August adores math. He's an all-around nerd, but just blows us away in math. Remember, he's six and only in the first half of first grade.
Tonight, as we were getting ready for bed, we got on the topic of square roots (as one does) and those are his favorite things for now. Knowing how much he loves them, and how his current favorite thing is to figure out some really big ones (like 3600) by seeing that it's really 36*100 so 6 and 10 = 60...I had thought today about helping him learn to estimate roots between two perfect squares. So, for example, the square root of 55 is something between 7 (49) and 8 (64). But then I wondered to myself as I was driving home if there was a way to be more precise. Could we easily estimate a benchmark?
As it turns out, yes. Here comes algebra, but hang on, because this is what Auggie is doing in his head. I figured I'd start by seeing if there was an easy way to identify when a square was more or less than half way between the two obvious roots. That is, could we quickly find the square of the value half way between two roots.
(n+.5)(n+.5) = n*n + n + 0.25.
So basically, if I know that 5 squared is 25 and 6 squared is 36, I can easily figure out that 5.5 squared is 30 (5-squared plus another 5) or 6.5 squared is 42 (6 squared plus another 6). Technically both are also plus a 0.25, but close enough for an estimate. So now if I want to estimate, say, the square root of 140, I might say "well, 121 is 11 and 144 is 12. 121+11 is 133 (my estimate for 11.5-square), so it's more than 11.5. Something between 11.5 and 12."
Cool enough trick to impress the kid, right? So I try to explain it to him and he keeps pointing out that the result I'm getting is the same as the average of the lower square and the upper square. Like,
(121 + 144) = 265 / 2 =133.5
Or, more generically, that is calculating:
[n*n + (n+1)(n+1)]/2
Well that does seem like a coincidence, right? Oh, except I then did the algebra, and the kid is right. You get:
[2n*n+2n+1]/2 = n*n + n + .5
His method was always giving an answer 0.25 higher than accurate, mine was 0.25 lower. Crazy!
(As I side note, I have always loved that (n+1)-squared is n-squared + (2n+1) so the next square is just this root times 2 plus 1 higher. And this solution above is a direct relative to that but I'd never fleshed it out.)
So then (then!) he starts speculating about what 1/4 of the way through would be and was it always a close estimate and I figured, probably not, because the slope probably skews somehow..? But then I sat down at excel and figured why not run some numbers?
Right? Those are close and what's more, the errors follow a very specific pattern. So out came the algebra again.
Auggie's method takes the lower number + some percentage of the range. Like, 3.4 is 40% between 3 and 4. So to estimate the square of it, it's 40% of the way between 3 squared (9) and 4 squared (16). That difference is 7, 40% of that is 2.8, so 9+2.8 = 11.8. The answer is 11.56.
Going the other way, the square root of 40: 40 is between 36 (6) and 49 (7). It's 4/13 of the way between, which is a little under 0.25. So it's around 6.25. It's 6.32. Not a bad quick estimate!
I had to play with the algebra to see what was going on. let "f" be the decimal on the lower number.
Auggie's formula uses the squared values:
lower number + f (upper -lower)
nn + f [(n+1)(n+1) - nn]
=nn +2fn +f
The real answer when you square (n+f) is
(n+f)(n+f) = nn + 2fn +ff.
So the answer is always off by the difference between f and f-squared, or (f-ff) or (f*(1-f)), which is what's up with that pattern on the errors. That whole last column is the percentage times (1-%). .1*.9; .2*.8, etc.
So when estimating something 40% between two roots, you just do 40% between the squares, then subtract back off (.4*.6) = .24 which is the exact adjustment I needed to get from my quick estimate of 3.4 squared (11.8) to the real value (11.56) above.
Dude. I am going to look so smart when I can estimate squares and roots of any number to 2 decimal places in my head (or at least quickly on paper!)
Are you bored and lost and wondering where the cute stuff is today? Yea, this is 100% how my brain works all.the.time.
So after this amazing conversation, we start digging on cubes and cube roots and Auggie, entirely on his own, connects that 4-cubed and 8-squared are both 64 and I ask why he thinks that is and he goes on a 3-minute speech of pure poetry about how the numbers connect and fit and patterns and ratios and I just gobbled that boy up whole. Every bit of it was accurate and complex and I wish I had my phone with me to record it because that speech will someday be the backdrop to a Field's medal slide show.
He's six. I love everything about him, but this just makes me want to hang out with him all night long doing algebra puzzles.