# Solving Problems Involving Proportions

## Overview

# Lesson Overview

Students solve problems using equations of the form *x* + *p* = *q* and *px* = *q*, as well as problems involving proportions.

# Key Concepts

Students will extend what they know about writing expressions to writing equations. An equation is a statement that two expressions are equivalent. Students will write two equivalent expressions that represent the same quantity. One expression will be numerical and the other expression will contain a variable.

It is important that when students write the equation, they define the variable precisely. For example, *n* represents the number of minutes Aiko ran, or *x* represents the number of boxes on the shelf.

Students will then solve the equations and thereby solve the problems.

Students will solve proportion problems by solving equations. This makes sense because a proportion such as $\frac{x}{a}=\frac{b}{c}$ is really just an equation of the form *xp* = *q* where $p=\frac{1}{a}$ and $q=\frac{b}{c}\text{.}$

Students will also compare their algebraic solutions to an arithmetic solution for the problem. They will see, for example, that a problem that might be solved arithmetically by subtracting 5 from 78 can also be solved algebraically by solving *x* + 5 = 78, where 5 is subtracted from both sides—a parallel solution to subtracting 5 from 78.

# Goals and Learning Objectives

- Use equations of the form
*x*+*p*=*q*and*xp*=*q*to solve problems. - Solve proportion problems using equations.

ELL: ELLs may have difficulty verbalizing their reasoning, particularly because word problems are highly language dependent. Accommodate ELLs by providing extra time for them to process the information. Note that this problem is a good opportunity for ELLs to develop their literacy skills since it incorporates reading, writing, listening, and speaking skills. Encourage students to challenge each others' ideas and justify their thinking using academic and specialized mathematical language.

# Cost of a Pizza

# Lesson Guide

Have students work in pairs on the task.

Point out the statement *p* = the price in dollars of one Super Combo pizza. Explain that this statement is called “defining the variable” and should always be the first step when using an equation to solve a word problem.

Ask students:

- How can you use the question in the word problem to help you define the variable? (
*Answer: Let the variable*p*equal the price in dollars of one Super Combo pizza.*) - What equation can you write to solve the problem? (
*Answer: 4*p*= 63*) - The equation shows that the two expressions are equivalent. What does each expression represent?
*(Answer: the cost of 4 Super Combo pizzas*)

Have a student show the solution to the equation using the multiplication property of equality. Then ask:

- How could you solve the problem using arithmetic?
- How is the algebraic solution like the arithmetic solution?

## Opening

# Cost of a Pizza

Let *p* equal the price in dollars of one Super Combo pizza.

- Write an equation that you could use to find that price.
- How much does each pizza cost?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will write and solve equations in order to solve real-world mathematical problems.

## Opening

Write and solve equations in order to solve real-world mathematical problems.

# The Track Team

# Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, 5, and 6.

# Mathematical Practices

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

Listen for student justifications for why the answers to the problems make sense.

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

Identify any student errors or misconceptions that would be useful to discuss.

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

Look for students who define the variable precisely (for example, *b* = the number of boys on the track team) and for students who could be more precise (for example, *b *= boys).

# Interventions

**Student does not know how to define the variable.**

- What does the problem ask you to find out?
- Choose a variable. Let it represent what the problem asks you to find out.

**Student does not know where to begin to write an equation for the track team problem.**

- You need two equivalent expressions to make an equation.
- Does the problem provide two ways to represent the number of girls on the team?

SWD: During this time, if you are the only teacher in the classroom, you will need to be available to confer with students who need additional prompting. If you have a professional who supports students with disabilities in the classroom during this time, have him or her pull a small group. There may be struggling students who are unable to revise their work without re-teaching or guided support.

# Answers

- Let
*b*= the number of boys on the track team. *b*+ 5 = 78*b*+ 5 − 5 = 78 − 5*b*= 73

There are 73 boys on the track team.- Explanations will vary. Possible explanation: 78 is 5 more than 73.

## Work Time

# The Track Team

There are 78 girls on the track team. This is 5 more than the number of boys on the team. How many boys are on the track team?

- Choose a variable to represent what you want to find out.
- Write an equation that you can use to solve the problem.
- Solve the equation.
- Think about your answer. Does it make sense? Explain.

## Hint:

Let b = the number of boys on the team. What expression has the same value as 78?

# Write and Solve Equations

# Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, 5, and 6.

# Mathematical Practices

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

Listen for student justifications for why the answers to the problems make sense.

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

Identify any student errors or misconceptions that would be useful to discuss.

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

Look for students who define the variable precisely (for example, *n* = the unknown number) and for students who could be more precise (for example, *n* = a number).

# Interventions

**Student does not know how to define the variable.**

- What does the problem ask you to find out?
- Choose a variable. Let it represent what the problem asks you to find out.

**Student does not understand why the answer to the unknown number problem is greater than 14.**

- Think about the product of $\frac{2}{3}$ and a number as being $\frac{2}{3}$ of a number.
- What happens when you multiply a number by 0? By 1? By a number between 0 and 1?

# Answers

- Let
*n*= the unknown number. - $\frac{2}{3}$
*n*= 14 - $\frac{3}{2}$ • $\frac{2}{3}$
*n*= $\frac{3}{2}$ • 14*n*= 21

The number is 21. - Explanations will vary. Possible explanation: 21 is greater than 14. 14 is $\frac{2}{3}$ of 21.

## Work Time

# Write and Solve Equations

The product of $\frac{2}{3}$ and a number is 14.

- Choose a variable to represent what you want to find out.
- Write an equation that you can use to solve the problem.
- Solve the equation.
- Think about your answer. Does it make sense? Explain.

## Hint:

- Choose a variable such as n to represent the number.
- What equation can you write?
- Think about whether your answer should be greater than or less than 14.

# Mixing Paint

# Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, 5, and 6.

# Mathematical Practices

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

Listen for student justifications for why the answers to the problems make sense.

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

Identify any student errors or misconceptions that would be useful to discuss.

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

Look for students who define the variable precisely (for example, *x* = the amount of blue paint in cups) and for students who could be more precise (for example, *x* = paint).

# Interventions

**Student does not know how to define the variable.**

- What does the problem ask you to find out?
- Choose a variable. Let it represent what the problem asks you to find out.

**Student does not know how to set up a proportion for the paint problem.**

- Let the cups of blue be in the numerators.
- Let the cups of yellow be in the denominators.
- Be sure the cups from the same mixture are on the same side of the equal sign.

# Answers

- Let
*x*= the number of cups of blue paint in the new mixture. - $\frac{4}{3}=\frac{x}{5}$
- $\frac{4}{3}\cdot 5=\frac{x}{5}\cdot 5$
*x*= 6$\frac{2}{3}$

Carlos will need 6$\frac{2}{3}$cups of blue paint in the new mixture. - Explanations will vary. Possible explanation: Carlos used more blue paint than yellow paint in a ratio of 4 to 3. The answer of 6$\frac{2}{3}$cups of blue paint is more than 5 cups of yellow paint and maintains the ratio of 4 to 3.

## Work Time

# Mixing Paint

Carlos mixed some green paint by using 4 cups of blue paint and 3 cups of yellow paint. He used all of the green paint and wants to make some more. How many cups of blue paint should he mix with 5 cups of yellow paint to get the same color green?

- Choose a variable to represent what you want to find out.
- Write an equation that you can use to solve the problem.
- Solve the equation.
- Think about your answer. Does it make sense? Explain.

## Hint:

- What proportion represents the problem?
- Where will you put the 5 in the proportion?
- What will you multiply or divide by to make the equation easier to work with?

# Cost of Cheese

# Lesson Guide

Have students work in pairs on the problems in Tasks 3, 4, 5, and 6.

# Mathematical Practices

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

Listen for student justifications for why the answers to the problems make sense.

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

Identify any student errors or misconceptions that would be useful to discuss.

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

Look for students who define the variable precisely (for example, *x* = the number of pounds of cheese for $25) and for students who could be more precise (for example, *x* = cheese).

# Interventions

**Student does not know how to define the variable.**

- What does the problem ask you to find out?
- Choose a variable. Let it represent what the problem asks you to find out.

**Student does not know where to begin to write an equation for the cheese problem.**

- You need two equivalent expressions to make an equation.
- Can you use a ratio to represent the cost of the cheese?

# Answers

- Let
*x*= the number of pounds of cheese for $25. - $\frac{3}{7}=\frac{x}{25}$
- $\frac{3}{7}\cdot 25=\frac{x}{25}\cdot 25$
*x*= 10$\frac{5}{7}$

You can buy 10$\frac{5}{7}$lb of cheese for $25. - Explanations will vary. Possible explanation: 10$\frac{5}{7}$lb for $25 maintains the ratio of 3 lb for $7.

## Work Time

# Cost of Cheese

If 3 lb of cheese cost $7, how many pounds of the same cheese can you buy for $25?

- Choose a variable to represent what you want to find out.
- Write an equation that you can use to solve the problem.
- Solve the equation.
- Think about your answer. Does it make sense? Explain.

## Hint:

- What proportion represents the problem?
- Where will you put the 25 in the proportion?

# Prepare a Presentation

# Preparing for Ways of Thinking

Identify students who have different solution methods to share during Ways of Thinking.

# Mathematical Practices

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

Listen for student justifications for why the answers to the problems make sense.

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

Identify any student errors or misconceptions that would be useful to discuss.

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

Look for students who define the variable precisely (for example, *b* = the number of toppings on the pizza) and for students who could be more precise (for example, *b* = toppings).

# Challenge Problem

## Answers

- Let
*t*= the number of additional toppings on the pizza.

$\begin{array}{c}13+2.25t=24.25\\ 13-13+2.25t=24.25-13\\ 2.25t=11.25\\ \frac{2.25t}{2.25}=\frac{11.25}{2.25}\\ t=5\end{array}$

There are 5 additional toppings on the pizza.

## Work Time

# Prepare a Presentation

Choose one of the word problems from this lesson to present.

- Explain how you came up with your equation or inequality.
- Explain the steps you took to solve it.

# Challenge Problem

Several friends share a pizza. The price of a plain cheese pizza is $13.00. Each additional topping costs $2.25. The friends pay $24.25 for their pizza.

- Write an equation to represent this situation and then solve it to find out how many additional toppings are on the pizza.

# Make Connections

# Lesson Guide

Discuss each problem, one at a time.

- For the track team problem, allow students who wrote an incorrect equation to share their solution. Allow other students to ask questions and lead the student to the correct equation and its correct solution. Relate the algebraic solution to the arithmetic solution in which you would subtract 5 from 78.

Choose several students to give an explanation for why the answer, 73 boys, makes sense.

Then ask students: Which explanation seems the clearest to you? Why?

- For the unknown number problem, have a student who used multiplication and a student who used division present their solutions to the equation.

Ask for students to explain why the number is greater than 14.

Then ask students: Which explanation seems the clearest to you? Why?

- For the paint problem and the cheese problem, students will need to solve a proportion. For the paint problem, if students used the proportion $\frac{4}{3}=\frac{x}{5}$, they should be able to solve it since they have solved proportions like this one before.

Look for students who wrote the proportion as $\frac{3}{4}=\frac{5}{x}$ with the variable in the denominator. Allow students to discover how they might solve the equation.

Solution 1

Rewrite the ratios as blue to yellow instead of yellow to blue. This equation, $\frac{3}{4}=\frac{5}{x}$, is equivalent to $\frac{4}{3}=\frac{x}{5}$. Students should remember from their study of ratios that the ratios can be written in either order, as long as both ratios are written in the same order.

$\begin{array}{cc}\frac{4}{3}\cdot 3=\frac{x}{5}\cdot 3& \text{Multiplication property of equality}\\ 4=\frac{3x}{5}& \\ 4\cdot 5=\frac{3x}{5}\cdot 5& \text{Multiplication property of equality}\\ 20=3x& \\ \frac{20}{3}=\frac{3x}{3}& \text{Multiplication property of equality}\\ x=6\frac{2}{3}& \end{array}$

Solution 2

$\begin{array}{cc}\frac{3}{4}=\frac{5}{x}& \\ \frac{3}{4}\cdot x=\frac{5}{x}\cdot x& \text{Multiplication property of equality}\\ \frac{3}{4}x=5& \\ \frac{4}{3}\cdot \frac{3}{4}x=5\cdot \frac{4}{3}& \text{Multiplication property of equality}\\ x=6\frac{2}{3}& \end{array}$

Have students discuss which method they prefer and why.

Have a similar discussion for the cheese problem.

For the Challenge Problem, allow students who attempted the problem to share their solutions. If no one attempted it, work through the solution as a class, eliciting as much as possible from the students.

SWD: Help students break down the problems into smaller parts, and assist them in identifying relevant information. Use a think aloud to help students understand how you created the proportion for the paint problem and the cheese problem.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the approaches your classmates used to write equations to solve the problems.

## Hint:

As your classmates present, ask questions such as:

- Describe the process that you used to write the equation and solve it.
- How did you decide where to put the 25 in the proportion?
- Is that the only correct way to write the proportion?

# Reflect On Your Work

# Lesson Guide

Have each student write a quick reflection before the end of the class. Review the reflections to learn if students understand how to write equations to solve problems.

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 primary language to discuss in their preferred language.

## 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.

**When I write an equation for a problem, I …**