In this section, students work with linear expressions and justify whether they are equivalent. Having learned to perform arithmetic with rational numbers, students are now prepared to examine more complex expressions.

First, students see that rewriting subtraction as adding the opposite makes it possible to apply properties of addition, such as the associative or commutative property, to generate equivalent expressions....

In this section, students focus on writing equivalent equations. They build a list of moves that can be used to write equivalent equations. Students begin by examining hanger diagrams as an intuitive way of understanding equivalence and the moves that maintain equivalence. These moves are described by labeling arrows that connect equivalent representations.

Then students reach the limits of hanger...

In this section, students are shown equations that do not have a single solution. Students recognize that some equations have no solutions and others have infinitely many solutions, then find conditions that indicate how many solutions an equation has without completely solving it. This work also serves as a transition to examining systems of equations in which similar situations arise...