Lesson 12 Represent Division of Unit Fractions by Whole Numbers

- Let’s make sense of diagrams that represent division of a unit fraction by a whole number.

Warm-up Estimation Exploration: How Much is Shaded?

How much is shaded?

Diagram. 4 equal sized parts. About 1 fifth of 1 part shaded. Total length, 1.

Record an estimate that is:

too low	about right	too high

Activity 1 Diagrams, Equations, Situations

Priya and Mai used the diagrams below to find the value of $\frac{1}{3} \div 4$ .

Priya’s diagram

Diagram. 12 equal parts. 1 part shaded. Total length, 1.

Mai’s diagram

Diagram. 3 equal parts. 1 of the 3 parts partitioned into 4 equal parts with 1 shaded. Total length, 1.

What is the same about the diagrams?
What is different?
Find the value that makes the equation true.

$\frac{1}{3} \div 4 =$
Han drew this diagram to represent $\frac{1}{3} \div 3$ . Explain how the diagram
shows $\frac{1}{3} \div 3$ .
Find the value that makes the equation true. Explain or show your reasoning.

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

Activity 2 Priya’s Work

Problem 1

Find the value of $\frac{1}{3} \div 2$ . Explain or show your reasoning.

Problem 2

This is Priya’s work for finding the value of $\frac{1}{3} \div 2$ :

Diagram. 3 equal parts. 1 part divided into 2 equal sized parts, 1 part shaded. Total length, 1.

$\frac{1}{3} \div 2 = \frac{1}{2}$ because I divided $\frac{1}{3}$ into 2 equal parts and $\frac{1}{2}$ of $\frac{1}{3}$ is shaded in.

What questions do you have for Priya?
Priya’s equation is incorrect. How can Priya revise her explanation?

Activity 3 Look for Patterns

Problem 1

Find the value that makes each equation true. Use a diagram if it is helpful.

$\frac{1}{4} \div 2 =$
$\frac{1}{4} \div 3 =$
$\frac{1}{4} \div 4 =$

Problem 2

What patterns do you notice?
How would you find the value of $\frac{1}{4}$ divided by any whole number? Explain or show your reasoning.

Practice Problem

Problem 1

Use the diagram to represent the expression $\frac{1}{5} \div 2$ .
Explain how the diagram shows $\frac{1}{5} \div 2$ .
What is the value of $\frac{1}{5} \div 2$ ?