This lesson is optional. The goal of this lesson is to apply students’ understanding of circumference to calculate how far a wheel travels when it rolls a certain number of times. This relationship is vital for how odometers and speedometers work in vehicles.
In previous lessons, students saw that the relationships between radius, diameter, and circumference of different circles are proportional relationships. In this lesson, they notice that the circumference of a circle is the same as the distance a wheel rolls forward as it completes one rotation. Next, they see that there is also a proportional relationship between the number of times a wheel rotates and the distance the wheel travels. The last activity examines the relationship between the speed a vehicle is traveling and the number of rotations of the tires in a given amount of time.
Students make use of the structure of a proportional relationship as they work toward describing the relationship between the number of rotations of a wheel and the distance the wheel travels with an equation (MP7).
Lesson overview
 5.1 Warmup: A Rope and a Wheel (5 minutes)
 5.2 Optional Activity: Rolling, Rolling, Rolling (15 minutes)

5.3 Optional Activity: Rotations and Distance (15 minutes)
 Includes "Are you Ready for More?" extension problem
 5.4 Optional Activity: Rotations and Speed (15 minutes)
 Lesson Synthesis
 5.5 Cooldown: Biking Distance (5 minutes)
Learning goals:
 Compare wheels of different sizes and explain (orally) why a larger wheel needs fewer rotations to travel the same distance.
 Generalize that the distance a wheel rolls in one rotation is equal to the circumference of the wheel.
 Write an equation to represent the proportional relationship between the number of rotations and the distance a wheel travels.
Learning goals (student facing):
 Let’s explore how far different wheels roll.
Learning targets (student facing):
 If I know the radius or diameter of a wheel, I can find the distance the wheel travels in some number of revolutions.
Required materials:
 Blank paper
 Cylindrical household items
 Receipt tape
 Rulers
Required preparation:
 You can reuse the same cylindrical household items from a previous lesson.
 Again, each group needs 3 items of relatively different sizes; however, it is not as important to include a wide variety of sizes.
 Because of the restrictions of paper size, you may want to forego using the larger objects (such as the paper plate) in this activity.
 Prepare to distribute blank paper that is long enough for students to trace one complete rotation of their cylindrical object.
 For objects with a diameter greater than 4 inches, receipt tape may be better.
Glossary:
 Access the complete Grade 7 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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