Here is something I worked out while teaching math classes as a substitute teacher. It works well on 4th graders through advanced or gifted classes.
I start with showing two ways to add on their fingers. The first is an abacus approach with the thumb counting 5 and the fingers counting 1 on 1 hand, on the other the thumb counts 50 and the fingers count 10. Holding up the fingers and adding them up can show any number less than 99.
The second way is the maximum number that you can count on your fingers (there's another proof that the advanced class can work on). The first finger counts as 1, the second counts as 2, 3 is both first and second fingers held up and counted, 4 is the third finger (I say "Don't laugh to head off any disruptive behavior), 5 is the 3rd finger and 1st finger, 6 is the second and third finger and so on. This is the binary number system, with the first finger being 2^{0}, second finger is 2^{1}, third finger is 2^{2} and so on.
Now I tell the following story:
A king and a gambler were playing poker one night and the king has lost all he could lose. He said, "That's it, I quit."
The gambler said "One more game. I'll bet all you lost against this: take a checker board and on the first square put 1 grain of wheat, on the second square put 2 grains, on the 3rd square put 4 and double it for each square. Of course the king lost and the next day he found out he did not have that much wheat in his kingdom. How many grains of wheat did he lose?
The fourth graders are happy to write 1 on a sheet of paper, copy the number under it and add the two together, copy that number and add those numbers, until they get a number that goes across the paper.
For the advance class I tell them that they have all the information to give me the formula for a checkerboard of n squares. Each figure is one more than the sum of all the fingers less than it. And that gives X = 2^{n}  1 (that's X=2 to the n power minus 1) where n is number of squares on the board.
