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 TITLE : Unit 07: One-Variable Inequalities SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
This unit bundles student expectations that address modeling, writing, solving, and representing solutions for one-variable, one-step inequalities. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology."

Prior to this Unit
In Unit 06, students generated equivalent numerical expressions using order of operations, including whole number exponents and prime factorization. Students were formally introduced to algebraic expressions and expected to generate equivalent algebraic and numeric expressions using properties of operations. Students were also expected to determine if two expressions were equivalent using concrete models, pictorial models, and algebraic representations. Students distinguished between expressions and equations verbally, numerically, and algebraically. They analyzed a problem situation and wrote a one-variable, one-step equation as well as a corresponding real-world problem when given a one-variable, one-step equation. Students modeled and solved one-variable, one-step equations from problems, including those involving geometric concepts, and represented those solutions on number lines. Students also determined if a given value(s) made a one-variable, one-step equation true.

During this Unit
Students transition from one-variable, one-step equations to one-variable, one-step inequalities. This is the first time students are formally introduced to algebraic inequalities. Constants or coefficients of one-variable, one-step inequalities may include positive rational numbers or integers. Students are expected to analyze constraints or conditions within a problem situation and write a one-variable, one-step inequality to represent the situation. Students are also expected to write a corresponding real-word problem when given a one-variable, one-step inequality. Concrete models, pictorial models, and algebraic representations are used again as students model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Although certain models, such as algebra tiles, may limit a student’s ability to model solving an inequality with whole number or integer constants or coefficients, students should also solve inequalities with positive rational number constants or coefficients with an algebraic model. Students are expected to represent their solution on a number line as well as determine if a given value(s) make(s) the one-variable, one-step inequality true.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 6

After this Unit
In Unit 08, students will examine two-variable relationships within algebraic representations, and in Unit 09, students will extend geometric inequality relationships to include the Triangle Inequality Theorem. In Grade 7, students will model, write, solve, and represent solutions for one-variable, two step equations and inequalities. Grade 8 students will write one-variable equations and inequalities with variables on both sides that represent problems with rational number coefficients and constants. Additionally, students will write a corresponding real-world problem situation when given an equation or inequality with variable on both sides.

In Grade 6, writing one-variable inequalities to represent constraints or conditions within problems and representing solutions on a number line, and writing corresponding real-world problems given a one-variable, one-step inequality are identified as STAAR Supporting Standard 6.9A, 6.9B, and 6.9C. Modeling and solving one-variable, one-step inequalities and determining if given values make an inequality true are identified as STAAR Readiness Standard 6.10A and STAAR Supporting Standard 6.10B. These standards are part of the Grade 6 STAAR Reporting Category: Computations and Algebraic Relationships. All of these standards are subsumed under Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Using expressions and equations to represent relationships in a variety of contexts. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning A1, C1, C2, C3, D1, D2; III. Geometric and Spatial Reasoning C1; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (2000), “most students will need extensive experience in interpreting relationships among quantities in a variety of problem contexts before they can work meaningfully with variables and symbolic expressions. An understanding of the meanings and uses of variables develops gradually as students create and use symbolic expressions and relate them to verbal, tabular, and graphical representations” (p. 225). Van de Walle, Karp, and Bay-Williams (2010) assert that, “inequalities are also poorly understood and have not received the attention that the equal sign has received, likely because inequalities are not as prevalent in the curriculum or in real like, Still, understanding and using inequalities is important and will require significant time and experiences to develop” (p. 227).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Inequalities can be modeled, written, and solved using various methods to gain insight into the context of the situation and make critical judgments about algebraic relationships and flexible, efficient strategies.
• Why are expressions considered foundational to inequalities?
• How are constraints or conditions within a problem situation represented in an inequality?
• How does the context of a problem situation, relationships within and between operations, and properties of operations aid in writing an inequality to represent the problem situation?
• How can a(n) …
• concrete model
• pictorial model
• algebraic representation
… be used to represent and solve an inequality?
• What models effectively and efficiently represent how to solve inequalities?
• What is the process for solving an inequality, and how can the process be …
• described verbally?
• represented algebraically?
• When considering inequalities, …
• why is the variable isolated in order to solve?
• how are negative values represented in concrete and pictorial models?
• how does a negative coefficient affect the equality or inequality symbol when solving?
• why must the solution be justified in terms of the problem situation?
• why does equivalence play an important role in the solving process?
• Why is it important to understand when and how to use standard algorithms?
• If the equality symbol of an equation is changed to an inequality symbol that includes equal to, why is the solution to the equation always included in the solution to the inequality?
• How does knowing more than one solution strategy build mathematical flexibility?
• How can a solution to an inequality be represented on a number line?
• What is the process for writing a real-world problem to represent constraints or conditions within an inequality?
• What is the process for determining if an inequality is true for a given value?
• Illustrating and analyzing geometric relationships in models and diagrams aid in representing attributes of geometric figures with quantifiable measures and equations in order to generalize geometric relationships and solve problems.
• How can problem situations involving the Triangle Inequality Theorem be represented and solved using an inequality?
• Why is the sum of two side lengths of a triangle always greater than the third side length?
• What model can be used to represent the Triangle Inequality Theorem, and how can the model lead to a generalization that can be represented with an inequality?
• Expressions, Equations, and Relationships
• Geometric relationships
• Measure relationships
• Geometric properties
• Operations
• Properties of operations
• Order of operations
• Numeric and Algebraic Representation
• Expressions
• Inequalities
• Equivalence
• Representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that anytime a negative is involved, the inequality switches and not just when multiplying or dividing by a negative.
• Some students may think that answers to both equations and inequalities are exact answers instead of correctly identifying the solutions to equations as exact answers and the solutions to inequalities as range of answers.
• Some students may shade in the wrong direction when they attempt to graph the solution set of an inequality on a number line.
• Some students may have difficulty determining which inequality symbol to use.
• Some students may think an inequality can only be written one way rather than understanding that an inequality can be written two ways by transposing the expressions and inequality symbol (e.g., 3x > 7 may also be written as 7 < 3x).

Underdeveloped Concepts:

• Some students may think variables are letters representing an object as opposed to representing a number or quantity of objects.
• Some students may think that the order of terms is not important in subtraction expressions (e.g., x – 7 is not equal to 7 – x unless x = 7).
• Some students may think that the order of terms is not important in division expressions (e.g., 20 ÷ y is not equal to y ÷ 20 unless y = 20).
• Some students may confuse the symbols (>, <, ≥, ≤) when working with inequalities.

#### Unit Vocabulary

• Coefficient – a number that is multiplied by a variable(s)
• Constant – a fixed value that does not appear with a variable(s)
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Variable – a letter or symbol that represents a number

Related Vocabulary:

 Condition Constraint Equivalent Evaluate Properties of operations Simplify Solution Solve
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 6 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
6.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
6.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
6.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.
Process Standard

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
6.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.
Process Standard

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
6.1D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
6.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
6.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
6.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
6.9 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to represent situations. The student is expected to:
6.9A

Write one-variable, one-step equations and inequalities to represent constraints or conditions within problems.

Supporting Standard

Write

ONE-VARIABLE, ONE-STEP INEQUALITIES TO REPRESENT CONSTRAINTS OR CONDITIONS WITHIN PROBLEMS

Including, but not limited to:

• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on one side of the inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• One-step inequalities
• A "step" only refers to an action involving both sides of the inequality (combining like terms on a single side of the inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an inequality
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Limited to positive whole number exponents
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• One-variable, one-step inequalities from a problem situation

Note(s):

• Grade 6 introduces writing one-variable, one-step equations and inequalities to represent constraints or conditions within problems.
• Grade 7 will write one-variable, two-step equations and inequalities to represent constraints or conditions within problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
6.9B

Represent solutions for one-variable, one-step equations and inequalities on number lines.

Supporting Standard

Represent

SOLUTIONS FOR ONE-VARIABLE, ONE-STEP INEQUALITIES ON NUMBER LINES

Including, but not limited to:

• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on one side of the inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• One-step inequalities
• A "step" only refers to an action involving both sides of the inequality (combining like terms on a single side of the inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Representations of solutions to one-step inequalities on a number line
• Closed circle
• Greater than or equal to, ≥
• Less than or equal to, ≤
• Open circle
• Greater than, >
• Less than, <
• Not equal to, ≠
• Solutions to real-world situations represented on a number line
• Situations may require determining if a particular value is part of the solution set
• Value is considered part of the solution if the value makes the inequality true.

Note(s):

• Grade 6 introduces representing solutions for one-variable, one-step equations and inequalities on number lines.
• Grade 7 will represent solutions for one-variable, two step equations and inequalities on number lines.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.1. Describe and interpret solution sets of equalities and inequalities.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
6.9C

Write corresponding real-world problems given one-variable, one-step equations or inequalities.

Supporting Standard

Write

CORRESPONDING REAL-WORLD PROBLEMS GIVEN ONE-VARIABLE, ONE-STEP INEQUALITIES

Including, but not limited to:

• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on one side of the inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• One-step inequalities
• A "step" only refers to an action involving both sides of the inequality (combining like terms on a single side of the inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an inequality
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Limited to positive whole number exponents
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Corresponding real-world problem situation from a one-variable, one-step inequality

Note(s):

• Grade 6 introduces writing corresponding real-world problems given one-variable, one-step equations or inequalities.
• Grade 7 will write a corresponding real-world problem given a one-variable, two-step equation or inequality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
6.10 Expressions, equations, and relationships. The student applies mathematical process standards to use equations and inequalities to solve problems. The student is expected to:
6.10A

Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts.

Model, Solve

ONE-VARIABLE, ONE-STEP INEQUALITIES THAT REPRESENT PROBLEMS, INCLUDING GEOMETRIC CONCEPTS

Including, but not limited to:

• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on one side of the inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Positive or negative decimal values
• Fractions
• Positive or negative fractional values
• One-step inequalities
• A "step" only refers to an action involving both sides of the inequality (combining like terms on a single side of the inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an inequality
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Limited to positive whole number exponents
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Models and solutions to solve one-variable, one-step inequalities from problem situations (concrete, pictorial, algebraic)
• Solutions to one-variable, one-step inequalities from geometric concepts

Note(s):

• Grade 4 introduced geometric concepts such as geometric attributes, parallel and perpendicular lines, and angle measures including complementary and supplementary angles.
• Grade 5 represented and solved multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
• Grade 7 will model and solve one-variable, two-step equations and inequalities.
• Grade 7 will write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.1. Describe and interpret solution sets of equalities and inequalities.
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.1. Make connections between geometry and algebraic equations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
6.10B

Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

Supporting Standard

Determine

IF THE GIVEN VALUE(S) MAKE(S) ONE-VARIABLE, ONE-STEP INEQUALITIES TRUE

Including, but not limited to:

• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Variable – a letter or symbol that represents a number
• One variable on one side of the inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Products of integers limited to an integer multiplied by an integer
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Limited to positive decimal values
• Fractions
• Limited to positive fractional values
• One-step inequalities
• A "step" only refers to an action involving both sides of the inequality (combining like terms on a single side of the inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the inequality
• Constraints or conditions
• Distinguishing between equations and inequalities
• Characteristics of equations
• Equates two expressions
• Equality of the variable
• One solution
• Characteristics of inequalities
• Shows the relationship between two expressions in terms of >, ≥, ≤, or ≠
• Inequality of the variable
• One or more solutions
• Equality and inequality words and symbols
• Equal to, =
• Greater than, >
• Greater than or equal to, ≥
• Less than, <
• Less than or equal to, ≤
• Not equal to, ≠
• Relationship of order of operations within an inequality
•  Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Exponents: rewrite in standard numerical form and simplify from left to right
• Limited to positive whole number exponents
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Evaluation of given value(s) as possible solutions of one-variable, one-step inequalities
• Value is considered part of the solution if the value makes the inequality true.

Note(s):

• Grade 5 represented and solved multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
• Grade 7 will determine if the given value(s) make(s) one-variable, two-step equations and inequalities true.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to represent relationships in a variety of contexts
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution. 