This straight-forward launch is used to make a very important point:
I ask students to factor the number 30 individually, then I ask them to compare their outcome with a partner.
After the pairs have a few moments to compare, I call on a student to explain his/her process. I expect that my students may say something like, "I factored 30 into 10 and 3 and then I broke 10 into 5 and 2." After listening, I will ask some follow-up questions that will help students reflect on what it means to factor completely:
All of the above questions set the stage for helping students learn to factor polynomials completely.
When teaching the factoring of polynomials, I have found that my students often try to factor a polynomial completely in one step, often incorrectly.
Ex: 2x^2+4x+2 -->2(x+2)(x+1)
The important point that I want students to realize is that when factoring, they are always breaking an expression into two factors at a time. Then, if one of those factors is composite and can be factored again, that factor can be broken into two factors, and so on.
After the launch, I try to keep the main idea in play as we discuss these examples as a class: factor_completely1_direct. I explain to students that when factoring polynomials they should always look for a greatest common factor first. Then, once they either factor using a greatest common factor or one of the other two structures, they should examine the two new factors to determine if either can be factored again. I find that the best way to illustrate this process is to "think out loud" for students as I am factoring an expression. An interactive dialogue can also be effective:
Teacher: I know I want to look for a greatest common factor first.
Teacher: Does anyone see a greatest common factor here?
Student: Yes, 2 is a greatest common factor because it is the largest number that can divide evenly into 2, 10, and 8. Not all of the terms have a variable so only the 2 can be factored out.
Teacher: So my factors would be 2 and what?
Student: 2 and x^2+5x+4 because those multiply to the original expression.
Teacher: Notice that we only factored the original expression into two factors. So, now I will look at these two factors and see if either one can be factored again. Does anyone think one of these expressions can be factored further? Take a minute because I want you to be able to explain why.
Through dialogue like this, I continue to progression ideas back to the launch. For example, when students factored the number 30, they factored it into 10 and 3 or 15 and 2. Then, they continued to factor the 10 or the 15 because both were still composite.
I plan to go through the second example in a similar way, letting students do most of the thinking. I will model (and steer) the train of thought that the group follows as they factor the polynomial.
The purpose of this scaffolded practice is for students to, once again, take an expression apart (factor it) and then put it back together (multiply it). This fluency with working in both directions will give students a better sense of why the polynomial expressions factor they way they do (MP8).
I have students work in pairs on this activity. I encourage students to focus on the quality of their strategy, more than the quantity of problems completed. The practical purpose of asking students to work backwards and forwards is to ensure that the factorization is correct. I encourage students to go back and check their factorization if they do not end up with the original expression after multiplying. Parenthesis are included for students on this worksheet as a scaffold for helping students remember to take out the greatest common factor before factoring the resulting trinomial or difference of perfect squares.
Teaching Note: I have included an original in word format (factor_completely_scaffolded.doc) so that the parenthesis can be removed. This way, students could factor the trinomial first and then factor the greatest common factor out of the remaining factor. Example: 2x^2+4x+2 --> (2x + 2)(x + 1) -->2(x+1)(x+1). For my students, I anticipate that most will need the more scaffolded version. The goal of the scaffolding is not to make the problem easier, it is to make it easy enough for students that they can reflect on their process and understand why these expressions can be factored in this way.
The factor_completely1_close is designed to get student's thinking down on paper. Often, students can follow a procedure without really understanding why they are doing what they are doing. A writing prompt like this will give me some insight into students' conceptual knowledge at the conclusion of today's lesson. It also gives students an opportunity to practice constructing a coherent mathematical explanation (MP3).