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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 12: Cost Comparison Analysis SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This is a project-based unit where students apply bundled student expectations that address formulating, solving, and applying linear equations and inequalities and systems of linear equations and inequalities. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, 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.

Prior to this Unit
In Units 01 – 05, students solved linear equations and inequalities in one and two variables. Students solved systems of linear equations and inequalities in two variables. Students applied linear equations, inequalities, and systems in both mathematical and real-world problems.

During this Unit
Students apply prior knowledge to compare products or services for two companies. Student groups formulate a real-world problem design to conduct a cost comparison analysis that can be modeled by linear functions. Students collect and analyze data for the two companies, make predictions and draw conclusions, and justify conclusions about the cost comparisons. Students present a written report and an oral presentation both including displays of project data, representations, analysis, equations/inequalities and systems of equations/inequalities, calculations, summary of predictions and conclusions, and justification in terms of their problem situation.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool HS Algebra I

After this Unit
Students will apply the concepts of linear equations, inequalities, functions, and systems of linear equations and inequalities in subsequent courses in mathematics. The concepts in this unit will also be applied in future career choices.

In Algebra I, analyzing linear equations, inequalities, systems, functions and their key attributes, and applications are identified as STAAR Readiness Standards A.2A, A.2C, A.2I, A.3B, A.3C, A5.A and A.5C and STAAR Supporting Standards A.2H, A.3H, and A.4C. These Readiness and Supporting Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities; and STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, C1, C2, C3, D1, D2; III. Geometric and Spatial Reasoning C2; V. Statistical Reasoning A1, C2; VI. Functions B1, C1, 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 (NCTM, 2000), Principles and Standards for School Mathematics, students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12 from the National Council of Teachers of Mathematics, “High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions” (NCTM, 2002, p. 3). According to Algebra in a Technological World (NCTM, 1995, p. 2),

Graphing tools influence the content of algebra in a technological world in the following ways:

• They allow a visualization of relationships.
• They allow the accurate solution of equations and inequalities not possible through symbolic manipulation alone.
• They provide numerical and graphical solutions that support solutions found using algebraic manipulation.
• They promote exploration by students and their understanding of the effect of change in one representation on another representation.
• They encourage the exploration of relationships and mathematical concepts.
• They promote “what if” modeling of realistic situations.

National Council of Teachers of Mathematics. (1995). Curriculum and evaluation standards for school mathematics: Algebra in a technological world. Reston, VA: National Council of Teachers of Mathematics, Inc
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments 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)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Linear functions are characterized by a constant rate of change and can be used to describe, model, and make predictions about situations.
• How can the rate of change of a linear function be determined?
• What kinds of mathematical and real-world situations can be modeled by linear functions?
• What graphs, key attributes, and characteristics are unique to linear functions?
• What pattern of covariation is associated with linear functions?
• How can the key attributes of linear functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a linear function?
• What are the real-world meanings of the key attributes of a linear function model?
• How can the key attributes of a linear function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can equations and inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write linear equations and linear inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing linear equations and linear inequalities?
• What methods can be used to solve linear equations and linear inequalities?
• How does the structure of the equation influence the selection of an efficient method for solving linear equations?
• How can the solutions to linear equations and linear inequalities be determined and represented?
• How are properties and operational understandings used to transform linear equations and linear inequalities?
• Systems of equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structures of the equations in the system.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can systems of equations be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write systems of equations?
• What methods can be used to solve systems of equations?
• How does the structure of the system influence the selection of an efficient method for solving the system of equations?
• How can the solutions to systems of equations be determined and represented?
• How are properties and operational understandings used to transform systems of equations?
• What kinds of algebraic and graphical relationships exist between equations in a system with …
• one solution?
• no solutions?
• infinitely many solutions?
• Systems of inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structures of the inequalities in the system.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can systems of inequalities be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write systems of inequalities?
• What methods can be used to solve systems of inequalities?
• How can the solution set of systems of inequalities be determined and represented?
• How do solution sets of systems of inequalities differ from solution sets of systems of equations?
• How are properties and operational understandings used to transform systems of inequalities?
• What kinds of algebraic and graphical relationships exist between inequalities in a system with …
• no solutions?
• infinitely many solutions?
• What relationships exist between the solution set of each inequality in a system and the solution set of the system of inequalities?
• How can the boundaries and the vertices of the solution set of a system of inequalities be used to make predictions and critical judgments in problem situations?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Functions, Equations, Inequalities
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Linear Equations and Inequalities
• Linear
• Statistical Representations
• Regression methods
• 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:

• None identified

#### Unit Vocabulary

• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Inequality notation – notation in which the solution is represented by an inequality statement
• Linear equation in one variable –a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may be not included in the solution, and the set of points above or below the line
• Range – set of output values for the dependent variable over which the function is defined
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or  or , denoted as m in y = mx + b
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Continuous function Correlation coefficient Direct variation Discrete function Domain Function notation Independent/dependent variables Inequality notation Linear equation in one variable Linear equation in two variables Linear function Linear inequality in two variables Range Reasonableness of solutions Regression equation Representations Scatterplot Slope of the line System of linear equations System of linear inequalities Trend line x-intercept y-intercept Zeros
System Resources Other Resources

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 – 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 Algebra I 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
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.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.
• 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.
A.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.
• 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.
A.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.
• 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.
A.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.
• 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.
A.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.
• 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.
A.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.
• 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.
A.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.
• 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.
A.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:
A.2A Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.

Determine

THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REAL-WORLD SITUATIONS, BOTH CONTINUOUS AND DISCRETE

Represent

THE DOMAIN AND RANGE OF A LINEAR FUNCTION USING INEQUALITIES

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Domain and range of linear functions in mathematical problem situations
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Inequality representations
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, yΖ
• Domain and range of linear functions in real-world problem situations
• Reasonable domain and range for real-world problem situations
• Comparison of domain and range of function model to appropriate domain and range for a real-world problem situation

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I introduces the concept of domain and range of a function.
• Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
• Algebra II will introduce representing domain and range using interval and set notation.
• Precalculus will introduce piecewise functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• 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.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.
• 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.
A.2C Write linear equations in two variables given a table of values, a graph, and a verbal description.

Write

LINEAR EQUATIONS IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Various forms linear equations in two variables
• Slope-intercept form, y = mx + b
• m is the slope.
• b is the y-intercept.
• Point-slope form, y – y1 = m(xx1)
• m is the slope.
• (x1, y1) is a given point
• Standard form, Ax + By = C; A, B, CΖ, A ≥ 0
• x and y terms are on one side of the equation and the constant is on the other side.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School introduced using multiple representations for linear relationships.
• Grade 8 represented linear proportional and non-proportional relationships in tables, graphs, and equations in the form y = mx + b.
• Algebra I introduces the use of standard form and point-slope form to represent linear relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a 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.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.
A.2H Write linear inequalities in two variables given a table of values, a graph, and a verbal description.
Supporting Standard

Write

LINEAR INEQUALITIES IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may not be included in the solution, and the set of points above or below the line
• Inequality notation
• Less than, <, dashed line with shading below the graph of the line
• Greater than, >, dashed line with shading above the graph of the line
• Less than or equal to, ≤, solid line with shading below the graph of the line
• Greater than or equal to, ≥, solid line with shading above the graph of the line
• For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School used multiple representations for linear relationships.
• Grade 8 solved problems using one-variable inequalities.
• Algebra I introduces linear inequalities in two variables given various representations.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.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.
• 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.
A.2I Write systems of two linear equations given a table of values, a graph, and a verbal description.

Write

SYSTEMS OF TWO LINEAR EQUATIONS GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Characteristics of 2 x 2 systems of linear equations
• Two equations
• Two variables
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School used multiple representations for linear relationships.
• Algebra I formally introduces systems of two linear equations in two variables.
• Algebra II will introduce systems of three linear equations in three variables and systems of one linear equation and one quadratic equation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.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.
• 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.
A.3 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:
A.3B Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.

Calculate

THE RATE OF CHANGE OF A LINEAR FUNCTION REPRESENTED TABULARLY, GRAPHICALLY, OR ALGEBRAICALLY IN CONTEXT OF MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Connections between slope and rate of change
• Rate of change by various methods
• Tabular method by analyzing rate of change in x and y values: m = = or m =
• Graphical method by analyzing vertical and horizontal change: slope =
• Algebraic method by analyzing m in y = mx + b form
• Solve equation for y
• Slope is represented by m
• Rate of change from multiple representations
• Tabular
• Graphical
• Algebraic
• Calculuation and comparison of the rate of change over specified intervals of a graph
• Meaning of rate of change in the context of real-world problem situations
• Emphasis on units of rate of change in relation to real-world problem situations

Note(s):

• Grade 8 introduced the concept of slope as a rate of change, including using the slope formula.
• Precalculus will introduce piecewise functions and their characteristics.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• 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.
• 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.
• 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.
A.3C Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.

Graph

LINEAR FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations

Identify

KEY FEATURES OF LINEAR FUNCTIONS, INCLUDING x-INTERCEPT, y-INTERCEPT, ZEROS, AND SLOPE, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations
• Characteristics of linear functions
• x-intercept – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• Zeros – the value(s) of x such that the y value of the relation equals zero
• y-intercept – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or = or
• denoted as m in y = mx + b
• denoted as m in f(x) = mx + b
• Notation of linear functions
• Equation notation: y= mx + b
• Function notation: f(x) = mx + b

Note(s):

• Grades 7 and 8 introduced linear relationships using tables of data, graphs, and algebraic generalizations.
• Grade 8 introduced using tables of data and graphs to determine rate of change or slope and y-intercept.
• Algebra I introduces key attributes of linear, quadratic, and exponential functions.
• Algebra II will continue to analyze the key attributes of exponential functions and will introduce the key attributes of square root, cubic, cube root, absolute value, rational, and logarithmic functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.
• 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.
A.3H Graph the solution set of systems of two linear inequalities in two variables on the coordinate plane.
Supporting Standard

Graph

THE SOLUTION SET OF SYSTEMS OF TWO LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE

Including, but not limited to:

• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may not be included in the solution, and the set of points above or below the line
• Systems of inequalities
• Two unknowns
• Two inequalities
• Graphical analysis of the system of inequalities
• Graphing of each function
• Solid line
• Dashed line
• Shading of inequality region for each
• Representation of the solution as points in the region of intersection
• Justification of solution to systems of inequalities
• Substitution of various points in the solutions region into original functions

Note(s):

• Algebra I introduces linear inequalities in two variables given various representations.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.
• II.C.2. Explain the difference between the solution set of an equation and the solution set of an inequality.
• 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.
A.4 Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. The student is expected to:
A.4C Write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.
Supporting Standard

Write

LINEAR FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, WITH AND WITHOUT TECHNOLOGY

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Function models for sets of data
• Trend line by manipulating slope and y-intercept
• Regression equation, y = ax + b, using the graphing calculator

To Estimate, To Make

SOLUTIONS AND PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Function models for sets of data
• Trend line by manipulating slope and y-intercept
• Regression equation, y = ax + b, using the graphing calculator
• Correlation coefficient as an indicator of reliability of regression equations

Note(s):

• Grade 8 graphed scatterplots of bivariate data and used trend lines to analyze the correlation as linear, non-linear, or no association.
• Algebra I introduces calculation and interpretation of the correlation coefficient between two quantitative variables.
• Algebra I introduces the use of algebraic strategies and regression technology to determine the line of best fit.
• Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.2. Make connections between geometry, statistics, and probability.
• 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.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• 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.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.
• 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.
A.5 Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:
A.5A Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.

Solve

LINEAR EQUATIONS IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES

Including, but not limited to:

• Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Linear equations in one variable including parentheses and variables on both sides of the equation
• Mathematical problem situations
• Real-world problem situations
• Multiple representations of mathematical and real-world problem situations
• Algebraic generalizations
• Missing coordinate of a solution point to a function
• Verbal
• Methods for solving equations
• Concrete and pictorial models (e.g., algebra tiles, etc.)
• Tables and graphs with and without technology
• Transformation of equations using properties of equality
• Distributive property
• Operational properties
• Possible solutions, including special cases
• No solution, empty set, ∅
• Infinite solutions, all real numbers, ℜ
• Relationships and connections between the methods of solution
• Justification of solutions to equations
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations

Note(s):

• Grade 5 used equations with variables to represent missing numbers.
• Grade 6 solved one-variable, one-step equations.
• Grade 7 solved one-variable, two-step equations.
• Grade 8 solved one-variable equations with variables on both sides.
• Algebra I introduces solving one-variable equations that include those for which the application of the distributive property is necessary and for which variables are included on both sides.
• Algebra II will introduce solving absolute value linear equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• 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.
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
A.5C Solve systems of two linear equations with two variables for mathematical and real-world problems.

Solve

SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES FOR MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Systems of 2 x 2 linear equations
• Two equations
• Two variables
• Solutions to systems of equations
• One common point of intersection, (x, y)
• Infinite set of points on a line
• Empty set, Ø
• Methods for solving systems of linear equations with and without technology
• Tables
• Graphs
• Concrete models
• Algebraic methods
• Substitution
• Linear combination (elimination)
• Special cases for empty set, Ø, and all real numbers, ℜ
• Relationships and connections between the methods of solution
• Justification of solutions to systems of equations with and without technology
• Systems of linear equations as models for real-world problem situations
• Interpretation of a solution point in terms of the real-world problem situation
• Justification of reasonableness of solution in terms of the real-world problem situation or data collection

Note(s):

• Algebra I formally introduces systems of two linear equations in two variables.
• Algebra II will introduce systems of three linear equations in three variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• 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.
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
•  VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• 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.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.