the school marm teaches math but never mentions what math is for.
(1) communication.
it is actually easier to communicate some things in math that with english language. (and some things are communicated well with pictures.) the nice thing about math is that if often gives you a way to prove things, and the proof is a form of communication. you can prove things with pictures too, that's pretty much what geometry is about - a formal study of proving things in pictures.
(2) simulation.
if you want to simulate something in virtual reality, like a ball that falls and hits the ground and bounces, often with the aid of a computer, you use math to describe the system, and then you can run that model on a computer. the example i give about the ball is a physics simulation.
(3) checking things.
you may want to be sure of something. like maybe somebody tells you if you rotate a die 90 degrees around a vertical pole and then 90 degrees around a horizontal pole, it will end up on a different face that if you rotate it 90 degrees around a horizontal pole and then 90 degrees around a vertical pole. you can try this with a real die. or you can represent the rotations as matrices and prove in with math. the nice thing about math is when you come across a problem that isn't easy to check by hand, sometimes it's easier to use math.
when you are using math to check things, you might want to use approximations. in the example above, if the angles are very small, like 1 degree rather than 90 degrees, you can show that the way the dice changes orientation (after both rotations) is pretty much the same whether you rotate it around the vertical pole (the z axis) or the horizontal pole (the x axis) first.
Simplifications
sometimes simplifications, like:
"sin(x) is about the same as x, when the angle is small and you are using radians to measure the angle"
are useful to check things out without dealing with the exact answer, which may take much more work to write down. when doing simulations, computers are fast so you can often just given them the whole formula without simplification and let them deal with it. but sometimes they are not fast enough and we make approximations so that the computer compute things more quickly.
