HCSSiM workshop day 1
So I’ve decided to try to explain what we’re doing in class here at mathcamp. This is both for your benefit and mine, since this way I won’t have to find my notes next time I do this.
We started the math, after intros, by assigning note-takers. In one row we wrote down the students’ names (14 of them), and in the other we wrote down the numbers 1 through 14. We drew lines from names down to numbers. These were the assigments for the days they’d take notes.
But to make it more interesting, we added pipes between different vertical lines. The pipes can be curly (my favorite ones were loopedy-loop) but have to start at one vertical line and end at another at “T” crosses.
Then the algorithm to get from a name to a number was this: start at the name, go down the vertical line til you hit a “T”, follow the “T” pipe til you hit another vertical line, and then go down.
This ends up matching people with numbers in a one-to-one fashion, but why? We promised to prove this by the end of the workshop.
Map of Math
We next had the kids talk about what “math” is. We had them throw up terms and we drew a collage on the board with everything they said. We circled the topics and connected them with lines if we could make the case they were related fields. We drew lines from the terms to the topics that used that a lot – like the symbol got pointed at Trigonometry and Geometry, for example. I think it was useful. Lots of terms were clarified or at least people got told they would learn stuff about it in the next few weeks.
Next, we asked the kids how many pieces you can cut a watermelon into with 17 cuts. Imagine the watermelon plays nice and stays the shape of a watermelon as you continue cuts, and you can’t rearrange the watermelon’s pieces either.
If you do a few cuts it quickly gets hard to imagine.
So go down to a 2-dimensional watermelon, which could be called a pizza or a flattermelon. We called it a flattermelon. In this case you’re trying to see how many pieces you can achieve with 17 cuts. But also you may notice that a single slice of a 3-d watermelon looks, to the knife’s edge, like you are spanning a flattermelon.
Similarly, you may notice that a cut of the flattermelon looks like a 1-dimensional watermelon, otherwise known as a flatterermelon. And there the problem is easy: if you have a one dimensional watermelon, i.e. a line, then n cuts gives you maximum n+1 pieces. But going back to a pizza a.k.a. flattermelon, any cut looks from the point of view of the knife like a 1-d watermelon, which is to say it is cutting n+1 regions into half assuming the lines are in general position. So we get a recursion. If we denote by the max number of pieces you can get in dimensions with cuts, then we can see that
Since we know this recursion relation generates everything, although not in closed form.
Next, I went on at length about the utility and frustration of notation. Namely, notation is only useful if everyone agrees on what it means. I like standard notation because it’s more, well, useful, but Hampshire is a place where kids absolutely adore making up their own notation. As long as we are consistent it’s ok with me, and I like the fact that they own it. So instead of the standard notation for “n choose k” we are using a pacman symbol with n inside the pacman and k being eaten by the pacman. We call it “n chews k”.
We talked about putting balls in baskets, and defined that pacman figure to be the number of ways we can do it. Then we proved the pascal’s triangle recursion relation using the argument where you isolate one basket and talk about the two cases, one where there’s a ball inside it and the other when there’s not. Then we identified Pascal’s triangle as being equivalent to this concept of counting. I described this as an example of a combinatorial argument, which I like because it doesn’t involve formulas and I’m lazy.
Finally, I introduced Mathematical Induction and did the standard first proof, namely to show the sum of the first n positive integers is