December 6, 2010

Super-Sized, Circumcised, Circumscribed

Filed under: High Effort/Low Payoff Ideas — Adam Glesser @ 10:02 pm
Tags: ,

Today was supposed to be observation day. At least once per semester since I arrived at Suffolk, I’ve been observed in the classroom by one of the department’s senior faculty. Just so that you don’t get the wrong idea, this isn’t some kinky thing where I prance in front of the examiner in a Speedo. No, outside of Europe, nobody wants to see that kind of nonsense. The faculty actually want to watch how I teach a class.

While I always try to outwardly act as if this is no big deal, the thought of being observed gets me super excited. It gives me a really good excuse to spend that extra hour preparing a special lecture. With that in mind, this observation session came at a pretty awkward time: we just finished the course material and this is review week with the final coming next Tuesday. This is usually when we kick back, tighten up the few topics that are hurting everyone and watch Raiders of the Lost Ark. Instead, though, I decided to review old material by introducing something slightly new.

Enter the Extended Law of Sines

It always surprises me how few people have heard of the extended law of sines. In short, it says that the quantity a/\sin(A) appearing in the law of sines is equal to the diameter of the circumcircle of the given triangle. If the radius of the circumcircle is r, then the theorem is usually written as:
\dfrac{a}{\sin(A)} = \dfrac{b}{\sin(B)} = \dfrac{c}{\sin(C)} = 2r
(where a,b,c and A,B,C represent the sides and angles of triangle, respectively, with a opposite angle A, etc.). Aside from giving a geometric interpretation of the terms in the law of sines, the result is also quite useful (I’ll give a fairly involved exercise at the bottom where it is possible to use it).

I wanted the students to prove the result, but I really was not confident in their elementary geometry skills, so I set things up for them. I told them to start from the triangle and draw the circumcircle. Naturally, they didn’t know what I was talking about and wondered if I’d said what they thought I’d said. Planning for this, I started explaining out loud how you physically construct the circumcircle. Eyes glaze. Planning for this, I opened up handy-dandy to show them what I can’t vividly describe verbally. Next came Geogebra where we found the circumcenter of the triangle and connected it to two of the vertices of the triangle. I now referenced the Central Angle Theorem which, again, nobody knew. Thankfully, Geogebra calculated the angles for us and since these students believe the convenient falsehood that one example proves that something always works, we were golden.

After connecting a few more dots, we were ready to prove the result. Here is the worksheet I gave them to work on in groups. It is nothing fantastic, but even some of the students whose attendance has been…spotty…worked out the answers. The beauty of the last question is that they need to shift gears and use the law of cosines before they can solve the problem.

Oh, and the professor who was coming to observe me? He didn’t show up. Go figure.


Exercise: With the notation as above, prove that the triangle is isosceles if and only if

\dfrac{a+b+c}{2} = a\cos(B) + b\cos(C) + c\cos(A).


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