### Why do this problem?

This problem offers the opportunity to practise the important skill
of ordering fractions while investigating and making
conjectures about an intriguing sequence of sequences. There is a
chance to work on a variety of questions at different levels.

### Possible approach

"This is the third Farey sequence. Can you work out what rules
have been used to generate it?"

Once students have identified all the criteria, ask them to
discuss with their partner what they think the fourth Farey
Sequence will look like. Then show them the

fourth sequence. Perhaps
clarify the rule about equivalent fractions by asking "Where is
$\frac{2}{4}$?"

Write up $F_2, F_3,$ and $F_4$ on the board, and challenge
students to work out $F_5, F_6,$ and $F_7$, using the agreed rules,
and think about what they will do next:

"When you've finished, I'll be asking you to investigate these
sequences, so think about questions you would like to ask and
things that you notice while you are working."

As students are working on the sequences, circulate to see if
everyone is getting the same results. If so, when the class is
ready to move on write the agreed results for $F_5, F_6,$ and $F_7$
on the board. If not, ask students with differing answers to write
their sequences on the board, and ask the class for their comments.
When consensus is reached, move on:

"Mathematicians often look for patterns to help them to
understand something better. What might mathematicians notice about
the Farey Sequences we have found? What questions might they want
to explore next?"

Take suggestions from the class and list them on the board.
There are some "questions to consider" at the bottom of the problem
which could be used to supplement the class's suggestions.

Allow the students to choose what they would like to explore.
They may wish to work with a partner. One nice way to feed back at
the end of this activity is for each student to work on paper and
for findings on similar conjectures to be displayed together on a
noticeboard.

### Key questions

When is $\frac{a}{b}< \frac{c}{d}$?

### Possible extension

Can you find an example where you put in an odd number of
fractions to get the next Farey Sequence? If not, why not?

### Possible support

Students need to be confident at comparing fractions by using
equivalent fractions. One way of supporting them in this is to use
a

fraction
wall.