In this lesson students build off their concrete calculations from the previous lesson to write a generalized formula for the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful so students can build their perseverance and sensemaking (MP1). Students should work with their groups to determine what information they need, how they calculated this information in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8).
Once students build a generalized formula, they apply the formula to approximate the value of \(\pi\). During the lesson synthesis students learn methods ancient mathematicians used to generate both upper and lower bounds of \(\pi\). This lesson presents an opportunity for seeing how mathematics has changed over history and how ancient techniques are still in use but with the added power of computers.
Lesson overview
 11.1 Warmup: More Sides (10 minutes)
 11.2 Activity: N Sides (15 minutes)

11.3 Activity: So Many Sides (15 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
Learning goals:
 Compare and contrast approximations of \(\pi\) (orally).
 Explain how to use regular polygons to approximate the value of \(\pi\) (using words and other representations).
Learning goals (student facing):
 Let’s approximate the value of pi.
Learning targets (student facing):
 I can explain how to use regular polygons to approximate the value of \(\pi\).
Required materials:
 Scientific calculators
Required preparation:
 Devices are required for the digital version of the activity N Sides.
 Acquire devices that can run the GeoGebra applet, ideally 1 per student.
 Be prepared to display an applet for all to see during the lesson synthesis.
Standards:
 This lesson builds towards the standard: CCSS.HSGGMD.A.1MS.GGMD.1MO.G.GMD.A.1
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