England · OCRSyllabus
Maths syllabus, dot point by dot point
Every dot point in the England Mathssyllabus, with a focused answer for each one. Click any dot point for a worked explainer, past exam questions, and links to related dot points. Written by Claude Opus 4.8, Anthropic's latest AI.
Mechanics
Module overview →- How do forces produce acceleration, and how do you analyse the forces on a particle including friction and connected systems?Newton's three laws of motion, weight, resolving forces, equilibrium of a particle, friction and the coefficient of friction, motion on an inclined plane, and connected particles.12 min answer →
- How do you describe motion in a straight line using the constant-acceleration equations, motion graphs and calculus?Displacement, velocity and acceleration, the constant-acceleration (suvat) equations, motion under gravity, displacement-time and velocity-time graphs, and using calculus when acceleration varies.12 min answer →
- How do you analyse the turning effect of forces and the conditions for a rigid body such as a beam to balance?The moment of a force about a point, the principle of moments, the equilibrium of a rigid body, and problems involving uniform and non-uniform rods, beams and reactions at supports.11 min answer →
- How do you analyse the motion of a projectile by treating its horizontal and vertical motion separately?Motion of a projectile under gravity, resolving the initial velocity into horizontal and vertical components, the independence of horizontal and vertical motion, and finding range, maximum height and time of flight.12 min answer →
- What are the base and derived quantities of mechanics, and how do you distinguish scalars from vectors and model a real situation?The SI base and derived units used in mechanics, the distinction between scalar and vector quantities, and the standard modelling assumptions such as particles, light strings and smooth surfaces.10 min answer →
Pure mathematics: advanced
Module overview →- How do you work with exponential and logarithmic functions, the number e, and use logarithms to solve equations and linearise models?Exponential functions and their graphs, the number e and the natural logarithm, the laws of logarithms, solving exponential and logarithmic equations, and using logarithms to estimate parameters in exponential and power-law models.12 min answer →
- How do you work with functions as mappings, compose and invert them, and handle the modulus function in graphs and equations?The language of functions (domain, range, composite and inverse functions), the modulus function and its graph, and solving modulus equations and inequalities.11 min answer →
- How do you split a rational expression into partial fractions, and why does that make integration and binomial expansion easier?Decomposing a proper algebraic fraction into partial fractions, including denominators with distinct linear factors and a repeated linear factor, and using partial fractions in integration and binomial expansion.10 min answer →
- How do you measure angles in radians, and how do you find arc lengths, sector areas and small-angle approximations?Radian measure, the relationship between radians and degrees, arc length and the area of a sector and segment, and the small-angle approximations for sine, cosine and tangent.10 min answer →
- How do you use the trigonometric identities, including the compound and double angle formulae and the R form, to prove results and solve equations?The Pythagorean and quotient identities, the reciprocal functions, the compound and double angle formulae, the R form for a sin theta plus b cos theta, and solving trigonometric equations over an interval.12 min answer →
- How do you work with the three trigonometric functions, the sine and cosine rules, and exact values to model and solve triangle problems?The sine, cosine and tangent functions and their graphs, the sine and cosine rules, the area of a triangle, and exact values of trigonometric ratios for standard angles.11 min answer →
- How do you represent quantities with both magnitude and direction, and use vectors in two and three dimensions for geometry and motion?Vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, position vectors and unit vectors, and geometric applications including collinearity and the midpoint.10 min answer →
Pure mathematics: calculus
Module overview →- How do you use the derivative to find tangents, classify stationary points, and solve optimisation and connected-rate problems?Tangents and normals, increasing and decreasing functions, stationary points and their nature using the second derivative, points of inflection, optimisation, and connected rates of change.11 min answer →
- How do you form a differential equation from a rate of change and solve it by separating the variables?Forming first-order differential equations from a context, solving them by separation of variables, finding particular solutions from initial conditions, and interpreting the solution in modelling.11 min answer →
- How do you find the rate at which a quantity changes, from first principles and using the standard rules?Differentiation from first principles, the power rule, the chain, product and quotient rules, derivatives of standard functions including exponentials, logarithms and trigonometric functions, and implicit and parametric differentiation.12 min answer →
- How do you integrate harder functions using substitution, integration by parts, and partial fractions?Integration by substitution, integration by parts, integration using partial fractions, and integrating expressions of the form f prime over f and products reducible by a trigonometric identity.12 min answer →
- How do you reverse differentiation to find areas under and between curves, and integrate the standard functions?Indefinite and definite integrals as the reverse of differentiation, the integrals of standard functions, the area under a curve and between two curves, and the trapezium rule for numerical integration.12 min answer →
- How do you locate and approximate the roots of an equation when an exact solution is not available?Locating roots by change of sign, iterative methods of the form x equals g of x, the Newton-Raphson method, and the conditions under which these numerical methods fail.11 min answer →
Pure mathematics: foundations
Module overview →- How do you solve quadratics, simultaneous equations and inequalities, and read information from the discriminant?Quadratic functions, completing the square, the quadratic formula and the discriminant, simultaneous equations (linear and quadratic), and linear and quadratic inequalities.11 min answer →
- How do you find equations of straight lines and circles, and use the geometry of tangents and chords?Straight lines, gradients, parallel and perpendicular conditions, the equation of a circle, the relationship between a tangent and the radius, and parametric equations of curves.11 min answer →
- How do you sketch standard curves and predict the effect of translating, stretching or reflecting a graph?Sketching curves including polynomials, the reciprocal function and its variations, intersections of graphs, and the transformations y equals f(x) plus a, f(x plus a), f(ax) and af(x).10 min answer →
- How do you manipulate powers and irrational roots exactly, without a calculator?Laws of indices for all rational exponents, surd manipulation and rationalising denominators, and the meaning of negative and fractional indices.9 min answer →
- How do you factorise and divide polynomials, and how do you expand a bracket raised to a power?Polynomial manipulation, the factor theorem and algebraic division, and the binomial expansion of (a plus b) to the power n for positive integer n using binomial coefficients.10 min answer →
- How do you prove a mathematical statement is always true, and how do you disprove one that is not?Methods of proof: proof by deduction, proof by exhaustion, disproof by counter-example, and proof by contradiction, including the irrationality of root 2 and the infinitude of primes.10 min answer →
- How do you describe and sum arithmetic and geometric sequences, and when does an infinite series converge?Arithmetic and geometric sequences and series, sigma notation, sum formulae, recurrence relations, increasing, decreasing and periodic sequences, and the sum to infinity of a convergent geometric series.11 min answer →
Statistics
Module overview →- How do you summarise, display and compare data, and identify outliers and correlation?Measures of central tendency and spread, histograms, box plots and cumulative frequency, identifying outliers, comparing distributions, and correlation and the regression line.12 min answer →
- How do you use sample evidence to decide whether a claim about a population proportion, mean or correlation is supported?Null and alternative hypotheses, one- and two-tailed tests, significance levels and critical regions, hypothesis tests for a binomial proportion, for a Normal mean, and for a correlation coefficient.12 min answer →
- How do you calculate probabilities for combined events, including mutually exclusive, independent and conditional cases?Probability of events, mutually exclusive and independent events, Venn diagrams, tree diagrams and two-way tables, the addition and multiplication laws, and conditional probability.11 min answer →
- How do you model a count of successes with the binomial distribution and continuous data with the Normal distribution?Discrete random variables and probability distributions, the binomial distribution as a model and its probabilities, the Normal distribution, standardising, the inverse Normal, and the Normal approximation to the binomial.12 min answer →
- How do you choose a sample that fairly represents a population, and what are the trade-offs of each sampling method?Populations and samples, the census, sampling methods (simple random, systematic, stratified, quota and opportunity), their advantages and disadvantages, and the role of the large data set.10 min answer →