Lesson 4 | Linear Expressions & Single-Variable Equations/Inequalities | 9th Grade Mathematics

Objective

Solve equations with a variable in the denominator.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Interpreting Functions

F.IF.A.1 — Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

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

The foundational standards covered in this lesson

8.EE.C.7.B

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.
8.F.A.1

Functions

8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.

Criteria for Success

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

Describe that dividing by zero produces an expression that is undefined.
Describe that when a variable is the divisor, there is a “domain restriction” on possible values for the variable so the expression is defined.
Identify the domain restriction when there is a variable in the denominator.
Identify equations with no solutions, based on a domain restriction of one side of the equation.
Efficiently solve an equation with a variable in the denominator.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

A common misconception can arise from students treating terms in the numerator or denominator separately instead of as grouped expressions when operating. Ensure students use parentheses around expressions in the numerator or denominator of an expression when solving to minimize this misconception.

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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

Solve for $$x$$, if possible.

a. $$\frac{x}{4}=5$$

b. $$\frac{x}{\frac{1}{2}}=6$$

c. $$\frac{x}{0}=10$$

Guiding Questions

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Problem 2

Solve both of the equations below.

$${10=\frac{x-2}{5}}$$

$${10=\frac{5}{x-2}}$$, where $${x\neq2}$$

Guiding Questions

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Problem 3

Identify any domain restrictions and solve the equation below.

$${\frac{3}{x}=\frac{2}{x+2}}$$

Guiding Questions

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Problem Set

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

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

Problem 1

Solve for $$x$$.

$$\frac{x-2}{x-9}=2$$

References

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 18 — Exit Ticket

Algebra I > Module 1 > Topic C > Lesson 18 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Write an equation that would have the restriction $${x\neq-3}$$.

References

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 18 — Exit Ticket

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 problems where students need to identify domain restrictions in an expression but not solve an equation.
Include problems where the domain restriction is the only solution to the equation. Ensure students are identifying that the solution is undefined.

EngageNY Mathematics Algebra I > Module 1 > Topic C > Lesson 18 — All Exercises and Problem Set problems (do not encourage students to solve equations as a system of equations).
Illustrative Mathematics Increasing or Decreasing? Variation 1. — This problem is repeated from Lesson 2
Illustrative Mathematics How Does the Solution Change? — This problem is repeated from Lesson 2. Ask students about domain restrictions for variables in this problem.

Lesson 3

Lesson 5

Lesson Map

Topic A: Properties and Solutions of Single-Variable Linear Expressions and Equations

Identify properties of operations that result in equivalent linear expressions.

A.REI.A.1 A.SSE.A.1 A.SSE.A.2

Use properties of equations to analyze and write equivalent equations.

A.REI.A.1

Solve single-variable linear equations using properties of equality.

A.REI.A.1 A.REI.B.3

Solve equations with a variable in the denominator.

A.REI.B.3 F.IF.A.1

Solve for a variable in an equation or formula.

A.CED.A.4

Topic B: Modeling with Single-Variable Linear Equations

Write equations using defined variables to represent a contextual situation.

A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Define variables; write and solve equations to represent a contextual situation.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Write and solve equations to represent contextual situations where estimations and unit conversions are required.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

2 days

Model a contextual situation and make an informed decision based on the model.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Topic C: Properties and Solutions of Single-Variable Linear Inequalities

Solve unbounded single-variable inequalities in contextual and non-contextual situations.

A.CED.A.3 A.REI.A.1 A.REI.B.3 A.SSE.B.3

Write and graph compound single-variable inequalities to describe the solution to contextual and non-contextual situations.

A.CED.A.3 A.REI.B.3

Solve and graph compound inequalities where algebraic manipulation is necessary in contextual and non-contextual situations.

A.CED.A.3 A.REI.A.1 A.REI.B.3

Linear Expressions & Single-Variable Equations/Inequalities

Objective

Common Core Standards

Core Standards

Interpreting Functions

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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