A1.a: Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; use methods of proof, including proof by deduction, proof by exhaustion

A1.b: Disproof by counter example

A1.c: Proof by contradiction (including proof of the irrationality of √2 and the infinity of primes, and application to unfamiliar proofs)

B1: Understand and use the laws of indices for all rational exponents

B2: Use and manipulate surds, including rationalising the denominator

B3: Work with quadratic functions and their graphs; the discriminant of a quadratic function, including the conditions for real and repeated roots; completing the square; solution of quadratic equations including solving quadratic equations in a function of the unknown

B4: Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation

B5.a: Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including inequalities with brackets and fractions

B5.b: Express solutions through correct use of 'and' and 'or', or through set notation

B5.c: Represent linear and quadratic inequalities such as y > x + 1 and y > ax² + bx + c graphically

B6.a: Manipulate polynomials algebraically, including expanding brackets and collecting like terms, factorisation and simple algebraic division; use of the factor theorem

B6.b: Simplify rational expressions including by factorising and cancelling, and algebraic division (by linear expressions only)

B7.a: Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations

B7.b: Understand and use proportional relationships and their graphs

B8: Understand and use composite functions; inverse functions and their graphs

B9: Understand the effect of simple transformations on the graph of y = f(x) including sketching associated graphs: y = af(x), y = f(x)+a, y = f(x+a), y = f(ax), and combinations of these transformations

B10: Decompose rational functions into partial fractions (denominators not more complicated than squared linear terms and with no more than 3 terms, numerators constant or linear)

B11: Use of functions in modelling, including consideration of limitations and refinements of the models

D1.a: Understand and use the binomial expansion of (a + bx)ⁿ for positive integer n; the notations n! and nCr; link to binomial probabilities

D1.b: Extend to any rational n, including its use for approximation; be aware that the expansion is valid for |bx/a| < 1. (proof not required)

D2: Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form x_n+1 = f(x_n); increasing sequences; decreasing sequences; periodic sequences

D3: Understand and use sigma notation for sums of series

D4: Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms

D5: Understand and work with geometric sequences and series including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r | < 1; modulus notation

D6: Use sequences and series in modelling

F1.a: Know and use the function a^x and its graph, where a is positive

F1.b: Know and use the function e^x and its graph

F2: Know that the gradient of e^kx is equal to ke^kx and hence understand why the exponential model is suitable in many applications

F3.a: Know and use the definition of log_ax as the inverse of a^x, where a is positive and x ≥ 0

F3.b: Know and use the function lnx and its graph

F3.c: Know and use lnx as the inverse function of e^x

F4: Understand and use the laws of logarithms: log_ax + log_ay = log_axy; log_ax − log_ay = log_a(x/y); klog_ax = log_ax^k (including, for example, k = -1 and k = -½)

F5: Solve equations of the form a^x = b

F6: Use logarithmic graphs to estimate parameters in relationships of the form y = axⁿ and y = kb^x, given data for x and y

F7: Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models

C1.a: Understand and use the equation of a straight line, including the forms y − y₁ = m(x − x₁) and ax + by + c = 0; gradient conditions for two straight lines to be parallel or perpendicular

C1.b: Be able to use straight line models in a variety of contexts

C2: Understand and use the coordinate geometry of the circle including using the equation of a circle in the form (x − a)² + (y − b)² = r²; completing the square to find the centre and radius of a circle; use of the following properties: the angle in a semicircle is a right angle, the perpendicular from the centre to a chord bisects the chord, the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point

C3: Understand and use the parametric equations of curves and conversion between Cartesian and parametric forms

C4: Use parametric equations in modelling in a variety of contexts

E1.a: Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine rules; the area of a triangle in the form ½absinC

E1.b: Work with radian measure, including use for arc length and area of sector

E2: Understand and use the standard small angle approximations of sine, cosine and tangent: sinθ ≈ θ, cosθ ≈ 1 − θ²/2, tanθ ≈ θ where θ is in radians

E3.a: Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity

E3.b: Know and use exact values of sin and cos for 0, π/6, π/4, π/3, π/2, π and multiples thereof, and exact values of tan for 0, π/6, π/4, π/3, π and multiples thereof

E4: Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains

E5.a: Understand and use tanθ = sinθ/cosθ

E5.b: Understand and use sin²θ + cos²θ = 1; sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ

E6.a: Understand and use double angle formulae; use of formulae for sin(A±B), cos(A±B) and tan(A±B); understand geometrical proofs of these formulae

E6.b: Understand and use expressions for acosθ + bsinθ in the equivalent forms of rcos(θ ± α) or rsin(θ ± α)

E7: Solve simple trigonometric equations in a given interval, including quadratic equations in sin, cos and tan and equations involving multiples of the unknown angle

E8: Construct proofs involving trigonometric functions and identities

E9: Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces

G1.a: Understand and use the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point (x, y); the gradient of the tangent as a limit; interpretation as a rate of change; sketching the gradient function for a given curve; second derivatives; differentiation from first principles for small positive integer powers of x and for sinx and cosx

G1.b: Understand and use the second derivative as the rate of change of gradient; connection to convex and concave sections of curves and points of inflection

G2.a: Differentiate xⁿ, for rational values of n, and related constant multiples, sums and differences

G2.b: Differentiate e^kx and a^kx, sinkx, coskx, tankx and related sums, differences and constant multiples

G2.c: Understand and use the derivative of lnx

G3.a: Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection

G3.b: Identify where functions are increasing or decreasing

G4: Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions

G5: Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only

G6: Construct simple differential equations in pure mathematics and in context, (contexts may include kinematics, population growth and modelling the relationship between price and demand)

H1: Know and use the Fundamental Theorem of Calculus

H2.a: Integrate xⁿ (excluding n = -1), and related sums, differences and constant multiples

H2.b: Integrate e^kx, 1/x, sin kx, cos kx and related sums, differences and constant multiples

H3: Evaluate definite integrals; use a definite integral to find the area under a curve and the area between two curves

H4: Understand and use integration as the limit of a sum

H5: Carry out simple cases of integration by substitution and integration by parts; understand these methods as the inverse processes of the chain and product rules respectively (Integration by substitution includes finding a suitable substitution and is limited to cases where one substitution will lead to a function which can be integrated; integration by parts includes more than one application of the method but excludes reduction formulae)

H6: Integrate using partial fractions that are linear in the denominator

H7: Evaluate the analytical solution of simple first order differential equations with separable variables, including finding particular solutions (Separation of variables may require factorisation involving a common factor)

H8: Interpret the solution of a differential equation in the context of solving a problem, including identifying limitations of the solution; includes links to kinematics

I1.a: Locate roots of f(x) = 0 by considering changes of sign of f(x) in an interval of x on which f(x) is sufficiently well-behaved

I1.b: Understand how change of sign methods can fail

I2.a: Solve equations approximately using simple iterative methods; be able to draw associated cobweb and staircase diagrams

I2.b: Solve equations using the Newton-Raphson method and other recurrence relations of the form x_n+1 = g(x_n)

I2.c: Understand how such methods can fail

I3: Understand and use numerical integration of functions, including the use of the trapezium rule and estimating the approximate area under a curve and limits that it must lie between

I4: Use numerical methods to solve problems in context

K1.a: Understand and use the terms 'population' and 'sample'

K1.b: Use samples to make informal inferences about the population

K1.c: Understand and use sampling techniques, including simple random sampling and opportunity sampling

K1.d: Select or critique sampling techniques in the context of solving a statistical problem, including understanding that different samples can lead to different conclusions about the population

L1.a: Interpret diagrams for single-variable data, including understanding that area in a histogram represents frequency

L1.b: Connect to probability distributions

L2.a: Interpret scatter diagrams and regression lines for bivariate data, including recognition of scatter diagrams which include distinct sections of the population (calculations involving regression lines are excluded)

L2.b: Understand informal interpretation of correlation

L2.c: Understand that correlation does not imply causation

L3.a: Interpret measures of central tendency and variation, extending to standard deviation

L3.b: Be able to calculate standard deviation, including from summary statistics

L4.a: Recognise and interpret possible outliers in data sets and statistical diagrams

L4.b: Select or critique data presentation techniques in the context of a statistical problem

L4.c: Be able to clean data, including dealing with missing data, errors and outliers

M1.a: Understand and use mutually exclusive and independent events when calculating probabilities

M1.b: Link to discrete and continuous distributions

M2.a: Understand and use conditional probability, including the use of tree diagrams, Venn diagrams, two-way tables

M2.b: Understand and use the conditional probability formula P(A|B) = P(A∩B)/P(B)

M3: Modelling with probability, including critiquing assumptions made and the likely effect of more realistic assumptions

N1: Understand and use simple, discrete probability distributions (calculation of mean and variance of discrete random variables is excluded), including the binomial distribution, as a model; calculate probabilities using the binomial distribution

N2.a: Understand and use the Normal distribution as a model; find probabilities using the Normal distribution

N2.b: Link to histograms, mean, standard deviation, points of inflection and the binomial distribution

N3: Select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or Normal model may not be appropriate

O1: Understand and apply the language of statistical hypothesis testing, developed through a binomial model: null hypothesis, alternative hypothesis, significance level, test statistic, 1-tail test, 2-tail test, critical value, critical region, acceptance region, p-value; extend to correlation coefficients as measures of how close data points lie to a straight line and be able to interpret a given correlation coefficient using a given p-value or critical value (calculation of correlation coefficients is excluded)

O2.a: Conduct a statistical hypothesis test for the proportion in the binomial distribution and interpret the results in context

O2.b: Understand that a sample is being used to make an inference about the population and appreciate that the significance level is the probability of incorrectly rejecting the null hypothesis

O3: Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context

J1: Use vectors in two dimensions and in three dimensions

J2: Calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form

J3: Add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations

J4: Understand and use position vectors; calculate the distance between two points represented by position vectors

J5: Use vectors to solve problems in pure mathematics and in context, including forces and kinematics

P1: Understand and use fundamental quantities and units in the S.I. system: length, time, mass

P2: Understand and use derived quantities and units: velocity, acceleration, force, weight, moment

Q1: Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration

Q2: Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph

Q3: Understand, use and derive the formulae for constant acceleration for motion in a straight line; extend to 2 dimensions using vectors

Q4: Use calculus in kinematics for motion in a straight line: v = dr/dt, a = dv/dt = d²r/dt², r = ∫v dt, v = ∫a dt; extend to 2 dimensions using vectors

Q5: Model motion under gravity in a vertical plane using vectors; projectiles

R1: Understand the concept of a force; understand and use Newton's first law

R2: Understand and use Newton's second law for motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors); extend to situations where forces need to be resolved (restricted to 2 dimensions)

R3: Understand and use weight and motion in a straight line under gravity; gravitational acceleration, g, and its value in S.I. units to varying degrees of accuracy (The inverse square law for gravitation is not required and g may be assumed to be constant, but students should be aware that g is not a universal constant but depends on location)

R4: Understand and use Newton's third law; equilibrium of forces on a particle and motion in a straight line (restricted to forces in two perpendicular directions or simple cases of forces given as 2-D vectors); application to problems involving smooth pulleys and connected particles; resolving forces in 2 dimensions; equilibrium of a particle under coplanar forces

R5: Understand and use addition of forces; resultant forces; dynamics for motion in a plane

R6: Understand and use the F ≤ μR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics

S1: Understand and use moments in simple static contexts