Exploring Interesting Math Topics: A Grade 12 Class Project

[Parth Dave]

I'm a grade 12 student and have been given the task of teaching our algebra and geometry class for a day . There are absolutely no restrictions on what I can teach about. It does not in any way have to be related to algebra or geometry, just mathematics in general. What I really need is suggestions as to what topic to teach. Heres a brief outline of what we have covered in the course:

Vector Algebra
Lines, Planes
Matrices - Determinants, inverses etc.
Linear Transformations
Translations
Rotations
Proof by Induction
Binomial/Multinomial Theorem
Complex number algebra
Chromatic polynomials
Eigenvalues/Eigenvectors - diagonalization, linear recurrences
Generating Functions
Proof and Cantor Set Theory
Basic graph theory - isomorphism, circuits/paths
Combinations/Permutations
Markov Chains

The idea is for me to teach something interesting and mathematical. The difficulty level isn't too big an issue. As long as the information presented isn't loaded with notation we haven't seen before I'm sure the class can manage. Does anyone have suggestions for what I could teach? (Something that I can learn in a day, since I do have to go teach this on Thursday)

 

[courtrigrad]

How about the Simplex Method?

 

[Curious3141]

How about "Unsolved and recently solved problems in Mathematics" ?

You can talk about Hilbert's problems, mentioning the ones that have been resolved. You can expand on the stuff you know, for example when you talk about the resolution of Hilbert's 7th problem, you can introduce the concept of a non-constructive proof in doing stuff like “Prove that there are irrational numbers x and y such x^y is rational.” (using x = y = $\sqrt{2}$), then talk about transcendental numbers and the importance of Gelfond's theorem. You should include the Riemann hypothesis, Goldbach conjecture, Collatz problem, the P vs NP problem and then briefly review the successful proof of FLT by Wiles (not as a lecture on modular functions, which might be too tough, but as a profound application of proof by contradiction).

It's just a thought.

 

[vincentchan]

and the 4 color map problem, do its history background and how people prove in by computer first and later by hand,this should be fun

 

[Parth Dave]

How about the Simplex Method?

I did a quick check on Wolfram to see what that was and I get something about linear programming and optimization. However, I'm not sure what this means. Could you possibly give me an example of a problem that would require you to use the Simplex Method?

You can talk about Hilbert's problems, mentioning the ones that have been resolved. You can expand on the stuff you know, for example when you talk about the resolution of Hilbert's 7th problem, you can introduce the concept of a non-constructive proof in doing stuff like “Prove that there are irrational numbers x and y such x^y is rational.” (using x = y = ), then talk about transcendental numbers and the importance of Gelfond's theorem. You should include the Riemann hypothesis, Goldbach conjecture, Collatz problem, the P vs NP problem and then briefly review the successful proof of FLT by Wiles (not as a lecture on modular functions, which might be too tough, but as a profound application of proof by contradiction).

I think it would have been nice of me to mention that I only have a period (90 mins) to do whatever I can. And we have already done similar proofs when we did our proof section. But I'm just abit concerned that your suggestions are going to require a lot more time than I am given.

and the 4 color map problem, do its history background and how people prove in by computer first and later by hand,this should be fun

I forgot to mention that. We actually did it when we were dealing with chromatic polynomials.


Thx for all the suggestions guys. Right now I'm making a list and I'm going to start checking it twice .

 

[Curious3141]

And we have already done similar proofs when we did our proof section.

What ?! You proved Riemann, Goldbach, Collatz and FLT in your Grade-12 class ? :rofl:

Just kidding. Seriously, I don't think it'll take that much time to run through those things. I mean you're not going to dwell on each one, just give a flavour of each.

But the choice is yours, of course.

 

[courtrigrad]

Maximize $$p = 10x + 7y$$ subject to $$x\geq 0 , y \geq 0$$

$$x \leq 200$$