A-level maths, animated.

Written explainers for all 64 core topics, organised by the maths rather than by any exam board.

Further Maths

The Further Mathematics core and its option papers, all 103 topics written.

Every topic

PROOF

• The structure of proof: deduction and exhaustion
• Disproof and proof by contradiction

ALGEBRA AND FUNCTIONS

• Indices and surds
• Quadratic functions
• Simultaneous equations and inequalities
• Polynomials and the factor theorem
• Graphs, proportion and transformations
• Functions, inverses and the modulus
• Partial fractions
• Functions in modelling

COORDINATE GEOMETRY

• Straight lines
• Circles
• Parametric equations

SEQUENCES AND SERIES

• The binomial expansion
• The general binomial expansion
• Sequences and sigma notation
• Arithmetic series
• Geometric series

TRIGONOMETRY

• Triangles and the sine and cosine rules
• Trigonometric graphs and equations
• Radians, arcs and small angles
• Reciprocal and inverse trigonometric functions
• Compound angles and the harmonic form
• Trigonometric modelling

EXPONENTIALS AND LOGARITHMS

• Exponential functions and e
• Logarithms and their laws
• Log graphs and exponential models

DIFFERENTIATION

• The derivative from first principles
• Differentiating powers of x
• Tangents, turning points and curve behaviour
• Differentiating trig, exponentials and logs
• The product, quotient and chain rules
• Implicit and parametric differentiation
• Rates of change and building differential equations

INTEGRATION

• Integration as antidifferentiation
• Definite integrals and areas
• Integrating standard functions
• Integration by substitution and by parts
• Integrating rational functions
• Areas, parametric curves and the limit of a sum
• Solving differential equations

NUMERICAL METHODS

• Locating roots and iteration
• Newton-Raphson and the trapezium rule

VECTORS

• Vectors in two dimensions
• Vectors in three dimensions

STATISTICS

• Sampling and the large data set
• Measures of location and spread
• Representing and interpreting data
• Correlation and regression
• Probability and Venn diagrams
• Conditional probability
• The binomial distribution
• The normal distribution
• Hypothesis testing with the binomial
• Hypothesis testing: correlation and the normal

MECHANICS

• Modelling, quantities and units
• Kinematics with constant acceleration
• Kinematics with variable acceleration
• Forces and Newton's laws
• Connected particles and pulleys
• Projectiles
• Friction and inclined planes
• Statics of a particle
• Moments

FURTHER PROOF

• Proof by induction: sums and series
• Induction: divisibility and matrices

COMPLEX NUMBERS

• Complex arithmetic and the Argand diagram
• Modulus, argument and loci
• De Moivre's theorem and trigonometric identities
• Roots of unity and complex roots

MATRICES

• Matrix algebra and transformations
• Determinants and inverses
• Systems of equations and invariance

FURTHER ALGEBRA AND SERIES

• Roots of polynomials
• Summing series and the method of differences
• Maclaurin series

FURTHER VECTORS

• Lines and planes in three dimensions
• Intersections, angles and distances

FURTHER CALCULUS

• Volumes of revolution
• Mean values and improper integrals
• Calculus with inverse trigonometric functions

HYPERBOLIC FUNCTIONS

• Hyperbolic functions and identities
• Calculus with hyperbolic functions

POLAR COORDINATES

• Polar curves
• Areas with polar coordinates

DIFFERENTIAL EQUATIONS

• First order equations and integrating factors
• Second order equations
• Modelling with differential equations

FURTHER PURE 1

• The t-formulae
• Taylor series
• Limits and L'Hospital's rule
• Leibnitz's theorem and the Weierstrass substitution
• Series solutions of differential equations
• Conic sections
• Tangents, normals and loci of conics
• The vector product and the scalar triple product
• Numerical methods for differential equations
• Inequalities and inequations

FURTHER STATISTICS 1

• Discrete random variables and expectation
• The Poisson distribution
• Geometric and negative binomial distributions
• Hypothesis tests for Poisson and geometric models
• The Central Limit Theorem
• Goodness-of-fit tests
• Contingency tables
• Probability generating functions
• The quality of tests

FURTHER PURE 2

• Groups and their axioms
• Subgroups, Lagrange's theorem and isomorphism
• Reduction formulae
• Arc length and surface area
• Eigenvalues and eigenvectors
• Diagonalisation and the Cayley-Hamilton theorem
• Further loci and regions in the Argand diagram
• Transformations of the complex plane
• The Euclidean algorithm and Bezout's identity
• Modular arithmetic and Fermat's little theorem
• Combinatorics
• Recurrence relations

FURTHER STATISTICS 2

• Least squares regression and residuals
• Continuous random variables: density and distribution functions
• Mean, variance and skewness of continuous variables
• The continuous uniform distribution
• Correlation coefficients: product moment and Spearman
• Testing a correlation coefficient
• Combinations of normal random variables
• Estimators, standard error and confidence intervals
• Comparing two normal means
• Testing variances: chi-squared and the F-distribution
• Confidence intervals and tests with the t-distribution

FURTHER MECHANICS 1

• Momentum and impulse
• Impulse and momentum as vectors
• Work, energy and power
• Hooke's law and elastic strings
• Elastic potential energy
• Direct impact and Newton's law of restitution
• Successive impacts and impacts with a wall
• Oblique impact and impact with a smooth surface

FURTHER MECHANICS 2

• Angular speed and horizontal circular motion
• Motion in a vertical circle
• Centre of mass of a discrete distribution
• Centres of mass of plane figures and frameworks
• Centres of mass by integration
• Equilibrium, suspension, toppling and sliding
• Newton's laws with a variable force
• Simple harmonic motion
• Oscillations on strings and springs
• Further kinematics: acceleration as a function of x, t or v

DECISION MATHEMATICS 1

• Algorithms, sorting and bin packing
• Graphs: order, Eulerian paths and planarity
• Minimum spanning trees: Prim and Kruskal
• Shortest paths: Dijkstra and Floyd
• Route inspection
• The travelling salesman problem
• Critical path analysis
• Float, Gantt charts and scheduling
• Linear programming: formulation and graphical solution
• The Simplex algorithm

DECISION MATHEMATICS 2

• Transportation problems
• The stepping-stone method
• Allocation and the Hungarian algorithm
• Flows in networks: cuts and capacity
• Maximum flow and the labelling procedure
• Dynamic programming
• Game theory: play safe and stable solutions
• Mixed strategies
• Decision analysis