# Divide a Fraction by a Fraction

## Overview

Students explore methods of dividing a fraction by a unit fraction.

# Key Concepts

In this lesson and in Lesson 5, students explore dividing a fraction by a fraction.

In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding $\frac{3}{4}\xf7\frac{1}{5}$ means finding the number of fifths in $\frac{3}{4}$. In this lesson, students will see that this is $\frac{3}{4}$ × 5.

Students learn and apply several methods for dividing a fraction by a unit fraction, such as $\frac{2}{3}\xf7\frac{1}{4}$.

- Model $\frac{2}{3}$. Change the model and the fractions in the problem to twelfths: $\frac{8}{12}\xf7\frac{3}{12}$. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.
- Reason that since there are 4 fourths in 1, there must be $\frac{2}{3}$ × 4 fourths in $\frac{2}{3}$. This is the same as using the multiplicative inverse.
- Rewrite both fractions so they have a common denominator: $\frac{2}{3}\xf7\frac{1}{4}=\frac{8}{12}\xf7\frac{3}{12}$. The answer is the quotient of the numerators. This is the numerical analog to modeling.

# Goals and Learning Objectives

- Use models and other methods to divide fractions by unit fractions

# Mia’s Method

# Lesson Guide

Have students work with a partner for a few minutes to read and discuss Mia’s method before discussing with the entire class.

# Mathematics

Have students explain how Mia’s method is similar to the ones she used in Lessons 2 and 3. How has it been modified?

ELL: Modeling is essential for ELL students. It helps them clarify difficult concepts and directions. It also gives students the opportunity to ask questions and it gives you, as instructor, the opportunity to model effective learning strategies. Some students may not recognize a general pattern for dividing fractions.

SWD: These types of concrete and basic examples will be helpful for students with more literal thinking skills.

## Opening

# Mia's Method

Mia uses the following method to find $\frac{2}{3}\xf7\frac{1}{4}$:

“To find $\frac{2}{3}\xf7\frac{1}{4}$, I need to find the number of fourths in $\frac{2}{3}$. I can make a model of $\frac{2}{3}$, but I think it would be difficult to figure out the number of fourths in the model.

“I think the problem would be easier if both fractions had the same denominator. I can change the denominator of each fraction to 12 and rewrite the problem as $\frac{8}{12}\xf7\frac{3}{12}$.

“Now I can make a model of $\frac{8}{12}$ and then find the number of groups of 3 twelfths in my model.

“In my model there are 2 groups of 3 twelfths, with 23 of a group of 3 twelfths left over.

So, $\frac{2}{3}\xf7\frac{1}{4}=2\frac{2}{3}$.”

- Discuss Mia’s method with your partner and then with the class.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore methods for dividing a fraction by a unit fraction.

## Opening

Explore methods for dividing a fraction by a unit fraction.

# Explore Dividing Fractions by Unit Fractions

# Lesson Guide

Students should work with a partner on the first two problems. Help students who have difficulty. As students work, encourage them to look for a general pattern they can use to divide any fraction by a unit fraction.

SWD: When listening to students' responses, give students with disabilities advance notice of when they will be presenting their work on a specific problem during the Ways of Thinking section. This will give them ample time to prepare a thoughtful response.

# Interventions

**Student has difficulty starting $\frac{4}{5}\xf7\frac{1}{3}$.**

- How did Mia start her method?
- What does the problem $\frac{4}{5}\xf7\frac{1}{3}$ mean? What are you trying to find?
- Try drawing a model for $\frac{4}{5}$. How could you find the number of thirds inside of $\frac{4}{5}$?

**Student finds the wrong solution.**

- How can you check to make sure your answer is correct?
- Think about the sizes of the numbers in the problems. Does your answer seem reasonable?

# Answers

- $\frac{4}{5}\xf7\frac{1}{3}$
Change the denominator of each fraction:

$\frac{4}{5}\times \frac{3}{3}=\frac{12}{15}$ $\frac{1}{3}\times \frac{5}{5}=\frac{5}{15}$Rewrite the problem: $\frac{12}{15}\xf7\frac{5}{15}$

Now make a model of $\frac{12}{15}$, then find the number of groups of 5 fifteenths in the model:

In this model there are 2 groups of 5 fifteenths, with $\frac{2}{5}$ of a group of 5 fifteenths left over, so:

$\frac{4}{5}\xf7\frac{1}{3}=2\frac{2}{5}$

- Check: $2\frac{2}{5}\times \frac{1}{3}=\frac{12}{5}\times \frac{1}{3}=\frac{12}{15}=\frac{4}{5}$

## Work Time

# Explore Dividing Fractions by Unit Fractions

- Use Mia’s method to find $\frac{4}{5}\xf7\frac{1}{3}$.
- Check your answer.

Ask yourself:

- Did you change $\frac{4}{5}$ and
to fractions that have the same denominator?1 3 - How many five-fifteenths (
) are in5 15 ?12 15

- How many five-fifteenths (

# Carlos’s Method

# Lesson Guide

Students learn Carlos's method of dividing a fraction by a unit fraction by multiplying the reciprocal or multiplicative inverse of a fraction.

# Interventions

**If students are having difficulties:**

- What was Carlos dividing by and what are you dividing by?
- What did Carlos do?
- What is the inverse of $\frac{1}{5}$?

# Answers

- $\frac{8}{3}\xf7\frac{1}{5}=\frac{8}{3}\times 5=\frac{40}{3}=13\frac{1}{3}$

## Work Time

# Carlos’s Method

Carlos uses the following method to find $\frac{7}{2}\xf7\frac{1}{4}$:

“I can change this problem to a multiplication problem by multiplying by the inverse. The multiplicative inverse, or reciprocal, of $\frac{1}{4}$ is 4.”

Here is Carlos’s solution:

$\frac{7}{2}\times 4=\frac{28}{2}$ or 14

So, $\frac{7}{2}\xf7\frac{1}{4}$ = 14.

- Use Carlos’s method to find $\frac{8}{3}\xf7\frac{1}{5}$.

How can you rewrite the division problem as a multiplication problem using the reciprocal of 15?

# Prepare a Presentation

# Preparing for Ways of Thinking

Choose solutions that use a variety of models and methods to be presented during Ways of Thinking. Choose both incorrect and correct solutions. You can use the incorrect solutions to clear up misconceptions students may have.

Select students who did the Challenge Problem to present during Ways of Thinking. If possible, choose one student for each method.

# Challenge Problem

## Answers

- Students will choose either Method 1 or Method 2. Explanations will vary. Possible explanations: Method 1: For $\frac{4}{5}\xf7\frac{1}{3}$, the denominator of $\frac{1}{3}$, which is 3, is the number of thirds in 1 whole. The number of thirds in $\frac{4}{5}$ is $\frac{4}{5}$ of the number of thirds in 1 whole, or $\frac{4}{5}$ of 1 x 3. This is the same as $\left(\frac{4\times 3}{5}\right)$. We are multiplying the numerator of the fraction by the denominator of the unit fraction. Method 2: We can rewrite $\frac{4}{5}\xf7\frac{1}{3}$ as $\frac{12}{15}\xf7\frac{5}{15}$. We must find the number of groups of 5 fifteenths in 12 fifteenths. This is just 12 ÷ 5.

## Work Time

# Prepare a Presentation

Explain how you divided a fraction by a unit fraction. Use your work to support your explanation.

# Challenge Problem

Below are two methods for dividing a fraction by a unit fraction.

**Method 1**

To divide a fraction by a unit fraction, multiply the numerator of the fraction by the denominator of the unit fraction.

**Method 2**

To divide a fraction by a unit fraction, rewrite both fractions so they have a common denominator. The answer is the quotient of the numerators.

- Choose one of the methods. Use the division problem $\frac{4}{5}\xf7\frac{\text{1}}{3}$ to explain why the method works.

# Make Connections

# Lesson Guide

Have students present a variety of models and methods for solving the problems.

ELL: When giving directions for this discussion, be sure that students understand that they can complement their oral explanation with pictures, number lines, and drawings. Provide sentence frames that will help ELLs explain their thinking.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Encourage students to clearly explain the steps in their solutions using correct mathematical language. Other students can ask questions and critique the presenter’s method and reasoning, using the hints as prompts.

**Mathematical Practice 6: Attend to precision.**

If possible, have at least one student explain each method from the Challenge Problem. If no students did the Challenge Problem, present and explain the methods yourself.

## Performance Task

# Ways of Thinking: Make Connections

As your classmates present, take notes to clarify your understanding of how to divide a fraction by a unit fraction.

As your classmates present, ask questions such as:

- Do all of the methods make sense?
- Which methods did you find easier to use? Why?
- Which methods did you find more difficult to use? Why?
- How are the two models you just looked at alike? How are they different?
- Where do you see the unit fraction in your model?
- Why can multiplication be used to solve a division problem?
- Is there a general pattern you can use to divide any fraction by a unit fraction?

# Servings of Rice

# Lesson Guide

Challenge students to use a method that they had not used in the Work Time problems. For students who are fluent in multiple methods, challenge them to create real-world problems. Have students work alone.

# Mathematics

This problem has a dividend that is a mixed number. You may need to remind students that a mixed number can be rewritten as a fraction.

# Answers

- $\frac{7}{3}\xf7\frac{1}{2}$
- Methods will vary. $\frac{14}{3}$ or 4$\frac{2}{3}$

Possible methods:

$\frac{7}{3}\xf7\frac{1}{2}=\frac{7}{3}\times 2=\frac{14}{3}$

$\frac{7}{3}\xf7\frac{1}{2}=\frac{14}{6}\xf7\frac{3}{6}=\frac{14}{3}$

The model shows $2\frac{1}{3}\xf7\frac{1}{2}=4\frac{2}{3}$

- Check: $\frac{1}{2}\times \frac{14}{3}=\frac{7}{3}$

## Work Time

# Servings of Rice

A pot contains 2$\frac{1}{3}$ cups of rice. How many $\frac{1}{2}$-cup servings does the pot contain?

- Write a division problem to represent this situation.
- Solve the problem using any method you wish.
- Check your answer.

Ask yourself:

- What is the total amount of rice in the pot? You should divide the total amount of rice into equal groups of what size? How many equal-sized groups are there?
- Did you write 2 1 3 as a fraction? Would rewriting both fractions so they have a common denominator help you find the answer?

# Divide a Fraction by a Unit Fraction

# A Possible Summary

To divide a fraction by a unit fraction, you must first determine the number of unit fractions inside the fraction. One way to do this is to draw a model. Sometimes it is hard to see the number of unit fractions inside the model, so you need to change both fractions (and the model) so they have a common denominator.

We learned two shortcut methods:

- Multiply the numerator of the fraction (the dividend) by the denominator of the unit fraction.
- Rewrite the fractions in the problem so they have a common denominator. The answer will be the quotient of the numerators.

## Formative Assessment

# Summary of the Math: Divide a Fraction by a Unit Fraction

Write a summary about how to divide a fraction by a unit fraction.

Check your summary.

- Do you describe at least two methods for dividing a fraction by a unit fraction?

# Reflect On Your Work

# Lesson Guide

Have students write a brief reflection before the end of class. Review the reflections to find out their understanding of fraction division.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**One way I can figure out how to solve a division problem when I am dividing by a unit fraction is...**