92nd Carnival of Mathematics

Ninety-two isn't any ordinary number; it is associated with "royalty". How does that work out? It represents the number of solutions to which 8 queens can be placed on a 8 × 8 chessboard such that no two queens can attack each other. If you are familiar with the rules of English Chess, then it shouldn't take you long to figure out that any feasible solution would only arise if no two queens are positioned in the same row, column or diagonal. A possible answer is depicted here:

For those who are interested in learning more about the strategy involved in cracking this conundrum, Wolfram MathWorld has a rather comprehensive page addressing THIS.

92 is also the atomic number of uranium, and an Erdős–Woods number. Say the latter again?

(Credits to PlanetMath.org for the above definition)

Before we begin things proper, let's admire the beauty of the Snub dodecahedron, an Archimedean solid comprising 92 faces, of which 80 are triangular and 12 are pentagonal:

And so the carnival shall commence. Welcome to the 92nd edition.

At CTK Insights, Alexander Bogomolny discusses a property of the fifth powers of integers in great detail. In his own words:

"Surprisingly, the fifth powers of the integers may end in one of only 15 2-digit numbers. This property provides an elementary way of proving that in any solution of x⁵ + y⁵ = z⁵ in integers, one of x, y, z is divisible by 5,confirming a more general theorem of Sophie Germain in this case."

Over at the Aperiodical, Christian Perfect provides laugh out loud humor in the form of a certain Mathgen Program and one fictional Professor Marcie Rathke all neatly packaged into his piece titled Advances in Pure Nonsense.Whilst at that, do check out his other article where he gives his take on the weird and unusual in Interesting Esoterica Summation, Vol 5.

On a more serious note, Peter Rowlett writes about seeking a decent explanation as to why surds are being featured within the GCSE syllabus. In Surds, What Are They Good For? ,he also highlights interesting snippets of his exchanges with other educators on Twitter. In his afterthoughts, he says:

"People seem to like this. Overtly, it's about an attemptto discover why surds are a valuable addition to the secondary school curriculum and particularly anyone uses it outside of education. Beyond this, there are some issues touched upon about utility in education generally."

Mathalicious provides an enlightening dissection of the benefits of Doubling Down when playing blackjack, while Patrick Honner laments the existence of terrible math questions set by teachers, highlighting it in Another Embarrassingly Bad Math Exam Question at his Math Appreciation blog.

Tim Gowers goes into a lengthy yet worthy discourse about How  Should Mathematics Be Taught To Non-Mathematicians? in his weblog. A word of thanks to Robin Whitty (who helms the site Theorem of the Day) for bringing this post to my attention. He explained:

"Tim Gowers' call for 'real world problems' which offer natural and accessible scope for application of school-level mathematics. There appears to be a genuine window of opportunity to influence UK maths education and Gowers appears to be in a good position to make it happen."

Lastly, in view of the fact the 2012 Singapore H2 Maths A level examinations are literally just round the corner next week (on the 7th and 9th of November) , I have penned a personal piece concerning Question Spotting Thoughts to assist students in their preparations.

This therefore concludes the current Carnival of Mathematics. I shall pass the baton to Tosin who will be hosting the 93th edition at X In Vogue.

Peace.

(PS: I would like to accord a sincere thank-you to Katie Steckles for giving me the opportunity to contribute to this blossoming math blogging community. )

2 November 2012

Maclaurin's Series

A level H2 maths question spotting thoughts for AY2012

Last year's papers delivered a nasty surprise in the form of 2 questions which were truly old-school style. Many teachers were caught off-guard, me included. It would seem that examination setters are favoring a return to the 90s, when the old Further Maths standard was in place over here in Singapore. Back then a completely different style of problem solving techniques were imparted, skills which H2 mathematics students are neither taught nor exposed to. So, should anyone be overly concerned? I would say no, not at least for the present moment. While it is rather likely the papers will contain an increasing quantity of F-maths flavored question structures,majority of the problems issued would still remain as common place ones. Which means it is highly advisable students attempt and be well acquainted with previous editions of the A Level H2 math examination papers (and yes that includes the tremendously easy 2010 ones).

For regular items on the menu, you can refer to writings of mine in previous years to capture the essence of what to expect in general, therefore I won't waste time reproducing these here.

Instead, I will touch upon what I have mentioned earlier with regards to the board embracing ancient stuff, as this is where things are clearly filed in the "unpredictable" cabinet. And most students fear the unpredictable, however small/trivial a portion of the big picture they occupy. I cannot say for certain my forecasts will be dead-on accurate, but hopefully they can help provide some measure of confidence. Here goes my set of anticipatory advice:

1. Complex Numbers

Binomial series expansion coupled with trigonometry could be rigorously implemented within a complex numbers question. Take a look at this problem (Q1 on the page) and study it thoroughly. I have been getting my students to repeatedly solve this bad-ass here, telling them: " It could save your life." (Not to be taken literally of course).

2. Binomial/Poisson Distributions

You can discover the mode of both distributions very easily with the use of the graphic calculator, but what if the setters desire to make your beloved GC redundant? And how can this be achieved? View this problem (Q10 which deals with Binomial distribution on the page) to get a better understanding of what I am talking about. Should a Poisson distribution be tested instead,here is the modified formula: P(X=k+1)/P(X=k) = λ / (k+1) , where λ denotes the Poisson parameter.

3. Calculus

What could happen here? The usage of the R-formula could pop out and bite you, in the form of an integral waiting to be evaluated. View this document I wrote for my supplementary site to learn more. And if you must know, the R-formula isn't present in the current version of the MF15.

For solving first order differential equations, while you were taught the variables separable and substitution techniques, there is actually one more within the family, called the integrating factor method. While its relevance lies outside the core syllabus, an existing question could be modified and adapted based on exam requirements. To attain a better appreciation of things, examine the handout on this page (see no. 3 under Supplements).

This concludes the discussion (there is only so much I can share within a single page), hopefully I have helped addressed some pressing doubts. Study hard and stay healthy for this very last lap. All the best for your A levels kids. Peace.