In this activity, students roll circular objects across a paper and mark how far they travel in one complete rotation. They divide the distance traveled by the diameter of the circle and see that the quotient is close to . They relate this to what they previously learned about the relationship between the diameter of a circle and its circumference.

As students investigate the distance an object rolls, by drawing, measuring, calculating, and comparing, they persevere in making sense of circumference (MP1).

Monitor for students who recognize that the distance the object rolls is equal to the circumference of the circle.

The goal of this discussion is for students to understand that the distance a wheel travels in one complete rotation is equal to the circumference of the wheel.

Poll the class on the quotients they found for the distance their object rolled divided by the object’s diameter. Connect the different responses to the learning goals by asking questions such as:

• “What do you notice about these values?” (They are close to .)
• “Why does it make sense for this quotient to equal pi?” (The distance an object rolls in one complete rotation is equal to the object’s circumference, and .)
• “Why aren’t all our measurements exactly equal to pi?” (measurement error)

Remind students of the activity from the other day when they measured the circumference of circular objects: “When we measure the circumference by wrapping a measuring tape around the circle, the circle stays in place while the measuring tape goes around it. When we roll the circle, we can imagine the measuring tape unwinding while the circle moves.”

If desired, discuss which method of measuring the circumference was more precise (rolling the circle or wrapping a measuring tape around it)? Some reasons why measuring the circumference of the circle directly may be more precise include:

• When you roll the circular object, it is hard to keep it going in a straight line.
• It is difficult to mark one rotation precisely.

In this activity, students investigate proportional relationships between the number of rotations a wheel makes and the distance that wheel travels. Students make repeated calculations with explicit numbers and then write an equation representing the proportional relationship (MP8). When students write an equation to represent the relationship and then use the equation to solve problems, they reason quantitatively and abstractly (MP2).

The goal of this discussion is to make sense of the equations that represent the situations. When looking at the relationship between the number of rotations and the distance a wheel travels, the constant of proportionality is the circumference of the wheel.

Ask students:

• “What is the constant of proportionality for each relationship? What does that tell you about the situation?” (65.3 and 81.7, the number of inches each wheel travels per rotation, which is also the circumference of each wheel.)
• “How do the two wheels compare? How can you see this in the equations?” (The bike wheel is larger. The constant of proportionality is larger in the equation representing the bike.)
• "Which wheel makes fewer rotations to travel one mile? (The bike does. Its wheels are larger, so it moves farther in one rotation.)

In this activity, students investigate the relationship between the rotational speed of a wheel (in rotations per second) and the linear speed of the vehicle (in miles per hour).

First, students must apply their recent learning to recognize that the circumference of the wheel equals the distance the car travels for one complete rotation of the wheel. Then, students work through a series of unit conversions to get from rotations per second to miles per hour. As students choose steps for converting the rates into different units, they reason quantitatively and abstractly (MP2).

Monitor for students who use the following strategies to solve the last problem:

• Reverse all the conversion steps they used to solve the previous problems, so 65 miles per hour is 343,200 feet per hour or 1,144 inches per second or 17.6 rotations per second
• Compare the new speed to their previous calculations for 1 rotation per second, such as
  • Compare 65 miles per hour to the approximation 3.7 miles per hour gets
  • Convert 65 miles per hour to 343,200 feet per hour and then comparing to the previous rate of 19,500 feet per hour gets .

The goal of this discussion is to highlight students’ use of multi-step ratio reasoning in solving these problems.

Invite students to share their reasoning for the first two questions. Emphasize the different rates that come up while solving these problems, such as:

• 65 inches of distance traveled per rotation of the wheels
• 1 rotation of the wheel per second
• 65 inches of distance traveled per second
• 60 seconds per minute
• 3,900 inches of distance traveled per minute
• 12 inches per foot
• 325 feet of distance traveled per minute
• 60 minutes per hour
• 19,500 feet of distance traveled per hour
• 5,280 feet per mile
• about 3.7 miles of distance traveled per hour

If time permits, invite students to share their reasoning for the last problem. Consider displaying this table with no rates in the right column except for the bottom row.

As students share their reasoning, record equivalent rates in the right column working from the bottom of the chart to the top.