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Czarist-era Russians who had to take long trips by sled, particularly at night, would often gather up several large chunks of meat in a sack before starting out. If the sled happened across a pack of wolves, the driver would throw out the meat a piece at a time in the hope that the wolves would stop for a few moments to fight over the food.
Even the best teachers have the best teachers will have a Russian sled moment now and then, when the wolves are circling your desk and searching diligently for your last nerve, so it's always a good idea to keep a few sacks of wolfmeat on hand just in case.
Hex – Probably my first choice for a "here, do this" moment. A fast, simple strategy game with a great pedigree. You can easily fit two hex boards on one side of a sheet of letter paper.
Checkers – Don't disrespect the lowly checker. Players who have mastered both chess and checkers often argue that checkers is the more challenging game, particularly the Spanish version with long jumps. A few cheap chess/checkers sets are a great classroom investment.
Chess – There are two contenders for the world's most popular game, Chess and Go, but only one is available for five dollars at your local discount store.
Chess Board Games – A chessboard is probably the most versatile playing surface ever invented. There are countless games that can be played on all or part of a chessboard. Here are a few good ones to start with:
Pencil and Paper Games – The only thing wrong with pencil and paper games is that people usually play the wrong one. Tic Tac Toe is the least interesting member of a distinguished family of row games (click here, here, and here for a few examples) many of which can be played with pencil and paper. In addition to row games there are Nim, Tac Tix, Dots and Boxes, Sprouts, Hangman and the very entertaining Racetrack.
Kruzno – Wolfmeat Born and Bred
Kruzno was designed and developed in the classroom to be an entertaining fast-paced game that would keep students thinking, but it was only of many games that can be played on a hexboard. After the chessboard, hexagonal tiling supports more games than any other pattern. Hexboards are a vital part of a math teacher's bag of tricks. You can get a hexagonal game set here or you can make your own hexboard, but remember to make sure that your board is at least 16" by 16".
Teaching across the Curriculum -- Yes, it's a buzzword, but as buzzwords go it's not bad. Here's a table to help you get started.