A tease

I keep forgetting about all the things I want to write about. So I will make a public list and hopefully these posts will come in quick succession.

• Tchebyshev Polynomials (Thanks to Al Cuoco again)
• Combinatorial Game Theory textbook
• Blogging as an assignment
• The genius of Euler

Feel free to write in the comments that one of these is more interesting than the next.

More Determinants

I was at the Joint Mathematics Meeting in January, and their I met an author that I really enjoy. His name is Al Cuoco. He does a lot with mathematics education and his book on Mathematical Connections is a must for any high school math teacher who wants to give their students a hint of the beauty of mathematics beyond high school math. I highly recommend it.

Since he was there, and he was speaking a little bit about linear algebra, I decided to ask him the question that has been bothering me for at least a year. What is the the determinant? Remember from my previous posts that it is not that I don't know how to calculate a determinant, or when to use a determinant. What I don't know is how to explain it to students. How to give it meaning. Al's opinion appears to be that it is most clearly connected to volume. To help prove his point he sent me a graphic which previous appeared on the cover of a math journal. The graphic does a pretty good job of explaining how determinants can be used to find the coordinates in a different coordinate system. It basically comes down to projections.

Pretty cool. I think that I will try to tie together the different determinant thoughts that I have had over the year in one post. I will hopefully be able to add some of the things that I have learned in Al's papers.

Calculator programming

While covering the quadratic formula with my 8th grade algebra students, I decided to show them how to program their calculator to do the quadratic formula. Students had a surprisingly strong response to it. They loved it.

Certainly they love the power of having the quadratic formula in their calculator. And their may be less chance that they will memorize the quadratic formula because they don't have to do it as often as they would have had to if I hadn't given them the formula. But it was a great way to discuss the ideas of input and output. I drew a function machine on the board, and we discussed how other machines can be seen as functions. So this connects this fundamental mathematical concept of function for them to their lives, and it allows them a way to construct a function of their own. We did a problem of finding how long it takes for an object to fall a certain distance. Students really started to get it.

In fact, after we programmed that we wrote a program to find the vertex, both the x and y coordinates. This was great for them because it emphasized that this process of writing a computer (calculator) program is really whatever you want. Got a process that you are doing over and over then make a program. I also indicated that these two separate programs might be merged together to create a bigger more complex program that could do both things. Other teachers told me later that the students were still talking about it several periods later.

On the technology side, it really helps to do this with a smartboard and TI-smartview. I was asked so rarely about what did I type or where that button is on the calculator. It is such a relief to have these tools.

Bear hunting Algebra

I am finding myself more of a shameless shill for mathematics in my 8th grade Algebra class. Want me to jump around like an idiot? I will do it. Want me to make a snowball and bring it in to the classroom? Good as done. So you can see I am interested doing anything that will meet my students where they are...

One more crazy thing I have been doing is trying to connect the kids books I read to my math class. And I think I found out one that actually worked a little bit. It is Going on a Bear Hunt...

You may the know story, you may have been to camp, you may have never heard of it, which may mean that you have never had a child, been to any place where teenagers are asked to supervise young children. If that is you click here to get a sense of what it means to go on a bear hunt.

I use this to remind/reinforce the idea about doing and undoing to solve an algebraic equation. So with the equation:

3x+1=10

I say we are going on bear hunt we are going catch a big one we are not scared. Oh no a times 3!. Can't go over it can't go under it... Oh no a plus 1, etc. And then I said a 10! AAAAAAHHHHHH! Back through the 1 (minus 1), back through the 3 (divide by three). These are the steps (minus 1 and divide by 3) that you need to solve the problem.

Now as many of surely know, much of this is not "necessary", but my students need the background of reading through the problem the first time to understand the order of operations. In fact, many of my students still don't understand that the 3x cannot be undone by subtracting 3.

I hope that this helps them see not only what to do, but why they are doing it.

At the very least I am exposing them to classic literature, well literature, well words.