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.

Write the first, second and third Farey Sequences on the board:

$$\frac01 \qquad \frac11$$ $$\frac01 \qquad \frac12 \qquad \frac11$$ $$\frac01 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac11$$

"Here are the first three Farey sequences. What do you think a Farey sequence is? What might the next one look like?"

Give students a short time to look at the sequences on their own, and then encourage them to discuss with their partners. Then bring the class together and share ideas. Clarify that a Farey sequence contains all the fractions with a denominator up to a particular number, in their simplest form, in order, and write up $F_4$ on the board: $$\frac01 \qquad \frac14 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac34 \qquad \frac11$$

This is a good opportunity to clarify the rule about equivalent fractions by asking "Where is $\frac{2}{4}$?"

Give students a short time to look at the sequences on their own, and then encourage them to discuss with their partners. Then bring the class together and share ideas. Clarify that a Farey sequence contains all the fractions with a denominator up to a particular number, in their simplest form, in order, and write up $F_4$ on the board: $$\frac01 \qquad \frac14 \qquad \frac13 \qquad \frac12 \qquad \frac23 \qquad \frac34 \qquad \frac11$$

This is a good opportunity to clarify the rule about equivalent fractions by asking "Where is $\frac{2}{4}$?"

Now challenge students to work out $F_5, F_6,$ and $F_7$. "As you are working, think about what questions a mathematician might ask and make a mental note of anything interesting 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. Here are some questions which could be used to supplement the class's suggestions:

- How many extra fractions are there in $F_{11}$ that aren't in $F_{10}$?
- How many extra fractions are there in $F_{12}$ that aren't in $F_{11}$?
- Is every Farey Sequence longer than the one before? How do you know?
- Is there a way of working out how many fractions there will be in the next sequence?
- So far, all the Farey Sequences except $F_{1}$ have contained an odd number of fractions. Can you find a Farey Sequence with an even number of fractions?

Encourage students to pick a question to explore, perhaps with a partner. While students are working, circulate and listen for interesting insights. At the end of the lesson, you could have a discussion about the different questions and invite students to share their insights. Alternatively, students could prepare a presentation or piece of work for display, outlining their findings.

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.

Students could go on to explore Mediant Madness and then Farey Neighbours.