In this unit, students work with writing equivalent equations and use reasoning to solve equations for a variable. Then students solve systems of linear equations using graphic and algebraic methods.

The unit begins with a focus 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.

Next, 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. Students finish the unit by examining systems of equations graphically and then finding solutions algebraically. They build on their understanding that the line representing an equation with 2 variables is made up of coordinate pairs that make the equation true. They find that the intersection of 2 lines is the point that makes both equations for the system true. Students also recognize when systems have no solution or infinite solutions based on the graphs and the slope and intercept.

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

• Strategies for writing equivalent equations (Lesson 1).
• Reasoning about maintaining balance in equations (Lesson 3).
• Solutions of linear equations (Lessons 4 and 5).
• Reasoning about structures of systems of equations (Lesson 14).
• Explanations of solutions (Lesson 16).

Justify

• Strategies for writing equivalent equations (Lessons 1 and 5).
• Predictions about maintaining balance (Lesson 2).
• Predictions about solutions of linear equations (Lesson 6).

Generalize

• About the structures of equations that have one, infinite, and no solutions (Lessons 7 and 8).
• About the structures of systems of equations (Lessons 14 and 15).

In addition, students are expected to use language to represent and interpret situations involving systems of linear equations, compare solutions of linear equations, and describe graphs of systems of linear 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.