# Solving One-Variable Equations

Students hone their skills of solving multi-step equations and inequalities, redefining their definition of "solution" to include cases such as infinite solutions, and interpreting solutions in context.

## Unit Summary

In Unit 2, eighth-grade students hone their skills of solving equations and inequalities. They encounter complex-looking, multi-step equations, and they discover that by using properties of operations and combining like terms, these equations boil down to simple one- and two-step equations. Students also discover that there are many different ways to approach solving a multi-step equation, and they spend time closely looking at their own work and the work of their peers. When solving an equation with variables on both sides of the equal sign, students are challenged with results such as 4=5, and they refine their definition of “solution” to take into account such examples. Throughout this unit, students use equations as models to capture real-world applications. They reason abstractly and quantitatively as they de-contextualize situations to represent them with symbols and then re-contextualize the numbers to make sense in context (MP.2).

In sixth grade, students developed the conceptual understanding of how the components of expressions and equations work. They learned how the distributive property can create equivalent forms of an expression and how combining like terms can turn an expression with three terms into an expression with one term. By the end of seventh grade, students fluently solved one- and two-step equations with rational numbers and used equations and inequalities to represent and solve word problems.

Students’ ability to manipulate and transform equations will be required again in Unit 5: Linear Relationships and Unit 6: Systems of Linear Equations. Furthermore, these skills will be needed throughout high school as students are introduced to new types of equations involving radicals, exponents, multiple variables, and more.

Pacing: 16 instructional days (12 lessons, 3 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 8th Grade Scope and Sequence Recommended Adjustments. • Unit Launch
• Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

## Assessment

This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

?

#### Internalization of Standards via the Post-Unit Assessment

• Take the Post-Unit Assessment. Annotate for:
• Standards that each question aligns to
• Strategies and representations used in daily lessons
• Relationship to Essential Understandings of unit
• Lesson(s) that Assessment points to

#### Internalization of Trajectory of Unit

• Read and annotate the Unit Summary.
• Notice the progression of concepts through the unit using the Lesson Map.
• Essential Understandings
• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

### Essential Understandings

?

• Solving an equation or inequality involves changing the equation into equivalent, simpler forms in order to reveal possible values for the variable.
• Solving an equation or inequality involves using properties of operations, combining like terms, and using inverse operations.
• An equation in one-variable may have a unique solution for the variable, no possible solutions for the variable, or all possible values may be solutions for the variable.

### Materials

?

• Calculators (1 per student)

### Vocabulary

?

infinite solutions

combine like terms

term

distribute a negative

properties of operations

inequality

expression

solution

unique solution

equation

distributive property

no solution

break even

To see all the vocabulary for this course, view our 8th Grade Vocabulary Glossary.

# 1

8.EE.C.7

Write equivalent expressions using properties of operations and verify equivalence using substitution.

# 2

8.EE.C.7

Define a solution to an equation. Solve and check solutions to 1 and 2 step equations.

# 3

8.EE.C.7.A

8.EE.C.7.B

Justify each step in solving a multi-step equation with variables on one side of the equation.

# 4

8.EE.C.7.B

Write and solve multi-step equations to represent situations, with variables on one side of the equation.

# 5

8.EE.C.7

Model with equations using a three-act task.

# 6

8.EE.C.7

Solve equations with variables on both sides of the equal sign.

# 7

8.EE.C.7

Write and solve multi-step equations to represent situations, including variables on both sides of the equation.

# 8

8.EE.C.7.A

Understand that equations can have no solutions, infinite solutions, or a unique solution; classify equations by their solution.

# 9

8.EE.C.7.A

Solve and reason with equations with three types of solutions.

# 10

8.EE.C.7

Use equations to model a business plan and determine the break-even point.

# 11

A.REI.B.3

Solve and graph inequalities with variables on one side of the inequality.

# 12

A.REI.B.3

Solve and graph inequalities with variables on both sides of the inequality.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

?

##### Expressions and Equations
• 8.EE.C.7 — Solve linear equations in one variable.

• 8.EE.C.7.A — Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

• 8.EE.C.7.B — Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

##### Reasoning with Equations and Inequalities
• A.REI.B.3 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

?

• 6.EE.A.2

• 6.EE.A.3

• 6.EE.B.5

• 6.EE.B.7

• 7.EE.A.1

• 7.EE.B.4

• 7.EE.B.4.A

• 7.EE.B.4.B

?

• A.CED.A.1

• A.CED.A.3

• A.CED.A.4

• 8.EE.C.8

• A.REI.A.1

• A.REI.B.3

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.