Wednesday, February 11, 2015

Section 13-8: Exterior Angles of Polygons (Day 110)

Last night I tutored my geometry student. As I said I would, I gave him yesterday's worksheet on the Triangle, SSS, and SAS Inequalities, to match up with Section 5-5 of the Glencoe text.

He was still having trouble with the proofs, though. There was one question from the Glencoe text which directed the student to use the SSS Inequality to compare two angle measures. The problem was that there appeared to be insufficient information to conclude that two of the sides are congruent (for one of the S's in the SSS Inequality).

Here is the question: AC and BD intersect at O. We are given the lengths AB = BC = 8, AD = 8.11, and DO = 3, and angle measures ABO = 60 and CBO = 40. We are supposed to use an inequality to compare the angles AOB and AOD.

The intended answer -- and since this was an odd problem, I confirmed it in the back of the text -- is that AOB is less than AOD, since the former is opposite a side of length 8 and the latter is opposite a side of length 8.11. This follows from the SSS Inequality -- at least it would, if we knew that BO and DO were congruent. The third sides are already known to be congruent because AO is the side common to both triangles.

But there's no way to prove that BO is congruent to DO. We know that DO = 3, but the length of BO is not given. There's no way to conclude that BO = DO. It's not as if ABCD is said to be a kite (since then we could use the Kite Symmetry Theorem). Indeed, the Glencoe text doesn't even cover quadrilaterals until Chapter 6, so the students wouldn't know the Kite Symmetry Theorem anyway.

Furthermore, we can actually prove that BO can't possibly be 3. Here is an indirect proof:

Indirect Proof:
Assume that BO = 3. Notice that ABC is an isosceles triangle (since AB = BC = 8), and so angles BAC and BCA are congruent. Since angle ABC is 100 degrees (since ABO = 60 and CBO = 40 and ABO + CBO = ABC), it follows that angle BCA (that is, BCO) is 40 degrees.

Now triangle BCO has two 40-degree angles, CBO and BCO. It follows that BCO is also an isosceles triangle, and since we are assuming that BO = 3, CO would also be 3.

Since we are given that BC = 8, it follows that BCO is a triangle with sides 3, 3, and 8. But this violates the Triangle Inequality -- since 3 + 3 is not greater than 8. This is a contradiction. Thus, BO can't possibly be 3 -- it must be at least 4. The Glencoe text made an error here! Not only can we not prove that BO is 3, but the assumption that it is 3 leads to a contradiction!

After seeing this problem, my student was still feeling frustrated about proofs. My story about how for centuries, Euclid's Elements was the gold standard of logical thinking failed to convince him that he should learn proofs -- as far as he was concerned, only mathematicians should have to learn proofs and everyone else should have a proof-free curriculum.

There are several problems with this belief -- one I know is common among geometry students today. First, if only mathematicians had to learn proofs, there would be no more mathematicians, for every single mathematician today had to endure his or her teen years, when he or she had to sit through these same frustrating proofs. Second, this belief, that anything that isn't easy shouldn't be learned at all, makes our generation look bad. And I emphasize our generation, because it begins with those born around the year that I was born, 1980. The "dumbest generation" of Mark Bauerlein refers to anyone who was under 30 ("Don't trust anyone under 30!") the year he wrote his book, namely 2008. So anyone born after 1978 -- which includes the author of this blog -- is part of the dumbest generation.

From a mathematical perspective, members of older generations complain that math is being watered down when proofs are omitted. Even when high-level proofs are taught, medium-level proofs (such as proofs of the major named theorems) almost never appear in texts as exercises, and when they do, they're usually skipped by the teacher. This was the Dr. David Joyce's biggest complaint. Dr. Franklin Mason is working to restore medium-level proofs in his text, and I've included some on worksheets.

Of course, I didn't tell my student any of this, but what he is advocating is watering down math classes even further, and making our generation look even worse in the eyes of those like Bauerlein and Joyce. Then again, I doubt that any teacher has ever convinced any student that a good reason to teach a some topic in math is that omitting it would be watering down the math class.

I've stated before that one way to convince students to learn a topic is to make it fun. Since I invoked the argument that one group who needs to learn proofs is lawyers -- I mentioned this in a worksheet near the first day of school, and I mentioned it again to my student last night -- why not come up with an activity where a geometry proof is compared to a courtroom proof? Here is a link to an activity:

I may consider using such an activity myself when I give that unit on non-Euclidean geometry -- which would be after the PARCC/SBAC testing.

Section 13-8 of the U of Chicago text is on the Exterior Angles of Polygons -- not just triangles. This lesson is mostly straightforward. The way I like to introduce exterior angles is by showing a triangle, square, and pentagon with its exterior angles, and then imaging shrinking the polygon so that only the exterior angles remain. Then it becomes obvious that these angles add up to 360.

We notice that this lesson, just like Section 2-3, refers to computer programming. This time, the programming language is not BASIC, but Logo. Speaking of my generation once again, it's noted that elementary schools in the 1980's often taught Logo to us students.

Here's a link to a website that discusses the Logo programming language:

As it turns out, Logo is a very sophisticated programming language. The author of the link above, Brian Harvey (another Berkeley professor, just like Dr. Wu), points out that Logo originally had nothing to do with turtle graphics:

"Historically, this idea that Logo is mainly turtle graphics is a mistake. As I mentioned at the beginning of Chapter 1, Logo's name comes from the Greek word for word, because Logo was first designed as a language in which to manipulate language: words and sentences. Still, turtle graphics has turned out to be a very powerful addition to Logo."

Indeed, here's a link to another page on Harvey's website. The author uses Logo to solve a common logic problem:

"You are at the side of a river. You have a three-liter pitcher and a seven-liter pitcher. The pitchers do not have markings to allow measuring smaller quantities. You need two liters of water. How can you measure two liters?"

Here are the programs given in the U of Chicago text. The first one draws a regular 18-gon:


And the other draws a regular 180-gon -- which ends up looking more like a circle:


Here's how Harvey writes a more complicated polygon program:

to poly :size :angle
forward :size
right :angle
poly :size :angle

I decided to keep the Logo problems in my worksheet. Unfortunately, unlike BASIC, Logo isn't easy to convert into TI-BASIC. Logo is mentioned in this section because the angles mentions in the RIGHT commands are in fact exterior angles. If the students don't have access to Logo (Harvey discusses how one can download Berkeley logo on his webpage), one can change it to simple angle questions -- for example, my Question #3 becomes, "What is the exterior angle of a regular octagon?" (rather than draw one in Logo).

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