# Lesson 12

Exploring Circumference

### 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 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).

Teacher Notes for IM 6–8 Accelerated
The lesson narrative refers to “knowledge from the previous unit on proportionality.” In IM 6–8 Math Accelerated, that learning is from a previous section of this unit.

### Learning Goals

Teacher Facing

• 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.

### Student Facing

Let’s explore the circumference of circles.

### 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 warm-up as a demonstration tool.

You will need one measuring tape per group of 2--4 students. Alternatively, you could use rulers and string.

### Student Facing

• I can describe the relationship between circumference and diameter of any circle.
• I can explain what $\pi$ means.

Building On

Building Towards

### Glossary Entries

• pi ($\pi$)

There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is $$\pi$$.

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.