# Lesson 3

Solve Multiplicative Comparison Problems

## Warm-up: Number Talk: Find the Unknown Factor (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for finding a unknown factor and relating multiplication and division. These understandings help students develop fluency and will be helpful later in this lesson when students represent and solve multiplicative comparison problems with unknown factors.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each unknown mentally.

• $$8 \times {?} = 16$$
• $$3 \times {?} = 24$$
• $${?} \times 8 = 32$$
• $$40 \div 8 = {?}$$

### Activity Synthesis

• “Does $$40 \div 8$$ belong in this number talk? Why or why not?” (It does belong because division is like finding an unknown factor.)

## Activity 1: A Book Drive (20 minutes)

### Narrative

In this activity, students are provided a discrete tape diagram to represent the first problem in which the multiplier (the quantity indicating $$n$$ times as many) is unknown. They also rely on what they know about the relationship between multiplication and division to represent and solve each problem.

When students create their representations for the books, whether a diagram or an equation, they reason abstractly and quantitatively (MP2).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the strategies have in common?”, “How were they different?”, and “Why did the different approaches lead to the same outcome?”

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “What do you know about a book drive? Have you participated in a book drive or book exchange?”
• “Sometimes schools organize a book drive where students donate books they are done reading or no longer need so others can enjoy them.”

### Activity

• 3 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who represent Noah and Priya's books with:
• 3 books at a time for Noah and stop at 21
• all of Noah’s books and circle groups of 3
• a tape diagram
• a multiplication equation with a symbol for unknown
• a division equation

### Student Facing

This diagram shows the books Lin and Diego donated for the school book drive.

1. Lin donated 16 books. Diego donated 4 books. How many times as many books did Lin donate as Diego did? Explain or show your reasoning. Use the diagram if it is helpful.
2. Priya donated 3 books. Noah donated 21 books. How many times as many books did Noah donate as Priya did?

Draw a diagram to show your reasoning.

3. Mai made a pile of 27 donated books. Tyler made his own pile of 3 books. How many times as many books did Mai stack as Tyler did?

### Student Response

Students may lose track of the group being multiplied and the factor serving as the multiplier and begin to represent, for example, 16 times as many as 4 books. Consider asking:

• “What is being compared in this problem?”
• “How many times as many books are being described in this problem?”

### Activity Synthesis

• Invite selected students to share their representation of Priya and Noah's books in the order described above.
• If no students created an equation, ask, “How could we represent Priya and Noah’s books with an equation with a symbol for the unknown?” ($${?} \times 3 = 21$$ or $$7 \times {?} = 21$$)
• “How did you use multiplication to solve the problem?” (I knew $$7 \times 3 = 21$$. So, Noah has 7 times as much.)
• “How did you use division to reason about the problem?” (I knew 21 divided by 3 is 7.)

## Activity 2: Represent a Missing Amount (15 minutes)

### Narrative

The purpose of this activity is for students to make sense of and represent multiplicative comparison problems in which a factor is unknown. Students use the relationship between multiplication and division to write equations to represent multiplicative comparisons. These problems have larger numbers than in previous lessons in order to elicit the need for using more abstract diagrams, which are the focus of upcoming lessons.

When students analyze Han's and Tyler's claims they construct viable arguments (MP3).

Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from starting with more accessible questions. For example, ask, “Who donates more books?”
Supports accessibility for: Conceptual Processing, Organization

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Take a moment to read the first problem to yourself.”
• “Explain to a partner what you are asked to do or find out.”

### Activity

• 7 minutes: independent work time
• Monitor for students who:
• draw a tape diagram with 48 rectangles and then guess and check by circling equal groups until finding an amount with no leftover.
• attempt to draw out all books and revise to create a more abstract diagram.
• write multiplication and division equations with a symbol for the unknown value.
• 2 minutes: partner discussion

### Student Facing

1. Clare donated 48 books. Clare donated 6 times as many books as Andre.

1. Draw a diagram to represent the situation.
2. How many books did Andre donate? Explain your reasoning.
2. Han says he can figure out the number of books Andre donated using division.

Tyler says we have to use multiplication because it says “times as many”.

1. Do you agree with Han or Tyler? Explain your reasoning.
2. Write an equation to represent Tyler’s thinking.
3. Write an equation to represent Han’s thinking.
3. Elena donated 9 times as many books as Diego. Elena donated 81 books.

Use multiplication or division to find the number of books Diego donated.

### Student Response

If students write equations that represent a different situation than presented, consider asking:

• “How does your equation show Clare’s (or Andre’s) books?”
• “Can a division equation represent each situation?”

### Activity Synthesis

• Focus the discussion on why Han is correct that we can use division. (Han is correct because when a factor is missing, we can use division to find out what was being multiplied.)
• “Why is it possible to use a multiplication or a division equation to represent the same situation?” (Because a multiplication equation with an unknown factor is the same as a division equation with an unknown quotient.)

## Lesson Synthesis

### Lesson Synthesis

“In today’s lesson you solved multiplicative comparison problems in which different pieces of information were missing.”

Display:

$$32 = {?} \times 8$$

32 is _____ times as much as 8.

32 is 8 times as much as _____.

$${?} = 7 \times 5$$

_____ is 7 times as much as 5.

“How would you complete the equations and comparison statements to make them true? Explain how you know.” (In the first set, 4, because 8 four times is 32, and in the second set, 35, because 7 times 5 is 35.)