It's Good to be Perplexed in Math
N came up to the board a number of times as we continued last week’s probability problem.* D had planned to start class with a new approach, but didn’t because his ideas had changed over the week. N first came up to question the possibility of actually choosing chords randomly. He suggested an experiment where a random-selection method would actually show bias. We started our question list with “how can we make it random and be consistent?”
The group discussion then yielded a diagram with an “infinite” number of chords drawn in green, starting at a single vertex and passing through the triangle, and another “infinite” number of chords drawn in blue, starting at that same vertex, totally outside of the triangle. The group agreed that all of the green chords were longer than a side of the triangle, and that all of the blue ones were shorter. Therefore, posited C, since each color contains the same number of chords, that is, infinity chords, there’s a 50% chance of choosing a longer one. Faces lit up as we apparently solved the problem. I began our conjecture list on the board: “50/50.” Then I saw a few frowns. While everyone agreed that each section contained infinity chords, not everyone agreed that each group of infinity was the same size. Could there be different sizes of infinity? I explained that this is no simple question, that we could talk about it for weeks.** So we added it to our question list, and amended our first conjecture with the qualifier “assuming infinities the same size.”
The students added two other questions to the running list on the board:
- Can we measure the areas (taken up by the blue and green chords)?
- Should we experiment (to determine the answer for sure)?
The students felt certain that the areas taken up by the blue and green chords could be measured, but no one wanted to. Actually, they did want to, very badly, but no one knew how to. (R had a method, but the others didn’t quite understand it.) They were faced with two weeks of work stymied because of a lack of geometry experience. What to do? Should I show them how to figure it out? To not show them would be cruel. But to show them would violate the Prime Directive of Math Circles: Tell Kids Nothing.
I decided that the humane tack would be to nudge. I drew a tangent line at the vertex-of-interest. Then I waited. “How is that supposed to help us?” asked P.
“How many degrees are in a line?” I asked back.
“180,” answered C. I wrote that in, and waited. Still silence.
“Is there anything else here that we know angle measures for?” I asked. This was the watershed question. A conversation began that ended with R and P bouncing in their seats, eyes afire. I signaled for them to remain silent until everyone else was aboard. Soon, most people agreed that of the 180 degrees in that tangent line, 60 degrees was taken up by green “longer” chords, and twice that many by blue “shorter” ones.
“So there’s a one-third chance of randomly choosing a longer chord!” announced R gleefully. We listed this conjecture on the board, with the qualifier that it holds only if there are different-sized infinities. The general mood in the room was celebratory. But then C put forth a dissenting opinion: maybe all the chords that emanated from the vertex and filled that 180-degree tangent line were not actually within the circle. Others joined the dissent. Some opinions vacillated. This question was discussed, debated, and drawn on paper for quite some time. I asked the class for the definition of an “angle.” The pursuant discussion led a few more members of the group return to the original majority opinion (R’s “one-third” conjecture).
But not all were convinced. Both D and A had been silent during this debate. I asked for their opinions. A said she had no opinion one way or the other, but commented that she really was enjoying hearing everyone else’s conjectures. When I asked D, he said something like, “I don’t know why we are getting so caught up in this debate. If you just look at the other end of each chord (the non-vertex endpoints), you can see that 2/3 of the chords are out of the circle and 1/3 go through it.” That was all it took. Boom. Done. Everyone was convinced immediately.
So, our emotional roller-coaster of a Math Circle was up on high again, convinced that the answer is “50% chance if all infinities are the same size, but 1/3 chance if different-sized infinities exist.”
Sadly, I had to burst their bubble (again). I explained that their methods and calculations were mathematically sound, and that many mathematicians have “solved” the problem in this very same way, but that other methods yield different results. I demonstrated another theoretical method (yielding an answer of 50% with different-sized infinities), and explained another experimental approach (yielding a 25% chance, and not involving infinity at all).
“So we were wrong!” someone lamented loudly. I told them that they are no more right or wrong than any of the people who took these other approaches; all the methods that have come up are sound in their calculations. Disappointment morphed into confusion.
“I’m perplexed,” someone said. We were out of time at this point.
“I will leave you perplexed. It’s good to be perplexed in math.” I promised more discussion of this problem next week. I’m looking forward to hearing what goes on in the students’ minds while we are away from the problem for a week.
*“If an equilateral triangle is inscribed in a circle, what is the probability that a randomly selected chord is longer than a side of the triangle?”
** In fact, we will have a chance to talk about infinity for weeks in our upcoming math circle for 9- and 10-year-olds.
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