# Lesson 15

More Factors, More Problems

## Warm-up: Number Talk: Tens (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplication by 10. These understandings help students develop fluency and will be helpful later in this lesson when students need to be able to represent and solve a problem involving groups of 10.

When students reason why as one factor increases by 1, the product increases by 10, they look for and express the regularity they notice in the expressions (MP8).

### 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.

• $$1\times10$$
• $$2\times10$$
• $$3\times10$$
• $$4\times10$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What pattern do we see as we look at all of the problems? Why is that happening?” (The factor that isn’t 10 goes up by 1 each time. The products increase by 10 because I have one more group of 10.)
• “Did anyone use a different strategy”?
• “Did anyone have the same strategy but would explain it differently?”

## Activity 1: Represent Situations with Equations (15 minutes)

### Narrative

The purpose of this activity is for students to represent a situation with a multiplication equation including a symbol for the unknown, and find the number that makes the equation true. Students are able to use an earlier representation to help them solve the problem, however some students may just write the equation and skip-count to find the product. Either is okay. In the synthesis, share different ways students represented the problem beyond the equation. If students used repeated addition, avoid saying ‘multiplication is repeated addition’ because while repeated addition is one way to find the product, it is not the meaning of multiplication.

To add movement to this activity, students can work in groups of 4 to make a poster for one of the problems. After each group is done, they can do a gallery walk to look for things that are the same or different in the posters.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Memory

### Launch

• Groups of 2
• “We are going to solve some problems about equal groups that you may see when you are making or eating a meal.”
• “What are some equal groups that you might see when making or eating a meal?”
• Share responses.
• “Think about how you could represent these problems in a way that could help you write an equation with an unknown number for each problem.”
• 1 minute: quiet think time
• 1 minute: partner discussion

### Activity

• “Now, independently work on these problems.”
• 5–7 minutes: independent work time
• As you circulate, consider asking:
• “How could you represent this situation?”
• “What information is missing from the situation?”

### Student Facing

For each problem:

• Write an equation with a symbol for the unknown to represent the situation.
• Find the number that makes the equation true. Show your reasoning.
1. There are 15 plates. Han placed 5 plates on each table. How many tables have plates on them?

1. equation:
2. solution:
2. Lin made 6 sandwiches. She used 2 slices of bread for each sandwich. How many pieces of bread did she use?

1. equation:
2. solution:
3. Han has 60 ice cubes. The ice cubes are in trays of 10. How many trays of ice cubes does Han have?

1. equation:
2. solution:

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display samples of student work for each problem next to each other, including a sample of a drawing of equal groups and a sample of a tape diagram.
• “Where do we see the parts of the problem in the drawing and the diagram?” (The number of objects in each group are the dots in the drawing, but the number is written in each part of the diagram.)
• “How did you use the factors in each equation to find the product?” (The factors told me how many groups there were and how many were in each group.)
• “How are drawings and diagrams useful for finding the solution to the problem?” (You can count the dots in the drawing. The diagram can be used to count by 10.)

## Activity 2: Multiplication Mashup (20 minutes)

### Narrative

The purpose of this activity is for students to practice solving multiplication problems in which the unknown amount can be the number of groups, the number in each group, or the total. The first three problems have the unknown in each of those locations. The sequence of these problems, the context, and the use of the same factors and product encourages students to use a known fact to find the unknown factor in the “how many in each group” problem. Students will make the connection between this problem type and division in a future unit. Students are able to choose the representation they use to represent and solve the problems.

MLR8 Discussion Supports. Monitor and clarify any questions about the context. As students look over the problems, ask, “Are there any words that are unfamiliar or that you have questions about?”

### Launch

• Groups of 2
• “Take a minute to look over these problems. What representations or strategies might be helpful to you as you solve these problems?”
• 1 minute: quiet think time
• Share and record responses.

### Activity

• “Work with your partner to solve each problem.”
• 8-10 minutes: partner work time
• Circulate and consider the following questions to focus students on the structure of the situations:
• “What information is missing in the situation?”
• “How could you represent this situation?”

### Student Facing

Solve each problem. Explain or show your reasoning.

1. Clare has 16 socks. She puts them in piles of 2. How many piles can she make?
2. Diego has 8 piles of socks. Each pile of socks has 2 socks. How many socks does Diego have?
3. Andre has 16 socks. He puts them in 8 groups that are the same size. How many socks are in each group?
4. The store has 9 boxes. Each box has 5 shirts. How many shirts are there?
5. A store has 80 sweaters. There are 8 sweaters in each pile on a shelf. How many piles of sweaters are on the shelf?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Share student work for each problem and ask students to explain their reasoning. Be sure to share a variety of strategies and representations.
• As students share, consider asking the class:
• “Why does this strategy make sense?”
• “Why does this representation make sense?”
• “Did anyone solve this problem in a different way?”
• “What do you notice is the same about the representations and strategies that we are using to solve these problems?”

## Lesson Synthesis

### Lesson Synthesis

“Today we solved multiplication problems using any strategy or representation that we wanted.”

“What strategy or representation do you find most helpful when you are solving these types of problems? Why?” (I like to draw equal groups so I can see how many groups there are and how many are in each group. I think a diagram is nice to draw because I don’t have to draw all the things, but I can still see the groups. I like to use an equation so I can see where the unknown number is.)

“What are the most important things to remember when you are solving multiplication problems?” (There are always groups that are the same size. You could be looking for the number of groups, how many things are in each group, or the total number of things in all the groups.)

## Cool-down: Solve the Problem (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

In this section, we learned about equal groups. We created drawings and diagrams to represent situations that involve equal groups.

situation

Diego has 8 piles of socks. Each pile of socks has 2 socks.

drawing

diagram

We wrote multiplication expressions and equations to represent equal groups.

expression

$$8\times2$$

equation

$$8\times2=16$$

We learned that the numbers that are multiplied are called factors and the number that is the result of multiplying is called a product. In the equation $$8\times2=16$$, the numbers 8 and 2 are the factors and 16 is the product.