A level Mathematics course

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. Leaving the most feared to last, 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 (sometimes associated with madness). To be fair, 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.

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 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 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. For example, between the ages of zero to puberty, height is positively correlated to age; the older you are the taller you are. Does this mean that age causes height? If you have a good teacher (all at Ashbourne) they will help you understand 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?

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.

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, sitting on your sofa watching the X factor). 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. 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 Aristotle’s idea.

You will learn a little bit more of everything you covered in your first year. Geometry will include circles and parametric equations. 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.

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 all recent elections in the United States.

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 differential equations. In resolving forces you will also find another application of trigonometry.

Number, the Language of Science by Tobias Danzig
This is easy-to-read classic was recommended by Albert Einstein.

Against the Gods: The Remarkable story of Risk by Peter Bernstein
This great book deals with the application of mathematics, in particular probability, to economics, risk and investment.

A Brief History of Infinity by Brian Clegg
For students beginning their study of mathematics this is a wonderful introduction to one of the most important and controversial topics.

The Man Who Knew Infinity: The Life and Genius of Ramanujan by Robert Knigel
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.

A Mathematician’s Apology by GH Hardy
This is a very brief autobiography of one of the best mathematicians of the last century.

Spacetime Physics: An Introduction to Special Relativity by John Archibald Wheeler
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 John Barrow, eg The Book of Nothing. This is a great book about the relationship between zero and infinity.

Title – Edexcel AS and A Level Pure Mathematics 1
Authors – Greg Attwood, Jack Barraclough et al.

Title – Edexcel AS and A level Mathematics Statistics and Mechanics 1
Authors – Greg Attwood, Ian Bettison et al.

Title – Edexcel A level Mathematics Pure Mathematics 2
Authors – Greg Attwood, Jack Barraclough et al.

Title – Edexcel A level Mathematics Statistics and Mechanics 2
Author – Harry Smith