What we had in mind
Goals of this activity
Students will be able to...
The task bounces the student from a table to a graph to symbols in five steps—without words describing why one abstraction is more useful than another. Students need to understand those differences. A table is great because it lets us forget about the physical pennies. A graph is great because it shows us the shape of the model. And the algebraic function is great because it lets us compute and predict. Penny Circle highlights these strengths.
This activity is a combination of cooperation and competition — "coopetition." Students will work with their classmates to collect data that each student will use to estimate the answer to a problem involving circles, pennies, and carpets. Then you'll all get to see who came closest!
When it comes time to select a model – linear, exponential, or quadratic – make sure students fully understand the implications of that model and how it relates to their initial guess. If a student selects an exponential model, for example, say something like, "So your guess earlier was 500 pennies. The exponential model grows so fast it says the answer is millions of pennies. Either your earlier guess was way off or the exponential model isn't the right model. You decide which." Ask a similar question in the other direction if a student selects a linear model.
While students are working, use the teacher view to identify students to talk with. For example:
Students need to understand why a quadratic is the best model here. It isn't enough to say that its prediction was closest to the actual answer. We need to look for the structure that leads to a good model.
Have students think about what happens to the area of a circle when the diameter goes from 3 inches to 6 inches. Does it double? (No, so it isn’t proportional.) How does it change? (It increases by a factor of four—which suggests it’s quadratic.) How can we rule out an exponential model? (It doesn’t have a zero y-intercept.)