# Lesson 10

Multiplication and Division

## Warm-up: Which One Doesn’t Belong: Multiplication and Division (10 minutes)

### Narrative

This warm-up prompts students to compare four representations. The reasoning here prepares students to connect the previous multiplication work to the division work of this lesson. It gives students an opportunity to use precise terms such as “factors,” “product,” and “quotient” in making comparisons (MP6). During the synthesis, ask students to explain the meaning of any terminology they use.

### Launch

• Groups of 2
• Display images and expression.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2-3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn’t belong?

### Activity Synthesis

• “What equation can each diagram represent?” (The array represents $$4\times6=24$$ because there are 4 rows of 6 dots, and there are 24 dots in the array. The area diagram could represent $$4\times7=28$$ or $$28\div4=7$$ since the sides are 4 and 7, and the area is 28. The tape diagram could represent $$24\div3={?}$$ or $$3\times{?}=24$$ because we know the total is 24 and there are 3 groups, but we don’t know how many are in each group.)

## Activity 1: Card Sort: Find the Match (25 minutes)

### Narrative

The purpose of this activity is for students to relate multiplication and division using a variety of representations. Students are given a card with a base ten diagram, tape diagram, area diagram, multiplication equation with a missing factor, or division equation. Students need to find the other student who has the card that matches their card. Each pair of cards includes a division equation. After students find the student with the matching card, they work together to create another diagram and a division situation that their cards could represent (MP2).

Here are images of the cards for reference:

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Organization, Social-Emotional Functioning

### Required Materials

Materials to Gather

Materials to Copy

• Find the Match

### Required Preparation

• The blackline master has 24 cards. Copy and cut enough cards so that each student can have one card.

### Launch

• Groups of 2
• Give one card to each student.
• Display an example of each type of representation shown on the cards (division equation, multiplication equation with missing factor, area diagram, tape diagram, and base ten diagram).
• “We saw in the warm-up that different diagrams and equations can represent the same situation such as 4 groups of 6 or 4 times 6.”
• “Think about what situation or quantities your card represents. Then, think about what another representation of the situation or quantities might look like.”
• 1 minute: quiet think time

### Activity

• “Find a student whose card represents the same situation or quantities as your card does. Be ready to explain why your cards belong together.”
• 2-3 minutes: partner work time
• Invite 2-3 groups to share their matches and how they knew they matched.
• “Work with your partner to create a poster that includes your cards and a diagram and situation that match your division equation. We’ll use these for a gallery walk, so organize your work so others can understand it.”
• Give students glue or tape and tools for creating a visual display.
• 10-15 minutes: partner work time

### Student Facing

Your teacher will give you a card that shows an equation or a diagram.

1. Find a classmate whose card represents the same situation or quantities as your card does. Be prepared to explain why your cards belong together.
2. Work with your partner to create a poster that includes:

2. a different diagram that your division equation could represent
3. a situation that your division equation could represent

Show your thinking and organize it so it can be followed by others.

### Activity Synthesis

• Display students' posters around the room.

## Activity 2: Find the Match Gallery Walk (10 minutes)

### Narrative

The purpose of this activity is to reinforce students' understanding of the relationship between multiplication and division by examining different representations of that relationship.

MLR7 Compare and Connect. Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations. “What did the representations have in common?” “How did the relationship between multiplication and division show up in each representation?” To amplify student language, and illustrate connections, follow along and point to the relevant parts of the displays as students speak.

### Required Preparation

• Keep posters from the previous activity displayed.

• Groups of 2

### Activity

• Arrange for half of the groups to stand at their poster and answer questions while the other half visit their posters.
• 8–10 minutes: gallery walk
• Ask groups to switch roles after 4–5 minutes.

### Student Facing

1. Describe a representation you saw that was different from the ones you showed in your poster.
2. Choose a poster that is not yours. Describe one way that it shows the relationship between multiplication and division.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we created posters that showed ways to represent division.”

“How does an area diagram show us the relationship between multiplication and division?” (It shows that multiplying is like finding the area of a rectangle when the two side lengths are known, and dividing is like finding a side length when we know the area and the other side length.)

“How does a tape diagram or equal-groups diagram show multiplication and division?” (Both show multiplying as a way to find the total when we know the number of groups and how many in each group, and dividing as a way to find either the number of groups or the size of each group when the total is known.)

“What were some aspects of the posters you saw that helped make the math your classmates used clear for you?” (Clear labels on diagrams that helped me understand their thinking. Units on their answers. When other students wrote their explanations, it helped me understand their thinking.)