This set of problems make a really great task because students can begin at the most concrete and basic level by simply building solids and literally counting, using the working definitions of volume and surface area. At the higher end, students can attempt to write function rules for the volume and the surface area of the solids without even doing any calculations at all. There are many different ways in which students can engage with these problems and this gets students to be the ones actually doing the practice standards--they ask if there are shortcuts, and they ask if they can make generalizations. The answer is--yes. I just tell them, "See what you can figure out." Sometimes they want some help figuring out how to think about the generalizations, but the point is that they are the ones asking for them.
Students can come into class and get started immediately. I put interlocking cubes and isometric dot paper on each table and tell them to have at it.
I also display on the projector the definitions of volume and surface area. (This document also includes the lesson closing.)
This extension is very interesting and really fun. I had many students arguing at lunch about how to find the function rule for the volume of sequence 3. Many students also connect these problems to the linear and quadratic tile patterns we did at the beginning of the year, which is awesome. I give these problems to any students for whom the basic surface area and volume problem set will be too easy or not challenging.
I explain to students the big metacognitive idea here: I cannot actually monitor everyone's progress all the time. The only way that they are going to maximize their learning each day is by tracking their own progress. I provide this list of sub-skills and components of these problems, and ask them to think carefully about the progress they have made in each of these areas.
I explain clearly that an expert in the surface area and volume functions we are exploring this week would be able to do all of the skills listed here, and that they will have the entire week to master these skills. Then, I ask them to write a brief reflection.