In this Warm Up, I ask students to consider binomial expressions in which the following methods or patterns can be used to factor the expression:
I expect the Warm Up to take students about 10 minutes to complete and for me to review with the class. Yesterday, my students worked with multiplying a Sum and Difference of Binomials to produce a Difference of Squares. To review what we learned, I have my students complete Problem 1, then they are to try to factor a Difference of Squares in Problem 2.
As students work, the expressions become progressively more difficult to factor. Two of the problems are a difference of squares, with the possibility of factoring out a common factor before applying this pattern to the expressions. I think that it is important to continually encourage my students to seek common factors before making a first attempt at using a special polynomial pattern.
As they work today, I am hoping to hear my students talking about how multiplying and factoring polynomials are related skills that can be used to undo each other.
Moving on from the Warm Up, I have students create four categories on their desk. The four categories are listed below. The categories are:
1. Greatest Common Factor Only
2. Difference of Squares Only
3. Greatest Common Factor and Difference of Squares
I discuss with students the understanding that the term "prime" means that the binomial is not factorable. We also practice using the language of polynomials. For example, I will say something like, "When I can only factor out a Greatest Common Factor, I am writing the expression as a product of a monomial and a polynomial."
Then I hand students a set of Cards with a binomial expression on each card. I instruct each student to work with their assigned Table Partner to place each card in the appropriate category. The goal of this activity is not for students to factor these binomials, but to recognize the category in which they belong. This activity allows students to focus on the structure of the expression (MP7) instead of focusing on factoring it correctly. I encourage students to talk in their pairs and to write on the cards. My students will factor the binomials in each category during the Guided Practice section of this lesson.
In this Guided Practice, I provide students an answer sheet for them to factor each binomial expression on the cards. The answer sheet is divided into the four categories from the Partner Activity so that students can factor problems of the same structure in the appropriate section. I have organized the practice in this way so that my students can focus on the structure of the problem (MP7) and develop deeper understanding through the application of repeated reasoning (MP8).
For this task I encourage my students to write the categories on their own paper and to show their work so that they create a useful reference. The majority of the Binomial Expressions will factor using the method of Difference of Squares only. Numbers one, three, five, ten, eleven, twelve, thirteen, fourteen, fifteen, and sixteen are in this category. Fourteen and sixteen require using the Difference of Squares twice to completely factor. The other categories have less. Number 9 is the only prime. Numbers two, four, six, and seven factor by using the Greatest Common Factor only. Finally, number eight is the only one that requires finding the Greatest Common Factor and factoring by using the Difference of Squares.