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 and 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 sensible real-world restrictions, read intercepts, vertices, asymptotes and intersections from a graph, and find them on the GDC. Build the two-way habit of moving between the algebra of a function and the picture of it.

Connects to — Every later family — linear, quadratic, exponential, sinusoidal — is a special case read through this lens; the graphical solution of equations recurs in modelling, calculus and statistics.

Calculator & technology — The graphing screen is the central tool of AI: graph a function, set a sensible window, and use the zero, intersect, maximum, minimum and value routines to read key features rather than solving by hand.

IA & investigative angle — A defensible domain and range, and a clear reading of what each graphical feature means in context, are the foundation of every modelling exploration.

Teach to this depth — Go well past y = mx + c. Treat the gradient as a rate of change with units, move fluently between gradient-intercept and general forms, use parallel and perpendicular conditions, distance and midpoint as reasoning tools, and fit a straight line to real bivariate data, interpreting slope and intercept in context.

Connects to — Gradient previews the derivative; the intersection of two lines is the simplest system; a fitted line is exactly the regression line of the statistics strand.

Calculator & technology — Plot data as a scatter, fit and graph a line, and use the value and intersect routines to make and compare predictions directly on the screen.

IA & investigative angle — A clean entry point for model criticism: fit a linear model to real data, read the slope as a rate, and discuss residuals and the honest limits of the fit.

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 with sigma notation, derive rather than memorise the sum formulae, and model growth, depreciation and repeated processes.

Connects to — The sequence-as-function idea ties to the functions strand; geometric sequences are the mathematics behind compound interest and financial modelling; recursion underlies population and savings models.

Calculator & technology — Generate sequences and partial sums with the list and table features, and verify long-run behaviour numerically rather than by inspection.

IA & investigative angle — A rich IA vein: financial modelling, geometric and growth patterns, and recursive population models.

Teach to this depth — The signature applied topic of AI. Cover compound growth and depreciation, real versus nominal interest and inflation, and loan, mortgage 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, then introduce the logarithm as the tool that solves an exponential equation. Keep the treatment applied — base 10, base e and the use of logs to find an unknown exponent — and motivate e through continuous growth.

Connects to — Exponents and logarithms are an inverse pair from the functions strand; logs are the engine behind linearising data in regression and the algebraic basis for exponential models.

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

IA & investigative angle — Linearising nonlinear data with logarithms to test an exponential model 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. Read turning points and intercepts in context rather than as abstract features.

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 growth and decay families, their horizontal asymptotes and the meaning of the rate parameter, fit a model with a non-zero limit, and use logarithms to solve for time or rate. Touch on the logistic shape informally as a capped-growth alternative.

Connects to — Builds on the exponent and log laws and on the sequences strand; feeds the rate models of the calculus core and the e-based normal curve of statistics.

Calculator & technology — Fit exponential regressions, graph a model against data, and use intersect to solve growth-and-decay questions for an unknown time.

IA & investigative angle — Growth and decay modelling against real data — cooling, populations, radioactivity — and comparing an exponential fit with a capped-growth alternative.

Teach to this depth — Cover direct and inverse variation and simple rational models, the meaning of their asymptotes, the domain restrictions a real context imposes, and the reading of a rate or concentration that levels off. Sketch from a full analysis and interpret each parameter.

Connects to — Extends the function-families work; asymptotic behaviour is an informal first encounter with limiting behaviour; rate models recur in the calculus strand.

Calculator & technology — Graph rational models, find asymptotic behaviour numerically with a table, and use intersect to solve in context.

IA & investigative angle — Rate, concentration and inverse-proportion models — fuel economy, intensity and dilution problems.

Teach to this depth — Work with three-dimensional coordinates and distances, the angle between a line and a plane resolved through right triangles, and the volume and surface area of solids and composite shapes. Keep the emphasis on real measurement contexts, not formula substitution.

Connects to — Three-dimensional distance and angle lean on triangle trigonometry; volume and surface area feed optimisation and real design problems.

Calculator & technology — Use the calculator for the heavier arithmetic and store intermediate results, reporting final answers to a justified accuracy.

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

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

Connects to — Underlies the periodic models of the next phase and the angle work of three-dimensional geometry; bearings and indirect measurement recur in vector-free navigation contexts.

Calculator & technology — Confirm the calculator is in degree mode, and use it to solve triangles and recover angles with the inverse trigonometric keys.

IA & investigative angle — Surveying, navigation and indirect measurement — heights, distances and bearings recovered from accessible field data.

Teach to this depth — Develop midpoint, distance and gradient, then the perpendicular bisector as the set of points equidistant from two sites — the geometric idea behind the Voronoi diagram. Construct and interpret Voronoi diagrams, add a site, and solve nearest-neighbour and largest-empty-circle (toxic-waste-site) problems.

Connects to — Builds on the straight-line work; the perpendicular bisector is the equidistance condition that defines every Voronoi edge; nearest-neighbour reasoning anticipates real spatial decision problems.

Calculator & technology — Use the calculator to compute distances and midpoints accurately, and to test which site a query point is nearest to.

IA & investigative angle — The richest geometry IA at SL: facility siting, catchment analysis and the toxic-waste-site problem applied to real local map data.

Teach to this depth — Treat amplitude, period, principal axis and horizontal shift fully, in degrees, and fit a sinusoidal model to real periodic data, interpreting each parameter in context and solving graphically for when a target value is reached.

Connects to — Draws on the triangle-trigonometry ratios and the transformation thinking of the modelling phase; the parameter-from-data skill is among the most productive IA contexts in the course.

Calculator & technology — Fit a sinusoidal regression where available, graph the model against the data, and use intersect to solve for the times a value is attained.

IA & investigative angle — Tides, daylight hours, temperature and other seasonal data — estimating the parameters from real data is what lifts the exploration from descriptive to analytic.

Teach to this depth — Establish the derivative as a gradient function and as a rate of change with units. Develop the power rule for polynomial models, tangent and normal equations, intervals of increase and decrease, and the location and classification of stationary points, always interpreting the derivative in context.

Connects to — Grows from the gradient idea of straight lines; the rate-of-change reading recurs in optimisation, kinematics and every dynamic model.

Calculator & technology — Use the numerical derivative at a point and graph the gradient function, reading where it is zero to locate turning points.

IA & investigative angle — Rates of change in real models — cost, speed and growth — and the groundwork for an optimisation exploration.

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

Connects to — Uses stationary-point analysis inside a genuine modelling context; the constraint-and-objective structure reappears throughout applied work.

Calculator & technology — Confirm the optimum with the maximum and minimum routines on the graph as a check against the analytic result.

IA & investigative angle — A classic structure for a strong SL IA — optimise a real quantity such as cost, area or volume with a carefully justified model.

Teach to this depth — Establish the antiderivative as the reverse of differentiation, the indefinite integral and the constant of integration, the definite integral as area, and numerical area by the trapezoidal rule. Read area as an accumulated total in context.

Connects to — Integration is the inverse of differentiation and the continuous counterpart of a running sum; area is its first and most useful application.

Calculator & technology — Use the numerical definite-integral routine and the area-under-a-curve feature to evaluate accumulations the trapezoidal rule only approximates.

IA & investigative angle — Accumulation models — total distance, total volume of flow or total quantity produced — read from a rate.

Teach to this depth — Relate displacement, velocity and acceleration through differentiation and integration, interpret the sign of each, distinguish distance from displacement, and read information from a motion graph.

Connects to — Bridges differentiation and integration in a single concrete setting and reinforces the rate-of-change and accumulation readings of the calculus core.

Calculator & technology — Graph displacement, velocity and acceleration, and use the numerical derivative and definite-integral routines to move between them.

IA & investigative angle — Motion modelling, including sport and vehicle trajectories analysed from real timing or video data.

Teach to this depth — Cover sampling methods and the bias they can 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. Make honest interpretation, not decoration, the goal.

Connects to — Standard deviation feeds the normal distribution; the data-handling habits underlie any data-driven IA; summary statistics are the inputs to the later tests.

Calculator & technology — Enter data into lists and produce one- and two-variable statistics, box plots and histograms directly, reading the summary values from the screen.

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, the interpretation and limitations of Pearson's r (causation, outliers, non-linearity), the least-squares line, prediction with the dangers of extrapolation, and the use of logarithms to linearise a non-linear relationship before fitting.

Connects to — Regression connects to the linear-models work; linearisation reaches back to logarithms; this is the immediate precursor to model-fitting IAs.

Calculator & technology — Compute r and the regression line, plot the line on the scatter, and make predictions with the value routine — noting where prediction becomes extrapolation.

IA & investigative angle — The mainstay of data explorations — teaching residual thinking and model criticism is what raises the depth of knowledge above mere curve-fitting.

Teach to this depth — Cover sample spaces, combined events, mutually exclusive and independent events kept clearly distinct, conditional probability, and the Venn-diagram, tree-diagram and table methods. Ground every rule in a real counting or frequency context.

Connects to — Conditional probability and independence set up the distributions; the frequency reading connects to the data strand; tree diagrams model staged real processes.

Calculator & technology — Use the calculator for the heavier combinatorial and fractional arithmetic, keeping the reasoning on the diagram rather than the keypad.

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

Teach to this depth — Establish the random variable, the probability distribution table, and expected value and its meaning, and construct a distribution from a real context. Read expectation as a long-run average, not a single outcome.

Connects to — Expectation generalises the weighted mean from the data strand; this is the direct precursor to the binomial distribution.

Calculator & technology — Use list operations to compute an expected value from a distribution table efficiently and reliably.

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

Teach to this depth — Cover the conditions for a binomial model, the mean and variance, and exact and cumulative probabilities, with a clear judgement of when the binomial model genuinely applies and when it does not.

Connects to — Rests on the independence idea from probability and the expectation idea from discrete distributions; it is a standard model for counts of successes.

Calculator & technology — Use the binomial pdf and cdf routines for exact and at-most or at-least probabilities, reading the parameters straight from the context.

IA & investigative angle — Quality control, success rates and sporting outcomes — any fixed number of independent trials with a constant success probability.

Teach to this depth — Cover the properties of the normal curve, the z-score and standardisation, probabilities and the inverse-normal by technology, 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 normal model underlies the inferential tests that follow and is the most widely used continuous model in AI.

Calculator & technology — Use the normal cdf and inverse-normal routines for probabilities and for the value at a given percentile, reading the parameters from the context.

IA & investigative angle — Modelling real measurement data — heights, masses, times — and assessing honestly whether the normal model fits.

Teach to this depth — The inferential capstone of AI SL. Establish hypotheses, the significance level and the p-value, then the chi-squared test for independence and goodness-of-fit and the two-sample t-test for a difference in means. Stress the conditions for each test and the careful, non-overclaiming interpretation of a result.

Connects to — Draws every earlier statistical strand together: data summaries set up the samples, probability supplies the logic of the p-value, and the normal distribution underlies the t-test. Choosing the right test for the question is the genuine synthesis.

Calculator & technology — Run the chi-squared and t-test routines directly, reading the statistic, the degrees of freedom and the p-value, and interpret rather than merely report them.

IA & investigative angle — The strongest inferential SL IA: a real, properly designed comparison or association investigated with the correct test and an honest conclusion about what it does and does not show.