Does a problem situation or context matter? Can it be used to engage (or disengage) students? A recent example comes to mind where students seemed to be more interested in one problem situation than another. While students responded differently, there is no way to know whether it is the students or the context that caused the resulted in the different responses. I tend to think it was the context that did matter.
Textbook example: Domingo decided to save $0.03 the first day and to triple the amount he saves each day. How much will he save on the seventh day?
Variation: Would you work for someone if they paid you $0.03 on the first day and tripled the amount for each additional day you worked? Would you work for them for a week? For two weeks?
In the textbook example, there is a single answer: $21.87. Students calculated the amount and it seemed to end the discussion. It is realistic for someone (or, if they personalize the situation, themselves) to have about twenty bucks to stash away at the end of a week. Extending the problem into the second week becomes unrealistic because it is very unlikely anyone could continue saving at that rate.
When I varied the problem, however, it provided a lot of discussion. While the context is unrealistic (no one would ever hire someone with this scheme), it did provide a connection to the students. They had to decide whether they would work for this employer... if so, how long? Clearly, it would not be advantageous to work for only a few days. However, someone would would continue to work would be rewarded generously. This seemed to really illustrate the point of exponential growth!
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