Lesson 9 | Solving One-Variable Equations | 8th Grade Mathematics

Objective

Solve and reason with equations with three types of solutions.

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

Determine, without completely solving, if an equation has a unique solution, no solution, or infinite solutions.
Change aspects of an equation to change the type of solution it has.
Solve one-variable equations of all types.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In this lesson, students use what they know about equations with no, infinite, or unique solutions to reason about solutions without having to completely solve the equation. For example, if they can simplify each side of an equation to one or two terms, they should be able to determine if a solution is possible and if so, if that solution is unique (MP.7).

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Anchor Problems

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

Problem 1

The equation below has a unique solution.

$${2(3x-4)-(x-7)=-6(x+1)+7x}$$

a. Change the equation so that it has no solution.

b. Change the original equation so that it has infinite solutions.

Guiding Questions

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Problem 2

The equation below is incomplete.

$${{1\over3}(2{x+}18-3)+x=}$$ _____ $${x+}$$ _____

Complete the equation, by filling in the blanks, to create an equation with:

a. A unique solution

b. No solution

c. Infinite solutions

Guiding Questions

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Problem 3

Without solving them, say whether these equations have a positive solution, a negative solution, a zero solution, or no solution.

a. $${3x=5}$$

b. $${5z+7=3}$$

c. $${7-5w=3}$$

d. $${4a=9a}$$

e. $${y=y+1}$$

Guiding Questions

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References

Illustrative Mathematics The Sign of Solutions

The Sign of Solutions, accessed on Aug. 31, 2017, 3:33 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

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

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Target Task

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

The left side of an equation is shown below.

$${-x-7+6\left ( {1\over2}x+2{1\over2} \right )=}$$ _________________

Each expression below can be placed on the right side of the equation to complete the equation.

$${2(x+4)}$$

$${2({x+8})}$$

$${2x+15}$$

$${-2{x+8}}$$

$${x+8}$$

$${8+2x}$$

Which expressions, when placed on the right side of the equation, create an equation with:

No solution
Infinite solutions
A unique solution

Student Response

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

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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.

Include reasoning problems similar to Anchor Problems.
Include procedural practice with solving all types of equations with any type of solution.

RDA Performance Task Bank SBAC Item MAT.08.ER.3.000EE.D.137 Claim 3

Lesson 8

Lesson 10

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

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

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