Math is cool, right?
March 27, 2008 7:33 AM   Subscribe

[Math filter] What fascinatingly cool mathematical topics do you wish you knew about in high school?

I'm giving a talk to a group of students (grades 10, 11, 12) at the end of April, who are attending a Math Camp at my University. The talk is called "What I Wish I Knew About Math In High School" and I'd like to discuss things that I've discovered about math that I think are really cool, but you would never have been encouraged to learn about while you were in high school. The current high school curriculum focuses on algebra, trig, and (in the later years) pre-calculus, which is an amazingly small portion of what math has to offer.

I'm interested in hearing what type of things the hive mind finds fascinating about math. These topics should be relatively easy for anyone to understand (after a brief explanation), and have some sort of "wow" factor to them. Bonus points if you can apply it somehow to "real" life.

An example could be the Golden Ratio, since it has roots which are purely mathematical, but is debated to be found in nature, art, and music.
posted by tomcochrane to Science & Nature (101 answers total) 166 users marked this as a favorite
I have always found the mathematical elements of cryptography very interesting. Attacks used against polyalphabetic substitution ciphers could be explained at a high school level.
posted by sindark at 7:38 AM on March 27, 2008 [1 favorite]

Probability, especially as it relates to things like lottery tickets and unprotected sex.
posted by box at 7:39 AM on March 27, 2008 [8 favorites]

I find the topic and application of Vedic Mathematics fascinating. There are many sites on the web regarding this topic, but I like this one.
posted by mcarthey at 7:42 AM on March 27, 2008 [2 favorites]

When I was in college I took a class called "Math for liberal arts majors" and my friends and I would just get high and go to that class because it was so cool. We didn't have to do any actual math, with numbers and stuff. The teacher would just go "So like, some infinities are bigger than others!" and would then prove it somehow and we would just sit in the back of the class and freak out. It was awesome. So tell the kids about how some infinities are bigger than others. That is really cool. I think the proof had something to do with a hotel. I don't remember it very well. I was pretty fucking high.
posted by ND¢ at 7:42 AM on March 27, 2008 [55 favorites]

Compound interest, especially as it relates to things like making the minimum monthly payment and starting to save for retirement.
posted by box at 7:42 AM on March 27, 2008 [1 favorite]

Graph theory, starting with the Konigsberg bridge problem.

(Also, I think I might've been friends with ND¢ in college.)
posted by box at 7:44 AM on March 27, 2008

I think the proof had something to do with a hotel. I don't remember it very well.
Sounds like "Hilbert's paradox". It's pretty neat (though this may not be the best explanation).

posted by inigo2 at 7:45 AM on March 27, 2008

I have to agree with sindark: cryptography is a really cool field. One thing still completely "freaks my trip" is the Diffie-Hellman exchange protocol. You'll definitely need concrete examples and make sure to pace it slowly to get everyone to understand, but grades 10-12 should be able to understand it. Good luck to you!
posted by trim17 at 7:47 AM on March 27, 2008 [1 favorite]

Here are a few I've used in teaching Math for Lib Arts:

Fractals, (mandelbrot/julia sets for those with backgrounds in complex, line generated fractals/cantor set for those without). I've actually used the Koch Snowflake in liberal arts math classes as an example of a figure with bounded area and unbounded perimeter, and they got it and dug it. Here's an shameful link to my YouTube video I used in my fractal unit for upper division Modern Geometry, which addressed similar topics but much deeper: Mandelbrot Set. Song by Jonathan Coulton, clips by me and loads of people I don't know.

Set cardinality and Cantor's diagonalization argument.

Anything combinatorial, like discrete probability. I usually run a casino day where students make up games, including payouts, and determine expectation. They then use chips and walk around the room playing everyone's games. It's loads of fun. This isn't something you can simply talk about, but I'm hoping someone else searching this thread may use it as an idea in their own class.
posted by monkeymadness at 7:48 AM on March 27, 2008 [3 favorites]

posted by orthogonality at 7:48 AM on March 27, 2008 [2 favorites]

Knot theory is awesome.

1. Mathematicians are untangling knots... WITH THEIR MINDS.

2. Applications include quantum field theory and the actions of enzymes on DNA.
posted by prefpara at 7:49 AM on March 27, 2008

the hilbert hotel has an infinite number of rooms. all of them are full.

a guest comes to the front desk and asks for a room. the clerk says, "ok".

he moves the guy in room one into room two, the guy in room two into room three, and the guy in room n into room n + 1. then he gives the guest room one.

suddenly, an infinite number of guests arrive. they want rooms. the clerk says, "ok"
posted by stubby phillips at 7:49 AM on March 27, 2008 [3 favorites]

oh, and prove that .9999 (with a bar over the 9) == 1. that one is kinda fun.
posted by stubby phillips at 7:51 AM on March 27, 2008

Non-Euclidean geometry! Like how you can have a triangle with 3 90 degree corners if you draw it on the surface of a sphere. This also fits in kind of with fractals with the Reimann Sphere.

I was actually really into this stuff in high school though I have to admit most of my peers were not...
posted by GuyZero at 7:55 AM on March 27, 2008 [1 favorite]

Game theory (stripped of all the complex formulas and theorems) because it can help explain human/animal decision making, economics to some degree, and how gambling/lotteries are really nothing more than special taxes for the mathematically challenged.

Also, within game theory you could focus on the prisoner's dilemma and have them do exercises to see how certain behaviors can maximize your payoff, while others do not. Try to have them predict beforehand which they think will benefit them most.
posted by subajestad at 8:03 AM on March 27, 2008

Yeah I remember that 1= .99999 one too. That class would let out at the same time as a bunch of others and I would always see friends milling about outside afterwards, just getting out of Poetry 101 or some shit, and I would run up to them with my face lit up going "You'll never guess what I learned in math class today!" and I would try to convey to them how cool some mathematical principle was and I would always fail and people would start avoiding me at that time of day, because they knew I was just going to bore them with some crap about math. I have rarely had an experience like that in education. It was incredible. It made me want to become a mathematician, and I had hated math my entire life up to that point. So, take heart. Math can indeed be really cool. I have experienced it.
posted by ND¢ at 8:04 AM on March 27, 2008 [3 favorites]

OMG, stubby, this was my favorite proof in high school! I was so excited about it! MATH IS SO COOL!
posted by prefpara at 8:05 AM on March 27, 2008

Mathematics in high-school is still geared to the cold war, specifically geometry, trig, and calculus. These are great for training physicist and other "hard" scientists, but there's a whole world mathematics that gets dismissed that is arguably more relevant to the modern world, specifically to computers.

Probability theory
Set theory
Graph theory
Game theory
Propositional & 1st order logic
Linear algebra (okay, maybe some schools teach this)

If you want some motivational examples for kids, I'd recommend paging through Martin Gardner's "The Colossal Book of Mathematics"
posted by qxntpqbbbqxl at 8:05 AM on March 27, 2008 [2 favorites]

How the change in volume of a solid (sphere?) relates to changes in the surface area.

Sorting algorithms, and how certain algorithms are better for certain data sets - how you can estimate and compare the average times that certain algorithms take to work through a data set. This is one side door to realizing how small decisions can add up to huge differences in the amount of time it takes to live.

I don't know much about chaos theory, but it sounds cool and I think it has some relationship to our weather models.

Weather modeling in general; what "30% chance of rain" really means.

Queing theory. So simple, but again, makes you aware of how arranging things can have deep implications for how time is spent.
posted by amtho at 8:08 AM on March 27, 2008

Seconding topology.
posted by dfan at 8:10 AM on March 27, 2008 [1 favorite]

Mathematical relationships that humans consider significant are insignificant to the universe. So choosing the numbers 3,4,5,6,7,and 8 for a lottery ticket will not increase or decrease your chances of winning, but will increase the chances that you will not have to share the prize money, because most people--including a mathematics teacher I know--will think you're crazy.
posted by weapons-grade pandemonium at 8:16 AM on March 27, 2008 [1 favorite]

Compound interest. Seriously.
Basic probability -- Monty Hall problem, why you don't need many people to expect two of them to have the same birthday, why seemingly remarkable coincidences often aren't that remarkable in context of their domain, application to gambling -- why you can't beat the house (over any long term), what the odds of winning the lottery are
How to Lie with Statistics
Basic logic (and basic fallacies)
Ponzi schemes and other economic bubbles (see Extraordinary Popular Delusions and the Madness of Crowds)

How failing to understand the above will get you pwned over and over. Provide examples from the claims of politicians and con artists.

Numeracy -- putting large numbers in context
Dimensions -- Flatland (and what our world would be like from a 4-D perspective)
Fractals, how they can uncannily resemble things in nature, how they can be described as being fractionally-dimensional
Some basic computational theory, why we still can't come up with an optimal solution to the travelling salesman problem (for a non-trivial data set)
Pigeonhole principle
Cryptography, with a discussion of the Enigma machine and its being cracked (Stephenson's Cryptonomicon includes an extraordinarily lucid explanation of cryptomath)

The Teaching Company's Joy of Thinking, based on the textbook The Heart of Mathematics, is all about a set of answer to this question. Don't balk at the DVD price -- they have this annoying pricing model where everything has an inflated price some of the time, and the rest of the time is available at its regular price that's supposed to look like a great bargain, relatively. Just get it on Ebay if you want it during the inflated price (disclosure -- I plan to sell one there soon, but probably won't get around to listing it until this thread is dead.)
posted by Zed_Lopez at 8:18 AM on March 27, 2008 [6 favorites]

Bah. High schoolers get enough lectures about compound interest and safe sex. Neither subject is going to get them excited, trust me.

What I wanted more of as a high school geek was the real brain-melting stuff. I didn't care about real-world applications, I wanted stuff that gave you glimpses out of the real world. Cryptography and game theory left me cold back then for the same reason — they were about COMPUTERS and MONEY and GETTING STUFF DONE and PEOPLE IN SUITS liked them and fuck that, right? — but I knew other people who liked 'em, so YMMV.

Definitely do transfinite numbers and topology. Both have the interesting property of breaking all your normal intuitions, but being accessible with some basic physical analogies — things like the hotel problem for the transfinites, or the donut-is-really-a-coffee-cup business for topology. So without too much work — not much notation or calculation, just a little visualizing — you can pull your brain up into this realm where all the rules are different. (Also, there's the hairy ball theorem. What high schooler wouldn't like a hairy ball theorem?)

If they've got some logic and set theory, give 'em Russell's paradox and maybe the incompleteness theorem. Both are wonderfully head-explodey, and they're pretty important historically. Again, they get you into that everything-you-know-is-wrong territory that's so appealing when you're just getting old enough to know all the rules. You need a little more theoretical baggage, but it's worth it.
posted by nebulawindphone at 8:21 AM on March 27, 2008 [4 favorites]

Just remembered Fibonacci numbers and the golden ratio. They dovetail well together and offer interesting examples of how math influences art and architecture, as well as helping you appreciate recurring patterns in nature (leaves of century plants, nautilus shell chambers, ratio of successive phalangeal bones of the digits and the metacarpal bone), and is even used in certain areas of investment and stock forecasting.

You can also use them as an example of things that when taken to an extreme can degenerate into sketchier subjects such as numerology.
posted by subajestad at 8:25 AM on March 27, 2008 [1 favorite]

Oh! If they're musically inclined, the use of ratios in harmony could be interesting. It's less abstract than the other stuff, and it doesn't really get interesting unless you already understand some things about chord changes, but if you've got the background it's lots of fun.

(I guess technically music counts as a practical application. I sure wouldn't have thought about it that way in high school, though — everybody knew it was more important than health or money. Dear lord we were young back then.)
posted by nebulawindphone at 8:26 AM on March 27, 2008

How about Number Theory, especially the prime number theorem? It would be interesting to talk about how it relates to encryption and how difficult it is to find large primes.
posted by Alison at 8:39 AM on March 27, 2008

Response by poster: @nebulawindphone: After doing a bit of thinking, I think you might be on the right track. If I can get them to come out of the talk with their minds completely blown, that would probably be more fun.

Alright, so Hilbert's Hotel paradox is definitely in. As well as some topology and non-Euclidean geometry. I'll probably touch on multiple dimensions (like Flatland and such).

Thanks everyone, keep the ideas rolling.
posted by tomcochrane at 8:40 AM on March 27, 2008

In high school, mathematics feels of rote learning and execution. I loved math. I was good at it and I felt its power. It wasn't until University that I discovered it is also a creative endeavor. You need creativity to make a difficult proof go through. You need to invent notation, and to break the sequence into lemmas the way an author would organize chapters.

In high school, problems always tell you which tool to use to solve a problem. Real life math isn't like that. Faced with a problem, you choose from your bag of mathematical techniques which want to apply and how to combine them, hoping that one approach will work. That's creative.

The following presentation by Scott Aaronson is absolutely brilliant. It touches lots of subjects mentioned in this thread, and it does it with a historical perspective that represents well the creativity of mathematicians working on the edge.
Who Can Name the Bigger Number?
posted by gmarceau at 8:43 AM on March 27, 2008 [2 favorites]

Pretty much anything John Allen Paulos talks about in his books. Here is a small collection of snippets of his writings.
posted by Jaltcoh at 8:44 AM on March 27, 2008

The number Zero (0) and how it was regarded in different civilizations. I thought "Zero: The Biography of a Dangerous Idea" by Charlies Seife was excellent. It's short and easy to read and covers several different civilizations how some used Zero and how some essentially banned the idea.
posted by Science! at 8:47 AM on March 27, 2008 [1 favorite]

Benford's Law. This has interesting applications to financial fraud, which may prove to be a useful hook to grab their attention. This could serve as an introduction to power laws and self-similarity.
posted by UrineSoakedRube at 8:57 AM on March 27, 2008

Game Theory and Negotiating Strategies using probabilities.
posted by JohnnyGunn at 9:01 AM on March 27, 2008

Bridges to Infinity by Michael Guillen will have plenty of ideas for you, and is written specifically for non-mathematicians.
posted by Zenabi at 9:05 AM on March 27, 2008

Graph Theory is really cool.
posted by chunking express at 9:13 AM on March 27, 2008

Graph theory: Small worlds, the oracle of Kevin Bacon, snippets from books like Linked or more recently Ghost Map.

Discrete Math: Please, please introduce this. I found discrete math, linear algebra, etc relatively easy, enjoyed the problem-solving, and struggled with calculus. If I hadn't realized this was an entire other side I wouldn't have continued taking math in college. Possible activities.

Fractals: Great for bringing in images. Mention "chaos theory"; being introduced to the idea was very cool for me. Mandelbrot set.

Calculus: Epidemics and disease I found very illustrative in understanding calculus. See Calculus in Context, Chapter 1, which is online.

Economics/Money: Seconding compound interest, strongly. Use any 401k calculator and explain. Or a amortization table on a house mortgage.
posted by ejaned8 at 9:20 AM on March 27, 2008

Knot theory and basic algebraic topology. High school kids can understand the concept of the fundamental group if you're good at drawing pictures (or if you just copy pictures from a book). I think that stuff is good because it's very visual and it doesn't rely on a lot of abstract machinery.
posted by number9dream at 9:41 AM on March 27, 2008

This is a really, really basic skill, which makes it all the more baffling that I didn't learn about it until I was 24: Lattice multiplication. All of a sudden, I was able to multiply, in my head, numbers with more than two digits.

Why the hell don't they teach that in Grade 3?
posted by Sys Rq at 9:45 AM on March 27, 2008 [2 favorites]

Euler's identity.
posted by futility closet at 9:50 AM on March 27, 2008

Wow this is interesting, and I would have to say, I wish I would have known about almost all of them in high school.

I'd really emphasize how to lie with statistics as an exercise in critical thinking as others have mentioned.

The Monty Hall Problem would've flipped me out then, still does.

I think fractals, and chaos theory as well.
posted by xetere at 9:56 AM on March 27, 2008

Sys Rq: lattice multiplication is controversial.
posted by prefpara at 9:58 AM on March 27, 2008

Nthing topology. Start with mobius strips, then progress to knots and mechanical puzzles.

And algorithm analysis. Show different sort strategies and discuss NP problems.
posted by DU at 9:59 AM on March 27, 2008

Seriously, if some of this sort of stuff was taught in high schools, maybe I wouldn't be so full of math fail.

Yeah, I mean, it sounds cheesy, but math seriously is amazing, and they take this incredible mind-blowing discipline and poison children against it. They suck every bit of cool out of it and make people hate it. They should teach these things in kindergarten before 2+2=4. Let children learn the hows after they have seen what the hows can do. The way they teach math now would be like requiring a child to know all the rules of grammar before they were introduced to their first book.
posted by ND¢ at 10:01 AM on March 27, 2008 [7 favorites]

Show them some successful young mathematicians and remind them that many of the best jobs for smart people now go to people who can deal with and apply math creatively.

Show them that seemingly simple problems (such as in graph theory) are not so simple and yet are very important. Many favorite problems, as you can see from comments above, can be shown with very tangible or simple examples -- traveling salesmen, knots, primes, etc. Give them a list of unsolved problems and teach them how they might go about solving them (and becoming rich and famous and attractive) by first learning about certain branches of mathematics and solving somewhat similar problems. Remind them that mathematics is a young person's game.
posted by pracowity at 10:03 AM on March 27, 2008

"How to Lie with Statistics"

Yes! I enjoyed math, most of the time, all the way into college (Calc 2!), but never did anything useful (or even particularly mind-blowing) with it. Then when mr. epersonae went back to school, he had to take statistics, so I learned some reading over his shoulder, reviewing homework, etc. I desperately wished I'd learned stats 10 years earlier.

Hitting the skepticism angle is probably the best idea; "how might this study be biased?" is probably the best question to have in your head after learning statistics. (IMHO)
posted by epersonae at 10:08 AM on March 27, 2008

PLEASE do not teach compound interest. I remember learning that in high school and wanting to jump out the window-- and I LIKE math!

I was really into fractals and the golden ratio back then. Now, I believe that everyone should know probability. Spectral analysis is also really cool-- how there is this WHOLE OTHER SECRET DOMAIN called frequency where you can visualize all these otherwise invisible relationships.
posted by hybridvigor at 10:16 AM on March 27, 2008 [1 favorite]

Is this just one lecture? If so, I have to agree with the compound-interest detractors (despite having put it first on my list) -- better to go for maximum cool.
posted by Zed_Lopez at 10:23 AM on March 27, 2008

e^(pi * i) + 1 = 0.

One formula that describes the relationship between all the important numbers in math. Suddenly, the entire course of mathematical study that they've been on for 10-odd years is summed up in one equation. (arithmetic to geometry to algebra to [statistics and probability] to trigonometry to calculus. And you need ALL of them to truly grok the significance of the above.)

And the neat thing is that what at first seems like something of coincidental curiosity turns out to have pretty handy application in solving complex integrals.

Anyone with the basics of calc can follow the proof, and its best if you don't tell them where you're going. It takes like 4 chalk boards worth of Taylor expansions, but then at the end everything cancels out in a really satisfying way, and you are left with what is, in my opinion, the single most beautiful thing in math.
posted by ChasFile at 10:30 AM on March 27, 2008

Musinum generates fractal music using the self-similarity of the Thue-Morse sequence.
posted by plexi at 10:38 AM on March 27, 2008

Show how a getting a positive result on a medical test having 99% accuracy doesn't mean there is a 99% chance you have the disease - good explanation here.
posted by Wet Spot at 10:41 AM on March 27, 2008 [1 favorite]

modal logic? the idea of non-classical logics in general? I actually learned some classical propositional logic in a geometry class, but I'm not sure if most people get even that.
posted by advil at 10:50 AM on March 27, 2008

Fibonacci numbers & their red-headed stepsibling, the Lucas Numbers. Don't touch the closed form part of it, that's only interesting to a few.

Strongly agreeing with gmarceau's Scott Aaronson suggestion, it is a very cool thing. Transcendental numbers in general can be awesome, as can recursion/Godelian work.

There is a Hydra game that a professor showed us in 1st-year calc which was awesome, even though he got lost halfway through the proof. He also showed us Hilbert's Hotels, and I keep that one in my head as 'awesome math'.

Just remembered the best recursive question I can ask anyone, it's the Blue Eyes problem. Make sure you practice your explanation of how you're right before teaching it.

Although I love crypto and security in general, I kind of disagree with people suggesting it to be taught. It's just a bit too far out from the realm of intuitive understandability, it'll be cool at the time but it's too hard to keep your mind in it as a HS-student afterwards. Due to the anyone can design a system that they can't break idea.

Good luck, and post back about how it went, I'm very curious!

@ stubby: As soon as those guests are roomed, Cantor's Bus Tours(tm) shows up with an infinite number of buses. Each bus has terrible conditions, with an infinite number of people on each, and they all want rooms. Clerk says 'yeah, no problem.'
posted by Lemurrhea at 10:53 AM on March 27, 2008

I wish my maths teacher had taught me how to play the ancient Japanese/Chinese board game 'Go' (known as 'Igo' in Japan). The game teaches you how to intuit numbers, game theory and the enormity of permutations that reality rests upon. Consider this, taken from the link below:

"In the course of a chess game, a player has an average of 25 to 35 moves available. In Go, on the other hand, a player can choose from an average of 240 moves. A Go-playing computer would take about 30,000 years to look as far ahead as Deep Blue can with chess in three seconds, said Michael Reiss, a computer scientist in London." - link

Go builds from a wonderfully simple game of numbers to an intense insight into the nature and complexity of human consciousness.

(Apologies if you know all about Go already - I tend to get a little over-enthusiastic about the subject.)
posted by 0bvious at 11:01 AM on March 27, 2008 [1 favorite]

I teach mathematics (mostly calculus courses) at a state university, and what I find to be the most unnerving and disturbing is that the majority of my students have absolutely no background in logic. It makes it very difficult to show them a proof (or get them to write one of their own) when their logic skills are hurting.

I suggest that things like set theory and combinatorics should be taught in high schools. Combinatorics is the most cannibalistic of mathematical disciplines, as it borrows results from seemingly unrelated fields all the time. For this reason, students should be introduced to it earlier.
posted by King Bee at 12:05 PM on March 27, 2008

The math required for 3D computer games is very cool (trig, quaternions, projection, etc). Also the geometry required for polygonal collision detection (again in computer games) and other physics is pretty neat.
posted by sandking at 12:05 PM on March 27, 2008

Half of what I wanted to suggest has already been suggested. But one more: talk about proofs. Proof by induction, proof by invariants and state machines, proof by contradiction, non-constructive proofs ...

Yes, I loved proofs. This class has some really cool examples of proofs (and yeah, other things that fall into that Cool Shit They Don't Teach In High School category).
posted by spaceman_spiff at 12:12 PM on March 27, 2008

Show them the Borda Voting method. Then, assume an election between, say, bush, Gore, Nader and Buchanan. (Just making up names at random)

Assume 50% of the population are democrats.
Assume 50% of the population are republicans.

25% of democrats like Gore.
25% of democrats want to send a message by voting for Nader.
25% of republicans like bush.
25% want to try to make sure Gore can't win.

First Groups Preferences: Gore > Nader > Buchanan > bush
Second Groups Preferences: Nader > Gore > Buchanan > bush
Third Groups Preferences: bush > Buchanan > Nader > Gore
Fourth Groups Preferences: bush > Nader > Buchanan > Gore

Nader gets (using 0, 1, 2, 3 points) 50 + 75 + 25 + 50 = 200 points and
wins the election!

(this is because I am reading william poundstone's Gaming the Vote now. )
posted by wittgenstein at 12:18 PM on March 27, 2008

Anything that involves Cantor's cardinalities. The fact that the cardinality of the natural numbers is equal to the cardinality of the real numbers, but less than the cardinality of the integers completely blew me away when I learned about it. The various diagonalization proofs are all pretty understandable for the layman, as long as they can see the logic behind them. Also, I found it very entertaining to learn about the history of infinite cardinalities, and how Cantor was shunned by many mathematicians of the time.

Beginning abstract algebra and automorphisms are also quite interesting. We did a thing in one of my classes recently about tic-tac-toe automorphisms (3x3, 4x4, 4x4x4) which yielded some very nontrivial and nonintuitive results.

My theory is, most people get turned off to mathematics during the tedious computational portions of it such as calculus and trig, but once they can push through that, the entire world opens up
posted by Geppp at 12:30 PM on March 27, 2008

Geppp - The cardinality of the natural numbers is strictly less than that of the real numbers.
posted by King Bee at 12:49 PM on March 27, 2008

Explaining how the complex numbers are constructed from the real numbers by adjoining the root of x^2 + 1 = 0 would make a nice 50 minute talk.

5 x 5 = 25.
25 ends in 5.
25 x 25 = 625.
625 ends in 25.
625 x 625 = 390,625.
390,625 ends in 625.

This would be an excellent problem with which to end the talk. Most likely, they will wonder about it for months, possibly years. WIN.
posted by proj08 at 1:32 PM on March 27, 2008

3^2 + 4^2 = 5^2.
5^2 + 12^2 = 13^2.

Find ALL integer solutions.

Solving this problem would make a great 50 minute talk.

The Sylow subgroup theorems. (they might not be able to appreciate this in highschool, but it's a personal fav.)

Draw some 3-dimensional commutative diagrams on the boards. They'll love it.

Last but definitely not least: Start with a well-known fact: "N equations in M unknowns. If N > M, there need not be a solution. If N < M, there might be infinitely many solutions." BUT! These are linear equations. What about polynomial equations? Example: x^2 + yz = 0 and xy^2 -z^3 = 0 and xyz = 0. Is the same thing true? This leads to really complicated things, and you can sample some of them. It's a natural question, with which they are familiar, so I think they will like it.
posted by proj08 at 1:59 PM on March 27, 2008

King Bee

Oops, hehe, that was abit of a typo on my part, thanks for catching that.
posted by Geppp at 2:45 PM on March 27, 2008

n-thing Cantor's diagonalization this is really easy to understand and mind-blowing.

If you want to talk about Godel, take a look at Raymond Smullyan's What is the name of this book. It presents Godel's incompletness theorem as a Knights and Knaves problem (ie knights always tell the truth, knaves always lie and you have to figure out which is which based on their answers to a question).

Godel really blew my mind when I first saw it (and still continues to do so).
posted by fingo at 3:28 PM on March 27, 2008

Another vote for statistics here. Especially the history of Statistics. I mean,William Sealy Gosset... who can resist beer?
posted by Megafly at 3:55 PM on March 27, 2008

One of my favourite moments in first year university maths was when I geometrically proved that the bisectors of any quadrilateral form a parallelogram.
It was quite early on, and the actual Maths bit was very simple, but when we had done it, it didn't make any sense to us.
So we started drawing them.
And they did!

Even the really weird ones.


I think this might be similarly amazing to anybody (if they think in exactly the same way I and my friends used to...).
posted by fizban at 5:43 PM on March 27, 2008

I'm going to go with Quantitative Finance. While the upper level stuff can get mathematically complex, a lot of the introductory stuff is simply an application of basic algebra, linear regression, and linear equations.

For example, the security characteristic line is just a version of the simple equation y = mx + b equation, SCL = αx + β.

I personally think this one is also a good choice because here's a field outside of what people normally associate with mathematics (engineering and the hard sciences) and also one that challenges the old notion that majoring in math won't make you baller-rich. Desk quants, derivative and bond traders, risk managers / analysts at a large bank or hedge fund roll seriously deep. For an example of a successful investment company built on these principles, you can point to Renaissance Technologies (even though they haven't been doing so hot lately).

Topics like Black-Scholes or Yield Curves could be topics for interested folk to explore further, but will require a bit too much background for a one-time talk.
posted by C^3 at 6:41 PM on March 27, 2008 [1 favorite]

Steinhaus polygon notation, for the representation of unimaginably large numbers. This absolutely blew my mind back in school.
posted by deadmessenger at 8:19 PM on March 27, 2008

Anything to do with numbers and counting, which nearly anyone can understand. Prime numbers are good for hours of fun.
posted by joeclark at 8:46 PM on March 27, 2008

Seconding game theory; I quite enjoyed The Compleat Strategyst as a beginner's guide.

Also, anything along the lines of lambda calculus, combinatorial calculus, or their stacktastic sibling, concatenative combinators. Would've thrilled me in high school. Still does.
posted by eritain at 9:41 PM on March 27, 2008

Lockhart's Lament [pdf] is a great rant on how poor math education is and how awesome it could be, if only... Also relavent is Cliff Stoll's TED talk [video, autoplays], both of which have been bouncing their way around my friends' and family's email boxes for a week or two.

I'd generally go with the mind-blowing ideas over the useful or critical ones, you're not going to really 'teach' anything in one lecture but a really brain-expanding concept might just stick. I did non-euclidian geometry and probability theory in HS math and enjoyed both, but neither broke my head in the same way something like Gödel's theorem does. Stay away from anything algorithmic unless they've had previous CS courses, there's too much background to cover in one lecture to get to anything really interesting IMO.
posted by Skorgu at 5:20 AM on March 28, 2008 [1 favorite]

Yikes, for some reason my earlier comment came to mind as I was taking a shower this morning and I realized that I screwed something up. SCL = βx + α, and not the other way that I had up earlier.
posted by C^3 at 5:27 AM on March 28, 2008

How about all those Gaussian iterative algorithms that were just waiting for a digital computer to be invented to put them to use?
posted by klarck at 5:49 AM on March 28, 2008

I teach high school. For my non-AP physics class I gave them a problem similar to the hydra problem (boxes and a warehouse, but same idea) as extra credit. I had no idea of its implications. Which is fantastically cool.

Anyway, cardinalities of infinity, Bayesian statistics (which still blow my mind and I majored in math), some basic number theory (infinite number of primes, Euclidean algorithm, etc.), some basic probability (chances of winning the lottery, danger of a car crash vs. a plane crash). A notion of what very large numbers actually mean.

For those who are interested in this sort of thing, I highly encourage you to check out The Math Circle in Boston, MA. There are other math circles out there, but this is the only one (that I know of) that doesn't just focus on teaching advanced topics to high school students who are already good at math. (Disclaimer: I went there as a middle school student, it's where I probably developed my love of math.)

Also, their books are really cool.
posted by Hactar at 6:21 AM on March 28, 2008

Non-transitive dice [1, 2] are non-intuitive and could make for a good demo - you have 3 dice on the table with different numbers on each, your opponent can choose any one and you choose one of the remaining two. Whichever die they choose, you can choose one which will beat it over repeated rolls.

The psychology of maths is pretty interesting too - this article from the New Yorker is a good intro:

According to Stanislas Dehaene, humans have an inbuilt “number sense” capable of some basic calculations and estimates. The problems start when we learn mathematics and have to perform procedures that are anything but instinctive.

posted by jamespake at 6:58 AM on March 28, 2008

I loved geometry in 8th or 9th grade, when it was all about logic and coming up with your own proof about the way actual shapes related to each other, but got bored of math in Algebra 2 & pre-Cal because it just turned into a bunch of rules that seemed meaningless. If someone could have taught math in a way that demonstrated how it was relating to real discoveries and properties of things, I think I would have enjoyed it.

Representations of numbers might be interesting - how zero and one weren't even considered numbers (zero was nothing, and one was just the unit) before arabic numerals; using decimal vs binary vs hexadecimal (since those are all in regular use somewhere) vs other base systems (just for fun) - kinda relates to set theory since it's about grouping, but ultimately it helps to stop thinking about the symbols and really get thinking about the amounts they refer to.

Speaking of symbols vs actual insight, that "lattice multiplication" thing doesn't seem interesting - it's just a slanty way to write long multiplication...

THis thread isn't exactly about how to make math cool, but it could have some useful ideas anyway (I made a similar comment there)
posted by mdn at 8:53 AM on March 28, 2008

Along similar lines to ND¢ 's hilarious answers...

I have a good friend who just couldn't wrap her head around math in high school. She's really smart, but for whatever reason just couldn't get into Algebra2/Trig etc. She ended up going to St. John's College and was taught math in a "liberal arts way," such going through Euclid's theorems and proving them all over again. Approaching math from a historical perspective, she discovered that she does have an aptitude and now she's a science teacher.

I wish I had learned math historically.
posted by frecklefaerie at 12:12 PM on March 28, 2008 [1 favorite]

Another vote for Godel's Incompleteness Theorem, maybe throwing in the Entscheidungsproblem for an encore.
posted by yetanother at 1:41 PM on March 28, 2008

Pascal's Triangle and coin flipping probabilities. When I learned it, it was cool to see binomial multiplication applied to something real word.
posted by nonmyopicdave at 1:57 PM on March 28, 2008

e to the pie-eye, plus one, equals zero.

Nothing more beautiful in all of mathdom.
posted by Civil_Disobedient at 4:03 PM on March 28, 2008

Concepts of dimension. There's a ton you can do with this. If I were giving a talk like this, I'd use an outline like this (though this may be too much for an hour). Note: I'm going to explain this in the shorthand I'd use with a mathematician, and not explain everything. Feel free to MeFiMail me if you want a longer explanation of anything. 2nd Note: You've gotten so many replies to your question that I doubt you'll care about mine, so lest anyone think I have too much time on my hands, I plan to come back to these notes the next time I'm writing a math camp talk.

0. Flatland stuff (already suggested by someone else)

1. Extra dimensions aren't just for crazy physicists -- we can use them to graph equations involving lots of variables, then use the "geography" of the graph (hills, valleys, saddles, etc.) to figure out stuff about the solutions. Like what you do in middle school algebra, only in n dimensions (that's just an inherently cool phrase...).

2. There are higher-dimensional objects that we can never build in our 3-dimensional world, but that doesn't mean we can't figure out what they're like. Examples I'd talk about in depth: the hypercube, the Klein bottle. I'd have them try to describe the pattern in the sequence "point, segment, square, cube, ..." and infer what the next term should be like. How many vertices, edges, square faces, cubic faces? Can we draw a 2-d shadow of it on the board? Can we build a better 3-d model out of sticks and gumdrops? How are these models imperfect? How would a 4-d universe dweller assemble a hypercube?

3. Regular polygons, Platonic solids, then what? What would a "regular" shape in n dimensions mean? Are there any interesting ones? (Yes, but only in 4 dimensions. After that, there are still regular polytopes, but they're kinda boring.)

4. What dimension is a curve? How about a sphere or a doughnut -- but the surface, not the solid? I'd discuss the difference between intrinsic dimension and the dimension of the ambient space. OK, so that's easy to grasp... or is it? A line is 1-dimensional. The red stripe on a candy cane, for instance, is 1-dimensional (if you ignore the thickness). Can we wrap a red stripe around a doughnut so that the whole thing turns red? Answer: kind of. This is starting to make them nervous. But relax, the image of the stripe is only "dense" on the doughnut. But wait! There's this thing called a space-filling curve, and it really does fill space! Zounds! Is a curve 1-dimensional, or can it be more than 1-dimensional? We need a better definition of dimension!

5. Hausdorff dimension, explained with copious fractal examples (e.g., Sierpinski gasket). I wouldn't dwell on the precise definition; I'd just explain it as "When you scale an n-dimensional object up by k, its 'size' scales by k^n. This is what it means for something to be n-dimensional." Attention-grabbing problem: What's the area of the Sierpinski gasket? Zero! So what's the length? Infinity! What kind of measurement would give a finite answer?
Real-world application (?): Mandelbrot's notion of a "science of roughness," though admittedly there isn't much there yet. Can also give a brief mention to quarter-power scaling laws here, which illustrate the concept of Hausdorff dimension nicely.

Another real-world app that could be worked in somewhere: Tomography, the methods used to reconstruct the "shape" of a 3-d object from 2-d projections. This is the science behind CAT scans. Projection is generally a good thing to talk about...

Other (separate) syllabi that could be a lot of fun:

-- Curvature of surfaces. This leads intuitively into non-Euclidean geometry, but also has immediate relevance to stuff like knitting and crocheting, or to understanding the growth of leaves, curly hair, etc. You can also talk about why long-distance flights often fly over the Arctic circle, why all flat maps of the world are distorted in one way or another, why you can't smooth out the wrinkles in a badly knit hat (even though there's no local rigidity), etc. Can also talk about the Gauss-Bonnet formula, and how it could be used to experimentally measure the radius of the earth. I saw a great talk at a math camp a few years back. The speaker pondered what it would be like to go bowling in an alley shaped like a lettuce leaf. (A memorable line: "In hyperbolic space, pizza has lots of crust and there's always free parking.")

-- Iteration. This is an entry point into chaos and fractals, with the key idea being that simple rules can produce astonishing complexity. Some of the pictures in Wolfram's A New Kind of Science are great for this (whatever the other merits/demerits of the book may be). You can tie iteration in with things the kids have probably seen (e.g.: Pascal's triangle is built by iterating a simple rule, and not coincidentally, it also contains a hidden image of the Sierpinski gasket...). The Sierpinski gasket can also be built via the "chaos game," and if you change the parameters, you get lots of pretty pictures. (This is probably a better starting point than complex dynamics!) If you really finesse this topic, you can tie in a shit ton of fundamental ideas about dynamical systems, such as local-vs.-global, stable-vs.-unstable equilibrium, and critical values of the parameters (what corresponds to "tipping points" in systems ecology).

-- Invariance. So, so many ways you can go. I like to talk about affine geometry, because it touches on a theme most kids will appreciate: reducing hard problems to easy ones. Why do the medians of a triangle always meet? Because they do for an equilateral triangle (duh), and every triangle is the same as the equilateral triangle in every way that counts! Why is the volume of a pyramid equal to 1/3 the volume of a prism with the same base and height? Well, if you cut a unit cube (centered at the origin) along the planes x=y, y=z, z=x, you get 6 identical pyramidal pieces (there are various ways to see this). Each of those has 1/3 the volume of a prism with the same base and height. Now explain why if it's true for those pyramids, it's true for all pyramids.

-- Symmetry, both geometric and formal. (Another great line I heard growing up: "WLOG is the most powerful function in mathematics." I thought it was some kind of logarithm...)

If I continue, I'll never stop.
posted by aws17576 at 5:50 PM on March 28, 2008

e to the pie-eye, plus one, equals zero.

Nothing more beautiful in all of mathdom.

True, that. But if the kids have had trig in the last 1-2 years, they may become very resentful when they realize that all the identities they had to memorize could have been summed up in just this one!
posted by aws17576 at 5:52 PM on March 28, 2008

Ditto-ing subajestad et al -- I'm into Fibonacci sequences for designing striped patterns in knits. And Diana Eng, who was on Project Runway, has done a whole series of math-inspired knits (you can see them on her website).
posted by at 9:15 PM on March 28, 2008

Third or fourth on Gödel's incompleteness theorem.

Skip 0.9999... = 1.0. If these kids are going to math camp, they know it.

Non-standard analysis is actually much easier than that Wikipedia article and can definitely be taught to kids that age if you can understand it yourself.

The Banach-Tarski paradox. It's a little tricky for this age but cool.

Surreal numbers, which is also an entertaining novel in its own right.

Or go way out... there's a terrific play called Proofs and Refutations that's very entertaining too and also has a lot of great ideas about the nature of proof. You could put the play on! (We did this in University).\

I have tons of other examples, drop me a line....
posted by lupus_yonderboy at 5:05 PM on March 29, 2008

In retrospective, I would have LOVED to have a good teacher with a good math background teaching me:

1) the concept of limit and the demonstration that
lim x->L f(x) = L ,with any given number "epsilon" no matter how little (but not zero) , is true if f(x) <> as a value a function can approximate, but never reach.

2) demonstate that sqrt(2) isn't a real number

3) logic, logic, logic applied to rethoric , connecting words and verbalization of reasoning with "science"

4) get us out of that simplistic stupidifying binary thinking most people spend a lifetime into
posted by elpapacito at 5:09 AM on March 30, 2008

Euler & Hamiltonian circuits

posted by saxamo at 5:09 PM on March 30, 2008

I wanted to propose exactly the two things Lupus_yonderboy just proposed.

Nonstandard analysis might sound daunting, but if they're already doing any calculus it might be a lot of fun. (I am no mathematician. Allow me to shame myself by relaying my recollection of it.)

It's involves a substantially different way to approach calculus due originally to Leibniz, rather than the approach we use due originally to Newton. No bloody limits and sequences! What's infinity plus three? Why: infinity plus three! 7+1/0 <>
And if like me the kids rather a good story than good math, it's romantic since it has this alien road-not-taken quality.

If the Banach-Tarski paradox isn't fun, I don't know what it is. The perhaps too fun version of the story is that one can cleverly slice up a pea into a finite number of parts, and then just by moving and rotating the parts reassemble it into the Moon (eh - the whole solid thing).

*shakes head* Math!
posted by ~ at 6:29 PM on March 30, 2008

(Oh drat. Fail at math and HTML. I meant: in the middle paragraph to write 7+1/0 < 8+1/0. But no matter.)
posted by ~ at 6:30 PM on March 30, 2008

Modular arithmetic, especially pointing out that the integers mod n form a field iff n is prime, and that in that case (as with any field) there is a single number which generates the field via multiplication. If you could go on to group theory from there, my 11th grade self would have worshipped you.

Graph theory. No one has mentioned Erdos numbers yet! Or Paul Erdos for that matter. Or you could just talk about the whole 6 degrees of separation concept.

One of my favorite graph algorithm problems (with a real world application) is the following: Suppose you have a matrix showing conversions between n currencies, where the [i, j]th entry is what currency i is worth in terms of currency j. Is it possible, for arbitrage purposes, to detect if there is a positive cycle in the graph? Meaning, is there an algorithm which detects a "cycle" of currencies (c1, c2, ..., cN, c1) such that if you follow it, converting from one currency to the next, you end up with more of currency c1 than you started with? (It's a matter of applying an all-pairs shortest-paths algorithm in an interesting way.)

When I was younger I derived a ridiculous amount of satisfaction from proving to myself that the sum of the digits of any multiple of 3 would be a multiple of 3 (in base 10 of course). But I don't know what practical purpose that serves, apart from being able to check in a fairly small amount of time if 3 divides a number.
posted by A dead Quaker at 6:44 PM on March 30, 2008

The Logistic Map
I'd suggest doing them up using processing or something similar, and point them to a website with your code so that they can play with it when they go home.
posted by sebastienbailard at 12:36 AM on March 31, 2008

Prove that irrational numbers exist...and talk about what makes them numbers.
posted by milestogo at 10:16 AM on March 31, 2008

Bayesian statistics. A diagnostic test that has a 95% accuracy rate indicates that you have a particular disease, yet it may still be more likely than not that you are completely healthy! Maybe link this to the prosecutor's fallacy, flawed statistical reasoning that has led to some real life miscarriages of justice, as an example of the importance of knowing how to interpret statistical data. Some of these concepts superbly presented here.
posted by genesta at 11:59 AM on March 31, 2008

I had some great math and computer science teachers in high school. For ideas of what to cover, I recommend flipping to a random chapter of Gödel, Escher, Bach.

Some of my favorites (from GEB and elsewhere)

- the halting problem

- back of the envelope calculations (ex: how many gas stations are there in this town). Combines probability with intuition about the real world - quite a useful skill

- algorithms and big O notation (work through 2 or more sorting algorithms)

- probability and combinatorics grounded in real world examples (no colored socks in drawers problems, please). Use combinatorics to explain the strength of passwords in the face of a brute force attach

- boolean logic. Work up to an adder composed of and, or, and not gates.

- 3d vector graphics and matrix transforms. work through some problems on paper.

- probabilistic algorithms applied to something interesting (like testing for primes)
posted by zippy at 8:06 PM on March 31, 2008

Already been said, but I have to second topology. In my 8th grade math class, we had to do a presentation on a math topic we knew nothing about, I think choosing from a list she gave us. (It was an honors class.) I picked topology, pretty much randomly, and as soon as I started reading about it, you could not shut me up. I was drawing little Klein bottles all over my notes and eagerly explaining to my dad that you could color in a map with JUST FOUR COLORS and have them NEVER TOUCH omg how cool is that. And I didn't even like math, although I wasn't that bad at it.

In college I've had three different classes that dealt with game theory (Econ, International Poli, Human Evolution), and there was a similar situation of not being able to shut me up. ("See, but if the cost of defecting is higher than- hey, where are you going?!") I actually wanted to take an entire game theory class after learning a little about it, but my school doesn't seem to have one.
posted by showbiz_liz at 9:57 AM on April 1, 2008

zippy - I tend to think that the probabilistic method is outside the range of most high school students. What exactly were you suggesting (certainly you don't mean Lovasz Local Lemma type stuff)?
posted by King Bee at 11:21 AM on April 2, 2008

Skorgu, that was a terrific video lecture from Clifford Stoll.

Also Lockhart's Lament - two thumbs up!

My vote for strange/cool math topic - Benford's Law after I learned about it just now from USR's posting above.
posted by storybored at 7:02 PM on April 6, 2008

I've been slowly working my way through this thread and looking up everything I didn't understand and then looking up the parts of the explanations that I didn't understand until I felt I had a basic grasp of everything suggested here and now think that I may have brain damage, but that brain damage is now in a larger world than I knew existed before I started reading the thread so I welcome it.
posted by Molesome at 6:08 AM on April 9, 2008

Oh, and just to add something to the thread, I always found the concept of imaginary numbers to have a sort of secret-cool to it.
posted by Molesome at 6:09 AM on April 9, 2008

I didn't know about nontransitive dice. I used to play a dice game called Stack. Nontransitive dice would make an interesting complication --- several possibilities come to mind.
posted by fantabulous timewaster at 8:22 PM on April 9, 2008

If you're going to touch algorithms at all, I recomment
Subset Sum. It's an exceptionally easy problem to state, which helps with explaining that verifying if any given answer is correct is very easy, but finding a correct answer when you have none is very, very time consuming.

In general, NP problems have a delicious threesome of mental rending goodness:
a) easy to verify
b) intractable to solve
c) unproven if we'll ever be able to do it better (nice lead in to Godel)
posted by enkiwa at 6:03 PM on April 11, 2008

As someone who sucks at math so much that I never made it to pre-calc, I'm fascinated by Bringing Down the House and the idea of professional card counters at blackjack tables in Vegas. Now that they've made 21, chances are a lot of high schoolers will probably find the idea fascinating as well.

But I also really love statistics and logic. Statistics as used by journalists, school boards, city councils -- sometimes read well and sometimes completely misunderstood. Logic a la Lewis Carroll.

I've always thought that I might have been good at math if I'd had better foundations, and there are times when I find myself struggling with some silly mathematical problem I would have been able to solve in high school. (Example: I have a round table and want to paint a checker board in the center, but I can't find the exact center of the table, although I'm fairly sure there is a very easy way to do so. Still, my boyfriend and I spent a good hour trying to remember how this works.)
posted by brina at 8:55 AM on April 19, 2008

Don't do stuff like compounding interest and Euler's identity. I was taught both of those in high school; I don't look back on either of them and think "WOW! I'm so lucky I was taught compounding interest - it's awesome!"

Do stuff that can be easily explained, hopefully in not-very-mathematical language, in some neat way.

These suggestions, or at least some of them, have already been made, but I'm going to back them up:

Graph Theory. It's awesome, and easy to get an initial grasp on. Talk about the Konigsberg Bridge problem (which was essentially the birth of graph theory), the Four Color Theorem (which is easily explained, pretty cool, and will get some kids very interested when they hear that it has been proven, but no "short proof" has ever been found), and maybe what a planar graph is and how to tell if a graph is planar (preceding this by showing the manual method of dragging vertices around willy-nilly).

Gödel's incompleteness theorem. Either there are true mathematical statements that cannot possibly be proven mathematically, or there are false mathematical statements that can be proven mathematically. That's totally mind-blowing.

The Collatz conjecture. I guarantee you that the idea that there is something that can be so easily explained, and so easily understood, requiring only basic arithmetic, which is true for every single number that's ever been checked, out to unimaginably huge numbers, yet has never been proven to be true in general, will get at least some kids hooked.
posted by Flunkie at 9:59 AM on April 22, 2008

Un Día Vi Una Vaca Vestida De Uniforme

Or, if you never took Spanish in high school (are there any other romance languages this could work for?):

Uncle David Visits Uncle Victor Vargas Down Under

That being said, I can't stand either of those mnemonics, don't usually use mnemonics, and usually just memorize formulas the brute-force way. Understand, memorize, drill, understand. That's the way I go about it.
posted by Precision at 3:08 PM on August 28, 2008

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