There is also a lot of good probability theory and combinatorics in family poker games. At the beginning of 6th grade, we discovered mathematics competitions. The best contests are those which require creative problem solving and which draw from a broad gamut of mathematics rather than just memorizing mental arithmetic tricks. MathCounts for middle schoolers, and the AMC exams leading to the International Mathematics Olympiad for high schoolers are good examples. There are books of problems and solutions from prior year's contests to help students prepare.
Similar contests exist in other areas such as computer programming, chemistry, and physics.
Learning mathematics by preparing for challenging contests is a little like learning a foreign language by living in the country. You are learning everything at once without having material compartmentalized into chapters or courses. Useful math concepts, like useful phrases, come up again and again as you go about your business of solving problems or surviving in another culture. At first, you do not understand much of what is whirling around you. But as you see more and more problems, and hear more and more phrases, more of it becomes comprehensible.