Home > math education > HCSSiM workshop day 1

HCSSiM workshop day 1

July 3, 2012

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.

Notetakers

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 \pi 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.

Cutting Watermelons

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 M_d(n) the max number of pieces you can get in d dimensions with n cuts, then we can see that

M_d(n) = M_d(n-1) + M_{d-1}(n-1).

Since we know M_1(n) = n+1, this recursion relation generates everything, although not in closed form.

Notation

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”.

Combinatorial Argument

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.

Induction

Finally, I introduced Mathematical Induction and did the standard first proof, namely to show the sum of the first n positive integers is \frac{n (n+1)}{2}.

Categories: math education
  1. suevanhattum
    July 3, 2012 at 1:06 pm

    The notetaker thing sounds intriguing, but I don’t really get it. Could you post a diagram showing how it works?

    The watermelon thing is fabulous! I’ve done the 2D version (calling it the magic pancake), but never considered generalizing to other dimensions – I’m eager to think about that.

  2. July 3, 2012 at 1:39 pm

    Cathy. Thank you so much for blogging this! Now I can pretend I’m actually present for this summer, which I was sadly unable to do.

    Also, I really like “flattermelon,” and absolutely love “n chews k.” That is simply brilliant.

    • joelibacsi
      July 4, 2012 at 7:48 am

      n chews k is indeed brilliant. i wonder what chewbacca would mean.

  3. Scott Carnahan
    July 3, 2012 at 3:19 pm

    Amidakuji (your notetaker selection method) is a standard way to make roughly random permutations in Japan. You’re supposed to sing the Amidakuji Song as you trace paths, although I’ve seen some people who were just too cool for that and stayed silent.

    It might be interesting to look at how Amidakuji is biased toward certain permutations.

  4. KCd
    July 3, 2012 at 9:16 pm

    While you’re letting students make up their own notation for concepts that already have a standard notation, consider passing along to them the story of Feynman’s creation of his own notation for trigonometric functions instead of sin, cos, tan, and their inverses. The story can be found in Surely You’re Joking, Mr. Feynman and excerpts come up on various web pages. The main point is the end of his story: “I thought my symbols were just as good, if not better, than the regular symbols — it doesn’t make any difference what symbols you use — but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, “What the hell are those?” I realized then that if I’m going to talk to anybody else, I’ll have to use the standard symbols, so I eventually gave up my own symbols.”

    • July 3, 2012 at 9:32 pm

      I agree. Which is why I basically use standard symbols. But the kids absolutely adore this kind of thing, so what are you gonna do. I don’t wanna be a curmudgeon.

      • Clark Thornton
        July 4, 2012 at 1:54 am

        I’m afraid I don’t understand your recursion equation. If Msub1(n) = n + 1, then by substitution into the formula, n+ 1 = Msub1(n-1) + Msub0(n-1). if n=17, then it’s 18 = 17 + . . . what? 17 cuts of d-1=0 dimension? If this latter value is zero then we have 18 = 17.

        • July 4, 2012 at 6:39 am

          M_0(n) = 1 for all n. Which is to say, no matter how many times you try to cut a point you always end up with one piece, namely your point.

        • Clark Thornton
          July 4, 2012 at 9:25 am

          Thanks! I love your blog.

        • July 23, 2012 at 1:26 pm

          you’re welcome! and thanks!

  5. None
    July 23, 2012 at 1:10 pm

    Having trouble following the watermelon “proof”, if thats what it is. I think I understand the specific statements, but then out of nowhere springs the recursive formula. Is the statement of the formula completely independent of the description you give, and the description only for context ?

    Looks like a fantastic experience for the kids, well done.

    • July 23, 2012 at 1:22 pm

      The recursion follows from a generalization of the two previous paragraphs – the idea is that, on the nth cut in dimension d you make a slice that, along the slice itself, looks like a dimension (d-1) problem with n-1 cuts. And you’ve cut each region in half, so you’ve altogether added M_{d-1}(n-1) regions from what you had previously.

      Does that help?

  6. None
    July 23, 2012 at 3:03 pm

    Thanks for replying. I think I understand what you are saying, but from that I can’t see the guarantee (even though its perhaps intuitively probable) that in order to have maximum total pieces after all cuts have been performed, each most effective nth cut in dimension d is going to necessarily add M_{d-1}(n-1) regions (ie maybe some “preparatory less effective cuts” set the scene for some later record breaking cuts). And even more difficult to see in my minds eye, and related to the previous point, is the guarantee that after X cuts it will still always be possible at dimension 2 (flattermelons) to make a new cut that can intersect the previous X-1 cuts – ie that the formula would be giving a theoretical maximum which was not always practically achievable. Sounds ridiculous I know, but I can’t easily convince myself.

  1. July 4, 2012 at 6:58 am
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