This post has nothing to do with geometry. I guess I can’t say that exactly (because of the possible geometric representations), but I am not dealing directly with these. I am always intrigued when I think like I want my students to think. It is these moments that keep me going into the classroom hoping for new understandings. There have been times this year where students have made connections that I never have. These innocent realizations are mathematics manifested in its purest form. A similar experience happened to me this morning.
I had been mulling over a problem posed by the NCTM about Pythagorean Triples. I am very familiar with the theory around these special sets of numbers, but have fallen slightly rusty over the years since the number theory classroom. The problem was:
My mind immediately wandered to the idea of being relatively prime (or coprime). Two numbers, although they may not be prime, are coprime if they share no common factor. I knew that new triples could be “built” simply by multiplying each entry (a,b, and c) by some constant ‘m’ because:
m(a^2 +b^2) = m(c^2)
The existence of a formula paralyzed me. I new it was out there and that Euclid had done all the work for me. I could not bring myself to embrace the simplicity of the problem; I was attempting to find the general formula instead of showing that one entry must be even. Not only did the formula kill my desire to derive it, it closed down all other pathways to an answer.
Then, this morning, I had a spark of childlike thinking. It was so elementary that I smiled when it came to mind. (Elementary when compared to the derivation of Euclid’s Formula). I tell my students that mathematicians are lazy; I also make it quite clear that this doesn’t mean they do no work. It means that they always try to find the quickest and most succinct path from hypothesis to proof. Ideas should not be disposed as trivial because they are simple. Some of the greatest theorems of all time come from modest assumptions. With this spirit in mind, I sat down and answered the question.
My new angle was this:
a = 2x+1, b = 2y+1, c=2z+1 | x,y,z are integers
(2x+1)^2 + (2y+1)^2 = (2z+1)^2
4x^2 + 4x +1 + 4y^2 + 4y + 1 = 4z^2 +4z + 1
collect terms and factor out a 2…
2[2x^2 + 2x + 2y^2 + 2y + 1] = 2[2z^2 + 2z] + 1
If we clean it up a bit by letting ‘m’ and ‘n’ represent general integers:
What other problems can be posed from this fact?
Can the Even be a,b, or c? Does it matter?
Can a triple be all Evens? Can I find one?
What organizations of Evens and Odds satisfy the equation:
a^2 + b^2 + c^2 = d^2 ??
I didn’t need Euclid’s formula to engage in meaningful mathematics. In fact, the knowledge of a formula acted as a roadblock. The feeling of enlightenment is one that I want my students to experience when they do mathematics. It is these “aha” moments that fuel my drive to create meaningful lessons. They invigorate me even when a students make a connection that I have already made, but I assure you this is not always the case. In this situation, life without Euclid provided me the opportunity to make numerous connections.