The Cornbreak Task has been on my to-do list for quite some time. It first came to my attention last year when I saw the @illustrativemath team (including Kristin Gray @mathminds, Jennie Beltramini @JennieBeltro, and Alicia Farmer) present their implementation of this task at NCSM 2016. I was impressed by the level of problem solving and math application required of students by the task, but even more impressed by the implementation, reflection, revision of this task by the @illustratemath team. You can see the teaching video here and the reflection video here.
As luck would have it, I was invited to model a task lesson in a fifth grade classroom who was currently studying fraction multiplication. Jackpot!
In preparing for this task, I decided to adopt the 5 Practices model for planning. For me, this can be quite challenging, as I am a meticulous lesson planner. However, Smith and Stein (the authors of 5 Practices for Orchestrating Productive Mathematics Discussions) state that a task can viewed as planned improvisation. Setting a goal and anticipating student responses are the first of the 5 practices, while the rest of the task planning will happen on the spot. For that reason, my lesson plan looked like this:
Game time! To get our fractional thinking started the class began with counting by fourths and noting as we made a whole. We spent some time thinking about all the patterns that existed in the recording of our count, including the diagonal pattern of the “wholes” and the difference of 5 fourths in each column.
Next, it was time for some story problem theatre. I asked the class to close their eyes while I told them the story of a school that needed help with their cornbread fundraiser. During this time, I read the introduction to the task twice, asking students to visualize as I read. After the second read, I had them discuss with their partner what they knew about the fundraiser. As a class, we came up with this collective information:
Once we agreed on what we knew, I set the students off to work independently answering the questions. The only parameter I gave was that their solutions must include pictures, numbers, and words. As they worked, I monitored the room (Practice Number 2), looking for evidence of student understanding. Here are some of the things I found:
This student showed evidence of partitioning a whole into fourths, but wasn’t sure how to model the 1/3 of the fourth that remained in the pan. This was a common issue among the students.
7/12 was a very common answer, as many students sensed a need for a common denominator, but weren’t sure what to do with it once they had it. I hadn’t anticipated that they would add it! I will be sure to add that to my anticipations for the next time I facilitate this task. However, a strength in the representation is that you can see the 1/3 of the 1/4 in the model. A great progression from the former representation.
As I monitored, I noticed there were a few in the room who presented the correct solution with a correct model and the selecting (Practice 3) and sequencing (Practice 4) were beginning to take shape in my mind. Under the document camera, both solutions above were presented by the respective students, as the class listened and worked to make sense of these two different ideas. After each student presented, I asked students in the class to repeat the information that had been shared with their partner. I then asked students to discuss what they noticed and wondered about these solutions. From this talk, a student raised his hand to ask why they had chosen a tape diagram rather than an area model. Neither student could articiulate the reason for their model choice, so the questioning student asked to share his model, which he had done with folded patty paper.
This student said that because the pan was square, the area model seemed the closest to the picture in his mind. He then explained how he showed a third of a fourth and how he determined that each square was worth $1. At this point, many heads began nodding and I knew we were on the right track.
Next, a student raised her hand and asked, “Wait. Isn’t this a multiplication problem? Since we need 1/3 of 1/4, doesn’t that mean we need to multiply to find our answer?” Again, nods of approval swept across the room.
As I recorded the student’s thinking, I asked students to look at the equation and notice and wonder once again. After some partner discussion, a student offered the idea that you can multiply the numerator with numerator and denominator with denominator in order to find the product. I recorded the thinking as he spoke:
At this moment, I saw the final chapter of this story nearing an end. Students had grappled through some tough application of their fraction knowledge, both finding success and struggle. Through this jouney, students pressed forward, carrying each other along to our final destination. The goal of the lesson had been to help students see that (a/b) × (c/d) = ac/bd. To see why the rule worked. I was thrilled with the sense making these students had engaged in to understand this concept.
I knew that I was going to enjoy this task, but I was thrilled to see that the students were just as excited as I was to solve the problem.