## Great Circles and Spherical Triangles

In the geometry of mariners, where straight lines are Great Circles, the angle sum of a triangle is more than 180 degrees.

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right  into the 21 century classroom.

The sine function starts life in a triangle (year 9?) then becomes intimate with a circle at A level, as y-coord of a point rotating on a unit circle.  Later it is about infinite series or, even, differential equations.

Have got absorbed by the National Cipher Challenge.  Not sure what this animation varying the multiplier in an ‘Affine Shift’ add 5.  Functional grid shows transition from Caesar Shift 5, multiply by 1, through to multiply by 13, which is a useless cipher.  In fact any multiplier not co-prime with 26 can be seen to be not one-one, and therefore useless too.  Modular arithmetic, straight line graphs, ciphers, dotty patterns, ….

Have been playing with Python to work on the National Cipher Challenge.  The string function translate seems to be tailor made for this sort of thing.

Russian peasants and Egyptians use magic of the distributive laws to multiply faster.  Two recursive algorithms, coded in Python, do different amounts of work.

Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams.  School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.

Why do we need Z as an axiom; can’t we conclude it from the others? If a is non-zero, then it has an inverse by M4, and then (1/a) ∙ a ∙ b = (1/a) ∙ 0, so b = 0; thus either a or b must be 0. (We need the fact that x ∙ 0 = 0 for any x, but this follows by observing that x ∙ 0 = x ∙ (0+0) = x ∙ 0 + x ∙ 0, and adding -(x ∙ 0) to both sides.)

Yes you are quite correct.  I suppose that it is in my list above to enable the axioms to be switched on and off in different combinations to allow for, say, a ring with no zero divisors; but I know that that is not what I am taking about here.  And yet I like it there from a school maths point of view.  Pupils should use this property when solving quadratic equations by factorization and, along with say -1x-1=+1, this may be one of the opportunities for a more profound algebraic discussion in the class room.  In my experience they may understand the algebra, or just do what they are told without being bothered why, or stumble over the fact not finding it obvious.  It is this third set who might be deep thinkers and could be very rewarding to teach maths to.

Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams.  School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.

Mathematics tells us that the winds can never blow in a continuous pattern across the surface of the Earth – a hairy ball can never be combed flat without partings or tufts.

FP3 2014 gives students the chance to prove the surface area of a sphere is 4 pi r squared.  Interestingly four time the area of a cross section through the centre and the derivative of the volume.  Pupils with strong maths general knowledge from GCSE would know what is coming.  TOP TIP do as much maths challenge and extension work as possible, the general knowledge thus gained can only help.  This FP3 2014 paper and many others are at my Exam Bucket: http://jped.co.uk/ExamBucket/exambucket.php.