Mathematics

What kind of number you are holding, which operations keep it that kind, and how much of a written figure is real. All eleven chapters of the module are here: the number classes and the line they live on, indices and standard form, ratio and commercial arithmetic, primes, and the error arithmetic every measured result depends on.

The point where arithmetic becomes symbolic — what a letter means, how expressions are collected and expanded, and how equations, inequalities and formulae are solved by doing the same thing to both sides rather than by moving letters across an equals sign.

The module where a symbolic expression acquires a shape. A quadratic is an equation to solve, a curve to draw, a factorisation to find and a trajectory to predict, and all four are one object here. Degree and end behaviour, the remainder and factor theorems, multiplicity, the three forms, the formula derived rather than handed over, the discriminant read exactly, Vieta both ways, the Physics bridge, and the excluded values that survive cancelling. All eleven chapters are authored.

The module the rest of the volume leans on. Sets and the regions a Venn diagram is a picture of, relations and what makes one a function, domain as something you discover by probing, composition and inverses, and the exponentials and logarithms that live here because a logarithm is an inverse before it is an arithmetic trick. All eleven chapters are authored, including the functions gate that stands before calculus.

The module where infinity first becomes computable. A pattern described by a rule rather than a list, arithmetic and geometric progressions with their sums derived rather than recalled, an infinite total that is finite, the three means and the inequality between them, the special series read off block models, recurrences whose ratios settle on a number no fraction reaches, and an explicit handoff to Limits & Continuity. All ten chapters are authored.

The module where a figure stops being a drawing and becomes a set of relationships that must hold. Angle chasing with reasons attached, the congruence criteria and the one that is not a criterion, similarity and what it does to area, two proofs of Pythagoras, the circle theorems with every one of them open to attack by dragging, constructions checked on method rather than on accuracy, and the point at which dragging stops being enough. All thirteen chapters are authored.

The module where geometry and algebra stop being two subjects. A point becomes a pair of numbers, a line becomes an equation with five spellings, and every conic is traced from the condition that defines it before its equation is shown. Distance as Pythagoras, gradient as a first derivative met three modules early, the circle both ways round, the four conics as one family with one number changed, loci predicted before they are drawn, and choosing better coordinates as a strategy. All fourteen chapters are authored.

The module usually reduced to a formula sheet, built the other way round: every formula is the end of an argument you can operate. The circle is unrolled into a strip, the cone is poured into its cylinder three times, the sphere arrives by Cavalieri's comparison, and the frustum is a subtraction of two similar cones. Solids are rotatable objects with nets you can flatten and sections you can cut, and slicing the cone produces all four conics before Coordinate Geometry gives them equations. Every answer is exact: a volume of thirty-six pi is shown as thirty-six pi. All ten chapters are authored.

Ratios in a right triangle, the two triangles that generate every exact value, and the unit circle that turns those ratios into waves — and then everything that follows from them. The compound-angle formula is built from a construction you can drag rather than quoted, and every other identity in the module descends from it by substitution. Identities are checked at hundreds of angles and the checker will reject a false one, because a verifier that only ever agrees has verified nothing. Equations are treated as having infinitely many solutions with an interval making the list finite, and the solver tells you which ones you missed rather than only that you were wrong. The ambiguous case shows every triangle that fits, not the first. All fourteen chapters are authored.

Quantities that carry a direction, and the operations that only make sense once they do. The dot product returns a number and the cross product returns a vector — sine where the other has cosine, which is why one vanishes for perpendicular vectors and the other for parallel ones. Every predicate is exact: perpendicular at exactly zero, coplanar at exactly zero volume, and direction cosines whose squares add to exactly one rather than to 0.9999999999999998. The second half moves into space proper, where two lines can be neither parallel nor intersecting — a thing that cannot happen on a page and is the ordinary case once there is a third dimension. Lines are stored as a point and a direction, so the Cartesian form that breaks for every horizontal line is a view that explains its own absence. All fourteen chapters are authored.

The derivative built three ways at once — as a limit, as a slope and as a rate — and then every rule derived from the first of them rather than handed over. Five functions are differentiated from the definition in-product, with the step where h leaves the denominator marked, because doing it once looks like a ritual and doing it five times is where the power rule stops looking like a coincidence. The naive (fg)′ = f′g′ is tested against a numerical derivative of the actual product and loses visibly. sin′ = cos is proved from the compound-angle formula and two standard limits, and a degree toggle breaks it — which is a better argument for radian measure than any assertion. e arrives as the answer to "which exponential is its own derivative?", having already appeared in module 07 as a compounding limit, and the two are cross-referenced as one constant rather than two. Implicit and parametric differentiation report a vertical tangent as undefined rather than as a large number. All thirteen chapters are authored.

What the derivative is for, taught as modelling rather than as differentiation. The optimisation workbench is deliberately not a solver: it scaffolds the constraint and the objective as separate gradeable steps, and the calculus does not run until they are done, because a learner who can differentiate and cannot turn a paragraph into an objective is stuck exactly where the marks are. Every rate carries units composed from its quantities and a sentence generated from them, so a rate cannot be labelled cm/s while being cm³/s. The closed-interval method puts both endpoints in the candidate list rather than trusting anyone to remember them, and the default example is one where an endpoint wins. Two traps are shown rather than mentioned: a stationary point that is not an extremum, and a vanishing second derivative that is not an inflection. Related rates are animated on the shared loop with a scrub, so any instant can be held. All twelve chapters are authored.

Accumulation as a computed thing before it is a symbol. The Riemann-sum chapter comes before the definite integral has a name, because the definite integral is defined as the limit of those sums — and the Fundamental Theorem is built around watching area accumulate rather than around the evaluation formula, so that F′ = f is something a learner notices rather than something they are told. Signed area and total area are shown as two numbers at once wherever a curve crosses the axis, because both are correct answers to different questions and a learner made to choose will lose the distinction. The area between two curves splits automatically where they cross, which is the error that produces a plausible wrong answer. The symmetry shortcuts are pictures before they are rules. And one integral here has no elementary antiderivative at all — not a hard one, an impossible one — which is the honest reason numerical methods exist and why every probability table was computed rather than derived. All fifteen chapters are authored.

An equation whose unknown is a function, and whose subject is a rate. The slope field comes second, before any technique, because a learner who has seen the field knows what solving means and one who meets separation of variables first has learnt a manipulation — and the click that picks a solution curve out of the field is module 20's constant of integration, arriving with a job. Every model carries its assumption beside its formula: N = N₀e^(kt) is something to substitute into, and dN/dt = kN is a claim about the world that can be argued with. The exponential law is derived rather than assumed, the separable technique's illegal-looking step is justified by the chain rule rather than by the notation, and the integrating factor is derived rather than produced from nowhere. Euler's method is shown drifting off the true curve, because each step starts from the last step's answer, and its poor error trade is what motivates everything better. The last chapter asks where each model stops being true, which is part of the answer and not a caveat appended to it. All twelve chapters are authored.

The one mathematical topic where the most consequential skill is judgement rather than computation. Every measure is computed on a dataset you can edit and break, so robustness is something you cause rather than something you are told about: drag one value to an extreme and the mean follows it while the median barely stirs. Three chapters exist purely to teach scepticism — sampling and bias, misleading graphs, outliers and skewness — and they are not enrichment. You will build a misleading chart yourself before you are shown the honest one, fail to beat the least-squares line by hand, watch a fitted line break the moment it leaves its data, and meet significance as a count of shuffles rather than a formula. All fourteen chapters are authored.

Counting, done as argument rather than as formula recall. Every count starts as a list you can actually see — the enumeration table shows every outcome under all four selection models, and the formula arrives afterwards as the shortcut it is, with the two numbers computed separately and shown agreeing. Raise the numbers and the list refuses to be drawn, which is the moment a formula stops being optional. The division by r! is a picture of grouping rather than a rule; nCr is read off a handshake grid and off the routes across a lattice; every restriction names the technique it used; and Pascal's triangle, the binomial expansion and the combination formula are driven by one control because they are one object. All counts are exact integers, so a poker hand is 2 598 960 and not approximately that. All ten chapters are authored.

The subject where intuition is wrong most often, taught by putting the intuition on the record first and then testing it. You predict before the Monty Hall simulation runs, and before the birthday curve is drawn. Every probability is an exact fraction — one third is one third, never 0.333 — and every simulation is seeded, so a result can be reproduced and checked rather than taken on trust. Conditional probability is performed as a restriction, with the excluded outcomes crossed out on screen; independence is tested rather than assumed, and shown to be the opposite of mutual exclusivity rather than a cousin; and base-rate neglect is corrected by counting a hundred thousand people rather than by restating Bayes' theorem. The card, dice and coin apparatus is imported from the counting module rather than defined again. Every simulated number carries the same sentence: it is evidence, not a proof. All fourteen chapters are authored.

The gateway to calculus, opened at a point where substitution fails rather than at one where it works. A limit is what a function heads for, which is a different question from what it equals — and the module's first widget shows them disagreeing before any technique is taught. Indeterminate forms are named and never evaluated: 0/0 is the signal that the algebra has not been done, not an answer. Every cancellation carries the sentence that makes it legal, which is that the approach never visits the point. The hole-versus-asymptote distinction opened in the polynomials module is closed here, on a plotter that refuses to draw a line across a pole. sin x / x has a degree toggle, and it reports π/180 — the single strongest argument for radian measure there is. The squeeze theorem is shown needing its bounds to meet at the point and nowhere else, with a case where genuine bounds fail to. Continuity is a three-condition diagnostic that names the condition that broke, and the module closes on the corner that is continuous, is not smooth, and is the picture module 18 begins from. All twelve chapters are authored.

The objects are easy to compute with and hard to believe in, so every complex number here is a point and every operation is a motion: addition translates, conjugation reflects, and multiplication rotates and scales. The Argand plane arrives third, before modulus and argument, because once multiplication is visibly a rotation De Moivre's theorem is a sentence about repeating a turn rather than a formula to memorise. i is defined by i² = −1 rather than as √−1, and the surd trap that notation licenses is shown and dissected. The argument widget prints arctan(b/a) beside the true principal value and names the quadrant correction rather than hiding it; the roots-of-unity widget adds the roots up rather than asserting their sum is zero; exponential form is labelled as notation whose justification is a power-series argument this module does not have. Module 06 raised the negative-discriminant question and declined to answer it — this module answers it, and the conjugate-root theorem's hypothesis can be removed so a learner can watch the symmetry fail. All nine chapters are authored.

Taught as an arithmetic of grids this is the most mechanical module in the senior syllabus, so it is taught here as transformation and area instead. A 2×2 matrix is something that moves the plane; its determinant is the factor by which it scales area, and the sign is whether it flipped orientation. A singular matrix collapses the plane onto a line — which is simultaneously why its determinant is zero, why it has no inverse, and why the matching system has no unique solution, three facts usually memorised separately. The transformation grid arrives fourth, seven chapters before its place in the syllabus table, because doc 11 requires it before the determinant formula: a learner who meets ad − bc first has been given a recipe. Every determinant, adjoint and inverse here is computed on exact fractions, because a widget reporting det = 0.0000000001 for a singular matrix has taught the wrong lesson. Zero determinant does not mean no solution — it means no unique solution, and the ranks are what tell parallel lines from the same line twice. All twelve chapters are authored.

Constrained optimisation where everything is linear and the answer is always at a corner — and one of the few school topics that is unambiguously a professional tool. The module is built on the premise that linear programming is 90% modelling and 10% method: the corner-point procedure takes ten minutes to teach, and turning a paragraph about a factory into four inequalities is where every mark is lost. So the workbench grades the variables, the constraints and the objective as separate steps and refuses to draw a graph until they are done, and a learner who mis-signs an inequality traces the wrong region back to the sentence that produced it. Vertices are solved for exactly and print as fractions, because reading 6.7 off an axis loses the meaning as well as the mark. The iso-profit slider shows the corner-point result before the theorem is stated, and rotating the objective reaches the multiple-optima case rather than being told about it. An unbounded region has no maximum and does have a minimum, both computed over the same region; infeasibility is reported with the conflicting constraints named; slack says which resource to buy more of and which not to. The last chapter's deliverable is a written recommendation, because a linear-programming answer that is not interpreted has not been given. All ten chapters are authored.

The module that formalises how mathematical argument works, and the module whose content underpins computation. Fifteen of its sixteen chapters are on no shipped syllabus — only modular arithmetic is routed, on the Applied Mathematics Program — and that is deliberate rather than an oversight: this module serves the Olympiad and undergraduate Tracks, and it serves every other module by making explicit the reasoning they all use. Module 12 taught proof with diagrams; here the same technique is formalised without them, in statements and quantifiers that no diagram can carry. Truth tables are generated rather than asserted, so a tautology is a column that never varies rather than a word to remember. The implication explorer shows a statement agreeing with its contrapositive in every row and disagreeing with its converse, which is what justifies proof by contrapositive. The induction widget requires both halves and demonstrates the failure when either is missing — including a step that never uses the hypothesis, which is the commonest induction failure there is and which no numerical check can see, because the claim it proves is true. The counterexample hunter's centrepiece survives forty consecutive tests before failing. The pigeonhole widget lets a learner try to beat the bound and watch the attempt fail, because that is the only convincing way to teach a non-constructive argument. Graph algorithms are stepped by learner action and never autoplayed. Growth orders are measured by running the algorithms and counting, on logarithmic axes with the decades drawn. All sixteen chapters are authored.

The volume's synthesis module, and the one that asks what all of it was for. Four of its twelve chapters are on the Applied Mathematics Program — estimation, linear models, financial mathematics and the proportional reasoning that arrives through them — and the other eight are routed by no shipped syllabus, which doc 02's coverage table marks as deliberate rather than as a gap. The order here is doc 26's learning flow rather than its chapter table: estimation comes first because it is the one chapter with no prerequisites at all. The modelling-cycle board is the frame every task runs inside, and modules 21 and 24 hand their final chapters to it rather than reimplementing it. The sensitivity bench is the module's most important widget, because it converts "state your assumptions" from advice into a measured number — and the assumptions that dominate turn out to be the unstated ones. Dimensional analysis shares the Physics volume's unit layer rather than defining a second one, and it carries its own counterexamples: two equations whose units balance perfectly and which are false. The amortisation schedule is computed exactly, rounded at the point of creation, and reaches zero on the final instalment because that instalment is whatever is left. The fit-versus-predict bench withholds the test range in the data rather than in the widget's discipline, so there is genuinely nothing to peek at. Estimation propagates ranges rather than point values, and counts landing inside the range as success. The population chapter fits its exponential to the early data only, which is the position a modeller is actually in, and then shows the capacity a logistic fit reports swinging by a factor of forty as points are added. All twelve chapters are authored, and the six case studies each name the volume that owns their facts.