# A Level Mathematics Course

Mathematics is the largest and one of the most successful faculties at Ashbourne. Around half of our students study Maths in one form or another and 55% achieve A*A grades on average.

Mathematics is the largest and one of the most successful faculties at Ashbourne. Around half of our students study Maths in one form or another and 55% achieve A*A grades on average.

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.

Ashbourne’s ambitious A level Maths and Further Maths students are strong contenders for the annual senior UKMT Maths Challenge, having won gold certificates in the past for individual and group entries. The college runs a Maths club to help students train for this extremely tough national competition which pushes students far beyond the scope of the A level syllabus.

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.

Which syllabus do we follow?

Ashbourne follows the Edexcel specification for Mathematics AS level and A level.

What is covered in the course?

The syllabus comprises three important elements of mathematics: pure mathematics, applied mathematics (mechanics) and statistics. Very briefly pure mathematics at A level introduces and explores calculus, arguably the most powerful tool of modern mathematics. Mechanics develops an understanding of Newton’s laws (how many?) and, more interestingly, of mathematical models used in analysing the forces of nature. Statistics develops a deeper understanding of probability.

**Pure Mathematics**

This begins with a very interesting topic – mathematical proof – and so introduces students to the basis of logic, philosophy and for believing that any mathematical statement is true or false. Although not on the A level syllabus, a classic example is proving that there are an infinite number of prime numbers. Students extend their knowledge of algebra, inequalities, functions and the graphs of the fundamental functions of straight lines and parabolas. This includes learning how to quickly write down the terms of such expressions as: (1 + 2x)^{12}. You will extend your understanding of sine, cosine and tangent so far that you will not recognise them and, as a bonus, get to know their graphs as well. You will learn about a new type of equation: a trigonometric equation of course. Trigonometry forms the basis for much of our understanding of the world, including MRI scanners at hospitals (x-rays too), sound boards in recording studios (equalisers) and starlight. Spinach is good for you and so is trigonometry.

Logarithms, natural logarithms and the mysterious number ‘e = 2.718626…’ are next along with exponential functions. They vie with trigonometry as the most hated topic in maths but, as you might guess, they are extremely important. 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.

We finally arrive at calculus and learn about differentiation and integration which are related by the fundamental theorem of calculus. Despite its grand introduction earlier on, this is perhaps the easiest topic for this part of the course!

We complete this section with a review and extension of students’ understanding of vectors. This is fun and gives you an imaginative new way of looking at geometry.

**Statistics**

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.

We examine the analysis of data and will help you answer the following questions: if a billionaire walks into a pub, what happens to the mean (average) income of those present? 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 stock markets. In fact they will even sell you volatility if you have the appetite! This section also deals with the very important area of correlation and regression. Correlation is used a lot in the social sciences such as sociology, psychology or economics. For example life expectancy is positively correlated to average income (both go up at the same time) but negatively correlated to birth rates (go down as income goes up). You might like to think about whether correlation between things means the one thing causes the other.

We review and advance your understanding of probability and in particular extend your facility with tree diagrams by teaching you binomial probability; for example how to calculate the probability of 17 heads in 20 tosses of a coin. This may sound boring but at least you will get to apply your knowledge of (1 + 2x)^{12} from the pure maths part of this course. But more importantly you will have a basis for deciding whether the coin is fair or not (probably not). This is called hypothesis testing and forms the basis of the scientific method for medicine, psychology and economics.

**Mechanics**

Once you ‘get it’ this section becomes a bit easy and mechanical. However, it is fair to say that students find mechanics very difficult; and the reason they find it difficult is that they don’t understand what forces act on everyday objects (say a book on a table or, on yourself). Our job is to make mechanics a breeze (we can do it).

Principally mechanics is about the mathematical model used in analysing the forces of nature and describing how objects move through space and time. Understanding why it is wrong to think that bodies require a force to keep them moving in a straight line. This leads your understanding from ancient Greece, to renaissance Italy (Gallileo), enlightenment Europe (Newton) and Einstein.

**Pure Mathematics**

You will learn a little bit more of everything you covered in your first year. Geometry will include circles and parametric equations used to describe them. You will extend your understanding of the binomial theorem to include fractional and negative indices; eg (1 + 2x)^{-1/2}. Of course there is more trigonometry, equations and identities.

Work on series continues with the study of both arithmetic and geometric series. This is very 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.

You will learn a different approach to functions and how to transform the graphs of related functions, each to the other. You will learn many more interesting and challenging methods of differentiation and integration and so be able to solve many types of differential equations, the foundation of the physical sciences. You will carry forward your study of vectors and learn a few really clever things you can do with them.

**Statistics**

What is important in this module is perhaps best illustrated by an example. Suppose one person in 1000 actually has a particular disease. A pharmaceutical company develops a test which is 100% effective in identifying the 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 is a question in conditional 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 which will help you make loads of money in the City should you choose. It is also an example of discrete probability and you will use your understanding of calculus to understand continuous probability. A good example of continuous probability is the normal or bell curve which, for example, tells you about how ‘intelligence’ is distributed among any population as well as the distribution of marks of students in the UK on any maths exam.

The most important part of this module for A level Mathematics is the study of hypothesis testing and confidence intervals which are fundamental for psychologists, economists and medicine. Used properly these tests are extremely powerful and have, with near perfection, predicted the results of national elections.

**Mechanics**

You will extend your understanding of mechanics by applying your knowledge of vectors and calculus. In particular you will be able to cope with simple problems of bodies moving in two dimensions as well as problems involving variable forces and introduce you an application of differential equations. In dealing with forces that act in 2 dimensions, you will learn two important things: first how to use trigonometry to analyse the effect of a force in two different directions and second through this understanding, come to know the important physical principle of superposition. In resolving forces you will also find another application of trigonometry.

Who teaches this course?

MMaths integrated masters (Bath University); PGCE Secondary Maths (Oxford University)

Richard has experience teaching Maths from pre-GCSE level all the way up to first year undergraduate. He has taught in both state and private sector schools in the UK and Australia. Richard also worked as a software developer for a company providing services for major UK retailers. He currently teaches GCSE and A level Maths at Ashbourne.

BA (Hons) Mathematics (Warwick University); PGCE Secondary Mathematics (University of East London)

Phil has a first class degree in Maths with experience teaching up to A level.

BSc Pure and Applied Mathematics (Sheffield University); PGCE (Sheffield University); NPQH (Lancaster University)

Luke has more than 30 years’ experience in education as a teacher, teacher trainer, head of department and vice and acting principal. He brings a wealth of experience and expertise to help his students achieve the best results.

BSc Physics, Pure Mathematics and Statistics (University of Jaffna, Sri Lanka); MSc Mathematics (The Open University); PGCE (University of Greenwich); BEng (Hons) Electronics and Communication (University of North London)

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.

MA Civil Engineering with Business Management (Warwick University)

Sean plays a very active role at Ashbourne both in terms of helping students reach their academic potential and in enjoying their experience at the college. He began his career as an investment banker but quickly realised that teaching was his preferred profession.

BSc Physics (Imperial College, London)

Imran is an experienced Maths and Physics tutor who has been teaching at independent colleges in London since 1998.

Textbooks

Edexcel AS and A Level Pure Mathematics 1

Greg Attwood, Jack Barraclough *et al.*

Edexcel AS and A level Mathematics Statistics and Mechanics 1

Greg Attwood, Ian Bettison *et al.*

Edexcel A level Mathematics Pure Mathematics 2

Greg Attwood, Jack Barraclough *et al.*

Edexcel A level Mathematics Statistics and Mechanics 2

Harry Smith

Reading and resources

** Number, the Language of Science** by Tobias Danzig

This is easy-to-read classic was recommended by Albert Einstein.

This great book deals with the application of mathematics, in particular probability, to economics, risk and investment.

For students beginning their study of mathematics this is a wonderful introduction to one of the most important and controversial topics.

This is a fascinating story of a self-taught Indian mathematician whose genius was discovered and developed by GH Hardy, one of the best mathematicians of the last century.

This is a very brief autobiography of one of the best mathematicians of the last century.

Mr Wheeler was one of the most brilliant physicists of the last century. This book does not require a background of complex mathematics but

does require a determined imagination.

Any books by

Why unsolved mathematical conundrums matter.

Sweeping history of mathematical ideas and keys thinkers.

Exploration into the nature of links between formal systems that underlies all mental activity.