Lesson 8 | Solving One-Variable Equations | 8th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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).

Foundational Standards

The foundational standards covered in this lesson

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Understand that if an equation has no solution, then there is no value for $$x$$ that will make the equation true; for example $${ x+2=x+3}$$ has no solution.
Understand that if an equation has infinite solutions, then $$x$$ can represent any value and it will make the equation true; for example $$ x+1=x+1$$ has infinite solutions.
Understand that if an equation has a unique solution, then there is one value for $$x$$ that will make the equation true; for example $$ x+1=3$$ has a unique solution of $$x=2$$.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Lessons 8 and 9 introduce students to the three types of solutions that an equation can have. This lesson focuses on students understanding what each solution means about the equation, and on becoming familiar with arriving at solutions that look like $${4=2}$$ or $${-8=-8}$$ (MP.2). In the next lesson, students will reason with all three types of solutions to further internalize what they mean. The concept of no solution, infinite solutions, and unique solution will re-appear when students study systems of linear equations in Unit 6.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

These scales are all currently balanced. You must choose one number to fill into the boxes in each problem that will keep them balanced. In each individual box you may only use one number, and it must be the same number in each box for that problem.

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Everybody is a Genius Solving Special Case Equations

Solving Special Case Equations is made available on Everybody is a Genius under the CC BY-NC-SA 3.0 US license. Accessed Aug. 31, 2017, 2:41 p.m..

Problem 2

Sort the equations below into the three categories.

a. $${3x=0}$$

b. $${3x=2x}$$

c. $${3x=2x}+x$$

d. $${2x+1=2x+2}$$

e. $${3x-1=2x-1}$$

f. $${x+2=2+x}$$

g. $${x+2=x-2}$$

h. $${2x-3=3-2x}$$

Infinite Solutions

Unique Solution

No Solution

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem 3

Solve the equations below and explain what each solution means.

a. $${5x-2\left ({1\over2}+x \right ) + 8 = - {1\over4}(16-12x)}$$

b. $${1.6(x+4)-3(0.2x+1)=2x-{1\over2}(18.6)}$$

c. $${{{{1\over2}x +4}\over-1} = {{-x-8}\over2}}$$

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Fishtank Plus Content

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Todd and Jason both solved an equation and ended up with this final line in their work: $${{-2}x=4x}$$.

Todd says, “This equation has no solution because $${{-2}\neq 4}$$.”
Jason says, “The solution is $${-2}$$ because $${-2}({-2})=4$$.”

Do you agree with either Todd or Jason? Explain your reasoning.

Student Response

An example response to the Target Task at the level of detail expected of the students.

Create a free account or sign in to view Student Response

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

MARS Formative Assessment Lesson for Grade 8 Solving Linear Equations in One Variable — This lesson has a great sorting activity, as well as problems where students examine and critique the reasoning of others.
EngageNY Mathematics Grade 8 Mathematics > Module 4 > Topic A > Lesson 7 — Exercises and Problem Set

Lesson 7

Lesson 9

Lesson Map

Topic A: Simplifying Expressions and Verifying Solutions

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

8.EE.C.7

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

8.EE.C.7

Topic B: Analyzing and Solving Equations in One Variable

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

8.EE.C.7.A 8.EE.C.7.B

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

8.EE.C.7.B

Model with equations using a three-act task.

8.EE.C.7

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

8.EE.C.7

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

8.EE.C.7

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

8.EE.C.7.A

Solve and reason with equations with three types of solutions.

8.EE.C.7.A

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

8.EE.C.7

Topic C: Analyzing and Solving Inequalities in One Variable

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

A.REI.B.3

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

A.REI.B.3

Solving One-Variable Equations

Objective

Common Core Standards

Core Standards

Expressions and Equations

Foundational Standards

Expressions and Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy