Further Mathematics is not that difficult but poses a problem even for those who enjoy maths: you have to study. With AS Maths, once you have confidently mastered a technique, say max/min problems, it is not really necessary to ‘revise’. For A level Maths you must spend a little time memorising trigonometric identities (at least you should) or refreshing your understanding of parametric integration. But in general, provided you practise a lot you will be fine come the exam. For Further Mathematics this process of refreshing is essential.

You will learn more about vectors, transformations and, if you choose to, hyperbolic functions which together provide a foothold on the mountain leading to Einstein’s general theory of relativity. You will also delve more deeply into the theory of complex numbers, culminating with an elegant introduction to complex transformations. Interestingly there are only two compulsory modules, core further mathematics 1 and 2; you have a wealth of choice and can avoid mechanics or statistics should one or the other not appeal.

Generally studying Further Maths is a good idea if you like maths but a nightmare if you don’t.

## Why study Further Mathematics?

If you are applying to mathematically related courses (engineering for example) at one of the top universities you are certain to require Further Maths. Such advice is, perhaps, a bit superfluous because if you are considering such courses you MUST enjoy Mathematics. If you do enjoy Maths why not study it in more depth? At the very least it will give you an insight into the difficulties, challenges and beauty of advanced Mathematics.

## Which syllabus do we follow?

Ashbourne follows the Edexcel specification for Further Mathematics A level.

## What is covered in the course?

Further Mathematics has four components: two compulsory core modules in pure mathematics and two optional modules chosen from further pure mathematics, further mechanics and further statistics. There is a final examination of 1 hour 30 mins for each of the four modules taken.

### Core modules

**mathematical proof**; eg why is 2n+1 always an odd number?

Next comes one of the most exciting ideas, constructs or acts of imagination in mathematics: **complex numbers**. You may recall your introduction to negative numbers many years ago and eventually being persuaded that -2 x -3 = + 6. Now you will be able to solve x^{2}= -1 using the invention, complex numbers. You will learn how to use them to solve equations, sketch beautiful curves and regions in the plane and solve otherwise impossible trigonometric equations and general series.

You will take forward your understanding of another fundamental of modern maths, **matrices**. You will learn how they can be used to transform one vector into another and solve simultaneous linear equations in an interesting and insightful way.

You will extend your understanding of series and will be able to show that sin(x) = x – x^{3} / 3! + x^{5} / 5! – x^{7} / 7! … and so be able to find its value if you happen to land on a desert island without a calculator.

Not surprisingly you will extend your knowledge of calculus, learning more techniques for integration.

You will learn a lot more about geometry. **Vectors** become a lot more engaging when you learn how to find the equation of a plane and line in three dimensions (or more if you wish). You will also learn a different way to label points in the plane called **polar coordinates**. These enable you to define and trace very fascinating curves and strengthen your understanding of complex numbers and **parametric equations**.

You will be introduced to hyperbolic functions. For example, you will recall that the sine of a function has a period of 2π; however, the hyperbolic sine of a function (sinh(x)) has a period of 2πi where i = (-1)^{1/2} the first complex number.

Finally you will extend your understanding of differential equations with a particular emphasis on simple harmonic motion.

### Optional modules

**Option 1**

This module is a bit of a jumble of topics including: a little bit more trigonometry; some theoretical aspects of calculus which will introduce you to the very modern and very important field of mathematical analysis; parametric equations for the fundamental trio of curves, the circle, the ellipse and the hyperbola. You will learn that they are fundamentally the same! (Very elegant and very surprising.) You will learn another way to multiply two vectors and how to use numerical methods to solve equations and inequalities.

**Option 2**

This module tests how sincere is your love of mathematics.

In group theory and isomorphisms you will learn about another elegant and entrancing aspect of modern mathematics. You will encounter a fascinating tool for integration and series, the reduction formula. You will extend your knowledge of matrices and learn about yet another fundamental concept of modern mathematics: eigenvalues and eigenvectors. You will extend your power in the field of complex numbers with more complicated curves and the crucial aspect of transformations using complex numbers. You will learn the elegant and fundamental proof by induction; eg 4^{2} -1 is divisible by 7 so 4^{2n} -1 is divisible by 7 for all integers n. Finally you will commence your study of number theory, the queen of the science of mathematics.

**Option 1**

You will learn about the theory which validates all statistical results: the central limit theorem. This says that the larger the sample from a population the more it behaves like a normal (bell) curve (as found in the distribution of exam results for example). Therefore this is the foundation of hypothesis testing; for example how can you use statistical mathematics to decide if a new drug is effective? This leads to a discussion of type 1 and type 2 errors in your testing: type 1 you reject a perfectly good drug and type 2 you accept a drug which is perfectly useless.

This highlights the basis of this module in probability theory and so should appeal to anyone who enjoys pure mathematics. You will extend your knowledge of probability through the study of the Poisson, geometric and negative binomial distributions which are all related to the simple binomial case of tossing coins. You will develop your facility with Σ notation, expected values and probability generating functions.

**Option 2**

In studying continuous random variables, probability density functions and distribution functions, you will have the pleasure of applying your knowledge of calculus.

You will also study correlation and regression which will equip you to understand much of modern economics, finance and the social sciences. There is more on hypothesis testing and you will learn about the closely related topic of confidence intervals. Finally you will be introduced to the very important topic of analysis of variance.

**Option 1**

You will study two of nature’s most important laws: conservation of energy and conservation of momentum. Students will deepen their understanding of gravitational potential energy and learn about elastic potential energy. Momentum and the related idea of impulse are studied in some depth, including the oblique collision of objects and coefficient of restitution.

**Option 2**

You will deepen your understanding of the relevance of vectors to mechanics, in particular the topic of circular motion. The important idea of centres of mass is introduced and you will be able to apply calculus to quite general areas and volumes to find their centres. Through investigating toppling and sliding you will be introduced to the idea of stability. Finally you will study differential equations of higher order derivatives which have a particular relevance to simple harmonic motion.

## 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 many 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 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)

Seema has been teaching Maths for more than 25 years in the UK, Netherlands, Canada and Mexico and has also been an A level examiner for more than 15 years. She joined Ashbourne in September 2016.

### Madeeha Saad

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

Madeeha has been teaching Maths for more than 17 years in London and has been an assistant examiner for Edexcel.

### Imran Shah

BSc Physics (Imperial College, London)

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

Authors – Greg Attwood, Ian Bettison

*et al*

Title – Edexcel A level Further Mathematics Core Pure Mathematics 2

Title – Edexcel AS and A level Further Mathematics Further Pure Mathematics 1 (FP1)

Authors – Greg Attwood, Tom Begley

*et al*

## Scheme of work

## Any other information

Students intending to study Further Mathematics at AS or A level should possess at least a B grade at A level. Because of the structure of the course at Ashbourne, further maths students must complete the entire A level mathematics syllabus in their first year and so the entire further mathematics syllabus in year 2. For international students with the appropriate background, it is possible to cover both mathematics and further mathematics in one year.

## Beyond A level Further Mathematics

After studying 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.