What really is "Factoring"? - Factor by Grouping

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Objective

SWBAT redefine their understanding of factoring. SWBAT factor polynomials by grouping.

Big Idea

From the entry poll to the final bell, the students will redefine their knowledge of factoring!

Entry Poll

5 minutes

As the students enter the classroom, I have a poll projected on the screen that says "Define Factoring."  These two words strike fear in the eyes of even my best and brightest students.  Many students tell me "Sure, I can do it.  But I just can't explain it!"

The responses that you get from the poll will likely be mathematically weak.  I use this lack of an ability to explain as a launch-pad for my lesson.  I tell the students that they have very clearly mastered a skill, but have not grasped the fundamental mathematical CONCEPT.  Today, I tell them, we will work to reveal the mystery of factoring. 

I use www.polleverywhere.com to set up my classroom polls.  I highly recommend giving it a try as an instructional tool in your classroom.  Creating an account is quick, and free.  Educators enjoy the benefit of exporting the poll results to a PowerPoint slide and saving them from year to year.  This will be a potentially powerful source for measuring the impact of the Common Core transition.  As we begin to ask these questions of the students more often, and the students are further exposited to the Common Core curriculum, we should see more well developed answers as the years progress. 

 

Opening Problem - The WHY?

10 minutes

I begin the lesson by writing a large, bold WHY? on the board.  This, I tell the students, is one element of factoring that we will seek to answer today.  Below this, I draw a rectangle and write inside of the rectangle that A=3x²+3x.

Next, I instruct the students to collaborate with their peers to find the dimensions of both sides in terms of x.  

The majority of students will be able to recognize from Algebra I, that a GCF of 3x can be pulled out of the expression, yielding 3x(x+1).  Therefore, one side is 3x and the other side is x+1.  An excellent extension question for the students is to ask them which side is the longest?  Most students will quickly say that 3x is the longest.  However, this opens up the window for the critical thinkers to think of situations when 3 times something would actually be smaller than something plus 1.  The students will then be able to come up with a set of negative values and fractions that are all counterexamples to the initial claim. 

If mistake have be made in factoring out a GCF, these are usually easily remedied by showing the students that their answer would distribute incorrectly if re-multiplied, that is, not produce the desired 3x²+3x.

Homework

15 minutes

The homework assignment that I give for this lesson is attached.  The first three problems ask the students to connect their learning about factoring to what is happening on the graph of a function.  The final 10 questions ask the students to factor by grouping.  A key thing to note is that they may have to take out a GCF first!