As always, students can get directly started with this warm-up. My hope that is that by starting the exploration of cubic and polynomial functions with a context that students are somewhat familiar with, students will be able to use their prior knowledge to help get started.
I wasn’t sure whether or not students had memorized formulas for surface area and volume. Rather than provide these formulas, which I think takes away the chance for students to rediscover them or remember them in context, I posted these definitions with the projector, and when students asked me how to calculate them, I referred to these.
Some students built several cubes and counted in order to fill out the table. I did not discourage this--to me, MP1 is all about using whatever strategies make sense to you to figure things out. I did not push these students to find the formulas today, unless they wanted to. Other students were able to immediately find the formulas without thinking too much about the cubes.
I had students work on all the levels today, and many students wanted to spend more time to figure out the patterns in Level D. I was happy that these problems were engaging for them, and all of the problems more or less got to my goal of having students generate polynomial functions to fit patterns. My vision was that if students created the functions themselves, they would be interested in figuring out how these functions worked in the long run throughout this unit.
The Painted Cube Problem is a great problem which offers students the opportunity to think both abstractly and concretely. The problem is so rich and useful that I want all students to understand it to varying degrees. For students who are least able to think abstractly, I hope that they learn to visualize the problem and to complete enough rows of the data table that they are able to see different numerical patterns. For students who are able to think more abstractly, I want them to be able to find a function rule for each column of the data table and to explain how these rules relate to the cube. Finally, for more advanced students, I created different versions of the problem involving rectangular solids with different proportions.
To get the problem started, I tell it to students out loud, while displaying different images of cubes being dropped into paint, along with this animation. My goal is for students to understand the problem in any way that works for them, and then I ask them to figure out how many cubes there will be in each category (painted on 0 sides, 1 side, 2 sides, and 3 sides) for a 3-by-3-by-3 cube. Once students have figured out answers to this question, I ask them to expand this to a 10-by-10-by-10 cube. I had copies of the data table available for students who needed this scaffold as they were working (this was most of my students--some were able to set this up themselves, but this problem offered so many other cognitive demands that I didn't want to overload them.)
I found that many student attempted to build really big cubes--like a 6-by-6-by-6 which ended up taking a really long time. While I didn't want them to spend the entire class period fussing with blocks, I also didn't want to discourage any approaches. I tried to suss out which students seemed to be just wasting time (such as advanced students who weren't thinking about the problem at all, just building with blocks) from students who really needed the 3D visual in order to think about the problem.
It was important while framing this problem to tell the students that they would have several days to solve it, so they didn't have to rush, but could think thoroughly about the relationships in the problem.