[Makeover] Penny Circle Preview

Our final #MakeoverMonday task this summer is also the most tragic, where tragedy is measured in wasted potential. There’s lots to love here. Lots to chew on. Lots to improve on. If you’ve tuned into this series this summer, you can probably anticipate 90% of “What I Did.” What would you do?

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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. Step one is just bad specification; where’s the “no stacking” rule?

    Personally I’d get rid of this “draw some circles” nonsense. It feels like busy work and makes the task feel arbitrary. Straws and paper cups. Or pringles cans and boxes. Or any tactile thing that we don’t spend setup time on and which we might ever want to put inside other things.

  2. The problem as set is tragic!! I would turn it inside out, starting each pair of pupils with a handful of pennies, say 25, and for various numbers of pennies they can try to estimate the size of the smallest circle which will hold each of those numbers. The real situation arises in package design for expensive chocolate mint creams (for example), where there are at most two layers and rarely as many as 25 in a box. Further relevant activities could be about choosing the box shape (round, square, oblong, hexagonal,…), and consideration of the trade-off between cost and convenience of packing many boxes in a carton.
    The hope that pupils will be able to assess the value of n in ax^n from the data (your edit box wont do superscripts !) is wishful thinking indeed, and if some computer program came up with n=2.02351 they would have trouble explaining what it meant.
    The use of the term quadratic function when one is hoping for a choice of the “less bad” from y=x squared and y=x cubed or multiples of them is too heavy, and step 5 as written scares me.
    I really admire what you are doing !!!!!!!!!!!!!!!!!!!!

  3. Inside out is good. Maybe a pizza parlor is selling birthday pizzas with pepperonis instead of candles? If I’m 30 years old, then how big is my pizza going to be? Will it feed the class?

  4. Present a large circular area to the class, and tell them you want to cover it with pennies. This could be a circular table, or a circular board from a board game, or you could even go to the gym and look at the center circle. It should be small enough so that it’s practical to cover it in pennies, but large enough so the students want to find a quicker way.

    Have the students guess how many pennies it will take. Record the guesses.

    Now, collect some data for smaller cases. Either the students can cut out circles of different sizes, or use common items such as coasters or paper plates. Count how many pennies it takes to cover these areas.

    Plot these points on a graph. Discuss the case 0 radius, 0 pennies. So , the graph should pass through the origin.

    At this point, rather than using the calculator to find the power law, I would prefer to get them thinking that it looks like a quadratic graph of the form y=ax^2. Have them discuss why this is reasonable (ie link between radius and area). Although not as accurate as using their calculators, I think each time students plug numbers in and the calculator spits something out, they lose some ownership of the problem, and they lose some sense that they are solving it themselves. Sometimes this is unavoidable, but I think we can avoid it here.

    So, what is the value of a in this case? Use one of the points to find it. But each point gives a slightly different value of a. Which value should we use? Should we average them? Should we use the point corresponding to the largest circle? Does it depend on what we are using the model for?

    Once we have our y=ax^2 model, use it to estimate the pennies needed for the original problem. Everyone now gives refined guesses.

    Now we actually do it! Cover the circle in pennies, and count them. Give a prize to whoever is closest (maybe the pennies could be the prize?!)

    As an extension, swap the variables. I have 1 million pennies. How large a circle can I cover with them? Also, you could ask how the model will change if we use a different type of coin. Or how the model will change if we fill a square instead of a circle.

  5. I would use this lesson as a start to quadratics. I would start the class with a question like, “How many pennies can we fit into a circle with a diameter of 50 inches?” Or some other large number. I would have at the ready some pennies, but certainly not enough to fit into a circle that big. The students would then predict their answers. We would then move onto a discussion on how we could figure this out with the limited number of resources currently at our disposal. With such a wide open lesson, it allows for a wide range of discussion such as recognizing patterns, independent vs dependent variables, recording data, and making better predictions. Each class would be different, but as long as I keep the end game in sight, each class would arrive at similar conclusions.

  6. William Carey

    August 23, 2013 - 3:29 am -

    This is a nice reminder that the transition from heuristic, particular solutions to a closed form solution is always weird and needs motivation. Instead of going through all this rigamarole trying to come up with an (inexact?) equation for a 6″ circle, why not just draw the circle and count?

    Finding a closed for solution should never be an answer in and of itself. It needs something *other* question to motivate it.

  7. Glad to find this one provoking so many interesting ideas. FWIW, Stuart is in my head in a scaaaary way, minus the Dave Major / Desmos material.

  8. I’d like to add a better context to this problem . Perhaps consider the construct of an amphitheater – stage is a semi circular arrangement, audience is the other half of the semi circle. Model with students in the class, if the stage was 1m in diameter, how many students could attend while standing ? 2 m in diameter ? 3m in diameter ? Build a model to fit the data, extend to ” Your city needs to build a new semi circular amphitheater to accommodate 50,000 spectators, what dimensions would the architect need to propose for this build ? What changes to your model would be necessary if everyone brought a lawn chair ?

  9. Suppose you filled a circle with pennies, and then used those pennies to form a square (as best you can). Which of the following graphs do you think might be the most similar or the most different? In what ways might they be the same or different?

    radius in pennies vs # pennies
    radius in inches vs # pennies
    radius in pennies vs side length of square in pennies
    radius in inches vs side length of square in inches
    radius in pennies vs side length in inches

    Go ahead and collect the data. Again discuss similarities and differences. What is the form of each equation? Why does that form make sense? Find each equation. What accounts for the differences in the coefficients?

    One issue is that the percentage of the circle covered by the pennies (packing efficiency) varies a lot between a 1”, 2”, …, 5” diameter circle. Maybe use something smaller, like MMs, so the packing efficiency is more consistent.

    I did the original activity fairly carefully. Regression produced a power model of: y = 1.02x^2.23 (not quadratic) and a quadratic model of: y = 1.8x^2 — 1.8x + 1.2. Neither model holds up well when extrapolating.

  10. I vaguely remember a problem like this from somewhere years ago. It had to do with predicting the number of organisms in a circular colony in a petri dish. Seems a bit more relevant than pennies in a circle and little more cross-curricular. As for the original problem I predict the two smart kids will do the problem and the rest will just go “Huh?” Step 5 and the conclusions seem way out there for most kids and a good number of math teachers. It also seems if the problem is going to go to this level it should ask why the numbers grow in this manner.