Use counting numbers, place value, number lines, factors, multiples, primes, and common number systems to compare and classify quantities.

Calculate with addition, subtraction, multiplication, and division using mental strategies, written algorithms, estimation, and order of operations.

Work with negative numbers, absolute value, signed operations, and directed quantities such as temperature, elevation, debt, and change.

Represent fractions as numbers, ratios, and division; compare them and perform all four operations with mixed numbers and improper fractions.

Convert among decimals, fractions, and percentages, then apply them to discounts, taxes, interest, error, and percentage change.

Use ratios, unit rates, proportions, scaling, and direct or inverse variation to solve comparison and conversion problems.

Choose units, convert dimensions, estimate magnitude, track significant figures, and use dimensional analysis to check calculations.

Trace the development of numerals, geometry, algebra, calculus, probability, formal proof, computation, and the institutions that shaped modern mathematics.

Evaluate and simplify expressions containing variables, constants, coefficients, parentheses, and operation properties.

Solve linear equations and inequalities, manipulate formulas, represent solution sets, and check for equivalent or invalid steps.

Represent relations with tables, graphs, formulas, and mappings; identify domain, range, intercepts, transformations, inverses, and composition.

Model constant rates with lines, interpret slope and intercepts, and solve systems by graphing, substitution, and elimination.

Add, multiply, divide, graph, and factor polynomials; use roots, multiplicity, and the remainder and factor theorems.

Simplify rational expressions, solve rational equations, analyze asymptotes, and identify excluded values and extraneous solutions.

Apply exponent laws, scientific notation, radicals, rational exponents, and radical equations while checking domain restrictions.

Calculate with imaginary and complex numbers, use polar form, and interpret roots and multiplication on the complex plane.

Model repeated growth and decay with exponential functions, solve with logarithms, and apply compound interest and half-life formulas.

Describe arithmetic, geometric, and recursively defined sequences; calculate finite sums and use sigma notation and the binomial theorem.

Use propositions, quantifiers, sets, counterexamples, and direct, contrapositive, contradiction, induction, and existence proofs.

Reason with points, lines, angles, polygons, circles, congruence, constructions, and the axioms behind Euclidean proofs.

Use similarity, scale factors, the Pythagorean theorem, and special right triangles to derive lengths and prove geometric relationships.

Calculate perimeter, area, surface area, and volume, including composite figures and solids, while tracking units and approximation error.

Represent geometry with coordinates, vectors, transformations, distance, midpoint, slope, and equations of lines and circles.

Use sine, cosine, tangent, radians, identities, inverse functions, and the laws of sines and cosines to solve triangles and periodic problems.

Analyze parabolas, ellipses, hyperbolas, parametric curves, and polar curves from equations, geometric definitions, and graphs.

Summarize distributions with tables, charts, center, spread, percentiles, and outlier measures while detecting misleading displays.

Count arrangements and selections with permutations, combinations, inclusion–exclusion, the pigeonhole principle, and generating arguments.

Calculate conditional probability, independence, Bayes’ theorem, expected value, variance, and common discrete and continuous distributions.

Design samples and experiments, estimate parameters, construct confidence intervals, test hypotheses, and interpret statistical significance and practical effect.

Work with relations, recurrence equations, Boolean algebra, finite-state models, and invariants used in algorithms and digital systems.

Model networks with vertices and edges; analyze paths, trees, connectivity, coloring, matching, flows, and shortest-path problems.

Study divisibility, congruences, prime factorization, Diophantine equations, modular arithmetic, and classical results such as Fermat’s and Euler’s theorems.

Analyze limits, one-sided behavior, asymptotes, continuity, and indeterminate forms as the foundation of calculus.

Compute derivatives from definitions and rules, including implicit, logarithmic, and higher derivatives, and interpret them as rates and local change.

Use derivatives for optimization, related rates, curve sketching, linear approximation, error estimates, and Newton’s method.

Interpret definite integrals as accumulation and signed area, compute antiderivatives, and connect differentiation and integration through the fundamental theorem of calculus.

Apply substitution, integration by parts, partial fractions, trigonometric methods, numerical quadrature, and tests for improper integrals.

Determine convergence of infinite sequences and series, then build power, Taylor, and Fourier series approximations with error control.

Represent systems with matrices, perform elimination and factorization, and determine rank, null spaces, consistency, and numerical sensitivity.

Work with vector spaces, subspaces, bases, dimension, coordinates, inner products, orthogonality, and least-squares approximation.

Treat matrices as linear transformations and use determinants, eigenvalues, eigenvectors, diagonalization, and singular value decomposition.

Differentiate and integrate functions of several variables using partial derivatives, gradients, Jacobians, multiple integrals, and constrained optimization.

Analyze vector fields with line and surface integrals, divergence, curl, and the Green, Stokes, and divergence theorems.

Solve first- and higher-order ordinary differential equations using qualitative, exact, transform, series, and numerical methods.

Build and analyze systems that evolve over time using phase portraits, stability, bifurcations, iteration, and deterministic chaos.

Turn a real question into variables, assumptions, equations, data needs, and validation tests; revise the model and communicate uncertainty and limitations.

Track rounding, truncation, conditioning, stability, and convergence so computer-generated numerical answers can be trusted.

Solve large matrix problems with direct and iterative algorithms, sparse representations, preconditioning, and numerical eigenvalue methods.

Approximate roots, derivatives, integrals, differential equations, and interpolating functions while measuring convergence and computational cost.

Formulate objective functions and constraints, then solve linear, integer, network, and combinatorial optimization problems.

Use convexity, gradients, Lagrange multipliers, duality, and first- and second-order algorithms for continuous optimization.

Analyze queues, inventories, schedules, resource allocation, simulation, and decision processes used in logistics and service systems.

Construct rigorous definitions of limits, continuity, differentiation, and integration using completeness, sequences, compactness, and uniform convergence.

Define measurable sets and functions, build the Lebesgue integral, and apply convergence theorems and product measures.

Analyze complex differentiable functions through contour integration, residues, analytic continuation, conformal mappings, and complex series.

Study symmetry through groups, subgroups, homomorphisms, quotient groups, group actions, and classification examples.

Work with rings, ideals, polynomial rings, fields, extensions, and finite fields, leading to the main ideas of Galois theory.

Classify spaces through open and closed sets, continuity, compactness, connectedness, quotient constructions, and separation properties.

Use homotopy, fundamental groups, homology, and cohomology to distinguish spaces through algebraic invariants.

Study polynomial solution sets, coordinate rings, projective space, varieties, singularities, and the algebra–geometry correspondence.

Analyze curves, surfaces, manifolds, tangent spaces, metrics, geodesics, curvature, and differential forms.

Extend linear algebra and analysis to infinite-dimensional normed, Banach, and Hilbert spaces, including bounded operators and spectral ideas.

Represent functions through frequencies and use Fourier transforms, convolution, orthogonal expansions, and wavelets in analysis and signal processing.

Classify and solve elliptic, parabolic, and hyperbolic equations; examine boundary conditions, weak solutions, stability, and numerical approximation.

Model random evolution with Markov chains, Poisson processes, martingales, Brownian motion, and stochastic differential equations.

Derive estimators and tests using likelihood, sufficiency, Bayesian inference, asymptotics, decision theory, and regression models.

Quantify uncertainty and communication with entropy, mutual information, channel capacity, compression, and divergence measures.

Construct error-detecting and error-correcting codes using linear algebra, finite fields, Hamming distance, and modern coding principles.

Connect modular arithmetic, finite fields, probability, and computational hardness to encryption, signatures, key exchange, and cryptographic protocols.

Analyze strategic choices through normal and extensive forms, Nash equilibrium, repeated games, cooperative games, and mechanism design.

Model rational choices under certainty and uncertainty using utility, risk, Bayesian decisions, multicriteria methods, and value of information.

Use geometry and algorithms to compute convex hulls, intersections, triangulations, nearest neighbors, and spatial partitions.

Model interest rates, bonds, derivatives, portfolios, arbitrage, risk, and stochastic pricing while stating assumptions and market limitations.

Represent populations, epidemics, biochemical systems, evolution, and ecological interactions with differential equations, probability, and networks.

Connect linear algebra, probability, optimization, geometry, and information theory to regression, classification, neural networks, generalization, and high-dimensional data.

Use simplicial complexes, persistent homology, filtrations, and persistence diagrams to identify stable geometric structure in noisy data.

Represent quantum states, measurements, entanglement, channels, and quantum algorithms with complex vector spaces, tensor products, and probability.

Study axiomatic set theory, cardinal and ordinal numbers, computability, incompleteness, independence, and the limits of formal mathematical systems.

Use objects, morphisms, functors, natural transformations, limits, and adjunctions to express structures shared across mathematical fields.

Use computer algebra systems for exact calculation, symbolic manipulation, equation solving, differentiation, integration, and hypothesis testing without treating output as proof.

Implement algorithms, simulations, visualizations, and numerical experiments with mathematical software while testing code and preserving reproducible data and results.

Write machine-checkable definitions and proofs in a proof assistant, manage dependencies, and distinguish formal verification from informal mathematical argument.

Use language models and automated reasoning tools to suggest examples, code, proof strategies, and literature leads while independently checking every claim and citation.

Carry a mathematical problem from observation or conjecture through examples, definitions, proof or computation, peer checking, revision, and final presentation.

Write clear proofs, prepare notation and diagrams, typeset with LaTeX, present results, cite sources, and adapt explanations for technical and public audiences.

Map major roles and specialties in research, education, data, finance, engineering, software, and public service; build proof, modeling, and computation portfolios and plan further study or credentials.