**Grade 11-12:Trains**

Fibonacci numbers and Pascal’s triangle both appear in the Grade 11U curriculum but my experience is that most teachers do very little with them, certainly little in the way of mathematical thinking.

In this unit we introduce the “train numbers” t_n, the number of trains of length n that can be built out of cars of length 1 and 2. Unexpectantly, these turn out to be a realization of the Fibonacci numbers, a realization that is powerful enough to give us a“train-theoretic” proof of some of the remarkable properties of those numbers, for example that the sum of the squares of two consecutive Fibonacci numbers is always a Fibonacci number, e.g. 5^2+8^2=89.

Teacher manualStudent workbook