Course Specifications IB Math AI HL

Course Specifications for IB Math AI HL

1. Number & Algebra

  • Number Toolkit

    • Standard Form

    • Approximation

    • Upper & Lower Bounds

    • Percentage Error

    • Accuracy & Estimation

    • Solving Equations using a GDC

  • Exponentials & Logs

    • Laws of Indices

    • Introduction to Logarithms

    • Laws of Logarithms

  • Sequences & Series

    • Language of Sequences & Series

    • Sigma Notation

    • Arithmetic Sequences & Series

    • Geometric Sequences & Series

    • Applications of Sequences & Series

  • Financial Applications

    • Compound Interest & Depreciation

    • Amortisation

    • Annuities

  • Complex Numbers

    • Introduction to Complex Numbers

    • Operations with Complex Numbers

    • Complex Roots of Quadratics

    • Modulus & Argument

    • Introduction to Argand Diagrams

  • Further Complex Numbers

    • Geometry of Complex Numbers

    • Modulus–Argument (Polar) Form

    • Exponential (Euler’s) Form

    • Conversion between Forms of Complex Numbers

    • Frequency & Phase of Trig Functions

  • Matrices

    • Introduction to Matrices

    • Operations with Matrices

    • Determinants & Inverses

    • Solving Systems of Linear Equations with Matrices

  • Eigenvalues & Eigenvectors

    • The Characteristic Polynomial, Eigenvalues & Eigenvectors

    • Diagonalisation & Powers of Matrices

2. Functions

  • Linear Functions & Graphs

    • Equations of a Straight Line

    • Parallel & Perpendicular Lines

  • Further Functions & Graphs

    • Functions & Mappings

    • Graphing Functions & Their Key Features

    • Intersecting Graphs

    • Quadratic Functions & Graphs

    • Cubic Functions & Graphs

    • Exponential Functions & Graphs

    • Sinusoidal Functions & Graphs

  • Modelling with Functions

    • Linear Models

    • Quadratic Models

    • Cubic Models

    • Exponential Models

    • Direct & Inverse Variation

    • Sinusoidal Models

    • Strategy for Modelling Functions

  • Functions Toolkit

    • Composite Functions

    • Inverse Functions

  • Transformations of Graphs

    • Translations of Graphs

    • Reflections of Graphs

    • Stretches of Graphs

    • Composite Transformations of Graphs

  • Modelling with Logarithmic, Logistic & Piecewise Functions

    • Natural Logarithmic Models

    • Logistic Models

    • Piecewise Models

3. Geometry & Trigonometry

  • Geometry Toolkit

    • Coordinate Geometry

    • Perpendicular Bisectors

    • Arcs & Sectors Using Degrees

    • Radian Measure

    • Arcs & Sectors Using Radians

  • Geometry of 3D Shapes

    • 3D Coordinate Geometry

    • Volume & Surface Area

  • Trigonometry

    • Pythagoras & Right-Angled Trigonometry

    • Sine Rule, Cosine Rule & Area of a Triangle

    • Angles of Elevation & Depression

    • Bearings & Constructions

  • Trigonometric Identities & Equations

    • The Unit Circle

    • Simple Identities

    • Graphs of Trigonometric Functions

    • Solving Equations Using Trigonometric Graphs

  • Voronoi Diagrams

    • Drawing Voronoi Diagrams

    • Interpreting Voronoi Diagrams

    • Toxic Waste Dump Problem

  • Matrix Transformations

    • Matrix Transformations

    • Matrices of Geometric Transformations

    • Matrices of Composite Transformations

    • Determinant of a Transformation Matrix

  • Vector Properties

    • Introduction to Vectors

    • Parallel Vectors

    • Adding & Subtracting Vectors

    • Position & Displacement Vectors

    • Magnitude of a Vector & Unit Vectors

    • The Scalar Product

    • Angle Between Two Vectors & Perpendicular Vectors

    • The Vector Product

    • Areas using the Vector Product

    • Components of Vectors

    • Geometric Proof with Vectors

  • Vector Equations of Lines

    • Equation of a Line in Vector Form

    • Equation of a Line in Parametric Form

    • Angle Between Two Lines

    • Shortest Distance Between a Point and a Line

    • Shortest Distance Between Two Lines

  • Modelling with Vectors

    • Kinematics with Vectors

    • Constant & Variable Velocity

  • Graph Theory

    • Introduction to Graph Theory

    • Walks & Adjacency Matrices

    • Minimum Spanning Trees (Kruskal’s Algorithm)

    • Minimum Spanning Trees (Prim’s Algorithm)

    • Chinese Postman Problem

    • Travelling Salesman Problem

    • Nearest Neighbour & Deleted Vertex Algorithms

4. Statistics & Probability

  • Statistics Toolkit

    • Sampling

    • Reliability & Validity of Data Collection Methods

    • Measures of Central Tendency

    • Measures of Dispersion

    • Frequency Tables

    • Linear Transformations of Data

    • Outliers

    • Box & Whisker Diagrams

    • Cumulative Frequency Graphs

    • Histograms

    • Interpreting Data

  • Correlation & Regression

    • Scatter Diagrams & Correlation

    • Pearson’s Product-Moment Correlation Coefficient

    • Spearman’s Rank Correlation Coefficient

    • Comparison of Correlation Coefficients

    • Linear Regression

  • Non-linear Regression

    • Least Squares Regression Curves & Coefficient of Determination

    • Logarithmic Scales

    • Linearising using Logarithms

  • Probability

    • Probability & Types of Events

    • Independent & Mutually Exclusive Events

    • Conditional Probability

    • Venn Diagrams

    • Tree Diagrams

  • Probability Distributions

    • Discrete Probability Distributions

    • Expected Values

  • Random Variables

    • Linear Combinations of Random Variables

    • Unbiased Estimates

  • Binomial Distribution

    • The Binomial Distribution

    • Calculating Binomial Probabilities

  • Normal Distribution

    • The Normal Distribution

    • Calculations with Normal Distribution

  • Combinations of Normal Distributions & Sample Mean Distributions

    • Sample Mean Distribution

    • Central Limit Theorem

    • Confidence Interval for the Mean

  • Poisson Distribution

    • Poisson Distribution

    • Calculating Poisson Probabilities

  • Hypothesis Testing using the Chi-squared Distribution

    • Introduction to Hypothesis Testing

    • Chi-squared Test for Independence

    • Goodness of Fit Test

  • Hypothesis Testing for Population Parameters

    • Hypothesis Testing for Mean (One Sample)

    • Hypothesis Testing for Mean (Two Sample)

    • Binomial Hypothesis Testing

    • Poisson Hypothesis Testing

    • Hypothesis Testing for Correlation

    • Type I & Type II Errors

  • Transition Matrices & Markov Chains

    • Markov Chains

    • Transition Matrices

    • Powers of Transition Matrices

    • Steady State & Long-term Probabilities

5. Calculus

    • Differentiation

      • Introduction to Derivatives

      • Differentiating Powers of x

      • Gradients, Tangents & Normals

      • Increasing & Decreasing Functions

      • Local Minimum & Maximum Points

      • Modelling with Differentiation

    • Further Differentiation

      • Differentiating Special Functions

      • Chain Rule

      • Product Rule

      • Quotient Rule

      • Related Rates of Change

      • Second Order Derivatives

      • Stationary Points

      • Concavity & Points of Inflection

    • Integration

      • Numerical Integration using the Trapezoidal Rule

      • Introduction to Integration

      • Integrating Powers of x

      • Finding the Constant of Integration

      • Finding Areas Using a GDC

    • Further Integration

      • Integrating Special Functions

      • Reverse Chain Rule

      • Integration by Substitution

      • Definite Integrals

      • Negative Integrals

      • Area Between Curve & y-axis

      • Area Between a Curve and a Line

      • Volumes of Revolution

    • Kinematics

      • Displacement, Velocity & Acceleration

      • Calculus for Kinematics

    • Differential Equations

      • Separation of Variables

      • Modelling with Differential Equations

      • Slope Fields

      • Approximate Solutions to Differential Equations using Euler’s Method

    • Coupled & Second Order Differential Equations

      • Coupled Differential Equations

      • Phase Portraits

      • Equilibrium Points

      • Sketching Solution Trajectories

      • Second Order Differential Equation

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