A level Mathematics

Mathematics is seen as a key skill at Ashbourne College and the department is fully resourced with an incredible teaching team. Specialist staff teach each strand of the course: physicists teach our students the mechanics modules; statisticians and ex-city workers teach Statistics; computer scientists teach the Decision Maths and pure mathematicians teach the Core Maths modules. This has led to consistently outstanding results with 80% of our students achieving A grades and 60% going onto achieve A grades at Further Maths.

Why study Mathematics?

Mathematics is the bedrock of nearly all science and economic-based disciplines, from physicists and geographers to bankers and engineers. A student should always consider taking Mathematics to at least AS level to prove to future employers that they are comfortable dealing with many different types of figures and calculations.

a level mathematics london

Which syllabus do we follow?

We follow Edexcel specification for Mathematics.

How many units are there?

There are six units in total. Mathematics works in a different way to most other A levels. All students must obtain the four Core Mathematics modules listed below. Ordinarily they will study for and sit C1 and C2 in the AS year and C3 and C4 in the A2 year. Students must also study for and sit two additional units. These could be both from the same area, for example S1 and S2, or from two different areas, for example S1 and M1.

What is each unit about?

Core Mathematics

C1C2C3C4
This is the first of the modules which focus on Pure Mathematics and in particular introduces students to Calculus, arguably the most powerful tool of modern mathematics. Before this happens students will extend their understanding of algebra, equations, inequalities, geometry and straight lines as well as advance their understanding of quadratic equations and their associated graphs. Students will also learn how to ‘transform’ one graph into another by making simple alterations to the equation of the original. Finally they will learn how to apply algebra and formulas to Arithmetic Series such as 1, 4, 7, 10 …
This module begins gently enough with a little more algebra (the Remainder Theorem since you ask), followed by the binomial theorem which will help you quickly express the terms of (1 + x)10 . This develops further in C3, the next module. After that you will extend your understanding of Calculus and learn how to prove things such as: if you are given a certain length of fencing the maximum rectangular area you can enclose is a square. These problems involving maximum and minimum values are interesting and challenging.

You will take your understanding of circles from GCSE a little further and learn about a different way to measure angles called Radians – very important. In addition to lines and parabolas, you will also add another equation to your arsenal. Work on series continues with the Geometric variety. This actually is interesting and will introduce two crucial elements of Pure Mathematics: limits and infinity. With your understanding of an Infinite Geometric Series you have pretty well all you need to understand interest rates, the value of stocks and bonds and how to make a fortune in the City.

Leaving the most feared to last: Trigonometry. You will extend your understanding of sine, cosine and tangent so far that you will not recognize them and as a bonus get to know their graphs as well. There is a kind of new equation waiting for you (trig equations of course) which has been known to drive some students potty. To be fair, Trigonometry forms the basis for our understanding of so much in the world right up to Quantum Mechanics. So the next time you visit the recording studio and observe the sound engineer chilling at the soundboard, know that they are employing harmonic analysis which derives from Fourier analysis (after the late, great French mathematician) which is founded on trigonometric functions. Spinach is good for you and I am sure so is Trigonometry.

This unit more or less extends your understanding of fundamental topics covered in C1 and C2, with one exception. So, there is to be found: a little more calculus, a little more on exponentials and logarithms, and a lot more on trigonometry (ouch). The only really new topic is functions. Although a bit disjointed it does introduce students to this extremely important idea which is, sadly, quickly orphaned with respect to the entire mathematics syllabus including further mathematics.

If you are a clever scientist you will see that the work on exponential functions is dominated by its application to rates of change. You may thus relate it to processes such as radioactive decay in physics, rates of reaction in chemistry and the growth of populations in biology. If you are really enthusiastic, you can use this study to introduce yourself to chaos theory!

Naturally enough students find this, the final module of pure maths, the most challenging. Interesting new topics include the extension of the Binomial Theorem (useful if you are on a desert island without a calculator) and the use of parameters and vectors. The difficulty lies in the new work involving calculus, which is a little disjointed. Whereas the new topics can be grasped logically and fundamentally, with the calculus, like mastering scales in music, students must steel themselves to practise, practise and practise.

Mechanics

M1M2
Once you ‘get it’ this module, M1, is for the most part quite easy. However, it is fair to say that students find M1 very difficult; and the reason they find it difficult is that their understanding of what forces act on everyday objects (say a book on a table or, more challenging, yourself, sitting on your sofa watching the X factor) is insecure – perhaps terrifying. A bit like Ghostbusters our job is to restore this confidence and make M1 a breeze (we can do it!).

Principally, M1 is about elementary applications of Newton’s laws (how many?) but, more interestingly, it is about the mathematical model used in analysing the forces of nature. Understanding the model is challenging but, if taken a little further than A level, will give anyone an appreciation of why Aristotle was wrong in thinking that all bodies require a force to keep moving in a straight line (the law of inertia). Equally you should wonder at the genius of Galileo, then Newton, then Einstein in overturning the Greek’s idea. The struggle to understand the model in M1 schools students in the patience necessary to understand Einstein’s Special Theory of Relativity.

Once you have mastered the M1 syllabus you can strut your stuff in M2 which by and large poses questions of technique. In other words how to solve more complex problems involving forces, turning forces (moments), momentum and motion under constant acceleration (projectiles). The questions are challenging but illustrate two ideas important for exam success:

1. Practice makes perfect.

2. Proficiency in exams depends on speed; the faster you are the more likely will be the reward of your efforts an A*. Speed depends on pattern recognition; you should be able to identify each and every problem you encounter as a ‘type’ which you have met before. You should rarely have to think in an exam because you have done your thinking beforehand.

M2 also introduces two new topics: centres of mass and conservation of energy. Of course all of these are related to physics. Sadly, much as we would otherwise prefer, most human beings seem genetically wired to silo mentalities: energy in mathematics is somehow different from energy in physics (where there is any energy at all). Still hope springs eternal.

Statistics

S1S2
Many people think the study of A level statistics is a sign of a lousy social life. They are wrong! Sixth form statistics is an essential tool for subjects as diverse as biology, psychology, economics and finance.

For example, suppose one person in 1000 actually has a particular disease. A pharmaceutical company develops a test which is 100% effective in identifying a disease; ie if you’ve got it, the technique will never fail to indicate that you’ve got it. Unfortunately, if you don’t have the disease, the test is wrong 1% of the time; ie even if you don’t have the disease, the test will say you’ve got it anyway one time in 100. You visit to your doctor’s surgery and test positive: what is the chance that you actually have the disease?

This example, of course, is a question in probability. There is an easy way to solve it and a hard way; and of course we teach you the hard way. The study of probability supplies you with an important way to understand risk. With little more than A level understanding of probability, it will certainly help you understand why Lehman Brothers’ bank collapsed as well as appreciating how careful you must be interpreting the results of trials for a new drug.

S1 also deals with the very important area of correlation and regression. If you have a good teacher (all at Ashbourne!) they will help you understand the true meaning of ‘regression to the mean’ and why correlation is not causation. An example of the latter is (ironically?): the more hours of private tuition students receives the worse their grades (negative correlation). Does this mean that teaching makes students stupid?

Finally S1 examines the analysis of data and will help you answer the following questions: 1. If Bill Gates walks into a pub, what happens to the mean (average) income of the patrons there? What happens to the modal income? Importantly it also deals with the concept of variance or standard deviation, both of which are used by financial boffins to measure risk and volatility in the markets. In fact they will even sell you volatility if you have the appetite!

P.S. You also get to learn quite a bit about bell (normal) distributions which apply to many things like IQs, exam results etc, but regrettably for bankers and the rest of us, not others like the price of shares and derivatives.

The most important part of the S2 module for A level Mathematics is the CLT (central limit theorem). This says that as the size of a sample from a population increases the mean (average) value of the sample follows the same distribution as for example, IQs (the bell or normal distribution). This means that if you could choose about 1000 adults at random (this random bit is not easy) whether or how they voted in the prelude to an election the result will with extremely high probability give you an excellent idea of the behaviour of all adults. This is obviously a powerful tool and in the right hands has with near perfection predicted the results of all recent elections in the United States.

This idea leads naturally to the topics of confidence intervals and hypothesis testing which are fundamental for psychologists, economists and doctors.

In S2 Mathematics you will also meet two very important distributions, namely the Poisson and Binomial. These are both extremely useful. For example, the Poisson arose from observing the number of officers in the Prussian cavalry in the nineteenth century who died each year because they were kicked by their horse. If someday you are bored waiting for your bus you might amuse yourself by counting the number of buses (not yours) that arrive at the bus stop each minute. More or less this illustrates a Poisson distribution.

Finally you will apply your knowledge of calculus to continuous distributions.

This module is not dense but is conceptually quite challenging. I suppose it indicates a general principle: in applied maths once you establish a rock solid understanding of the fundamental first principles you should have little difficulty afterwards.

Decision mathematics

D1D2
Algorithms, algorithms on graphs, the route inspections problems; critical path analysis; linear programming; matchings.
Transportation problems; allocation (assignment) problems; the travelling salesman, game theory, further linear programming, dynamic programming; flows in networks.

How is each unit examined?

All examinations last 1 hour and 30 minutes and have 75 marks. C1 is a non-calculator paper. Students can use calculators in all the other papers.

How is the course structured?

AS students will study for C1 between September and December of the AS year, with a view to taking the C1 examination in January. They will then begin studying for C2 with a view to taking the C2 examination in June. Students also study for M1 and S1. In the A2 year students will begin studying for C3 between September and December with a view to taking the exam in January. They will then begin studying for C4. Student may continue with S1 and M1, alternatively they may study for S2, M2 or D1 and D2. To gain an A level all students must sit all four Core Mathematics modules and two further modules.

When do the exams take place?

Students sit C1 in January and the remainder of the AS units in the May/June examination period. There will be opportunities to resit AS units or sit C3 in January of the A2 year. The remaining units will be sat in May/June of the A2 year.

Who teaches this course?

 

Pete Franklin

BSc, Actuarial Science (London School of Economics); PGCE, Secondary Mathematics (University of London).

Before retraining as a teacher in 2002, Pete worked at Channel 4 for 11 years, first as Marketing Manager and later writing and directing TV trailers. He joined Ashbourne in 2010.

Chella Nathan

BEng Hons, (North London); BSc Hons (Sri Lanka) Mathematics and Physics

Chella has been teaching Mathematics and Physics for more than twenty years. His research interests are number theory and the development of renewable energy sources.

Sean Pillai

MA Civil Engineering with Business Management (Warwick University)

Sean began his career as an investment banker but quickly realised that teaching was his preferred profession. Sean is a keen sportsman who plays lacrosse, football and tennis in his spare time.

Dr Barry Rhule

PhD (London); BSc Hons (CNAA), Mathematics, Physics

Barry is a member of the London Mathematical Society and has been teaching Maths and Physics at Ashbourne for more many years.

Dr Abdul Sami

BSc Hons, Pure Maths; MSc, Pure Maths (Queen Mary University of London); PhD, Algebra (Imperial College London)

Abdul taught Pure Maths at Michigan State University for three years before returning to the UK. Then he transcribed parts of Newton’s Principia at Sussex University before joining Ashbourne College. Some of his research is published in leading journals and he has given seminars and conference speeches in the UK, USA and Germany.

Seema Ali

PGCE (University of Durham), BSc Maths (University of Durham)

Zahra Behrang

MSc Mathematical Sciences (Tabriz University, Iran), PhD Pure Mathematics (University of Birmingham, UK)

Madeeha Saad

BSc Mathematical Sciences (Open Univerisity), MSc Medical Statistics (University of London)

Carolina Emmanuel

BSc Mathematics (University of Nottingham), Diploma Actuarial Techniques (Institute & Faculty of Actuaries)
 

Beyond A level for Mathematics and Further Mathematics

After studying Maths and Further Maths many of our students have chosen to read degrees in Engineering, Physics, Economics, Accountancy and Pure Mathematics. Our students have gone on to study both in London at the prestigious University College London and Imperial College London and further afield with many students being placed at Warwick, Nottingham and Cambridge.

Any other information

Students intending to study Mathematics at AS and A2 level should ideally possess A*, A or B at GCSE. However, it is possible for a student with a grade C at intermediate level to be successful in their study of A level Mathematics.

Mathematics at AS and A2 level is both challenging and demanding. It is fundamentally different from the study of Mathematics at GCSE requiring strong skills in algebraic manipulation and a disciplined approach to study. Success is achieved by students who adopt a consistent work pattern. Students who have a tendency to ‘cram’ at the last minute may find this strategy unsuccessful in this subject.

Textbooks

(Books are provided by the college)

AS levelA2 levelOptional modules
Title – Edexcel AS and A Level Modular Mathematics: Core Mathematics 1
Author – Keith Pledger and Mr Dave Wilkins
Weblink – http://www.amazon.co.uk/Edexcel-AS-Level-Modular-Mathematics/dp/0435519107/ref=cm_cr_pr_product_top

Title – Edexcel AS and A Level Modular Mathematics: Core Mathematics 2
Author – Keith Pledger and Mr Dave Wilkins
Weblink – http://www.amazon.co.uk/Edexcel-AS-Level-Modular-Mathematics/dp/0435519115/ref=sr_1_1?ie=UTF8&s=books&qid=1275642513&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Core Mathematics 3
Author – Keith Pledger
Weblink – http://www.amazon.co.uk/Edexcel-AS-Level-Modular-Mathematics/dp/0435519093/ref=sr_1_1?ie=UTF8&s=books&qid=1275642552&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Core Mathematics 4
Author – Keith Pledger
Weblink – http://www.amazon.co.uk/Edexcel-AS-Level-Modular-Mathematics/dp/0435519077/ref=sr_1_1?ie=UTF8&s=books&qid=1275642640&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Mechanics 1
Author – Ms Susan Hooker et al.
Weblink – http://www.amazon.co.uk/Edexcel-Level-Modular-Mathematics-Mechanics/dp/0435519166/ref=sr_1_1?ie=UTF8&s=books&qid=1275644435&sr=8-1

Title – Edexcel AS and A Level Modular Mathematics: Mechanics 2
Author – Keith Pledger
Weblink – http://www.amazon.co.uk/Edexcel-Level-Modular-Mathematics-Mechanics/dp/0435519174/ref=sr_1_1?ie=UTF8&s=books&qid=1275644518&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Statistics 1
Author – Keith Pledger et al.
Web link – http://www.amazon.co.uk/Edexcel-Level-Modular-Mathematics-Statistics/dp/0435519123/ref=sr_1_1?ie=UTF8&s=books&qid=1275644563&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Statistics 2
Author – Keith Pledger et al.
Weblink – http://www.amazon.co.uk/Edexcel-Level-Modular-Mathematics-Statistics/dp/0435519131/ref=sr_1_1?ie=UTF8&s=books&qid=1275644613&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Decision Mathematics 1
Author – Susie Jameson
Weblink – http://www.amazon.co.uk/Edexcel-AS-Level-Modular-Mathematics/dp/0435519190/ref=sr_1_1?ie=UTF8&s=books&qid=1275644705&sr=1-1

Title – Edexcel AS and A Level Modular Mathematics: Decision Mathematics 2
Author – Susie Jameson
Weblink – http://www.amazon.co.uk/Edexcel-Level-Modular-Mathematics-Decision/dp/0435519204/ref=sr_1_1?ie=UTF8&s=books&qid=1275644757&sr=1-1

A level Maths A2/AS scheme of work