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.
Algebraic Transformations Applet
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
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.)
Rectangle Activity Sheet
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
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:
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.
1. Ask students to answer the following question in their journal: From the original position, (N), a first move is made. Describe what happens to the rectangle if a second move is made that is the inverse of the first move.
2. Ask several questions to ensure that students understand the chart
and how it represents the # operation. You may ask questions such as:
- What is the inverse of H?
- What is (H # V) # H?
3. Observe student participation during the group discussion about
identity, inverse, commutativity, and associativity. Use a system to
randomly select students to answer your questions. Much can be learned
about students’ conceptual understanding during class discussion. In
particular, use the class discussion to review pairs that students
think are not commutative and triples that students think are not
1. What can you say about the relationship between the possible first
moves followed by the second moves for the rectangle and each row or
column of the results in the table?
[Each move is represented as the result of a first move followed by a
second move in each row and column of the table. No move is repeated in
any row or column.]
2. Examine the completed 4 × 4 table of results. Identify any lines of
symmetry or points of rotational symmetry. Make a conjecture relating
the results in the completed table to an operation being commutative or
[The diagonals of the table are lines of symmetry. In addition, there
is 180° rotational symmetry about the center of the 4 × 4 table. These
observations lead to two conjectures: 1) If both diagonals of the table
are lines of symmetry, then the operation is both commutative and
associative; and, 2) If there is 180° rotational symmetry about the
center of the table, then the operation is commutative and
Questions for Students
1. Is there an identity move in the system? An identity move is a move that does not change the position of the rectangle. Identify any move that does not change the position of the rectangle.
[The "do nothing" move is the identity move for the # operation. Following any move by an N move does not change the orientation of the rectangle.]
2. Does every move have an inverse? A move followed by an inverse move returns the rectangle to the original position, with N at the top front. Do a first move and then determine if there is a second move that can be done to return the rectangle to the original (N) position. What can be said about the inverse of each move?
[Each move is its own inverse.]
3. The commutative property states that the order in which an operation is performed does not affect the outcome. For instance, the operation of addition is commutative because 1 + 2 = 2 + 1 and, in general, a + b = b + a for all a and b. Is the # operation commutative? That is, does a # b = b # a?
[It can be shown that, in every case, a # b = b # a. For instance, ½ # V = V # ½, because both result in H.]
4. The associative property states that the way the objects are grouped when an operation is performed does not affect the outcome. For example, the operation of addition is associative because (1 + 2) + 3 = 1 + (2 + 3) = 6 and in general a + (b + c) = (a + b) + c for all a, b, and c. Is the # operation associative? That is, does (a # b) # c = a # (b # c)?
[Note that (½ # V) # H = ½ # (V # H), because (½ # V) # H = H # H = N and ½ # (V # H) = ½ # ½ = N. This suggests that the operation may be associative, but a full proof would require checking every possibility. Keep in mind that if any sequence does not hold true, then the operation is not associative.]
- What learning styles does this lesson address?
- How did you address the transition between known operations —
addition, subtraction, multiplication, and division — and the new
- How did the students perform in relation to the stated behavioral objectives?
- What advantages are there in presenting mathematical properties in an atypical setting for your students?
- What were some of the ways your students illustrated that they were actively engaged in the learning process?