Categories
Algebra

Why Do Kids Need to Factor?

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:

phetprojectiles

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?

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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.

Categories
Algebra

Factoring – Sending Out the Bat Signal!

One of the joys of my job is having mathematically interesting chats with my colleagues about how they approach  specific problems with their classes.  These conversations often begin as one-on-one discussions, but usually evolve into calling multiple people into the fray to give their two cents.  This semester, a teacher in my department is tackling an Accelerated Algebra II class for the first time.  Having taught academically talented kids for many years, my advice to him was to constantly challenge his students, perhaps using problems like those from the American Mathematics Competitions as openers.  But while offering up academic challenges can keep a teacher’s mind sharp, there is the risk of having that “hmmmm…” moment…..that uncomfortable feeling where you’re not quite sure what the correct response to a student question is.

The discussion today came from a review of factoring, and a problem which seems innocent enough:

Factor x6– 64

Take a moment and think about how you would factor this….show all work for full credit…

Enjoy a few lines of free space as you consider your work….

And…time….pencils down….

The interesting aspect of x6– 64 is that it is both a difference of cubes and a difference of squares.  I used the neat algebraic interface on purplemath.com to do some screen captures and make the algebra look pretty here.  In this case, the calculator factors this expression as a difference of squares, (x3– 8 ) (x3 + 8),  which then become both a sum and difference of cubes and can both factor further:

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But, the initial expression is also a difference of cubes, and can be factored as such.  It is verified below:

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The plot thickens as the discussion then centers about the “remnants” we get when we factor a difference of cubes.  We can verify that the two “remnants” (underlined in red) from the first factorization are factors of the remnant of the second method (underlined in green):

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So, what’s happening here?

The extra, messy, factor we get when we factor a sum or difference of cubes is up for discussion here.
According to Purplemath:

The quadratic part of each cube formula does not factor, so don’t attempt it.

But we don’t have a quadratic here (though we could perform a quick substitution and consider it is one), we have a 4th degree polynomial.  Even the algebra calculator on the site doesn’t care for this quirky 4th power expression:

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So, I am looking to my math peeps for some thoughts:

  1. Is there an order to consider when a polynomial meets 2 special cases?  Should we look at sum of cubes or squares first?
  2. Does anyone have any insight on x4+4x2 + 16?

Good night, and good factoring…