A tree of fractions

Have you ever tried to list out all of the rational numbers? In this lesson we explored a nifty way of writing all rational numbers called the Calkin—Wilf tree.

But first we explored something seemingly completely different. Suppose that a triangle represents 1, a diamond represents 2, a square represents four, a rhombus represents 8, and a trapezoid represents 16. For each number N, how many ways are there to represent N using at most two of each shape? The answers to this question form the hyperbinary sequence.

In the second part of the class we visited the Calkin—Wilf tree. It is a complete binary tree with a fraction on every node. If some node has the fraction i/j, then its two children will have the fractions i/(i+j) and (i+j)/j. Once this is completed, we found that the numerators and denominators of all the fractions are taken from the hyperbinary sequence!

We spend the rest of the time asking (and mostly answering): why?

To start playing, check out the two worksheets: