# Circumference-Diameter Ratio and Arc Length

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## Objective

Students will be able to explain pi as the ratio of a circle's circumference to its diameter.

#### Big Idea

Through a hands-on task involving circular objects, string, and measuring tools, students will make sense of pi!

## What is Pi?

15 minutes

Since the Common Core requires students to see that an arc intercepted by an angle is proportional to the radius of the circle, it is necessary to ensure students build on their understanding that all circles are similar and that the ratio between the circumference of any circle to its own diameter is constant.  While students may have done or seen a demonstration of this task prior to this lesson, I have seen that students have difficulty articulating what pi represents, which is why we do this quick investigation.

In this investigation, students work in their groups to collect data by measuring the circumferences and diameters of lids of varying sizes.  After groups have finished collecting their data, I call on groups to share their data and as a class, we express regularity in repeated reasoning and determine that pi is approximately 3.14 (MP8).

## Notes: Circumference and Arc Length

20 minutes

Like most lessons, we formally debrief the investigation by formally taking notes in our note takers. Here is the Notes Template.

At this time, I make sure to incorporate whole-class examples through which we can all practice circumference and arc length problems since there is a wide range of algebraic skills in my geometry classroom.  For example, some students easily see 144/360 as 2/5 while others get lost in the mechanics of simplifying, making it hard for them to grasp the proportionality of these situations.  I use this time to allow for questions, interpretations of the problem, and sense making to ensure that all of my students will have at least one approach to take when solving these kinds of problems.

## Practice: Circumference and Arc Length

30 minutes

After we take notes, I want to make sure students have time to practice solving arc length problems on their own.  During this time, students practice individually, but have the option to check in with other students in their group or with me.

I debrief the practice worksheet by posting an answer key and listening in on groups' discussions as they make corrections to their work.  If there are common errors or confusions that appear to arise, I make sure to highlight these particular problems with the whole class.