ProgramsMath Core III

Math Core · Logic & Proof

Learn to see the structure—and prove why it works.

Math Core III develops the habits of rigorous mathematical thought: precise definitions, valid deduction, proof, counterexamples, and strategic reasoning. Geometry provides a rich setting for proof, while competition-style problems extend those habits beyond familiar procedures.

16 sessions1 hr 45 min eachOnce weekly* school year$300* per month

*Summer terms meet twice weekly and complete the same 16-session course in eight weeks. Summer tuition is $600 per month.

Course fit

For students ready to move from calculation to mathematical argument.

This course is designed for students who

  • have dependable arithmetic and algebra foundations and are ready to reason abstractly
  • can calculate accurately but want to explain why a result must be true
  • want experience with definitions, counterexamples, deduction, and proof
  • are preparing for proof-based mathematics, geometry, or competition problem solving

Curriculum

Five phases. From valid statements to complete proofs.

Students learn the language of logic, build proof methods, apply them in geometry, and extend them through advanced and competition-style reasoning.

  1. 01
    The Language of LogicSessions 1–3
    1. Statements, Truth & Counterexamples
    2. Conditions, Equivalence & Quantifiers
    3. Definitions & Mathematical Precision
  2. 02
    Methods of ProofSessions 4–6
    1. Direct Proof
    2. Contrapositive & Contradiction
    3. Proof Structure & Communication
  3. 03
    Geometry as a Proof SystemSessions 7–10
    1. Diagrams, Axioms & Deduction
    2. Congruence & Similarity Proofs
    3. Coordinate Proof
    4. Transformations & Invariants
  4. 04
    Advanced ReasoningSessions 11–14
    1. Cases & Exhaustion
    2. Parity & Invariants
    3. Construction & Strategy
    4. Competition-Style Reasoning
  5. 05
    IntegrationSessions 15–16
    1. Proof Workshop & Synthesis
    2. Final Examination

How students learn

Understand it. Practice it. Keep it.

01Concept

See the mathematical structure clearly.

02Guided practice

Apply the idea with instructor support.

03Independent practice

Build reliable execution.

04Mastery

Work through substantial targeted volume.

05Retention

Revisit prior skills as new ideas are added.

Practice connects claims to reasons.

Targeted problem sets vary assumptions, counterexamples, diagrams, and proof structures so students learn to justify each step instead of relying on a familiar-looking procedure.

Recommended progression

Continue into advanced mathematics.

After Math Core III, students may continue into AMC preparation, proof-based coursework, or the appropriate SAT pathway based on grade, prior preparation, and goals.

See the complete pathway