# Penny Circle + Modeling With Functions

So this is happening in Boston right now:

School on Monday, then at least two days off (right now we have off at least through Wednesday) meant I didn’t feel confident with the day I had planned writing exponential functions from tables. Figured our time would be better spent waiting until after the storm to dive deeper into exponential functions, and instead do some modeling on a one-off day. I’d been keeping the Desmos Penny Circle lesson in my back pocket for awhile, and now seemed like the right time for it.

Quick summary: In Penny circle, students watch a video, where first someone fills a small circle with pennies:

Then, they start filling a large circle with pennies:

The question is simple — how many pennies will fit in the big circle? We could count, but that would take a long time. Instead, students gather data using this slick interface to sample lots of small circle sizes:

Then, they fit a model — choosing from linear, quadratic, or exponential, to all of the data the class collected to make a prediction about how many pennies will fit in the circle:

This lesson is awesome. It has a ton of great mathematical thinking built into it, and it gives a lot of meaning to using functions to model situations. In no way is this a real world question, but students don’t care — they want to know the answer.

Also, this is not 50 minutes of class time. Maybe I could stretch it with a ton of discussion, but I found the most value with lessons like this is the individual conversations students have as they move through the lesson at their own pace.

What I did: My lesson had three parts. First, students played with some graphs to warm up. Then, they did penny circle. Then, we looked at a few more graphs to test their knowledge.

1. I gave them these graphs:
McDonald’s
Gateway Arch
Gateway Arch #2
Crazy cathedral in Iceland

Their job  was to see how well different functions fit physical objects. My goal was pretty simple — scaffold the idea of dragging around a model and evaluating whether it fits, as well as to get kids engaged and make sure everyone’s computer was working. They really enjoyed it, and I had to drag a bunch of kids away from the Iceland cathedral. Check it out! It’s called Hallgrimskirkja, and it’s really beautiful.

2. Penny circle. I farmed the lesson pretty briefly, and got them going. I required that each kid add 5 data points to the graph before moving into the modeling to make sure there was plenty of data and prevent kids from rushing through. I also think the tactile experience of dragging the pennies around reinforces the quantities that students are working with in the lesson.

Almost all kids figured out the model was quadratic — 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. I think this is an important moment of learning — they know that one of the defining characteristics of quadratics is their symmetry, but this data isn’t symmetric — they’re only looking at one half of the quadratic, and that tripped a bunch of kids up the first time around. 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).

My one criticism of the lesson is that the video of the answer didn’t work for everyone — I played the version from Dan Meyer’s Threeacts site for the class at the end so that kids whose videos didn’t work got the chance to see the answer.

3. This is the part I was most interested in. I wrapped up by showing them two more graphs,

I asked them whether they thought this data would best be modeled with a linear, quadratic, or exponential function.

This was my quick check to see how well they can apply what they were working with — to me, the big idea here is deciding which model is most appropriate for a data set. I want them to be able to look at women’s marathon records and say that the data looks exponential — and also, if they’re paying attention, catch that the asymptote is not at y=0, and name what that means about marathon times.
The second graph is a bit trickier — they didn’t think it was linear or exponential, but there was disagreement over whether it’s quadratic. My point here was pretty simple — I want my students to recognize that a quadratic has some things in common with this graph, and also to recognize that it’s not quadratic because it keeps alternating between increasing and decreasing — linear, quadratic, and exponential models are really powerful, and model lots of things in the world, but there’s plenty of other math out there. This was a really useful assessment for me of whether a student was “answer getting” — if their approach was to just grab the first answer that looked close to being right and try to move on, or if they thought deeply and carefully about whether the answer makes sense.

Anyway, you should check out Penny Circle. It’s awesome, and there are lots of opportunities for some great mathematical thinking. Do it!