Mathematical thinking and STEP

In my last blog I highlighted how establishing patterns in Maths is often the route to discovering theorems and solving complex looking problems.

STEP questions often challenge the student to recognise an approach established in an early part of a question and apply it to a more complex problem posed in later parts of the same question. This is demonstrated by the following STEP II question and solution.

In this question, you may assume without proof that any function f for which f'(x) \geq 0 is increasing, that is f(x_2) \geq f(x_1) if x_2 \geq x_1

(i)   (a) Let f(x) = \sin x -x\cos x. Show that f(x) is increasing for 0\leq x \leq\dfrac{1}{2} \pi and deduce that f(x) \geq 0 for 0 \leq x \leq \dfrac{1}{2} \pi.

f'(x) = \cos x - \cos x +x\sin x = x\sin x \geq0 for 0\leq x\leq\dfrac{1}{2}\pi

Also f(0)=0 \Rightarrow f(x)\geq0 for 0\leq x\leq \dfrac{1}{2}\pi

(b) Given that \dfrac{d}{dx}(\arcsin x) \geq 1 for 0\leq x < 1, show that

    \[\arcsin x \geq x\qquad  (0\leq x<1).\]

Let f(x)= \arcsin x-x then f'(x)=\dfrac{d}{dx}(\arcsin x)-1 \geq 0 \qquad (0\leq x<1)

f(0)=0 \Rightarrow f(x)=\arcsin x-x\geq 0 \qquad \Rightarrow \arcsin x \geq x \qquad (0\leq x<1)

(c) Let g(x)=x\csc x for 0<x<\dfrac{1}{2}\pi. Show that g is increasing and deduce that

    \[(\arcsin x)x^{-1} \geq x\csc x \qquad (0<x<1)\]

g'(x)=\csc x - x\csc x\cot x = \dfrac{\sin x - x\cos x}{\sin ^2 x} \geq 0 \quad (0<x<\dfrac{1}{2}\pi )\quad from (a).

(Note that the re-arrangement of g'(x)allows the use of (a) to prove the result).

It is now tempting to rewrite the result from (b) as (\arcsin x)x^{-1} \geq 1 to prove the desired inequality. However this would require showing that x\csc x\leq 1 \quad (0<x<1) which is not true! So we need to look further!

This is essentially the crux of the whole question. Experience suggests that we look at results demonstrated previously to give  clues as to how best to proceed, but our first attempt to use (b) has proved unsuccessful.

However, the moment of inspiration arrives when we spot that g(\arcsin x) yields the left hand side of the inequality.

Since g(x) is increasing

    \[\arcsin x\geq x \qquad (0\leq x<1)\]

    \[\Rightarrow g(\arcsin x)\geq g(x)\qquad (0<x<1)\]

    \[\Rightarrow (\arcsin x)x^{-1}\geq x\csc x\qquad (0<x<1)\]

(ii) Given that \dfrac{d}{dx}(\arctan x)\leq 1 for x\geq 0, show that

    \[(\tan x)(\arctan x)\geq x^2 \qquad (0<x<\dfrac{1}{2}\pi )\]

Again we now look carefully at previous results. We know that the given inequality yields \arctan x \leq x so we look for an increasing function h(x) such that h(\arctan x)\leq h(x) will yield the desired result. At first this does not seem obvious, but a simple re-arrangement of the inequality as follows provides the solution.

    \[\dfrac{x}{\arctan x} \leq \dfrac{\tan x}{x}\]

So letting h(x)=\dfrac{\tan x}{x} it is merely required to show that h(x) is increasing.

    \[h'(x)=\dfrac{x\sec ^2x - \tan x}{x^2} \]

and 

    \[\dfrac{d}{dx}(x\sec ^2x - \tan x) = 2x\sec ^2x\tan x \geq 0 \quad (0<x<\dfrac{1}{2}\pi )\]

The rest of the proof follows trivially.

I hope this has provided some insight into developing Mathematical thinking and how many STEP questions are designed to test the ability to build on previously established results.

After all, that is how the whole of Mathematical theory has been developed.

 

STEP or MAT or both?

MAT (Mathematics Admissions Test) and STEP (Sixth Term Examination Paper) are used by leading universities to help them assess students applying to read Maths courses.

Candidates for the University of Oxford and Imperial College London are required to take the MAT in early November prior to being selected for interview. It consists of a single 2.5 hour paper. There is no official grading of MAT, though clearly students achieving higher marks are more likely to be short-listed.

Candidates for the University of Cambridge and University of Warwick are usually required to take two of the three STEP papers in June. Each paper is three hours long and divided into sections on Pure Maths (eight questions), Mechanics (three questions) and Probability and Statistics (two questions). All questions carry equal marks but only the best five answers count towards the Grades which are often form part of a conditional offer.

So what are the differences?

On the face of it MAT is more approachable by all students, including those who may not be taking Further Maths or are from other educational systems. One could say it aims to test aptitude rather than the highest knowledge. On review students are likely to find it less daunting than STEP.

In contrast STEP is more demanding in both the mathematical knowledge required and its applications. It is more akin to the type of examination students would sit at leading universities. It is noted that while only Cambridge and Warwick usually require students to sit STEP, many other leading universities encourage applicants to sit the exams as preparation for their courses.

But regardless of which (or both) exams students take focused preparation is vital to achieving outstanding performance. Specialised assistance either within the school and/or through external tutoring is recommended.

Building your online presence

So you’ve taken on board all the advice about deciding on your niche and establishing an online presence and its time for action. But where do you start? Here’s my story so far.

As a Maths tutor primarily targeting GCSE and A-level students I found myself competing with an increasing number of tutors and agencies in the same space. It was becoming hard to get noticed and I wasn’t prepared to drop my fees to compete solely on price. So I needed to establish a niche!

I have chosen the Maths STEP (Sixth Term Examination Papers) which are used by the top Universities to assess applicants wishing to study Maths.

The next step was to create a website so I needed to choose a domain name, a web hosting platform, and a web builder. I chose a domain name that best matched my niche and researched hosting platforms and site creation tools over the internet. With such a wide variety available this wasn’t easy but I eventually settled on FreshSites as my host and WordPress as my website creation tool. My choices were driven by cost, reviews and availability of support.

So having purchased my domain and installed WordPress I set about creating my site. The first step was to select a ‘Theme’ from the many available, which sets the basic structure of the pages on your site. I chose a fairly simple theme ‘Twenty Twelve’. Then its on to customising your site, creating menu structures and content. It took me quite a bit of time to familiarise myself with system, but every time I got stuck I found my question answered on the support forum.

As well as basic content I learned how to create internal and external links and embed a youtube video I had made explaining how I go about online tuition. Then it was on to plugins. Plugins are essentially bits of packaged software that make it easy integrate common functions. And there are loads available for free! I installed a plugin that creates a Contact Form, one that links to Social Media and another for SEO. The latter submits your site to Google and other search engines, builds a sitemap hat helps them index your site and gives you hints on optimising your site for SEO.

That’s as far as I have got. There is more to do in creating Landing Pages and getting my blog up and running, but its been an exciting and challenging journey so far.

Maths Theorems and Proofs

Theorems (and their proofs) are the building blocks of all Mathematics – in fact they ARE the whole of Mathematics.

But the concept of proof is barely mentioned in the school curriculum until Further Maths. Maybe it is thought to be too difficult or complex. It does not have to be.

Every GCSE Maths student is expected to learn Pythagoras’ Theorem. ‘In a right angle triangle the square of the hypotenuse (the longest side or side opposite the right angle) is equal to the sum of the squares of the other two sides.’  It is often written in shorthand as a² = b² + c².

But how many students (even at A-level) can prove it? There are many simple and elegant proofs. Have a go.