Lesson 9

Previously, students used area diagrams to expand expressions of the form \((x+p)(x+q)\) and generalized that the expanded expressions take the form of \(x^2 + (p+q)x +pq\). In this lesson, they see that the same generalization can be applied when the factored expression contains a sum and a difference (when \(p\) or \(q\) is negative) or two differences (when both \(p\) and \(q\) are negative).

Although they have encountered an algebraic approach, students still benefit from drawing diagrams to expand unfamiliar factored expressions. Area diagrams are intuitive for visualizing the product of two sums, but they are less intuitive for visualizing the product of two differences (for example, \((x-5)^2\)) or of a sum and a difference (for example, \((x+3)(x-4)\)). Subtraction can be represented by removing parts of a rectangle and finding the area of the remaining region, but this strategy can get complicated when both factors are differences.

At this point, students transition from thinking about rectangular diagrams concretely, in terms of area, to thinking about them more abstractly, as a way to organize the terms in each factor. (Students made similar transitions from area diagrams to abstract diagrams in middle school, for example, when they learned to distribute the multiplication of a number or a variable—positive and negative—over addition and subtraction.)

Students also learn to use the terms standard form and factored form. When classifying quadratic expressions by their form, students refine their language and thinking about quadratic expressions (MP6). In an upcoming lesson, students will graph quadratic expressions of these forms and study how features of the graphs relate to the parts of the expressions.