Seeing Structure in Factoring the Difference of Squares
Lesson 16 of 18
Objective: SWBAT factor polynomial expressions that describe the difference between two perfect squares.
During lesson leading up to this Warm Up, most students have begun to notice how the structure of trinomials with a leading coefficient of "1" supports the factoring process. Today, we will build on their observations, first by reviewing them.
In the first slide, we go through the two examples as a class without using the box method of factoring. I tell students that if they still like to use the box method they certainly can. However, I let them know that for the sake of efficiency, when factoring, using the more abstract method will be better.
In the second slide, students have an opportunity to practice factoring using the traditional method. I encourage students to check their answer by multiplying the factors together (MP6).
With a brief review of factoring trinomials complete, I plan to lead my students through an introduction to factoring difference of square polynomials. We will follow this presentation: Factoring_Difference_of_Squares.
Slide 3 (is page 1 of the .pdf file)
After students multiply all three pairs of binomials, they quickly notice that #2 and #3 are different from question #1. At this point I ask students to look at the binomials and try to determine why #2 and #3 ended up with binomial answers rather than producing a trinomial. I want students to think about this individually first. Then, I will have them Turn and Talk with a partner.
While students are discussing, I move around the room to identify students that are making a cogent argument. I plan to call on several groups to share their thinking. When I think students have a firm grasp on their own ideas, I will ask students to make up their own pair of binomials that will result in a binomial product. As students volunteer their answers, I make a list the expressions on the board so that all students can try to identify a pattern.
With the pattern fresh in their minds, I will take a few minutes to discuss the terminology "Difference of Perfect Squares". Since the term "difference" has many uses, students may need prodding to see that both #2 and #3 result in a difference because of the opposite signs.
Instructional Note: Some teachers use abbreviations. For example, an acronym for the difference of two perfect squares might be D2PS. I feel that if students use the vocabulary words to describe the structure of these expressions it leads to more understanding. If students use abbreviations, my experience is that they forget what the abbreviation stands for.
I have noticed over the years that my students are less familiar with numerical perfect squares than I would like them to be. On this slide, I ask students to make a list of all perfect squares from 1-144. Sometimes, students really struggle and have to think about each product (7*7, 8*8, 9*9) in order to make their list. If I observe this to be the case, I will periodically stop in the middle of class over the next few lessons and ask a student to reproduce the list. I think it is crucial for students to recognize the perfect squares when they see them (MP2)
With the foundation laid so far, students usually do fine with questions 1-3 on this slide. They begin to recognize how to quickly factor these types of expressions (MP7). On question #4, let students work independently and then share their ideas with a partner. Usually, students will say (1) it cannot be factored because 6 is not a perfect square or (2) they will factor it as (x-3)(x+2) (or some variation). Put this second idea up on the board and ask student to either prove or disprove this idea by multiplying the factors (MP3). Students will quickly see why this cannot work. (NOTE: while x^2-6 can be factored using irrational numbers, our focus in this lesson is on factoring using integers. A student could certainly suggest to factor the expression into (x-sqrt(6))(x+sqrt(6)) which would be legitimate. It will be left up to the teacher's discretion whether or not to take the conversation in this direction. This concept will come up when students solve equations using square roots in the next unit).
Let students try each of these one at a time and have them prove their answer is correct by multiplying the factors. Encourage them to generalize the structure they have used so far to these more difficult expressions.
This portion of the lesson (factoring_difference_of_squares_practice.pdf) allows students to develop some fluency with both factoring and multiplying. Students will continue to develop their understanding of both the structure of the original expression and the factors.
Students should work with their partner on this portion of the lesson. Encourage them to focus on quality over quantity. The reason behind multiplying to check is that students can change their factorization if it is not correct. Look for examples of students who are being precise and correcting their mistakes. Share these with the class using a document camera or other means as exemplars. After about 5 minutes, have students begin to post their work on the board so that others can check their work.
NOTE: Watch as students are working on factoring the expression 49-y^6. Often, students who are used to seeing these expressions written with the variable first will inadvertently factor it as (y^3+7)(y^3-7). If this happens, see if students catch their mistake with their check. If not, you can intervene to talk them through factoring the expression correctly.
In this closure (factoring_difference_of_squares_closure.pdf), students will take their understanding of the difference of two perfect squares to the next level.
In the first question, once students factor the polynomial using the box method (I still have students use this method for "factoring by grouping" type questions) one of the factors is a difference of two perfect squares that needs to be factored again.
In the second expression, once the students factor the difference of two perfect squares, one of the factors is another difference of two perfect squares. This closure gets students thinking about the idea of factoring completely (factoring an expression more than once).
With both of these questions I will use a guided practice model. For example, I may have students factor the initial expression and then write that expression on the board to ask students what they notice about each factor. If you feel like your students are prepared to attempt these on their own you could also use these two questions as a ticket out.