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Difference of Squares

  • Lesson
6-8
2
Algebra
Unknown
Location: Unknown

This activity uses a series of related arithmetic experiences to prompt students to generalize into more abstract ideas. In particular, students explore arithmetic statements leading to a result that is the factoring pattern for the difference of two squares. A geometric interpretation of the familiar formula is also included. This lesson plan was adapted from an article by David Slavit, which appeared in the February 2001 edition of Mathematics Teaching in the Middle School.

The idea of algebra as generalized arithmetic involves structural understanding of arithmetic operations when acting on arbitrary quantities. This aspect of algebra requires a general understanding of arithmetic that is grounded in a series of computations, then abstracted into a single idea. In any prealgebra or beginning-algebra course, plenty of opportunities exist to discuss this kind of algebraic reasoning, because factoring patterns are abstractions of arithmetic operations.

Teachers can design tasks that allow students to develop both a knowledge of common factoring patterns and related symbolic manipulations and an understanding of algebra as generalized arithmetic. The following equation is the basis for designing such tasks:

Concrete Computations + Thinking = Algebra as Generalized Arithmetic

The activity described in this lesson shows how this transition can be accomplished in middle-grades classrooms.

Using the Difference of Squares 

Ask each student to do the first three steps individually and without sharing with their neighbors. Step 4 will be discussed after some experimenting with various numbers.

Step 1: Pick any two consecutive numbers.
Step 2: Square each, and find the difference of the larger square minus the smaller square.
Step 3: Add the two original numbers.
Step 4: Explain why Steps 2 and 3 give the same result.

One possible reaction when students complete Steps 2 and 3 is often amazement, followed by confusion. Allowing students to communicate with fellow students is an invaluable tool. Students who are comfortable with variables and symbolic manipulation usually solve the problem using an appropriate algebraic equation, for example,

(n +1)2 - n2 = (n2 + 2n + 1) - n2 = 2n + 1 = (n + 1) + n

The figure below illustrates how a student's computations may lead him or her to develop a symbolic explanation. This example revolves around the numbers 3 and 4, from which one can derive a "general computation" using x and x + 1.

1581 fig1

A solution such as that in the above figure can set the stage for a discussion of the power of algebraic symbols; however, you may wish to delay this discussion while other students explore the problem using a more arithmetic approach, one that is also at the heart of algebra as generalized arithmetic. This approach usually involves repeating steps 1-3 for several pairs of consecutive numbers.

The figure below shows a sample of the results of this approach after the students have organized their data. Invariably, several students try to find a pattern in the various computations that they used to test the validity of Step 4. The students often have some interesting observations, but they usually cannot adequately explain the underlying reasons for the similarities in the results of the computations contained in Steps 2 and 3.

42 - 32 = 16 - 9 = 7 4 + 3 = 7 
52 - 42 = 25 - 16 = 9 5 + 4 = 9 
62 - 52 = 36 - 25 = 11 6 + 5 = 11 
102 - 92 = 100 - 81 = 19 10 + 9 = 19 

Because further investigation of patterns may be necessary for these students, pose a similar task which is designed to supply scaffolding for the students' thinking. However, instead of using two consecutive numbers, use two numbers that differ by 2 (e.g., 4 and 6). Their investigations may lead to information similar to that in the previous figure. If necessary, repeat this process for numbers that differ by 3, but many students will attempt this exploration on their own.

42 - 22 = 16 - 4 = 12
 
4 + 2 = 6
 
12 ÷ 2 = 6
 
52 - 32 = 25 - 9 = 16
 
5 + 3 = 8
 
16 ÷ 2 = 8
 
62 - 42 = 36 - 16 = 20
 
6 + 4 = 10
 
20 ÷ 2 = 10
 

At this stage, the students are in a position to cross the algebraic divide. Encourage a discussion in which students examine a holistic view of all the computations and attend to the patterns found in them. Ask students what specific observations they made that allowed them to express the general rule. In response, such comments may include "The second one was the sum of the numbers times 2, so I guessed that the third one is the sum of the numbers times 3."

The discussion can continue using such language, and the students can discover the general pattern in the computation. At times, students may invent their own language to discuss their discoveries. Students may use the notion of "start-off numbers" to explain how they understood the general computational aspects of the difference-of-squares factoring pattern.

1581 figure4 

Although algebraic understanding has emerged, the need to express the ideas with formal algebraic symbols may still exist. If so, ask the students to use variables to write the results of the three investigations. Their equations for each investigation, respectively, may resemble the following:

Case 1: When b = a – 1, then a2b2 = a + b

Case 2: When b = a – 2, then a2b2 = 2(a + b)

Case 3: When b = a – 3, then a2b2 = 3(a + b)

Appropriate discussion or student insight can guide students to realize that the coefficient outside the parentheses in those three cases is equal to the difference between a and b. That is, a - b is equal to 1, 2, and 3, respectively. Eventually, this may lead students to form a conjecture about the general factoring pattern:

a2 - b2= (a + b)(a - b)

Slowly and with guidance, the students make an algebraic leap. Further, they require very little knowledge of symbolic manipulation to produce the expression; the construction is grounded in their arithmetic experiences.

  • Scissors (optional)

Extensions 

  1. Find a prime number that is one less than a cube. Find another prime number that is one less than a cube. Explain.

    Solution: Any number that is one less than a cube can be represented algebraically as x3 - 1. Consider the factoring pattern
    x3 - 1 =  (x -   l)(x2 + x + 1). This number is prime only if either one of the factors is equal to 1, but x2 + x + 1 cannot equal 1 for any value for which x > 0. The only way for one of the factors to be equal to 1 is when x - 1 = 1, which is true only when x = 2. Therefore, 23 - 1 = 7 is the only prime number that is one less than a perfect cube.

  2. A geometric interpretation can introduce greater depth to the task. Have the students cut an arbitrarily sized square with a side of length a, then draw in its lower-left portion a smaller square with a side of length b, shading all the figure except the smaller square of side b (see the figure below). Students could use graph paper, but doing so may lead to investigations dealing with specific calculations rather than to a more general approach to the task. First, ask the students how this object relates to the results of their arithmetic investigations of this pattern. With guidance, the discussion will turn to the fact that the area of the shaded region can be found by performing the calculation a2 - b2.

    1581 figure6 

    How could this region be shown to have an area of (a + b) (a - b)?  Give students time to analyze their drawings, and provide scissors to allow them to manipulate portions of the figure and investigate the situation dynamically. By sliding and rotating the larger shaded rectangle, we see that the shaded portion of the square is a rectangle of dimension (a + b) by (ab). If given time, one or more of your students may make this exciting observation. (See below.)

    1581 figure7 

     
 
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Learning Objectives

Students will:

  • Analyze and represent patterns with symbolic rules
  • Represent and compare quantities with integers
  • Write about their experiences with these patterns