What is taught,
and what “complete” means.

Lessons

1. Numbers, direction, and order
2. Fractions, ratios, and percentages
3. Powers, roots, logs, and scale
4. Sets, logic, and careful language
5. Units, estimation, and proof habits
6. Remainders and clock arithmetic

Mastery outcomes

• Compare signed numbers and calculate with correct operation order.
• Move confidently among fractions, ratios, decimals, rates, and percentages.
• Use powers, roots, logarithms, and scientific notation to reason about scale.
• Read set and logic notation, and distinguish a claim from its converse.
• Check answers with units, estimates, counterexamples, and simple proof patterns.
• Compute standard remainders and use congruences to reason about repeating cycles.

Lessons

1. Variables and expressions
2. Equations and inequalities
3. Functions and graphs
4. Polynomials and quadratics
5. Composed, inverse, exponential, and piecewise functions
6. Angles, radians, and the unit circle
7. Trigonometric graphs, identities, and inverse angles
8. Complex numbers as plane coordinates
9. Polar form, Euler's formula, and frequency waves
10. Sequences, sums, and long-run behaviour

Mastery outcomes

• Translate precise verbal rules into algebraic expressions.
• Solve linear equations, systems, and inequalities and verify the result.
• Move between degrees, radians, unit-circle coordinates, trigonometric graphs, identities, and inverse angles.
• Calculate with complex numbers in rectangular and polar form and connect Euler's formula to Fourier waves.
• Factor polynomials and solve quadratics over the real and complex numbers.
• Compose and invert functions, and analyse arithmetic, geometric, recurrent, and convergent sequence models.

Lessons

1. The addition and multiplication rules
2. Permutations: when order matters
3. Combinations: when order does not matter
4. Repetition, identical objects, and circles
5. Restrictions, inclusion–exclusion, and distributions

Mastery outcomes

• Choose correctly between addition and multiplication counting rules.
• Recognize whether order matters and use permutations or combinations.
• Handle repeated selections, identical objects, and circular symmetry.
• Count restricted outcomes with cases, complements, and inclusion–exclusion.
• Model identical-item distributions with stars and bars and prove guaranteed crowding with the pigeonhole principle.

Lessons

1. From repeated multiplication to Pascal's triangle
2. The theorem and its terms
3. Specific coefficients and identities
4. Binomial probability
5. Approximation, bounds, and multinomials

Mastery outcomes

• Build and read Pascal's triangle and explain its counting meaning.
• Expand positive-integer binomial powers with correct coefficients, signs, and exponents.
• Find a requested term, power, or coefficient without full expansion.
• Use coefficient identities and solve binomial probability questions.
• Apply truncated generalized expansions and multinomial coefficients with their conditions.

Lessons

1. Coordinates and vector operations
2. Length, distance, and unit vectors
3. Dot products, angles, and projections
4. Lines, planes, and distances
5. Cross products and high-dimensional geometry
6. Orthogonal bases and barycentric coordinates

Mastery outcomes

• Distinguish points from vectors and calculate vector combinations.
• Compute norms, distances, and normalized directions.
• Use dot products for angles, similarity, orthogonality, and projections.
• Represent lines and planes and calculate their intersections or distances.
• Build orthonormal directions with Gram–Schmidt and locate points with barycentric weights.
• Use cross products in 3D and reason carefully about high-dimensional embeddings.

Lessons

1. Matrices as data and transformations
2. Matrix multiplication and composition
3. Linear systems, elimination, and inverses
4. Span, independence, basis, and rank
5. Eigenvectors, least squares, and principal directions
6. Quadratic forms and positive-definite matrices
7. SVD, low-rank approximation, and PCA

Mastery outcomes

• Read matrix shapes and treat a matrix as data and a linear map.
• Multiply matrices in the correct order and use identity and transpose rules.
• Solve linear systems by elimination and identify unique, absent, or infinite solutions.
• Reason about span, independence, basis, dimension, rank, and nullity.
• Classify quadratic forms with positive-definite, semidefinite, and indefinite curvature.
• Compute small SVDs, pseudoinverses, low-rank approximations, and PCA explained-variance ratios.

Lessons

1. Limits: getting as close as we want
2. Derivatives: an instant speedometer
3. Integrals: adding infinitely many tiny pieces
4. Series: building curves from simple powers
5. Differential equations: rules for evolving systems
6. Implicit curves and the Mean Value Theorem
7. Related rates and arc length
8. Integration by parts and improper integrals
9. Convergence tests for infinite series
10. Power series, radius, and safe algebra
11. Linear first- and second-order differential equations
12. ODE systems, phase pictures, and Fourier modes

Mastery outcomes

• Explain a limit from a graph, table, and formula.
• Differentiate compositions and interpret units and signs.
• Turn rates into totals with definite integrals.
• Choose convergence tests, find power-series intervals, and manipulate valid power series with error control.
• Use implicit differentiation, the MVT, related rates, arc length, integration by parts, and improper integrals.
• Solve basic first- and second-order linear ODEs and interpret vector-system eigenmodes and Fourier modes.

Lessons

1. Partial derivatives: one direction at a time
2. Gradients: the arrow of steepest rise
3. Jacobians: local maps between spaces
4. Multiple integrals and vector fields
5. Hessians: measuring curvature
6. Tensor shapes, broadcasting, and contraction
7. Matrix calculus and backpropagation
8. Log-sum-exp, softmax, and cross-entropy
9. Trace rules, log-determinants, and Hessian products

Mastery outcomes

• Compute and interpret partial derivatives and tangent planes.
• Use gradients and directional derivatives geometrically.
• Build Jacobians with correct shapes and compose them.
• Set up double and triple integrals, apply Jacobian factors, and interpret line integrals, divergence, curl, Green, Stokes, and divergence theorems with their assumptions.
• Use Hessians to describe curvature and stationary points.
• Track batch, sequence, and feature axes; apply broadcasting rules; and shape-check tensor contractions and einsum expressions.
• Differentiate vector and matrix expressions and trace a backward pass.
• Derive stable log-sum-exp, softmax Jacobians, and the fused softmax–cross-entropy gradient.
• Use trace differentials, log-determinant rules, JVPs, VJPs, and Hessian-vector products with correct shapes.

Lessons

1. Outcomes, sample spaces, and events
2. Counting without listing everything
3. Conditional probability and independence
4. Total probability and Bayes' rule
5. Repeated trials and dependence

Mastery outcomes

• Define experiments, outcomes, sample spaces, and events
• Use complement, union, intersection, and counting rules
• Distinguish conditional probability from its reverse
• Test independence and handle without-replacement dependence
• Use total probability and Bayes' rule with base rates

Lessons

1. Random variables, PMFs, and expectation
2. Variance, covariance, and correlation
3. Bernoulli, binomial, geometric, and Poisson
4. A map of category, count, and waiting distributions
5. Continuous variables and density
6. A map of bounded, positive, and Gaussian-related distributions
7. Normal distributions and limit theorems
8. Probability spaces and almost-sure statements
9. Convergence modes and honest limit theorems

Mastery outcomes

• Read PMFs, PDFs, and CDFs
• Build probability statements from (Ω,𝓕,P) and interpret almost-sure claims
• Compute expectation, variance, covariance, and correlation
• Choose among categorical, multinomial, binomial, negative-binomial, hypergeometric, and Poisson stories
• Match beta, gamma, lognormal, chi-square, Student-t, and multivariate normal models to their supports and assumptions
• Use family relationships such as beta-Bernoulli updating, chi-square as gamma, and normal projections
• Find continuous interval probabilities
• Distinguish almost-sure, probability, Lp, and distributional convergence
• State the assumptions and different conclusions of the classical LLN and CLT

Lessons

1. Data, summaries, and good sampling
2. Estimators, bias, variance, and standard error
3. Confidence intervals
4. Hypothesis tests, errors, and power
5. Likelihood and simple regression
6. Experiments, confounding, and causal comparisons
7. Robust, nonparametric, and resampling inference

Mastery outcomes

• Separate populations, samples, parameters, and statistics
• Distinguish random sampling, randomized assignment, controls, blocking, blinding, and observational confounding
• Diagnose bias, variance, and sampling uncertainty
• Use robust summaries without hiding scientifically important extremes
• Explain bootstrap and permutation assumptions for independent, paired, clustered, or time-dependent data
• Construct and interpret basic confidence intervals
• Explain p-values, error types, power, and multiple testing
• Use likelihood and linear regression without causal overclaiming

Lessons

1. Joint, marginal, and conditional laws
2. Transformations, inequalities, and concentration
3. Bayesian models and posterior prediction
4. Conjugate updating and Bayesian decisions
5. Monte Carlo, importance sampling, and MCMC
6. Hierarchical Bayes and variational inference

Mastery outcomes

• Move among joint, marginal, and conditional distributions
• Transform variables and apply honest tail bounds
• Build and critique a Bayesian model
• Perform conjugate updates and posterior prediction
• Use hierarchical partial pooling while checking exchangeability and hyperprior assumptions
• Explain ordinary and self-normalized importance sampling, MCMC dependence, and diagnostics
• Derive the ELBO identity and name variational approximation, optimization, and model errors

Lessons

1. Objectives and gradient descent
2. Convex optimization: bowl-shaped problems
3. Constrained optimization: best among allowed choices
4. Nonconvex landscapes: saddles, basins, and restarts
5. Stochastic optimization: learning from noisy clues
6. Nonsmooth penalties, proximal steps, and coordinate descent
7. Line search and the Armijo condition

Mastery outcomes

• Translate a real goal into variables, an objective, and constraints.
• Run and diagnose gradient descent with a sensible stopping rule.
• Recognise convex structure and explain conditioning.
• Solve small Lagrange and KKT problems.
• Reason about nonconvex landscapes and stochastic training methods.
• Use subgradients, soft-thresholding, proximal gradient, and coordinate descent on simple nonsmooth objectives.
• Choose and verify step lengths with Armijo backtracking and descent-direction checks.

Lessons

1. Floating point: a finite number line
2. Conditioning and stability: problem versus algorithm
3. Numerical differentiation and integration
4. Linear systems: solve, do not invert
5. Root finding and reliable iteration
6. Richardson extrapolation and stable numerical kernels
7. DFT, FFT, and fast convolution
8. Sampling, Nyquist, and spectral differentiation

Mastery outcomes

• Explain rounding, cancellation, overflow, and relative error.
• Separate conditioning of a problem from stability of an algorithm.
• Choose sensible steps for numerical derivatives and quadrature.
• Use Richardson extrapolation only when the observed error follows its assumed order, and evaluate log-sum-exp and related kernels stably.
• Construct and invert a DFT, explain FFT speed and zero-padded convolution, and convert bins to physical frequencies.
• Apply Nyquist's limit and perform correctly scaled spectral differentiation on smooth periodic data.
• Solve linear systems with factorizations or appropriate iterations.
• Use safeguarded root finding and defend a stopping tolerance.

Lessons

1. Surprise and entropy
2. Joint, conditional, and mutual information
3. Cross-entropy and KL divergence
4. Compression and noisy channels
5. Information theory in ML and LLMs
6. Continuous entropy and maximum-entropy models

Mastery outcomes

• Compute discrete entropy and explain why continuous differential entropy depends on units
• Identify maximum-entropy distributions only after support and constraints are stated
• Use conditional entropy and mutual information
• Relate cross-entropy to entropy and KL
• Explain source coding, capacity, and data processing
• Interpret log loss and perplexity with their limitations

Lessons

1. Random walks and Brownian motion
2. Markov chains and long-run behavior
3. Martingales and stopping
4. The Itô integral and Itô's lemma
5. SDEs, geometric Brownian motion, and simulation
6. Stopping times and martingale convergence
7. Multidimensional Itô calculus and Stratonovich form
8. Strong and weak SDE solutions and numerical error
9. Generators and the Fokker–Planck equation
10. Girsanov and Feynman–Kac

Mastery outcomes

• Relate random walks to Brownian scaling
• Compute Markov transitions and stationary distributions
• Recognize stopping times and state valid optional-stopping and martingale-convergence conditions
• Apply one- and multidimensional Itô formulas and convert scalar Stratonovich equations
• Distinguish strong from weak solutions and strong from weak numerical error
• Use generators and the Fokker–Planck equation to connect paths with densities
• Explain the assumptions and roles of Girsanov and Feynman–Kac
• Read, simulate, solve simple SDEs, and critique GBM assumptions

Lessons

1. Prediction, loss, gradients, and optimization
2. Embeddings and vector geometry
3. Attention step by step
4. Normalization, residuals, and stable deep networks
5. Sampling, temperature, and the full LLM loop

Mastery outcomes

• Connect linear and logistic regression to logits, softmax, cross-entropy, gradients, and updates
• Explain learned embeddings and similarity choices
• Compute scaled dot-product attention with masks
• Explain layer normalization and residual gradient paths
• Use temperature and truncation sampling, and audit an end-to-end LLM evaluation