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Numbers and Arithmetic

The Integers

Arithmetic
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The Number Line
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Negative Numbers
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Absolute Value
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Properties of Zero
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Place Value
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Fractions

Introduction
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Fraction Arithmetic
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Mixed Numbers
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Dividing Fractions
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Decimals

Introduction
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Adding and Subtracting Decimals
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Ordering Decimals
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Multiplying and Dividing Decimals
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Converting Decimals and Fractions
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Rounding
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Rates, Percentages and Ratios

Ratios and Mixtures
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Percentages
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Percentage Increase and Decrease
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Interest
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Ratios and Rates
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Equations and Functions

Introduction to Algebra

Proportional Relationships
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Graphs an Variables
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Manipulating Expressions
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Modelling
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Linear Equations

Weighing and Balancing
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Tape Diagrams
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Solving Linear Equations
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Inequalities
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Linear Functions

Input, Output and Graphs
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Slope and Intercept
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Parallel and Perpendicular Lines
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Systems of Equations
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Roots and Exponents

Square and Cube Roots
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Rational and Irrational Numbers
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Powers and Exponents
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Scientific Notation
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Geometry

Area and Shapes

Introduction
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Parallelograms
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Triangles
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Polygons
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Circles and Circumferences
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Area of Circles
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Angles and Polygons

Angles
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Angles in Polygons
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Drawing Triangles
Pythagoras’ Theorem
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The Coordinate Plane
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Transformations and Congruence
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3D Solids

Introduction
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Nets and Surface Area
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Prisms and Pyramids
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Cylinders and Cones
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Spheres
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Units and Measuring

Measuring
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Units and Conversion
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Scale Drawings
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Scaling and Dimensions
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Estimation
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Probability and Statistics

Introduction to Probability

Introduction
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Computing Probabilities
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Probability Trees
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Venn Diagrams
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Data and Statistics

Introduction
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Center and Spread
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Visualising Data
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Sampling
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Scatter Plots and Linear Models
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Geometry

Euclidean Geometry

IntroductionEuclid’s AxiomsRuler and Compass Construction
Even More Constructions
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Angles and Proofs
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Origami and Paper Folding

Transformations and Symmetry

IntroductionRigid Transformations
Congruence
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SymmetrySymmetry Groups and WallpapersSymmetry in PhysicsDilations
Similarity
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Triangles and Trigonometry

IntroductionProperties of Triangles
Midsegments and Similarity
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Triangle CongruencePythagoras’ Theorem
Isosceles and Equilateral Triangles
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TrigonometrySine and Cosine Rules

Polygons and Polyhedra

PolygonsQuadrilateralsTessellationsPolyhedra
Nets and Cross Sections
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Scaling Shapes and Solids
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Platonic Solids

Circles and Pi

IntroductionDegrees and RadiansTangents, Chords and Arcs
The Circle Theorems
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Cyclic Polygons
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Spheres, Cones and CylindersConic Sections

Non-Euclidean Geometry

Spherical GeometryMap ProjectionsHyperbolic GeometryMetric SpacesTopology
Higher Dimensions
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Algebra

Functions

Relations and Functions
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Graphing and Interpreting Functions
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Piecewise Functions
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Absolute Value Functions
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Inverse Functions
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Rates of Change
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Sequences and Patterns

IntroductionArithmetic and Geometric SequencesFigurate Numbers
Sequences as Functions
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Fibonacci NumbersSpecial SequencesPascal’s Triangle
Limits and Convergence
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Quadratic Equations

Introduction
Binomial Expressions
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Solving Quadratic Equations
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The Quadratic Formula
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Graphing Quadratics
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Projectile Motion
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More Applications
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Inequalities and Systems of Equations

Systems of Linear Equations
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Row Operations and Elimination
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Linear Inequalities
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Systems of Inequalities
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Quadratic Inequalities
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Exponential Functions

Carbon Dating
Exponential Growth and Decay
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Comparing Models
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Compound Interest
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Population Dynamics
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Probability and Discrete Mathematics

Probability

Introduction
Probability Trees and Venn Diagrams
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Conditional Probability
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The Monty Hall Problem
The Birthday Problem
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True Randomness

Statistics and Data

Casino Mathematics
Data Visualisation
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Center and Spread of Data
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Sampling and Estimation
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The Wisdom of Crowds
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Spreadsheets and Frequency Tables
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Linear Models
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Codes and Ciphers

Introduction
Binary Numbers
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Error Detection
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Secret Codes
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The Enigma
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Public Key Cryptography
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Game Theory

The Prisoners’ Dilemma
Cards, Coins and Dice
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The Winning Move
Random Walks
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Algebra and Analysis

Polynomials

Introduction
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Zeros of Polynomials
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Sketching Polynomial Functions
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The Factor and Remainder Theorems
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Systems of Equations
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Function Transformations

Combining and Composing Functions
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Translating Functions
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Reflecting Functions
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Scaling Functions
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Inverse functions
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Rationals and Radicals

Rational and Irrational Numbers
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Rational Functions and Expressions
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Solving Rational Equations
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Exponent Laws
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Radical Functions
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Solving Radical Equations
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Exponentials and Logarithms

Exponential Growth and Decay
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Exponential Functions
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Introduction to Logarithms
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Laws of Logarithms
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The Number e
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Logarithmic Functions
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Sequences and Series

Sequences
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Series and Sigma Notation
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Arithmetic and Geometric Series
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The Binomial Theorem
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Logic, Sets and Proof

Logic and Paradoxes
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Axioms and Proof
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Proof by Induction
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Infinity and Hilbert’s Hotel
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Geometry and Linear Algebra

Coordinate Geometry

Equations of Lines
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Parallel and Perpendicular Lines
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Equations of Circles
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Properties of Polygons
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Transformations
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Trigonometry

The Unit Circle Definition
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Graphs of Trigonometric Functions
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Amplitude, Frequency and Transformations
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Pythagorean Identities
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More Trigonometric Identities
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Inverse Trigonometric Functions
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Circular Motion
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Conic Sections and Polar Coordinates

Parametric Curves
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Circles and Ellipses
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Parabolae
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Hyperbolae
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Polar Coordinates
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Vectors

Introduction
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Vector Arithmetic
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Scalar Products and Equations of Planes
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Cross Products and Equations of Lines
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Geometry Problems
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Matrices

Transformations
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Matrix Arithmetic
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Determinants
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Matrix Inverses
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Cramer’s Rule and Gaussian Elimination
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Eigenvalues and Eigenvectors
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Complex Numbers

Introduction
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Complex Arithmetic
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Euler’s Formula
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Solving Polynomials
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De Moivre’s Theorem and Roots of Unity
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Fractals

IntroductionThe Sierpinski TriangleThe Mandelbrot Set
Space Filling Curves
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Calculus

Differentiation

Introduction
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Limits and Gradients
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Differentiation Rules I
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Differentiation Rules II
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Optimisation problems
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Integration

Introduction
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Integration Rules
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Definite Integrals and Areas under a Curve
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Improper Integrals
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Solids of Revolution
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Numerical Methods

Solving Equations Numerically
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The Newton-Raphson Method
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Numerical Integration
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Maclaurin and Taylor series
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Differential Equations

Simple Differential Equations
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First Order Separable Equations
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Second Order Differential Equations
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Homogenous Equations and Particular Integrals
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Simple Harmonic Motion
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Coupled Differential Equations
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Chaos Theory

Introduction
Mathematical Billiard
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The Three Body Problem
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Phase Space and Strange Attractors
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The Logistic Map
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Probability and Statistics

Random Variables

Introduction
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Discrete Random Variables
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Binomial and Poisson Distribution
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Continuous Random Variables
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The Normal Distribution
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The Central Limit Theorem
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Statistics and Hypothesis Tests

Sampling and Estimation
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Hypothesis Tests and Confidence Intervals
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Linear Models and Correlation Coefficients
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Contingency tables and Chi Squared Tests
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Bayesian Statistics
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Algorithms

Introduction to Computing
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Complexity and O Notation
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Sorting Algorithms
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Linear Programming and the Simplex Algorithm
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Graphs, Trees and Networks
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Machine Learning

Introduction
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Linear Regression
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Support Vector Machines
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Neural Networks
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Unsupervised Learning
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