I came home from work this evening and Kyra immediately attacked me and said, “Can you help me with my math homework? There’s this one problem that I can’t figure out, and mom’s no help.” Of course she’s been on vacation for two and a half weeks and hasn’t done this homework, and the night before she goes back to school suddenly it pops into her head. But that’s neither here nor there.
Anyway, I sat down with her to see what I could do. The problem was to write a function that would tell you the number of dots in a square or a triangle as you added dots to the bottom. In other words, two horizontal dots makes a square of four dots, three makes nine, and so forth. Obviously the function for that is y=x^2, easy cheesy. But the triangle one was a bit harder: two horizontal dots is a triangle of three, three is six, four is ten, etc. Each time, for a row of x, you add the number x to whatever your last number was. It’s easy to see the pattern but I couldn’t figure out the function for it.
I told her the answer was y=sigma(1->x) x which is technically correct… but I suspect summation notation is beyond seventh-grade math. I scribbled a few equations but didn’t have an answer. Dang.
Enter a YouTube video about the 12 Days of Christmas, which Mel sent to me today. It explains about halfway through how to sum the so-called “triangular numbers”. After some scribbles in her notebook, the narrator Vi demonstrates that the equation is (x^2+x)/2. And she does it in a way that’s simple but elegant. Nice!
So I thought that was pretty cool. I watched a few other videos from Vi’s collection and they’re fun. I think I like them because that’s exactly the way I was as a kid… I would play with numbers and graphs and my calculator and find all kinds of fascinating patterns and rules. I found stuff like the base of the natural logarithms and the relationship between odd numbers and primes and even the spatial dimension where the volume of a hypersphere is maximized in relation to its surface area. I’m not even kidding.
Once a math geek, always a math geek.