Daily Archives: January 30, 2008

I am teaching exponential functions in my Algebra II classes this week. And I just came back from this teaching conference, where one of the sessions included a few handouts of the types of problems that this one charter school uses. And lucky for me, one was on exponential growth and decay.

I wholesale retyped this activity-based lesson up and gave it to my students. I can’t say it was the “most awesome thing ever,” but I can say that it got students to think for themselves instead of being spoon fed everything. What it had students do is to:

Fold a piece of paper in half and record: (1) the number of folds made, (2) the number of regions the paper is divided into, and (3) the fractional area of each region. Then fold the paper again and record those numbers again. (So after 1 fold, there are 2 regions, each with fractional area 1/2; after 2 folds, there are 4 regions, each with fractional area 1/4; …).

What students discovered was exponential growth and decay. What was interesting i&s that when I had them try to come up with a function relating fold number to the number of regions (y=2^x), many of them couldn’t do it. They would try thinks like y=2x, or y=x^2, but it wasn’t until I reminded them that the number of regions (2, 4, 8, 16, 32, …) could be re-written as (2^1, 2^2, 2^3, 2^4, 2^5, …) that the majority of them could figure it out.

In any case, it took a good 20 – 30 minutes for them to finish the activity (which included some plotting, and some discussion of independent and dependent variables), but overall, I’d like to think they got more out of it than me simply explaining in words what an exponential function is.

Not that I have the time to come up with a bunch more of these, nor the classtime to implement them, but I think having one or two per chapter up my sleeve would be perfect.

I told my calculus class, in the last 15 minutes of class on Monday, that:

“Somebody told me something, and I don’t know if it’s true. They said that if there’s a cubic equation that hits the x-axis three times, then there’s a point of inflection, and it will be the average of these three x-intercepts. I don’t know if I believe them. It sounds plausible, but I’m skeptical. Because why would the zeros be related to the inflection point? And why would they be related to it in such an elegant way? Crazy talk, I said.”

Of course, they called me on it, saying that of course I knew if it was true or not. So I chuckled, and said that they got me, and of course I know if it’s true or not. But I wanted them to figure it out. So I asked them to figure out what the problem is, and guess how we would solve it.

By the end of the class, we together (but letting them lead the discussion) determined that our general equation for the cubic would be f(x)=k(x-a)(x-b)(x-c), where the x-intercepts are at a, b, and c.

Then for homework I let them loose on figuring out if what I was told was true.

The next day (today), I asked them how far they got. One person solved it, and a few had the right idea, but got frustrated with the algebra. No one “checked” to see if the inflection point truly was an inflection point (if at that point, the function switched curvature from being concave up to concave down or vice versa). But going over the solution together was awesome because:

1. I got to reinforce that a, b, and c in this equation are constants, not variables (a few were confused about that)
2. I got to show them a quick way to “foil” out (x-a)(x-b)(x-c)
3. I got to remind them how to prove something is an inflection point
4. I got to show them what a formal proof looks like

and most importantly,

I got a few of them to see how cool it was. I basically told them why I loved problems like this… because even though the algebra can get hairy, even though you might make a wrong turn somewhere along the line, we were able to show something that is totally not intuitive. To use the words I used in class, that “the payoff is worth so much more than the work.” And even though only one person solved it on their own, I think a few of my students felt that ownership as we solved it together in class.

In theory and practice it was 30 minutes of class well spent. I should do more of these sorts of problems. Hard things we do together in class.