Study plan
How do you study Further Maths without a teacher at school?
Backwards from the exam date, in blocks that follow the core. Here is a 30-week outline we use as a starting point, and three starter problems.
A 30-week Further Maths outline for one academic year of the core, ours (adjust to the board's specification)| Weeks | Core block | Checkpoint |
|---|
| 1 to 4 | Proof by induction; roots of polynomials | Induction proofs written with base case, step and conclusion |
| 5 to 9 | Complex numbers: arithmetic, Argand diagrams, modulus and argument | Loci sketched and equations solved |
| 10 to 14 | Matrices: arithmetic, determinants, inverses, transformations | Transformations described from a matrix and back |
| 15 to 18 | Further algebra and functions; series | Method of differences used confidently |
| 19 to 23 | Further calculus and further vectors | Volumes of revolution; lines and planes in three dimensions |
| 24 to 27 | Polar coordinates and hyperbolic functions | Polar areas and hyperbolic identities |
| 28 to 30 | Differential equations, then a mixed review | A full timed core paper from the board |
Our starter problem 1, proof by induction. Prove that 5n + 3 is divisible by 4 for all positive integers n. For n = 1, 5 + 3 = 8. Assume 5k + 3 = 4m. Then 5k+1 + 3 = 5(5k + 3) − 12 = 20m − 12 = 4(5m − 3), a multiple of 4. So the result holds for every positive integer.
Our starter problem 2, roots of polynomials. The roots of x3 − 6x2 + 11x − 6 = 0 are α, β and γ. Without solving, find α2 + β2 + γ2. The sum of the roots is 6 and the sum of products in pairs is 11, so the answer is 62 − 2 × 11 = 14. Checking: the roots are 1, 2 and 3, and 1 + 4 + 9 = 14.
Our starter problem 3, matrices. The matrix with rows (0, 1) and (1, 0) reflects the plane in the line y = x. Squaring it gives the identity matrix, which matches the geometry: reflecting twice returns every point to where it started. Ideas like this, a picture behind each calculation, are what make self-study stick.
The outline and problems were written by us on 31 October 2026. Each board publishes its own specification and past papers; follow the content list of the specification you are entered for.
