11 core topics + 29 prerequisite topics taught as needed · approximately 10 hours of instruction including spaced review
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.
| Propositional Logic [E] | T/F values combined with AND, OR, NOT, and IMPLIES. |
| Set Operations [M] | Union, intersection, difference — and inclusion–exclusion. |
| Functions Between Finite Sets [M] | Counting maps: every input independently picks an output. |
| The Product & Sum Rules [E] | Multiply independent stages; add exclusive alternatives. |
| Permutations & Combinations [M] | Ordered vs. unordered selections. |
| The Pigeonhole Principle [M] | More pigeons than holes forces a crowded hole. |
| Recurrence Relations [M] | Sequences defined by how each term follows from the last. |
| Graph Theory Basics [M] | Vertices, edges, and the handshake lemma. |
| Trees [M] | Connected, cycle-free graphs: n vertices, n − 1 edges. |
| Modular Arithmetic [M] | Arithmetic on remainders. |
| Induction & Closed-Form Sums [M] | Classic sums with formulas provable by induction. |
| 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. |
| 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. |
| 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. |
| Evaluating Expressions | Substituting a value for a variable. |
| The Coordinate Plane | Locating points with (x, y) pairs. |
| Slope of a Line | Rise over run between two points. |
| 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. |
| Exponential Growth & Decay | Quantities that multiply by the same factor each time step: y = a·bᵗ. |
| Function Notation & Evaluation | Reading f(x) notation and plugging in inputs. |
| Arithmetic Sequences | Sequences that grow by a constant difference each step. |
| Geometric Sequences | Sequences that grow by a constant ratio each step. |
| Sigma Notation & Series | Σ compresses a sum: read the limits, add up the terms. |
| Modular Arithmetic | Working with remainders directly. |
| Permutations | Ordered arrangements. |
| Combinations | Unordered selections: n choose k. |
| The Pigeonhole Principle | Guaranteeing a repeat. |