More with Factoring Completely

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SWBAT identify the structure of various polynomial expressions in order to determine how to factor them completely.

Big Idea

Which of these things are not like the other? Students compare the structure of polynomial expressions.


10 minutes

I ask students to work on the launch individually today.  I plan to give students about 2-3 minutes to determine which expressions have a greatest common factor.  Then,  I use  a non verbal cue to determine which expressions they have chosen.  Typically, there will be at least one vote for every expression, but more votes for the two expressions that have a greatest common factor.  

I try to use the results of this quick poll productively by turning it into an opportunity for students to conduct mathematical arguments (MP3).  First, I have students turn and talk with a partner to give a reason why they chose the expression(s) they did. I encourage students to listen with a critical ear.  I say, "Either agree or disagree with your partner, but offer evidence as to why."

Once students have had a few minutes to discuss the expressions, I plan to do another non-verbal cue to determine if anyone has changed their mind.  Then, I allow a few students to offer their arguments to the class. I say, "Which expressions have a greatest common factor? Explain it for us."

Before proceeding with the lesson, I will take a minute to review the second slide with the class. This slide reviews the three types of structures (MP7) students are looking for when factoring. My goal for this lesson is for students to continue to practice recognizing each of these structures.  I remind students that this is not a list but rather a hierarchy.  Students should always look for a greatest common factor first, and then look for one of the other two structures.

3's a Crowd Activity

25 minutes

In the 3's_a_crowd activity the emphasis is on the front of the worksheet. The back side can be used for an extension or for extra practice if time permits.  I like to approach this activity in a structured way by having students first go through all 24 expressions on the front to determine which expressions have a greatest common factor.  I plan to let students work on this with a partner. I will remind them to provide a verbal rationale (MP3) or each expression they identify as having a greatest common factor.

Once students have identified the problems that have a GCF, I ask them to go through the remaining expressions and determine which are the difference of two perfect squares and which are trinomials.  Students should then factor only the expression that is different in each row (for example, in row 1 (a) and (b) are both difference of two perfect squares and (c) is a greatest common factor.  The students would factor (c) only. My rationale is that in order to factor completely, students need to be able to identify each type of factoring structure. By using the structure of the expression as a filter, this activity encourages students to identify when an expression is factored completely, versus when it needs to be factored further.

If my students complete the activity early, I will ask them to attempt the questions on the reverse of the page. 

Teaching Note: The final question on the page deals with a type of trinomial that often trips students up.  This is because both 3*2 and 6*1 will multiply to 6 and both will combine to give 5. The correct answer comes from using the signs of the numbers appropriately.  Let students analyze this solution and explain why they feel that the factorization is correct or incorrect. x^2+13x+30 is another of these trinomials that sometimes trip students up.


5 minutes

The factoring_3s_a_crowd_close asks students to design expressions that fit into the three categories identified in the table headinds.  This task raises the cognitive level because identifying the type of factoring is not as challenging as designing an expression that falls into a particular category.  I plan for students work on this ticket out individually.  If time permits, I will have students trade papers with a partner and let the partner attempt to factor each of their expressions.