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
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.
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
|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.
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 a2 – b2 = a + b
Case 2: When b = a – 2, then a2 – b2 = 2(a + b)
Case 3: When b = a – 3, then a2 – b2 = 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