Math / 8th Grade / Unit 2: Solving One-Variable Equations

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.

Math

Unit 2

8th Grade

Unit Summary

Please Note: In April 2025, this unit and its lesson plans received a round of enhancements. Teachers should pay close attention as they intellectually prepare to account for the updated pacing and content.

In Unit 2, 8th 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 are equivalent 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 Reason abstractly and quantitatively. .

In 6th 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 7th grade, students fluently solved one- and two-step equations with rational numbers and used equations and inequalities to represent and solve word problems.

Students will manipulate and transform equations again in Unit 5 and Unit 6. Furthermore, these skills will continuously be used 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)

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Assessment

The following assessments accompany Unit 2.

Pre-Unit

Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.

Mid-Unit

Have students complete the Mid-Unit Assessment after Lesson 7.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Expanded Assessment Package

Use student data to drive instruction with an expanded suite of assessments. Unlock Pre-Unit and Mid-Unit Assessments, and detailed Assessment Analysis Guides to help assess foundational skills, progress with unit content, and help inform your planning.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Unit Launch

Before you teach this unit, unpack the standards, big ideas, and connections to prior and future content through our guided intellectual preparation process. Each Unit Launch includes a series of short videos, targeted readings, and opportunities for action planning to ensure you're prepared to support every student.

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.
Do all Target Tasks. Annotate the Target Tasks for:
- 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.

Unit-Specific Intellectual Prep

Read the Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations for standards relevant to this unit, pages 171–184.

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

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

The materials, representations, and tools teachers and students will need for this unit

Calculators (1 per student)

To see all the materials needed for this course, view our 8th Grade Course Material Overview.

Vocabulary and Models

Unit Vocabulary

infinite solutions

no solution

unique solution

Foundational Vocabulary

distributive property

equation

expression

inequality

like terms

solution

term

To see all the vocabulary for Unit 2, view our 8th Grade Vocabulary Glossary.

Access foundational vocabulary for Unit 2 in the same 8th Grade Vocabulary Glossary.

Lesson Map

Topic A: Simplifying Expressions and Verifying Solutions

Lesson

Write equivalent expressions using properties of operations and verify equivalence.

Standards

8.EE.C.7

Lesson

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

Standards

8.EE.C.7

Topic B: Analyzing and Solving Equations in One Variable

Lesson

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

Standards

8.EE.C.7.A8.EE.C.7.B

Lesson

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

Standards

8.EE.C.7.B

Lesson

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

Standards

8.EE.C.78.EE.C.7.B

Lesson

Model with equations using a three-act task.

Standards

8.EE.C.7

Lesson

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

Standards

8.EE.C.7

Lesson

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

Standards

8.EE.C.78.EE.C.7.A

Lesson

Solve and reason with equations with three types of solutions.

Standards

8.EE.C.78.EE.C.7.A

Lesson

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

Standards

8.EE.C.7

Topic C: Analyzing and Solving Inequalities in One Variable

Lesson

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

Standards

A.REI.B.3

Lesson

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

Standards

A.REI.B.3

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

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).

Expressions and Equations

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.

Expressions and Equations

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.

Reasoning with Equations and Inequalities

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

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Expressions and Equations

6.EE.A.1

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2

Expressions and Equations

6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3

Expressions and Equations

6.EE.A.3 — Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
6.EE.B.5

Expressions and Equations

6.EE.B.5 — Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.EE.B.7

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
7.EE.A.1

Expressions and Equations

7.EE.A.1 — Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.B.4

Expressions and Equations

7.EE.B.4 — Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.A

Expressions and Equations

7.EE.B.4.A — Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4.B

Expressions and Equations

7.EE.B.4.B — Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Future Standards

Standards in future grades or units that connect to the content in this unit

Creating Equations

A.CED.A.1

Creating Equations

A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.A.3

Creating Equations

A.CED.A.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
A.CED.A.4

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Expressions and Equations

8.EE.C.8

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Reasoning with Equations and Inequalities

A.REI.A.1

Reasoning with Equations and Inequalities

A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
A.REI.B.3

Reasoning with Equations and Inequalities

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

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.

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