Two of my favorite problems: Einstein's Puzzle and the Locker Problem

As school is now in full swing, I am finding that I don't have time to blog as much about the "play" and the "eat." But truthfully, the thing that brought me to blogging is the "math." I was going to skip blogging last week because I didn't have the "perfect" eat play math combination...but I have heard if you have a blog, you should blog at least once a week...so if I have to skip a little eating and playing, so be it!!

This week, I want to blog about two of my favorite problems/investigations. I just wrapped up teaching the first unit in my Algebra 2 Honors class, and I decided to take two days on our special schedule days to allow students to "play." I think they were relieved that they could attempt some difficult "famous" problems and not be graded on them. Here was their homework the night after their test...with a 40 minute time limit:

Einstein's Puzzle.

Students came in very excited to present. One came to the board and got stuck, and then another came to help him finish. We crossed out everything we used until we realized we needed to write all remaining possibilities on the board. Once they realized through process of elimination that there was only one possibility for one of the houses, everything started to fall in place. One student said she solved hers by guessing at one point, going the wrong way, and then finishing by choosing the other method. Explaining to each other in class was hard for the kids to do, but it is early in the year, and they will soon come to realize that explaining to me is not what matters, it's explaining to those who do not yet understand, and getting them to see the light that matters! After all, that's the best part of teaching!!

Once the explanation was done, I handed out the next investigation. Students who did not get Einstein's riddle the night before were extra determined to solve this puzzle. Students went up to the board in pairs to try to solve this puzzle for about the last 15 minutes of class.

The Locker Problem:

On a recent problem from the first day of class (I will have to blog about that another time), the problem solving method was "the method of exhaustion"...writing down every possibility. Obviously that was not the best method here. So the only hint I gave them was to start with a smaller, simpler problem and generalize a pattern. Some asked how many should they start with, and I told them it was up to them.

I walked around, watching students and advising them if they made a mistake, but not telling them much. Finally, a group discovered the answer. I did not want them to give it away to the others, so we talked quietly...but the next, perhaps more important question was WHY  did these locker numbers that remained open. Another group got the pattern, and again, I asked why...then the bell rang. The assignment for the night was, for some, to find the pattern, and for all, to find why the pattern. Again the next day, the students who discovered the answer were excited to share.

SPOILER ALERT: DO NOT READ IF YOU WANT TO SOLVE THE PROBLEM YOURSELF. It turns out that lockers are touched by the number of factors the locker number has...and the pattern that remain open are perfect squares...because perfect squares have an odd number of factors. We talked about why that is...for locker 6, for example, the factors are 1,2,3, and 6...and so they pair up...1x6 and 2x3 = 6. But for locker 4, the factors are 1,2,4. The factor of 2 repeats, which results in an odd number of factors, and this only happens with perfect squares. An extension could be which lockers are only touched twice? (the lockers that are prime.)

Finally, each student was given an index card with a locker number on it and they "acted" as lockers...opening and closing by turning away and toward the front of the classroom so any who did not understand could visually see which lockers remained open.

We ended class with a few more brain teasers...most notably Crossing the river with a goat, a wolf, and a cabbage. Students acted it out, and they were very cute. Their homework was to come in with their favorite (or newly discovered) brain teaser that we will solve throughout the year.