I just took a quick peek at wikipedia's article on continued fractions and I found that it doesn't mention (at least explicitly, in a 1 min skim) something that you may assign as an exercise to your students, namely (in pseudo-Octave notation here):

Prove that the recurrence relation of the p_i's and q_i's that we mentioned before can be obtained via matrix multiplication. More precisely, prove that:

[a_0, 1; 1, 0] * [a_1, 1; 1, 0] * ... * [a_n, 1; 1, 0] = [p_n p_{n-1}; q_n q_{n_1}].

As a corollary, derive Cassini's identity for the Fibonacci Numbers.