What Can You Do With This: Pocket Change

[followed up here]

Let’s push this forward. The question is “how much cash?” The reference point is CoinCalc.

Your challenge is to outline the supporting materials so that this activity will a) scale from easy to hard, b) throw a few curveballs at the students who figure out its mechanics quickly, and c) offer visual confirmation of the answer to provoke a discussion of sources of error.

If you then consider the fact that a) it’s easier to mix coins than unmix them and b) it’s easier to tally the value of a roll of coins than a pile of loose change, you’ll understand why producing this unit took a week of detailed planning and an afternoon of careful shooting.

[click for high-res]

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. Easy would be if the money in the picture was only pennies. (That would be a great lead-in even to the hard version.)

    I’m not sure how to make the picture above easy.

  2. Right, but Russel, do you see how this photo alone won’t cut it? You need more material to flesh the lesson out along the lines I illustrated above:

    1. scaled difficulty
    2. interesting outlying cases
    3. visual confirmation

    So tell me what you need from me to make this work for your students.

    Jason’s suggestion is right on: start with pennies, or just one kind of coin.

  3. Easy would be start with a few coins spread out on a table (2D). Then put those same coins into a jar/glass so students can get a handle on what that starts to look like. (“Four rolls of 50 pennies each in a circular cylinder looks like this. Three rolls of 40 nickels each looks like this.” etc.)

    From there, I’d go to the 3D and, as Jason Dyer just said, you could also start with a single-value of coins–or a jar of M&Ms as is often popular for contests (especially around Halloween time of year). With the M&Ms you could add some sort of value to them so that you can ask how many reds or blues in the jar of all sorts mixed together. You’ve introduced estimation of a part from a whole, but they are all the same size.

    Ramp it up with just layers of coins (pennies on bottom, nickels on top of those, then dimes, etc.). Then shake the container to make it even harder.

    I would use a coin sorter to unmix them. Or unmix them first and THEN mix them (they don’t know what order the pictures were taken in). Could even be cool to take a video of your dumping the coins into the container and shaking it around, then playing the video backwards.

    Curveballs could be things like: oddly shaped containers, put some bills in there, Australian money, etc.

  4. Other notes:

    I remember a Duck Tales episode where one of Uncle McDuck’s workers was a “bean counter” and could count the change when you flung it in the air. I have no more information that that, but tracking down that clip could be a fun way to kick things off. Dustin Hoffman in Rainman did something similar with toothpicks, I think.

    Non-glass containers could also be interesting. “Here’s a bulging pocket (or coin-purse) full of change. How much does that guy have?” Hopefully you don’t breed thieves that way.

  5. @Dan Campbell…

    Hmmm…you could tell them how many total coins are in the jar, how much the coins weigh together, and how much money is in the jar. Then they could tell you how many of each coin there are.

    I like the CalcDave’s idea of leading in with a cartoon. Always fun. I also like the idea of starting simple with one type of coin and having them work with ratios based on the weight of each type of coin as listed by the treasury department (it’s gotta be somewhere on the web but I’m too lazy to look it up).

    I would lead into this by having the jar on my desk, asking the kids to submit their guess for how many coins are in the jar. Closest to the right amount gets a prize of some sort. After I’ve got their attention, and I name a winner, then we can go back, look at some pics, weight different jars, etc.

  6. Dan

    Of course I would need more than the photo. I was referring to the activity. Sorry

    I was working on something similar with Australian coins the other day in a workshop with other teachers. I have the notes on that tucked away as well.

    Thanks for your good work. You and your work are a very real encouragement

  7. I would add questions about estimating the value of the coins without a representative sample from the jar, based on how the change was gathered through hundreds of transactions:

    If you assume that the totals of each transaction are evenly distributed between ending in $.00-$.99, could you use the weight of each type of coin to estimate the value per pound of a typical jar of change?

    Possibly give them a hint/more straightforward question: if you made a hundred purchases, and each of them ended with a different number of cents (therefore representing all possible endings equally $.00-$.99), and you made all of those purchases with paper money, how much change would you have, and how much of each coin?

    What are the pros and cons of using this as an estimation of a typical coin jar? (pro: simpler; cons: purchases probably aren’t evenly distributed over $.00-$.99, representative sample is very likely more accurate)

    I might steer the conversation a little towards the idea that quarters are probably more likely to be picked out of the jar. I’d consider a question like “Is there any way you could tell if someone was stealing from the jar?”…but I can’t come up with an elegant mathematical solution to that which doesn’t involve a lot of work. Taking samples from different layers? Lots of work, and assumes that your purchase patterns have been consistent for years. I also wouldn’t want to plant the idea that students could get away with stealing from their parent’s coin jars. :)

    For some reason, I’m interested in exploring the physical relationship of the coins a little, too: Your purchasing patterns haven’t been consistent, so one layer might have more pennies, and one might have more quarters, etc. If you shake the jar a bunch before you take a representative sample from the top, will the representative sample be better? (Will heavier, smaller, denser coins sink towards the bottom?) Is there a way you can “shake” or mix the jar that wouldn’t hurt the representative sample? (Maybe involving rotation? Let’s try it!)