The purpose of this discussion is to elicit the idea that the more sides an inscribed or circumscribed polygon has the closer it comes to approximating a circle, which will be useful when students calculate perimeter and eventually in later activities. Tell students they will have the opportunity to test their predictions by calculating soon.

Once all groups have completed the Card Sort, discuss the following:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

During this activity students are asked to compute the perimeter of inscribed regular polygons given only the radius. They will need to figure out that drawing in the altitude will form a right triangle. Drawing this auxiliary segment shows that students recognize the important structure of right triangles to calculate unknown information (MP7). In this activity students are building towards an equation to approximate the value of , which they will complete in the next lesson.

The digital version of this activity includes an applet that may be helpful to display for the class during the Synthesis.

If students are stuck after sketching a diagram, ask them what helpful auxiliary lines they could draw. (Connect the center to the other vertices to make triangles. The altitude would create right triangles, so I can use trigonometry.)

This activity requires more interpretation than a basic right triangle problem and requires students to use skills that will help them in mathematical modeling (MP4). First, students need to draw a diagram and figure out how to label it. Second, students need to recognize that the units do not match and figure out how to convert to a common unit. At this point the problem looks as if there are several basic right triangles, and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.

Encourage students to draw a new triangle for each row of the table and put their calculations next to it. This will support organization and seeing structure to write the equation.

The purpose of this discussion is for students to consider different strategies for finding side lengths of right triangles. 

Invite groups to share their strategy for finding the hiking distances. Ask if they would prefer to substitute values into and then solve, or write an equation solved for first and then substitute. Acknowledge that both methods are valid.