Consider the problem of how one can use an 11 minute egg timer and a 6 minute egg timer to cook a 3 minute egg. The timers I mean are like the timers in Pictionary. They look like an hour glass, like this:

Write a 1 or 2 page essay which explains how to do it, but the real point of your essay is to also explain to a teacher how she or he can relate solving this problem to finding the greatest common divisor of 6 and 11 and writing it as an integer linear combination of 6 and 11. (If you can write the greatest common divisor of 6 and 11 as an integer linear combination of 6 and 11, how does that help you to solve the egg timer problem? And how does one figure out how to write the greatest common divisor of 6 and 11 as an integer linear combination of 6 and 11? Don't aim your essay at me; pretend you're explaining this to a teacher for possible use in his or her classroom.)