14 core topics + 26 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.
| Modular Arithmetic [M] | Working with remainders directly. |
| Counting Divisors [M] | Divisor count from the prime factorization. |
| GCD & LCM Relationships [M] | The identity gcd(a,b) · lcm(a,b) = a · b. |
| Digit Problems [M] | Reasoning about the digits of a number. |
| Units Digit of Powers [M] | Combine last-digit cycles, then add. |
| Permutations [M] | Ordered arrangements. |
| Combinations [M] | Unordered selections: n choose k. |
| The Pigeonhole Principle [M] | Guaranteeing a repeat. |
| Complementary Counting [M] | Count the opposite, then subtract. |
| Counting & Probability [H] | Favorable outcomes over total outcomes. |
| Vieta's Formulas [H] | Relating roots to coefficients without solving. |
| Telescoping Sums [H] | A sum that collapses to its endpoints. |
| Sums, Parity & Invariants [M] | Elegant closed forms and even/odd reasoning. |
| Recursion & Sequences [M] | Each term built from earlier ones. |
| 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. |
| Equivalent Fractions | Different fractions can name the same amount. |
| Simplifying Fractions | Reducing a fraction to lowest terms. |
| Adding Fractions (Like Denominators) | Same-denominator addition. |
| Adding Fractions (Unlike Denominators) | Rewrite over a common denominator first. |
| 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. |
| 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. |
| Systematic Counting | Counting without missing or double-counting. |
| Number Patterns | Finding the rule behind a sequence. |