Note: This article will not be part of the book “Traditional Math”. I include it here only for your interest. The following problem appears in an algebra 1 textbook (Dolciani, et al, 1973): Two boys are camped at a spot where a river enters a big lake. One boy is injured so severely that every minute counts. His companion can use an outboard motorboat to get a doctor by going 3 miles down the lake and back, or by going 3 miles up the river and back. Show that even though the boy does not know the speed of either the boat or the current, he should choose the lake.
I think this problem is a great example because it challenged my common sense intuition that the current just cancels. To understand in a more general way why this is not the case, this same problem could also be introduced again when covering topics such as linear vs. non-linear equations, convex functions, etc. Since speed is distance over time, we can also say that time is distance over speed, and therefore non-linear w.r.t. speed. And since functions of the form 1/x are convex (x>0), it follows that it is faster to travel at an average speed both ways rather than fast one way followed by slow the other way.
See, we’ve had “rich” problems for decades.
I think this problem is a great example because it challenged my common sense intuition that the current just cancels. To understand in a more general way why this is not the case, this same problem could also be introduced again when covering topics such as linear vs. non-linear equations, convex functions, etc. Since speed is distance over time, we can also say that time is distance over speed, and therefore non-linear w.r.t. speed. And since functions of the form 1/x are convex (x>0), it follows that it is faster to travel at an average speed both ways rather than fast one way followed by slow the other way.