# Lesson 5

Circumference and Wheels

### Lesson Narrative

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).

### Learning Goals

Teacher Facing

• 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.

### Student Facing

Let’s explore how far different wheels roll.

### 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.

### 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.

### Glossary Entries

• circle

A circle is made out of all the points that are the same distance from a given point.

For example, every point on this circle is 5 cm away from point $$A$$, which is the center of the circle.

• circumference

The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius $$r$$ then the circumference is $$2\pi r$$.

The circumference of a circle of radius 3 is $$2 \boldcdot \pi \boldcdot 3$$, which is $$6\pi$$, or about 18.85.

• diameter

A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.

• pi ($\pi$)

There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is $$\pi$$.

We can represent this relationship with the equation $$C=\pi d$$, where $$C$$ represents the circumference and $$d$$ represents the diameter.

Some approximations for $$\pi$$ are $$\frac{22}{7}$$, 3.14, and 3.14159.

For example, $$r$$ is the radius of this circle with center $$O$$.