# Lesson 11

Divide Unit Fractions by Whole Numbers

## Warm-up: Number Talk: Double the Divisor (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for dividing whole numbers. These understandings help students develop fluency and will be helpful later in this lesson when students divide a fraction by a whole 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

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

### Student Facing

Find the value of each expression mentally.

• $$72 \div 4$$
• $$36 \div 4$$
• $$4 \div 4$$
• $$1 \div 4$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What patterns do you notice in the quotients?” (When the dividend is split in half, the quotient is also split in half.)
• “Why does that happen?” (There are half as many to start with, so there will be half as many in each group.)

## Activity 1: More Macaroni and Cheese (15 minutes)

### Narrative

The purpose of this activity is for students to solve a contextual problem about dividing a fractional amount by a whole number. Students draw a diagram to represent the situation and relate the diagram to a division expression. Because of earlier work in this unit, students may draw one of the familiar square area diagrams showing the product $$\frac{1}{3} \times \frac{1}{2}$$. Other students may make a diagram resembling the macaroni and cheese pan and divide it appropriately. The focus in the synthesis is on bringing out how the diagram shows $$\frac{1}{2} \div 3$$ and how it allows students to answer the question. The relationship between $$\frac{1}{3} \times \frac{1}{2}$$ and $$\frac{1}{2} \div 3$$ will be brought out in later lessons. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

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

Representation: Develop Language and Symbols. Represent the problem in multiple ways to support understanding of the situation. For example, a cut-out or another physical manipulative to represent the pan of macaroni and distribute the “macaroni” to three students or a video that shows the distribution of macaroni and cheese between three people.
Supports accessibility for: Attention, Conceptual Processing

### Launch

• Groups of 2
• Display and read: “Last night, Jada’s aunt baked a pan of macaroni and cheese for dinner. Today, she brought the leftovers to Jada’s home for Jada and her sisters to share.”
• “What do you notice? What do you wonder?” (We solved problems about macaroni and cheese before. I wonder how much macaroni and cheese Jada’s aunt brought.)
• 1–2 minutes: partner discussion

### Activity

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
• draw diagrams like the ones in the student responses
• explain the situation as $$\frac {1}{2}$$ divided into 3 equal pieces
• recognize each person will get $$\frac{1}{6}$$ of the whole pan of macaroni and cheese

### Student Facing

Jada and her 2 sisters equally share $$\frac{1}{2}$$ a pan of macaroni and cheese.
1. Draw a diagram to represent the situation.
2. Explain how this expression represents the situation: $$\frac {1}{2} \div 3$$
3. How much of the whole pan of macaroni and cheese will each person get?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

MLR7 Compare and Connect
• “Create a visual display that shows your thinking about the problems. 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
• “How does each representation show $$\frac {1}{2}$$ a pan of macaroni and cheese?”
• “How does each representation show 3 equal pieces?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “How do we know that each person got the same amount of macaroni and cheese?” (I divided the half into 3 equal shares.)
• “How much of the whole pan of macaroni and cheese did each person get?” ($$\frac {1}{6}$$)
• “How do the diagrams show $$\frac {1}{6}$$?” (Each $$\frac {1}{2}$$ of the pan is divided into 3 equal pieces, so each of those pieces is $$\frac {1}{6}$$ of the whole pan.)

## Activity 2: More People Share (20 minutes)

### Narrative

The purpose of this activity is for students to continue to solve problems about dividing a unit fraction by a whole number. The unit fraction in both problems is $$\frac{1}{2}$$ so that students will consider the relationship between the number of people sharing the macaroni and cheese and the size of the serving each person gets. When students connect the quantities in the story problem to an equation and a diagram representing the story, they reason abstractly and quantitatively (MP2).

• Groups of 2

### Activity

• 1–2 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who :
• draw diagrams like the ones in the student responses
• describe the amount each person gets as a fraction of the whole pan
• describe a relationship between the number of people sharing and the size of the serving each person gets

### Student Facing

1. 4 people equally share $$\frac {1}{2}$$ a pan of macaroni and cheese.
1. Draw a diagram to represent the situation.
2. Explain how your diagram represents $$\frac {1}{2} \div 4$$.
3. How much of the whole pan of macaroni and cheese did each person get? Be prepared to explain your reasoning.
2. 5 people equally share $$\frac {1}{2}$$ a pan of macaroni and cheese.
1. Draw a diagram to represent the situation.
2. Explain how your diagram represents $$\frac {1}{2} \div 5$$.
3. How much of the whole pan of macaroni and cheese did each person get? Be prepared to explain your reasoning.
3. How are the problems the same? How are they different?

### Student Response

For access, consult one of our IM Certified Partners.

If students do not identify or explain how much macaroni and cheese each person gets, consider asking, “How much of the whole pan of macaroni and cheese did each person get?”

### Activity Synthesis

• “How are the situations the same? How are they different?” (Both situations are about $$\frac {1}{2}$$ a pan of macaroni and cheese, but this one has more people sharing it, so each person gets less macaroni and cheese. When 4 people share, each person gets $$\frac {1}{8}$$ of the whole pan. When 5 people share, each person gets $$\frac {1}{10}$$ of the whole pan.)

## Lesson Synthesis

### Lesson Synthesis

If you taught the previous optional lesson, display the poster from the previous lesson’s synthesis.
“What can we add to our poster to show what we learned about division today?” (We can divide unit fractions by whole numbers.)
“How can we show examples of what we learned?” (We can show equations and representations.)

If you did not teach the previous optional lesson, ask: “What did you learn about division today? How can we show examples of what we learned?” Record responses on a poster to be used in future lessons.

“What do you still wonder about division?” (Can you divide fractions? When would you ever need to divide a fraction? Does the answer get smaller or bigger when you divide fractions?)

Record student responses for all to see. Keep the display visible. Refer back to it in future lessons.

## Cool-down: Share Macaroni and Cheese (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.