Okay, disclaimer: I'm way too drunk for this. Probably regret it all in the morning. Sorry..

I'd start by showing proofs of basic geometric theorems and trigonometric identities using the unit circle (you'll need more sophisticated algebra very shortly, unfortunately. Might be best to wait just a little.). Here's why, for someone who might be interested in maths purely for the applications: (I think I said something along these lines when I was sober once)

Trigonometry and basic geometry are a lot cooler than people make them out to be. The way the Greeks could make measurements with basically sticks and stones shows how much power that elementary knowledge gave them. To them it wasn't elementary, it was cutting edge; they didn't have good notation and language for it all, so everything was more difficult than it is today. And no one was there to tell them what else they could do with their ideas, they were among the first to work on it all.

In the same way, today we have our formulations of Newton's laws as given by Lagrange and Hamilton, and game theory and methods of optimization and methods to predict the weather by simulation and ways of recovering an accurate image of the brain from magnetic resonance, ...

It's incredibly useful. It's made us all richer, in a way.

For someone with some basic ability geometry and trigonometry are good, because it's obvious why they're useful. It doesn't have to be boring! The sequence of proofs that gets you from understanding arithmetic to the more modern stuff takes a while, but every part of it is beautiful, or can be presented in an aesthetically pleasing way.

Oh, and there's another thing: it has a different flavor from what he's done so far at school. That might be an advantage, but if you do teach it you have to be very careful to do things in the right order and in a nice way so he doesn't get frustrated. Missing a crucial piece of background information is the worst..