Wednesday, July 9, 2008
"Any Way You Slice It" - A Classic Cube Dissection Problem to the Nth!
The following series of questions was inspired by a recently released SAT question. The first two levels are appropriate for middle or secondary students. Level III requires more algebraic background or strong visualization skills. I could have attempted to include a graphic for some of this but I'll leave that to the experts out there!
LEVEL I
A cube is cut into 8 equal cubes by dividing each edge in half with three planes which are parallel to the faces of the original cube. Show that the total surface area of the 8 smaller cubes (when separated) is TWICE the surface area of the original cube.
Note: Most secondary students would attempt this algebraically or substitute particular values. To develop spatial sense, encourage them to find another solution, which is purely visual and elegant! Middle school students (or younger children) would greatly benefit from constructing a physical model of the cube from modeling clay (or something equivalent) and slicing it with appropriate tools. Better yet, one can avoid slicing by constructing the bigger cube from 8 smaller cubical blocks (There are many sets of plastic or wooden blocks available from catalogs).
LEVEL II
A cube is cut into 27 equal cubes by dividing each edge into 3 equal parts with planes parallel to the faces of the original cube. Show that the total surface area of the 27 cubes is THREE times the surface area of the original cube.
LEVEL III
Generalize the above relationships by dividing each edge of a cube into N equal parts with planes parallel to the faces of the original cube (N is an integer greater than 1). State a conclusion and explain! Again, try to find both an algebraic and a visual explanation.
Posted by Dave Marain at 6:58 PM 9 comments
Labels: dissection, geometry, spatial sense