GCSE Maths → A-level Maths

You can already

Solve and factorise quadratics; complete the square; rearrange formulae.

Now you'll

Complete the square as a tool (turning points, the discriminant condition $b^2 - 4ac$ for the number of roots), algebraic division and the factor theorem for cubics, and partial fractions.

You can already

Use $f(x)$ notation and read values off graphs.

Now you'll

Domain and range, composite functions $fg(x)$, inverse functions $f^{-1}(x)$, the modulus function $|f(x)|$, and combining graph transformations.

You can already

Straight lines $y = mx + c$, and the equation of a circle centred on the origin with the tangent at a point.

Now you'll

The general circle $(x-a)^2 + (y-b)^2 = r^2$, normals as well as tangents, and parametric equations of curves.

You can already

SOHCAHTOA, the sine and cosine rules, exact values in degrees.

Now you'll

Radians, the identities $\sin^2\theta + \cos^2\theta \equiv 1$ and $\tan\theta \equiv \frac{\sin\theta}{\cos\theta}$, solving trig equations over a range, the graphs of all the functions, and small-angle approximations.

You can already

Find the $n$th term of a linear (arithmetic) sequence.

Now you'll

Sigma notation, sums of arithmetic and geometric series, sum to infinity, recurrence relations, and the binomial expansion of $(a + bx)^n$.

You can already

Laws of indices; simplify surds.

Now you'll

The exponential function $e^x$ and natural logarithm $\ln x$, the laws of logs, solving $a^x = b$, and exponential growth/decay models.

You can already

Estimate a gradient by drawing a tangent, and an area under a graph by counting squares (the informal GCSE precursors).

Now you'll

Formal differentiation — rates of change, tangents, stationary points and optimisation — and integration — the reverse process and the area under a curve.

You can already

Algebraic proof and disproof by counterexample.

Now you'll

Named, structured proof — writing deductive proofs formally, proof by exhaustion, and (later in the course) proof by contradiction.

You can already

Averages, box plots, scatter graphs, and probability with tree and Venn diagrams (including conditional probability).

Now you'll

Work with a large pre-released data set, the binomial and normal distributions, and hypothesis testing.

You can already

Speed = distance ÷ time; simple vectors.

Now you'll

The kinematics (SUVAT) equations, $F = ma$ and Newton's laws, and forces on a particle — a whole applied strand.