We have been learning about mathematical models, and I came across this very nice activity put together by Dan Meyer, and tweaked by Andrew Shauver.

After showing this brief video Dan made:

we talked about what questions it raises, which eventually led to the main one, “How many pennies will it take to fill the big circle?” Then we discussed what information we needed to answer that, namely see how many pennies it takes to fill some smaller circles. I handed out sheets with 1 inch, 2 inch, 3 inch, 4 inch, 5 inch, and 6 inch diameter circles, lots of pennies, and let them go at it:

We plotted the data using the online grapher Desmos and decided that a quadratic function would model it best:

Using the QuadReg feature on our graphing calculators, we found the quadratic model for the data, and used it to predict how many pennies would fill the big circle. Here’s Dan’s answer:

We had varying results, which led to a good discussion of how small changes in data can lead to big differences in results, but several students were within 10 pennies of the answer.