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

Prime numbers represent both the most basic properties and the most complex unsolved problems. Here you can learn about the building blocks of mathematics.

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Fractions

Comparing fractionsSimplifying fractionsAddition and subtractionMultiplication and divisionMixed numbers and improper fractions

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Decimals

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

Geometry

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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

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Combinatorics

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

Geometry

Euclidean Geometry

Geometry is one of the oldest parts of mathematics – and one of the most useful. Its logical, systematic approach has been copied in many other areas.

Transformations and Symmetry

Symmetry can be seen everywhere in nature – but it also underlies completely invisible laws of nature. Mathematics can explain why that is the case.

Triangles and Trigonometry

Triangles are some of the most important shapes in geometry: they have countless interesting properties and appear everywhere in engineering and technology.

Polygons and Polyhedra

Geometric shapes are everywhere around us. In this course you will learn about angels, polygons, tessellations, polyhedra and nets.

Circles and Pi

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

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Non-Euclidean Geometry

Metric spacesSpherical geometryHyperbolic geometryProjectionsHigher dimensionsTopologyMöbius strip and Klein bottle

Algebra

Sequences and Patterns

Learn about some of the most fascinating patterns in mathematics, from triangle numbers to the Fibonacci sequence and Pascal’s triangle.

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Functions

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

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Quadratics

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

Probability

Cards, dice, roulette and game shows: probability is one of the most fun areas of mathematics, full of surprises and real life applications.

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Statistics

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

Graphs and Networks

Discover the mathematical principles that connect our world: from shaking hands to travel and navigation, colouring maps and social networks.

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Codes and Ciphers

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

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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

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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

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Infinity

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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Trigonometry

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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Vectors

IntroductionMagnitude and directionVector arithmeticPosition vectorsScalar productEquation of lines and planes

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Matrices

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

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Fractals

Dimension of fractalsThe Sierpinski gasketMandelbrot setFractals in nature and technology

Calculus

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Differentiation

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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Integration

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

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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