Mathematics lessons
A level & IB Maths: pure techniques, statistics, and mechanics with worked methods and exam practice.
Key topics
- Algebra, functions & graphs
- Differentiation & integration
- Trigonometry & sequences
- Statistics & probability
- Mechanics: kinematics & forces
- Proof & problem solving
Lesson course
Work through each lesson, run the interactive practice, and tick it off as you go.
1. Algebra & functions
Pure
Indices, surds & quadratics
Manipulate powers and surds; solve and sketch quadratics.
A quadratic ax²+bx+c=0 has roots from x=(-b±√(b²-4ac))/2a. The discriminant b²-4ac tells you how many real roots: >0 two, =0 one, <0 none.
Solve 10 quadratics by factorising, completing the square, and the formula.
Polynomials & the factor theorem
Divide polynomials and factorise using the factor theorem.
Factor theorem: if f(a)=0 then (x−a) is a factor of f(x). Test small integer/fraction values of a to find a root, then divide out.
Factorise three cubics using the factor theorem and verify by expansion.
Graphs & transformations
Sketch curves and apply translations, stretches, and reflections.
f(x)+a shifts up by a; f(x+a) shifts left by a; af(x) stretches vertically by a; f(ax) stretches horizontally by 1/a.
Sketch y=f(x), then y=f(x)+a, f(x+a), af(x), f(ax) for one curve.
2. Calculus
Pure
Differentiation from first principles
Differentiate powers of x and interpret the gradient function.
The derivative gives the gradient of the tangent. For y=xⁿ, dy/dx=nxⁿ⁻¹. f'(x)>0 means increasing; f'(x)=0 at stationary points.
Differentiate 8 expressions and find gradients at given points.
Applications of differentiation
Find and classify stationary points; solve optimisation.
At a stationary point dy/dx=0. Use the second derivative: d²y/dx²>0 → minimum, <0 → maximum. Optimisation = model, differentiate, solve, justify.
Optimise one real-world quantity (max area/volume) end-to-end.
Integration
Integrate powers of x and find areas under curves.
Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c (n≠−1). A definite integral ∫ₐᵇ gives the signed area under the curve.
Evaluate 6 definite integrals and one area between a curve and the x-axis.
3. Trigonometry & sequences
Pure
Trig ratios & identities
Use exact values and the identities sin²+cos²=1 and tan=sin/cos.
Key identity: sin²θ + cos²θ = 1. Also tanθ = sinθ/cosθ. Learn exact values for 0,30,45,60,90°.
Prove three identities and solve two equations over 0–360°.
Solving trig equations
Solve equations using the CAST diagram and periodicity.
Find the principal value, then use symmetry/period (360° for sin/cos, 180° for tan) to get all solutions in the interval.
Solve four trig equations giving all solutions in a range.
Sequences & series
Use arithmetic and geometric sequences and their sums.
Arithmetic: term aₙ=a+(n−1)d. Geometric: aₙ=arⁿ⁻¹; sum to infinity a/(1−r) exists only when |r|<1.
Find sums of one arithmetic and one geometric series; test convergence.
4. Statistics
Applied
Data presentation & measures
Calculate and interpret mean, median, spread, and outliers.
Mean is sensitive to outliers; median is not. Spread: range, interquartile range (IQR=Q3−Q1), standard deviation. Outlier if beyond Q1−1.5·IQR or Q3+1.5·IQR.
Summarise one data set; identify outliers using the 1.5×IQR rule.
Probability
Apply the addition and multiplication rules and tree diagrams.
P(A∪B)=P(A)+P(B)−P(A∩B). Independent events: P(A∩B)=P(A)P(B). Conditional: P(A|B)=P(A∩B)/P(B).
Solve three probability problems including conditional probability.
Distributions & hypothesis testing
Use the binomial/normal models and test a hypothesis.
Binomial models fixed n trials with constant p. A hypothesis test compares an observed result against H0 using a significance level (e.g. 5%); reject H0 if the result is in the critical region.
Run one binomial hypothesis test stating H0, H1, and a conclusion.
5. Mechanics
Applied
Kinematics
Use the suvat equations for constant acceleration.
For constant acceleration: v=u+at, s=ut+½at², v²=u²+2as. On a velocity–time graph, gradient=acceleration and area=displacement.
Solve four motion problems and sketch velocity–time graphs.
Forces & Newton's laws
Resolve forces and apply F = ma.
Newton's 2nd law: resultant force F=ma. Draw a free-body diagram, resolve into perpendicular directions, then form equations of motion.
Solve two connected-particle problems with free-body diagrams.
Variable acceleration
Use calculus to link displacement, velocity, and acceleration.
When acceleration varies, use calculus: v=ds/dt, a=dv/dt; integrate to reverse. Constant-acceleration suvat no longer applies.
Differentiate/integrate one motion model to find v and a.
6. Proof & exam technique
Skills
Methods of proof
Use deduction, exhaustion, and counter-examples.
Proof by deduction builds from known facts. Proof by exhaustion checks all cases. One counter-example disproves a universal claim.
Prove two statements and disprove one with a counter-example.
Problem solving
Plan multi-step problems and check answers for sense.
Read carefully, define variables, choose a method, show working, and sanity-check units and magnitude at the end.
Tackle two unstructured problems, showing every step.
Mock exam paper
Sit a timed mixed paper and review every error.
Time per mark ≈ exam length ÷ total marks. Attempt every question, show method for method marks, and review mistakes by topic afterwards.
Complete one past paper to time, then mark and correct it.
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