Linear Expressions & Single-Variable Equations/Inequalities

Students become proficient at manipulating and solving single-variable linear equations and inequalities, and using them to model and interpret contextual situations.

Math

Unit 3

9th Grade

Unit Summary

In Unit 3, students become proficient at manipulating and solving single-variable linear equations and inequalities, as well as using linear expressions to model contextual situations. Domain and range are introduced again through the lens of a “constraint” with inequalities. The understanding students develop in this unit builds the foundation for Unit 4, Unit 5, and Unit 6, as well as provides an algebraic outlet for modeling contextual situations started in Unit 1 and continued in Unit 2.

Unit 3 begins in Topic A with a review of creating equivalent expressions and equations, as well as solving equations using the properties of operations and equality. Students are introduced to the concept of a domain restriction by understanding that dividing by zero produces an undefinable expression. Students develop their understanding of how to construct viable arguments in producing equivalent equations and expressions, as well as making use of structure to efficiently manipulate and solve expressions and equations, using repeated reasoning to develop properties, and attend to precision in solutions. Topic B builds on this conceptual understanding and fluency with expressions and equations to develop algebraic models for contextual situations. Students begin by using pre-defined variables to model simple contextual situations and build up to identifying quantities, estimating constants, defining variables, and writing and revising algebraic models to illustrate and make generalizations about a contextual situation. This section of the unit has several opportunities to model contextual situations. Topic C expands students’ understanding of constraints through single-variable inequalities and requires students to make strong connections between algebraic, contextual, and graphical representations.

Pacing: 15 instructional days (12 lessons, 1 flex day, 1 assessment day)

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Assessment

The following assessments accompany Unit 3.

Post-Unit

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

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- 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

Essential Understandings

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

Expressions, equations, and inequalities are effective tools to generalize the relationships in contextual situations. Modeling a situation by defining and relating variables in the context situation provides tools to understanding and drawing conclusions about a contextual situation.
Properties of expressions, equations, and inequalities can be used to reveal solutions, structure, and relationships between quantities in contextual and non-contextual situations. Establishing equivalence between expressions and equations is the key to revealing these features.
Constraints on a model can be determined algebraically, by understanding what produces undefined solutions, or within context, by understanding what solutions are not viable in real life.

Vocabulary

Terms and notation that students learn or use in the unit

Commutative property of addition and multiplication	Associative property	Distributive property
domain restriction	coefficient	variable
term	equivalence	properties of operations
Proeprties of Equality	solve for [variable/quantity]	compound inequality (and/or)
solution

Lesson Map

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

Lesson

Identify properties of operations that result in equivalent linear expressions.

Standards

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

Lesson

Use properties of equations to analyze and write equivalent equations.

Standards

A.REI.A.1

Lesson

Solve single-variable linear equations using properties of equality.

Standards

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

Lesson

Solve equations with a variable in the denominator.

Standards

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

Lesson

Solve for a variable in an equation or formula.

Standards

A.CED.A.4

Topic B: Modeling with Single-Variable Linear Equations

Lesson

Write equations using defined variables to represent a contextual situation.

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

2 days

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

Standards

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

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Lesson

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

Standards

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

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Building Functions

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Creating Equations

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.
Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
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.
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.

High School — Number and Quantity

High School — Number and Quantity

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Interpreting Functions

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).
Interpreting Functions

F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Reasoning with Equations and Inequalities

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.
Reasoning with Equations and Inequalities

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

Seeing Structure in Expressions

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

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

Expressions and Equations

Expressions and Equations

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

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

Functions

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.
Functions

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

High School — Number and Quantity

High School — Number and Quantity

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Future Standards

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

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Building Functions

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Creating Equations

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.
Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
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.
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.

Interpreting Functions

Interpreting Functions

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions

F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

A.REI.C.5 — Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Reasoning with Equations and Inequalities

A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Reasoning with Equations and Inequalities

A.REI.D.10 — Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Reasoning with Equations and Inequalities

A.REI.D.11 — Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

Seeing Structure in Expressions

A.SSE.A.1.B — Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

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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