# Lesson 13

Divide Whole Numbers by Unit Fractions

## Warm-up: Notice and Wonder: Quilt (10 minutes)

### Narrative

The purpose of this warm-up is for students to describe the rectangles in the representation of a quilt, which will be useful when students divide strips of paper into unit fraction sized pieces in a later activity. While students may notice and wonder many things about this image, the variety of lengths and colors of fabric strips is the important discussion point.

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “These pictures show women from Gee's Bend, Alabama, who have been making quilts for generations. How would you describe the quilt they are working on?” (It is colorful. There are rectangles. There are different colored pieces of fabric.)
• Consider showing students examples of abstract or improvised quilts by Gee’s Bend Quiltmakers from the website of Souls Grown Deep.

## Activity 1: Paper Strips (20 minutes)

### Narrative

The purpose of this activity is for students to solve problems about dividing a whole number by a unit fraction in a way that makes sense to them. The context of quilt making is used so students can visualize a strip of paper that is a whole number length being cut into fractional sized pieces. As students describe how the problems are similar and different, listen for the authentic language they use to describe division. The paper strip, or tape, is a helpful diagram to use when dividing a whole number by a unit fraction because students recognize important relationships between the divisor, dividend, and quotient (MP7). For example, if the length of the strip stays the same, but the size of the piece gets smaller, then the number of pieces will get bigger.

### Launch

• Groups of 2
• Refer to the picture from the warm up.
• “If the blue strip of fabric under the woman’s chin is 1 meter long, about how long is the short gray strip next to it?” ($$\frac{1}{6}$$ meter)

### Activity

• 1–2 minutes: independent think time
• 8–10 minutes: partner work time
MLR2 Collect and Display
• Circulate, listen for, and collect the language students use to describe what was the same and different about the strategies they used to determine the number of pieces of paper for each color. Listen for:
• The size of the piece changed.
• The pieces were shorter.
• There were more pieces.
• Record students’ words and phrases on a visual display and update it throughout the lesson.

### Student Facing

Below are diagrams that show strips of different colored paper. Each strip is 2 feet long. The paper strips will be cut into different sized pieces.
1. The red strip will be cut into pieces that are $$\frac{1}{2}$$ foot long. How many pieces will there be?
2. The green strip will be cut into pieces that are $$\frac{1}{3}$$ foot long. How many pieces will there be?
3. The yellow strip will be cut into pieces that are $$\frac{1}{4}$$ foot long. How many pieces will there be?
4. Describe what was the same about the problems you solved. Describe what was different.

### Student Response

Students may not immediately make a connection with the situation and division, since the number of pieces that results (the quotient) is greater than the number of feet being divided (the dividend). Consider asking, “How does this situation represent division?” Allow students to recognize that the cuts are similar to partitions in a diagram, or sharing. They may also recognize that the quotient represents the number of groups, while the fraction being divided is the size of each group. Have them write a division equation for each situation, if it helps. They will write division equations in the next activity.

### Activity Synthesis

• “Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display:
$$2 \div \frac {1}{2} = 4$$
$$2 \div \frac {1}{3} = 6$$
$$2 \div \frac {1}{4} = 8$$
• “How do these equations represent the problems about the paper strips?” (The 2 is for 2 feet of paper, and the fractions show the size of the pieces that the paper is being cut into. The 4, 6, and 8 are the number of pieces of each color of paper.)

## Activity 2: More Paper Strips (15 minutes)

### Narrative

The purpose of this activity is for students to represent division of a whole number by a unit fraction with diagrams and equations. The context is the same as the previous activity so students can use a tape diagram to solve the problem, if they choose. In the previous activity, students recognized that when the length of paper stays the same and the size of the piece gets smaller, there are more pieces of paper. In this activity, students will consider what happens when the length of the paper changes, but the size of the pieces stays the same.

Representation: Access for Perception. Provide access to strips of paper for students to cut and fold. Ask students to identify correspondences between the number of pieces/folds and the fraction they represent.
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 1–2 minutes: independent think time
• 6–8 minutes: partner work time
• Monitor for students who:
• determine that Kiran will have 12 pieces of paper
• can explain how the equation $$2 \div \frac{1}{6} = 12$$ represents the yellow strip of paper being cut into $$\frac16$$-foot strips
• describe the equation $$3 \div \frac {1}{6} = 18$$ as representing a 3 foot strip of paper being cut into 18 pieces that are each $$\frac {1}{6}$$ of a foot long

### Student Facing

Kiran has a yellow strip of paper that is 2 feet long. He wants to cut the strip into pieces that are $$\frac{1}{6}$$ foot long.
1. How many pieces will Kiran have? Explain or show your reasoning.
2. Write a division equation to represent the situation.
3. Describe how the equation $$3 \div \frac{1}{6} = 18$$ represents a strip of paper that is 3 feet long being cut into equal-sized pieces.

### Student Response

If students do not write an equation that represents the situation, show them $$2 \div \frac{1}{6}$$ and ask, “How does this expression represent the situation?”

### Activity Synthesis

• Ask previously identified students to share their solutions.
• Display: $$2 \div \frac {1}{6} = 12$$
• “How does this equation represent the yellow strip of paper?” (The strip of paper is 2 feet long and it is cut into pieces that are $$\frac {1}{6}$$ of a foot long so there will be 12 pieces.)
• Display: $$3 \div \frac {1}{6} = 18$$
• “How does this equation represent a different strip of paper being cut into equal sized pieces?” (A 3 foot piece of paper is cut into pieces that are $$\frac {1}{6}$$ of a foot long so there are 18 pieces.)
• “Why is the quotient larger than the dividend in both of these equations?” (Because you are cutting a whole number length into fractional sized pieces, so there will be more pieces than when you started.)

## Lesson Synthesis

### Lesson Synthesis

"Today, we solved problems about cutting strips of paper into small pieces. We wrote equations to represent dividing a whole number by a unit fraction.”

Display:

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

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

$$2 \div \frac {1}{4} = 8$$

$$2 \div \frac {1}{6} = 12$$

“These are some of the equations we discussed today. Why is the quotient getting larger in each equation?” (Because the size of the piece is getting smaller, so there will be more pieces.)

Display: $$3 \div \frac {1}{6} = 18$$

“Here is another equation we discussed. In this equation, the size of the piece is the same as the equation above it. Why is the quotient larger than when 2 is divided by $$\frac16$$?” (3 is being divided into smaller pieces, instead of 2, so you get more pieces.)

“We are going to learn more about the relationships between the numbers in division equations with unit fractions in the next lesson.”