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