Further Mathematics Year 13 course 2

This course by Imperial College London is designed to help you develop the skills you need to succeed in your A-level further maths exams.

You will investigate key topic areas to gain a deeper understanding of the skills and techniques that you can apply throughout your A-level study. These skills include:

* Fluency – selecting and applying correct methods to answer with speed and efficiency

* Confidence – critically assessing mathematical methods and investigating ways to apply them

* Problem solving – analysing the ‘unfamiliar’ and identifying which skills and techniques you require to answer questions

* Constructing mathematical argument – using mathematical tools such as diagrams, graphs, logical deduction, mathematical symbols, mathematical language, construct mathematical argument and present precisely to others

* Deep reasoning – analysing and critiquing mathematical techniques, arguments, formulae and proofs to comprehend how they can be applied

Over eight modules, you will be introduced to

* Simple harmonic motion and damped oscillations.

* Impulse and momentum.

* The work done by a constant and a variable force, kinetic and potential energy (both gravitational and elastic) conservation of energy, the work-energy principle, conservative and dissipative forces, power.

* Oblique impact for elastic and inelastic collision in two dimensions.

* The Poisson distribution, its properties, approximation to a binomial distribution and hypothesis testing.

* The distribution of sample means and the central limit theorem.

* Chi-squared tests, contingency tables, fitting a theoretical distribution and goodness of fit.

* Type I and type II errors in statistical tests.

Your initial skillset will be extended to give a clear understanding of how background knowledge underpins the A -level further mathematics course. You’ll also be encouraged to consider how what you know fits into the wider mathematical world.

What you'll learn

• How to derive and solve a second order differential equation that models simple harmonic motion.
• How to derive a second order differential equation for damped oscillations.
• The meaning of underdamping, critical damping and overdamping.
• How to solve coupled differential equations.
• How to calculate the impulse of one object on another in a collision.
• How to use the principle of conservation of momentum to model collisions in one dimension.
• How to use Newton’s experimental law to model inelastic collisions in one dimension.
• How to calculate the work done by a force and the work done against a resistive force.
• How to calculate gravitational potential energy and kinetic energy.
• How to calculate elastic potential energy.
• How to solve problems in which energy is conserved.
• How to solve problems in which some energy is lost through work against a dissipative force.
• How to calculate power and solve problems involving power.
• How to model elastic collision between bodies in two dimensions.
• How to model inelastic collision between two bodies in two dimensions.
• How to calculate the energy lost in a collision.
• How to calculate probability for a Poisson distribution.
• How to use the properties of a Poisson distribution.
• How to use a Poisson distribution to model a binomial distribution.
• How to use a hypothesis test to test for the mean of a Poisson distribution.
• How to estimate a population mean from sample data.
• How to estimating population variance using the sample variance. How to calculate and interpret the standard error of the mean.
• How and when to apply the Central Limit Theorem to the distribution of sample means.
• How to use the Central Limit Theorem in probability calculations, using a continuity correction where appropriate.
• How to apply the Central Limit Theorem to the sum of n identically distributed independent random variables.
• How to conduct a chi-squared test with the appropriate number of degrees of freedom to test for independence in a contingency table and interpret the results of such a test.
• How to fit a theoretical distribution, as prescribed by a given hypothesis involving a given ratio, proportion or discrete uniform distribution, to given data.
• How to use a chi-squared test with the appropriate number of degrees of freedom to carry out a goodness of fit test.
• How to calculate the probability of making a Type I error from tests based on a Poisson or Binomial distribution.
• How to calculate probability of making Type I error from tests based on a normal distribution.
• How to calculate P(Type II error) and power for a hypothesis test for tests based on a normal, Binomial or a Poisson distribution (or any other A level distribution).