Farey Fibonacci
Investigate Farey sequences of ratios of Fibonacci numbers.
Age
16 to 18
Challenge level
Problem
Denoting the Fibonacci numbers $1,1,2,3,5,8,...$ by $f_n$ where $f_n=f_{n-1}+ f_{n-2}$ prove for all positive integer values of $n$ that $\frac{f_n}{f_{n+2}}$ and $\frac{f_{n+1}}{f_{n+3}}$ are Farey neighbours, that is $|f_{n+1}f_{n+2}-f_nf_{n+3}|=1$.
Show that the mediant of $\frac{f_n}{f_{n+2}}$ and $\frac{f_{n+1}}{f_{n+3}}$ is $\frac{f_{n+2}}{f_{n+4}}$.
See the problem Farey Neighbours.
Getting Started
Test the result for several small values of $n$ and try to decide why the absolute value appears in the formula.
What values of $n$ are you asked to prove the result for and what method of proof does this suggest?
Teachers' Resources
Why do this problem?
Here you have a result that must be proved for all positive whole numbers $n$ which suggests that proof by induction is the obvious method to try. The result involves ratios of Fibonacci numbers so it is certainly non-standard. Even so the algebra involved is elementary and so the problem is a good exercise in using proof by induction. It also links apparently unrelated mathematical topics such as Fibonacci numbers and mediants and the geometry of Ford Circles. Fibonacci numbers should be part of the repertoire of every mathematician as they are useful in many appllications in higher mathematics.
Possible approach
How about using the 'Moore' method here? In other words tell your students that the class must work out the solution together, that you will stand at the back of the room and you will observe and just ask a few questions. If you have not taught in this way before explain why you are choosing this approach and how you believe it will benefit them.
Invite the students to decide between them who will record on the board what needs to be recorded. They should first decide on what method to use and exactly what is involved in using that method of proof. Then it works well for the learners to work individually to complete the proof before someone writes it up on the board for discussion. When the class gets stuck, or when they think that they have completed the proof successfully, then they need to check each step carefully and everyone should agree that it is correct before they proceed.
The teacher will need to exercise restraint so as not to make suggestions or point out mistakes even if there are periods when the class is stuck and does not seem to be making progress. This is an exercise in learning how to learn and this method of teaching can be very effective once the students realise that they have to think for themselves and the teacher is not going to lead them. If absolutely necessary the teacher should ask a question that prompts the learners to consider issues that are conducive to finding a way forward or to spotting their own mistakes or omissions.
Before teaching in this way it is essential for the teacher to have a good appreciation of all the alternative methods of solution and to have worked through the question themselves so that they know the likely pitfalls.
Key questions
What questions should we be asking ourselves at this stage? [Keep asking this question to encourage the learners to ask themselves the key questions. The teacher should only ask the following questions as a last resort if the class are not making progress.]
Have we understood what the question is asking us to do?
Have we investigated the result for some small values of $n$ and checked them numerically?
Why does the absolute value appear here?
What values of $n$ are we asked to prove the result for and what method of proof does this suggest?
Exactly what do we have to do to give a proof by induction?
In proving that the truth of the result for $n=k+1$ follows from the truth of the result for $n=k$ what formula should we work with?
In trying to solve this problem, have we used all the information we were given?
If $a+b=c$ then what do we know about $c-b$?
Possible extension
Try the problems Farey Neighbours and Ford Circles .
Possible support
Try the problem Farey Sequences.