Foundations

• Axioms & Definitions
• Contradiction & Game semantics & True induction
• Circularity in mathematics & 'Paradoxes' resolved
• Building blocks & Incompleteness theorems
• Logic texts

Natural deduction

• Rules & Examples

Rigour with intuition

• Naturals
• Reals & Real exponentiation & $\cos,\sin$ & $\exp(iπ)+1=0$
• General differentiation & applications
• Matrix multiplication

Beautiful stuff

Puzzles:

• Rubik's cube & other permutation puzzles

Finite differences:

• Indefinite summation (like sum of cubes)
• Linear recursive sequences & their indefinite sum

Asymptotic expansions:

• For k-times-differentiable functions
• Central difference formulae
• Newton-Raphson with convergence criteria
• Systematic computation of limits

Smoothing:

• To prove AM-GM & Jensen's inequality
• Maximum area of polygon given perimeter
• Maximum expression value using brackets

Multi-dimensional analysis:

• Multi-dimensional IVT, which proves:
  • Poincaré–Miranda theorem
  • Ham sandwich theorem [consider all hyperplanes that bisect one piece]

Inversion:

• Apollonius' Problem

Common oversights

• Teaching separating variables
• 'One-step' analysis to find expectation
• Most explanations of induction