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*.