IB Mathematics

Teach to this depth — Go well past y = mx + c. Treat the gradient as a rate of change (the first quiet appearance of the derivative), move fluently between point-slope, gradient-intercept and general forms, and use the parallel and perpendicular conditions, distance and midpoint as reasoning tools rather than recall items. Read a line as a function with a domain and range, and interpret its parameters inside a real context.

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

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 residuals and the limits of the fit.

Teach to this depth — This is the grammar of the whole course. Establish the function as a mapping, determine domain and range with restrictions, and 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). Classify functions as odd, even, periodic or self-inverse and justify it.

Connects to — Inverses recur as logarithms (inverse of exponentials), inverse trigonometric functions, and the derivative of an inverse; composition is the conceptual root of the chain rule.

IA & investigative angle — Every modelling exploration needs a defensible domain and range; iterated or staged processes rely on composition.

Teach to this depth — Teach all three forms — standard, vertex and factored — and fluent conversion between them. Use completing the square as a structural tool, not just an equation solver, and treat the discriminant as a classifier used in tangency and intersection arguments and in proving an expression is sign-definite. Cover quadratic inequalities by sign analysis and introduce the sum and product of roots as a first taste of root–coefficient relationships.

Connects to — The discriminant counts intersections of graphs and, later, distinguishes real from complex roots; completing the square reappears in vertex form, optimisation and integration set-ups; sum and product of roots generalise to polynomials.

IA & investigative angle — Projectile and area problems, parabolic model fitting, and tangency conditions framed through the discriminant.

Teach to this depth — Pulled forward from its usual late slot because it is a reading lens, not a finishing topic. Cover translations, stretches and reflections, 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, trigonometric and rational graphs that follow, and reflection in y = x is exactly the inverse-function relationship.

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 — 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 functions. Build real fluency with sigma notation, including shifting the index, splitting sums and factoring constants; derive (not memorise) the sum formulae; establish the convergence condition for an infinite geometric series; and model compound interest, depreciation and annuities. Contrast recursive and closed forms.

Connects to — The sequence-as-function idea ties directly to the Functions unit; sigma notation foreshadows the Riemann sum behind integration; closed forms are the natural first targets of proof by induction; and series are the backbone of Maclaurin expansions.

IA & investigative angle — Among the richest IA veins: financial modelling, geometric and fractal patterns, convergence investigations, and recursive population models.

Teach to this depth — Derive the logarithm laws as consequences of the exponent laws rather than presenting them as rules, justify change of base, and solve exponential and logarithmic equations including those that reduce to quadratics. Motivate e and ln, and introduce logarithmic scales and orders of magnitude.

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

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

Teach to this depth — Study the graph families, their transformations and asymptotic behaviour, determine domain and range, and model growth and decay — touching on logistic behaviour to foreshadow differential equations. Solve in context and interpret every parameter.

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

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

Teach to this depth — Develop both Pascal’s triangle and the nCr formula, the general term, and the extraction of a specific coefficient. Extend (HL) to fractional and negative indices — the binomial series — and establish its validity range, then use truncation to approximate.

Connects to — The nCr here is literally the nCr of the binomial distribution and of combinations — make that identity explicit. The binomial series is the Maclaurin expansion of (1 + x)^n, and truncation links to approximation and error.

IA & investigative angle — Approximation-error analysis, probability through the binomial model, and bridging algebra to combinatorics.

Teach to this depth — Build from the fundamental counting principle, sharply distinguishing arrangements from selections. Develop factorial reasoning and handle restrictions (items together or apart, fixed positions), circular arrangements and repetition, and prove combinatorial identities such as Pascal’s rule.

Connects to — Factorials are a feeder for inductive divisibility and factorial proofs; the counting here is the counting behind probability; nCr ties straight back to the binomial expansion.

IA & investigative angle — Combinatorial probability, analysis of games and lotteries, and counting in real-world structures.

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

Connects to — Three-dimensional distance and angle anticipate the rigorous vector treatment (the dot product for angles); the right-triangle angle work leans on trigonometry.

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

Teach to this depth — Define the radian as a ratio rather than treating it as 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 all later trigonometry and calculus — the result that the derivative of sin x is cos x holds only in radians, a point worth stressing explicitly.

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

Teach to this depth — Treat the unit circle as the single source of all values, signs and symmetries. Establish exact values for the standard angles, the periodicity and symmetry relationships, the Pythagorean identity as a direct consequence of the circle, and the reciprocal ratios.

Connects to — This is the foundation for trigonometric graphs, equations, compound angles, De Moivre’s theorem and the integration of trigonometric functions; the Pythagorean identity surfaces almost everywhere.

IA & investigative angle — The conceptual bedrock for any periodic modelling.

Teach to this depth — Derive the sine and cosine rules and the area formula, and treat the ambiguous (SSA) case with genuine geometric reasoning about why two triangles can arise. Cover bearings, three-dimensional triangle problems, and angles of elevation and depression.

Connects to — Applies the exact values; the triangle area formula reappears as the magnitude of a vector cross product; this work precedes the geometric treatment of vectors.

IA & investigative angle — Surveying, navigation and indirect measurement.

Teach to this depth — Treat amplitude, period, phase shift and vertical shift fully, apply the transformation toolkit to the trigonometric functions, solve equations graphically, and model real periodic phenomena while interpreting each parameter in context.

Connects to — Draws directly on the transformations unit and precedes trigonometric calculus; the modelling here is among the most productive IA contexts in the course.

IA & investigative angle — Tides, daylight hours, sound and seasonal data — emphasise estimating parameters from real data, which lifts an IA from descriptive to analytic.

Teach to this depth — Restrict domains to define arcsin, arccos and arctan, work with principal values and their graphs, and handle the reciprocal functions (secant, cosecant, cotangent), their graphs and identities, and equations involving them.

Connects to — Uses the inverse-function machinery from the Functions unit; arctan reappears as the result of a standard integral, and the reciprocal identities feed advanced trigonometric work.

IA & investigative angle — Contexts where an angle is recovered from a ratio.

Teach to this depth — Derive the compound- and double-angle formulae rather than only applying them, prove identities, solve quadratic-in-trigonometry equations over given intervals, and use the R-method for a sin x + b cos x. Make simplification-before-solving a habit.

Connects to — Compound angles are exactly what De Moivre’s theorem reproduces; the double-angle and power-reduction forms are essential for integrating trigonometric functions; proving identities builds the discipline used later in formal proof.

IA & investigative angle — Any periodic model that needs simplification, and explorations that prove a trigonometric relationship.

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 proof using vectors.

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

IA & investigative angle — Geometric proof, the resolution of forces and velocities, and three-dimensional modelling.

Teach to this depth — Cover the vector, parametric and Cartesian forms of a line, the relationships between lines (parallel, intersecting and skew), the angle between two lines, the kinematic reading of the vector equation, and shortest-distance problems.

Connects to — Builds on vector operations; the kinematic reading links to the calculus of motion, and the analysis of intersections previews the linear-systems work.

IA & investigative angle — Trajectory and navigation modelling and closest-approach problems.

Teach to this depth — Cover the vector and Cartesian equations of a plane, the normal vector, the intersection of a line with a plane and of planes with one another, the angle between planes and between a line and a plane, and geometric applications.

Connects to — Uses the dot product for normals and angles and the systems-of-equations technique for intersections; it completes the geometry strand.

IA & investigative angle — Three-dimensional structural and architectural modelling, and intersection geometry.

Teach to this depth — Cover algebraic and synthetic division, the factor and remainder theorems, finding roots, and the relationship between roots and coefficients generalised from the quadratic case. Connect end behaviour and root multiplicity to graph shape, and preview the Fundamental Theorem of Algebra by counting complex and real roots.

Connects to — Extends quadratics; the root–coefficient relationships carry into complex roots; polynomial division is the doorway to partial fractions in integration; the root count anticipates complex numbers.

IA & investigative angle — Polynomial modelling, root-finding investigations, and the error of polynomial approximations — which links forward to Maclaurin series.

Teach to this depth — Find vertical, horizontal and oblique asymptotes and explain how each arises, identify holes, determine domains, sketch from a full analysis, and solve rational equations and inequalities. Introduce partial fraction decomposition as deliberate preparation.

Connects to — Uses transformations; partial fractions are needed for integration; asymptotic analysis is an informal first encounter with limits.

IA & investigative angle — Rate and concentration models and optimisation with rational cost functions.

Teach to this depth — Develop sign diagrams, critical values, polynomial and rational inequalities, modulus inequalities, and the expression of solution sets in interval notation, with a parallel graphical interpretation throughout.

Connects to — Inequality technique is reused directly in inductive inequality proofs (for example Bernoulli’s inequality), in domain analysis, and in stating optimisation constraints.

IA & investigative angle — Feasible regions and constraint analysis.

Teach to this depth — Distinguish carefully between |f(x)| and f(|x|), solve modulus equations and inequalities both algebraically and graphically, handle the reciprocal transformation 1/f(x), and combine transformations, reading modulus functions as piecewise objects.

Connects to — Caps the transformations strand; the piecewise viewpoint connects to continuity and limits; the modulus as a distance underlies several proof arguments.

IA & investigative angle — Piecewise modelling and tolerance or error-band analysis.

Teach to this depth — Cover arithmetic and the conjugate, the Argand plane, modulus and argument, the polar and Euler (e^{iθ}) forms and conversion between them, De Moivre’s theorem, integer and fractional powers, the nth roots of a complex number and their arrangement as a regular polygon, the roots of unity, solving z^n = a, and loci in the complex plane. Use De Moivre to derive multiple-angle identities.

Connects to — Depends on trigonometric modulus–argument form and the compound-angle identities; De Moivre is a flagship application of proof by induction and a generator of trigonometric identities; the Euler form ties to exponentials; the roots tie to polynomials and the Fundamental Theorem of Algebra; loci tie to coordinate geometry.

IA & investigative angle — Alternating-current circuits, fractals through iteration on the Argand plane (Mandelbrot and Julia sets), signal phase, and plane transformations expressed as complex multiplication — exceptionally fertile.

Teach to this depth — Solve two- and three-variable systems by elimination and row reduction, interpret them geometrically as intersecting lines or planes, and analyse the conditions for a unique solution, no solution or infinitely many, expressing the last as a parametric set.

Connects to — The geometry of planes connects to vectors; the count of solutions previews determinant reasoning; systems underpin the normal equations behind regression.

IA & investigative angle — Network and flow problems, balancing systems, and intersection geometry.

Teach to this depth — Develop the intuitive limit, one-sided limits and limits at infinity, an intuition for indeterminate forms, the definition of continuity, and the relationship between continuity and differentiability — with the limit established as the foundation of the derivative.

Connects to — Underlies the derivative from first principles, the asymptotes of rational functions, and the convergence of series; it is the prerequisite for L’Hôpital’s rule.

IA & investigative angle — Convergence investigations and the numerical estimation of limits.

Teach to this depth — Establish the derivative as a limit (first principles for polynomials), as a gradient function and as a rate of change. Develop the power rule, tangent and normal equations, intervals of increase and decrease, and basic stationary points, always interpreting the derivative in context.

Connects to — Grows from the gradient idea of straight lines and from limits; the rate-of-change reading recurs throughout the course.

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

Teach to this depth — Build fluency with the chain, product and quotient rules and their combinations, the derivatives of trigonometric, exponential, logarithmic and inverse-trigonometric functions, and higher derivatives. Use the second derivative for concavity and points of inflection, stressing that f”(x) = 0 is necessary but not sufficient, and sketch curves from f, f’ and f” together.

Connects to — The chain rule is composition differentiated; higher derivatives are the engine of the Maclaurin series; curve sketching reunites the functions and transformations work with calculus.

IA & investigative angle — The toolkit behind any serious calculus exploration.

Teach to this depth — Differentiate new functions from first principles, differentiate inverse functions and use the result, and develop implicit differentiation, related rates and logarithmic differentiation.

Connects to — Implicit differentiation handles curves that are not functions; related rates model dynamic systems; the inverse-derivative result connects back to inverse functions.

IA & investigative angle — Related-rates scenarios and curves defined implicitly.

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 and justifying with the second derivative.

Connects to — Uses stationary-point analysis and inequality-based constraints inside a genuine modelling context.

IA & investigative angle — A classic structure for a strong IA — optimise a real quantity with a carefully justified model.

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 vectors for two-dimensional motion, and reinforces rate interpretation.

IA & investigative angle — Motion modelling, including sports trajectories.

Teach to this depth — Establish the antiderivative, indefinite and definite integrals, the constant of integration, the Fundamental Theorem of Calculus as the link between the two operations, and area found by integration, using both analytic methods and technology.

Connects to — Integration is the inverse of differentiation and the continuous counterpart of the sigma sum; area is its first application.

IA & investigative angle — Accumulation models.

Teach to this depth — Cover the reverse chain rule, substitution, trigonometric integrals using power-reduction identities, exponential and reciprocal integrals, the properties of definite integrals, and the area between curves.

Connects to — Substitution is the chain rule reversed; the trigonometric integrals depend on the advanced-trigonometry identities; area extends to accumulation.

IA & investigative angle — Area and accumulation modelling.

Teach to this depth — Develop integration by parts (including repeated and cyclic cases), integration by partial fractions, integrals producing arctangent and logarithm, areas measured about the y-axis, and volumes of revolution.

Connects to — Parts mirrors the product rule; partial fractions reuse rational-function algebra; volumes connect to three-dimensional geometry. This topic is a prerequisite for both differential equations and probability density functions.

IA & investigative angle — The volume of a real solid, accumulation problems, and the normalisation of a probability model.

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, 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; data handling underlies any data-driven IA.

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, the meaning of r-squared, and the use of logarithms to linearise non-linear relationships.

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

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

Teach to this depth — Cover sample spaces and the axioms, combined events, conditional probability and independence (kept clearly distinct), Venn diagrams, tree diagrams and tables, Bayes’ theorem and Bayesian reasoning, and counting-based probability.

Connects to — The counting reaches back to permutations and combinations; conditional probability sets up the distributions; Bayes’ theorem models real inference.

IA & investigative angle — Risk analysis, medical testing through Bayes’ theorem, and games of chance.

Teach to this depth — Establish the random variable, the probability mass function, expectation 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 binomial distribution.

IA & investigative angle — Expected value in decision-making and in games.

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

Connects to — The nCr in the formula is exactly the binomial-expansion coefficient — make the link explicit, as it is one of the clearest cross-topic connections in the course. The conditions rest on independence.

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

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 unit; the curve is itself a probability density function, the bridge to the next topic.

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

Teach to this depth — Establish the continuous random variable, the properties of a density function (it integrates to one), probability as area, the cumulative distribution function, and the median, mode, mean and variance computed by integration, including piecewise densities.

Connects to — This topic requires definite integration — the dependency that fixes its position after the calculus core; the mean and variance mirror their discrete analogues, and the normal curve is a particular density.

IA & investigative angle — Continuous modelling such as waiting times and lifetimes.

Teach to this depth — Cover forming a differential equation from a rate description, separation of variables, the integrating factor for first-order linear equations, the homogeneous substitution, Euler’s numerical method, an intuition for slope fields, and modelling of growth, cooling and mixing.

Connects to — The integrating factor needs the product rule and separation needs the integration techniques; the solutions are exponential and logarithmic functions; Euler’s method is the continuous analogue of a recursive sequence.

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

Teach to this depth — Derive a series from repeated differentiation, establish the standard series (e^x, sin x, cos x, ln(1 + x) and the binomial series), build composite series by substitution, form products and quotients of series, differentiate and integrate series, generate a series from a differential equation, and discuss the interval of validity, approximation and error behaviour.

Connects to — This is where higher derivatives, the binomial series, integration and differential equations all converge; the validity discussion reaches back to limits. It is the true capstone of the course.

IA & investigative angle — Approximation accuracy, computing values of functions, and analysing truncation error — a genuinely advanced exploration.

Teach to this depth — Cover indeterminate forms, L’Hôpital’s rule with its conditions and pitfalls, the evaluation of limits using Maclaurin expansions, and a comparison of the two approaches.

Connects to — A synthesis of limits, differentiation and the Maclaurin series.

IA & investigative angle — Limit and approximation investigations.

Teach to this depth — Cover logical structure, direct proof, disproof by counterexample, and proof by contradiction (the irrationality of certain numbers, the infinitude of primes), and then induction across its full range — summation closed forms, divisibility results, inequalities such as Bernoulli’s, factorial and product statements, De Moivre’s theorem, the sum of a series, and even formulae for an nth derivative. Distinguish strong from weak induction and dissect the common errors in the inductive step.

Connects to — This topic deliberately pulls from sequences (sums), permutations and combinations (factorials), complex numbers (De Moivre), inequalities and calculus (the nth derivative). Placed last, the inductive step finally has a full toolkit to draw on — the central argument of this entire framework.

IA & investigative angle — Proof-based explorations in number theory and combinatorial identities, and conjecture-and-prove investigations.