The goal of this lesson is to have students explore what it means to have an identity element, what it means for an element to have an inverse, and what it means for a system to be commutative or associative under a binary operation. These concepts are typically introduced using integers and the four basic operations of addition, subtraction, multiplication, and division. Students are often told that a system is commutative if they arrive at the same value when the integers are reversed. For instance, they see that 2 + 3 = 3 + 2, so the operation of addition is commutative, but 3 – 2 ≠ 2 – 3, so the operation of subtraction is not commutative. When taught this way, a deep conceptual understanding of commutativity is often not developed, because the examples rely on self-evident facts that students have accepted for many years.
In addition, this lesson allows students to see that a relationship is not commutative using a symmetry argument. The table below shows that the operation of addition is commutative because there is symmetry across the main diagonal.
By comparison, students can use the Algebraic Transformations applet to create a similar table. The table created with the applet, however, will not have symmetry across the main diagonal, indicating that the two transformations with a plus sign are not commutative.
This lesson moves the student from a familiar environment to one that requires an investigation of a well-defined yet unknown operation, and the concepts of identity, inverse, commutativity, and associativity are examined.
Each of the four basic operations — addition (+), subtraction (‑), multiplication (×), and division (÷) — is well defined. For example, the + symbol represents addition and is defined as the sum of a and b.
For this lesson, the symbol # will be used to represent an operation on a rectangle. The operation a # b indicates that one move of the rectangle (a) is followed by a second move (b).
Have each student cut a rectangle from an 8½" × 11" piece of paper to make a model of a shape sorter. (The last page of the Rectangle Activity Sheet can be used to create a rectangle and shape sorter.)
The first step is to determine the different moves that can be performed on the rectangle so that the end result appears to have the same orientation as the original and can then be passed through the rectangular hole in the shape sorter in the same way as the original rectangle. The moves occur about lines or points of rotational symmetry. A rectangle has two lines of symmetry and 180° rotational symmetry. The moves that can be done and the labels that are used to identify them are as follows:
| N | do nothing
|
| ½ | one-half turn in a clockwise direction
|
| V | rotation about the vertical line of symmetry
|
| H | rotation about the horizontal line of symmetry
|
Have students label the upper end of the rectangle with an N to represent the "do nothing" move.
The other labels are then placed on the rectangle to mark the top front after a move has been made from the original position. For instance, the V is at the same end as the N but on the other side, indicating a rotation about the vertical axis; the graphic below shows a rotation about the vertical axis (red dashed line) and makes clear why the V appears on the back side of the rectangle behind the N:
The other labels are placed similarly. The label ½ appears at the end opposite the N, indicating the top front after a half-turn (180° rotation) about the center of the rectangle, and the H occurs at the end opposite the V, indicating a rotation about the horizontal axis.
A correctly labeled rectangle will look like this:
Define the operation a # b to be one move of the rectangle (a) followed by a second move (b). Instruct students that each time the operation is performed, they must begin with the rectangle in the original position (with N at the top front), perform the first move (a), and then perform the second move (b).
It may be necessary to give students an example, such as ½ # ½ = N, because two half turns return the rectangle to its original position:
You may wish to show additional examples, such as
V # ½ = H
and
H # ½ = V.
Be careful, however, not to show too many examples; there are only 16 possible combinations, and you want to leave some of the discovery for students.
As you show examples, also give students an understanding of how to read the statements. For instance, read ½ # ½ = N as "a half turn followed by a half turn is N," and read V # ½ = H as "a vertical flip followed by a half turn is H." Students will start to realize that the symbol # indicates the "followed by" operation.
Students should perform two moves in succession and note the label that appears at the top front of the rectangle after both moves. Results should be recorded in the 4 × 4 table on the worksheet. (Note the organization of the table. Second moves occur in the same order as the first moves. This organization allows students to discover symmetry in the table when the results are analyzed later in the lesson.)
The table should be completed as follows:
| |
2nd Move |
1st
Move |
# |
½ |
V |
H |
N |
½ |
N |
H |
V |
½ |
V |
H |
N |
½ |
V |
H |
V |
½ |
N |
H |
N |
½ |
V |
H |
N |
|
When students are ready to move on, conduct a class discussion that focuses on the questions in the Questions For Students section below. Effective use of those questions should solidify student understanding as well as correct any misconceptions.
