Engaging students: Proving that the angles of a convex n-gon sum to 180(n-2) degrees

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Victor Acevedo. His topic, from Geometry: Proving that the angles of a convex $n$-gon sum to $180(n-2)$ degrees.

How could you as a teacher create an activity or project that involves your topic?

A great activity to try with students would be to look at regular and irregular polygons and the triangles “inside” of them. Using some string and a few tacks, students could “construct” regular polygons on pegboard or foam. They could then measure the angles made by the string using a protractor and find the sum. After doing a the first few regular polygons, the students could do the same with irregular convex polygons and notice that the sum of the angles is the same for regular and irregular polygons with the same number of sides. At this point the students might have established a pattern for the sum of all the interior angles of a polygon as the number of sides (n) increases. The next task would be to go back to the regular polygons and make the triangles inside. This would be done by picking one of the vertices as the starting point and connecting that point to all the other vertices. Since the starting point is already connected to two of the other vertices, we wouldn’t have to make those connections again. The students would then see that inside of the regular polygons there are n-2 triangles. Since the sum of every triangle’s interior angles is 180°, the sum of the regular polygons’ interior angles would be 180(n-2). To further prove our original statement, the students would repeat the process of creating triangles with the irregular convex polygons.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This interactive Desmos program helps students work through proving that the sum of the interior angles of convex n-gons is 180(n-2). The program starts with a review of the sum of the angles in triangles. The students would then look at the diagonals of polygons and count the triangles formed. The students get the opportunity of deriving the formula for the sum of interior angles by continuing patterns as the number of sides increase. This program also encourages students to think about the “limit” to the interior angles of a polygon and why it approaches 180º but will never actually reach it. There is also a link to an extension of this activity to looking at the exterior angles of a polygon as well.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

René Descartes did not necessarily contribute to the discovery of the sum of the interior angles of convex polygons, but he was able to apply some of the outcomes to philosophy. Descartes uses the regular chiliagon (1,000-sided polygon) to demonstrate the differences between intellection and imagination. While we can clearly picture understand a triangle, a chiliagon is not quite as simple to picture due to the large number of angles and edges. To the naked eye, a chiliagon would look nearly identical to a circle. The only possible way to discern any difference would be to zoom in until you can possibly see different vertices. This application to philosophy is great for students to begin thinking about the limit that the interior angles of regular polygons reach as the number of edges increases.

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