I will begin with the essential question: How can we use the distributive property to factor expressions? Here it would be worth while to discuss factors by asking simple questions like: What are the factors of 6? -8? etc.... The idea here is for students to see a "complex" structure like 4 * 2 being the same thing as 8. This will lead students to see the factored version of 8x + 12 --> 4(x + 3) as the product of two factors 4 and (x+3). This mathematical practice (MP7) goes throughout the lesson.
The video here (minutes 1:40 - 3:20) presents a good visualization of the area model and how it relates to factoring. I will either show the video or make the model myself for the first couple of examples. In my opinion, the model works very well for positive values but gets a little trickier with negative values. I will not show the model for negatives.
I will then present two examples. I will encourage students to factor using the GCF or the common factor with the greatest magnitude, but I will not insist on it. The point is for students to apply properties to generate equivalent expressions as the standard states.
To check our work, we will substitute an "easy" number - 0 or 1 - into each expression. If we have factored correctly, each expression should evaluate to the same value.
The first problem uses all positive integers so it might be considered a 6th grade item. The remainder of the problems use rational numbers that are not all positive so these speak to 7th grade work. At this point in my students progress with the common core, I am most concerned with their ability to factor expressions involving negative integers, but I have included a fraction and decimal expression.
When there is a negative coefficient, I will encourage my students to factor out a negative factor. I will also have my students write any differences as sums before factoring. Again, this is more clear to me as opposed to referring to 18x as a negative value in 12 - 18x. I know many mathematicians and teachers are comfortable with this but I am not.
I love the last problem. Students are given an area (15x + 40) and a width of a rectangle 5. They must find the length. As students struggle, it may help to subdivide the rectangle into two rectangles.
Students now work independently. I'll pull aside students who were still struggling with the previous two sections of work. This first 7 problems are virtually identical to the last 7 problems. Students who are working independently should first refer to the previous section before asking me for help. That being said, I will need to make sure students are showing work as I model it so that they have a good reference.
Problem #8 is similar to problem 7 but it deals with money as opposed to area.
The last problem mirrors a task that students will see on the final assessment. There are many ways to solve this and I hope to see many strategies put to use: 1) factoring 2) expanding 3) substituting values. When reviewing this problem, I will make sure to point out these various strategies.
The exit ticket has 3 problems. Before we begin, we should discuss how do we make sure our work is correct. The method used throughout the lesson is to substitute values into each expression and evaluate.
Every student should be able to answer the first question; it involves positive numbers and the only common factor greater than 1 is 5. The second problem involves a negative value. I will accept any factored expression that can be derived. The third problem is similar to the last problem of the last section. It can be solved in a variety of ways. Students who do not read carefully may overlook the work "NOT" even though it is bold and capitalized! It may be worth pointing this out to students before beginning the exit ticket.