Pathway B
Advanced Mathematical Thinking™
Move beyond grade level through algebra, geometry, functions, number theory, modeling, proof, abstraction, and mathematical discovery.
Algebraic Structure
Moving beyond solving equations to understanding algebraic structure — groups, rings, and fields begin as patterns you already know.
Functions & Abstraction
Function families as objects to compose, invert, and transform — the bridge from arithmetic to higher mathematics.
Number Theory
Divisibility, primes, modular arithmetic, and the structure of the integers — the foundation of rigorous proof.
Definition
Mathematical Proof
P ⇒ Q ≡ ¬Q ⇒ ¬P (contrapositive)
A proof is a finite sequence of logical inferences, each following from axioms, definitions, or previously established results, that establishes the truth of a statement.
A proof is not persuasion — it is a chain of unavoidable conclusions.
Theorem
Fundamental Theorem of Arithmetic
- If (hypothesis)
- n is an integer greater than 1.
- Then (statement)
- n can be written uniquely (up to order) as a product of primes.
Proof sketch
Prove existence by strong induction and uniqueness by showing any two factorizations share the same prime multiset via Euclid's lemma.
AMC 10
Competition connection
Modular arithmetic — Remainder reasoning underpins cyclic counting problems, last-digit problems, and divisibility arguments throughout AMC 10.
What is the remainder when 7¹⁰⁰ is divided by 25?
Where this leads next
Linear equations→Vector spaces
Once you treat solution sets as objects with structure, linear equations become the entry point to linear algebra.
Advanced challenges
AdvancedChallenging
The Two-Color Grid
How many rectangles on a 4×4 grid can be tiled by 1×2 dominoes? Generalize to an m×n grid and find when tiling is impossible.
CountingParity
AdvancedApproachable
The Function Family
Describe every transformation that maps f(x) = x² to g(x) = −(x−3)² + 4, then find all x where f(x) ≥ g(x).
FunctionsTransformations
AdvancedOlympiad
Prime Gap Conjecture
Prove there are arbitrarily long sequences of consecutive composite integers. What does this say about prime gaps?
Number theoryProof