11 core topics + 40 prerequisite topics taught as needed · approximately 13 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.
| Vectors in Rⁿ [E] | Ordered lists of numbers, added and scaled componentwise. |
| Dot Product & Norm [M] | Multiply matching components and add; lengths and angles follow. |
| Matrix Addition & Scalar Multiples [E] | Matrices add entry by entry; scalars multiply every entry. |
| Matrix Multiplication [M] | Row times column: each entry of AB is a dot product. |
| Determinants [M] | A single number that measures how a matrix scales area or volume. |
| Inverse of a 2×2 Matrix [H] | Swap the diagonal, negate the off-diagonal, divide by the determinant. |
| Solving Ax = b [H] | A linear system is one matrix equation. |
| Linear Independence [M] | Vectors are dependent when one is a combination of the others. |
| Linear Transformations [M] | Matrices are functions: they move every point of space linearly. |
| Eigenvalues of a 2×2 Matrix [H] | The scaling factors along a matrix's special directions. |
| Eigenvectors [H] | The directions a matrix merely stretches. |
| 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. |
| Multiplying Fractions | Multiply straight across. |
| Dividing Fractions | Multiply by the reciprocal. |
| Ratios & Proportions | Two quantities that scale together. |
| 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. |
| Multi-Step Equations | Equations needing distribution or variables on both sides. |
| 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. |
| Systems of Equations (Substitution) | Two equations, two unknowns. |
| 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. |
| 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. |
| Systems by Elimination | Adding or subtracting equations to cancel a variable. |
| Right-Triangle Trigonometry | SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle. |
| 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. |