Thursday, June 30, 2005

Contest Problems

I'm only doing half-hearted preparations for teaching the honors section of the gen-ed course in the fall. My enrollment has dropped from eight students down to six since the last time I'd checked, and I need 15 for it to run. (As predicted, my calculus class has filled up and is now at 30 students with a cap of 32 -- the cap will be raised as the course continues to fill.)

What I am doing is looking through the pile of stuff that my dad gave me when I saw him in Florida. In addition to old issues of The Mathematics Teacher (which I will promply throw away, as I do not keep printed copies "just in case" of any journal that can be easily be found in a library or that is automatically mailed to every member of a professional association), I've also been given a book produced at the University of Delaware in 1986 called "Resource Problems to Enhance the Teaching of Mathematics." Here's a randomly selected problem:
In a 5 by 12 rectangle, one of the diagonals is drawn and circles are inscribed in both right triangles thus formed. Find the distance between the centers of the two circles.
I can't use problems like this. Most of the problems in the book are what I would categorize as contest problems for general-population high school students. These might be fine for high school students who already like math, but those are not my students. I'll definitely have a bunch of pre-engineers who need to learn an ambitious amount of calculus in a short period of time (dictated by the departmental syllabus), and the other bunch is an unknown. The regular gen-ed class would mutiny over this sort of problem (these are the students so averse to doing unfamiliar problems that they will leave questions blank on the exam and leave early). The honors version of the gen-ed class does not have a math placement prerequisite -- just an overall standardized test score -- so I have no information about how good they are at math. I'm guessing, though, that if I were to give them puzzle problems that they'd be more appreciative of the LSAT puzzle problems or problems from the quantitative section of the GMAT.

This book also features word problems about completely fanciful nonsense situations:
Sally is having a party. The first time the doorbell rings, one guest enters. If on each successive ring a group enters that has two more persons than the group that entered on the previous ring, how many guests will have arrived after the 20th ring?
It's problems like this that make my weaker students hate word problems.

And then there's problems like this:
A comet moves in a parabolic orbit with the sun at the focus. When the comet is 4 million miles from the sun, the line from the sun to the comet makes an angle of 60 degrees with the axis of symmetry of the orbit (down in the direction in which the orbit opens). Find how near the comet comes to the sun.
Aren't orbits elliptical?