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Ordering fractions can seem like quite a mundane and routine task. This problem encourages students to take a fresh look at the process of comparing fractions, and offers lots of opportunities to practise manipulating fractions in an engaging context where students can pose questions and make conjectures.
This printable worksheet may be useful: Farey Sequences
If $\frac1n<\frac1m$ what can you say about $n$ and $m$?
Why is each Farey Sequence longer than the last?
When does a Farey Sequence have lots of extra entries?
When does it have only a few?
When is $\frac{a}{b}< \frac{c}{d}$?
Tumbling Down uses fraction walls to provide a possible route into understanding Farey Sequences.