# Maths and Further Maths

At Imperial College London Mathematics School you will study A Level Maths and Further Maths together. They will not be treated as separate subjects, but the total teaching hours will equate to that of two full A Levels. Treating these two A Levels as a whole means we are able to map a coherent journey through mathematics, deepening your understanding of the subject and helping you connect different areas of mathematics. However, at the end of Year 13 you will sit two sets of examinations – one in A Level Mathematics and one in A Level Further Mathematics – which assess the different areas of mathematics discussed below.

Mathematics

• How can we find the area between the graph of y = sin x and the x-axis?
• Why does 0. really equal 1?
• If I roll a dice 1000 times, and obtain 186 sixes, is it reasonable to conclude that the dice is biased?
• If I throw a ball off the top of a building, can I predict where it will land?
These are the types of questions we try to answer in A Level Mathematics. Two-thirds of the course is pure mathematics. This builds on your understanding of algebra, graphs, functions, sequences, trigonometry and vectors that you have acquired in GCSE mathematics, and introduces concepts such as logarithms, calculus (differentiation and integration) and formal methods of proof.

The remaining third of the course consists of mechanics and statistics. In mechanics, we investigate how Newton’s Laws of Motion allow us to make predictions about the behaviour of objects such as falling objects, connected objects such as pulleys or trailers attached to vehicles, sliding objects or ladders propped against walls, and how calculus can be used in these predictions.

In statistics, we build on your understanding of probability, and discuss how we can use data from a sample to make inferences about a population through hypothesis testing.

Further Maths

• Why does eπi + 1 = 0? What do e and i even mean?
• What does the equation 4x5 – 3 = 0 have to do with a pentagon?
• How can we use a matrix to rotate objects in 3D space? What is a matrix?
• How can I draw the optimum line of a best fit on a scatter graph?
The pure maths component of A Level Further Maths focuses on topics such as matrices and complex numbers which you may not have met before. We look at what complex numbers are, how they fit into the numbers we already know about, why they are useful, and how they are related to trigonometry. We define a different coordinate system called Polar Coordinates. We learn what matrices are and how they are related to vectors and transformations in 2D and 3D space. We also expand our work on calculus and introduce a new set of functions called the Hyperbolic Functions.

All students will study some further statistics and mechanics. In statistics we examine discrete probability distributions such as the Poisson distribution, and we analyse how well different distributions fit on to real events. We discuss how best to measure the strength of a correlation on a scatter graph, and whether we can infer if two variables are related.

In mechanics we introduce concepts such as work, energy, power, impulse and momentum to predict the motion of objects in various scenarios. We consider the impact of frictional forces, or the size and shape of the object, on the motion of these objects.

Students may then continue to study additional pure mathematics, mechanics or statistics, or perhaps study a unit on discrete mathematics and algorithms, or the use of technology with mathematics.