Mathematics

Module 1: Algebra, Lines, Curves & Trigonometry

• Revise GCSE algebra (indices, expanding/factorising, surds, rationalising) via a baseline and independent home tasks.
• Solve quadratics (factorising, formula, completing the square); use the discriminant and graph features for modelling.
• Inequalities (linear/quadratic), transformations of functions; cubic, quartic and reciprocal graphs.
• Trigonometry: sine/cosine rules, triangle area, graph interpretation and transformations.
• Coordinate geometry: gradients, straight-line forms, parallels/perpendiculars, midpoints and perpendicular bisectors.
• Circles: equation, intersections with lines, tangents and chords.
• Mechanics introduction: modelling assumptions; scalars vs vectors; representing vectors for magnitude/direction.

Module 2: Trigonometry in Degrees & Radians, Intro to Statistics, Differentiation

• Degrees ↔ radians; arc length and sector area; exact values; identities; multi-angle equations; small-angle approximations.
• Statistics: populations vs samples; sampling methods; Large Data Set; types of data; measures of location and spread (including coding), variance and standard deviation.
• Kinematics graphs (distance–time, velocity–time); suvat for constant acceleration; gravity in vertical motion.
• Differentiation basics: gradients of tangents/normals; increasing/decreasing functions; second derivative and stationary points; modelling with rates of change.

Module 3: Graphical Representations, Probability, Integration, Forces

• Statistics visuals: box plots, cumulative frequency, histograms; compare distributions; correlation and linear regression.
• Probability from tables/lists; Venn diagrams; independence/mutual exclusivity; tree diagrams for multi-stage problems.
• Integration of powers of x (indefinite/definite); areas under/ between curves; find functions from derivatives.
• Vectors in geometry and simple models; force diagrams; Newton’s laws; 2D acceleration and motion; connected particles and pulleys.

Module 4: Probability Distributions, Hypothesis Testing, Calculus in Mechanics, Binomial Expansion

• Binomial expansion via Pascal/factorials; applications to integer, negative and fractional powers and estimation.
• Binomial distribution; cumulative probabilities; one- and two-tailed hypothesis tests using critical values.
• Calculus in mechanics: velocity/acceleration from differentiation; displacement from integration; max/min motion problems; constant-acceleration applications.

Module 1: Trigonometry & Differentiation (Advanced) | 3D Vectors & Applied Rates

• Trig: reciprocal functions (sec, cosec, cot), inverse trig, addition/double-angle formulae, solving over intervals, rewriting acos⁡x+bsin⁡xa\cos x + b\sin xacosx+bsinx, proofs and modelling.
• Differentiation: chain/product/quotient rules; trig/exponential/log functions; implicit differentiation; second derivative for curve shape; applied rates of change.
• 3D coordinates & vectors for geometric problems and mechanics (particle motion and forces in space).

Module 2: Integration Techniques, Functions & Moments

• Integration techniques: trig identities, reverse chain rule, substitution, integration by parts, partial fractions.
• Areas under curves, trapezium rule, differential equations for growth/decay and motion modelling.
• Functions: modulus, composite and inverse functions; solving modulus equations; sketching transformations.
• Mechanics: moments, resultant moments, equilibrium conditions, centre of mass and tilting/ stability.

Module 3: Parametric Equations | Correlation & Probability | Friction, Projectiles & Connected Particles

• Parametric curves: sketching, intersections, modelling motion; differentiate/integrate parametrics.
• Statistics: exponential models; correlation; hypothesis testing; conditional probability with Venns/trees/formulae.
• Mechanics: resolving forces (including on inclines), friction and resistance; projectile motion (horizontal and angled); statics of particles/rigid bodies; connected particles and pulleys; dynamics on slopes.

Module 4: Normal Distribution, Proof, Variable Acceleration, Sequences & Numerical Methods

• Normal distribution: probabilities, inverse normal, standard normal; find μ/σ from probabilities; binomial ≈ normal; hypothesis testing with the normal.
• Proof: direct methods and proof by contradiction.
• Mechanics with vectors: variable acceleration in 1D/2D; velocity/displacement/acceleration via differentiation/integration.
• Sequences & series: arithmetic/geometric (including sum to infinity), sigma notation, recurrence relations, modelling with series.
• Numerical methods: locating roots by iteration and Newton–Raphson.

Module 5: Revision & Examinations

• Synoptic practice, interleaving of topics and challenging exam questions under timed conditions; teacher-led modelling and targeted reteach of priority areas.