How many pennies fit in the circle?

What we had in mind - Algebra Students
- One 45-minute class period
- Students who are familiar with linear, quadratic, and exponential functions
| Goals of this activity Students will be able to... - Use smaller things to make predictions about bigger things
- Understand the difference between linear, quadratic, and exponential models
| Supported Devices |

About this activity

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.

How the activity works:

1. Guess

Students watch a video of someone filling a smaller circle, then starting to fill a larger circle. They guess how many pennies fit in the bigger circle.

2. Data Collection

Students will drag pennies into different sized circles and record their data in a table. Data is aggregated across the entire class creating, a large, very useful set of data.

3. Model and Extrapolate

Students choose a model for the data. Is it linear, quadratic, or exponential? Then we show the implications of this choice.

The Student Experience

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.

The Teacher Experience

Ask students:

- "Did anybody change their model? Like from linear to quadratic or anything else. Why did you change?"
- To use their model in reverse; "What if there were 2,000 pennies? What's the smallest circle that'd fit those pennies?"
- To think of other circumstances where this kind of modeling would be useful.

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.)*

Tips from Teachers

Penny Circle + Modeling with Functions "My favorite moment here is where students say they think it’s exponential, and then see that their model predicts that 21 million pennies will fit in the big circle. . . . We had fun looking at which students’ predictions were the closest, and talking about why the function has to be quadratic (because the area of a circle is quadratic)."