Cambridge A-Level Further Mathematics Tuition Online (9231)
Cambridge 9231 Further Pure, Further Mechanics and Further Probability & Statistics, for students who find Mathematics comfortable and want more.
Founder & Head of School, Afterkelas
Afterkelas teaches Cambridge International AS & A Level Further Mathematics (9231) online to pre-university students in Malaysia. Tutors teach live in English, 1-to-1 or in small groups, across both Further Pure papers, Further Mechanics and Further Probability & Statistics.
What Cambridge 9231 covers
The topics of each 9231 paper. Further Pure Mathematics 1 underpins the rest, so it is usually taught first.
Further Pure Mathematics 1
7 topics- Roots of polynomial equations
- Rational functions and graphs
- Summation of series
- Matrices
- Polar coordinates
- Vectors
- Proof by induction
Further Pure Mathematics 2
6 topics- Hyperbolic functions
- Matrices
- Differentiation
- Integration
- Complex numbers
- Differential equations
Further Mechanics
6 topics- Motion of a projectile
- Equilibrium of a rigid body
- Circular motion
- Hooke's law
- Linear motion under a variable force
- Momentum
Further Probability & Statistics
5 topics- Continuous random variables
- Inference using normal and t-distributions
- Chi-squared tests
- Non-parametric tests
- Probability generating functions
Who takes Further Mathematics
Further Mathematics suits students who find 9709 Mathematics comfortable and are heading for mathematics, physics, engineering, economics or computer science at university. Some university courses ask for it or prefer it, and students who take it arrive at university having already met matrices, complex numbers and differential equations.
It is taken alongside Mathematics, not instead of it, and the two compete for time. Tutors plan around the school's order, so a Further Pure topic is never taught before the Mathematics it builds on.
Proof and algebra under pressure
Further Pure papers reward rigour. Proof by induction, for example, earns its marks only when the base case, the assumption, the inductive step and the conclusion are all written out; a correct idea with a missing conclusion loses marks. Tutors teach the structure first and then practise it until it is automatic.
The questions are long and chain topics together, so an early algebra slip can cost the rest of an answer. Students practise checking intermediate results, the sum of roots, a determinant, a derivative, before building on them.
Further Mechanics or Further Probability & Statistics
For the AS, students take Further Pure Mathematics 1 with either Further Mechanics or Further Probability & Statistics; the full A Level takes both. The choice usually follows the applied paper the student is strongest in from Mathematics, since Further Mechanics builds on Mechanics and Further Probability & Statistics on the statistics papers.
A question, worked through
A Further Pure Mathematics 1 proof by induction. The algebra is short; the marks go to writing every stage of the argument.
Question
Prove by induction that, for all positive integers n, 1 × 2 + 2 × 3 + … + n(n + 1) = n(n + 1)(n + 2) ÷ 3.
- Base case, n = 1: the left side is 1 × 2 = 2, and the right side is 1 × 2 × 3 ÷ 3 = 2. The result holds for n = 1.
- Assume the result holds for n = k: the sum up to k(k + 1) is k(k + 1)(k + 2) ÷ 3.
- Add the next term, (k + 1)(k + 2): k(k + 1)(k + 2) ÷ 3 + (k + 1)(k + 2) = (k + 1)(k + 2)(k ÷ 3 + 1) = (k + 1)(k + 2)(k + 3) ÷ 3, which is the formula with n = k + 1.
- Conclusion: the result is true for n = 1, and if it is true for n = k it is true for n = k + 1, so by induction it is true for all positive integers n.
Answer The formula holds for every positive integer n, by induction.
Frequently asked questions
Is Further Mathematics worth taking?
For students heading to mathematics, physics, engineering or computer science, usually yes: some courses ask for it, and it covers material those degrees start with. For students who find Mathematics a stretch, a stronger grade in Mathematics is often the better use of the time.
Can Further Mathematics be taken as an AS only?
Yes. The AS is Further Pure Mathematics 1 with either Further Mechanics or Further Probability & Statistics, and students sometimes take it alongside a full A Level in Mathematics.
How is it different from Edexcel IAL Further Mathematics?
The mathematics overlaps, but Cambridge sets four papers, while Edexcel builds the qualification from Further Pure units FP1 to FP3 and applied units sat across exam series. Tutors plan for the board on the student's entry.
How much does Cambridge A-Level Further Mathematics tuition cost?
The fee depends on the format (1-to-1 or a small group), how many classes a week and the tutor, so there is no single figure. Send an academic advisor the papers the student is entered for, the exam series and the latest Mathematics result on WhatsApp, and you will get the current packages and a quote, usually within one working day.