未就学ごろ（パート A） → 上級 · レベル 4
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未就学ごろによく学ぶ内容（パート A） · Ages 3–4 · about 2.5 hours
Numbers to five, comparing groups, basic shapes, patterns and sorting, for children who cannot read yet: every prompt is read aloud and every answer is a picture or a single number.
未就学ごろによく学ぶ内容（パート B） · Ages 4–5 · about 4 hours
Numbers to ten, one more and one fewer, comparing and ordering, more shapes and positions, comparing lengths, weights and capacity, three-part patterns and sorting with counts. Every prompt is read aloud.
Take first: Early mathematics A
幼稚園年長ごろによく学ぶ内容 · Ages 5–6 · about 4.5 hours
Counting to one hundred, counting up to twenty objects, comparing numbers, adding and subtracting within ten, teen numbers as ten and ones, flat and solid shapes, measuring words, and sorting. Prompts are read aloud.
Take first: Early mathematics B
2 年生ごろによく学ぶ内容 · Ages 7–8 · about 16.5 hours
Fluency within twenty, even and odd, arrays, place value to 1000, adding and subtracting to 1000, money and time to five minutes, measuring length, picture graphs and line plots, shapes by sides, and thirds.
Take first: Mathematics foundations
3 年生ごろによく学ぶ内容 · Fractions · Ages 8–9 · about 9 hours
What a fraction names, fractions as numbers on a line, equivalent fractions, comparing fractions, whole numbers as fractions, the words numerator and denominator, fractions as decimals, and adding fractions with the same denominator.
Take first: Mathematics level 2
4 年生ごろによく学ぶ内容 · Ages 9–10 · about 19 hours
Multiplicative comparison, factors and primes, numbers to a million, multi-digit multiplication and division with remainders, equivalent fractions and operations, decimals to hundredths, unit conversion, angles, lines and symmetry.
Take first: Fractions, level 3
6 年生ごろによく学ぶ内容 · Ages 11–12 · about 20 hours
Ratios, rates and percent, dividing fractions, negative numbers and the full coordinate plane, exponents and algebraic expressions, one-step equations and inequalities, area and volume, and statistics.
Take first: Mathematics level 4
7 年生ごろによく学ぶ内容 · Ages 12–13 · about 14 hours
Proportional relationships and percent problems, operations with rational numbers, linear expressions and two-step equations and inequalities, scale drawings, triangles, circles and angles, sampling and comparing populations, and probability.
Take first: Mathematics level 6
8 年生ごろによく学ぶ内容 · Ages 13–14 · about 18.5 hours
Irrational numbers, exponent rules and scientific notation, slope and linear equations and systems, functions, transformations and similarity, angle facts, the Pythagorean theorem, volumes of curved solids, and bivariate data.
Take first: Mathematics level 7
Algebra I · Ages 14–18 · about 6 hours
Quantities and structure, exponent rules and polynomials, linear equations with justification, formulas, systems and inequalities, functions and their features, sequences and transformations, linear and exponential models, quadratics and parabolas, and statistics with lines of fit and correlation.
Take first: Mathematics level 8
Geometry · Ages 14–18 · about 5.5 hours
Rigid motions and congruence, proofs about triangles, similarity and right-triangle trigonometry, circle theorems and arcs, coordinate geometry with circles and parabolas, volume and modelling, constructions, and conditional probability.
Take first: Algebra I
Algebra II · Ages 14–18 · about 6.5 hours
Polynomials and the remainder theorem, rational and radical equations, complex numbers, exponential models and logarithms, trigonometric functions on the unit circle, inverse and composite and piecewise functions, geometric series, the normal distribution and inference.
Take first: Geometry
Precalculus · Ages 14–18 · about 5 hours
Rational functions and transformations, logarithms and exponential models, trigonometric identities and inverse functions, the laws of sines and cosines, polar and parametric forms, vectors and matrices, conic sections, sequences and the binomial theorem, induction, limits, and the complex plane.
Take first: Algebra II
Statistics · Ages 14–18 · about 4 hours
Describing distributions, study design, probability rules and random variables, binomial and geometric models, sampling distributions, confidence intervals, hypothesis tests, chi-square, and regression inference.
Take first: Algebra II
Calculus AB · Ages 14–18 · about 3.5 hours
Limits and continuity, the derivative and its rules, the chain rule and implicit differentiation, applications to rates and extrema, Riemann sums and the fundamental theorem, antiderivatives and substitution, area and volume, separable differential equations, and motion on a line.
Take first: Precalculus
Calculus BC · Ages 14–18 · about 9.5 hours
Integration by parts and partial fractions; improper integrals at an infinite limit and at a singular endpoint; parametric and polar curves with their slopes, lengths and areas; the convergence tests and the hypotheses each one needs; power and Taylor series with the radius and the interval they converge on; the alternating and Lagrange error bounds; and Euler's method with the logistic model.
Take first: Calculus AB
上級 · レベル 1 · Ages 18+ · about 10 hours
The real numbers and completeness; functions, domains and inverses; limits with epsilon and delta, the limit laws, the squeeze theorem and continuity with the intermediate and extreme value theorems; the derivative from its definition and from the rules, including the chain rule, implicit differentiation and the exponential and logarithmic derivatives; related rates, linearisation, extrema, the mean value theorem, curve shape, optimisation and L'Hopital's rule; and the definite integral from Darboux sums through both parts of the fundamental theorem to substitution.
Take first: Precalculus
上級 · レベル 1 · Ages 18+ · about 11.5 hours
Integration by parts, reduction formulas, trigonometric integrals and substitutions, completing the square and partial fractions, with a strategy for choosing between them; area, volume by slicing and by shells, arc length, surface area, work and average value; improper integrals at an infinite limit and at a singular endpoint, with comparison; sequences, partial sums, and the divergence, integral, comparison, ratio, root and alternating series tests, with absolute and conditional convergence; power series, term-by-term calculus, Taylor series and the Lagrange remainder; and parametric and polar curves, with their slopes, lengths and areas.
Take first: Calculus I
上級 · レベル 1 · Ages 18+ · about 13.5 hours
Linear equations and the augmented matrix; Gaussian elimination, the reduced row echelon form and the solution set; rank, consistency and homogeneous systems; matrix arithmetic, multiplication as composition, inverses by row reduction and the LU factorisation; determinants by cofactor expansion and by row operations, with Cramer's rule and the reading of a determinant as an area; vector spaces and subspaces, span, independence, bases, dimension, the four fundamental subspaces and the rank-nullity theorem; linear maps and their matrices, kernel and image, coordinates and change of basis; the characteristic polynomial, eigenvectors, diagonalisation and matrix powers; the dot product, projections, Gram-Schmidt and least squares; and the spectral theorem with an introduction to the singular value decomposition.
Take first: Precalculus
上級 · レベル 1 · Ages 18+ · about 11 hours
Propositional and predicate logic: the connectives, the converse and the contrapositive, logical equivalence, quantifiers and how to negate them. The proof techniques: direct, contrapositive, contradiction, cases and induction. Sets, power sets and products; relations, equivalence relations and partitions; injections, surjections and bijections. Counting with the product and sum rules, permutations and combinations, binomial coefficients and the pigeonhole principle. Graphs: degrees and the handshake lemma, trees, Euler circuits, planarity and bipartite graphs. Number theory: divisibility and factorisation, the Euclidean algorithm and Bézout's identity, and modular arithmetic. And recurrence relations, solved by running them forward and by their characteristic equations.
Take first: Precalculus
上級 · レベル 2 · Ages 18+ · about 13 hours
Vectors in space with the dot, cross and scalar triple products, and the lines, planes and quadric surfaces they describe; vector functions, velocity and acceleration, arc length, curvature and the moving frame of a curve; functions of several variables, limits along paths, partial derivatives, tangent planes, the chain rule and its Jacobian, directional derivatives and the gradient, and extrema both free and constrained by Lagrange multipliers; double integrals over rectangles and general regions, polar coordinates, triple integrals, cylindrical and spherical coordinates and the change-of-variables formula; and vector calculus — fields, divergence and curl, line integrals, conservative fields and potentials, Green's theorem, surface integrals and flux, and Stokes' and the divergence theorem.
Take first: Calculus II
上級 · レベル 2 · Ages 18+ · about 8.5 hours
Sample spaces and Kolmogorov's axioms, counting for equally likely models, permutations and combinations, inclusion-exclusion, and probability as length and area; conditional probability, the multiplication rule, independence, the law of total probability and Bayes' theorem with its base rates; discrete random variables, mass and cumulative distribution functions, the Bernoulli, binomial, geometric and Poisson families, densities, and the uniform, exponential and normal families; expectation, variance, linearity and moment generating functions; joint and marginal distributions, conditional expectation and the tower property, covariance and correlation; and the limit theorems — Markov and Chebyshev, the law of large numbers, and the central limit theorem.
Take first: Statistics and probability, Calculus II
上級 · レベル 2 · Ages 18+ · about 9 hours
Estimators and their sampling distributions, bias, mean squared error, and estimates built by the method of moments and by maximum likelihood; confidence intervals for a mean, for a mean with the spread estimated from the data, and for a proportion, with the width, the confidence level and the sample size traded against one another; hypothesis tests from the null and the alternative through the test statistic, the rejection region and the P-value to the two kinds of error and the power, and on to paired and two-sample comparisons; simple linear regression by least squares, with fitted values, residuals, the explained fraction of the variation, inference for the slope and the diagnostics that decide whether the line should have been fitted at all; one-way analysis of variance, the split of the total sum of squares and the F test, with the pairwise comparisons that follow it; and a first look at Bayesian inference — priors, likelihoods and posteriors, conjugate updating, and credible intervals set beside confidence intervals.
Take first: Probability
上級 · レベル 2 · Ages 18+ · about 15.5 hours
First-order equations — reading a slope field, separation, integrating factors, exact equations, existence and uniqueness, and the models they answer; linear second-order equations from the characteristic roots through the Wronskian, undetermined coefficients, variation of parameters and boundary value problems; the Laplace transform, what it does to an initial value problem, and how a switched forcing is handled; autonomous equations on a phase line, systems in matrix form and the classification of a phase portrait; series solutions at ordinary and at regular singular points; and the numerical methods a modeller reaches for, with the error and stability facts that decide the step size.
Take first: Calculus II, Linear algebra
上級 · レベル 3 · Ages 18+ · about 11 hours
The completeness of the real numbers — suprema, the approximation property, the Archimedean property and density, and estimating with the triangle inequality; sequences, from the epsilon-N definition through the algebra of limits, monotone convergence, Bolzano-Weierstrass and the Cauchy criterion, and then series and the tests that settle them; limits of functions and continuity, with the intermediate and extreme value theorems and uniform continuity; differentiation, the mean value theorem and what it settles, and Taylor's theorem with remainder; the Riemann integral by Darboux sums, which functions are integrable, and the fundamental theorem; and sequences of functions, where pointwise convergence keeps almost nothing and uniform convergence keeps continuity, the integral and — with care — the derivative.
Take first: Calculus II, Introduction to proof and discrete mathematics
上級 · レベル 3 · Ages 18+ · about 12 hours
Binary operations and the group axioms; Cayley tables; what the axioms already force; the order of an element, subgroups and the subgroup test, and the classification of cyclic groups; permutations in cycle notation, composition and inverses, order and parity, and the alternating group; cosets, Lagrange's theorem with its corollaries and normal subgroups; homomorphisms, kernels and images, quotient groups and the first isomorphism theorem; rings, integral domains and fields, units and zero divisors, ideals and quotient rings, prime and maximal ideals; and polynomial rings over a field — division, roots, irreducibility and the construction of the finite fields.
Take first: Introduction to proof and discrete mathematics
上級 · レベル 3 · Ages 18+ · about 10 hours
Complex numbers in the plane, with modulus, conjugate, polar form, de Moivre's theorem and the roots of unity; the exponential and the logarithm with its branches; limits along every direction at once, the complex derivative, the Cauchy-Riemann equations, harmonic functions and harmonic conjugates; contours, contour integrals and the estimation bound, Cauchy's theorem, Cauchy's integral formula and the theorems Liouville's argument buys; Taylor and Laurent series, the classification of isolated singularities, residues and the residue theorem, with the real integrals it settles; and conformal maps — angle preservation, Möbius transformations, the cross ratio, circles and lines, and the mapping of regions that turns a hard boundary value problem into an easy one.
Take first: Calculus III, Real analysis I
上級 · レベル 3 · Ages 18+ · about 8 hours
What finite arithmetic does to mathematics: how a number is stored and which values are exact, absolute against relative error, catastrophic cancellation and the rewrites that remove it, and an ill-conditioned problem against an unstable method; root finding by bracketing, fixed-point iteration, Newton and the secant method, with the order each converges at and the hypothesis each order rests on; interpolation by Lagrange and divided differences, its error term, Runge's phenomenon and the splines that answer it; numerical differentiation, the step size that balances truncation against rounding, and the trapezoid, midpoint and Simpson rules with their composite error orders; linear systems by elimination, partial pivoting and LU factorisation, and by Jacobi and Gauss-Seidel iteration; and solvers for initial value problems from Euler through Heun and Runge-Kutta to absolute stability and stiffness.
Take first: Differential equations
上級 · レベル 3 · Ages 18+ · about 8.5 hours
The language of propositional logic: formulas and their main connectives, symbolising English, truth tables, tautology and contradiction, logical equivalence, normal forms and complete sets of connectives. Validity as the absence of a countermodel, the named argument forms and the fallacies they are confused with, consistent sets and what follows from them. Natural deduction: the basic rules, disjunction and the chain rule, strategy, conditional proof and reductio. Predicate logic: predicates and quantifiers, mixed quantifiers and the order they come in, negation, interpretations and truth in a structure, and identity with the counting claims it makes possible. And the metatheory: what soundness and completeness say about the relation between a derivation and an entailment, which questions have a decision procedure, and what the incompleteness theorems state about a formal system that meets their conditions.
Take first: Introduction to proof and discrete mathematics
上級 · レベル 3 · Ages 18+ · about 11 hours
Divisibility and the division algorithm, Euclid's algorithm and Bézout's identity, unique factorisation and the divisor functions; congruences, inverses, linear congruences and the Chinese remainder theorem, with the divisibility tests they explain; Fermat's little theorem, Euler's totient and theorem, Wilson's theorem, and the arithmetic public-key cryptography rests on; orders and primitive roots, quadratic residues, the Legendre symbol, Euler's criterion and quadratic reciprocity; linear Diophantine equations, Pythagorean triples, sums of two squares, continued fractions and Pell's equation; and the distribution of primes, from Euclid's proof and the sieve to the prime number theorem and Dirichlet's theorem on arithmetic progressions.
Take first: Abstract algebra I
上級 · レベル 4 · Ages 18+ · about 9 hours
What a computed answer is worth: the condition number of a problem, backward error and what backward stability does and does not promise, the condition number of a matrix and why a small residual is not a small error, and the digits no method can recover; orders of convergence read from the errors, the contraction mapping theorem with its hypotheses and its two bounds, why Newton's method is quadratic at a simple root and linear at a repeated one, and what acceleration assumes; approximation theory from least squares and orthogonal families through the Chebyshev polynomials, the nodes that minimise the interpolation error factor, and the equioscillation theorem that recognises a best uniform approximation; quadrature by degree of exactness, Gauss rules and the bound they meet, Romberg extrapolation and adaptive subdivision; numerical linear algebra by cost, growth factor and conditioning, QR and least squares, the spectral radius that decides an iteration, Krylov methods and preconditioning, and the power method's rate; and the stability theory of solvers for differential equations — consistency and convergence, the stability function and its region, A-stability, stiffness and Dahlquist's two theorems.
Take first: Real analysis I, Numerical methods
上級 · レベル 4 · Ages 18+ · about 13 hours
Group actions, orbits and stabilisers, counting with Burnside's lemma, the class equation and the Sylow theorems with the counting arguments that settle the structure of small groups; normal and composition series, Jordan-Hölder, solvable groups, the derived series and the simplicity of the alternating group; divisibility in an integral domain, Euclidean domains, principal ideal domains, unique factorisation and Noetherian rings; field extensions and degree, minimal polynomials, splitting fields, the classification of the finite fields and the classical impossible constructions; and Galois theory — Galois groups, normality and separability, the fundamental correspondence, cyclotomic extensions, and why the quintic has no formula in radicals.
Take first: Abstract algebra I
上級 · レベル 4 · Ages 18+ · about 9 hours
Modelling linear programs, the graphical method, the simplex method, duality and sensitivity, integer programming with branch and bound, and network flows.
Take first: Linear algebra, Introduction to proof and discrete mathematics
上級 · レベル 2 · Ages 18+ · about 11 hours
Formulating a model — decision variables, an objective, constraints, and the four things a feasible region can turn out to be; convexity and why it, rather than linearity, is the dividing line; optimality conditions from stationarity through Lagrange multipliers to the KKT conditions and duality; the continuous methods (gradient descent, line search and trust regions, Newton, quasi-Newton, the simplex idea geometrically, interior point, quadratic programming); integer programming with relaxation, branch and bound, cutting planes, network flows and honest heuristics; and optimization under uncertainty — multiple objectives, sensitivity, stochastic and robust formulations, and what a model is and is not.
Take first: Calculus AB, Linear algebra, Calculus III
上級 · レベル 4 · Ages 18+ · about 15.5 hours
Entropy, relative entropy and mutual information with their inequalities; source coding from Kraft's inequality through Huffman, typical sets and arithmetic coding; discrete and Gaussian channels, capacity and the channel coding theorem with its converse; linear block codes, syndrome decoding, Hamming and Reed-Solomon codes and the bounds that limit them.
Take first: Linear algebra, Probability, Abstract algebra I