A-level
Guides for Years 12 and 13 — core topics explained with interactive examples.
Integration: The Three Main Tricks
Substitution, integration by parts, and changing the form — the three techniques that unlock the vast majority of integrals. Includes partial fractions, trig identities, and De Moivre's theorem for Further Maths.
Read →The Power of Checking
Mathematics is almost unique among school subjects in letting you verify your own answers. A systematic guide to twelve checking techniques — from back-substitution to dimensional analysis to recovering simpler cases.
Read →Pythagorean Triples
Discover the infinite family of Pythagorean triples and how a 4,000-year-old Babylonian tablet fits into modern matrix algebra. Includes a live Python code editor.
Read →Improper Integrals
When a limit of integration is infinite or the integrand blows up, replace the problem point with a parameter and take a limit. Convergence, divergence, the p-integral, and where L'Hôpital's rule appears.
Read →The Handshaking Lemma and Graph Traversability
The sum of all vertex degrees equals twice the number of edges — and this one fact immediately tells you whether a graph can be traversed without repeating an edge, and whether the journey must start and end at the same point.
Read →1D and 2D Collisions: Avoiding Double-Counted Signs
The most common error in collision problems is mixing signed velocities with unsigned speeds. Use the velocity form of Newton's Experimental Law throughout and the algebra handles the signs automatically.
Read →Centres of Mass: Integrals as Limits of Sums
The integral formula for centre of mass is the natural limit of a discrete weighted average. Laminae, regions between curves, and solids of revolution — all follow from the same idea.
Read →Statistics: All Roads Lead to Hypothesis Testing
Binomial, normal, t, Poisson, chi-squared and more — every test asks the same question. See how eleven distributions and tests are unified by one central idea.
Read →Generating Functions are Maclaurin Series
Polynomials, binomial expansions, and standard series (eˣ, sin x, cos x, ln) are all the same thing. One formula — differentiate, evaluate at zero, divide by n! — derives them all.
Read →L’Hôpital’s Rule: a rule for determining limits
The rule is formally examined only in optional Further Pure, yet the limits it resolves appear throughout A-level Maths and compulsory Further Pure. Includes the rewriting trick for products and recursive application.
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