What this resource is & the design principle
This framework presents the Analysis & Approaches Standard Level course as a single, prerequisite-ordered teaching sequence — not five syllabus units taught in numerical order. It is built on three convictions: that topics are taught in the order their dependencies require, that each is taught to its full depth rather than its minimum, and that the connections between topics are made explicit so students see mathematics as one connected subject. The aim is a student taught deeply and connectedly enough to design and write their own exploration — not one who, never shown the depth, outsources it.
Foundational — self-contained, and a prerequisite for later topics.
Developmental — extends one or more foundations.
Synthesis — applied topics that draw several earlier strands together, taught once those strands are secure. Optimisation, kinematics, the applications of integration and the normal distribution all sit here.
Standard Level has no late-stage ‘capstone’ topics of the kind found at Higher Level; instead the synthesis happens in these applied topics, where the foundations are brought to bear on a real problem. The dependency-ordered path below makes sure those foundations are always in place first.
The Teaching Spine
The full 30-topic sequence. Teach top to bottom; each phase is a prerequisite for the next.
Colour shows each topic’s role; the pill shows the phase’s teaching hours.
Where the Strands Come Together
Each applied topic rests on the foundations to its left. A gold arrow means ‘is a prerequisite for’ — so teach the feeders first.
Each applied topic rests on the foundations to its left; teach the feeders first.
The sequence at a glance
Every topic in teaching order, with its role and teaching hours.
| # | Topic | Phase | Role | Hours |
|---|---|---|---|---|
| Phase A — Algebraic & Functional Foundations (40 h) | ||||
| 1 | Straight-Line Functions & Coordinate Geometry | Phase A | Foundational | 3 h |
| 2 | Functions: Domain, Composite & Inverse | Phase A | Foundational | 4 h |
| 3 | Quadratic Functions & Equations | Phase A | Foundational | 4 h |
| 4 | Transformations of Graphs | Phase A | Foundational | 3 h |
| 5 | Sequences & Series | Phase A | Foundational | 7 h |
| 6 | Exponents & Logarithms | Phase A | Foundational | 7 h |
| 7 | Exponential & Logarithmic Functions | Phase A | Developmental | 4 h |
| 8 | Reciprocal & Rational Functions | Phase A | Developmental | 3 h |
| 9 | Binomial Theorem | Phase A | Foundational | 5 h |
| Phase B — Geometry & Trigonometry (25 h) | ||||
| 10 | 3D Geometry: Volume, Surface Area & Angles | Phase B | Foundational | 5 h |
| 11 | Right-Triangle Trig, Sine & Cosine Rules | Phase B | Foundational | 5 h |
| 12 | Radians, Arc Length & Sectors | Phase B | Foundational | 3 h |
| 13 | The Unit Circle, Exact Values & Identities | Phase B | Foundational | 5 h |
| 14 | Trigonometric Functions, Graphs & Modelling | Phase B | Developmental | 4 h |
| 15 | Solving Trigonometric Equations | Phase B | Developmental | 3 h |
| Phase C — Calculus (28 h) | ||||
| 16 | Limits & the Derivative Concept | Phase C | Foundational | 3 h |
| 17 | Differentiation: Power Rule, Tangents & Normals | Phase C | Foundational | 5 h |
| 18 | Differentiation Rules — Chain, Product & Quotient | Phase C | Developmental | 5 h |
| 19 | Graph Analysis — Stationary Points & Concavity | Phase C | Developmental | 4 h |
| 20 | Optimisation | Phase C | Synthesis | 3 h |
| 21 | Kinematics | Phase C | Synthesis | 3 h |
| 22 | Integration — Antiderivatives & Definite Integrals | Phase C | Developmental | 3 h |
| 23 | Areas & Applications of Integration | Phase C | Synthesis | 2 h |
| Phase D — Statistics & Probability (27 h) | ||||
| 24 | Exploring & Summarising Data | Phase D | Foundational | 5 h |
| 25 | Correlation & Regression | Phase D | Developmental | 4 h |
| 26 | Probability | Phase D | Foundational | 5 h |
| 27 | Probability Diagrams & Conditional Probability | Phase D | Developmental | 4 h |
| 28 | Discrete Random Variables & Expectation | Phase D | Developmental | 3 h |
| 29 | The Binomial Distribution | Phase D | Developmental | 3 h |
| 30 | The Normal Distribution & z-Scores | Phase D | Synthesis | 3 h |
| Total taught content | 120 h | |||
The 30 topics in depth
Each topic carries its teaching depth, its interconnections, and the investigative angle through which it prepares a student for the exploration.
Phase A — Algebraic & Functional Foundations
40 hTeach to this depth — Move fluently between the gradient-intercept, point-gradient and general forms; use the parallel and perpendicular conditions, distance and midpoint as reasoning tools; and read a line as a function whose gradient is a rate of change in context.
Connects to — The gradient is the first quiet appearance of the derivative, and a fitted straight line is the regression line of the statistics unit.
IA & investigative angle — Linear modelling of real data and interpreting a slope as a rate.
Teach to this depth — Establish the function as a mapping, determine domain and range, and build fluency with composite functions (and why order matters) and inverse functions (one-to-one behaviour and reflection in y = x). Read graph features confidently.
Connects to — Inverses return as logarithms (the inverse of exponentials); composition is the idea behind the chain rule; a defensible domain underpins every model.
IA & investigative angle — Every modelling exploration needs a justified domain; staged processes use composition.
Teach to this depth — Teach the three forms and conversion between them, completing the square as a structural tool, the discriminant as a classifier of roots and intersections, and the solution of quadratic equations and simple quadratic models.
Connects to — The discriminant counts intersections of graphs; completing the square gives the vertex used in optimisation; quadratics model projectile and area problems.
IA & investigative angle — Projectile motion, area optimisation and parabolic model fitting.
Teach to this depth — Cover translations, stretches and reflections and the careful decoding of combined transformations, and their effect on key features. Build the two-way habit of predicting a graph from algebra and the algebra from a graph.
Connects to — It is the lens for reading exponential, logarithmic and trigonometric graphs, and reflection in y = x is exactly the inverse relationship.
IA & investigative angle — Parameter-driven modelling depends on knowing how each coefficient moves a curve.
Teach to this depth — Establish that a sequence is a function of its position; develop arithmetic and geometric sequences and series, sigma notation, the sum to infinity of a convergent geometric series, and financial applications such as compound interest.
Connects to — The sequence-as-function idea ties to the Functions unit, and geometric growth is the discrete cousin of the exponential function.
IA & investigative angle — Financial modelling, geometric patterns and convergence investigations — fertile IA ground.
Teach to this depth — Derive the logarithm laws from the exponent laws, solve exponential and logarithmic equations including those that reduce to quadratics, and introduce e and ln and the idea of a logarithmic scale.
Connects to — Exponentials and logarithms are an inverse pair; the log laws are the engine behind linearising data; this is the basis for exponential calculus.
IA & investigative angle — Linearising data with logarithms to test exponential or power models — a hallmark of a strong SL exploration.
Teach to this depth — Study the graph families, their transformations and asymptotes, and model growth and decay, interpreting each parameter in context.
Connects to — Builds on the laws and on transformations; feeds the calculus of exponential functions and real-world modelling.
IA & investigative angle — Growth and decay modelling against real data.
Teach to this depth — Study the reciprocal function and the simple rational function (ax + b)/(cx + d): asymptotes, domain and sketching from a full analysis.
Connects to — Uses transformations, and its asymptotic behaviour is an informal first meeting with limits.
IA & investigative angle — Rate and concentration models.
Teach to this depth — Develop Pascal’s triangle and the nCr formula, the general term, and the extraction of a specific coefficient (positive integer index).
Connects to — The same nCr appears in the binomial distribution of the statistics unit — make the link explicit.
IA & investigative angle — Counting, approximation and probability links.
Phase B — Geometry & Trigonometry
25 hTeach to this depth — Work with three-dimensional coordinates and distances, the volume and surface area of solids and composites, and angles between lines and planes resolved through right triangles, in real measurement contexts.
Connects to — The right-triangle angle work leans on trigonometry, and the spatial reasoning supports later modelling.
IA & investigative angle — Packaging and volume optimisation, and surveying.
Teach to this depth — Cover right-angled trigonometry, the sine and cosine rules and the area formula — including the ambiguous case with genuine geometric reasoning — and applications such as bearings and angles of elevation and depression.
Connects to — Applies the trigonometric ratios and precedes the unit circle and identities.
IA & investigative angle — Surveying, navigation and indirect measurement.
Teach to this depth — Define the radian as a ratio and compute arc length and sector area, combining sectors and segments.
Connects to — Radians are required for trigonometric graphs and for the calculus of trigonometric functions.
IA & investigative angle — Circular contexts — gears, clocks and design.
Teach to this depth — Treat the unit circle as the source of all values and signs; establish exact values, the Pythagorean identity, and the double-angle identities for sine and cosine.
Connects to — The foundation for trigonometric graphs, equation solving and the calculus of trigonometric functions.
IA & investigative angle — The conceptual bedrock for periodic modelling.
Teach to this depth — Treat amplitude, period, phase and vertical shift, apply the transformation toolkit to the trigonometric functions, and model real periodic phenomena, interpreting each parameter.
Connects to — Draws directly on the transformations unit and precedes trigonometric calculus.
IA & investigative angle — Tides, daylight hours and seasonal data — a rich SL context; emphasise estimating parameters from real data.
Teach to this depth — Solve trigonometric equations over a given interval, using identities to simplify and the unit circle to find every solution.
Connects to — Brings together the unit circle, the identities and the graphs.
IA & investigative angle — Where a periodic model must be solved for a specific value.
Phase C — Calculus
28 hTeach to this depth — Develop an intuitive limit, the derivative as a gradient function and as a rate of change, and the link between the two.
Connects to — The foundation of all differentiation; the gradient idea comes straight from the work on lines.
IA & investigative angle — Numerical exploration of rates and gradients.
Teach to this depth — Develop the power rule and the derivatives of standard functions, the equations of tangents and normals, and intervals of increase and decrease.
Connects to — Applies limits and function notation, and the rate-of-change reading recurs throughout the course.
IA & investigative angle — Rates in real models.
Teach to this depth — Build fluency with the chain, product and quotient rules and the derivatives of sin x, cos x, e^x and ln x, and their combinations.
Connects to — The chain rule is composition differentiated; this is the toolkit for every later calculus application.
IA & investigative angle — Any model needing the rate of change of a composite quantity.
Teach to this depth — Use the first and second derivatives to find and classify stationary points, determine concavity and points of inflection, and sketch curves from f, f’ and f”.
Connects to — Reunites the functions and transformations work with calculus and sets up optimisation.
IA & investigative angle — Analysing the behaviour of a modelled quantity.
Teach to this depth — Run the full modelling cycle — variable, constraint, objective, domain, classify the extremum and interpret — justifying with the second derivative.
Connects to — Pulls together the differentiation rules, stationary-point analysis and quadratic or area modelling.
IA & investigative angle — A classic strong-SL structure: optimise a real quantity with a justified model.
Teach to this depth — Relate displacement, velocity and acceleration through differentiation and integration, interpret signs, and distinguish distance from displacement.
Connects to — The one topic where differentiation and integration meet in a single context.
IA & investigative angle — Motion modelling, including sport.
Teach to this depth — Establish the antiderivative, the definite integral, the constant of integration, and the basic integration of standard functions.
Connects to — The inverse of differentiation and the foundation of area.
IA & investigative angle — Accumulation models.
Teach to this depth — Find the area under a curve and between a curve and the x-axis, and apply integration to accumulation problems.
Connects to — Builds on integration and on reading functions and their graphs; area is accumulation made visible.
IA & investigative angle — Area and accumulation modelling.
Phase D — Statistics & Probability
27 hTeach to this depth — Cover sampling and the bias it can introduce, measures of centre and spread, standard deviation, quartiles and the interquartile range, outliers, box plots, cumulative frequency and histograms, and the effect of a linear transformation on the mean and standard deviation.
Connects to — Standard deviation feeds the normal distribution, and this work underlies any data-driven IA.
IA & investigative angle — The foundation for a data IA — but go beyond the survey-and-bar-chart; depth means honest analysis and interpretation.
Teach to this depth — Cover scatter plots, the interpretation and limitations of Pearson’s r, the least-squares regression line, and the dangers of extrapolation.
Connects to — The regression line is a straight-line function, and correlation does not imply causation — a key reasoning point.
IA & investigative angle — The mainstay of data explorations; teach interpretation and caution to add depth.
Teach to this depth — Cover sample spaces, combined events, mutually exclusive and independent events, and the basic probability rules.
Connects to — The basis for all the distributions that follow, with counting linking back to the binomial coefficient.
IA & investigative angle — Risk analysis and games of chance.
Teach to this depth — Use Venn diagrams, tree diagrams and tables, and compute conditional probabilities, keeping independence and exclusivity clearly distinct.
Connects to — Conditional reasoning sets up the distributions, and the diagrams make problems tractable.
IA & investigative angle — Medical-testing and decision scenarios.
Teach to this depth — Establish the random variable, the probability distribution and the expected value, and build a distribution from a real context.
Connects to — Expectation generalises the weighted mean and is the precursor to the binomial distribution.
IA & investigative angle — Expected value in decisions and games.
Teach to this depth — Cover the conditions for a binomial model, the probability calculation built on nCr, the mean and variance, and cumulative probabilities.
Connects to — The nCr here is exactly the binomial-expansion coefficient — one of the clearest cross-topic links in the course.
IA & investigative angle — Success rates and quality control.
Teach to this depth — Cover the properties of the normal curve, standardisation and the z-score, probabilities by technology, the inverse normal, and finding an unknown mean or standard deviation.
Connects to — Standardisation uses the standard deviation from the data unit; it brings probability and continuous data together.
IA & investigative angle — Modelling real measurement data and judging the fit of a normal model.
Time allocation & two-year pacing
The IB recommends 150 teaching hours for a Standard Level subject — for AA SL, 120 hours of taught content plus 30 hours for the toolkit and exploration. The hours are reconciled to the official per-unit totals and paced so that teaching is complete by the end of January in Year 2.
Reconciliation to the official IB allocation
| IB syllabus unit | Official IB hours | Allocated here |
|---|---|---|
| Number & Algebra | 19 h | 19 h |
| Functions | 21 h | 21 h |
| Geometry & Trigonometry | 25 h | 25 h |
| Statistics & Probability | 27 h | 27 h |
| Calculus | 28 h | 28 h |
| Taught content subtotal | 120 h | 120 h |
| Toolkit + Mathematical Exploration (IA) | 30 h | 30 h |
| SL course total | 150 h | 150 h |
The two-year pacing plan
Built on roughly three to three-and-a-half teaching hours per week; the cumulative column tracks progress toward the 150-hour total. Gold rows fall outside the teaching budget.
| Period | Focus | Hours | Cumul. |
|---|---|---|---|
| YEAR 1 | |||
| Autumn term | Phase A — Algebraic & Functional Foundations | 40 | 40 |
| Spring term | Phase B — Geometry & Trigonometry · begin the toolkit | 30 | 70 |
| Summer term | Phase C — Calculus (Limits → Optimisation) · launch the IA exploration | 28 | 98 |
| YEAR 2 (to end January) | |||
| Autumn term | Finish Phase C (Kinematics, Integration & applications) · Phase D — Statistics & Probability · write & submit the IA | 47 | 145 |
| To end of January | Synthesis review, exam-style consolidation & IA finalisation | 5 | 150 |
| FEBRUARY – APRIL | Dedicated revision: past papers, Paper 1 / 2 drills and timed mocks (additional to the 150 teaching hours) | — | — |
| MAY | IB examinations | — | — |
Revision time (February–April) is additional to the 150 teaching hours, in line with the subject guide’s reminder to set aside time for examination revision.