The following example is not completely elementary, in that it requires the notion of sum of a family of non-negative real numbers (defined as supremum of the finite sub-sums), and some topology of the real line. But it allows a nice picture, and leads to a visualization of ordinal numbers as subsets of $\mathbb{R}$, so that it could be understood and appreciated by well-motivated high-school students, if you take the time to explain the problem, possibly skipping the technicality, and solving particular cases first. Moreover, it is close to the birth-place of the ordinal numbers. 

*The additivity of the length on the family of all left-closed, right-open intervals  of the real line.*

Let $P$ be the family of all left-closed, right-open intervals of the real line. Suppose $I=:[a,b)\in P$ is partitioned into a family elements $J_x:=[x,x')$ of $P$, that we may indicize by the left end-point $x\in S$, that is,   $I=\cup_{x\in S} J_x $. Then,    
$$|I|=\sum_{x\in S} |J_x|\, .$$ 

*Proof:* You  may first consider and solve the case of finitely many intervals, and the case of a sequence of intervals accumulating at $b$, that gives rise to a telescopic sum. For the general case, the key point is  that $S$ is well-ordered by the natural order of $\mathbb{R}$. This allows to prove  
$$\Big|\bigcup_{x\in S\atop x \le u} J_x \Big|= \sum_{x\in S\atop x \le u} |J_x|  \, .$$
by transfinite induction on $u\in S$.