One of the joys of my job as a math coach is having conversations with colleagues about the hows and whys of math class. Often, the best conversations come from unplanned meetings, just chewing the fat about what is happening in classrooms. Earlier this month, I visited a second-year high school teacher with a quick question that turned into a deeper conversation after I noticed her posted objective: “factor polynomials where a > 1”. The objective was for our academic-level algebra class. I asked how things were going with the class, and the teacher expressed the usual frustration of teaching kids the process of factoring, which led to her asking what I would do differently…what did I have in my “bag of tricks” to teach factoring?

I had to think hard about my answer. Like most math teachers, I had taught Algebra I many times, and had gone through the process struggles students have with traditional factoring. I’ve never really subscribed to a “trick” for factoring: I have some colleagues who use a chart method, while others attempt grouping. But it occurred to me that the entire premise here was flawed…just how important is factoring to teach and learn? My second-year teacher friend thought for a minute, then gave the answer many teachers would give:

They’ll need factoring in Algebra II and other classes they take down the road.

*Sigh…*

Is that really the best answer we, as math teachers, can give for learning factoring? It’s just a cog in the polynomial machine: add – subtract – FOIL – factor – simplify rationals – graph. How exciting!

In a previous chat with a different teacher, I suggested that we upset the entire process. Why do we save fun, neat stuff like projectile motion for “later”? Dive right in, look at some non-linear graphs, and develop new ideas about quadratic functions, symmetry, intercepts, and vertices right away. If you have never been to the PhET (interactive science simulations from the University of Colorado), go there now. Check out the projectile motion applet:

Have fun tossing Buicks across the sky.

Think about the number of kids who never get to experience these great math models, because we beat them with a stick with pages of FOIL and factoring worksheets, before they ever get to see a parabola.

So, what of factoring? Let’s say I wanted to re-write the following function in factored form:

Is it wrong to have students graph the function and look for intercepts?

If we have an intercept at x = 2, then we know that (x-2) must be a factor. From there, we can piece our way to the second factor of (3x-7). Is it “wrong” to teach students to look at polynomials this way? I suspect many would call it blasphemous, but somehow I know that my kids know more about the inter-connectedness of functions, intercepts, and polynomials than theirs.

I want my students to utilize and switch between multiple representations of all families of functions: equation, table, graph, context. Unfortunately, most textbook ensure that these topics will continue to be taught in a linear fashion.

Let’s take that old objective of “factor polynomials” and change it to “develop and understand quadratic models for natural phenomena”

And if the reason we teach anything in math is because “they’ll need it for their next class”, then we are doing math wrong.

I’ve taken some heat for asking the same question. Thanks for throwing this out there.

It’s a pretty phony thing we ask our kids to do – similar to rationalizing denominators. I was at a conference at Philips Exeter a number of summers ago and during a presentation on quadratics the instructor mentioned that he had a written a program to analyze quadratics with the a, b, and c coefficients ranging from – 20 to 20 and something on the order of 7% or so were factorable at all over the integers. Tying this in with graphs and teaching the factor theorem is far more powerful than an strategy for quadratic factoring!

I’m actually surprised that the proportion of factorable quadratics is as high as 7%. This is a questions I have brought up with classes: given a random 1,b and c..what is the probability that we can factor it? But I never pursued it. I’d be interested in seeing the work on it. I suppose it comes down to finding if the discriminant is a perfect square, but that’s no easy task.

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Why Factor Polynomials?

Date: 02/20/98 at 18:52:05

From: Alexandra Wrigley

Subject: Factoring polynomials

Why is factoring polynomials important – how do we use it in our

everyday lives?

Date: 02/21/98 at 13:42:31

From: Doctor Sam

Subject: Re: Factoring polynomials

Alexandra,

Well, factoring ISN’T important to most people in everyday life. I

mean shopping and cleaning and cooking and going to the movies. But

many occupations use different kinds of mathematics, ranging from

accountants to carpenters to scientists and engineers to people who

work to protect the environment.

Many of them will sometimes need to use factoring, but factoring isn’t

a goal in itself. Factoring is used to solve different kinds of

problems. I think you might want to know why you should learn

factoring if you aren’t ever going to use it “in real life.”

One answer to your question is that most of us don’t know what we will

be doing in real life until it happens to us. Sometimes we plan for it

and sometimes it takes us by surprise. But it is a good idea to be

prepared. If you don’t know ANY mathematics then there are hundreds,

maybe thousands of jobs that you won’t be able to do. For most of

these jobs mathematics isn’t the main point of the job, it is just one

of the many tools that are used. So if you don’t know mathematics you

may be losing the opportunity to do something that you would find

exciting and worthwhile.

And now to the mathematical part of your question: how is factoring

used?

I can think of two important uses of factoring.

One is to make complicated things look simpler. For example,

I don’t have any idea what this fraction 1848 / 16632 means.

The numbers are too large. But if I factor the numerator and

denominator I get: (3)(8)(7)(11) / (8)(11)(7)(9) and I can

see that the 3, the 8, the 7, and a 5 are factors of both the

numerator and denominator. Since 3/3, 8/8, 7/7 and 11/11 are all

equal to 1 the fraction reduces to 1/3, which is a lot easier to

understand and to compute with.

It’s the same with complex algebraic fractions. (x^4 – x^2)/(x^3+x^2)

looks REALLY complicated, but the numerator factors into x^2(x-1)(x+1)

and the denominator factors into x^2(x+1), and since x^2/x^2 and

(x+1)/(x+1) are both equal to 1, this fraction simplifies to x-1 . . .

a LOT easier to work with.

A second important use of factoring is in solving equations. You don’t

need to factor to solve 2x+3 = 5 … linear equations use a different

method. And you don’t need to factor second degree equations because

you can use the Quadratic Formula (although factoring is often MUCH

easier!). But if you need to solve equations where the degree of the

highest term is more than 2 then you really have no choice at all

because you don’t have formulas for most of them.

Here is an example of a hard equation to solve:

x^5 – 4x^4 – 12x^3 = 0

But if you factor it completely you get (x^3)(x-6)(x+2) = 0 and now it

is easy, because this says that a product of things turns out to be

equal to zero. If you multiply, the only way to get zero as an answer

would be if you multiplied by zero. So one of the three factors has

to be zero.

If x^3 = 0 then x = 0

If x-6 = 0 then x = 6

If x+2 = 0 then x = -2

So the solutions to this equation are x = 0 or 6 or -2.

I hope that helps.

-Doctor Sam, The Math Forum

Check out our Web site http://mathforum.org/dr.math/

I like how you didn’t truly answer her question.

I’m sorry but I’m a firm believer that there’s no need for maths to be taught after primary school because the degree of mathematics used by 999 in 1000 people is no more than simple adding subtracting etc quadratic factoring algebra all a complete and utter waste of education unless a child shows promise or interest from the age of nine then I would say the billions of hours wasted are really scandalous and nothing more than posturing by academics who have nowhere else to preen their feathers