# Lesson 5

More Multiples

## Warm-up: Estimation Exploration: Banquet Seating (10 minutes)

### Narrative

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it. For example, students may reason that there are 10 chairs at each table and then make an estimate for the number of tables (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems, and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).

### Launch

• Groups of 2
• Display the image.

### Activity

• “What is an estimate that’s too high? Too low? About right?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.

### Student Facing

About how many chairs are in the room?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Activity Synthesis

• “What part of the picture did you use to make your estimate?” (The number of seats at one table. The number of rows and tables in each row.)
• “Which estimates would have been unreasonable?” (353, 22, or 198, and other numbers not ending in a zero. Because there are ten chairs at a table, the total number has to be a multiple of 10 and end in a 0.)

## Activity 1: Choose the Right Tables (20 minutes)

### Narrative

The purpose of this activity is to find multiples of two different numbers in context. Students decide possible table sizes for a party based on whether or not a given number of people is a multiple of 6, 8, both, or neither. In situations where a given number is not a multiple of 6 or 8, they reason about what it means in context (MP2).

Focus the synthesis on how the number of seats at the two table sizes relate to multiplication and what it means when the number of people is not a multiple of the number of seats at a table.

MLR2 Collect and Display. Collect the language students use as they discuss the different options for tables. Display words and phrases such as: no empty seats, 6 times a number, 8 times a number, multiplied by, multiples of 6 or 8, and $$\underline{\hspace{.7cm}} \times 6$$ added to $$\underline{\hspace{.7cm}} \times 8$$ for combinations of table sizes. During the synthesis, solicit ways to update the display: “What are some other words or phrases we should include?” etc. Invite students to borrow language from the display as needed.
Representation: Access for Perception. Provide access to materials to help students represent the problem, such as inch tiles, connecting cubes, a blank or partially completed set diagram, and a blank or partially completed tape diagram. Invite students to identify correspondences between these representations, multiplication equations, and the term “multiple” throughout the lesson.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Groups of 2
• “Work with your partner to choose a type of table to seat different groups of students.”

### Activity

• 8–10 minutes: partner work time
• Monitor for students who:
• draw pictures to highlight why the number of students at tables of 6 is a multiple of 6 and that at tables of 8 is a multiple of 8
• list the multiples of 6 and multiples of 8 to determine which table to use
• use multiplication facts to explain their reasoning

### Student Facing

Students are preparing for a party. The school has tables where 6 people can sit and tables where 8 people can sit.

The students can only choose one type of table and they want to avoid having empty seats.

1. Jada’s class has 18 students. Which tables would you choose for Jada’s class? Explain or show your reasoning.
2. Noah’s class has 30 students. Which tables would you choose for Noah’s class? Explain or show your reasoning.
3. Which tables would you choose for Noah’s and Jada’s classes together? Can you find more than one option? Explain or show your reasoning.
4. If you also want places for Noah’s teacher and Jada’s teacher to sit, which tables would you choose? Explain or show your reasoning.

### Activity Synthesis

• Invite students who used drawings, the idea of multiples, or multiplication facts to share while discussing the following questions.
• “Can Noah’s class fit exactly in tables for 6? How do you know?” (Yes. $$30 = 5 \times 6$$, so Noah’s class will fill 5 tables exactly.)
• “Can Noah’s class fit exactly in tables for 8? How do you know?” (No, 30 is not a multiple of 8. They would need at least 4 tables but that’s too many seats, so 2 of the seats would be empty.)
• “How do multiplication facts or the idea of multiples help us think about this problem?” (We know that 24 and 32 are products of 8 and another number, but 30 is not.)
• “For the problem with both classes and the teachers, could they all fit at tables of 6? Why or why not?” (Yes, they could get 9 tables of 6, but there will be 4 empty seats. If they don’t want empty seats, tables of 6 won’t work as 50 is not a multiple of 6.)
• “Could they all fit at tables of 8? Why or why not?” (Yes, they could get 7 tables of 8, but there will be 6 empty seats. If they don't want empty seats, tables of 8 won't work as 50 is not a multiple of 8.)

## Activity 2: Hot Dogs and Buns (15 minutes)

### Narrative

In this activity, students solve problems that involve finding multiples that are shared by two different numbers: the number of hot dogs in a package and the number of hot dog buns in a package. They reason about how many of each package to get to make a certain number of servings of hot dogs. As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).

To solve the problems, students may decontextualize the situation and reason about factors and multiples, and then recontextualize the solutions in terms of servings of hot dogs. As they do so, they practice reasoning quantitatively and abstractly (MP2). During the synthesis, analyze different solutions and discuss why numbers that are multiples both of 8 and 10 are useful in this situation.

### Launch

• Groups of 2–4
• Read the opening paragraph as a class.
• 1 minute: quiet think time
• Share responses.
• “Let’s solve problems about hot dogs and hot dog buns, which come in different-size packages.”
• “For each problem, look for possible numbers of packages that are needed to serve hot dogs at a picnic.”
• “Take a minute to read the problems and think about how you would solve them.”
• 1 minute: quiet think time

### Activity

• “Now work with your group to solve these problems.”
• 8–10 minutes: group work time
• Monitor for students who:
• use the term “multiple” in their explanations
• try different possibilities for the number of packages to buy
• try to find numbers of packages that would give the exact (or close to the exact) numbers of servings

### Student Facing

Each package of hot dogs has 10 hot dogs. Each package of hot dog buns has 8 buns.

1. Lin expects to need 50 hot dogs for a class picnic.

1. How many packages of hot dogs should Lin get? Explain or show your reasoning.
2. Can Lin get exactly 50 hot dog buns? How many packages of hot dog buns should Lin get? Explain or show your reasoning.
2. Diego expects to need 72 hot dogs for a class picnic.

1. How many packages of hot dogs should Diego get? Explain or show your reasoning.
2. How many packages of hot dog buns should Diego get? Explain or show your reasoning.
3. Is it possible to buy exactly the same number of hot dogs and buns? If you think so, what would that number be? If not, explain your reasoning.

### Student Response

If students are using guess and check, consider asking: “How are you choosing which numbers to try?” or “How can thinking about multiples help you solve the problem?”

### Activity Synthesis

• Invite students to share their responses to the problem about 72 hot dogs, including those who get 70 hot dogs and 72 buns or 80 hot dogs and 80 buns. Encourage students to highlight how they are using multiples in their reasoning.
• “Why are there different answers to this question?” (The numbers do not work out exactly to get 72 hot dogs and 72 buns, so we need to look at different alternatives.)
• “Is it possible to buy exactly the same number of hot dogs and buns? If so, how many of each?” (Yes, 40, 80.)
• “Why is it possible to buy exactly the same number of hot dogs and buns when the packaging is different?” (Some multiples of 10 are also multiples of 8, such as 40 and 80.)

## Lesson Synthesis

### Lesson Synthesis

Math Community

Create a T-chart titled “Ways to Disagree” and two columns labeled “sounds like” and “looks like”.

Review the list of what you are doing and what students are doing in math class.

“Is making mistakes a part of math class and doing math together?” (Yes.)

“What does it feel like when someone points out a mistake or disagrees with your idea?” (It’s embarrassing. It hurts my feelings.)

“What might it look like or sound like to challenge each other’s ideas without hurting feelings?” (Ask questions to better understand, say I disagree, ask “can you please explain what you mean by…?”)