Further Mathematics
Complex numbers, matrices, further calculus, advanced proofs and university entrance preparation.
Official detailed content statements
Advanced maths.
One clear pathway.
Pure Mathematics, Statistics and Mechanics organised from the UK Advanced Level mathematics curriculum, with visual lessons, worked examples, quizzes and future AI tutor support.
19 curriculum topics3 connected strands1 animated lesson available
A-LEVEL MATHEMATICS
Detailed
curriculum map.
Every topic is prepared for video explanations, visual notes, practice, progress tracking and AI tutor support.
01
Pure Mathematics
A
Proof
Proof, contradiction, counterexamples and logical mathematical arguments.
6 lesson unitsB
Algebra & Functions
Indices, surds, quadratics, inequalities, graphs, functions and modelling.
12 lesson unitsC
Coordinate Geometry
Straight lines, circles, parametric equations and geometric models.
8 lesson unitsD
Sequences & Series
Binomial expansion, sigma notation, arithmetic and geometric series.
8 lesson unitsE
Trigonometry
Radian measure, identities, trig graphs, equations and modelling.
12 lesson unitsF
Exponentials & Logarithms
Exponential models, logarithms, growth, decay and graph transformations.
9 lesson unitsG
Differentiation
Derivatives, tangents, stationary points, product, quotient and chain rules.
13 lesson unitsH
Integration
Fundamental theorem, definite integrals, substitution, parts and differential equations.
11 lesson unitsI
Numerical Methods
Root finding, iteration, Newton-Raphson and trapezium rule.
7 lesson unitsJ
Vectors
2D and 3D vectors, position vectors, magnitude and applications.
8 lesson units02
Statistics Mathematics
K
Statistical Sampling
Populations, samples, sampling methods and critique.
5 lesson unitsL
Data Presentation
Histograms, scatter diagrams, regression, variation and outliers.
9 lesson unitsM
Probability
Independent events, conditional probability and modelling.
8 lesson unitsN
Statistical Distributions
Binomial and Normal distributions with calculator technology.
5 lesson unitsO
Hypothesis Testing
Null hypothesis, significance, critical regions and interpretation.
8 lesson units03
Mechanics Mathematics
P
Quantities & Units
SI units, velocity, acceleration, force, weight and moments.
5 lesson unitsQ
Kinematics
Motion graphs, SUVAT, calculus in motion and projectiles.
8 lesson unitsR
Forces & Newton’s Laws
Newton’s laws, equilibrium, connected particles and friction.
10 lesson unitsS
Moments
Moments in simple static contexts.
3 lesson unitsANIMATED LESSON
Structure of
mathematical proof.
Observe how every proof begins with an assumption, follows logical reasoning and ends with a justified conclusion.
PROOF / A1–A6
Reason it.
Prove it.
Six structured proof lessons with key ideas, outcomes and worked examples.
A1 · 12 minutesStructure of Mathematical Proof+
Understand how mathematicians construct rigorous proofs using assumptions, logical deductions and conclusions.
Key idea
A mathematical proof demonstrates that a statement is true for every possible case, not just one example.
Learning outcome
Understand assumptions, identify logical deductions and recognise valid mathematical conclusions.
Worked example
If n is even, write n = 2k. Then n² = 4k² = 2(2k²), so n² is even.
Watch animated lesson →A2 · 12 minutesProof by Deduction+
Start with known facts and use valid logical deductions to prove a mathematical statement.
Key idea
Accepted facts, definitions and algebraic reasoning show that a conclusion must be true.
Learning outcome
Identify assumptions, justify each step and state a clear final conclusion.
Worked example
For consecutive integers n and n + 1, their sum is 2n + 1 and is therefore odd.
Watch animated lesson →A3 · 12 minutesProof by Exhaustion+
Prove a statement by checking every possible case in a finite set.
Key idea
A proof by exhaustion is valid only when every possible case has been considered.
Learning outcome
List cases systematically, verify each one and write a complete conclusion.
Worked example
For n = 1, 2, 3 or 4, checking n² + n gives 2, 6, 12 and 20—each is even.
Watch animated lesson →A4 · 11 minutesDisproof by Counterexample+
Use one valid counterexample to disprove a universal mathematical statement.
Key idea
A universal statement is false when one case satisfies its conditions but not its conclusion.
Learning outcome
Recognise universal statements and verify an appropriate counterexample.
Worked example
The statement ‘all prime numbers are odd’ is false because 2 is prime and even.
Watch animated lesson →A5 · 14 minutesProof by Contradiction: Infinite Primes+
Follow Euclid’s classic proof that there are infinitely many prime numbers.
Key idea
Assume the opposite and show that the assumption leads to an impossibility.
Learning outcome
State the opposite assumption, construct a new number and identify the contradiction.
Worked example
If p₁…pₙ were all primes, N = p₁p₂…pₙ + 1 has a prime factor absent from the list.
Watch animated lesson →A6 · 14 minutesThe Irrationality of √2+
Use contradiction to show that √2 cannot be written as a fraction of integers.
Key idea
A lowest-terms fraction cannot have both numerator and denominator even.
Learning outcome
Assume √2 is rational, manipulate the equation and expose the contradiction.
Worked example
From √2 = a/b follows a² = 2b², making both a and b even—contradicting lowest terms.
Watch animated lesson →FUTURE PATHWAYS
Built to grow.
GRE Quantitative
Arithmetic, algebra, geometry, statistics and problem solving.
GCSE Higher
An expansion route after the advanced mathematics platform is established.