By now students have encountered a variety of situations that can be modeled with quadratic functions. They are also familiar with some features of the expressions, tables, and graphs that represent such functions. This lesson transitions students from reasoning concretely and contextually about quadratic functions to reasoning about their representations in ways that are more abstract and formal (MP2).
In earlier grades, students reasoned about multiplication by thinking of the product as the area of a rectangle where the two factors being multiplied are the side lengths of the rectangle. In this lesson, students use this familiar reasoning to expand expressions such as \((x+4)(x+7)\), where \(x+4\) and \(x+7\) are side lengths of a rectangle with each side length is decomposed into \(x\) and a number. They use the structure in the diagrams to help them write equivalent expressions in expanded form, for example, \(x^2 +11x + 28\) (MP7). Students recognize that finding the sum of the partial areas in the rectangle is the same as applying the distributive property to multiply out the terms in each factor.
After this lesson, students move to think more abstractly about such diagrams. Rather than reasoning in terms of area, they use the diagrams to organize and account for all terms when applying the distributive property.
The terms “standard form” and “factored form” are not yet used and will be introduced in an upcoming lesson, after students have had some experience working with the expressions.
- Use area diagrams to reason about the product of two sums and to write equivalent expressions.
- Use the distributive property to write equivalent quadratic expressions.
- Let’s use diagrams to help us rewrite quadratic expressions.
- I can rewrite quadratic expressions in different forms by using an area diagram or the distributive property.