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Farey Sequences

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

Display the third Farey Sequence on the board and ask students:
"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.