Tag Archives: factoring

Breaking Apart Sums and Differences of Perfect Cubes

The first few days of math class…an awkward time for both students and teacher.  The kids haven’t picked up on my mannerisms yet, aren’t sure why I fly around the room like a maniac, and worse, they aren’t laughing at the jokes.  I tend to use the first few days of any class seeing how far I can put my foot on the gas…what do my students understand?  Where are there gaps?  Who will I need to sit on during year?  Without exception, I dedicate at least part of the 1st day with students at boards, shouting out review problems.  The problems are strategically chosen to allow for initial success, dust off some cobwebs, provide for discussion when we hit some road blocks, and most importantly let students know that it will be perfectly acceptable to struggle in my class….as long as you keep trying.

I’m trying a differenmt apporach with an Algebra 2 course I am teaching this semester, and hoping to build some discovery and communication moments right in the first few days.  As their first day assignment, students fill out a Google form with information about themselves: hobbies, goals, clubs, etc.  As part of the form, I am adding this task: “Tell me everything you know about this graph:”


For day 2, I’m hoping the responses will provide a review of vocabulary (intercepts, roots, solutions, even rotational symmetry perhaps?), and some table-setting for what’s to come.  In our district, I can expect that Algebra 2 students will have a solid background in linear functions and basic polynomial operations, mostly limited to quadratics.  Cubics for the most past have not been explored yet.  And while polynomial multiplication and factoring are not new, rarely do students see polynomial division before algebra 2, so I will bring this into the discussion as a new idea.  NOTE: our district uses the Everyday Math program in the early grades, which stresses partial quotients.  Wondering how this will play when I attempt polynomial division…update may be coming!


Starting algebra 2, students should be able to “read and recognize” the following polynomial patterns: difference of squares and perfect-square trinomials.  But beyond this, I want students to be able to relate factored form to graphs, which often seems to be marginalized in the drive to practice process.  So, one of my first lesson openers will be a short and sweet challenge.  Does the following polynomial factor?

In their teams (my students always sit in groups), I will provide some time for students to consider this problem, and observe their trials.  I expect that will have a few groups attempt (x-2)^3, which will end badly, but hopefully lead to more trials.

CubicSo, how do we cross the bridge to the formulas for differences, and sums, of perfect cubes.  Time to start looking at some graphs, in particular the functional form of the given expression:

What do we notice with this graph?  And what characteristics will be helpful with the factoring problem at hand?  Here is where I hope students drive the discussion:

  • This graph has an x-intercept of 2.
  • This means that x-2 is a factor.
  • There are no other obvious intercepts, but we can employ long division here.

So, x^3-8 DOES factor.  Do other cubics factor?  How?

With their teams, students will now be given a few more cubics to factor:


What patterns do we notice?  Can we develop a general rule for factoring difference of cubes, and even sums of cubes?

Guiding the discussion towards a generalization, without students feeling forced-fed, is part of the art of teaching.  Hoping these first day discussions tie together lots of previous knowledge with a discovery moment.  I am not sure how it will go, but I hope to set the table that nothing is given for free.  Show me what you know!

Why is “Simplify” So Damn Complicated?

Making my classroom rounds this week, I came across a class reviewing concepts for the upcoming Pennsylvania Keystone Exams in Alegebra 1.  The PA Department of Education provides an eligible contect document with sample items on its website, and the class was working on the following question:


Pretty standard problem.  Factor the numerator and denominator, cancel common factors, and you’re home.  But this class was struggling with the factoring review, so I stepped in with a different approach.  How about taking the given expression, and using a graphing calculator to evaluate it?  Sadly, the class was not familiar with the Table on their TI-84’s, but understood what it did right away:

Calc 1

Some nice discussions emerge here.  What’s with that “error”?  Is our calculator broken?  And some evidence over this function’s behavior emerges.  Note the slowly increasing values of y.

But how does this help us with the question at hand?  A number of students recognized that the correct answer would be the expression which had the same Y-values.  In essence, simplfying produces a different-looking expression with the same outputs as the original.  So, let’s try the answer choices.  Here’s A:

Calc 2

No dice.  Values are much different.  And a fantastic opportunity to discuss the difference between an output of zero, and an undefined output.  But eventually we get to D, and can check the tables:

Calc 3

Looks pretty good, butttt……..what’s with the errors?  And they seem different for some inputs.  But now we can review and discuss domain, and look at those pesky domain restrictions in a new light.

So, am I a bad person for bypassing the factoring review, and encouraging calculator use?  After the discussion, I reminded the class that factoring is a skill they need to have in their toolbox, but the alternate discussion of equivalent forms and assessing values was also worthwhile.  I feel good.

This classroom visit got me thinking about the nature of the word “simplify” in math class.  How often do we ask students to “simplify” in math class, and in what contexts?

Sometimes we want to simplify an expression:

Or maybe we want to simplify a rational expression:

Or perhaps we want so simplify a radical expression:

And make sure you simplify when there is a radical in the denominator (unless you are taking AP Calc, in which case we don’t care about such silliness)

For different situations, we have subtle differences in what it means to simplify, but is there a common goal of simplifying?  Is it just to make things look pretty?    And is a simplified expression always the most useful?  When is it not?

I’m curious if anyone has a short and snappy answer to “what does it mean to simplify an expression?”.  I invite you to participate and contribute your response on Todays Meet (click to participate).  If you have never used Today’s Meet, it is a nice, free way to gather responses.  Simply provide the link and start a conversation!  Feel free to share the link with your students a “bell ringer” activity.  If we get some responses, I’ll make a later blog post about them.


Look Through the Eyes of Your Students

Here in Pennsylvania, many high schools are gearing up for the upcoming Keystone Exam in Algebra 1.  In this first year of Keystones, the Algebra 1 exam is being given not only to students as they complete Algebra 1, but also to 10th and 11th graders who have passed the course in the past.  The state has provided a number of sample items, which we have been using in math classes to help our students prepare.

A discussion of one of these released items not only revealed a common algebra misconception, but also generated thoughts of how teachers may see problems differently that their students.  Here is the question:


So, how could teachers and students view this problem differently?


Teacher Problem

My teacher eyes tell me immediately that this is a factoring problem.  I’m not sure how it is going to factor yet, but I am pretty confident that the answer will be C or D.  Choices A and B are not really even up for debate here.

My radar perked up when a colleague shared that a handful of students in one of her classes chose B.  B???  How the heck did they get B????


Problem Student

My old nemesis….cancelling across addition and subtraction signs, how nice to see you again!

So, while I immediately see the problem as two expressions which will separately require factoring, I need to remember that students don’t always view problems the same way.  Being able to identify, discuss, and personalize these common errors are all part of the wonderful art of teaching.

And while illegal cancelling will be a struggle for students long after I retire, I often use the exercise below to generate discussion.



So, either I have made a math error here, or I have a great method for generating some side income here (and why am I sharing it with you, anyway???).

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.


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.

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:


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


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):


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:


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…