Solving One-Variable Equations

Students hone their skills of solving multi-step equations and inequalities, redefining their definition of "solution" to include results such as 4 = 5, 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 include 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 2020-2021 school year due to school closures, see our 8th Grade Scope and Sequence Recommended Adjustments.

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Texts and Materials

Supporting Materials

Inequalities
Equations

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 Unit Assessment

Take 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 “Unit Summary.”
Notice the progression of concepts through the unit using “Unit at a Glance”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions
Identify key opportunities to engage students in academic discourse. Read through our Guide to 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.

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.

Unit Materials, Representations and Tools

calculators

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

Related Teacher Tools:

8th Grade Vocabulary Glossary

Lesson Map

Topic A: Simplifying Expressions and Verifying Solutions

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.

Topic B: Analyzing and Solving Equations in One Variable

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.

Topic C: Analyzing and Solving Inequalities in One Variable

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.

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

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

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.

Solving One-Variable Equations

Unit Summary

Texts and Materials

Supporting Materials

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Unit Materials, Representations and Tools

Vocabulary

Related Teacher Tools:

Lesson Map

Topic A: Simplifying Expressions and Verifying Solutions

1

2

Topic B: Analyzing and Solving Equations in One Variable

3

4

5

6

7

8

9

10

Topic C: Analyzing and Solving Inequalities in One Variable

11

12

Common Core Standards

Core Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Future Standards

Creating Equations

Creating Equations

Creating Equations

Creating Equations

Expressions and Equations

Expressions and Equations

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Standards for Mathematical Practice

Unit 1

Exponents and Scientific Notation

Unit 3

Transformations and Angle Relationships