By Angela T. Barlow, Posted August 4, 2014 –   

I hope that you and your students or colleagues enjoyed discussing the Counterfeit Bill problem. I suspect that a variety of solutions were offered, including these:

$40—The shoe owner gave $20 to the grocer and $20 (counterfeit) to the FBI.

$55—The shoe owner gave $15 to the customer, $20 to the grocer, and $20 (counterfeit) to the FBI.

To think about whether these solutions are correct, let’s start by trying the act-it-out strategy.

Although the shoe-store owner is not able to make change for the $20, you can assume that he has some money in his cashbox (just like you can assume that other pairs of shoes are in the store). Let’s say, for the sake of argument, that he has $40 in his cashbox and that he has the slippers on his sales rack. 

Transaction 1
The customer hands the fake $20 to the shoe-store owner. The shoe-store owner hands the fake $20 to the grocer. The grocer hands the shoe-store owner four $5 bills. The shoe-store owner hands the customer $15 and the slippers.
Result of transaction 1: The shoe-store owner has $45 in his cashbox.

Transaction 2
The shoe-store owner gives the grocer $20 in exchange for the fake $20.
Result of transaction 2: The shoe-store owner has $25 in his cashbox and a fake $20.

Transaction 3
The shoe-store owner gives the fake $20 to the FBI.
Result of transaction 3: The shoe-store owner has $25 in his cashbox.

Compare
The shoe-store owner had $40 and a pair of slippers to start with, and then he ended with $25. He lost $15 and a pair of slippers—or $20, if you assume the value of the slippers is $5.

If you don’t believe me, act it out!

Alternatively, you might use the strategy look at the problem from a different view. With this in mind, consider the following argument. By handing the shoe-store owner a counterfeit bill, what did the customer receive free of charge? That’s right, $15 and a pair of shoes. So the shoe-store owner lost what was taken from him: $15 and a pair of shoes.

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I have used this problem in a variety of settings, and it is always interesting that students expect me to tell them who is right. If you tried this problem in your classroom, I suspect the same thing was true with your students. Not telling them, though, supports students in making sense of the problem, listening to one another, and considering others’ justifications. And by doing so, this problem supports the establishment of your classroom norms. I hope you’ll share your students’ experiences of the Counterfeit Bill problem with us.

Did you or your students use a different strategy? You are welcome to share photos or work samples. We hope to hear from you.

Angela T. Barlow is a Professor of Mathematics Education and Director of the Mathematics and Science Education Ph.D. program at Middle Tennessee State University. During the past fifteen years, she has taught content and methods courses for both elementary and secondary mathematics teachers. She has published numerous manuscripts in Teaching Children Mathematics, among other journals, and currently serves as the editor for the NCSM Journal of Mathematics Education Leadership.

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