I really enjoyed doing this activity with my students. It starts with a background story of a housing developer wanting to change the lot sizes for a new housing development. Instead of all the lots being square the city planner wants some of other types of rectangles. The questions tell the students exactly how to change the dimensions of the lots. The first 4 are situations where they increase the size of the lots and the last 2 involve a decrease in the length of at least one side.
Mrs. New had already taught this lesson and showed me the way the teacher's guide recommended the sketch of the lots be drawn. These diagrams will look like "the box method" that they will use to multiply and factor polynomials. The activity asks the students to express the area as the product of the length and the width (which will be binomials) or as a sum of smaller areas. Since the 2 areas are equivalent the students are led to realize that the 2 expressions are equal. I aksed them to look for connections between the sum and the product and a few of them saw it. I love that the authors have once again provided a context for the formal math to make sense to them!
I led the students through #1 so I could model how to sketch the diagram with the original side length of X. In #5 and #6 I let them come up with their own diagrams to represent the situation. Also, just for the sake of organizing, I labeled the bullets as A, B, and C so that it would be a little easier to organize and discuss.