(May 21, 2013) There’s more than one way to prove the vertical angle theorem: verbally, numerically, algebraically (with various approaches), intuitively, visually. Our Math Circle participants debated and attempted them all. When the theorem was finally proven algebraically, everyone smiled. It was a satisfying proof. And it was fun. And we learned on the way. Some highlights:

- N asked “What does 180 degrees mean?” Great question. How do you define a degree? How do you define an angle? (The definition that I like and shared is that an angle is an amount of turning.)
- R and P were so excited to contribute steps that they were, well, squealing. With their hands waving in the air. One student commented to them, “You have an epiphany every Tuesday!” This was meant – and taken – kindly. I had them play Rock/Paper/Scissors to determine speaking order. Soon, P walked up to the board, picked up the chalk, and began to explain her reasoning to the class. I moved to my more appropriate place in the classroom: in a chair to the side with my mouth shut.
- Someone clarified the group’s algebraic method with the comment, “You can do the same thing to both sides” of an equation. I, the secretary, wrote this on the board. N immediately corrected me. “You MUST do the same thing to both sides. This is math. We have to be precise. We don’t want the Algebra Devils to get us.”
- At one point, we were really struggling with the algebra. Fortunately, P suggested 2 equations, then R refined them, and then C suggested using the substitution method. Each suggestion built upon the prior, in true collaboration.
- At the end of the proof, 5 students strongly agreed that the proof was clear and direct. One student, A, said “no comment,” but agreed that we should put the proof to bed and move on.
- We contrasted this lovely proof with the proof by contradiction we had done last week. C pointed out how great it was that this week “
*we*got to prove this. You didn’t just demonstrate it for us.” We talked about how we could have taken multiple approaches. Our ability to choose makes this an art. “We could hear the same song everyday – and we do – and get something different out of it every time,” to quote Adam Hoopengardner, professional tango dancer.^{1,2}

After the angle proof, we revisited the Pythagorean theorem. N had been absent when we worked with visual Pythagorean proofs. I drew a right triangle on the board, with squares adjacent to each side. N’s eyes lit up. He approached the diagram, kneeled in front of it, and pressed his hands to the board. “I see it!” Then he silently stared at the diagram, drawn in by the human architectural instinct. “No, I lost it,” he said disappointedly. Then he redrew the figure in another attempt, stared, tried to make sense of it, but finally sat down in contemplation.

What impressed me was the reaction of the other students. N was basically staring at the board, thinking. Everyone else sat in rapt silence, staring at N. For quite some time. We were mesmerized – rooting for him, trying to see what he was seeing. I feel chastised for any insinuation I may have made about “watching golf” in my first report on this proofs circle. Today I saw the contagious power of concentration. Math *can* be a spectator event.

Speaking of my first post on this circle, I want to let you know that one of my favorite math blogs, Let’s Play Math, is hosting this month’s Math Teachers at Play Blog Carnival, which includes that post. For those of you looking for challenging and fun math activities to do with your students and children of all ages, look no further.

And finally, you may have noticed that N seemed to take center stage in today’s Circle. In the five prior weeks, he didn’t have all that much to say. Today has been a nice reminder to me of the fact that just because a student is quiet does not mean that the student is not fully present. My experience continues to be that every person *will* find their own breakout math topics. Have you found yours?

Rodi

^{1} Crossover, WRTI, May 18, 2013.

^{2} I have been mentally chewing on the idea of math as an art for the past 2 years. I get it intuitively but haven’t been able to easily verbalize an explanation until now. This tango reference helped, as did a reference from The Story of Ain’t: dictionary editor Philip Gove described lexicography as “an overpowering art, requiring subjective analysis, arbitrary decisions, and intuitive reasoning. It often uses analogy, precedent, and probability.” (p233) Might this describe at least some aspects of mathematics too?

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