Maths Lessons
Concise notes and worked examples for every point of the A-Level specification. Pick a topic to see its lessons.
Year 1 · AS content
Pure Mathematics
01
ProofDeduction, exhaustion and disproof by counter-example
Proof by deduction
Proof by exhaustion
Disproof by counter-example
02
Algebra & FunctionsIndices, surds, quadratics and inequalities
Indices & surds
Quadratic functions
Modelling with quadratics
Simultaneous equations
Inequalities
Inequalities on graphs & regions
Polynomials & the factor theorem
03
Straight LinesEquations, parallel/perpendicular lines, length, area and modelling
Equation of a straight line
Parallel & perpendicular lines
Length & area
Modelling with straight lines
04
Circle GeometryEquations of circles, midpoints, intersections, tangents and triangles
The equation of a circle
Midpoints & perpendicular bisectors
Intersections of lines & circles
Tangent & chord properties
Circles & triangles
05
Graphs & TransformationsCubics, quartics, reciprocals, intersections and transforming curves
Cubic graphs
Quartic graphs
Reciprocal graphs
Points of intersection
Translating & stretching graphs
Transforming functions
06
Binomial ExpansionPascal's triangle, factorial notation, and the binomial expansion for positive integer n
Pascal's triangle
Factorial notation & nCr
Binomial expansion (positive integer n)
Solving binomial problems
Binomial estimation
07
Trigonometric RatiosSine and cosine rules, area, and the graphs of sin/cos/tan
Cosine rule
Sine rule
Area of triangles
Problems with triangles
Sine, cosine & tangent graphs
Transforming trigonometric graphs
08
Trigonometric Identities and EquationsQuadrant signs, exact values, identities and solving equations
Angles in all four quadrants
Exact trig values
Trigonometric identities
Trigonometric equations
09
Exponentials & Logarithmse^x, ln x, laws of logs and exponential models
Exponential functions & e^x
Logarithms & their laws
Natural logarithms
Solving equations using logarithms
Exponential growth & decay
Linearising with logs
10
DifferentiationFirst principles, rules and applications
Differentiation from first principles
The power rule & gradient functions
Tangents & normals
Increasing & decreasing functions
Sketching gradient functions
Stationary points
Optimisation
11
IntegrationIndefinite and definite integration
Integration as the reverse of differentiation
Finding the constant of integration
Definite integrals
Area under a curve
Area between curves and lines
12
Vectors2D vectors and their applications
Introduction to vectors
Magnitude & direction
Position vectors
Solving geometric problems
Statistics
13
Data CollectionPopulations, sampling methods, data types and the large data set
Populations & samples
Sampling
Non-random sampling
Types of data
The large data set
14
Measures of Location and SpreadCentral tendency, quartiles, spread, variance and coding
Measures of central tendency
Other measures of location
Measures of spread
Variance & standard deviation
Coding
15
Representations of DataOutliers, box plots, cumulative frequency, histograms and comparing data
Outliers
Box plots
Cumulative frequency
Histograms
Comparing data
16
CorrelationScatter diagrams, correlation and linear regression
Correlation
Linear regression
17
ProbabilityCalculating probabilities, Venn diagrams and tree diagrams
Calculating probabilities
Venn diagrams
Mutually exclusive & independent events
Tree diagrams
18
Statistical DistributionsDiscrete distributions and the binomial model
Discrete probability distributions
The binomial distribution
Cumulative probabilities
19
Hypothesis TestingTesting a binomial proportion
Hypothesis testing
Critical values
One-tailed binomial tests
Two-tailed binomial tests
Mechanics
20
Modelling in MechanicsSI units, modelling assumptions and vectors
Quantities & SI units
Modelling assumptions in mechanics
Working with vectors
21
Constant AccelerationMotion graphs and the SUVAT equations
Displacement–time graphs
Velocity–time graphs
Constant acceleration formulae
Vertical motion under gravity
22
Forces and MotionForces as vectors, F = ma, and connected systems
Force diagrams
Forces as vectors
Forces and acceleration
Motion in 2 dimensions
Connected particles (introduction)
Pulleys
23
Variable AccelerationCalculus applied to displacement, velocity and acceleration
Functions of time
Using differentiation
Maxima and minima problems
Using integration
Proving the SUVAT formulae
Year 2 · full A-level content
Pure Mathematics
24
ProofProof by contradiction
Proof by contradiction
25
Algebraic MethodsAlgebraic fractions, partial fractions, repeated factors and algebraic division
Algebraic fractions
Partial fractions
Repeated factors
Algebraic division
26
Functions and GraphsMapping functions & domain/range, modulus function, composite & inverse functions, sketching y=|f(x)| and y=f(|x|)
The modulus function
Mapping functions
Composite functions
Inverse functions
Sketching y = |f(x)| and y = f(|x|)
27
Sequences & SeriesArithmetic, geometric series, sigma notation, recurrence and modelling
Arithmetic sequences & series
Geometric sequences & series
Sigma notation
Recurrence relations
Modelling with series
28
Binomial ExpansionExpanding (1+x)^n, (a+bx)^n, and using partial fractions
Expanding (1+x)^n
Expanding (a+bx)^n
Binomial expansion using partial fractions
29
RadiansRadians, arc length and small angle approximations
Radians and arc length
Areas of sectors and segments
Solving trig equations with radians
Small angle approximations
30
Trigonometric FunctionsReciprocal graphs, identities and inverse trig functions
Graphs of sec x, cosec x and cot x
Using reciprocal trigonometric functions
Trigonometric identities
Inverse trigonometric functions
31
Trigonometric ModellingAddition & double angle formulae, R cos/sin form and modelling real-world oscillation
Addition formulae
Double angle formulae
Solving trigonometric equations
R cos/sin form
Proving identities
Modelling with trigonometric functions
32
Parametric EquationsTrig-based curves, sketching, points of intersection and modelling motion
Parametric equations
Parametric equations using trig identities
Sketching curves
Points of intersection
Modelling with parametric equations
33
DifferentiationStandard derivatives, the chain/product/quotient rules, and further techniques
Differentiating sin and cos functions
Differentiating exponentials and logarithms
Differentiating trig functions (reciprocal & inverse)
The chain rule
The product rule
The quotient rule
Implicit differentiation
Parametric differentiation
Concave and convex functions
Rates of change
34
IntegrationAdvanced techniques and differential equations
Area between curves
Integrating standard functions
Integrating f(ax + b)
The reverse chain rule
Integrating using trigonometric identities
Integration by substitution
Integration by parts
Integrating using partial fractions
The trapezium rule
Differential equations
Areas between curves and lines
35
Numerical MethodsLocating roots and iterative techniques
Locating roots by change of sign
Iteration & staircase/cobweb diagrams
The Newton–Raphson method
Modelling with numerical methods
36
Vectors3D vectors
Coordinates and vectors in 3D
Unit vectors
Solving geometric problems with vectors
Application to mechanics
Statistics
37
Regression and CorrelationCoding non-linear data and testing for correlation
Exponential models
Measuring correlation
Hypothesis testing for zero correlation
38
Conditional ProbabilitySet notation, conditional probability and tree diagrams
Set notation
Conditional probability
Conditional probability in Venn diagrams
Probability formulae
Tree diagrams
39
The Normal DistributionContinuous data, probabilities, and tests on the mean
The normal distribution
Finding probabilities
Inverse normal distribution function
The standard normal distribution
Finding μ and σ
Approximating a binomial distribution
Hypothesis testing with the normal distribution
40
Hypothesis TestingTesting correlation
Testing for correlation
Mechanics
41
MomentsTurning effects, equilibrium and tilting
Moments & turning effects
Resultant moments
Equilibrium of rigid bodies
Centres of mass
Tilting
42
Forces and FrictionResolving forces, inclined planes and friction
Resolving forces
Inclined planes
Friction
43
ProjectilesMotion under gravity, launch angles and range
Horizontal projection
Horizontal and vertical components
Projection at any angle
Projectile motion formulae
44
Applications of ForcesStatic particles, rigid bodies, friction and dynamics on inclined planes
Static particles
Modelling with statics
Friction and static particles
Static rigid bodies
Dynamics and inclined planes