# Lesson 2

Interpret Representations of Multiplicative Comparison

## Warm-up: How Many Do You See: Times as Many (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to use grouping strategies to describe the images they see.

In the synthesis, students describe how two images can be used to describe a multiplicative comparison and connect the images to a multiplication equation.

### Launch

• Groups of 2
• “How many do you see and how you do see them?”
• Flash the image.
• 30 seconds: quiet think time

### Activity

• Display the image.
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses. Use multiplication equations when appropriate.
• Repeat for each image.

### Student Facing

How many do you see? How do you see them?

### Activity Synthesis

• “How does this show that the second rectangle has 2 times as many as the first rectangle?”
• “How could we write an equation that shows this comparison?” ($$6 = 2 \times 3$$ or $$2 \times 3 = 6$$ or $$3 \times 2 = 6$$)

## Activity 1: Represent “Times as Many” (20 minutes)

### Narrative

The purpose of this activity is for students to analyze and describe how images and diagrams can show “$$n$$ times as many”. Students generate ideas for how to use a multiplication equation to represent the comparison.

Students begin by interpreting an image in which the multiplier (3) and the numbers are given. They explain how some number of times of the smaller amount can be seen in the larger amount. Next, they create their own diagram and see different ways of representing the iterations (groups of) the smaller amount to create the larger amount.

During the activity, make connecting cubes accessible for students who may choose to use them for problem solving—either to reason about the quantities or to explain their reasoning.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give students access to connecting cubes.
• Display the image of Mai's cubes and Kiran's cubes.
• “How do these cubes represent 3 times as many?” (Mai has 6 cubes and Kiran has 2. Mai has 3 groups of 2 cubes. Mai has 6 cubes and Kiran has 2. Three times as many as 2 is 6, or 3 times 2 is 6.)
• Read the statement about Jada and Kiran’s cubes as a class.
• 1 minute: quiet think time

### Activity

MLR7 Compare and Connect
• “Create a visual display that shows your thinking about the cubes in each problem and include details to help others understand your thinking.”
• 6–8 minutes: independent or group work
• 3 minutes: gallery walk
• “How does each representation show ‘times as many’?”
• 30 seconds quiet think time
• 1 minute: partner discussion
• Monitor for students who create diagrams that are similar to connecting cube images and discrete tape diagrams to share in the synthesis.

### Student Facing

1. Jada has 4 times as many cubes as Kiran. Draw a diagram to represent the situation.
2. Diego has 5 times as many cubes as Kiran. Draw a diagram to represent the situation.
3. Lin has 6 times as many cubes as Kiran. How many cubes does Lin have? Explain or show your reasoning.

### Advancing Student Thinking

Students may compare additively instead of multiplicatively when using cubes or drawing diagrams. Consider asking: “How would you use the cubes to show 3 more?” and “How would you use the cubes to show 3 times as many?”

### Activity Synthesis

• Have selected students share diagrams and explain how they show “times as many”.
• If needed, use cubes to represent statements.
• “How could you write an equation to compare Kiran’s and Jada’s cubes?”
• “What do the numbers in the equation represent in the situation?” (Four is the “4 times as many”. Two is how many Kiran had. Eight is how many Jada had.)
• Write equations for each situation and ask about what students notice about the relationships.

## Activity 2: Diagrams to Solve Multiplicative Comparison Problems (15 minutes)

### Narrative

The purpose of this activity is for students to deepen their understanding of how diagrams and multiplication equations can represent “$$n$$ times as many”. Students explain how the diagrams and equations represent the situation. In order to match situations, diagrams, and equations, students reason abstractly and quantitatively (MP2).

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances: Conversing, Representing, Speaking
Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to look and listen for examples of multiplicative comparison in their own lives. Encourage them to share these throughout the unit.
Supports accessibility for: Conceptual, Processing, Language, Attention

### Launch

• Groups of 2
• “Take turns reading a description and finding a diagram and an equation that also represent the situation. Explain your reasoning to your partner.”

### Activity

• 5–7 minutes: partner work time
• Monitor for students who connect the factors and product of the equation to the situation and diagram
• If students finish early, give them blank index cards. Ask them to make several sets of matching representations, shuffle the cards, and trade them with another group that is also creating their own representations.

### Student Facing

Here are four sets of descriptions, diagrams, and equations that compare pairs of quantities.

Match each description to a diagram and an equation that represent the same situation. Be prepared to explain your reasoning.

Record your matches here:

Set 1: _____, _____, _____

Set 2: _____, _____, _____

Set 3: _____, _____, _____

Set 4: _____, _____, _____

### Advancing Student Thinking

Students may consider the total amount represented on a card rather than making comparisons. For example, on card K, they might see 4 groups of 3 in the two quantities combined. Consider asking:

• “How would you compare the two quantities shown on the card?”
• “Where might you see two quantities being compared in the equation on card E?”

### Activity Synthesis

• Select students to share their matches.
• Record student explanations to show how they connected the diagrams and equations (display or draw the diagrams and equations and annotate).

## Lesson Synthesis

### Lesson Synthesis

“Today we looked at a new way to use multiplication equations. Multiplication equations can describe equal groups, but they also represent multiplicative comparison.”

Display a student's representation of Kiran's cubes and Jada's cubes from the first activity.

“Explain how you see $$4 \times 2 = 8$$ in this diagram.” (There are 4 groups of 2 cubes each or Jada has 4 times as many as Kiran does)

“In this case, the value of the product is 8. How is one of the factors being compared to 8 in this diagram?”

If not mentioned by students, highlight that:

• “In the first case, the multiplication equation represents equal groups of objects.”
• “In the second case, the multiplication equation represents multiplicative comparison. It allows us to see how many times as many objects one person has compared to another person.”