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# Expressing Ratios As A Unit Rate

## Overview

Students write the relationship between two fractions as a unit rate and use unit rates and the constant of proportionality to solve problems involving proportional relationships.

# Key Concepts

- In situations where there is a constant rate involved, the unit rate is a constant of proportionality between the two variable quantities and can be used to write a formula of the form
*y*=*kx*. - A given constant rate can be simplified to find the unit rate by expressing its value with a denominator of 1.
- The ratios of two fractions can be expressed as a unit rate.

# Goals and Learning Objectives

- Express the ratios of two fractions as a unit rate.
- Understand that when a constant rate is involved, the unit rate is the constant of proportionality.
- Use the unit rate to write and solve a formula of the form
*y*=*kx*.

# Construct and Critique Arguments

# Mathematical Practices in Action

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

Have students watch the video and listen to the dialogue between Karen and Sophie as they work together to figure out how many miles a person runs in an hour if that person runs $\frac{1}{4}$ of a mile in $\frac{1}{2}$ of an hour at a constant rate. This video shows students engaged in Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Karen and Sophie first try solving the problem with decimals and they each make arguments at various points along the way. For example, Karen makes the argument that you would divide 60 by 8 to find the number of minutes in $\frac{1}{8}$ of an hour.

Karen and Sophie then decide that using decimals was a bit complicated and decide to try solving the problem with fractions. Again we see Karen and Sophie making arguments. For example, Karen makes an argument for setting up the problem as $\frac{\text{\hspace{0.17em}}\frac{1}{4}\text{\hspace{0.17em}}}{\text{\hspace{0.17em}}\frac{1}{8}\text{\hspace{0.17em}}}$.

Have students talk about the arguments that Karen and Sophie made throughout the video as they solved the problem with decimals and then with fractions. Also, have students point out places where either Karen or Sophie helped to correct each other.

In the discussion, elicit that it is important to make arguments and to listen and critique the arguments of others when solving problems. Point out how Karen and Sophie did this so nicely throughout the video.

## Opening

# Construct and Critique Arguments

Watch the video that shows Karen and Sophie constructing and critiquing arguments as they solve a problem.

- What arguments were made in this video as Karen and Sophie solved the problem?
- At what points did either Karen or Sophie critique themselves and realize they needed to adjust their thinking?

VIDEO: Mathematical Practice 3

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will write the relationship between two fractions as a unit rate, and use unit rates to solve problems.

## Opening

Write the relationship between two fractions as a unit rate, and use unit rates to solve problems.

# Marcus’s and Lucy’s Work

# Lesson Guide

Have students work in pairs for all problems and the presentation.

SWD: If students seem unsure of the task, model the steps for solving the problem with them before asking them to do the problems themselves.

# Mathematics

In each problem, students simplify a given rate to find the unit rate and then use the unit rate to find a corresponding value of one quantity for a given value of the other quantity. In each case, one quantity is time (i.e., minutes, hours).

# Interventions

**Student sees simplifying the rate as a division problem involving fractions and is confused about how to divide fractions.**

- A unit rate is a rate per single unit, so it should have a value of 1 in the denominator.
- What do you need to do to the denominator to make 1? Do the same to the numerator, and you’ll have the unit rate—the amount per single minute or hour.

**Student operates on the given values in an intuitive way to get an answer that makes sense without any explicit connection with the mathematics of proportional relationships.**

- What you did worked! Now describe to your partner what you did, including where you got the values you used.
- How does the unit rate show up in your work?
- What two quantities are proportional to each other in the situation, and what is the constant of proportionality? How do you know?

**[common error] Student operates on the given values in an intuitive way and gets an incorrect answer.**

- Does your answer make sense in terms of the situation?
- Can you use the unit rate to write a formula that relates the two quantities in the situation?
- Write an equation showing what you did to get your answer. Now use letters instead of values to represent the two quantities in this situation. What answer do you get if you solve your equation using the new value given in the problem?

SWD: Partner Work is designed to help students consolidate their learning by justifying their solutions to their peers. Students have an opportunity to collaborate with peers as they summarize, analyze, problem solve, explain, and apply new knowledge. Identify students who can explain their thinking, and have them present during the Ways of Thinking section.

# Answers

$\frac{\frac{1}{2}}{\frac{1}{4}}=\frac{4\left(\frac{1}{2}\right)}{4\left(\frac{1}{4}\right)}=\frac{2}{1}$

The unit rate is 2 miles per hour.

Possible answer: Marcus thought of the problem as a proportion with a missing value. He used the ratios between miles and hours to set up the proportion: $\frac{1}{2}$ mile to $\frac{1}{4}$ hour and y miles to 3 hours. Lucy thought of the situation as a proportional relationship between distance and time. She found the constant of proportionality by simplifying the given rate to a unit rate. She used the unit rate to write a formula in the form *y* = *kx* for the proportional relationship.

## Work Time

# Marcus's and Lucy’s Work

Marcus and Lucy worked on the following problem:

Rosa can run $\frac{1}{2}$ mile in $\frac{1}{4}$ hour. If she runs 3 hours at this rate, how many miles will she run?

Marcus solved the problem by setting up a proportion:

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

$\frac{1}{2}=\frac{1}{4}\cdot \frac{y}{3}$ (multiply each side by $\frac{1}{4}$)

$\frac{1}{2}=\frac{1}{12}y$

$12\cdot \frac{1}{2}=y$ (multiply each side by 12)

*y* = 6

Lucy solved the problem by using this equation:

*y* = 2 ⋅ *x*

*y* = 2 ⋅ 3

*y* = 6

- Write the unit rate for the problem in miles per hour.
- Then explain each student’s approach to the problem.

## Hint:

- Why does Marcus write the fraction as follows? $\frac{\frac{1}{2}}{\frac{1}{4}}$
- Why does Marcus divide
*y*by 3? - Why does Lucy use the equation
*y*= 2*x*? What does the 2 in the equation represent? - Why does Lucy substitute 3 for
*x*?

# Drink Milk

# Lesson Guide

Have students work in pairs for all problems and the presentation.

# Interventions

**Student sees simplifying the rate as a division problem involving fractions and is confused about how to divide fractions.**

- A unit rate is a rate per single unit, so it should have a value of 1 in the denominator.
- What do you need to do to the denominator to make 1? Do the same to the numerator, and you’ll have the unit rate—the amount per single minute or hour.

**Student operates on the given values in an intuitive way to get an answer that makes sense without any explicit connection with the mathematics of proportional relationships.**

- What you did worked! Now describe to your partner what you did, including where you got the values you used.
- How does the unit rate show up in your work?
- What two quantities are proportional to each other in the situation, and what is the constant of proportionality? How do you know?

**[common error] Student operates on the given values in an intuitive way and gets an incorrect answer.**

- Does your answer make sense in terms of the situation?
- Can you use the unit rate to write a formula that relates the two quantities in the situation?
- Write an equation showing what you did to get your answer. Now use letters instead of values to represent the two quantities in this situation. What answer do you get if you solve your equation using the new value given in the problem?

# Answers

- $\frac{\frac{1}{4}}{2\frac{1}{2}}=\frac{\frac{1}{4}}{\frac{5}{2}}=\frac{\frac{2}{5}\left(\frac{1}{4}\right)}{\frac{2}{5}\left(\frac{5}{2}\right)}=\frac{1}{10}$
The unit rate is 0.1 gallon per minute.

- 3 ⋅ 0.1 = 0.3

Marcus will drink 0.3 gallon of milk.

## Work Time

# Drink Milk

Marcus was thirsty. He drank $\frac{1}{4}$ gallon of milk in $2\frac{1}{2}$ minutes.

- Write the unit rate in gallons per minute.
- If Marcus drinks for 3 minutes at this rate, how much milk will he drink?

## Hint:

How can you use the unit rate to set up an equation that represents the amount of milk Marcus can drink in 3 minutes?

# Skateboarding

# Lesson Guide

Have students work in pairs for all problems and the presentation.

# Interventions

**Student sees simplifying the rate as a division problem involving fractions and is confused about how to divide fractions.**

- A unit rate is a rate per single unit, so it should have a value of 1 in the denominator.
- What do you need to do to the denominator to make 1? Do the same to the numerator, and you’ll have the unit rate—the amount per single minute or hour.

**Student operates on the given values in an intuitive way to get an answer that makes sense without any explicit connection with the mathematics of proportional relationships.**

- What you did worked! Now describe to your partner what you did, including where you got the values you used.
- How does the unit rate show up in your work?
- What two quantities are proportional to each other in the situation, and what is the constant of proportionality? How do you know?

**[common error] Student operates on the given values in an intuitive way and gets an incorrect answer.**

- Does your answer make sense in terms of the situation?
- Can you use the unit rate to write a formula that relates the two quantities in the situation?
- Write an equation showing what you did to get your answer. Now use letters instead of values to represent the two quantities in this situation. What answer do you get if you solve your equation using the new value given in the problem?

# Answers

- $\frac{3\frac{1}{3}}{\frac{1}{2}}=\frac{\frac{10}{3}}{\frac{1}{2}}=\frac{20}{3}$
The unit rate is approximately 6.66 miles per hour.

- $1\frac{1}{2}\times \frac{20}{3}=10$
The skateboarder will travel 10 miles.

## Work Time

# Skateboarding

A skateboarder traveled $3\frac{1}{3}$ miles in $\frac{1}{2}$ hour.

- Write the unit rate in miles per hour.
- If the skateboarder rides for 1$\frac{1}{2}$ hours at this rate, how far will he or she travel?

## Hint:

How can you use the unit rate to set up an equation that represents the number of miles the skateboarder can travel in 1 1/2 hours?

# Prepare a Presentation

# Preparing for Ways of Thinking

Listen and look for students who:

- [common error] Have trouble dividing fractions, leading to incorrect unit rates
- Discuss and debate about whether the unit rate and the constant of proportionality are the same thing
- Discuss and debate about whether the quantities are proportional to one another and then refer to the rate as evidence that they are
- Are clear about the role of the unit rate in relating the quantities
- Recognize the quantities in the situation as variable

# Challenge Problem

## Answers

Answers will vary. Possible methods are likely to include identifying a constant of proportionality and using it to write a formula, setting up a proportion, or simply multiplying to “get the answer.”

## Work Time

# Prepare a Presentation

Choose one of the problems you worked on today. Prepare a presentation in which you describe and explain the steps you used to find the solution.

# Challenge Problem

Solve one of the problems you did today in a different way. Describe each method you used to solve the problem.

# Make Connections

# Mathematics

Show work from students who used a direct computational approach to find out how much milk Marcus drinks and how far the skateboarder rides. Have other students describe what is happening in each step, including what the elements and operations mean. Ask students the following:

- Why did [student] multiply here?
- Where did this value come from?
- Could there be a different value here? What is another value that would make sense?
- Is there a unit rate in this computation?

Then choose as presenters one or more students who wrote a formula to solve the same problems. Ask where their k-value came from, and then ask if the unit rate and the constant of proportionality play the same role in the relationship between the quantities.

ELL: Show these questions in writing and consider providing sentence frames such as the following:

- “I multiplied because ... .”
- “The value came from ... .”
- “It shows ... .”
- “(Name) organized his or her thoughts in the following way: He or she ... .”
- “That makes sense to me because ... .”
- “The structure of the mathematics was brought up by ... .”

Post these sentence frames as a reference for future lessons.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ descriptions and explanations of the methods they used to solve the problems.

## Hint:

As your classmates present, ask questions such as:

- How did you decide what to do first?
- How does your method compare to the methods that Marcus and Lucy used to solve the problem about Rosa?
- Why did you choose that method to solve the problem?
- What is the unit rate?
- How did you use the unit rate to solve the problem?
- What equation could you write if you wanted to find out how far the skateboarder can travel in
*x*hours—that is, in any number of hours?

# Unit Rates and the Constant of Proportionality

# A Possible Summary

In situations where there is a constant rate involved, the unit rate is a constant of proportionality between the two variable quantities and can be used to write a formula of the form *y* = *kx*. A given constant rate can be simplified to find the unit rate by expressing its value with a denominator of 1.

# Additional Discussion Points

Discuss the following points if time allows:

- The ratios of two fractions can be expressed as a unit rate.
- The unit rate is equivalent to the constant of proportionality.

ELL: Ask some questions to all students, but especially to ELLs, to check for understanding before moving on to the summary.

## Formative Assessment

# Summary of the Math: Unit Rates and the Constant of Proportionality

Write a summary about using unit rates and the constant of proportionality to solve problems involving proportional relationships.

## Hint:

Check your summary.

- Do you explain how to find a unit rate?
- Do you describe the relationship between the unit rate and the constant of proportionality in a proportional relationship?
- Do you describe at least two different ways to solve problems involving proportional relationships?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out what connections students see between the unit rate and the constant of proportionality.

## 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 think about problems involving proportional relationships, I see these connections between the unit rate and the constant of proportionality …**