Do you feel like you learned some real mathematics here today?
Before showing Vi Hart’s “Binary Trees” short film, we did a few function machines as we waited for everyone to arrive. (This session was class 1 of a new session for middle-school kids.) The rule y=2x was obvious. The next rule, y=x squared, was obvious as well, but the terminology was not (“I never knew that little number was called an exponent,” said R). Other terms that a handful of kids knew were “squaring a number,” and “taking it to the power of 2.” For now, we did not use variables to describe the functions, and instead used operational symbols with the words “in” and “out”. So we described “y=x to the second” as “out=in to the second”).
I then modified the function: in went 1, out came 2; in went 2, out came 4, in went 3, out came 8 (“what?!”) and so on. M asked, “Is it Fibonacci?” (No.) P said, “I think I see a pattern in the out numbers: they keep doubling.” Faces lit up, but I burst that bubble by switching the order of the pairs of “in” and “out” numbers. I told them that functions totally depend upon the number that goes in.
After a lot of staring and thinking, and the hint that this function was similar to the previous one, L noticed that the in number dictated how many 2s there were in the out number. Then M sat straight up and announced, “I get it! You make the ‘in’ number the exponent!” With a little more work, the group worked out that the function was “out=2 to the in power,” (y=2 to the x).
By this time, everyone had arrived, so we moved to the floor and watched the video. Everyone was rapt. When it ended, K asked, “Can we watch it again?” So we did. Afterwards, I asked whether it made them feel like doodling, and of course the answer was yes, so we moved to the table.
“What was that name again, so I can google it on my tablet in the car?” asked M. He was talking about Sierpinski. Most were captivated by Sierpinski’s Triangle, and set to work attempting to create it. While they doodled, I asked what they heard in the video that aroused their curiosity. Fractals had, so I showed an artistic looking one and as a group we tried to define a fractal. Vi Hart herself had, so I gave a biographical sketch. The Hydra had, so N (with a bit of help from L and K) told the group the story from Greek mythology. We reviewed the arithmetic of the hydra’s knack: when 1 head got chopped off, 2 new necks/heads grew. In chorus, we got up to 7 cuts (128 new heads). I then jumped and asked them how many heads if 32 cuts? They remembered from the video that it was a really huge number. I wrote down 4,294,967,296. In chorus again, we named the number.
“How did that number get so big so fast,” I asked? We discussed it in layman’s terms, and then I gave them the mathematical term: exponential growth. “And what does this have to do with those function machines we did at the beginning?” R realized that the hydra head regrowth function was taking 2 to a power.
As we discussed exponential growth, people were doodling. J and S were drawing the hydra. Others were still working on, or struggling with, Sierpinski’s Triangle. “How do you do that?” X volunteered to demonstrate its construction on the board, and I helped people individually. L thought that Sierpinski’s Triangle might be related to Pascal’s Triangle (it is – another thing to put on the To Do list). People asked again how to spell Sierpinski, and wrote it down.
We were nearing our time limit, so I quoted Hart on doodling: “But the point is not to learn about fractals or cellular automata or Sierpinski, but to show that simple doodle games can lead to mathematical results so cool and beautiful that they’re famous. At least, famous to people like me. And, if you’re good at inventing doodle games, you might even end up doing some real mathematics during your math class.”
“Do you feel like you learned some real mathematics here today?” I queried.
“No,” said J.
“Cause it was fun,” replied N, with others nodding their heads.
I reminded them about fractals and exponential growth and the students grudgingly agreed that we had covered some actual mathematics. I promised that we will soon delve more deeply into these concepts and expand upon them. Finally, I kicked them out, promising more Vi Hart next time.