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Crazy Curves

Except for the circle (which is shown for reference), on the other three curves, except where the curves cross an axis, all the points (x,y) have an interesting property. In fact this property holds for all the curves of this family for any power of x and y above 2. What is so special about the co-ordinates (x,y) of these points and why?

Pencil in the time

You can see the real time clock here

1950s Maths O level

RAQ is a tangent at A to the circle ABC. The lines BQ and CR are perpendicular to the tangent and AP is perpendicular to BC.

Show that: BQ/AP = AP/CR

Tweet from @SofARMaths (Christmas puzzle)
@SofARMaths: Let's do the Christmas puzzle again this year... #maths

Not the appropriate place for a rant? But I cannot resist listing these annoyances:

Using the train as an office, Using the train as a music centre, Any form of music in the office, Any form of musac anywhere.

Dress down Friday, Dress down any day, A shirt and suit without a tie, Trousers worn around the knees, Trainers worn in the office.

Inane conversation, Like, Kinda like, You know what I mean, Affectation - talking loudly.

Open plan offices, Open plan houses.

Eating in the street, Drinking on the pavement.

International call centres.

Discretionary tips.

"No thinking required" approach.

Making lists!

Four interlocking wheels

I hope this is a well-defined question!

I'm interested in a general formula for the area shaded in blue as you roll four interlocking (perhaps geared) wheels as shown in the diagram above. Sounds like a nice advanced GCSE or O level question? Assume the wheels have the same radius r. What is the minimum and maximum of the shaded area?

This will stop me playing around with biscuits. Perhaps I need to buy some Meccano.

Curious numbers

I saw a tweet on my @SofARMaths account yesterday from Nalini Joshi @monsoon0 which read "I love all numbers great & small. Why are there so many anthropomorphic adjectives for them: odious, evil, perfect, nice, pernicious?" This prompted a frivolous exchange of tweets during which I asked whether there are any "curious" numbers.

I decided to call a curious number one that belongs to the set {Cn} where Cn is the product of the first n prime numbers +1. This is a definition that has been buzzing around my head for many months, probably years, since I first encountered Euclid's proof that the number of primes is infinite.

When I asked myself whether 200560490131 is prime I came across a link to primorial numbers which shows that this area is already well researched. Oh well – perhaps we can classify these and other numbers by the number of prime factors they have. Is there a name for numbers which are the product of exactly 2 prime numbers such as 30031?

Su Doku solver

My Su Doku solver is here for you to try.

It does not just present you with the solution but rather it leads you through solving it yourself by suggesting numbers for each square for which it has determined there is a unique answer.  As set up it has a default puzzle but you can change any and all of the numbers to start the solution of your own puzzles.  Give it a go!  It is a useful aid to teaching and having “maths fun” – or should I say logic?

You do not need a logon to download the file and save it as your own copy.  Note however that in Internet Explorer (because of a known problem with its interface to sites such as this one) you may see a request to provide an ID and password.  Press cancel, perhaps twice, and it will open the file quite happily.  If you are using firefox or chrome you will not get this problem.

Reflections on tutoring 2011- 2012

Perhaps it is too early to be writing on this subject since the examination results will not be published for another month or so. However tutoring in my view has never been solely about preparing for public examinations even though they are important to students and parents alike. On this front I do expect to see them all achieve the results that they need.

Equally I hope they have enjoyed the work we have done together and had a measure of fun along the way. I certainly did. Over the past 12 months I have worked with eight students on a regular basis at least once a week.  Two students were working with the GCSE syllabus and four with the A level syllabus.  The remaining two are following an IB based syllabus.  I find this wide range of topics and levels very satisfying to tutor. 

Of particular interest this year has been the A level S1 and S2 Statistics modules which subjects I had not covered for some time and the Geometry course in the American system.  I found the rigorous approach in the McDougal Littell Jurgensen Geometry book both challenging and stimulating.

There was also a high demand for tutoring in Mechanics M1 and M2 as well as the core modules C1 to C4 which I have covered for many years and which together with Mechanics is my speciality.  Sadly I have not been asked to cover Further Maths this year.

On the materials front I have used Wolfram Mathematica, Wolfram Alpha, Cabri and a plotting application on a mobile phone - but for me there is no substitute for paper and pencil which are the mainstay tools of all Mathematicians!

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