In this unit, students apply their understanding of arithmetic to reason about algebraic expressions and equations.

In the first section, students work with equations of the form and where and are positive rational numbers. They use tape diagrams and hanger diagrams to reason about the meaning of equations, and to develop an understanding that to solve an equation is to find a value that would make the equation true. Students end the section by identifying, interpreting, and writing equations to represent and solve real-world problems.

In the last two sections, students analyze real-world relationships between two quantities where one quantity depends on the other. They use tables, graphs, and equations to represent and reason about such relationships.

The work here prepares students to represent quantities and relationships involving all rational numbers in a later unit, as well as to solve equations that are more complex and work with proportional relationships in grade 7.

Progression of Disciplinary Language

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

Interpret

• Tape diagrams involving letters that stand for numbers (Lesson 1).
• The parts of an equation (Lesson 5).
• Numerical expressions involving exponents (Lesson 12).
• Different representations of the same relationship between quantities (Lesson 17).

Describe

• Solutions to equations (Lesson 2).
• Stories represented by given equations (Lesson 5).
• Patterns of growth that can be represented using exponents (Lesson 12).
• Relationships between independent and dependent variables using tables, graphs, and equations (Lesson 16).

Explain

• The meaning of a solution using hanger diagrams (Lesson 3).
• How to solve an equation (Lesson 4).
• How to use equations to solve problems involving percentages (Lesson 6).
• How to determine whether two expressions are equivalent, including with reference to diagrams (Lesson 8).
• Strategies for determining whether expressions are equivalent (Lesson 14).

In addition, students are expected to compare descriptions of situations, expressions, equations, diagrams, tables, and graphs. They generalize about properties of operations and strategies for solving equations. Students also justify claims about equivalent expressions and justify reasoning when evaluating expressions.

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.

lesson	new terminology
	receptive	productive
6.6.1	value (of a variable)	operation
6.6.2	variable coefficient solution to an equation true equation false equation	value (of a variable)
6.6.3	each side (of an equal sign) balanced hanger diagram
6.6.4	solve (an equation)	each side (of an equal sign)
6.6.6		true equation false equation
6.6.8	equivalent expressions commutative property
6.6.9	distributive property area as a product area as a sum
6.6.10	term	equivalent expressions
6.6.12	to the power
6.6.13		to the power exponent
6.6.15		solution to an equation
6.6.16	independent variable dependent variable horizontal axis vertical axis	variable relationship
6.6.17	coordinates
6.6.18	plot