# Two Methods for Dividing Fractions by Fractions

## Overview

Students explore methods of dividing a fraction by a fraction.

# Key Concepts

Students extend what they learned in Lesson 4 to divide a fraction by any fraction. Students are presented with two general methods for dividing fractions:

- Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.
- Multiply the dividend by the reciprocal of the divisor.

These two methods will work for all cases, including cases in which one or both of the numbers in the division is a fraction or whole number.

# Goals and Learning Objectives

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

# Emma Makes Soup

# Lesson Guide

Give students a minute or two to read the word problem and write the corresponding division problem. Be sure all students write the correct division problem, $1\frac{3}{4}\xf7\frac{3}{8}$, before moving on to Work Time.

## Opening

# Emma Makes Soup

Emma made $1\frac{3}{4}$ quarts of soup. She is serving the soup in bowls that hold $\frac{3}{8}$ quart each.

How many bowls can she fill?

- Write a division expression to represent this situation, but do not solve the problem yet.

# Math Mission

# Lesson Guide

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

## Opening

Explore methods for dividing a fraction by a fraction.

# Emma’s Soup Problem

# Lesson Guide

Students should work with a partner on the problems. If students have trouble getting started, encourage them to look back at the methods used in earlier lessons.

# Mathematics

If students are confused by the fact that the dividend, $1\frac{3}{4}$, is a mixed number, remind them that any mixed number can also be written as a fraction.

# Interventions

**Student has difficulty starting.**

- Look back at the methods in Lesson 4. Can you use a method similar to one of those to solve this problem?
- What does the problem $1\frac{3}{4}\xf7\frac{3}{8}$ mean? What are you trying to find?
- Would it help if you wrote $1\frac{3}{4}$ as a fraction?
- Would rewriting the fractions so they have a common denominator help you think about what the problem means?

**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

- $1\frac{3}{4}$ rewritten is $\frac{7}{4}\xf7\frac{3}{8}=\frac{7}{4}\times \frac{8}{3}=\frac{56}{12}=4\frac{8}{12}=4\frac{2}{3}$ Emma can fill $4\frac{2}{3}$ bowls.
- Check: $4\frac{2}{3}\times \frac{3}{8}=\frac{14}{3}\times \frac{3}{8}=\frac{7}{4}=1\frac{3}{4}$

## Work Time

# Emma’s Soup Problem

Emma made $1\frac{3}{4}$ quarts of soup. She is serving the soup in bowls that hold $\frac{3}{8}$ quart each.

How many bowls can she fill?

- Solve this problem using any method you wish. Use the division expression you wrote in the Opening.
- Check your answer.

Did you rewrite 134 as a fraction? Would rewriting both fractions so they have a common denominator help you find the answer?

# Common Denominators

# Mathematics

This problem asks students to apply and explain the common-denominator method of division. Many students prefer this method to the traditional “multiply by the reciprocal” method.

ELL: Write the key points on a poster so that students can refer back to them throughout the unit. When working with ELLs, provide supplementary materials, such as graphic organizers, to illustrate new concepts and vocabulary necessary for mathematical learning. Have students record all information in their Notebook.

# Interventions

**Student is having difficulty reasoning about the common-denominator method.**

- You rewrote $\frac{5}{6}\xf7\frac{2}{5}$ as $\frac{25}{30}\xf7\frac{12}{30}$. What does the rewritten problem mean? What are you trying to find?
- You are trying to find the number of groups of 12 thirtieths in 25 thirtieths.
- How does this compare to finding the number of groups of 12 apples in 25 apples?
- What division would you solve to find the number of groups of 12 apples in 25 apples?
- So, what division would you solve to find the number of groups of 12 thirtieths in 25 thirtieths?

**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

- $\frac{5}{6}\xf7\frac{2}{5}=\frac{25}{30}\xf7\frac{12}{30}=25\xf712=2\frac{1}{12}$
- Check: $2\frac{1}{12}\times \frac{2}{5}=\frac{25}{12}\times \frac{2}{5}=\frac{50}{60}=\frac{5}{6}$
- Answers will vary. Possible explanation: Once we have rewritten the fractions with a common denominator, the numerator represents units of the same size. We can see this in a bar model: Since the fractions now represent units of the same size, we can disregard the denominator to complete the division: $25\xf712=\frac{25}{12}=2\frac{1}{12}$.

## Work Time

# Common Denominators

One method for dividing two fractions is to first write them with a common denominator. The answer will be the quotient of the numerators.

- Use this method to solve $\frac{5}{6}\xf7\frac{2}{5}$.
- Check your answer.
- Explain why this method works.

To explain why this method works, think about what happens to the common denominators when you change the division problem into a multiplication problem using the reciprocal.

# Fraction Problem

# Mathematics

Many students find the “multiply by the reciprocal” method difficult to understand. It is fine if students cannot explain it.

To solve the problems, students may use methods similar to those in Lesson 4, including:

- Draw a model of the dividend, change the fractions and the model to show a common denominator, and circle groups represented by the divisor.
- Rewrite the fractions so they have a common denominator and find the quotient of the new numerators. Students should be able to explain why this method works.
- Rewrite the problem to multiply by the reciprocal.
- Reason about the number of groups of unit fractions in the dividend.

# Interventions

**Student has difficulty starting.**

- Look back at the methods in Lesson 4. Can you use a method similar to one of those to solve this problem?
- What does the problem $\frac{2}{3}\xf7\frac{3}{5}$ mean? What are you trying to find?
- Would rewriting the fractions so they have a common denominator help you think about what the problem means?

**Student is having difficulty reasoning about the common-denominator method.**

- You rewrote $\frac{2}{3}\xf7\frac{3}{5}$ as $\frac{10}{15}\xf7\frac{9}{15}$. What does the rewritten problem mean? What are you trying to find?
- You are trying to find the number of groups of 9 fifteenths in 10 fifteenths.

**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?

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students must find ways to make sense of dividing a fraction by a fraction and find a way to represent and solve the given problems.

**Mathematical Practice 4: Model with mathematics.**

The Opening presents students with a real-world problem that they must model with a mathematical expression or equation.

# Answers

- 1$\frac{1}{9}$
- Check: $1\frac{1}{9}\times \frac{3}{5}=\frac{10}{9}\times \frac{3}{5}=\frac{30}{45}=\frac{2}{3}$

## Work Time

# Fraction Problem

- Solve the problem $\frac{2}{3}\xf7\frac{3}{5}$ using any method.
- Check your answer.

Can you use multiplication to find the answer?

# 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 solved the Challenge Problem to present during Ways of Thinking.

# Mathematical Practices

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

- Students must be able to explain why the common-denominator method works. In the Challenge Problem, some students will explain using the “multiply by the reciprocal” method.

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

- Encourage students to use precise language to explain the methods. In Ways of Thinking, allow other students to critique the reasoning of presenters and to help clarify confusion and correct misconceptions.

SWD: During student presentations, provide some students with a note-taking organizer that includes a space for the presenters’ answer to the problem, the presenters’ explanation of the mathematics, etc. This will help students when they reflect on the presentation during the Ways of Thinking discussion.

# Challenge Problem

## Answers

- The second option, $\frac{5}{7}\xf7\frac{1}{4}$, will be greater. Explanations will vary. Possible explanation: These problems can be rewritten, so instead of dividing by $\frac{1}{2}$ or $\frac{1}{4}$, we can multiply by 2 or 4. Since multiplying a number by 4 will result in a larger number than multiplying the same number by 2, $\frac{5}{7}\xf7\frac{1}{4}$ will yield the larger number.

## Work Time

# Prepare a Presentation

Explain the method you used to divide a fraction by a fraction. Use your work to support your explanation.

# Challenge Problem

Look at the two problems below:

$\frac{5}{7}\xf7\frac{\text{1}}{2}$

$\frac{5}{7}\xf7\frac{\text{1}}{4}$

- Explain how you can tell which answer will be greater without actually solving the problems.

# Make Connections

# Mathematics

Have a student demonstrate the common-denominator method to solve a division problem and then explain why the method works. You might suggest that when students apply this method they visualize a bar model or area model. Discuss the meaning of *reciprocal* and then have a student present who used the "multiply by the reciprocal" method for any of the problems.

If no one did the Challenge Problem, present an explanation yourself.

Point out to students that both the common-denominator and "multiply by the reciprocal" methods work even if one of the numbers is a whole number or a unit fraction. So, these methods will work for all the cases students have considered in previous lessons. Have volunteers demonstrate how to use these methods to solve the problems below. Remind students that a whole number *n* can be written as the fraction $\frac{\mathrm{n}}{1}$.

# Mathematical Practices

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

Have students ask questions and critique the presenter’s method and reasoning.

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

Encourage students to clearly explain the steps in their solutions as they present using correct mathematical language.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ methods for dividing a fraction by a fraction.

As your classmates present, ask questions such as:

- What does dividing a fraction by a fraction mean?
- What happens when the fractions are written with a common denominator?
- How is dividing a fraction by a fraction similar to dividing a fraction by a unit fraction? How is it different?

# Methods for Dividing a Fraction by a Fraction

# A Possible Summary

To divide a fraction by a fraction, you can use some of the methods we used in earlier lessons, such as making a model or solving a related multiplication equation.

There are two shortcut methods for dividing a fraction by a fraction:

- Rewrite the fractions so they have a common denominator. Then you can just ignore the denominators and divide the numerators.
- Multiply by the reciprocal of the divisor.

ELL: When writing the summary, provide ELLs access to a dictionary, and give them time to discuss their summary with a partner before writing to help them organize their thoughts. Allow ELLs who share the same language of origin to discuss in their language of choice.

SWD: Struggling students may still need explicit instruction and guided scaffolding to recognize the relationships for dividing fractions. Provide small group instruction to help those students recognize the relationship between the two methods.

## Formative Assessment

# Summary of the Math: Methods for Dividing a Fraction by a Fraction

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

Check your summary.

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

# Reflect On Your Work

# Lesson Guide

Have students write a brief reflection before the end of class. Review the reflections to find out what they still find confusing about 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 thing that still confuses me about dividing with fractions is …**