Tuesday, 2 October 2012
Student's Difficulty w/ Geometry
In 30+ years of teaching advanced HS math, I frequently opined to other math teachers that one of the problems many students had with problems and proofs in mathematics, especially geometry, was that they saw too "literally"; that is, they saw what the problem stated or the image showed, but could not see the un-drawn lines and un-stated relations that were essential, and often almost obvious to some others (those who saw geometry as "easier").
I mention this now not because I ever found a teaching solution for the problem, but because I was just reminded with a beautiful problem that is in "The Lighter Side of Mathematics", Edited by Guy and Woodrow.
The Problem, illustration at top, simply asks, Given two intersecting circles what line segment passing through an intersection (F) with endpoints on the two circles would be the longest. While many students will flounder others will immediately see that the only useful tool at hand is another point of intersection which gives the triangle GEI shown.
Since FE is a chord of each circle, as G and I are moved around (but still passing through F) the angle FGE will always be the same. In the same way the angle FIE will also remain of a fixed size. We don't know how big they are, but the size of the angles not changing assures us that whatever triangle we draw under the given constraints will ALL be Similar... and our seemingly unsolvable problem of finding a maximum length for IG, can be solved by finding a length for either GE or IE since the ratio of the sides of similar triangles are constant also.
The obvious longest side for GE is to make it a diameter, and that will force EI to be a diameter also (why students?). When these two legs are the longest, the third and proportional side GI, will also be at its longest.
Student's exploring this w/ an interactive geometry software may wish to confirm, and try to justify why this means that the final position of GI must be perpendicular to the line joining the two intersections of the circles, and is thus parallel to the line through the two circle centers.
Whether they can work this out for themselves, the student who can see the mathematical "art" in this kind of problem, will begin to see mathematics a little more deeply.
If you try this in your classes, I would love your comments.