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.