I wanted to find ways to keep students engaging with the big ideas of this unit. I am trying to develop a curriculum where students move towards abstraction really slowly, by providing them many opportunities to think concretely about actual problems, and then letting them push themselves to more abstract thinking.
My hope with this warm-up was to get them to continue thinking about the idea of generating data based on a concrete representation, and then trying to find patterns and rules for this data.
I had many blocks available, and I also provided lots of isometric dot paper for students to draw the solids as well. During my first rounds of circulating, I asked students if they wanted me to show them how to draw solids on the dot paper. My hope was that all students would find some way to fill out the data table themselves, without just waiting for other people to share their answers.
As students started to look for rules, I asked them to think about how to represent their calculations algebraically. It was nice to see students looking for vertical patterns in the data tables, and noticed that there were linear and quadratic relationships. They also realized that the volume column showed neither of these types of relationships, which is what I wanted them to start to think about.
I used the closing as a chance to show this image of the painted cube to all students, because I think it can be quite helpful for students at different points in solving the problems. For some students who are still struggling to understand the problem, the image can help them visualize the situation. For students who already understand, the image can help them generalize to create a formula.
I leave the questions open-ended, but this makes it really important to hold students accountable for giving substantial and thoughtful answers. I often have students write things like “Everything” and “Nothing” as answers to the first two questions. To set a higher standard, I explicitly tell students that these are not satisfactory answers. One strategy to address this is to simply stand at the door and read students’ exit tickets as they leave. This can be a bit time-consuming, so I do it more as an audit, just checking to make sure that they wrote something substantial, and I send them back to write more extensive answers. Another strategy is to grade exit tickets periodically so that students realize that they are accountable to meeting the expectations here.
The purpose of exit tickets like these is to help students become aware of their own progress and to see that they don’t need to fully master each problem each day. As much as possible I try to explicitly say these things over and over again—I want them to hear the message clearly that the emphasis is on making progress each day, and on identifying areas of growth.