Geometry of the Cube
Tonya C. Brooks
Goal: To better understand the symmetry of the cube
Day 1: The first day should be set aside to review symmetry of two dimensions, making sure to discuss rotations about different axes and mirror reflections across a mirror as well as classifying the symmetry of shapes. Find the axes and mirror lines on an equilateral triangle and determine that the equilateral triangle has classification D3. Using the mirror lines and axes of rotation for the equilateral triangle, figure out the multiplication table for the equilateral triangle. Do the students see any patterns? What patterns do they see?
Let the students work on finding the axes of rotation and the
mirror lines for the square. What classification would we give the square? Let
students work on multiplication tables for the square. Do they see any patterns
in this table? What do the students notice about the properties of
multiplication in working with the symmetries?
Day 2: Pick up with a discussion on what the students found for the
square. Follow this with an activity based on finding the 35 different
hexaminoes possible. Allow students to work in groups using 1Ó square graph
paper to find the different forms that they can make using 6 squares and having
each square butt up against at least one other square. If we take the
hexaminoes and fold along the edges, which ones give us a closed cube?
Day 3 Ð 4: The students should have determined which hexaminoes
give a net for the cube. Allow the students to choose a net and make a larger
cube (preferably two) so that they end with a cube whose sides are at least 4Ó
long. Once each student has one or two cubes for themselves, set the students
into groups to work on finding all the different two-dimensional shapes that
arise when the cube intersects a plane. There are several different shapes, and
students might find them easier if given other manipulatives such as clay to
model with or cube models that they can cut apart.
Day 5 Ð 7: Begin working on the different symmetries of the cube.
Allow students to determine axes of rotation and mirror lines, as well as the
degrees of rotation for the different axes. They can relate several of these to
the square that they worked on in two-dimensions. There are 24 of these. There
are also 24 other symmetries, sometimes called turn-reflections. These are a
little harder for students to see due to the fact that they are a combination
of both a rotation and a reflection. You might help the students with these by
giving a few hints or showing one using a tetrahedron. If they havenÕt gotten
around to figuring out these symmetries by the end of the second day, you might
want to show one on the cube to see if that helps them. You want the students
to find all symmetries of the cube by the end of the third day.
Day 8 + : Allow the students to work together on figuring out the
multiplication table for the cube. This is a very intense activity that
requires quite a bit of manipulation of the cubes. It is definitely easier to
allow the students to work with two cubes. When I worked on this, I found it to
be easier to keep everything straight if I labeled the sides, edges, and
vertices on my two cubes so that I could see before I manipulated one cube and
after I did the manipulation.
The students should look for patterns within the multiplication
table for the cube. What do they get when they do (rotation)(rotation)?
(rotation)(reflection)? (reflection)(rotation)? Others? What is an easy rule
that they can use to determine what each multiplication will give?
Following this, other activities (or assessment activities) could
include extending these ideas to a rectangular prism (or rectangular box).