Course contents document · Foundations Series · generated 2026-07-15
Mathematical Foundations I
37 core topics
· approximately 9 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
Arithmetic Foundations
· 8 topics
Adding & Subtracting Whole Numbers[E]
Multi-digit addition and subtraction.
Multiplication[E]
Multiplying whole numbers.
Division[E]
Dividing whole numbers.
Order of Operations[M]
Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting[M]
Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing[M]
Sign rules for products and quotients.
Exponents[M]
Repeated multiplication in shorthand.
Square Roots[M]
Undoing a square.
Fractions
· 7 topics
Equivalent Fractions[M]
Different fractions can name the same amount.
Simplifying Fractions[M]
Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)[M]
Same-denominator addition.
Adding Fractions (Unlike Denominators)[M]
Rewrite over a common denominator first.
Multiplying Fractions[M]
Multiply straight across.
Dividing Fractions[M]
Multiply by the reciprocal.
Mixed Numbers & Improper Fractions[M]
Converting between forms.
Decimals, Percents & Ratios
· 5 topics
Decimal Addition & Subtraction[M]
Line up the decimal points.
Fractions ↔ Decimals[M]
Converting between the two notations.
Percent of a Number[M]
Percent means per hundred.
Percent Increase & Decrease[M]
Applying a percent change to a quantity.
Ratios & Proportions[M]
Two quantities that scale together.
Expressions & Equations
· 7 topics
Evaluating Expressions[M]
Substituting a value for a variable.
Combining Like Terms[M]
Adding the coefficients of matching variable parts.
The Distributive Property[M]
Multiplying across a sum.
One-Step Equations[M]
Undoing a single operation.
Two-Step Equations[M]
Undo addition/subtraction first, then multiplication.
Multi-Step Equations[H]
Equations needing distribution or variables on both sides.
Linear Inequalities[M]
Solving with <, >, ≤, ≥.
Linear Functions
· 10 topics
The Coordinate Plane[M]
Locating points with (x, y) pairs.
Slope of a Line[M]
Rise over run between two points.
Slope-Intercept Form[M]
y = mx + b describes a whole line.
Finding a Line from Points[H]
Reconstructing y = mx + b from data.
Systems of Equations (Substitution)[H]
Two equations, two unknowns.
Elimination: A First Look[H]
Add or subtract equations so one variable cancels.