This lesson is the first of two that apply new understanding of algebraic expressions and equations to represent relationships between two quantities. Students use tables, graphs, and equations that represent these relationships and make connections across these representations. 

In the first activity, students analyze a relationship that can be defined by addition or subtraction. The context, about the difference of two measurements, is one that students encountered earlier in the unit. At that point, students wrote expressions in one variable that can be used to find a quantity when the relationship to the other quantity is known. Here, they write equations that relate two quantities and examine the graphs that can represent the relationship.

In the second activity, students revisit and extend their understanding of equivalent ratios. A familiar scenario of mixing paints in a given ratio provides the context for writing equations that represent a multiplicative relationship between two quantities. Students then create a table of values that shows how changes in one quantity affect changes in the other, and graph the points from the table on a coordinate grid.

Students learn that relationships between the two quantities can be described by two different but related equations with one quantity, the dependent variable, affected by changes in the other quantity, the independent variable. In mathematical modeling, which variable is considered independent and which is considered dependent is often the choice of the modeler, although sometimes the situation suggests choosing one way over the other. The contexts in this lesson were chosen because they do not suggest which quantity should be selected as the independent variable.