Unit 3, Lesson 3
Exploring Circumference
Grade 7
Preparation
Lesson Narrative
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 circumference given the diameter (and vice versa) for different circles. Students learn that pi () is the value of the constant and discuss various commonly used approximations. Students will relate the circumference to the radius in a future lesson, so it is not necessary to do so in this 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 and repeated reasoning (MP7, MP8).
- 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.
Let’s explore the circumference of circles.
Required Materials
Activity 1
Required Preparation
The empty toilet paper roll is for optional use as a demonstration tool.
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.
Provide one measuring tape per group of 2--4 students. Alternatively, use rulers and string.
To get a good spread of points on the graph, it is important to use circles with a wide variety of diameters, from 3 cm to 25 cm.
For the digital version of the activity, acquire devices that can run the applet.
There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is .
This relationship can be represented with the equation , where represents the circumference and represents the diameter. In the graph, pi can be seen as the value of when the value of is 1.
Some approximations for are , 3.14, and 3.14159.