- #1

Diaz Lilahk

- 27

- 8

I have a handful of high school students that are all prospective math/physics majors and have pooled their resources to hire me to teach them a proof based math course because it has become apparent to them in my physics class that proof and derivations are important. Basically I meet with them 2 hours a week and run it like a socratic method or students have to prove the theorems with my limited guidance and so far I have covered the following topics:

1) GCD

2) Euclid's Lemma

3) Well Ordered Principle and Mathematical Induction

4) Fundamental Theorem of Arithmetic

5) Theorems of Elementary Arithmetic a*0=0, (-a)(-b)=ab etc.

6) Arithmetic mean - Geometric Mean Inequality

7) Pythagorean Theorem

8) Cauchy Schwartz Inequality

9) Irrationality of sqrt(p) where p is prime

I will have only a limited amount of time with these students and I need to decide on what topics would be most valuable for them to be exposed to, here is the list of topics, which do you think would be most valuable:

- Conic Sections - going from the geometric definitions to the algebraic representations of conics
- Proofs of Archimedes: Areas of Circle, Quadrature of the Parabola, On the Sphere and Cylinder

- Exploring the completeness property of reals
- Sets, Nested Intervals and the Uncountability of the Reals

- Exploring sequences and series - in particular using telescoping series to derive Σi, Σi^2 , etc
- Area under curves Riemann Sums
- Counting and Binomial Theorem
- Sets and the Axioms of Probability
- Limits, Continuity (delta-epsilon proofs)
- Differentiability
- Properties of Exponential and Logarithmic Function
- Vectors, Vector Spaces, Linear Operators

Zaid Khalil