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.)

*"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:

*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!"*

Pascal, Blaise (1623-1662) Pensees, 1670

**Go To Homepage** **Go To Introduction**
**1) Constructions** **2) Clock Problem** **3) Test Corrections** **4) ASN Explain** **5) Thoughts About Slope** **6) What is Proof?**

**7) Similar Triangles** **8) Homework Corrections** **9) Quads Midpoints** **10) Quads Congruence** **11) Polygons**

**12) Polygons Into Circles** **13) Area and Perimeter** **14) Writing About Grading** **15) Locus** **16) Extra Credit Projects**

**17) Homework Reflections** **18) Students' Overall Reflections** **19) Parents' Evaluate Method** **20) In Conclusion**