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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:

  1. Conic Sections - going from the geometric definitions to the algebraic representations of conics
  2. Proofs of Archimedes: Areas of Circle, Quadrature of the Parabola, On the Sphere and Cylinder
  3. Exploring the completeness property of reals
  4. Sets, Nested Intervals and the Uncountability of the Reals
  5. Exploring sequences and series - in particular using telescoping series to derive Σi, Σi^2 , etc
  6. Area under curves Riemann Sums
  7. Counting and Binomial Theorem
  8. Sets and the Axioms of Probability
  9. Limits, Continuity (delta-epsilon proofs)
  10. Differentiability
  11. Properties of Exponential and Logarithmic Function using power series definition of the exponential function
  12. Vectors, Vector Spaces, Linear Operators
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    $\begingroup$ Depends on your program. One that jumps out at me is "limits, continuity (delta-epsilon proofs" because that's incredibly fundamental to the calculus course, but today commonly cut from the course and many textbooks. $\endgroup$
    – Daniel R. Collins
    Commented Dec 29, 2015 at 18:49
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    $\begingroup$ This is a wonderful "list of topics." My only advice would be to cut the topics in half (or less) and spend twice as long (or more) on each. It would be better that they understand a few topics thoroughly than many topics somewhat superficially. $\endgroup$
    – Joseph O'Rourke
    Commented Dec 30, 2015 at 1:58
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    $\begingroup$ This is why i kind of need help because there are definitely more topics than I will be able to cover in the time that we have. What I feel is that many of these are topics that get a short shrift in the k-12 curricula. So I am more inclined to cut out vectors, vector spaces than I am to cut out conic sections. But some of these depend on knowledge of other topics. So it really comes down to what is best for these kids, and that I am not so sure of. $\endgroup$
    – Zaid Khalil
    Commented Dec 31, 2015 at 4:22
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    $\begingroup$ I don't know that they will run into much proof-based math in their first year. Check into it. I would concentrate on problem solving in calculus --the hardest part of first year math IIRC. If you want to introduce them to proof-based math and they are having any trouble with the basic methods of proof, you might consider some software I have developed to help learn these methods. It includes an interactive, self-study tutorial. They should be able to work through it on their own. Download it free at my website dcproof.com $\endgroup$
    – Dan Christensen
    Commented Dec 31, 2015 at 6:07
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    $\begingroup$ Check what they will run into later on, and concentrate on giving them a leg up on that. E.g. look at contents of a sample of freshman math courses at colleges. $\endgroup$
    – vonbrand
    Commented Jan 18, 2016 at 11:38

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I have experience from my high school that the most helpful preparation for majoring in math in college was not at all learning the college math beforehand. It was the reasoning skills I learned while solving hard high school problems (somewhat under IMO level, but hours and hours of time spent on them). Some problems that you could give your students from are from Art of Problem Solving Website.

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