Multiplication | Math Catalyst by General

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Represent Contextual Multiplication

Multiplication

For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Concept Guide | Represent Contextual Multiplication Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide) • Counters

• Print copies of the Progress Check Tool and the Pause and Monitor Tool. • Gather the counters (20 per student).

• Concept Mini Lessons Teacher Guide • Counters (12)

• Objectives 1–3 Student Pages • Personal whiteboard • Container of 100 counters

• Print copies of the Objectives 1–3 Student Pages. • Gather the counters.

• Practice Teacher Guide

• Practice Pages • Practice Helpers • Counters (20 per student)

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Equal Groups Match Cards • Game Instruction Card • Counters (20 per student) • Study a Solution Student Page • Solve a Task Student Page • Solve a Problem Recording Page (optional) • Highlighters and other annotation tools (optional) • Read–Draw–Write Tool (optional)

• Ready the following materials: - Application Word Problem Cards (1 set per student or pair) - Game Instruction Card (1 per student or pair) - Equal Groups Match Cards (1 per group) • Print copies of the following: - Solve a Problem Recording Page (1 per student or pair; optional) - Study a Solution Student Page (1 per student) - Solve a Task Student Page (1 per student pair) - Read–Draw–Write Tool (optional) • Gather the following: - Personal whiteboard (1 per student or pair) - Highlighters and other annotation tools (optional) - Counters (20 per student)

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• Application Teacher Guide

Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Represent Contextual Multiplication

Addressing Student Misconceptions How to Address Misconception

Students think they can switch the number of groups and the number of objects in each group.

Have students use counters to model groups of different sizes and different numbers of equal groups.

For example, if students are shown 3 vases with 2 flowers in each, they might feel they can say 2 groups of 3 instead of 3 groups of 2.

Student Misconception

Ask students to model the 3 vases with 2 flowers.

• How many groups do you have? How do you know? • How many are in each group? How do you know? • How is this different than 2 groups of 3?

Language Support

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Emphasize that groups are collections of objects.

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping. • 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The chart should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Represent Contextual Multiplication Dear Family,

Your student is working on representing multiplication situations. Students do this by using concrete objects and by showing equal groups. You can support your student’s progress by asking the questions in the table below as your student represents a multiplication situation. Describe the equal groups. Complete each statement.

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4 groups of

How many groups are there? How many

How can you use repeated addition to

How does the repeated addition

are in each group? How can that help you

find the total?

equation match the equal groups?

The repeated addition equation is

In the repeated addition equation, the addends

3 + 3 + 3 + 3 = _____.

are the same number, 3. There are 3 in each

write a repeated addition equation? There are 4 groups of 3.

I know 3 + 3 is 6. 6 + 3 is 9. 9 + 3 is 12.

I can add 4 threes.

equal group. In the repeated addition equation, there are 4 addends. This means there are 4 equal groups.

Concept Guide | Teacher Guide

Progress Check | Represent Contextual Multiplication

Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

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• Can the student model multiplication situations by using concrete objects? | Objective 1

• Can the student represent and describe multiplication situations by using repeated addition? | Objective 2

• Can the student create stories to represent multiplication situations involving equal groups?

Teacher Tip

| Objective 3

Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress.

Progress Check | Teacher Guide

Progress Check | Represent Contextual Multiplication

Item 1

Item 2

Items 3 and 4

Objective 1

Objective 2

Objective 3

Not Yet Proficient

The student may show evidence of beginning to understand modeling multiplication situations by using concrete objects but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand using repeated addition to represent and describe multiplication situations but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand creating stories to represent multiplication situations involving equal groups but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly models the multiplication situation:

The student correctly represents and describes The student correctly creates stories to multiplication situations: represent situations involving equal groups:

1. 3 circles with 4 pears in each; 3

2. 5 + 5 + 5 + 5 = 20; 5, 20

3. 5; 3; 5, 3 4. 3; 2; 3, 2

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Progress Check Tool Item(s)

Progression Toward Proficiency Rubric

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Represent Contextual Multiplication

Use counters to model groups of 4 pears. Circle them. Complete the statement.

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There are

groups of 4 pears.

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Progress Check | student Page

NAME

DATE

Progress Check Tool | Represent Contextual Multiplication

Describe the equal groups. Complete the statement.

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4 groups of

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Progress Check | student Page

NAME

DATE

Progress Check Tool | Represent Contextual Multiplication

Create a story to represent equal groups. Complete each statement.

There are

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There are

bags. marbles in each bag.

groups of

There are

marbles.

fishbowls.

There are

fish in each fishbowl.

There are

groups of

fish.

This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Represent Contextual Multiplication Progression of Mini Lesson Objectives 2 Represent and describe multiplication

3 Create stories to represent multiplication

concrete objects.

situations by using repeated addition.

situations involving equal groups.

1 Model multiplication situations by using

2 + 2 + 2 + 2 + 2 = 10 5 groups of

Start here if students

There are

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• can model multiplication situations by using concrete objects, but • need support representing and describing multiplication situations by using repeated addition.

There are There are There are

beds.

cats on each bed. groups of

cats.

Start here if students • can represent and describe multiplication situations by using repeated addition, but • need support creating stories to represent multiplication situations involving equal groups.

groups of 2 fish.

Start here if students

• can apply doubles strategies to find a total and • can skip-count to find a total efficiently, but • need support modeling multiplication situations by using concrete objects.

Concept Mini Lessons | Teacher Guide

Objective 1 | Model multiplication situations by using concrete objects. 10 M I NU T ES

Show 8 counters organized into 4 groups of 2 on a personal whiteboard. H Are these groups equal or unequal? How do you know? They are equal. There are 2 counters in each group.

H Let’s pretend you have 12 marbles. Use all your marbles and organize them into equal groups to share.

Give students time to organize their counters. H How did you organize your marbles (counters)? I put 6 marbles into each group. I made 2 groups.

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With a dry erase marker, circle each of the 4 groups of 2 and say the following.

Materials • Personal whiteboard • Container of 100 counters • Objective 1 Student Page

Summary Students articulate how they know groups are equal, and they model equal groups situations with counters.

H There are 2 counters in each group. There are 4 groups of 2 counters. When each group has an equal amount, they are called equal groups.

Rearrange the counters to show 2 groups of 3 and 1 group of 2 counters.

H Are these groups equal or unequal? How do you know? Unequal; one group has 2 counters.

They’re unequal because the groups don’t have the same number of counters.

H For groups to be equal there needs to be the same number of counters in each group.

Have students take 12 counters from the container and place them onto their personal whiteboard.

I have 6 groups and 2 marbles in each group. I have 4 groups of 3 marbles.

H There can be more than one way to make equal groups. Try organizing your marbles (counters) in another way.

Give students time to try a different way. H Turn and tell your partner how many marbles (counters) are in each group. Language Support Provide students with a sentence frame to support them in describing the situation. Begin with the first sentence frame and progress to the second frame. • There are

counters in each group.

• There are

groups of

counters.

Distribute the Objective 1 Student Page and direct students to problem 1. Read the directions aloud. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 1 | Model multiplication situations by using concrete objects. 10 M I NU T ES

H How many fish are there?

Invite students to turn and talk about how they know groups are equal.

10 fish

H I can ask myself, Do groups of 2 tell me that there are 2 groups of fish or that there are 2 fish in each group? What do you think? It tells you there are 2 fish in each group.

H Let’s use counters to model groups of 2 fish.

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Give students time to make groups of 2 on their personal whiteboard and to circle each group.

• Model groups of 5 blueberries. Circle them. • Model groups of 6 cars. Circle them. • Model groups of 4 plums. Circle them.

Teacher Tip

It is common for students to confuse the number of groups with the number in each group. For example, students may hear “groups of 2 fish” and make 2 groups of fish. Consider creating an anchor chart with a colorcoded illustration to support students with understanding the difference between these phrases.

Analyze Student Progress

Monitor: • Can the student articulate the difference between equal and unequal groups? • Can the student use counters to model equal groups? • Can the student tell how many groups? How many in each group?

Number of groups: There are 5 groups.

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Number in each group: There are 2 in each group.

Questions to Advance Student Thinking: • How do you know these groups are equal? • How can you use counters to show equal groups? • How many groups do you have? How many are in each group?

H Now, circle groups of 2 fish. H Complete the statement: There are ____ groups of 2 fish. There are 5 groups of 2 fish.

Concept Mini Lessons | Teacher Guide

Objective 2 | Represent and describe multiplication situations by using repeated addition. 10 M I NU T ES

Materials • Container of 60 counters • Personal whiteboard • Objective 2 Student Page

Show 3 groups with 4 counters in each group on a personal whiteboard. Have students do the same.

Point to each 4 from left to right.

H Are the groups equal? How do you know?

Yes. There are the same number of counters in each group.

Add them all up

H How many are in this group?

H 8 + 4 is? 12

H We just wrote a repeated addition equation. Repeated addition is when the same addend is added over and over again. Here we added 4 over and over again. H How many fours are there?

Write 4 below the first group, followed by a plus sign. Point to the next group and repeat the process for the remaining groups.

4 + 4 + 4 = 12

H So what equation represents the equal groups? 4 + 4 + 4 = ____

H What do you notice about the addends? They are the same number.

Write 12 to complete the equation.

Draw a line below each group. Point to the counters in the first group.

H 4 + 4 is?

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H How can we find the total number of counters?

Summary Students describe equal groups situations and write repeated addition equations to represent equal groups abstractly.

3 fours

H Yes, we have 3 equal groups. Complete the statement: 3 groups of ____ is ____. 3 groups of 4 is 12.

Write the statement 3 groups of 4 is 12. Distribute the Objective 2 Student Page and direct students to problem 1. Read the directions aloud.

Concept Mini Lessons | Teacher Guide

Objective 2 | Represent and describe multiplication situations by using repeated addition. 10 M I NU T ES

H I can ask myself, What repeated addition equation represents the equal groups? What do you think?

H What’s the total? How do you know? 10. I counted by twos: 2, 4, 6, 8, 10.

10. I know 2 + 2 is 4. 4 + 2 is 6. 6 + 2 is 8. 8 + 2 is 10.

2 + 2 + 2 + 2 + 2 = ____

Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Analyze Student Progress

Teacher Tip

Consider showing students that they can group the addends to show a more efficient way to add (i.e., using doubles).

+ 2

H Complete the statement: 5 groups of ____ is ____. 5 groups of 2 is 10.

Monitor: • Can the student write a repeated addition equation to represent the equal groups? • Can the student find the total by using repeated addition? • Can the student describe how the repeated addition equation matches the equal groups?

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2 + 2 + 2 + 2 + 2 = 10 4

• Describe the 4 equal groups of 3 tomatoes. • Describe the 4 equal groups of 5 bagels. • Describe the 5 equal groups of 4 bananas.

Invite students to turn and talk about how the repeated addition equation matches the equal groups.

Revoice key learning from the problem with a description such as the following.

Questions to Advance Student Thinking: • How many groups of ___ are there? How can that help you write a repeated addition equation? • How can you use repeated addition to find the total? • How does the repeated addition equation match the equal groups? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

H The 2 tells how many are in each group. There are 5 groups of 2, so we add 5 twos. The groups are equal, so instead of counting one by one, we can use repeated addition.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 3 | Create stories to represent multiplication situations involving equal groups. 10 M I NU T ES

Materials • Objective 3 Student Page • Personal whiteboard

Summary Students create contextual multiplication situations and record repeated addition equations to represent equal groups.

Distribute personal whiteboards and the Objective 3 Student Page. Direct students to problem 1.

I notice there are some beds. The beds are the groups. What is on the beds? Cats

Let’s complete these statements together. There are ____ beds.

Teacher Tip: Differentiation

H How many groups of 2 are there? Complete the statement: There are ____ groups of ____. There are 4 groups of 2.

Language Support

Students may misrepresent and incorrectly describe equivalent sets of objects. For example, students may say, “2 groups of 4” instead of “4 groups of 2.” Advance understanding by chunking the phrase and using repetition, such as the following: • How many groups are there? • We have 4 groups.

There are 4 beds.

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If students would benefit from concrete support, provide counters and small plates or cups to represent equal groups.

• How many in each group? • 4 groups of …?

H I can ask myself, Can I create a story to match 4 groups of 2? What do you think?

There are ____ cats on each bed. There are 2 cats on each bed.

There are There are There are

beds.

cats on each bed. groups of

cats.

There are ____ groups of ____ cats. There are 4 groups of 2 cats.

Have students turn and retell the story to a partner. Then have them complete problem 1. H What repeated addition equation can we write to represent the equal groups in the story? Write it on your whiteboard. 2+2+2+2=8

H How many cats are there in all? 8 cats

Yes! For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 3 | Create stories to represent multiplication situations involving equal groups. 10 M I NU T ES

• Create a story to represent 3 equal groups of 4 birds. • Create a story to represent 4 equal groups of 5 ducks.

H Complete the statement: There are ____ groups of ____. There are 5 groups of 3.

H We can create a story to match 5 groups of 3. H What will be our groups?

Analyze Student Progress

Monitor: • Can the student describe equal groups in a context? • Can the student tell how many groups? How many in each group? • Can the student write a repeated addition equation to represent the equal groups? Questions to Advance Student Thinking: • What are the groups in each picture? What is in each group? • How many groups of _____ are there? How many are in each group? • How does the repeated addition equation match the equal groups in your story?

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Nests

Direct students to problem 2.

H What is in each group, or nest? Eggs

Have students complete the statements together or with a partner. Then have students retell the story and write a repeated addition equation to match it.

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

15 eggs

H How many eggs are there altogether?

Invite students to turn and talk about what they need to create a story that matches a repeated addition equation. Listen for equal groups or the same number in each group. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Represent Contextual Multiplication Answer Key Objective 2

1. Models with counters; circles

1. 2 + 2 + 2 + 2 + 2 = 10; 2, 10

1. 4; 2; 4, 2

2. 3 + 3 + 3 + 3 = 12; 3, 12

2. 5; 3; 5, 3

3. 5 + 5 + 5 + 5 = 20; 5, 20

3. 3; 4; 3, 4

5 groups of 2 fish; 5

2. Models with counters; circles

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3 groups of 5 blueberries; 3

3. Models with counters; circles 2 groups of 6 cars; 2, 6

Objective 1

4. 4 + 4 + 4 + 4 + 4 = 20; 4, 20

Objective 3

4. 4; 5; 4, 5

4. Models with counters; circles

5 groups of 4 plums; 5, 4

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Represent Contextual Multiplication Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Represent Contextual Multiplication Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Model multiplication situations by using concrete objects.

Model groups of 2 fish. Circle them.

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Model equal groups with counters and circle them. Complete each statement.

There are

Model groups of 5 blueberries. Circle them.

groups of 2 fish.

There are

groups of 5 blueberries.

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Concept Mini Lessons | Student Page

NAME

DATE

Objective 1 | Model multiplication situations by using concrete objects.

Model groups of 6 cars. Circle them.

groups of

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There are

cars.

Model groups of 4 plums. Circle them.

Model equal groups with counters and circle them. Complete each statement.

There are

groups of

plums.

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Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Represent and describe multiplication situations by using repeated addition. 1

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Describe the equal groups. Complete each statement.

5 groups of

4 groups of

= .

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Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Represent and describe multiplication situations by using repeated addition.

Describe the equal groups. Complete each statement.

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4 groups of

+ 5 groups of

+ is

= .

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Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Create stories to represent multiplication situations involving equal groups.

Create a story to represent equal groups. Complete each statement.

There are

cats on each bed.

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There are

beds.

groups of

There are

There are There are

cats.

nests. eggs in each nest.

groups of

eggs.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Create stories to represent multiplication situations involving equal groups.

Create a story to represent equal groups. Complete each statement.

There are

birds on each branch.

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There are

branches.

There are

groups of

There are

There are There are

birds.

ponds. ducks in each pond.

groups of

ducks.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

Practice | Represent Contextual Multiplication Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers during Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Represent Contextual Multiplication Concept Mini Lessons Objectives 1–3. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Practice Page 2

Practice Page 3

Objective 1 Model multiplication

Objective 2 Represent and describe

Objective 3 Create stories to represent

situations by using concrete objects.

multiplication situations by using repeated addition.

multiplication situations involving equal groups.

Look for...

• Can the student write a repeated addition equation to represent the equal groups? • Can the student find the total by using repeated addition? • Can the student describe how the repeated addition equation matches the equal groups?

• Can the student describe equal groups in a context? • Can the student tell how many groups? How many in each group? • Can the student write a repeated addition equation to represent the equal groups?

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Practice Page 1

Look for...

• Can the student articulate the difference between equal and unequal groups? • Can the student use counters to model equal groups? • Can the student tell how many groups? How many in each group?

Practice | Teacher Guide

Practice | Represent Contextual Multiplication

Answer Key Practice Page 2

Practice Page 3

1. Models with counters; circles

1. 4 + 4 + 4 + 4 = 16; 4, 16

1. 2; 5; 2, 5

2. 6 + 6 = 12; 6, 12

2. 3; 6; 3, 6

3. 5 + 5 + 5 = 15; 5, 15

3. 4; 3; 4, 3

2 groups of 4 pinecones; 2

2. Kate makes 2 groups, not groups of 2 cherries. There are 4 groups of

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2 cherries.

3. Models with counters; circles 6 groups of 3 bananas; 6

Practice Page 1

4. Sal confused the number of groups

4. 5; 2; 5, 2

with the number in each group. Sal said there are 3 groups of 5, but his work shows 5 groups of 3.

4. Models with counters; circles

3 groups of 5 cows; 3

Practice | Teacher Guide

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Student Edition | Printable Pages for Students

Practice | Teacher Guide

NAME

DATE

Practice Page 1 | Model multiplication situations by using concrete objects. Model equal groups with counters. Model groups of 4 pinecones. Circle them. Complete the statement.

Kate tried to model groups of 2 cherries. She circles them. Look at Kate’s work.

groups of 4 pinecones.

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There are

groups of 2 cherries.

What mistake did Kate make?

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Practice | Student Page

NAME

DATE

Practice Page 1 | Model multiplication situations by using concrete objects.

How does the work show what is known?

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Model equal groups with counters and circle them. Complete each statement.

There are

Model groups of 5 cows. Circle them.

groups of 3 bananas.

There are

groups of 5 cows.

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

and describe multiplication situations by using Practice Page 2 | Represent repeated addition.

Describe the equal groups. Complete each statement.

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4 groups of

= .

+ 2 groups of

= is

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Practice | Student Page

NAME

DATE

Represent and describe multiplication situations by using Practice Page 2 | repeated addition.

Describe the equal groups. Complete each statement.

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3 groups of

= is

Sal tried to describe the equal groups. Look at Sal’s work.

3 + 3 + 3 + 3 + 3 = 15 3 groups of 5 is 15.

What mistake did Sal make? For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

stories to represent multiplication situations involving Practice Page 3 | Create equal groups.

Create a story to represent equal groups. Complete each statement. There are

stacks.

There are

plates in each stack.

There are

groups of

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plates.

There are

cartons.

There are

eggs in each carton.

There are

groups of

eggs.

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Practice | Student Page

NAME

DATE

Create stories to represent multiplication situations involving Practice Page 3 | equal groups.

Create a story to represent equal groups. Complete each statement.

There are

peanuts in each bag.

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There are

bags.

There are

groups of

There are

peanuts.

lily pads.

There are

frogs on each lily pad.

There are

groups of

frogs.

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Practice | Student Page

NAME

DATE

Practice Helper 1

Look at the problem. Then look at the work. It shows how to model equal groups by using counters. Model equal groups with counters. Model 4 jellyfish. Circle them. Complete the statement.

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There are

How do you know these groups are equal?

How can you use counters to show the equal groups?

There are the same number of jellyfish in each group.

groups of 4 jellyfish.

How many groups do you have? How many are in each group? I have 3 groups. There are 4 in each group.

3 There are of 4 jellyfish.

groups

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Practice | Student Page

NAME

DATE

Practice Helper 2

Look at the problem. Then look at the work. It shows how to represent and describe equal groups by using repeated addition. Describe the equal groups. Complete each statement.

4 groups of

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How many groups are there? How many are in each group? How can that help you write a repeated addition equation?

There are 4 groups of 3.

How can you use repeated addition to find the total? The repeated addition equation is . 3+3+3+3= I know 3 + 3 is 6. 6 + 3 is 9. 9 + 3 is 12.

How does the repeated addition equation match the equal groups? In the repeated addition equation, the addends are the same number, 3. There are 3 in each equal group. In the repeated addition equation, there are 4 addends. This means there are 4 equal groups.

I can add 4 threes.

4 groups of

12 .

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Practice | Student Page

NAME

DATE

Practice Helper 3

Look at the problem. Then look at the work. It shows how to create stories to represent equal groups. There are boxes. There are

crayons in each box.

There are

How many groups are there? How many are in each group?

Each group is a box.

There are 4 groups.

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What are the groups in each picture? What is in each group?

There are 5 crayons in each group.

What repeated addition equation represents the equal groups? How does the repeated addition equation match the equal groups in your story?

There are 5

There are There are

crayons.

5 + 5 + 5 + 5 = 20 There are 4 groups of 5 crayons. I can add 4 five.

There are 5 crayons in each box, or group.

groups of

boxes.

crayons in each box. groups of

crayons.

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Practice | Student Page

Application | Represent Contextual Multiplication Read–Draw–Write

Use the chart below to determine how to use the Application resources to best meet the needs of your students. The activities provide opportunities for partner or independent work. Use any combination of the activities and structures to engage students in applying their understanding of representing contextual multiplication.

Support students as they use this simple, repeatable process to solve problems. The more students participate in reasoning through problems with this systematic approach, the more they internalize these practices and thought processes. The Read–Draw–Write (RDW) process is a way students can make sense of problems, choose and apply mathematical strategies, and solve. Here are the steps students take when using the RDW process.

Activities, Structures, and Considerations

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• Read the problem all the way through. Then reread a chunk at a time. As you reread, ask yourself, “Can I draw something?” Then ask, “What can I draw?” • Draw to represent the problem as you reread. Add to or revise your drawing as you uncover new information or discover what is unknown. As you draw, label what is known and what is unknown. When you finish rereading and drawing, ask yourself, “What does my drawing show me?” Let your drawing help you find a way to solve. • Write number sentences or equations to represent your thinking. Solve. Then use your solution to write a statement that answers the original question.

Structure(s)

Considerations

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving contextual multiplication. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a personal whiteboard. • Consider printing the cards on cardstock and laminating them for long-term use.

Activity

Play a Game

Partner Work

Study a Solution

Independent Work

Solve a Task

Partner Work

• Consider providing highlighters and other tools for students to use to annotate the sample solution. • Consider providing highlighters and other tools for students to use to annotate the task.

Application | Teacher Guide

Application | Represent Contextual Multiplication

Solve a Problem Materials

Play a Game: Equal Groups Match

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

Materials

Students use the Read–Draw– Write process to solve word problems involving representing contextual multiplication. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problems 1 and 2 lend themselves to using repeated addition to represent equal groups and find the total. Problem 3 involves determining the number of groups and the total.

Students work with a partner to play a game involving multiplication situations and equal groups.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Study a Solution

Students work with a partner to solve a multi-part task involving representing contextual multiplication. They are given important information about the problem to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Solve a Task

Materials

• Study a Solution Student Page • Highlighters (optional) Students work independently or with a partner to analyze a correct solution to a word problem involving representing contextual multiplication. Students answer questions about how the known and unknown information in the problem is represented in the sample solution. They also analyze how the sample drawing provides a solution path. Finally, they consider whether the sample statement answers the question in the word problem.

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Preparing to Play • Print and cut out the Equal Groups Match Cards. • Players shuffle the cards and arrange them facedown into rows of six.

Playing the Game • The players take turns flipping over two equal groups match cards. • A matching pair of cards is a picture of equal groups and a description of the groups. If the cards make a matching pair, the player models the multiplication situation with counters, takes the cards, and turns over two new cards.

Teacher Tip

• Equal Groups Match Cards • Game Instruction Card • Personal whiteboard • Counters (20 per student)

• If the cards do not make a matching pair, the player flips the cards back over and their turn is over. • The player with the most cards at the end of the game wins.

Materials

• Solve a Task Student Page • Counters (20 per student) • Highlighters (optional)

Application | Teacher Guide

Application | Represent Contextual Multiplication

Answer Key Study a Solution

1. Lan puts 12 cherries into cups.

1. The counters represent the

Solve a Problem

7 groups of 5 pencils Kate has.

2. The choir teacher hangs up

2. The total number of counters

10 notes.

Solve a Task 1. Counters to show 4 groups of 5; drawing of 4 equal groups of 5

2. 12 balloons; 4 + 4 + 4 = 12

represents the number of pencils Kate has.

3. 5; 3; 5, 3; 15

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3. They have 20 strawberries in all.

3. It shows I need to add 5 fives.

4. Yes, the sentence tells how many pencils Kate has. I know because

5 + 5 is 10. 10 + 5 is 15. 15 + 5 is 20. 20 + 5 is 25.

Application | Teacher Guide

Application | Solve a Problem Word Problem Cards

1 Lan has 4 cups. He puts 3 cherries

into each cup. How many cherries does Lan put into cups?

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2 A choir teacher hangs up 5 groups of 2 music

notes in her classroom. How many notes does the choir teacher hang up?

3 Matt, Lee, Jade, and Pam each have a basket

of 5 strawberries. How many strawberries do they have in all? For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Application | Teacher Guide

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Application | Play a Game • Equal Groups Match Cards

Application | Teacher Guide

Application | Play a Game • Equal Groups Match Cards

3+3+3=9

2 + 2 + 2 + 2 + 2 = 10

3 groups of 4 is 12.

3 groups of 3 is 9.

5 groups of 2 is 10.

4 + 4 + 4 + 4 = 16

2+2+2+2=8

5 + 5 + 5 = 15

4 groups of 4 is 16.

4 groups of 2 is 8.

3 groups of 5 is 15.

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4 + 4 + 4 = 12

5 + 5 + 5 + 5 = 20

3 + 3 + 3 + 3 + 3 = 15

2+2+2=6

4 groups of 5 is 20.

5 groups of 3 is 15.

3 groups of 2 is 6.

Application | Teacher Guide

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Student Edition | Printable Pages for Students

Application | Teacher Guide

NAME

DATE

Application | Solve a Problem

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Problem Number

This page may be reproduced for classroom use only.

Application | Student Page

Application | Play a Game Game Instruction Card

Equal Groups Match

4. If the cards make a matching set, model the equal groups with counters. Then take the cards and turn over two new cards.

What You Need • Equal Groups Match Cards • Personal whiteboard • Counters

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How to Play

1. Mix up the cards. Place the cards facedown into rows of six.

2. Take turns turning over two cards.

3. Look for a matching set of two cards that shows equal groups and how to describe the equal groups.

5 + 5 + 5 = 15 3 groups of 5 is 15.

5. If the cards do not make a matching set, flip the cards back over. It is the next player’s turn.

How to Win The player with the most cards at the end of the game wins.

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Study a Solution

Tim solved the problem. Read the problem and look at Tim’s work. Then answer the questions. Kate has 5 cups. There are 5 pencils in each cup. How many pencils does Kate have?

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Tim’s Work

5 + 5 + 5 + 5 + 5 = 25 5 groups of 5 is 25. Kate has 25 pencils.

How does the work show what is known?

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Study a Solution How does the work show what is unknown?

How does the drawing help you find the unknown?

Does the sentence answer the question? How do you know?

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This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Solve a Task Bookstore

1 bunch of balloons

Nick puts 4 packages of flowers onto a table to give to customers. Model the packages of flowers with counters. Draw to show the equal groups.

1 stack of books

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1 package of flowers

Nick opens a bookstore. The table shows the items Nick uses on the first day.

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Solve a Task Bookstore Nick places 3 bunches of balloons around the store. How many balloons does he use? Show how you know with a repeated addition equation.

Nick puts 2 stacks of books onto a shelf. Then he puts 3 stacks of books onto another shelf. Create a story to represent equal groups. Complete each statement.

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There are

There are Nick puts

stacks.

books in each stack. groups of

books.

books onto shelves.

This page may be reproduced for classroom use only.

Application | Student Page

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Representations of Multiplication

Multiplication

Concept Guide | Representations of Multiplication Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool

• Print copies of the Progress Check Tool and Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard • Square inch tiles

• Personal whiteboard and Student Pages • Square inch tiles

• Print copies of Student Pages as needed. • Place copies of Student Pages in whiteboards for objectives 1, 2, and 3. • Gather 15 square inch tiles per student.

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Application Teacher Guide

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • Study a Solution Student Page • Solve a Task Student Page • Square inch tiles (optional) • Grid paper (optional) • Highlighters (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards • Gather optional tools such as square inch tiles and grid paper. • Print copies of the following: - Solve a Problem Recording Page (optional) - Study a Solution Student Page - Solve a Task Student Page

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Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Representations of Multiplication

Addressing Student Misconceptions How to Address Misconception

Students do not represent the factors in the multiplication expression as rows and columns in a rectangular array.

Invite students to use concrete materials, such as square inch tiles, to make equal groups to represent 3 × 2. Then have students move the equal groups into equal rows with gaps between the tiles. Finally, direct students to push the tiles together to make a rectangular array. Have students write a multiplication equation to represent the rectangular array. Use the following prompts to help students make connections between the multiplication equation and the rectangular array.

Student Misconception

Language Support

3×2=6

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• Where do you see the factor 3 represented in the rectangular array? • Where do you see the factor 2 represented in the rectangular array? • How can you use the rectangular array to help you find the product?

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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.

• key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Representations of Multiplication Dear Family,

Your student is working on representations of multiplication. They connect their understanding of equal groups, repeated addition, arrays, and equal jumps on a number line to multiplication. They verbalize how the factors, or numbers they multiply, are represented in different models, and they use the model to help them find the product, or total. You can support your student’s progress by asking the questions in the table below as your student learns the foundations of multiplication.

4×3=

3 3

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How can you represent the number in

equal groups?

each group?

How can you represent the number of I can draw 4 circles to show 4 groups.

I can show 4 rows.

I can make 4 equal jumps on the number line.

I can put 3 tiles into each group.

I can put 3 tiles into each row.

I can make jumps of 3 on the number line.

How can you find the product?

I can skip-count by 3s.

3, 6, 9, 12

3 + 3 + 3 + 3 = 12

I can use repeated addition.

4 × 3 = 12

I can multiply.

Concept Guide | Teacher Guide

Progress Check | Representations of Multiplication

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before, during, or after providing direct instruction. It is intended to collect data about students’ proficiency with representing multiplication facts with various models and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problems 1 and 2 involve drawing equal groups to find a product, problems 3 and 4 involve drawing arrays to find a product, problem 5 involves representing multiplication with a rectangular array, problem 6 involves using a number line to find a product, and students self-select a strategy for problems 7 and 8. Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

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• Can the student represent multiplication with equal groups? | Objective 1

• Can the student represent multiplication with arrays? | Objective 2

• Can the student represent multiplication with area models?

| Objective 3

• Can the student represent multiplication with number lines? | Objective 4

• Can the student use a model to correctly determine a product? | Objectives 1–4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Representations of Multiplication

Progression Towards Proficiency Rubric Items 1 and 2

Items 3 and 4

Item 5

Item 6

Items 7–8

Objective 1

Objective 2

Objective 3

Objective 4

Objectives 1–4

Not Yet Proficient

The student may show evidence of beginning to understand representing multiplication with equal groups but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand representing multiplication with arrays but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly draws the equal groups and finds the products:

The student correctly draws the arrays and finds the products:

Progress Check Tool Item(s)

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The student may show evidence of beginning to understand representing multiplication facts with rectangular arrays, but the answer is incorrect.

1. Draws 4 groups of 2; 8

2. Draws 7 groups of 5; 35

3. Draws an array with 5 rows of 2; 10 4. Draws an array with 6 rows of 3; 18

The student correctly circles option B.

The student may show evidence of beginning to understand representing multiplication facts with number lines but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand representing multiplication with equal groups, arrays, rectangular arrays, or number lines but makes more than one error that leads to an incorrect answer.

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

8 hops of 3 and finds 24 as

The student correctly finds the products and shows their work:

The student correctly draws

the product.

7. 12; shows 4 groups of 3. 8. 45; shows 9 groups of 5.

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Representations of Multiplication Draw equal groups to find the product.

4×2=

Draw an array to find the product.

5×2=

6×3=

7×5=

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This page may be reproduced for classroom use only.

Progress Check | student Page

NAME

DATE

Progress Check Tool | Representations of Multiplication Which rectangular array represents 3 × 2? Circle the letter of the correct answer.

8×3=

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Use the number line to find the product.

4×3=

Multiply. Show your work.

9×5=

This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Representations of Multiplication Progression of Mini Lesson Objectives 2 Represent multiplication facts

3 Represent multiplication facts

4 Represent multiplication facts

with equal groups.

with arrays.

with area models.

with equal jumps on a number

Start here if students

• can add more than two single‑digit addends, • can write a repeated addition equation to represent an equal groups model, but • need support relating equal groups to multiplication.

• can represent multiplication facts with equal groups; • can skip‑count by 2s, 3s, and 5s; but • need support representing multiplication with an array.

3 6 9 12 15

Start here if students

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5 10 15

1 Represent multiplication facts

• can represent multiplication facts with equal groups and arrays but • need support identifying the size of equal groups and the number of equal groups in an area model and • need support representing multiplication with an area model.

line. +3 +3 +3 +3 +3 +3

9 12 15 18

6 × 3 = 18 Start here if students • can represent multiplication facts with equal groups, arrays, and area models; • can represent addition on a number line; but • need support identifying the size of equal groups and the number of equal groups on a number line and • need support representing multiplication on a number line.

Concept Mini Lessons | Teacher Guide

Objective 1 | Represent multiplication facts with equal groups. 10 M I NU T ES

Distribute 12 tiles and the Student Page to each student. Language Support

Consider providing sentence frames to support students with verbalizing how many equal groups and how many in each group. • There are

equal groups of

• There are

in each group.

groups.

Students write 3 + 3 + 3 + 3 = 12 in the table.

H 4 threes is 12. The unit is 3 and there are 4 of them. Let’s write that in unit form: 4 threes is 12.

Write 4 threes is 12 and direct students to do the same. Make equal groups.

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Ask students to use their tiles to make equal groups of 3 in the space provided on the Student Page. • There are

Materials • Square inch tiles • Objective 1 Student Page in a personal whiteboard

Summary Students make equal groups to represent multiplication facts.

H How many equal groups did you make? Circle the groups as you count them. 4

H How many tiles are in each group? Write the number of tiles under each group as you count them. 3

H 4 equal groups of 3 tiles is how many tiles altogether? We can skip-count by 3s to find the answer. 3, 6, 9, 12. There are 12 tiles altogether.

H Write a repeated addition equation that shows 4 threes equals 12.

Write a repeated addition equation. Write in unit form. Write a multiplication equation.

3 + 3 + 3 + 3 = 12 4 threes is 12 4 × 3 = 12

Multiply.

H 3 + 3 + 3 + 3 is a total of 12 tiles. 4 threes is a total of 12 tiles. Both of these statements represent equal groups.

H Multiplication is another way to write repeated addition.

Instead of writing addition of the same number over and over, we can write the number of groups times the number of objects in each equal group.

H Let’s write the multiplication equation together.

Concept Mini Lessons | Teacher Guide

Objective 1 | Represent multiplication facts with equal groups. 10 MINUTES

Model writing 4 × 3 = 12 as students write the equation on their Student Page.

H Where do you see the factors 2 and 6 in your equal groups?

H We wrote a multiplication equation. We say 4 times 3 equals 12.

Make equal groups.

Invite students to say the multiplication equation to a partner. Then point to the 4, the 3, and the multiplication symbol.

Write a repeated addition equation. Write in unit form.

Multiply.

2 × 6 = 12

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at the bottom of your paper.

Students write the multiplication equation at the bottom of the Student Page. H Use your tiles to make 2 groups of 6. Circle the groups.

Make 2 groups of 6 tiles as students do the same.

H I can ask myself: How many equal groups did I make? What do you think? 2

H I can ask myself: What is the size of each group? What do you think? 6

Label each group with the size of the group as students do the same.

6 + 6 = 12

Write a multiplication equation.

H The 4 and 3 are called factors, and we call × the multiplication symbol. Ask students to erase their whiteboards but to keep their 12 tiles in the workspace. H Write 2 × 6 =

The 2 in the problem are the 2 circles.

The 2 equal groups of tiles

The 6 means that there are 6 tiles in each group.

H How can we find the total number of tiles? We could count all the tiles. We can add 6 and 6.

I know it’s 12 because I know my doubles.

H So 2 times 6 equals? 12

Have students fill in the blank in 2 × 6 = the multiplication equation to a partner.

. Then invite them to read

Invite students to turn and talk about how they can represent multiplication with equal groups.

Concept Mini Lessons | Teacher Guide

Objective 1 | Represent multiplication facts with equal groups. 10 M I NU T ES

Notes

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. If time is limited, instruct students to draw equal groups rather than use tiles. Consider providing the Objective 1 Practice Helper and supporting students in using the worked‑out example to guide their own work.

• 6×2 • 3×4

Analyze Student Progress

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Have students fill out all or parts of the table on the Student Page depending on the time available and the needs of your class.

Monitor: • Can the student create equal groups? • Can the student explain how each factor is represented in the equal groups? • Does the student correctly find the product?

Questions to Advance Student Thinking: • How can you draw equal groups to find the total? • How are the factors represented in the equal groups? • What is the total? How do your equal groups help you determine the total?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 2 | Represent multiplication facts with arrays. 10 M I NU T ES

Materials • Square inch tiles • Objective 2 Student Page in a personal whiteboard

Summary Students make arrays to represent multiplication facts.

Set out the square inch tiles and distribute the Student Page to each student. Write the expression 3 × 5. H Let’s make equal groups to represent the multiplication expression.

Guide students to place each straight line of tiles under the previous line until they form an array.

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Demonstrate making 3 equal groups of 5 tiles as students make the same groups.

H Let’s arrange each group of tiles into a straight line but leave space between each tile.

H We represented 3 × 5 with 3 equal groups of 5 tiles.

H The tiles are arranged into equal rows. When we use rows and columns to arrange equal groups, it is called an array.

Draw a circle around each group and write the repeated addition expression underneath. H We can think of multiplication as repeated addition, so we know that 3 × 5 = 5 + 5 + 5.

H How many rows, or equal groups, do we have in our array? 3

H How many tiles are in each row? 5

H What is the total number of tiles in the array? How do you know? There are 15 tiles altogether. I know because I counted them all.

Concept Mini Lessons | Teacher Guide

Objective 2 | Represent multiplication facts with arrays. 10 MINUTES

Teacher Tip

• How can we write 3 fives is 15 as a multiplication equation?

groups of

fives

The objectives scaffold the meaning of multiplication for students by consistently using the first factor to represent the number of groups (or rows) and the second factor to represent the number in each group. In the future, this scaffold can be removed as students come to understand that factors may be written in any order and that the number of groups may be represented by the first or the second factor. For now, continue to probe students on what each factor represents and consider having students draw boxes around each row to highlight the equal groups.

H Let’s draw circles to make an array with the same number of rows and number in each row.

H Each row has 5 circles. Let’s skip-count by fives to find the total number of circles in the array. Write the skip-count down the side of the array. Ready? 5, 10, 15

Record the skip‑count down the side of the array as students do the same. Then use the following questions to support students in completing the sentence frames to describe the array. • • • •

How many groups of 5 do we have? What is the total of 3 groups of 5? How can we describe the array in unit form? How can we write 3 fives as a repeated addition equation?

5 10 15

repeated addition: 5 + 5 + 5 = 15

= 15

H The factors in the multiplication equation are 3 and 5. I can ask myself: Where is the factor 3 represented in the array?

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If students need more room to draw, have them move the array of tiles above their Student Page. Demonstrate drawing 3 rows of 5 circles and have students draw the same.

What do you think? There are 3 rows of circles in the array.

H I can ask myself: Where is the factor 5 represented in the array? What do you think? There are 5 circles in each row of the array.

H In multiplication, the total number is called a product. Where do you see the product, 15, represented in the array? There is a total of 15 circles in the array.

Language Support Consider asking students the following questions to have them revoice their understanding of multiplication and of the terms factors and product: What do the 3, the 5, and the symbol × mean in the expression 3 × 5? What are the 3 and the 5 called in the equation 3 × 5 = 15? What is the 15 called in the equation 3 × 5 = 15?

Concept Mini Lessons | Teacher Guide

Objective 2 | Represent multiplication facts with arrays. 10 M I NU T ES

Notes

Invite students to turn and talk about how multiplication can be represented with an array.

•4×2 •5×5 •2×9

Analyze Student Progress

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Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice with drawing arrays to represent multiplication expressions as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked‑out example to guide their own work.

Monitor: • Can the student create an array? • Can the student explain how each factor is represented in the array? • Does the student correctly find the product?

Questions to Advance Student Thinking: • How can you draw an array to find the product? • How are the factors represented in the array? • What is the product? How does your array help you determine the product? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Represent multiplication facts with area models. 10 M I NU T ES

Set out the square inch tiles and distribute the Student Page in a personal whiteboard to each student.

Write the expression 4 × 3.

Materials • Square inch tiles • Objective 3 Student Page in a personal whiteboard

Summary Students use square inch tiles to make area models to represent multiplication facts.

H What shape did we make with our tiles? We made a rectangle.

H When we arrange our array into rows and

H Let’s make an array to represent this multiplication expression. We can think of multiplication as repeated addition, so we know that 4 × 3 = 3 + 3 + 3 + 3.

Language Support

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Use square inch tiles to make an array with 4 rows of 3. Have students do the same on their Student Page.

columns without gaps and overlaps to form a rectangle, it is called a rectangular array.

Consider creating a chart with pictures and labels of an array and a rectangular array to help students differentiate between the two representations.

Array

Rectangular Array

Guide students to verify the number of equal groups by pointing to each row and counting while you point.

H Look at your array. How many equal groups do you see? 4

Guide students to verify the number of square inch tiles in each group by pointing to each column and counting while you point. H How many square inch tiles are in each group? 3

Guide students to see the number of groups by pointing to each row and counting while you point.

Push the square inch tiles together to form an area model. Have students do the same. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTaLysT | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 3 | Represent multiplication facts with area models. 10 M I NU T ES

H We can think of the number of columns as the number of square inch tiles in each group.

Guide students to see the number of square inch tiles in each group by pointing to each column and counting while you point. H How many square inch tiles are in each group? 3

Write 4 groups of 3 is 12 and direct students to do the same.

H We can also describe the rectangular array in unit form: 4 threes is 12.

Write 4 threes is 12 and direct students to do the same.

H I can ask myself: How can I write 4 threes is 12 as a multiplication equation? What do you think? 4 × 3 = 12

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H Because 4 × 3 = 3 + 3 + 3 + 3, we can skip-count by threes to find the total number of square inch tiles in the rectangular array. 3, 6, 9, 12 Teacher Tip: Differentiation

H We have 4 groups of 3. What is the total of 4 groups of 3? 12

H How many groups are in the rectangular array? 4

Some students may benefit from using repeated addition to find the total. Instead of skip-counting 3, 6, 9, 12, students may choose to add 3 + 3 + 3 + 3 to find the total number of tiles.

This method provides a clear pathway to the total and helps reinforce the connection between addition and multiplication.

Record the skip‑count down the side of the rectangular array and have students do the same. Refer to the sentence frames on the Student Page and guide students to complete the frames when prompted.

Write 4 × 3 = 12 and direct students to do the same.

groups of

threes

Gesture to the 4 in the multiplication equation.

H Where do you see the factor 4 represented in the rectangular array? There are 4 rows.

Concept Mini Lessons | Teacher Guide

Objective 3 | Represent multiplication facts with area models. 10 M I NU T ES

Gesture to the 3 in the multiplication equation.

Gesture to the 12 in the multiplication equation.

Questions to Advance Student Thinking: • How can you draw a rectangular array to find the product? • How are the factors represented in the rectangular array? • What is the product? How does your rectangular array help you determine the product?

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H Where do you see the product 12 represented in the rectangular array? There are 12 tiles altogether.

Monitor: • Can the student create a rectangular array? • Can the student explain how each factor is represented in the rectangular array? • Does the student correctly find the product?

H Where do you see the factor 3 represented in the rectangular array? There are 3 tiles in each row.

Analyze Student Progress

Invite students to turn and talk about how they can represent multiplication facts with rectangular arrays.

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

•9×2 •3×5 •4×4

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Encourage students to begin to transition from building a rectangular array with square inch tiles to drawing the rectangular array. Consider providing the Objective 3 Practice Helper and supporting students in using the worked‑out example to guide their own work.

Concept Mini Lessons | Teacher Guide

Objective 4 | Represent multiplication facts with equal jumps on a number line. 10 M I NU T ES

Display the number line showing 3 equal jumps of 5. +5 +5 +5

Materials • Objective 4 Student Page in a personal whiteboard

Summary Students use equal jumps on a number line to represent multiplication facts and to make connections to other representations of multiplication.

Gesture to the 5 in the multiplication equation.

H Where do you see the factor 5 represented on the number Each equal jump on the number line represents a group of 5.

line?

5 10 15

H What repeated addition equation can we write to represent the jumps on the number line?

5 + 5 + 5 = 15

Write the repeated addition equation and direct students to do the same.

H What multiplication equation can we write to represent the repeated addition equation? 3 × 5 = 15

Gesture to the 15 in the multiplication equation.

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Write 3 × 5 = 15 and direct students to do the same. Gesture to the 3 in the multiplication equation.

H Where do you see the factor 3 represented on the number There are 3 equal jumps on the number line.

line?

H Where do you see the product 15 represented on the number line? The last jump lands on the number 15.

Display the expression 6 × 3 and the number line.

H Let’s represent this expression on the number line and then use the number line to help us find the product. We know that 6 × 3 = 3 + 3 + 3 + 3 + 3 + 3. I can ask myself: How can I use the number line to show the repeated addition? What We can draw 6 jumps of 3 on the number line.

do you think?

Draw a jump from 0 to 3 on the number line and label the jump as + 3. Have students do the same.

Concept Mini Lessons | Teacher Guide

Objective 4 | Represent multiplication facts with equal jumps on a number line. 10 MINUTES

Direct students to label the tick mark 3.

Repeat the process of representing each addend in the repeated addition expression as an equal jump on the number line.

+3 +3 +3 +3 +3 +3

9 12 15 18

H What is 6 × 3? How do you know?

I know 6 × 3 = 18 because the last jump lands on the number 18.

Record 6 × 3 = 18 and direct students to do the same.

H The factors in the multiplication equation are 6 and 3. Where do you see the factor 6 represented on the number line?

There are 6 equal jumps on the number line.

H Where do you see the factor 3 represented on the number line? Each equal jump on the number line represents a group of 3.

H Where do you see the product, 18, represented on the number line? The last jump lands on the number 18.

9 12 15 18

H How do the jumps on the number line represent 6 equal groups of 3?

EV I

+3 +3 +3 +3 +3 +3

Teacher Tip: Differentiation

Consider having students draw all the tick marks on the number line to support skip-counting by 3s.

H We need to label the tick mark where our jump landed on the number line. What number did we land on when we added 3? How do you know? We landed on 3 because we started at 0 and added 3.

We can think of the 6 equal jumps as 6 equal groups. Each equal

jump on the number line adds 3, so the size of each equal group is 3.

H We can also represent multiplication with skip-counting. Where do you see skip-counting in the number line representation?

The equal jumps on the number line are like skip-counting. lands are the skip-count: 3, 6, 9, 12, 15, 18.

The labels on the tick marks that show where each equal jump

Invite students to turn and talk about how multiplication can be represented on a number line. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked‑out example to guide their own work.

Concept Mini Lessons | Teacher Guide

Objective 4 | Represent multiplication facts with equal jumps on a number line. 10 M I NU T ES

•8×5 •2×7 •9×4

Teacher Tip: Differentiation

If students are still developing proficiency skip-counting by larger numbers, consider relating 2 groups of 7 to the doubles fact 7 + 7.

Analyze Student Progress

Notes

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Monitor: • Can the student draw a number line to represent the multiplication expression? • Can the student explain how each factor is represented on the number line? • Does the student correctly find the product?

Questions to Advance Student Thinking: • How can you use the number line to find the product? • How are the factors represented on the number line? • What is the product? How does drawing equal jumps on the number line help you determine the product?

Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons| Representations of Multiplication Answer Key Objective 2

Objective 3

Objective 4

1. 3 + 3 + 3 + 3 = 12; 4 threes is 12; 4 × 3 = 12

1. 3 groups of 5 is 15; 3 fives is 15; 5 + 5 + 5 = 15; 3 × 5 = 15

1. 4 groups of 3 is 12; 4 threes is 12; 4 × 3 = 12

1. 5 + 5 + 5 = 15; 3 × 5 = 15

3. 2 + 2 + 2 + 2 + 2 + 2 = 12; 6 twos is 12; 6 × 2 = 12

3. 5 groups of 5 is 25; 5 fives is 25; 5 + 5 + 5 + 5 + 5 = 15; 5 × 5 = 25 4. 2 groups of 9 is 18; 2 nines is 18; 9 + 9 = 18; 2 × 9 = 18

4. 4 + 4 + 4 = 12 3 fours is 12; 3 × 4 = 12

2. 4 groups of 2 is 8; 4 twos is 8; 2 + 2 + 2 + 2 = 8; 4×2=8

2. 9 groups of 2 is 18; 9 twos is 18; 9 × 2 = 18 3. 3 groups of 5 is 15; 3 fives is 15; 3 × 5 = 15

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2. 6 + 6 = 12; 2 sixes is 12; 2 × 6 = 12

Objective 1

4. 4 groups of 4 is 16; 4 fours is 16; 4 × 4 = 16

2. The number line shows 6 jumps, each labeled + 3. The tick marks are labeled 0, 3, 6, 9, 12, 15, 18; 18

3. The number line shows 8 jumps, each labeled + 5. The tick marks are labeled 0, 5, 10, 15, 20, 25, 30, 35, 40; 40

4. The number line shows 2 jumps, each labeled + 7. The tick marks are labeled 0, 7, 14; 14

5. The number line shows 9 jumps each labeled + 4. The tick marks are labeled 0, 4, 8, 12, 16, 20, 24, 28, 32, 36; 36

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Representations of Multiplication Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Representations of Multiplication Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Represent multiplication facts with equal groups.

EV I

Make equal groups.

Write a repeated addition equation.

Write in unit form.

Write a multiplication equation.

Multiply.

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Concept Mini Lessons | student Page

NAME

DATE

EV I

Objective 2 | Represent multiplication facts with arrays.

groups of

is is

. .

repeated addition: ×

⁼

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Concept Mini Lessons | student Page

NAME

DATE

EV I

Objective 3 | Represent multiplication facts with area models.

groups of

is is

. .

⁼

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Concept Mini Lessons | student Page

NAME

DATE

Objective 4 | Represent multiplication facts with equal jumps on a number line. Write the repeated addition equation and the multiplication equation that are represented on the number line . +5

Multiplication equation:

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Repeated addition equation:

6×3=

Use the number line to find the product. Then write the product.

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Concept Mini Lessons | student Page

NAME

DATE

Objective 4 | Represent multiplication facts with equal jumps on a number line. 8×5=

2×7=

EV I

9×4=

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

Practice | Representations of Multiplication Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers during Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Represent

Objective 2 Represent

Objective 3 Represent

Objective 4 Represent

multiplication facts with area models.

multiplication facts with equal jumps on a number line.

Look for...

• Can the student draw a rectangular array? • Can the student identify how each factor is represented in the rectangular array? • Can the student use a rectangular array to find a product?

• Can the student draw a number line to represent a multiplication fact? • Can the student identify how each factor is represented on the number line? • Can the student use a number line to find a product?

Look for...

multiplication facts with arrays.

Look for...

• Can the student draw an array? • Can the student identify how each factor is represented in the array? • Can the student use an array to find a product?

• Can the student identify the size of equal groups and the number of equal groups in a situation? • Can the student represent multiplication with equal groups and repeated addition?

EV I

multiplication facts with equal groups.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Representations of Multiplication Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Practice | Teacher Guide

Practice | Representations of Multiplication

Answer Key

each group; 6 groups groups and labels of

of eggs; 3 eggs in each group; 18 total eggs;

6 × 3 = 18

2. Appropriate circled each group; 4 groups groups and labels of

of crayons; 5 crayons

in each group; 20 total

crayons; 4 × 5 = 20

1. 4 groups of 5 is 20; 4 fives is 20; 5 + 5 + 5 + 5 = 20; 4 × 5 = 20; The product is 20.; 10, 15, 20.

1. 5 groups of 3 is 15; 5 threes is 15; 5 × 3 = 15; 6, 9, 12, 15

2. An array with 5 rows and 3 columns; 15; 3 + 3 + 3 + 3 + 3 = 15

3. An array with 2 rows and 8 columns; 16; 8 + 8 = 16

4. Gabe incorrectly

6 rows of 6 circles and drew an array with

3. Equal groups show 3 groups with 6 in each group; 3 × 6 = 18

Practice Page 3

1. Appropriate circled

Practice Page 2

4. Equal groups show 8 groups with 2 in each group; 8 × 2 = 16

2. A rectangular array with 4 rows and 2 columns; 8

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Practice Page 1

skip‑counted by sixes

down the array instead

3. A rectangular array with 8 rows and 3 columns; 24 4. A rectangular array with 2 rows and 9 columns; 18

Practice Page 4 1. The number line shows 9 jumps, each labeled + 2; the tick marks are labeled 0, 2 , 4, 6, 8, 10, 12, 14, 16, 18; 18 2. The number line shows 2 jumps, each labeled + 8; the tick marks are labeled 0, 8, 16; 16 3. The number line shows 4 jumps, each labeled + 3; the tick marks are labeled 0, 3, 6, 9, 12; 12 4. B

6 rows of 3 circles and

of drawing an array with skip‑counting by threes down the array.

Practice | Teacher Guide

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Student Edition | Printable pages for students

Practice | Teacher Guide

NAME

DATE

Practice Page 1 | Represent multiplication facts with equal groups. 1

There are

groups of eggs.

There are

eggs in each group.

Number of groups

Total

Number in each group

groups of crayons.

There are There are

Number of groups

eggs.

EV I

There is a total of

Circle and count equal groups. Count and label how many are in each group. Fill in the blanks.

crayons in each group.

There is a total of

crayons.

Number in each group

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Practice | student Page

NAME

DATE

Practice Page 1 | Represent multiplication facts with equal groups. Draw equal groups to find the total. Fill in the blanks.

EW Number of groups

8×2=

Number of groups

Total

Number in each group

3×6=

EV I

Number in each group

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Practice | student Page

NAME

DATE

Practice Page 2 | Represent multiplication facts with arrays. Use the array to fill in the blanks. groups of

fives is

⁼

The product is

EV I

Draw an array to find the product. Write a repeated addition equation to match the array.

5×3=

2×8=

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Practice | student Page

NAME

DATE

Practice Page 2 | Represent multiplication facts with arrays. Gabe incorrectly drew an array to find 6 × 3. Look at Gabe’s work. What mistake did Gabe make?

12 18

30 36

EV I

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Practice | student Page

NAME

DATE

Practice Page 3 | Represent multiplication facts with area models. Use the rectangular array to fill in the blanks. Skip‑count by 3. groups of

threes is

⁼

Draw a rectangular array to find the product.

4×2=

8×3=

2×9=

EV I

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 4 | Represent multiplication facts with equal jumps on a number line.

2×8=

EV I

9×2=

4×3=

Use the number line to find the product.

Which representation does not represent the multiplication shown on the number line? A

4+4+4+4+4

B 4 groups of 5

C 4, 8, 12, 16, 20

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Practice | student Page

NAME

DATE

Practice Helper 1 Look at the problem. Then look at the work. It shows how to draw equal groups to find the total.

4×6=

How are the factors represented in the

What is the total? How do your equal

the total?

equal groups?

groups help you determine the total?

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How can you draw equal groups to find

I know that 4 × 6 means 6 + 6 + 6 + 6, so

4×6=

I know the total is 24 because I can use my

equal groups to help me skip‑count by 6s,

add, or multiply to find the total.

of groups. There are 4 groups.

The factor 6 is represented by the number

in each group. There are 6 in each group.

4 × 6 = 24

6 + 6 + 6 + 6 = 24

Number of groups

6 + 6 + 6 + 6 = 24

I draw 4 groups with 6 in each group.

The factor 4 is represented by the number

4 × 6 = 24

Number in each group

Total

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Practice | student Page

NAME

DATE

Practice Helper 2 draw an array to find the product.

6×5=

Look at the problem. Then look at the work. It shows how to

How can you draw an array to find the

How are the factors represented in the

What is the product? How does your

product?

array?

array help you determine the product?

5 10 15 20 25 30

5 in

each row

EV I 6 rows

I know that 6 × 5 means 6 groups of 5, or

There are 6 rows of circles. Each row has

6 fives, so I can draw an array that has 6 rows of 5 circles.

I know the product is 30 because I can

skip‑count by fives to find the total number of circles.

6×5=

5 circles.

5 10 15 20 25 30

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Practice | student Page

NAME

DATE

Practice Helper 3 Look at the problem. Then look at the work. It shows how to draw a rectangular array to find the product.

7×3=

How can you draw a rectangular array to

How are the factors represented in the

What is the product? How does the

find the product?

rectangular array?

rectangular array help you determine

EV I

3 in

7 rows

I know that 7 × 3 means 3 + 3 + 3 + 3 + 3 +

There are 7 rows of squares. Each row has

3 + 3, or 7 threes, so I can draw a rectangular array that has 7 rows of 3 squares.

7×3=

each row

3 squares.

the product?

3 6 9 12 15 18 21

I know the product is 21 because I can skip‑count

by threes to find the total number of squares.

3 6 9 12 15 18 21

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Practice | student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to use a number line to find the product.

6×2=

How are the factors represented on the

What is the product? How do the equal

the number line?

number line?

jumps on the number line help you

+2 +2 +2 +2 +2 +2

EV I

How can you find the product by using

+2 +2 +2 +2 +2 +2

10 12

I know that 6 × 2 means 2 + 2 + 2 + 2 + 2 + 2,

6×2=

10 12

There are 6 equal jumps. Each jump is 2.

so I can draw 6 jumps on the number line and label each one + 2.

determine the product?

10 12

I know the product is 12 because the last jump

lands at 12.

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Practice | student Page

Application | Representations of Multiplication Read–Draw–Write

Support students as they problem solve by using a simple, repeatable process. The more students participate in reasoning through problems with this systematic approach, the more they internalize these practices and thought processes. The Read–Draw–Write (RDW) process is a way students can make sense of problems, choose and apply mathematics, and solve.

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplication. The Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

Study a Solution

Independent Work Partner Work

Solve a Task

Partner Work

Activity

• A standard deck of playing cards can be used instead of Eureka Math2 cards. • Consider providing students with square inch tiles or grid paper to support them with creating a model to represent multiplication facts. • Consider providing highlighters and other tools for students to use to annotate the sample solution. • Consider providing manipulatives, such as square inch tiles, for students to use to represent and solve the multiplication problems.

Application | Teacher Guide

Application | Representations of Multiplication

• Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • square inch tiles (optional) • grid paper (optional)

Preparing to Play • Remove the J, Q, K, and Joker cards from the deck. Aces can represent 1. • Shuffle the remaining cards. Divide the cards equally among the players. Each player keeps their cards in a single facedown pile. • Consider providing tools such as square inch tiles or grid paper to support students. Playing the Game • Each player takes two cards off the top of their pile and places them faceup. • The players each use the values of their cards as factors and find the product. Then they check the other player’s product. • The player with the greater product takes all the cards

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Materials

played and places them at the bottom of their pile. • If the products are the same, a Top It round ensues: A second round is played, and the player with the greater product takes all the cards played from both rounds. • The player with the most cards at the end of the designated time wins.

Materials

Play a Game: Multiplication Top It

Teacher Tip: Differentiation Consider removing select cards from the deck for students who may not yet have the fluency needed to fully participate in the game.

EV I

Solve a Problem

Study a Solution Materials

• Study a Solution Student Page • highlighters (optional)

a solution path. Finally, they consider whether the sample statement answers the question in the word problem.

Solve a Task Materials

• Solve a Task Student Page • square inch tiles (optional) Students work with a partner to solve a multi-part task involving multiplication. They are given important information about a context and an image to support their understanding. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Students work independently or with a partner to analyze a correct solution to a word problem involving multiplication. Students answer questions about how the known and unknown information in the problem is represented in the sample solution. They also analyze how the sample drawing provides

Application | Teacher Guide

Application | Representations of Multiplication

Answer Key Study a Solution

1. An accurate picture is drawn to

1. There are 8 rows of 3 circles in the

work shows 6 × 2 = 12; there are represent the problem; student

12 muffins in the pan.

array to represent the equal groups of fish in the tanks.

2. The unknown information is

2. An accurate picture is drawn to work shows 5 × 10 = 50; Miss Diaz

3. An accurate picture is drawn to

work shows 4 × 3 = 12; there are represent the problem; student

12 pepper plants in the garden.

Solve a Task 1. Miss Wong has 6 pencils left; student work shows 3 × 10 = 30 and 30 − 24 = 6. 2. Accurate arrays are drawn

represented by the total number

to represent the problem;

of circles in the array.

multiplication equations match

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represent the problem; student has 50 markers altogether.

Solve a Problem

3. To find the unknown, I can skip‑count by 3, which is the number in each group, 8 times because there are 8 groups. 4. Yes. The statement tells the total

the arrays drawn.

3. No. The rest of the students cannot be put into equal groups of 4. There will be 5 groups of 4 and 1 group of 3.

number of fish in all the tanks.

Application | Teacher Guide

Application | solve a Problem

A muffin pan has 6 rows. Each row has 2 muffins.

Word Problem Cards

How many muffins are in the pan?

Miss Diaz has 5 boxes of markers. There are

10 markers in each box. How many markers

EV I

does Miss Diaz have altogether?

A garden has 4 rows of pepper plants.

There are 3 plants in each row. How many

pepper plants are in the garden?

Application | Teacher Guide

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Student Edition | Printable pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

This page may be reproduced for classroom use only.

Application | student Page

Application | Play a Game Game Instruction Card

Multiplication Top It

4. If you have the greater product, take all the cards.

What You Need

Put them at the bottom of your stack.

• Eureka Math2 cards (or a standard deck of playing cards) represent 1.

with the J, Q, K, and joker cards removed. Aces can • Square inch tiles (optional) • Grid paper (optional)

1. Mix up the cards. Deal the same number of cards to each player. Put the cards into a stack facedown.

2. At the same time as the other player, turn over 2 cards.

R 7 × 2 = 14

Player B

5. If the products are equal, it is time to Top It! Play another round. The player with the greater product takes the cards from both rounds.

How to Win The player with the most cards at the end of the game wins.

3. Multiply the numbers. Say the product.

Player A

Player B

EV I

How to Play

Player A

18 is greater than 14.

6 × 3 = 18

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Application | student Page

NAME

DATE

Application | study a solution David solved the problem below. Read the problem and look at David’s work. Then answer the questions.

The nature center has 8 fish tanks. There are 3 fish in each tank. How many fish does the nature center have? David's Work

3 6 9

EV I

12 15 18 21

The nature center has 24 fish.

How is the known information in the problem represented?

How is the unknown information in the problem represented?

How does the drawing help you see a solution path for finding the unknown?

Does the statement answer the question? How do you know?

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Application | student Page

NAME

DATE

Application | solve a Task There are 24 students in Miss Wong’s class.

How many pencils does Miss Wong have left? Show how you know.

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Miss Wong buys 3 packs of 10 pencils. She gives each student 1 pencil.

Show two different ways Miss Wong can arrange 24 desks into an array.

Write a multiplication equation to represent each array.

Miss Wong’s Class

equal groups of 4? How do you know?

One student leaves school early. Can the rest of the students be put into

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Application | student Page

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Multiplication as Multiplicative Comparison

Multiplication

Concept Guide | Multiplication as Multiplicative Comparison Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and Pause and Monitor Tool

• Concept Mini Lessons Teacher Guide • Personal whiteboard • Linking cubes (6 of the same color) • Square sticky notes

• Personal whiteboard or Student Pages • Linking cubes (12 same-color cubes per student) • Square sticky notes

• Print copies of Student Pages as needed. • Gather the following materials: - Linking cubes - Square sticky notes

• Practice Teacher Guide

• Practice Pages • Practice Helpers • Linking cubes (16 per student or pair)

• Print copies of Practice Pages and corresponding Practice Helpers. • Gather linking cubes.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Game Instruction Card • • Solve a Task Student Page • Manipulative tools such as cubes and sticky notes (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Multiplicative Comparison Match Cards • Print copies of the following pages: - Solve a Problem Recording Page (optional) - Solve a Task Student Page • Gather tools such as cubes and sticky notes (optional).

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• Application Teacher Guide

Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Multiplication as Multiplicative Comparison

Addressing Student Misconceptions How to Address Misconception

Students do not understand the relationship between the product and the factors as a multiplicative comparison. For example, given the equation 57 = 3 × 19, students are not able to describe the relationship as 57 is 3 times as much as 19.

Help students notice how tape diagrams can be used to represent multiplication as a comparison. Until now, students have understood multiplication as repeated addition. Multiplicative comparison is another way to describe multiplication. Encourage students to identify what the product and each factor represent in the tape diagram. Consider providing sentence frames to support students with comparison language. The second factor represents the unit that is repeated.

19 19

? 57 = 3 × 19

The tape diagram shows that the unit of 19 is repeated 3 times. The first

factor represents how many times the unit is repeated.

The product is 57 . 57 is 3 times as much as 19 .

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Language Support

Student Misconception

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The charts should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Multiplication as Multiplicative Comparison Dear Family,

Your student is working on describing and representing multiplicative comparisons, or describing multiplication as the comparison of two numbers. This involves students’ prior learning about multiplication as repeated addition. Now students are learning to describe multiplication as a comparison with times as many or times as much language. The sample work shows a description of a multiplicative comparison. You can support your student’s progress by asking the questions in the table below as your student makes multiplicative comparisons.

= 3 × 28

28 28

Which factor represents the unit that is repeated?

= 3 × 28

the unit that is repeated.

times as many as

28 .

Which factor represents

How can you represent

How can you describe the

how many times the unit

the comparison by using

multiplication equation as

is repeated?

a tape diagram?

a comparison?

The second factor, 28, represents

EV I

84 = 3 × 28

= 3 × 28

The first factor, 3, represents how

many as 28

3 times as 28 .

many times the unit is repeated.

? 84 = 3 × 28

I can draw the unit 28 and repeat it

3 times. The product is 84.

Concept Guide | Teacher Guide

Progress Check | Multiplication as Multiplicative Comparison

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before, during, or after providing direct instruction. It is intended to collect data about students’ proficiency with describing multiplication expressions as multiplicative comparisons and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problems 1–5 have one-digit factors, and problem 6 has a two-digit factor and a one-digit factor. Students draw arrays in problems 1 and 2, draw a tape diagram in problems 3–6, and solve a word problem in problems 5 and 6. Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

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• Can the student represent multiplication expressions as multiplicative comparisons by using arrays? | Objective 1 • | Objectives 1–3 • | Objectives 2–4

• Can the student describe multiplication expressions as multiplicative comparisons by using story contexts? | Objectives 3 and 4

• Can the student describe two-digit by one-digit multiplication expressions as multiplicative comparisons? | Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Multiplication as Multiplicative Comparison

Progression Towards Proficiency Rubric Progress Check Tool Item(s)

Items 3 and 4

Item 5

Item 6

Objective 1

Objective 2

Objective 3

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand describing multiplication expressions as comparisons but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand describing multiplication expressions but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student demonstrates solid reasoning but makes one error that leads to an incorrect answer, or the student demonstrates some reasoning and circles the correct answer.

The student demonstrates solid reasoning but makes one error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly draws arrays to represent the expression and circles the correct answers.

The student correctly draws a tape diagram to represent the equation, finds the product, and completes the sentences.

The student correctly fills in the blanks, draws a tape diagram, and solves.

1. D

4. 35; 35; 5; 7

EV I 3. 32; 32; 4; 8

2. C

Items 1 and 2

5. 18; 6; 3; pencils

6. 132; 6; 22; Jayla

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication as Multiplicative Comparison Draw arrays to represent the expression as a comparison. Then circle the letter of the statement that

2 × 4

A 4 is 2 more than 2. B 6 is 2 more than 4.

C 4 is 2 times as much as 2.

A 5 is 3 and 2.

B 5 is 3 more than 2.

C 6 is 3 times as much as 2.

D 9 is 3 times as much as 3.

EV I

D 8 is 2 times as much as 4.

3 × 2

correctly describes the expression.

Draw a tape diagram to represent the equation as a comparison and find the product. Then complete the sentence.

= 4 × 8

= 5 × 7

times as much as

This page may be reproduced for classroom use only.

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication as Multiplicative Comparison Fill in the blanks to describe the multiplication equation as a comparison word problem.

= 3 × 6

Casey has

erasers. Casey has does Casey have?

= 6 × 22

Jayla and David collect coins. Jayla has David has

times as many pencils as erasers.

EV I

How many

Draw a tape diagram to represent the problem. Solve and complete the equation.

coins. How many coins does

times as many coins as David. have?

This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Multiplication as Multiplicative Comparison Progression of Mini Lesson Objectives 1 Describe one-digit

2 Represent multiplicative

3 Describe one-digit

4 Describe two-digit by

multiplication expressions as comparisons when using arrays.

comparison statements by using tape diagrams and multiplication equations.

multiplication expressions

one-digit multiplication expressions as comparisons by using story contexts.

2 T

Eva

Pablo

? 45 = 5 × 9

Start here if students

• can use times as many language to describe multiplication equations, but • need support representing multiplicative comparisons by using tape diagrams.

David

Mia

? 57 = 3 × 19

Start here if students

• can use concrete objects and a tape diagram to represent a multiplication equation as a multiplicative comparison, but • need support representing and understanding a multiplicative comparison in story contexts.

• can multiply two-digit by one-digit numbers, but • need support understanding two-digit by one-digit multiplication expressions as multiplicative comparisons.

EV I

• can describe additive comparisons, • can draw arrays, and • can understand multiplication as repeated addition, but • need support understanding multiplication as comparison and • need support using times as many to describe a multiplication expression.

as comparisons by using story contexts.

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons when Objective 1 | Describe using arrays. 10 M I NU T ES

Display the multiplication expression 2 × 3.

2 times 3 3+3

2 groups of 3

Invite students to use their cubes to build an array to represent 2 × 3.

H How does your array show 2 × 3? The array shows 2 rows of 3.

Count out 3 cubes.

H I have 3 cubes. Watch as I connect them. Do I still have 3 cubes? Yes.

H We both have units of 3. We can also think about multiplication as a comparison. When I compare my cubes with your cubes, I can say that you have 2 times as many cubes as I do. Or I can say that you have 2 times as many as 3.

EV I

Materials • Personal whiteboard • Linking cubes (12 of the same color, student) • Objective 1 Student Page • Linking cubes (6 of the same color, teacher)

Summary Students build arrays with cubes to model and describe multiplication expressions as comparisons.

Guide each student to say, “I have 2 times as many as 3.” Language Support

Consider providing sentence frames to support students with discussing multiplicative comparisons. The students are the “I” and the teacher is “you.” • You have cubes. I have times as many cubes as you. (You have 3 cubes. I have 2 times as many cubes as you.) • I have

Put your three-stick next to a student’s array.

times as many as

. (I have 2 times as many as 3.)

Write the equation 6 = 2 × 3. Point to each number as you ask the following questions.

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons when Objective 1 | Describe using arrays. 10 M I NU T ES

It’s the total number of cubes I have. Two sticks of 3, 3 + 3.

It’s the number that tells us how many times we have to make a stick equal to yours.

It tells us how many times to repeat the unit, 2 threes. That tells how many in each of our sticks.

H 6 is 2 times as many as … 3 H

EV I

It tells us the unit that is being repeated.

Model how to draw the cube comparison while students do the same on the Student Page.

H When we describe multiplication as a comparison, the first factor tells us how many times to repeat the unit. The second factor tells us which unit is being repeated.

Direct students to problem 1 on their Student Page. NAME

DATE

6 is 2 times as many as 3.

Teacher Tip This mini lesson targets students describing multiplicative comparison expressions. The first factor indicates how many of the second factor are represented in the relationship. This order allows students to see the connection between the times as many language and the multiplication expression. The multiplicative comparison equation is lightly touched in this lesson and continues in Objective 2.

| Describe one-digit multiplication expressions as comparisons when using arrays. 2×3 6 is

3×2

times as many as

Use cubes to represent the expression. Then draw arrays to represent the cubes. Complete and read the sentence.

3 .

6 is

times as many as

10 is

times as many as

12 is

times as many as

2×5

4×3

Invite students to turn and talk about how they can describe multiplication expressions as comparisons with cubes, arrays, and words. Teacher Tip The phrase times as many is used throughout Objectives 1–4 when students are describing multiplicative comparison relationships between objects (e.g., cubes, blocks). Students should also practice describing multiplication as a comparison without a context. Provide multiplication equations, such as 18 = 3 × 6, and model describing the equation as 18 is 3 times as much as 6. Invite students to repeat the multiplicative comparison statement.

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons when Objective 1 | Describe using arrays. 10 M I NU T ES

• 3 × 2; 6 is • 2 × 5; 10 is • 4 × 3; 12 is

times as many as times as many as times as many as

. . .

EV I

Analyze Student Progress

Notes

Monitor: • Can the student correctly represent the unit that is repeated? • Can the student correctly represent how many times the unit is repeated? • Does the student correctly describe the multiplication expression as a comparison?

Questions to Advance Student Thinking: • Which factor represents the unit that is repeated? How can you show that with your cubes? • Which factor represents how many times the unit is repeated? How can you show that with your cubes? • How can you use times as many to describe the multiplication expression?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

epresent multiplicative comparison statements by using tape diagrams Objective 2 | Rand multiplication equations. 10 M I NU T ES

Give each student 8 cubes. Connect 2 of your cubes to create a two-stick.

Teacher

Student

H What multiplication equation represents the relationship between your cubes and my cubes? 8=4×2 Teacher Tip

EV I

H I have 2 cubes. If you have 4 times as many cubes, how many cubes do you have? Show with your cubes. Students show 4 sticks of 2 cubes.

Materials • Personal whiteboard • Objective 2 Student Page • Linking cubes • Sticky notes

Summary Students draw tape diagrams to relate multiplication equations to times as many comparisons.

H How many cubes do I have in all? 2

H How many times did you repeat the unit of 2? 4 times

H How can we describe the relationship between your cubes and my cubes using times as many? 8 is …? 4 times as many as 2.

In this mini 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 are 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.

H Let’s use sticky notes to represent this same comparison.

Teacher

Label one sticky note with 2.

H One sticky note represents 2 cubes. How many sticky notes do you need to show 4 times as many as 2? 4 sticky notes

Student

Concept Mini Lessons | Teacher Guide

epresent multiplicative comparison statements by using tape diagrams Objective 2 | Rand multiplication equations. 10 M I NU T ES

Write the equation 8 = 4 × 2.

H What does the product, 8, represent? It’s the total value of sticky notes I have.

We can draw 1 unit of 2.

Model drawing and labeling a tape with 1 unit of 2 . Label the tape with the letter T. Direct students to do the same.

It’s what I get if I skip-count the 4 sticky notes by their value of 2. It’s the number of sticky notes I have.

It’s the number of times the unit of 2 is repeated.

8=4×2

EV I

Direct students to create the comparison by using 4 sticky notes labeled with 2 each.

It’s the value of each sticky note. It’s the unit that is repeated.

Direct students to problem 1 on the Student Page. H How can we use the sticky notes to help us draw a tape diagram to represent 8 is 4 times as many as 2? We can use the value of your sticky note and my sticky notes to help us draw a tape diagram.

H I can ask myself, What can I draw to represent your sticky notes? What do you think? We can draw 4 units of 2.

Model drawing and labeling a tape with 4 units of 2 . Label the tape with the letter M. Direct students to do the same. H What is the total of the 4 units of 2? 8

Model labeling 8 as the total for M. Direct students to do the same.

Concept Mini Lessons | Teacher Guide

epresent multiplicative comparison statements by using tape diagrams Objective 2 | Rand multiplication equations. 10 M I NU T ES

Analyze Student Progress Monitor: • Can the student correctly represent the unit that is repeated? • Can the student correctly represent how many times the unit is repeated? • Does the student correctly write a multiplication equation to match the comparison statement?

H How can you tell that each unit in M has a value of 2 even though I didn’t label each unit? We know that T has a value of 2 and each unit in M is the same size

as the unit in T, so each unit in M has the same value as the unit in T. It’s like when we labeled our sticky notes all with 2 because they

each represented the 2 cubes, but the drawing is just those sticky notes all labeled with 8 as the total.

H How does the tape diagram show that 8 is 4 times as many as 2? There is 1 unit of 2 in T, and M shows that 4 units of 2 make 8.

EV I

There are 4 times as many units of 2 in M than in T, and the total

Questions to Advance Student Thinking: • Which number in the comparison statement represents how many times the unit is repeated? How can you show that in your tape diagram? • Which number in the comparison statement represents the unit that is repeated? How can you show that in your tape diagram? • How can you write an equation to represent the comparison statement?

for M is 8.

Invite students to turn and talk about how comparisons involving multiplication can be represented by using tape diagrams.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice with using sticky notes and drawing tape diagrams to represent multiplicative comparisons, as necessary. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

• 15 is 5 times as many as 3. • 40 is 8 times as many as 5. 40 = • 24 is 6 times as many as 4.

=5× × =

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons by using Objective 3 | Describe story contexts. 10 M I NU T ES

Direct students to problem 1 on the Student Page: Gesture to the expression to the right of the equal sign.

Materials • Objective 3 Student Page

Summary Students describe multiplication expressions as multiplicative comparison situations, represent them by using tape diagrams, and multiply to find the product.

= 4 × 3.

H Say the multiplication expression. 4 times 3

Language Support

Consider labeling a multiplication equation and a multiplicative comparison statement to clarify the meaning of each number.

EV I

H Another way to say this as a comparison is 4 times as much as 3. Repeat that. 4 times as much as 3

H What does the factor 4 represent? The 4 represents how many times the unit is repeated.

Teacher Tip

The phrase times as much is used in problem 1 because the multiplicative comparison does not include a context. As students describe multiplication expressions as multiplicative comparisons, listen for understanding of the meaning of each factor rather than focusing on the use of times as many versus times as much.

The second factor, 3, tells the unit being repeated.

Draw and label a unit of 3. Direct students to do the same.

Product

Unit being repeated How many times the unit is repeated

28 is 4 times as many as 7. Product

H Let’s use a tape diagram to represent the comparison. H

28 = 4 × 7

How many times the unit is repeated

Unit being repeated

The unit 3 being repeated 4 times

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons by using Objective 3 | Describe story contexts. 10 M I NU T ES

H How does your tape diagram show the unit being repeated? 3 is the unit being repeated because there are 4 units of 3. H How does your tape diagram show how many times the unit is repeated? The unit is repeated 4 times because there are 4 units of 3. H

I know it is 12 because 12 = 4 × 3.

12 is 4 times as many as 3.

3 3

Have students fill in the blanks of the multiplicative comparison statement in problem 1.

? 12 = 4 × 3

= 5 × 9.

H Say the multiplication expression. 5 times 9

H How can we say the expression as a comparison? 5 times as many as 9

Read aloud the word problem with the blanks: Eva has blocks. times as many blocks as Eva has. How many blocks Pablo has have? does

EV I

Write 12 = 4 × 3. Direct students to do the same.

Direct students to problem 2 on the Student Page: Gesture to the expression to the right of the equal sign.

Draw and label the unit 3 being repeated 4 times. Direct students to do the same.

Invite students to turn and talk about how to fill in the blanks. H

The second factor, 9, tells the unit being repeated. Eva has 9 blocks.

H Which factor represents how many times the number of Eva’s blocks is repeated? The first factor, 5, tells how many times the unit gets repeated. Pablo has 5 times as many blocks as Eva. H What will the product represent?

The number of blocks that Pablo has

Pablo will have more blocks than Eva. Pablo will have 5 times as

many blocks as Eva. 5 times 9 is more than 5.

Concept Mini Lessons | Teacher Guide

one-digit multiplication expressions as comparisons by using Objective 3 | Describe story contexts. 10 M I NU T ES

Eva

Pablo

Zara has times as many stickers as Ivan has. Ivan has have? stickers. How many stickers does

=4×8

•

Jayla reads times as many books as Ray reads. Ray reads read? _______ books. How many books does

45 = 5 × 9

Eva has 1 unit of 9, Pablo has 5 times as many units as Eva, and the

total amount Pablo has is 45.

Analyze Student Progress

Monitor: • Does the student correctly describe the multiplication expression as a comparison word problem? • Can the student correctly represent the unit that is repeated? • Can the student correctly represent how many times the unit is repeated?

EV I

=3×7

•

Fill in the blanks as students follow along. Read aloud the word problem with the numbers filled in. Have students use a tape diagram to represent the problem. Remind them to label each tape and find the product.

Invite students to turn and talk about how they can draw tape diagrams to represent times as many word problems. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work.

Questions to Advance Student Thinking: • How can you use the expression to help you fill in the blanks in the word problem? • Which factor represents the unit that is repeated? How can you show that in your tape diagram? • Which factor represents how many times the unit is repeated? How can you show that in your tape diagram? Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

two-digit by one-digit multiplication expressions as comparisons Objective 4 | Describe by using story contexts. 10 M I NU T ES

Direct students to problem 1 on the Student Page: Gesture to the expression to the right of the equal sign.

Materials • Objective 4 Student Page

Summary Students describe multiplication expressions as multiplicative comparison situations, represent them by using tape diagrams, and multiply to find the product.

= 4 × 23.

H Say the multiplication expression. 4 times 23

EV I

H Just as we can with one-digit multiplication expressions, we can say this as a comparison: 4 times as much as 23. Repeat that. 4 times as much as 23

The second factor, 23, tells the unit being repeated.

Draw and label a unit of 23. Direct students to do the same. H What does the factor 4 represent? The 4 represents how many times the unit is repeated. H What should we draw? The unit 23 being repeated 4 times

Draw and label the unit 23 being repeated 4 times. Direct students to do the same.

? 92 = 4 × 23 23 is the unit being repeated because there are 4 units of 23.

H How does your tape diagram show how many times the unit is repeated? The unit is repeated 4 times because there are 4 units of 23. H

The total of 4 twenty-threes is unknown.

Draw a bracket and label the unknown with a question mark. Have students do the same. H

We can multiply 4 and 23.

Concept Mini Lessons | Teacher Guide

two-digit by one-digit multiplication expressions as comparisons Objective 4 | Describe by using story contexts. 10 M I NU T ES

H Would it be more efficient to use repeated addition, an array, or the standard algorithm to solve this problem? How do you know? An array would take a long time to draw and count! The standard algorithm would be the most efficient.

Direct students to problem 2 on the Student Page: Gesture to the expression to the right of the equal sign.

= 3 × 19.

Have students turn and talk about the strategy they will use to find 4 × 23.

H Say the multiplication expression. 3 times 19

H How can we say the expression as a comparison? 3 times as many as 19

Read aloud the word problem with the blanks. Invite students to turn and talk about how to fill in the blanks.

Teacher Tip

Direct students to find 4 × 23.

H Which factor represents the number of points David scores? The second factor, 19, tells the unit being repeated. David scores 19 points.

EV I

Students might solve multiplication problems by using strategies such as repeated addition and drawing arrays. Although these strategies can be used to multiply accurately, they are usually not the most efficient choice. Support students in advancing their strategies. Ask them to identify similarities between their strategies and more efficient options.

H What is the value of the unknown? How do you know? I know it is 92 because 92 = 4 × 23.

Write 92 = 4 × 23. Direct students to do the same.

H I can ask myself, How can I use the words times as many to describe the equation? What do you think? 92 is 4 times as many as 23.

Have students fill in the blanks of the multiplicative comparison statement.

H Which factor represents how many times David’s number of points is repeated? The first factor, 3, tells how many times the unit gets repeated. Mia scores 3 times as many points as David. H

The number of points that Mia scores

H Will the number of points Mia scores be more or less than the number of points David scores? How do you know? Mia will score more points than David. Mia will score 3 times as

many points as David. 3 times 19 is more than 19.

Concept Mini Lessons | Teacher Guide

two-digit by one-digit multiplication expressions as comparisons Objective 4 | Describe by using story contexts. 10 M I NU T ES

Mia

Oka’s book has book has

= 6 × 18

times as many pages as Shen’s book. Shen’s pages. How many pages does book have?

•

Luke picks times as many apples as Liz picks. Liz picks apples. How many apples does pick?

? 57 = 3 × 19

Analyze Student Progress

EV I

= 5 × 44

•

David

David has 1 unit of 19, Mia has 3 times as many units as David, and

the total number Mia has is 57.

Invite students to turn and talk about how they can draw tape diagrams to represent times as many word problems. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work.

Questions to Advance Student Thinking: • How can you use the expression to help you fill in the blanks in the word problem? • Which factor represents the unit that is repeated? How can you show that in your tape diagram? • Which factor represents how many times the unit is repeated? How can you show that in your tape diagram? Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Multiplication as Multiplicative Comparison Answer Key Objective 2

1. Correctly drawn comparison; 6 is 2 times as many as 3

1. Correctly drawn tape

3. Correctly drawn comparison; 10 is 2 times as many as 5

3. Correctly drawn tape diagram. 8; 5

Objective 4 1. Correctly drawn tape diagram; 92; 4; 23

2. 45; 9; 5; Pablo

2. 57; 19; 3; Mia

4. 32; 4; 8; Jayla

4. 108; 6; 18; Luke

3. 21; 3; 7; Zara

3. 220; 5; 44; Oka’s

4. Correctly drawn tape diagram. 24; 6; 4

4. Correctly drawn comparison; 12 is 4 times as many as 3

2. Correctly drawn tape diagram. 15; 3

1. Correctly drawn tape diagram; 12; 12; 4; 3

EV I

2. Correctly drawn comparison; 6 is 3 times as many as 2

diagram

Objective 3

Objective 1

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication as Multiplicative Comparison Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication as Multiplicative Comparison Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

one-digit multiplication expressions as comparisons when Objective 1 | Describe using arrays. Use cubes to represent the expression. Then draw arrays to represent the cubes. Complete and read the sentence.

3×2

times as many as

EV I

6 is

2×3

6 is

times as many as

10 is

times as many as

2×5

4×3 12 is

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

epresent multiplicative comparison statements by using tape diagrams Objective 2 | Rand multiplication equations. Draw a tape diagram to represent 8 is 4 times as many as 2.

Draw a tape diagram to represent 15 is 5 times as many as 3. Then complete the equation.

Draw a tape diagram to represent 40 is 8 times as many as 5. Then complete the equation.

40 =

EV I

=5×

Draw a tape diagram to represent 24 is 6 times as many as 4. Then complete the equation.

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

one-digit multiplication expressions as comparisons by using Objective 3 | Describe story contexts. Draw a tape diagram to represent the equation and find the product.

=4×3

times as many as

Then fill in the blanks to describe the multiplication equation as a comparison.

Fill in the blanks to describe the multiplication equation as a comparison word problem.

2 Eva has

=5×9

EV I

Draw a tape diagram to represent the problem. Solve and complete the equation.

blocks. Pablo has

times as many blocks

as Eva has. How many blocks does

Zara has

=3×7

times as many stickers as Ivan has. Ivan has

have?

stickers. How many stickers does

have?

=4×8 Jayla reads

times as many books as Ray reads. Ray reads

books. How many books does

read?

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

two-digit by one-digit multiplication expressions as comparisons Objective 4 | Describe by using story contexts. Draw a tape diagram to represent the equation and find the product.

= 4 × 23

times as many as

Then fill in the blanks to describe the multiplication equation as a comparison.

Fill in the blanks to describe the multiplication equation as a comparison word problem. Draw a tape diagram to represent the problem. Solve and complete the equation.

EV I

= 3 × 19

David scores

points during the basketball game. Mia scores

times as many points as David. How many points does

= 5 × 44

3 has

times as many pages as Shen’s book. Shen’s book

Oka’s book has

score?

pages. How many pages does

book have?

= 6 × 18 Luke picks

times as many apples as Liz picks. Liz picks

apples. How many apples does

pick?

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

Practice | Multiplication as Multiplicative Comparison Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Describe

Objective 2 Represent

Objective 3 Describe

Objective 4 Describe

one-digit multiplication expressions as comparisons by using story contexts.

two-digit by one-digit multiplication expressions as comparisons by using story contexts.

Look for... • Can the student correctly

multiplicative comparison statements by using tape diagrams and multiplication equations.

Look for...

• Can the student correctly

represent the unit that is repeated?

EV I

one-digit multiplication expressions as comparisons when using arrays.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication as Multiplicative Comparison Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

• Can the student correctly

represent how many times the unit is repeated?

• Does the student correctly describe the multiplication

expression as a comparison?

• Can the student correctly represent how many times the unit is repeated?

• Does the student correctly write a multiplication equation to match the comparison statement?

Look for...

• Does the student correctly describe the multiplication expression as a comparison word problem? • Can the student correctly represent the unit that is repeated?

• Can the student correctly represent how many times the unit is repeated?

Look for... • Does the student correctly describe the multiplication expression as a comparison word problem? • Can the student correctly represent the unit that is repeated?

• Can the student correctly represent how many times the unit is repeated?

Practice | Teacher Guide

Practice | Multiplication as Multiplicative Comparison

Answer Key Practice Page 2

Practice Page 3

Practice Page 4

1. Correctly drawn comparison; 14 is 2 times as many as 7

1. Correctly drawn tape diagram; 20; 5

1. Correctly drawn tape diagram; 24; 24; 6; 4

1. Correctly drawn tape diagram; 98; 98; 7; 14

2. Correctly drawn tape diagram; 3; 4

2. Correctly drawn tape diagram; 18; 18; 2; 9

Correctly drawn tape;

3. Correctly drawn comparison; 16 is 2 times as many as 8

4. B, E

4. Correctly drawn comparison; 15 is 3 times as many as 5

3. Correctly drawn tape diagram; 54; 9; 6

3. 7; 4; spoons; Correctly drawn tape diagram; 28

EV I

2. Correctly drawn comparison; 12 is 4 times as many as 3

Practice Page 1

4. 3; 6; Sunday; Correctly drawn tape diagram; 18 5. Mia said that 5 is the product and 15 is how many times the unit, 3,

2. 6; 31; community garden; 186

3. 89; 2; the store; Correctly drawn tape; 178

4. 45; 5; the school; 225

Correctly drawn tape;

product is 15 and the

is repeated. The correct unit, 3, is repeated 5

times. 15 is 5 times as many as 3.

Practice | Teacher Guide

EV I

Student Edition | Printable Pages for students

Practice | Teacher Guide

NAME

DATE

one-digit multiplication expressions as comparisons when Practice Page 1 | Describe using arrays. Use cubes to represent the expression as a comparison. Then draw arrays to represent the cubes.

2×7

4×3 12 is

times as many as

2×8 16 is

3×5 15 is

EV I

14 is

Complete and read the sentence.

times as many as

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

multiplicative comparison statements by using Practice Page 2 | Represent tape diagrams and multiplication equations.

20 is 4 times as many as 5.

12 =

54 is 9 times as many as 6.

EV I

=4×

12 is 3 times as many as 4.

Draw a tape diagram. Then complete the equation.

Circle all the choices that describe the tape diagram.

A 56 = 7 × 8

B 56 = 8 × 7

C 56 is 7 times as many as 8. D 7 is 8 times as many as 56.

E 56 is 8 times as many as 7.

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

one-digit multiplication expressions as comparisons by Practice Page 3 | Describe using story contexts.

Draw a tape diagram to represent the equation and find the product.

Then fill in the blanks to describe the multiplication equation as a comparison.

=6×4

EV I

times as much as

=2×9

times as much as

Fill in the blanks to describe the multiplication equation as a comparison word problem. Draw a tape diagram to represent the problem. Solve and complete the equation.

=4×7

=3×6

Amy went fishing. She caught There are

times as many

spoons. The number of forks is

fish on Sunday as she caught on Saturday. She caught

times as many as the number of spoons.

fish on Saturday. How many fish did she catch

How many

are there?

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

one-digit multiplication expressions as comparisons by Practice Page 3 | Describe using story contexts. Mia incorrectly described the multiplication equation

as a comparison. Look at Mia’s work. What mistake did Mia make?

Mia’s Work

times as much as

EV I

=5×3

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

two-digit by one-digit multiplication expressions as Practice Page 4 | Describe comparisons by using story contexts. Draw a tape diagram to represent the equation and find the product.

= 7 × 14

times as many as

Then fill in the blanks to describe the multiplication equation as a comparison.

Fill in the blanks to describe the multiplication equation as a comparison word problem. Draw a tape diagram to represent the problem. Solve and complete the equation.

= 6 × 31

EV I

Mr. Davis has

plants in his garden. There are

times as many plants in the

community garden as there are in Mr. Davis’s garden. How many plants does

= 2 × 89

times as many baseball cards as Gabe has. Gabe has

The store has

baseball cards. How many baseball cards does

have?

= 5 × 45 The school sold Carla sold

times as many candy bars for the fundraiser as Carla sold. candy bars. How many candy bars did

sell?

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Practice | student Page

NAME

DATE

Practice Helper 1 describe multiplication expressions as comparisons.

3×5=

Look at the problem. Then look at the work. It shows how to

Use cubes to represent the expression as a comparison.

Then draw arrays to represent the cubes. Complete and read the sentence.

times as many as

Which factor represents

How can you use an array to

How can you use the

unit that is repeated?

how many times the unit is

represent the multiplication

words times as many as to

3×5

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Which factor represents the

repeated?

The second factor, 5,

represents the unit that is repeated.

expression as a comparison?

3×5

The first factor, 3, represents

how many times the unit is repeated.

that is repeated, 5. Then I can

I can use cubes to show the unit

unit 3 times. I can draw arrays to

3×5 15

use more cubes to repeat the

times as many as

describe the multiplication expression? times as many as

15 is 3 times as many as 5.

represent the cubes.

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Practice | student Page

NAME

DATE

Practice Helper 2 Look at the problem. Then look at the work. It shows how to

Draw a tape diagram. Then complete the equation.

28 is 7 times as many as 4.

draw a tape diagram and complete an equation to represent a comparison statement.

Which number in the comparison

How can you write an equation to

statement represents the unit that is

statement represents how many times

represent the comparison statement?

repeated? How can you show that in

the unit is repeated? How can you show

your tape diagram?

that in your tape diagram?

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Which number in the comparison

I can draw 1 unit of 4 on the first tape to show the

unit that is repeated.

28 = 7 × 4

I can draw 7 units of 4 on the second tape to

show how many times 4 is repeated.

28 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 3 =4×3

Look at the problem. Then look at the work. It shows how to based on a story.

describe one-digit multiplication expressions as comparisons

Miss Diaz and Mr. Lopez are planting trees. Miss Diaz

Fill in the blanks to describe the multiplication equation as a

comparison word problem. Draw a tape diagram to represent the problem. Solve and complete the equation.

has

seedlings. Mr. Lopez has

times as

many seedlings as Miss Diaz has. How many seedlings does

have?

Which factor represents

How can you use the

How can you represent the

unit that is repeated?

how many times the unit is

expression to help you fill

comparison using a tape

repeated?

in the blanks?

diagram?

=4×3

The second factor, 3, represents

the unit that is repeated.

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Which factor represents the

=4×3

The first factor, 4, represents how

many times the unit is repeated.

The second factor, 3, represents

Miss Diaz has. The first factor, 4, the number of seedlings that

seedlings Mr. Lopez has than Miss

the number of seedlings that

=4×3

Mr. Lopez has.

3 seedlings.

4 times as many seedlings as Miss Diaz has. How many

Miss Diaz and Mr. Lopez are planting trees. Miss Diaz has

Mr. Lopez has

Mr. Lopez

represents how many times more Diaz has. The product will represent

Miss Diaz

seedlings does Mr. Lopez have?

? 12 = 4 × 3

I can draw the unit 3 repeated

4 times. The product is 12. Mr. Lopez has 12 seedlings. Miss Diaz

Mr. Lopez

? 12 = 4 × 3 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

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Practice | student Page

NAME

DATE

Practice Helper 4 = 3 × 28

Look at the problem. Then look at the work. It shows how to comparisons based on a story.

describe two-digit by one-digit multiplication expressions as

The aquarium has

Fill in the blanks to describe the multiplication equation as a

has. Toby has

fish. How many fish does the

have?

comparison word problem. Draw a tape diagram to represent the problem. Solve and complete the equation.

times as many fish as Toby

Which factor represents

How can you use the

How can you represent the

unit that is repeated?

how many times the unit is

expression to help you fill

comparison using a tape

repeated?

in the blanks?

diagram?

= 3 × 28

The second factor, 28, represents

the unit that is repeated.

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Which factor represents the

= 3 × 28

The first factor, 3, represents how

= 3 × 28

many times the unit is repeated.

The aquarium has

The second factor, 28, represents

factor, 3, represents how many

Toby

Aquarium

how many fish Toby has. The first

times as many fish the aquarium has as Toby has. The product will

represent the number of fish the

84 = 3 × 28

aquarium has.

3 times as many fish as Toby has. Toby has 28 fish.

I can draw the unit 28 and repeat

it 3 times. The product is 84. The

aquarium has 84 fish. Toby

Aquarium

How many fish does the aquarium have?

? 84 = 3 × 28

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Practice | student Page

Application | Multiplication as Multiplicative Comparison Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw-Write Tool to support students as they solve problems involving multiplicative comparison. Two printable versions of the Read–Draw-Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

• Consider printing the cards on cardstock and laminating them for long-term use.

Solve a Task

Partner Work

• Consider providing cubes or sticky notes for students to use to represent the multiplicative comparison problems.

Activity

Application | Teacher Guide

Application | Multiplication as Multiplicative Comparison

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

• Game Instruction Card • Multiplicative Comparison Match Cards

Students match various representations of multiplicative comparison situations.

Preparing to Play • Print and cut out Multiplicative Comparison Match Cards. • Shuffle the cards and arrange them facedown in rows of six. Playing the Game • The players take turns flipping over three cards. • If the cards make a matching set with a multiplication equation, a tape diagram, and a times as many statement that all represent the same multiplicative comparison, the player takes the cards and goes again, turning over three more cards.

Students use the Read–Draw– Write process to solve word problems involving multiplicative comparisons. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problems 1 and 2 involve understanding times as many language to multiply one-digit numbers. Problem 3 involves understanding times as many language to multiply a two-digit number by a one-digit number and gives the unit first and then the number of times the unit is repeated.

Materials

• If the cards do not make a matching set, the player flips the cards back over. • The player with the most cards at the end of the game wins.

Materials

Play a Game: Multiplicative Comparison Match

Language Support

When a player finds a matching set of cards, consider having the player use the following sentence frames while pointing to the corresponding place on the cards. Players can then read aloud the times as many statement together. The unit being repeated is

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Solve a Problem

The unit is repeated The product is

times.

Solve a Task Materials

• Solve a Task Student Page • Cubes (optional) • Sticky notes (optional) Students work with a partner to solve a multi-part task involving multiplicative comparisons. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution. Teacher Tip Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Application | Teacher Guide

Application | Multiplication as Multiplicative Comparison

Answer Key Solve a Task

1. Accurate picture drawn to represent the problem; 3 × 7 = 21; Adam reads 21 poetry books.

2. Accurate picture drawn to represent the problem; 4 × 6 = 24; Mia scores 24 points in her last basketball game.

1. Each student needs 4 cups of sugar for the first cake.

Solve a Problem

2. No, I do not agree with Mr. Lopez. Students’ work and 2 cups of butter, is repeated one more time, so Mr. Lopez would add 2 cups to the number of cups of sugar to find explanations may vary but should show that the unit,

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the amount of flour each student needs.

3. Mr. Lopez needs to buy 48 eggs.

3. Accurate picture drawn to represent the problem; 8 × 14 = 112; There are 112 chairs in the cafeteria.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

Adam reads 3 times as many poetry books as chapter books. Adam reads 7 chapter books.

How many poetry books does Adam read?

Mia scores 4 times as many points in her last basketball

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scored 6 points in her first basketball game. How game as she did in her first basketball game. Mia

many points does Mia score in her last basketball game?

Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Application | Student Page

Application | Play a Game Multiplicative Comparison Match What You Need

• Multiplicative Comparison Match Cards How to Play

1. Mix up the cards. Place the cards facedown in rows of six.

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2. Take turns turning over three cards.

3. Look for a matching set of three cards that represent the

same multiplicative comparison: a multiplication equation, a tape diagram, and a times as many statement.

4. If the cards make a matching set, take the cards and go again, turning over three more cards.

15 is 3 times as many as 5.

15 = 3 × 5

5. If the cards do not make a matching set, flip the cards back over. It is the next player’s turn. How to Win The player with the most cards at the end of the game wins.

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Application | Student Page

12 = 4 × 3

15 6

15 = 3 × 5

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18 = 3 × 6 24 = 6 × 4

12 is 4 times as many as 3. 15 is 3 times as many as 5. 18 is 3 times as many as 6. 24 is 6 times as many as 4.

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Application | Student Page

45 = 5 × 9

42 8

42 = 6 × 7

EV I

56 = 7 × 8

70 = 7 × 10

45 is 5 times as many as 9. 42 is 6 times as many as 7. 56 is 7 times as many as 8. 70 is 7 times as many as 10.

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Application | Student Page

32 = 2 × 16

60 34

100

60 = 5 × 12

EV I

68 = 2 × 34

100 = 4 × 25

32 is 2 times as many as 16. 60 is 5 times as many as 12. 68 is 2 times as many as 34.

100 is 4 times as many as 25.

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Application | Student Page

NAME

DATE

Application | solve a Task Baking Cakes

Ingredient butter sugar flour eggs

Mr. Lopez teaches cake-baking classes. He uses the table to find the amount of ingredients for each cake. Amount cups

2 times as many cups of sugar as cups of butter 3 times as many cups of flour as cups of butter

4 times as many eggs as the number of cups of butter

EV I

To make the first cake, each student needs 2 cups of butter.

How many cups of sugar does each student need for the first cake?

Mr. Lopez thinks he can find the amount of flour each student needs by

adding 1 cup to the number of cups of sugar each student needs. Do you agree with Mr. Lopez? Use a tape diagram to explain your answer.

There are 6 students signed up for the first cake-baking class. How many eggs does Mr. Lopez need to buy so that each student can make their own cake?

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Application | student Page

Multiplication

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Multiplication of Two-Digit Numbers by One-Digit Numbers

Concept Guide | Multiplication of Two-Digit Numbers by One-Digit Numbers Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and Pause and Monitor Tool. • Ready place value disks as an optional manipulative.

• Concept Mini Lessons Teacher Guide • Personal whiteboard •

• • • • Base ten blocks (optional)

• Print copies of Student Pages for objectives 1, 2, and 3. Objective 4 Student Page is optional. • Gather: - Sets of 1 hundred, 20 tens, and 20 ones place value disks - Craft stick bundles or base ten blocks (optional)

• Application Teacher Guide

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•

Teacher Materials

• • Practice Helpers

•

• Personal whiteboard • • • Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • Two-color counters • Three in a Row Game Board • Study a Solution Student Page • Highlighters (optional) • Solve a Task Student Page • Manipulatives and tools such as place value disks or place value charts (optional)

• - Application Word Problem Cards - Eureka Math2 cards or a standard deck of playing cards - Game Instruction Card - Highlighters (optional) - Three in a Row Game Board in a personal whiteboard • Print copies of the following: - Solve a Problem Recording Page (optional) - Study a Solution Student Page - Solve a Task Student Page • Gather the following: - two-color counters - tools such as place value disks or place value charts (optional)

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Two-Digit Numbers by One-Digit Numbers

Addressing Student Misconceptions How to Address Misconception

Students do not use place value understanding when multiplying a two-digit number by a one-digit number (e.g., students incorrectly find the product of 47 and 2 because when they multiply the tens, they multiply 2 and 4 instead of multiplying 2 and 4 tens).

Unit form is used throughout Concept Mini Lessons to model how to attend to place value. For example, in objective 3 when recording partial products in vertical form, the teacher restates the process in unit form: “We multiplied 7 ones by 2. Now we need to multiply 4 tens by 2.”

Student Misconception

When students refer to the two-digit factor, encourage them to use unit form. For example, when referring to the digit 4 in 47, ask questions such as the following:

• What place is the 4 in? • How can you use unit form to think about the value of the 4?

× +

7 2

2 × 7 ones

2 × 4 tens

2 × 4 tens + 2 × 7 ones

Language Support

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Consider asking students to record the multiplication expression in unit form next to each partial product.

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The chart should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Multiplication of Two-Digit Numbers by One-Digit Numbers Dear Family,

by first?

8 × 3 ones = 24 ones

Can you rename a partial

What unit will you multiply

How many tens do we have in

product using a larger unit?

by next?

all? Can we rename the tens

How can you represent any

23 × 8 2 4

using a larger unit?

EV I

What unit will you multiply

23 8 2 184

Your student is working on multiplying two-digit numbers by one-digit numbers. This involves putting together a few skills: multiplication facts; decomposing—or breaking apart—numbers; finding partial products; and knowing when and how to compose a new unit. The sample work shows how to multiply 23 × 8 by using the standard algorithm. You can support your student’s progress by asking the questions in the table below as your student multiplies by using the standard algorithm.

renamed units?

23 × 8

24 ones can be renamed as 2 tens

I multiply the ones first.

23 8 2 4

4 ones.

I write regrouped units on the line under the correct place value.

23 × 8 2 184

Then I multiply the tens.

8 × 2 tens = 16 tens

the 2 tens on the line means it will

I add partial products together, so get added to any other tens.

16 tens + 2 tens = 18 tens

18 tens can be renamed as 1 hundred 8 tens.

Concept Guide | Teacher Guide

Progress Check | Multiplication of Two-Digit Numbers by One-Digit Numbers

Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

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• Can the student find the partial products and use them to determine the product? | Objectives 1–4

• Does the student use place value disks to find the product? | Objective 1

• Does the student use place value drawings to find the product?

| Objective 2

• Does the student record partial products in vertical form? | Objective 3

• Does the student use the standard algorithm to find the product? | Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Multiplication of Two-Digit Numbers by One-Digit Numbers

Progression Towards Proficiency Rubric Items 1 and 2

Item 3

Item 4

Items 5 and 6

Objectives 1–4

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand multiplication of two-digit numbers by one-digit numbers but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and circles the correct answer.

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly finds the products:

The student correctly multiplies and circles option A.

The student correctly multiplies and circles option H.

The student uses the standard algorithm to find the correct products:

Teacher Tip

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2. 65

1. 63

Progress Check Tool Item(s)

5. 126

6. 603

Students self-select strategies for problems 1–4 and are prompted to use the standard algorithm for problems 5 and 6. Observe the strategies that students use to multiply two-digit numbers by one-digit numbers to help you determine their proficiency level.

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Two-Digit Numbers by One-Digit Numbers Multiply. Use place value disks or the place value chart as needed.

tens

ones

Multiply. Show your work. Circle the letter of the correct answer.

42 × 3 =

A 126 B 128 C 138

D 192

5 × 13 =

3 × 21 =

EV I

tens

ones

4 × 26 = F

G 84

H 104 J

824

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Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Two-Digit Numbers by One-Digit Numbers Multiply. Use the standard algorithm.

7 × 18 =

67 × 9 =

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Progress Check | student Page

of Two-Digit Numbers by One-Digit Concept Mini Lessons | Multiplication Numbers Progression of Mini Lesson Objectives 2 Multiply by using place value

models and the distributive property.

drawings and partial products.

tens 1

ones

1 10

Start here if students

partial products

2 4

2 × 7 ones

2 × 4 tens

2 × 4 tens + 2 × 7 ones

Start here if students • can draw models to represent multiplication, but • need support recording partial products in vertical form.

Start here if students

• use concrete models to represent multiplication, but • need support drawing models to represent multiplication, and • need support using partial products to determine the product.

4 Multiply by using the standard algorithm.

36 × 4 2 144 Start here if students • can record partial products in vertical form, but • need support multiplying using the standard algorithm.

• can multiply single digit factors, • can interpret the meaning of multiplication as “groups of,” but • need support using concrete models to represent multiplication, and • need support using equations to record the distributive property.

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3 Multiply by recording partial products in vertical form.

1 Multiply by using concrete

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply by using concrete models and the distributive property. 10 M I NU T ES

Invite students to draw a three-column unlabeled chart on their whiteboards. Direct students to problem 1 on the Student Page. Teacher Tip: Differentiation

H Let’s use place value disks to find the product of 2 and 36. 3 tens 6 ones

H How do you see the factor 36 represented on the chart? Each group has 3 tens 6 ones.

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If using place value disks poses a challenge for students, have them use a proportional model such as craft stick bundles or base ten blocks.

H Say 36 in unit form.

Materials • Personal whiteboard • Place value disks • Objective 1 Student Page • Craft stick bundles (optional) • Base ten blocks (optional)

Summary Students use place value disks and complete equations to use the distributive property to multiply.

Represent 3 tens 6 ones with place value disks as students do the same. Encourage students to organize their disks in 5-groups. H How many groups of 36 do we need to show? How do you know? We need to show 2 groups, because 2 × 36 means 36 + 36.

Represent another group of 3 tens 6 ones with place value disks as students do the same.

Point to the second factor, 36, in the expression as you ask the following question. Encourage students to verbalize or use gestures to answer.

1 10

1 1

H The size of each group is 36. How did we use place value units to decompose 36? We decomposed 36 into 3 tens and 6 ones.

Direct students’ attention to the Student Page. Fill in the blanks for tens + ones) to show how 36 is decomposed into (2 × 3 tens and 6 ones. Direct students to do the same.

Point to the first factor, 2, in the expression as you ask the following question. Encourage students to verbalize or use gestures to answer. H How do you see the factor 2 represented on the chart? There are 2 groups.

H The place value disks show 2 groups of 3 tens and 2 groups of 6 ones.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply by using concrete models and the distributive property. 10 M I NU T ES

ones) with

Point to the expression 2 × 3 tens.

Language Support

H In unit form, what is 2 × 3 tens? 6 tens

Write 6 in the blank in (

tens), as students do the same.

Point to the expression 2 × 6 ones.

Consider highlighting the word part in partial products to help students recognize a familiar word that can support their understanding of partial products.

Write 60 and 12 in the blanks in H How do we use the partial products to get the product? We add 60 and 12 to get 72.

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H In unit form, what is 2 × 6 ones? 12 ones

Write 12 in the blank in (

H What is 12 ones in standard form? 12

Model filling in the blanks in (2 × tens) + (2 × 3 tens and 6 ones. Direct students to do the same.

ones), as students do the same.

H When we decompose a factor and multiply each part by another factor, we get partial products. Our equations show the partial products of 6 tens and 12 ones. On the chart I see the partial product of 60 because there are 2 groups of 3 tens, which is 6 tens, or 60. H Where do you see another partial product on the chart? I see the partial product of 12 ones, or 12, because there are 2 groups of 6.

H Let’s record the partial products in standard form in the equation. What is 6 tens in standard form? 60

H We add the partial products to get the product.

Write 72 in the blank in =

as students do the same.

2 × 36 = 2 × ( 3 tens + 6 ones)

= (2 × 3 tens) + (2 × 6 ones) = ( 6 tens) + ( 12 ones) = 60 + 12 = 72

as students do the same.

H What is 2 × 36? 72

Invite partners to think–pair–share about the similarities and differences between the chart and the equations they recorded.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply by using concrete models and the distributive property. 10 M I NU T ES

36 into tens and ones. 3 tens 6 ones by 2.

The chart and the equations both show how we multiplied

The chart and the equations both show the partial products.

H We can see the break apart and distribute strategy and the partial products on the chart and in our equations. They show the same partial products, but the chart uses objects, and the equations use numbers.

H How do you see the product, 72, on the chart? There are 7 tens and 2 ones, which is 72.

Invite students to turn and talk about how they can use place value disks and the break apart and distribute strategy to multiply a two-digit number by a one-digit number.

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Teacher Tip

Invite students to make that exchange with their place value disks. Encourage students to place the new unit of 1 ten below the 6 tens.

The chart and the equations both show how we decomposed

The application of the distributive property is referred to as the break apart and distribute strategy. Naming the distributive property in this way helps students remember the process of breaking apart a factor and distributing the other factor to both parts.

Return students’ attention to the chart.

H I can ask myself: Can I compose a larger place value unit with either of the partial products? What do you think? Yes. We can rename 12 ones as 1 ten and 2 ones. Yes. We can regroup 10 ones to compose 1 ten.

If students need support understanding what to do when there are 0 ones left in the ones place after composing a ten, give them the problem 2 × 35.

Teacher Tip: Differentiation

H I have enough ones to compose a larger unit. I can exchange 10 ones for 1 …? Ten

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply by using concrete models and the distributive property. 10 M I NU T ES

Notes

Analyze Student Progress

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Questions to Advance Student Thinking: • How can you represent both factors with your disks? • Where do you see the partial products? • What can you do with the partial products to find the product?

Monitor: • Can the student correctly represent both factors on the chart? • Can the student identify the partial products? • Can the student use the partial products to determine the product? • Can the student use place value disks and complete the equations to use the distributive property to multiply?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply by using place value drawings and partial products. 10 M I NU T ES

Write 36 × 2. Draw a labeled two-column place value chart.

The expression 36 × 2 is intentionally used as a bridge from objective 1 where students represented 2 × 36 concretely with place value disks.

Teacher Tip

Materials • Personal whiteboard • Objective 2 Student Page

Summary Students draw on a place value chart and write equations with partial products to multiply.

Consider creating an anchor chart that highlights the commutative property of multiplication by showing that changing the order of the factors results in the same product.

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H I can represent this expression by drawing on the place value chart. How many groups of 2 should I draw?

Language Support

36 groups of 2

Start drawing groups of 2 in the ones column. After drawing 4 groups of 2, pause.

H It is going to take a while to draw 36 groups of 2. The commutative property of multiplication tells me that I can change the order of the factors, 36 and 2, and still get the same product. How can the commutative property of multiplication help me? You can draw 2 groups of 36 instead.

2 4 6 8 10

5 10

5×2=2×5

Direct students to problem 1 on the Objective 2 Student Page. H Say 36 in unit form. 3 tens 6 ones

H Represent one group of 36 on your place value chart.

Draw a group of 3 tens 6 ones as students do the same. H How many groups of 36 do we need to draw? How do you know? We need to draw 2 groups because 2 × 36 means 36 + 36.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply by using place value drawings and partial products. 10 M I NU T ES

Draw another group of 3 tens 6 ones as students do the same.

Teacher Tip

H I can ask myself: Where do I see the partial products on the place value chart? Where do you see them? In the ones place, I see 12 ones. There are 2 groups of 6 ones. In the tens place, I see 6 tens. There are 2 groups of 3 tens.

H How do the partial products help us find the product of 2 and 36? We can add the partial products together. 60 + 12 = 72 H So, 2 × 36 is …? 72

Have students complete the equation. Return students’ attention to the place value chart.

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H Yes, in the ones place, we see the partial product of 12. In the tens place, we see the partial product of 60.

Write 12 below the ones column on the place value chart and 60 below the tens column on the place value chart as students do the same. Label them with the term partial products.

H Look at our place value chart. Can we compose a larger place value unit with either of the partial products? How? Yes. We can regroup 10 ones to make 1 ten.

Circle 10 ones, draw an arrow pointing to the tens column, and draw a new unit of 1 ten below the 6 tens as students do the same.

ones

tens

The decision to record ones before tens helps students to move toward the standard algorithm. If students prefer to work from left to right, they may choose to record partial products in the tens place first.

H On the place value chart, we used place value units to decompose one factor, 36, into 3 tens 6 ones. When we multiply each of those parts by the other factor, 2, we get partial products.

tens

ones

partial products For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply by using place value drawings and partial products.. 10 M I NU T ES

Analyze Student Progress

Invite students to turn and talk about how they can draw on a place value chart to find partial products.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper to support students in understanding the worked-out example.

Questions to Advance Student Thinking: • How can the commutative property help you represent the problem in a different way? • How can you represent both factors on the place value chart? • Where do you see the partial products on the place value chart? • What can you do with the partial products to find the product?

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• 52 × 3 • 4 × 42 • 34 × 4

Monitor: • Can the student use the commutative property to represent the problem in a different way • Can the student correctly represent both factors on the place value chart? • Can the student identify the partial products on the place value chart? • Can the student use the partial products to determine the product?

H How do you see the product of 72 on the place value chart now? I see 7 tens and 2 ones.

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply by recording partial products in vertical form. 10 M I NU T ES

Materials • Personal whiteboard • Objective 3 Student Page

Summary Students use vertical form to record partial products.

Distribute the Objective 3 Student Page. Direct students to problem 1.

Teacher Tip: Differentiation

tens

ones

H What is the value of 14 ones? 14

If students need a different pictorial support, provide an area model and ask the following questions to support students in using place value units to find and record partial products:

40 80

7 14

H 2 × 7 ones equals how many ones? 14 ones

Record 14 as the first partial product as students do the same. Invite students to make connections between the partial product in vertical form and the partial product on the place value chart.

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H How is 2 × 47 represented with a place value drawing? There are 2 groups of 4 tens 7 ones.

Gesture to vertical form. Have students point to the unit form.

• How does the area model show how we break apart, or decompose, one factor? • How can you think about that same decomposition when recording partial products in vertical form? • Where do you see the partial products represented in the area model? In vertical form? • How can you find the product in the area model? In vertical form?

H Let’s use vertical form to record the partial products.

H I can ask myself: What unit is next to multiply by 2? What do you think? Tens

H We multiplied 7 ones by 2. Now we need to multiply 4 tens by 2.

Gesture to vertical form. Have students point to the unit form. H 2 × 4 tens equals how many tens? 8 tens

4 × +

7 2

2 × 7 ones

2 × 4 tens

2 × 4 tens + 2 × 7 ones

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply by recording partial products in vertical form. 10 M I NU T ES

H What is the value of 8 tens? 80

H I can ask myself: Are there any units left to multiply by 2? No, tens is the largest unit in 47.

H What can we do with the partial products to find 2 × 47? We can add them.

Direct students to add the partial products.

H What is 14 + 80? 94

ones

H I do not see enough

tens in the tens place to regroup or make a larger unit. How do you see the product of 94 on the place value chart? I see 9 tens and 4 ones.

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H We multiplied 7 ones by 2 and 4 tens by 2 to get the partial products. What are the partial products? 14 and 80

tens

Record 80 as the second partial product as students do the same. Invite students to make connections between the partial product in vertical form and the partial product on the place value chart.

Circle 10 ones, draw an arrow pointing to the tens column, and draw a new unit of 1 ten below the 8 tens as students do the same.

H So, what is 2 × 47? 94

Invite students to turn and talk about how they can use place value units to record partial products in vertical form. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work. • 28 × 3 • 2 × 65 • 41 × 5

Invite students to look at the place value chart to confirm their answer. Model a think aloud. H When I look at our place value chart, I see enough ones in the ones place to regroup or make a larger unit.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply by recording partial products in vertical form. 10 M I NU T ES

Teacher Tip: Differentiation

8 ×

6 4 4

4 × 6 ones

+ 3

4 × 8 tens

Monitor: • Can the student use place value understanding to find the partial products? • Can the student correctly record the partial products in vertical form? • Can the student use the partial products to determine the product? Questions to Advance Student Thinking: • What unit will you multiply by first? Next? • How can you record the value of ones in vertical form? tens? • What can you do with the partial products to find the product?

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Analyze Student Progress

If students need more support with aligning place value units vertically, encourage them to use grid paper to record partial products

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 4 | Multiply by using the standard algorithm. 10 M I NU T ES

Show the solution to 36 × 4 using partial products.

4 × 6 ones 4 × 3 tens 4 × 3 tens + 4 × 6 ones

Teacher Tip

For the standard multiplication algorithm, digits that represent newly composed/regrouped units are written on the line. This means that each two-digit partial product is written with the digits close together. Students multiply the original digits first and then add the new units.

H We write renamed units on the line under the correct place value. We add partial products together, so the 2 tens on the line means it will get added to any other tens.

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36 × 4 24 +120 144

Materials • Personal whiteboard • Objective 4 Student Page (optional)

Summary Students use the standard algorithm to multiply.

H The work shows how to record partial products in vertical form. Now, let’s use the standard algorithm to show how to combine partial products and record them on one line.

Write 36 × 4 vertically and have students do the same. Gesture to 4 × 6 ones.

H Let’s record the partial product for the ones. 4 × 6 ones is 24 ones. When we show our work using the standard algorithm, we can only write one digit in each place. We can rename 24 ones with larger units as 2 tens 4 ones. Watch how I record 2 tens 4 ones to leave space for recording the partial product for the tens on the same line.

Write a small 2 to represent 2 tens on the line under the tens place. Write 4 below the line in the ones place. Direct students to do the same.

36 × 4 2 4

H Now we have room to record the partial product for the tens.

Gesture to 4 × 3 tens.

H 4 × 3 tens is 12 tens. We also have 2 tens we regrouped from multiplying the ones. How many tens do we have in all? 14 tens

H Yes. There are 14 tens, because 12 tens plus 2 tens is 14 tens. Can the 14 tens be renamed with a larger unit? Yes, 14 tens can be renamed as 1 hundred 4 tens. H Watch how I record 1 hundred 4 tens to show that I already added the renamed 2 tens.

Concept Mini Lessons | Teacher Guide

Objective 4 | Multiply by using the standard algorithm. 10 M I NU T ES

H I can ask myself: Starting with the ones place, what should I do first? What do you think? Multiply 5 times 8 ones to get 40 ones.

H How does the recording of the standard algorithm show the partial products 24 ones and 12 tens? 24 ones is recorded as 2 tens 4 ones. We wrote the 2 on the line in

Write a small 4 to represent 4 tens on the line under the tens place. Write 0 below the line in the ones place. Direct students to do the same.

H What is 36 × 4? 144

the tens place to show that we renamed it.

36 × 4 2 144

H 40 ones is 4 tens 0 ones, so we record 4 tens on the line in the tens place. We write 0 ones in the ones place, so it still shows 40 ones.

H We add partial products together, so the 4 tens on the line means to add it to any other tens. I can ask myself: Moving to the tens place, what do I do next? What do you think? Multiply 5 times 1 ten and get 5 tens.

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We wrote the partial product for the tens on the same

Cross out the 2 on the line. Below the line in the hundreds place, write 1 to represent 1 hundred. Below the line in the tens place, write 4 to represent 4 tens. Direct students to do the same.

line as the partial product for the ones.

When we multiplied by tens, we added 12 tens to the

2 tens from the other partial product.

H We recorded the partial products on the same line, except for the renamed 2 tens. We added the partial product of 12 tens to the renamed 2 tens to get 14 tens. When we record with the standard algorithm, it takes fewer steps to solve.

Write the expression 18 × 5 vertically. Instruct students to do the same. Language Support

Students are already familiar with the standard algorithms for addition and subtraction. Consider asking them what step-by-step procedure they already know for adding: “What is the same about the algorithm for addition and the algorithm for multiplication?”

18 × 5 4 0

H How many tens do we have in all? How do you know? 9 tens. We have 5 tens and then 4 more tens.

H 5 × 1 ten is 5 tens. Remember that we have 4 tens we renamed from multiplying the ones. So 5 tens plus 4 tens is 9 tens. H How do we show that we already added the renamed 4 tens? We can cross out the 4 because we already added it in.

Concept Mini Lessons | Teacher Guide

Objective 4 | Multiply by using the standard algorithm. 10 M I NU T ES

18 × 5 4 90

H What is 18 × 5? 90

Analyze Student Progress

Cross out the 4 on the line. Below the line in the tens place, write 9 to represent 9 tens. Direct students to do the same.

Invite students to turn and talk about how they can use the standard algorithm to multiply.

Questions to Advance Student Thinking: • What unit will you multiply by first? Next? • Can you rename a partial product with a larger unit? • How can you represent any renamed units? • How many tens are there in all? Can you rename the tens with a larger unit?

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Repeat the process: Use the following problems during Concept Mini Lessons or come back at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work.

Monitor: • How does the student represent the partial products using the standard algorithm? • How does the student represent renaming units? • Does the student add any renamed tens to the partial product for the tens place?

• 4 × 72 • 53 × 6

Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

of Two-Digit Numbers by One-Digit Concept Mini Lessons | Multiplication Numbers Answer Key Objective 2

Objective 3

Objective 4

1. 3, 6 3, 6 6, 12 60, 12 72

1. 72; 60 + 12 = 72

1. 94

1. 144

3. 130

3. 288

2. 156; 150 + 6 = 156 3. 168; 160 + 8 = 168

4. 136; 120 + 16 = 136

4. 205

2. 90

4. 318

3. 3, 1 3, 1 18, 6 180, 6 186

2. 84

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2. 2, 4 2, 4 8, 16 80, 16 96

Objective 1

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Two-Digit Numbers by One-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Two-Digit Numbers by One-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Multiply by using concrete models and the distributive property. 2 × 36 = 2 × ( = (2 × =( =

4 × 24 = 4 × ( = (4 × =( =

6 × 31 = 6 × ( = (6 × =( =

ones)

tens +

tens) + (4 ×

tens) + (

ones)

tens) + (2 ×

tens) + (

tens +

EV I

tens +

tens) + (6 ×

tens) + (

Multiply. Use place value disks and complete the equations.

ones) ones)

ones)

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Multiply by using place value drawings and partial products. 36 × 2 =

tens

ones

EV I

Represent the problem by drawing dots on the place value chart. Then, complete the equations.

52 × 3 =

tens

ones

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Multiply by using place value drawings and partial products. 4 × 42 =

ones

tens

EV I

34 × 4 =

tens

ones

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Multiply by recording partial products in vertical form. Multiply. Write the problem vertically and record the partial products.

2 × 47 4

tens

7 2 × 7 ones

8 3

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2 × 4 tens

2 × 65

41 × 5

ones

28 × 3

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 4 | Multiply by using the standard algorithm. Multiply. Use the standard algorithm.

36 × 4

4 × 72

53 × 6

4 × 6 ones 4 × 3 tens 4 × 3 tens + 4 × 6 ones

18 × 5

36 4 24 +120 144

EV I

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

Practice | Multiplication of Two-Digit Numbers by One-Digit Numbers Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Multiply by

Objective 2 Multiply by

Objective 3 Multiply by

Objective 4 Multiply by using

using place value drawings and partial products.

recording partial products in vertical form.

the standard algorithm.

Look for...

• Can the student use the

• Can the student use place value understanding to find the partial products? • Can the student correctly record the partial products in vertical form? • Can the student use the partial products to find the product?

Look for...

commutative property to

represent the problem in a different way? • Can the student correctly represent both factors on the place value chart? • Can the student identify the partial products on the place value chart? • Can the student use the partial products to find the product?

• Can the student use concrete models to represent multiplication? • Can the student identify the partial products on the place value chart? • Can the student use place value disks and complete the equations to use the distributive property to multiply? • Can the student use the partial products to find the product?

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using concrete models and the distributive property.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of Two-Digit Numbers by One-Digit Numbers Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for... • Can the student represent the partial products using the standard algorithm? • Can the student correctly represent renaming units? • Does the student add any renamed tens to the partial product for the tens place?

Practice | Teacher Guide

Practice | Multiplication of Two-Digit Numbers by One-Digit Numbers

Answer Key Practice Page 2

Practice Page 3

Practice Page 4

1. 2, 4 2, 4 6, 12 60, 12 72

1. 56; 40 + 16 = 56

1. 96

1. 63

3. 208

3. 364

2. 153; 150 + 3 = 153

3. 140; 120 + 20 = 140

2. 2, 1 2, 1 12, 6 120, 6 126

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2. 81

4. Ray incorrectly multiplied the tens. 3 × 6 tens = 18 tens, which is 180, not 18.

2. 84

4. C

3. 3, 5 3, 5 12, 20 120, 20 140

4. 215; 200 + 15 = 215

Practice Page 1

Practice | Teacher Guide

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Student Edition | Printable pages for students

Practice | Teacher Guide

NAME

DATE

Practice Page 1 | Multiply by using concrete models and the distributive property. 3 × 24 = 3 × ( = (3 × =( =

6 × 21 = 6 × ( = (6 × =( =

4 × 35 = 4 × ( = (4 × =( =

ones)

tens +

tens) + (6 ×

tens) + (

ones)

tens) + (3 ×

tens) + (

tens +

EV I

tens +

tens) + (4 ×

tens) + (

Multiply. Use place value disks and complete the equations.

ones) ones)

ones)

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 2 | Multiply by using place value drawings and partial products. 2 × 28 =

tens

ones

EV I

Represent the problem by drawing dots on the place value chart. Then, complete the equations.

51 × 3 =

tens

ones

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 2 | Multiply by using place value drawings and partial products. 4 × 35 =

ones

tens

EV I

43 × 5 =

tens

ones

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 3 | Multiply by recording partial products in vertical form. Multiply. Write the problem vertically and record the partial products.

2 × 48 =

tens

ones

27 × 3 =

4 ×

8 2

EV I

2 × 8 ones +

2 × 4 tens

2 × 4 tens + 2 × 8 ones 4 × 52 =

Ray found an incorrect product of 3 and 64. Look at

Ray’s work. What mistake did Ray make? Ray’s Work

3 2

1 1 3

8 0

× +

3 × 64 = 30

3 × 4 ones 3 × 6 tens 3 × 6 tens + 3 × 4 ones

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Practice | student Page

NAME

DATE

Practice Page 4 | Multiply by using the standard algorithm. 21 × 3

6 × 14

52 × 7

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Multiply. Use the standard algorithm.

5 × 96 A 75

B 450 C 480

Multiply. Use the standard algorithm. Circle the letter of the correct answer.

D 750

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Practice | student Page

NAME

DATE

Practice Helper 1 3 × 25 = 3 × (

Look at the problem. Then look at the work. It shows how

= (3 ×

to use place value disks and the break apart and distribute

strategy to multiply.

How can you represent both factors

Can you make a new unit?

with your disks? 10

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I can exchange 10 ones for 1 ten.

I can show 25 + 25 + 25, or 3 groups of

2 tens 5 ones.

7 tens

5 ones 75

tens +

tens) + (3 ×

tens) + (

ones) ones)

ones)

Where do you see the partial products? What can you do with them to find the product?

3 × 25 = 3 × ( 2 tens + 5 ones)

= (3 × 2 tens) + (3 × 5 ones) = ( 6 tens) + ( 15 ones) = 60 + 15 = 75

The partial products are 60 and 15.

60 + 15 = 75, so 3 × 25 = 75.

3 × 25 = 3 × ( 2 tens + 5 ones)

= (3 × 2 tens) + (3 × 5 ones) = ( 6 tens) + ( 15 ones) = 60 + 15

= 75

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Practice | student Page

NAME

DATE

Practice Helper 2 65 × 2 =

Look at the problem. Then look at the work. It shows how

to draw on the place value chart and use partial products to determine the product.

How can the commutative property

How can you represent both factors on

Where do you see the partial products?

help you to represent the problem in a

the place value chart?

What can you do with them to find the

different way?

65 × 2 =

1 group of 6 tens 5 ones

2 × 65 means 65 + 65, or 2 groups of 65.

ones

130

product? tens

ones

2 groups of 6 tens 5 ones

I can switch the order of the factors.

ones

EV I

65 × 2 = 2 × 65

tens

120

130

I see the partial product of 10 in the ones

place. I see the partial product of 120 in the

tens place. I add the partial products to find the product.

130

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 3 use vertical form to record partial products.

46 × 2 =

Look at the problem. Then look at the work. It shows how to

What unit will you multiply by first?

What unit will you multiply by next?

How can you record the partial products

46 × 2 12

46 × 2 12 +80

in vertical form? What can you do with

2 × 6 ones = 12 ones

I multiply the ones first.

2 × 4 tens = 8 tens

Then I multiply the tens.

46 × 2 12 +80 92

2 × 6 ones 2 × 4 tens 2 × 4 tens + 2 × 6 ones

I can add the partial products to find the product.

46 × 2 =

2 × 6 ones 2 × 4 tens

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2 × 6 ones

them to find the product?

2 × 6 ones 2 × 4 tens 2 × 4 tens + 2 × 6 ones

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to

use the standard algorithm to multiply.

23 8

What unit will you multiply by

Can you rename a partial

What unit will you multiply by

How many tens do we have in

first?

product as a larger unit?

next?

all? Can we rename the tens

How can you represent any

23 × 8 2 4

as a larger unit?

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8 × 3 ones = 24 ones

renamed units?

23 × 8

I multiply the ones first.

23 8 2 4

8 × 2 tens = 16 tens

Then I multiply the tens.

24 ones can be renamed as 2 tens 4 ones. I write the

renamed units on the line

under the correct place value.

23 × 8 2 184 the 2 tens on the line means it will

I add partial products together, so get added to any other tens.

16 tens + 2 tens = 18 tens

18 tens can be renamed as 1 hundred 8 tens.

23 × 8 2 184 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Practice | student Page

Application | Multiplication of Two-Digit Numbers by One-Digit Numbers Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw-Write Tool to support students as they solve problems involving multiplying two-digit numbers by one-digit numbers. Two printable versions of the Read–Draw-Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

• Consider using a standard deck of playing cards if you do not have Eureka Math2 cards.

Study a Solution

Independent Work Partner Work

• Consider providing highlighters and other tools for students to use to annotate the sample solution.

Solve a Task

Partner Work

Activity

• Consider providing manipulatives, such as place value disks, for students to use to represent and solve the multiplication problems.

Application | Teacher Guide

Application | Multiplication of Two-Digit Numbers by One-Digit Numbers

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) Students use the Read–Draw –Write process to solve word problems involving multiplying two-digit numbers by one-digit numbers. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problems 1 and 2 involve regrouping once. Problem 3 involves regrouping twice. Teacher Tip

Playing the Game • Players take turns turning over cards to generate one-digit and two-digit numbers for the game board. They write the two-digit numbers in the left column and the three one-digit numbers in the top row. A player turns over an additional card if any of the one-digit numbers are the same. • Player A places a counter in an empty space on the game board. The player multiplies the number in the left column by the number in the top row and shows their work and answer below the game board.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor students’ progress and provide support as needed.

Preparing to Play • Have students play the game with a partner. • Remove the J, Q, K, and Joker from the deck. Aces represent 1. • Shuffle the remaining cards. Place in a single facedown pile. • Each player selects one side of the two-color counters to use as their game pieces or chooses unique objects as counters.

Play a Game: Three in a Row Materials

• Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card

• Player B checks Player A’s work. If both players agree on the answer, the counter stays in the space. If the players disagree, they share their work, identify the error, and the counter is returned to the player. • Player B erases the work, chooses a new space, and multiplies. Player A checks Player B’s work, following the same process. • Players continue taking turns until a player announces Three in a Row or until the game board is filled.

Materials

• Two-color counters • Three in a Row Game board in a personal whiteboard

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Solve a Problem

Teacher Tip: Differentiation

If students need either concrete or pictorial support, provide place value disks or encourage them to make place drawings or use an area model.

Study a Solution Materials

• Study a Solution Student Page • highlighters (optional) Students work independently or with a partner to analyze a correct solution to a word problem involving multiplying a two-digit number by a one-digit number.

Students answer questions about how the known and unknown information in the problem are represented in the sample solution. They also analyze how the sample drawing provides a solution path. Finally, they are asked to consider whether the sample statement answers the question in the word problem.

Solve a Task Materials

• Solve a Task Student Page Students work with a partner to solve a multi-part task involving multiplying two-digit numbers by one-digit numbers. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Application | Teacher Guide

Application | Multiplication of Two-Digit Numbers by One-Digit Numbers

Answer Key Study a Solution

1. Accurate picture drawn to represent the problem; 3 × 16 = 48; Robin uses 48 beads in all.

1. The known information is

2. Accurate picture drawn to represent the problem; 4 × 52 = 208; James saves $208 to buy a bike.

2. The unknown information is

Solve a Problem

of 48 in the tape diagram.

represented by the labeled part

partitioned into 3 equal parts. They

EV I

represented by the tape diagram

3. Accurate picture drawn to represent the problem; 5 × 35 = 175; Pablo practices for 175 minutes each

week.

are labeled with a question mark.

3. The drawing helps me see that if I multiply the known part by 3, I can find the unknown product.

4. Yes. The question is about how far Deepa bikes in all, and the

Solve a Task 1. Yes, the family has enough money student work shows 3 × $35= $105 for every person to get a ticket; and 3 × $27 = $81, then

$105 + $81 = $186.

2. No, the family does not have enough money for each person to get a snack and a drink; student family needs $30.

work may vary and shows that the

3. Answers may vary and depend on how many drinks and/or snacks are purchased and shared.

144 kilometers.

statement says that Deepa bikes

Application | Teacher Guide

Application | Solve a Problem Word Problem Cards

Robin makes a bracelet using 16 beads.

She makes a bracelet for each of her 3 friends.

How many beads does she use in all?

James saves $52 dollars every month to buy a bike. If James saves money for 4 months,

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how much money does he have to buy a bike? Pablo practices violin for 35 minutes a day. He practices 5 days a week. How many

minutes does Pablo practice each week? For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Application | Teacher Guide

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Student Edition | Printable pages for students

Application | Teacher Guide

NAME

DATE

Application | Solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Application | Student Page

Application | Play a Game Game Instruction Card

Three in a Row

What You Need

Three in a Row Multiplication Game Board

• Eureka Math2 cards (or a standard deck of playing cards) represent 1.

with the J, Q, K, and joker cards removed. Aces can • Two-color counters

• Three in a Row Game Board in a personal whiteboard

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How to Play

1. Mix up the cards. Put the cards into a stack facedown.

Take turns turning over cards to choose numbers for the game board. Write two-digit numbers in the left column and three one-digit numbers in the top row. Turn over

another card if any of the one-digit numbers are the same.

The left column and top row must be completely filled before placing a counter on the board.

2. Player A places a counter in an empty space on the game board. Multiply the number in the left column by the number in the top row. Show your work and answer below the game board.

Three in a Row Multiplication Game Board

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Application | Student Page

Application | Play a Game Game Instruction Card 3. Player B checks Player A’s work. If both players agree on

How to Win The first player with three counters in a row (vertically,

disagree, they share their work with each other,

horizontally, or diagonally) wins. If the game board is filled

find the mistake, and the counter is returned to the player.

25 × 7 3 175

4. Player B erases the work, chooses a new space, and

without a player getting three in a row, clear the board and play another round.

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multiplies. Player A checks their work.

the answer, the counter stays in the space. If the players

5. Continue taking turns choosing a space and multiplying until a player gets three in a row or until the game board is filled.

Three in a Row Multiplication Game Board

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Application | Student Page

NAME

DATE

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Three in a Row Multiplication Game Board

Application | Play a Game • Three in a Row Game board

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Study a Solution Ivan solved the problem below. Read the problem and look at Ivan’s work. Then answer the questions.

Deepa bikes 48 kilometers each day for 3 days. How many kilometers does Deepa bike? Ivan's Work

48 ?

3 × 48 = ?

48 × 3 2 144

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Deepa bikes 144 kilometers.

How is the known information in the problem represented?

How is the unknown information in the problem represented?

How does the drawing help you see a solution path for finding the unknown?

Does the statement answer the question? How do you know?

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Solve a Task Let’s Go to the Theater! A family has $200 to spend at the theater. The family has 3 adults and 3 children.

Adults Children

Ticket Prices

$35 $27

Does the family have enough money for every person to get a ticket?

At the theater, snacks are $3 and drinks are $2. Does the family have

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Show how you know.

enough money for each person to get a snack and a drink? Show how

C oconut Wate r

you know.

IT ADM E ON

How do you think the family should spend their leftover money on snacks and drinks? What advice would you give them?

This page may be reproduced for classroom use only.

Application | Student Page

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Relate Multiplication to Area

Multiplication

Concept Guide | Relate Multiplication to Area Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide) • Square tiles, 1 inch (8) • Square tiles, 1 centimeter (20)

• Ready the following materials: - Square tiles, 1 inch (8) - Square tiles, 1 centimeter (20) • Print copies of the Progress Check Tool and the Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Square tiles, 1 inch (12) • Square tiles, 1 centimeter (24) • Ruler

• Student Pages • Square tiles, 1 inch (12) • Square tiles, 1 centimeter (24) • Ruler

• Print copies of the Student Pages as needed.

• Application Teacher Guide

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• Practice Teacher Guide

Teacher Materials

• Practice Pages • Practice Helpers • Square tiles, 1 inch (24) • Square tiles, 1 centimeter (25)

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • Solve a Task Student Page • Square tiles, 1 inch (optional) • Grid paper (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards - Square tiles, 1 inch (optional) - Grid paper (optional) • Print copies of the following: - Solve a Problem Recording Page (optional) - Solve a Task Student Page

Concept Guide | Teacher Guide

Concept Guide | Relate Multiplication to Area

Addressing Student Misconceptions How to Address Misconception

Some students may believe that they must complete the entire array by drawing in all the missing unit squares to find the area of a rectangle. They might think that without a fully drawn grid, they do not have enough information to solve the problem.

Have students identify the length and width of the rectangle by sliding their finger along the top and side while saying the terms aloud. Emphasize that the width is represented by the number of unit squares in each column. The length is represented by the number of unit squares in each row. Guide students to recognize that they can multiply these two numbers to find the area without needing to draw every square. Model this by writing the multiplication equation for the given dimensions and explaining that the equation represents the total number of unit squares in the array.

Language Support

Student Misconception

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To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The chart should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding.

Concept Guide | Teacher Guide

Family Math | Relate Multiplication to Area Dear Family,

4 units

Your student is working on using what they know about multiplication to find the area of a rectangle. They tile a rectangle with square units and count the units to determine the rectangles’ area. Your student recognizes that they can count the unit squares in one row and one column to label the side lengths of a rectangle. Then they multiply the length and the width to find the area. You can support your student’s progress by asking the questions in the table below as your student relates multiplication to area.

4 units

3 units

3 × 4 = 12

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3 units

Area: 12 square units

What do you need to know to find the

How can you find the side lengths of the

How can you use the side lengths of

area of a rectangle?

rectangle?

the rectangle to find the area of the

4 units

I need to know the side lengths of the rectangle.

3 units

rectangle?

4 units 3 units

3 × 4 = 12

I can multiply the side lengths to find the area.

I can count the number of rows to find one side length. I can count the number of columns to find the other side length.

Concept Guide | Teacher Guide

Progress Check | Relate Multiplication to Area

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before, during, or after providing direct instruction. It is intended to collect data about students’ proficiency with relating area to multiplication and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problem 1 involves tiling a rectangle with square-inch tiles to find the area. Problem 2 involves tiling a rectangle and drawing an array to find the area of a rectangle. Problems 3 and 4 involve using an area model to find the area of a rectangle. Problem 5 involves finding the area of rectangle when given an incomplete array. Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions: • Can the student find the area by tiling a rectangle with squares?

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| Objective 1 • Can the student relate side lengths to area? | Objective 2

• Can the student find side lengths of area models and use the side lengths in an equation to find area? | Objective 3

| Objective 4 Teacher Tip

• Can the student find the area of a rectangle without the support of an array or grid?

Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress.

Progress Check | Teacher Guide

Progress Check | Relate Multiplication to Area

Progression Toward Proficiency Rubric Item 1

Item 2

Items 3 and 4

Item 5

Objective 1

Objective 2

Objective 3

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand relating multiplication to area but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly uses squareinch tiles to find the area:

The student correctly draws an array:

The student correctly labels the side lengths and finds the area:

The student correctly multiplies to find the area and selects D.

1. 8 square inches

2. Side lengths labeled 4 cm and 5 cm; 20 square centimeters

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Progress Check Tool Item(s)

3. 3 units and 6 units; 3 × 6 = 18; 18 square units 4. 3 centimeters and 8 centimeters; 3 × 8 = 24; 24 square centimeters

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Relate Multiplication to Area Use square-inch tiles to find the area of the rectangle.

Use square centimeter tiles to cover the rectangle. Draw an array inside the rectangle to represent the

centimeter tiles. Label the side lengths. Find the area

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of the rectangle.

Area: ___________________________

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Progress Check | Student Page

NAME

DATE

Progress Check Tool | Relate Multiplication to Area Each

represents 1 square unit. Label the side

lengths of each rectangle. Write a multiplication

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equation to represent the area. Then write the area.

Equation: ___________________________

side lengths of the rectangle. Write a multiplication

equation and find the area.

Equation: ___________________________

Area: ___________________________

The grid represents square centimeters. Label the

This page may be reproduced for classroom use only.

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Relate Multiplication to Area Find the area of the rectangle. Circle the letter of the correct area.

1 square unit 5 square units 6 square units 30 square units

A B C D

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This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Relate Multiplication to Area Progression of Mini Lesson Objectives 2 Tile a rectangle with squares to

3 Use rectangular arrays to

4 Determine the area of

to find its area.

make arrays and relate the side lengths to area.

interpret area models.

rectangles in problems by using multiplication related to the

7 cm

• can find the total number of objects in an array by counting or using repeated addition, but • need support finding area by tiling a rectangle with squares.

3 cm

Start here if students

• can tile a rectangle with unit squares concretely, but • need support relating side lengths to area.

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Start here if students

1 Tile a rectangle with squares

Start here if students • can find the side lengths of a rectangle by counting the unit squares in an array, but • need support finding side lengths of area models and using the side lengths in an equation to find area.

number of rows times the number of square units in each

row. 7 ft

8 ft

Start here if students • can find the area of a rectangle when supported by an array or grid, but • need support finding the area of a rectangle without the support of an array or grid.

Objective 1 | Tile a rectangle with squares to find its area. 10 M I NU T ES

Give each student 12 square-inch tiles.

Materials • Square tiles, 1-inch (12) • Objective 1 Student Page

Summary Students use inch tiles to tile rectangles and find their areas.

H These are unit squares. Let’s stack our unit squares on top of each other to make sure they are the same size and shape.

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Model stacking the tiles to confirm they are the same size and shape. Distribute the Objective 1 Student Page. Direct students to rectangles A and B in problems 1 and 2.

H What do you notice about the number of unit squares needed to cover rectangle A and the number of unit squares needed to cover rectangle B? They each need 6 unit squares to cover the shape.

H Do you think rectangle A or rectangle B takes up more space? Why?

I think rectangle A takes up more space because it is longer than rectangle B.

I think rectangle B takes up more space because it is taller and wider.

Invite students to use the tiles to cover rectangles A and B. Model incorrect and correct methods of tiling as you give the following instructions.

H Be sure the unit squares do not have gaps between them, do not overlap, and do not go beyond the sides of the rectangle.

H Once we decide on a unit square, the number of those unit squares that tile a shape without any gaps or overlaps is called the shape’s area.

Invite students to turn and talk about how they found the area of rectangles A and B. Teacher Tip Consider providing a visual for the term area by inviting students to lightly shade inside one of the rectangles and write area inside the shaded space.

H What shape did we tile with? Squares

Observe students to ensure they are tiling correctly.

Concept Mini Lessons | Teacher Guide

Objective 1 | Tile a rectangle with squares to find its area. 10 M I NU T ES

Teacher Tip Consider inviting students to precisely name rectangles C and D as squares. Then ask students what they notice about the areas of the squares. Guide students to notice that polygons can have the same shape but different areas.

H When we measure area, we fill up the space inside a shape with unit squares. We measure in square units. What is the area of rectangles A and B in square units? 6 square units

Direct students to write the area of rectangles A and B in problems 1 and 2 as 6 square units.

H Why is it possible for rectangles A and B to have equal areas? The 6 unit squares can be arranged into different shapes. No matter how you arrange the 6 unit squares, there are still 6 of

Monitor: • Can the student describe what area is? • Can the student tile a rectangle with no gaps, no overlaps, and without going over the sides? • Can the student use the term square units to correctly name the area of a rectangle?

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them.

Analyze Student Progress

H Because the same number of identical unit squares is used to tile rectangles A and B, the rectangles have equal areas.

Invite students to turn and talk about how they can use unit squares to find the area of a rectangle.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Questions to Advance Student Thinking: • What is area? • How can you use unit squares to find the area of a rectangle? • What is the area in square units? How do you know? Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

• Rectangle C: a rectangle that is 3 inches by 3 inches • Rectangle D: a rectangle that is 2 inches by 2 inches

Concept Mini Lessons | Teacher Guide

Objective 2 | Tile a rectangle with squares to make arrays and relate the side lengths to area. 10 M I NU T ES

Give each student one square-inch tile. Invite students to measure the lengths of the sides of the tile. Gesture to the square-inch tile. H What do you notice about the side lengths? They are each 1 inch.

Units to Measure Area

Units to Measure Length

We must use identical unit squares when we compare areas.

Distribute the Objective 2 Student Page and provide each student with 24 centimeter tiles. Direct students to the rectangle in problem 1. Have students tile the rectangle with centimeter tiles.

Gesture to the side length that has 7 centimeter tiles as you ask the following question.

1 inch

1 square inch

H How many centimeter tiles are on this side length of the rectangle? 7

1 inch 1 centimeter 1 square centimeter

1 centimeter

1 inch

Consider creating a poster to define square inch and square centimeter. Include a labeled picture of each, such as the following. Later in the mini lesson, add to the poster to define each measurement unit.

compare to a square centimeter?

H Why is it important to name units precisely to describe area?

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Give each student one square-centimeter tile. Repeat the process of measuring the side lengths and naming the area of the square as a square centimeter.

H How does a square inch

H What do you think the units to measure area are called when a rectangle is tiled with squares that measure 1 foot on each side? What about 1 mile on each side? Square feet; square miles

H Because each side of the square measures 1 inch, we call the area of one of these squares a square inch.

Language Support

Materials • Square tile, 1-inch • Square tiles, 1-centimeter (24) • Objective 2 Student Page • Ruler

Summary Students relate the number of square tiles used to form the side of a rectangle to that side’s length.

1 centimeter

A square centimeter is a smaller unit of area than a square inch.

H We know that each centimeter tile has side lengths of 1 centimeter. You used 7 centimeter tiles along this side length, so what do you think the length of this side is? 7 centimeters

Concept Mini Lessons | Teacher Guide

Objective 2 | Tile a rectangle with squares to make arrays and relate the side lengths to area. 10 M I NU T ES

Model using a centimeter tile to draw an array inside the rectangle. Invite students to do the same. Gesture to each side length as you ask the following question.

H What do you notice? The side length is 7 centimeters.

Label the side length 7 cm and have students do the same. Gesture to the side length with 3 centimeter tiles as you ask the following question.

Have students measure the side length in centimeters.

length

Invite students to measure the side length in centimeters. Then label the side length 3 cm and have students do the same.

H What is the area of the rectangle? How do you know? The area is 21 square centimeters because we used 21 centimeter tiles to tile the rectangle.

Teacher Tip

The goal of this mini lesson is for students to see the connection between the side lengths and the number of rows and columns in a rectangle. The intent is not for students to only find the area by multiplying the length and width of a rectangle. They might multiply, but they could also count each tile, skip-count by the number of tiles in each row, or skip-count by the number of tiles in each column. Look for and encourage efficient and accurate counting strategies.

H What are the different ways you know to find the area of a rectangle?

Record the area as students do the same. H Watch as I draw an array inside the rectangle to represent the centimeter tiles.

H The number of rows and columns in the array helps us know the lengths of the sides.

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H What do you think the length of this side is? Why? 3 centimeters, because we used 3 centimeter tiles on this side

H How does the array represent each side length? The array shows 3 rows of 7 centimeter tiles. The side lengths are 3 centimeters and 7 centimeters.

3 cm

7 cm

I can fill the rectangle with unit squares and count them. I can draw a grid inside the rectangle representing unit squares and skip-count them.

Invite students to turn and talk about how the side lengths of rectangles are related to the tiles along its sides.

Concept Mini Lessons | Teacher Guide

Objective 2 | Tile a rectangle with squares to make arrays and relate the side lengths to area. 10 M I NU T ES

• Problem 2: a rectangle that is 4 cm by 6 cm • Problem 3: a rectangle that is 5 in by 2 in • Problem 4: a rectangle that is 3 in by 4 in Analyze Student Progress

Notes

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

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Monitor: • Can students draw an array to represent the square tiles? • Can students relate the number of columns and rows to the side lengths of the rectangle? • Can the student use square inches or square centimeters to correctly name the area of a rectangle?

Questions to Advance Student Thinking: • How can you draw an array to represent the square units? • How many columns are in the rectangle? How many rows are in the rectangle? What does the number of columns and rows tell you about the side lengths of the rectangle? • What units can you use to describe the area of the rectangle? Why? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Use rectangular arrays to interpret area models. 10 M I NU T ES

Materials • Objective 3 Student Page

Distribute the Objective 3 Student Page. Direct students to the rectangle on the left in problem 1.

Gesture to the top of the array.

Invite students to think–pair– share about how the array can help them find the area of the rectangle.

We can count the unit squares.

We can use the number of rows and columns to skip-count the unit squares.

H How many columns of unit squares are there? What is the length of this side of the rectangle? There are 6 columns. The length of the rectangle is 6 units long.

Label the side length of 6 units in both rectangles and have students do the same. Gesture to the left side of the array.

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H This rectangle shows an array of unit squares.

Summary Students find the area of a rectangle without using concrete tiles or drawing the grid of unit squares inside.

We could multiply the number of rows and the number of columns.

Invite students to find the area of the rectangle. H What is the area of the rectangle? 18 square units

Gesture to the rectangle on the right in problem 1.

H How many rows of unit squares are there? What is length of this side of the rectangle? There are 3 rows. This side of the rectangle is 3 units long.

Label the side length of 3 units in both rectangles and have students do the same. Teacher Tip Students will likely see that multiplication can be helpful when finding the total number of unit squares that tile a rectangle. Keep the emphasis on the relationship between side lengths and area rather than teaching the formula length × width = area.

Gesture to the rectangle on the right in problem 1.

H The rectangles have equal areas. This rectangle doesn’t have an array and we don’t have any tiles. Since the rectangles have equal areas, we can use the array in the other rectangle to help us label the side lengths of this rectangle. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Objective 3 | Use rectangular arrays to interpret area models. 10 M I NU T ES

Teacher Tip Consider using the following questions to support students with using the grid to label the side lengths in problems 3 and 4 on the Objective 3 Student Page:

H This rectangle is an area model. It shows the side lengths without an array of squares inside. We can use an area model to help us find the area of a rectangle by multiplying the side lengths.

• What do we need to know to find the area of a rectangle?

H What multiplication equation can we use to represent the area? 6 × 3 = 18

• What is the side length of one of the unit squares in the grid?

Analyze Student Progress

Monitor: • Does the student know what information is needed to find the area of a rectangle? • Can the student use a visual support, such as an array or a grid, to find the side lengths of the rectangle? • Can the student find the area of a rectangle by multiplying the side lengths?

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H What is the area of the rectangle? 18 square units

• How can we use the grid to help us label the side lengths of the rectangle?

Have students record the multiplication equation and the area in problem 1.

Invite students to turn and talk about how they can use an area model to find the area of a rectangle.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work. • Problem 2: a rectangle that is 4 units by 5 units • Problem 3: a rectangle that is 3 centimeters by 8 centimeters • Problem 4: a rectangle that is 9 centimeters by 2 centimeters

Questions to Advance Student Thinking: • What do you need to know to find the area of a rectangle? • How can you find the side lengths of the rectangle? • How can you use the side lengths of the rectangle to find the area of the rectangle? Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

the area of rectangles in problems by using multiplication related Objective 4 | Determine to the number of rows times the number of square units in each row. 10 M I NU T ES

Distribute the Objective 4 Student Page. Direct students to the rectangle in problem 1.

Language Support

H How many unit squares would be in each row? 7 unit squares

H Do we need to complete the array to find the area of the rectangle? Why? No. We already know there would be 4 rows of 7 unit squares.

No. The array shows us the side lengths we need to find the area.

Slide your finger along the top and side of the rectangle to demonstrate as you say length and width. H We can find the area of the rectangle if we know its length and width. The number of rows is the width in square units. The number of square units in each row is the length in square units. So we can multiply the number of rows by the number of square units in each row to find the area of the rectangle.

length

Consider creating a visual to support students in using the terms length and width appropriately.

width

H What multiplication equation can you use to find the area? 4 × 7 = 28

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H The rectangle has a unit square array that is incomplete. Imagine if the array was complete. How many rows would be in the array? 4 rows

Materials • Objective 4 Student Page

Summary Students multiply side lengths to determine the area of rectangles.

H What is the area? 28 square units

7 ft

Have students record the multiplication equation and area for problem 1. Direct students to the rectangle in problem 3.

8 ft

H What do you notice about this rectangle? The side lengths are labeled but there is no grid inside.

Concept Mini Lessons | Teacher Guide

the area of rectangles in problems by using multiplication related Objective 4 |Determine to the number of rows times the number of square units in each row. 10 M I NU T ES

Yes. We know the length and the width.

• Problem 2: a rectangle that is 6 units by 9 units • Problem 4: a rectangle that is 6 feet by 6 feet

H Is there enough information to find the area? H We can find the area of the rectangle if we know its length and width. We can multiply the length and the width to find the area. What multiplication equation can we use to find the area? 7 × 8 = 56

Monitor: • Does the student know what information is needed to find the area of a rectangle? • Can the student determine the side lengths when given an incomplete array? • Can the student find the area of a rectangle by multiplying the side lengths?

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H What unit is used to measure the length and the width of this rectangle?

Analyze Student Progress

Feet

H What unit is used to measure the area of this rectangle? Square feet

H What is the area of the rectangle? 56 square feet

Plan Future Practice: Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Have students record the multiplication equation and area for problem 3.

Questions to Advance Student Thinking: • Is there enough information to find the area? • How can the number of rows and the number of square units in each row help you label the length and width? • What multiplication equation can you use to find the area?

Invite students to turn and talk about how multiplication relates to a rectangle’s side lengths and its area. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Relate Multiplication to Area Answer Key Objective 1

Objective 2

1. Square-inch tiles used

1. Correctly drawn array;

Objective 3

correctly to find the area;

side lengths labeled

6 square units

7 cm and 3 cm; 21 square centimeters

2. Square-inch tiles used correctly to find the area;

6 square units

2. Correctly drawn array;

3. Square-inch tiles used correctly to find the area;

9 square units

2. Correctly labeled side lengths; 4 × 5 = 20; 20 square units

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side lengths labeled

1. Correctly labeled side lengths; 6 × 3 = 18; 18 square units

4 cm and 6 cm; 24 square centimeters

3. Correctly drawn array;

3. Correctly labeled side lengths; 3 × 8 = 24; 24 square centimeters

Objective 4 1. 4 × 7 = 28; 28 square units 2. 9 × 6 = 54; 54 square units 3. 8 × 7 = 56; 56 square feet 4. 6 × 6 = 36; 36 square feet

side lengths labeled

correctly to find the area;

4 square units

5 in and 2 in; 10 square inches

4. Square-inch tiles used

4. Correctly labeled side lengths; 9 × 2 = 18; 18 square centimeters

4. Correctly drawn array; side lengths labeled

3 in and 4 in; 12 square inches For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATh CATALyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Relate Multiplication to Area Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Relate Multiplication to Area Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Tile a rectangle with squares to find its area. 1 A

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Area: _______________________________

Use square-inch tiles to find the area of the rectangle.

Area: _________________________

Area: ______________________________

Area: __________________________

This page may be reproduced for classroom use only.

Concept Mini Lesson | Student Page

NAME

DATE

Objective 2 | Tile a rectangle with squares to make arrays and relate the side lengths to area. Use square centimeter tiles to cover the rectangle. Draw an array inside the rectangle to represent the centimeter tiles. Label

the side lengths. Find the area of the rectangle.

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Area: _______________________________

This page may be reproduced for classroom use only.

Concept Mini Lesson | Student Page

NAME

DATE

Objective 2 | Tile a rectangle with squares to make arrays and relate the side lengths to area. 3

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Area: _______________________________

This page may be reproduced for classroom use only.

Concept Mini Lesson | Student Page

NAME

DATE

Objective 3 | Use rectangular arrays to interpret area models. Each

represents 1 square unit. Label the side lengths of each rectangle. Write a multiplication equation to represent the area.

Then write the area.

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Equation: ___________________________

Area: ___________________________

The grid shows square centimeters. Label the side lengths of the rectangle. Write a multiplication equation and find the area.

Equation: _________________________

Equation: ___________________________

Area: ___________________________

This page may be reproduced for classroom use only.

Concept Mini Lesson | Student Page

NAME

DATE

Determine the area of rectangles in problems by using multiplication related Objective 4 | to the number of rows times the number of square units in each row.

Write a multiplication equation to find the area of the rectangle.

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Equation: _________________________

Area: ___________________________

3 8 ft

7 ft

Equation: _________________________

6 ft

Equation: _________________________

Area: ___________________________

This page may be reproduced for classroom use only.

Concept Mini Lesson | student Page

Practice | Relate Multiplication to Area Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers during Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Tile a rectangle

Objective 2 Tile a rectangle

Objective 3 Use rectangular

Objective 4 Determine the

with squares to make arrays and relate the side lengths to area.

arrays to interpret area models.

area of rectangles in problems by using multiplication related to the number of rows times the number of square units in each row.

Look for ...

• Can the student draw an array to represent the square tiles? • Can the student relate the number of columns and rows to the side lengths of the rectangle? • Can the student use square inches or square centimeters to correctly name the area of a rectangle?

• Can the student describe what area is? • Can the student tile a rectangle with no gaps, no overlaps, and without going over the sides? • Can the student use square units to correctly name the area of a rectangle?

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with squares to find its area.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Relate Multiplication to Area Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for ...

• Does the student know what information is needed to find the area of a rectangle? • Can the student use a visual support, such as an array or a grid, to find the side lengths of the rectangle? • Can the student find the area of a rectangle by multiplying the side lengths?

Look for ... • Does the student know what information is needed to find the area of a rectangle? • Can the student determine the side lengths when given an incomplete array? • Can the student find the area of a rectangle by multiplying the side lengths?

Practice | Teacher Guide

Practice | Relate Multiplication to Area

Answer Key Practice Page 2

Practice Page 3

1. Square-inch tiles used

1. Correctly drawn array;

1. Correctly labeled side lengths; 7 × 4 = 28; 28 square units

Practice Page 1

correctly to find the area;

side lengths labeled

12 square inches

3 cm and 4 cm; 12 square centimeters

2. Square-inch tiles used 10 square inches

3. Square-inch tiles used correctly to find the area;

16 square inches

2. Correctly drawn array; side lengths labeled

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correctly to find the area;

2. Correctly labeled side lengths; 5 × 6 = 30; 30 square units

5 cm and 5 cm; 25 square centimeters

3. Correctly drawn array;

3. Correctly labeled side lengths; 2 × 7 = 14; 14 square centimeters

side lengths labeled

4. Amy counted only 1 row

6 cm and 2 cm; 12 square centimeters

of unit squares.

Practice Page 4 1. 6 × 3 = 18; 18 square units 2. 8 × 5 = 40; 40 square units 3. 4 × 4 = 16; 16 square feet 4. Shen added the side lengths instead of multiplying the side lengths.

4. D

4. C

Practice | Teacher Guide

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Student Edition | Printable Pages for Students

Practice | Teacher Guide

NAME

DATE

Practice Page 1 | Tile a rectangle with squares to find its area. Use square-inch tiles to find the area of the rectangle.

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Area: ___________________________

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Practice | Student Page

NAME

DATE

Practice Page 1 | Tile a rectangle with squares to find its area. 4

Amy uses unit squares to tile a rectangle. She says the area of the rectangle is 3 square units. What mistake

did Amy make?

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Amy’s Work

Area: ___________________________

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Practice | Student Page

NAME

DATE

a rectangle with squares to make arrays and relate the side Practice Page 2 | Tile lengths to area.

the side lengths. Find the area of the rectangle.

Use square centimeter tiles to cover the rectangle. Draw an array inside the rectangle to represent the centimeter tiles. Label

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Area: ___________________________

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Practice | Student Page

NAME

DATE

a rectangle with squares to make arrays and relate the side Practice Page 2 | Tile lengths to area.

What is the area of the rectangle? Circle the letter of the correct area.

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A B C D

Area: ___________________________

9 square units 10 square units 14 square units 16 square units

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Practice | Student Page

NAME

DATE

Practice Page 3 | Use rectangular arrays to interpret area models. Each

represents 1 square unit. Label the side lengths of each rectangle. Write a multiplication equation to represent the

area. Then write the area.

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Equation: ___________________________

Area: ___________________________

Equation: ___________________________

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Practice | Student Page

NAME

DATE

Practice Page 3 | Use rectangular arrays to interpret area models. The grid shows square centimeters. Label the side lengths of the rectangle. Write a multiplication equation

What is the area of the rectangle? Circle the letter of the correct area.

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and find the area.

A B C

7 square units 9 square units 10 square units

D 12 square units

Equation: ___________________________ Area: ___________________________

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Practice | Student Page

NAME

DATE

Practice Page 4

Determine the area of rectangles in problems by using multiplication related to the number of rows times the number of square units in each row.

Write a multiplication equation to find the area of the rectangle.

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Equation: ___________________________ Area: ___________________________

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Practice | Student Page

NAME

DATE

Practice Page 4 3

Determine the area of rectangles in problems by using multiplication related to the number of rows times the number of square units in each row.

Shen incorrectly finds the area of a square. Look at

4 ft

Shen’s work. What mistake did Shen make?

6 ft

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6 ft

Equation: ___________________________

Area: ___________________________

6 + 6 = 12

Area: 12 square feet

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Practice | Student Page

NAME

DATE

Practice Helper 1

Look at the problem. Then look at the work. It shows how to find the area of a rectangle. Use square-inch tiles to find the area of the rectangle.

How can you use unit squares to find

What is the area in square units? How

the area of a rectangle?

do you know?

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What is area? Area is the amount of space a shape takes up.

The area is 3 square units. I know because I counted the 3 unit squares inside the rectangle.

I can cover the rectangle with unit squares, making sure there are no gaps or overlaps. Then I can count the tiles.

Area: 3 square units

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Practice | Student Page

NAME

DATE

Practice Helper 2 draw an array to find the area of a rectangle.

Look at the problem. Then look at the work. It shows how to

Use square-centimeter tiles to cover the rectangle. Draw an array inside the rectangle to represent the centimeter tiles.

Label the side lengths. Find the area of the rectangle.

How many columns are in the rectangle?

What units can you use to describe the

the square units?

How many rows are in the rectangle?

area of the rectangle? Why?

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How can you draw an array to represent

What do the numbers of columns and rows tell you about the side lengths of the rectangle?

5 cm

I can tile the rectangle with square centimeters. Then I can draw an array with 2 rows and 5

columns to represent the tiles.

Area: 10 square centimeters I can describe the area in square centimeters because I used 10 centimeter tiles to tile the rectangle.

2 cm

The rectangle has 2 rows and 5 columns. The number of rows tells me one side length is 2 cm. The number of columns tells me the other side length is 5 cm.

5 cm 2 cm

Area: 10 square centimeters

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Practice | Student Page

NAME

DATE

Practice Helper 3 label a rectangle on a grid and find its area. Each

Look at the problem. Then look at the work. It shows how to

represents 1 square unit. Label the side lengths of each rectangle. Write a

multiplication equation to represent the area. Then write the area.

How can you find the side lengths of the

How can you use the side lengths of

area of a rectangle?

rectangle?

the rectangle to find the area of the

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What do you need to know to find the

4 units

I need to know the side lengths of the rectangle.

3 units

rectangle?

4 units 3 units

3 × 4 = 12

I can multiply the side lengths to find the area.

I can count the number of rows to find one side length. I can count the number of columns to find

4 units

the other side length.

4 units

3 units

3 × 4 = 12

Area: 12 square units

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Practice | Student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to

write an equation to find the area of a rectangle.

Write a multiplication equation for finding the area of the rectangle.

How can the number of rows and the

What multiplication equation can you

area?

number of square units in each row help

use to find the area?

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Is there enough information to find the

you label the length and width?

Yes. I can use the number of rows and the

5 units

number of square units in each row to determine the side lengths of the rectangle.

3 units

5 units 3 units

5 × 3 = 15

Area: 15 square units I can multiply the length and the width to find the area.

The full column shows 3 squares, which tells me

the width is 3 units. The full row shows 5 squares, which tells me the length is 5 units.

5 × 3 = 15

Area: 15 square units

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Practice | Student Page

Application | Relate Multiplication to Area Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Considerations

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving relating multiplication to area. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a personal whiteboard. • Consider providing tools such as square tiles or grid paper to support students.

Play a Game

Activity

Solve a Task

Partner Work

• Consider providing manipulatives, such as square-inch tiles, for students to use to build a model to support their understanding of the problem.

Application | Teacher Guide

Application | Relate Multiplication to Area

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) Students use the Read–Draw– Write process to solve word problems involving relating multiplication to area. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. All problems involve multiplying the width times the length to find the area of a rectangle. Teacher Tip

• Eureka Math2 cards or a standard deck of playing cards • Personal whiteboard • Game Instruction Card • Square tiles (optional) • Grid paper (optional) Students work with a partner to play a game involving relating multiplication to area.

Preparing to Play • Remove the jack, queen, king, and joker cards from the deck of cards. Aces can represent 1. • Shuffle the remaining cards. Divide the cards equally among the players. Each player keeps their cards in a single facedown pile. • Consider providing tools such as square-inch tiles or grid paper to support students.

Solve a Task Materials

• Solve a Task Student Page • Square-inch tiles (optional) Students work with a partner to solve a multi-part task involving relating multiplication to area. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Materials

Playing the Game • Each player takes two cards off the top of their pile and places them faceup. • The players each use the values of their cards as side lengths for an area model. • The players multiply to find the area. The player whose rectangle has the largest area wins all the cards in play and places them at the bottom of their pile. • If players create rectangles that have the same area, a Top It round ensues. The player whose rectangle has the largest area in the Top It round wins all the cards from both rounds. • The player with the most cards at the end of the game wins.

Materials

Play a Game: Area Top It

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Solve a Problem

Application | Teacher Guide

Application | Relate Multiplication to Area

Answer Key Solve a Task

Solve a Problem 1. Accurate picture drawn to represent the problem; equation shows the product of 3 and 5; the area of the rug is 15 square feet.

2. The area of the wall covered by picture 2 is 25 square inches.

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2. Accurate picture drawn to represent the problem; equation shows the product of 7 and 4; Gabe’s bed covers 28 square feet of his bedroom floor.

1. The area of the wall covered by picture 1 is 80 square inches.

3. The total area of the wall covered by all 3 pictures is 140 square inches.

3. Accurate picture drawn to represent the problem; equation shows the product of 4 and 9; the area of Robin’s garden is 36 square feet.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

Mr. Endo buys a rectangular rug that has a

width of 3 feet and a length of 5 feet. What is the area of the rug? Show your work.

Gabe’s rectangular bed is 7 feet long and 4 feet

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wide. What is the area of Gabe’s bedroom floor that is covered by the bed? Show your work. Robin plants a rectangular garden. The

garden has a width of 4 feet and a length of

9 feet. What is the area of Robin’s garden? Show your work. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

Application | solve a Problem Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Problem Number

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Application | Student Page

Application | Play a Game Game Instruction Card

Area Top It

6. If the areas are equal, it is time to Top It! Play another round. The player with the larger area takes the cards from

What You Need

both rounds.

• Eureka Math cards or a standard deck of playing cards 2

with the jack, queen, king, and joker cards removed. Aces can represent 1.

The player with the most cards at the end of the game wins.

• Personal whiteboard

Player A

• Square tiles (optional)

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player. Put the cards into a stack facedown.

2. At the same time as the other player, turn over two cards. 3. Draw and label an area model for a rectangle. Use the

numbers on the cards to represent the side lengths of

1. Mix up the cards. Deal the same number of cards to each

8 units

2 units 2 × 8 = 16 Area: 16 square units

4 units 3 units 3 × 4 = 12 Area: 12 square units

your area model.

Player B

• Grid paper (optional) How to Play

How to Win

4. Find the area of your rectangle. Show your work. Record the area in square units.

5. Compare your area to the other player’s area. If you have the larger area, take all the cards. Put them at the bottom of your stack.

Application | Student Page

NAME

DATE

Application | solve a Task Mrs. Smith’s Pictures

• Mrs. Smith hangs 2 rectangular pictures on a wall.

• Picture 1 has a width of 8 inches and a length of 10 inches.

What is the area of the wall covered by picture 1?

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• Picture 2 is a square. It has a width of 5 inches.

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Application | Student Page

NAME

DATE

Application | Solve a Task What is the area of the wall covered by picture 2?

Mrs. Smith buys another rectangular picture. The picture has a width of 5 inches and a length of 7 inches. She hangs the

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2 1

new picture next to pictures 1 and 2. What is the total area of the wall covered by all 3 pictures?

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Application | student Page

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Applications of Multiplication and Area

Multiplication

Concept Guide | Applications of Multiplication and Area Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and the Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard

• Personal whiteboard or Student Pages

• Print copies of Student Pages as needed. • Place copies of Student Page in whiteboards for Objectives 1, 2, 3, and 4. • Prepare the Composite Shapes template. • Prepare the Sample Solutions template.

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Game Instruction Card • Three in a Row Game Board • Grid paper (optional) • Solve a Task Student Page • Highlighter (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Grid paper (optional) • Print copies of the following: - Solve a Task Student Page - Three in a Row Game Board - Solve a Problem Recording Page (optional) • Gather the following materials: - Highlighters (optional)

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• Application Teacher Guide

Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Applications of Multiplication and Area

Addressing Student Misconceptions How to Address Misconception

Students do not correctly find the unknown side lengths when finding the area of composite shapes.

Invite students to trace the rectangles and to use highlighters so that they see one rectangle at a time. When students are looking for unknown side lengths, remind them that one of the properties they know about rectangles is that opposite sides are the same length. Students can trace or highlight opposite sides of each rectangle to help them distinguish which side lengths are known and which side lengths are unknown.

Language Support

Student Misconception

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

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• 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.

• key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding.

Concept Guide | Teacher Guide

Family Math | Applications of Multiplication and Area Dear Family,

Your student is working on finding the area of shapes that are composed of rectangles, which are known as composite shapes. Your student decomposes the shapes into two rectangles, subtracts from a larger area, and uses the properties they know about rectangles to find unknown side lengths. You can support your student’s progress by asking the questions in the table below as your student finds the area of composite shapes. (6 × 5) − (2 × 3) = 30 − 6

3 2

= 24

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Area: 24 square units

Where do you see a large rectangle?

Where do you see a smaller rectangle?

How can you use the area of the large

How can you find the area of the large

How can you find the area of the smaller

rectangle and the area of the unshaded

rectangle?

rectangle to find the area of the shaded

I see a large rectangle around the outside of both the shaded and unshaded rectangles. I can find

I see a smaller rectangle in the unshaded part.

the area of the large rectangle by labeling its side

I can find the area of the smaller rectangle by

lengths and then multiplying.

labeling its side lengths and then multiplying.

shape?

(6 × 5) − (2 × 3) = 30 − 6 = 24

I can subtract the area of the unshaded rectangle from the area of the large rectangle to find the area of the shaded shape.

Concept Guide | Teacher Guide

Progress Check | Applications of Multiplication and Area

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Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

• Can the student decompose a composite shape into rectangles and apply the additive property of area? Objective 1

• Can the student subtract from a larger area to find the area of a composite shape? Objective 2

• Can the student find the area of composite figures with unknown side lengths? Objective 3

• Can the student decide which strategy to use to find the area of a composite shape? Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATh CATAlysT | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Applications of Multiplication and Area

Progression Toward Proficiency Rubric Progress Check Tool Item(s)

Item 2

Item 3

Item 4

Item 5

Objective 1

Objective 2

Objective 3

Objective 4

Objectives 1–4

Not Yet Proficient

The student may show evidence of beginning to understand finding the area of composite shapes on a grid by decomposing but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand finding the area of composite shapes on a grid by subtracting from a larger area but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand finding the area of composite shapes with unknown side lengths by subtracting but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand finding the area of composite shapes by using rectangles but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand how to show their work when finding the area of composite shapes by using a method of their choice but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and circles the correct answer.

Proficient

The student correctly finds the area of composite shapes on a grid by decomposing:

The student correctly finds the area of composite shapes on a grid by subtracting from a larger area:

The student correctly finds the area of composite shapes with unknown side lengths by subtracting:

The student correctly finds the area of composite shapes by using rectangles:

The student correctly uses a model of their choice to find the area of a composite shape and circles option B.

2. Correctly labeled side lengths; 30

3. 46 square centimeters

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1. Correctly drawn line; correctly labeled length and width; 42

Item 1

4. 60 square feet

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Applications of Multiplication and Area. Draw a line in the shape to break it apart into two rectangles. Label the length and width of each rectangle. Then find the area of the shape. represents 1 square unit.

Label the unknown side lengths. Then subtract from the larger area to find the area of the shaded shape. Each

represents 1 square unit.

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Each

Area: ________ square units

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Progress Check | student Page

NAME

DATE

Progress Check Tool | Applications of Multiplication and Area. Subtract from the larger area to find the area of the shaded shape.

3 cm

Find the area of the shaded shape. Show your strategy.

10 ft

9 ft

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7 cm

5 cm

5 ft

2 ft

8 cm

Area: ________ square centimeters

Area: ________ square feet

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Progress Check | student Page

NAME

DATE

Progress Check Tool | Applications of Multiplication and Area. Find the area of the shaded shape. Show your strategy. Circle the letter with the correct area.

9 in 2 in

2 in

1 in

38 square inches

B C

43 square inches 48 square inches

7 in

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2 in

D 51 square inches

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Progress Check | student Page

Concept Mini Lessons | Applications of Multiplication and Area Progression of Mini Lesson Objectives 2 Find the area of composite

3 Find the area of composite

4 Find the area of composite

shapes on a grid by decomposing.

shapes on a grid by subtracting from a larger area.

shapes with unknown side lengths by subtracting.

shapes by using rectangles.

1 Find the area of composite

Eva’s Work 8 in

10 cm

2 in 4 in

3 cm

3 (4 × 2) + (5 × 3) = 8 + 15 = 23 Area: 23 square units Start here if students

8 cm

5 cm

6 cm

4 cm

= 50 Area: 50 square centimeters

= 55 Area: 55 square units

Start here if students

• can determine the side lengths of a rectangle on a grid, • can find the area of a rectangle by multiplying its side lengths, and • can find the area of a composite shape on a grid by decomposing, but • need support subtracting from a larger area to find the area of a composite shape.

• can determine the side lengths of a rectangle on a grid and • can find the area of a rectangle by multiplying its side lengths, but • need support decomposing a composite shape into rectangles and • need support applying the additive property of area.

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David’s Work 8 in

Start here if students • can find the area of a composite shape on a grid and • can use the attributes of rectangles to determine side lengths, but • need support finding the area of composite figures with unknown side lengths.

6 in 4 in

2 in 2 in 8 × 2 = 16 4×2=8 4×2=8 16 + 8 + 8 = 32 Area: 32 square inches

4 in 2 in 4 in 2 in 8 × 6 = 48 4 × 4 = 16

Area: 32 square inches

Start here if students • can find the area of composite shapes with unknown side lengths, but • need support deciding which strategy to use to find the area of a composite shape.

6 in

Concept Mini Lessons | Teacher Guide

Objective 1 | Find the area of composite shapes on a grid by decomposing. 10 M I NU T ES

Show shapes A, B, and C from the Composite Shapes template. H Each square in the

Shape B

Shape A

shapes represents

lengths of 4 units and

3 units. H What do you notice about shape B?

H How can we find the area of shape C? How do you know?

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Shape C

rectangle. It has side

squares in the rectangle. length times the width to find the area: 8 × 2 = 16.

do you notice about Shape A is a

H What is the area of shape B? How do you know? The area of shape B is 16 square units because I counted 16 The area of shape B is 16 square units because I can multiply the

1 square unit. What shape A?

Materials • Composite Shapes • Objective 1 Student Page

Summary Students find the area of a composite shape by breaking it apart into smaller rectangles.

Shape B is a rectangle. It has side lengths of 2 units and 8 units.

H What do you notice about shape C?

Shape C is made up of shapes A and B.

H What is the area of shape A? How do you know? The area of shape A is 12 square units because I counted 12 squares in the rectangle. The area of shape A is 12 square units because I can multiply the

length times the width to find the area: 4 × 3 = 12.

I can count the squares inside shape C to find the total area.

Shape C is made up of shapes A and B, so we can add the areas of shapes A and B to find the area of shape C.

H Shapes A and B were composed, or put together, to make shape C. The area of shape C is equal to the area of shape A plus the area of shape B.

Teacher Tip Consider cutting out shapes A and B and placing them on top of shape C to support students in understanding that area is additive.

Invite students to find the area of shape C. H What is the area of shape C? The area of shape C is 28 square units because 12 + 16 = 28.

Concept Mini Lessons | Teacher Guide

Objective 1 | Find the area of composite shapes on a grid by decomposing. 10 M I NU T ES

H Let’s decompose, or break apart, this shape into two

area of this rectangle? How do you know? We can write 4 × 2. The length of the rectangle is 4 units and the width is 2 units. The area of the rectangle is equal to the length times the width.

rectangles to help us find the area. Draw a vertical dotted line to

Write (4 × 2) and have students do the same.

Gesture to the second rectangle.

decompose the shape into

2 rectangles. Direct students

H What are the side lengths of this rectangle? 5 units and 3 units

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to do the same.

Teacher Tip: Differentiation

H What multiplication expression can we write to represent the

Distribute the Objective 1 Student Page. Direct students to the shape in problem 1.

Consider providing students with manipulatives to decompose the shapes instead of having them draw dotted lines. For example, students can use square tiles to form the composite shape; then they can physically separate the smaller rectangles and find the areas.

H To find the area of the shape, we can find the area of each rectangle and then add the areas together.

Gesture to the rectangle with side lengths of 4 units and 2 units. H What are the side lengths of this rectangle? 4 units and 2 units

Label the side lengths and have students do the same. H What multiplication expression can we write to represent the area of this rectangle? How do you know? We can write 5 × 3. The length of the rectangle is 5 units and the width is 3 units. The area of the rectangle is equal to the length times the width.

Write + (5 × 3) and have students do the same. H What is 4 × 2? 8

H What is 5 × 3? 15

Label the side lengths and have students do the same.

Write = 8 + 15 and have students do the same.

Concept Mini Lessons | Teacher Guide

Objective 1 | Find the area of composite shapes on a grid by decomposing. 10 M I NU T ES

H What is 8 + 15? 23

Gesture to the rectangle with side lengths of 7 units and 2 units.

Write = 23 and have students do the same.

7 units and 2 units

H What multiplication expression can we write to represent the

H The sum of the areas of the

area of this rectangle? 7×2

rectangles is …? 23

(4 × 2) + (5 × 3) = 8 + 15 = 23 Area: 23 square units

Record the area and have students do the same.

Direct students to shape D from the Composite Shapes template. H Shape D is the same shape

from problem 1. What do you notice?

It’s broken apart differently.

Write (7 × 2) below the shape.

Gesture to the rectangle with side lengths of 3 units and 3 units.

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H What is the area of the shape? 23 square units

H What are the side lengths of this rectangle?

H Shape D shows that we can

find the area by decomposing in a different way. Do you still

see shape D decomposed into rectangles? How? Yes. I see a long rectangle on top and a smaller rectangle on the bottom. Yes. I see a rectangle and a square.

H What are the side lengths of this rectangle? 3 units and 3 units

H What multiplication expression can we write to represent the area of this rectangle? 3×3

Write + (3 × 3).

H What is 7 × 2? 14

H What is 3 × 3? 9

Write = 14 + 9.

H The sum of the areas of the rectangles is …? 23

H What is the area of shape D? 23 square units

Analyze Student Progress

H What do you notice about the area of shape D and the area of the shape in problem 1?

In both shapes, the sum of the areas of the rectangles is the same. The total square units in each shape is the same.

Questions to Advance Student Thinking: • How can you break apart the shape into two rectangles? • How can you find the length and width of each rectangle? • How can you find the area of each rectangle? The total area?

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H We can decompose shapes that are made up of rectangles

Monitor: • Can the student decompose the shape into two rectangles? • Can the student identify the length and width of each rectangle? • Can the student find the area of each rectangle and add to find the total area?

to help us find the area. Sometimes we can decompose the shape into rectangles in more than one way.

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Invite students to turn and talk about how decomposing a shape into rectangles can help them find the area of that shape.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 1 Practice Helper and supporting students in using the worked-out example to guide their own work. • A composite figure with an area of 28 square units • A composite figure with an area of 30 square units • A composite figure with an area of 31 square units

Objective 2 | Find the area of composite shapes on a grid by subtracting from a larger area. 10 M I NU T ES

Distribute the Objective 2 Student Page. Direct students to the shape in problem 1. H How could you decompose this shaded shape to find the area? rectangles. You could make two long

rectangles across the top and bottom and one smaller rectangle in the middle.

Trace around the outer rectangle that surrounds the shape and have students do the same. H What are the length and width of the large rectangle? 8 units and 8 units

Label the side lengths and have students do the same.

EV I

You could decompose it into three

Materials • Objective 2 Student Page

Summary Students find the area of a composite shape by subtracting the area of a smaller rectangle from the area of a larger rectangle.

You could make one long rectangle down the left side and two smaller rectangles on the other side.

Invite students to work with a partner to decompose the shape and find the area. H What is the area of this shape? How do you know? The area is 55 square units. We broke apart the shape into three

rectangles and found the area of each rectangle. Then we added the areas to find the total area of the shape.

H Instead of decomposing, let’s think about a different way to find the area of the shaded shape, a way with fewer steps.

H What multiplication expression represents the area of this rectangle? 8×8

H What is the area of the large rectangle? 64 square units H Is 64 square units the area of the shaded shape? No.

H What rectangle could we remove from the large rectangle so that we are left with the area of the shaded shape? The unshaded rectangle

Gesture to the unshaded 3 by 3 rectangle. H What are the length and width of the unshaded rectangle? 3 units and 3 units

Concept Mini Lessons | Teacher Guide

Objective 2 | Find the area of composite shapes on a grid by subtracting from a larger area. 10 M I NU T ES

Write (8 × 8) as students do the same.

Label the side lengths and have students do the same.

H Then we found the area of the smaller, unshaded rectangle.

expression represents

H What multiplication

How did we find that? We multiplied the side

the area of the unshaded

(8 × 8) − (3 × 3) = 64 − 9 = 55 Area: 55 square units

lengths to find the area of the

rectangle? 3×3

H What is the area of the

the area of the large rectangle to find the area of the shaded

H So, what is the area of the shaded shape? 55 square units

Invite students to model the following sequence by tracing and covering the rectangles with their fingers when appropriate. Record the equations while asking the following questions. H Let’s record our thinking. First, we found the area of the large rectangle. What expression did we use to find the area of the large rectangle? (8 × 8)

We multiplied: 3 × 3.

H We subtracted the area of the smaller, unshaded rectangle

H We can subtract the area of the unshaded rectangle from shape. What is 64 − 9? 55

unshaded rectangle.

from the area of the large rectangle.

Write − (3 × 3) as students do the same.

EV I

unshaded rectangle? 9 square units

Gesture to each expression as you ask the following questions. H What is 8 × 8? 64

H What is 3 × 3? 9

Write = 64 − 9 as students do the same.

H What is the area of the shaded shape? 64 − 9 = 55

The area of the shaded shape is 55 square units.

Record the area and have students do the same.

Concept Mini Lessons | Teacher Guide

Objective 2 | Find the area of composite shapes on a grid by subtracting from a larger area. 10 M I NU T ES

H Does the area we found by subtracting match the area you found earlier by decomposing the shape into rectangles?

Yes.

Teacher Tip: Differentiation

Consider providing a more concrete representation of subtracting the area of the smaller, unshaded rectangle from the area of the large rectangle. Draw the shape on grid paper and give students scissors to cut out the smaller rectangle.

• A composite figure with an area of 18 square units • A composite figure with an area of 14 square units • A composite figure with an area of 30 square units Analyze Student Progress

Monitor: • Can the student identify and find the area of the large rectangle? • Can the student identify and find the area of the unshaded rectangle? • Can the student subtract to find the area of the shaded shape?

EV I

Invite students to turn and talk about how they can find the area of a shape composed of rectangles by subtracting from a larger area. Language Support

Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Consider providing sentence frames and starters to support students as they talk about how to find the area of composite shapes by subtracting from a larger area.

• The area of the large rectangle is ______ because ______ × ______ = ______.

• The area of the unshaded rectangle is _____ because _____ × _____ = _____. • To find the area of the shaded shape, we subtract ________ from ________. • The area of the shaded shape is _________ square units because _________.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed.

Questions to Advance Student Thinking: • Where do you see a large rectangle? How can you find the area of the large rectangle? • Where do you see a smaller rectangle with an area you can subtract from the large rectangle? How can you find the area of the smaller rectangle? • How can you use the area of the large rectangle and the area of the unshaded rectangle to find the area of the shaded shape? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Find the area of composite shapes with unknown side lengths by subtracting. 10 M I NU T ES

Distribute the Objective 3 Student Page. Direct students to the shape in problem 1.

10 cm

Trace your finger along the boundary of the unshaded rectangle and have students do the same.

3 cm

8 cm

H What do you notice

a large rectangle.

We don’t know.

EV I

rectangle cut out of

H What are the side lengths of the unshaded rectangle? They aren’t labeled.

about the shape? It shows a small

Materials • Objective 3 Student Page

Summary Students use attributes of rectangles to find unknown side lengths of a shape and then to find the area of the shape.

4 cm

H How can we find the area of the shaded shape?

H The side lengths of the unshaded rectangle aren’t labeled, but we can use what we know about rectangles to help us find the side lengths.

Gesture to the length of the large rectangle.

We can break apart the shaded shape into two rectangles.

We can subtract the unshaded area from the area of the large

rectangle.

H Let’s subtract the area of the unshaded rectangle from the area of the large rectangle.

Trace your finger along the boundary of the large rectangle and have students do the same. H What are the side lengths of the large rectangle? 10 centimeters and 8 centimeters

H What is the length of the large rectangle? 10 centimeters H We know that opposite sides of a rectangle are the same length. Since we know the length of the large rectangle is 10 centimeters, what is the opposite side length? 10 centimeters

Gesture to the side length of 4 cm.

Concept Mini Lessons | Teacher Guide

H We know this part of the side length is 4 centimeters and the the unknown side length of the unshaded rectangle. H What is 10 − 4?

To support students in discussing and describing relationships between side lengths, consider using color to highlight opposite sides, known parts, and unknown parts.

whole side length is 10 centimeters. We can subtract to find

Language Support

10 cm

3 cm

Gesture to the length of the unshaded rectangle.

H So the length of the unshaded rectangle is 6 centimeters.

10 cm

3 cm

5 cm Use the same strategy to find the width of the unshaded rectangle. Label 6 cm the width of the unshaded rectangle 5 cm and have students do the same.

6 cm

4 cm

H Can we now find the area of the shaded shape? How do you

EV I

Label the length of the unshaded rectangle 6 cm and have students do the same.

8 cm

5 cm

know?

8 cm

4 cm

Yes. We can find the area of the large rectangle and then subtract the area of the unshaded rectangle.

Guide students in writing equations to find the area of the shaded shape as you ask the following questions. H What multiplication expression represents the area of the whole rectangle? 10 × 8 H What multiplication expression represents the area of the unshaded rectangle? 6×5

shaded shape? How do you know?

(10 × 8) − (6 × 5) = 80 − 30 = 50 Area: 50 square centimeters

The area of the shaded shape is 50 square centimeters because

80 − 30 = 50.

Record the area and have students do the same.

Analyze Student Progress

H What is the area of the

Questions to Advance Student Thinking: • How can you find the area of the large rectangle? • How can you use what you know about rectangles to find the unknown side lengths? • How can you find the area of the unshaded rectangle? • How can you use the area of the large rectangle and the area of the unshaded rectangle to find the area of the shaded shape?

EV I

Invite students to turn and talk about how they can find the area of a shape composed of rectangles with unknown side lengths by subtracting.

Monitor: • Can the student find the area of the large rectangle? • Can the student use what they know about rectangles to find the unknown side lengths? • Can the student find the area of the unshaded rectangle? • Can the student subtract to find the area of the shaded shape?

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

• A composite figure with an area of 33 square inches • A composite figure with an area of 31 square feet • A composite figure with an area of 42 square centimeters

Objective 4 | Find the area of composite shapes by using rectangles. 10 M I NU T ES

Summary Students self-select a strategy to use to find the area of a composite shape.

Materials • Sample Solutions • Objective 4 Student Page

H How can we use rectangles to help us find the area of a shape that is composed of rectangles?

H What strategy did Eva use to find the area of the shaded shape?

We can break the shape apart into smaller rectangles and then add

She broke apart the shape into three rectangles, found the area of

the areas of the smaller rectangles together.

each rectangle, and then added the areas.

We can find the area of the larger rectangle and subtract the area of the unshaded part.

EV I

H What strategy did David use to find the area of the shaded

Show the Sample Solutions template. Eva’s Work

8 in

2 in

6 in 4 in

2 in

6 in

He found the area of the large rectangle and then subtracted the area of the unshaded rectangle.

H What is similar about their work? They both found the area to be 32 square inches.

4 in

shape?

David’s Work

8 in

Gesture to David’s work.

2 in

8 × 2 = 16 4×2=8 4×2=8 16 + 8 + 8 = 32 Area: 32 square inches

Gesture to Eva’s work.

2 in

H Both strategies give the same area. You can choose a

4 in

8 × 6 = 48 4 × 4 = 16 Area: 32 square inches

2 in

strategy to find the area of a shape that is composed of rectangles.

Distribute the Objective 4 Student Page. Direct students to the shape in problem 1. Invite students to turn and talk about which strategy they will use to find the area of the shape.

Concept Mini Lessons | Teacher Guide

Objective 4 | Find the area of composite shapes by using rectangles. 10 M I NU T ES

H Which strategy will you use to find the area of this shape? Why? of the rectangles because I prefer to add.

I will break apart the shape into rectangles and then add the areas

I will subtract the area of the unshaded rectangle from the area of

the large rectangle because that’s fewer steps. If I add the areas of the rectangles inside, I will have to add three different areas.

• Where are the rectangles you used to find the area of the shaded shape? • How did you know the side lengths of each rectangle? • What did you do with the areas of those rectangles? Why? • Which strategy is more efficient? Why? Language Support

Consider providing a word bank to support student explanations of their strategies. Include words and phrases such as rectangle, area, shaded shape, unshaded shape, unknown, length, and width.

EV I

Invite students to choose a strategy to use to find the area of the shaded shape. As students work, encourage them to label any unknown side lengths that they will use in their strategy Teacher Tip

Consider providing highlighters and other tools for students to use to label and annotate on the shape.

Select work samples that show each strategy.

7 cm

8 cm

5 cm

3 cm 1 cm 3 cm 8 × 7 = 56 5×1=5 Area: 51 square centimters

3 cm

1 cm

7 cm

8 cm

Invite students to share their work one at a time. As students share, use questions such as the following to highlight how the areas of different rectangles were part of their strategies:

5 cm

3 cm

Invite students to turn and talk about how to use rectangles to find the areas of shapes composed of rectangles. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work. • A composite figure with an area of 19 square inches • A composite figure with an area of 20 square feet • A composite figure with an area of 42 square inches

8 × 3 = 24 8 × 3 = 24 3×1=3 24 + 24 + 3 = 51 Area: 51 square centimters

Concept Mini Lessons | Teacher Guide

Objective 4 | Find the area of composite shapes by using rectangles. 10 M I NU T ES

Notes

Analyze Student Progress

Monitor: • Can the student select a strategy to find the area of the shaded shape? • Can the student correctly determine the unknown side lengths? • Can the student correctly find the area of the shaded shape?

EV I

Questions to Advance Student Thinking: • Which strategy will you use to find the area of the shaded shape? • How can you use what you know about rectangles to find the unknown side lengths? • How can you use rectangles to find the area of the shaded shape?

Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 1 | Find the area of composite shapes on a grid by decomposing. Composite Shapes Shape A

Shape D

EV I

Shape C

Shape B

Concept Mini Lessons | Teacher Guide

Objective 4 | Find the area of composite shapes by using rectangles. Sample Solutions David’s Work

8 in

2 in

6 in 4 in

2 in

6 in

4 in

EV I

4 in

Eva’s Work

2 in

8 × 2 = 16 4×2=8 4×2=8 16 + 8 + 8 = 32 Area: 32 square inches

2 in

4 in

2 in

8 × 6 = 48 4 × 4 = 16 Area: 32 square inches

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Applications of Multiplication and Area Answer Key Objective 1

Objective 2

1. Correctly drawn line;

1. Correctly labeled side lengths; 55

correctly labeled length and width; 23

2. Correctly drawn line;

2. Correctly labeled side lengths; 18

3. Correctly labeled side lengths; 14

3. Correctly drawn line; correctly labeled length and width; 30

1. 51 square centimeters

2. 33 square inches

2. 19 square inches

3. 31 square feet

3. 20 square feet

4. 42 square centimeters

4. 42 square inches

4. Correctly labeled side lengths; 30

4. Correctly drawn line;

Objective 4

1. 50 square centimeters

EV I

correctly labeled length and width; 28

Objective 3

correctly labeled length and width; 31

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Applications of Multiplication and Area Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Applications of Multiplication and Area Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Find the area of composite shapes on a grid by decomposing. Draw a line in each shape to show how to break it apart into two rectangles. Label the length and width of each rectangle. Then represents 1 square unit.

find the area of the shape. Each

EV I

Area: ______________ square units

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 1 | Find the area of composite shapes on a grid by decomposing. 4

EV I

Area: ______________ square units

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 2 | Find the area of composite shapes on a grid by subtracting from a larger area. Each

represents 1 square unit.

Label the unknown side lengths. Then subtract from a larger area to find the area of the shaded shape.

EV I

Area: ______________ square units

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 2 | Find the area of composite shapes on a grid by subtracting from a larger area. 3

EV I

Area: ______________ square units

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 3 | Find the area of composite shapes with unknown side lengths by subtracting Subtract from a larger area to find the area of the shaded shape.

10 cm

1 3 cm

9 in 2 in

5 in

8 cm

5 in

EV I

4 cm

Area: _____________________________________

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 3 | Find the area of composite shapes with unknown side lengths by subtracting 7 ft

1 cm 2 cm

4 cm

5 ft 5 ft

8 ft

10 cm

EV I

5 cm

Area: _____________________________________

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 4 | Find the area of composite shapes by using rectangles. Find the area of each shaded shape. Show your strategy.

7 cm

1 8 cm

5 cm 3 cm

4 in 4 in

EV I

3 cm

1 in

7 in

Area: _____________________________________

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

NAME

DATE

Objective 4 | Find the area of composite shapes by using rectangles.

4 ft

2 in

1 in

6 ft

6 in

1 in

1 ft

9 in

4 in

EV I

6 ft

Area: _____________________________________

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

Practice | Applications of Multiplication and Area Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers during Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Find the area of

Objective 2 Find the area of

Objective 3 Find the area of

Objective 4 Find the area

composite shapes on a grid by

subtracting from a larger area.

composite shapes with unknown side lengths by subtracting.

of composite shapes by using rectangles.

Look for ...

• Can the student identify and find the area of the large rectangle? • Can the student identify and find the area of the unshaded rectangle? • Can the student subtract to find the area of the shaded shape?

• Can the student find the area of the large rectangle? • Can the student use what they know about rectangles to find the unknown side lengths? • Can the student find the area of the unshaded rectangle? • Can the student subtract to find the area of the shaded shape?

• Can the student select a strategy to find the area of the shaded shape? • Can the student correctly determine the unknown side lengths? • Can the student correctly find the area of the shaded shape?

Look for ...

• Can the student decompose the shape into two rectangles? • Can the student identify the length and width of each rectangle? Can the student find the area • of each rectangle and add to find the total area?

EV I

composite shapes on a grid by decomposing.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Applications of Multiplication and Area Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Practice | Teacher Guide

Practice | Applications of Multiplication and Area

Practice Page 1

Practice Page 2

1. Correctly drawn line;

1. D

correctly labeled length and width; 38

Practice Page 3

Practice Page 4

Answer Key

1. A

1. 36 square inches

2. 34 square inches

2. 32 square feet

3. 26 square centimeters

3. 34 square centimeters

2. Correctly labeled side lengths; 41

2. Correctly drawn line; and width; 49

3. Correctly drawn line; correctly labeled length and width; 34

4. Correctly labeled side lengths; 52

4. Correctly drawn line;

3. Correctly labeled side lengths; 65

EV I

correctly labeled length

4. Casey subtracted

4. Pablo incorrectly found

the side length of the

the area of the unshaded

unshaded rectangle

rectangle. The side

from the large rectangle

lengths of the unshaded

instead of subtracting

rectangle are 4 inches

the area of the unshaded

and 3 inches, not

rectangle.

5 inches and 3 inches.

correctly labeled length and width; 20

Practice | Teacher Guide

EV I

Student Edition | Printable Pages for students

Practice | Teacher guide

NAME

DATE

Practice Page 1 | Find the area of composite shapes on a grid by decomposing. Draw a line in each shape to show how to break it apart into two rectangles. Label the length and width of each rectangle. Then represents 1 square unit.

find the area of the shape. Each

EV I

Area: ________ square units

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 1 | Find the area of composite shapes on a grid by decomposing. 3

EV I

Area: ________ square units

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

the area of composite shapes on a grid by subtracting from a Practice Page 2|Find larger area. Find the area of the shaded shape. Circle the letter of the correct answer.

Label the unknown side lengths. Then subtract from a larger area to find the area of each shaded shape. Each

represents 1 square unit.

EV I

Each

represents 1 square unit.

A 40 square units B 32 square units Each represents 1 square unit. units 28 square C D 24 square units

Area: ________ square units

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

the area of composite shapes on a grid by subtracting from a Practice Page 2|Find larger area.

EV I

Area: ________ square units

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

the area of composite shapes with unknown side lengths by Practice Page 3|Find subtracting.

Circle the letter of the correct answer.

10 in

6 in

4 ft 3 ft 1 ft

4 in

17 square feet 18 square feet 27 square feet 36 square feet

1 in

EV I

9 ft

A B C D

Subtract from a larger area to find the area of the shaded shape.

Area: ________________________

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

the area of composite shapes with unknown side lengths by Practice Page 3|Find subtracting. 6 cm 2 cm

3 cm

5 cm

Casey incorrectly found the area of the shape below. Look at Casey’s work. What mistake did Casey make?

1 cm

Casey’s Work

3 feet

2 feet

EV I

2 feet 6 feet

8 feet

6 × 8 = 48

48 − 2 = 46 Area: 46 square feet

Area: ________________________

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 4 | Find the area of composite shapes by using rectangles. Find the area of each shaded shape. Show your strategy.

9 in

2 in

8 ft

4 ft

3 in

5 in

2 ft

5 ft

EV I

1 ft

Area: ________ square inches

Area: ________ square feet

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Page 4 | Find the area of composite shapes by using rectangles. 7 cm

2 cm

Pablo incorrectly found the area of the shaded shape. Look at Pablo’s work. What mistake did Pablo make?

Pablo’s Work

2 cm

1 cm

6 cm

2 in

EV I

2 cm

3 in

5 in

1 in 8 in

8 × 5 = 40

3 × 5 = 15

40 − 15 = 25 Area: 25 square inches

Area: ________ square centimeters

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 1 Area: ________ square units

Look at the problem. Then look at the work. It shows how to find the area of a shape on a grid by decomposing.

How can you find the length and width

How can you find the area of each

two rectangles?

of each rectangle?

rectangle? The total area?

EV I

How can you break apart the shape into

I can draw a dotted line to decompose the shape

into 2 rectangles.

7 3

I can find the length and width of each rectangle

(3 × 7) + (3 × 5) = 21 + 15 = 36

I can find the area of each rectangle by multiplying its length and the width. I can find the total area of the shape by adding the areas of the rectangles.

by counting the squares on its sides.

Area: 36 square units 3 5

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Practice | student Page

NAME

DATE

Practice Helper 2

Area: ________ square units

Look at the problem. Then look at the work. It shows how to find the area of a shape on a grid by subtracting from a larger area.

Where do you see a smaller rectangle?

How can you use the area of the large

How can you find the area of the large

How can you find the area of the smaller

rectangle and the area of the unshaded

rectangle?

rectangle to find the area of the shaded

EV I

Where do you see a large rectangle?

I see a large rectangle around the outside of both the shaded and unshaded rectangles. I can find

the area of the large rectangle by labeling its side lengths and then multiplying.

I see a smaller rectangle in the unshaded part.

shape?

(6 × 5) − (2 × 3) = 30 − 6 = 24

I can subtract the area of the unshaded rectangle from the area of the large rectangle to find the area of the shaded shape.

I can find the area of the smaller rectangle by labeling its side lengths and then multiplying.

Area: 24 square units

6 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATh cATAlysT | © 2025 Great Minds PBc

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 3 Look at the problem. Then look at the work. It shows how to find the area of a shape with unknown side lengths by subtracting.

9 cm

10 cm

Area: ________ square centimeters

4 cm 2 cm

1 cm

the large rectangle? The side lengths of the large rectangle are 10 centimeters and

9 centimeters. I can multiply to find

the area. 10 × 9 = 90.

How can you find the area of

How can you use the area

know about rectangles

the unshaded rectangle?

of the large rectangle and

to find the unknown side

lengths?

I can find the area of the unshaded

rectangle by finding the unknown

I know that the opposite sides of a

rectangle are the same length. If I

side length and then multiplying the

side lengths.

9 cm

the area of the unshaded rectangle to find the area of the shaded shape? I can find the area of the shaded

know the whole side length and a

shape by subtracting the area of the

part of that side length, I can subtract

unshaded rectangle from the area

to find the unknown side length.

of the large rectangle.

10 cm

7 cm

How can you use what you

EV I

How can you find the area of

Area: 62 square centimeters

4 cm 2 cm

1 cm

This page may be reproduced for classroom use only.

Practice | student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to find the area of a shape by using rectangles.

2 ft

5 ft

7 ft

10 ft

1 ft

Area: ________ square feet

EV I

3 ft

Which strategy will you use to find the

How can you use what you know about

How can you use rectangles to find the

area of the shaded shape?

rectangles to find the unknown side

area of the shaded shape?

I can subtract the area of the unshaded rectangle from the area of the large rectangle.

lengths?

I can find the area of the shaded shape by

I know that the opposite sides of a rectangle are

finding the area of the large rectangle and then

the same length. If I know the whole side length

subtracting the area of the unshaded rectangle.

and a part of that side length, I can subtract to

10 ft

1 ft

2 ft

3 ft 3 ft

find the unknown side length.

5 ft

7 ft

Area: 61 square feet

3 ft

This page may be reproduced for classroom use only.

Practice | student Page

Application | Applications of Multiplication and Area Read–Draw–Write

Activities, Structures, and Considerations

EV I

Structure(s)

Considerations

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplication and area. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting student to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a personal whiteboard. • Consider providing grid paper for students to use to draw the composite figures and find their areas. • Consider providing highlighters and other tools for students to use to annotate the information provided in the task.

Play a Game Solve a Task

Activity

Partner Work

Independent Work Partner Work

Application | Teacher Guide

Application | Applications of Multiplication and Area

Solve a Problem Materials

Play a Game: Three in a Row

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

Materials

Students use the Read–Draw– Write process to solve word problems that involve finding the area of composite shapes. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. For all three problems, students make decisions about decomposing the shape into rectangles or subtracting from a larger area.

Students work with a partner to play a game involving finding the area of composite shapes.

Playing the Game • Players take turns choosing a space on the game board and finding the area of the shaded shape.

Solve a Task Materials

• Solve a Task Student Page • Highlighters (optional) Students work independently or with a partner to solve a multipart task that involves finding the area of composite shapes. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Preparing to Play • Players decide who will mark an X and who will mark an O. • Players decide who will go first.

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Teacher Tip

• Personal whiteboard • Game Instruction Card • Three in a Row Game Board • Grid paper (optional)

• Players check each other’s work. If they agree, the player marks an X or an O in that space. If they disagree, they share their work with each other and find the mistake. The space gets marked if the player who chose it got it correct. • The first person to get three in a row vertically, horizontally, or diagonally wins.

Application | Teacher Guide

Application | Applications of Multiplication and Area

Answer Key Solve a Task

Solve a Problem 1. Mr. Endo needs 60 square feet of carpet to cover the floor.

1. Luke could paint 48 square feet of wall with the paint that is left.

2. The area of the wall Jayla paints is 63 square feet.

2. 46 square feet of the wall are not covered by the painting

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or the fireplace.

3. I know Luke has tiles left because the area he needs to cover is less than 20 square feet.

3. The area of Shen’s garden is 37 square feet.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

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Mr. Endo wants new carpet for his office. How many square feet of carpet does he need to cover the floor?

10 ft 3 ft

Shen plants a garden. What is the area of his garden?

8 ft

4 ft

9 ft

Jayla paints a wall that has a window. What is the area of the wall she paints?

3 ft

Window

2 ft

1 ft

9 ft

3 ft

2 ft

1 ft

3 ft

Garden

5 ft

8 ft

Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Problem Number

This page may be reproduced for classroom use only.

Application | student Page

Application | Play a Game Game Instruction Card

Three in a Row

How to Win

What You Need

The first player with three marks in a row (vertically, horizontally, or diagonally) wins. If the game board is filled

• Personal whiteboard

without a player getting three in a row, clear the board and

• Three in a Row Game Board

play another round.

• Grid paper (optional) How to Play

1. Decide who will mark an X and who will mark an O.

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Player A, choose a problem.

2. Player A, find the area of the shaded shape.

3. Player B, check player A’s work. If you both agree on the

answer, tell player A to mark their X or O in that space. If

you disagree, share your work with each other and find the mistake. If player A had the right answer, tell them to mark

the space with their X or O.

4. Player B, choose a new space and solve. Player A, check their work. If you both agree on the answer, tell player B to mark their X or O in that space. If you disagree, share your work with each other and find the mistake. If player B had the right answer, tell them to mark the space with their X or O.

This page may be reproduced for classroom use only.

Application | student Page

Application | Play a Game | Three in a Row Game Board 4 cm

9 ft

6 ft

2 ft

4 ft

8 ft

2 ft

3 ft 4 ft

8 cm 3 cm 9 cm 2 in 3 in

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8 feet

2 feet

4 feet

2 in

6 in

10 feet

9 ft

3 cm

9 in

7 in

8 in

3 ft

10 cm

3 in

7 ft

2 ft

5 cm 8 cm For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATh CATAlysT | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Application | student Page

NAME

DATE

Application | solve a Task • Luke wants to paint a wall.

Luke’s Fireplace • He wants to put new stone tile on the fireplace. • Luke measures the wall and the fireplace.

9 ft

8 ft

5 ft 4 ft

• He buys 1 can of paint that covers 100 square feet.

Luke paints the wall. He does not paint the fireplace. How many more square feet of wall could Luke paint with the paint he has left?

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• Luke buys 2 boxes of stone tiles. Each box covers 10 square feet.

This page may be reproduced for classroom use only.

Application | student Page

NAME

DATE

Application | solve a Task

not covered by the painting or the fireplace?

Luke puts the stone tile around the fireplace. Does he have any tiles left? How do you know?

Luke hangs a painting above the fireplace. The painting measures 2 feet by 3 feet. How many square feet of the wall are

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This page may be reproduced for classroom use only.

Application | student Page

Multiplication

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Multiplication of Multi-Digit Numbers by One-Digit Numbers

Concept Guide | Multiplication of Multi-Digit Numbers by One-Digit Numbers Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard

• Personal whiteboard and Student Pages

• Print copies of Student Pages as needed.

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Application Teacher Guide

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards • Gather 9 two-color counters for each student pair. • Print copies of the following: - Solve a Problem Recording Page (optional) - Read–Draw–Write Tool (optional) - Three in a Row Game Board - Solve a Task Student Page • Place copies of Three in a Row Game Board into whiteboards.

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Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Multi-Digit Numbers by One-Digit Numbers

Addressing Student Misconceptions How to Address Misconception

Students misunderstand how the partial products of multi-digit multiplication are related and recorded with the standard algorithm.

Have students notice how multi-digit partial products are written in vertical form and then related to the standard algorithm notation when placed side by side. This observation allows students to attend to place value when recording each partial product in the standard algorithm.

478 ×1 12 956

Language Support

8 2

1 4 8 0

0 0

2 2 2

× × ×

8 7 4

ones tens hundreds

478 ×1 12 956

Encourage students to point to and verbalize where they see each factor and partial product in each notation. Remind students to rename partial products with larger units when possible (e.g., 16 ones can be renamed as 1 ten 6 ones).

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478 × 2 81416

Student Misconception

As applicable, complement any of these groupings by pairing students who speak the same home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. These charts should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Multiplication of Multi-Digit Numbers by One-Digit Numbers Dear Family, Your student is working on using the standard algorithm to multiply multi-digit numbers by one-digit numbers. This involves putting together a few skills: multiplication facts, finding partial products, composing new units, and understanding place value concepts when using the standard algorithm. The sample work shows how to use partial products to understand the standard algorithm notation. You can support your student’s progress by asking the questions in the table below as your student multiplies by using the standard algorithm.

2,5 7 6 × 5 5 × 6 ones = 30 ones

I will multiply the ones first.

2, 5 + 1 0, 0

0 0

1 2, 8

EW 3

3 5

5 0 0

5 × 6 ones 5 × 7 tens 5 × 5 hundreds 5 × 2 thousands

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What will you multiply by first?

2, 5

2,5 7 6 5 2 3 3 1 2,8 8 0

Can you rename a partial product as a

What unit will you multiply by next?

larger unit? How do you show how to

Did you remember to rename units

rename units?

if possible?

2,5 7 6 5 3 0

2,5 7 6 5 2 3 3 12,8 8 0

30 ones can be renamed as 3 tens 0 ones. I write

Next, I will multiply the tens, then the hundreds,

the renamed units on the line under the correct

and then the thousands. I crossed out the

place value.

renamed ones, tens, and hundreds to show I added them to the other like units.

Concept Guide | Teacher Guide

Progress Check | Multiplication of Multi-Digit Numbers by One-Digit Numbers

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Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions: • Can the student find the partial products and use them to determine the product? | Objectives 1–3

• Does the student use area models to find the product? | Objective 1

| Objective 2

• Does the student record partial products in vertical form?

• Does the student use the standard algorithm to find the product? | Objective 3 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress.

Progress Check | Teacher Guide

Progress Check | Multiplication of Multi-Digit Numbers by One-Digit Numbers

Progression Towards Proficiency Rubric Progress Check Tool Item(s)

Items 3–4

Items 5–6

Objective 1

Objective 2

Objective 3

Not Yet Proficient

The student may show evidence of beginning to understand multiplication of multi-digit numbers by one-digit numbers but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes an error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly completes an area model to represent the problem and finds the correct products:

The student correctly records the partial products in vertical form and finds the correct products:

2. 27,492

4. 19,620

The student correctly multiplies by using the standard algorithm and circles the letter of the correct answer:

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1. 1,588

Items 1–2

5. B 6. J

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Multi-Digit Numbers by One-Digit Numbers Complete the area model to represent the problem. Then write the product.

4 × 397 =

Record the partial products vertically. Then write the product.

836 × 7 =

5 × 3,924 =

6 × 4,582 =

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This page may be reproduced for classroom use only.

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Multi-Digit Numbers by One-Digit Numbers Multiply. Show your work by using the standard algorithm. Circle the letter of the correct answer.

3 × 617 =

5,248 × 9 =

F 45,862

A 1,831 B 1,851

G 46,908

C 2,031

47,232

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D 2,051

H 47,160

This page may be reproduced for classroom use only.

Progress Check | student Page

of Multi-Digit Numbers by One-Digit Concept Mini Lessons | Multiplication Numbers Progression of Mini Lesson Objectives 2 Multiply multi-digit numbers by one-digit

3 Multiply multi-digit numbers by one-digit

numbers by using area models and partial products.

numbers by relating area models to recording partial products in vertical form.

numbers by using the standard algorithm.

4 × 532

500

2,000

120

Start here if students

5, 6

1 4

6 8 0

3, 6 + 3 0, 0

0 0

3 3, 8

6 × 3 ones 6 × 4 tens 6 × 6 hundreds 6 × 5 thousands

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2,000 + 120 + 8 = 2,128

1 Multiply multi-digit numbers by one-digit

• can draw a rectangular array to represent multiplication and • can decompose three- and four-digit numbers into place value units, but • need support using an area model to multiply multi-digit numbers by one-digit numbers and • need support using the distributive property and mental math to find and add partial products.

2,5 7 6 5 2 3 3 12,8 8 0

Start here if students • can record partial products in vertical form but • need support multiplying three- and four-digit numbers by one-digit numbers by using the standard algorithm.

Start here if students

• can use an area model to multiply multi-digit numbers by one-digit numbers and • can use mental math to find partial products, but • need support recording partial products in vertical form.

Concept Mini Lessons | Teacher Guide

multi-digit numbers by one-digit numbers by using area models Objective 1 | Multiply and partial products. 10 M I NU T ES

Materials • Personal whiteboard • Objective 1 Student Page

Summary Students use area models to multiply and relate partial products to the distributive property. Direct students’ attention to the Student Page. Point to the 4 by 532 rectangle as you ask the following question.

H Let’s decompose and label the area model.

Draw lines to decompose the length of 532 and label the parts as 500, 30, and 2. Have students do the same.

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500

532

We can multiply the side lengths.

H We know that 4 × 532 = 532 + 532 + 532 + 532. We could add 532 four times, or we could draw an area model to help us find the product of 4 and 532.

Gesture to the 532 in the expression as you ask the following question. H How can we decompose 532 into place value units? 5 hundreds, 3 tens, and 2 ones

H What are 5 hundreds, 3 tens, and 2 ones in standard form? 500, 30, and 2

H To find the area of the whole rectangle, we can find the areas of the smaller rectangles and add them together.

Gesture to the 4 by 500 rectangle.

H How can we find the area of this rectangle? We can multiply the side lengths.

Direct students to skip-count or to use repeated addition to find the product. H What is 4 × 500? 2,000

H How can we find the area of this rectangle?

Gesture to the expression 4 × 532.

Concept Mini Lessons | Teacher Guide

Objective 1 | 10 M I NU T ES

Write 2,000 inside the rectangle and direct students to do the same.

Repeat the process to find the area of the two remaining rectangles in the area model.

500

2,000

120

H Let’s use equations to show the partial products we found by using the area model.

Model completing the first equation below the area model to show how 532 is decomposed into 500 + 30 + 2. Direct students to do the same. H This equation represents the area of the rectangle. In this equation, the length, 532, is represented as 500 + 30 + 2. The length is multiplied by the width, 4.

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532

products. When we add the partial products, we get the product of the original two factors.

H Now that we have found the area of each of the smaller rectangles, how can we find the area of the whole rectangle? We can add the areas of the smaller rectangles.

H What is the area of the whole rectangle? How do you know? The area of the whole rectangle is 2,128. I know because 2,000 + 120 + 8 = 2,128.

H What does the area of the whole rectangle tell us about 4 × 532? 4 × 532 = 2,128

Gesture to the values 2,000, 120, and 8.

H Let’s imagine we have two factors. When we decompose one of the factors and multiply each part by the other factor, we get partial products. In this case, 2,000, 120, and 8 are partial

H We can use the distributive property to rewrite the equation as the sum of the areas of the smaller rectangles.

Model completing the second equation to show how 500, 30, and 2 are multiplied by 4. Direct students to do the same.

4 × 532 = 4 × ( 500 + = (4 × 500 ) + (4 × = 2,000 + 120 + = 2,128

+ 2 ) ) + (4 × 2

H What are the partial products? 2,000, 120, and 8

Have students complete the third row of blanks to show the partial products.

)

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by using area models Objective 1 | and partial products. 10 M I NU T ES

Have students complete the recording on the Student Page. Teacher Tip: Differentiation

If connecting the different representations presents a challenge for students, write the equations in unit form and engage students in comparing the standard form equations and the unit form equations.

Consider providing sentence frames to support students as they discuss how to use an area model to multiply. • We can decompose

into

• We can use

to find the partial products.

• We can add

to find the product.

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4 × 532 = 4 × (5 hundreds + 3 tens + 2 ones) = (4 × 5 hundreds) + (4 × 3 tens) + (4 × 2 ones) = 20 hundreds + 12 tens + 8 ones = 2,000 + 120 + 8 = 2,128

Language Support

H How can we use the partial products to find the product? We can use mental math to find 2,000 + 120 + 8. We get a product of 2,128.

4 × 532 = 4 × (500 + 30 + 2) = (4 × 500) + (4 × 30) + (4 × 2) = 2,000 + 120 + 8 = 2,128

Invite students to turn and talk about how they can use an area model and partial product equations to multiply multi-digit numbers by a one-digit number.

Students may ask about the area model for the 0 in the hundreds place in 8 × 3,045. Mention that either including a space for 0 hundreds or not including a space for 0 hundreds in the area model is correct. Consider discussing the efficiency of not including the space.

Teacher Tip

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by using area models Objective 1 | and partial products. 10 M I NU T ES

Notes Analyze Student Progress

Monitor: • Does the student correctly decompose the multi-digit number into place value units? • Does the student correctly represent both factors on an area model and in an equation? • Can the student find the partial products and use them to find the product?

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Questions to Advance Student Thinking: • How can you represent both factors on an area model? • How can you partition your model to help you decompose a factor? • Where do you see partial products in the area model? Where do you see partial products in the equation? How can you use partial products to find the product?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

multi-digit numbers by one-digit numbers by relating area models Objective 2 | Multiply to recording partial products in vertical form. 10 M I NU T ES

Language Support

Display the area model.

5,000

600

30,000

3,600

240

Consider creating a chart that labels the representations of an area model and vertical form to help students relate the spoken words to the visual recording.

Area Model

Vertical Form

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Materials • Personal whiteboard • Objective 2 Student Page

Summary Students relate area models to recording partial products in vertical form.

H What multiplication expression does the area model represent? 6 × 5,643

H How can we use the area model to find the product of 6 and 5,643? We can find the sum of the areas of the smaller rectangles.

We can add the partial products.

Gesture to the partial products.

H Finding 30,000 + 3,600 + 240 + 18 is difficult for me to do in my head. Let’s use vertical form to record the partial products.

Write 6 × 5,643 in vertical form and direct students to do the same. H When we use vertical form, we usually start with the smallest unit.

Gesture to the ones part of the area model.

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by relating area models Objective 2 | to recording partial products in vertical form. 10 M I NU T ES

Gesture to the vertical form as you say the following.

H Here I see 6 × 3 ones in the vertical form. I can ask myself, What is 6 × 3 ones? What do you think? 18 ones H What is 18 ones in standard form? 18

6 8

6 × 3 ones

H What multiplication equation represents this part of the area model? 6 × 4 tens = 24 tens

Gesture to the vertical form as you say the following. H Here I see 6 × 4 tens in the vertical form. I can ask myself, What is 6 × 4 tens? 24 tens

H What is 24 tens in standard form? 240

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Record 18 ones as 18 in the vertical form and direct students do the same.

5, 6

Gesture to the tens part of the area model.

H What multiplication equation represents this part of the area model? 6 × 3 ones = 18 ones

Record 24 tens as 240 in the vertical form and direct students to do the same. H I am also going to record that the 24 tens represent 6 × 4 tens to help me remember how we found this partial product.

Draw an arrow to the right of 240 and write 6 × 4 tens. Direct students to do the same. Repeat the process to represent the remaining partial products in vertical form.

H I am also going to record that the 18 represents 6 × 3 ones to help me remember how we found this partial product.

Draw an arrow to the right of 18 and write 6 × 3 ones. Direct students to do the same.

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by relating area models Objective 2 | to recording partial products in vertical form. 10 M I NU T ES

1 4

6 8 0

3, 6 3 0, 0

0 0

Gesture to the area model and invite students to add the partial products in the area model.

6 × 3 ones 6 × 4 tens 6 × 6 hundreds 6 × 5 thousands

5, 6

No. Thousands is the largest unit.

H How can we use the partial products to find 6 × 5,643? We can add them.

Invite students to add the partial products.

5, 6 ×

3, 6 + 3 0, 0

0 0

3 3, 8

Invite students to turn and talk about how they can relate area models to recording partial products in vertical form. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work. • 7 × 349 • 2,348 × 4 • 678 × 8

Teacher Tip

1 4

6 8 0

The products are equal.

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H I can ask myself, Are there any units left to multiply? What do you think?

H How does 33,858 compare to the product we found by using the area model?

6 × 3 ones 6 × 4 tens 6 × 6 hundreds 6 × 5 thousands

H What is the sum of the partial products? 33,858

Consider discussing the commutative property to help students think about 2,348 × 4 as 4 × 2,348 and 678 × 8 as 8 × 678. This discussion can assist students as they think about how to represent these expressions in vertical form. If necessary, show both versions of vertical form and discuss the usefulness of writing the larger factor at the top.

2, 3 ×

8 4

4 × 2, 3

Concept Mini Lessons | Teacher Guide

two-digit numbers by two-digit numbers by using an area model Objective 2 | Multiply and recording in vertical form. 10 M I NU T ES

Notes Analyze Student Progress

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Questions to Advance Student Thinking: • What unit will you multiply by first? Next? • How can you use standard form to record ones in vertical hundreds? thousands? form? tens? • How can you use the partial products to find the product?

Monitor: • Does the student find the partial product for each place value unit in the multi-digit factor? • Does the student correctly record the partial products in vertical form? • Can the student use the partial products to correctly find the product?

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

multi-digit numbers by one-digit numbers by using the Objective 3 | Multiply standard algorithm. 10 M I NU T ES

Materials • Personal whiteboard • Objective 3 Student Page

Display the partial products of 2 × 478 in vertical form. Direct students to the same problem on their Student Page.

Language Support

2 1

1 4 8 0

0 0

2 × 8 ones 2 × 7 tens 2 × 4 hundreds

Consider revoicing the standard form of the partial product after students say the multiplication expression it corresponds to.

2 times 8 ones equals 16. 2 times 7 tens equals 140. 2 times 4 hundreds equals 800.

H Complete the expressions for the partial products. Then add the partial products to find the product.

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Summary Students multiply three- and four-digit numbers by one-digit numbers by using the standard algorithm.

H Look at the partial products recorded for 2 × 478. What is the multiplication expression for the first partial product, 16? 2 × 8 ones

H What is the multiplication expression for the next partial product, 140? 2 × 7 tens H What is the multiplication expression for the last partial product, 800? 2 × 4 hundreds

H There is a way to record the partial products all in one line. This way is called the standard algorithm. Let’s see how it works using the same problem.

Display 2 × 478 in vertical form. Direct students to the same problem on their Student Page. H We multiplied 2 by 8 ones to get the first partial product, 16. When we show our work with the standard algorithm, we can only write one digit in each place. We can rename 16 ones with larger units as 1 ten 6 ones. Watch how I record 1 ten 6 ones to leave space for recording the partial product for the tens on the same line.

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

2 1

1 4 8 0

0 0

2 2 2

× × ×

8 7 4

ones tens hundreds

Demonstrate crossing out the renamed ten as students do the same.

Write a small 1 to represent 1 ten on the line under the tens place. Write 6 below the line in the ones place. Direct students to do the same.

478 2 1 6

1 4 8 0

0 0

2 2 2

× × ×

8 7 4

ones tens hundreds

478 ×1 12 56

H Can the 15 tens be renamed as larger units? Yes. 15 tens can be renamed 1 hundred 5 tens.

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H Remember, we add partial products together, so the 1 ten on the line means it will get added to any other tens.

H Point to where you see 2 × 8 in the partial products problem. Point to where you see 2 × 8 in the standard algorithm.

H Point to where you see 16 in the partial products problem. Point to where you see 16 in the standard algorithm. H Now we go to the next-largest place value. What partial product do we find next? We multiply 2 by 7 tens and get 14 tens.

H Watch how I record 1 hundred 5 tens to leave space for recording the partial product for the hundreds on the same line.

Write a small 1 to represent 1 hundred on the line under the hundreds place. Write 5 below the line in the tens place. Direct students to do the same. H What does the 1 on the line in the hundreds place represent? That there is another hundred we need to remember to add to any other hundreds

H Remember to add the renamed ten. 14 tens plus the renamed ten is 15 tens. Cross out the renamed ten to show you remembered to add it to the other tens. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Multiply multi-digit numbers by one-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

2 1

1 4 8 0

0 0

2 2 2

× × ×

8 7 4

ones tens hundreds

Consider acknowledging and praising students’ effort. Identify opportunities when you can offer feedback such as the following:

478 ×1 12 956

Sometimes when I work through a problem, I think it won’t ever end. I tell myself to keep going with my strategy and eventually I will get to the answer.

Analyze Student Progress

Monitor: • How does the student show renaming units when using the standard algorithm? • Does the student remember to add renamed units? • Does the student correctly relate the partial products and product to the recording of the analogous partial product and product in the standard algorithm?

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Teacher Tip

Direct students to point to where they see the partial product in both problems. Then repeat the process with 2 × 4 hundreds, demonstrating how to record each step as students do the same on the Student Page.

H The partial products help us understand what the numbers represent when we record multiplication with the standard algorithm. The standard algorithm gives us a way to keep track of how many we have of each unit while allowing us to just think about single-digit multiplication.

Invite students to turn and talk about how they can use the standard algorithm to multiply.

Questions to Advance Student Thinking: • What will you multiply by first? Next? • Can you rename a partial product as a larger unit? • How can you show that you added the renamed units? • Can you identify the partial products in the vertical form and the standard algorithm? Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

of Multi-Digit Numbers by One-Digit Concept Mini Lessons | Multiplication Numbers Answer Key Objective 2

1. Area model correctly represents the expression; 4 × 532 = 4 × (500 + 30 + 2); (4 × 500) + (4 × 30) + (4 × 2); 2,000 + 120 + 8; 2,128

1. Partial products are written vertically; 33,858

2. Area model correctly represents written vertically; 2,443

the problem; partial products are

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2. Area model correctly represents the expression; 3 × 731 = 3 × (700 + 30 + 1); (3 × 700) + (3 × 30) + (3 × 1); 2,100 + 90 + 3; 2,193

Objective 1

3. Area model correctly represents the expression; 5 × 2,124 = 5 × (2,000 + 100 + 20 + 4); (5 × 2,000) + (5 × 100) + (5 × 20) + (5 × 4); 10,000 + 500 + 100 + 20; 10,620 4. Area model correctly represents the expression; 8 × 3,045 = 8 × (3,000 + 40 + 5); (8 × 3,000) + (8 × 40) + (8 × 5); 24,000 + 320 + 40; 24,360

3. Area model correctly represents written vertically; 9,392

the problem; partial products are

Objective 3 1. 956

2. 16,345 3. 1,757

4. 19,520

4. Area model correctly represents written vertically; 5,424

the problem; partial products are

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Multi-Digit Numbers by One-Digit Numbers Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Multi-Digit Numbers by One-Digit Numbers Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

multi-digit numbers by one-digit numbers by using area models Objective 1 | Multiply and partial products. Complete the area model to represent the expression. Then complete the equations to find the product.

4 × 532

= (4 × =

) + (4 ×

×(

EV I

532 4 × 532 =

3 × 731

) + (4 ×

)

3 × 731 =

×(

= (3 ×

) + (3 ×

+ +

) + (3 ×

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Concept Mini Lessons | Student Page

)

NAME

DATE

multi-digit numbers by one-digit numbers by using area models Objective 1 | Multiply and partial products. 5 × 2,124

5 × 2,124 =

×(

= (5 ×

) + (5 ×

EV I

8 × 3,045

8 × 3,045 =

×(

= (8 ×

) + (5 ×

) + (8 ×

+ +

) + (5 ×

)

) + (8 ×

)

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Concept Mini Lessons | Student Page

)

NAME

DATE

multi-digit numbers by one-digit numbers by relating area models Objective 2 | Multiply to recording partial products in vertical form. 6 × 5,643 =

5,000

600

30,000

3,600

240

7 × 349 =

5, 6

EV I

Complete the area model to represent the problem. Record the partial products in vertical form. Then write the product.

3 6

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Concept Mini Lessons | Student Page

NAME

DATE

multi-digit numbers by one-digit numbers by relating area models Objective 2 | Multiply to recording partial products in vertical form. 2,348 × 4 =

678 × 8 =

EV I

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Concept Mini Lessons | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using the Objective 3 | Multiply standard algorithm.

Multiply in vertical form by using partial products.

+ 8

ones

tens

× ×

hundreds

251 × 7 =

Multiply by using the standard algorithm.

3,2 6 9 × 5

EV I

3, 2

478 × 2

ones

× ×

tens

thousands

hundreds

2 × 9,760 =

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Concept Mini Lessons | student Page

Practice | Multiplication of Multi-Digit Numbers by One-Digit Numbers Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of Multi-Digit Numbers by One-Digit Numbers Concept Mini Lessons Objectives 1–3. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Practice Page 2

Practice Page 3

Objective 1 Multiply multi-digit numbers

Objective 2 Multiply multi-digit numbers

Objective 3 Multiply multi-digit numbers

by one-digit numbers by using area models and partial products.

by one-digit numbers by relating area models to recording partial products in vertical form.

by one-digit numbers by using the standard algorithm.

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Practice Page 1

Look for...

• Can the student decompose a multi-digit number into place value units? • Can the student represent both factors on an area model and in an equation? • Can the student find the partial products and use them to find the product?

Look for...

• Can the student find the partial product for each place value unit in the multi-digit number? • Can the student correctly record the partial products in vertical form? • Can the student use the partial products to correctly find the product?

Look for... • Can the student correctly represent renaming units in the standard algorithm? • Can the student remember to add the renamed units? • Can the student relate the partial products and product to the recording of the partial products and product in the standard algorithm?

Practice | Teacher Guide

Practice | Multiplication of Multi-Digit Numbers by One-Digit Numbers

Answer Key Practice Page 2

Practice Page 3

1. Area model correctly represents the expression; 5 × 245 = 5 × (200 + 40 + 5); (5 × 200) + (5 × 40) + (5 × 5); 1,000 + 200 + 25; 1,225

1. Area model correctly represents

1. 10,152

written vertically; 3,808

the problem; partial products are

2. Area model correctly represents written vertically; 32,532

the problem; partial products are

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2. Area model correctly represents the expression; 4 × 5,132 = 4 × (5,000 + 100 + 30 + 2); (4 × 5,000) + (4 × 100) + (4 × 30) + (4 × 2); 20,000 + 400 + 120 + 8; 20,528

Practice Page 1

3. Area model correctly represents the expression; 3 × 6,035 = 3 × (6,000 + 30 + 5); (3 × 6,000) + (3 × 30) + (3 × 5); 18,000 + 90 + 15; 18,105

3. Area model correctly represents

2. 2,274

3. 17,952 4. 36,540

written vertically; 18,028

the problem; partial products are

4. B

4. Mia incorrectly decomposed 642 into 6 ones, 4 ones, and 2 ones instead of decomposing 642 into 6 hundreds, 4 tens, and 2 ones.

Practice | Teacher Guide

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Student Edition | Printable pages for students

Practice | Teacher Guide

NAME

DATE

multi-digit numbers by one-digit numbers by using area Practice Page 1 | Multiply models and partial products. Complete the area model to represent the expression. Then complete the equations to find the product.

5 × 245

200

5,000

1,000

5 × 245 = 5 × (200 +

= (5 × 200) + (5 ×

)

) + (5 ×

= 1,000 +

EV I

4 × 5,132

)

4 × 5,132 = 4 × (5,000 +

= (4 × 5,000) + (4 × = =

+ 2)

) + (4 × +

) + (4 × 2)

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Practice | Student Page

NAME

DATE

ultiply multi-digit numbers by one-digit numbers by using area Practice Page 1 | M models and partial products. 3 × 6,035

Mia incorrectly found the product of 5 and 642. Look at

Mia’s work. What mistake did Mia make?

3 × 6,035 = 3 × ( = =

) + (3 ×

)

) + (3 ×

5 × 642 = 5 × (6 + 4 + 2) = (5 × 6) + (5 × 4) + (5 × 2) = 30 + 20 + 10 = 60

)

= (3 ×

EV I

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Practice | Student Page

NAME

DATE

multi-digit numbers by one-digit numbers by relating area Practice Page 2 | Multiply models to recording partial products in vertical form.

8 × 476 =

400

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5,422 × 6 =

Complete the area model to represent the problem. Record the partial products in vertical form. Then write the product.

5, 4 ×

6 8 8 × 6 ones 8 × 7 tens 8 × 4 hundreds

2 6

6 × 2 ones 6 × 2 tens 6 × 4 hundreds 6 × 5 thousands

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Practice | Student Page

NAME

DATE

multi-digit numbers by one-digit numbers by relating area Practice Page 2 | Multiply models to recording partial products in vertical form. 2 × 9,014 =

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Which of the following does not correctly represent 497 × 4?

400 1,600

360

B 4

7 4

3 + 1, 6

6 0

0 0

1, 9

D (4 × 7 ones) + (4 × 9 tens) + (4 × 4 hundreds)

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Practice | Student Page

NAME

DATE

multi-digit numbers by one-digit numbers by using the Practice Page 3 | Multiply standard algorithm.

2, 5 ×

4 × × ×

ones

thousands

tens hundreds

2,5 3 8 4

EV I

Multiply in vertical form by using partial products.

3 × 758 =

Multiply by using the standard algorithm.

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Practice | Student Page

NAME

DATE

7,308 × 5 =

EV I

2 × 8,976 =

ultiply multi-digit numbers by one-digit numbers by using the Practice Page 3 | M standard algorithm.

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Practice | Student Page

NAME

DATE

Practice Helper 1 2 × 4,342 =

Look at the problem. Then look at the work. It shows how to

2 × 4,342 = 2 × ( = (2 × = +

+ + + ) + (2 × ) + (2 × + +

use an area model and partial products to multiply.

)

) + (2 ×

)

How can you represent both

How can you find partial

How can you use partial

the multi-digit numbers into

factors on an area model?

products?

products to find the product?

place value units?

4,000

4,342 is 4 thousands, 3 hundreds, 4 tens, and 2 ones.

300

4,000

300

8,000

600

I can represent the factor 2 as

I can find the area of each of the

decompose the length, 4,342, the width of a rectangle. I can

smaller rectangles by multiplying

into 4,000, 300, 40, and 2.

the length times the width.

I can add the partial products to get

8,000 + 600 + 80 + 4 = 8,684

the product.

2 × 4,342 = 8,684

2 × 4,342 = 2 × (4,000 + 300 + 40 + 2) = (2 × 4,000) + (2 × 300) + (2 × 40) + (2 × 2) = 8,000 + 600 + 80 + 4

2 × 4,342 = 8,684

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How can you decompose

4,000

300

8,000

600

2 × 4,342 = 2 × (4,000 + 300 + 40 + 2) = (2 × 4,000) + (2 × 300) + (2 × 40) + (2 × 2) = 8,000 + 600 + 80 + 4 = 8,684

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Practice | Student Page

NAME

DATE

Practice Helper 2 Look at the problem. Then look at the work. It shows how to use vertical form to record partial products.

3 × 795 =

What unit will you multiply by

What can you do with the

first?

next?

partial products to find the

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What unit will you multiply by

3 1

3 × 5 ones

3 × 5 ones is 15.

3 × 5 ones

3 × 9 tens

Then I will multiply the tens.

3 × 9 tens is 270.

3 × 795 = 2,385

I will multiply the ones first.

product?

5 3

3 × 5 ones

3 × 9 tens

2, 1

3 × 7 hundreds

3 × 7 hundreds is 2,100.

Then I will multiply the hundreds.

5 3

3 × 5 ones

3 × 9 tens

+ 2, 1

3 × 7 hundreds

2, 3

I can add the partial products to find the product.

700

2,100

270

3 5

+ 2, 1

2, 3

3 × 5 ones 3 × 9 tens 3 × 7 hundreds

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Practice | Student Page

NAME

DATE

Practice Helper 3 multiply by using the standard algorithm.

2,5 7 6 5

Look at the problem. Then look at the work. It shows how to

What will you multiply by first?

Can you rename a partial product as a

What unit will you multiply by next?

2,5 7 6 × 5

larger unit? How do you show how to

Did you remember to rename units

rename units?

if possible?

5 × 6 ones = 30 ones

2,5 7 6 5 3 0

EV I

I will multiply the ones first.

Next, I will multiply the tens, then the hundreds,

the correct place value.

I crossed out the renamed ones, tens, and

and then the thousands.

hundreds to show I added them to the other like units.

2,5 7 6 5 2 3 3 12,8 8 0

30 ones can be renamed as 3 tens 0 ones.

I write the renamed units on the line under

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Practice | student Page

Application | Multiplication of Multi-Digit Numbers by One-Digit Numbers Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplying three- and four-digit numbers by one-digit numbers. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Activity

Play a Game

Partner Work

• Consider using a standard deck of playing cards if you do not have Eureka Math2 cards.

Solve a Task

Partner Work

• Consider providing grid paper for students to draw area models on or to align place value units when solving using the standard algorithm.

Application | Teacher Guide

Application | Multiplication of Multi-Digit Numbers by One-Digit Numbers

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) Students use the Read–Draw– Write process to solve word problems involving multiplying multi-digit numbers by one-digit numbers. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problems 1 and 2 involve multiplying a three-digit number by a one-digit number. Problem 3 involves multiplying a four-digit number by a one-digit number. Teacher Tip

• Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • Two-color counters (9) • Three in a Row Game Board in a personal whiteboard Students work with a partner to play a game involving multiplying multi-digit numbers by one-digit numbers.

Preparing to Play • Remove the J, Q, K, and Joker cards from the deck. Aces can represent 1. • Shuffle the remaining cards and place them in a single facedown pile. • Each player selects one side of the two-color counters to use as their game piece. Alternatively, each player may choose a set of objects as counters.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Materials

one-digit numbers in the top row. A player turns over a replacement card if any of the one-digit numbers are the same. • Player A places a counter in an empty space on the game board. The player multiplies the number in the left column by the number in the top row. The player shows their work and product below the game board. • Player B checks player A’s work. If the answer is correct, the counter stays in the space. If the answer is incorrect, the counter is returned to the player. • Player B erases the work, chooses a new space, and multiplies. Player A checks player B’s work, following the same process. • Players continue taking turns choosing a space and multiplying until a player announces “Three in a Row” or until the game board is filled.

Materials

Play a Game: Three in a Row

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Solve a Problem

Playing the Game • Players take turns turning over sets of three cards to generate numbers for the game board. They write three-digit numbers in the left column and three

Variations • Engage students in multiplying four-digit numbers by one-digit numbers by having students

turn over four cards to generate numbers for the left column of the game board. Teacher Tip: Differentiation Consider having partners turn over two cards to generate two‑digit numbers in the left column. Consider offering the use of place value disks, place value charts, or area models as concrete and pictorial tools.

Solve a Task Materials

• Solve a Task Student Page Students work with a partner to solve a multi-part task involving multiplying three- and four-digit numbers by one-digit numbers. Students are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution and then use the solution to solve the next problem.

Application | Teacher Guide

Application | Multiplication of Multi-Digit Numbers by One-Digit Numbers

Answer Key Solve a Task

1. Accurate picture drawn to represent the problem; equation shows 144 × 7 = 1,008; Mr. Lopez has 1,008 pencils.

2. Miss Diaz plants 6,225 seeds in the fields; student work shows 1,245 × 5 = 6,225.

3. Miss Diaz plants 12,450 seeds combined; student work shows 6,225 × 2 = 12,450.

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2. Accurate picture drawn to represent the problem; equation shows 925 × 6 = 5,550; The theater sells 5,550 tickets.

1. Miss Diaz plants 1,245 seeds in each acre of the field; student work shows 415 × 3 = 1,245.

Solve a Problem

3. Accurate picture drawn to represent the problem; equation shows 2,650 × 4 = 10,600; Jayla runs 10,600 feet.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

Mr. Lopez has 7 boxes of pencils. There are

144 pencils in each box. How many pencils does Mr. Lopez have?

A theater sells 925 tickets for each of 6 shows.

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How many tickets does the theater sell for the shows?

Jayla runs 4 laps around the lake. Each lap

is 2,650 feet. How many feet does Jayla run?

Application | Teacher Guide

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Student Edition | Printable pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Application | Student Page

Application | Play a Game Game Instruction Card

Three in a Row

2. Player A, place a counter in an empty space on the game

What You Need

board. Multiply the number in the left column by the number in the top row. Show your work and the product

• Eureka Math cards (or a standard deck of playing cards) 2

represent 1.

with the J, Q, K, and Joker cards removed. Aces can

below the game board.

Three in a Row

• Two‑color counters (9)

• Three in a Row Game Board in a personal whiteboard

318

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How to Play

425

1. Mix up the cards. Put the cards into a stack facedown.

Take turns turning over 3 cards to choose numbers for

739

the game board. Write 3 three‑digit numbers in the left

column and 3 one‑digit numbers in the top row. Turn over

another card if any of the one‑digit numbers are the same.

Three in a Row

318

425 × 1 37 2,9 7 5

3. Player B, check player A’s work. If the answer is correct, the counter stays in the space. If the answer is incorrect, return the counter to the player.

425

739

4. Player B, erase the work, choose a new space, put your counter into it, and multiply. Player A, check their work.

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Application | Student Page

Application | Play a Game Game Instruction Card 5. Continue taking turns choosing a space and multiplying

until a player gets three in a row or until the game board is filled.

Three in a Row

318

EV I

425

739

How to Win

The first player with three counters in a row (vertical,

horizontal, or diagonal) wins. If the game board is filled without another round.

either player getting three in a row, clear the board and play

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Application | Student Page

NAME

DATE

EV I

Three in a Row

Application | Play a Game • Three in a Row Game Board

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Application | Student Page

NAME

DATE

Application | solve a Task

Planting Seeds Miss Diaz buys packets of seeds to plant. Each packet contains 415 seeds.

Miss Diaz plants 3 full packets of seeds in each acre of a field. How many

Miss Diaz’s field is 5 acres. How many seeds does Miss Diaz plant in

Miss Diaz has another field that is the same size. She plants the same

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seeds does Miss Diaz use for each acre of the field?

the field?

number of seeds in the second field. How many seeds does Miss Diaz plant in the two fields combined?

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Application | student Page

Multiplication

EV I

Multiplication of Two-Digit Numbers by Two-Digit Numbers

Concept Guide | Multiplication of Two-Digit Numbers by Two-Digit Numbers Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard • Area Model Template • Array Template

• Personal whiteboard and Student Pages

• Print copies of the Student Pages for Objectives 1–4. • Place the Area Model Template in a whiteboard for teacher use in Objective 1. • Place the Array Template in a whiteboard for teacher use in Objective 2.

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Game Instruction Card • Play a Game Recording Page • Eureka Math2 cards or a standard deck of playing cards • Grid paper (optional) • Study a Solution Student Page • Solve a Task Student Page • Blank calendar grid (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards - Grid paper (optional) - Blank calendar grid (optional)

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• Application Teacher Guide

Teacher Materials

• Print copies of the following: - Solve a Problem Recording Page (optional) - Play a Game Recording Page - Study a Solution Student Page - Solve a Task Student Page

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Two-Digit Numbers by Two-Digit Numbers

Addressing Student Misconceptions How to Address Misconception

When multiplying two-digit numbers by two-digit numbers, students multiply tens by tens and ones by ones rather than using place value understanding to break apart both factors and multiply each part separately. For example, a student may represent 16 × 32 incorrectly by writing (10 × 30) + (6 × 2).

Have students relate pictorial representations, such as an area model, to abstract representations of multiplication. The structure of the area model supports students in breaking apart each two-digit factor into tens and ones and representing them as side lengths. Students find four partial products by multiplying each part of one factor by each part of the other factor, recording the four partial products in vertical form, and finding the sum of the partial products.

Student Misconception

6 × 30 = 180

6 × 2 = 12

10 × 30 = 300

10 × 2 = 20

32 × 16 12 180 20 +300 1 512

Language Support

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By having students relate the work they did with the area model to working in vertical form, they begin to conceptualize the connection between the representation and the mathematics.

To support multilingual students in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual students in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The chart should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Multiplication of Two-Digit Numbers by Two-Digit Numbers Dear Family,

36 × 21 =

6 ones × 1 one 6 ones × 2 tens 3 tens × 1 one 3 tens × 2 tens

EV I

21 × 36 6 120 30 +600 756

756

Your student is working on multiplying two-digit numbers by two-digit numbers. They draw area models and break apart each factor into tens and ones before multiplying each part separately to find the partial products. Students record the four partial products in vertical form and find the sum of the partial products. You can support your student’s progress by asking the questions in the table below as your student multiplies two-digit numbers by two-digit numbers in vertical form.

How do you know you found all the

How can you use the partial products to

you think about how to break apart the

partial products?

find the product?

factors?

Can you picture an area model to help

21 is 2 tens 1 one

36 is 3 tens 6 ones I break apart the factors into tens and ones.

6 120 30 600

6 ones × 1 one 6 ones × 2 tens 3 tens × 1 one 3 tens × 2 tens

I multiply each part of 21 by each part of 36 and

label each partial product.

6 120 30 +600 756 I add the four partial products.

Concept Guide | Teacher Guide

Progress Check | Multiplication of Two-Digit Numbers by Two-Digit Numbers

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before starting, while providing, or after providing direct instruction. It is intended to collect data about students’ proficiency with multiplying two-digit numbers by two-digit numbers and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problems 1 and 2 have a factor that is a multiple of 10, problems 3 and 4 have minimal regrouping, and problems 5 and 6 have at least two instances of regrouping. Students self-select strategies for problems 1–4 and use the standard algorithm for problems 5 and 6. Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

| Objective 1

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• Can the student multiply two-digit numbers by two-digit multiples of 10? • Does the student solve by using an area model? | Objectives 1 and 2

| Objective 3

• Does the student solve by using vertical form with four partial products?

• Does the student solve by using the standard algorithm? | Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress.

Progress Check | Teacher Guide

Progress Check | Multiplication of Two-Digit Numbers by Two-Digit Numbers

Progression Towards Proficiency Rubric Item 1

Item 2

Item 3

Item 4

Items 5 and 6

Objective 1

Objectives 2 and 3

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand multiplying two‑digit numbers by two‑digit numbers but makes more than one calculation error that leads to an incorrect answer.

The student may show evidence of beginning to understand multiplying two‑digit by two‑digit numbers but makes more than one calculation error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes one calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student correctly demonstrates solid reasoning but makes one calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and circles the correct answer.

The student correctly demonstrates solid reasoning but makes one calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly uses a strategy or the standard algorithm to multiply and finds the correct product:

The student correctly uses a strategy or the standard algorithm to multiply and circles option D.

The student correctly uses a strategy or the standard algorithm to multiply and finds the correct product:

The student correctly uses the standard algorithm to multiply and finds the correct products:

EV I 2. 1,380

1. 680

Progress Check Tool Item(s)

4. 1,302

5. 4,823 6. 1,674

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Two-Digit Numbers by Two-Digit Numbers Multiply. Show your work.

34 × 20 =

23 × 60 =

EV I

Multiply. Show your work. Circle the letter of the correct answer.

14 × 24 = A 84

B 216 C 236

D 336 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

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Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Two-Digit Numbers by Two-Digit Numbers Multiply. Show your work.

42 × 31 =

EV I

Multiply by using the standard algorithm. Show your work.

91 × 53 =

27 × 62 =

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Progress Check | Student Page

Multiplication of Two-Digit Numbers by Two-Digit Concept Mini Lessons | Numbers Progression of Mini Lesson Objectives

an area model.

2 Multiply two-digit numbers

3 Multiply two-digit numbers

by two-digit numbers by using an area model and recording in vertical form.

by two-digit numbers by using

two-digit numbers by using the

vertical form with four partial products.

standard algorithm.

1 Multiply two-digit numbers by two-digit multiples of 10 by using

10 40

40 × 70 = 2,800

40 × 9 = 360

4 × 10 = 40

4 × 5 = 20

10 × 10 = 100

10 × 5 = 50

Start here if students

• can multiply two-digit numbers by two-digit multiples of 10 by using an area model, but • need support representing the multiplication of two-digit numbers by two-digit numbers by using an area model, and • need support relating the area model to recording four partial products in vertical form.

• can multiply one-digit numbers by a two-digit multiple of 10, and • can represent multiplication of one-digit numbers by a two-digit multiple of 10 with an area model, but • need support multiplying two-digit numbers by two-digit multiples of 10, and • need support representing multiplication of two-digit numbers by two-digit multiples of 10 with an area model.

EV I

Start here if students

31 × 27 7 210 20 +600 837

• can multiply two-digit by two-digit numbers by using an area model, but • need support multiplying two-digit by two-digit numbers by recording partial products in vertical form.

4 Multiply two-digit numbers by

54 × 26 2 324 +1 0 8 0 1 1,4 0 4 Start here if students • can multiply two-digit numbers by two-digit numbers in vertical form with four partial products, but • need support multiplying two-digit numbers by two-digit numbers by using the standard algorithm.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 10 M I NU T ES

Materials • Personal whiteboard • Area Model Template (Teacher) • Objective 1 Student Page

Summary Students draw an area model and record in vertical form to multiply by a multiple of 10. Teacher Tip

H How does the rectangular array show 10 × 15? There are 10 rows of 15 broken apart into 10 rows of 10 and 10 rows of 5. There are 10 rows of 10 shown in one color and 10 rows of 5 in

another color.

EV I

The factors in a multiplication expression can be represented by either side length of an area model. To support students as they multiply two‑digit numbers by two‑digit numbers and record partial products in vertical form, encourage them to consistently represent the first factor by using the vertical side length and the second factor by using the horizontal side length.

Show the Area Model Template in a personal whiteboard.

H The area model also represents 10 × 15 but it doesn’t include unit squares as the rectangular array does. How can we use what we already know about finding the area of a rectangle to find 10 × 15 by using the area model? We can multiply 10 and 10 to find the area of the larger rectangle and 10 and 5 to find the area of the smaller rectangle. Then we can add the areas to find the total area of the rectangle.

Direct students to the area model. Point to the smaller rectangle and then the larger one.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 10 M I NU T ES

1 ten × 1 ten =

1 ten × 5 ones =

H Where do we see the partial products in the area model? How can we use the partial products to determine the product? When I add the partial products together, the total area is 1 hundred 5 tens, or 150.

EV I

If students need support understanding why 1 ten × 1 ten = 1 hundred, have them use base‑ten blocks to represent the ones, tens, and hundreds in each multiplication problem. Students may build a rectangular array using the manipulatives and count together. Alternately, students may place manipulatives on top of centimeter grid paper and shade the squares in the array.

Teacher Tip: Differentiation

H What is the product of 10 × 15? 150

H Let’s multiply tens by ones first. What is 1 ten × 5 ones in unit form? 5 tens

H What is 1 ten × 1 ten in unit form? 1 hundred

H I noticed we had 1 ten multiplied by 1 ten and our answer was in hundreds, not tens. When I think of multiplying two multiples of 10 in unit form, I can multiply by using a familiar multiplication fact and then think of the product in hundreds.

Direct students to problem 1 on the Student Page.

H How should we break apart 79 on this area model? We should break 79 into 70 and 9.

Record 70 and 9 on the top of the area model and invite students to do the same. Point to the area model as you ask the following question.

40 × 70 = 2,800

40 × 9 = 360

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 10 M I NU T ES

Write 2,800 in the area model and in vertical form.

H What is 4 tens × 9 ones? 36 tens

Invite students to add the partial products to find the product.

H What is 40 × 9? 360

H What is 40 × 79? 3,160

Language Support

Encourage students to respond with their answer in unit form when prompting them with unit form. This helps reinforce place value understanding.

Invite students to turn and talk about how they can use an area model to find and record partial products.

The follow‑up in standard form is to support writing the partial products in vertical form.

4 tens × 9 ones

+ 2

4 tens × 7 tens

EV I

Write 360 in the area model.

H Let’s also record in vertical form. We multiplied 4 tens by 9 ones and got 36 tens, or 360.

Write 4 tens × 9 ones and 360 in vertical form. Direct students to do the same.

Point to the area model and vertical form as you ask the following questions. H 4 tens × 7 tens is…?

Record 4 tens × 7 tens beside the second partial product line. 28 hundreds

H How do we record 28 hundreds in standard form? 2,800

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 10 M I NU T ES

Notes Analyze Student Progress

Monitor: • Does the student use place value to break apart one of the factors? • Does the student multiply a unit of ten by a unit of ten to get a unit of hundreds? • Does the student add the partial products to find the product?

EV I

Questions to Advance Student Thinking: • How can you break apart one of the factors to help you multiply? • How does the unit change when tens are multiplied by tens? • How can you use the partial products to find the product?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

1 ten × 1 ten =

1 ten × 5 ones =

EV I

Objective 1 | Area Model Template

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using an area model Objective 2 | M and recording in vertical form. 10 M I NU T ES

Show the Array Template in a personal whiteboard. Point to the smaller array and expression. 10

Materials • Array Template (Teacher) • Personal whiteboard • Objective 2 Student Page

Summary Students multiply two-digit numbers by two-digit numbers by using the area model and relating it to the vertical form.

H The smaller array represents the expression 10 × 15. Which factor is broken apart to help us find two partial products? 15 is broken apart into 10 and 5.

Point to the larger array and expression.

10 × 15

EV I

H What is different about this array? There are 4 parts in the array.

H What is different about the multiplication expression it represents? The first number is 14. You broke that number apart too.

H The larger array shows how we can break apart both factors to find four partial products.

14 × 15

Point to the factors 14 and 15.

H I can ask myself, How can I break apart the factors to help me multiply?

Invite students to break apart both factors and help label the side lengths of the larger array. On the Objective 2 Student Page, guide students to draw and label an area model representing 14 × 15.

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using an area model Objective 2 | M and recording in vertical form. 10 M I NU T ES

5 4

4 × 10 = 40

4 × 5 = 20

14 × 15

EV I

Point to the 4 by 5 rectangle in the area model and model a think-aloud.

H After we break apart the factors, we can distribute or multiply each part of these factors separately. I can ask myself, How can I find the partial product represented by the 4 by 5 rectangle? What do you think? You can multiply. 4 × 5 = 20

Write the equation in the rectangle as students record the partial product. Point to the 4 by 10 rectangle and repeat the process.

H So, we distributed the 4 to each part of 15. First, we found

4 × 5. Then, we found 4 × 10.

Guide students to record the first row of partial products in vertical form on the Student Page. H Now, let’s record the partial products in vertical form. We will use unit form to say the expressions that represent the partial products. This helps us record the place value units correctly. We can start with the ones.

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using an area model Objective 2 | M and recording in vertical form. 10 M I NU T ES

4 ones × 5 ones

4 ones × 1 ten 1 ten × 5 ones 1 ten × 1 ten

H What is 4 ones × 5 ones? 20 ones

H How do we write 20 ones? 20 H What is 4 ones × 1 ten? 4 tens

Teacher Tip

H How do we write 4 tens? 40

To minimize language demands when relating the distributive property to the area model and vertical form, encourage students to use gestures and annotations such as arrows, circles, and color‑coding. Support students’ language development by revoicing the relationships they show visually.

Draw students’ attention to the bottom row of the area model. H What parts of the factors do we still need to multiply?

EV I

Language Support

The factors in a multiplication expression can be written in any order when using vertical form. To align the partial products of the area model in vertical form, the first factor is consistently written below the second factor in the vertical form. Students may record partial products in any order; however, encouraging students to start with the ones mirrors the work of the standard algorithm.

We need to multiply 10

4 × 10 = 40

4 × 5 = 20

10 × 10 = 100

10 × 5 = 50

and 5 by the 10 in 14.

Point to the 10 by 5 rectangle in the area model and model a think-aloud.

H I can ask myself, How can I find the partial product represented by the 10 by 5 rectangle? What do you think? You can multiply. 10 × 5 = 50

Write the equation in the rectangle as students record the partial product. Point to the 10 by 10 rectangle and repeat the process.

H This time, we distributed the 10 to each part of 15. First, we found 10 × 5. Then, we found 10 × 10.

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using an area model Objective 2 | M and recording in vertical form. 10 M I NU T ES

H So, what is 14 × 15? 210

4 ones × 5 ones

4 ones × 1 ten 1 ten × 5 ones

Questions to Advance Student Thinking: • How can you draw and label an area model to show how to break apart both factors? • How can you use the area model to help you find the partial products? • How can you record the partial products in vertical form? • How can you use the partial products to find the product? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Repeat the process: Use the + 1 0 0 1 ten × 1 ten following problems during 1 2 1 0 Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper to support students in using the worked-out example. • 26 × 31 • 17 × 43 • 58 × 32

Monitor: • Does the student use place value to break apart both factors in the area model? • Does the student multiply a unit of ten by a unit of ten to get a unit of hundreds? • Can the student record the partial products in vertical form? • Does the student add the partial products to find the product?

EV I

Invite students to turn and talk about what they notice about using the area model and vertical form to record partial products.

Analyze Student Progress

Guide students to record the bottom row of partial products in vertical form on the Student Page. Continue to say the unit form. Then have students add the four partial products to find the final product.

Concept Mini Lessons | Teacher Guide

EV I

Objective 2 | Array Template

10 × 15

14 × 15

Concept Mini Lessons | Teacher Guide

two-digit numbers by two-digit numbers by using vertical form Objective 3 | Multiply with four partial products. 10 M I NU T ES

Materials • Personal whiteboard • Objective 3 Student Page

Summary Students use unit form to identify place value units as they record four partial products in vertical form. Write 31 × 27.

Language Support

H Let’s record the partial products in vertical form without drawing an area model. We can still picture it in our heads without drawing it. Language Support

First, we multiplied

ones by

Next, we multiplied

Then, we multiplied

Finally, we multiplied

EV I

The word picture has multiple meanings. In this context, picture is used to prompt students to create a mental image in their minds. In other contexts, it is used to refer to a concrete image of something. Consider highlighting the different meanings of picture.

Consider providing scaffolded sentence frames to support students as they explain their calculations.

Direct students to write 31 × 27 in vertical form and to picture an area model that represents 31 × 27. Invite students to turn and talk about how the factors would be broken apart in the area model.

H Let’s use unit form to say the expressions that represent the partial products to make sure we record the place value units correctly.

ones.

We added the partial products and got

Point to the numbers as you refer to them. H Picture the top row of the area model. We need to multiply each part of 31 by 7 ones. First, we find 7 ones × 1 one. What is 7 ones × 1 one? 7 ones

Record the partial product 7 and direct students to do the same. H I can ask myself, What do I multiply by 7 ones next? 3 tens 7 ones × 3 tens is 21 tens.

H How do we write 21 tens in standard form? 210 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using vertical form Objective 3 | M with four partial products. 10 M I NU T ES

Record the partial product 210 and direct students to do the same.

H Next, we find 2 tens × 1 ones. What is 2 tens × 1 ones? 2 tens

H Now picture the bottom row of the area model. What do we have left to multiply? We need to multiply each part of 31 by 2 tens.

Invite students to turn and talk about how they used place value as they multiplied. H How do we know that we found and recorded all the partial products? We multiplied each part of 31 by each part of 27. There are four partial products.

We started with 7 ones and multiplied each part of 31 by 7 ones.

Record the partial product 20 and direct students to do the same.

H When we multiply two multiples of ten, what unit do we use for the product? Hundreds

H What is 2 tens × 3 tens? 6 hundreds

Record the partial product 600 and direct students to do the same. H How do we find the product?

Add together all the partial products.

H What is 31 × 27? 837

31 × 27 7 210 20 +600 837

matched what we would find if we represented 31 × 27 with an

We pictured the area model and made sure our partial products

EV I

H What expression represents the final partial product? 2 tens × 3 tens

Then we multiplied each part of 31 by 2 tens.

area model.

Invite students to turn and talk about how they can record partial products in vertical form. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work. • 19 × 43 • 83 × 62 • 54 × 32

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using vertical form Objective 3 | M with four partial products. 10 M I NU T ES

Notes Analyze Student Progress

EV I

Monitor: • How does the student break apart one of the factors to multiply? • Does the student multiply a unit of ten by a unit of ten to get a unit of hundreds? • Does the student find and record all the partial products in vertical form? • Does the student add the partial products to find the product?

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

two-digit numbers by two-digit numbers by using the Objective 4 | Multiply standard algorithm. 10 M I NU T ES

Materials • Personal whiteboard • Objective 4 Student Page

Direct students to the image of the two area models on the Student Page.

3 × 40 = 120

3×1=3

20 × 40 = 800

41 3

20 × 1 = 20

The area model with four partial products broke apart both factors, and the area model with two partial products only broke apart one

3 × 41 = 123

20 × 41 = 820

Gesture to each area model and say how many partial products are represented by the area models. Invite students to think–pair–share to compare the area model with four partial products and the area model with two partial products. Both area models represent 23 × 41.

3 × 41 is represented in the top row of each area model. The 123

the partial products 120 and 3 in the area model with four partial in the area model with two partial products is represented by

products.

by the partial products 800 and 20 in the area model with four

partial products.

of the factors.

H Both area models represent the same multiplication problem but break apart the factors differently. The second area model has fewer partial products to add.

EV I

Summary Students use the standard algorithm to multiply.

20 × 41 is represented in the bottom row of each area model. The 820 in the area model with two partial products is represented

Write 26 × 54.

H Let’s record the partial products in vertical form with two partial products. We can still picture an area model in our heads without drawing it.

Direct students to write 26 × 54 in vertical form and to picture an area model with two partial products that represents 26 × 54. Invite students to turn and talk about how one of the factors would be broken apart in the area model with two partial products. H Let’s use unit form to say the expressions that represent the partial products to make sure we record the place value units correctly.

Point to the numbers as you refer to them.

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using the Objective 4 | M standard algorithm. 10 M I NU T ES

Language Support

H We need to multiply 54 by each part of 26. Picture the top row of the area model. I can ask myself, Which part of 26 do I multiply by first? What do you think? 54 6 ones

26 2 4

H Picture the bottom row of the area model. I can ask myself, Which part of 26 do I multiply by next? What do you think? 2 tens H To multiply 54 by 2 tens, first we find 2 tens × 4 ones. What is 2 tens × 4 ones in unit form? 8 tens

H Can I rename 24 ones using a larger unit? How? Yes, you can rename 24 ones as 2 tens 4 ones.

H Are there only going to be 2 tens after we multiply 54 by 6 ones, or might there be more? There could be more.

Cross out the 2 on the line. Below the line in the hundreds place, write 3 to represent 3 hundreds. Below the line in the tens place, write 2 to represent 2 tens. Direct students to do the same.

EV I

H To multiply 54 by 6 ones, first, we find 6 ones × 4 ones. What is 6 ones × 4 ones? 24 ones

H Say 32 tens as hundreds and tens. 3 hundreds 2 tens

Point and use gestures to support students in tracking the factors they multiply in each step and in recognizing where they record the digits in the partial products.

H Let’s record those 2 tens and leave room to add more later. Watch how I record 24 ones, or 2 tens 4 ones.

Write a small 2 to represent 2 tens on the line under the tens place. Write 4 below the line in the ones place. Direct students to do the same. H What do we multiply by 6 ones next? 5 tens; 6 ones × 5 tens is 30 tens. H How many tens do we have in all? 30 tens plus 2 tens is 32 tens.

54 26 2 324

54 26 2 324 80

H Watch how I record 8 tens, or 80.

Write 8 below the first partial product in the tens place, and write 0 in the ones place. Direct students to do the same. H I ask myself, What do I still need to multiply? What do you think? 2 tens × 5 tens, which is 10 hundreds H How else can we say 10 hundreds? 1 thousand

Write 1 below the first partial product in the thousands place, and write 0 in the hundreds place. Direct students to do the same.

Concept Mini Lessons | Teacher Guide

ultiply two-digit numbers by two-digit numbers by using the Objective 4 | M standard algorithm. 10 M I NU T ES

Label each partial product to show that 324 is 6 × 54 and 1,080 is 20 × 54.

54 × 26 2 324 1080

6 × 54 20 × 54

• 52 × 63 • 81 × 14 • 37 × 39

H How do we know we found all the partial products? We multiplied 54 by each part of 26.

Teacher Tip: Differentiation Consider providing scaffolding for the steps in the standard algorithm.

H What is important to think about as we record two partial products on the same line in vertical form?

× +

63 52

2× 50 ×

We need to think about the units we are multiplying by and record the partial product with the correct units.

H When we combine partial products and record them on the same line, it’s called using the standard algorithm.

Model adding the partial products to find the product as students follow along.

H What is 26 × 54? 1,404

Analyze Student Progress

EV I

We may need to rename units and add new units after we multiply.

54 × 26 2 324 +1 0 8 0 1 1,4 0 4

Invite students to turn and talk about how they can find and record products by using the standard algorithm. Repeat the process: Use the following problems during 0 or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work.

Monitor: • How does the student break apart one of the factors to multiply? • Does the student multiply a unit of ten by a unit of ten to get a unit of hundreds? • Does the student record two partial products by using the standard algorithm? • Does the student add the partial products to find the final product? Questions to Advance Student Thinking: • Can you picture an area model to help you think about how to break apart the factors? • How can you record two partial products by using the standard algorithm? • How can you use the partial products to find the product? Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

of Two-Digit Numbers by Two-Digit Concept Mini Lessons | Multiplication Numbers

Objective 1

Objective 2

Objective 3

Objective 4

1. Area model completed

1. 210

Answer Key 1. 837

1. 1,404

3. 5,146 4. 1,728

3. 1,134 4. 1,443

2,800, sum of 3,160

2. Area model completed Partial products: 140, accurately.

200, sum of 340

3. Area model completed Partial products: 120, accurately.

3. 731

2. 817

4. 1,856

2. 3,276

900, sum of 1,020

2. 806

EV I

Partial products: 360, accurately.

4. Area model drawn and Partial products: 360,

completed accurately.

3,200, sum of 3,560

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Two-Digit Numbers by Two-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Two-Digit Numbers by Two-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 1

40 × 79

40 ×

EV I

20 × 17

40 ×

Complete the area model. Then multiply by recording the partial products in vertical form.

× =

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 1 | Multiply two-digit numbers by two-digit multiples of 10 by using an area model. 30 × 34

EV I

40 × 89

Draw an area model to represent the expression. Then multiply by recording the partial products in vertical form.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using an area model Objective 2 | Multiply and recording in vertical form. Draw an area model to represent the expression. Then multiply by recording the partial products in vertical form.

14 × 15

26 × 31

EV I

4 4 ones × 5 ones 4 ones × 1 ten 1 ten × 5 ones

1 ten × 1 ten

6 6 ones × 1 one 6 ones × 3 tens 2 tens × 1 one

2 tens × 3 tens

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

ultiply two-digit numbers by two-digit numbers by using an area model Objective 2 | M and recording in vertical form.

58 × 32

17 × 43

EV I

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using vertical form Objective 3 | Multiply with four partial products.

83 × 62

19 × 43

EV I

31 × 27

54 × 32

Multiply in vertical form using four partial products.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using the Objective 4 | Multiply standard algorithm. 1

3 × 40 = 120

3×1=3

3 × 41 = 123

20 × 40 = 800

20 × 1 = 20

20 × 41 = 820

EV I

Multiply by using the standard algorithm.

26 × 54

52 × 63

81 × 14

37 × 39

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

Practice | Multiplication of Two-Digit Numbers by Two-Digit Numbers Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Multiply two-digit

Objective 2 Multiply two-digit

Objective 3 Multiply two-digit

Objective 4 Multiply two-digit

numbers by two-digit numbers by using an area model and recording in vertical form.

numbers by two-digit numbers by using vertical form with four partial products.

numbers by two-digit numbers by using the standard algorithm.

Look for...

• Can the student use place value to break apart the factors in an area model? • Can the student use an area model to find the partial products? • Can the student record the partial products in vertical form? • Can the student add the partial products to find the product?

• Can the student mentally break apart the factors? • Can the student find and record all the partial products in vertical form? • Can the student add the partial products to find the product?

Look for...

• Can the student use place value to break apart one of the factors? • Can the student multiply a unit of ten by a unit of ten to get a unit of hundreds? • Can the student add the partial products to find the product?

EV I

numbers by two-digit multiples of 10 by using an area model.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of Two-Digit Numbers by Two-Digit Numbers Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for... • Can the student mentally break apart one of the factors? • Can the student record two partial products by using the standard algorithm?

• Can the student add the partial products to find the product?

Practice | Teacher Guide

Practice | Multiplication of Two-Digit Numbers by Two-Digit Numbers

Answer Key Practice Page 2

Practice Page 3

Practice Page 4

1. Area model completed

1. 169

1. 378

1. 399

3. 2,806

3. 1,333

sum of 360

2. Area model completed products: 360, 800, accurately; partial sum of 1,160

3. Area model completed products: 350, 1,500, sum of 1,850

3. 1,517 4. 2,170

2. 1,738

4. D

2. 2,624

4. When finding the Riley multiplied 64 by

second partial product,

2 ones. She should have multiplied 64 by 2 tens.

accurately; partial

2. 550

EV I

products: 160, 200, accurately; partial

Practice Page 1

4. Area model drawn and partial products: 630,

completed accurately;

1,400, sum of 2,030

Practice | Teacher Guide

EV I

Student Edition | Printable pages for students

Practice | Teacher Guide

NAME

DATE

two-digit numbers by two-digit multiples of 10 by using an Practice Page 1 | Multiply area model. Complete the area model. Then multiply by recording the partial products in vertical form.

20 ×

40 × 29

20 × 18

20 ×

EV I

× =

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 1 |

50 × 37

EV I

Draw an area model to represent the expression. Then multiply by recording the partial products in vertical form.

70 × 29

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using an area Practice Page 2 | Multiply model and recording in vertical form. Draw an area model to represent the expression. Then multiply by recording the partial products in vertical form.

13 × 13

25 × 22

EV I

3 3 ones × 3 ones 3 ones × 1 ten 1 ten × 3 ones

1 ten × 1 ten

5 5 ones × 2 ones 5 ones × 2 tens 2 tens × 2 ones

2 tens × 2 tens

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

ultiply two-digit numbers by two-digit numbers by using an area Practice Page 2 | M model and recording in vertical form. 37 × 41

62 × 35

EV I

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using vertical Practice Page 3 | Multiply form with four partial products. Multiply in vertical form by using four partial products.

34 × 52

27 × 14

61 × 46

EV I

98 × 73

A 1,484

B 6,640

Multiply in vertical form by using four partial products. Circle the letter of the correct answer.

C 6,911 D 7,154

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

two-digit numbers by two-digit numbers by using the Practice Page 4 | Multiply standard algorithm. Multiply by using the standard algorithm.

19 × 21 =

Riley multiplied 23 and 64 by using the standard algorithm. Riley did

82 × 32 =

43 × 31 =

94 × 12 =

EV I

not find the correct answer. What mistakes did Riley make? Riley’s Work

64 × 23 1 182 +128 1 1 310

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 1 40 × 76 =

Look at the problem. Then look at the work. It shows how to

multiply a two-digit number by a multiple of ten by using an area model.

How can you break apart one of the

How does the unit change when tens

How can you use the partial products to

factors to help you multiply?

are multiplied by tens?

find the product?

EV I

apart the factor that isn’t a multiple of 10.

2 8

4 0

0 0

3, 0

I add the two partial products.

familiar multiplication fact of 4 × 7 and then

think of the product in hundreds.

2 8

4 0

0 0

3, 0

× 40

+ 2

To multiply 4 tens and 7 tens, I can use the

3,040

40 × 6 = 240

I multiply 4 tens by 6 ones to get 24 tens.

I can draw an area model. Then I break

40 × 76 =

40 × 70 = 2,800

40 × 6 = 240

+ 2 1

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 2 23 × 36 =

Look at the problem. Then look at the work. It shows how to

multiply two-digit numbers by two-digit numbers by using an area model and recording in vertical form. How can you draw and label an area

How can you use the area model to help

How can you record the partial products

model to show how to break apart both

you find the partial products?

in vertical form? How can you find the

factors?

23 × 36 =

I break apart 23 into 20 and 3. I break apart

36 into 30 and 6.

3 × 30 = 90

3 × 6 = 18

EV I

828

20 × 30 = 600

product? I am careful to line up the place value units when

3 × 30 = 90

3 × 6 = 18

20 × 30 = 600

20 × 6 = 120

9 2

0 0

+ 6

in vertical form. I add the four partial products.

2 tens × 3 tens = 6 hundreds

I record partial products

20 × 6 = 120

3 ones × 6 ones

9 2

0 0

3 ones × 3 tens 2 tens × 6 ones

+ 6

2 tens × 3 tens

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Practice | Student Page

NAME

DATE

Practice Helper 3 Look at the problem. Then look at the work. It shows how vertical form with four partial products.

36 × 21 =

to multiply two-digit numbers by two-digit numbers, using

Can you picture an area model to help

How do you know that you found and

How can you use the partial products to

you think about how to break apart the

recorded all the partial products in

find the product?

factors?

vertical form?

36 is 3 tens 6 ones

6 120 30 600

6 ones × 1 one 6 ones × 2 tens 3 tens × 1 one 3 tens × 2 tens

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21 is 2 tens 1 one

I multiply each part of 21 by each part of 36.

I break apart the factors into tens and ones.

6 120 30 +600 756 I add the four partial products.

I record and label four partial products in vertical form.

756

36 × 21 =

21 × 36 6 120 30 +600 756

6 ones × 1 one 6 ones × 2 tens 3 tens × 1 one 3 tens × 2 tens

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to standard algorithm.

24 × 46 =

multiply two-digit numbers by two-digit numbers by using the

Can you picture an area model to help

How can you record two partial products

How can you use the partial products to

you think about how to break apart the

by using the standard algorithm?

find the product?

factors?

4 × 46 20 × 46

EV I

24 is 2 tens 4 ones

184 920

I break apart the factor into tens and ones.

I multiply 46 by each part of 24.

I record the partial product for 4 × 46 on

184 1 + 920 1 1,1 0 4 I add the two partial products.

one line.

Then I record the partial product for 20 × 46

on another line.

1,104

24 × 46 =

46 24 2 184 1 + 920 1 1,1 0 4 ×

4 × 46 20 × 46

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Practice | Student Page

Application | Multiplication of Two-Digit Numbers by Two-Digit Numbers Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplying two-digit numbers by two-digit numbers. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

Study a Solution

Independent Work Partner Work

• Consider providing highlighters and other tools for students to use to annotate the sample solution.

Solve a Task

Partner Work

• Consider providing a blank calendar grid for students to annotate.

Activity

• Consider using a standard deck of playing cards if you do not have Eureka Math2 cards. • Consider providing grid paper to support students with drawing an area model or using the vertical form to solve multiplication problems.

Application | Teacher Guide

Application | Multiplication of Two-Digit Numbers by Two-Digit Numbers

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

Teacher Tip

• Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card • Grid paper (optional) Students play a game involving multiplying two-digit numbers by two-digit numbers.

Preparing to Play • Remove the 10, Q, K, and Jokers from the deck. J can represent 0. Aces can represent 1. • Shuffle the cards. Divide the cards equally among the players. Each player keeps their cards in a single facedown pile. • Each player picks a card. The player with the lower number is Player A. For each round, Player A will get a point if the product of the two-digit numbers is less than 3,000. Player B will get a point if the product is greater than 3,000. • Consider providing tools such as grid paper to support students in drawing an area model or using vertical form.

Students use the Read–Draw –Write process to solve word problems involving multiplying two-digit numbers by two-digit numbers. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problem 1 involves multiplying a two-digit number by a two-digit multiple of ten. Problems 1 and 2 include contexts that lend themselves to drawing an area model to determine the product.

Materials

Before beginning small group rotations, consider spending a few minutes explaining and modeling the multiplication game with the whole class. This piques student interest and engagement and sets up all students for success when they play without teacher support.

Playing the Game • Each player takes two cards off the top of their pile, uses them to make a two-digit number, and records the number. • Each player records their partner’s number and multiplies the two-digit numbers to find the product. Players check each other’s work. • Player A gets a point if the product is less than 3,000. Player B gets a point if the product is greater than 3,000. • The player with more points after five rounds wins.

Materials

Play a Game: Products Greater or Less Than 3,000

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Solve a Problem

Study a Solution Materials

• Study a Solution Student Page Students work independently or with a partner to analyze a correct solution to a word problem involving multiplying two-digit numbers by two-digit numbers. Students answer questions about how the known and unknown information in the problem is represented in the sample solution. They also analyze how the sample drawing provides a

solution path. Finally, they are asked to consider whether the sample statement answers the question in the word problem.

Solve a Task Materials

• Solve a Task Student Page • Blank calendar grid (optional) Students work with a partner to solve a multi-part task multiplying two-digit numbers by two-digit numbers. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution. Teacher Tip Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor students’ progress and provide support as needed.

Application | Teacher Guide

Application | Multiplication of Two-Digit Numbers by Two-Digit Numbers

Answer Key Study a Solution

1. Accurate picture drawn to represent

1. The known information is

product of 30 and 28; There are

Solve a Problem

with 24 representing how many in

the problem; equation shows the

represented by a tape diagram

840 lemon trees on the farm.

each group and 68 representing the

2. Accurate picture drawn to represent product of 18 and 65; There are

2. The unknown information is

EV I

the problem; equation shows the

number of groups.

1,170 boxes in the warehouse.

represented by the total of the tape diagram and labeled with a w.

3. The diagram shows that the factors

the problem; equation shows the

product of 55 and 29; Jayla plays

are known and the product is

basketball for 1,595 minutes.

unknown. When the factors are

3. Accurate picture drawn to represent

Solve a Task 1. Oka spends 620 minutes walking her dog in May.

2. Oka practices soccer for a total of 975 minutes in May.

3. Oka spends 1,210 minutes walking her dog and using technology on school days in May.

known, we can multiply them to get the product.

4. Yes. The question asks how many windows there are in the building,

1,632 windows.

and the statement says there are

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

Smith’s Farm has 30 rows of lemon trees. There are 28 lemon trees in each row.

How many lemon trees are on the farm?

18 rows with 65 boxes in each row. How

The boxes in a warehouse are packed in

EV I

many boxes are in the warehouse?

In March, Jayla plays basketball for 55 minutes

every day for 29 days. What is the total number

of minutes that Jayla plays basketball in March?

Application | Teacher Guide

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Student Edition | Printable pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

This page may be reproduced for classroom use only.

Application | Student Page

Application | Play a Game Game Instruction Card

Products Greater or Less Than 3,000

I made the number 16. I want the product to be less than 3,000.

Player B

I made the number 87. I want the product to be greater than 3,000.

What You Need

Player A

with the 10, Q, K, and Jokers removed. J can represent 0.

• Eureka Math2 cards (or a standard deck of playing cards) Aces can represent 1.

• Play a Game Recording Page

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• Grid paper (optional)

How to Play

1. Mix up the cards. Deal the same number of cards to each player. Put the cards in a stack facedown. Before playing,

each player picks a card. The player with the lower number is Player A.

3. Record the other player’s number and the multiplication

2. Both players turn over a card at the same time. Make a

expression.

two-digit number and record it.

16 × 87

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Application | Student Page

Application | Play a Game Game Instruction Card

16 × 87 = 1,392

Player A

Okay, let’s play another round!

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1,392 is less than 3,000. I get a point!

Player B

4. Find the product and record it. Player A gets a point if the product is less than 3,000. Player B gets a point if the product is greater than 3,000.

How to Win

The player with more points after five rounds wins.

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Application | Student Page

NAME

DATE

Application | Play a Game Products Greater or Less Than 3,000 Round

Player A’s Number

Multiplication Expression

Product

Points

Player B’s Number

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Recording Page

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | study a solution Shen solved the problem below. Read the problem and look at Shen’s work. Then answer the questions.

How many windows are in the building? Shen's Work

68 × 24 = w w = 1,632

w 24

A tall apartment building has 68 floors. There are 24 windows on each floor of the building.

...

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68 floors

160

1,200

240

2 2

4 0

0 0

1, 6

+ 1

There are 1,632 windows in the building.

How is the known information in the problem represented?

How is the unknown information in the problem represented?

How does the drawing help you see a solution path for finding the unknown?

Does the statement answer the question? How do you know?

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Application | Student Page

NAME

DATE

Application | solve a Task • Oka walks her dog for 20 minutes every day.

• Oka spends 35 minutes using technology every

• Oka practices soccer for 75 minutes every Monday, Monday through Friday.

Wednesday, and Friday, except on holidays.

Oka's Busy Month

Sunday

• Oka has school every Monday through Friday, except

Tuesday

Wednesday Thursday

Friday

Saturday

Holiday

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on holidays.

Monday

May

How many minutes does Oka spend walking her dog in May?

What is the total number of minutes Oka practices soccer in May?

What is the total number of minutes Oka spends walking her dog and using technology on school days in May?

This page may be reproduced for classroom use only.

Application | student Page

Multiplication

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Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

Concept Guide | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and the Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard

• Personal whiteboard or Student Pages

• Print copies of Student Pages as needed. • Place copies of Student Pages into whiteboards for Objectives 1, 2, and 3.

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Game Instruction Card • Dice (2) • Products Greater Than or Less Than 10,000 Game Board • Solve a Task Student Page • Grid paper (optional)

• Ready the following materials: • Application Word Problem Cards • Game Instruction Card • Dice (2) • Grid paper (optional) • Print copies of the following: • Solve a Problem Recording Page (optional) • Products Greater Than or Less Than 10,000 Game Board • Solve a Task Student Page

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• Application Teacher Guide

Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

Addressing Student Misconceptions How to Address Misconception

Students do not record the partial products with the correct units.

Invite students to use unit form to say the expressions that represent the partial products. For example, when students are multiplying 345 and 24, use the following questions to help them make connections between unit form and the standard algorithm.

Student Misconception

• What is 5 ones × 4 ones? • Can we rename 20 ones by using a larger unit? • Are there only going to be 2 tens after we multiply 345 by 4 ones, or will there be more?

Language Support

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Demonstrate how to record the 2 below the line in the tens place to leave room to add more tens later. Then record the 0 in the ones place. Have students discuss how they see the partial product of 20 represented. Continue using unit form to say the expressions that represent the partial products.

Concept Guide | Teacher Guide

Family Math | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Dear Family,

6 ×

Can you picture an area model to help you think about how to break apart the factors?

1, 6

about how the factors would be

How can you record the partial products when you need

How can you use the partial

to rename as a larger unit?

products to find the product?

head without drawing it. I think

I can picture the area model in my

34 × 637 = 21,658

EV I

2 1

+ 1

Your student is working on multiplying multi-digit numbers by multi-digit numbers. Your student uses area models, the distributive property, vertical form, and partial products to multiply multi-digit numbers. All of this leads to multiplying multi-digit numbers by using the standard algorithm with and without regrouping. You can support your student’s progress by asking the questions in the table below as your student multiplies multi-digit numbers by multi-digit numbers.

I can add the partial products.

broken apart in the area model.

When I need to rename as a larger unit, I can use more than one place to record the product. For example, the product of 4 ones and 7 ones is 28 ones or 2 tens 8 ones. I can record a small 2 in the tens place and an 8 in the ones place. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Progress Check | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before, during, or after providing direct instruction. It is intended to collect data about students’ proficiency with multiplication of multi-digit numbers by multi-digit numbers and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problems 1 and 2 involve multiplying three-digit numbers by two-digit numbers by using the distributive property, problems 3 and 4 involve multiplying three-digit numbers by two-digit numbers by using vertical form and six partial products, problems 5 and 6 involve multiplying three-digit numbers by two-digit numbers by using the standard algorithm, and problems 7 and 8 involve multiplying three-digit numbers by two-digit numbers with multiple regroupings by using the standard algorithm. Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions:

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• Can the student multiply a three-digit number by a two-digit number by using the distributive property? | Objective 1

• Can the student multiply a three-digit number by a two-digit number by using vertical form and six partial products? | Objective 2

| Objective 3

• Can the student multiply a three-digit number by a two-digit number by using the standard algorithm when no regrouping is required in the partial products?

• Can the student multiply a three-digit number by a two-digit number by using the standard algorithm when regroupings are required in the partial products? | Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

Progression Toward Proficiency Rubric Items 1 and 2

Items 3 and 4

Item 5

Items 6 and 7

Objective 1

Objective 2

Objective 3

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand how to multiply three-digit numbers by two-digit numbers by using the distributive property but makes more than one calculation error that leads to an incorrect answer.

The student may show evidence of beginning to understand how to multiply three-digit numbers by twodigit numbers by using vertical form and six partial products but makes more than one calculation error that leads to an incorrect answer.

The student may show evidence of beginning to understand how to multiply three-digit numbers by twodigit numbers with no regrouping by using the standard algorithm but makes more than one calculation error that leads to an incorrect answer.

The student may show evidence of beginning to understand how to multiply three-digit numbers by two-digit numbers with multiple regroupings by using the standard algorithm but makes more than one calculation error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes one calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly uses the distributive property to multiply.

The student correctly uses vertical form and six partial products to multiply.

The student correctly uses the standard algorithm to multiply.

The student correctly regroups while using the standard algorithm to multiply.

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Progress Check Tool Item(s)

1. Area model completed; 25,848 2. Area model completed; 21,126

3. Partial products: 56, 160, 4,800, 70, 200, 6,000; Product: 11,286 4. Partial products: 24, 0, 2,100, 400, 0, 35,000; Product: 37,524

5. Correct standard algorithm;

6,908

6. Correct standard algorithm;

19,608

7. D

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Distribute to multiply.

36 × 718 = _______________

42 × 503 = _______________

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Multiply by recording the partial products in vertical form.

18 × 627 = _______________

53 × 708 = _______________

This page may be reproduced for classroom use only.

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Multiply by using the standard algorithm.

22 × 314 = _______________

What is the product of 65 and 807? Show your work. Circle the letter of the correct answer.

A B C D

41,455 52,055 52,077 52,455

43 × 456 = _______________

EV I

This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Progression of Mini Lesson Objectives 2 Multiply three-digit numbers

3 Multiply three-digit numbers

4 Multiply three-digit numbers

by two-digit numbers by using the distributive property.

by two-digit numbers by using vertical form and six partial products.

by two-digit numbers by using the standard algorithm.

by two-digit numbers with multiple regroupings by using the standard algorithm.

300

2 × 300 = 600

2 × 20 = 40

2×4=8

10 × 300 = 3,000

10 × 20 = 200

10 × 4 = 40

2 ones × 5 ones

2 ones × 1 ten

2 ones × 4 hundreds

5 tens × 5 ones

5 0

0 0

5 tens × 1 ten

0 1

1, 5

4, 1

3 1

5 tens × 4 hundreds

EV I

+ 2

Start here if students

• can multiply a two-digit number by a two-digit number by using four partial products and • can multiply a three-digit number by a two-digit number by using the distributive property, but • need support multiplying a three-digit number by a two-digit number by using vertical form and six partial products.

• can decompose a multi-digit number into place value units and • can multiply two-digit numbers by two-digit numbers by using an area model, but • need support multiplying a three-digit number by a two-digit number by using the distributive property.

Start here if students

1 Multiply three-digit numbers

Start here if students • can multiply a three-digit number by a two-digit number by using partial products and • can multiply a two-digit number by a two-digit number by using the standard algorithm, but • need support multiplying a three-digit number by a twodigit number by using the standard algorithm when no regrouping is required in the partial products.

5 ×

1 2, 5

+ 1

Start here if students • can multiply a two-digit number by a two-digit number by using the standard algorithm and • can multiply a three-digit number by a two-digit number by using the standard algorithm when no regrouping is required, but • need support multiplying a three-digit number by a two-digit number by using the standard algorithm when regroupings are required in the partial products.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 1 | distributive property. 10 M I NU T ES

Materials • Objective 1 Student Page

Distribute the Objective 1 Student Page. Direct students to problem 1: 12 × 324.

Gesture to the 2 by 4 rectangle in the area model as you ask the following question.

Summary Students multiply three-digit numbers by two-digit numbers by using an area model and relating it to the distributive property.

H I can ask myself, How can I break apart the factors to help me multiply? What do you think? 3 hundreds + 2 tens + 4 ones 1 ten + 2 ones

10 + 2

H Now we can use an area model to multiply.

Teacher Tip

300

Invite students to label the side lengths of the area model.

Write the equation in the rectangle as students record the partial product in their area model. Repeat the process with the 2 by 20 rectangle and the 2 by 300 rectangle.

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300 + 20 + 4

H How can I find the area of the 2 by 4 rectangle? You can multiply: 2 × 4 = 8.

A factor in a multiplication expression can be represented by either side length of an area model. To support students as they multiply multi-digit numbers and record partial products in vertical form, encourage them to consistently represent the first factor by using the vertical side length and the second factor by using the horizontal side length.

H We distributed 2 to the 4 ones, the 2 tens, and the 3 hundreds of the factor 324. First, we found 2 × 4. Next, we found 2 × 20. Then, we found 2 × 300. What do we still need to multiply? We need to multiply 10 by 300, 20, and 4.

Gesture to the 10 by 4 rectangle in the area model as you ask the following question. H I can ask myself, How can I find the area of the 10 by 4 rectangle? What do you think? 10 × 4 = 40

Write the equation in the rectangle as students record the partial product in their area model. Gesture to the remaining rectangles and

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 1 | distributive property. 10 M I NU T ES

300

2 × 300 = 600

2 × 20 = 40

2×4=8

10 × 300 = 3,000

10 × 20 = 200

10 × 4 = 40

H We used an area model to multiply. We can also use equations and the distributive property to organize the multiplication.

repeat the process. After each row is complete, discuss how 10 was distributed to 300, 20, and 4.

Write the following:

12 × 324 = (10 + 2) × (300 + 20 + 4)

Teacher Tip

thousands

hundreds

× 10

= (10 × 300) + (10 × 20) + (10 × 4) + (2 × 300) + (2 × 20) + (2 × 4)

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Consider drawing on a place value chart to support student understanding of 1 hundred × 1 ten = 1 thousand. It is important for students to be able to apply this understanding to multiply multiples of 10 by multiples of 100 (e.g., 10 × 300). tens

ones

1 hundred × 1 ten = 1 thousand 100 × 10 = 1,000

H The area model shows the partial products. How do we use the partial products to find 12 × 324? We add all the partial products.

Invite students to add the partial products to find the product. H What is the sum of the partial products? 3,888 H What is the product of 12 and 324? 3,888

= 3,000 + 200 + 40 + 600 + 40 + 8

= 3,888

Gesture to the multiplication expression 10 × 300.

H Where do you see 10 × 300 in the area model? It’s the 10 by 300 rectangle.

Repeat the process with the remaining multiplication expressions to help students make connections between the equations that show the distributive property and the area model. Invite students to turn and talk about how they can use the distributive property to multiply multi-digit numbers. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as necessary. Consider providing the Objective 1 Practice Helper and supporting students in using the worked-out example to guide their own work.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 1 | distributive property. 10 M I NU T ES

• 23 × 516 = ________ • 38 × 912 = ________ • 52 × 604 = ________

Notes

Analyze Student Progress

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Monitor: • Can the student use place value to break apart both factors in the area model? • Can the student correctly find the partial products? • Does the student add the partial products to find the product? • Can the student explain how the distributive property is used to find the product?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using vertical form and Objective 2 | six partial products. 10 M I NU T ES

Materials • Objective 2 Student Page

Summary Students use unit form to identify place value units as they record six partial products in vertical form.

Distribute the Objective 2 Student Page. Direct students to problem 1: 52 × 415.

Language Support

10 ones

Write the expression 2 ones × 5 ones near where the partial product will be recorded. Then record the partial product 10 as students do the same. Repeat the process with 2 ones × 1 ten and 2 ones × 4 hundreds.

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H Let’s record the partial products without drawing an area model. Try to picture the area model in your head without drawing it. In your head, how would the factors be broken apart in the area model? We would break apart 52 as 50 and 2. We would break apart 415 as 400, 10, and 5.

H Picture the top row of the area model. We need to distribute to multiply 5 ones, 1 ten, and 4 hundreds by 2 ones. First, we find 2 ones × 5 ones. What is 2 ones × 5 ones?

Write 52 × 415 in vertical form as students do the same.

H Let’s use unit form to say the expressions that represent the partial products to make sure we record the place value units correctly.

Point to the numbers in vertical form as you refer to them.

4 ×

2 ones × 5 ones

H We distributed 2 to the 5 ones, 1 ten, and 4 hundreds of the factor 415. First, we found 2 × 5. Next, we found 2 × 10. Then, we found 2 × 400. Picture the bottom row of the area model. What do have left to multiply? We need to multiply 5 ones, 1 ten, and 4 hundreds by 5 tens.

H Now we need to find 5 tens × 5 ones. What is 5 tens × 5 ones? 25 tens

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using vertical form and Objective 2 | six partial products. 10 M I NU T ES

Write the expression 5 tens × 5 ones near where the partial product will be recorded. Then record the partial product 250 as students do the same.

H When we multiply two multiples of ten, what unit do we use for the product? H What is 5 tens × 1 ten? 5 hundreds

Write the expression 5 tens × 1 ten near where the partial product will be recorded. Then record the partial product 500 as students do the same. H What is the final partial product? 5 tens × 4 hundreds

H When we multiply a multiple of ten by a multiple of one hundred, what unit do we use for the product? Thousands

Write the expression 5 tens × 4 hundreds near where the partial product will be recorded. Then record the partial product 20,000 as students do the same. H How do we find the product? We add all the partial products.

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H I can ask myself, What do I multiply by 5 tens next? What do you think? 1 ten

Hundreds

H What is 5 tens × 4 hundreds? 20 thousands

H How do we record 25 tens? 250

4 ×

H What is the sum of the partial products? 21,580 H

2 ones × 5 ones

2 ones × 1 ten

2 ones × 4 hundreds

5 tens × 5 ones

5 0

0 0

5 tens × 1 ten

1, 5

Invite students to add the partial products to find the product. + 2

0 1

5 tens × 4 hundreds

21,580

H How do we know that we found and recorded all the partial products? We multiplied each part of 415 by each part of 52. We started with 2 ones and multiplied each part of 415 by 2 ones. Then we multiplied each part of 415 by 5 tens.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using vertical form and Objective 2 | six partial products. 10 M I NU T ES

Language Support

Consider providing scaffolded sentence frames to support students as they explain their calculations. • First, we multiply ________ ones by ________ ones. • Next, we multiply ________ tens by ________ ones.

• Then, we multiply ________ by ________. (Repeat for remaining units.)

Questions to Advance Student Thinking: • Can you picture an area model to help you think about how to break apart the factors? • How do you know that you found and recorded all the partial products? • How can you use the partial products to find the product? Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work. • 45 × 281 = ________ • 16 × 836 = ________ • 34 × 201 = ________

Monitor: • Can the student use place value to break apart both factors? • Can the student find and record all the partial products in vertical form? • Does the student add the partial products to find the product?

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• Finally, we add the partial products and get ________.

Analyze Student Progress

Invite students to turn and talk about how they can record partial products in vertical form.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

Materials • Objective 3 Student Page

Summary Students use the standard algorithm to multiply multi-digit numbers that require no regroupings in the partial products.

The area model with six partial products broke apart both factors,

Distribute the Objective 3 Student Page. Direct students to the image of the two area models on the Student Page. 10

212

3 × 200 = 600

3 × 10 = 30

3×2=6

40 × 200 = 8,000

40 × 10 = 400 40 × 2 = 80

3 × 212 = 636

of the factors.

H Both area models represent the same multiplication problem, but they break apart the factors differently. The second area model has fewer partial products to add.

EV I

200

and the area model with two partial products only broke apart one

40 × 212 = 8,480

Gesture to each area model and say how many partial products are represented by the area models. Invite students to think–pair–share to compare the area model with six partial products to the area model with two partial products. Both area models represent 43 × 212.

3 × 212 is represented in the top row of each area model. The 636

in the area model with two partial products is represented by three separate partial products, 600, 30, and 6, in the area model with six partial products.

40 × 212 is represented in the bottom row of each area model. The 8,480 in the area model with two partial products is represented by three separate partial products—8,000, 400, and 80—in the area

Write 13 × 321.

H Let’s record the partial products in vertical form with two partial products. We can still picture an area model in our heads without drawing it.

Direct students to write 13 × 321 in vertical form and to picture an area model with two partial products that represent 13 × 321. Invite students to turn and talk about how one of the factors would be broken apart in the area model with two partial products. H Let’s use unit form to say the expressions that represent the partial products to make sure we record the place value units correctly.

Point to the numbers as you refer to them.

model with six partial products. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

Language Support

H We multiplied 3 ones by each digit of 321. Picture the bottom row of the area model. I can ask myself, Which part of 13 do I multiply by next? What do you think? We should multiply by the other part of 13, 1 ten.

Point and use gestures to support students in tracking the factors they multiply in each step and in recognizing where they record the digits in the partial products.

H We need to multiply 13 by each part of 321. Picture the top row of the area model. I can ask myself, Which part of 13 do I multiply by first? What do you think? 3 ones H To multiply 321 by 3 ones, we first find 3 ones × 1 one. What is 3 ones × 1 one? 3 ones

H We need to multiply 1 ten by 321. To do that we first multiply 1 ten by 1 one. What is 1 ten × 1 one in unit form? 1 ten

EV I

H Watch how I record 1 ten, or 10.

Write 3 below the line in the ones places as students do the same.

H What do we multiply by 3 ones next? 2 tens; 3 ones × 2 tens is 6 tens.

H We record the 6 tens to the left of the 3 ones.

Write 6 below the line in the tens places as students do the same. H What do we multiply by 3 ones next? 3 hundreds; 3 ones × 3 hundreds is 9 hundreds. H We record the 9 hundreds to the left of the 6 tens.

Write 1 below the first partial product in the tens place and write 0 in the ones place. Direct students to do the same.

× +

H I can ask myself, What do I still need to multiply? What do you think? We still need to multiply 1 ten × 2 tens, which is 2 hundreds.

Write 2 below the first partial product in the hundreds place, to the left of the 1 ten. Direct students to do the same. H Do I still need to multiply? What do you think? Yes. We need to find 1 ten × 3 hundreds, which is 3 thousands.

Write 3 below the first partial product in the thousands place, to the left of the 2 tens. Direct students to do the same.

Write 9 below the line in the hundreds places as students do the same. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

H Do I still need to multiply? What do you think? No. We have multiplied all the parts of 13 by 321. 3

3 ones × 321

1 ten × 321

× +

4, 1

× + 3 1

Invite students to turn and talk about how they can find and record partial products by using the standard algorithm. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 3 Practice Helper and supporting students in using the worked-out example to guide their own work.

EV I

H What is important to think about as we record three partial products on the same line in vertical form?

We need to think about the units we are multiplying by and record the partial product with the correct units.

H When we combine partial products and record them on the same line, it is called using the standard algorithm.

Model adding the partial products to find the product as students follow along.

H What is 13 × 321? 4,173

Students can record the factors in either order when recording in vertical form. Recording the number with more digits on top, as modeled in this Concept Mini Lesson, reduces the number of partial products.

Label each partial product to show that 963 is 3 × 321 and 3,210 is 10 × 321. Direct students to do the same.

Teacher Tip

4, 1

3 1

• 21 × 143 = ________ • 43 × 421 = ________ • 504 × 22 = ________ Teacher Tip

Problems 3 and 4 on the Student Page do not contain grids. Consider providing grid paper to support students in aligning the digits by place value units.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm. 10 M I NU T ES

Notes

Analyze Student Progress Monitor: • Can the student use place value to break apart both factors? • Can the student record two partial products by using the standard algorithm? • Can the student add the partial products to find the final product?

EV I

Questions to Advance Student Thinking: • Can you picture an area model to help you think about how to break apart the factors? • How can you record two partial products by using the standard algorithm? • How can you use the partial products to find the product?

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers with multiple regroupings Objective 4 | by using the standard algorithm. 10 M I NU T ES

Write 23 × 546 and direct students to do the same.

H What partial products could we use to find this product? 3 × 546 and 20 × 546 5

H To multiply 546 by 3 ones, we first find 6 ones × 3 ones. What is 6 ones × 3 ones? 18 ones H Can I rename 18 ones by using a larger unit? Yes. You can rename 18 ones as 1 ten 8 ones.

EV I

H Let’s use unit form to say the expressions that represent the partial products to make sure we record the place value units correctly. We can still picture an area model in our heads without drawing it, if needed.

Materials • Objective 4 Student Page (optional)

Summary Students use the standard algorithm to multiply multi-digit numbers that require regrouping in the partial products.

3 ones × 546

2 tens × 546

Direct students to write 23 × 546 in vertical form and to identify the partial products. Point to the numbers as you refer to them.

H We need to multiply 546 by 2 tens and 3 ones. Picture the top row of an area model. I can ask myself, Which part of 23 do I multiply by first? What do you think? 3 ones

H Is there only going to be 1 ten after we multiply 546 by 3 ones, or could there be more? There could be more.

H Let’s record that 1 ten and leave room to add more later. Watch how I record 18 ones, or 1 ten 8 ones.

Write a small 1 to represent 1 ten on the line under the tens place. Write 8 below the line in the ones place. Direct students to do the same.

5 ×

H What do we multiply by 3 ones next? 4 tens; 3 ones × 4 tens is 12 tens. H How many tens do we have in all? 12 tens plus 1 ten is 13 tens.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers with multiple regroupings Objective 4 | by using the standard algorithm. 10 M I NU T ES

H Is there only going to be 1 hundred after we multiply 546 by 3 ones, or could there be more? There could be more.

5 1

H To multiply 546 by 2 tens, first we find 2 tens × 6 ones. What is 2 tens × 6 ones in unit form? 12 tens H Say 12 tens as hundreds and tens. 1 hundred 2 tens

EV I

Cross out the 1 on the line. Write a small 1 to represent 1 hundred on the line under the hundreds place. Write 3 below the line in the tens place. Direct students to do the same.

H Picture the bottom row of the area model. I can ask myself, What do I multiply by next? What do you think? 2 tens

H Say 13 tens as hundreds and tens. 1 hundred 3 tens

H Will there be more tens after we multiply 546 by 2 tens?

H What do we multiply by 3 ones next? 5 hundreds. 3 ones × 5 hundreds is 15 hundreds.

H Is there going to be only 1 hundred after we multiply 546 by 2 tens, or could there be more?

H How many hundreds do we have in all? 15 hundreds plus 1 hundred is 16 hundreds.

H Say 16 hundreds as thousands and hundreds. 1 thousand 6 hundreds H Is there only going to be 1 thousand after we multiply 546 by 3 ones, or could there be more?

No. There will be no more tens.

There could be more.

H Watch how I record 1 hundred 2 tens.

1 6 3

There cannot be more, because we have multiplied each unit of 546 by 3 ones.

Cross out the 1 on the line. Write a 1 below the line in the thousands place and a 6 below the line in the hundreds place.

Write a small 1 to represent 1 hundred just under the 6 in the hundreds place. Write 2 below the 3 in the tens place. Write 0 below the 8 in the ones place. Direct students to do the same.

5 × 1 +

6 1

H I can ask myself, What do I still need to multiply? What do you think? We still need to multiply 2 tens by 4 tens, which is 8 hundreds.

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers with multiple regroupings Objective 4 | by using the standard algorithm. 10 M I NU T ES

H Watch how I record 9 hundreds.

5 × 1 +

6 1

Cross out the small 1. Below the 6 in the hundreds place, write 9.

H What is important to think about as we record the partial products in vertical form?

We need to think about the units we are multiplying by and record the partial product with the correct units. We may need to rename units and add new units after we multiply.

Model adding the partial products to find the product as students follow along.

H Is there more to multiply? What do you think? We still need to multiply 2 tens by 5 hundreds, which is 10

H What is 23 × 546? 12,558

EV I

thousands.

H How many hundreds do we have in all? 9 hundreds

H Say 10 thousands as ten thousands. 1 ten thousand H Will there be more ten thousands after we multiply 546 by 2 tens? No. There will be no more ten thousands.

+ 1

1 1

H Watch how I record 1 ten thousand.

Write a 1 in the ten thousands place and a 0 below the 1 in the thousands place. H How do we know we found all the partial products? We know because we multiplied 546 by each part of 23.

5 ×

1 2, 5

+ 1

1 1

Invite students to turn and talk about how they can find and record partial products by using the standard algorithm. Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work. • 38 × 529 = ________ • 63 × 804 = ________ • 624 × 83 = ________

Concept Mini Lessons | Teacher Guide

Multiply three-digit numbers by two-digit numbers with multiple regroupings Objective 4 | by using the standard algorithm. 10 M I NU T ES

Teacher Tip

The last problem, 624 × 83, is presented with the larger number first. Consider discussing how students can apply the commutative property of multiplication to think about the problem as 83 × 624.

Notes

Analyze Student Progress

EV I

Monitor: • Can the student use place value to break apart both factors? • Can the student record two partial products by using the standard algorithm, including regrouping units? • Can the student add the partial products to find the final product?

Questions to Advance Student Thinking: • Can you picture an area model to help you think about how to break apart the factors? • How can you record the partial products when you need to rename as a larger unit? • How can you use the partial products to find the product?

Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Answer Key Objective 1

Objective 2

1. Area model completed; 3,888

1. Partial products: 10, 20, 800, 250, 500, 20,000; Product: 21,580

3. Area model completed; 34,656

4. Area model completed; 31,408

2. Partial products: 5,400, 1,000, 40, 3,200, 8,000; Product: 12,645

1. Correct standard algorithm; 4,173

2. Correct standard algorithm; 3,003

EV I

2. Area model completed; 11,868

Objective 3

3. Partial products: 36, 180, 4,800, 60, 300, 8,000; Product: 13,376

3. Correct standard algorithm; 18,103

4. Correct standard algorithm; 11,088

Objective 4 1. Correct standard algorithm; 12,558 2. Correct standard algorithm; 20,102 3. Correct standard algorithm; 50,652 4. Correct standard algorithm; 51,792

4. Partial products: 4, 0, 800, 30, 0, 6,000; Product: 6,834

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Multiply three-digit numbers by two-digit numbers by using the Objective 1 | distributive property.

12 × 324 = _______________

38 × 912 = _______________

23 × 516 = _______________

EV I

Distribute to multiply.

52 × 604 = _______________

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Multiply three-digit numbers by two-digit numbers by using vertical form and Objective 2 | six partial products.

52 × 415 = _______________

EV I

45 × 281 = _______________

Multiply by recording the partial products in vertical form.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Multiply three-digit numbers by two-digit numbers by using vertical form and Objective 2 | six partial products.

16 × 836 = _______________

EV I

34 × 201 = _______________

Multiply by recording the partial products in vertical form.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm. 10

3 × 200 = 600

3 × 10 = 30

3×2=6

3 × 212 = 636

40 × 200 = 8,000

40 × 10 = 400 40 × 2 = 80

40 × 212 = 8,480

Multiply by using the standard algorithm.

13 × 321 = _______________

× +

212

EV I

200

21 × 143 = _______________

× +

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Multiply three-digit numbers by two-digit numbers by using the Objective 3 | standard algorithm 43 × 421 = _______________

504 × 22 = _______________

EV I

Multiply by using the standard algorithm.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Multiply three-digit numbers by two-digit numbers with multiple regroupings Objective 4 | by using the standard algorithm. 23 × 546 = _______________

63 × 804 = _______________

38 × 529 = _______________

EV I

Multiply by using the standard algorithm.

624 × 83 = _______________

This page may be reproduced for classroom use only.

Concept Mini Lessons | student Page

Practice | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Practice Helpers Practice Helpers can be used to support students working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers during Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Multiply

Objective 2 Multiply three-

Objective 3 Multiply three-

Objective 4 Multiply

digit numbers by two-digit numbers by using vertical form and six partial products.

digit numbers by two-digit numbers by using the standard algorithm.

Look for ...

three-digit numbers by twodigit numbers with multiple regroupings by using the standard algorithm.

• Can the student use place value to break apart both factors? • Can the students find and record all the partial products in vertical form? • Does the student add the partial products to find the product?

• Can the student use place value to break apart both factors? • Can the student record two partial products by using the standard algorithm? • Can the student add the partial products to find the final product?

Look for ...

• Can the student use place value to break apart both factors in the area model? • Can the student correctly find the partial products? • Does the student add the partial products to find the product? • Can the student explain how the distributive property is used to find the product?

EV I

three-digit numbers by twodigit numbers by using the distributive property.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for ... • Can the student use place value to break apart both factors? • Can the student record two partial products by using the standard algorithm, including regrouping units? • Can the student add the partial products to find the final product?

Practice | Teacher Guide

Practice | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

Answer Key Practice Page 2

Practice Page 3

Practice Page 4

1. Area model completed; 3,850

1. Partial products: 10, 100, 600, 150, 1,500, 9,000; Product: 11,360

1. Correct standard algorithm; 6,913

1. Correct standard algorithm; 21,658

2. Correct standard algorithm; 4,708

2. Correct standard algorithm; 33,652

3. Correct standard

3. Correct standard algorithm; 60,600

3. Area model completed; 28,032

4. Area model completed; 32,292

2. Partial products: 14, 70, 2,800, 80, 400, 16,000; Product: 19,364

EV I

2. Area model completed; 14,099

Practice Page 1

algorithm; C

3. Partial products: 49, 350, 6,300, 70, 500, 9,000; Product: 16,269

4. Partial products: 6, 0, 4,200, 20, 0, 14,000; Product: 18,226; C

4. To find the first partial product, 4 × 402, Kayla multiplied 4 ones by 4 tens. She should have multiplied 4 ones by 4 hundreds. To find the second partial product,

30 × 402, Kayla multiplied 3 tens by 4 tens. She

4. To find the second partial product, 60 × 752, Noah multiplied 6 tens by 2 ones and wrote the product as 12. 60 × 2 = 120.

should have multiplied

3 tens by 4 hundreds. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math catalyst | © 2025 Great Minds PBC

Practice | Teacher Guide

EV I

Student Edition | Printable Pages for students

Practice | Teacher Guide

NAME

DATE

three-digit numbers by two-digit numbers by using the Practice Page 1 | Multiply distributive property. 14 × 275 = _______________

32 × 876 = _______________

23 × 613 = _______________

EV I

Distribute to multiply.

46 × 702 = _______________

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

three-digit numbers by two-digit numbers by using vertical Practice Page 2 | Multiply form and six partial products.

32 × 355 = _______________

EV I

47 × 412 = _______________

Multiply by recording the partial products in vertical form.

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

three-digit numbers by two-digit numbers by using vertical Practice Page 2 | Multiply form and six partial products.

17 × 957 = _______________

Multiply by recording the partial products in vertical form.

What is the product of 26 and 701? Circle the letter of the correct answer.

EV I

A B C D

1,846 5,608 18,226 18,926

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

three-digit numbers by two-digit numbers by using the Practice Page 3 | Multiply standard algorithm.

31 × 223 = _______________

22 × 214 = _______________

Multiply by using the standard algorithm.

EV I

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

three-digit numbers by two-digit numbers by using the Practice Page 3 | Multiply standard algorithm.

Multiply by using the standard algorithm. What is the product of 23 and 213? Show your work. Circle the letter of the correct answer.

Kayla incorrectly multiplied by using the standard algorithm. Look at Kayla’s work. What mistake did Kayla make?

1,065 4,896 4,899 4,999

EV I

402 × 34 = 1,428

Kayla’s Work

1, 4

× +

A B C D

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

three-digit numbers by two-digit numbers with multiple Practice Page 4 | Multiply regroupings by using the standard algorithm. Multiply by using the standard algorithm.

34 × 637 = _______________

Noah incorrectly multiplied by using the standard algorithm. Look at Noah’s work. What mistake did Noah make?

47 × 716 = _______________

75 × 808 = _______________

EV I

Noah's Work

752 × 68 = 10,528 7 × + 1

0, 5

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Practice | Student Page

NAME

DATE

Practice Helper 1 Look at the problem. Then look at the work. It shows how to multiply three-digit

numbers by two-digit numbers by using the distributive property. Distribute to multiply.

37 × 528 = ________

How can you use the area

How can you use the partial

How do you see the

an area model to show how

model to help you find the

products to find the product?

distributive property

to break apart both factors? 5 hundreds + 2 tens + 8 ones 500

7 30

EV I

How can you draw and label

partial products?

The area model shows a partial

500

7 × 500 = 3,500

7 × 20 = 140

7 × 8 = 56

30 × 500 = 15,000

30 × 20 = 600

30 × 8 = 240

product in each rectangle. I can add all the partial products to find 37 ×

528.

I can find the area of each rectangle

3 tens + 7 ones

model? In the top row, I see 7 distributed to all three parts of 528. In the bottom row, I see 30 distributed to all three parts of 528.

in the area model.

represented in the area

I can break apart both factors by place value. Then I can label the side lengths of the area model.

37 × 528 = 19,536

500

7 × 500 = 3,500

7 × 20 = 140

7 × 8 = 56

30 × 500 = 15,000

30 × 20 = 600

30 × 8 = 240

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 2 Look at the problem. Then look at the work. It shows how to multiply three-digit

numbers by two-digit numbers by using vertical form and six partial products. Multiply by recording the partial products in vertical form.

32 × 355 = ________

How do you know that you have found

How can you use the partial products to

you think about how to break apart the

and recorded all the partial products?

find the product?

I know that I have found and recorded all the

I can add all the partial products.

EV I

Can you picture an area model to help factors?

I can picture the area model in my head without

partial products when I have multiplied each part

drawing it. I think about how the factors would be

of 355 by each part of 32.

broken apart in the area model.

32 × 355 = 11,360

5 0

0 0

1, 3

1 9

+ 1

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 3 Look at the problem. Then look at the work. It shows how to multiply three-digit

Multiply by using the standard algorithm.

31 × 233 = ________

numbers by two-digit numbers by using the standard algorithm.

How can you record two partial products

How can you use the partial products to

you think about how to break apart the

by using the standard algorithm?

find the product?

I can record the partial products on the same

I can add the partial products.

EV I

Can you picture an area model to help factors?

drawing it. I think about how the factors would be

multiplying by and record the partial product with

broken apart in the area model.

the correct units.

31 × 233 = 6,913

I can picture the area model in my head without

line. I need to think about the units we are

6, 9

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to multiply three-digit

Multiply by using the standard algorithm.

34 × 637 = ________

numbers by two-digit numbers with multiple regroupings by using the standard algorithm.

How can you record the partial products

How can you use the partial products to

you think about how to break apart the

when you need to rename as a larger unit?

find the product?

EV I

Can you picture an area model to help factors?

I can picture the area model in my head without

drawing it. I think about how the factors would be broken apart in the area model.

I can add the partial products.

When I need to rename as a larger unit, I can use more than one place to record the product.

For example, the product of 4 ones and 7 ones

34 × 637 = 21,658

× + 1 1

is 28 ones, or 2 tens 8 ones. I can record a small 2 in the tens place and an 8 in the ones place.

4 8

1, 6

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Practice | student Page

Application | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers Read–Draw–Write

Activities, Structures, and Considerations

EV I

Structure(s)

Considerations

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplication of multi-digit numbers by multi-digit numbers. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a personal whiteboard. • Consider providing tools such as grid paper to support students in drawing an area model or in using vertical form. • Consider providing grid paper to support students in aligning digits by place value units.

Play a Game

Activity

Solve a Task

Partner Work

Application | Teacher Guide

Application | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

Teacher Tip

Materials

• Dice (2) • Products Greater Than or Less Than 10,000 Game Board • Game Instruction Card • Personal whiteboard • Grid paper (optional) Students work with a partner to play a game involving multiplying three-digit numbers by two-digit numbers.

Preparing to Play • Before playing, each player rolls a die. The player with the lower number is player A. • Consider providing tools such as grid paper to support students in drawing an area model or in using vertical form.

Solve a Task Materials

• Solve a Task Student Page Students work with a partner to solve a multi-part task that involves multiplication of multi-digit numbers by multidigit numbers. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Students use the Read– Draw–Write process to solve word problems that involve multiplication of multi-digit numbers by multi-digit numbers. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problem 1 involves multiplying a number by a multiple of 10. Problems 2 and 3 involve multiplying a three-digit number by a two-digit number.

10,000

two times and records the twodigit number. • Each player records both multidigit numbers and multiplies them to find the product. Players check each other’s work. • Player A gets a point if the product is less than 10,000. Player B gets a point if the product is greater than 10,000. • The player with more points after five rounds wins.

Materials

Play a Game: Products Greater Than or Less Than

EV I

Solve a Problem

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Playing the Game • Player A rolls their die three times and records the three-digit number. Player B rolls their die

Application | Teacher Guide

Application | Multiplication of Multi-Digit Numbers by Multi-Digit Numbers

Answer Key

1. The warehouse has 6,750 glue sticks.

2. There are 4,920 staples in one box.

Solve a Task

Solve a Problem

1. The grocery store sells 1,500 ounces of peanut butter.

2. About 42,120 peanuts were used to make the peanut butter.

3. Adesh walks his dog for 16,425 minutes.

EV I

3. Grocery store A sells 1,344 more ounces of peanut butter.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

A warehouse has 225 packs of glue sticks. Each pack has 30 glue sticks. How many glue sticks does the warehouse have?

One box of staples has 24 strips. Each strip has 205 staples. How many staples are in one box?

Adesh walks his dog for 45 minutes each day. How many minutes does Adesh walk his dog in 365 days?

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Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

Application | solve a Problem Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Problem Number

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Application | student Page

NAME

DATE

Application | Play a Game 4. Find the product and record it. Check each other’s work. Player A gets a point if the product is less than 10,000. Player B gets a point if the product is greater than 10,000.

Game Instruction Card Products Greater Than or Less Than 10,000 What You Need

Player A

• Dice 2

• Products Greater Than or Less Than 10,000 Game Board • Personal whiteboard

Player B

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• Grid paper (optional) How to Play

3 6 4 2 6

1. Before playing, roll a die. The player with the lower number is player A.

5. Play again. This time, switch roles.

2. Player A, roll a die three times and record each number in the top row of the game board.

How to Win

The player with more points after five rounds wins.

3. Player B, roll a die two times and record each number in the bottom row of the game board.

This page may be reproduced for classroom use only.

Application | student Page

Score

Player B

Player A

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Application | Play a Game | Products Greater Than or less Than 10,000 Game Board

Application | student Page

NAME

DATE

Application | solve a Task That’s a Lot of Peanuts

• Peanuts grow on a plant. Each plant produces 25–50 peanuts.

A grocery store sells 125 12-ounce jars of peanut butter. How many ounces of peanut butter does the grocery store sell?

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• It takes about 540 peanuts to make one 12-ounce jar of peanut butter.

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Application | student Page

NAME

DATE

Application | solve a Task A grocery store sells 78 12-ounce jars of peanut butter. About how many peanuts were used to make the peanut butter?

Grocery store A sells 879 12-ounce jars of peanut butter. Grocery store B sells 767 12-ounce jars of peanut butter.

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How many more ounces of peanut butter does grocery store A sell than grocery store B?

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Application | student Page

Multiplication

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Representing Multiplication of Decimals by Whole Numbers

Concept Guide | Representing Multiplication of Decimals by Whole Numbers Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and the Pause and Monitor Tool.

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Read–Draw–Write Tool (optional) • Game Instruction Card • Eureka Math2 cards or a standard deck of playing cards • Place value chart (optional) • Square inch tiles (optional) • Grid paper (optional) • Multiplication Top It Game Mat Page (2 copies per student pair) • Study a Solution Student Page • Highlighters (optional) • Solve a Task Student Page • Play money (optional) • Place value disks (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards - Copies of Multiplication Top It Game Mat in whiteboards • Print copies of the following: - Solve a Problem Recording Page (optional) - Study a Solution Student Page - Solve a Task Student Page • Gather the following materials: - Place value charts, square inch tiles, or grid paper - Highlighters - Play money or place value disks

• Concept Mini Lessons Teacher Guide • Personal whiteboard • Play money (3 dollars, 10 dimes, 10 pennies)

• Application Teacher Guide

• Personal whiteboard or Student Pages • Play money (3 dollars, 10 dimes, 10 pennies)

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• Practice Teacher Guide

Teacher Materials

• Print copies of Student Pages as needed. • Gather play money (3 dollars, 10 dimes, 10 pennies) per student.

Concept Guide | Teacher Guide

Concept Guide | Representing Multiplication of Decimals by Whole Numbers

Addressing Student Misconceptions How to Address Misconception

Students name partial products and products with incorrect place value units when multiplying decimals by whole numbers (e.g., 7 × 3.12 = 2,184).

Notice how the area models are used to organize and label the partial products in unit form. Two area models with the same digits but different values are shown side by side with the factors and partial products written in unit form.

Encourage students to verbalize the entire equation, including the units being multiplied, as they record each partial product in unit form in the area model. For example, 7 times 3 hundreds is 21 hundreds.

3 hundreds

1 ten

2 ones

21 hundreds

7 tens

14 ones

7 × 3.12 3 ones

1 tenth

2 hundredths

21 ones

7 tenths

14 hundredths

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Language Support

7 × 312

Student Misconception

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The charts should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MaTh CaTalysT | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Representing Multiplication of Decimals by Whole Numbers Dear Family,

Your student is working on representing multiplication of decimals by whole numbers. This involves putting together a few skills: composing new units, finding partial products, and understanding decimal place value concepts when multiplying. The sample work shows three representations of 3 × 0.42 to support your student’s understanding of multiplication of decimals by whole numbers. You can support your student’s progress by asking the questions in the table below as your student multiplies with the standard algorithm.

3 × 0.42 =

Can you use objects or a drawing to represent the multiplication problem?

tenths

hundredths

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ones

1.26

6 hundredths

1.2

0.06

1.2 + 0.06 = 1.26

How can you use the area

How can you use the partial

dots on a place value chart

model to help you find and

products in the area model

to make a larger unit and find

record the partial products?

to find the product?

R area model.

12 tenths

Can you regroup objects or

ones

in a place value chart, or use an

2 hundredths

the product?

I can use coins in a chart, draw dots

4 tenths

tenths

4 ones

hundredths

I can trade 10 of a smaller unit for

1 of the next larger unit. I trade

10 dimes for 1 dollar or 10 tenths for

3 tenths

1 hundredth

36 ones 27 tenths 9 hundredths 36

2.7

0.09

36 + 2.7 + 0.09 = 38.79

I can add the partial products in the area model to find the product.

I can multiply the length and width of each small rectangle to find the partial products.

1 one. I count to find the product.

Concept Guide | Teacher Guide

Progress Check | Representing Multiplication of Decimals by Whole Numbers

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Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions: • Can the student represent multiplication of decimals by whole numbers with place value charts? | Objective 1

• Can the student represent multiplication of decimals by one-digit numbers and two-digit numbers with area models and partial products?

Teacher Tip

| Objectives 2–3

Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress.

Progress Check | Teacher Guide

Progress Check | Representing Multiplication of Decimals by Whole Numbers

Progression Towards Proficiency Rubric Progress Check Tool Item(s)

Item 2

Items 3–4

Items 5–6

Objective 1

Objective 2

Objective 3

Not Yet Proficient

The student may show evidence of beginning to understand representing multiplication of decimals by whole numbers with a place value chart but makes more than 1 error that leads to an incorrect answer.

The student may show evidence of beginning to understand representing multiplication of decimals by whole numbers with an area model but makes more than 1 error that leads to an incorrect answer.

The student may show evidence of beginning to understand multiplying decimals by whole numbers but makes more than 1 error that leads to an incorrect answer.

Partially Proficient

The student correctly demonstrates solid reasoning but makes 1 calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student correctly demonstrates solid reasoning but makes 1 error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student correctly demonstrates solid reasoning but makes 1 calculation error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly completes a place value chart to represent the problem and finds 6.24 as the product.

The student correctly completes an area model to represent the problem and finds 19.72 as the product.

The student correctly finds the products:

The student finds the correct products:

3. B

6. 350.4

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Item 1

4. H

5. 57.24

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Representing Multiplication of Decimals by Whole Numbers 2 × 3.12 = ones

tenths

hundredths

Multiply. Show your work. Circle the letter of the correct answer. 13 × 0.34 =

A 0.442

B 4.42

C 44.2

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Multiply. Show your work with an area model.

D 442

4 × 4.93 =

Multiply. Show your work with the place value chart.

4.68 × 37 =

F 1.7316

G 17.316

H 173.16

1,731.6

This page may be reproduced for classroom use only.

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Representing Multiplication of Decimals by Whole Numbers Multiply. Show your work.

73 × 4.8 =

1.59 × 36 =

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This page may be reproduced for classroom use only.

Progress Check | Student Page

Multiplication of Decimals by Concept Mini Lessons | Representing Whole Numbers Progression of Mini Lesson Objectives 2 Multiply decimals by one-digit whole

3 Multiply decimals by two-digit whole

numbers by using objects in a chart.

numbers by using area models and partial products.

1 Multiply decimals by one-digit whole

7 × 3.12 =

3 ones

2 × 1.54 =

21 ones 21

1 tenth

2 hundredths

7 tenths 0.7

14 hundredths 0.14

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21 + 0.7 + 0.14 = 21.84

Start here if students

• can use repeated addition to multiply and • can represent money as cents and as dollars and cents, but • need support making new units when multiplying decimals by one-digit whole numbers on a chart and • need support correctly determining the product from representations of multiplication on a chart.

3.54 × 13 =

21.84

Start here if students

• can decompose decimals into place value units, • can add decimals to the hundredths, and • can multiply whole numbers by using an area model, but • need support multiplying decimals to the hundredths by one-digit whole numbers by using an area model.

46.02

3 ones

5 tenths

4 hundredths

9 ones 9

15 tenths 1.5

12 hundredths 0.12

30 ones 30

50 tenths 5

40 hundredths 0.40

9 + 30 + 1.5 + 5 + 0.12 + 0.40 = 46.02

Start here if students

• can multiply decimals by one-digit whole numbers by using an area model, but • need support multiplying decimals by two-digit whole numbers by using an area model.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart. 10 M I NU T ES

Tell the story.

$ 0. 23

H That’s repeated addition. That means we can find 3 × 0.23. H Let’s show 3 equal groups of $0.23 with our coins. Make a chart by drawing 2 lines on your whiteboard. Then show

3 groups of $0.23.

Demonstrate as students follow along. H Can you compose to make a larger unit with the pennies? How do you know? No. There are not 10 or more pennies.

H Can you compose to make a larger unit with the dimes? How do you know? No. There are not 10 or more dimes. H How much money do I need to buy 3 pencils? How do you know?

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H The pencils at the student store are $0.23 each. I want to buy 3 of them. How can we figure out how much money I need? We can add 0.23 + 0.23 + 0.23.

Materials • Personal whiteboard • container of play money (dollars, dimes, and pennies) • Objective 1 Student Page

Summary Students use money and pictorial representations on a chart to multiply decimals by one-digit whole numbers.

69 cents. I counted by tens for the dimes and then added the 9 pennies.

Direct students to problem 1 on Objective 1 Student Page. H How is the place value chart different from the chart you drew on your whiteboard? The place value chart has the headings of ones, tenths, and hundredths. The place value chart has a decimal point.

H A dollar represents 1 one. Which coins represent tenths? How do you know? The dimes represent tenths because it takes 10 dimes to make 1 dollar.

a dime as $0.10, which is the same as 1 tenth.

The dimes represent tenths because we can write the value of

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart. 10 MINUTES

The pennies represent hundredths because it takes 100 pennies to

H Which coins represent hundredths?

make 1 dollar.

of a penny as $0.01.

The pennies represent hundredths because we can write the value

H We used money to represent 3 × 0.23. We can also draw dots on a place value chart to represent the multiplication expression.

H What is 3 × 0.23? 0.69

Invite students to think–pair–share about whether 0.69 is a reasonable answer. Consider inviting students to use the context of buying pencils to consider the reasonableness of their answer.

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Draw to represent 3 × 0.23 on the place value chart as students do the same.

3 × 0.23 =

H How do you see the factor 0.23 represented on the place value chart? Each group has 2 tenths 3 hundredths.

0.69

0.69 is a reasonable answer because when we think about the story with buying pencils, each pencil costs about 1 quarter and 3 quarters

have a value of 75 cents. Our answer of 0.69 is close to 0.75.

Teacher Tip

tenths

ones

hundredths

Direct students to clear their coins and get ready to listen to another problem.

H How do you see the factor 3 represented on the place value chart? There are 3 groups.

Consider acknowledging and praising students’ effort. Identify opportunities when you can offer feedback such as the following: Your effort really shows in this work. I notice you are organizing your dots in rows, and that is helpful when you are counting to find the product. You must be proud of yourself.

H The notebooks at the student store are $1.54 each. I want to buy 2 of them. How can we represent that with money on

4 $1.5

We can show $1.54 on our chart 2 times.

a chart?

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart. 10 MINUTES

H Can you compose to make larger units? How? Yes. There are 10 dimes. 10 dimes for a dollar.

3 dollars and 8 cents

3 ones and 8 hundredths, $3.08

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Guide students to exchange the 10 dimes for 1 dollar. 2 notebooks?

3.08

ones

Yes, 10 tenths is equal to 1 one.

do I need to buy

place value chart to represent 2 × 1.54.

2 × 1.54 =

We can exchange

H How much money

H Let’s record our work by drawing dots on the

Instruct students to model 2 groups of $1.54 with the play money.

Direct students to write a multiplication equation that represents the problem. Then direct students to problem 2 on the Student Page.

tenths

hundredths

Guide students as they draw to represent 2 × 1.54 on the place value chart. H How do you see the factor 2 represented on the There are 2 groups.

place value chart?

H How do you see the factor 1.54 represented on the Each group has 1 one 5 tenths 4 hundredths.

place value chart?

H What is 2 × 1.54? 3.08

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart. 10 M I NU T ES

Analyze Student Progress Monitor: • How does the student represent the multiplication expression on a chart? • Does the student correctly regroup on a chart as needed? • Does the student correctly use the chart to find the product?

H Is our answer reasonable? How do you know? Yes. If we think about money, $1.54 is about 1 dollar and 50 cents. I know that 2 groups of 1 dollar and 50 cents is $3.00. Our answer is close to $3.00.

Invite students to turn and talk about how they can use objects and drawings on a place value chart to multiply decimals by a one-digit whole number. Language Support

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As students share their ideas about how they composed units, revoice their responses with precise language. For example, if a student says, “These 10 dots were moved to this column to make 1,” respond by saying, “Yes, you composed 10 tenths to make 1 one.”

Questions to Advance Student Thinking: • What can you draw to represent the multiplication expression on a chart? • Do you need to regroup? How do you know? • How can you use the chart to find the product?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

• Each large eraser at the student store costs $1.13. What is the total cost of 3 large erasers? • Each small eraser at the student store costs $0.65. What is the total cost of 2 small erasers?

Concept Mini Lessons | Teacher Guide

decimals by one-digit whole numbers by using area models and Objective 2 | Multiply partial products. 10 M I NU T ES

Materials • Personal whiteboard • Objective 2 Student Page

Write 7 × 312 and 7 × 3.12. Display the area model for 7 × 312 and the blank area model.

Invite students to turn and talk to estimate the product of 7 × 3.12.

7 × 312

tenths, and hundredths are in 3.12? What do you think?

1 ten

21 hundreds

H Let’s break apart 3.12. I can ask myself, How many ones,

2 ones

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3 hundreds

Summary Students represent multiplication of a decimal by a one-digit whole number with an area model and use partial products to find the product.

7 tens

14 ones

7 × 3.12

H How will the area model for 7 × 3.12 be the same as and

different from the area model for 7 × 312? Why?

The digits in the area model for 7 × 3.12 will be the same as the

digits in the area model for 7 × 312 because the digits in the factors of the two expressions are the same.

The units will be different because the digits 3, 1, and 2 have

different place values in 312 than they do in 3.12.

3 ones, 1 tenth, and 2 hundredths

Label the columns of the area model. Direct students to do the same. 3 ones

1 tenth

2 hundredths

H What are we multiplying 3.12 by? 7

Label 7 on the area model and have students do the same.

H We can use unit form thinking to help us find the area of each small rectangle. What is 7 × 3 ones? 21 ones

Label 21 ones in the first rectangle and have students do the same.

Concept Mini Lessons | Teacher Guide

Multiply decimals by one-digit whole numbers by using area models and Objective 2 | partial products.

10 MINUTES

H What is 7 × 1 tenth? 7 tenths

H What expression can we write to represent the product of

H What is 7 × 2 hundredths? 14 hundredths

Label 14 hundredths in the third rectangle and have students do the same.

1 tenth

2 hundredths

Write the expression 21 + 0.7 + 0.14 below the area model. Direct students to do the same.

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3 ones

21 + 0.7 + 0.14

Label 7 tenths in the second rectangle and have students do the same.

7 and 3.12 by using the partial products?

21 ones

7 tenths

14 hundredths

3 ones

1 tenth

2 hundredths

21 ones

7 tenths

14 hundredths

0.7

0.14

21 + 0.7 + 0.14 = 21.84

H When we add decimals, it’s important to add like units. Let’s rewrite the problem vertically to help us line up the place value units and find the sum.

Teacher Tip

Consider demonstrating how the area model and partial products relate to the distributive property by writing the following below the area model:

7 × (3 ones + 1 tenth + 2 hundredths) = (7 × 3 ones) + (7 × 1 tenth) + (7 × 2 hundredths) = 21 ones + 7 tenths + 14 hundredths

H We have three partial products in unit form. What is each partial product in standard form? 21, 0.7, and 0.14

Write 21, 0.7, and 0.14 in the appropriate parts of the area model and have students do the same.

Rewrite the expression vertically and direct students to do the same. Then give students time to find the sum. H What is 21 + 0.7 + 0.14? 21.84

Write = 21.84 and have students do the same. H So, what is 7 × 3.12? 21.84

Direct students to write the product. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

Concept Mini Lessons | Teacher Guide

decimals by one-digit whole numbers by using area models and Objective 2 | Multiply partial products. 10 M I NU T ES

Invite students to compare the product with their estimate to determine whether the product is reasonable.

Monitor: • Does the student correctly draw and label an area model to represent the multiplication expression? • Does the student find and record each partial product in the area model? • Can the student correctly add the partial products to find the product?

Direct students back to the area model for 7 × 312 and invite them to use the partial products to find the product.

Analyze Student Progress

H What do you notice about the partial products and the product of 7 and 312 compared to the partial products and the product for 7 and 3.12?

The partial products and the products have the same digits, but the

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units are different.

Questions to Advance Student Thinking: • How can you draw and label an area model to represent the multiplication expression? • How can you use the area model to help you find and record the partial products? • How can you use the partial products to find the product?

H Just as we observed earlier, the digits are the same in both problems, but the place value units are different. This is true for the partial products and the product as well.

Invite students to turn and talk about how they can represent multiplication of decimals by one-digit whole numbers with an area model and use partial products to find the product.

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

• 2.6 × 9 • 0.52 × 3 • 5 × 6.08

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Concept Mini Lessons | Teacher Guide

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products. 10 M I NU T ES

Write 354 × 13 and 3.54 × 13. Display the area model for 354 × 13 and the blank area model. 354 × 13 4 ones

5 tens

9 hundreds

15 tens

12 ones

30 hundreds

50 tens

40 ones

3.54 × 13 =

H How will the area model for 3.54 × 13 be the same as and

different from the area model for 354 × 13? Why? The digits in the area model for 3.54 × 13 will be the same as the digits in the area model for 354 × 13 because the digits in the

factors of the two expressions are the same.

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3 hundreds

Materials • Personal whiteboard • Objective 3 Student Page

Summary Students represent multiplication of a decimal by a two-digit whole number with an area model and use partial products to find the product.

The units will be different because the digits 3, 5, and 4 have

different place values in 354 than they do in 3.54.

Invite students to turn and talk to estimate 3.54 × 13.

H How should we label the parts in the area model to show how to break apart 3.54? 3 ones, 5 tenths, and 4 hundredths

Label the columns of the area model and have students do the same. H How should we label the parts in the area model to show how to break apart 13? 3 ones and 1 ten 3 and 10

Concept Mini Lessons | Teacher Guide

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products. 10 M I NUTES

Label the rows of the area model and have students do the same. 5 tenths

4 hundredths

5 tenths

4 hundredths

9 ones 9

15 tenths 1.5

12 hundredths 0.12

30 ones 30

50 tenths 5

40 hundredths 0.40

3 ones

9 + 30 + 1.5 + 5 + 0.12 + 0.40 = 46.02

If students do not suggest breaking apart 13, ask them why it might be helpful to break apart 13. Expect them to say that breaking apart 13 allows them to use familiar facts to find partial products. Although it is mathematically valid to use a single row, the multiplication is more complex. Encourage students to use the area model in a way that helps them most.

H We can use unit form thinking to help us find the area of

each small rectangle. What is 3 × 3 ones? 9 ones

H What is 9 ones in standard form? 9

H What is an addition expression we can write to represent the total area? 9 + 30 + 1.5 + 5 + 0.12 + 0.40

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Teacher Tip

Label the parts and have students do the same. Then continue to find each partial product and label each part with the unit form and the standard form. Have students do the same.

Write the expression below the area model and have students do the same. H When we add decimals, it’s important to add like units. Let’s rewrite the problem vertically to help us line up the place value units and find the sum.

Rewrite the expression vertically and direct students to do the same. Give students time to find the sum. H What is the sum of the partial products? 46.02

Concept Mini Lessons | Teacher Guide

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products. 10 M I NU T ES

Write = 46.02 and have students do the same.

Direct students to write the product.

Invite students to compare the product with their estimate to determine whether the product is reasonable.

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Direct students’ attention back to the area model for 354 × 13 and invite students to use the partial products to find the product.

H What do you notice about the partial products and the product of 354 and 13 compared to the partial products and the product for 3.54 and 13?

The partial products and the products have the same digits, but the units are different.

Consider supporting students in the discussion by posting a word bank to encourage the use of precise language. Include words such as decimals, digits, factors, place, place value, and product.

H So, what is 3.54 × 13? 46.02

Language Support

Students may ask about the 0 ones column in the area model for 64 × 20.8. Mention that either including the column or not including the column is correct.

Teacher Tip

H Just as we observed earlier, the digits are the same in both problems, but the place value units are different. This is true for the partial products and the product as well.

Invite students to turn and talk about how they can represent multiplication of decimals by two-digit whole numbers with an area model and use partial products to find the product.

Concept Mini Lessons | Teacher Guide

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products. 10 M I NU T ES

Notes

Analyze Student Progress

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Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Multiplication of Decimals by Concept Mini Lessons | Representing Whole Numbers Answer Key Objective 2

1. Play money and place value chart correctly represent the problem;

1. Area model correctly represents the problem; 21.84

1. Area model correctly represents the problem; 46.02

2. Play money and place value chart correctly represent the problem;

2. Area model correctly represents the problem; 23.4

2. Area model correctly represents the problem; 715.5

3. Play money and place value chart

3. Area model correctly represents the problem; 1.56

3. Area model correctly represents the problem; 90.78

4. Area model correctly represents the problem; 30.4

4. Area model correctly represents the problem; 1,331.2

0.69

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3.08

3.39

correctly represent the problem;

4. Play money and place value chart 1.30

Objective 1

Objective 3

correctly represent the problem;

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Representing Multiplication of Decimals by Whole Numbers Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Representing Multiplication of Decimals by Whole Numbers Objective 1

Objective 2

Objective 3

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Student

Concept Mini Lessons | Teacher Guide

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Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart. value chart. Then write the product.

2 × 1.54 =

ones

tenths

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3 × 0.23 =

ones

tenths

$ 0. 23

hundredths

$1.54

Use play money to represent the multiplication equation. Draw to represent the multiplication equation on the place

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 1 | Multiply decimals by one-digit whole numbers by using objects in a chart.

2 × 0.65 =

tenths

ones

hundredths

$1.13

hundredths

$ 0.6 5

ones

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3 × 1.13 =

tenths

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

decimals by one-digit whole numbers by using area models and Objective 2 | Multiply partial products. Use the area model to find 7 × 3.12.

7 × 3.12 =

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Multiply. Show your work with an area model.

2.6 × 9 =

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

5 × 6.08 =

EV I

0.52 × 3 =

decimals by one-digit whole numbers by using area models and Objective 2 | Multiply partial products.

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Concept Mini Lessons | Student Page

NAME

DATE

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products. Use the area model to find 3.54 × 13.

3.54 × 13 =

EV I

Multiply. Show your work with an area model.

27 × 26.5 =

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Concept Mini Lessons | Student Page

NAME

DATE

64 × 20.8 =

EV I

5.34 × 17 =

decimals by two-digit whole numbers by using area models and Objective 3 | Multiply partial products.

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Concept Mini Lessons | Student Page

Practice | Representing Multiplication of Decimals by Whole Numbers Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Representing Multiplication of Decimals by Whole Numbers Concept Mini Lessons Objectives 1–3. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Practice Page 2

Practice Page 3

Objective 1 Multiply decimals by

Objective 2 Multiply decimals by

Objective 3 Multiply decimals by

one‑digit whole numbers by using objects in a chart.

one‑digit whole numbers by using area models and partial products.

two‑digit whole numbers by using area models and partial products.

Look for...

• Can the student draw and label an area model to represent a multiplication expression? • Can the student find and record each partial product in an area model? • Can the student correctly add partial products to find a product?

EV I

Practice Page 1

• Can the student make new units when representing the multiplication of decimals by one-digit whole numbers with objects in a chart? • Can the student correctly determine a product by counting objects in a chart?

Practice | Teacher Guide

Practice | Representing Multiplication of Decimals by Whole Numbers

Answer Key Practice Page 2

Practice Page 3

1. Play money and place value chart correctly represent the problem;

1. Area model correctly represents the problem; 12.96

1. Area model correctly represents the problem; 13.02

2. Play money and place value chart

2. Area model correctly represents the problem; 14.35

2. Area model correctly represents the problem; 338.4

3. Area model correctly represents the problem; 27.2

3. Area model correctly represents the problem; 149.8

0.39

1.24

EV I

correctly represent the problem;

3. Play money and place value chart 3.96

correctly represent the problem;

4. Play money and place value chart 4.92

Practice Page 1

correctly represent the problem;

4. 0.32 and 0.072 are circled; 40 + 3.2 + 0.72 = 43.92; 43.92

4. Area model correctly represents the problem; 194.48 5. B

Practice | Teacher Guide

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Student Edition | Printable pages for students

Practice | Teacher Guide

NAME

DATE

decimals by one-digit whole numbers by using objects Practice Page 1 | Multiply in a chart.

on the place value chart. Then write the product.

4 × 0.31 =

ones

tenths

EV I

3 × 0.13 =

ones

tenths

hundredths

Use play money to represent the multiplication equation. Draw to represent the multiplication equation

$0.13

$0.31

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Practice | Student Page

NAME

DATE

decimals by one-digit whole numbers by using objects Practice Page 1 | Multiply in a chart.

4 × 1.23 =

tenths

ones

hundredths

$1.3 2

ones

EV I

3 × 1.32 =

tenths

hundredths

$1.23

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

decimals by one-digit whole numbers by using area models Practice Page 2 | Multiply and partial products.

6 × 2.16 =

ones

3.4 × 8 =

hundredths

7 × 2.05 =

tenths

EV I

Multiply. Show your work with an area model.

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Practice | Student Page

NAME

DATE

decimals by one-digit whole numbers by using area models Practice Page 2 | Multiply and partial products.

Circle two mistakes in the partial products.

5 ones

4 tenths

40 ones 40

32 tenths 0.32

An area model for finding 8 × 5.49 was incorrectly completed.

9 hundredths

72 hundredths 0.072

EV I

Add the correct partial products and find the product.

8 × 5.49 =

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Practice | Student Page

NAME

DATE

decimals by two-digit whole numbers by using area models Practice Page 3 | Multiply and partial products. Multiply. Show your work with an area model.

tenths

hundredths

EV I

35 × 4.28 =

3.74 × 52 =

36 × 9.4 =

0.31 × 42 =

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Practice | Student Page

NAME

DATE

decimals by two-digit whole numbers by using area models Practice Page 3 | Multiply and partial products. Which addition expression does not represent the partial products in the area model?

6.23 × 32

6 ones 12 ones

180 ones

2 tenths

3 hundredths

4 tenths

6 hundredths

60 tenths

90 hundredths

EV I

A 12.46 + 186.90

B 12 + 4 + 6 + 180 + 60 + 90

C 12 ones + 4 tenths + 6 hundredths + 180 ones + 60 tenths + 90 hundredths

D 192 ones + 64 tenths + 96 hundredths

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Practice | Student Page

NAME

DATE

Practice Helper 1 Look at the problem. Then look at the work. It shows how to

3 × 0.42 =

multiply a decimal by a one-digit whole number with objects in a chart.

$0.42

Can you represent the

Can you regroup to make

What can you draw to

How can you show making

multiplication problem with

a larger unit and find the

represent multiplication

a larger unit and finding the

objects in a chart?

product?

in a place value chart?

product from your drawing?

EV I

ones

tenths

show 0.42 three times.

hundredths

I draw dots on a place value chart to

I use objects to show 0.42 three times.

I trade 10 of a smaller unit

ones

tenths

hundredths

I circle 10 tenths and draw an arrow

to show I am regrouping 10 tenths

to make 1 one. I count to find the product.

for 1 of the next larger unit.

I trade 10 dimes for 1 dollar.

I count to find the product.

3 × 0.42 =

1.26

ones

tenths

hundredths

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Practice | Student Page

NAME

DATE

Practice Helper 2 9 × 4.31 =

Look at the problem. Then look at the work. It shows how to model and partial products.

multiply a decimal by a one-digit whole number with an area

How can you draw and label an area

How can you use the area model to

How can you use the partial products to

model to represent the multiplication

help you find and record the partial

find the product?

equation?

products? 3 tenths

4 ones

1 hundredth

3 tenths

1 hundredth

36 ones

27 tenths

9 hundredths

2.7

0.09

EV I

4 ones

I can break apart 4.31 into 4 ones, 3 tenths, and

I can multiply the length and width of each

rectangle and label each part.

write the partial products in standard form.

1 hundredth. Then I can draw and partition a

36 + 2.7 + 0.09 = 38.79

I can add the partial products to find the product.

rectangle to find the partial products. Then I can

9 × 4.31 = 38.79

4 ones

3 tenths

1 hundredth

36 ones

27 tenths

9 hundredths

2.7

0.09

36 + 2.7 + 0.09 = 38.79 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

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Practice | Student Page

NAME

DATE

Practice Helper 3 0.82 × 13 =

Look at the problem. Then look at the work. It shows how to model and partial products.

multiply a decimal by a two-digit whole number with an area

How can you draw and label an area model to

How can you use the area model to help you

How can you use the partial

represent the multiplication equation?

find and record the partial products?

products to find the product?

8 tenths

2 hundredths

24 tenths

6 hundredths

2.4

0.06

80 tenths

20 hundredths

0.20

EV I

8 tenths

I can break apart 0.82 into 8 tenths and 2 hundredths.

I can multiply the length and width of each

and partition a rectangle and label each part.

write the partial products in standard form.

I can break apart 13 into 3 and 10. Then I can draw

2.4 + 8 + 0.06 + 0.20 = 10.66 I can add the partial products to find the product.

rectangle to find the partial products. Then I can

0.82 × 13 = 10.66

8 tenths

2 hundredths

24 tenths

6 hundredths

2.4

0.06

80 tenths

20 hundredths

0.20

2.4 + 28 + 0.06 + 0.20 = 10.66 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. MATH CATALYST | © 2025 Great Minds PBC

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Practice | Student Page

Application | Representing Multiplication of Decimals by Whole Numbers Read–Draw–Write

Activities, Structures, and Considerations

EV I

Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplying decimals by whole numbers. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a personal whiteboard.

Play a Game

Partner Work

Study a Solution

Independent Work Partner Work

• Consider providing highlighters and other tools for students to use to annotate the sample solution.

Solve a Task

Partner Work

• Consider providing manipulatives to support students with representing the multiplication problems.

Activity

• Consider using a standard deck of playing cards if you do not have Eureka Math2 cards. • Consider providing manipulatives to support students with representing the multiplication problems.

Application | Teacher Guide

Application | Representing Multiplication of Decimals by Whole Numbers

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Read–Draw–Write Tool (optional) Students use the Read–Draw– Write process to solve word problems involving multiplying decimals by whole numbers. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problem 1 involves multiplying a decimal to hundredths by a two-digit number. Problems 2 and 3 involve multiplying money amounts by two-digit numbers.

Students work with a partner to play a game involving multiplying a decimal to the hundredths place by a one-digit whole number.

Preparing to Play • Remove the 10, J, Q, and K cards from the deck. Aces can represent 1. • Shuffle the remaining cards. Divide the cards equally among the players. Each player keeps their cards in a single facedown pile. • Consider providing tools such as square inch tiles or grid paper to support students.

Play a Game: Multiplication Top It (Decimal by Whole Number) Materials

• Eureka Math cards or a standard deck of playing cards • Game Instruction Card 2

• If the products are the same, a Top It round ensues: A second round is played, and the player with the greater product takes all the cards played from both rounds. • The player with the most cards wins.

Materials

• 2 copies of Multiplication Top It Game Mat • place value chart (optional) • square inch tiles or grid paper (optional)

Teacher Tip

Consider having students think– pair–share about how they can use mental math and estimation to decide how to arrange the digits to create the greatest product.

EV I

Solve a Problem

Playing the Game • Each player takes three cards off the top of their pile to fill in the unknown digits on the Multiplication Top It Game Mat and finds the product. • The player with the greater product takes all the cards played and places them at the bottom of their pile.

Variations Players can build fluency by turning over two cards to multiply a decimal to tenths by a one-digit whole number. Or for a challenge, they can turn over four or more cards to multiply a decimal to hundredths by a multi-digit whole number.

Study a Solution Materials

• Study a Solution Student Page • highlighters (optional) Students work independently or with a partner to analyze a correct solution to a word

problem involving multiplying decimals by whole numbers. Students answer questions about how the known and unknown information in the problem is represented in the sample solution. They also analyze how the sample drawing provides a solution path. Finally, they are asked to consider whether the sample statement answers the question in the word problem.

Solve a Task Materials

• Solve a Task Student Page • play money (optional) • place value disks (optional) Students work with a partner to solve a multi-part task involving multiplying decimals by whole numbers. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Application | Teacher Guide

Application | Representing Multiplication of Decimals by Whole Numbers

Answer Key Study a Solution

1. Accurate picture drawn to represent

1. The known information is

product of 25 and 0.75; The scientist

Solve a Problem

of 6.25 quarts and 45 cars in the

the problem; equation shows the

represented by the labeled parts

uses 18.75 milliliters of water in all.

drawing.

2. Accurate picture drawn to represent product of 12 and 4.82; Mr. Evans

pays $57.84 for the ribbon.

3. Accurate picture drawn to represent the product of 28 and 2.25; Lisa the problem; equation shows

earns $63.00 selling lemonade on Saturday.

represented by the question mark

EV I

the problem; equation shows the

2. The unknown information is in the drawing.

3. The drawing shows the number of groups and the size of each group.

Solve a Task 1. It costs $36.27 to buy 9 gallons of unleaded gas at gas station B.

2. Yes, the driver has enough money; 12 × 4.05 = 48.60 3. The total amount of money spent by is $211.72.

the two customers at gas station A

When the number of groups and the size of each group are known, we can multiply to find the unknown total.

4. Yes. The question is about how many total quarts of oil the auto shop recycles at the end of the week, and the statement answers that question.

Application | Teacher Guide

Application | Solve a Problem Word Problem Cards

A scientist puts 0.75 milliliters of water into each of 25 test tubes. How many milliliters

of water does the scientist use in all?

Mr. Evans buys 12 spools of ribbon. Each spool costs $4.82. How much does Mr. Evans pay

EV I

for the ribbon?

Lisa earns $2.25 for each cup of lemonade she sells.

She sells 28 cups of lemonade on Saturday. How

much money does Lisa earn selling lemonade on Saturday?

Application | Teacher Guide

EV I

Student Edition | Printable pages for students

Application | Teacher Guide

NAME

DATE

Application | Solve a Problem

EV I

Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Application | Student Page

Application | Play a Game Game Instruction Card

Multiplication Top It (Decimal by Whole Number)

3. Multiply the numbers. Show your work below the

What You Need

multiplication expression and write the product.

• Eureka Math2 cards (or a standard deck of playing cards) represent 1.

with the 10, J, Q, K, and Joker cards removed; Aces can • 2 copies of Multiplication Top It Game Mat • Place value chart (optional)

How to Play

Put them at the bottom of your stack.

Player A

EV I

• Grid paper (optional)

4. If you have the greater product, take all the cards.

Player B

1. Mix up the cards. Deal the same number of cards to each player. Put the cards into a stack facedown.

2.48 is greater than 1.32.

3 tenths

1 hundredth

24 tenths 2.4

8 hundredths 0.08

2.4 + 0.08 = 2.48 8 × 0.31 = 2.48

5. If the products are equal, it is time to Top It! Play another

2. At the same time as the other player, turn over

three cards. Use the cards to fill in the unknown digits in

round. The player with the greater product takes the cards from both rounds.

the multiplication expression. Choose where to write the digits to create the greatest product.

How to Win The player with the most cards at the end of the game wins.

× 0. 3 1

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Application | Student Page

NAME

DATE

EV I

Application | Play a Game • Multiplication Top It Game Mat

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Application | Student Page

NAME

DATE

Application | Study a Solution Lisa solved the problem below. Read the problem and look at Lisa’s work. Then answer the questions.

An auto shop drains and recycles 6.25 quarts of oil from each car when they change the car’s oil. This week, the

auto shop changes the oil in 45 cars. How many quarts of oil does the auto shop recycle at the end of the week? Lisa's Work

? qt …

45 cars

5 hundredths

30 ones

10 tenths

25 hundredths

80 tenths

200 hundredths

240 ones

30 + 1 + 0.25 + 240 + 8 + 2 = 281.25 45 × 6.25 = 281.25 The auto shop recycles 281.25 quarts of oil at the end of the week.

45 × 6.25 = ?

2 tenths

EV I

6.25 qt

6 ones

How is the known information in the problem represented?

How is the unknown information in the problem represented?

How does the drawing help you see a solution path?

Does the statement answer the question? How do you know?

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Application | Student Page

NAME

DATE

Application | Solve a Task • There are 3 gas stations at an intersection.

Gas Stations • Each gas station has two options: unleaded gas and diesel. • The table shows the gas prices at each station.

$3.98

$4.12

Unleaded Gas

Diesel

$4.03

$4.14

Gas Station B Gas Station C

$3.95

$4.05

EV I

Gas Station A

How much does it cost to buy 9 gallons of unleaded gas at gas station B?

A driver has $50 to spend on gas and wants 12 gallons of diesel at gas station C.

Gas station A sells 18 gallons of unleaded gas to one customer and 34 gallons

Does the driver have enough money? How do you know?

of diesel to another customer. What is the total amount of money that the two customers spend on gas?

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Application | Student Page

EV I

Multiplication of Decimals by Decimals

Multiplication

Concept Guide | Multiplication of Decimals by Decimals Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide)

• Print copies of the Progress Check Tool and Pause and Monitor Tool.

• Concept Mini Lessons Teacher Guide • Personal whiteboard

• Personal whiteboard or Student Pages • Colored pencils (2)

• Print copies of Objectives 1–4 Student Pages

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

• Application Teacher Guide

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Game Instruction Card • Multiplication Top It Game Mat • Eureka Math2 cards or a standard deck of playing cards • Hundredths grids (optional) • Study a Solution Student Page • Highlighter (optional) • Solve a Task Student Page

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards - Hundredths grids (optional) • Print copies of the following: - Solve a Problem Recording Page (optional) - Multiplication Top It Game Mat - Study a Solution Student Page - Solve a Task Student Page • Gather the following: - highlighters (optional)

EV I

Teacher Materials

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Decimals by Decimals

Addressing Student Misconceptions Student Misconception

How to Address Misconception

When multiplying decimals by decimals, students think that multiplying two numbers always results in a product that is greater than either factor (e.g., “The product of 0.2 and 0.7 is greater than 0.7 because the product is always larger than the factors.”)

Consider using the following sequence to reason about the relationship between the factors and the product.

• Write a decimal to represent the part of the whole pizza that Tara has left. (0.8) • Write a decimal to represent 5 tenths. • Use the word of to describe how much of the leftover pizza Tara’s brother eats. • Write a multiplication expression to represent how much of the leftover pizza Tara’s brother eats. • Ask if 5 tenths of 8 tenths is more than or less than 1 group of 8 tenths.

Restate the learning: “When we multiply with decimals, the product might be less than one or both factors. If we think about equal groups, a factor that is less than 1 means less than 1 group. So, if one or both factors are less than 1, the product is smaller than one or both factors.”

EV I

Language Support

Tara has a pizza cut into 10 equal slices. Tara eats some pizza and has 8 tenths of the pizza left. Tara’s brother eats 5 tenths of the leftover pizza. Will Tara’s brother eat more than or less than 8 tenths of the pizza?

To support multilingual learners in making cross-linguistic connections through oral discourse, consider using strategic, flexible grouping.

• 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 home language. Encourage students to use their home language alongside English to make sense of the directions and the mathematics. To support multilingual learners in making cross-linguistic connections through written discourse, consider creating cross-linguistic connections anchor charts. The charts should include • key terminology in the student’s home language that is related to the current concept, • the same key terminology in English, and • images to support understanding. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Concept Guide | Teacher Guide

Family Math | Multiplication of Decimals by Decimals Dear Family,

6.9

EV I

4.6 × 1.5 =

Your student is working on multiplying decimals by decimals. They multiply decimals with tenths by shading a grid to represent each factor. Each square in the grid represents 1 hundredth, which helps students understand that the unit of the product when multiplying tenths by tenths is hundredths. Then they use an area model to break apart decimals before multiplying each part separately to find the partial products. Students record the partial products and add them to find the product. The sample work below shows how students use vertical form and what they know about multiplying whole numbers to multiply decimals. Instead of multiplying 46 × 15, students multiply 46 tenths × 15 tenths and record the partial products of 230 and 460. Then they add the partial products to determine the product. They record the unit of the product as hundredths. You can support your student’s progress by asking the questions in the table below as your student uses vertical form and place value thinking to multiply decimals.

4 6 tenths × 1 5 tenths 3 230 +460 6 9 0 hundredths

What is the unit of the product when

How can you write the product in

unit form?

multiplying tenths by tenths?

standard form ?

4 6 tenths × 1 5 tenths 3 230 +460 6 9 0 hundredths

I write 690 hundredths as a decimal, 6.9.

4 6 tenths × 1 5 tenths

How can you write the problem using

4.6 is equal to 46 tenths. 1.5 is equal to 15 tenths.

4.6 × 1.5 = 6.9

When I multiply tenths by tenths, the unit of the

46 tenths × 15 tenths = 690 hundredths product is hundredths.

Concept Guide | Teacher Guide

Progress Check | Multiplication of Decimals by Decimals

EV I

Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions: • Can the student use a hundredths grid to multiply decimals? | Objective 1

• Can the student use an area model to multiply decimals? | Objective 2

| Objective 3

• Does the student use place value reasoning to multiply decimals?

• Does the student use the relationship between whole number multiplication and decimal multiplication to multiply decimals? | Objective 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Multiplication of Decimals by Decimals

Progression Towards Proficiency Rubric

Not Yet Proficient

Item 1

Items 2 and 3

Item 4

Items 5–8

Objective 1

Objective 2

Objectives 2–4

The student may show evidence of beginning to understand multiplying decimals by decimals, but the answer is incorrect.

The student may show evidence of beginning to understand multiplying decimals by decimals but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand multiplying decimals by decimals, but the answer is incorrect.

The student may show evidence of beginning to understand multiplying decimals by decimals but makes more than one error that leads to an incorrect answer.

Partially Proficient

EV I

The student has the correct answer but is unable to show evidence of accurately modeling the multiplication by shading the hundredths grid; or the student shows evidence of correctly modeling multiplying decimals by decimals, but the answer is incorrect.

The student selects the correct model to represent the context and circles option D.

The student correctly shades the hundredths grids to represent the multiplication problems, and finds the products:

Proficient

Progress Check Tool Item(s)

2. 0.14 3. 0.24

The student selects the correct model to represent the context and circles option B.

The student has the correct answer but is unable to show evidence of accurately modeling the multiplication by using diagrams, numbers, or words; or the student shows evidence of correctly modeling multiplying decimals by decimals, but the answer is incorrect. The student correctly finds the products: 5. 0.48 6. 0.26 7. 7.25

8. 10.88

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Decimals by Decimals Which hundredths grid represents 0.5 × 0.3? Circle the letter of the correct answer.

EV I

Shade the hundredths grid to multiply. Then complete the equation.

0.7 × 0.2 =

0.4 × 0.6 =

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Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Decimals by Decimals 3 tenths

2 tenths

4 tenths

12 hundredths

hundredths

1 tenth

3 hundredths

hundredths

C 4 tenths

10 tenths

30 tenths

2 tenths

4 tenths

120 hundredths

hundredths

10 tenths

300 hundredths

hundredths

3 tenths

2 tenths

EV I

Which area model represents 1.4 × 3.2? Circle the letter of the correct answer.

30 tenths

2 tenths

120 tenths

8 tenths

4 tenths

12 tenths

8 tenths

20 tenths

1 tenth

3 tenths

2 tenths

300 tenths

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Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Decimals by Decimals Multiply. Show your work.

1.3 × 0.2 =

2.9 × 2.5 =

6.4 × 1.7 =

0.8 × 0.6 =

EV I

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Progress Check | student Page

Concept Mini Lessons | Multiplication of a Decimal by a Decimal Progression of Mini Lesson Objectives 2 Multiply decimals with tenths

3 Multiply decimals with tenths

4 Relate decimal multiplication

by using a hundredths grid.

by using area models.

by using place value reasoning.

to the multiplication of whole numbers.

10 tenths 4 tenths

0.1

0.4

40 hundredths

1 Multiply decimals with tenths

6 tenths 24

0.3 × 0.1 =

0.03

3 tenths × 1 tenth = 3 hundredths

hundredths

Start here if students

0.04 10 tenths

• can efficiently rename ones and tenths as tenths, and • can multiply one-digit by up to two-digit whole numbers, but • need support determining the product by using place value reasoning.

Start here if students

• can apply the break apart and distribute property to decompose factors, and • can use an area model to multiply whole numbers, but • need support multiplying tenths by using area models.

Start here if students • can multiply whole numbers by using the standard algorithm, but • need support relating decimal multiplication to the multiplication of whole numbers.

• can represent tenths on a hundredths grid, but • need support representing the multiplication of tenths on a hundredths grid, and • need support determining the product represented on a hundredths grid.

hundredths

EV I

Start here if students

100 hundredths

3 6 tenths × 2 5 tenths 3 180 1 +7 2 0 1 9 0 0 hundredths

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals with tenths by using a hundredths grid. 10 M I NU T ES

Write 1 × 0.4 =

H Do you think the product will be more than or less than 4 tenths? How do you know? The product will be less than 4 tenths, because we don’t need 1 group of 4 tenths.

Direct students to problem 1 on the Student Page.

EV I

H 1 times 4 tenths. We can use groups of to think about how to determine the product. What is 1 group of 4 tenths? 4 tenths

Write 0.1 × 0.4 =

Materials • Personal whiteboard • Colored pencils (2) • Objective 1 Student Page

Summary Students shade a hundredths grid to represent multiplying tenths by tenths.

H 1 tenth times 4 tenths. Let’s use groups of to think about how to determine the product. How many groups of 4 tenths do we need, more than 1 group or less than 1 group? How do you know? We need less than 1 group because 1 tenth is less than 1 one.

H We need less than 1 group of 4 tenths. We can use the word of to help us think about how to determine the product. We need 1 tenth of 4 tenths. I can ask myself: What is 1 tenth of 4 tenths?

Direct students to say 0.1 × 0.4 in unit form and use the word of instead of times.

Point to the hundredths grid. H How many columns do you see? How many rows? 10 columns and 10 rows

H How many squares are in the grid? How do you know? There are 100 squares because 10 × 10 = 100.

H There are 100 squares. If the grid represents 1 one and there are 10 columns, what does 1 column represent? 1 tenth H What does 1 row represent? 1 tenth

H What does each square represent? 1 hundredth

0.4

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals with tenths by using a hundredths grid. 10 M I NU T ES

H I can ask myself: How can I shade the rows to represent 1 tenth?

Color any row as students do the same. Then point to the double-shaded part of the grid.

Have students choose a colored pencil. Gesture to the columns on the hundredths grid. H Let’s begin with 4 tenths. I can ask myself: How can I shade the columns to represent 4 tenths?

Color any group of 4 columns as students do the same. Then have students choose a different colored pencil. Gesture to the rows on the same hundredths grid. Teacher Tip: Differentiation

EV I

If students need support visually decomposing the hundredths grid into 1 . Have tenths, provide a same-sized tenths grid with one part labeled __ 10 4 ___ students shade . Transition to the hundredths grid with each tenth 10 partitioned into 10 equal parts horizontally. Explain that we can partition

H We shaded the hundredths grid to find 0.1 × 0.4, or 1 tenth of 4 tenths. Where 0.1 0.4 0.04 do you see 1 tenth of 4 tenths in the area model? I see 1 tenth of 4 tenths where both colors are in the same squares. It’s 4 squares.

each of the 4 tenths into 10 equal parts and then shade 1 tenth of 4 tenths.

H Yes, the double-shaded part of the hundredths grid shows 1 tenth of 4 tenths. If each square represents 1 hundredth, then the area of the double-shaded part is 4 …? Hundredths

H When we multiply tens by tens, we get hundreds. When we multiply tenths by tenths, we get hundredths.

Language Support

1 10

Consider posting an anchor chart to help students make connections between multiplying tens by tens and multiplying tenths by tenths.

1 ten × 1 ten = 1 hundred 10 × 10 = 100 1 tenth × 1 tenth = 1 hundredth 0.1 × 0.1 = 0.01

Have students outline the 4 hundredths on their grid.

Concept Mini Lessons | Teacher Guide

Objective 1 | Multiply decimals with tenths by using a hundredths grid. 10 M I NU T ES

H So, 0.1 × 0.4 is …? 0.04

Have students complete the equation on their Student Page.

Analyze Student Progress Monitor: • Can the student use the word of to think about how to find the product? • Can the student shade a hundredths grid to represent each factor? • Can the student identify the product in the shaded grid? • Does the student correctly write the product as hundredths?

H When we shade 1 tenth of 4 tenths, the result is 4 hundredths. How can we write that as a decimal? 0.04

Invite students to turn and talk about how they can use a hundredths grid to represent multiplying tenths by tenths.

EV I

Questions to Advance Student Thinking: • How can you use the word of to help you think about how to determine the product? • How can you shade the grid to represent each factor? • Where do you see the product on the hundredths grid? • How does the unit change when tenths are multiplied by tenths?

• 0.3 × 0.3 • 0.9 × 0.3 • 0.7 × 0.5

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply decimals with tenths by using area models. 10 M I NU T ES

Direct students to problem 1 on the Student Page.

Materials • Personal whiteboard • Objective 2 Student Page

Summary Students use an area model to multiply tenths by tenths.

H What is the product? 64 hundredths 0.64

H Let’s use an area model to represent this product. Teacher Tip

EV I

Prompting students to think about area models for multiplying whole numbers can help them make connections to the new learning. Consider asking students the following questions:

H What do you notice about the factors in the problem 0.4 × 1.6? One factor is less than 1, and the other factor is greater than 1.

H How is 1.6 represented on the grid? 1.6 is represented by the shading of a whole grid, 10 tenths, and shading 6 tenths in the other grid. H How is 0.4 represented? 4 rows in both grids are shaded.

H Why are 4 rows shaded in both grids? There are 4 rows shaded in each hundredths grid because 1.6 means 1 one 6 tenths. So, there are 4 rows shaded in the grid that represents 1 one, and there are 4 rows shaded in the grid that represents 6 tenths.

• How can you use what you know about multiplying whole numbers to multiply with decimals? • How can you decompose 16 tenths into parts that would help you multiply efficiently?

H How can we say 0.4 × 1.6 in unit form? 4 tenths times 1 one 6 tenths 4 tenths of 1 one 6 tenths

Gesture to the decomposed area model in problem 1. H I notice the area model shows the length decomposed. Let’s decompose the larger factor. How can we decompose 1 one 6 tenths using only tenths in unit form? 1 one 6 tenths is 16 tenths, so we can decompose 1.6 into 10 tenths and 6 tenths.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply decimals with tenths by using area models. 10 M I NU T ES

Label the length of the area model as 10 tenths and 6 tenths as students do the same.

We need to add the two partial products together.

We need to add the areas of the two smaller rectangles to get the

H What should we label the width? 4 tenths

area of the larger rectangle.

Label the width of the area model as 4 tenths as students do the same.

10 tenths

6 tenths

40 hundredths

hundredths

Invite students to add the partial products to find the product. H So 0.4 × 1.6 is …? 64 hundredths 0.64

Invite students to turn and talk about how the area model shows the same partial products as the hundredths grids. Encourage students to gesture to the hundredths grids and the area model as they discuss what they notice.

EV I

4 tenths

H How will we find the product?

H How can we find the area of each rectangle in the area model, or the partial products? We can multiply 4 tenths and 6 tenths. Then we can multiply 4 tenths and 10 tenths. H What is 4 tenths times 6 tenths? 24 hundredths

Label the area as 24 hundredths as students do the same. H Why is the unit hundredths?

When we multiply tenths by tenths, the unit changes to hundredths.

H What is 4 tenths times 10 tenths? 40 hundredths

Label the area as 40 hundredths as students do the same.

Direct students to problem 2 on the Student Page. H How are the factors in problem 2 different than the factors in problem 1? This time both factors are greater than 1. H How is our area model going to change from problem 1 to problem 2? We will need to break apart, or decompose, both factors. Our area model will have more parts, so we will have more partial products.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply decimals with tenths by using area models. 10 M I NU T ES

Draw to decompose the length of the area model. Then label the decomposed length as 10 tenths and 6 tenths.

10 tenths

6 tenths

40 hundredths

hundredths

H After we break apart the factors, we can distribute or multiply each part of these factors separately. I can ask myself: How can I find the partial product represented by the 4 tenths by 6 tenths rectangle? What do you think? You can multiply. 4 × 6 = 24

H What will our unit be? 24 …? Why? 24 hundredths because when we multiply tenths by tenths, we get

EV I

4 tenths

10 tenths

Gesture to the area model as you walk through finding the first partial product.

H Let’s label the length as 1.6. How can we decompose 1.6? 10 tenths and 6 tenths because we can rename 1.6 as 16 tenths. Then we can decompose 16 tenths into 10 tenths and 6 tenths.

100 hundredths

hundredths.

Record the partial product as students do the same. Direct students to find the rest of the partial products.

hundredths

Language Support Consider partnering students to have them discuss how to determine the partial products. This provides students an opportunity to hear, refine, and use math language with peers.

H Let’s label the width as 1.4. What is 1 one 4 tenths using only tenths? 14 tenths H How can we decompose 14 tenths? We can decompose 14 tenths into 10 tenths and 4 tenths.

Draw to decompose the width of the area model. Then label the decomposed width as 10 tenths and 4 tenths.

H How can we find the product?

We can add all the partial products.

100 hundredths + 40 hundredths + 24 hundredths + 60 hundredths = 224 hundredths H What is the sum of the partial products? 224 hundredths

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply decimals with tenths by using area models. 10 M I NU T ES

H So 1.4 × 1.6 is …? 224 hundredths

Analyze Student Progress

Invite students to turn and talk about how they multiplied tenths by tenths by using an area model.

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

Questions to Advance Student Thinking: • How can you decompose the factors to label the sides of the area model? • How can you use the area model to help you find the partial products? • How can you use the partial products to find the product? • How can you write the product in standard form?

EV I

• 2.3 × 3.7 • 8.1 × 4.2

Monitor: • How does the student decompose the factors in the area model? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Can the student use the partial products to determine the product? • Does the student correctly write the product in standard form?

2.24

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply decimals with tenths by using place value reasoning. 10 M I NU T ES

Write the equations 4 × 2 = 8, 40 × 20 = 800, and 0.4 × 0.2 = 0.08. Ask students what connections they notice among the equations. Highlight thinking that shows the relationship between place value units of the factors and the product. Each equation has factors with the digits 4 and 2, but in different

place values.

H We can use unit form and place value thinking to help us find products.

Write the expression 30 × 10. 300

H What is 3 tens times 1 ten? 3 hundreds

H How is problem 1 different than 30 × 10? The place value units are different.

We are still multiplying 3 units by 1 unit.

H How is it similar?

H How can we write 0.3 × 0.1 using unit form? 3 tenths times 1 tenth

EV I

Each product has the digit 8, but in a different place value.

Materials • Personal whiteboard • Objective 3 Student Page

Summary Students use place value reasoning to determine the unit of a product.

H How do you know? I know 3 × 1 = 3 and when we multiply two multiples of ten, the unit of the product is hundreds. So 3 tens × 1 ten = 3 hundreds.

Write the equation in unit form: 3 tens × 1 ten = 3 hundreds.

Record 3 tenths × 1 tenth as students do the same.

0.3 × 0.1 = 0.03 3 tenths × 1 tenth = 3 hundredths

Gesture to 3 and 1.

H What is 3 × 1? 3

H We are multiplying units of tenths by units of tenths. I can ask myself: What unit will I use for the product? What do you think? Hundredths

Direct students to problem 1 on the Student Page.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply decimals with tenths by using place value reasoning. 10 M I NU T ES

Teacher Tip: Differentiation

H How can we rename 1.3 or 1 one 3 tenths as tenths? How do you know? 13 tenths because 1 one is the same as 10 tenths and 10 tenths + 3 tenths = 13 tenths.

If students need more support with understanding that multiplying tenths by tenths gives hundredths, show a hundredths grid that represents 0.1 × 0.1 = 0.01.

1.3 × 0.1 = 0.13

13 tenths × 1 tenth = 13 hundredths

Write the product in unit form and direct students to do the same.

Gesture to 13 and 1.

EV I

H When we multiply tenths in unit form, we can multiply by using a familiar multiplication fact and then think of the product in hundredths. We know 3 × 1 = 3 and the unit of the product is hundredths. So 3 tenths × 1 tenth = 3 hundredths.

Record 13 tenths × 1 tenth as students do the same.

Have students turn and talk about how to write the product in standard form. Direct them to record their answer on the Student Page.

Direct students to problem 2 on the Student Page. Have students think–pair–share about how 1.3 × 0.1 is similar to and different from problem 1.

There is a digit other than zero in the ones place in the first factor. We are still multiplying tenths by tenths.

H How can we write 1.3 × 0.1 using unit form? 1 one 3 tenths times 1 tenth

H What is 13 × 1? 13

H We are multiplying units of tenths by units of tenths. I can ask myself: What unit will I use for the product? What do you think? Hundredths

Write the product in unit form and direct students to do the same. Have students turn and talk about how to write the product in standard form. Direct them to record their answer on the student page. Language Support Consider providing sentence stems for students who need support showing their work by using unit form. Have them focus on the place value units. Consider providing a word bank of place value units.

1.3 × 0.1 = ×1

= 13

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply decimals with tenths by using place value reasoning. 10 M I NU T ES

Direct students to Problem 3 on the Student Page.

Toby multiplied 0.7 and 0.1. Toby’s product is incorrect.

Look at Toby’s work. What mistake did Toby make? What is the correct product?

Toby’s Work

0.7 × 0.1 = 0.7

EV I

7 tenths × 1 tenth = 7 tenths

Invite students to turn and talk about how they can use place value thinking and unit form to multiply decimals without the use of a hundredths grid or area model.

H How did Toby show his thinking?

He re-wrote the problem in unit form.

Invite students to work with a partner to determine why Toby’s product is incorrect.

H What mistake did Toby make?

Toby wrote the product as tenths.

H What unit should Toby use for the product? Hundredths

Have students answer parts (a) and (b) on the Student Page.

Analyze Student Progress

Monitor: • Can the student write the problem using unit form? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Does the student correctly write the product in standard form?

Questions to Advance Student Thinking: • How can you write the problem using unit form? • What is the unit of the product when multiplying tenths by tenths? • How can you write the product in standard form? Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 4 | Relate decimal multiplication to the multiplication of whole numbers. 10 M I NU T ES

Direct students to solve problem 1 on the Student Page using vertical form. H What is 19 × 6? 114

H We know that 19 × 6 = 114 and that 1 9 tenths × 6 tenths when multiplying tenths by tenths, we get hundredths. So 19 tenths × 1 1 4 hundredths 6 tenths = 114 hundredths. We can use unit form when we’re multiplying decimals by decimals, but we also need to write the product in standard form.

EV I

Write tenths next to the factors in the problem. Direct students to do the same. Have students think–pair–share about how the units change the problem. At first, we found 1 ten 9 ones times 6 ones.

1 9 tenths × 6 tenths

H What unit do we get when we multiply tenths by tenths?

Invite students to turn and talk about how to write the equation in standard form. Have students record the equation on the Student Page. Direct students to problem 2 on the Student Page. Have them rewrite the problem using unit form.

Now, we are multiplying 19 tenths by 6 tenths.

Hundredths

Materials • Personal whiteboard • Objective 4 Student Page

Summary Students use vertical form and place value thinking to multiply two decimals.

H So, if we know 19 × 6 = 114, what do we get when we multiply 19 tenths by 6 tenths? 114 hundredths

Write hundredths next to the product in the problem. Direct students to do the same.

H What do we already know that will help us find the product? We know how to multiply 36 by 25. When we multiply tenths times tenths, we get hundredths.

H What do we need to know? The product of 36 and 25.

Have students multiply to find the product of 36 and 25. H What is 36 × 25? 900

Concept Mini Lessons | Teacher Guide

Objective 4 | Relate decimal multiplication to the multiplication of whole numbers. 10 M I NU T ES

H What is 900 hundredths in standard form? 9

3 6 tenths × 2 5 tenths 3 180 1 +7 2 0 1 9 0 0 hundredths

If students need support renaming 900 hundredths in standard form, consider reviewing some of the place value units and relationships for students to reference.

1 one = 10 tenths 1 one = 100 hundredths

Monitor: • Can the student write the problem using unit form? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Can the student correctly write the product in standard form?

Questions to Advance Student Thinking: • How can you write the problem using unit form? • What is the unit of the product when multiplying tenths by tenths? • How can you write the product in standard form? Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

EV I

Teacher Tip

Analyze Student Progress

H So 36 tenths × 25 tenths is …? 900 hundredths

Have students record the product on the student page.

Invite students to turn and talk about how they can use what they know about whole number multiplication and place value to multiply decimals.

• 8.3 × 4.6 • 12.3 × 1.4

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 4 Practice Helper and supporting students in using the worked-out example to guide their own work.

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Multiplication of a Decimal by a Decimal Answer Key

accurately shaded; 0.09

2. Hundredths grid

accurately shaded; 0.27

3. Hundredths grid

accurately shaded; 0.35

Objective 4

1. Hundredths grid

1. 0.03

1. 114

3. Toby wrote the incorrect

3. 38.18

accurately shaded; completed; 0.64

area model accurately

2. Area model accurately completed; 2.24 3. Area model accurately completed; 8.51

4. Area model drawn and

4. Hundredths grid

Objective 3

accurately shaded; 0.04

1. Hundredths grid

Objective 2

2. 0.13

unit with the product.

EV I

Objective 1

34.02

accurately completed;

When we multiply tenths by tenths, we get product is 7 hundredths, hundredths. The correct

2. 9

4. 17.22

or 0.07.

4. 0.28

5. 0.96

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of a Decimal by a Decimal Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of a Decimal by a Decimal Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable Pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

Objective 1 | Multiply decimals with tenths by using a hundredths grid.

0.9 × 0.3 =

0.3 × 0.3 =

EV I

0.1 × 0.4 =

0.7 × 0.5 =

Shade the hundredths grid to find the product. Each grid represents 1.

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Multiply decimals with tenths by using area models.

1.4 × 1.6 =

EV I

0.4 × 1.6 =

Complete the area model to multiply by using the distributive property..

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Multiply decimals with tenths by using area models.

2.3 × 3.7 =

EV I

Draw an area model to multiply.

8.1 × 4.2 =

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Multiply decimals with tenths by using place value reasoning. Multiply. Show your work in unit form.

0.3 × 0.1 =

Toby multiplied 0.7 and 0.1. Toby’s product is incorrect.

1.3 × 0.1 =

EV I

Look at Toby’s work. What mistake did Toby make?

What is the correct product?

Toby’s Work

0.7 × 0.1 = 0.7

7 tenths × 1 tenth = 7 tenths For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Multiply decimals with tenths by using place value reasoning. Multiply. Show your work in unit form.

0.2 × 1.4 =

3.2 × 0.3 =

EV I

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 4 | Relate decimal multiplication to the multiplication of whole numbers. Use what you know about multiplying whole numbers to multiply. Show your work in unit form.

8.3 × 4.6 =

3.6 × 2.5 =

EV I

19 × 6

12.3 × 1.4 =

This pag page e ma may y be rrepr eproduced oduced for classr classroom oom use only only..

Concept Mini Lessons | student Page

Practice | Multiplication of a Decimal by a Decimal Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Multiply

Objective 2 Multiply decimals

Objective 3 Multiply decimals

Objective 4 Relate

with tenths by using an area model.

with tenths by using place value reasoning.

Look for...

decimal multiplication to the multiplication of whole numbers.

• Can the student decompose the factors in the area model? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Can the student use the partial products to determine the product? • Does the student correctly write the product in standard form?

• Can the student write the problem using unit form? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Does the student correctly write the product in standard form?

Look for...

• Can the student use the word of to think about how to determine the product? • Can the student shade a hundredths grid to represent each factor? • Can the student identify the product in the shaded grid? • Can the student correctly write the product as hundredths?

EV I

decimals with tenths by using a hundredths grid.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of a Decimal by a Decimal Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for... • Can the student write the problem using unit form? • Does the student multiply a unit of tenths by a unit of tenths to get a unit of hundredths? • Can the student correctly write the product in standard form?

Practice | Teacher Guide

Practice | Multiplication of a Decimal by a Decimal

Answer Key

accurately shaded; 0.18

2. Hundredths grid

accurately shaded; 0.25

3. Hundredths grid

4. Hundredths grid 0.40 or 0.4

accurately shaded;

Practice Page 4

1. Area model accurately

1. 0.21

1. 0.78

3. 9.09

3. 43.68

180 hundredths, 45 hundredths; 2.25

labeled; partial products:

2. Area model accurately 180 hundredths, 45 hundredths, 200 hundredths, 50 hundredths; 4.75

labeled; partial products:

3. Area model drawn and 8.32

accurately completed;

5. A

Practice Page 3

accurately shaded; 0.08

1. Hundredths grid

Practice Page 2

2. 0.84

EV I

Practice Page 1

4. Scott got the correct

2. 45.14

4. B

answer in unit form but made a mistake when rewriting the answer correct product is 0.4

in standard form. The or 0.40.

4. Area model drawn and 31.92

accurately completed;

Practice | Teacher Guide

EV I

Student Edition | Printable Pages for students

Practice | Teacher Guide

NAME

DATE

Practice Page 1 | Multiply decimals with tenths by using a hundredths grid. Shade the hundredths grid to find the product. Each grid represents 1.

0.2 × 0.4 =

0.5 × 0.5 =

Complete the statement. Circle the letter of the correct answer.

0.6 × 0.3 =

EV I

0.8 × 0.5 =

The product of 0.3 × 0.8 is

A less than 0.8

C more than 0.8

B equal to 0.8

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 2 | Multiply decimals with tenths by using area models. Complete the area model to multiply by using the distributive property.

1.9 × 2.5 =

tenths

5 tenths

0.9 × 2.5 =

EV I

tenths

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 2 | Multiply decimals with tenths by using area models.

4.2 × 7.6 =

EV I

3.2 × 2.6 =

Draw an area model to multiply.

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 3 | Multiply decimals with tenths by using place value reasoning. 1

0.3 × 0.7 =

Scott multiplied 0.5 and 0.8. Scott did not find the

0.2 × 4.2 =

10.1 × 0.9 =

EV I

Multiply. Show your work in unit form.

correct product. Look at Scott’s work. What mistake

did Scott make? What is the correct product?

Scott’s Work

0.5 × 0.8 = 0.040

5 tenths × 8 tenths = 40 hundredths For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 4 | Relate decimal multiplication to the multiplication of whole numbers. Use what you know about multiplying whole numbers to multiply. Show your work using vertical form.

7.4 × 6.1 =

2.6 × 0.3 =

8.4 × 5.2 =

EV I

Multiply. Show your work using vertical form. Circle the letter

3.5 × 9.6 = A 3.36 B 33.6

of the correct answer.

C 336

D 3,360

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 1 Look at the problem. Then look at the work. It shows how to

0.3 × 0.5 =

How can you shade the

to help you think about how to find the product?

0.3 × 0.5

as 3 tenths of 5 tenths.

I can think about the problem

Where do you see the

How can you write

hundredths grid to represent

product on the hundredths

hundredths as a decimal?

each factor?

grid?

EV I

How can you use the word of

multiply decimals with tenths by using a hundredths grid.

0.3

0.5

show 5 tenths.

First, I shade the columns to

3 tenths.

0.3 × 0.5 =

Then I shade the rows to show

0.5

0.3 × 0.5 = 0.15

I write 15 hundredths as a decimal,

0.15.

The product is the double-shaded part of the hundredths grid. I count 15 small squares.

That’s 15 hundredths.

0.15

0.3 0.5

0.15

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 2 multiply tenths by tenths by using an area model.

4.8 × 3.9 =

Look at the problem. Then look at the work. It shows how to

How can you decompose the

How can you use the area

How can you use the partial

How can you write the

factors to label the sides of

model to help you find the

products to find the product?

product in standard form?

the area model?

partial products? 9 tenths

8 tenths

30 tenths 8 tenths

9 tenths

8 tenths

9 tenths

240 hundredths

hundredths

EV I

30 tenths

4.8 × 3.9 = 18.72

I write 1,872 hundredths as a decimal, 18.72.

360 40 tenths 1,200 hundredths hundredths

40 tenths 1,200 hundredths

40 tenths

4 ones 8 tenths = 48 tenths I decompose 48 tenths into

40 tenths and 8 tenths.

I decompose 39 tenths into

30 tenths and 9 tenths.

I multiply the side lengths of the

smaller rectangles to find the partial products.

40 tenths × 30 tenths = 1,200 hundredths

3 ones 9 tenths = 39 tenths

I add the four partial products to find the product.

That’s 1,872 hundredths.

4.8 × 3.9 = 18.72

1,200 hundredths + 360 hundredths + 240 hundredths + 72 hundredths = 1,872 hundredths

8 tenths

30 tenths

9 tenths

240 hundredths

hundredths

360 40 tenths 1,200 hundredths hundredths

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Practice | Student Page

NAME

DATE

Practice Helper 3 use unit form to multiply tenths by tenths.

2.4 × 0.2 =

Look at the problem. Then look at the work. It shows how to

How can you write the problem using

What is the unit of the product when

How can you write the product in

unit form?

multiplying tenths by tenths?

standard form?

2.4 is equal to 24 tenths. 0.2 is equal to 2 tenths.

2.4 × 0.2 =

0.48

24 tenths × 2 tenths = 48 hundredths

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24 tenths × 2 tenths =

When I multiply tenths by tenths, the unit of the product is hundredths.

2.4 × 0.2 = 0.48

I write 48 hundredths as a decimal, 0.48.

24 tenths × 2 tenths = 48 hundredths

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Practice | Student Page

NAME

DATE

Practice Helper 4 Look at the problem. Then look at the work. It shows how to

relate decimal multiplication to the multiplication of whole numbers.

4.6 × 1.5 =

What is the unit of the product when

How can you write the product in

unit form?

multiplying tenths by tenths?

standard form?

4 6 tenths × 1 5 tenths

4 6 tenths × 1 5 tenths 3 230 +460 6 9 0 hundredths

I write 690 hundredths as a decimal, 6.9.

4.6 is equal to 46 tenths. 1.5 is equal to 15 tenths.

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How can you write the problem using

4.6 × 1.5 = 6.9

When I multiply tenths by tenths, the unit of the

4.6 × 1.5 =

product is hundredths.

6.9

46 tenths × 15 tenths = 690 hundredths

4 6 tenths × 1 5 tenths 3 230 +460 6 9 0 hundredths

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Practice | student Page

Application | Multiplication of Decimals by Decimals Read–Draw–Write

Activities, Structures, and Considerations

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Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplying decimals by decimals. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

Study a Solution

Independent Work Partner Work

• Consider providing highlighters and other tools for students to use to annotate the sample solution.

Solve a Task

Partner Work

• Consider providing hundredths grids to support students with representing the multiplication problems.

Activity

• Consider using a standard deck of playing cards if you do not have Eureka Math2 cards. • Consider providing hundredths grids to support students with representing the multiplication problems.

Application | Teacher Guide

Application | Multiplication of Decimals by Decimals

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) Students use the Read–Draw– Write process to solve word problems involving multiplying decimals by decimals. Problem 1 involves multiplying tenths when both factors are less than 1, problem 2 when one factor is greater than 1, problem 3 when both factors are greater than 1. Teacher Tip

Students work with a partner to play a game involving multiplying a decimal by a decimal.

Variations Players can turn over two, instead of three cards. Consider modifying the Multiplication Top It Game Mat by writing 0 in the ones place of the second factor.

Preparing to Play • Remove the 10, J, Q, K, and Joker cards from the deck. Aces can represent 1. • Shuffle the remaining cards. Divide the cards equally among the players. Each player keeps their cards in a facedown pile. • Consider providing tools such as hundredths grids for support. Playing the Game • Each player takes three cards off the top of their pile and fills in the unknown digits on the Multiplication Top It Game Mat. Players try to make a multiplication expression that will result in the greatest product. • The player with the greater product places all the cards at the bottom of their pile. • If the products are the same, a Top It round ensues: A second round is played, and the player with the greater product takes all the cards from both rounds.

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

• The player with the most cards at the end of the time wins.

Materials

• Multiplication Top It Game Mat • Hundredths grids (optional)

Play a Game: Multiplication Top It (Decimal by Decimal) Materials

• Eureka Math2 cards or a standard deck of playing cards • Game Instruction Card

Teacher Tip

Students may incorrectly think that the largest digit should always go in the ones place. Consider modeling how the placement of the digits changes the product. Read the expression using the word of and unit language, for example, “7 tenths of 61 tenths” or “1 tenth of 76 tenths."

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Solve a Problem

Study a Solution Materials

• Study a Solution Student Page • Highlighter (optional)

Students work independently or with a partner to analyze several correct solutions to a problem involving multiplying decimals by decimals. Students answer questions about how the partial products and product are represented in the sample

solutions. They analyze how the models show the product in hundredths. Finally, they explain how they know the solution is written correctly in standard form.

Solve a Task Materials

• Solve a Task Student Page • Hundredths grids (optional) Students work with a partner to solve a multi-part task multiplying decimals by decimals. They are given important information about the problem and an image to support their understanding of the context. Then students solve three problems related to the given context. The problems require students to think critically about how to determine a solution. Language Support The term proceeds has multiple meanings. In Solve a Task, proceeds refers to the amount of money each pizza restaurant receives from the sale of a small cheese pizza. Consider relating proceeds to a context that is familiar to your students.

Application | Teacher Guide

Application | Multiplication of Decimals by Decimals

Answer Key Study a Solution

1. Accurate picture drawn to represent

1. The partial products are shown in

the problem; equation shows the

product of 0.4 and 0.8; Leo needs

both hundredths grids with light

0.32 pounds of blueberries.

shading of 7 rows. The partial

2. Accurate picture drawn to represent product of 2.5 and 0.9; Yuna drinks

2.25 liters of water.

shading of 18 columns and darker

products are shown in the area

3. Accurate picture drawn to represent product of 2.4 and 3.4; The area of the problem; equation shows the

the rug is 8.16 square meters.

4. The product is shown as 126 hundredths. There are 100 hundredths in 1, so the decimal is 1.26.

model by the labels on the two sides.

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the problem; equation shows the

Solve a Problem

2. The product is shown in both hundredths grids by the

double‑shaded part. The product is shown in the area model by adding the areas of both parts.

3. Hundredths are shown in the hundredths grids by the small squares. In the area model, hundredths are shown as the area because tenths times tenths gives hundredths. In vertical form, hundredths are shown in the partial

Solve a Task 1. Riley’s Pizza donates $3.21 from donates $3.36 from each pizza

each pizza to charity. Blake’s Pizza to charity. Accurate drawings are shown.

2. Riley’s Pizza would donate $5.35. Blake’s Pizza would donate $4.20. 3. Riley’s Pizza donates $4.86 to the charity from each special order.

products after tenths are multiplied by tenths. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

Leo needs 0.8 pounds of mixed berries for a muffin recipe.

0.4 pounds of the mixed berries are blueberries.

How many pounds of blueberries does Leo need?

Yuna drinks 2.5 bottles of water. Each bottle

of water contains 0.9 liters of water. How many

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liters of water does Yuna drink in all?

is 2.4 meters wide and 3.4 meters long. Mr. Perez buys a rectangular rug that What is the area of the rug?

Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

This page may be reproduced for classroom use only.

Application | Student Page

Application | Play a Game Multiplication Top It (Decimal by Decimal)

Player B

What You Need

• Eureka Math2 cards (or a standard deck of playing cards) represent 1.

with the 10, J, Q, K, and Joker cards removed. Aces can

0. 8 × 3 . 1

• Multiplication Top It Game Mat in a personal whiteboard • Hundredths grids (optional)

The product is 2.48.

4. If you have the greater product, take all the cards. Put them at the bottom of your stack.

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How to Play

1. Mix up the cards. Deal the same number of cards to each player. Put the cards into a stack facedown.

Player A

4.27 is greater than 2.48.

Player B

2. At the same time as the other player, turn over three cards. 3. Use the numbers to make a multiplication expression on your game mat. Write a number in each space to find the

greatest product. Multiply. Say the product. Player A

If the products are the same, it is time to top it! Play another round. The player with the greater product takes the cards

The product is 4.27.

0. 7 × 6 . 1

from both rounds. How to Win The player with the most cards at the end of the game wins.

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Application | Student Page

NAME

DATE

Application | Play a Game • Multiplication Top It Game Mat

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0. × . For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | study a solution Shen correctly found the product three different ways. Look at Shen’s work. Then answer the questions. Shen’s Work

1.26

1.8 × 0.7 =

7 tenths

56 hundredths

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8 tenths

10 tenths

70 hundredths

1 8 tenths × 7 tenths 8 tenths × 7 tenths 56 10 tenths × 7 tenths + 70 1 2 6 hundredths

How are the partial products represented in the hundredths grids? In the area model?

Where do you see the product on the hundredths grid? In the area model?

How do you see the unit of hundredths represented in the models?

How do you know that the product is correctly written in standard form?

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Application | Student Page

NAME

DATE

Application | solve a Task Cheese for a Cause

• Two pizza restaurants sell small cheese pizzas and donate part of the proceeds from • Riley’s Pizza sells a small cheese pizza for $10.70 and donates 0.3 of the proceeds to each pizza to a charity.

• Blake’s Pizza sells a small cheese pizza for $8.40 and donates 0.4 of the proceeds to the charity. the charity.

Which restaurant donates more to the charity for each cheese pizza it

How much money would each restaurant give to charity if they donate

Riley’s Pizza has a weekly special: buy one cheese pizza and get another cheese

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sells? Show how you know.

0.5 of the proceeds from each cheese pizza sold?

pizza for $5.50. For the special, 0.3 of the proceeds from each pizza is donated to

charity. How much money does Riley’s Pizza donate to charity for both pizzas sold if the customer takes part in the weekly special?

This page may be reproduced for classroom use only.

Application | student Page

Multiplication

EV I

Multiplication of Whole Numbers and Fractions

Concept Guide | Multiplication of Whole Numbers and Fractions Materials and Preparation Student Materials

Suggested Preparation

• Progress Check Teacher Guide

• Progress Check Tool • Pause and Monitor Tool (found in the Implementation Guide) • Fraction tiles (optional)

• Print copies of the Progress Check Tool and the Pause and Monitor Tool. • Gather fraction tiles as an optional manipulative.

• Concept Mini Lessons Teacher Guide • Personal whiteboard • Fractions Strips template or fraction tiles

• Personal whiteboard or Student Pages • Fraction Strips template or fraction tiles

• Print copies of the following: - Student Pages as needed - Fraction strips as needed

• Practice Teacher Guide

• Practice Pages • Practice Helpers

• Print copies of the Practice Pages and the corresponding Practice Helpers.

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• Application Teacher Guide

Teacher Materials

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional) • Eureka Math2 cards or a standard deck of playing cards • Fraction Cards • Game Instruction Card • Two-Color Counters • Three in a Row Game Board • Solve a Task Student Page • Manipulative tools such as fraction tiles (optional)

• Ready the following materials: - Application Word Problem Cards - Game Instruction Card - Eureka Math2 cards or a standard deck of playing cards - Fraction Cards • Print copies of the following: - Solve a Problem Recording Page (optional) - Three in a Row Game Board - Solve a Task Student Page • Gather the following materials: - Two-Color Counters - Tools such as fraction tiles (optional)

Concept Guide | Teacher Guide

Concept Guide | Multiplication of Whole Numbers and Fractions

Addressing Student Misconceptions How to Address Misconception

Students multiply the numerator and the denominator by the whole number.

Encourage students to write the multiplication equation in unit form. Using unit form shows the distinction between the unit and the number of units. The denominator identifies the unit. The numerator tells how many of that unit. When we make copies of a fractional unit, the denominator stays the same and the numerator increases. 3 × 1 fourth = 3 fourths = __ 3 4 3 × __ 1 = __ 3 4 4 Restate the learning: When we are multiplying a whole number by a fraction, the whole number tells us how many copies of the fraction to add together. Rather than use repeated addition, we can multiply the whole number and the numerator of the fraction to find how many units we have.

1 3 = 4 12

Language Support

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3×

Student Misconception

Concept Guide | Teacher Guide

Family Math | Multiplication of Whole Numbers and Fractions Dear Family,

__ 28

? 4 12

How can you draw a tape diagram to represent the multiplication expression?

4 12

4 . I can draw 7 copies of __ 12

4 12

(

)

1 = (7 × 4) × __ 12

= 28 × __ 1 = __ 28 12

How can you rewrite the

How can you move the

How can you find the

fractional factor as a whole

parentheses to group the

product?

number times a unit fraction?

whole-number factors?

4 12

4 = 7 × 4 × __ 1 7 × __ 12 12

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7 × __ 4 =

Your student is working on multiplying whole numbers and fractions. This involves using their prior learning about representing multiplication as repeated addition. Now they are learning to multiply fractions through models and the associative and commutative properties of multiplication. They decompose non-unit fractions into a whole number times a fraction and then multiply the whole numbers. You can support your student’s progress by asking the questions in the table below as your student multiplies whole numbers by fractions.

1 12

1 4 × __ 12

The fraction is 4 twelfths, which is 1 . 4 copies of __ 12

(

)

4 = 7 × 4 × __ 1 7 × __ 12 12

1 = (7 × 4) × __ 12

The associative property of multiplication says the product is the same if I multiply the

I can multiply the whole-number factors first.

7 × 4 = 28

Then I can multiply by the unit fraction.

28 1 = __ 28 × __ 12 12

whole numbers first. I move the parentheses to group the whole-number factors.

Concept Guide | Teacher Guide

Progress Check | Multiplication of Whole Numbers and Fractions

About the Progress Check Tool The Progress Check Tool is an assessment that can be used before, during, or after providing direct instruction. It is intended to collect data about students’ proficiency with multiplying whole numbers and fractions and is not designed to be graded. The Progress Check Tool has problems that are sequenced from simple to complex. Problems 1–4 involve multiplying a whole number by a fraction, and problems 5–8 involve multiplying a fraction by a whole number. Students can use a concrete model or number line in problems 1 and 2, a pictorial model or the associative property of multiplication in problems 3 and 4, and self-selected strategies in problems 5 and 6. Students apply the commutative property of multiplication to choose an efficient way to model and find the product in problems 7 and 8.

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Using the Progress Check Tool to Inform Instruction When making instructional decisions based on the data collected with the Progress Check Tool, consider the following questions: • Can the student represent the multiplication of a whole number by a fraction by using a concrete model or number line and find the product? | Objective 1

• Can the student represent the multiplication of a whole number by a fraction or a fraction by a whole number by using a tape diagram and find the product?

| Objectives 2–4

• Can the student multiply a whole number by a fraction? | Objectives 1, 2, and 4

• Can the student multiply a fraction by a whole number | Objectives 3 and 4 Teacher Tip Consider acknowledging and praising students’ achievements. Share the results of the Progress Check Tool with students and celebrate their progress. For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Progress Check | Teacher Guide

Progress Check | Multiplication of Whole Numbers and Fractions

Progression Towards Proficiency Rubric Progress Check Tool Item(s)

Items 3 and 4

Items 5 and 6

Items 7 and 8

Objective 1

Objective 2

Objective 3

Objective 4

Not Yet Proficient

The student may show evidence of beginning to understand multiplying whole numbers by fractions but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand multiplying fractions by whole numbers but makes more than one error that leads to an incorrect answer.

The student may show evidence of beginning to understand multiplying fractions and whole numbers but makes more than one error that leads to an incorrect answer.

Partially Proficient

The student demonstrates solid reasoning but makes one error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

The student demonstrates solid reasoning but makes one error that leads to an incorrect answer, or the student demonstrates some reasoning and circles the correct answer.

The student demonstrates solid reasoning but makes one error that leads to an incorrect answer, or the student demonstrates some reasoning and has the correct answer.

Proficient

The student correctly uses fraction tiles or the number line, if necessary, and finds the product. 7 3 1. __

The student correctly draws a tape diagram or uses the associative property of multiplication, if necessary, and finds the product: 6 3. _8_

The student correctly multiplies and circles the correct answer:

The student correctly draws a tape diagram, finds the product, and circles the equation they modeled:

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2. __ 15 4

Items 1 and 2

85 4. __

5. D

6. H

7. 4; circled equation

8. 15; circled equation

Progress Check | Teacher Guide

NAME

DATE

Progress Check Tool | Multiplication of Whole Numbers and Fractions 1

7× __13 =

5× _34_ =

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Multiply. Use fraction tiles or the number line as needed.

` 6× _18_=

Multiply. Draw a tape diagram or use the associative property of multiplication as needed.

4× __ 25 =

Progress Check | Student Page

NAME

DATE

Progress Check Tool | Multiplication of Whole Numbers and Fractions Multiply. Show your work. Circle the letter of the correct answer.

_2_×9= 3

__4 ×21= 7

2 A __ 27 11 B __ 3

4 F ___ 147

25 G __ 7

C 3

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D 6

H 12 J

Use a tape diagram to find the product. Circle the equation you modeled with the tape diagram.

12×__ 13 =

20×__ 34 =

1 ×12= OR __ 3

3 ×20= OR __ 4

This page may be reproduced for classroom use only.

Progress Check | student Page

Concept Mini Lessons | Multiplication of Whole Numbers and Fractions Progression of Mini Lesson Objectives 2 Multiply a whole number by a

3 Multiply a fraction by a whole

4 Multiply fractions and

by a fraction by using concrete objects and the number line.

fraction by using tape diagrams.

number by using tape diagrams.

whole numbers by using the commutative property of multiplication.

1 4

0 4

1 4

3 4

Start here if students

1 4

2 4

3 4

Start here if students

3 4

• can multiply whole numbers by using tape diagrams, • can apply the associative property of multiplication with whole numbers, and • can multiply a whole number by unit and non-unit fractions by using a number line, but • need support multiplying a whole number by unit and non-unit fractions by using tape diagrams.

• can represent multiplication as repeated addition, but • need support representing multiplication of whole numbers and fractions by using concrete objects and the number line.

3 4

Start here if students • can find equivalent fractions and • can multiply a whole number by a fraction by using tape diagrams, but • need support multiplying a fraction by a whole number by using tape diagrams.

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? 3 4

3×1=1+1+1=3

1 Multiply a whole number

2 3

Start here if students • can find equivalent fractions and • can model the multiplication of a whole number by a fraction as repeated addition and multiply, but • need support modeling the multiplication of a fraction by a whole number and multiplying.

Concept Mini Lessons | Teacher Guide

a whole number by a fraction by using concrete objects and Objective 1 | Multiply the number line. 10 M I NU T ES

Direct students to problem 1 on the Student Page and distribute the 1 fourth fraction strips.

Have students read problem 1 aloud.

3×1=1+1+1=3 4

1 4

Write __1 + __ 14 + __14 = as students do the same. 4

Write __3 as students do the same. 4

• Unit fraction: a fraction with a numerator of 1

• Interval: the distance between two points on a number line

1 4

Consider reviewing some terminology introduced in third grade to support students with this objective.

• Partition: to divide a whole into equal parts

H Multiplication is another way to write repeated addition. What repeated addition expression is equal to 3 × 1 fourth? __1 + __ 1 + __ 1 4

Language Support

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Three times 1 fourth equals

Materials • Personal whiteboard • Objective 1 Student Page • Fraction Strips or fraction tiles

Summary Students use repeated addition to represent and solve multiplication of whole numbers by fractions with concrete objects and the number line.

3 H Let’s show __as repeated addition on the number line with 4 our fraction strips. What unit fraction are we using? 1 fourth

H Look at the 0 on the number line. Write the denominator, 4, to show that each interval will represent a fourth. Then place three 1 fourth fraction strips on the number line with no gaps or overlaps. 3×1=1+1+1=3 4

1 4

0 4

1 4

2 4

3 4

1 4

Concept Mini Lessons | Teacher Guide

ultiply a whole number by a fraction by using concrete objects and Objective 1 | M the number line. 10 M I NU T ES

3 H Write a repeated addition expression that is equal to 2 × __. 4

Support students as they work.

2×3=3+3=6

H Show the intervals with tick marks at the ends of the fraction 1 and label each tick mark as you strips. Count on to find 3 × __ 4 count. 1 fourth, 2 fourths, 3 fourths

Direct students to show where 1, or _4_, is on the number line. 4

Have students remove the fraction strips. Show 3×__ 14 = __34 with hops on the number line as students do the same. 3×1=1+1+1=3 4

1 4

2 4

1 4

3 4

1 4

Direct students to problem 2 on the Student Page and have them read the problem aloud. 2 times 3 fourths equals

Write __3 + __ 3 as students do the same. 4

H What is 2 × 3 fourths? __6

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H We can represent 3 × _1_= __ 3 with hops between the tick 4 4 marks instead of the fraction strips.

H What is 3 × 1 fourth? __34

3 H We used fraction strips to show __on the number line. Now 4 3 3 we need to show 2 × __ , two copies of __ , on the number line. 4 4 Will the product be greater than or less than 1? Greater than 1

H How many intervals should we partition each whole into on the number line? 4

H The denominator lets us know how to partition each whole.

Invite students to watch as you model writing the denominator, 4, and marking the intervals between 0 and 1 on the number line.

Concept Mini Lessons | Teacher Guide

ultiply a whole number by a fraction by using concrete objects and Objective 1 | M the number line. 10 M I NU T ES

Support students as they work. Teacher Tip

2×3=3+3=6 4

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H Good job. Let’s label each tick mark as we count together. 1 fourth, 2 fourths, 3 fourths, 4 fourths, or 1

Because the product is greater than 1, we need to partition the next interval into fourths. How should you label the next fourth on the number line? How do you know?

2×3=3+3=6 4

H How many intervals did you make between 0 and 1? 4

H I’ll write the denominator, 4, here. That tells me I need to partition each whole equally into fourths. I’ll make 3 tick marks—here, here, and here. That gives me 4 sections, or fourths. You try.

Consider acknowledging and praising students’ effort. Identify opportunities when you can offer feedback such as the following:

Sometimes when I work through a problem, I think it won’t ever end. There are a lot of parts to this problem. I like how you kept going with your strategy. It helped you persevere and get the number line labeled correctly. Nice job!

H How many tick marks did you put onto the number line? 3

0 1 2 3 1 4 4 4 4 4 4

I should label it _5_. If I keep counting fourths, the next point will be 4 5 fourths.

H Partition and label the number line between 1 and 2 into fourths.

Support students as they work.

3 3 3 __ H Now, show 2 × __= __+ __ = 6 with hops on the number line. 4 4 4 4 3 6 H Nice job. You represented 2 × __= __ on a number line. 4 4

Concept Mini Lessons | Teacher Guide

ultiply a whole number by a fraction by using concrete objects and Objective 1 | M the number line. 10 M I NU T ES

2×3=3+3=6 3 4

Analyze Student Progress

3 4

0 1 2 3 1 5 6 7 2 4 4 4 4 4 4 4 4 8 4 4

Support students as they work.

8 • 4×__ 38

Questions to Advance Student Thinking: • How can you write a repeated addition expression that is equal to the multiplication expression? • How can you use fraction strips to help you partition intervals on the number line? What part of the fraction tells you how to partition the number line? • How can you count on and label fractions on the number line? • How can you show multiplication of fractions as hops on the number line? What part of the expression tells you what size to make the hops? What part of the expression tells you how many hops to make? Where do you find the product on the number line?

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Invite students to turn and talk about how they can multiply a whole number by a fraction by using concrete objects and hops on a number line.

Monitor: • How does the student represent multiplication of fractions as repeated addition? • How does the student partition intervals on the number line? • How does the student count on and label fractions on the number line? • Does the student correctly show multiplication of fractions with fraction strips and hops on the number line?

Plan Future Practice: Use Practice Page 1 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 1 | Fraction strips

1 4

1 8

1 4

1 8

1 3

1 2

1 4

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1 8

1 6

1 2

1 6

1 8

1 3

1 6

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply a whole number by a fraction by using tape diagrams. 10 M I NU T ES

Materials • Personal whiteboard • Objective 2 Student Page

Gesture to the tape diagram for problem 1(a) on the Student Page.

Gesture to problem 1(b) on the Student Page.

? 1 4

1 4

11 44

3 H How is 8 × __similar to 8 × _1_? And how are they different? 4 4 1, We still have 8 copies of a fraction. But instead of 8 copies of __ 4 3 . we now have 8 copies of __ 4

3 H How is our tape diagram for 8 × __ going to look different 4 1? from the tape diagram representing 8 × __ 4

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H How many copies of _1_do you see? 4 8 copies

Summary Students use tape diagrams to multiply a whole number by unit and non-unit fractions.

H What multiplication expression does the tape diagram show? 8×_14_

Write the expression in the blanks and instruct students to do the same.

1 ? How do you know? H What is the product of 8 × __ 4 8 __ 4 . I can count the parts of the tape diagram by fourths:

__1 , _2_, __3 , _4_, __5 , __6 ,_7_, __8 . 4 4 4 4 4 4 4 4

41 + __ 41 + __ 41 + __ 41 + __ 41 + __ 41 + __ 41 . __84 . It is the same as adding __14 + __

__84 . It is 8 groups of __14 . And 8 groups of 1 is 8, so 8 groups of __14 is __ 84 .

Write _8_as the product and direct students to do the same. 4

? 3 4

3 4

13 44

3 instead of __ 1 . Each part is going to have a value of __ 4 4

Model drawing one part of the tape diagram labeled __3 . Invite students 4

to do the same and then have them complete their tape diagram.

Point to 8×_3_. 4

H Let’s say the expression. 8 times 3 fourths

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply a whole number by a fraction by using tape diagrams. 10 M I NU T ES

Write 8 in the first blank.

3 H How can we represent __ as a whole number times a unit 4 fraction? 3×__14

Write 3×__ 1 in the parentheses. 4

H Let’s use the associative property of multiplication to group the whole-number factors. Language Support

6 × 70 = 6 × (7 × 10) = (6 × 7) × 10 = 42 × 10 = 420

Ask students how grouping 6 × 7 helps them multiply. Then invite students to identify the similarities and differences between the whole-number equation and the fractional unit form equation when the associative property is applied.

Write (8×3)×__ 1 on the second line. 4

1? H What is 24 × __ 4 Twenty-four 1 fourths

24 fourths

3 H What is 8 × __ ? How do you know? 4 24 because 8×3=24 and 24×__ 24 . 1 = ___ I know it is ___ 4

H Would using a number line be an efficient way to multiply 8 by __34 ? 3 No. The number line would have to be really long to show __

EV I

Consider activating students’ prior knowledge of the associative property of multiplication.

6 × 70 = 6 × (7 tens) = (6 × 7) tens = 42 tens

H By moving the parentheses, we get 8 and 3 to multiply first. What is 8 × 3? 24

eight times.

No. It would take a long time to partition 8 wholes into 4 parts.

Invite students to think–pair–share about how the tape diagram and associative property of multiplication help them multiply 8 by __3 . 4

By using the tape diagram, I can see the copies of the fractions,

so I can skip-count, use repeated addition, or multiply to find the product. The associative property helps me multiply. I can think about the fraction as a whole number times a unit fraction. When I move the parentheses, I end up multiplying two whole numbers and then multiplying by a unit fraction.

Concept Mini Lessons | Teacher Guide

Objective 2 | Multiply a whole number by a fraction by using tape diagrams. 10 M I NU T ES

Analyze Student Progress Monitor: • How does the student represent the multiplication expression with a tape diagram? • Does the student rewrite the fractional factor as a whole number times a unit fraction and move the parentheses to group the whole-number factors? • Does the student correctly find the product of a whole number and a fraction?

Repeat the process: Use the following problems during Concept Mini Lessons or at another time to provide additional practice as needed. Consider providing the Objective 2 Practice Helper and supporting students in using the worked-out example to guide their own work.

3 • 2×__ 4 • 4×_5_ 8 • 6×_3_ 7

EV I

Questions to Advance Student Thinking: • How can you draw a tape diagram to represent the multiplication expression? • How can you rewrite the fractional factor as a whole number times a unit fraction? • How can you move the parentheses to group the whole-number factors? • How can you find the product?

Plan Future Practice: Use Practice Page 2 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply a fraction by a whole number by using tape diagrams. 10 M I NU T ES

1 . Write 6×__ 2

H How can we think about this expression? __ 1 + __ 1 + __ 1 + __ 1 + __ 1 + _1_ 2

2 1 6 groups of __2

Draw semicircles to represent 6×__ 1 .

H If you have 6 halves of a cookie, how many whole cookies do you have? 3

Write __1 × 6.

1 × 6, we can H To find __ 2 partition 6 into 2 equal

parts. One of those equal 1 × 6. parts represents __ 2 1 of a The product is __ 2 group of 6.

Draw a partitioned set of circles 1 . to represent 6×__

EV I

Materials • Objective 3 Student Page

Summary Students use tape diagrams to multiply a fraction by a whole number.

H How is this expression different from the first expression? The order of the factors is different.

H How we interpret this expression is also different because 1 of a copy of 6 does not make sense. adding __ 2

H If you have 1 half of 6 cookies, how many cookies do you have? 3

H Let’s look at an area model that shows a fraction of a whole number.

Write __1 × 12. 4

1 × 12, or __ 1 of a group of 12, needs to H A model that shows __ 4 4 have 12 units that are partitioned into fourths.

Direct students to problem 1 on the Student Page. Gesture to the area model that shows __1 of a group of 12. 4

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply a fraction by a whole number by using tape diagrams. 10 M I NU T ES

H What does each column of this model represent? 1 H Where do you see fourths in this model?

12 as 3. Consider grouping the shaded units to show renaming ___ 4

Teacher Tip: Differentiation

H Where do you see a group of 12 in this model? There are 12 columns.

1 4

Direct students to problem 2 on the Student Page.

Each column has been partitioned into 4 rows to make fourths.

H To show _14_× 12, we partitioned 12 into 4 equal parts. What does each unit in this model represent? __1 4

Gesture to the area model that shows _3_of a group of 12.

EV I

1 4

H How many are shaded? 12 fourths are shaded because there are 12 columns that each have 1 unit shaded.

What equation does this area model represent? __ 14 × 12=___ 12 = 3 4

12_ H __ 14 of a group of 12 is __ , which we can rename as 3. 4

H How are this model and the first model similar? Both models show a group of 12, each column represents 1, both 1 . models are partitioned into fourths, and each unit represents __ 4

H What has changed from the first model to this model?

3 4 3 rows are shaded.

More units are shaded.

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply a fraction by a whole number by using tape diagrams. 10 M I NU T ES

H What equation does this area model represent? __ 3 × 12=___ 36 = 9 4

3 36 H _ _of a group of 12 is ___ , which we can rename as 9. 4 4

Divide the tape into thirds as students follow along. H We know 3 parts have a value of 6. How do we find the value of 1 part? It’s like 6÷3, which is 2. 6

H With 3 rows shaded, how many units are shaded? How do you know? I know 36 fourths are shaded because there are 12 columns that each have 3 units shaded, and 12×3=36.

Gesture back to the original expression, __2 × 6. Have we answered the question?

EV I

H Let’s use tape diagrams to find a fraction of a whole number.

Label each part of the tape diagram as students follow along.

Direct students to problem 3 on the Student Page.

H How can we interpret the expression _2_× 6? 3 __ 2 of a group of 6 3

H Let’s use a tape diagram instead of drawing all the rows and columns and counting the units. Start with a tape that represents the whole. What is the whole? 6

H How many parts should we partition the whole into? How do you know? We should partition it into 3 parts.

The denominator tells us we want thirds.

The fraction names thirds as the unit.

We are asked about thirds, so we need 3 equal parts.

No. We found 1 part is equal to 2, but we need to find 2 parts.

2+2=4

Have students record the answer.

Invite students to turn and talk about the steps they need to take to use a tape diagram to model a fraction of a whole number.

1 × 100 • __ 2

• __ 43 × 20

H How can we find the value of 2 parts? 2×2=4

Concept Mini Lessons | Teacher Guide

Objective 3 | Multiply a fraction by a whole number by using tape diagrams. 10 M I NU T ES

Language Support

Notes

• I knew I needed to find a fraction of • Because I wanted to find into .

, so I drew

, I partitioned the tape diagram

Analyze Student Progress

EV I

Monitor: • Does the student’s tape diagram accurately represent the problem? • Does the student correctly find the product of a fraction and a whole number?

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

Questions to Advance Student Thinking: • How can we represent the whole? • How many parts should you partition the whole into? How do you know? • What is the value of 1 part? • Did we find the product?

Plan Future Practice: Use Practice Page 3 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

Concept Mini Lessons | Teacher Guide

fractions and whole numbers by using the commutative property Objective 4 | Multiply of multiplication. 10 M I NU T ES

Materials • Objective 4 Student Page

Direct students to problem 1 on the Student Page.

Have students record the equation.

2 3

Summary Students use the commutative property of multiplication to efficiently model and multiply fractions and whole numbers.

2 3

Have students think–pair–share to compare the two models. The products are equal.

or 4.

EV I

H What equation does this model represent? How can you tell? __total, It represents 6×__ 2 = 4. I see 6 groups of __2 . 6 groups of _2_is _12 3

2 as 6×__ We write 6 groups of __ 32 , so the equation is 6×__ 32 = 4. 3

Have students record the equation.

H The commutative property of multiplication allows us to multiply the factors in any order. When we need to find a product that can be difficult to model with one interpretation, we can reverse the order and model it another way. The product will be the same. Let’s try it.

Gesture to problem 3.

Direct students to problem 2 on the Student Page.

2 , and The representations look different. One shows 6 copies of __ 3 the other shows _2_of a group of 6. 3

H What equation does this model represent? How can you tell? 2 2 2 2 × 6=4. I see a tape It represents __ 3 diagram split into 3 equal parts, so the ? denominator is 3. The bracket shows that 2 of the 3 equal parts represent the product, so the numerator is 2. The whole tape represents 6. That tells us the whole number. 2 of the 3 equal parts is equal to 4, so the product is 4.

48×__34 =

3 ×48= OR __ 4

H Which interpretation would help you model and solve this problem? Why? 48 copies of __34 would take a long time to draw. We would need to 3 . draw 48 units of __ 4

3 of a group of 48. We should think of this problem as __ 4

Concept Mini Lessons | Teacher Guide

ultiply fractions and whole numbers by using the commutative property Objective 4 | M of multiplication. 10 M I NU T ES

Analyze Student Progress Monitor: • Can the student explain the difference between the models when the commutative property of multiplication is applied? • Can the student use the commutative property of multiplication to choose an interpretation of the equation that can be efficiently modeled? • Does the student correctly model and find the product of a fraction and a whole number?

3 H How could we model __of a group of 48 with a tape diagram 4 and use the tape diagram to help us find the product? We could draw one tape that represents 48 and partition the whole into 4 equal units. Then we could find the value of 3 of the units.

Direct students to draw the tape diagram and to find the product. 3 H What is __× 48? 4 36

3 = • Model and solve 2×_ 8

• Model and solve 20×_2 =

OR _3 × 2= 8

Plan Future Practice: Use Practice Page 4 to support students who need additional practice. Structure the additional practice strategically to allow for teacher support or peer support.

OR _2 × 20= 5

Questions to Advance Student Thinking: • Which interpretation of the equation would help you find the product efficiently? How do you know? • What equation does your model represent?

EV I

Invite students to turn and talk about when they might want to use the fraction of a set representation instead of repeated addition.

Concept Mini Lessons | Teacher Guide

Concept Mini Lessons | Multiplication of Whole Numbers and Fractions Answer Key __ 14 + __41 + __14 ; correctly

modeled fraction strips, 3 number line showing __ 4 as the product

hops, and labeled

2. __ 34 + __43 ; correctly modeled fraction strips, hops,

and labeled number line 6 as the product showing __ 4

3. __18 + __81 + __18 + __81 + __81 ; correctly modeled

fraction strips, hops.

1 ; tape diagram; 8; 3; __ 4 24 8; 3; __14 ; 24; __14 ; ___ 4

b. Correctly drawn

2. Correctly drawn tape 6 1 ; 3; __ 1 ; 6; __ 1 ; __ diagram; __ 4 4 4 4 3. Correctly drawn tape 1 ; 4; 5; __ 1 ; diagram; 5; __ 8 8 20 20; __18 ; ___ 8

4. Correctly drawn tape 1 ; 6; 3; __ 1 ; diagram; 6; 3; __ 7 7 18 18; __17 ; ___ 7

and labeled number line 5 as the product showing __ 8

1. a. 8; __14 ; _48_

4. __ 38 + __83 + __38 + __83 ; correctly

Objective 3 1. __ 14 × 12 = 3

Objective 2

__ 34 × 12 = 9

3. 4; correctly drawn tape diagram

4. 50; correctly drawn

EV I

Objective 1

tape diagram

5. 15; correctly drawn tape diagram

Objective 4 1. 6 × _23_ = 4 2. __ 23 × 6 = 4

3. 36; correctly drawn

tape diagram; Equation is circled.

4. __68 ; correctly drawn tape diagram; Equation is circled.

5. 8; correctly drawn tape diagram;

Equation is circled.

modeled fraction strips,

12 number line showing ___ 8 as the product

hops, and labeled

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Whole Numbers and Fractions Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

Observational Data Recording Sheet Multiplication of Whole Numbers and Fractions Objective 1

Objective 2

Objective 3

Objective 4

EV I

Student

Concept Mini Lessons | Teacher Guide

EV I

Student Edition | Printable pages for students

Concept Mini Lessons | Teacher Guide

NAME

DATE

a whole number by a fraction by using concrete objects and Objective 1 | Multiply the number line. Write a repeated addition expression. Use fraction strips to find the product. Then represent the product on the number line.

3×__ 14 =

2×__ 34 =

EV I

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Concept Mini Lessons | Student Page

NAME

DATE

ultiply a whole number by a fraction by using concrete objects and Objective 1 | M the number line. 5×__ 18 =

4×__ 38 =

EV I

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 2 | Multiply a whole number by a fraction by using tape diagrams. Complete parts A and B.

a. 1 4

1 4

11 44

EV I

1 4

Draw a tape diagram to represent the expression. Then use

the associative property of multiplication to find the product.

2×__ 34

8×__34

⎜ ⎛

8×__ 34 =

⎝

⎜

⎝

⎠

⎟ ⎠

⎟ ⎞

=(2×

)×

⎛

2×__ 34 =2× 3×

⎞

)×

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Concept Mini Lessons | Student Page

NAME

DATE

ultiply a whole number by a fraction by using tape diagrams. Objective 2 | M

EV I

6×__37

⎝

⎜ ⎛

6×__ 37 =

⎜

4×__ 58 =4×

4×__ 58

⎛

⎝

⎟ ⎞ ⎠

)×

⎟ ⎞ ⎠

This page may be reproduced for classroom use only.

Concept Mini Lessons | Student Page

NAME

DATE

Objective 3 | Multiply a fraction by a whole number by using tape diagrams. Write the equation that each model represents.

3 4

EV I

1 4

__2 ×6= 3

__1 ×100= 2

__3 ×20= 4

Use a tape diagram to find the product.

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Concept Mini Lessons | Student Page

NAME

DATE

fractions and whole numbers by using the commutative property Objective 4 | Multiply of multiplication. Write the equation that each model represents. 2 3

2 3

6 2

Use a tape diagram to find the product. Circle the equation you modeled with the tape diagram.

3 ×48= OR __ 4

2×__ 38 =

3 ×2= OR __ 8

20×__ 25 =

EV I

48×__ 34 =

2 ×20= OR __ 5

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Concept Mini Lessons | student Page

Practice | Multiplication of Whole Numbers and Fractions Practice Helpers Practice Helpers can be used to support students who are working independently or with a partner to complete the Practice Pages. Each Practice Page has a corresponding Practice Helper with one solved problem that is similar to a problem on the Practice Page. The solutions include sample work and guiding questions that represent the thinking required to approach the problem. Consider providing the Practice Helpers in Concept Mini Lessons and supporting students in using the worked-out examples to guide their own work.

Practice Page 1

Practice Page 2

Practice Page 3

Practice Page 4

Objective 1 Multiply a whole

Objective 2 Multiply a whole

Objective 3 Multiply a

Objective 4 Multiply

number by a fraction by using tape diagrams.

fraction by a whole number by using tape diagrams.

Look for...

fractions and whole numbers by using the commutative property of multiplication.

• Can the student correctly represent an expression showing copies of a fraction on a tape diagram? • Can the student correctly apply the associative property of multiplication? • Can the student correctly use tape diagrams and the associative property to multiply whole numbers by fractions?

• Can the student correctly represent an expression as a fraction of a set on a tape diagram? • Can the student correctly use tape diagrams to multiply fractions by whole numbers?

Look for...

• Can the student correctly use fraction strips to represent multiplication of whole numbers by fractions? • Can the student correctly hop on the number line to represent multiplication of whole numbers by fractions?

EV I

number by a fraction by using concrete objects and the number line.

Practice Pages The Practice Pages are sequenced from simple to complex and align with Multiplication of Whole Numbers and Fractions Concept Mini Lessons Objectives 1–4. Consider providing students with the answer key for the Practice Pages. Then students can check their work and make corrections if necessary.

Look for... • Can the student use the commutative property of multiplication to choose an interpretation of the equation that can be

efficiently modeled? • Can the student identify which equation their model represents?

Practice | Teacher Guide

Practice | Multiplication of Whole Numbers and Fractions

Answer Key

modeled fraction strips,

3 as number line showing __ 3 the product

hops, and labeled

2. __46 + __ 46 + _46_; correctly

modeled fraction strips,

12 number line showing ___ 6 as the product

hops, and labeled

3. __32 + _32_; correctly modeled fraction strips, hops,

2. Correctly drawn tape 1 ; 3; ___ 1; diagram; 3; ___ 10 10 18 1 ; ___ 18; ___ 10 10

3. Correctly drawn tape 1 ; 4; 8; ___ 1; diagram; 4; 8; ___ 12 12 32 1 ; ___ 32; ___ 12 12 4. D

and labeled number line 6 as the product showing __ 2

1. Correctly drawn tape 9 diagram; __ 8

Practice Page 3

Practice Page 4

1. 30; correctly drawn

1. a. 6

1. __ 13 + __ 13 + _13_; correctly

Practice Page 2

tape diagram

2. 21; correctly drawn tape diagram

3. 10; correctly drawn tape diagram

EV I

Practice Page 1

4. 28; correctly drawn tape diagram

5. 18; correctly drawn tape diagram

b. Correctly drawn tape diagrams

2. a. 3

b. Correctly drawn tape diagrams

3. 4; correctly drawn

tape diagram; Equation is circled.

4. 16; correctly drawn

tape diagram; Equation is circled.

4. Lee’s work is incorrect. The intervals are partitioned incorrectly.

Practice | Teacher Guide

EV I

Student Edition | Printable Pages for students

Practice | Teacher Guide

NAME

DATE

a whole number by a fraction by using concrete objects and Practice Page 1 | Multiply the number line. Write a repeated addition expression. Use fraction strips to find the product. Then represent the product on the number line.

3×__ 13 =

3×__ 46 =

EV I

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Practice | Student Page

NAME

DATE

2×__ 32 =

EV I

ultiply a whole number by a fraction by using concrete objects and Practice Page 1 | M the number line.

Lee represented 3×__ 34 = __ 94 on a number line. Look at Lee’s work. Is he correct? Explain.

1 4

3 4

2 4

3 4

4 1 5 4 4

6 7 4 4

8 2 4

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Practice | Student Page

NAME

DATE

Practice Page 2 | Multiply a whole number by a fraction by using tape diagrams. 9×__ 18 =

Use a tape diagram to find the product.

EV I

Draw a tape diagram to represent the expression. Then use the associative property of multiplication to find the product. 3 6×__ 10

⎜ ⎛

3 =6× 6×__ 10

⎝

=(6× = =

)×

⎟ ⎞

⎠

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Page 2 | Multiply a whole number by a fraction by using tape diagrams.

⎜

8 = 4×__ 12

EV I

8 4×__ 12

⎝

)×

⎟ ⎞

⎠

Which expression do both models represent? Circle the letter of the correct answer.

A 2×__ 45

8 B 2+__ 10

C 2+__ 45

8 D 2×__ 10

⎛

? 8 10

8 10

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Practice | Student Page

NAME

DATE

Practice Page 3 | Multiply a fraction by a whole number by using tape diagrams.

__3 ×28= 4

__2 ×15= 3

__7 ×32= 8

__2 ×54= 6

EV I

__1 ×60= 2

Use a tape diagram to find the product.

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Practice | Student Page

NAME

DATE

fractions and whole numbers by using the commutative Practice Page 4 | Multiply property of multiplication. Use a tape diagram to find each product.

Use a tape diagram to find each product.

b. __13 ×9=

EV I

a. 9×__13 =

b. __ 34 ×8=

a. 8×__34 =

Use a tape diagram to find the product. Circle the equation you modeled with the tape diagram.

16×__ 14 =

24×__ 23 =

1 ×16= OR __ 4

2 ×24= OR __ 3

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 1 2×__ 34 =

Look at the problem. Then look at the work. It shows how

to multiply a whole number by a fraction and represent the answer on the number line.

How can you write a repeated

How can you use fraction

How can you represent the

addition expression that is

strips to help you partition

answer on the number line?

equal to the multiplication

intervals on the number line?

expression?

__ 34 + __ 34

I think about 2 copies of

3 fourths, and then I write 3 fourths in the addition expression 2 times.

EV I

3 4

I write the denominator, 4, so

I know each interval will be partitioned into fourths.

I can place fraction strips on the number line to help me

1 4

How do you find the product? 3 4

3 4

2 4

3 4

1 4 4

5 4

6 4

0 7 4

2 8 4

I hop to show 2 copies of 3 fourths, 34 . or 2×__

1 4

3 4

2 4

3 4

1 4 4

5 4

6 4

7 4

2 8 4

6 . The last hop ends at the product, __ 4

partition each interval. I am

careful not to overlap or leave gaps between fraction strips.

2×__ 34 =__43 + __ 43

3 4

0 4

1 4

2 4

3 4

1 4 4

5 4

6 4

7 4

2 8 4

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 2 to use a tape diagram and the associative property of multiplication to multiply a whole number by a fraction.

4 = 7×__ 12

Look at the problem. Then look at the work. It shows how

How can you draw a tape

How can you rewrite the

How can you move the

How can you find the

diagram to represent the

fractional factor as a whole

parentheses to group the

product?

multiplication expression?

number times a unit fraction?

whole-number factors?

? 4 12

4 12

4 . I can draw 7 copies of __ 12

4 12

1 12

4 =7× 4×__ 1 7×__ 12 ( 12 )

EV I

4 12

The fraction is 4 twelfths, which is 1 . 4 copies of __ 12

1 =(7×4)×__ 12

The associative property of

multiplication says the product is the same if I multiply the

I can multiply the whole-number factors first.

7×4=28

Then I can multiply by the unit fraction.

1 = __ 28×__ 28 12 12

whole numbers first. I move the parentheses to group the

? 4 12

4 12

whole-number factors.

4 12

4 =7× 4×__ 1 7×__ 12 ( 12 ) 1 =(7×4)×__ 12 1 =28×__ 12 28 =__ 12

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Practice | Student Page

NAME

DATE

Practice Helper 3 __ 23 ×12=

Look at the problem. Then look at the work. It shows how to

use a tape diagram to multiply a fraction by a whole number.

How can we represent the

How many parts should we

whole?

partition the whole into?

__ 23 ×12=

How do you know?

__ 32 ×12=

__ 23 ×12=

need 3 equal parts.

We are asked about thirds, so we

3 equal parts. Each part has a value of 12÷3, or 4.

__ 23 ×12=8

The whole, 12, is divided into

EV I

the whole 12.

Did we find the product?

I can draw a tape diagram and label

What is the value of 1 part?

? The problem asks for 2 thirds of 12.

Each part is 1 third of the whole, so

2 parts is 2 thirds. 2 parts is 4+4, or 8.

? For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Practice | Student Page

NAME

DATE

Practice Helper 4 40×__ 34 =

Look at the problem. Then look at the work. It shows how to fractions and whole numbers.

Which interpretation would help you

use the commutative property of multiplication to multiply

What is the product?

find the product efficiently? How do

What equation does your model represent?

you know? Repeated addition would take too long. 3 . I would need to draw 40 groups of __ 4

40×__ 34 =

Instead, it would help me to think about 3 of a group of 40. finding __ 4

3 ×40= OR __ 4

My model does not show repeated addition;

EV I

3 ×40= OR __ 4

it shows a fraction of a group.

__ 34 of a group of 40 is 30.

40×__ 34 =

3 ×40= OR __ 4

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Practice | student Page

Application | Multiplication of Whole Numbers and Fractions Read–Draw–Write

Activities, Structures, and Considerations

EV I

Structure(s)

Consideration

Solve a Problem

Independent Work Partner Work

• Consider providing the Read–Draw–Write Tool to support students as they solve problems involving multiplication of whole numbers and fractions. Two printable versions of the Read–Draw–Write Tool can be found in the Implementation Guide. • Consider inviting students to share their work with a partner. Students can compare solution paths and make connections between different representations. • Consider providing students with the Solve a Problem Recording Page as an alternative to working on a whiteboard.

Play a Game

Partner Work

• Consider printing the Fraction Cards on cardstock and laminating them for long-term use. • Consider using a standard deck of playing cards if you do not have Eureka Math2 cards.

Solve a Task

Partner Work

• Consider providing fraction tiles for students to use to represent the problems involving multiplication of whole numbers and fractions.

Activity

Application | Teacher Guide

Application | Multiplication of Whole Numbers and Fractions

• Personal whiteboard • Application Word Problem Cards • Solve a Problem Recording Page (optional)

Teacher Tip

• Eureka Math2 cards or a standard deck of playing cards • Fraction Cards • Game Instruction Card • Two-color counters (9) • Three in a Row Game Board in a personal whiteboard Students work with a partner to play a game involving multiplying whole numbers by fractions.

Preparing to Play • Remove the 10, J, Q, K, and Joker cards from the Eureka Math2 card deck. Aces represent 1. • Cut out the Fraction Cards. • Shuffle each type of card and place them facedown into separate piles. • Have each player select one side of the two-color counters to use as their game pieces. Alternatively, have each player choose a set of objects to use as counters.

Students use the Read– Draw–Write process to solve word problems involving multiplication of whole numbers and fractions. Students can record solutions on a whiteboard or on the Solve a Problem Recording Page. Problem 1 involves multiplying a whole number by a unit fraction. Problem 2 involves multiplying a whole number by a non-unit fraction. Problem 3 involves multiplying a fraction by a whole number.

Materials

Consider facilitating one of the Application activities with a small group of students. Facilitating an Application activity enables you to informally monitor progress and provide support as needed.

Playing the Game • Players take turns turning over three Eureka Math2 cards and three Fraction Cards to generate numbers for the game board. They write one-digit numbers in the left column and fractions in the top row. A player turns over a replacement card if any of the one-digit numbers is repeated. • Player A places a counter into an empty space on the game board. The player multiplies the number in the left column by the fraction in the top row. The player shows their work and product below the game board. • Player B checks player A’s work. If the answer is correct, the counter stays in the space. If the answer is incorrect, the counter is returned to the player. • Player B erases the work, chooses a new space, and multiplies. Player A checks player B’s work, following the same process. • Players continue taking turns choosing a space and multiplying until a player gets

Materials

Play a Game: Three in a Row

EV I

Solve a Problem

three in a row (across, down, or diagonally) or until the game board is filled. The first player to get three in a row is the winner.

Solve a Task Materials

• Solve a Task Student Page • Fraction tiles (optional) Students work with a partner to solve a multi-part task involving multiplication of whole numbers and fractions. They are given important information about the problem and an image to support their understanding. Then students solve three problems related to the given context. The problems require students to think critically about how to use the given information to determine a solution.

Application | Teacher Guide

Application | Multiplication of Whole Numbers and Fractions

Answer Key Solve a Task

Solve a Problem

1. Mrs. Chan spends 2 __ 13 hours walking the dog.

1. An accurate picture is drawn to represent the problem; 6 × __14 = __ 64 = 1 __42 = 1 __21 ; Toby uses 1 __12 cups of cornstarch. 2. An accurate picture is drawn to represent the problem; 5 × __78 = __ 35 = 4 __38 ; Julie bikes 4 _38_miles in all. 8

3. Yes. Mrs. Chan appears to follow the instructions

about the treats. Mrs. Chan feeds the dog 72 = 9. The instructions say to feed 9 treats; __38 × 24 = __ 8 the dog 2 treats on 2 of the days and 1 treat on 5 of the

EV I

3. An accurate picture is drawn to represent the problem; 24 = 8; Gabe uses 8 balloons to make balloon __2 × 12 = __ 3 3

2. Mrs. Chan feeds the dog 10 _12_cups of food in all.

days, which is a total of 9 treats.

animals.

Application | Teacher Guide

Application | solve a Problem Word Problem Cards

1 cup of cornstarch for each cherry Toby uses __ 4

does Toby use to bake 6 cherry pies?

pie he bakes. How many cups of cornstarch

7 miles each day for 5 days. Julie bikes __ 8

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How many miles does Julie bike in all?

There are 12 balloons in each pack of balloons. Gabe uses _23_of a pack to make balloon animals. How many balloons does Gabe use to make

balloon animals? For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

Application | Teacher Guide

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Student Edition | Printable Pages for students

Application | Teacher Guide

NAME

DATE

Application | solve a Problem

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Problem Number _________________________

Use the Read–Draw–Write process to solve the problem on the card. Record the problem number and show your work.

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Application | Student Page

Application | Play a Game Three in a Row (Whole Number by Fraction)

2. Player A, place a counter into an empty space on the game board. Multiply the number in the left column by the fraction in the top row. Show your work and the product

What You Need

below the game board.

• Eureka Math cards (or a standard deck of playing cards) 2

represent 1.

with the 10, J, Q, K, and Joker cards removed. Aces • Fraction Cards

3 5

Three in a Row

1 8

1 3

• Two-color counters

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• Three in a Row Game Board in a personal whiteboard How to Play

1. Mix up each type of card separately. Put each type of

card into a stack facedown. Take turns turning over three Eureka Math2 cards and three Fraction Cards to choose numbers for the game board. Write one-digit numbers

in the left column and fractions in the top row. Turn over

3. Player B, check player A’s work. If the answer is correct, the counter stays in the space. If the answer is incorrect, return the counter to player A.

another card if any of the one-digit numbers is repeated.

Three in a Row

1 3

3 5

1 8

1 3

4 3

4×3 = 3

7 For review purposes only and not intended for distribution or reproduction. Unauthorized printing, reproduction, or distribution of this material is strictly prohibited. All rights reserved. Math Catalyst | © 2025 Great Minds PBC

This page may be reproduced for classroom use only.

Application | Student Page

NAME

DATE

Application | Play a Game

4. Player B, erase the work, choose a new space, put your counter into it, and multiply. Player A, check their work.

5. Continue taking turns choosing a space and multiplying until a player gets three in a row or until the game board is filled. 3 5

How to Win

1 3

1 8

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Three in a Row

The first player with three counters in a row (vertically, horizontally, or diagonally) wins. If the gameboard is filled without either player getting three in a row, clear the board and play another round.

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Application | Student Page

Application | Play a Game • Fraction Cards

__1

1 __ 3

__2

__3 5

1 __ 6

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__1

3 __ 4 5 __ 6

1 __ 4 1 __ 8 2 __ 5 3 __ 8

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Application | Student Page

NAME

DATE

EV I

Three in a Row

Application | Play a Game • Three in a Row Game Board

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Application | Student Page

NAME

DATE

Application | solve a Task Caring for a Friend’s Dog

Mrs. Chan takes care of her friend’s dog for 7 days. The list shows the instructions her friend left. 1 hour each day. • Walk the dog for __ 3 3 _ _ • Feed the dog cups of food 2 times each day. 4 • Give the dog 2 treats on the first day and on the

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last day. Give the dog 1 treat on the other days.

Mrs. Chan follows the instructions for walking the dog each day. How many hours does Mrs. Chan spend walking the dog?

Mrs. Chan follows the instructions for feeding the dog. How many cups

Mrs. Chan opens a bag of 24 dog treats. By the end of the week, she gives the 3 of the treats. Does Mrs. Chan follow the instructions about the treats? dog __ 8 Explain.

of food does Mrs. Chan feed the dog in all?

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Application | student Page