Mathematics for Machine Learning

10 core topics + 74 prerequisite topics taught as needed · approximately 21 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Mathematics for Machine Learning · 10 topics

Norms & Distances [M]	How far apart two feature vectors are.
Data Matrices & Predictions [M]	Rows are examples; Xw scores them all at once.
Linear Models [M]	ŷ = w · x + b — the workhorse of machine learning.
Loss Functions [M]	One number that says how wrong the model is.
Gradient Descent [H]	Step downhill: w ← w − η ∇L.
Sigmoid & Logistic Regression [H]	Squash a score into a probability.
Softmax & Argmax [M]	Turn scores into a distribution; pick the biggest.
Entropy & Information [M]	Expected surprise, measured in bits.
Chain Rule & Backpropagation [H]	Gradients flow backward through composed functions.
Variance Explained & PCA [H]	Principal components rank directions by variance.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction	Line up the decimal points.
Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 6 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 3 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.

Quadratics & Polynomials · 8 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving x² = k	Taking square roots of both sides — remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.

Radicals & Exponentials · 6 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Geometry · 5 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Distance & Midpoint	Measuring segments in the coordinate plane.
Similar Triangles	Same shape, different size: corresponding sides are proportional.

Functions & Algebra II · 4 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Function Composition	Feeding one function's output into another: f(g(x)).
Piecewise Functions	Functions defined by different rules on different intervals.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.

Trigonometry · 1 topics

Right-Triangle Trigonometry

SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.

Precalculus · 3 topics

Vectors: Components & Magnitude	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations	Scaling, adding, and dotting vectors — all component by component.
Average Rate of Change	The slope of the secant line: (f(b) − f(a)) / (b − a).

Limits & Continuity · 2 topics

Limits: Graphical & Numerical	What value a function approaches — which need not be the value it takes.
Evaluating Limits Algebraically	Direct substitution — and the factor-and-cancel fix for 0/0.

Differentiation · 5 topics

The Limit Definition of the Derivative	The derivative is the limit of average rates of change.
The Power Rule	d/dx xⁿ = n·xⁿ⁻¹ for any real n.
Sum & Constant-Multiple Rules	Differentiate term by term.
The Product Rule	(fg)′ = f′g + fg′.
The Chain Rule	d/dx f(g(x)) = f′(g(x)) · g′(x).

Math Olympiad: MOEMS & AMC 8 · 3 topics

Permutations	Ordered arrangements.
Combinations	Unordered selections: n choose k.
Counting & Probability	Favorable outcomes over total outcomes.

Linear Algebra · 6 topics

Vectors in Rⁿ	Ordered lists of numbers, added and scaled componentwise.
Dot Product & Norm	Multiply matching components and add; lengths and angles follow.
Matrix Addition & Scalar Multiples	Matrices add entry by entry; scalars multiply every entry.
Matrix Multiplication	Row times column: each entry of AB is a dot product.
Determinants	A single number that measures how a matrix scales area or volume.
Eigenvalues of a 2×2 Matrix	The scaling factors along a matrix's special directions.

Multivariable Calculus · 2 topics

Partial Derivatives	Differentiate in one variable while holding the others constant.
The Gradient	The vector of partials — it points uphill, fastest.

Probability & Statistics · 3 topics

Mean, Median & Range	Center and spread in one pass.
Variance & Standard Deviation	Average squared distance from the mean — then unsquare.
Probability Basics	Favorable over total, when outcomes are equally likely.

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