I'd present a juicy morsel of mathematics, say -- the Euler characteristics theorem for $\ \mathbb S^2.\ $ I'd adopt a classic proof.

Let $\ \mathbb S^2\ $ be divided into convex geodesic polyhedra, $\ P.\ $. (Allow some neighboring edges to extend one another so that occasionally they lie on the same large circle). Then the sum of the angles of a polyhedron $\ p\in P\ $ is equal to

$$ \pi\cdot(n_p-2)\ +\ A_p $$

where $\ n_p\ $ is the number of edges (or vertices) of $\ p,\ $ and $\ A_p\ $ is the area of $\ p.\ $ Then summing over $\ p\in P\ $ gives us the Euler formula rapidly:

$$ |V| - |E| + |P| = 2 $$

where $\ V\ E\ P\ $ are the sets of vertices, edges, and polyhedra of the given scheme.

The simple combinatorial argument must be satisfying to youngsters. On the other hand, the students get a feel for the place of the general theory since they would be pointed to the measure theory. Finally, they may appreciate the power of special examples, e.g. of surfaces of constant curvature. Indeed, one can go beyond $\ \mathbb S^2.\ $ One only needs surfaces for which the sum of areas of geodesic polyhedra would be, say, $\ -8\cdot\pi\ $ (instead of $\ +4\!\cdot\pi)\ $ and everything else would be the same.