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 TITLE : Unit 11: Essential Understandings of Algebra SUGGESTED DURATION : 13 days

#### Unit Overview

Introduction
This unit bundles student expectations that address one-variable, two-step equations and inequalities, and various representations of linear relationships. 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 02, students modeled and solved one-variable, two-step equations and inequalities. In Unit 03, students represented constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations.

During this Unit
Students revisit and solidify essential understandings of algebra. Students represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. Students model and solve one-variable, two-step equations and inequalities with concrete and pictorial models and algebraic representations. Solutions to equations and inequalities are represented on number lines and given values are used to determine if they make an equation or inequality true. Students are expected to write an equation or inequality to represent conditions or constraints within a problem and then, conversely, when given an equation or inequality out of context, students are expected to write a corresponding real-world problem to represent the equation or inequality.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 7

After this Unit
In Grade 8, students will use rational number coefficients and constants to write one-variable equations or inequalities with variables on both sides, as well as write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign. Students will model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants. Students will formally develop the concepts and applications of slope and y-intercept in functions. Additionally, students will identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.

In Grade 7, modeling and solving one-variable, two-step equations and inequalities is STAAR Readiness Standard 7.11A. Representing linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b is identified as STAAR Readiness Standard 7.7A. Writing one-variable, two-step equations and inequalities, representing their solutions on a number line, and determining if a given value(s) makes the equation or inequality true are identified as STAAR Supporting Standards 7.10A, 7.10B, and 7.11B. Writing real-world problems given a one-variable, two-step equation or inequality is STAAR Supporting Standard 7.10C. These standards are subsumed under the Grade 7 STAAR Reporting Category 2: Computations and Algebraic Relationships. All of these standards are a foundational block of the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Using expressions and equations to describe relationships in a variety of contexts, including geometric problems. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning C1, C2, D1, D2; V. Statistical Reasoning A1, C2; VI. Functions B1, 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), “Fluency with algebraic symbolism helps students to represent and solve problems in many areas of the curriculum…Students should be able to operate fluently on algebraic expressions, combining them and re-expressing them in alternative forms. These skills underlie the ability to find exact solutions for equations, a goal that has always been at the heart of the algebra curriculum” (p. 300 – 301). Algebraic expressions are the foundation for all equations and inequalities, therefore, “Students need an understanding of how to apply mathematical properties and how to reserve equivalence as [expressions] simplify” (Van de Walle, Karp, & Bay-Williams, 2010, p. 263). Van de Walle and Lovin (2006) note that, “Students in middle grades should develop an understanding of the multiple methods of expressing real-world functional relationships (words, graphs, equations, and tables). Working with these different representations of functions will allow students to develop a fuller understanding of functions.” (p. 284). Students need to develop the ability to move among algebraic representations flexibly, “Students are often not made aware of the power of mathematics. The realization of how the transfer of an equation to graphic form can reveal a whole set of possible solutions may be an eye-opening motivating factor for learning mathematics” (Solomon, 2006, p. 200).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
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
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 5 – 8. Boston, MA: Pearson Education, Inc.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.
Solomon, P. (2006). The math we need to know and do in grades 6 – 9. Thousand Oaks, CA: Corwin Press.

 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)
• Equations and 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 efficient strategies.
• What is the process of representing a linear relationship …
• verbally?
• with a table?
• with a graph?
• with an equation that simplifies to the form of y = mx + b?
• What is the purpose of using different representations, and how is the context of the problem highlighted in each representation?
• What are the characteristics of a linear relationship?
• How are independent and dependent quantities related in a linear problem situation?
• What is the meaning of each of the variables in the equation y = mx + b?
• How are the table and graph of a linear problem situation related to an equation that simplifies to the form of y = mx + b?
• Why are expressions considered foundational to equations and inequalities?
• How are constraints or conditions within a problem situation represented in an …
• equation?
• inequality?
• How does the context of a problem situation, relationships within and between operations, and properties of operations aid in writing an equation and/or inequality to represent the problem situation?
• How can a solution to an …
• equation
• inequality
… be represented on a number line?
• What is the process for writing a real-world problem to represent constraints or conditions within an …
• equation?
• inequality?
• How can a(n) …
• concrete model
• pictorial model
• algebraic representation
… be used to model and solve an …
• equation?
• inequality?
• What models effectively and efficiently represent how to solve equations and inequalities?
• What is the process for solving an …
• equation
• inequality
…, and how can the process be …
• described verbally?
• represented algebraically?
• When considering equations and 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?
• how are the solution processes alike and different?
• how are the solutions alike and different?
• 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?
• What is the process for evaluating an …
• equation
• inequality
… for a given value?
• Expressions, Equations, and Relationships
• Algebraic Relationships
• Linear
• Independent and dependent quantities
• Numeric and Algebraic Representations
• Expressions
• Equations
• Inequalities
• Equivalence
• Operations
• Properties of operations
• Order of operations
• 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

Underdeveloped Concepts:

• Some students may think that operations involving negatives when solving inequalities always require the inequality symbol to be reversed instead of applying the rule of reversing the inequality symbol when dividing or multiplying both sides of an inequality by a negative value.
• Some students may think that a constant term can be combined with a variable term (e.g., 2x + 5 = 7x).
• 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 a range of answers.
• Some students may not relate the constant rate of change to m in the equation y = mx + b.
• Some students may think that the ratio for rate of change in a linear relationship is m = , since the x-coordinate (horizontal) always comes before the y-coordinate (vertical) in an ordered pair, instead of the correct representation that rate of change in a linear relationship is m = .

#### Unit Vocabulary

• Coefficient – a number that is multiplied by a variable(s)
• Constant – a fixed value that does not appear with a variable(s)
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Inequality – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Solution set – a set of all values of the variable(s) that satisfy the equation or inequality
• Variable – a letter or symbol that represents a number

Related Vocabulary:

 Algebraic Condition Constraint Dependent Equation Equivalent Equality Equal to (=) Evaluate Expression Fraction Graphical Greater than (>) Greater than or equal to ( ≥) Independent Less than (<) Less than or equal to (≤) Linear Not equal to (≠) Number line Ordered pair Origin Parentheses/brackets Positive Negative Rise Run Simplify Slope-intercept form Solution Tabular Undefined x-coordinate x-value y-coordinate y-value
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 7 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.
TEKS# SE# TEKS SPECIFICITY
7.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
7.1A 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1B 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1C 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1D 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, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

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

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1E 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1F 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.1G 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• 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.
7.7 Expressions, equations, and relationships. The student applies mathematical process standards to represent linear relationships using multiple representations. The student is expected to:
7.7A Represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.

Represent

LINEAR RELATIONSHIPS USING VERBAL DESCRIPTIONS, TABLES, GRAPHS, AND EQUATIONS THAT SIMPLIFY TO THE FORM y = mx + b

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers as constants and coefficients
• Coefficient – a number that is multiplied by a variable(s)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear proportional relationship
• Linear
• Passes through the origin (0, 0)
• Represented by y = kx
• Constant of proportionality represented as • When b = 0 in y = mx + b, then k = m
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant rate of change represented as m = or m = • Rate of change is either positive, negative, zero, or undefined
• Various representations to describe algebraic relationships
• Verbal descriptions
• Tables
• Graphs
• Equations
• In the form y = mx + b(slope intercept form)

Note(s):

• Grade 6 identified independent and dependent quantities from tables and graphs.
• Grade 6 wrote an equation that represents the relationship between independent and dependent quantities from a table.
• Grade 6 represented a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
• Grade 8 will represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
• Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
• Grade 8 will distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• 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.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• 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.
7.10 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations and inequalities to represent situations. The student is expected to:
7.10A Write one-variable, two-step equations and inequalities to represent constraints or conditions within problems.
Supporting Standard

Write

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

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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 equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Two-step equations and inequalities
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation or 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 equation or 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 numer 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, two-step equations from a problem
• One-variable, two-step inequalities from a problem

Note(s):

• Grade 6 wrote one-variable, one-step equations and inequalities to represent constraints or conditions within problems.
• Grade 7 represents writing one-variable, two-step equations and inequalities to represent constraints or conditions within problems.
• Grade 8 will write one-variable equations or inequalities with variables on both sides that represent problems using rational number coefficients and constants.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• 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.
7.10B Represent solutions for one-variable, two-step equations and inequalities on number lines.
Supporting Standard

Represent

SOLUTIONS FOR ONE-VARIABLE, TWO-STEP EQUATIONS AND INEQUALITIES ON NUMBER LINES

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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 equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Two-step equations and inequalities
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation or 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 equations and inequalities on a number line
• Closed circle
• Equal to, =
• Greater than or equal to, ≥
• Less than or equal to, ≤
• Open circle
• Greater than, >
• Less than, <
• Not equal to, ≠

Note(s):

• Grade 6 represented solutions for one-variable, one-step equations and inequalities on number lines.
• Grade 7 represents 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 describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• 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.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.
7.10C Write a corresponding real-world problem given a one-variable, two-step equation or inequality.
Supporting Standard

Write

A CORRESPONDING REAL-WORLD PROBLEM GIVEN A ONE-VARIABLE, TWO-STEP EQUATION OR INEQUALITY

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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 equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Two-step equations and inequalities
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation or 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 equation or 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 numer 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 from a one-variable, two-step equation
• Corresponding real-world problem from a one-variable, two-step inequality

Note(s):

• Grade 6 wrote corresponding real-world problems given one-variable, one-step equations or inequalities.
• Grade 7 writes corresponding real-world problems given one-variable, two-step equations or inequalities.
• Grade 8 will write a corresponding real-world problem when given a one-variable equation or inequality with variables on both sides of the equal sign using rational number coefficients and constants.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• 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.
• 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.
• 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.
7.11 Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities. The student is expected to:
7.11A Model and solve one-variable, two-step equations and inequalities.

Model, Solve

ONE-VARIABLE, TWO-STEP EQUATIONS AND INEQUALITIES

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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 equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Two-step equations and inequalities
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation or 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 equation or 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 numer 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
• Model and solve one-variable, two-step equations (concrete, pictorial, algebraic)
• Model and solve one-variable, two-step inequalities (concrete, pictorial, algebraic)
• Solutions to one-variable, two-step equations from a problem situation
• Solutions to one-variable, two-step inequalities from a problem situation

Note(s):

• Grade 6 modeled and solved one-variable, one-step equations and inequalities that represented problems, including geometric concepts.
• Grade 8 will model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• 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.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.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.
7.11B Determine if the given value(s) make(s) one-variable, two-step equations and inequalities true.
Supporting Standard

Determine

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

Including, but not limited to:

• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• 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 equation or inequality
• Coefficient – a number that is multiplied by a variable(s)
• Integers
• Decimals
• Fractions
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Two-step equations and inequalities
• A “step” only refers to an action involving both sides of the equation or inequality (combining like terms on a single side of the equation or inequality does not constitute a step).
• Solution set – a set of all values of the variable(s) that satisfy the equation or 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 equation or 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 numer 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 a given value(s) as a possible solution to one-variable, two-step equations
• Evaluation of a given value(s) as a possible solution to one-variable, two-step inequalities

Note(s):

• Grade 6 determined if the given value(s) make(s) one-variable, one-step equations or inequalities true.
• Grade 8 will identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• 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. 