# 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.

# Engaging students: Perimeters of polygons

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 comes from my former student Nicholas Sullivan. His topic, from Geometry: perimeters of polygons. How could you as a teacher create an activity or project that involves your topic?
As a future educator teaching the subject of perimeter of a polygon I would suggest making a project for your students. The main materials needed would be poster board and duct tape, but its possible for the project to end up bigger than a poster board. Using the duct tape the students will fold it in half to make a small “fence”. The students will be able to choose from a variety of situations in which they need to create a fence for certain open areas. An example of situation would be a barn that needs sectioned off areas for chickens, cows, goats, and horses. The student using their own judgement would create the optimal fenced in area to separate the animals as necessary. Then once finished they would need to figure out how much fence to buy, first by converting the model to actual dimensions and then finding the total amount of fence. By the end of the lesson they will realize that no matter what kind of indents they made into the fenced in area they still had to count it as part of the fence, which relates to how perimeter works, you have to find the total amount of distance around any polygon. What are the contributions of various cultures to this topic?
How have different cultures throughout time used this topic in their society?
Ancient Egyptians and Babylonains used perimeter amongst other complex math calculations around 1800 B.C. Building the pyramids involved finding the perimeter of the different sections of the pyramids, such that the next layer be measured out and cut correctly. Perimeter breaks down to mean “around measure”. Many people were trying to efficiently and correctly compute the perimeter of a circle (we later came to know this as circumference). Knowing the perimeter of a wheel can help you know how much distance one full wheel rotation takes. Perimeter is a concrete subject and there was not any credit for anyone who “discovered” perimeter, because its something that people have always done, and needed to do. How can technology be used to effectively engage students with this topic?

This youtube video very clearly introduces the main topics related to perimeter. I would use it as  an introductory video to engage the students and get some of the vocabulary in their head. I really enjoy the way it talks about breaking a square and taking the edges off, laying them side by side and how that is also the perimeter. This video could set up an activity involving a similar activity to that, for example using string to create a square, and then measuring how long the string is, and comparing that to the perimeter. If everything is done correctly you will get the same answer doing both ways.

# Engaging students: Classifying polygons

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 Brianna Horwedel. Her topic, from Geometry: classifying polygons. How could you as a teacher create an activity or project that involves your topic?

A great activity for classifying polygons would be a card sort. Give students index cards with different ways to describe polygons on them. For example, the cards could say “has three sides”, “has five sides”, “has 4 equal sides”, “has four sides”, “triangle”, “quadrilateral”, “square”, “pentagon”. Also include cards with a pictorial representation of the polygons that you want them to identify. Have the students work in groups of three or four to match all of the cards. For example, “has three sides”, “triangle”, and the picture of a triangle would all be matched together. After about five to ten minutes of the students working in their groups, I would have a larger set of the index cards (probably on standard printer paper) that one person from each group would come up and place in the correct category/group. How can technology be used to effectively engage students with this topic?

There are tons of great polygon sorting games online. At the beginning of the unit I would have the students play Polygon Shape Game (http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/PolygonShapesShoot.htm). It is a great introductory game. It really helps the students understand what it means to be a polygon. The students have to pick out all of the shapes that are polygons on one round; on the next round, they have to pick out all of the shapes that aren’t polygons. Once students have a better understanding of what defines a polygon and different types of polygons, I would have them play Half a Min: Polygon (http://www.math-play.com/types-of-poligons.html). This game makes the students type in a type of polygon based on the hint given. This game is definitely harder than the first one; I would save it for maybe a review before a test. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Polygons occur frequently in abstract art, particularly the movement called De Stijl. It was inspired in part by the chaos of war. Dutch artists in 1917 wanted to contrast the messiness of war with art that consisted of balance, harmony, and the absence of individual expression. Piet Mondrian is one of the most famous artists to come from this particular movement. His work is created using grids, which creates various rectangles, and primary colors. Here is one of his paintings titles Broadway Boogie Woogie: It would be really fun for the students to then create their own artwork in the De Stijl style using only polygons that we have previously discussed.

References:

http://visionandverse.blogspot.com/2014/08/the-art-of-piet-mondrian.html

http://www.abstract-art-framed.com/mondrian.html

http://www.math-play.com/types-of-poligons.html

http://www.sheppardsoftware.com/mathgames/geometry/shapeshoot/PolygonShapesShoot.htm

# A Visual Proof of a Remarkable Trig Identity

Strange but true (try it on a calculator): $\displaystyle \cos \left( \frac{\pi}{9} \right) \cos \left( \frac{2\pi}{9} \right) \cos \left( \frac{4\pi}{9} \right) = \displaystyle \frac{1}{8}$.

Richard Feynman learned this from a friend when he was young, and it stuck with him his whole life.

Recently, the American Mathematical Monthly published a visual proof of this identity using a regular 9-gon: This same argument would work for any $2^n+1$-gon. For example, a regular pentagon can be used to show that $\displaystyle \cos \left( \frac{\pi}{5} \right) \cos \left( \frac{2\pi}{5} \right) = \displaystyle \frac{1}{4}$,

and a regular 17-gon can be used to show that $\displaystyle \cos \left( \frac{\pi}{17} \right) \cos \left( \frac{2\pi}{17} \right) \cos \left( \frac{4\pi}{17} \right) \cos \left( \frac{8\pi}{17} \right) = \displaystyle \frac{1}{16}$.

# My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Here’s my series on the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, 2b, and 2c: The 1089 trick.

Part 3a, 3b, and 3c: A geometric magic trick (see also here).

Part 4a, 4b, 4c, and 4d: A trick using binary numbers.

Part 5a, 5b, 5c, 5d: Predicting a digit that’s been erased from a number.

Part 6: Finale.

Part 7: The Fitch-Cheney 5-card trick.

Part 8a, 8b, 8c: A trick using Pascal’s triangle.

# My Mathematical Magic Show: Part 3c

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else. In the last couple of posts, I discussed a trick for predicting the number of triangles that appear when a convex $x-$gon with $y$ points in the middle is tesselated. Though I probably wouldn’t do the following in a magic show (for the sake of time), this is a natural inquiry-based activity to do with pre-algebra students in a classroom setting (as opposed to an entertainment setting) to develop algebraic thinking. I’d begin by giving the students a sheet of paper like this: Then I’ll ask them to start on the left box. I’ll tell them to draw a triangle in the box and place one point inside, and then subdivide into smaller triangles. Naturally, they all get 3 triangles.

Then I ask them to repeat if there are two points inside. Everyone will get 5 triangles.

Then I ask them to repeat until they can figure out a pattern. When they figure out the pattern, then they can make a prediction about what the rest of the chart will be.

Then I’ll ask them what the answer would be if there were 100 points inside of the triangle. This usually requires some thought. Eventually, the students will get the pattern $T = 2P+1$ for the number of triangles if the initial figure is a triangle.

Then I’ll repeat for a quadrilateral (with four sides instead of three). After some drawing and guessing, the students can usually guess the pattern $T=2P+2$.

Then I’ll repeat for a pentagon. After some drawing and guessing, the students can usually guess the pattern $T=2P+3$.

Then I’ll have them guess the pattern for the hexagon without drawing anything. They’ll usually predict the correct answer, $T = 2P+4$.

What about if the outside figure has 100 sides? They’ll usually predict the correct answer, $T = 2P+98$.

What if the outside figure has $N$ sides? By now, they should get the correct answer, $T = 2P + N - 2$.

This activity fosters algebraic thinking, developing intuition from simple cases to get a pretty complicated general expression. However, this activity is completely tractable since it only involves drawing a bunch of figures on a piece of paper.

# My Mathematical Magic Show: Part 3b

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else. This is a magic trick that my math teacher taught me when I was about 13 or 14. I’ve found that it’s a big hit when performed for grade-school children. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child: (gives a number, call it $x$)

Magician: On a piece of paper, draw a shape with $x$ corners.

Child: (draws a figure; an example for $x=6$ is shown) Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child: (gives a number, call it $y$)

Magician: Now draw that many dots inside of your shape.

Child: (starts drawing $y$ dots inside the figure; an example for $y = 7$While the child does this, the Magician calculates $2y + x - 2$, writes the answer on a piece of paper, and turns the answer face down. Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other.

Child: (connects the dots until the shape is divided into triangles; an example is shown) Magician: Now count the number of triangles.

Child: (counts the triangles)

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$, and there are indeed $18$ triangles in the figure.

This trick works by counting the measures of all the angles in two different ways.

Method #1: If there are $T$ triangles created, then the sum of the measures of the angles in each triangle is $180$ degrees. So the sum of the measures of all of the angles must be $180 T$ degrees. Method #2: The sum of the measures of the angles around each interior point is $360$ degrees. Since there are $y$ interior points, the sum of these angles is $360y$ degrees. The measures of the remaining angles add up to the sum of the measures of the interior angles of a convex polygon with $x$ sides. So the sum of these measures is $180(x-2)$ degrees. These two different ways of adding the angles must be the same. In other words, it must be the case that $180T = 360y + 180(x-2)$,

or $T = 2y + x - 2$. I’m often asked why it was important to choose a number between 5 and 10. The answer is, it’s not important. The trick will work for any numbers as long as there are at least three sides of the polygon. However, in a practical sense, it’s a good idea to make sure that the number of sides and the number of points aren’t too large so that the number of triangles can be counted reasonably quickly.

After explaining how the trick works, I’ll again ask a child to stand up and play the magician, repeating the trick that I just did, before I move on to the next trick.

# My Mathematical Magic Show: Part 3a

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else. For my second trick, I’ll show something that my math teacher taught me when I was about 13 or 14. Everyone in the audience has a piece of paper and a pen or pencil. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child #1: (gives a number, call it $x$)

Magician: On a piece of paper, draw a shape with $x$ corners. Don’t draw something really, really tiny… make sure it’s big enough to see well.

Audience: (draws a figure; an example for $x=6$ is shown) The Magician also draws this figure on the board. Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child #2: (gives a number, call it $y$)

Magician: Now draw that many dots inside of your shape.The Magician also draws $y$ dots inside the figure on the board, an example for $y = 7$ is shown. Audience: (starts drawing $y$ dots inside the figure) The Magician also calculates $2y + x - 2$ and says, “Now while you’re doing that, I’m going to write a secret number on the board,” discreetly writes the answer on the board, and then covers up the answer with a piece of paper and some adhesive tape.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other. For example, your figure could look like this: Audience: (quietly connects the dots until the shape is divided into triangles)

Magician: Now count the number of triangles.

Audience: (counts the triangles)

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$, and there are indeed $18$ triangles in the figure.

In tomorrow’s post, I’ll explain why this trick works.

# The worst math education video on YouTube

We now have a winner for the worst math education video on YouTube:

My personal favorite part is demonstrating that 140*9 is a multiple of 9 by casting out nines.

Why is this so awful? There are two essential ideas that make this work:

1. Humans have chosen a convention that there are 360 degrees in a circle. There’s nothing particularly magical about 360; that’s just the number that humanity has chosen for measuring angles with degrees. Notice that 360 happens to be a multiple of 9.
2. In base-10 arithmetic, one can check an integer is divisible by 9 by checking if the sum of the digits is a multiple of 9.

The first part of the video shows that, when 360 degrees is successively bisected, the digits of the resulting angle still sum to 9. That’s because dividing by 2 is the same as multiplying by 5 and then dividing by 10. Dividing by 10 is unimportant for the purpose of adding digits, so the only operation that’s important is multiplying by 5. And of course, if a multiple of 9 is multiplied by 5, the product is still a multiple of 9.

Notice that’s important that the angles are successively bisected. If the angles were trisected instead, this would fail (360/3 = 120, which is not a multiple of 9.)

The second part of the video notes that the sum of the angles in a convex polygon is a multiple of 9. That’s because the sum of the angles is (in degrees) $180(n-2)$, which of course is a multiple of 9. Furthermore, this formula is a consequence of the human convention of choosing 360 degrees to measure a complete rotation. From this number, the measure of a straight angle is 180 degrees. From this, the sum of the angles in a triangle is determined to be 180 degrees, and from this the sum of the angles in a convex polygon is found to be $180(n-2)$ degrees. All this to say, there’s nothing mystical about this. The second part of the video is a logical consequence of choosing 360 degrees for measuring circles.

The third part of the video is utter nonsense.

# Proving theorems and special cases (Part 10): Angles in a convex n-gon

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve seen that a conjecture could be true for the first 40 cases or even the first $10^{316}$ cases yet not always be true. We’ve also explored the computational evidence for various unsolved problems in mathematics, noting that even this very strong computational evidence, by itself, does not provide a proof for all possible cases.

However, there are plenty of examples in mathematics where it is possible to prove a theorem by first proving a special case of the theorem. For the remainder of this series, I’d like to list, in no particular order, some common theorems used in secondary mathematics which are typically proved by first proving a special case.

1. Theorem. The sum of the angles in a convex n-gon is $180(n-2)$ degrees.

This theorem is typically proven after first proving the following lemma:

Lemma. The sum of the angles in a triangle is $180$ degrees.

Clearly the lemma is a special case of the main theorem: for a triangle, $n=3$ and so $180(n-2) = 180 \times 1 = 180$. The proof of the lemma uses alternate interior angles and the convention that the angle of a straight line is 180 degrees. Using this, the main theorem follows by using diagonals to divide a convex n-gon into $n-2$ triangles. (For example, drawing a diagonal divides a quadrilateral into two triangles.) The sum of the angles of the n-gon must equal the sum of the angles of the $n-2$ triangles.  So it is possible to prove a theorem by proving a special case of the theorem. Using the sum of the angles of a triangle to prove the formula for the sum of the angles of a convex n-gon is qualitatively different than the previous computational examples seen earlier in this series.