Circumference and Wheels

5 min

Teacher Prep
Setup
1 minute quiet think time. 1 minute small group discussion. Whole-class discussion.

Narrative

In this Warm-up, students analyze a statement comparing the distance around a wheel with the length of a rope. They apply their understanding of the meaning and rough value of π\pi to determine whether the rope is long enough to go around the wheel. As students analyze Han’s statement, they critique the reasoning of others (MP3).

Launch

Give students 1 minute of quiet think time followed by 2 minutes of partner discussion.

Student Task

Han says that you can wrap a 5-foot rope around a wheel with a 2-foot diameter because 52\frac52 is less than pi. Do you agree with Han? Explain your reasoning.

Sample Response

Han is not correct. Sample reasoning: The circumference of the wheel is 2π2\pi feet. Since π\pi is a little bit larger than 3, this is more than 6 feet, and the 5-foot rope will not fit all the way around. 

Activity Synthesis (Teacher Notes)

The goal of this discussion is for students to articulate that Han’s calculation is correct, but his conclusion is incorrect. Invite several students to share their reasoning. After each response, ask the class if they agree or disagree.

The key takeaways are:

  • The circumference of the wheel is 2π2\pi feet.
  • π\pi is larger than 3, so the circumference of this wheel is more than 6 feet.
  • Han is right that 52<π\frac{5}{2} \lt \pi, but this means that the rope will not make it all the way around. 

If time permits, extend the discussion by asking:

  • "Would a 6-foot rope be long enough to go around the wheel?" (No because 62\frac{6}{2} is still less than π\pi.)
  • "What about a 7-foot rope?" (Yes, because the circumference of the wheel is 2π2\pi feet, and this is less than 7.)

Students may observe that it is possible to wrap the rope around the wheel going around a diameter twice as opposed to going around the circumference.

MLR8 Discussion Supports. Display sentence frames to support whole-class discussion: “I agree because . . . .” or “I disagree because . . . .”
Advances: Speaking, Conversing
Standards
Addressing
  • 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
  • 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

15 min

15 min