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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.

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

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

Circle 1:

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has centre:

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and radius:

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Circle 2:

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has centre:

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and radius:

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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:

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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 Fsequence.  So do other consecutive terms in the Farey sequence also generate circles touching at a point?

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

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

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

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

Essential Resources for IB Teachers

1) Intermathematics.com

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If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

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There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.