Algebra II : Unit #3 - Polynomials : Lesson #8

Special Factoring Situations

Objective: Students will be able to factor differences of squares, and sums and differences of cubes.
Standards: HSA-SSE.A.2 HSA-APR.C.4 MP3 MP4 MP7
Subject(s): Math
60 minutes
1 Warm up and Homework Review - 10 minutes

I include Warm ups with a Rubric as part of my daily routine.  My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogenous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Special Factoring Situations which asks students to describe how they factor a trinomial.

I also use this time to correct and record any past Homework.

Explaining a Process in Writing
Writing Across the Disciplines

Explaining a mathematical process in words is often one of the most difficult tasks for many students.   We had recently worked on the box method of factoring and so this was what most of them attempted to describe.  Many answers I received were incomplete and/or unclear.  A lot were similar to Student Work 3 which offers a short description that will only make sense to some one with prior experience with the box method.  Student Work 2 offers a slightly more in-depth description but falls apart, and is incorrect, toward the end.  Student Work 1 was one of the few descriptions that was reproducible by itself.  

This inexperience with verbally describing a process has a lot to do with the transition to the Common Core standards.  This warm up reaffirmed that I need to do more activities that require students to explain a process or their reasoning.  Fortunately, this doesn't have to be a big stand alone activity.  It can be incorporated inside any lesson as a portion of the notes, warm up, or exit ticket.  

2 Factoring Difference of Squares - 20 minutes

This lesson is a foundational lesson to ensure student success in the Common Core Algebra 2 standards. 

I give the students the expression m2 – 9 to factor without any introduction.  Once they have struggled with it a bit, I may give them the hint that this is really a trinomial.  A further hint is that we need to add the m term but there is no m term.  They should get the point where they rewrite this m2 +0m – 9 and then factor it.  I give them the next two. 

At this point, I ask them to talk to their partner and determine the pattern (Math Practice 7) .  We then make a list as a class.  The goal is that the students recognize.

  • Both terms are perfect squares.
  • There are only two terms.
  • They form a difference.
  • The factored form is the sum and difference of their square roots.

The next portion of this lesson involves a set of differences of squares, a few of which are prime. This is my favorite part of this lesson.  I tease the students that I am going to try to trick them.  For a lesson like this, this is a relatively brisk guided practice.  We run through the problems and whenever we get to a problem that is prime, we discuss the issue after they have tried it.  With a bit of tease, I have found that the students are motivated to really try and not get tricked.  

3 Factoring Sum and Difference of Cubes - 18 minutes

Sums and differences of cubes are a challenge to teach.  I have found that focusing on the pattern is the best method.  We begin by looking at the volume model of a3 – b3 (Math Practice 4).  We look at this model and then I ask the students to find the volume of each of the three prisms.  It may help more concrete learners to have a model build out of unit blocks or legos to model taking the b3 out of the a3.  We then re-factor those to become (a – b) (a2 + ab + b2).

We now look at the pattern formulas for both sums and differences of cubes.  I ask the students to do a think-pair-share on the similarities and differences between these two patterns.    The remainder of the lesson uses Guided Practice  for factoring cubes.  This is one of those places where students need to practice several times to really get the pattern.  

4 Exit Ticket - 2 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

Today's Exit Ticket asks students to factor a perfect cube.

5 Homework - 0 minutes

The goal of this homework is to solidify the patterns for factoring squares and cubes.  The cubes in particular need repetition for students to become confident using it as a tool.  

This assignment was created using Kuta Software, a product I would highly recommend to any mathematics teacher.