Race Games
Who will finish first? Many classic children's games, like "Chutes and Ladders" or "Sorry!", involve moving one's pieces around a pre-determined course; the player who first arrives at the goal wins. A terrific math-oriented race games is the German game "Rechenpiraten" ("Arithmetic Pirates"). The very first game I designed for our home school math class - and still my favorite - is called "Race to Altair."
Simulations
Simulation games can be highly complex or quite simple. For example, "Monopoly" is a simple simulation, while the "Railroad" games are much more involved. Typically, simpler ones are more appropriate for kids.
The best math simulation games involve real-life situations that children can relate to, like the innumerable variations on the "Store" theme. This lets children not only practice arithmetic, but also become familiar with money. My version of this is "Cha Ching! Salesman of the Week."
Strategy Games
Strategy games are among the most ancient and popular games. The basis of most strategy games involves capturing enemy pieces while defending one's own. This includes chess and checkers, modern classics like "Risk," "Stratego," and "Axis and Allies," and also more recent games like "Heroscape."
The most math-intensive strategy games are Role Playing Games like "Dungeons and Dragons," but these are probably too complicated for kids in this age range (5-10).
If you want an explicitly math-based strategy game for kids, you'll probably have to design one yourself. I made up one that we like quite a bit; it's called "Space Battle." My son came up with the idea for "Catman vs. Daredoggie," which we still play often.
Cooperative Games
Cooperative games are nice for creating a relaxed and - not surprisingly - cooperative mood among the players. Younger kids can find the competition inherent in more traditional games (especially race and strategy games) emotionally trying. Because they are so invested in games, it's hard for them to be good winners and losers. Here is where cooperative games shine!
In completely cooperative games, players all work toward a common goal. For instance, in "Shadows over Camelot" players all belong to the Round Table and complete individual quests for the good of the whole kingdom.
Other games combine both cooperation and competition. For example, in the immensely popular "Settlers of Catan," players compete in building cities, town, and roads. The brute-force approach rarely succeeds. Rather, the winning player often trades resources with others, avoids building his cities too close other players', and generally tries to minimize conflict.
"Shadows over Camelot" and "Settlers of Catan" are known as "Euro games" because they were designed in Europe. There are many cooperative Euro games, and these are also among the most aesthetically pleasing games. The pieces are generally wooden rather than plastic, the art work is incredible, and all the components are extremely durable. If you are interested in Euro games, you should keep an eye out for games that won (or were nominated for) Germany's "Spiel des Jahres" (Game of the Year) award, which is "Academy Award" equivalent for board games.
"King Richard's Ransom" is my foray into the cooperative-game genre.
Card Games
Several card games are easy to use in math class. Some, like the old family favorite "Casino," use addition as well as matching for capturing cards (e.g., a seven can capture a five and a two, or a four and a three). An excellent, addition and subtraction-based card game with a mythological theme is "Zeus on the Loose."
Although it's not a game I designed, I've included a variant of the solitaire "Pyramid" in the Game Ideas section.