Tutoring Small Groups of Children (for money)

If their homework involves tedious memorization, introduce them to spaced repetition (begin with physical flashcards).

My uninformed opinion is that games should be a good medium for teaching young children. (I would be very interested in hearing opinions about this from people who actually work in education.) I think a surprising amount can be learned by carefully inspecting even a game as simple as Minesweeper. There are various "lemmas" you can prove in Minesweeper about how various configurations uniquely pin down the location of mines or non-mines and I suspect this could be an accessible introduction to mathematical proof (the advantage being that all of the relevant proofs can be done by analyzing a finite number of cases). I haven't actually tried this though.

What's your level of math and science knowledge? Can we recommend that you teach the kids group theory?

Also, what's the goal for the kids? You'll want to teach different things depending on whether they want to keep up with schoolwork, skip ahead in schoolwork (and hopefully proceed to skip grades), or just learn more about math and science. Fir the first two you'll need to cover a known curriculum, for the latter you need to avoid that curriculum like the plague of boredom.

For math resource, you might check out Vi Hart's videos and A Mathematician's Lament.

For science resources, check out museums :P Also, I highly recommend pH paper as a powerful experimental tool an 8 year old can use. Lots of cool chemistry experiments out there. For biology, growing plants and growing germs on agar and taking nature walks with a book about trees and making Punnett squares are good activities. Kid physics is mostly like engineering - making telescopes from lenses and water bottle rockets and freezing rubber bands on dry ice and having a competition over who can make something out of paper and tape that will catch dropped eggs from the highest height.

I drew on How to Solve It for teaching bare bones rationality skills to grade and middle schoolers.

TBH the curriculum is easy, the real trick is getting the kids to perceive you as high status so they actually pay attention.

If you are being paid to improve their grades, warning them against password-guessing is a breach of contract.

If you are being paid to educate them, it's not really low-hanging fruit, although it is a very tasty one. Most people are more likely to learn the wrong lesson (beware of trick questions) than the right lesson.

Materials-wise, I can't recommend the Murderous Maths books by Kjartan Poskitt enough. They're what got me into maths, and introduce topics at a basic level while leaving the top end open for further development. They've got lots of fun puzzles and activities. I think they're aimed at 10-14 year olds, so would be towards the top of your tutee's age range.

Less paper-based and more free is NRICH (http://nrich.maths.org/), which is run by a group at Cambridge (I've done some holiday work writing questions for them). It contains materials for all ages (pre-school up to sixth form! (4-or-5-ish to 8)) and is intended as simultaneous for teachers and keen students. It's got a good selection of mathematical games as well as themed monthly questions, and keen students can type up their solutions and submit it to them for a chance of being featured on the website. The games in particular will be a good way of introducing lots of mathematical concepts, since that's how they're designed.

In fact, teaching your kids basic game theory is probably a good idea. It teaches you them to think rationally and is easy to motivate (beat all your school friends!). Many rationality concepts and techniques are founded upon the basics of game theory!

One interesting kid science experiment is to dissolve an egg's calcium shell in vinegar. Naked egg experiments can make a good demonstration of osmosis. (Osmosis comes up a lot in cryobiology, so it is of considerable interest for cryonics.)

Thank you for trying to impart useful, compounding knowledge, even when selling opium is almost certainly more lucrative in a middle-class neighborhood.

Use fun experiments to teach the scientific method instead of trying to impart on them a superficial understanding of chemistry and geology (in the case of the volcano).

For math, try to address any form of mathematical anxiety. I think that is more important than whatever knowledge you could teach them, but if you can make it engaging I recommend introducing some logic and naive set theory. Combinatorial problems are also easily illustrated with physical objects and can serve as an introduction to probability theory.