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 1” 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 – 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 4” 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 – 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 haven’t 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).