**The Locker Problem**

Last week, we went off on a tangent in our grade 5 classroom at YIS. It happens a lot in a PYP classroom. We were searching our class Twitter account @yis5b and ran across a posting from some fellow 5th graders in New York City. They were working on a math problem about 100 lockers, and it involved factors and multiples, which is our focus at this time.

So, I decided to set my students up for an inquiry into this problem. My initial thoughts were that I would give my class the problem, give them some time every day to work on it, and see what happens. I would ask questions along the way, observe and ask them to keep reflecting on their work. I had no idea what the answer was when I began except that there was a pattern. I’d listen to the students and work it out along with them. I had no idea how far the problem would go.

Here is the site where we found the problem online. Thank you for sharing, Jee, on “Teach to Inspire.” Basically, the problem is that there are 100 lockers. The first person opens every one of the lockers. The second person opens every 2nd lockers, or multiples of 2. The third person starts at 3 and reverses every 3rd (if open, then closed), the 4th person hits every 4th locker and reverses, and so on until the 100th person finishes. **The problem is to figure out how many lockers will remain open after 100 students have opened or closed them and why**. **That was the first question.**

**Our Inquiry Takes its First Twist:**

Just as we were about to start our journey, Craig, @CraigDwyer, wrote from Tohoku International School in Japan and wanted to collaborate on a math problem. So, I thought this problem would be perfect for differentiation for several of my students. We discussed with four of our “needing a challenge” students, and they were enthusiastic about collaborating on Skype to solve the problem together. We had 45 minutes of time that the four of them could solve the problem together.

**Many Ways to Solve a Problem **

While the two from my class who were going to collaborate on Skype worked on creating an i-movie about a different puzzle they had solved, the rest of the class began their inquiry into the “locker problem.” I grouped students in mixed ability groups, based more on language skills and ability to communicate their thinking. Already in our class, we’ve focused on showing our thinking–visually and orally. Students know an answer isn’t good enough, and that it’s more about the process.

Off the students went. Without any tips on how to go about the problem, one group took out colored tiles and started building a row of 100 yellow tiles. They had 100 blue tiles in reserve so they could make a visual change every time the locker changed from open to closed. I liked that one. If I had been in their shoes, I probably would have done something similar because I’m a visual learner. When I asked the students why they had chosen that method, they said they all like to “see” things. Visual learners.

Other groups armed themselves with whiteboard markers, hand sanitizer (for erasing) and 100 charts in front of them. They started by crossing out every even number (because of the 2nd person closing every 2nd locker). They proceeded to 3, 4, 5, 6, etc..and it started getting messy. A few groups were writing down what they were doing, but many had forgotten that they needed to show their thinking by the end. They quickly wrote their names on the 100 charts, wrote a few things down in their notebooks so they could come back to it. The group with the tiles took a photo of their tiles so they could redo the next day.

I felt bad about stopping them on a problem, but I figured it was reality. Most of the time, I never have enough time in one sitting to finish something (even this blog post).

**Returning to the Problem**

Day Two. Students took out all of the materials again. The visual learner group started laying out their tiles again. Other groups stared at their messy 100 boards. A few erased the entire board and started again. **My question today for the students was: Is this the most effective way to solve the problem?**

A few groups with whiteboards said yes, and they kept crossing and erasing their crosses as they proceded through the problem. They were looking for visual patterns, but not finding any.

One group decided that crossing out and deleting wasn’t helping, so they moved on to paper. By now, one girl had figured out it was all about factors of a number, so she started writing x’s and o’s on paper for each number. 4, for example, was opened (o) then closed by 2 (x) because 2 is a factor and then opened again by 4(o).

The group with the tiles on the floor reflected and said it was taking too long, and they needed another way. Seeing that everyone else was using whiteboards, they tried the whiteboard method for the rest of day two. I liked how they were able to reflect and change plans.

**Getting Closer to the Answer **

Day Three: By the third day, one group had found the answer through methodically going through each number. They showed me a whiteboard of square numbers to 100–open. The girl who had started writing every number with x’s and o’s realized also that it had to do with square numbers. **My question for both of them, since they were correct, was “Why are only square numbers opened?”** Both groups started discussing and working on that next challenge.

The group who had moved from colored tiles laid out in a row to whiteboards were now in the hallway posting post-it notes on lockers–marking the lockers which were closed. I had shared some ideas from the other class, who were also working on this problem, and they liked the idea of seeing the real lockers with post-it notes. I again complimented them on their flexibility and thinking skills–how they could change when needed.

**Skype Collaboration**

And the two students from my class got on Skype and met students from Sendai for the first time. The students from Sendai were grade 6 students, and they too were needing a challenge. Craig and I set them up with the problem, some expectations about collaborating and then let them at it. We talked about how they could communicate by talking, showing on whiteboards and 100 charts and could chat through Skype chat if needed.

The collaboration went great. My students were quick, and Craig’s students were thoughtful and reflective. Within 10 minutes, they had solved the problem and were able to explain why. While my students shared their thinking and the Sendai students shared theirs, they also asked each other questions to clarify. They bounced back and forth. Craig quickly jumped in to move the problem to 1000 lockers and how that would look. Through questioning, they all were able to explain themselves.

**Reflecting**

By Thursday, students either had the answer, wanted to work more, or were tired. A few groups kept doing the same thing they had done on day one. I took some mental notes of these students. They needed some prompting to move on and to stay with it and probably would in other tasks as well. Other groups were writing up their process in color on a poster. We had a class discussion **with the question: How does looking for an answer and not finding it better than finding the answer. **

Students said: We had to learn different ways to solve the problem. We had to learn how to communicate better (the Skype group). They asked us questions that helped clarify our thinking. We explained our thinking. We had to learn to record our thinking.

**The End**

Day Five: Students were finishing up their posters on the process. Others were explaining to groups how they got the answer. **My question for them today was: How did it feel to work on one problem all week?**

Their response:

- hard
- boring
- weird
- uh…really fun but hard.

They said it was the first time they had spent so long on something. I understand their answers. This is new for them. They’re 10. They like bee-bopping around. These are the kids who, when on the computer, are on 5 different things at once. Sure, I wanted them to say this was the best experience they’d ever had, but I did see growth. I also was able to observe where they were developmentally in solving harder problems. One girl spent a lot of time walking around and needing direction. She didn’t know where to begin. Others were so thoughtful and quick to adapt. The group on Skype really got some good differentiation, and I was glad they weren’t in the classroom. They went through the same steps as everyone in the classroom except at lightning speed, and for the rest of us, that might make us feel bad. And they really felt special, important, challenged.

Overall, it was a great exercise. I think we’ll come back to final reflections about their posters, but because they’re 10, I know it’s time to move on.

I love the way students were engaged in solving the problem, the diversity of pathways and reactions.

If you teach in a PYP school I think you should send the link to this article so it can be published on http://blogs.ibo.org/sharingpyp/ . Thanks for sharing it!

I love that you were able to collaborate with another class through Skype! I haven’t been able to do this problem this year, because my new school uses Everyday math and I haven’t had time to incorporate it into the curriculum. I’m hoping that next year I can do the problem with my students! If you are interested in collaborating with my class sometime, please let me know! I teach 5th grade at Singapore American School!