Farey Sequences printable worksheet
If I gave you a list of decimals, you might find it quite straightforward to put them in order of size. But what about ordering fractions?
A man called John Farey investigated sequences of fractions in order of size - they are called Farey Sequences.
The third Farey Sequence, $F_3$, looks like this:
$$\frac01 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac11$$
It lists in order all the fractions between $0$ and $1$, in their simplest forms, with denominators up to and including $3$.
$F_4$ lists all the fractions between $0$ and $1$, in their simplest forms, with denominators up to and including $4$. Can you write it out? Click below to check once you've had a go.
Now that you've got the hang of it, write $F_5$.
Which extra fractions are in $F_5$ which weren't in $F_4$?
Which extra fractions will be in $F_6$ which aren't in $F_5$?
Where will they appear in the sequence?
There are lots of questions you could explore about Farey Sequences. Here are just a few that we thought of:
Send us your solutions to these and any other questions you decide to explore!
There is a curious link between the mediant of two fractions and Farey Sequences. See the problem Mediant Madness to learn more about mediants, and Farey Neighbours to apply it to Farey Sequences. Finally, the problem Ford Circles shows the relationship between these fractions and some beautiful geometrical patterns.