# Using Rates to Determine Efficiency

## Overview

Students watch a video in which two students discuss the problem of how to compare fuel efficiency. Students then analyze the work of the two students as they use rates to determine fuel efficiency in two different ways.

# Key Concepts

Fuel efficiency is a rate. Fuel efficiency can be expressed in miles per gallon (mpg). This rate is useful for determining how far a vehicle can travel using any number of gallons of gas. Fuel efficiency can also be expressed in gallons per mile (gpm). This rate is useful for determining how many gallons of gas a vehicle uses to travel any number of miles.

The rates miles per gallon and gallons per mile are inverse rates—they both describe the same relationship. For example, the rates 20 miles per gallon and 0.05 gallon per mile both describe the relationship between 300 miles and 15 gallons. The greater the rate in miles per gallon, the better the fuel efficiency. The smaller the rate in gallons per mile, the better the fuel efficiency.

SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. This will help to highlight for students the critical features and/or concepts and will help them to pay close attention to salient information.

# Goals and Learning Objectives

- Explore rate in the context of fuel efficiency.
- Express fuel efficiency as the rate miles per gallon (mpg) and as its inverse, gallons per mile (gpm).
- Use the rate miles per gallon to find the number of miles a car can travel on a number of gallons of gas.
- Use the rate gallons per mile to find the number of gallons of gas used for a number of miles driven.

# Denzel and Jason Discuss Cars

# Lesson Guide

Have students watch the video.

Tell students to watch the video more than once, if needed, to identify the answers to the following questions:

- What quantities do you see in the video?
- What rates might Denzel and Jason be interested in finding?

Expect answers such as price per amount of gas (dollars per gallon), amount of gas per price (gallons per dollar), distance per amount of gas (miles per gallon), and amount of gas per distance (gallons per mile).

ELL: When showing the video, monitor that ELLs are following the meaning of what is presented. If necessary, pause the video and allow them to ask clarifying questions. Alternatively, ask questions to check for understanding of what they are watching.

# Mathematics

Briefly discuss the term *fuel efficiency*. Make sure students understand that fuel efficiency is a measure of the relationship between the amount of gas consumed and the distance traveled. The video shows the price of the gas, but this information is not relevant to the fuel efficiency of the car.

Point out that the mathematics students used in the previous lesson can help them with today's lesson.

## Opening

# Denzel and Jason Discuss Cars

Watch the video.

- What information does the video tell you about the situation?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will use data from the video to express a car's fuel efficiency as a rate. They will think about two different rates that can be used for one situation.

## Opening

Express the fuel efficiency of a car as a rate.

# Find Fuel Efficiency

# Lesson Guide

Draw students' attention to Denzel's and Jason's work. Tell students that their job is to finish the work and that their answers should include both a number and a unit.

# NOTE:

Most countries use the unit kilometers per liter to represent fuel efficiency. In the United States, however, we usually use miles per gallon.

# Mathematical Practices

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

- As students work, pay attention to how students respond to the question “Who is right?” Some students will choose Denzel or Jason rather than realizing that both of the students are correct. Identify students who choose either Denzel or Jason and have them use mathematical reasoning to defend their thinking in Ways of Thinking.

# Interventions

**Student has difficulty getting started.**

- Describe the problem in your own words to your partner.
- What are you trying to find?
- What information do you know?
- How can you use what you learned in the previous lesson about unit price to help you?

**Student has a solution.**

- Explain your strategy for solving the problem.
- How can you define the fuel efficiency of a car? Explain your thinking.
- Is there more than one way to define fuel efficiency? Explain.

# Answers

- Denzel: 20 mpg
- Jason: 0.05 gpm
- Both Denzel and Jason have correctly calculated the fuel efficiency of Jason's father's car.
- Jason's dad could travel 320 miles using 16 gallons of gas.
- After traveling 150 miles, Jason's dad's car would use 7.5 gallons of gas.
- Denzel's way of expressing the rate is most helpful in solving the first problem. Jason's way of expressing the rate is most helpful in solving the second problem.

## Work Time

# Find Fuel Efficiency

Denzel and Jason took different approaches to solving the fuel efficiency problem. Look at each approach and then answer the following questions. Remember to specify the appropriate unit, such as miles per gallon or gallons per mile.

- What is the unit rate in Denzel’s calculation?
- What is the unit rate in Jason’s calculation?
- Who calculated the fuel efficiency correctly?
- How far could Jason’s dad travel using 16 gallons of gas?
- Jason’s dad drove the car 150 miles. How much gas did the car use?
- Which rate—Denzel’s or Jason’s—was most helpful in answering each question?

## Hint:

- What are the quantities involved in the problem?
- What operation do you need to use in order to find the rate?
- In the last lesson, to find the unit price you divided the total price by a quantity. To determine the fuel efficiency of a car, should you divide miles by gallons or gallons by miles?

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for students who find the following rates:

Denzel:

$\frac{\text{300 miles}}{\text{15 gallons}}$= 20 mpg

Jason:

$\frac{\text{15 gallons}}{\text{300 miles}}$ = 0.05 gpm

Also look for responses in which students confuse the units in the answers—for example, writing 20 gallons per mile or 0.05 miles per gallon.

# Challenge Problem

## Answers

- Yes. Pounds per dollar is the inverse rate of dollars per pound.
- Answers will vary, but students should recognize that in looking at a rate and its inverse, one of the rates will be a decimal less than one and the other will be a number greater than one (unless both numbers are exactly 1). Usually we use the rate that has a number greater than one rather than using the decimal rate. Thus, we use miles per gallon rather than gallons per mile.

## Work Time

# Prepare a Presentation

Summarize Denzel’s and Jason’s thinking about fuel efficiency. Use your work to support your summary.

# Challenge Problem

Miles per gallon and gallons per mile are called *inverse rates*.

- Does dollars per pound have an inverse rate? If so, what is it?
- Write several rates and their inverse rates. Why do you think one form of the rate is usually used?

# Make Connections

# Lesson Guide

Have students share their thinking about Denzel's and Jason's solutions. Let students debate which solution is correct. If you have any students who say both solutions are correct, let them present their work.

Students should understand that fuel efficiency can be represented using miles per gallon (mpg) or gallons per mile (gpm). Point out that miles per gallon is more commonly used than gallons per mile. Tell students that liters per kilometer is the common rate used to describe fuel efficiency in Europe.

Reinforce the fact that a rate compares two quantities, and that the rates can be expressed in two ways:

- Dividing distance (in miles) by gas (in gallons) gives the rate distance per gas used in miles per gallon.
- Dividing gas (in gallons) by distance (in miles) gives the rate gas is used per distance in gallons per mile.

Discuss the fact that students need to look at what a problem is asking in order to decide the best way to express the rate. To find the number of miles, Denzel's way of expressing the rate is more helpful. To find the number of gallons, Jason's way of expressing the rate is more helpful.

During the discussion, look for students who clearly explain why they chose certain rates to compare fuel efficiency, and for students who used units correctly. These students are attending to precision.

SWD: Students with disabilities may need additional support seeing the relationships among problems and strategies. Throughout this unit, keeping anchor charts available and visible will assist them in making connections and working toward mastery. Provide explicit think-alouds comparing strategies and making connections. In addition, ask probing questions to get students to articulate how a peer solved the problem or how one strategy or visual representation is connected or related to another.

# Mathematics

Encourage students to draw conclusions from the math. Ask questions such as the following:

- Is a rate of 22.5 miles per gallon more or less fuel efficient than a rate of 20 miles per gallon? Explain.
- Is a rate of 0.04 gallons per mile more or less fuel efficient than a rate of 0.05 gallons per mile? Explain.
- Why can't you compare only the amount of gas that two cars hold to determine which car is more fuel efficient?
- Why can't you compare only the distance each car travels?
- How did [Name] make sense of the problem? Do you think this is a good approach? Why or why not?
- Why are miles per gallon and gallons per mile called inverse rates?

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ thinking with regard to expressing rates for gallons and miles.

## Hint:

As your classmates present, ask questions such as:

- What are the quantities in this situation?
- What operation did you use to find the unit rate? Explain your thinking.
- Does your answer make sense? Explain.
- When would you use Denzel’s way of expressing the rate?
- When would you use Jason’s way of expressing the rate?

# Comparing Fuel Efficiency

# Lesson Guide

# Interventions

**Student has difficulty getting started.**

- Describe the problem in your own words to your partner.
- What are you trying to find?
- What information do you know?
- How can you use what you learned in the previous lesson about unit price to help you?

# Answer

- Denzel's mom's car is more fuel efficient than Jason's dad's car because Denzel's mom's car gets 22.5 mpg, while Jason's dad's car gets 20 mpg.

## Work Time

# Comparing Fuel Efficiency

Denzel’s mom’s car can travel 450 miles using 20 gallons of gas. You know that the fuel efficiency of Jason’s dad’s car is about 20 miles per gallon.

- Which car is more fuel efficient? Defend your answer.

## Hint:

- What are the quantities in the problem?
- How can you express the fuel efficiency of Denzel’s mom’s car in miles per gallon?
- How can you compare the two rates?

# Motorcycle Fuel Efficiency

# Lesson Guide

# Interventions

**Student has difficulty getting started.**

- Describe the problem in your own words to your partner.
- What are you trying to find?
- What information do you know?
- How can you use what you learned in the previous lesson about unit price to help you?

# Answers

- The motorcycle is more fuel efficient. Explanations will vary.
- The rate is 60 miles per gallon.
- The rate is 0.0167 gallons per mile.

## Work Time

# Motorcycle Fuel Efficiency

Jason’s brother has a motorcycle.

The motorcycle can travel 210 miles using 3.5 gallons of gas.

- Do you think its fuel efficiency will be more or less than that of Jason’s dad’s car? Explain your thinking.
- Express the fuel efficiency rate in miles per gallon.
- Express the fuel efficiency rate in gallons per mile.

## Hint:

- What are the two quantities in the problem?
- If you want to find the rate miles per gallon (mpg), or miles divided by gallons, what quantity must be in the numerator?
- If you want to find the rate gallons per mile (gpm), or gallons divided by miles, what quantity must be in the numerator?

# Characteristics of Rate

# Lesson Guide

Be sure students understand the following information from the summary.

- Fuel efficiency in miles per gallon (mpg) is a rate. This rate is useful for determining how many miles a vehicle can travel using any number of gallons of gas.
- Fuel efficiency in gallons per mile (gpm) is also a rate. This rate is useful for determining how many gallons of gas a vehicle needs to travel any number of miles.
- Miles per gallon and gallons per mile are inverse rates.
- The larger the rate in miles per gallon, the better the fuel efficiency.
- The smaller the rate in gallons per mile, the better the fuel efficiency.
- The rates 20 miles per gallon and 0.05 gallon per mile describe the same situation. They both describe the relationship between 300 miles and 15 gallons.

## Formative Assessment

# Summary of the Math: Characteristics of Rate

**Read and Discuss**

- Fuel efficiency can be described using two rates: miles per gallon and gallons per mile. More miles per gallon means greater fuel efficiency; but fewer gallons per mile also means greater fuel efficiency.
- Fuel efficiency in miles per gallon (mpg) is a rate. You will find mpg useful for finding how far you can go on any number of gallons of gas.
- Fuel efficiency in gallons per mile (gpm) is also a rate. You will find gpm useful for finding how many gallons of gas you need to go any number of miles.
- The smaller the rate in gpm, the greater the fuel efficiency.
- The rates 20 mpg and 0.05 gpm describe the same situation. They both describe the relationship between 300 miles and 15 gallons.

## Hint:

Can you do the following?

- Explain what
*miles per gallon*and*gallons per mile*mean. - Include the term
*rate*. - Discuss how to define the fuel efficiency of a car.

# Reflect On Your Work

# Lesson Guide

Have each student do a quick reflection before the end of class. Review the reflections.

If any reflections look interesting enough to pursue later, make a note of them and consider sharing them with the class if you have time.

## 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.

**Something new I learned today about rates is …**