Topic for a lecture intended for High School students I am not sure if this is the right place to post this, but here is the situation.
In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore high school kids. I am required to find some interesting topic, but that doesn't require any advanced mathematics. I believe, all I can use is up to, but not including, Calculus. 
So, I was wondering if any of you guys could help me brainstorm some interesting topics that I could talk about, without using any advanced mathematics.
Thank you in advance!
 A: Hopefully you don't want to actually lecture for 2-3 hours. I suggest you have some worksheets or some small group project that the students can work on. I would guess that even you would have a hard time just listening for 3 hours. 
You should make sure that the students will have fun.
Here are some ideas:


*

*Modular arithmetic maybe leading to cryptography. This is great because you can have the students encrypt/decrypt messages

*Do something on the history of mathematics. Maybe you can have the students complete calculations using old methods. There are a lot of ideas hidden here. You could go back to Egyptian mathematics, or do something on polynomial equations. One benefit of this is that you can keep it simple and you get to tell stories.

*The mathematics of the Rubik's cube is always fun. There works well as a gentle introduction to group theory. The only thing is that you should get a bunch of Rubik's cubes so that the students get to try them out.

*Related to the one on the Rubik's cube, you can basically take a game and do some math with it. A good example would by the $15$ puzzle: http://en.wikipedia.org/wiki/15_puzzle.

*How about something related to the pigeonhole principle principle? Maybe do something related to logic and counting. A lot of fun little problems can be made here.

*My last idea is something on number theory. That is, something with prime numbers. This has the risk that some might find it boring, but many people find questions/problems related to prime numbers fascinating. 

A: One rather non-traditional topic for a high-school talk that I would certainly find interesting if I didn't know it (this is coming from essentially a high school student) is the basic ideas of infinity. This essentially requires no high-school math background if done correctly, and I think is a fascinating topic (at least if these students have some sort of mathematical interest).
You could start with some motivating statement, such as: "Which of these sets is bigger: $\{1,3,5,\dots\}$ or $\{1,2,3,4,5,\dots\}$?" and proceed to discuss how "actually, they're the exact same size!" You could then go into things like one-to-one correspondence, the "real" notion of different types of infinity, Cantor's diagonal argument, Hilbert's hotel, ad infinitum (all in a non-rigorous fashion where the audience is as actively involved as possible).
Of course, there are two things that are important in a 2-3 hour maths talk. First, you need lots of interesting pictures, and unfortunately this topic doesn't immediately cater to that. However, there are lots of cool diagrams you can dream up about 1-1 correspondence, you can add a picture of Hilbert's hotel, blah blah. Also, I'm sure there are plenty of ways to find some sort of engaging activity to go with the "lecture" part of the talk.
A: How about some game theory? You could teach an hour or so of a game theory topic and get the students to play some of the games with different strategies themselves to see how they work (and win!) E.g. the game of nim, take away combinatorial games, graph games etc. This way they get to see some nice mathematics with quite a fun application.  
A: We are bombarded with statistics all the time. Whether its from politicians and economists or sociologists, or whatever. Statistics is the basis for science as well.  Its unfortunate that so many people recite statistics and base their judgments on it, without having a grasp of how it works or what it even means.  We teach our students science, and even how to employ the scientific method, but all without teaching them how to critically think about statistics.
I think it would be not only interesting but very practical, in all aspects of life and for all paths young students might choose to take, professionally or academically, to confront statistics now.
Teach students how to interpret statistics. Common fallacies. Common methods used to manipulate people with statistics.  Teach them the questions they need to be asking.  That skepticism and critical thinking they should have.
Basically, I think you should teach about the maths of life. Logic, statistics, the principles of critical thinking... from a mathematical perspective.  Game theory included. The math of democratic voting, even. There is a lot to learn about everything wrong with American democracy, for example, and plenty of never-attempted reconciliations backed by mathematics.
A: Cryptography or fractals/(concept of $\infty$)
