In my experience as someone with a lot of math in my toolkit, many, perhaps most, of the people I work with end up wishing they had similar skills. It's not just me saying math is useful, it's the people who have seen it used.
I agree with that, but it's hard to believe something is important if you have no concept of it. (People know that "math" exists, but until they've studied specific areas, they probably will have never been exposed to those concepts or ways of thinking.)
What year? What experience do you have? What math have you done? What are you interested in?
Number theory.
+ What is modulo arithmetic?
+ If p is prime and 1 <= a < p, why is a^(p-1)-1 a multiple of p?
+ If a prime is conguent to 3 mod 4, why is it never the sum of two squares?
+ If a prime is conguent to 1 mod 4, why is it always the sum of two squares?
Topology
+ Find examples of why path-wise connected is stronger than "not disconnected".
+ Show that in 2D if you consider parallel lines to converge at infinity, and that they all converge to the same point, then what you have is a 2-sphere
+ Show that in 2D if you consider parallel lines to converge at infinity, and that non-parallel lines converge to different points at infinity, then what you have is a Moebius strip with its edge glued to a disk. (projective plane, or RP2)
Analysis
+ Understand why the sum 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... diverges (and what that means)
+ Understand why the sum 1/2 + 1/4 + 1/8 + 1/16 + ... converges, what that means, and what it converges to.
++ Note: most undergraduate mathematicians get this wrong.
There's a bunch of stuff, and this is all straight from the top of my head. It's not necessarily good advice, but they are a few things I found interesting when I was 12 or 13.
