# Penny Circles

This discussion represents the final Makeover Monday Problem of the Year. I must admit, I found this problem to be easily the most engaging for me simply as a curious math learner. (Here’s  the original problem posted by Mr. Meyer.)

Now, I will admit that this is going to be more of a discussion of my experiences exploring this problem and, if you get lucky, I might come up with a recommendation somewhere toward the end.

The problem presents a rather interesting set up: filling up circles with pennies, making predictions, modeling with data, best fit functions. Lots of different entry points and possible rabbit holes to get lost in.

First change was that my circles were not created by radii of increasing inches. For the record, I can’t quite explain why I chose to make that change. Instead, I used penny-widths. Circle one had a radius of one penny-width. Circle two’s radius of two penny-widths. Circle three’s radius… you get the idea.

It looked like this:

I started with 4 circles with radii based on penny width.

Note: I understand the circles look sloppy. I had wonderful, compass-drawn circles ready to go and my camera wasn’t playing nice with the pencil lines. So, I chose to trace them with a Sharpie freehand, which… yeah.

Next, I kept the part of the problem that included filling each circle up with pennies.

It looked like this:

Then I filled them with as many pennies as I could

Here’s where the fun began.

I saw a handful of different interesting patterns that were taking place. If we stay with the theme of the original problem, then we’d be comparing the radius of each circle to the penny capacity, which would be fairly satisfying for me as an instructor watching the students decide how to model, predict, and then deciding whether or not it was worth it to to build circles 5, 10, or 20, or if they can develop a way to be sure of their predictions without constructing it.

But, I also noticed that by switching to the radius measured in penny-widths, the areas began to be covered by concentric penny-circles. So, we could discuss how predicting the number of pennies it would take to create the outside layer of pennies. Or we could use that relationship as a method for developing our explicit formula (should the students decide to do that).

So, if I am designing this activity, I feel like the original learning targets are too focused and specific. I want the students to be able to use inductive reasoning to predict. Simple as that. If they find that using a quadratic model is the most accurate, then cheers to them, but this problem has so much more to offer than a procedural I-do-you-do button pushing a TI-84.

First, I’d start with them building their circles using a compass and some pennies. Build the first four and fill them in. I’m quite certain that the whole experience of this problem changes if I leave the students to explore some photos of the my circles. They need to build their own. Plus, that presents a low entry point. Hard to get intimidated lining up pennies on a paper (although, I have plenty of student intimidated by compasses).

Next, I think that I would be if I simply asked them to tell me how many pennies they would need to build circle five and to “prove” their answer, I believe we would see a fair amount of inductive reasoning arguments that don’t all agree. The students battle it out to either consensus or stale-mate and then we build it and test. It should be said, though, that as they are building the first four and filling them in, I’d be all ears wandering about waiting for the students to make observations that can turn into hooks and discussion entry points.

Then I would drop circle ten on them and repeat the process. This is where the students might try to continue to use iterations if they can figure out a pattern. Some will model with the graphing calculators (in the style the original problem prefers). I’m not sure that I have a preference, although I would certainly be strategic in trying to get as many different methods discussed as possible.

Then, when they have come to an agreement on circle ten, I’d move to circle 20 and offer some kind of a reward if the class can agree on the right answer within an agreed margin of error. Then, we’d build it and test (not sure of the logistics of that, but I think it’s important).

All that having been said, I see an opportunity to engage this in a different manner by lining the circumference of the increasingly-bigger circles with pennies so that the circle was passing though the middle of the pennies and seeing how, as the circles got bigger, how the relationship of the number of pennies in the radius compared to the number of pennies in the circumference. I suspect it could become an interesting study on asymptotic lines.

It just seems like this particular situation has, as Dan put it, “lots to love” and “lots to chew on.”