I have been interested recently in the mathematics lying just under the surface of middle school math problems (see also “How many ways to roll a 10?” ). I enlisted a former student, Ben Harste, to help me write an article on such an investigation. The premise is below; until and unless it is accepted for publication, you can read the full article by downloading it from my Papers page.
In the Connected Mathematics unit Filling and Wrapping, students find all of the rectangular prisms that can be made using whole number side lengths and 24 unit cubes. Then they find the surface areas of these prisms and notice that surface area can vary greatly for prisms with fixed volumes.
Commonly, teachers have their students conjecture how to tell which of two prisms will have a smaller surface area based on appearances alone, and a frequent observation is that the closer a prism is to being a cube, the smaller its surface area will be (in comparison to other prisms of the same surface area). This property could be called cubeyness.
It is easy to judge the relative cubeyness of a 1 by 1 by 24 prism against a 2 by 3 by 4 prism. But what about when two prisms appear similarly cubey? How can we measure cubeyness without calculating the surface area? And furthermore, how can we compare the cubeyness of two prisms that have different volumes (so that, for example, a 2 by 3 by 4 prism would be just as cubey as a 4 by 6 by 8 prism, or so that the cubeyness of a prism does not depend on the units we use to measure it?)
These turn out to be much harder questions than they appear to be on the surface, and answering them calls into question a lot of what many teachers know about measurement, statistics and geometry.