In this unit, students work with writing equivalent expressions and use reasoning to solve equations, including equations that have a variable on both sides of the equal sign. This builds on students’ previous work solving equations of the form or . Students will build on this work in future units when they solve systems of linear equations.

First, students work with equivalent linear expressions that are more complex due to having more terms, more parentheses, and negative rational numbers. Students use properties of operations to justify why the expressions are equivalent.

Next, the unit focuses on moves that can be done to write equivalent equations. At first, students use hanger diagrams as an intuitive representation of equality and represent their reasoning by labeling arrows that connect equivalent representations. With the reintroduction of negative values, students move away from hanger diagrams to algebraic equations and writing equivalent equations with the intention of solving for a variable.

Lastly, students examine the conditions under which equations could have 0, 1, or infinite solutions as a transition to thinking about similar situations involving systems of equations.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as critiquing, justifying, and generalizing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Critique

• Reasoning of peers about expressions and corresponding diagrams (Lesson 1).
• Reasoning about equivalent expressions (Lesson 4).
• Reasoning about maintaining balance in equations (Lesson 6).
• Solutions of linear equations (Lessons 7 and 8).

Justify

• Reasoning about the distributive property (Lesson 2).
• Strategies for writing equivalent equations (Lesson 8).
• Predictions about maintaining balance (Lesson 5).
• Predictions about solutions of linear equations (Lesson 9).
• Whether different sequences of calculations give the same result (Lesson 12).

Generalize

• About when expressions are equivalent (Lesson 3).
• About the structures of equations that have one, infinite, and no solutions (Lessons 10 and 11).

In addition, students are expected to use language to explain strategies for identifying and writing equivalent expressions, represent situations using equations, compare solutions of linear equations, and compare features of equations.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.