In this unit, students deepen their algebraic reasoning as they write and solve equations of the forms and and inequalities of the forms and \(p(x+q). Students also work with equivalent expressions that are more complex than what they have seen previously. This builds on grade 6 work with equations of the form or and with simpler equivalent expressions. Students will build on this work in grade 8 when they solve equations that have a variable on both sides of the equal sign and when they work with systems of equations.

Students begin the unit by making sense of situations that involve both multiplication and addition. They represent such situations with tape diagrams and with equations. They see that different diagrams and equations can represent the same situation, and they use diagrams to find solutions to equations.

Next, students consider hanger diagrams as another way to represent equations. The diagrams help students understand solving equations in terms of “doing the same thing to each side of the equation.” Students examine different pathways for solving the same equation and consider whether one method is more efficient than another.

Balanced hanger. Left side, circle labeled x and square labeled 2, the square appears to be loose from the hanger. Right side, rectangle labeled 4 and square labeled 2, the square appears to be loose from the hanger. To the side, an equation says x = 4.

Then students apply what they have learned about equations to inequalities. They write inequalities to represent situations and solve inequalities by reasoning about the related equation. The inequality symbols and are introduced.

Lastly, 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.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as comparing, explaining, and justifying. 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:

Compare

• Stories with corresponding tape diagrams (Lesson 2).
• Tape diagrams with corresponding equations (Lesson 3).
• Hanger diagrams and equations (Lesson 7).
• Solution pathways (especially Lesson 10).
• Descriptions of situations with corresponding inequalities (Lesson 16).

Explain

• Strategies for using hanger diagrams to solve equations (Lesson 8).
• Different strategies for solving equations (Lesson 9) and inequalities (Lesson 14).
• Reasoning about situations, tape diagrams, and equations (Lesson 12).
• Strategies for identifying and writing equivalent expressions (Lesson 21).

Justify

• Reasoning about inequalities (Lesson 13).
• Reasoning about solutions to inequalities (Lesson 15).
• The need for specific information in order to write and solve inequalities (Lesson 17).
• Reasoning about the distributive property (Lesson 19).
• Whether different sequences of calculations give the same result (Lesson 22).

In addition, students are expected to interpret solutions to equations, interpret and represent nonproportional situations with constant rates of change, represent nonproportional situations using tape diagrams, describe the structure of equations and tape diagrams, critique reasoning of peers about expressions and corresponding diagrams, critique reasoning about solving equations, critique reasoning about equivalent expressions, and generalize about solving equations and about when expressions are equivalent.

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.