# Fractions and Division in Word Problems

## Overview

Students solve word problems that require dividing and multiplying with fractions and mixed numbers.

# Key Concepts

Students apply their knowledge about multiplying and dividing fractions to solve word problems. This includes applying the general methods for dividing fractions learned in previous lessons:

- 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.

# Goals and Learning Objectives

- Apply knowledge of fraction multiplication and division to solve word problems.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will solve word problems involving fractions.

ELL: Have students create an organizer that summarizes the conventions for multiplication and division of fractions. Have students create their own organizers because it is important for ELLs to organize information in a way that makes sense to them.

## Opening

Solve word problems involving fractions.

# Dog Play Area

# Lesson Guide

Students should work with a partner to solve the problems in Tasks 2–4. You might tell students that not all of the problems involve division, so they should think carefully about the operations they need to use.

# Mathematics

If students struggle with a problem, suggest that they try writing an equation or drawing a diagram. If students have difficulty determining which operation to use, suggest that they think about a simpler problem with whole numbers instead of fractions. Once they have determined the appropriate operations, they can go back to the original problem.

Encourage students to check that their answers are reasonable in the context of the problem. If not, they will need to determine whether they made an error in their calculations or in interpreting or setting up the problem.

# Interventions

**Student has difficulty starting a problem.**

- Use a variable to represent what you are trying to find.
- The problem involves three quantities—the two numbers in the problem and the variable. How are these three quantities related?
- Can you write an equation to show this?
- Did you try drawing a diagram?
- Start by thinking of a simpler problem. Try putting whole numbers in the problem instead of fractions or mixed numbers. Does that help you figure out what operation to use?

**Student uses the wrong operation.**

- If the numbers in the problem were whole numbers, what operation would you use to solve it?
- What equation could you write for this problem? Would you solve that equation by multiplying or dividing?

**Student gets the wrong solution.**

- Go back and read the problem. Does your answer make sense?
- Explain the strategy you used to solve the problem.
- How can you check that your calculation is correct?

ELL: Have students list the key information. Use diagrams or visuals to assist them with these word problems.

# Answers

- 6 yards wide (50 ÷ $8\frac{1}{3}$)

## Work Time

# Dog Play Area

Denzel wants to fence in a rectangular play area for his dog. The play area will extend the entire length of his backyard, which is $8\frac{1}{3}$ yards.

- If he wants his dog to have 50 square yards to play in, how wide does he need to make the play area?

Ask yourself:

- What is the formula for the area of a rectangle?
- What measurements does the problem provide?
- What is the unknown measurement?

# Walking to School

# Interventions

**Student has difficulty starting a problem.**

- Use a variable to represent what you are trying to find.
- The problem involves three quantities—the two numbers in the problem and the variable. How are these three quantities related?
- Can you write an equation to show this?
- Did you try drawing a diagram?
- Start by thinking of a simpler problem. Try putting whole numbers in the problem instead of fractions or mixed numbers. Does that help you figure out what operation to use?

**Student uses the wrong operation.**

- If the numbers in the problem were whole numbers, what operation would you use to solve it?
- What equation could you write for this problem? Would you solve that equation by multiplying or dividing?

**Student gets the wrong solution.**

- Go back and read the problem. Does your answer make sense?
- Explain the strategy you used to solve the problem.
- How can you check that your calculation is correct?

# Answer

- $\frac{2}{3}$ mile ($\frac{5}{6}$ × $\frac{4}{5}$)

## Work Time

# Walking to School

Martin walks $\frac{4}{5}$ of a mile to school each day. The distance Emma walks to school is $\frac{5}{6}$ of the distance Martin walks.

- How far does Emma walk?

Ask yourself:

Can you use a number line to model the situation?

# Hippopotamus

# Mathematics

**[common error]** Students may multiply the two numbers because they see the words “times as much.”

Point out that the female hippopotamus must weigh less than the male. Suggest that students let $f$ represent the weight of the female and write an equation to represent the problem. The equation is $1\frac{1}{2}\times f=2\times f=2$, which is solved by dividing 2 by $1\frac{1}{2}\times f=2$.

# Mathematical Practices

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

Students must make sense of each problem. They should be able to explain it to themselves and their partner. They need to determine what information is given, what information they need to find, and how these things are related. Finally, they need to devise an approach or strategy for solving the problem.

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students must make sense of the problem and represent it symbolically with an equation or expression. Once a solution is found, students should again consider the context of the problem to make sure that it makes sense.

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

To solve word problems, students must use mathematics to model the situation. These models might be equations, expressions, or diagrams.

# Interventions

**Student has difficulty starting a problem.**

- Use a variable to represent what you are trying to find.
- The problem involves three quantities—the two numbers in the problem and the variable. How are these three quantities related?
- Can you write an equation to show this?
- Did you try drawing a diagram?
- Start by thinking of a simpler problem. Try putting whole numbers in the problem instead of fractions or mixed numbers. Does that help you figure out what operation to use?

**Student uses the wrong operation.**

- If the numbers in the problem were whole numbers, what operation would you use to solve it?
- What equation could you write for this problem? Would you solve that equation by multiplying or dividing?

**Student gets the wrong solution.**

- Go back and read the problem. Does your answer make sense?
- Explain the strategy you used to solve the problem.
- How can you check that your calculation is correct?

# Answer

- $1\frac{1}{3}\times f=2$ tons ($2\xf71\frac{1}{2}\times f=2$)

## Work Time

# Hippopotamus

The male hippopotamus at the zoo weighs 2 tons. He weighs $1\frac{1}{2}\times f=2$ times as much as the female hippopotamus weighs.

- How much does the female hippopotamus weigh?

Does the female hippopotamus weigh more or less than the male?

# Prepare a Presentation

# Preparing for Ways of Thinking

Select solutions to be presented during Ways of Thinking. Try to choose solutions that involve different methods or reasoning.

Choose some Challenge Problems to be presented as well.

# Challenge Problem

## Answers

- Multiplication problems will vary.
- Division problems will vary.
- Possible answers for multiplication: $4\frac{2}{3}\times \frac{1}{3}=\frac{14}{9}$ or $1\frac{5}{9}$

Possible answers for division: $4\frac{2}{3}\xf7\frac{1}{3}=14$$\frac{1}{3}\xf74\frac{2}{3}=\frac{1}{14}$

## Work Time

# Prepare a Presentation

Choose one of the problems you solved. Prepare a solution that you can share with your classmates. Include a drawing and an explanation of what you did to solve the problem.

# Challenge Problem

Write two word problems.

- Write one word problem that uses multiplication and the numbers $4\frac{2}{3}$ and $\frac{1}{3}$.
- Write a second word problem that uses division and the numbers $4\frac{2}{3}$ and $\frac{1}{3}$.
- Solve both problems.

# Make Connections

# Lesson Guide

Have students present and explain their solutions.

# Mathematical Practices

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

- Students should ask questions and critique the presenters’ reasoning.

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

- Encourage presenters to explain their solutions and their reasoning using clear and precise language.

# Mathematics

Encourage students’ critical thinking by posing the following questions:

- What information are you given in this problem? What information do you need to find?
- How did you decide which operation to use? What are the relationships between the quantities that are given and the quantity you need to find?
- How did you come up with that equation?
- How does your diagram represent the problem?
- How can you tell that your answer is reasonable?

## Performance Task

# Ways of Thinking: Make Connections

Take notes on your classmates’ approaches to solving and writing fraction word problems.

As your classmates present, ask questions such as:

- How are the quantities in the problem related?
- What is the unknown quantity in the problem?
- How does your equation represent the problem situation?
- Is your answer reasonable? How do you know?
- Where are the known and unknown quantities in the problem you wrote?
- How do you know that your problem is a multiplication or a division situation?

# Reciprocals and Dividing by Fractions

# Lesson Guide

Have students discuss the summary with a partner before turning to a whole class discussion. Use this opportunity to review terminology and correct or clarify misconceptions.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students should be able to connect the summary to the models and numerical methods from previous lessons.

## Formative Assessment

# Summary of the Math: Reciprocals and Dividing by Fractions

**Read and Discuss**

- The fractions $\frac{a}{b}$ and $\frac{b}{a}$, with numerators and denominators inverted, are called
*reciprocals*of each other. The key property of reciprocals is that their product is always 1.

$\frac{a}{b}\times \frac{b}{a}=\frac{a\times b}{b\times a}=1$Reciprocals are also called

$\frac{3}{4}\times 5=\frac{15}{4}$*multiplicative inverses*. This name refers to the fact that if you multiply a fraction by a number, and then multiply the result by the reciprocal of the fraction, the result is “undone.” For example:

$\frac{15}{4}\times \frac{4}{3}=\frac{15}{3}=5$- You can use the properties of reciprocals to help you divide by fractions. The general rule is:
*To divide by a fraction, multiply by the reciprocal of the fraction*. Algebraically, the rule looks like this: $\frac{a}{b}\xf7\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}$

Can you:

- Solve a problem that involves dividing a fraction by a fraction?
- Determine which operation is needed to solve a word problem?
- Explain what reciprocals are and how to use them to help you solve division problems involving fractions?

# 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 the everyday situations that involve 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.

**An example of an everyday situation that involves fraction division is …**