Today could have many different focuses, depending on the amount of time you have to allot to this extension activity. If you must choose which of the questions to focus on, I suggest focusing attention towards the interior angle sum of polygons discussed in challenge 1 and the area of regular polygons discussed in challenge 2. Rotational symmetry is a nice connection but not as essential in later math content as interior angle sums and area.
Clarify your Learning Intentions and Criteria for Success by telling students that today we will focus on two important concepts that will also be important to high school geometry - the interior angle sum of any polygon and the area of any regular polygon. Both of these big ideas begin with triangles.
Quickly review the three compound rotation figures and ask students questions about the triangles such as, "What is the relationship between each of the triangles within the hexagon? (They are congruent). How do you know they are congruent? Can you prove they are all the same triangle? (Rotations are rigid motion and therefore produce congruent image and pre-image). It is essential that students remember this relationship to begin working towards the interior angle sum of their hexagon, pentagon, and octagon. Students must have a correctly measured and labeled diagram to begin the work today as well. It would not hurt to place an example student paper under the document camera for students to use as a guide for checking their own work.
Challenge question 1 relies on students’ ability to read patterns and generalize those patterns with an algebraic expression. You may need to review the vocabulary interior angle sum. I would expect most students to already know the interior angel sum of a triangle is 180 degrees (7th grade standard). I also expect them to already know the interior angle sum of a square is 360 degrees if by no other means than four times 90 degrees. Using their own compound rotations, students should be able to complete the table for a pentagon, hexagon, and octagon by simply adding all the interior angles they have labeled. I sometimes offer a reward for this challenge. I allow students 10 minutes to work on the table and reading the pattern. I then offer a soda to the first group who can generalize the pattern in the table into a function rule for the interior angle sum of any polygon with n sides. I ask students to put their rule on a post-it note and I walk about the room collecting notes.
During this Mini Wrap-Up I usually have had two most common versions of this rule. I expect to see the common rule 180(n – 2); however, what I get most often is 180n – 360. If I happened to have a class where both rules are turned in, then I put both on the board and ask groups to spend time deciding which rule is correct. After about five minutes we share out as a class. In the end, I reward both groups with a soda. I usually follow this rule with the question, if you have a regular polygon, how can you adapt this rule to find the measure of just one interior angle? As you conduct this whole group discussion be sure to Script important information on the board for students to keep as a guide to help them take good notes and work productively throughout the rest of the class period.
Challenge 2 is the fastest and easiest for students to grasp. Visually, it is simple to see the concept of area as area of one triangle multiplied by the number of triangles. I usually prepare students by saying that one day they will need to find the area of regular polygons beyond a square. I ask if they know the formula for area of a regular hexagon or pentagon. When no one can say yes, I say, “Actually, you already know all the formulas you need in order to find the area of this hexagon. I allow groups about 10 minutes to work on the challenge question. I walk about the room Providing Feedback that Moves Learning Forward and allowing Students to Be Resources for One Another. As I walk about the room, I choose students who will present their thinking during the Mini Wrap-Up.
After discussing both area questions in challenge 2 create a comprehensive explanation of finding area in any regular polygon and assign the homework page area of regular polygons – either version depending on the comfort level of your students using radicals.