# Summer PD: The Fair Share Task

For those of you who think I spend the summer sipping pina coladas on the beach, you’re right—I do. Don’t be jealous. However, I also spend some of the summer going to and running professional development.

I had the chance to work with some amazing teachers this week on the fair share task I talked about last time. Here’s how it went.

Warm Up : Today’s Number: Write or draw ¾ in as many ways as you can.

I started with this for two reasons. One was because I wanted to start a conversation about what types of tasks help engage students at the start of class. I introduced High Yield Routines as one place to get ideas from.

The other reason was that it allowed me to assess what the group’s understanding of fractions was before we went on to the fair share task. I charted out the responses the group shared out:

Pretty impressive, no? We had lots to talk about in terms of the math and in terms of using this type of activity with kids.

After some mandatory discussions about the Danielson framework, we got to work on the task. We started with reading the problem together and making a prediction. The predictions gave me an idea of whether the group understood the task. I also was able to clarify some parts of the task, like the sandwiches being equal sizes. However, next time I think I’d change this to a notice/wonder protocol.

Teachers then worked on the task in groups of 2 or 3. Some struggled with how to start the task. I encouraged using pictures. I also had some linking cubes and paper and scissors for those who wanted to use them.

I walked around as everyone worked. An interesting misconception came up when trying to figure out how much each student ate:

One group said that when sharing 3 subs among 4 people, each person would get 3/12 of a sub.  I tried to recreate the picture they drew below:

After the group explained why it was 3/12, I asked them to make a new picture that showed the reverse in order to prove their answer works. If four people each ate 3/12 of a sub, how much did they eat all together?  I left them to work this out and when I came back, they changed their answer.

This misconception led to an interesting conversation about the whole. I posed the question: each student did eat three-twelfths, but 3/12 of what? A discussion about the whole had come up earlier during the warm up when someone said they we could represent three-fourths with money and use 3 quarters. Another teacher brought up the idea that when we talk about money, 3 quarters could be 3 wholes or three-fourths depending whether the whole was a quarter or a dollar. This connected nicely to our conversation here about whether one sandwich or three sandwiches were the whole.

I love the idea of using the chart as a way to prove. Here’s an example:

Gallery walk: The plan was to do a gallery walk and have everyone walk around and note whether the other groups solved the task in a similar or different way and why. Because of time and because we didn’t have a large group, we shared out the charts. An interesting conversation came up here as a group shared one of their mistakes and how they realized their strategy of how they drew the sandwiches needed to change. It was so helpful to have their thinking visible.

I shared how I often tell students they don’t have to use pencil in class. And if they do use a pencil, they should NOT use an eraser. This is always a controversial statement but the reason is that I want to see what the student is thinking at all points of the problem solving process. Plus, it encourages us to all make mistakes.

I’m still looking through the reflections that were filled out by the teachers, but I wanted to capture my reflection first. I loved the task and I think it allowed different learners to access it. Of course, I ran out of time for the rest of the activities I had planned, but I think it was worth it.

Here are some questions I’m wresting with right now:

• What’s the best way to group teachers? One participant felt uncomfortable with her math skills and being in a group of “stronger” math students didn’t help this. I had to come over several times and check in and at one point redirect the other members of the group so that her and I could chat about the math without the other group members telling her what to do. There are also teachers who do not want to work in groups and would rather work alone. I’m not sure how to address these grouping issues. Thoughts?
• How do I help teachers generalize from this task? What did we learn about engaging students in problem solving tasks? How do we transfer what we learned from this experience of doing math to teaching math?