The Locker Problem

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SWBAT work in groups to solve a famous math problem.

Big Idea

Group problem solving comprises half the final exam in this course. Today's lesson is a high-spirited introduction to what I'll ask students to do.

Opener: New Random Groups

3 minutes

The last three class periods have been open-ended work periods in which students have worked individually to complete assignments of their choice.  Within that structure, there has been plenty of impromptu group work and conversations as students discuss their work and help each other out.  

There will be a half-dozen or so similar days as the year comes to a close, but today's lesson is a break from that.  Today, we're going to focus specifically on group work and problem solving, which are as fundamental to success as any algebra skills my students have learned this year.  Today's activity is a preview of what students will do on one part of the final exam, and we'll do this at least one more time between now and the exam.

As students arrive, I tell everyone that today we're taking a break from individual work time, and I post this Random Group Generator on the front board.  This is an Excel spreadsheet that I use to group students randomly.  The name of each student in class is written in the "Name" column, where for now, you see the letters of the alphabet as placeholders.  Of course, I have to adjust the length of the list and the number of groups to the size of the class.  I use as many groups as necessary to ensure that everyone will work in a group of 3 or 4.  (There are advantages to groups of either size.  Depending on the size of the class and how productively I know students to work, I'll adjust to have more groups of one size or the other.)

I tell students not to get too comfortable because they're going to be placed in a new group right now.  Then I tell everyone to watch as I randomly sort the class, which is done by clicking the sort button on the "Rand" column.  It's a brief opportunity to show students another feature of Excel, and if anyone is interested, I'll show them the "RANDBETWEEN()" function that I used to make this.  I ask for a volunteer to tell me how many times I should sort the list, and then I press the sort button that many times.

"Find your group quickly," I say, "and decide where you'll sit.  Then I'll tell you about the problem you're going to work on today."

Set Up & The Problem

10 minutes

When a group finds each other and a place to sit, I provide one sheet of 11x17 ledger paper (a ream of which I recommend heartily as a classroom resource - useful in so many ways!).  I say that when everyone is seated with their group, I'll explain the task.

Students will attempt to solve the famous Locker Problem today.  There are no visual notes.  I tell the class that I'm going to share the instructions verbally, and that it's up to everyone to pay attention to the details.  Watch this video to hear how I provide the instructions to the class.

When I'm done sharing the problem, throughout which I answer any and all clarifying questions, I tell the class that from here on, I'm only going to answer "yes" and "no" questions.  I also point to the colored pencils (another indispensable resource!) and rulers available at the front of the room.

Check out some student work to see how they record the problem.  This example is pretty much word for word, even though they didn't record that last detail about each subsequent student.

As I note at the end of the video, I'll scaffold this problem for some groups of students, by posing two questions for getting started.  Here's an example from a group in a class where I first asked about the 10th and 100th lockers.  Notice also that we talked about student #1000, because a student asked, "So will the 1000th student only touch the 1000th locker?" to which I was able to respond, "Yes!"

Group Work Time

30 minutes

From there, the remainder of the period is for students to work.  I walk around to answer yes and no questions, which always leads to some funny conversations as students try to game the system.  Once they get started, however, their search for an easy way out gives way to genuine curiosity and a drive to push themselves and their group to figure things out.

I usually have to try hard to keep my mouth shut - mostly because I'm delighted at what I see and hear - but I make sure not to jump into any conversations.  The most I'll do is point to something on the page and say, "ooh I like that," or "that's interesting," particularly if I think a group should pay more attention to something they've already written down.  I'll also nod my approval as I listen in on a great conversation.

Here's an example of why it's hard to stay quiet.  Spend a couple of minutes with this group's work to see the great ideas they're starting to have.  What I see makes me want to dance around, because they're so close to getting it.  Can you see where?  Like I said, I'll work hard to stay quiet.  If I see the group get stuck, I'll point to the middle of their page, just to the right of the crease and ask what they've noticed...

After students have worked for 10 minutes or so, I tell them that when there are 15 minutes left in class, I'm going to provide a hint.  By the time I share the hint, some groups have already had the idea I'm about to share (like this group, who focused on the first 20 lockers) and I start by saying, "I saw that Ben's group did something like this, and I'd like everyone to think about this now."

The hint is to try solving the same problem for a smaller number of students and lockers.  I'll tell groups to try to solve the problem as if there are only ten students with a total of ten lockers.  I call out any groups who have tried the problem with 10, 20, or 30 lockers, I commend them on a good idea, and I thank them for letting me steal it.  I might also provide the hint visually - note that this hint is for an even smaller version of the problem, just five students.

The other way I provide hints is by answering some yes/no questions for the entire class to hear.  On this student work, for example, you'll see two additional "hints" both of which came by way to student questions:

  • "Can we do this with real lockers?"
  • "Do the lockers that are open have something in common?"

Finally, as with everything else we do, this can take more than today, if kids choose.  Some groups will be very close to getting it as time runs out, and they'll be excited to ask if they can continue tomorrow.  I think that's a great use of time, and I'll certainly let groups go that route if that's how they want to spend it!