Not this guy

The #1 rule of the game was that you couldn't tell anyone the rules of the game. The only way know the rules of the game were to figure it out by playing (back with the people I played against at that time, you weren't allowed to say why you were giving someone 2 extra cards for not following a certain rule and you weren't really allowed to watch a game without playing). Last year, a group of my students found this game again and were playing it all the time.

It's awesome for math class because you have to figure out the rules by noticing patterns. You're confident there are actual, logical rules out there, but you can only learn them by testing your theories in laying down certain cards or doing certain actions and then getting rewarded or punished based on the rules.

This week, I'm going to be doing a similar method with my classes in calculus. We've just "finished" learning about finding slope of tangent lines using a limit definition. I told them there was a "secret trick" so that you could just look at most of these problems and know the answer pretty quickly, but we wouldn't get to it until the next week. So, when they asked what the answer was, I could spout off the answer pretty quickly and not because I am so smart that I expanded (3+h)^4 in my head really quickly and then computed the limit, but because I knew the trick. Immediately they were intrigued. Some were guessing at what it could be and I just told them to either work it out to confirm their suspicions or wait until next week.

So, tomorrow starts "next week." It would be fun to do this with wolfram|alpha or something where they could come up with their own theories and test them individually, but we don't have the tech for that at our school. So, I'm going to be their wolfram|alpha.

I'm King Mao and I know the secret rules. I will have students throw out polynomials and points and I'll give them the slopes of the tangent lines at those points fairly quickly using the secret rule. When you think you've figured out the pattern, don't say the rule out loud. I'll throw out a problem for you to test if you're right. If you've got it right, you can sit out and the rest of the class can try to figure it out. If you're not right, keep playing until we all get it.

Oooh, damn. About halfway through the post my head was screaming, "USE WOLFRAM ALPHA!! USE WOLFRAM ALPHA!!" Do your kids have internet at home? Maybe you could make a widget and have them explore on their own the night before?

ReplyDeleteIn summer school I used Wolfram Alpha a couple of times for some exploration stuff, and its great. I was actually planning on doing exactly what you are doing, but with Wolfram. Let us know how they deal with it.

Well, here's how it went in the one class where we had enough time for them to actually get anywhere with it:

ReplyDeleteThey started with throwing out really long problems. So, I had to reassure them that this wasn't a challenge to get my to show some "math magic." Even problems like 2x^2 + 3x - 4 at x = 5 has like 9 different "variables" that could each factor in to the answer (each of the coefficients, each x, each + or -, and the 5). So, treat it like a science experiment and start small and only change one thing at a time.

Then, they thought they got it too early when they threw out y = x + c at various x-values and various constants (since they are all 1 for the slope of the tangent). They also assumed the linearity of the operator pretty quickly--which is ok in this case because derivatives work like that, but it's something to keep an eye on later.

They only needed to get to cubics to think they knew the pattern and it checked out. Then I proved it using synthetic division for (x^n - a^n)/(x-a). They weren't so interested in that since they already knew the "trick" and don't care where it comes from. (perhaps another blog post there)