Spring break is next week for us. Algebra finished up multiplying polynomials and after the break we will start factoring. Today, students worked through one of my favorite problems The Locker Problem, which is out of Fostering Algebraic Thinking by Mark Driscoll.
The Locker Problem
There are 20 lockers in one hallway of the King School. At the start of the school year, the janitor closed all the lockers and put a fresh coat of paint on the doors, which are numbered from 1 to 20.
When the 20 students from Mrs. Mahoney’s class returned from summer vacation, they decided to burn off some energy. They came up with a plan: The first student ran down the row of lockers and opened every door. The second student started with locker #2 and closed every second door. The third student started with locker #3 and changed the state of every third locker door. The fourth student started with locker #4 and changed the state of every fourth locker, and so on, until all 20 students had passed by the lockers.
Which lockers are open after the twentieth student is finished? Which locker or lockers changed the most?
I like this problem so much because it has a very low point of entry and investment for students. I had each group work on the problem by themselves for about 10 minutes then they broke into groups.
The problem also has various exit points depending on the direction each group decided to take. I had some students looking at differences of squares while others were using pictures, and another group was looking at factors. Either way, all students were able to engaged in the problem.