It's the final week of classes, and we'll revisit each of the lesson structures that I've put in place over the last few weeks. There will be more student-directed work time happening in the classroom, and a final online practice session in the computer lab. Today is the second of two group problem solving activities. Half of the final exam will follow a structure similar to this.
The set up for today's group problem solving session is just like it was last time. As students arrive, I tell everyone that today we're doing another round of group problem solving, and again, I post this Random Group Generator on the front board.
Students will wonder if they're working with the same problem solving group as last time, and I tell them that, no, we're going to re-shuffle the class now. I tell everyone to watch as I randomly sort the class, and I ask for a volunteer to tell me how many times I should sort the list. I press the sort button that many times, and everyone knows what to do. I like how the computer helps to ensure that these groups are indeed random, and pushback is very rare from students.
"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."
Like last time, groups find each other and a place to sit, and I provide one sheet of 11x17 ledger paper. I say that when everyone is seated with their group, I'll reveal the task.
Students know the drill, because they've already done this. They can only ask yes or no questions, they must show me everything they can on the ledger paper, and midway through the work session I'll give a hint. There are colored pencils and rulers available for everyone who needs a hint.
My delivery of the problem is different. Last time, I only said the problem aloud, with no visuals. Today, I'll post the problem on the board while saying as little as possible. I'll only provide help in the form of my answers to yes/no questions and in the hint I provide.
Here is today's problem, which I encountered while working with Math for America. It's fun, and I recommend giving it a try before you read further. Don't worry, I won't give anything away in the next section, but if you'd like to approach it with a clear mind, now is your chance. After playing with the problem on my own, I knew I had to put it front of my students, and today was a perfect chance to do so!
I think that a good problem is easy to get started on and hard to solve completely. This one is probably more of both than the Locker Problem - easier to start, and harder to solve. That's ideal for the purposes of today's class. I want kids to have a real chance to PLAY! It's the kind of problem that everyone can jump into, and it's fun to watch what kids do.
The most common sort of questions students will ask are about whether or not "this train is ok" or whether or not two trains are the same. I'm happy to answer these questions within reason, but I make it clear to each group that I'm not going to stand there and answer dozens of such questions. It's appropriate to wonder if the train consisting of 11+1 works for our purposes (no, it does not - there are no 1's allowed - which means, by extension that, yes, 11 rods are out of the question) or whether 3+3+6 is the same thing as 3+6+3 (no, they're not, and each should be counted, as stated in the problem).
Students work, I circulate and do my best to keep my mouth shut, and I experience a lot of joy in seeing how well students can work together and hearing their conversations.
With about 20 minutes left, I give a hint similar to last time, about trying some smaller problems first. There are actually a few approaches for applying this strategy, and I usually start by addressing short trains. "How many trains of length 1 can we make?" None. "How about length two?" One, and so one.
You can also start by counting all the solutions that include a rod of length 12, then all that include a rod of length 11, and so on. Or, you can consider all trains consisting of exactly one rod, two rods, or three, etc.
I won't share the solution here, but it is satisfying, so go for it! I've also included some student work here, but at the risk of saying too much, I'll withhold my detailed assessment of it. For now, I'll just say that a lot of it makes me proud.