In reading through the activity, I found that it lacked a tie-in into the real world. For example, living near Puget Sound, the idea of surface-area to volume ratio is very important with regard to hypothermia. I never ask that question until students have done a science experiment with the science teacher on heat conductivity through different diameter metal rods. I normally prefer sugar cubes, but we have already worked with sugar cubes this year once, and I wanted their experience to be different. If students even think they are doing something again, they will not be engaged.
Variety is so important.
This activity may take a few class periods, but the idea is to start noticing patterns. I do not like to use the word cube with cubes until the students discover a need for the word. But before I do that, the students need some experience with the blocks, the word cube and box. Along the way I can ask questions to get students to notice things like prime numbers, volume, and rotation of boxes in space. It is the same type of thinking as area under the curve in calculus.