## Sunday, August 24, 2014

### "What Do You Mean, There's No Right Answer?!?"

Wanting to start the school year off with a bang, I decided to introduce my students to my classroom culture through a mathematical modeling problem that I did not have an answer to.  The Laptop Battery Task from Illustrative Mathematics (www.illustrativemathematics.org) is one that I've used with teachers during professional development, as well as with students, and I like that it makes everyone uncomfortable.  Teachers and students alike have something to gain from engaging in this problem, and other problems of its kind.  Students are uncomfortable because this is possibly their first encounter with a math problem where their teacher doesn't know the answer, and teachers are uncomfortable with presenting a problem that is so open ended.  It's scary stuff!!!
On the first day of school, I thought I would take a risk and see how my students would handle The Laptop Battery Task.  I went over several things on the first day, laying groundwork for classroom culture, and with 15 minutes left in the 90 minute period, I put this task in front of them.  I asked them to work independently and brainstorm a solution for when Jerry might have a fully charged battery. Students exited the class without discussing their work and their homework was to continue working on the task.

The next day, as students entered, I asked them to take out their work, being thoughtful to not use words like "answer" because I wanted to emphasize that this was a work in progress and their work could be revised.  Students began bouncing ideas off of one another, comparing strategies and initial solutions.  It became immediately clear that no two students arrived at the same answer, or if they did, they got there in different ways.
As I circulated the room, I recorded times that students predicted Jerry's battery to be fully charged, as well as what methods students used to arrive at their solution.  I then wrote the times on the whiteboard and students saw that there was a large range of solutions from 10:28 to 11:23.  As far as a teacher move, at this point I wanted to make sure that students saw the two main approaches that students took in solving this problem (graphing and average rate of change), but that we also looked at student work from an individual on the 10:28 end, one on the 11:23 end, and one in the middle somewhere.
I selected students whose work was to be shown to the rest of the class, but it was the task of their partner to articulate their reasoning and process.  I took this as an opportunity to build collaboration and a culture of being responsible for understanding the reasoning, not just a passive audience member.  It seemed to work very well!
As the first two students presented, the class saw that graphing would have been a useful option/tool to solve this problem (only one or two students in each class chose to graph), but also that even though their own solution did not match exactly, their reasoning was very similar to those of their peers.  The final student presentation was of the solution of 11:23.  The time of 11:23 didn't sit well with the class, but they had a difficult time articulating why.  They knew in their gut that 11:23 was too late (even the students who had this solution knew something was off), but I loved seeing the perseverance and the unwillingness to back down, as opposed to admitting defeat in arriving at an "incorrect" answer.  What came next was simply magical.
The partner of a student who had a solution of 11:23 came to the document camera and displayed the work for us to interpret.  Students were instantly engaged and curious to know how their peer arrived at 11:23.  The student work showed proportions, which many students had used, so there was a certain timidness for students to question the solution.  Essentially, this is what the student work showed:

Students were able to make sense of the proportion and found the setup to be quite useful.  They understood the meaning of the 132 minutes, but they still weren't buying the 11:23 final charge time.  Finally, one student raised their hand and asked the question, "But why are they adding the 132 minutes to 9:11?"  Silence.  Nobody answered...they just looked at me, waiting for me to answer.  I took this opportunity to have them turn to each other and brainstorm their answer to that question.
Ultimately, we came to the understanding that THIS was the issue with the 11:23 charge time.  The 132 minutes of total time made sense to them, but they grappled with where that fit in the context of this problem.
A student asked if they could come to the board to draw something that they thought might help make sense of this particular issue.  After picking my jaw up off the ground, I said of course and handed over the whiteboard marker.  This is what she drew:

Without me saying anything, students looked at the image and began digesting its meaning.  Students were arguing, the discussion was getting interesting.  Finally, I asked if someone could articulate their connection to the diagram and how it helped them understand the problem?  The original student who arrived at a solution of 11:23 raised her hand and gave a thoughtful response to how she had added the 132 minutes to 9:11, which would have meant that the battery charge was at 0% at that time, which was not true.  That was the crux of her issue, and she was able to work through it without me saying a thing.  Other students articulated their thoughts of where the other 66 minutes should go or even proposing another proportion we could set up from here.  They were making sense of the argument of a peer to help build their own understanding.
Once the excitement had died down a bit, I brought closure to the discussion by highlighting some of the elements of this lesson.  I even asked students to share with me some of their observations of the process as a whole, and they said things like, "You didn't lecture on this topic before giving us this problem", or "You placed a lot of emphasis on how we got an answer, not on the answer itself".  Bingo.
Of course, it would have been too good to be true if it ended there.  Before excusing them for the day, I had a student raise their hand and ask, "So, what's the answer?"  With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged).  Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class.
Mission accomplished.