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# TCM Blog

### The Counterfeit Bill Problem

I am often asked what the best way is to start the school year. My answer is always, “With a problem, of course!” Not just any problem will do, though, as I want a problem that will spark discussion by eliciting a variety of solutions and/or solution strategies. One problem that I have found to be particularly fun is the Counterfeit Bill problem (Sobel and Maletsky 1999).

A customer enters a store and purchases a pair of slippers for \$5, paying for the purchase with a \$20 bill. The merchant, unable to make change, asks the grocer next door to change the bill. The merchant then gives the customer the slippers and \$15 change. After the customer leaves, the grocer discovers that the \$20 bill is counterfeit and demands that the shoe-store owner make good for it. The shoe-store owner does so, and by law is obligated to turn the counterfeit bill over to the FBI. How much does the shoe-store owner lose in this transaction?

In the past, I have asked students to work collaboratively in groups to solve this problem and represent their work on poster paper. The mathematics in the problem is limited to addition and subtraction, thus allowing engagement of a wide range of students in terms of both grade level (grades 3 and up) and ability level. The power of the problem lies in its ability to support students in recognizing the need to understand the problem rather than rushing to compute with numbers and to elicit the act-it-out strategy, a strategy often forgotten as students get older. In using this problem, typically three or four different solutions surface, which selected groups can then present for discussion. In doing so, students engage in justifying their solution processes to the class. Of equal importance, however, are critically listening to and critiquing the arguments of others, which are necessary for the class to move forward in agreeing on the solution. By engaging in these processes, students are able to begin establishing classroom norms that will support their mathematical adventure.

Try the Counterfeit Bill problem. Here’s a hint: Two problem-solving strategies that you might find useful are act it out and look at the problem from a different view. Note that for younger students, the problem may be modified to involve a \$10 counterfeit bill, and you may want to provide counters or other manipulatives to support student engagement with the problem.

Reference

Sobel, Max A., and Evan M. Maletsky. 1999. Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies. 3rd ed. Boston, MA: Allyn and Bacon.

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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