of place value remains as students round mentally. These experiences with the vertical number line prepare students for representing ratios with vertical double number lines and graphing pairs of values in the coordinate plane.

Why are metric units of measurement addressed in this module? When are customary units of measurement addressed?

Work with metric units of length, mass, and liquid volume in topic E provides an opportunity for students to apply their place value understanding to a measurement context. Students convert metric units that have relationships involving hundreds (e.g., meters to centimeters) and thousands (e.g., kilograms to grams). They apply multi-digit addition and subtraction strategies, including the standard algorithm, to add and subtract mixed-unit measurements. Introducing metric units in module 1 also provides the opportunity to use the units in word problem contexts throughout the rest of the year.

Customary units are included within modules 2 and 3 because the relative sizes of customary units of measurement do not align with the place value unit structure. Customary units of length are addressed in module 2 when students work with two-digit multiplication, area, and perimeter. Additionally, units of time and customary units of weight and liquid volume are addressed in module 3 alongside multiplication and problem solving.

Achievement Descriptors: Overview

Place Value Concepts for Addition and Subtraction

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 twelve ADs listed.

4.Mod1.AD1

Create two comparison statements, given a multiplication equation.

Write multiplicative comparison statements as multiplication equations.

Solve word problems involving multiplicative comparison by using multiplication or division within 100.

Assess reasonableness of estimates when using rounding as an estimation strategy.

4.Mod1.AD5

Solve multi-step word problems by using addition and subtraction, represent these problems by using equations, and assess the reasonableness of the answers.

4.Mod1.AD6

Explain the relationship between a digit in a multi-digit whole number and the same digit in the place to the right.

Read and write multi-digit whole numbers in unit, standard, word, and expanded form.

Compare two whole numbers by using >, =, or <.

Round multi-digit whole numbers.

Add and subtract multi-digit whole numbers by using the standard algorithm.

Express larger units in terms of a smaller unit within the metric system in a table.

Solve addition and subtraction word problems that require expressing measurements of larger units in terms of given smaller units.

4.Mod1.AD7

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 4 module 1 is coded as 4.Mod1.AD1.

• AD Language: The language is crafted from standards and concisely describes what will be assessed.

• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.

• Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.

Achievement Descriptors: Proficiency Indicators

AD Code: Grade.Module.AD#

AD Language

4.Mod1.AD1 Create two comparison statements, given a multiplication equation.

RELATED CCSSM

4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

Partially Proficient Proficient Highly Proficient Create a comparison statement, given a multiplication equation.

Fill in the blanks to complete a statement that represents the equation 35 = 5 × 7 is times as much as

Create two comparison statements, given a multiplication equation.

Fill in the blanks to complete two statements that represent the equation 35 = 5 × 7 is times as much as is times as much as

Related Standard

AD Indicators

Topic A Multiplication as Multiplicative Comparison

In topic A, students use models and multiplicative comparison language to represent multiplicative relationships.

In prior grades, students compare quantities through additive comparison where one quantity is more than or less than another quantity and use addition or subtraction to find the unknown quantity or difference. Multiplicative comparison presents a new way to relate quantities. Students recognize that a figure in a number or shape pattern does not increase by the same amount each time. Rather, the increase is a result of multiplying by the same factor each time. Students use visual models, including tape diagrams, as tools to demonstrate the multiplicative relationship between quantities. Once they understand the multiplicative relationship, students find an unknown quantity by using multiplication or division.

Multiplicative comparison gives students another way to interpret multiplication. For example, they see 15 = 3 × 5 as 15 is 3 times as many as 5. This interpretation of multiplication is foundational throughout grade 4 as students describe place value relationships, identify multiples of whole numbers and fractions, and convert measurement units. It also prepares students for multiplication as scaling in grade 5.

Students apply multiplicative comparison to the contexts of measurements and units of money. They interpret and represent a variety of measurement comparisons and use language specific to the contexts such as times as long as, times as heavy as, and times as far as. Students also compare units of money—pennies, dimes, and dollars—by using times as much as relationships. Students recognize similarities between place value units and the relationship of increasing units of money (i.e., each increasing unit from pennies to dimes to dollars is 10 times as much as the previous unit).

In topic B, students use multiplicative comparisons to relate place value units up to 1,000,000.

Progression of Lessons

Lesson 1

Interpret multiplication as multiplicative comparison.

Lesson 2

Solve multiplicative comparison problems with unknowns in various positions. ?

Lesson 3

Describe relationships between measurements by using multiplicative comparison.

I notice that some shape and number patterns have rules that use addition and some have rules that use multiplication. I can describe the multiplication patterns by using times as many. For example, I can read a multiplication equation such as 16 = 2 × 8 as 16 is 2 times as many as 8.

I can use the relationship between multiplication and division to help me solve a multiplicative comparison problem where the factor or product is unknown. I can draw a tape diagram to identify the known and unknown information and then use multiplication or division to find the unknown.

I can use language such as times as long as and times as heavy as to describe how measurements are related. Tape diagrams and pictures of objects with measurement tools such as scales, rulers, and beakers can show the relationships between different measurements. I can describe the relationships by using words and equations.

Amy’s Tower Gabe’s Tower

Lesson 4

Represent the composition of larger units of money by using multiplicative comparison.

dollars dimes pennies

The relationship between pennies, dimes, and dollars is like the relationship between place value units. I can use multiplicative comparison to relate units of money. For example, 1 dime is worth 10 times as much as 1 penny.

Interpret multiplication as multiplicative comparison.

Lesson at a Glance

Students create a pattern by using a multiplication rule. They learn how to use the language of times as many to describe the relationship between the number of objects in consecutive figures in a multiplicative pattern. Students also write multiplication equations and draw tape diagrams to represent multiplicative comparison situations.

Key Questions

• How can you describe a multiplication relationship between numbers?

• How can multiplication equations and tape diagrams represent times as many situations?

Achievement Descriptors

4.Mod1.AD1 Create two comparison statements, given a multiplication equation. (4.OA.A.1)

4.Mod1.AD2 Write multiplicative comparison statements as multiplication equations. (4.OA.A.1)

4.Mod1.AD3 Solve word problems involving multiplicative comparison by using multiplication or division within 100. (4.OA.A.2)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Multiplication and Times as Many

• Multiplicative Comparison and Tape Diagrams

• Multiplicative Comparison Match

• Problem Set

Land 10 min

Teacher

• Sticky notes (5)

• Multiplicative Comparison Match Cards (in the teacher edition)

• Computer or device*

• Projection device*

• Teach book*

Students

• Sticky notes (5 per student pair)

• 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.

Lesson Preparation

Print or copy and cut out Multiplicative Comparison Match Cards.

Fluency

Choral Response: Multiply and Divide Whole Numbers

Students find a product or quotient to prepare for multiplicative patterns and multiplicative comparisons.

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 multiplication equation and question.

What is 2 groups of 5?

10

Display the product.

Repeat the process with the following sequence:

= 2 × 5 10

What is 2 groups of 5?

Teacher Note

Use hand signals to introduce a procedure for answering choral response questions. For example, cup your hand around your ear for listen, lift your finger to your temple for think, and raise your own hand to remind students to raise theirs.

Teach the procedure by using general knowledge questions.

• What grade are you in?

• What is the name of our school?

• What is your teacher’s name?

Teacher Note

Display the division equation and question.

6 is 2 groups of what?

3

Display the quotient.

Repeat the process with the following sequence:

6 is 2 groups of what?

Establish a signal (e.g., show me your boards) to introduce a procedure for showing whiteboard exchange responses.

Practice with basic computations until students are accustomed to the procedure.

• What is 2 × 3?

• What is 10 ÷ 5?

Establish a procedure for providing feedback on whiteboard exchanges. Consider circulating to give hand signals—thumbs-up or try again.

Whiteboard Exchange: Interpreting Tape Diagrams

Students write and complete an equation to represent a tape diagram to prepare for similar work with multiplicative comparisons.

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.

What does the tape diagram show? Tell your partner.

Provide time for students to think and share with their partners. The total is unknown. There are 2 equal parts. Each part has a value of 6.

Write and complete a multiplication equation to represent the tape diagram.

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

Teacher Note

Validate all correct responses that may not be displayed on the image. For example, students may write 12 = 2 × 6 as a correct equation to represent the tape diagram.

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 of students to form small groups of four.

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

Launch

Students examine and describe additive and multiplicative shape patterns. Direct students to problems 1 and 2 in their books and chorally read the directions. Write a rule for each pattern.

Teacher Note

In grade 3, students use the term pattern to describe the relationship between numbers in input–output tables.

Divide the input by 8

3

In this lesson, pattern refers to a collection of figures that follow a rule. A rule describes the relationship between consecutive figures in the pattern.

1.

Figure A

Figure B

Figure C

Figure D

Rule: Add

Invite students to turn and talk about the shape pattern and try to identify how each figure is different from the one before it.

What is the relationship between the number of squares in figure B and the number of squares in figure A?

Figure B has 3 more squares than figure A. There are twice as many squares in figure B than in figure A.

Is that same relationship true for the number of squares in figure C compared to figure B? Figure D compared to figure C? How do you know?

Yes, there are 3 more squares in figure C than in figure B, and there are 3 more squares in figure D than in figure C.

No, there are not twice as many squares in figure C compared to figure B or figure D compared to figure C.

Which relationship tells you how each figure in the shape pattern changes in the same way?

Each figure has 3 more squares than the figure before it.

3 squares are added each time.

Add a column of 3 squares

The rule for the pattern is add 3. If you know the rule, you can make more figures in the shape pattern. The rule tells how you can make the next figure.

Direct students to write the rule for the pattern in problem 1 in their books.

Invite students to turn and talk about how knowing the rule can help them figure out how many squares would be in figure E.

If the pattern continues, how many squares would be in figure E? How do you know? There would be 15 squares in figure E. I followed the rule. I added 3 more to the number of squares in figure D.

UDL: Representation

Consider annotating the figures to emphasize the patterns as students describe how each figure is different from the one before it.

Direct students to the pattern in problem 2. 2.

Invite students to turn and talk about the shape pattern and try to identify how each figure is different from the one before it.

What is the relationship between the number of circles in figure M and the number of circles in figure L?

Figure M has 2 more circles than figure L.

It’s a double. Figure M has double the number of circles as figure L.

Is that same relationship true for the number of circles in figure N compared to figure M? How do you know?

No, it’s not true because figure N has 4 more circles than figure M, not 2 more.

Yes, there are double the number of circles in figure N than figure M, so 2 × 4 = 8.

The relationship is different from the relationship in problem 1, so the rule can’t be add 2.

Invite students to think–pair–share about the rule.

It’s multiply by 2, so 2 × 2 = 4.

It’s double, so 2 × 2 = 4.

Language Support

The term figure has multiple meanings in mathematics and everyday life. In this lesson, figure refers to a collection of objects or shapes in a pattern. Consider using pictures to highlight some different meanings of figure.

• Figures A, B, and C are part of a shape pattern.

• A cube and a triangle are examples of geometric figures.

• Gabe figures out the answer to the problem.

Figure M

Figure L

Figure N Figure O Rule: Multiply by 2

Figure A Figure B Figure C

Is the same relationship true for the number of circles in figure O compared to figure N? How do you know?

Yes. 2 × 8 = 16.

What is the rule for this shape pattern?

Multiply by 2

Direct students to write the rule for the pattern in problem 2.

Invite students to turn and talk about how knowing the rule can help them figure out how many circles would be in the next figure.

If the pattern continues, how many circles would be in the next figure? How do you know?

There would be 32 circles in the next figure because I multiply the number of circles in figure O by 2.

2 × 16 = 32. I can just think about it as 16 + 16 = 32.

Transition to the next segment by framing the work.

Today, we will learn how to describe a relationship between numbers by using multiplication.

Learn

Multiplication and Times as Many

Materials—T/S: Sticky notes

Students use sticky notes and multiplicative patterns to relate multiplication to times as many.

Pair students and give each pair five sticky notes.

Teacher Note

In this lesson, the multiplication equation is consistently written with the total first, followed by the equal sign and the multiplicative comparison expression. The first factor indicates how many of the second factor is represented in the relationship. This order allows students to see the connection between the times as many language and the multiplication equation.

3 = 3 × 1

3 is 3 times as many as 1.

Direct partner A to place a sticky note on their whiteboard. Direct partner B to place 2 sticky notes below partner A’s sticky note.

How many sticky notes does partner A have? Partner B?

What multiplication equation represents the relationship between partner B’s and partner A’s sticky notes?

2 = 2 × 1

Write the equation 2 = 2 × 1. Point to each number as you ask the following questions.

What does the product, 2, represent?

It’s the number of sticky notes partner B has.

What does the first factor, 2, represent?

It’s the number we multiplied partner A’s sticky notes by to get the number of sticky notes partner B has.

What does the second factor, 1, represent?

It’s the number of sticky notes partner A has.

We can say that partner B has 2 times as many sticky notes as partner A. 2 is 2 times as many as 1. Write the statements below the equation. Invite students to turn and talk about how the equation and statements represent the relationship between partner B’s and partner A’s sticky notes.

Use a similar process for partner B to represent 3 and 4 times as many sticky notes as partner A.

Direct students to their books and chorally read the directions for problem 3. Invite students to complete problem 3 with their partners by recording how they represented 4 times as many.

Draw sticky notes to represent 4 times as many. Then fill in the blanks.

3. Partner A

Partner A

Partner A

Partner B Partner B

Partner B

4 = 4 × 1

Partner B has 4 times as many sticky notes as partner A.

4 is 4 times as many as 1 .

Invite students to think–pair–share about why we can use times as many to describe the relationship between the number of sticky notes partners A and B have.

When we read the multiplication equation, we say 4 times. Using the word times is another way to show multiplication.

To get the number of sticky notes partner B has, we multiply the number of sticky notes partner A has by 4. A word we use when we multiply is times, so we can use 4 times to show how they are related.

UDL: Representation

Consider using different-colored highlighters in the equation and statements to help students make connections between the various representations.

Partner A

Partner B ×

Direct partners to write the number 4 on each sticky note.

What is the total value of the numbers on partner A’s sticky note? Partner B’s?

What multiplication equation represents the relationship between the total value of the numbers on partner B’s sticky notes compared to the value of the number on partner A’s sticky note?

16 = 4 × 4

Write the equation.

How did changing the value of each sticky note to 4 change the equation 4 = 4 × 1?

It’s still 4 times as many, but we started with 4 instead of 1 and the total is 16 instead of 4.

Direct students to problems 4–6. Invite students to complete problems 4–6 with their partners.

Use the pictures to fill in the blanks.

4. Partner A Partner B

4 4444

16 = 4 × 4

16 is 4 times as many as 4 .

5. Partner A

Partner B

7 7777

28 = 4 × 7

28 is 4 times as many as 7 .

6. Partner A

Partner B

9 9999

36 = 4 × 9

36 is 4 times as many as 9 .

What do you notice about the relationship between the total value of partner B’s sticky notes and partner A’s sticky notes in problems 3 through 6?

The relationship is the same in all the problems.

The relationship is 4 times as many. The total value of the numbers on partner B’s sticky notes is always 4 times as many as the value of partner A’s sticky note.

Invite students to turn and talk about how multiplication and times as many are related.

Multiplicative Comparison and Tape Diagrams

Students draw and interpret tape diagrams that represent multiplicative comparisons.

Direct students to problem 7 and chorally read the directions.

7. Draw a tape diagram to represent 36 is 4 times as many as 9. Then complete the equation.

Teacher Note

In grade 3, students use a variation of brackets when drawing tape diagrams. This variation enables students to label the tape without the added complexity of drawing the brackets. In grade 4, students see tape diagrams labeled with brackets but continue to draw arms. Students may transition to drawing brackets as they are ready.

Students might draw the following tape diagrams to represent the multiplicative relationships.

How can we use the picture in problem 6 to help us draw a tape diagram to represent 36 is 4 times as many as 9?

The picture shows that 36 is 4 times as many as 9.

We can use the value of partner A’s and partner B’s sticky notes to help us draw a tape diagram.

What can we draw to represent partner A’s sticky note?

We can draw 1 unit of 9.

Model drawing and labeling a tape with 1 unit of 9. Label the tape with the letter A. Direct students to do the same.

What can we draw to represent partner B’s sticky notes?

We can draw 4 units of 9.

Model drawing a tape with 4 units of 9 and label it with the letter B. Direct students to do the same.

What is the total of 4 units of 9?

Model labeling 36 as the total for B. Direct students to do the same.

How can you tell that each unit in B has a value of 9 even though I didn’t label each unit?

We know that A has a value of 9 and each unit in B is the same size as the unit in A, so each unit in B has the same value as the unit in A.

Invite students to think–pair–share about how the tape diagram shows that 36 is 4 times as many as 9.

There is 1 unit of 9 in A, and B shows that 4 units of 9 make 36.

There are 4 times as many units of 9 in B than in A, and the total for B is 36.

Invite students to complete the equation in their books.

Direct students to problems 8–10. Chorally read the directions.

Use the tape diagram to fill in the blanks. Then complete the equation and statement. 8.

Promoting the Standards for Mathematical Practice

Students look for and make use of structure (MP7) when they see the tape diagram as being composed of units and relate this to a multiplication equation and multiplicative comparison situation.

Ask the following questions to promote MP7:

• How can what you know about the tape diagram help you write a multiplication equation using 30 and 6?

is 5 times as many as 6 .

• How are the tape diagram and the times as many statement related? How can that help you write a multiplication equation?

What do you notice about the tape diagram in problem 8?

There is 1 unit of 6 and 5 units of 6.

The total for the 5 units of 6 is 30.

What multiplication equation represents the relationship between 30 and 6? 30 = 5 × 6

How can you use times as many to describe the relationship between 6 and 30? 30 is 5 times as many as 6. Invite students to complete problem 8. Use a similar process to complete problems 9 and 10. 9.

8

.

42

42 = 6 × 7

42 is 6 times as many as 7 .

Invite students to turn and talk about how they can use tape diagrams and multiplication equations to represent times as many relationships.

Multiplicative Comparison Match

Materials—T: Multiplicative Comparison Match Cards

Students match various representations of multiplicative comparison situations.

Distribute one Multiplicative Comparison Match card to each student.

Direct students to move around the room to find other students with cards that match the multiplicative situation on their card. Students should look for a tape diagram, a statement that uses the phrase times as many, and a multiplication equation that all represent the same multiplicative comparison.

Example matching cards: 28

Teacher Note

4

28 is 7 times as many as 4. 28 = 7 × 4

Students should form groups of three once they have found their matches. Use the following prompts to engage groups in a discussion about their matching cards:

• How do you know each card represents the same situation?

• How are the representations on each card similar?

• How are the representations on each card different?

As time allows, shuffle the cards, redistribute them, and invite students to repeat the activity with a new card.

Problem Set

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

The 24 Multiplicative Comparison Match cards are designed to result in eight groups with three matches in each group. Each group should contain a tape diagram, a times as many statement, and a multiplication equation. If you have more or fewer than 24 students in your class, consider using one of the following modifications to ensure that all students have at least one match.

• Reduce the number of cards used by removing a tape diagram, times as many statement, or equation from sets of matching cards.

• Reduce the number of cards used by removing entire matching sets.

• Give some students more than one card but be sure the cards belong to a matching set.

• Create a shape pattern card for each matching set to allow more than 24 students to participate. The following is an example shape pattern for 28 = 7 × 4:

Figure R

Figure S

Land

Debrief 5 min

Objective: Interpret multiplication as multiplicative comparison. Display the pattern that contains figures L through O and facilitate a discussion about using times as many to describe multiplication.

Earlier, we determined that the rule for this pattern was multiply by 2. How can you describe the relationship between the number of circles in figure M and the number of circles in figure L?

We can say that there are 2 times as many circles in figure M than in figure L.

How can you use a multiplication equation to represent the relationship between the number of circles in figure N compared to figure M?

8 = 2 × 4

How can you use a model to represent the relationship between the number of circles in figure N compared to figure M?

I could draw a tape diagram to show 1 unit of 4 to represent the number of circles in figure M. Then I could draw 2 units of 4 to represent the number of circles in figure N.

I could draw a tape diagram with 1 unit of 4 and 2 units of 4 to show that 8 is 2 times as many as 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.

Teacher Note

Students might describe the relationship between the number of circles in each figure in the pattern as twice as many instead of 2 times as many. Recognize that this is a valid way to describe the relationship, but facilitate a conversation about the convention of using times as many language.

Figure M

Figure L

Figure N

Figure O

Sample Solutions

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

1. Liz draws circles by using this rule: Multiply the number of circles by 2

a. How many circles should Liz draw for figure D? How do you know?

Liz should draw 40 circles because 2 × 20 = 40

b. Complete the statements and equation to match the figures.

There are 2 times as many circles in figure B than in figure A.

10 is 2 times as many as 5

10 = 2 × 5

There are 2 times as many circles in figure C than in figure B.

20 is 2 times as many as 10

20 = 2 × 10

Figure A Figure B Figure C Figure D

9. There are 9 tables in the cafeteria. There are 8 times as many chairs as tables. How many chairs are in the cafeteria?

There are 72 chairs in the cafeteria.