But, how do I introduce that sense of play on a daily basis? I can't help but feel that I should be doing more of it in Geometry, since Geometry is such a visual subject.

I've noticed time and again that there are problems that showcase how even many of my honors Geometry kids lack basic intuition when looking at a diagram, and they end up way over-complicating the situation. Take a look at the following problem, given on my most recent H. Geometry exam. The directions were deceptively simple -- to find the area of the concave quadrilateral ABCD.

I made this problem and envisioned that many of the kids would quickly get it solved, after our foray into quadrilateral areas. To hint at the fact that they didn't need to cut ABCD further into smaller parts, I even led into the problem with, "Given that Area of ABCD + Area of ADC = Area of ABC, find the area of Quadrilateral ABCD." ...A dead giveaway?!

Anyway, perhaps predictably, some of the kids struggled on this problem. (To their credit, many others did brilliantly, including a girl who was absent for several days during the unit and had to catch up belatedly on all of the heavy-duty trig content.) Mostly because those kids made some false assumptions, such as assuming that Segment BD would bisect Angle ABC. But, even some others who were able to solve the problem correctly took some detours to get there, such as cutting ABCD up into a right triangle and a scalene triangle. It's clear to me that most of them still lack the ability to zoom in and zoom back out on a diagram -- which is what Geometry is all about!!

I've certainly given them "similar" problems to struggle through in class, but especially as honors students, I also expect them to have a level of ability to apply that knowledge to new situations, on the day of a test. But, how do I

*teach*that flexibility?

Anyway, I really feel like I should be doing a better job helping them develop a better sense of spatial intuition. I just don't know how. :(

I have the same problem with my "honors" kids (we don't really have honors math classes). I don't see them having the problem solving skills that I think at this point they should have. They want to be told how to do EVERYTHING. These are the same kids that complain that I over teach, but will be the first to say "I don't get this!"

ReplyDeleteHow do you solve this problem? All I know to do is use the Pythagorean Theorem to find AC.

ReplyDelete1. Find AC using Pythagorean Theorem.

ReplyDelete2. Find angle BAC using arctan, which will then allow you to find angle DAC.

3. Now use law of sines to find remaining sides and angles inside triangle ADC. (Until you've found enough info to move on to #4, anyway.)

4. Use Area=1/2*ab*sin(C) to find area of scalene triangle ADC.

5. Subtract areas ABC and ADC to get area of ABCD.

Cheers,

Mimi