The Cosines Group
Let's use this discussion to brainstorm about possible interesting topics that you would like to write about / learn more about.
As mentioned by Akshat, we will be more inclined towards 'basic math' and hence will have a much larger audience. I will be spending most of my time here, to help brainstorm / guide / explain. This group has a lot of promise, to help other students to learn about the interesting aspects of math.
Here are some suggestions:
1) Bob - fun explorations into different math topics like the golden ratio, infinity, cross sections, etc.
2) Simple math games - The n-gon game, The 4 number game, Tic-Tac-Toe strategy
3) Lateral Thinking puzzles
Interesting math concepts like absolute value, greatest integer function.
4) Real world applications of certain concepts, like prime factorization, finding the maximum/minimum, etc.
Note by
Calvin Lin
7 years, 6 months ago
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For brainstorming, what are some topics that you think will be interesting to our members?
Just list out as many as you can think of.
Basic Infinite Sequences, Interesting Geometric Shapes, Tessellations, Penrose Tiling, Martin Gardner's Games, Chinese Remainder Theorem, Platonic Solids and Euler's Equation, Fractals, Pascal's Triangle, Other Number Triangles (and developing them), The Parallel Postulate, Proofs of the Pythagorean Theorem, OEIS, Ulam Spiral, Various Number Theoretic Tricks, Sums of Powers
Recreational Mathematics Puzzles that could include: tic-tac-toe, Minesweeper, Sudoku, Pentominoes
The Origin of Math Concepts: What is Addition? Why Multiply? Zero, Negative Numbers, Rational Numbers, Imaginary Numbers, Vectors, Exponents
Fibonacci numbers
Prime numbers and primality testing
Heron's formula ?
Circle properties - Power of a point
Factoring polynomials
Complex numbers and the relation to trigonometry
Basic counting exercises (which generalize) - how many handshakes amongst 20 people, how many paths from (0,0) to (10,15), etc
Fermat's Little theorem, Euler's theorem (the one which uses the totient function). A question based on the Euler's Theorem (Taken from RMO 1990):
I think that the main goal of this group is to educate those interested, but not adept, in math. We should show them its beauty that, while on a basic level, is still amazing. These could be little things, like the topics of Numberphile videos or other various recreational mathematics magazines. We should also take care to develop explicitly our heuristics, the things that we recognize to make a problem easier. This could be accomplished with practice problems, preferably in areas less taught in school (e.g. combinatorics), and their accompanying solution(s).
As far as games, puzzles, and real world applications, these would fall under the first category (the little things) above. However, I think much of our time should be spent on readying their minds for higher level problem solving. So we would develop skills like pattern recognition, choosing the correct variables, and looking at the problem in the right way. While I know that many of these ideas are captured in the techniques trainer, I think that this would be a nice aid. Just an idea. What do you guys think? Is it overkill? Should we just stick with games and puzzles and other quirky concepts? And if we do venture into developing these skills, how should we go about doing it?
I like your idea of focusing on a technique, and then supplying problems which use that over the following week. This is akin to my weekly Wednesday Discussions, which introduce a topic, add some problems with working to demonstrate its usefulness, and also ask others to contribute problems towards it. It would be great if you have similar write-ups, which I can easily incorporate into the Techniques Trainer.
I'd like to add something else that I've been reflecting on. I don't want to just present a topic or technique and some problems. I want to develop the topic from intriguing questions (to be answered by the technique) or already known knowledge. Since we are working with those that don't know a whole lot of Olympiad mathematics, I think that this is the most effective approach.
After obtaining feedback from various members, here are more suggestions on how to make math more interesting / engaging.
There is a high likelihood that the more mathematically inclined members of our community will be tempted to immediately announce the answer to questions / discussions posed by The Cosines Group. As such, I would encourage you to add something along the lines of
Alternatively, if you see a question posted by the other groups that you find interesting and want to present a simplified version, you could say
I can write up those explorations any time we need them, I just need a couple days of advance notice. If you guys are interested in any of those topics yourself, inquire further.
What do you mean by 'explorations'?
They are guided lectures that I have taught at some of my Math Club meetings. I would just have to transcribe my notes into latex writing and then add commentary. The whole idea is not just to present the information, such as a source like Wikipedia would, but guide the reader through discovering and realizing the information themselves. It is much more satisfying for the reader. My lecture on the Golden Ratio is the best, and I will be typing that one up soon. For example, we don't start off with the number. We start off with a rectangle that can be divided into a square and a rectangle similar to the original rectangle.
I would suggest breaking up the notes into several bite-sized chunks, which would make it easier to digest (and also allow you to post more). For example, just talking about the rectangle decomposition and resulting quadratic equation will make an interesting introduction post.
You could tag your posts "Golden Ratio 1", "Golden Ratio 2", or simply add a tag of "Golden Ratio", so that your other posts can be easily found.
Good idea. That would make it more sizable.
Posts will happen from an individuals account, with the tag of "The Cosines Group". This way, others will be able to identify who the originator is, and also find similar interesting material through the tags.
As such, you do not need to choose 1 area to focus on. In fact, having a variety of posts would be preferable, so that you get a broad coverage. I anticipate that different people will focus on specific topics that interest them, and there isn't a requirement for cohesion as a group.
It would be good to think about how you want to structure the timing of the posts. Having 1-2 posts a day would be great, but would of course depend on your availability as a group. I would also expect that other members would be interested in contributing similar posts, though at a lower frequency.
I never thought that my suggestion would transform into reality! Thanks! Regarding the posts, I need to think, as now a days I am able to denote less time for Brilliant as compared to earlier.
Don't worry about it, that's the reason why we are working in a large group. Even if you just make 1 small post every week, there will be a lot of different material for others to think about.
The posts don't need to be extremely long either. In fact, short insightful posts can be very meaningful, like "Why is the product of 2 negative numbers a positive number?" or " Does ii make sense? Why? What is it equal to?"
Thanks for the idea--I just made a post on ii and how to derive its value.
Some things I'm interested in writing is those intriguing scenarios where there is an pleasant unexpected result: for example, my post on the are of an annulus depending only on the chord between the two circles. However, I'm having trouble thinking up more of these types of problems. Any ideas?