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The first suggestions were games of NIM, which is a game played with stones. Any sort of counter will do and they don’t have to be uniform. Basically the game is played by making a pile of stones then picking up a number of stones in turn eventually forcing your opponent to pick up the last stone. Rules can include putting the stones in various sized groups and picking from one group at a time. Having a minimum and maximum number of stones that can be picked up, or really anything you can think of.
The second suggestion was playing “20 questions”. The answer can be as simple as a number and increase in difficulty such as rules or functions, to equations of lines, or just about any sort of concept in math. Imagine guessing a number but not being allowed to ask if it is higher or lower.
When I teach 8th grade math I basically like to make sure my students can recognize each function from the graph, the equations, and the table. So this fits in nicely. Actually anything we define in terms of properties should, theoretically, be a good answer for a 20 questions game. The game can and should be a vehicle for teaching students how to think critically about the properties of an object.
The last suggestion was Bizz Buzz. I’ve played Buzz a lot, which is a simple game. The rules are: students line up or sit in a circle and count up saying Buzz when they reach the number or its multiple. Bizz Buzz is a variation using two numbers and their multiples. Too add even more difficulty try using numbers from different bases. After playing this in the classroom a few times I increased the difficulty one my time by asking students to say Bang when they reach a number that is a common multiple. Playing with factors and common factors should also work.
I might also recommend ideas such as http://www.mathinyourfeet.com/ which I think is a great method to learn math. Creating patterns of dance or stomps with your feet.
I was also talking to a music teacher a few weeks ago. He was trying to teach his students the relationship between fractions and notes using the old pizza method. I suggested he stay with what is natural and use the timing of the notes. Whole notes, half notes, quarter and eights are fractions of time not pizza. Sustained notes are simply adding fractions. Students would obviously practice with their instruments, but drums can be easily created. I would assume that difficulty could be increased with various time measures.
If you have any other suggestions please add them to the comments below.