iTunes Song Sales and Exponential Functions

Earlier in the year I was looking for some information about using iPods in the classroom. At the same time I was teaching the exponential functions unit in my grade 12 course. One of the pages I came across was this iTunes Song Sales graph. Despite my excitement for iPods I stopped what I was doing and integrated the data I had found into my class. My thinking was that this is data that students can relate to and hopefully it would keep them interested and help them see how math is used in the 'real world'.

Here's what I had my students do:

a) Describe the shape of the curve.
b) Why does it look the way it does?
c) Model it with a function.
d) Can we expect this trend to continue?
e) If not what can Apple do to increase its revenue. This question led to some interesting discussion about what Apple should be doing in the future (including selling books) before the iPad was announced.

There's not much to it but it just gets students thinking about where exponential functions appear in real life and who might care about them. Next time I might get students to do some research to find their own 'real life' exponential situation.

One of the interesting tangents of this lesson arose when a student asked "Why would anyone buy music when you can just download it for free?". We had a good discussion about why it's important to compensate content creators for the enjoyment of their work. We also talked about Creative Commons and how some creators choose to release their products for free and encourage others to use it or even change it. I love days when I end up teaching more than just math.

Quadratic Functions With Video

There are some lessons that I know students are just going to hate. Often times they'll put up with it but I feel like I'm doing them a disservice. I want to find a way to make these lessons more engaging.

One such topic is having student determine the function that models some form of quadratic data. I find that many students get lost in the wording of such questions and get bogged down by the details. I thought I would introduce a video to see if it helped with their understanding. I showed my class the video below and asked them to determine the function that models the height of the ball at any given time.

I played the video on the interactive white board and had a student come up to the board and put dots on the board to show the location of the ball. Once the video was finished he drew a smooth curve through the points. My question to the class was how do we determine the function that models the situation? I was amazed how engaged the students were. I received a ton of answers: We need to find the vertex. We need a set of axes. We need the 'a' value.

The video and the one question I asked produced the buy-in I was after. It was great. The nice thing about this lesson is that later in the unit when students struggled and asked for help they would say things like "Oh right, this is what we did with the video isn't it?".

As an added bonus we were able to practice a couple of times just by shifting axes around, which also reinforced the idea that the 'a' value was the same because we were talking about the same curve.

I realize that the problem is still somewhat contrived. Ideally I'd like to have a situation where students feel compelled to solve the problem and end up using the math to do it. Nobody really wants to know the function that models the path of the ball, but it was a situation they could relate to and was a step in the right direction.