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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 05: Systems of Linear Equations and Inequalities SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address methods for writing and solving systems of two linear equations or inequalities in two variables that model problems and methods for justifying the solutions. Solving systems with and without technology is incorporated in the unit. 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 Grade 8, students identified and verified 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 Algebra I Unit 01, students solved linear equations and inequalities. In Algebra I Units 02 – 04, students represented and analyzed linear functions using tables and graphs. Included in these units, students formulated and solved linear equations and functions modeling real-world problems and justified the solutions.

During this Unit
Students analyze a table of values representing a system of two linear equations in two variables and determine the solutions, if they exist. Students graph systems of two linear equations in two variables on the coordinate plane and determine the solutions, if they exist. Students solve systems of two linear equations with two variables for mathematical problems, including substitution and elimination methods. Students formulate, estimate, and solve systems of equations in real-world problem situations and justify the solutions in terms of the situation. Students also graph the solution set of systems of two linear inequalities in two variables on the coordinate plane, and formulate and solve graphically two linear inequalities in two variables in real-world problem situations and justify the solution.

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

After this Unit
In Algebra II Unit 02, students will extend their study of systems of equations to include systems of equations in two or more unknowns, and in Algebra II Unit 04, students will include systems of linear and quadratic relationships. The concepts in this unit will also be applied in later units in Algebra I and subsequent mathematics courses.

In Algebra I, methods for solving systems of two linear equations in two variables are identified as STAAR Readiness Standards A.2I and A.5C and subsumed under STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Graphically solving systems of two linear equations or inequalities in two variables is identified as STAAR Supporting Standard A.3F, A.3G, and A.3H and subsumed under STAAR Reporting Category 2: Describing and Graphing 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, C1, C2, C3, D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Navigating through Algebra in Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), “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” (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. (2002). Navigating through algebra in grades 9 – 12. 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.
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)
• 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?
• Functions, Equations, and Inequalities
• Equations
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Equations
• Linear
• Associated Mathematical Processes
• 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.

 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)
• 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?
• 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?
• Functions, Equations, and Inequalities
• Equations
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Equations
• Linear
• 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.

 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)
• 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
• Inequalities
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Inequalities
• Linear
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the solution to a system of two linear equations in two variables is only the x-value rather than an ordered pair.
• Some students may think that the solution to a system of two linear inequalities in two variables is only one point rather than an infinite set of points in the identified region.

#### Unit Vocabulary

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• 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.

Related Vocabulary:

 Coinciding lines Elimination Feasible region Intersection Linear combination Ordered pair Parallel lines Solutions to systems of equations (empty set (no solution), one point solution, infinite points on coinciding lines) Solutions to systems of inequalities (no solution, feasible region of points) Substitution
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 – 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.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.3F Graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.
Supporting Standard

Graph

SYSTEMS OF TWO LINEAR EQUATIONS IN TWO VARIABLES ON THE COORDINATE PLANE

Determine

THE SOLUTIONS OF SYSTEMS OF TWO LINEAR EQUATIONS IF THEY EXIST

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
• Multiple representations
• Table of values
• Verbal descriptions
• Algebraic generalizations
• Graphical analysis of the system of equations
• Conversion to slope-intercept form
• Graph of each equation
• Identification of possible solutions
• (x, y), one point of intersection
• Ø, no points of intersection (parallel lines)
• Infinite number of points on the line y = mx + b, (coinciding lines)
• Application of systems of equations in real-world problem situations
• Justification of solution to systems of equations
• Substitution of point(s) in the solution into original functions

Note(s):

• Grade 8 introduced graphing two linear equations simultaneously to identify the values of x and y of the intersection point that satisfy the two linear equations.
• 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.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.3G Estimate graphically the solutions to systems of two linear equations with two variables in real-world problems.
Supporting Standard

Estimate

GRAPHICALLY THE SOLUTIONS TO SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES IN 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 equations to model real-world problem situations
• Two unknown variables
• Two equations
• Graphical analysis of the system of equations
• Graph of each equation
• Identification of solution
• (x, y), one point of intersection
• Ø, no points of intersection (parallel lines)
• Infinite number of points on the line y = mx + b, (coinciding lines)
• Justification of solution in terms of real-world problem situations

Note(s):

• Grade 8 introduced graphing two linear equations simultaneously to identify the values of x and y of the intersection point that satisfy the two linear equations.
• 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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• 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.
• 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.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.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.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. 