# Lesson 17

Fraction Multiplication and Division Situations

## Warm-up: Number Talk: Multiply and Divide (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying and dividing fractions. These understandings help students develop fluency and will be helpful later in this lesson when students solve problems about multiplying and dividing fractions.

### 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.

• $$5 \div \frac{1}{6}$$
• $$5 \times \frac{1}{6}$$
• $$\frac{1}{5} \div 6$$
• $$\frac{1}{5} \times \frac{1}{6}$$

### Activity Synthesis

• “How are the last two problems related?” (Dividing by 6 and multiplying by $$\frac{1}{6}$$ give the same result.)

## Activity 1: Info Gap: Tiles (20 minutes)

### Narrative

This Info Gap activity gives students an opportunity to determine and request the information needed to solve multi-step problems involving multiplication and division of unit fractions. In both cases, the student with the problem card needs to find out the side lengths of the area being covered and the size of the tiles and from there they can figure out how many tiles are needed. The numbers in the problems are chosen so that students can draw diagrams or perform arithmetic directly with the numbers.

In this Info Gap activity, the first problem encourages students to think about multiplying the given fractions. The second problem involves a given area and a missing side length, which may encourage students to represent and solve the problem with a missing factor equation.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

### Required Materials

Materials to Copy

• Info Gap: Tiles

### Required Preparation

• Create a set of cards from the blackline master for each group of 2.

• Groups of 2

### Activity

• Explain the Info Gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
• 15 minutes: partner work time
• After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
• Monitor for students who:
• Draw pictures to arrange the tiles and see how they can fit the tiles together to cover the area.
• Use division and multiplication in their calculations.

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think about what information you need to answer the question.
2. Ask your partner for the specific information that you need.
3. Explain to your partner how you are using the information to solve the problem.
4. Solve the problem and explain your reasoning to your partner.

If your teacher gives you the data card:

1. Silently read the information on your card.
2. Ask your partner, “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask, “Why do you need that information?”
4. After your partner solves the problem, ask them to explain their reasoning and listen to their explanation.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

### Activity Synthesis

• Invite students to share their responses to the problem about tiling the floor.
• “How did you know whether to use multiplication or division to solve the problem?” (I needed to use multiplication to find the area of the floor. I also used multiplication to figure out how many tiles I needed.)
• “How did you solve the problem?” (I found out how many tiles I needed for one square foot by drawing a picture and then multiplied that by the number of square feet in the floor.)
• Invite students to share their responses to the problem about the tiles for the wall.
• “How did you know whether to use multiplication or division?” (I used multiplication to figure out how many square feet of the wall I could cover with each box and how many square feet were in the wall.)
• “How did you solve the problem?” (I used division to estimate how many boxes of tiles I would need.)

## Activity 2: Multiplication or Division (15 minutes)

### Narrative

The goal of this activity is to solve a variety of different problems, all of which can be solved by multiplying two fractions though the first problem may also be represented as division of a unit fraction by a whole number. The numbers are complex, making drawings an unlikely strategy to solve the problems. This encourages students to use their understanding of how to multiply fractions or divide with a whole number and a unit fraction. The synthesis focuses on why students chose multiplication or division to solve the problems.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 2 out of the 3 to complete within the time allowed for the activity.
Supports accessibility for: Organization, Attention

• Groups of 2

### Activity

• 6 minutes: independent work time
• 4 minutes: partner discussion
• Monitor for students who:
• represent the situations with equations
• explain how their equations represent each situation

### Student Facing

Solve each problem. Explain or show your reasoning.
1. If 11 grains of rice weigh $$\frac{1}{3}$$ gram, how much does each grain of rice weigh?
2. Mai’s road is $$\frac {9}{10}$$ mile long. She ran $$\frac {3}{4}$$ of the length of her road. How far did she run?
3. If each tennis ball weighs $$2\frac{1}{16}$$ ounces, how much do 9 tennis balls weigh?

### Advancing Student Thinking

If students are not familiar with any of the contexts in the situations consider showing them pictures of rice or tennis balls or asking them to describe a landmark that is approximately a mile from the school.

### Activity Synthesis

• Ask previously selected students to share their solutions.
• Display:
$$\frac{1}{3} \div 11 = \underline{\hspace{1 cm}}$$
• "How does the equation represent the first situation?" (The 11 grains of rice together weigh $$\frac{1}{3}$$ gram so I need to divide $$\frac{1}{3}$$ by 11 to find out how many grams each grain weighs.)
• Dislay: $$\frac {3}{4} \times \frac {9}{10} = \underline{\hspace{1 cm}}$$
• “How does the equation represent Mai's run?” (The $$\frac{9}{10}$$ is the full length of the road but she only ran $$\frac{3}{4}$$ of it so I multiply $$\frac{9}{10}$$ by $$\frac{3}{4}$$.)

## Lesson Synthesis

### Lesson Synthesis

Display problems from Number Talk:

$$5 \div \frac{1}{6} = 30$$

$$5 \times \frac{1}{6} = \frac {5}{6}$$

$$\frac{1}{5} \div 6 = \frac {1}{30}$$

$$\frac{1}{5} \times \frac{1}{6} = \frac {1}{30}$$

“What relationships do you see between these problems?” (There is a 5 and a 6 in each expression. Some of the equations have multiplication and some have division. Some have a whole number value and some have a fraction.)

“Describe one relationship to your partner.”

“What do we know about the relationship between multiplication and division?” (I can sometimes find the value of a division expression using multiplication.)

Record responses for all to see.