# Lesson 14

Situations Involving Factors and Multiples

## Warm-up: Number Talk: Dividing by 7 (10 minutes)

### Narrative

This Number Talk encourages students to compose or decompose multiples of 7 and to rely on properties of operations to mentally solve problems. The ability to compose and decompose numbers will be helpful when students divide multi-digit numbers. It also promotes the reasoning that is useful when finding multiples of a number, or when deciding if a number is a multiple of another number.

### 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

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$21 \div 7$$
• $$35 \div 7$$
• $$140 \div 7$$
• $$196 \div 7$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What do the expressions have in common?” (They all involve division by 7. The dividends are all multiples of 7. The results have no remainders.)
• “How did the first three expressions help us find the value of the last expression?”
• Consider asking:
• “Who can restate _______ 's reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to____’s strategy?”

## Activity 1: Write Multiples (20 minutes)

### Narrative

This activity prompts students to use the relationship between multiplication and division and their understanding of factors and multiples to solve problems about an unknown factor (MP7). Students recognize such problems as division situations. Here the dividends are three-digit numbers and the divisors are one-digit numbers. Students use the context of factors and multiples to interpret division that results in a remainder.

One approach for solving these problems is by decomposing the dividend into familiar multiples of 10 and then dividing the remaining number (which is now smaller) by the divisor. Another is to find increasingly greater multiples of the divisor until reaching the dividend. Both are productive and appropriate. In the synthesis, help students see the connections between the two paths.

MLR8 Discussion Supports. Synthesis: Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Engagement: Develop Effort and Persistence. If students don’t recognize this as a division situation at first, they may do so when presented with a more accessible value. Consider offering this situation first: “Han starts writing multiples of a number. When he reaches 12, he has written 3 numbers.” Invite students to identify what number Han is writing multiples of (4), and how they know. Then ask how they might apply that reasoning to the task as presented here.
Supports accessibility for: Conceptual Processing

### Launch

• “Who can remind the class of the meaning of ‘multiple?’”

### Activity

• 4–5 minutes: quiet work time for the first set of questions
• Monitor for students who:
• decompose 104 into a multiple of 8 and another number
• compose 104 from increasingly larger multiples of 8
• use multiplication or division equations to show their reasoning
• Pause for a discussion before the second set of questions.
• Select students who use different decomposition strategies to share responses. Record and display their reasoning.
• “Let’s revisit each question we just answered. What equations could we write to represent them?”
• Display equations and highlight their connection to the questions:
1. $$8 \times {?} = 104$$ or $$104 \div 8 = {?}$$
2. $$15 \times 13 = {?}$$ or $$13 \times 15 = {?}$$
3. $${?} \times 13 = 286$$ or $$286 \div 13 = {?}$$
• 4–5 minutes: quiet work time for the second set of questions

### Student Facing

1. Han starts writing multiples of a number. When he reaches 104, he has written 8 numbers.

For each of the following questions, show your reasoning.

1. What number is Han writing multiples of?
2. What is the 15th multiple of this number?
3. Han gets to 286. How many numbers has he written at that point?
2. Kiran wants to know how many multiples of 7 are between 0 and 150.

1. He thinks he can use division to find out. Do you agree? Explain your reasoning.
2. How many multiples will he find? Show your reasoning.
3. Is 150 a multiple of 7? Show how you know.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share responses for the last set of questions.
• Highlight that to divide a number by a smaller number—say, divide 150 by 7, we can:
• Use familiar multiples or multiplication facts to help us. For example, if we know $$(20 \times 7) + (1 \times 7) = 147$$, we know the result is 21 with a remainder of 3, or $$150 = 21 \times 7 + 3$$.
• Think of the dividend in smaller chunks. For example: We can see the 150 as $$140 + 10$$ and divide each 140 and 10 by 7 separately, which gives $$20 + 1$$ with a remainder of 3.

## Activity 2: Jada’s Mystery Number (15 minutes)

### Narrative

In this activity, students continue to use the relationship between multiplication and division to reason about situations that involve division. Students divide three-digit numbers by single-digit divisors and find results with and without a remainder.

• Groups of 2

### Activity

• 5 minutes: independent work time on the first question
• 2 minutes: partner discussion
• 23 minutes: partner work time on the second question
• Monitor for students who clearly show how they use partial products or partial quotients to answer the questions.

### Student Facing

Jada is writing multiples of a mystery number. After writing a bunch of numbers, she writes out 126.

• Mai says 6 is the mystery number.
• Priya says 8 is the mystery number.
• Andre says 9 could be the mystery number.
1. Which student do you agree with? Show your reasoning and include equations.
2. Jada gives one more clue: “If I keep writing multiples, I’ll get to 153.”

What is the mystery number? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How could we use division to help us find out the mystery number?” (If the result has a remainder, then the divisor could not be the mystery number. 153 divided by 6 has 3 as a remainder. 153 divided by 9 has no remainder, so 9 is the mystery number.)

## Lesson Synthesis

### Lesson Synthesis

“Today we tried to find out if a number is a multiple or a factor of another number. For instance: Is 267 a multiple of 8?”

“Is this question a division problem?” (Yes)

“Why?” (We’re looking for how many 8s are in 267.)

“What are we dividing?” (267 by 8)

“One way to answer the question is by using familiar multiplication facts or by finding partial products. How would you start?” (One way is to start with 10 x 8 or its multiples, build the products up to 267 or close to it, and then try smaller multiples of 8.

$$10 \times 8 = 80$$
$$20 \times 8 = 160$$
$$30 \times 8 = 240$$
$$3 \times 8 = 24$$
$$33 \times 8 = 264$$

264 is 3 away from 267, not enough to make another 8, so 267 is not a multiple of 8.)

“Why might it be helpful to start with multiples of 10?” (They’re easy to find and easy to add.)

“Can we use division to answer the question?” (We can start a number close to 267 that is a multiple of 8, divide it by 8, see what is left, and find a multiple of 8 that is close to that number. For example:

• $$160 \div 8 = 20$$. After taking 160 away, there’s 107 left.
• $$80 \div 8 = 10$$. After taking 80 away, there’s 27 left.
• $$24 \div 8 = 3$$. After taking 24 away, there’s 3 left, which is not enough to make 8.)

“How are the two approaches alike?” (They involve using smaller multiples of a number to see if a larger number is a multiple of that number.)

## Cool-down: Reaching 161 with Multiples (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.