# Engaging students: Define the term angle and the measure of an angle

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 Perla Perez. Her topic, from Geometry: defining the term angle and the measure of an angle.

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

On teacherideas.co.uk, there is a very fun activity that helps students understand what an angle is. Karen Westley’s activity, Learning About Angles, involves the students forming a human angle by getting into two lines (Karen uses only one line) that connect at one point. The instructor then asks them to form angles of different degrees: 45º, 90º, 180º, etc. This activity is meant for ages five to eleven, but it can still be helpful for high school students, and can be modified to fit your class. For example, after finishing this activity have the students give a written definition of what an angle is based on their activity and their knowledge of line segments, vertices, rays, etc.

Resources:

How can this topic be used in your students’ future courses in mathematics or science?

Angles are a fundamental part of geometry. It is essential that students know how to measure them because when coupled with the right information angles can help determine many different things about a shape, such as area, length of an arc, etc. The students will return to these ideas in more advanced math courses, specifically trigonometry and pre-calculus. For example a problem given in inmath.com says: “Find the area of the sector with radius with 7cm and central angle of 2.5 radians.” In order to answer this question, a student must know the two common types of measurement, degrees and radians. The student will also need to differentiate between the radius and radians since they sound similar and can be easily misinterpreted. When it comes to polar coordinates students will need to convert from the measure of an angle to rectangular coordinates which are in the form (x,y) rather in (r,theta).

Resources:

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

The measure of an angle can be found in two common ways, through radians and through degrees. Dave Joyce from Clark University in his article “Measurements of Angles” says that even before Thomas Muir created the word radians, many mathematicians like Euler were using the idea long before him. That in fact was essential for his famous formula “e = cos θ + i sin θ”, which is true because “[you] measure angles by the length of the arc cut off in the unit circle”. Over time, mathematicians like Thomas Muir began to rediscover the measure of angles in the same way Euler did. An interesting fact about radians is that the angle is equal to the arc length divided by the radius. It was Euclid’s postulates that contributed to the finding the measures of angles without actually stating whether one uses a form of degrees or radians. This information can be found on the website: http://www.storyofmathematics.com/hellenistic_euclid.html and his book Euclid Elements.

Resources:

http://www.clarku.edu/~djoyce/trig/angle.html

http://www.storyofmathematics.com/hellenistic_euclid.html

Euclid’s Elements Book

# Engaging students: Proving that the measures of a triangle’s angles add to 180 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 comes from my former student A’Lyssa Rodriguez. Her topic, from Geometry: proving that the measures of a triangle’s angles add to 180 degrees.

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

People generally do not believe something until they can see it for themselves. So this activity can help do just that. Each student will receive a sheet of paper. They are then asked to draw a triangle on that sheet of paper and cut it out. Having each student draw their own triangle allows for many types of triangles and further proving the point later. Once the triangles are cut, each student will rip off each angle from the triangle. Next, they will arrange those pieces so that each vertex is touching the other. Once all the vertices are touching, they will notice that a straight line is formed and therefore proving that the sum of a triangles angles all add up to 180 degrees.

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

Euclid proves that the measures of a triangle’s angles add up to two right angles (I. 32) in the compilation geometrical proofs better known as Euclid’s Elements. This compilation was actually all the known mathematics at the time.  So not all of the theorems were written or discovered by Euclid, rather by several individuals such as Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. Euclid’s Elements actually consist of 465 theorems, all of which are proven with only a ruler (straight edge) and compass. This book was so important to the mathematical community that it remained the main book of geometry for over 2,000 years. It wasn’t until the early 19th century that non-Euclidean geometry was considered.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Students can be given a variety of images such as the Louvre, the pyramids in Egypt, certain types of sports plays, and the Epcot center in Disney World and then be asked what they all have in common. It may or may not be hard for them to notice but they all have triangles. Then, hand the students the same images but with the triangles outlined and with the measurement of all the angles. They can then compute the sum of the angles for each triangle. Each triangle obviously looks different and all the angles are different but the sum will always be 180 degrees.

# 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.

# Langley’s Adventitious Angles: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my short series on a couple of easily stated but remarkably difficult geometry problems.

Part 1: The world’s second hardest easy geometry problem.

Part 2: The world’s hardest easy geometry problem.

# Langley’s Adventitious Angles (Part 1)

Math With Bad Drawings had an interesting post about solving for $x$ in the following picture (this picture is taken from http://thinkzone.wlonk.com/MathFun/Triangle.htm):

I had never heard of this problem before, but it’s apparently well known and is called Langley’s Adventitious Angles. See Math With Bad Drawings, Wikipedia, and Math Pages for more information about the solution of this problem. Math Pages has a nice discussion about mathematical aspects of this problem, including connections to the Laws of Sines and Cosines and to various trig identities.

I’d encourage you to try to solve for $x$ without clicking on any of these links… a certain trick out of the patented Bag of Tricks is required to solve this problem using only geometry (as opposed to the Law of Cosines and the Law of Sines). I have a story that I tell my students about the patented Bag of Tricks: Socrates gave the Bag of Tricks to Plato, Plato gave it to Aristotle, it passed down the generations, my teacher taught the Bag of Tricks to me, and I teach it to my students. In the same post, Math With Bad Drawings has a nice discussion about pedagogical aspects of this problem concerning when a “trick” becomes a “technique”.

I recommend this problem for advanced geometry students who need to be challenged; even bright students will be stumped concerning coming up with the requisite trick on their own. Indeed, the problem still remains quite challenging even after the trick is shown.

# 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.

# Angular size

Source: http://www.xkcd.com/1276/