Exploring Circumference

5 min

Narrative

The purpose of this Warm-up is to help students visualize circumference as a linear measurement, in preparation for examining the relationship between diameter and circumference in the next activity. Some students may be able to imagine unrolling the tube into a rectangle in order to compare its length and width. Other students may benefit from hands-on experience with a real toilet paper tube.

Launch

Arrange students in groups of 2. Display the image for all to see. Ask students to indicate when they have reasoning to support their response. Give students 1 minute of quiet think time and then time to share their thinking with their group.

Student Task

Clare wonders if the height of the toilet paper tube or the distance around the tube is greater. What information would she need in order to solve the problem? How could she find this out?

<p>Three different views of a circular shaped toilet paper tube. </p>
Three different views of a circular shaped toilet paper tube. The first view is of the vertical height of the tube. The second view is of the circular base of the tube. The third view is of both the base and the height of the tube.

 

Sample Response

Clare needs to measure the length of the tube and the distance around. To find the distance around she could measure the tube with a flexible measuring tape, or cut and flatten the tube. 

Synthesis

The goal of this discussion is for students to internalize that the distance around a circle is a length, just like the height of an object is. As a one-dimensional measurement, it can be measured using a ruler.

Survey students on which length they think is greater. Consider displaying the image again for reference while students are explaining their reasoning. To involve more students in the conversation, consider asking questions like:

  • "What was important to you when making your decision?"
  • "Did anyone think about the measurements in a different way?"
  • "Do you agree or disagree? Why?"
  • "What information do you think would help you make a better decision? If an actual toilet paper tube is available, demonstrate measuring the tube by unrolling the tube to measure the circumference and height."

If an actual toilet paper tube is available, consider demonstrating unrolling the tube to measure the circumference in a straight line and compare it to the height.

Anticipated Misconceptions

Students may not understand what is meant by the height of the tube because it can sit in two different ways. Point these students to the first picture of the tube, and ask them to identify the height as the tube is sitting in that picture. 

Standards
Building On
  • 2.MD.A·Measure and estimate lengths in standard units.
  • 2.MD.A·Measure and estimate lengths in standard units.
Building Toward
  • 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.

20 min

10 min