# 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.

# 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 Samantha Smith. Her topic, from Geometry: classifying polygons.

D. 4. What are the contributions of various cultures to this topic?

# Tangrams

The tangram is a puzzle game that originated in China. It has been documented that this puzzle has been played since at least the early 1800s, and even before that. By around 1817, the tangram had gained popularity in Europe and America. Its components consist of seven pieces: one square, one parallelogram, two small isosceles triangles, one medium isosceles triangle, and two large isosceles triangles. Each piece is called a tan. The shapes can be arranged into different figures. As you can see in the picture below, these pieces can be arranged in many ways. For the classroom, the teacher can give the students tans to make their own figures, or the teacher can give them a silhouette of a figure and have the students create the tangram. This is just a fun way to have the students interact with the shapes they are learning about, and experience some world culture.

http://www.activityvillage.co.uk/tangram-black-and-white

http://www.logicville.com/tangram.htm

B.2. How does this topic extend what your students should have learned in previous courses?

# Geo-Boards

When I took Concepts of Algebra and Geometry last semester, we had a full lesson on polygons. The professor gave us a tool called a Geo-Board to model polygons using rubber bands (as pictured below). This was a really fun and short hands-on activity to engage us in the lesson. After we made our shape with the rubber band, we would go more in depth and triangulate it to find the sum of the angles in the polygon. The Geo-Board will be exciting for high school students. The teacher can name a familiar shape that the students can model on their board. Most of the shapes they will be modeling they will have worked with in many previous math courses. Another cool thing about the Geo-Board is that the students can see there is more than one way to make a polygon, as long as they have the right number of sides. Of course there are stricter rules for squares, equilateral triangles, etc.; however, students can still model those shapes on the Geo-Board while the teacher walks around and checks their work.

C. How has this topic appeared in culture?

# Traffic Signs

Every day people get into cars and drive. They are expected to follow the laws of the road. One of the first things you learn in Driver’s Ed. are the different traffic signs; their colors, their shapes and what they mean. What I notice is that traffic signs are in the shapes of polygons, and their shape is important to their meaning. A stop sign is an octagon, a yield sign is a triangle, and a pedestrian crossing sign is a pentagon. Knowing these shapes can help determine what a sign means, especially if the driver is too far away to read what it says.

This is an everyday use of classifying polygons. Students do it all the time; they just might not realize it. Engaging high school students with traffic signs could prove beneficial for them in more ways than one: not just learning about shapes, but about traffic signs they will be tested on before they get their driver’s license, which is what many students are doing in high school.

# 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 Jessica Trevizo. Her topic, from Geometry: Proving that the angles of a convex $n$-gon sum to $180(n-2)$ degrees.

E.1 How can technology be used to effectively engage students with this topic?

This website allows the students to see that any polygon, whether regular, concave, or convex, the sum of the interior angles will not change. The students are able to drag any angle of their choice and either enlarge, shrink, or rotate the figure. As the student is able to change the figure, the angles automatically change and are shown on the right hand side of the screen. All of the angles are color coordinated so students are able to easily observe which angle measure goes with the corresponding angle they are moving. Also, this activity allows the students to explore with six different polygons which include the triangle, quadrilateral, pentagon, hexagon, heptagon, and octagon. The triangle and the quadrilateral include an animated clip which consists of a visual proof for the value of the angle sum. It is a simple proof that students will be able to see and understand at their level.

http://illuminations.nctm.org/Activity.aspx?id=3546

A1. How could you as a teacher create an activity or project that involves your topic?

Using geoboards will help the students derive the sum of interior angles formula on their own. For the activity every student will need a geoboard and a couple of rubber bands. The students will be asked to create a specific shape on the geoboard using the rubber bands. Once every student has completed the figure they will be asked to dissect the figure into triangles. Whenever the teacher gives the students the task he/she needs to make sure to state the rules before they begin. The rules are that the rubber bands cannot cross each other, and the rubber bands must start and end at a vertex of the figure. The students will need to fill out the worksheet provided in the link below. The worksheet is arranged to help them see the pattern after they do a couple of examples with different shapes. The goal is to try to help the student realize that the number of triangles that can be created in a certain figure will be (n-2), n being the number of sides. A higher level question for the students could be, “Why are you only able to create (n-2) triangles?”

http://www.scribd.com/doc/60173215/2-4-Finding-the-sum-of-interior-angles-of-polygons-Worksheet

D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Euclid was a famous Greek mathematician that enjoyed the beauty of mathematics. He created a book called Euclid’s Elements where he gathered the knowledge of other famous mathematicians about the logical development of geometry. Pythagoras, Aristotle, Eudoxus, and Thales were some of the other men that influenced his work. Euclid’s Elements is compressed of 13 different volumes that are filled with geometrical theories. He proved the theories by using definitions as well as the axioms used in math.

Euclid was known as the “Father of Geometry” because he discovered geometry and gave it its value. The book contains over 467 propositions and they all include their proof. One of his propositions is about interior and exterior angles which is relevant to the sum of the interior angles topic. Proposition 32 states that an exterior angle is equal to the sum of the two opposite interior angles of a triangle, as well as the three interior angles of a triangle add up to two right angles. Since Euclid proves that a triangle is equal to 180ᵒ, it proves why we need to multiply (n-2)*180.

# Geometric magic trick

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.

Magician: Tell me a number between 3 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 3 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.

Why does this magic trick work? I offer a thought bubble if you’d like to think about it before scrolling down to see the answer.

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.

In other words, it must be the case that

$180T = 360y + 180(x-2)$, or $T = 2y + x - 2$.