Mathematics: Applications and Interpretation

Numbers — Scientific Notation and Approximation

Scientific Notation — taming very big and very small numbers · Significant Figures, Bounds, and Percentage Error

Sequences — Arithmetic and Geometric Patterns

Arithmetic Sequences and Series · Geometric Sequences and Series

Money — Compound Interest, Depreciation, Amortization

Compound Interest and Depreciation · Amortization and Annuities (using technology)

Solving with Technology — Exponents, Logs, and Equations

Exponents and Logarithms (base 10 and e) · Solving Systems and Polynomial Equations with Technology

Logarithms and Rational ExponentsHL

Laws of Logarithms · Rational Exponents

Infinite Convergent Geometric SeriesHL

When Geometric Series Converge · Sum to Infinity Formula

Complex Numbers — Cartesian FormHL

Real and Imaginary Parts, Cartesian Arithmetic · Argand Diagram and Complex Roots

Complex Numbers — Polar and Exponential FormHL

Modulus, Argument, and Form Conversions · Geometric Multiplication

Matrices — Operations, Determinants, Inverses, Linear SystemsHL

Matrix Operations · Determinant, Inverse, and Linear Systems

Eigenvalues, Eigenvectors, and DiagonalizationHL

Eigenvalues from the Characteristic Equation · Eigenvectors and Diagonalization

Lines and Functions

Linear Equations and Parallel/Perpendicular Lines · Function Notation, Domain, Range, Inverse

Graphing with Technology

Setting an Appropriate Window on a GDC · Reading Key Features and Solving by Intersection

Choosing a Model

Choosing and Fitting a Model · Direct vs Inverse Variation

The Modelling Process

Fit, Test, Refine — the Modelling Cycle · Extrapolation — When the Model Stops Working

Composite and inverse functionsHL

(f ∘ g)(x) · Domain restrictions for invertibility

Graph transformations — translations, reflections, stretchesHL

y = f(x − h) + k · y = af(bx) and reflections

Extended models — half-life, logarithmic, logistic, piecewiseHL

Exponential decay and half-life · Logistic growth · Piecewise definitions

Logarithmic scaling and linearizationHL

Linearizing y = a·bˣ via log y · Linearizing y = a·xⁿ via log–log

3D geometry — distance, midpoint, volume, surface area

3D distance and midpoint · Volume and surface area of standard solids · Angles in 3D

Sine, cosine, tangent ratios; sine and cosine rules; triangle area

SOH-CAH-TOA · Sine rule and cosine rule · Area = ½ab sin C

Triangle applications — elevation, depression, bearings

Angles of elevation and depression · Bearings

Circle arc length and sector area

Arc length L = θ/360 × 2πr · Sector area A = θ/360 × πr²

Perpendicular bisector equations

Midpoint and slope of perpendicular · Equation of the bisector

Voronoi diagrams and nearest-neighbour interpolation

Voronoi cells from perpendicular bisectors · Toxic-waste / nearest-shop / mobile-cell applications

Radian measureHL

Radian definition · Arc length and sector area in radians

Unit circle, Pythagorean identity, ambiguous caseHL

Unit-circle definitions · sin²θ + cos²θ = 1 · Ambiguous SSA case

Geometric transformations using matricesHL

Rotation, reflection, scaling matrices · Determinant as area-scale factor

Vectors — components, unit vectors, position vectorsHL

Vector addition and scalar multiplication · Magnitude and unit vectors · Position vectors

Vector line equations in 2D and 3DHL

r = a + λb · Parametric form

Vector kinematicsHL

Position, velocity, acceleration as vectors · Constant vs variable velocity

Scalar and vector productsHL

Dot product and angle between vectors · Cross product and area of parallelogram

Graph theory — vertices, edges, weighted graphsHL

Vertices, edges, degrees · Weighted vs unweighted graphs

Adjacency matrices, walks, transition matricesHL

Adjacency matrices and walks of length n · Transition matrices

Trees, MST, Eulerian and Hamiltonian paths, TSPHL

Minimum spanning tree (Prim, Kruskal) · Eulerian and Hamiltonian paths · Chinese postman and travelling salesman

Population, sampling, and bias

Population vs sample · Sampling techniques and their biases · Outliers

Frequency distributions, histograms, cumulative frequency, box plots

Frequency tables and histograms · Cumulative-frequency S-curves · Quartiles and box plots

Central tendency and dispersion

Mean, median, mode · Range, IQR, variance, standard deviation

Bivariate correlation and regression

Scatter plots and Pearson's r · Least-squares regression line · Predictions and reliability

Probability — outcomes, complement, expected value

Sample space and events · Complementary events · Expected value

Venn diagrams, tree diagrams, conditional probability, independence

Venn and tree diagrams · Conditional probability P(A|B) · Independence test

Discrete random variables and expected value

Probability distributions for discrete X · E(X) and Var(X)

Binomial distribution

Conditions for binomial · Mean np, variance np(1−p) · GDC binomial calculations

Normal distribution

Bell-curve properties · Z-scores and 68-95-99.7 rule · GDC inverse normal

Spearman's rank correlation

Rank-based correlation · When to use rₛ vs r

Hypothesis testing — χ² and t-tests

Null and alternative hypotheses · χ² goodness-of-fit and independence · Two-sample t-test on a GDC

Survey design and reliabilityHL

Validity and reliability of variables · Bias in survey design

Non-linear regression and R²HL

Non-Linear Regression & R²

Linear transformations of random variables; unbiased estimatesHL

Linear Transforms & Unbiased Estimates

Linear combinations of normals; central limit theoremHL

Linear Combinations of Normals · Central Limit Theorem

Confidence intervals for the population meanHL

Confidence Intervals for μ

Poisson distributionHL

Poisson Distribution

Critical regions, hypothesis tests; Type I/II errorsHL

Critical Region & Errors

Markov chains and steady-state probabilitiesHL

Transition Matrices · Steady-State Probability

The derivative as gradient and rate of change

Gradient at a Point · Derivative as a Rate

Increasing and decreasing functions; sign of f′

Sign of f' and Monotonicity

Power rule differentiation (integer exponents)

d/dx(axⁿ) = anxⁿ⁻¹ · Sums and constants

Tangent and normal lines

Tangent gradient = f′(a) · Normal gradient = −1/f′(a)

Anti-differentiation and definite integrals

Reverse of differentiation · Definite integral as area

Critical points where the gradient is zero

Local max, local min, stationary points · Sign-change test

Optimization in context

Setting up an objective · Solving f′(x) = 0 in real-world problems

Trapezoidal rule for area approximation

Trapezoidal rule formula · Over/under estimation depending on concavity

Derivatives of trig, exp, log; chain, product, quotient rulesHL

Standard derivatives · Chain, product, quotient rules

Second derivative test for max/min classificationHL

Concavity from f′′ · Classifying critical points

Integration of rational, trig, exp; substitutionHL

Standard integrals · u-substitution

Area and volume of revolutionHL

V = π ∫ y² dx (about x-axis) · Volume about y-axis

Kinematics — displacement, velocity, accelerationHL

v = ds/dt, a = dv/dt · Distance vs displacement

Differential equations by separation of variablesHL

Separating dy/dx · General and particular solutions

Slope fieldsHL

Visualising solution families · Reading slope fields

Euler's method for numerical solutionsHL

Step-by-step numerical solution · Step size and accuracy

Phase portraits for coupled differential equationsHL

Coupled DE systems · Eigenvalue analysis · Trajectories in phase space

Numerical solution of second-order differential equationsHL

Reducing to a first-order system · Numerical step-by-step solution