## Tuesday, December 7, 2010

### Functions and Uncountable Sets

So I've got no idea how to offer a scientific context for functions in general, which is what I'm working on now in my Algebra 2 class. So I gave myself a bit of a challenge. I spent today trying to get my class excited and confused about the idea that besides for the countable infinity there is a larger, uncountable infinity. Then I told them that once we learn functions they have everything that they need to understand the proof.

I might have dug myself into a hole here. My plan was that I could lay the groundwork for the proof as I introduce them to domain, range, one-to-one, onto and bijection. Then I figured at the end I'd devote a bunch of class time to trying to help them grasp Cantor's diagonalization proof. This is problematic, though, because I'm going to need to devote almost a full period to help them grasp the diagonalization argument, and the confusing parts aren't the Algebra 2 parts, and I'm already crunched for time with this curriculum.

But I'm a desperate guy. Almost all my students think that what we're learning is worthless. I need to do something!

UPDATE: This might help.

1. I caught your post on Dan's blog re: real-world topics beyond ratios & proportions. Have you seen this ? There's a lot of R&P stuff for sure, but hopefully it'll be helpful in other ways, too.

2. Thanks Karim!

I hadn't seen this resource, and it looks as if it's got some great stuff on there, especially for Algebra 1. I probably should have done a nice application of exponents while we were studying it--maybe I'll spend a day doing some applications when it comes time to review. They have a nice lesson or two on exponential growth that I think my 9th graders could handle.

The Algebra 2 stuff is kind of typical of what I've seen, though. Exponential growth and the pythagorean theorem are the ratios/rates of Algebra 2. They're the thing we all lean on when we think of great application problems, but they're the ONLY things that we have great application problems for. I've got a textbook that motivates rational equations by telling us that kayakers use them to figure out how fast the river is going. SIGH. Pseudocontext.

What I'm working out on this blog is that when it comes to Algebra 2 we need context, but it can't be the sort of "everyday real life" context we can pull of with a good deal of Algebra 1. We need to give the scientific context, because that's the truth.