Ford Circles

This carries on the previous investigation into Farey sequences, and is again based on the current Nrich task Ford Circles.  Below are the Farey sequences for F₂, F₃ and F₄. You can read about Farey sequences in the previous post.

This time I’m going to explore the link between Farey sequences and circles.  First we need the general equation for a circle:

This has centre (p,q) and radius r.  Therefore

Circle 1:

has centre:

and radius:

Circle 2:

has centre:

and radius:

Now we can plot these circles in Geogebra – and look for the values of a,b,c,d which lead to the circles touching at a point.

When a = 1, b = 2, c = 2, d = 3:

Do we notice anything about the numbers a/b and c/d ?  a/b = 1/2 and c/d = 2/3 ?  These are consecutive terms in  the F₃ sequence.  So do other consecutive terms in the Farey sequence also generate circles touching at a point?

a = 1, b = 1, c = 2, d = 3

Again we can see that the fractions 1/1 and 2/3 are consecutive terms in the F₃ sequence. So by drawing some more circle we can graphically represent all the fractions in the F₃ sequence:

So these four circles represent the four non-zero fractions of in the F₃ sequence!

and this is the visual representation of the non-zero fractions of in the F₄ sequence.  Amazing!

Essential resources for IB students:

1) Revision Village

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.