One of my favorite activities to use in the math classroom or as a MathCounts activity, is the locker problem. I believe I first found this problem when I was looking for riddles and puzzles to display at a booth for the University of Arkansas Math Club when I was president as an undergrad. If you google the locker problem you can find many resources but I generally just tell the kids the scenario and ask them to solve.
So… the locker problem:
At a school, there are 100 students and 100 lockers. All of the lockers are closed currently but the first student is going to go along and open each locker. The second student is going to go along and shut every second locker (2,4,6,8,…). Now the third student will go along the lockers and for every third locker (3,6,9,12,…) they will shut the locker if it is open and open it if it is closed. Now the fourth student will go along to every fourth locker (4,8,12,…) and open if closed and close is open… and so on. All 100 students do this. How many lockers are open after all 100 students?
I think what I love most about this activity is watching the kids work. Sometimes I put them in groups for a while before discussing it with them or helping them. I have done it with the whole class at once, assigning students to be student 1, student 2, student 3 and so on with at least 20 lockers drawn on the white board. Sometimes the students write O and C for open and closed, sometimes they do X and O, but they all walk through doing their job and the class discusses what they are noticing. Not telling the students to act it out this way also leads to the students coming up with some really creative ways to try to solve the problem or work it out. Great discussions happen and its so fun to watch.
Many times I have done this with kids who do not know much about square roots and perfect squares and it always leads to a good ah-hah! moment! I have also done this activity with college freshman in a college algebra class and they struggled more than my 7th graders!