# Lesson 4

Dive Back Into Division

## Warm-up: Notice and Wonder: Blank Spaces (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the relationship between multiplication and division, which will be useful when students identify missing dividends and divisors in a later activity. While students may notice and wonder many things about this equation, possible numbers to fill in the blanks are the important discussion points.

### Launch

• Groups of 2
• Display the equation.
• “What do you notice? What do you wonder?”

### Activity

• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

$$\underline{\hspace{1 cm}} \div \underline{\hspace{1 cm}} = 136$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What numbers could go in the blanks? How do you know?” (I could put in 136 and 1 or 272 and 2. The first number has to be 136 times the second number.)

## Activity 1: Reasonable Estimates (15 minutes)

### Narrative

The purpose of this activity is for students to estimate quotients of multi-digit numbers and reason about multiplication expressions that are helpful when dividing. In previous units, students learned a partial quotients algorithm to divide multi-digit whole numbers. This lesson prepares them to revisit this algorithm in the next lesson.

• Groups of 2

### Activity

• 5–10 minutes: independent work time
• 5 minutes: partner discussion

### Student Facing

1. Circle the most reasonable estimate. Show your reasoning.

1. $$364 \div 13$$

20

30

40

2. $$540 \div 12$$

40

50

60

3. $$1,\!008 \div 14$$

70

80

90

2. Find the value of each quotient.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How did multiplication help you with your estimates?” (Multiplying by tens is easy to do in my head so I could find the product that was closest to the dividend.)
• “How did your estimates help you find the quotient?” (My estimates gave me a good idea what multiple of the divisor to subtract to begin my calculation. They helped me check that my answer was reasonable.)

## Activity 2: Missing Dividends and Divisors (20 minutes)

### Narrative

The purpose of this activity is for students to use the relationship between multiplication and division to determine possible dividends and divisors that have a given value for their quotient (MP7). Monitor for students who:
• multiply the quotient by friendly numbers such as 1, 2, or 5.
• multiply the quotient by powers of ten and use place value understanding.
• can explain the relationship between multiplication and division.

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy to determine which whole number makes the equation true before they begin. Students can speak quietly to themselves or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

• Groups of 2

### Activity

• 3–5 minutes: independent work time
• “Share your responses with your partner. If you have any equations that are the same, write a new equation that is different. Together, work to find 10 different equations.”
• 3–5 minutes: partner discussion
• Give each group tools for creating a visual display.

MLR7 Compare and Connect

• “Work with your partner to create a visual display that shows your thinking about problems 1 and 2.”
• 2–5 minutes: independent or group work
• 3–5 minutes: gallery walk

### Student Facing

1. Write different numbers in the blanks that make the equations true.

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 700$$

$$\underline{\hspace{1 cm}}\div\underline{\hspace{1 cm}}= 78$$

2. What strategy did you use to choose numbers to write in the blanks?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display: $$\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 700$$
• Invite students to share the numbers they used to make the equation true.
• “How did you find numbers that make the equation true?” (I multiplied 700 by different numbers. I chose simple quotients, like dividing by 1 or 2 or 10.)
• Display: $$\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 78$$
• “Did the same strategies work to find numbers that make this equation true?” (Yes, but 78 is a more difficult number to multiply by in my head. So, 1 worked and 2 and 10, but it was harder to find a variety of numbers.)

## Lesson Synthesis

### Lesson Synthesis

Display or write for all to see.

$$\underline{\hspace{1cm}} \div \underline{\hspace{1cm}} = 25$$

“How can you use multiplication to find numbers that make the equation true?” (I know the number in the first blank has to be 25 times the number in the second blank.)

“What are some examples of numbers that make the equation true?” (25 and 1, 50 and 2, 100 and 4, 250 and 10)

“How did you choose the numbers?” (I looked for numbers that are easy to calculate in my head. I know lots of multiples of 25 that I can find in my head.)

## Cool-down: Estimate and Evaluate (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.