[Makeover] Penny Circle

TLDR: Check out Penny Circle, a digital lesson I built with Desmos based on material I had previously developed. Definitely check out the teacher dashboard, which I think is something special.

This is it, the last entry in our summer series of #MakeoverMonday. Thanks for pitching in, everybody.

The Task


Click for the PDF.

What Desmos And I Did

Lower the literacy demand of the task. The authors rattle off hundreds of words to describe a visual modeling task.

Clarify the point of the task. A great way to lower the literacy demand is to convey the point of the task quickly, concisely, informally, and visually, and then formalize, expand, and verbalize that point as students make sense of it. Here, the point isn’t all that clear and the central question (“How can you fit a quadratic function to a set of data?”) is anything but informal.

Add intellectual need. The task poses modeling as its own end rather than a means to an end. Models are useful tools for lots of reasons. Their algebraic form sometimes tells us interesting things about what we’re modeling (like when we learn the average speed of a commercial aircraft in Air Travel by modeling timetable data). Models also let us predict data we can’t (or don’t want to) collect. We need to target one of those reasons.

The most concrete, intellectually needy question, the one we’re going to pin the entire task on, pops its head up 80% of the way down the page, in question #2, and even then it needs our help.

Lower the floor on the task. We’re going to delay a lot of these abstractions — tables, graphs, and formulas — until after students know the point of the task. We’re going to add intuition also and ask for some guesses.

Motivate the different abstractions. The task bounces the student from a table to a graph to a power function in five steps without a word at any point to describe 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. If those advantages aren’t clear to students then they’re only moving between abstractions because grownups told them to.

Show the answer. We tell students that math models their world. We should prove it. The textbook does great work here, asking students in question #2 to “check your prediction by drawing a circle with a diameter of 6 inches and filling it with pennies.” Good move. But students have already drawn and filled circles with 1, 2, 3, 4, and 5-inch diameters. I’m guessing they would rather draw and fill a 6-inch circle than do all that math. The circles need to get huge to make the mathematics worth their while.

So here’s our new central question: how many pennies will fill a really big circle? We’re going to pose that question by showing someone filling a smaller circle, then cutting to the same person starting to fill a larger circle. It’ll be a video. It’ll take less than thirty seconds and zero words. It’ll look like this:

We’ll ask students to commit to a guess. We’ll ask for a number they know is too high, and too low, asking them early on to establish boundaries on a “reasonable” answer. The digital, networked platform here lets us quickly aggregate everybody’s guesses, pulling out the highs, lows, and the average.


Then we’ll talk about the process of modeling, of looking at little instances of a pattern to predict a larger instance. We’ll have them gather those little instances on their computers, drawing circles and filling them.


Those instances will be collected in a table which will then be aggregated across the entire class creating, a large, very useful set of data.


(Aside: of course a big question here is “should students be collecting that data live, on their desks, with real pennies?” Let’s not be simple about this. There are pros and cons and I think reasonable people can disagree. For my part, the pennies and circles are basically a two-dimensional experience anyway so we don’t lose a lot moving to a two-dimensional computer screen and we gain a much easier lesson implementation. However, if we were modeling the circumference of a balloon versus the breaths it took to blow it up, I wouldn’t want students pressing a “breathe” button in an online simulator. We’d lose a lot there.)

Next we’ll give students a chance to choose a model for the data, whereas the textbook task explicitly tells you to use a quadratic. (Selecting between linear, quadratic, and exponential models is work the CCSS specifically asks students to do.) So we’ll let students see that linears are kind of worthless. Sure a lot of students will choose a quadratic because we’re in the quadratics chapter, but something pretty fun happened when we piloted this task with a Bay Area math department: the entire department chose an exponential model.

Eric Berger, the CTO at Desmos, suspected that people decide between these models by asking themselves a series of yes-or-no questions. Are the data in a straight line? If yes, then choose a linear model. If not, do they curve up on one side of the graph (choose an exponential) or both sides of the graph (choose a quadratic)? That decision tree makes a lot of sense. But the domain here is only positive circle diameters so we don’t see the graph curve on both sides.

Interesting, right?

All this is to say, if you’re a little less helpful here, if you don’t gift-wrap answers like it’s math Christmas, students will show off some very interesting mathematical ideas for you to work with.

Once a student has selected a model, we’ll show her its implications. The exponential model will tell you the big circle holds millions of pennies. We’ll remind the student this is outside her own definition of “reasonable.” She can change or finish.

We’ll show the answer and ask some follow-up questions.

What We Didn’t Do

Change the context. It’s a totally fair point that packing pennies in a circle is a fairly pointless activity, one with no real vocational value. When I pose this task to teachers as an opportunity for task revision, they’ll often suggest changing the context from pennies and circles to a) pepperoni slices and pizza dough or b) cupcakes and circular platters or c) Oreos and circular plates, basically running a find-and-replace on the task, swapping one context for another.

I don’t think that does nothing but I don’t think it does a lot either. It’s adding a coat of varnish to a rotting shed. You’re still left with all the other issues I called out above.

We’ll talk about the teacher dashboard next.

What You Did

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Michael Pershan

    August 26, 2013 - 5:02 pm -

    Sweet. This is pretty freaking cool.

    I experienced a good deal of friction as I was dragging pennies into my circles, though. It felt slow and frustrating, and I really just wanted to get that part of the task over with.

  2. @Michael — yikes. What browser? We’ve seen some trouble with safari, but chrome / FF / iPad have been snappy for me.

  3. Well done, Dan and Desmos! What an improvement. It would be a joke to even ask my kids to read the original activity, but I definitely think that they would find your makeover engaging- it’s so visual and interactive. I love the graph of class data, and I love that the students can try out the different models. You guys do beautiful work.

  4. Awesome! If you are looking for a less pointless context, light collection and aperture area are related in the same way. Telescopes, cameras, and pupils — doubling the aperture size quadruples the amount of incoming light.

    In my astronomy unit last year, we fit M&Ms (photons of light) into different sized circles (telescope apertures). More info, including handout, here: http://noschese180.wordpress.com/2013/04/22/day-135-light-collection-the-cosmic-calendar-and-alternating-current/

    It’s not a 3 Act makeover, and the concept of photons is itself abstract, so your mileage may vary.

    Extension question for the pennies task: How might the model change if we used dimes? quarters?

  5. Very nice, although not technically working well on Nexus 7 with Chrome (which you say in the beginning). For example, video doesn’t start.

  6. I like the circles on the computer with the pennies, makes it easy to smash pennies in cause they can’t move back outside of the circle once in!

  7. I love this. I know we are going for efficiency and quickness, but can we have the student use their own calculator (or some other software like excel) to run the regression for a quadratic function, power function, exonential, ect. as well?

  8. The platform is really impressive and encouraging. Really nice to be able to aggregate data and dump it into a graph so seamlessly. But, …

    The pennies are arranged and counted for me. I record some data. The points are plotted for me. I click on a model and slide some sliders to get a good fit – easily done through trial and error. The graph is rescaled for me and the predicted number of pennies to fill the large circle is calculated for me. What is a student supposed to come away with from this task?

  9. This collaboration between you and Desmos has to be one of the coolest things I have seen in math education in a long time. Definitely where I hope technology enables us to go. Thank you both.

  10. I find the opening few seconds of Act 3 more compelling for a “pause” than what gets given at the end of Act 1, which lacks momentum and a compelling “penny count” meter.

    Also, why does the exponential model require e as its base? Just for simplicity?

    (I haven’t got a chance to try the other stuff which looks very slick, because my work computer is using IE 8, and no I can’t change that, and yes it means I can’t even use the regular Desmos site.)

  11. Greg Schwanbeck

    August 27, 2013 - 2:18 pm -

    Just wanted to say that I love the little touches–the pennies roll along the edge of the circle when dragged, they are randomly presented as heads or tails (not just alternating,) and all different pennies–even corroded pennies!

  12. I like the penny-adding part of the program too. It made me feel slightly competitive and try to fit as many pennies as possible; I can see students getting into it more than the real-penny version.

    I also hate to say I agree with l hodge above the slider part of the activity. I just twiddled knobs until it matched. It’s too easy to change settings without thinking about any of the math behind it. Interface-wise I’d think I’d prefer something like TI-Navigator, which makes the students type in the entire formula and then change the numbers manually; additionally it should allow them to compare prior test answers and new test answers for modeling the data.

  13. I’m sort of suspicious of sliders as well. I’ve tried them before via Geogebra and afterwards wasn’t sure if students really made the connection between the sliders and where the constants were in the eqn.

    Maybe if they had constructed the formula + sliders themselves, though?

    I’d also wonder if a Bret-Victor-style slide-the-actual-numbers would make the visual connection stick better.

    I have absolutely no real data on any of this though, just vague suspicions.

  14. Wow, this is a freakishly brilliant use for a ridiculously useless task (filling circles with pennies). I wish I taught quadratics in my core so I could use it.

  15. Personally, I love sliders. I think you are right Josh, having students perform the creation is where the real power is at. It is also relate-able to real life. In real life, we see the patterns, then make a program to do the calculations for us. After that, we use the program we made. I believe, if you can program it, then you know it.

  16. I want to do this in my class! I appreciate how the sliders strip away the fuss and let the IDEAS shine through. As a teacher I can walk around and ask students to explain why the sliders do certain things. Or to punch a calculator to see how far the model misses a data point.

    As a follow up I’d put a graph of y1=e^x and y2=x^2+1 (just Q1) the next day and ask students to write which is which and how they know.

    Regression equations are pretty ‘black box’ until you learn linear anyway. I think writing a function and having students make the transformations (with sliders) requires WAY MORE MATH than fitting a few regression models and choosing the one that minimizes 1-R^2.

    Is the Teacher Dashboard similar to that of TED-ED where I can create a class of my students and review their work online? If so, will there be a customizable template so I can start making some of my own?

  17. Slider regression is great. You get an easy to use tool that gives a nice visual of how the curves can change. It can help free up time and mental energy for more demanding tasks.

    My gripe is that the other parts of the task can be completed in a very passive manner as well. Why not ask the student to drag a point to the position that would represent the solution? Maybe cover up part of the circle they fill with pennies and make the student come up with a count using some proportional reasoning.

    We want to know how much stuff will fit into a large circle. Wouldn’t many kids think to find the area of the big circle and the area of a penny as a start? Why not go that route and explore the relationship between square inches and pennies? How does the formula A = pi * r^2 change if we measure area in pennies instead of square inches? The pure regression approach avoids all of the rich material in the question — how does the area of the pennies compare to the area of the big circle, how about the covered area to the uncovered area, radius in pennies to # pennies, etc.

  18. Awesome stuff! I love how intuitive the app is, how it manages to not get in the way of the user’s mathematical thinking at any point (this seems to be the biggest difficulty that maths apps have at the moment).

    In terms of the slider, maybe replacing it with something like this might make the affect on the equation more obvious:

    + + +
    y = 0 (x – 0)^2 + 0
    – – –

    where the pluses and minuses are buttons to increase or decrease the value between them.

  19. I love the visual queues and getting the student involved in math, more so than students parroting back equations, they put it to the test. Visual queues are always great for students to learn, seeing is believing.

  20. Re: sliders, all I can say is, from my perspective, I just slid the sliders around until something matched and didn’t even look at the numbers. I’m guessing this would be true for many of the students as well.

  21. I don’t comment on your blog posts because I don’t feel that I’m creative enough to have valuable input. I was trained very classically and that has always been my comfort zone for teaching.

    With that said, I LOVE this activity. My favorite part of it, beyond the hands-on nature and the lack of formulae and calculation is that students are able to see how their guesses tack up against other students who have completed the activity.

    I would like to do more of this but I don’t know how. Clearly I should be participating in this blog more often.

    Thank you for this make-over

  22. Thanks for pitching in, Justin and others.

    There’s enormous variety in how modeling tasks are structured on paper, in textbooks, and even less agreement on what those tasks might look like in a digital, networked medium. It’ll take all kinds of creative, constructive criticism to push these kinds of tasks forward and the Desmos team and I really can’t get enough of it.

  23. Dan
    I love this activity and want to use it NOW as we are doing quadratics. When I click the class activity link, it already tells me there are 16 entries of data.
    Is it supposed to just collate data from my class, and provide us with a clean graph or am I missing something?
    I love the modelling aspect and was wondering how to adapt it in Australia as we don’t have pennies so I am very grateful for the virtual penny scenario.

  24. Hi Sheila, when you go to labs.desmos.com/pennies and click on “Teacher,” it should give you a fresh class activity. From there, you can pass the link out to your students and the data will all go to your teacher page. If you have any trouble at all, please let me know.

  25. Dan, great work as usual.

    One point of criticism. Students might get the impression from the activity that the task of choosing a model is simply about “fitting the data.” I want to know (and my students to know) *why* a quadratic model is appropriate to the situation, and I’m not sure they’re going to get that from your implementation.

    i.e. “the area of a circle varies with the square of the radius” is a key understanding here, and that explains why the quadratic model works. I would think that would be a key outcome in all of this. How do you envision students reaching this conclusion?

  26. @James, totally valid criticism. I don’t know how well a tablet-based program can help students grapple with that idea but, at minimum, I should have included the question in the student response section.

  27. Dan, very nice. I’m super excited you and desmos are teamed up.

    It would be great to tie this into microbiology and cell culture populations in a petri dish. Scientists use the area of a growth circle to predict cell population. With time built in, they can determine growth rate of the culture.

    Going to try this project in my precalc class today.

  28. I’m using the Penny Circle project in my Algebra 2 classes this year but none of the Desmos stuff. I bought a couple of pizza screens, 10 inch and 18 inch diameter, because they keep the pennies in the circle. Students created data with their own compass drawn circles and we used the 10 inch screen for another data point.

    The target is the 18 incher. Low/Just Right/High guessing first. Make the graph with the intention of future use. Then . . . wait for a couple of months to find out the real answer because we need to learn some math first to figure it out beyond just faking where you think the graph will go.

    I’ll try to come back when we finish to say how it worked.

    By the way, when I went to the bank to get 1000 pennies, the teller thought I meant $1000 worth of pennies and had to have a discussion with the manager in the vault before clearing up the misunderstanding.

  29. I have an idea to answer the choosing contexts problem: have a brainstorming activity at the end where students think of real world contexts in which the general “circles in circles” rule might apply. Of course the teacher could have a list and get things started, but hopefully students could come up with ideas that aren’t on the list. Even if students don’t come up with ideas, if the teacher displays a long list of diverse examples, it would be more impressive than just one. It could demonstrate that a single seemingly trivial mathematical generalization applies to many concrete examples.

  30. This is beyond the scope of the problem, but if you have an advanced student who is intrigued by packing the pennies in as tightly as possible you might point them to the following resource:


    It contains the proven ideal arrangements for 1-11, 13, and 19 pennies, and the best known arrangements for any amount of pennies up to 20.

  31. I just worked through the activity again, and I’m pretty excited with how well the answer comes out analytically. My empirical model was approximately y = 1.3x^2, with small h and k constants.

    I tried deriving this, assuming that the number of pennies is simply (Area of circle)/(Area of each penny). This gives [pi(d/2)^2]/[pi(0.75/2)^2], with 0.75 being the diameter of a penny in inches, according to the U.S. Mint. But that gave me y=1.78x^2, which was way off.

    So I went back to the picture of the pennies in a circle, and decided that each penny basically takes up an area the size of the square that is circumscribed around it, because that better accounts for the empty spaces between pennies. In that case, the area taken up by the penny is just the area of the square, (0.75)^2.

    I did the calculation this time, and I got approximately y=1.33x^2. It was great! So this will be my follow-up to this lesson. WHY does the empirical equation make sense. And then I do plan to move into the blog post about composition of functions, which I linked to directly above this comment.

  32. Beautiful activity, I went through it all as a student and I love how i could keep going back to the actual circle and find and create more data points. I was using the activity on an iPad and it worked very well, the pennies moved fairly realistically.
    I truly loved how the students can see other students data for that “coopertition”! And the teacher can see the whole thing in real time on their computer…VERY WELL DONE!!

    A few improvements: can you move the model sliders to the bottom of the screen? I noticed on the ipad my hand was in the way so as I was sliding i couldn’t see the graph very well. Also, could you allow teachers to add different summary questions on the last step?