Course Library

Numbers and Arithmetic

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The Integers

Whole number arithmeticNegative numbersThe number lineAbsolute valueInverse operationsOrder of operationsAssociative, commutative and distributive lawDivision by zero

Divisibility and Primes

Factors and multiplesDivisibility rulesPrime numbers and prime factorisationnLowest common multiple and Greatest common divisor

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Comparing fractionsSimplifying fractionsAddition and subtractionMultiplication and divisionMixed numbers and improper fractions

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Decimal notationDecimals on the number lineAddition and subtractionMultiplication and divisionConverting between fractions and decimals

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Percentages and Ratios

PercentagesPercentage changeRatio notationCalculating ratios

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Exponents and Roots

ExponentsExponent lawsRoots, negative and fractional exponentsRationalising the denominatorRational and irrational numbersApproximating irrational numbers using rationals

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Place Value and Rounding

Place valueBig and small numbersOrderingRounding integers and decimalsScientific notationDifferent bases

Equations and Functions

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Proportional Relationships

Direct and inverse proportionsGraphs of proportional relationshipsScale factorsRate of change

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Introduction to Algebra

Introduction and historyVariablesWriting, manipulating and evaluating algebraic expressionsModelling using algebra

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Linear Equations

Equations and identitiesManipulating and rearranging equationsLinear equations in one variableSolve simultaneous linear equationsLinear inequalities

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Linear Functions

Introduction to functionsGraphs of linear equationsIntercepts and slopeParallel and perpendicular lines


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Lines and Angles

Points, lines, segments and raysParallel and perpendicular linesAngles and classification (acute, right, obtuse)Adjacent, opposite, alternate and corresponding anglesThe coordinate plane

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2D Shapes

Area and perimeter of triangles and quadrilateralsArea and circumference of circlesSum of angles in polygonsRuler and compass constructionPythagoras’ theoremCongruence and similarity

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3D Solids

Cuboids, prisms and pyramidsCones, cylinders and spheresCavalieri’s principleDensity and weightsNets of 3D solidsCross-sections of 3D objectsModelling the real world

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Units and Measuring

Space, time and weight unitsUnit conversionMeasuring angles distances in geometric figuresScale drawings

Probability and Statistics

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Basic Probability

IntroductionSimple probability experimentsUnions, intersections and Venn diagramsIndependent and mutually exclusive eventsCalculating probabilitiesTree diagrams



FactorialsPermutations and combinationsProbabilityRandom Walks

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Data and Statistics

Mean, meadian and modeRange, quartiles, standard deviation and outliersSampling and estimationBar charts, pie charts, scatter plots and frequency tablesStem-and-leaf plots, box plots and histogramsMisleading statistics


Euclidean Geometry

History and definitionsCongruenceEuclid’s postulatesStraight-edge and compass constructionsOrigamiApplications

Transformations and Symmetry

Reflections, rotations and translationsReflectional and rotational symmetrySymmetry groupsWallpaper symmetriesSimilarity

Triangles and Trigonometry

Triangle propertiesMedians, midsegments, incircle and circumcircleTriangle inequalityTriangle congruencePythagoras’ theoremIsosceles and equilateral trianglesTrigonometrySine and cosine rules

Polygons and Polyhedra

Angles in polygonsPolygons propertiesQuadrilateralsTessellationsPolyhedraEuler’s FormulaPlatonic and Archimedean solids

Circles and Pi

Circles and PiRadiansChords, arcs and tangentsCircle theoremsInscribed shapes and anglesSpheres, cones and cylindersConic sections


Non-Euclidean Geometry

Metric spacesSpherical geometryHyperbolic geometryProjectionsHigher dimensionsTopologyMöbius strip and Klein bottle


Sequences and Patterns

Triangle, square and cube numbersLinear and quadratic sequencesFibonacci numbersPerfect numbersArithmetic and geometric sequencesRecursive equationsPascal’s triangle

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Function notationDomain and rangeAbsolute value and piecewise functionsIncreasing, decreasing, maxima and minimaRecognising and interpreting functionsApproximate solutions to equationsGraphing two-variable equationsAverage rate of changeArea under a curve

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Polynomials and Factorisation

Adding, subtracting and multiplying polynomialsBinomialsDifference of squaresFactoring simple polynomials



Solving quadratic equationsCompleting the squareThe quadratic formulaParabolaeStandard and vertex formTransforming parabolae

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Inequalities and Systems of Equations

Linear inequalitiesQuadratic inequalitiesGraphing two-variable inequalitiesSystems of equations with two unknownsElimination and substitution methods

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Exponential and Rational Functions

Exponents, radicals and rational expressionsSquare and cube root equationsRadical functionsExponential growth and decayLinear vs exponential models

Probability and Discrete Mathematics


IntroductionCoins, dice and rouletteVenn diagrams and probability treesIndependent and mutually exclusive eventsConditional probabilityThe Monty Hall problemRandomness

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Sampling and estimationTables, graphs and histogramsCentral tendency and spreadScatter plots and regressionCorrelation vs causationComparing models and data sets

Graphs and Networks

IntroductionComplete and bipartite graphsKönigsberg bridgesPlanar graphsMap colouringTravelling salesman problemSocial networks and other applications

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Codes and Ciphers

Simple codes and MorseError detection and correctionOne time padsThe Caesar and Vigenère cipherThe EnigmaRSA cryptography


Game Theory

Combinatorial games and NimTree diagramsPrisoners’ paradoxGames based on chanceThe lottery

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Algorithms and Computing

Introduction to computingComputational complexity and the O() notationSorting algorithmsLinear programming and the simplex algorithmData structuresGraphs, trees and networks

The Language of Mathematics


Logic, Sets and Proof

Set theoryLogic and proofProof by contradictionProof by induction

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Imaginary and Complex Numbers

IntroductionComplex algebraThe complex planeModulus and argument formComplex conjugatesQuadratic, cubic and quartic equationsDe Moivre’s theoremEuler’s identityComplex roots of unity



Hilbert’s hotelCountabilityCantor’s diagonalThe continuum hypothesis

Algebra and Analysis

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Advanced Algebra

Manipulating rational expressionsSystems of quadratic equationsLinear and quadratic inequalitiesPartial fraction expansionPolynomialsPolynomial divisionFactor and remainder theoremBinomial theoremFundamental theorem of algebraSolve polynomial equations up to quartics

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Advanced Functions

Combining and composing functionsInverse functionsFunction transformations (shifting and stretching)Polynomial functionsRational functions and discontinuitiesLimit behaviour of polynomial and rational functionsGraph sketching

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Exponentials and Logarithms

Exponential equationsExponential growth and decayIntroduction to LogarithmsThe natural logarithm and eLogarithmic equationsGraphs of exponential and logarithmic functionsLogarithmic scale

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Numerical Methods

Change of sign methodsInterval bisectionNewton­-Raphson method

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Sequences and series

Binomial expansion of (a + bx)^nRecurrence relationsArithmetic and geometric seriesSigma notationMaclaurin seriesConvergence and divergence

Geometry, Trigonometry and Matrices

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Coordinate Geometry

Equation of lines and circlesDistances and midpointsParallel and perpendicular linesQuadrilaterals and polygons

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Conic Sections

Equation of a circleEllipsesParabolae and hyperbolaeKepler’s laws and planetary orbits

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Unit circle definitionsTrigonometric functions and their graphsAmplitude and frequencyPythagorean identitiesDouble angle and addition formulasSecant, cosecant and cotangentInverse trigonometric functionsSimple trigonometric equationsHyperbolic functions

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IntroductionMagnitude and directionVector arithmeticPosition vectorsScalar productEquation of lines and planes

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IntroductionMatrix arithmeticMatrix multiplictionMatrices as transformationsDeterminantsMatrix inversesLinear systems of equationsGaussian elimination

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Parametric and Polar Coordinates

Parametric curvesConversion between polar and cartesianSketching polar curvesArea enclosed by a polar curve



Dimension of fractalsThe Sierpinski gasketMandelbrot setFractals in nature and technology


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Differentiation as gradient of tangentsLimits and rate of changeSimple derivativesProduct rule, quotient rule and chain ruleImplicit differentiationIncreasing, decreasing, convex and concave functionsEquations of tangents and normalsStationary points and optimisation problemsDerivatives of inverse functionsMean value theoremL’Hôpital’s rule

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Integrals as anti-derivativesIntegrals as areas under a curveFundamental theorem of calculusTrapezium ruleIntegration by parts or substitutionIntegration using partial fraction expansionsIntegration using trigonometric identitiesImproper integralsArc lengthVolumes of revolution

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Differential Equations

First order differential equationsSeparable equationsLogistic mapSecond order differential equationsHomogenous equations and particular integralCoupled differential equationsSimple harmonic motion and dampened oscillators


Chaos Theory

Double PendulumsThree body problemLogistic MapStrange AttractorsButterfly effectChaotic scattering

Probability and Statistics

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Advanced Probability

Populations and sample spacesRandom variables and distributionsBinomial distributionPoisson distributionNormal distributionExpectation and varianceLaw of large numbersModelling

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Hypothesis Tests

Designing statistical experimentsHypothesis testsEstimating population mean

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Data Representation

Histograms and frequency tablesScatter plots and regressionMean and standard deviationData collectionData visualisation