When you were a kid, did you figure out anything in math before your teachers taught you?
Oh, yeah, lots of things. Usually really simple generalisations of stuff, though. For example, the moment I was taught how to add variables in algebra, it was immediately obvious how to subtract, multiply, and divide them.
(Admittedly, this sometimes went awry. When I was five I saw an episode of Cyberchase in which I learned what negative numbers were and how to add and subtract them, and I immediately generalised it to multiplication and division. However, I didn’t realise at the time that multiplying or dividing two negatives gave a positive, so that messed me up.)
Anyway, the instance of generalisation I’m most proud of from primary school is from grade 5, when we were taught simultaneous equations. It was always two equations of two variables each, and we were taught how to solve them via addition and substitution. After the teacher had done two of them on the blackboard, I asked her if you could solve simultaneous equations of three or more variables. She said “no”.
However, as she continued doing examples, I became more and more convinced that systems of n-variable equations must be solvable, if you have at least n equations. I had nothing else to go on, and had no idea if fewer than n equations could work, but my intuition was really strong on the idea that n was the upper bound.
So, in between solving each official class problem (because I could do them quickly), I experimented with different methods of solving 3-variable equations. I didn’t figure it out in that class, but I kept working on it during lunch time, and then the lunch times of the next two days, until suddenly I made a breakthrough and figured out how to solve them via substitution.
I did it like this:
a) 2x + 3y - z = 13
b) x - 2y + 3z = 9
c) -5x + 3y - 2z = -8
b) x = 2y - 3z + 9
a) 2(2y - 3z + 9) + 3y - z = 13
c) -5(2y - 3z + 9) + 3y - 2z = -8
a) 7y - 7z = -5
c) -7y + 13z = 37
a) y = -5/7 + z
c) -7(-5/7 + z) + 13z = 37
c) 6z = 32
c) z = 16/3
a) y = -5/7 + 16/3
a) y = 97/21
b) x = 2(97/21) - 3(16/3) + 9
b) x = 47/21
a) 2(47/21) + 3(97/21) - (16/3) = 13
b) (47/21) - 2(97/21) + 3(16/3) = 9
c) -5(47/21) + 3(97/21) - 2(16/3) = -8
I later learned there were other ways to do this, but this one has a special place in my heart because I discovered it after being told it was impossible. I didn’t share it with my 5th grade teacher, because she was kind of scary, but 10 year old me was so smug.
I’m also proud of two discoveries I made about squares when I was 7, at the beginning of third grade. My third grade classroom had a chart on the wall with the times tables up to 12x12. It looked pretty much like this:
I found this chart absolutely fascinating. I used to love tracing the diagonal with my finger and, at the time, I thought the etymology of “square root” was actually the diagonal route that was traced through the chart by the perfect squares.
And, while tracing, a couple patterns became obvious. One that hit me suddenly one day, while looking at the chart, is that if you draw a square whose corners are one of the perfect squares, the two instances of its root (on the top and the left of the chart), and 1, adding them all together gets you the next perfect square. For example, for 25, drawing a square that touches 25 + 5 + 5 + 1 would get you 36. I was able to generalise that rule to n2 + 2n + 1 = (n+1)2, though not in quite those terms.
Then I dashed outside and started running around the playground, challenging people to ask me for the square of any number. By this point, I’d memorised all the perfect squares from 22 to 122, and everyone knew those were on the chart, so they’d ask me for 13, 14, 15 and so on. Each time, I did it by starting at 144 and counting up from there.
144 + 24 + 1 = 169
169 + 26 + 1 = 196
196 + 28 + 1 = 225
A few days later, I was looking at the chart again to see if there were other patterns to multiplication. That’s when I noticed that there was a pattern to the diagonal lines that intersected the square line. For example, the one that intersects 49 goes …40 - 45 - 48 - 49 - 48 - 45 - 40…
I could see there was a clear pattern to the way the numbers get smaller as they move further away from the square, but I couldn’t quite grasp the description of it. Then it hit me: Multiplying the number less than the root by the number more than the root gets you 1 less than the square, multiplying the numbers two away from the root got you 4 less than the square, multiplying the numbers three away got you 9 less, and so forth. In algebraic terms:
(n + a)(n - a) = n2 - a2
For example, for 36 the perpendicular diagonal would contain:
7 * 5 = 36 - 1 = 35
8 * 4 = 36 - 4 = 32
9 * 3 = 36 - 9 = 28
10 * 2 = 36 - 16 = 20
(NB: I had still never heard of “quadratic equations” at this point.)
This was harder to show off to people, but I found it just as cool. At that point, I decided I wanted to study perfect squares when I grew up (in addition to curing malaria, of course).