- Author:
- Chris Adcock
- Subject:
- Algebra
- Material Type:
- Lesson Plan
- Level:
- Middle School
- Grade:
- 6
- Provider:
- Pearson
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Text/HTML

# Using Rate To Determine Speed & Distance

## Overview

In this lesson, students watch a video of a runner and express his speed as a rate in meters per second. Students then use the rate to determine how long it takes the runner to go any distance.

# Key Concepts

*Speed* is a rate that is expressed as distance traveled per unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed. The measures of speed, distance, and time are all related. The relationship can be expressed in three ways: *d* = *rt*, *r* = *dt*, *t* = *dr*.

# Goals and Learning Objectives

- Explore speed as a rate that measures the relationship between two aspects of a situation: distance and time.
- In comparing distance, speed, and time, understand how to use any two of these measures to find the third measure.

# Jason Running

# Lesson Guide

Have students watch the video and try to answer the questions. Students may need to watch it more than once to figure out the two aspects of the situation tracked by the double number line.

Discuss the questions as a class.

ELL: Help ELLs build their mathematical vocabulary by continually modeling the use of new terms in the context of classroom work and activities. If time allows, provide an opportunity for ELLs to build their own sentences using the new vocabulary.

# Mathematics

To successfully approach the Work Time problems, students need to be able to answer and understand all three of the following questions:

- What two quantities do you see in this video (tracked by the number lines)?
- One number line tracks the distance Jason runs.
- The other number line tracks the time that he runs.

- How does the double number line show the relationship between the two quantities?
- The double number line shows how much time has passed for each given distance, and it shows the distance Jason has run for each given time.
- Dividing any distance in meters on the double number line by the corresponding time in seconds gives us the following rate: Jason runs at a speed of 5 meters per second.

- What is Jason's speed in meters per second?
- 5 meters per second.

Discuss these questions with the class. Keep the discussion short, but make sure that all students understand that Jason's speed is 5 meters per second. They will use this rate during Work Time.

## Opening

# Jason Running

Watch the video and think about the following questions.

- What two quantities in this video can be measured?
- How does the double number line track the relationship between the quantities?

VIDEO: Jason Running

# Math Mission

# Lesson Guide

Discuss the Math Mission.

Students will use Jason's speed, expressed as the rate 5 meters per second, to find how long it takes him to go any distance.

SWD: Students with disabilities may have difficulty with the complexity of this Math Mission. Consider how you may provide all students access to this information in a manageable way. One option is to break down the task into discreet, sequential steps (first, students will find the ratio; then, they will use that ratio to help them calculate the time to travel a given distance).

## Opening

Explain how to use Jason’s speed to find how long it takes him to go any distance.

# Jason’s Running Speed

# Lesson Guide

Have students who finish early:

- Prepare to present their work to the class.
- Find another way to solve this problem, or show their solution using another representation.

ELL: For the Work Time activity, make sure you demonstrate and orally explain the activity step by step to ensure that ELLs understand what they are being asked to do.

# Mathematical Practices

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

- As students work, look for students who make sense of the problem situation and understand the relationship between speed, distance, and time.

**Mathematical Practice 5: Use appropriate tools strategically.**

- Identify students who appropriately use the double number line tool or a table to represent the relationship between distance and time.

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

- Watch for students who attend to precision when calculating or when labeling their double number lines and tables.

# Interventions

**Student has difficulty getting started.**

- Describe the problem in your own words to your partner.
- What information do you know?
- What are you trying to find?
- What does the rate 5 meters per second mean?
- Can you use a double number line or a table to help you?
- Can you try solving a simpler problem?

**Student has an incorrect solution.**

- Multiplies 5 meters per second by 240 meters:
- Does your answer 1,200 seconds make sense? How many minutes are equal to 1,200 seconds?
- Can you use a double number line or a table to support your answer?
- Explain why you multiplied the rate 5 meters per second by 240 meters.

**Student has a solution.**

- Explain your strategy for solving the problem.
- Where do you see the rate in your [diagram, double number line, table]?
- Where do you see the answer in your [diagram, double number line, table]?
- Could you have used a different method to solve this problem? Explain.

# Answers

- It will take Jason 48 seconds to run 240 meters.
- Jason can run 275 meters in 55 seconds.
- Students' answers should include an estimate, the quantities involved, a representation of their solution, equations, and two complete sentences.

## Work Time

# Jason’s Running Speed

Jason runs at a constant speed of 5 meters per second.

- How much time does it take Jason to run 240 meters?
- How many meters can Jason run in 55 seconds?
- Include the following in your answers:
- An estimate of what you think the answers might be
- The quantities involved in the problem
- A diagram, table, double number line, or other representation that shows why your solution makes sense
- Your equations, your work, and your solutions
- Two complete sentences that answer the questions

## Hint:

Multiply the rate 5 meters per second by the amount of time to find the distance traveled. For example, to find the distance traveled in 20 seconds, multiply 5 meters per second by 20 seconds:

20 seconds • 5 meters/second = 100 meters

# Prepare a Presentation

# Preparing for Ways of Thinking

As students work, look for responses in which students:

- Multiply speed by distance. 5 meters 1 second⋅240 meters = 1,200 seconds
- Correctly divide distance by speed. 240 meters ÷ 5 meters1 second = 48 seconds
- Find the rate 0.2 seconds per meter and use it to solve the first problem.

1 second 5 meters = 0.2 seconds per meter

0.2 seconds 1 meters⋅240 meters = 48 seconds

# Challenge Problem

## Answers

Michael Duane Johnson ran the 200-meter race at a speed of 10.35 meters per second and the 400-meter race at a speed of 9.20 meters per second.

- Possible solution: His 200-meter speed was faster because he ran a longer distance—10.35 meters compared to 9.20 meters—in the same time (1 second).

## Work Time

# Prepare a Presentation

Prepare a presentation about your method for solving the problem.

- Explain what you did differently to find the time compared with finding the distance.
- Use your work to support your explanation.

# Challenge Problem

At the 1996 Olympics, Michael Duane Johnson set world records for the 200-meter and 400-meter races. He ran the 200-meter race in 19.32 seconds and the 400-meter race in 43.49 seconds.

- Calculate his speed for each race using a rate.
- In which race did he have the fastest speed? Justify your thinking.

# Make Connections

# Lesson Guide

Select a variety of student work with an eye to drawing out correspondences between sensible methods. By the end of the discussion, students should understand each other's methods and why some methods did not work.

SWD: The idea that three different presentations of data represent the same information may be challenging for students with disabilities. Review and reinforce this idea whenever possible to help students comprehend and recall this important concept.

# Mathematics

Ask students who used different methods (for example, a table, a number line, or division by the rate) to present. As part of the discussion, challenge students to compare their representations of the problem.

- How did [Names] make sense of the problem, and how did they model the problem situation mathematically? What is similar about their approaches? What is different?
- Which method helps you to see the structure of the mathematics the most clearly? Which method makes the most sense to you?
- [Name] multiplied 0.2 seconds per meter by 240 meters. Why does this method work? How does this method compare to dividing 240 meters by 5 meters per second?
- How can you find the distance using the rate 0.2 meter per second? How does this approach compare to using the rate 5 meters per second to find the distance?
- If no students notice, you might point out that dividing any number on the distance number line by 5 results in the corresponding number on the time number line.

At the end of the discussion, remind students that when a rate is something we use many times, sometimes it has a special name. Speed is the rate that represents distance over time. Population density is a rate that represents population per area.

## Performance Task

# Ways of Thinking: Make Connections

- Take notes during the class discussion about how to solve for both distance and time.
- Answer questions about the methods you used to solve the problem.

## Hint:

As your classmates present, ask questions such as:

- Where do you see the rate 5 meters per second in your methods? Where do you see the answer?
- What operation did you use and why?
- Can you explain your solution in terms of the unit: what are the units of the rate, and how did the seconds cancel out?

# Distance Equals Rate Times Time

# Interventions

**Student has difficulty getting started.**

- Describe the problem in your own words to your partner.
- What information do you know?
- What are you trying to find?
- Can you use a double number line or a table to help you?
- Can you try solving a simpler problem?

**Student has a solution.**

- Explain your strategy for solving the problem.
- Where do you see the rate in your [diagram, double number line, table]?
- Where do you see the answer in your [diagram, double number line, table]?
- Could you have used a different method to solve this problem? Explain.

# Answers

- Emma's walking rate is 1/20 of a mile per minute, or 20 minutes per mile.
- Emma can walk 4.5 miles in 90 minutes.
- Emma can walk 0.75 mile in 15 minutes.
- Students will make a double number line to show the relationship between the numbers.

## Work Time

# Distance Equals Rate Times Time

Emma walked 3 miles in 60 minutes.

- What was her walking speed in terms of a rate?
- If she walks 4.5 miles at the same rate, how long will she walk?
- If she walks for 15 minutes at the same rate, how far will she walk?
- Make a double number line to show the relationships between the numbers.

## Hint:

- What are the quantities in the situation?
- How can you find the walking speed in terms of a rate?
- Did you use the expression
*d*=*rt*to help you solve for time or distance?

# Understanding Speed

# A Possible Summary

Speed is a rate that is calculated by dividing distance by time. You can multiply time by speed to find any distance. The quantities speed, distance, and time are all related. The relationship can be expressed in three ways: *d* = *rt*, *r* = *dt*,* t* = *dr*.

# Additional Discussion Points

Speed is a rate that is expressed as the distance traveled in a unit of time. Miles per hour, laps per minute, and meters per second are all examples of units for speed.

## Formative Assessment

# Summary of the Math: Understanding Speed

Summarize the mathematics of speed and how speed relates to rate.

## Hint:

Check your summary:

- Does your summary explain what
*speed*means, and does it give an example? - Does your summary include the term
*rate*? - Does your summary show the relationship among speed, distance, and time in three different ways?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review students' reflections.

If you find some reflections interesting enough to pursue later, you can save them and share them with the class when appropriate.

## 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 I learned about speed that really helps me solve problems is …**