Lots of approximation algorithms fall out of calculus pretty naturally. So they end up taught as examples or applications in that sort of class, rather as tricks when learning algebra or arithmetic.

Your solution must be exactly as it is taught, or the solution is worthless.

You're not supposed to think for yourself in school, just do what you are told.

I wasn't. At least, not officially. I struggled in math, though, and one of my teachers (not my math teacher) taught me several of these tricks on the sly. I remain grateful to him to this day.

I always had a calculator. If they taught me shortcuts, I didn't listen. But, I was not a good student in primary school.

I learned them for exams that involved lots of arithmetic. A lot of it was folk knowledge passed along the peer group partly by asking some of the kids who were one year ahead (and closer to the exam). There was some stuff like this but also other “tricks” (in physics problems, approximating sqrt(g) by pi and stuff like that), or calculating the integral of e^ax times sin bx by “completing” it to e^(a+ib)x, or the cube root technique here http://thinkinghard.com/blog/CubeRoots.html that we used mostly for modulo arithmetic.

I’ve forgotten nearly all the useful stuff and now only recall trivia like the base ten logarithm of a few small primes. I wonder if those school kids are still passing on this sort of cobbled together knowledge, now decades past.

I always considered this kind of trick stuff somewhat beneath me, as a child, since it did not discover some underlying truth of the universe like a mathematics proof in number theory would. But that was dumb because in the end I managed neither anything interesting in Number Theory nor complete mastery of thE tricks and at least the latter would have been entertaining to document.

As an aside, there was a kid in our group who could rapidly decompose numbers into sums of powers of two (a skill that turned out useless even in the constrained environment), essentially obtaining the binary representation as soon as you wrote the number out (or showed him the written one). He couldn’t explain how, though, because he was mentally not quite there and would frequently lose verbal skills. After the nth time he stood up on the desk and randomly peed while we were there he was out of the place. Besides it wasn’t that useful and it could just be high skill at the straightforward modulo and shift algorithm. Going the other way was also fast for him but useless for us, though more impressive since adding 2^64 (a number I could still tell you off the top of my head) to some other power of two seems less straightforward.