IB Mathematics

Teach to this depth — Treat number sense as a reasoning discipline, not warm-up arithmetic. Work fluently with standard form, significant figures and decimal places, and — distinctively for AI — make upper and lower bounds, percentage error and estimation into habits applied to every later result. Establish from the outset that an answer carries a precision and an uncertainty.

Connects to — Percentage error reappears whenever a model is compared against data; bounds underlie the honest reporting of any computed or measured quantity; estimation disciplines every modelling check.

Calculator & technology — Set up the GDC correctly from lesson one: mode, fixed decimal places, scientific notation and the answer/memory variables. Teach students to read calculator output critically rather than transcribe every displayed digit.

IA & investigative angle — The quiet backbone of IA rigour — quantifying and reporting the uncertainty in measured data and in a fitted model is exactly what separates a credible exploration from a naive one.

Teach to this depth — This is the grammar of the whole course. Establish the function as a mapping, determine domain and range with restrictions, read key features — intercepts, asymptotes, maxima and minima, intersections — and build genuine fluency moving between the algebra of a function and its graph. The GDC is the primary instrument for graphing and for solving, so disciplined window choice and feature-finding are taught as core skills.

Connects to — Domain and range discipline every model that follows; intersection-finding is the engine of equation solving across the course; the function-as-mapping idea underlies sequences, distributions and rates alike.

Calculator & technology — The heart of AI calculator technique: graphing in a sensible window, the zero/intersect/maximum/minimum routines, and the numeric solver. These routines are revisited in almost every later topic.

IA & investigative angle — Every modelling exploration needs a defensible domain and range and an honest reading of what the graph does and does not show.

Teach to this depth — Form equations of straight lines in every form, read the gradient as a rate of change and the intercept as an initial value, and use parallel and perpendicular conditions as reasoning tools. Extend to piecewise-linear models — tariffs, tax bands, step costs — which are a genuine AI staple, and interpret each parameter inside a real context.

Connects to — Gradient previews the derivative; a fitted line is the regression line of the statistics strand; the intersection of two lines is the simplest system and the break-even point of an applied model.

Calculator & technology — Plot, find intersections (break-even), and evaluate piecewise branches; later, the same line is produced automatically as a regression model.

IA & investigative angle — A clean entry point for model criticism: fit a linear model to real bivariate data, interpret the slope as a rate, and discuss where the linear assumption breaks down.

Teach to this depth — Open by establishing that a sequence is a function with domain the natural numbers — arithmetic and geometric sequences are the discrete cousins of linear and exponential models. Build fluency with the nth-term and sum formulae and sigma notation, and (HL) establish the convergence condition and the sum of an infinite geometric series. Contrast recursive and closed forms throughout.

Connects to — The sequence-as-function idea ties to the functions strand; geometric sequences are the mathematics behind compound growth in finance; recursion previews Euler's method and Markov chains.

Calculator & technology — Generate sequences and partial sums with list and table features; verify convergence numerically.

IA & investigative angle — A rich IA vein: financial modelling, recursive population models, and convergence investigations.

Teach to this depth — The signature applied topic of AI. Cover compound growth and depreciation, real versus nominal rates and inflation, and loan, mortgage, amortisation and annuity calculations. Emphasise reading the structure of a financial instrument and interpreting the output, not memorising formulae.

Connects to — Built directly on geometric sequences; the exponential model of growth and decay is the continuous limit of compound interest; percentage error frames the rounding of money.

Calculator & technology — The TVM (finance) solver is examined explicitly in AI — teach every field (N, I%, PV, PMT, FV, P/Y, C/Y), sign conventions and the amortisation schedule, with cross-checks against the sequence formulae.

IA & investigative angle — Loan comparison, investment strategies, and the true cost of credit — accessible, data-rich and personally relevant explorations.

Teach to this depth — Establish the laws of exponents including rational exponents (HL), introduce logarithms base 10 and e, and (HL) derive the laws of logarithms as consequences of the exponent laws. Motivate e and ln, and introduce logarithmic scales and orders of magnitude.

Connects to — Logarithms and exponentials are an inverse pair from the functions strand; the log laws are the engine behind linearising data in regression; this is the algebraic basis for exponential models and for exponential calculus.

Calculator & technology — Evaluate logarithms of any base by technology; use logarithmic axes when scaling wide-ranging data.

IA & investigative angle — Linearising nonlinear data with logarithms to test power or exponential models is a hallmark of a strong AI IA — the Richter scale, pH and decibels are natural contexts.

Teach to this depth — Teach quadratic and cubic models as fitting tools: the meaning of each coefficient, the vertex as an optimum, roots as solutions, and the use of given data points to determine an unknown model. Treat the discriminant as a classifier and read turning points in context.

Connects to — Quadratics underlie projectile and area modelling and reappear in optimisation; cubic models extend the same fitting logic; polynomial regression in the statistics strand is the data-driven version.

Calculator & technology — Fit quadratic and cubic regressions on the GDC, solve polynomial equations, and locate vertices and roots graphically.

IA & investigative angle — Projectile, profit and area problems; fitting a polynomial to real data and judging when its turning points are meaningful rather than artefacts.

Teach to this depth — Study the exponential and logarithmic graph families, their asymptotic behaviour, and growth and decay modelling with a horizontal asymptote (the AI form a*k^x + c). Solve in context and interpret every parameter, foreshadowing the logistic model and differential equations.

Connects to — Builds on the log laws and on transformations; feeds exponential calculus, the growth, cooling and decay models of differential equations, and the normal curve's e^x core.

Calculator & technology — Use exponential and logarithmic regression; find the asymptote and half-life or doubling time numerically.

IA & investigative angle — Growth and decay against real data — cooling, population, radioactive decay — and comparing an exponential fit with a logistic one.

Teach to this depth — Build genuine fluency with composite functions (including why order matters) and inverse functions (existence, one-to-one behaviour, reflection in y = x, restricting a domain to force invertibility). Interpret composition and inversion inside applied chains — unit conversions, staged costs, reversing a model to recover an input.

Connects to — Inverses recur as logarithms and as the back-transformation after linearising data; composition is the conceptual root of the chain rule met later in calculus.

Calculator & technology — Evaluate compositions numerically and graph a function with its inverse to check the reflection.

IA & investigative angle — Iterated or staged processes rely on composition; recovering an input from an output needs a defensible inverse.

Teach to this depth — Cover translations, stretches and reflections and the careful decoding of composite transformations such as a*f(b(x – c)) + d, and the effect of each on domain, range, asymptotes and key points. Build the two-way habit: predict the graph from the algebra and the algebra from the graph.

Connects to — It is the lens for exponential, logarithmic, rational and sinusoidal graphs; reflection in y = x is exactly the inverse-function relationship; parameter effects underpin all model fitting.

Calculator & technology — Use sliders or repeated re-graphing to see a parameter move a curve in real time before committing to a fit.

IA & investigative angle — Parameter-driven modelling depends on understanding precisely how each coefficient moves a curve — essential for fitting and interpreting any model.

Teach to this depth — Cover direct and inverse variation and rational models, locating vertical and horizontal asymptotes and explaining how each arises, identifying domains, and interpreting limiting behaviour in context (saturation, dilution, rate-versus-time).

Connects to — Uses transformations; asymptotic analysis is an informal first encounter with limits; inverse variation models appear again in physical and economic contexts.

Calculator & technology — Graph rational models with care over the window and asymptotes; solve rational equations numerically.

IA & investigative angle — Concentration, intensity and cost-per-unit models, and optimisation with a rational cost function.

Teach to this depth — Work with three-dimensional coordinates and distances, angles between lines and planes resolved through right triangles, and the volume and surface area of solids and composites. Keep the emphasis on reasoning and real measurement contexts, not formula substitution.

Connects to — Three-dimensional distance and angle anticipate the vector treatment; mensuration feeds optimisation and volumes of revolution in calculus.

Calculator & technology — Store and reuse intermediate values to avoid rounding error in multi-step solid calculations.

IA & investigative angle — Optimisation of packaging and volume, surveying, and spatial modelling of real structures.

Teach to this depth — Establish right-triangle ratios, then derive and apply the sine and cosine rules and the area formula, treating the ambiguous (SSA) case with genuine geometric reasoning. Cover bearings, three-dimensional triangle problems, and angles of elevation and depression.

Connects to — Feeds the unit-circle and identity work; the area formula reappears as the magnitude of a vector cross product; this precedes the geometric treatment of vectors.

Calculator & technology — Degree mode discipline, storing angles, and solving triangles efficiently while watching for the ambiguous case the calculator will not flag.

IA & investigative angle — Surveying, navigation and indirect measurement — classic accessible AI contexts.

Teach to this depth — Define the radian as a ratio rather than a conversion trick, then compute arc length and sector area in radians and combine sectors and segments. Connect everything back to the unit circle.

Connects to — Radians are a prerequisite for the identity and equation work, for sinusoidal modelling and for trigonometric calculus — the derivative of sin x is cos x only in radians, a point worth stressing.

Calculator & technology — Switch confidently between degree and radian modes and know which the problem demands.

IA & investigative angle — Any circular context — gears, clock mechanisms, and optimisation under a circular constraint.

Teach to this depth — Construct Voronoi diagrams from sites using perpendicular bisectors, interpret the cells as nearest-neighbour regions, find the equation of a boundary, and apply the toxic-waste-dump problem — the largest-empty-circle reasoning that maximises distance from all sites.

Connects to — Built on coordinate geometry, distance and the perpendicular bisector; the nearest-site idea previews the nearest-neighbour heuristic in networks and the notion of optimisation under a constraint.

Calculator & technology — Plot sites and bisectors accurately; compute distances and intersections of boundaries.

IA & investigative angle — Facility siting — schools, hospitals, mobile masts, fire stations — and catchment analysis from real coordinate data, an exceptionally accessible and original AI context.

Teach to this depth — Treat the unit circle as the single source of all values, signs and symmetries. Establish the Pythagorean identity, the periodicity relationships, and solve trigonometric equations both graphically and over a given interval. Keep the treatment applied and technology-supported.

Connects to — Foundation for sinusoidal modelling and for the modulus-argument form of complex numbers; feeds the integration of trigonometric functions later.

Calculator & technology — Solve trigonometric equations graphically over a stated interval, reading every solution in range.

IA & investigative angle — The conceptual bedrock for any periodic modelling and for the complex-number topic that follows.

Teach to this depth — Treat amplitude, period, phase shift and vertical shift fully, apply the transformation toolkit to the sine and cosine functions, and model real periodic phenomena while interpreting each parameter in context. Emphasise estimating parameters from real data.

Connects to — Draws on transformations and on radians; the parameters reappear as frequency and phase in the Euler form of a complex number; periodic data is among the richest AI contexts.

Calculator & technology — Use sinusoidal regression on real periodic data and interpret the reported amplitude, period and shift.

IA & investigative angle — Tides, daylight hours, temperature and sound — fitting a sinusoid to real data lifts an IA from descriptive to analytic.

Teach to this depth — Introduce the logistic model with its carrying capacity and inflection, contrast it with unbounded exponential growth, and — the synthesis — develop a principled procedure for choosing among the model families met so far, judging fit, residuals and the plausibility of extrapolation.

Connects to — Requires the full model toolkit (linear, polynomial, exponential, rational, sinusoidal); the logistic curve is the analytic solution of a differential equation met later; model criticism links directly to regression diagnostics.

Calculator & technology — Fit a logistic model on the GDC, locate the carrying capacity and inflection, and compare fits across candidate models using residuals.

IA & investigative angle — Constrained growth — disease spread, technology adoption, population under resource limits — with an explicit, defended comparison of competing models, which is exactly what raises an IA's depth.

Teach to this depth — Cover Cartesian arithmetic and the conjugate, the Argand plane, modulus and argument, the polar and Euler (re^{i*theta}) forms and conversion between them, and De Moivre's theorem for powers. Keep the AI emphasis applied: complex numbers as a tool for combining quantities with magnitude and phase.

Connects to — Depends on trigonometric modulus-argument form; the Euler form ties to exponentials and to sinusoidal phase; complex eigenvalues later explain oscillating and spiralling solutions in the phase-portrait capstone.

Calculator & technology — Use the GDC's complex mode for arithmetic, conversion between forms, and powers.

IA & investigative angle — Alternating-current circuits, the addition of waves with different phases, and impedance — genuinely applied complex-number contexts.

Teach to this depth — Establish matrix notation, addition, scalar multiplication and the meaning of matrix multiplication, the determinant and inverse of square matrices, and the solution of simultaneous systems by matrix methods. Stress that a matrix encodes a structured relationship, not just a grid of numbers.

Connects to — The doorway to geometric transformations, to eigen-analysis, to graph adjacency, and to the transition matrices of Markov chains; systems link back to intersecting lines and planes.

Calculator & technology — Enter matrices, compute products, determinants and inverses, and solve systems by technology — a heavily examined AI skill.

IA & investigative angle — Network flow, balancing equations, and any system with structured interacting quantities.

Teach to this depth — Apply matrices to geometric transformations — rotations, reflections, enlargements, shears — compose transformations by multiplication, and interpret the determinant as an area scale factor and its sign as orientation. Read a composite transformation as a single matrix.

Connects to — Uses matrix algebra; the determinant-as-area idea connects to the cross product and to probability area; composition mirrors function composition.

Calculator & technology — Multiply transformation matrices and apply them to coordinate sets; verify the area-scaling result numerically.

IA & investigative angle — Computer graphics, tessellation and fractal generation, and the geometry of repeated transformations.

Teach to this depth — Find eigenvalues from the characteristic equation and the corresponding eigenvectors, diagonalise a 2×2 matrix, and use the diagonal form to compute high matrix powers efficiently. Interpret an eigenvector as a direction the transformation merely scales.

Connects to — Built on matrix algebra and the determinant; this is the engine behind the long-run behaviour of Markov chains (the steady state is an eigenvector) and behind the stability of the phase-portrait capstone.

Calculator & technology — Use technology to find eigenvalues and eigenvectors and to verify the diagonalisation by reconstruction.

IA & investigative angle — Long-term behaviour of structured populations and the stability of dynamic systems.

Teach to this depth — Establish vector notation, magnitude and unit vectors, addition and scalar multiplication, the parallel and perpendicular conditions, the scalar (dot) product and its use in finding angles, and the vector (cross) product whose magnitude gives an area. Develop geometric reasoning with vectors.

Connects to — The dot product rests on the cosine relationship from trigonometry; the cross-product magnitude is the triangle-area formula; vectors extend 3D geometry and feed the kinematic reading of lines.

Calculator & technology — Compute magnitudes, dot and cross products, and angles between vectors with technology.

IA & investigative angle — Resolution of forces and velocities, geometric proof, and three-dimensional modelling.

Teach to this depth — Cover the vector and parametric forms of a line, the relationships between lines (parallel, intersecting and skew), the angle between two lines, and the kinematic reading of the vector equation — position, velocity and the constant-velocity model — including closest-approach problems.

Connects to — Builds on vector operations; the kinematic reading links forward to the calculus of motion; closest-approach reasoning is an optimisation in disguise.

Calculator & technology — Parametrise motion, evaluate position at given times, and minimise a distance function numerically.

IA & investigative angle — Trajectory and navigation modelling, collision and closest-approach problems for ships or aircraft.

Teach to this depth — Develop the language of graphs — vertices, edges, degree, walks, weighted and directed graphs — the adjacency matrix and its powers, minimum spanning trees by Kruskal's and Prim's algorithms, the Chinese postman problem, and the travelling-salesperson problem with nearest-neighbour and deleted-vertex bounds.

Connects to — The adjacency matrix and its powers reuse matrix algebra directly; minimisation echoes optimisation; weighted-graph reasoning underlies logistics and routing models.

Calculator & technology — Raise adjacency matrices to powers to count walks, and organise weight tables for the algorithms.

IA & investigative angle — Transport, delivery and utility networks; route optimisation; and scheduling — a distinctive and fertile AI HL context.

Teach to this depth — Establish the derivative as a gradient function and as a rate of change, develop the power rule, find tangent and normal equations, and identify intervals of increase and decrease and basic stationary points — always interpreting the derivative in an applied context.

Connects to — Grows from the gradient idea of straight lines; the rate-of-change reading recurs throughout the applied course and is the heart of differential equations.

Calculator & technology — Evaluate a numerical derivative at a point and find maxima, minima and gradients graphically.

IA & investigative angle — Rates in real models — the seeds of optimisation and of any dynamic exploration.

Teach to this depth — Run the full modelling cycle: define the variable, state the constraint and objective, set the domain, classify the extremum, and interpret the result, distinguishing boundary from interior optima.

Connects to — Uses stationary-point analysis inside a genuine modelling context; the constraint logic links back to mensuration and to rational cost models.

Calculator & technology — Locate optima graphically and confirm with the numerical derivative; useful when the algebra is heavy.

IA & investigative angle — A classic structure for a strong AI IA — optimise a real quantity (cost, volume, time) with a carefully justified model and an honest domain.

Teach to this depth — Build fluency with the chain, product and quotient rules and the derivatives of exponential, logarithmic and trigonometric functions, develop related rates, and use the second derivative for concavity and points of inflection.

Connects to — The chain rule is composition differentiated; higher derivatives describe concavity and feed the analysis of models; related rates model dynamic systems.

Calculator & technology — Cross-check analytic derivatives against numerical ones and locate inflection points graphically.

IA & investigative angle — Related-rates scenarios and the curvature analysis behind any serious calculus exploration.

Teach to this depth — Establish the antiderivative, the indefinite and definite integral, the constant of integration, the trapezoidal rule for numerical integration, and area under a curve by both analytic and technological means. Frame the definite integral as accumulated change.

Connects to — Integration is the inverse of differentiation and the continuous counterpart of the sigma sum; the trapezoidal rule is a genuinely applied AI technique; area is its first application.

Calculator & technology — Compute definite integrals and areas numerically — the standard AI route — and apply the trapezoidal rule to tabulated data.

IA & investigative angle — Accumulation from rate data — distance from speed, volume from flow — using real measured values.

Teach to this depth — Integrate exponential, reciprocal and trigonometric functions, use the reverse chain rule and substitution, evaluate definite integrals, find the area between curves and areas about the y-axis, and compute volumes of revolution.

Connects to — Substitution is the chain rule reversed; volumes connect to mensuration; this technique is a prerequisite for the differential equations that close the course.

Calculator & technology — Evaluate definite integrals and volumes of revolution by technology, reserving analytic methods for where they are required.

IA & investigative angle — The volume of a real solid of revolution, and accumulation problems grounded in measured rate data.

Teach to this depth — Relate displacement, velocity and acceleration through differentiation and integration, interpret signs, distinguish distance from displacement (the integral of speed), and read motion graphs.

Connects to — Bridges differentiation and integration in one setting, links to the vector treatment of motion, and reinforces the rate interpretation that drives differential equations.

Calculator & technology — Differentiate and integrate motion functions numerically and read areas and gradients from velocity-time graphs.

IA & investigative angle — Motion modelling, including sports trajectories and vehicle dynamics from real data.

Teach to this depth — Cover sampling methods and the bias they introduce, measures of centre and spread, the meaning of standard deviation, quartiles and the interquartile range, outlier rules, box plots, cumulative frequency, histograms and grouped data, and the effect of a linear transformation on the mean and standard deviation.

Connects to — Standard deviation feeds the normal distribution; the transformation effect anticipates standardisation; honest data handling underlies every data-driven IA.

Calculator & technology — Enter data into lists and produce one-variable statistics, box plots and histograms — the core AI statistics workflow.

IA & investigative angle — The foundation for any data IA — but warn against the shallow survey-and-bar-chart exploration; depth here means appropriate analysis and honest interpretation.

Teach to this depth — Cover scatter plots, Pearson's r and its interpretation and limits (causation, outliers, non-linearity), Spearman's rank coefficient and when to prefer it, the least-squares line, and prediction with the dangers of extrapolation.

Connects to — Regression connects to the linear model and to the systems behind the normal equations; the linearisation idea reaches back to logarithms; this is the immediate precursor to model-fitting IAs.

Calculator & technology — Produce the regression line, r and r-squared, and plot residuals — and resist over-reading a high r.

IA & investigative angle — The mainstay of AI data explorations — teaching residual analysis and model criticism is what raises the depth of knowledge.

Teach to this depth — Fit power, exponential and polynomial regressions, interpret the coefficient of determination across models, use logarithmic scales, and transform nonlinear relationships into linear form using logarithms to test the underlying model.

Connects to — Uses the log laws directly; r-squared comparison links to model selection in the functions strand; linearisation is a genuinely advanced AI data technique.

Calculator & technology — Apply the GDC's power, exponential and polynomial regressions and compare r-squared across candidates.

IA & investigative angle — Testing whether a relationship is truly a power law or exponential by linearising and judging the fit — a strong analytic move in a data IA.

Teach to this depth — Cover sample spaces and the basic rules, combined events, conditional probability and independence (kept clearly distinct), Venn diagrams, tree diagrams and tables, and Bayesian reasoning through the conditional formula.

Connects to — Conditional probability sets up the distributions and Markov chains; the counting reaches back to systematic listing; Bayesian reasoning models real inference and testing.

Calculator & technology — Organise outcomes in tables and trees; compute conditional probabilities reliably.

IA & investigative angle — Risk analysis, medical testing through conditional reasoning, and games of chance.

Teach to this depth — Establish the random variable, the probability distribution, expected value and variance, the behaviour of these under a linear transformation, and the construction of a distribution from a real context.

Connects to — Expectation generalises the weighted mean; the transformation rules echo the data-transformation rules; this is the precursor to the named distributions.

Calculator & technology — Tabulate a distribution and compute its expectation and variance with list operations.

IA & investigative angle — Expected value in decision-making, insurance and games — where the long-run average drives the choice.

Teach to this depth — (HL) Cover linear combinations of independent random variables, the mean and variance of a sum or difference, and the idea of an unbiased estimator of a population mean and variance from a sample.

Connects to — The combination rules underpin the distribution of a sample mean and the central limit theorem; unbiased estimators justify how a sample informs a population claim.

Calculator & technology — Simulate sums and differences of variables to see the variance-addition rule emerge.

IA & investigative angle — Aggregated risk and the propagation of variability through a combined quantity.

Teach to this depth — Cover Bernoulli trials and the conditions for a binomial model, the probability built on nCr, the mean and variance, cumulative probabilities, and a clear judgement of when the model is and is not appropriate.

Connects to — The nCr is exactly the binomial-expansion coefficient and the combinations count; the independence condition rests on the probability strand; the binomial seeds the binomial test.

Calculator & technology — Use the binomial pdf and cdf routines, watching the boundary conventions (at most / fewer than).

IA & investigative angle — Quality control and success rates, including in sport and manufacturing.

Teach to this depth — (HL) Establish the conditions for a Poisson model, calculate probabilities for counts in an interval, use the mean as the single parameter, and combine the means of independent Poisson variables.

Connects to — A counting-in-time companion to the binomial; the additivity of means links to linear combinations; the model feeds the Poisson test.

Calculator & technology — Use the Poisson pdf and cdf routines and read the parameter from rate data.

IA & investigative angle — Modelling arrivals — calls, customers, decay events, goals — and judging whether the Poisson conditions truly hold.

Teach to this depth — Cover the properties of the normal curve, standardisation and the z-score, probabilities by technology, the inverse normal, finding an unknown mean or standard deviation, and an informal check of whether data are approximately normal.

Connects to — Standardisation uses the standard deviation from the data strand; the curve is a continuous distribution and the basis of the sampling-distribution work and many tests.

Calculator & technology — Use normal cdf and inverse-normal routines fluently — the central AI normal-distribution skill.

IA & investigative angle — Modelling real measurement data — heights, masses, timings — and assessing the fit of a normal model.

Teach to this depth — (HL) Establish the distribution of a sample mean, the central limit theorem, and the construction and interpretation of a confidence interval for a population mean — what the interval does and does not claim.

Connects to — Synthesises linear combinations, estimators and the normal distribution; the inferential logic is the immediate foundation of hypothesis testing.

Calculator & technology — Compute confidence intervals by technology and explore the central limit theorem through simulation.

IA & investigative angle — Estimating a population quantity from a sample with an honest statement of precision.

Teach to this depth — Develop the logic of a hypothesis test, then the chi-squared test for independence (expected frequencies, degrees of freedom, the test statistic and p-value) and the goodness-of-fit test against a proposed distribution, with careful interpretation against a significance level.

Connects to — Rests on the probability and distribution strands; the goodness-of-fit test checks whether a binomial, Poisson or other model actually fits real data; it is the gateway to the wider test suite.

Calculator & technology — Run chi-squared tests on the GDC, reading the statistic, degrees of freedom and p-value, and checking the expected-frequency conditions.

IA & investigative angle — Testing association in survey or experimental data, and testing whether real counts fit a proposed model.

Teach to this depth — (HL) Bring the inferential strand together: one- and two-sample t-tests for a mean, the binomial and Poisson tests, the test for a correlation coefficient, paired versus unpaired designs, one- versus two-tailed tests, and Type I and Type II errors.

Connects to — Every named distribution and the sampling-distribution work feed into the appropriate test; choosing the right test for a context is the synthesis skill.

Calculator & technology — Select and run the correct test by technology, interpret the p-value against a stated significance level, and state the conclusion in context.

IA & investigative angle — A rigorous comparison — two treatments, two groups, before-and-after — built on a properly chosen and justified test, the analytic centrepiece of many statistics IAs.

Teach to this depth — (HL) Build a Markov model: the state vector, the transition matrix, the evolution by repeated matrix multiplication, matrix powers for multi-step transitions, and the long-run steady state found as the equilibrium distribution.

Connects to — Fuses matrices, eigen-analysis (the steady state is the eigenvector for eigenvalue 1) and probability; matrix powers reuse diagonalisation; it is a clear cross-strand synthesis.

Calculator & technology — Compute high powers of the transition matrix and solve for the steady state by technology.

IA & investigative angle — Long-run behaviour of systems that switch between states — weather, brand loyalty, populations, queues — with an interpretation of the equilibrium.

Teach to this depth — (HL) Form a differential equation from a rate description, solve by separation of variables, sketch and read slope fields, and apply Euler's numerical method to models of growth, cooling and mixing — comparing the numerical and analytic solutions.

Connects to — Separation needs the integration techniques; the solutions are exponential and logistic functions met earlier; Euler's method is the continuous analogue of a recursive sequence.

Calculator & technology — Implement Euler's method on lists or a short program and overlay the numerical and analytic solutions.

IA & investigative angle — Dynamic real systems — population, cooling, simple epidemics — with a comparison of numerical and analytic solutions, which makes outstanding IA material.

Teach to this depth — (HL) The capstone. Set up coupled first-order systems and second-order equations, find equilibrium points, use the eigenvalues of the system matrix to classify stability, read complex eigenvalues as oscillation or spiralling, sketch phase portraits and solution trajectories, and apply Euler's method to systems.

Connects to — This is where linear algebra, calculus and complex numbers converge: eigenvalues from the matrix strand classify the equilibrium, complex eigenvalues from that strand explain spirals, and the differential-equation machinery provides the dynamics. It is the true capstone of the AI HL course.

Calculator & technology — Compute system eigenvalues, run Euler's method for the coupled system, and plot the trajectory in the phase plane.

IA & investigative angle — Predator-prey, competing species, coupled tanks and oscillating systems — a genuinely advanced exploration that ties the whole course together.