I enjoy watching students exploring a problem that forces them to come up with their own structure for solving it. Today, a group got a chance to mess around with The Pencil Sharpener Problem, which is a problem I posted a month or so ago. (I'll leave you to read it if you are curious what the problem is.)
TheGeometryTeacher has here the four kinds of results gotten from a class given a word problem (about the time needed for a certain event to occur). I like not just the original problem but the different approaches taken to the answer. It seems to me often lost to students, or at least poorly communicated to them, that nearly any interesting problem can be solved several ways over. Probably that's a reflection of wanting to teach the most efficient way to do any particular problem, so showing more than one approach is judged a waste of time unless the alternate approach is feeding some other class objective.
Given the problem myself, I'd be inclined toward what's here labelled as the ``guessing and checking'' approach, as I find a little experimentation like that helps me get to understand the workings of the problem pretty well. If the problem is small enough this might be all that I need to get to the answer. If it's not, then the experience I get from a couple guesses and seeing why they don't work would guide me to a more rigorous answer and one that looks more like the graph depicted.
Guessing and checking gets little respect, probably because when you're trying to train the ability to calculate like ``what is eight times seven'' it's hard to distinguish informed guessing from a complete failure to try. (The correct answer is, of course, ``nobody knows''; the sevens and eights times tables are beyond human understanding.)
But when you're venturing into original work for which there may be no guidance what a correct answer is (or whether there is one), or when you're trying to do something for fun like figure out ``What are the odds my roller coaster car will get stuck at the top of a ride like Top Thrill Dragster?'' guessing and correcting from that original guess are often effective starting points.