Thursday, February 20, 2014

Problem Solving is Fun

There is a goat, says the question. The goat is tethered to one edge of a barn, on a rope 10 meters long. If the barn is 9 meters long and 5 meters wide, and the goat cannot enter the barn, what is the total area that the goat can cover?

To my eleven-year-old mind, this question brought up countless other questions: what, for example, would inspire the goat to traverse as much distance as possible in the first place? Would it not be more sensible for the goat to double 'round and chew off the rope, then escape the tyranny of the barn and traipse off to whatever happy meadows were certainly waiting in store? What exactly is the farmer hiding in the barn that is so important that he will not even allow an innocent goat to see it?

The problem with the stereotypical math questions posed in high school is that they all look something like this: If x^2 - 6x + 5 = 0, what is x? Such a question gives no incentive to solve it. There is no sympathy for 'x', no tears shed over the tragedy of a beleaguered animal. All the question does is to continue to drill factoring and quadratics into brains already well-tired of such things.

Stemming from this is my proposition on how to make math 'cool'. Rather than the continuous drilling of formulas and equations, problems in math classes should be set up like the problems offered on standard math competitions such as the MathCounts programs and the AMCs. The goat question is a perfect example - it provides the blending of a possible real-life situation (if any mathematicians are striving to become farmers with curious goats) with the concept of sectors of circles. Schools offer the basics; almost all high school-students know that the area of a circle is "pi-r-squared". However, I have offered the goat question - a standard on several MathCounts state tests - to a roomful of honors-level high-schoolers during my time as MathLeague coach, and been rewarded with a bunch of blank stares.

"We don't know how to do that," say my teammates. "Why is there a goat? The goat is confusing us!"

And here's the problem - schools force the memorization of formulas, but do nothing to teach applied math. In order to really understand math, students should be offered questions which challenge their imagination and understanding. It's not hard, either - after explaining the goat question once, all of the MathLeague students present at the lecture could solve any similar questions; all they needed was a moment's exposure to competitional math.

Additionally, several of the - well, granted, not-very-mature - students suddenly caught a glimpse of how I, in all my madness, could call math 'fun'. "This is kind of cool," one boy admitted, after a ten-minute discussion on how we could find the number of diagonals that a convex pentagon - whimsically named DUCKS - had. "I like ducks."

So it's easy, really, to make math fun. Amuse the students - put math in situations they've never seen before, with not only goats, but ducks, and cows, and even once, to the chagrin of our MathLeague supervisor who had already been losing faith in the human race, a llama. Offer new ways to play with probability - probability as used in gambling, for example, which tends to amuse students to no end.

And, finally, challenge the kids, because that's the only way they're going to go from memorizing formulas to applying them. After all, my teammates can now say quite confidently each time they see a question dealing with circles, "I've got your goat."

Monday, February 17, 2014

2014 Mathcounts State Prep : Inscribed Circle Radius and Circumscribed Circle Radius of an equilateral triangle

#1: P is the interior of equilateral triangle ABC, such that perpendicular segments from P to each of the sides of triangle ABC measure 2 inches, 9 inches and 13 inches. Find the number of square inches in the area of triangle ABC, and express your answer in simplest radical form.

#2: AMC 2007-B: Point P is inside equilateral  ABC. Points Q, R, and S are the feet of the
perpendiculars from P to AB, BC, and CA, respectively. Given that PQ = 1,PR = 2, and PS = 3, what is AB ?

This is an equilateral triangle. If the side is "S", the length of the in-radius would be\(\dfrac {\sqrt {3}} {6}\) of  S (or \(\dfrac {1} {3}\)of the height) and the length of the circum-radius would be\(\dfrac {\sqrt {3}} {3}\) of  S (or \(\dfrac {2} {3}\)of the height).

You can use 30-60-90 degree special right triangle angle ratio to get the length of each side as well as the height.

Solution I: Let the side be "s" and break the triangle into three smaller triangles.

\(\dfrac {9+13+2} {2}\)= 12s (base times height divided by 2)= \(\dfrac {\sqrt {3}} {4}\times s^{2}\)

s = 16 3 

Area of the triangle = 192 3


Solution II: Let the side be "s" and the height of the equilateral triangle be "h"
24* s (by adding 9, 2 and 13 since they are the height of each smaller triangle)  = s*h 
(Omit the divided by 2 part on either side since it cancels each other out.)
h = 24
Using 30-60-90 degree angle ratio, you get \(\dfrac {1} {2}\) s = 8 3  so  s = 16 3 
Area of the equilateral triangle = \(\dfrac {24\times 16\sqrt {3}} {2}\) = \(192\sqrt {3}\) 

#2: This one is similar to #1, the answer is 4 3