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Pure
Proof
Deduction, exhaustion and counterexample · Proof by contradiction2 subtopic lessonsAlgebra and functions
Includes quadratic functions and completing the squareCompleting the square and the vertex · Modulus graphs and equations · Solving quadratic equations · Modulus inequalities · The discriminant and the nature of roots · Polynomials: division, factor and remainder theorems · Partial fractions · Linear and quadratic inequalities · Simultaneous equations and lines meeting curves9 subtopic lessonsCoordinate geometry
Gradient, length, midpoint and equations of lines · Parallel and perpendicular lines · The equation of a circle · Tangents, chords and intersections4 subtopic lessonsSequences and series
Binomial expansion · Arithmetic progressions · Geometric progressions · Sum to infinity · Binomial expansion for any rational n5 subtopic lessonsTrigonometry
Radians · Exact values and the unit circle · Reciprocal functions and their identities · Arc length and sector area · Compound-angle formulae · Graphs and transformations · Inverse trigonometric functions · Double-angle formulae · Segments, chords and composite shapes · Trigonometric identities · The R-form (harmonic form) · Solving trigonometric equations12 subtopic lessonsExponentials and logarithms
Laws of logarithms · The functions eˣ and ln x · Exponential equations and inequalities · Logarithmic equations · Exponential models and linear forms5 subtopic lessonsDifferentiation
Derivatives of eˣ, ln x and trigonometric functions · Gradients and the power rule · The chain rule · The product and quotient rules · Tangents and normals · Implicit differentiation · Stationary points and increasing functions · Parametric differentiation · Stationary points and tangents · Optimisation and rates of change10 subtopic lessonsIntegration
Integrating eˣ, 1/x and trigonometric functions · Indefinite integration · Integration using trigonometric identities · Integrating (ax + b)ⁿ · Definite integrals · Exact definite integrals and logarithms · Areas under and between curves · Integration by substitution · Integration by parts · Forming differential equations · Separating the variables · Particular solutions with harder integrals · Modelling with differential equations13 subtopic lessonsNumerical methods
Locating roots: sketches and sign changes · Fixed-point iteration · Rearrangements, staircases and cobwebs · The trapezium rule · Newton–Raphson, bisection and interpolation5 subtopic lessonsVectors
2D and 3D vectors; vector equations of lines are in Further MathematicsPosition vectors, magnitude and unit vectors1 subtopic lessons