After a long weekend, some of my Geometry students moved on to Investigation #2, but others wanted to continue working on the challenges I posed to them on Friday in the whole-class test. They a chunk of the period discussing several issues, including the status of objects that you get when you glue two “ordinary” polyhedra together—like a pair of triangular prisms, as pictured above. Alternatively, the object can be seen as a cube with an “extra face” cutting both it and a pair of its opposite faces in half.

Is such a thing, considered as a whole, a polyhedron? Is the object even changed by the addition of that extra face (or three)? How many faces would we even say this has? Should the triangle pairs that lie flat in a plane together be counted as separate faces? Even though there’s not even a bend there? This led us back to the question of whether a triangle—if looked at a certain way—can be seen as a quadrilateral with one 180° angle. Also lurking here is the question of whether there can be a corner, edge, or face to a shape even if you can’t see it, and this undermines our sense that we have any idea of what’s going on with shapes.

So many questions. So much ambiguity. I love it!

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