(1) Stephen F. Austin State University (2) Mustang High School, Mustang, Oklahoma

Visual Factoring: Beyond Symbolic Success

Keith Hubbard(1), Lesa Beverly(1), Leah Handrick(1), & Meagan Habluetzel(2)

Quick! How do you factor f(x)=4x2+3x–10? How would you explain it to a 14-year-old?

An educator’s answers to these questions betray much of what she or he sees as significant in

mathematics pedagogy. A harder question is this: Why is it our goal to teach every student to

complete integer factorizations of polynomials? Clearly, looking at Common Core State

Standards (CCSS) A-APR3, A-REI4b, F-IF7c, and F-IF8a, this is a national goal. But authors

such as Anna Sfard (even in this journal’s editorial section) have challenged us not to continue

teaching a topic and continue teaching it the same way simply because it ‘always has been’

(2012). Certainly allowing teachers to continue to teach this slightly abstract topic with no

insight into why their students should understand this material and little connection to other ideas

encountered in school mathematics seems a recipe for futility. In this article, we make the case

for highlighting the connections between integer factorization and geometry, set diagrams, and

even history. Our goal is to make factorization more than a sterile algorithm applicable to two or

three exceedingly special cases and to offer the classroom teacher practical tips on how to

integrate geometric factoring into their curriculum.

Certain students will automatically be attracted

to any new nugget of mathematics their teacher

presents, but most students will need some

contextualizing, extrinsic motivation for factoring. In

the spirit of appealing to more than just ‘the usual

suspects’ teachers might begin with a brief review of

how we got to where we are in mathematics. (This might be particularly engaging for students

Teaching thought:

Mentioning that history of algebra

before the year 0, or of any other

part of mathematics for that matter,

makes a grand opportunity to look

at another practical representation

of the negative numbers.

whose ethnic heritage overlaps with some part of this history.) The motivation for endeavoring to

teach integer factorization arguably arises as much for historical reasons as from philosophical

ones. The elements of algebra date back at least to Ancient Babylonia, specifically the

Hammurapi dynasty between 1800 and 1600 BC. Ironically, although quadratic equations have

been studied for almost 4,000 years, it wasn’t until several Babylonian cuneiform writings were

deciphered around 1930 that modernity understood this (Bashmakova and Smirnova 2000). The

Egyptians recorded working with linear equations between 2000 and 1800 BC. The Greeks

carried on and extended these traditions, particularly between 550 and 300 BC, with the Indian,

Chinese, and Arab mathematical communities also ‘discovering’ various pieces of algebra

(Boyer and Merzbach 1989).

Though algebraic concepts pervaded early civilization, a fundamental issue challenged all

of these endeavors. When Al-Khwarizmi (from whose name the word ‘algorithm’ is thought to

have come) wrote an example of solving a quadratic equation around 800 AD, he had to write

“What must be the square, which, when increased by ten of its own roots, amount to 39?”

(Varadarajan 1998) How successful would your students be at parsing this sentence? Al-

Khwarizmi lacked a symbolic mathematical language; the very thing that peeves countless a

modern student makes such a problem vastly less ambiguous and easier to communicate

universally. In modern mathematics, Al-Khwarizmi is asking “What value(s) of x solve the

equation x2+10x=39?” We have the opportunity to educate students throughout the process of

teaching them algebraic symbol manipulation that the symbols were initially created as tools to

articulate problems rather than as an end in themselves.

Moving on from historical context, let us explore the factoring process with an explicit

view toward fostering connections across strands of mathematics. We believe such exploration

not only fosters mastery of symbol manipulation that is at least as robust and accurate as the

traditional algorithmic approach, but (more importantly in our minds) encourages connections

across content and validates exploration into mathematics beyond just ‘find the answer’.

Visualizing Algebraic Symbols as Areas

As far as the symbolic manipulation required for factoring, there seems to be a fair bit of

uniformity on how teachers teach factoring, say f(x)=x2+5x+6. Integer factorizations of 6 are

usually checked to determine which factorization adds to 5. The quadratic is then factored into

two monic linear terms with the factors of 6 as the constant terms. In our example, since 2•3=6

and 2+3=5, we conclude that f(x)=(x+2)(x+3). What varies more widely than finding such a

solution is the justification for such an approach. One approach uses this as an opportunity to

emphasize the distributive property and verification by ‘reverse engineering’. After all, if one

distributed, one would get

(x+2) (x+3)=x (x+3)+2(x+3)=x2+3x+2x+6=x

2+5x+6

Hence, students get to practice factoring and verifying an equality.

Another approach incorporates geometric reasoning. Since the very name “x squared”

connotes two-dimensional geometry, let’s consider x2 + 5x + 6 geometrically:

The goal is to form a rectangle from these shapes since rectangles are the fundamental visual

representation of multiplication. Thus, geometrically we can observe that x2+5x+6= (x+2)(x+3)

and we can argue that the reason the student should factor 6 is so we can form a rectangle from

the unit blocks.

We advocate both justifications, as there is evidence that multiple representations and

horizontally integrated curriculum improves quality of instruction (Hill and Ball 2009; Hill et. al.

2008).

Regardless of whether it is a method of

primary utility for students, we believe exposure to

manipulatives is a valuable experience for all

students. As one former high school teacher who

is now a university educator articulated, they “give

a very clear, vivid demonstration of the

connectedness between algebra and geometry. So

it gives the geometry of algebra; and students

enjoy seeing that they’re not disjoint.”

Ironically, as Phillips noted in 2012, “In every

decade since 1940, the National Council of

Teachers of Mathematics has encouraged the use

of manipulatives at all grade levels, yet many high school teachers are reluctant to use this type

of resource.” Unfortunately, there seems to be a prevalent view that manipulatives are a childish

approach (Swan and Marshall 2010). However, this view overlooks manipulatives utility in

Teaching thought:

One teacher the authors interviewed noted

that he saw the benefits of algebra tiles

primarily for “students who did not grasp

the concepts any other way.” He came to

use the visual approach as a sort of

remediation for students who were not

handling the symbolic manipulation

correctly. In this vein, one can go as far

back as the distributive property with

whole numbers and find a compelling

justification:

2•(3+1)=(2•3)+(2•1)

deepening students’ understanding of mathematics in addition to easing initial entry (Phillips

2012).

Visualizing Negative Areas

Explanations linking factoring with geometry must logically begin with positive

coefficients which correspond to ‘positive’ areas. It might be tempting to then simply relapse

into teaching a rule for solving when some

coefficients are negative. Certainly, it is grand

when students notice the pattern that when the

coefficient of the linear term is negative, at least

one of the factors is negative; and if the

constant is positive both linear factors must

have the same sign. However, the geometry

underlying these factorizations is a preview for

material to come – from polygons inscribed within polygons, to defining the definite integral, to

the washer method for calculating the volume of rotational solids, signed area is a vital topic in

mathematics. Linking to future content enhances student buy-in and is again tied to quality of

instruction (Ball et. al. 2008).

We suggest visualizing negatives as borrowing from the existing blocks. Consider the

function f(x)=x2–5x+6.

If we illustrate positive areas by blue and areas

to be removed by red, we

Teaching thought:

Although manipulatives are an excellent

way of engaging students (Ozel et. al.

2008), there are certainly pitfalls. One

teacher stated, “It’s very hard in a class of

20 or more to use [algebra tiles].

Everybody’s at a different place and their

tiles are at a different place [i.e. different

configurations].” A solution might be the

SMARTboard approach (cf. Kilgore and

Capraro 2010).

Teaching thought:

Describing positive and negative areas

as military troops or electrical charges

that neutralize each other can be

effective. After all, 4+(-3) might really

most accurately be interpreted as adding

3 electrons to a molecule with 4

unmatched protons.

have a chance to revisit the cancelation of negative and positive areas. Using manipulatives for

demonstration, one might lay the red strips over the blue square and recognize that the lower

right-hand is doubly covered.

This is a fantastic first opportunity to show students the significance of the need to add

back a copy of the overlap, the so-called Inclusion-Exclusion Principle which is prominent in

counting and probability. Visually, we might relate the two like this:

Factoring with Nontrivial Lead Coefficients: The Algebra

It would be natural to ask students if the

geometric interpretation still works with a nontrivial

lead coefficient. Consider f(x)=4x2+3x–10.

Perhaps the most common approach is to

consider factorizations of 4x2, and of -10, then pair-

wise multiply all combinations of them. Since lead

coefficients are changing also, the order of the second factorization now matters. So we have

Teaching thought:

For a different beneficial use of

manipulatives, one teacher observed:

“The ESL kids just ate it up …

because they didn’t have the language

skills.” We’re constantly looking for

effective teaching methods that

particularly target underserved

subpopulations.

Factorization of 1st term: Factorization of constant term:

(4x)(x)

(2x)(2x)

(10)(-1) (-1)(10)

(5)(-2) (-2)(5)

(2)(-5) (-5)(2)

(1)(-10) (-10)(1)

for a total of 16 different options. This selection process suggests calculating the coefficient of

the linear term and checking it against the original problem. Usually the thought process goes

something like this:

Let’s try (4x+10)(x-1). The coefficient of the linear term is (4)(-1)+(10)(1)=6.

Nope.

Let’s try (4x+5)(x-2). The coefficient of the linear term is (4)(-2)+(5)(1)=-3.

Nope; but close. Let’s swap the signs.

Let’s try (4x-5)(x+2). The coefficient of the linear term is (4)(2)+(5)(-1)=3. Yep.

Using progressions like this can build number sense. The fact that the distributive property is

employed, however, might be obscured. Consider this generalization of the factorization method

used above.

1. Multiply the lead coefficient and the constant.

2. Find a factorization of this number that adds to the coefficient of the linear term.

3. Decompose the linear term, and then use the associative and distributive properties to

remove common factors.

4. Use distribution again to complete the factorization.

In our example, the algorithm works as follows. Multiply the lead coefficient and constant:

4(-10)=-40. Find a factorization of -40 where the factors sum to 3.

40(-1)? 40-1=39 No.

20(-2)? 20-2=18 No.

10(-4)? 10-4=6 No.

8(-5)? 8-5=3 Yes.

We then decompose the linear term, gather common factors, and complete the factorization.

f(x)=4x2+8x–5x–10

f(x)=4x(x+2)–5(x+2)

f(x)=(4x–5)(x+2)

This method employs the same initial factoring

step as the factorization process usually taught

for quadratics with leading coefficient one, but

the distributive property is explicitly used 3

times. (All methods use distribution, only some

do it explicitly.)

The exacting reader will notice that our factorization of -40 as (8)(-5) does not require a

specific order. Had we chosen the other ordering, we would have

f(x)=x(4x–5)+2(4x–5)=(x+2)(4x–5).

The point of this article however is to push the discussion toward connections: Does the

geometry motivate, or even support, multiplying

and then factoring the product? Recall that, given

the factorization ( ) ( )( ),

and . So represents the product of the blue

sides in the upper left corner and the lower right

Teaching thought:

One master teacher implemented

manipulatives for factoring by using

math stations when she had a student

teacher with her in the classroom. One

of the stations involved using physical

manipulatives to factor. Another station

then asked students to draw

representations of the manipulatives.

This approach increases the level of

abstraction and student independence.

Teaching thought:

The question of representing the

“multiply a and c” method geometrically

was posed to multiple students before the

solution described was arrived at. This is

an example of an ‘extension question’ for

particularly motivated students.

corner of the diagram below. In order to complete the rectangle, we must add a -by- rectangle

in the upper right and a -by- rectangle in the lower left. The product of the areas of these two

rectangles is ( )( ), the same as . Moreover the sum of these areas is , which is

simply , the coefficient of the linear term.

Visualizing Factoring with Nontrivial Lead Coefficients

Geometric factoring problems are quite scalable. Let’s revisit f(x)=4x2+3x–10.

The circled region below represents addition of zero since the area of the 5 blue rectangles are

neutralized by the area of the 5 red rectangles.

Our goal is to create positive and negative rectangles

with matching heights. Observe in the next diagram

how both groupings have height x+1+1.

Finally, we combine the two rectangles to get a single rectangle. This rectangle’s dimensions are

the factorization of the original problem: 4x2+3x–10=(x+2)(4x–5).

Teaching thought:

Here we have a visual representation

of the additive inverse and additive

identity properties. Specifically, one

might ask students what 5x+(-5x) is,

then whether adding 0 to an equation

(or area) changes anything.

Notice that the width “4x-5” represents a width of 4 x’s with 5 units removed. Perhaps

an example which factors into two linear terms, both

with negative constants is also in order. Consider

f(x)=6x2–7x+2.

As before we:

1. Multiply. (6•2=12)

2. Find a factorization. (12=-3•-4; -3+-4=7)

3. Decompose and factor.

f(x)=6x2–3x–4x+2=3x(2x–1)–2(2x–1)

4. Complete factorization. f(x)=(3x–2)(2x–1)

Teaching thought:

These graphs assume that the area of x2 is positive,

but students might not immediately grasp why that

is reasonable without prompting. How would this

picture change if we knew x =-10?

One way to represent this subtlety is to use two-

sided manipulatives with different colors on

opposite sides (say blue and red). Place the

manipulative blue-side-up on the number line with

its left edge starting at 0. To demonstrate negative

values of x, flip the manipulative red-side-up

leaving the same edge at zero.

Extension 1: Graphing and the Vertex of a Parabola

A subtlety that hinders many of us in mathematics is transitioning from symbolic

function notation to graphical representations. If students have some familiarity with graphing

equations or functions on the Cartesian plane, a great extension would be to use a geometric

factorization of a function like f(x)=(x–2)(x+4) to investigate the graph of the function. Students

should see that when x=2, the area of the rectangle is exactly 0. This is a compelling visual

demonstration of the zero product property. Visualizing a specific negative value for x is also a

challenging exercise.

One application of visual factoring has to do with compound interest. Suppose one

invests a dollar and receives r1% interest the first year, then r2% interest the second year.

Algebraically we describe the return as (1+r1)(1+r2). Of course, we want to maximize our return.

If the sum of the interests, r1+r2, is fixed (which amounts to having the perimeter fixed), we see

that the following facts are actually one and the same:

1. For a fixed perimeter, the rectangle of maximum area is a square.

2. For a fixed average rate of return, an investment is maximized when there is no variation

in the annual rate.

Interested readers might enjoy a discussion of average annual return vs. annualized return (cf.

Baker, Jensen, and Harris 1998).

Extension 2: Cubic Polynomials and Three Dimensions

To push student learning and the algebra-geometry connection beyond the ordinary, try

generalizing to cubic polynomials (starting with a trivial lead coefficient). Consider f ( x ) = x3 +

5x2 + 8x + 4.

(Hopefully the connection to place value tiles is not lost on the reader.) One could form a

rectangular box with the cube at one corner and the 4 unit cubes at the opposite corner. Of

course, the 4 unit cubes need to be divided into three factors – either 4,1,1 or 2,2,1 (i.e. the

dimensions of box of unit cubes):

or

Observe that the x-by-x-by-1 squares are in one-to-one correspondence with each

dimension of the rectangular box created by the 4 unit cubes, so in actuality, we seek a

factorization whose factors sum to 5. Thus we use the factorization 2+2+1=5:

Many will read this section and salivate for the extension to higher degree polynomials and to

the Fundamental Theorem of Algebra. The

connection could be made that quadratic

polynomials have at most two linear factors, and

cubic polynomials have at most three linear

factors. Without digressing too far into the

complex numbers, we suggest a teaser

something like the following.

“Wouldn’t it be nice if a degree 2

polynomial always factored into 2 linear terms and a degree 4 polynomial always

factoring into 4 linear terms? It turns out that if you have enough numbers they

really do. The extra numbers we need are called the complex numbers. Just like

we added negative integers to the whole numbers, then rational numbers, then

irrational numbers, once we expand to the complex numbers every polynomial

will factor just like we wish it would. That actually comes from what’s called the

Fundamental Theorem of Algebra, but you’ll have to keep going in math to get

there.”

Conclusion

Let’s say you’re sold on incorporating geometry into the algebra of factoring. Where

should you start? We like the progression from actual manipulatives to drawing representations

of manipulatives as described above. However you do it, here is a progression you might find

useful:

Teaching thought:

Somewhere in the progression, one might

remember with students that “factoring” a

polynomial, by definition means rewriting it

as the product of polynomials of lower

degree. For quadratic polynomials this means

rewriting the polynomial as the product of

two linear terms. Although one might be able

to divide each monomial in a polynomial by 2

and hence “factor out a 2”, it is said that a

quadratic “does not factor” if it cannot be

written as the product of two linear terms.

x2+5x+6

x2+6x+8

x2+5x

x2+2x+2 (a reality check)

x2–5x+6

x2–7x+6

x2+2x–3

x2–2x–3

2x2+5x+2

4x2+10x+4

4x2+3x–10

8x2+6x–9

18x2 –39x–70

Part of the fun (and value) is recognizing that one question leads to another, then many

others. Use our extensions or, better yet, provoke your students into making their own and

challenge each other.

Integer factorization will likely remain a staple of high stakes testing as well as university

mathematics. It does have numerous applications. But greatest benefit can be gained by also

allowing students to explore the connections between algebraic symbolism and geometric

representation, developing a symbol sense through practical manipulation, and growing an

affinity for the mathematical practices that transcend content strand.

References

Teaching thought:

In the spirit of full disclosure, one might also

point out to students that if three integer

coefficients were chosen at random for a

quadratic polynomial, the chance of the

polynomial possessing an integer factorization

would be 0.

Students can establish this fact experimentally

using Excel by typing

“=RANDBETWEEN(0,1000000000)” into cells

A1, B1, and C1. Next type “=(-B1+(B1^2-

4*A1*C1)^0.5)/(2*A1)” into cell D1. Drag these

cells down to your heart’s content.

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