This past week was our TAKS week, and therefore I only saw some of my precalculus classes, and so I could not move on, and I did NOT want to have "free days". One day, I taught them about binary and counting and adding and subtracting and multiplying in different bases.
I made the connection to base 10 where you can think of each position as a bin where you drop marbles. You start filling them up from right to left. The capacity of each bin is 9, and once you try to put one more in a bin, it would overflow, and so you have to scoop the 10 you want to put in there out and place one marble in the next bin over to the left to designate a higher power of 10.
So we started counting in base 4. Each bin has capacity 3. We did the "scooping" thing. We counted to 20.
1, 2, 3,10,11,12,13,20,21,22,23,30, ..., 103, 110.
We made the connection that each bin represents (from right to left: 4^0, 4^1, 4^2, 4^3, ...). We did other bases. We then converted between base 4 (say) and base 10. And between base 10 and another base.
I also recalled a version of a game I played in 8th grade, that I still play when I need to occupy my mind and have time to kill.
Pick any 4 single-digit numbers. Then number your page from 1 - 10 (as one example), and you have to use every one of your chosen numbers once and any math operation to get all 10 numbers listed. I had them play and the first one done would get candy. Boy were they quiet and working. I'm SURE it was the math and not the candy.
For example: 5,2,8,9
1. 9 - 8/2 - 5 = 1
2. 2 + 5 * (9-8) = 2