visual factoring: beyond symbolic success · across content and validates exploration into...
TRANSCRIPT
(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.
Baker, Guy, Richard Jensen, and Ken Harris. 1998. Investment Alchemy: A Guide to Asset
Allocation. Newport Beach: Standel Publications.
Ball, Deborah Lowenberg, Mark Hoover Thames, and Geoffrey Phelps. 2008.“Content
knowledge for teaching: What makes it special?” Journal of Teacher Education 59, no. 5:
389-407.
Bashmakova, Isabella, and Galina Smirnova. 2000. The Beginnings and Evolution of Algebra,
Dolciani Mathematical Expositions 23. Translated by Abe Shenitzer. The Mathematical
Association of America.
Boyer, Carl, and Uta Merzbach. 1989. A History of Mathematics (Second Edition ed.). New
York: John Wiley & Sons, Inc.
Hill, Heather, and Deborah Lowenberg Ball. 2009. “The Curious - and Crucial - Case of
Mathematical Knowledge for Teaching.” Phi Delta Kappan 91, no. 2: 68-71.
Hill, Heather, Merrie Blunk, Charalambos Charalambous, Jennifer M. Lewis, Geoffrey
Phelps, Laurie Sleep, and Deborah Loewenberg Ball. 2008. “Mathematical Knowledge for
Teaching and Mathematical Quality of Instruction: An Exploratory Study.” Cognition and
Instruction 26: 430-511.
Kilgore, Kelly E., and Mary Margaret Capraro. 2010. “A Technological Approach to Teaching
Factorization.” Journal of Mathematics Education 3, no. 2: 115-125.
Ozen, Serkan, Zeynep Ebrar Yetkiner, and Robert M. Capraro. 2008. “Technology in K-12
Mathematics Classroooms.” School Science and Mathematics 108, no. 2: 80-85.
Phillips, Marilyn Curtain, www.mathgoodies.com/articles/manipulatives.html, 10 September
2012
Sfard, Anna. (2012). Why Mathematics? What Mathematics? The Mathematics Educator, 22 (1),
3-16.
Swan, Paul and Linda Marshall, (2010). “Manipulative Materials.” Australian Primary
Mathematics Classroom,15 (2), 13-19.
Varadarajan, V.S. 1998. Algebra in Ancient and Modern Times. Mathematical World 12.
American Mathematics Society.