# Lesson 14

Represent Division of Whole Numbers by Unit Fractions

## Warm-up: Number Talk: Increasing Quotients (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for dividing a whole number by a unit fraction. These understandings will be helpful later in this lesson when students match situations to equations and solve the equations.

### 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

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$6 \div 1$$
• $$6 \div \frac {1}{2}$$
• $$6 \div \frac {1}{3}$$
• $$6 \div \frac {1}{6}$$

### Activity Synthesis

• “Why is the quotient getting larger with each problem?” (Because we are dividing the same number into smaller sized groups so there are going to be more groups.)

## Activity 1: Notice Patterns (20 minutes)

### Narrative

The purpose of this activity is for students to find quotients of a whole number by a unit fraction and observe patterns in how the size of the numerator and denominator influence the size of the quotient. Whereas students were provided a tape diagram in the previous lesson, here they may draw a diagram but they may also reason about the size of the quotients in other ways. When students notice a pattern or repetitive action in computation, they are looking for and expressing regularity in repeated reasoning (MP8).

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

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding the value of each statement. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 2
• “Decide with your partner who will work on set A and who will work on set B.”

### Activity

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion

### Student Facing

Set A: Find the value that makes each equation true. Draw a diagram if it is helpful. What patterns do you notice?

• $$3 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$4 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$5 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$6 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$

Set B: Find the value that makes each equation true. Draw a diagram if it is helpful. What patterns do you notice?

• $$3 \div \frac {1}{2} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{3} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{5} = \underline{\hspace{1 cm}}$$

What is the same about problem set A and B? What is different?

### Student Response

If students don’t solve the equations correctly, prompt them to draw diagrams to represent each equation and ask: “How does each diagram represent the equation?”

### Activity Synthesis

MLR7 Compare and Connect
• “Work with your partner to create a visual display that shows your thinking about how the problem sets are the same and different. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
• 2–5 minutes: independent or group work
• 5–7 minutes: gallery walk
• “What is the same and what is different between the two sets of problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “What is changing in each problem set?” (The size of the number being divided is changing in set A and the size of the piece the number is being divided into is changing in problem set B.)
• “If the patterns continued, what would be the next equation in each set?”
• “Why is the quotient getting larger in both sets of problems?” (In set A, it is getting larger because you have 4 more $$\frac {1}{4}$$ size pieces in each additional whole. In set B, it is getting larger because the size of the piece is getting smaller, so you have more of them.)

## Activity 2: Match the Situation to the Expression (15 minutes)

### Narrative

The purpose of this activity is for students to match situations to expressions and then find the value of the expressions (MP2). Students see expressions that show both quotients of a whole number by a fraction and quotients of a fraction by a whole number. They need to think carefully about the situations to make sure the expression they choose matches the situation (MP2).

### Launch

• Groups of 2
• Display the image from student book:
• “What do you notice? What do you wonder?” (They look like seeds. How many are in the bowls? Why are there 2 bowls?)
• “The small bowl is filled with $$\frac {1}{4}$$ cup of kernels. That is one serving of popcorn kernels. What does ‘one serving’ mean?” (It is the amount you are supposed to eat at one time.)
• “About how many servings are in the big bowl?” (Answers vary. Sample responses range from 10 to 15.)

### Activity

• 1–2 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who:
• draw diagrams to represent the situations
• describe the relationship between the dividend and divisor using language such as “groups of”
• use the value of the expression to describe why the expression matches the situation

### Student Facing

1. Match each problem to an expression that represents the problem. Some expressions will not have a match. Be prepared to explain your reasoning.
1. One serving of popcorn is $$\frac {1}{4}$$ cup of kernels. There are 3 cups of kernels in the bowl. How many servings are in the bowl?

2. One serving of orange juice is $$\frac {1}{4}$$ liter. The container of juice holds 2 liters. How many servings are in the container?
3. One serving of granola is $$\frac {1}{2}$$ cup. The bag of granola holds 5 cups. How many servings are in the bag?

$$\frac {1}{4} \div 3$$

$$\frac {1}{2} \div 5$$

$$3 \div \frac {1}{4}$$

$$\frac {1}{4} \div 2$$

$$5 \div \frac {1}{2}$$

$$2 \div \frac {1}{4}$$

2. Find the value of each expression.

### Student Response

If students do not correctly match the situations to the expressions, refer to each situation and ask, “How does the situation represent division?”

### Activity Synthesis

• Ask previously selected students to share their solutions.
• “How did you decide which expression matched which situation?” (I thought about the number of things being divided and whether it was a fraction or a whole number. I thought about the size of the pieces.)
• Display: $$3 \div \frac {1}{4}= 12$$
• “Describe how this equation represents the situation in the first problem.” (There are 3 cups of kernels in the bowl and a serving is $$\frac {1}{4}$$ a cup, so there are 12 servings in the bowl.)
• Display: $$\frac {1}{4} \div 3 = \frac {1}{12}$$
• “How do you know that this equation does not match the situation?” (The answer is a fraction. That doesn’t make sense because there is more than 1 serving in 3 cups.)

## Lesson Synthesis

### Lesson Synthesis

“Today we used expressions to represent and solve problems involving the division of a whole number by a unit fraction.”

Display: “Jada says when you divide a whole number by a unit fraction, the answer will always be greater than 1.”

“Do you agree with Jada? Be prepared to explain your thinking.” (I think so, because there will always be more than 1 unit fraction in a whole number, so even if it was 1 divided by a unit fraction, there will be however many unit fractions make up 1 whole, so that will be a whole number of unit fractions.)