Illuminations: Algebraic Transformations

Algebraic Transformations

Unit Overview

Lesson 1

Lesson 2

The Plus Sign and Triangle

Using a scheme similar to the one in the Rectangle lesson of this unit, students will further explore the concepts of identity, inverse, commutativity, and associativity. Students investigate and analyze moves performed with a plus sign, and an online activity is available to help students with this investigation.

Learning Objectives

Students will:

Determine the identity element for a geometric operation
Decide if there is an inverse for each move
Determine if the operation is commutative or associative

Materials

Plus Sign activity sheet
Triangle activity sheet
A cardstock version of the Plus Sign (optional — see below)

Instructional Plan

Building on the Rectangle lesson, the goal of this lesson is to have students further explore the concepts of identity, inverse, commutativity and associativity.

The first shape in this lesson is a plus sign — the intersection of two rectangles meeting at a right angle — as shown below. This shape has four lines of reflective symmetry as well as 90° rotational symmetry.

For this lesson, the symbol * will be used to represent an operation on a plus sign. The operation a * b indicates that one move of the plus sign (a) is followed by a second move (b).

Allow each student to cut a plus sign from an 8½" × 11" piece of paper to make a model of a shape sorter. (The last page of the Plus Sign activity sheet can be used to create a plus sign and shape sorter.)

Plus Sign Activity Sheet

The first step is to determine the different moves that can be performed on the plus sign, so that the end result has the same orientation as the original. To make sure this is possible, students should use the remaining paper after the plus sign is cut out as a "shape sorter." Tell students that the shape should be able to pass through the sorter after moves are made. Each possible move occurs about a line of symmetry or around a point of rotational symmetry.

If you have done the Rectangle lesson with your students, then in this lesson allow them to discover the possible moves. They should have no difficulty identifying ½, V, and H, because they are the same moves as for the rectangle, but they may have difficulty with the ¼, ¾, R, and L moves.

In total, there are eight possible moves. The moves are as follows:

N do nothing
¼ one-quarter turn (or 90°) in a clockwise direction
½ one-half turn (or 180°) in a clockwise direction
¾ three-quarter (or 270°) turn in a clockwise direction
V rotation about the vertical line of symmetry
H rotation about the horizontal line of symmetry
R rotation about the right diagonal
L rotation about the left diagonal

The rotations about the right and left diagonals may be particularly difficult for students to identify. You may need to show them an example, such as the following. The animated drawing below shows a reflection over the right diagonal:

The sequence of pictures below shows a reflection over the left diagonal:

Allow students to work together to determine how the plus sign should be labeled. When labeled, the plus sign will look something like this:

Define the operation a * b to be one move of the plus sign (a) followed by a second move (b). Instruct students that each time the operation is performed, they must begin with the plus sign 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 some examples, such as ½ * ½ = N and ¼ * V = L.

You can also have students use the Algebraic Transformations applet to investigate this operation. (Note that the labeling used within the applet may be different from the labels that your students chose for the plus sign shape.)

Algebraic Transformations Applet

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 ¼ * V = L as "a quarter turn followed by a vertical flip is L." Students will start to realize that the symbol * is a binary operator and indicates the "followed by" operation.

To divide the work, allow students to work in groups to complete the chart on the activity sheet. Students should perform two moves in succession; they should then note the label that appears at the top front after both moves are completed. (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 later analyzed.)

The table should be completed as follows:

2nd Move

1st
Move

*	¼	½	¾	N	H	V	R	L
¼	½	¾	N	¼	R	L	V	H
½	¾	N	¼	½	V	H	L	R
¾	N	¼	½	¾	L	R	H	V
N	¼	½	¾	N	H	V	R	L
H	L	V	R	H	N	½	¾	¼
V	R	H	L	V	½	N	¼	¾
R	H	L	V	R	¼	¾	N	½
L	V	R	H	L	¾	¼	½	N

After students have explored operations with the plus sign and triangle, conduct a class discussion. Use the questions in the Questions For Students section below to keep the conversation focused. You may need to review the terms identity, inverse, commutative, and associative before beginning this discussion.

Questions for Students

Is there an identity move?

[Yes, the N move is an identity element.]

Does every move have an inverse? What can be said about the inverse of each move?

[The ¼ and ¾ moves are inverses of each other. For the other moves, each move is its own inverse.]

Is the * operation commutative? That is, does a * b = b * a?

[In most cases, but not all, a * b = b * a. For instance, ½ * V = V * ½, because both result in H. However, if any case does not hold true, the operation is not commutative; and, ¼ * H ≠ H * ¼ because ¼ * H = R but H * ¼ = L.
Therefore, the * operation is not commutative.]

Is the * operation associative? That is, does
(a * b) * c = a * (b * c)?

[It can be shown 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 complete proof would require checking every possibility. Unfortunately, there are 8³ = 512 possibilities to be checked! To expedite the process, you might want to have every student randomly choose 10 different combinations and test them. Although this would not constitute a full proof, it would give enough data to make a better conjecture.
As it turns out, the * operation is associative.]

Assessment Options

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 plus sign if a second move is made that is the inverse of the first move.

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?

Allow students to investigate the moves for another shape, such as a triangle. As with the rectangle and plus sign, have them record the results for combining two moves. They can use the Triangle activity sheet for this assessment.

Triangle Activity Sheet

Students should be required to identify the six possible moves on their own, and they should label the triangle something like this:

As with the rectangle and the plus sign, students should compile a chart of all possible combinations of moves. Students should notice that the chart does not have reflective symmetry, so operations on the triangle are not commutative; however, operations on the triangle are associative.

Extensions

What can you say about the relationship between the possible first moves followed by the second moves for the plus sign and each row or column of the results in the table?

[Each move occurs exactly once in each row and each column of the table. No move occurs more than once in any row or column.]

Examine the completed tables 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 associative.

[The diagonals of the table are not lines of symmetry nor does the table have rotational symmetry. These observations lead to two conjectures: 1) If the diagonals of the table are not lines of symmetry, then the operation is not commutative; and, 2) If there is no rotational symmetry about the center of the table, then the operation is not commutative. Both of these conjectures could be explored and proven true or false.]

Teacher Reflection

What learning styles does this lesson address? How does this lesson address auditory, tactile and visual learning styles?
How did students respond to the * operation? Did the lack of familiarity cause confusion, or were students intrigued by something new?
The concepts in this lesson are usually presented with the more familiar operations of addition, subtraction, multiplication and division. How might the method used in this lesson help students learn the concepts?
How did you deal with students who presented combinations that they thought were not commutative or associative?

NCTM Standards and Expectations

Algebra 6-8

Relate and compare different forms of representation for a relationship.
Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Geometry 6-8