Saturday, August 22, 2009

Sample Challenging Problems

I think it is important to challenge kids to think beyond mere formula recall. This is particularly the case with gifted learners because absorbing spoon-fed material is not a challenge. Acceleration merely increases the pace, but it does little to challenge the brain. I believe that most gifted educators without formal advanced training in mathematics fail to see the difference because most have no clue what real mathematicians do.

In keeping with last week's post about challenging vs. accelerating, here is an elementary problem requiring only simple mathematical facts while going significantly above simple formula recall.

1. Draw a rectangle. Draw a triangle such that its base is one side of the rectangle and the opposite vertex is chosen arbitrarily on the opposite side of the rectangle. Find a relationship between the areas of the rectangle and the triangle.

2. If your kid has a rudimentary understanding of probability, I would phrase the problem differently.

Repeat the rectangle / triangle construction above. Pick a point at random inside the rectangle. What is the probability that you picked a point inside the triangle?

Note that the triangle is picked before the random point. Hence, we asking for the probability of picking a point interior to the this particular triangle.

The answer to this problem is simple. The area of a triangle is 1/2 of its base times its height. Since the height and base are always the same regardless of the location of the triangle's apex, the area never changes. In fact, the area is 1/2 that of the rectangle. You will pick a point interior to the triangle with a 50% probability, and this independent of where the triangle's apex is located.

If your kid is advanced enough, you can ask him or her to generalize this problem to 3 dimensions. The simpliest, 3D generalization is to think of pyramids inside prisms. The answer is 1/3 probability of picking a point inside the pyramid. This is because the volume of a pyramid is 1/3 * area of base * height of pyramid.

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