Methods of Proof

8 core topics + 31 prerequisite topics taught as needed · 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

Methods of Proof · 8 topics

Statements, Quantifiers & Negation [M]	Negating ∀ gives ∃-not; negating ∃ gives ∀-not.
Direct Proof [M]	Assume the hypothesis, push definitions forward to the conclusion.
Contrapositive Proof [M]	Prove “if P then Q” by proving “if not Q then not P”.
Proof by Contradiction [M]	Assume the statement is false; derive an impossibility.
Proof by Induction [M]	Base case plus domino step proves it for all n.
Divisibility & Set Proofs [M]	Unfold definitions: a | b means b = ak; A ⊆ B means every element carries over.
Counterexamples [M]	One failing case disproves a universal claim.
Biconditionals [M]	An ⟺ needs both directions.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 7 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.

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 · 3 topics

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.

Expressions & Equations · 1 topics

Evaluating Expressions

Substituting a value for a variable.

Linear Functions · 2 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.

Radicals & Exponentials · 4 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.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Functions & Algebra II · 2 topics

Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Precalculus · 1 topics

Sigma Notation & Series

Σ compresses a sum: read the limits, add up the terms.

Math Olympiad: MOEMS & AMC 8 · 1 topics

Modular Arithmetic

Working with remainders directly.

Discrete Mathematics · 6 topics

Propositional Logic	T/F values combined with AND, OR, NOT, and IMPLIES.
Set Operations	Union, intersection, difference — and inclusion–exclusion.
The Product & Sum Rules	Multiply independent stages; add exclusive alternatives.
Recurrence Relations	Sequences defined by how each term follows from the last.
Modular Arithmetic	Arithmetic on remainders.
Induction & Closed-Form Sums	Classic sums with formulas provable by induction.

🖨 Print / save as PDF ← All courses