Lessons

Quadratics: Completing the Square, the Discriminant and Solving Quadratic Equations

be able to solve quadratic equations by factorising, by completing the square and by using the quadratic formula · be able to complete the square of a quadratic expression and use the result to sketch the graph and identify the vertex and line of symmetry · understand and use the discriminant b^2 - 4ac to determine the nature of the roots of a quadratic equation · be able to solve equations that can be reduced to a quadratic in a function of x (e.g. quadratic in x^2 or in sqrt(x))

Quadratic Inequalities and Linear-Quadratic Simultaneous Equations

be able to solve simultaneous equations in two unknowns (linear-linear and linear-quadratic) algebraically and graphically · be able to solve linear and quadratic inequalities, expressing solution sets using inequality and interval notation · understand how to represent linear and quadratic inequalities graphically, including regions defined by multiple inequalities

Functions: Domain, Range, Composition and Inverses

understand the definitions of one-one, many-one and inverse functions, and the domain and range of a function · be able to form composite functions fg(x) and inverse functions f^(-1)(x), and sketch the graph of an inverse as the reflection of the graph of f in the line y = x

Transformations of Graphs

be able to sketch graphs of quadratic, cubic, reciprocal and simple rational functions, identifying key features (intercepts, asymptotes, turning points) · understand the effect of the transformations y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax) on the graph of y = f(x), and apply these to sketch transformed curves

Straight Lines

understand and use the equation of a straight line in the forms y = mx + c, y - y1 = m(x - x1) and ax + by + c = 0 · be able to calculate the gradient of a line through two points and the distance between two points · be able to find the midpoint of a line segment and the coordinates of the point dividing it in a given ratio

Parallel and Perpendicular Lines

understand the conditions for two lines to be parallel (equal gradients) or perpendicular (product of gradients = -1) · be able to find the equation of a line parallel or perpendicular to a given line through a specified point

The Equation of a Circle

understand and use the equation of a circle (x - a)^2 + (y - b)^2 = r^2, including completing the square to find the centre and radius from a general second-degree form

Lines and Circles, and the Intersection of a Line and a Curve

be able to find equations of tangents and chords to a circle and solve problems using the perpendicularity of a tangent to the radius at the point of contact · understand and use the standard circle theorems: angle in a semicircle, perpendicular from centre bisects chord, tangent perpendicular to radius

Circular Measure: Radians, Arc Length and Sector Area

understand and use the definition of a radian, and convert between degrees and radians · be able to use the formulae for arc length s = rθ and area of a sector A = 1/2 r^2 θ where θ is in radians

Trigonometric Graphs, Exact Values and Inverse Functions

understand and use the definitions of sine, cosine and tangent for any angle, using the unit circle to interpret signs of trigonometric ratios in each quadrant · know and use exact values of sin, cos and tan of 0°, 30°, 45°, 60°, 90° and related angles · be able to sketch the graphs of y = sin x, y = cos x and y = tan x and recognise their periodicity and symmetries

Trigonometric Identities and Equations

understand and use the identities sin^2 θ + cos^2 θ = 1 and tan θ = sin θ / cos θ · be able to solve trigonometric equations of the form sin(kθ + α) = c, cos(kθ + α) = c and tan(kθ + α) = c within a given interval

Binomial Expansion of (a + b)^n for Positive Integer n

understand and use Pascal's triangle to expand (a + b)^n for small positive integer n · be able to use the binomial theorem to expand (a + b)^n for positive integer n, using the notation nCr (or (n r))

Finding a Specific Term or Coefficient in a Binomial Expansion

be able to identify and calculate specific terms or coefficients in a binomial expansion without expanding fully

Arithmetic Progressions

understand and use the formula for the nth term of an arithmetic sequence: u_n = a + (n - 1)d · be able to use the formula for the sum of the first n terms of an arithmetic series: S_n = n/2 [2a + (n - 1)d]

Geometric Progressions and Sum to Infinity

understand and use the formula for the nth term of a geometric sequence: u_n = a r^(n-1) · be able to use the formula for the sum of the first n terms of a geometric series and the sum to infinity for |r| < 1

Introduction to Differentiation: Gradients, Notation and the Power Rule

understand the derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a general point, and use the notations dy/dx and f'(x) · be able to differentiate x^n for any rational n, including sums and constant multiples of such terms · be able to differentiate from first principles for simple polynomial functions

Tangents, Normals and Increasing/Decreasing Functions

be able to find equations of tangents and normals to a curve at a given point · be able to locate stationary points of a curve and determine their nature using the second derivative or sign of the first derivative · be able to apply differentiation to problems involving rates of change, optimisation and curve sketching

Stationary Points, Their Nature and Curve Sketching

be able to find and classify stationary points (maxima, minima, points of inflection) using the second derivative test · be able to apply differentiation to optimisation problems involving maxima and minima in geometric and applied contexts

Connected Rates of Change

be able to use the chain rule to solve related-rates problems where one variable is changing as a known function of time and another depends on it

Indefinite Integration and the Constant of Integration

understand integration as the reverse of differentiation and use the notation ∫ f(x) dx, including the role of the constant of integration · be able to integrate x^n for n ≠ -1, including sums and constant multiples of such terms, and find a curve from its gradient function given a point

Definite Integration and Area Under a Curve

be able to evaluate definite integrals and interpret them as signed areas between a curve and the x-axis · be able to find the area between a curve and the x-axis, including regions that cross the axis (treating sign appropriately)

Area Between a Curve and a Line and Between Two Curves

be able to find areas between two curves, between a curve and a line, or between a curve and the y-axis using definite integration

Volumes of Revolution

be able to find the volume of revolution generated by rotating a curve about the x-axis or the y-axis through 2π radians, using V = π ∫ y^2 dx or V = π ∫ x^2 dy

The Modulus Function

understand the definition |x| of the modulus function and be able to sketch graphs of y = |f(x)| and y = f(|x|) · be able to solve equations and inequalities involving the modulus function algebraically and graphically

Polynomial Division, the Factor and Remainder Theorems

be able to divide a polynomial by a linear or quadratic divisor, expressing the result as quotient plus remainder · understand and apply the factor theorem and the remainder theorem to factorise cubic and quartic polynomials

Exponential Functions and Their Graphs

understand the function y = a^x and sketch its graph for a > 1 and 0 < a < 1 · understand the function y = e^x as the exponential function and recognise its special role in calculus

Logarithms: Laws, Exponential Equations and Linear Form

understand and use the laws of logarithms: log(ab), log(a/b), log(a^n), change of base · be able to solve equations of the form a^x = b using logarithms, including problems modelled by exponential growth and decay · be able to use logarithms to linearise relationships of the form y = a x^n and y = a b^x, and interpret the gradient and intercept of the linearised graph

e^x and ln x as Inverse Functions

understand the function y = e^(kx) and its derivative ky' = y, including modelling applications of exponential growth and decay · understand and use the natural logarithm function y = ln x as the inverse of y = e^x, including sketching its graph and noting its domain

Reciprocal Trigonometric Functions and the Pythagorean Identities

understand and use the reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ, including their graphs and domains · be able to use the identities 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = cosec^2 θ to prove identities and solve equations

Compound-Angle and Double-Angle Formulae, and the R-Form

be able to use the compound-angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B) · be able to use the double-angle formulae sin 2A = 2 sin A cos A, cos 2A = 1 - 2 sin^2 A = 2 cos^2 A - 1 and tan 2A = 2 tan A / (1 - tan^2 A) · be able to express a sin θ + b cos θ in the form R sin(θ + α) or R cos(θ - α) and use this form to find maxima, minima and to solve equations

Differentiating Exponential, Logarithmic and Trigonometric Functions

be able to differentiate e^x, ln x, sin x, cos x, tan x and their constant multiples and sums

The Chain, Product and Quotient Rules

be able to apply the chain rule to differentiate composite functions · be able to apply the product rule and the quotient rule to differentiate products and quotients of functions

Implicit and Parametric Differentiation

be able to differentiate implicitly to find dy/dx for relations between x and y · be able to use parametric differentiation: if x = x(t), y = y(t), then dy/dx = (dy/dt)/(dx/dt)

Integrating Standard Functions and Using Trigonometric Identities

be able to integrate e^x, 1/x, sin x, cos x, sec^2 x and related forms involving linear substitutions

The Trapezium Rule

understand and use the trapezium rule to estimate the value of a definite integral · use sketch graphs in simple cases to determine whether the trapezium rule gives an over-estimate or an under-estimate

Locating Roots of an Equation

be able to locate roots of f(x) = 0 by using a change of sign in f(x) over a suitable interval, recognising the conditions under which this method may fail

Iterative Methods

be able to solve equations approximately using an iteration of the form x_(n+1) = g(x_n), and understand the conditions for convergence

Partial Fractions

be able to express a proper rational function as a sum of partial fractions when the denominator factorises into distinct linear factors, repeated linear factors or a linear and a quadratic factor · be able to reduce an improper rational function to a polynomial plus a proper rational function before decomposing into partial fractions

The Binomial Expansion for Rational n

be able to expand (1 + x)^n for any rational n, stating the range of values of x for which the expansion is valid · be able to adapt the binomial series to expand (a + bx)^n by first factorising out a^n

Exponential Functions and Their Graphs

understand the function y = a^x and sketch its graph for a > 1 and 0 < a < 1 · understand the function y = e^x as the exponential function and recognise its special role in calculus

Logarithms: Laws, Exponential Equations and Linear Form

understand and use the laws of logarithms: log(ab), log(a/b), log(a^n), change of base · be able to solve equations of the form a^x = b using logarithms, including problems modelled by exponential growth and decay · be able to use logarithms to linearise relationships of the form y = a x^n and y = a b^x, and interpret the gradient and intercept of the linearised graph

e^x and ln x as Inverse Functions

understand the function y = e^(kx) and its derivative ky' = y, including modelling applications of exponential growth and decay · understand and use the natural logarithm function y = ln x as the inverse of y = e^x, including sketching its graph and noting its domain

Reciprocal Trigonometric Functions and the Pythagorean Identities

understand and use the reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ, including their graphs and domains · be able to use the identities 1 + tan^2 θ = sec^2 θ and 1 + cot^2 θ = cosec^2 θ to prove identities and solve equations

Compound-Angle and Double-Angle Formulae, and the R-Form

be able to use the compound-angle formulae for sin(A ± B), cos(A ± B) and tan(A ± B) · be able to use the double-angle formulae sin 2A = 2 sin A cos A, cos 2A = 1 - 2 sin^2 A = 2 cos^2 A - 1 and tan 2A = 2 tan A / (1 - tan^2 A) · be able to express a sin θ + b cos θ in the form R sin(θ + α) or R cos(θ - α) and use this form to find maxima, minima and to solve equations

Differentiating Exponential, Logarithmic and Trigonometric Functions

be able to differentiate e^x, ln x, sin x, cos x, tan x and their constant multiples and sums

The Chain, Product and Quotient Rules

be able to apply the chain rule to differentiate composite functions · be able to apply the product rule and the quotient rule to differentiate products and quotients of functions

Implicit and Parametric Differentiation

be able to differentiate implicitly to find dy/dx for relations between x and y · be able to use parametric differentiation: if x = x(t), y = y(t), then dy/dx = (dy/dt)/(dx/dt)

Integration by Substitution, by Parts and Using Partial Fractions

be able to integrate by substitution, including using a given substitution and choosing an appropriate one · be able to integrate by parts using the formula ∫ u dv = uv - ∫ v du, including repeated application · be able to integrate rational functions by first decomposing into partial fractions

Integration Using Trigonometric Identities

use trigonometrical relationships in carrying out integration · use the double-angle formulae to integrate sin^2 x and cos^2(2x), e.g. rewrite sin^2 x = (1-cos 2x)/2 then integrate term by term · use further trig identities (product-to-sum, sec^2 x = 1 + tan^2 x) to convert an integrand into a directly integrable form

Locating Roots of an Equation

be able to locate roots of f(x) = 0 by using a change of sign in f(x) over a suitable interval, recognising the conditions under which this method may fail

Iterative Methods

be able to solve equations approximately using an iteration of the form x_(n+1) = g(x_n), and understand the conditions for convergence

Vectors: Notation, Operations, the Scalar Product, Lines and Skew Lines in 3D

understand and use vectors in three dimensions, including position vectors, magnitude, unit vectors and the i, j, k basis · understand and use the scalar (dot) product a · b = |a||b| cos θ to find angles between vectors and to test for perpendicularity · be able to find the vector equation of a line in 3D, determine whether two lines are parallel, intersect or are skew, and find the angle between two lines

First-Order Separable Differential Equations

be able to formulate and solve first-order differential equations with separable variables, including those arising from contextual modelling problems · be able to find particular solutions of separable differential equations using given boundary conditions and interpret solutions in context

Complex Numbers in Cartesian Form

understand the idea of a complex number; recall real/imaginary part, modulus, argument and conjugate (Re z, Im z, |z|, arg z, z*); equality iff real and imaginary parts equal · addition, subtraction, multiplication and division of complex numbers in Cartesian form x+iy with full working · for a polynomial equation with real coefficients, non-real roots occur in conjugate pairs (e.g. solving a cubic/quartic given one complex root) · find the two square roots of a complex number in exact Cartesian form (e.g. square roots of 5+12i)

The Argand Diagram and Polar (Modulus-Argument) Form

represent complex numbers on an Argand diagram, plotting z, its conjugate and sums/differences · multiplication and division in polar form r(cos theta + i sin theta) / r e^(i theta), including |z1 z2|=|z1||z2|, arg(z1 z2)=arg z1+arg z2 and the division results · geometrical effects of conjugating and of adding, subtracting, multiplying and dividing complex numbers

Loci in the Argand Diagram

illustrate simple equations and inequalities involving complex numbers by loci in an Argand diagram, e.g. |z-a|<k (disc/circle), |z-a|=|z-b| (perpendicular bisector), arg(z-a)=alpha (half-line)

Forces, Components and Resultants

be able to add and subtract vectors, multiply a vector by a scalar, and resolve a vector into components parallel and perpendicular to a given direction · be able to find the magnitude and direction of a vector expressed in component form, and write a vector in i, j component form given its magnitude and direction

Equilibrium of a Particle

be able to solve problems involving the equilibrium of a particle under the action of two or more coplanar forces by resolving in two perpendicular directions

Friction and Limiting Equilibrium

understand the model of friction and use F ≤ μR for a rough contact, recognising that at the point of slipping F = μR

Velocity, Acceleration and the Constant-Acceleration Equations

be able to use the equations for uniformly accelerated motion in one dimension: v = u + at, s = (u + v)t/2, s = ut + 1/2 at^2 and v^2 = u^2 + 2as · be able to apply these equations to problems involving vertical motion under gravity, taking g = 9.8 m s^(-2) unless otherwise specified

Displacement-Time and Velocity-Time Graphs

be able to draw and interpret displacement-time and velocity-time graphs, including the use of gradients to find velocity and acceleration, and areas to find displacement and change in velocity

Kinematics Using Differentiation and Integration

understand the relationships v = dx/dt and a = dv/dt = d^2 x/dt^2, and use calculus to solve kinematics problems with variable acceleration in one dimension

Momentum and Conservation of Linear Momentum

understand the definition of momentum p = mv and impulse J = F t = Δp, and apply the principle of conservation of linear momentum to direct collisions between particles

Newton's Laws of Motion, F = ma, and Motion on Inclined/Rough Planes

understand and apply Newton's first, second and third laws of motion to particles moving in one dimension, including using F = ma · be able to draw force diagrams and write equations of motion for particles experiencing weight, normal reaction, tension, thrust and friction

Connected Particles: Pulleys and Tow-Bars

be able to apply Newton's laws to systems of connected particles, including particles linked by a light inextensible string passing over a smooth pulley

Work, Energy and Conservation of Energy

understand and use the definitions of work done by a force, kinetic energy KE = 1/2 m v^2 and gravitational potential energy PE = m g h · be able to apply the work-energy principle and the principle of conservation of mechanical energy to motion under gravity, friction and other forces

Power

understand and use the definition of power as P = Fv for a vehicle of constant driving force, and apply it to problems of motion on level ground and on an inclined plane

Representing Data: Stem-and-Leaf, Box Plots, Histograms and Cumulative Frequency

be able to construct and interpret stem-and-leaf diagrams and histograms (including histograms with unequal class widths, using frequency density = frequency / class width) · be able to construct and interpret cumulative-frequency diagrams and box plots, identify outliers using the 1.5 × IQR rule, and comment on skewness from a diagram

Measures of Central Tendency and Variation; Mean and SD from Totals

be able to calculate the mean, median, mode, range, and interquartile range (IQR) for raw and grouped data, and choose an appropriate measure of location and spread · be able to calculate the variance and standard deviation of raw and grouped data using Σ(x − x̄)²/n = Σx²/n − x̄²; understand coding (linear transformation) and its effect on mean and standard deviation

Permutations and Arrangements

understand the term permutation and apply the counting (multiplication) principle · arrangements of n distinct objects in a line (n!) and r from n (nPr = n!/(n-r)!) · arrangements involving repetition (e.g. arranging the letters of NEEDLESS) · arrangements involving restriction (e.g. people in a line where two must, or must not, be adjacent), including two or more rows

Combinations and Selections

understand permutation vs combination (order matters vs not) · use nCr = n!/(r!(n-r)!) to count selections of r from n · solve selection problems including selections subject to conditions/restrictions (9709 excludes circular arrangements)

Probability: Sample Spaces and the Rules of Probability

understand the language of probability: sample space, outcomes, mutually exclusive events, exhaustive events, complementary events · be able to use the addition rule P(A ∪ B) = P(A) + P(B) - P(A ∩ B) and the multiplication rule for independent events P(A ∩ B) = P(A) P(B)

Conditional Probability and Tree Diagrams

be able to use conditional probability P(A | B) = P(A ∩ B) / P(B), interpret the result and assess independence by comparing P(A | B) with P(A) · be able to draw and interpret tree diagrams and Venn diagrams to solve probability problems

Discrete Random Variables: Distributions, Expectation and Variance

understand the concept of a discrete random variable, its probability distribution and the requirement that probabilities sum to 1 · be able to calculate the expectation E(X), variance Var(X) and standard deviation of a discrete random variable, including E(aX + b) and Var(aX + b)

The Binomial Distribution

understand the conditions under which a random variable can be modelled by B(n, p): a fixed number of independent trials with two outcomes and constant probability of success · be able to calculate probabilities, expectation E(X) = np and variance Var(X) = np(1 - p) for a binomially distributed variable

The Geometric Distribution

use the formula for geometric probabilities Geo(p): P(X=r)=p(1-p)^(r-1), r=1,2,3,... · recognise situations modelled by the geometric distribution (trials up to and including the first success, constant p) · use the expectation E(X)=1/p (proof not required)

The Normal Distribution

understand the properties of the normal distribution N(μ, σ^2), including symmetry about μ and the inflection points at μ ± σ · be able to use standardisation Z = (X - μ) / σ and tables (or a calculator) of the standard normal distribution to find probabilities and inverse values · be able to solve problems where μ and/or σ are unknown by setting up and solving simultaneous equations from given probabilities

Normal Approximation to the Binomial Distribution

conditions for approximating B(n,p) by a normal distribution: n large enough that np>5 and nq>5 (q=1-p) · approximate B(n,p) by N(np, npq) and use the standard normal distribution · apply a continuity correction with full standardisation working

The Poisson Distribution

understand the conditions under which a random variable can be modelled by Po(λ): events occurring independently at a constant average rate λ in a fixed interval · be able to calculate Poisson probabilities, use E(X) = Var(X) = λ, and use the Poisson distribution as an approximation to the binomial when n is large and p is small

Normal Approximation to the Poisson Distribution

use the normal distribution, with continuity correction, as an approximation to the Poisson distribution where appropriate (m large; m >= 15 approximately)

Linear Combinations of Random Variables

understand and use the results E(aX + bY) = a E(X) + b E(Y) and, for independent X and Y, Var(aX + bY) = a^2 Var(X) + b^2 Var(Y) · understand that linear combinations of independent normal variables are themselves normally distributed, and use this to find probabilities

Continuous Random Variables and Probability Density Functions

understand the concept of a continuous random variable defined by a probability density function f(x) and use ∫ f(x) dx = 1 over the support · be able to find probabilities, the cumulative distribution function, the median and other percentiles, and the expectation and variance of a continuous random variable

Sampling: Populations, Samples and Randomness

understand the purpose of sampling and recognise common sampling methods (simple random, stratified, systematic, quota, cluster, opportunity), including their advantages and disadvantages

The Sample Mean as a Random Variable and the Central Limit Theorem

understand and use the result that if X has mean μ and variance σ^2, then the sample mean of n independent observations has mean μ and variance σ^2 / n, and is normally distributed when X is normal or n is large (central limit theorem)

Unbiased Estimates and Confidence Intervals for a Mean

be able to construct and interpret confidence intervals for a population mean when σ is known (using the normal distribution) and when σ must be estimated (using the t distribution)

Confidence Interval for a Population Proportion

determine, from a large sample, an approximate confidence interval for a population proportion (p-hat +/- z*sqrt(p-hat(1-p-hat)/n)) · interpret the proportion confidence interval in context and recognise the large-sample (normal approximation) validity conditions

Hypothesis Testing: Concepts and the Binomial Test

understand the language of hypothesis testing: null hypothesis, alternative hypothesis, test statistic, significance level, critical region, p-value, and one- and two-tailed tests · be able to carry out a hypothesis test for the parameter p of a binomial distribution, identify the critical region and state conclusions in context

Hypothesis Test for a Poisson Distribution

be able to carry out a hypothesis test for the parameter λ of a Poisson distribution, identify the critical region and state conclusions in context

Hypothesis Test for a Population Mean

be able to carry out hypothesis tests for the mean of a normal distribution and for the difference between two means using independent samples, applying both z- and t-tests where appropriate

Type I and Type II Errors

understand Type I and Type II errors in hypothesis tests · calculate P(Type I error) in normal/binomial/Poisson tests by direct evaluation or standard normal · calculate P(Type II error) for a specified alternative value in the same contexts · interpret Type I and Type II error probabilities in context