Chapter 11 - Polygons

When the subject of polygons came up in the geometry curriculum, my students did a number of projects that were not included in the text book, in addition to studying the usual topics.

One of these projects was one we called "Angles of a Polygon". I had prepared a chart, with the polygons in order of number of sides, and a list of properties to consider. The horizontal heading in the chart contained diagrams of the regular polygons we would explore: triangle, quadrilateral, pentagon, hexagon, heptagon (having 7 sides) and octagon. The students were asked to set their textbooks aside, and work with their group to fill in the chart with the following: number of triangles formed by dividing the figure into triangles, sum of the interior angles of the polygon, measure of each interior angle (one at each vertex), each exterior angle, and sum of the exterior angles. In the end, they would look at their conclusions and come up with a formula for calculating the number of triangles formed, and formulas for the sum of the interior and exterior angles of each of the polygon. Only after they had completed all of this were they allowed to open the textbook.

There is a "worksheet" which includes the introduction to the assignment, a copy of which was given to each student. Here is the worksheet: (You will find the answers at the end of this web page.)

Worksheet: Sum of the angles of a polygon:

"What is the sum of the interior angles of a polygon"? Let's explore this question, using inductive reasoning. Inductive reasoning is a type of reasoning in which we solve a series of related numerical problems, notice patterns, and make conclusions based on the patterns that we find. In Geometry, after coming to our conclusion or conclusions, we would then proceed to verify our findings using deductive reasoning, which would mean writing a geometric proof. (Inductive Reasoning is defined as making conclusions from observations or experiments. Deductive reasoning is making conclusions based on mathematical proof.) For example, what number comes next in this sequence: 2, 4, 8, 16 ,32 . . . ? (Answer the question before you read on.)

If you answered 64, then you are correct. The pattern is 'multiply each number by 2 to get the next number'. This pattern is called 'powers of 2'. And you probably figured it out by looking for a pattern, trying some things, then checking to see if what you tried worked. This is called inductive reasoning."

It is worth the time, before doing this project, to spend 10 or 15 minutes discussing the difference between deductive and inductive reasoning. In my writing-intensive math class, I asked them to find examples of inductive reasoning in "real life". I gave them an example, to "get the ball rolling": "People used to think the earth was flat, because that is how it appears to us from where we are standing. It is only when you get high above the earth, or far away from it, that you can see it's curvature."

The students worked together in groups of 3 or 4, exploring the polygons and making conclusions about their properties. In our class, the students did this project using pencil and paper, and then in later years, we used Geometer's Sketchpad,

There are a number of ways to approach this concept. One method is the Geometer's Sketchpad, an alternative is to use compass and straight-edge constructions. But there is a certain value to doing this project the "really old-fashioned way", using just pencil and paper. Using the pencil and paper method, these were the insructions to the students:

"In this project, we will use deductive reasoning to find out what the sum of the angles of a regular polygon might be, and derive a formula that will work for a polygon with any number of sides.

With a pencil, triangulate each one of the polygons drawn in the chart below. Triangulate means to connect vertices in such a way that the polygon is made up entirely of triangles; as few triangles as possible. It is easiest to start at one vertex and continue from there. For example. An eight-sided polygon is called an octagon, as in the diagram on the left below. If you were to triangulate this eight-sided polygon it would look like the drawing on the right below:

Once triangulated, the octagon is now composed of 5 triangles. Now, you know that the sum of the angles of a triangle is 180 degrees, so you can fill in the chart below, using the same method for each of the polygons."

The students entered their findings on the worksheet below, where "n" represents a polygon with "n" sides (this column would be the formula that they would derive.) You will find the answers at the end of this web page.

The next instruction was as follows: "Now, you probably already know that the sum of the angles of a triangle is 180 degrees, so you can go on to fill the chart below. Look for patterns in your answers and try to figure out, using inductive reasoning, what the sum of the interior angles of an "n-gon" would be (an n-gon is a polygon with "n" sides). By the time you finish, you should know a lot about the angles of polygons!"

In her reflections on this project, Lindsay said "This worksheet was a lot of fun. Instead of just reading the textbook or listening to the teacher, we were able to figure it out for ourselves! Learning in this way, we could understand it better because we knew exactly how we came across the answers, not just by memorizing some formula from the book. At first, I must admit I couldn't figure it out, and just wanted somebody to tell me the answer. However, the teacher wouldn't tell me! She just said 'Look at the relationships between all the numbers in the columns and rows.' and after a while I started to see the patterns. It was pretty cool to figure it out! Overall I got to learn things in a different way than usual, and found some interesting conclusions. I think this will help me remember this stuff, because if I forget the answer I can just figure it out again the way I did in this project."

Amelia made the following comments: "Now that I have done this, I think it is much easier to understand than the way it is written in the book. Because I was forced to figure out the relationships myself, I think my way is better than the way they showed it in the book, it's just not as not as good as doing it yourself."

And here are the answers to the polygons grid:

While we are on the subject of polygons, a related short writing project that I assigned was: "Consider the interior angles of a polygon, and the exterior angles. What is the relationship between them? As the number of sides of the polygon increases, what effect does this have on the interior angles, the sum of the interior angles, the exterior angles and the sum of the exterior angles? Do any of these quantities remain constant? Why? Which increase? Why? Which decrease? Why? " After much experimentation and deliberation, the students came to their conclusions, and wrote essays on this topic.

Ian answered these questions as follows: "An interior angle and its corresponding exterior angle of a regular polygon are supplementary. As the sides of a regular polygon increase, the measure of each interior angle increases, but the measure of the adjacent exterior angle decreases correspondingly. As each interior angle increases, the measure of it's adjacent exterior angles decreases. The sum of the interior angles goes up with the number of sides, but the sum of the exterior angles remains constant. (The sum of the exterior angles is always 360 degrees.) The exterior angle's sum remains constant because the sum of the interior angles increases but so does the number of sides, and the number of angles."

Remy's comments included the following reflections: "I really enjoy when we do activities like this as a class. I felt that this worksheet was the best worksheet so far, because it encouraged us to discover patterns and theorems ourselves. I also like the follow-up: it we didn't get all the patterns ourselves, we could discuss it with our classmates and ask the teacher for help - infinitely quicker than reading the book five million times trying to understand all of the big words. I especially liked the way that we were not given the patterns straight out and told to memorize them, because I find that incredibly boring and I have difficulty learning that way. I find that I remember more information when I have worked out the solutions and theorems myself, and this project encouraged me to do just that. I like this kind of hands-on approach to learning. It us much better than memorizing, and I also get a better understanding of how and why it works. I also found this worksheet VERY useful while I was doing my homework. I just looked back at this worksheet and it jogged my memory. I think that if we didn't do this as a class and discuss it, I would have been dead in the water and totally confused while doing my homework."

The diagram below shows an octagon with exterior angles, one at each vertex, constructed by Remy as part of her answer to the question:

In her reflections on this project, Lindsay wrote: "These worksheets were a lot of fun. Instead of just learning the proof in the book and having to memorize it, we were able figure it out for ourselves. We used inductive reasoning, which was different from the usual deductive reasoning (proving things with a mathematical proof). Learning things this way, we were able to understand concepts better because we knew exactly how we came across the answers. For the angles of a polygon worksheet, the beginning seemed easy but the farther down you went the harder you had to think about it. Once you figured out a system for deriving your conclusion, it made sense logically and I could take off from there. I that it was interesting (and surprising!) that the sum of all the exterior angles always is 360 degrees, but after I thought about it, it made sense logically."

In his solution to the question of interior and exterior angles of poygons, Mike created the following table:

. . . and added these comments: "As my chart shows, as the number of sides of a polygon increases, the size of the interior angles increases also. The sum of the intererior angles increases as well. But each exterior angle gets smaller as its interior angle gets bigger so the sum of the exterior angles remains the same."

In her reflections on this project, Miranda expressed her feelings about this project: "These worksheets were a lot of fun. Instead of just reading the book and memorizing a bunch of theorems, we were able to figure things out for ourselves. We used inductive reasoningin this project, which was different from the usual deductive reasoning. Learning this new way, we could understand concepts better because we knew exactly how we came across the answers. At first I couldn't figure it out, and just wanted someone to tell me the answer. But, fortunately, we were working in groups and Jake had a good idea as to how to start, and once we got started we really took off! We figured out a system, and our system worked. My biggest 'aha' moment was when I added down the rows and found that the first column times the second column equals the third column for each one. Overall, I got to learn things in a different way, and I think we did a good job of it, too!"

Then we went on to explore the polygons using coordinate geometry. The next question that I posed to the students was this: "Given the following coordinates, what is the most descriptive name for quadrilateral ABCD? Justify what it is and what is not." The coordinates were as follows: A(-4, 1), B(6,8), C(13,-2) and D(3,-9). The students answered the question not only with the coordinates and name, but also with proofs, as well as essays explaining how they found their answer, and why they believed their answer to be correct. Ian wrote:

"Here are the slopes of the 4 lines:

AB: 7/10 and DC: 7/10

DA: -10/7 and CB: -10/7

The slopes of AB and DC are the same, and the slopes of DA and CB are the same. This would make the quadrilateral at least a paralleogram (both pairs of opposite sides parallel.). But then we see that the slopes of AB and DC are opposite reciprocals of the slopes of DA and CB so both pairs of sides of the quadrilaterar are are perpendicular, so this makes it at least a rectangle.

And here are the lengths of the segments:

AB and DC both = the square root of 149 in length

DA and CB= the square root of 149 also

therefore AB = BC = CD = DA so it has to be a square!"

"We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others."

Pascal, Blaise (1623-1662) Pensees, 1670