A pursuit curve

Answering the problems you get in class is less difficult than posing them in the first place.

One of the local High Schools is a specialized science-mathematics school, whose students are not selected based on where they live but have to go through an admissions process and be chosen.  This includes an examination of grades, until recently a test of mathematical and verbal ability, an essay on themselves, and an essay on a math problem.  Our tutor has been tasked with preparing several students for the math problem.  This is difficult because the school does not publish its criteria for evaluating the essays, nor give any indication of what’s fair game for the question.  All we have is the question itself for the past several years.  At this point in the process, with a month or two to go until the actual essay, he has used up all of these.  He is faced with the problem of writing a new problem.

Of course he enjoys the opportunity to break out of the standardized-test word-problem routine, where everything is solved by checking “B” after a minute or two.  There is always a lingering doubt in his normal tutoring about teaching to the test, rather than having the student actually learn math or ways to attack a real situation.  But coming up with a problem that requires some creative thought, and is hard enough but not too hard, is, well, hard.

Consider: the students are in Middle School.  They are now taking Geometry, so we can’t assume any specific knowledge of that.  No Physics or Chemistry, apart from what ideas may have shown up in earlier Science classes, would be fair.  Basic probabilities can be used, along with time-speed-distance questions and per-cents.  No calculator is allowed.  And yet, the question must be demanding enough to require some creativity in solution.

Our tutor generated one he was happy with, a series of time-speed-distance questions involving escape from a pursuing lion.  Students could grapple with numbers that didn’t quite come out even, or show some insight that meant they didn’t have to.  In the final section, the student should realize that the lion is moving along a curve, and thus has too far to go to catch its prey.  The curve itself is a pursuit curve, something that would challenge an upper-level Calculus student to work out, and is well beyond the capabilities of our Algebra 1 veterans.  Unfortunately, when it was given to several students they all tried, and wasted a great deal of time doing so.

So, should our tutor ditch the pursuit-curve part of the problem?  It occurs to us that being able to recognize when a question is actually beyond your capabilities would be a very useful skill.  Unfortunately, it’s one that isn’t taught in school (beyond a single lesson we’ve seen in upper-level Calculus).  Students are always given something they can (in principle) do.  And recognizing that a curve, whatever it is, is going to be longer than a straight line with the same endpoints may be demanding too much.

But it was such a promising problem. . .