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.

Unit Summary

In Unit 3, Linear Expressions & Single-Variable Equations/Inequalities, 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. 

Unit Prep

Essential Understandings

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  • 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

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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    

Intellectual Prep

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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 

Assessment

This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.

Lesson Map

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

Topic B: Modeling with Single-Variable Linear Equations

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

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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

  • 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 — 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
  • 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
  • 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).

  • 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
  • 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 — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

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.

  • 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).

  • 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

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Expressions and Equations
  • 7.EE.A.1

  • 7.EE.B.4.B

  • 8.EE.C.7

  • 8.EE.C.7.A

  • 8.EE.C.7.B

Functions
  • 8.F.A.1

  • 8.F.B.4

High School — Number and Quantity
  • N.Q.A.1

Future Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.A.1

Building Functions
  • F.BF.A.1

Creating Equations
  • A.CED.A.1

  • A.CED.A.2

  • A.CED.A.3

  • A.CED.A.4

Interpreting Functions
  • F.IF.B.4

  • F.IF.B.5

  • F.IF.C.9

Reasoning with Equations and Inequalities
  • A.REI.C.5

  • A.REI.C.6

  • A.REI.D.10

  • A.REI.D.11

Seeing Structure in Expressions
  • A.SSE.A.1.B

  • A.SSE.A.2

  • A.SSE.B.3

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.