PROOFS #2: Exponents, Roots, Pythagoras, Theorems, Proofs, and The Kaplans
(April 23, 2013) Before continuing our TV problem, the students recapped last session for A, who had been absent last week. This week’s problem solving once again presented rich opportunities for delving deeply into arithmetic; it has not been the algebra and geometry that is challenging for these students so far.
The group wanted to apply the Pythagorean Theorem to their triangle to calculate the TV’s diagonal: 102+5.62=c2, but no one was positive about the best way to begin. I asked, “What are the friendliest looking numbers?” expecting to hear 10 in response.
“Those twos look pretty friendly,” replied N. I asked the group what those twos mean, and discovered that not everyone had seen exponents before, and those who had did not know much. I moved to the other board, wrote 32, and asked what it means. From there, the students thought and conversed about what they knew, generating an impressive amount of information. Even deeper understanding came from discussions of why exponents are called “powers” and why the powers of 2 and 3 are called “squares” and “cubes” of numbers. The students wanted to know how to calculate large powers, such as 2105, and came up with a multi-step method. R also wondered whether it can be done in one step without a calculator.
They then easily squared the 10, but hesitated on 5.6. “Couldn’t you just square 5 and get 25 and square .6 and get .36, for an answer of 25.36?” asked G. Great idea, of course, as is every math conjecture.
“I just got 28.36 multiplying it out the long way,” added R, with some doubt. “I wonder if I made a mistake in my multiplication.” What to do? No one besides R wanted to do long multiplication to double check.
“Do you know another method?” No one did. “How about the box method?” I asked. This sounded vaguely familiar to G and R, so we did it on the board. This method generated 31.36 and visually revealed what was missing in our first two attempts. I sang the praises of the “Babylonian Box,”explaining that this method can be used for many arithmetic and algebraic calculations. The students were intrigued, but didn’t really understand exactly why this method did not generate the same answer as G’s: where did those 2 little edge strips (representing 5 times .6) come from, and what do they mean? I started to discuss it, but most of the group wanted to get a trustworthy answer more quickly in order to return to the Pythagorean Theorem. I promised R a future, private, and deeper discussion of the box method.
Everyone wanted to figure out which of our 3 different answers was correct, and asked to do the long multiplication method collaboratively for confirmation. We did it, and got 31.36 - same as the box. R’s multiplication mistake was revealed and we moved on.
“So the diagonal of the TV we need is 131.36!” someone happily announced.
“Wait a minute, that doesn’t make sense,” said A.
“That’s a good strategy: ask whether your answer makes sense,” I said. The kids then realized that they needed to take the square root of the answer. Since not all students had done roots before, those who knew how taught the concept to those who didn’t, with me simply serving as secretary at the board. They debated whether they needed the exact root of 131.36. G suggested estimating it to the nearest 5. A and several others frowned at this. I asked N, who had been quiet for most of the class, for his opinion.
“Well, this is math,” he said. “Precision is important.” The rest agreed, but still did not want to engage in tedious calculations. Fortunately, A pointed out that TVs are most commonly sold in whole number measurements, and only occasionally in half-inch increments. He proposed estimating to the nearest half inch, and everyone agreed with a collective sigh of relief. The group reviewed perfect squares and quickly agreed that the answer was between 11 and 12.
“11.5,” announced several voices at once. C estimated that 11.4 would be closer since 112 is closer to 131 than is 122. R calculated 11.42 and reported that, at 129.96, it’s a bit smaller than what we want, but all agreed that this degree of precision was unnecessary for buying a TV. R noticed that squaring the numbers in a ratio does not preserve the ratio, and wondered why. No one knew, so we set aside that question on the board for another day.
The problem was solved. The kids needed the Pythagorean Theorem and had wielded it for results. They achieved mathematical success. But their brains were tired. “I think that now you should tell us about that guy who was killed because of a number,” declared G. I agreed that this was a good place for a history interlude.
“To understand that story, you’ll need to know about Pythagoras himself, and you’ll need an understanding of the Theorem. What does it mean? What does squaring those numbers do and why does it work?” The students discussed this, and ended up with a diagram of a triangle with attached squares. Kids could state geometrically what the Theorem was saying, but didn’t understand why it works. We set that part aside for next time, and talked about the Pythagorean Cult.
“Aren’t cults where people drink the kool-aid?” asked N. After others piped in their knowledge of cults, I read aloud a description of some of the cultish practices of Pythagoras and his followers. We also talked about trends in math history – about how views of historical figures change over time.
With 2 minutes left, I decided to ask a seemingly straightforward question: “What is the definition of a theorem?” The kids discussed this term along with the terms theory, conjecture, guess, and hypothesis to conclude that a theorem is a guess or theory backed up by a proof. Their definition is remarkably close to that of the Kaplans’: “A theorem is an insight shackled by a proof.”
“So, does a proof prove something without a doubt,” asked the kids,”or can you find exceptions to it?” This led to a heated debate among R, C, and G. I took the position that if you state all your assumptions, follow a logical train of thought, and no counter-examples exist, then you’ve got yourself a proof. “No one has found a counterexample to the Pythagorean Theorem yet,” I argued.
“Just how old is that theorem?” countered N. I answered that Pythagoras lived around 500 B.C., but information about the theorem’s precise origination is vague. Everyone mulled over whether two-thousand-plus years is enough time for counterexamples to emerge. Then I kicked the students out, with some of their voices still raised in debate over whether proofs can be trusted. It seems that everyone wants to trust them. G, however, announced defiantly upon exit, “I am determined to find a non-right-triangle that works with the Pythagorean Theorem!” Several others walked out discussing whether math is invented or discovered. I promised to continue this philosophical discussion, with examples of good proofs and bad proofs, next week.
NOTE: My primary research source for this lesson is Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, by Bob and Ellen Kaplan. Not only has the book helped, but so too have Bob and Ellen themselves. I received a generous travel grant from the MSRI and the NSA to visit the Kaplan’s Math Circles in Boston. I sat in on the sessions with my children; R (age 13) and J (age 8).
Afterward, R said to me, “I’ve always thought you are a great teacher, but I can see why Bob and Ellen are The Masters.” I think she wants to move to Boston to keep working with Ellen on calculating the exact square root of 7. And J was awed by Bob’s ability to work with a group of young children (and us) at a wide variety of math experience levels to explore questions ranging from “What is 3 times 3 times 3” to “How many corners does a 10-dimentional cube have?”
Bob and Ellen were generous with their time after the Circles. We talked about mathematics in general, Math Circles (theirs and mine), my specific interests in mathematics, R’s and J’s interests, and this particular Math Circle on proofs. I have been planning this Math Circle course on proofs for the past 2 years, with input from Bob and Ellen all along. Thanks to both of you, for everything, more than I can say.