In this lesson, students discover that there is a proportional relationship between the diameter and circumference of a circle. They use their knowledge from the previous unit on proportionality to estimate the constant of proportionality. Then they use the constant to compute the diameter given the circumference (and vice versa) for different circles. We define \(\pi\) as the value of the constant and discuss various commonly used approximations. In the next lesson, students use various approximations for pi to do computations. Also, relating the circumference to the radius is saved for the next lesson.
Determining that the relationship between the circumference and diameter of circles is proportional is an example of looking for and making use of structure (MP7).
Lesson overview
 3.1 Warmup: Which Is Greater? (5 minutes)

3.2 Activity: Measuring Circumference and Diameter (25 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.

3.3 Activity: Calculating Circumference and Diameter (10 minutes)
 Includes "Are you Ready for More?" extension problem
 There is a digital applet in this activity.
 Lesson Synthesis
 3.4 Cooldown: Identifying Circumference and Diameter (5 minutes)
Learning goals:
 Comprehend the word “pi” and the symbol \(\pi\) to refer to the constant of proportionality between the diameter and circumference of a circle.
 Create and describe (in writing) graphs that show measurements of circles.
 Generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.
Learning goals (student facing):
 Let’s explore the circumference of circles.
Learning targets (student facing):
 I can describe the relationship between circumference and diameter of any circle.
 I can explain what \(π\) means.
Required materials:
 empty toilet paper roll
 measuring tapes
 cylindrical household items
Required preparation:
 Household items: collect circular or cylindrical objects of different sizes, with diameters from 3 cm to 25 cm.
 Each group needs 3 items of relatively different sizes.
 Examples include food cans, hockey pucks, paper towel tubes, paper plates, CD’s.
 Record the diameter and circumference of the objects for your reference during student work time.
 The empty toilet paper roll is for optional use during the warmup as a demonstration tool.
 You will need one measuring tape per group of 24 students.
 Alternatively, you could use rulers and string.
Glossary:
 pi (\(π\))  There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is \(π\). 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.
 Access the complete Grade 7 glossary.
Standards
 This lesson builds on the standards:CCSS.2.MD.AMO.2.GM.B.4 CCSS.6.SP.B.5.cMS.6.SP.5cMO.6.DSP.B.5cMS.2.MD
 This lesson builds towards the standard:CCSS.7.G.B.4MS.7.G.4MO.7.GM.A.4b
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).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.