Every winter, a group of my college friends get together to hang out, catch up, and do a bunch of silly physical contests. One that was introduced last year was to do 50 push-ups as fast as you can, and then throw three darts at a dartboard. Your score is your time for the push-ups, with time subtracted based on how close your darts are to the bulls-eye. It’s a pretty fun event.

The catch was, some of us, especially me, took a pretty long time to do 50 push-ups. We were debating this week how many push-ups we should require for this year’s competition. One guy wanted to do 100. I wasn’t too excited about that. Sounds to me like an opportunity for some modeling.

After school on Thursday, I had another teacher time me, got data points for 5, 10, 15 and 30 push-ups, and put together this graph to illustrate my point.

**Quadratic Model:**

which predicts that it’ll take me about 10 minutes to do 50 push-ups and close to 40 minutes for 100 push-ups.

**Exponential Model:**

which predicts that it’ll take me about half an hour to do 50 push-ups and 4 days to do 100 push-ups.

**What’s your conclusion?** Obviously the predictions of the exponential function are pretty outlandish — seems to me that the function should become linear at some point. But the quadratic model seems a bit conservative — I’m pretty skeptical I could do 100 push-ups in under an hour. What are these models most useful for? What do they tell us?

I really want to use something like this later this year — it combines intuition and something concrete that students can see with some fascinating mathematics, and reinforces some ideas that they’ve struggled with — in particular, it underscores the usefulness of function notation, and what that pesky f(50) actually means in a problem. Need to keep thinking about whether this task or something like it will work in my classroom.

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howardat58You could try y = a*x^3 (a times x cubed)

Or more generally any power, but a log transformation of the data is needed before using a curve fitting routine ln(y) = ln(a) + n*ln(x), which now fits a straight line.

But there may be a physical maximum, so (it gets worse) try y = a*tan(kx).

!!!!!

AshliI very much like this for thinking about function notation. Possibly taking the angle that “what is f(45)?” is a waaaay shorter way to say “how long will it take to do 45 push ups according to the model?” It also gets into what is, for me, the point of functions: they have predictive power.