In this lesson, students see how to use the distributive property to write a compact expression for situations where one quantity is described in relation to another quantity in language such as “half as much again“ and “one third more than.“ If  is half as much again as , then . Using the distributive property, this can be written as . Students apply this sort of reasoning to various situations and other fractional amounts of increase or decrease.

As students investigate multiple examples to recognize the relationship , they are making use of repeated reasoning (MP8). As students look for opportunities to use the distributive property to write equations in a simpler way, they are making use of structure (MP7).

The last activity is optional because it provides an opportunity for additional practice relating descriptions, tape diagrams, and equations.