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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 03: Systems of Linear Equations and Inequalities SUGGESTED DURATION : 12 days

#### Unit Overview

This unit bundles student expectations that address formulating and solving systems of three linear equations in three variables and systems of at least two linear inequalities in two variables. 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 Algebra I Unit 05, students solved systems of two linear equations in two variables. Also, in Algebra 1, Units 03 and 05, students formulated and graphed linear inequalities in two variables and systems of two linear inequalities in two variables.

During this unit, students formulate and solve systems of three linear equations in three variables for mathematical and real-world problem situations using substitution, Gaussian elimination, and matrices with technology. Students also formulate and solve systems of at least two linear inequalities in two variables for mathematical and real-world problem situations, including linear programming. Students determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

After this unit, in Algebra 2 Unit 05 and Unit 12, students will continue to explore systems and combined systems, both linear and nonlinear. The concepts in this unit will also be applied in subsequent mathematics courses.

In Algebra II, formulating and solving systems of three linear equations in three variables are identified as STAAR Readiness Standards 2A.3A and 2A.3B. All STAAR Readiness Standards are subsumed under STAAR Reporting Category 3: Writing and Solving Systems of Equations and Inequalities. Formulating, solving, and determining the reasonableness to solutions of a least two linear inequalities in two variables are identified as STAAR Supporting Standards 2A.3E, 2A.3F, and 2A.3G. All STAAR Supporting Standards are subsumed under STAAR Reporting Category 3: Writing and Solving Systems of Equations and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning A1, C1, D1, D2; III. Geometric Reasoning C1; VII. Functions C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (NCTM), “Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone” (2009, p. 41). This unit places particular emphasis on multiple representations of functions, including matrices. As explained by the National Council of Teachers of Mathematics (NCTM) in their Developing Essential Understanding of Functions, Grades 9 – 12, (2010, p. 22), “Matrices represent an important family of functions that are not defined on the real numbers (Essential Understanding 1c). A major purpose of matrices is to solve systems of linear equations. To understand the theory of solving systems of linear equations, viewing matrices as representing functions is helpful.” Both state and national mathematics standards support such an approach. Foremost, the Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions. According to the Principles and Standards for School Mathematics (1989), instructional programs should enable all students to understand relations and functions and use and convert between various representations (NCTM). More recently, the NCTM highlighted the importance of multiple representations in their Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics document (2007). According to this resource, students should be able to translate among representations of functions and describe how aspects of a function appear in the various representations as early as the Grade 8. Such an approach also helps students developmentally. In Elementary and Middle School Mathematics: Teaching Developmentally (2004), Van de Walle documents the benefits of beginning with meaningful contexts and connecting the different representations of the situation.

National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2007). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2010). Developing Essential Understanding of Functions, Grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Van
de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston: Pearson Education, Inc.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Systems of equations and inequalities can model problem situations and be solved using various methods.

• Why are systems of equations and inequalities used to model problem situations?
• How are systems of equations and inequalities used to model problem situations?
• What methods can be used to solve systems of equations and inequalities?
• Why is it essential to solve systems of equations and inequalities using various methods?
• How can solutions to systems of equations and inequalities be represented?
• How do the representations of solutions to systems of equations and inequalities compare?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 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.

Algebraic Reasoning

• Equations
• Solve
• Systems of Equations

Functions

• Attributes of Functions
• Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Linear systems of equations can be formulated to represent mathematical and real-world problem situations and solved using a variety of methods, with and without technology, and the reasonableness of the solutions can be justified in terms of the problem situations.

• How can linear systems of three equations in three variables be used to represent problem situations?
• What methods can be used to solve linear systems of three equations in three variables?
• How does the solution for a linear system of three equations in three variables differ from the solution to a linear system of two equations in two variables?
• How can the reasonableness of the solutions to a system of three equations in three variables be justified in terms of the problem situation?
 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.

Algebraic Reasoning

• Expressions
• Inequalities
• Solve
• Systems of Inequalities

Functions

• Attributes of Functions
• Linear Functions

Associated Mathematical Processes

• Application
• Tools and Techniques
• Problem Solving Model
• Communication
• Representations
• Relationships
• Justification

Linear systems of inequalities in two variables can be formulated to represent problem situations and solved using a selected method, and the reasonableness of the solution can be justified in terms of the problem situation.

• What process is used to formulate the constraint inequalities in a problem situation?
• What is the feasible region of a system of inequalities?
• How does the feasible region of a system of linear inequalities differ from the solution to a system of linear equations?
• Why are the vertices of the feasible region of a system of linear inequalities important in a problem situation?

The solution to a system of linear equations is a specific point, whereas the solution to a system of linear inequalities is a region of points.

• How is the solution to a system of linear equations shown on a graph?
• How is the solution to a system of linear inequalities shown on a graph?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the x- and y-values in the solution to a system of two linear equations in two variables can never be equal in value.
• Some students may think that the x-, y-, and z-values in the solution to a system of three linear equations in three variables can never be equal in value.
• Some students may think that a system of three equations in three variables can be represented on a coordinate plane rather than on a three-dimensional coordinate system.

Underdeveloped Concepts:

• Some students may not understand the differences between linear equations in one variable and linear equations in two variables.

#### Unit Vocabulary

• Gaussian Elimination – sequence of elementary row operations on a matrix of coefficients and answers to transform the matrix into row echelon form (ref)
• Standard Form for Systems of Equations – variables on left side of the equal sign in alphabetical order with constant on the right side of the equal sign

Related Vocabulary:

 Augmented matrix Coinciding lines Column in a matrix Consistent lines Dependent lines Dimension of a matrix Element of a matrix Feasible region Identity matrix Inconsistent lines Independent lines Infinite number of solutions Intersection of lines Inverse matrix Linear equation in three variables Linear programming Matrix Maximize Minimize No solutions Ordered triple Reduced row echelon form (rref) Row echelon form (ref) Row of a matrix Vertices of a region
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 Creator 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 (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

• Bold black text in italics: Knowledge and Skills Statement (TEKS)
• Bold black text: Student Expectation (TEKS)
• Bold red text in italics:  Student Expectation identified by TEA as a Readiness Standard for STAAR
• Bold green text in italics: Student Expectation identified by TEA as a Supporting Standard for STAAR
• Strike-through: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)
• Blue text: Supporting information / Clarifications from TCMPC (Specificity)
• Blue text in italics: Unit-specific clarification
• Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)
2A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

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:
• X. Connections
2A.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.

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:
• VIII. Problem Solving and Reasoning
2A.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.

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:
• VIII. Problem Solving and Reasoning
2A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

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:
• IX. Communication and Representation
2A.1E Create and use representations to organize, record, and communicate mathematical ideas.

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:
• IX. Communication and Representation
2A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

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:
• X. Connections
2A.1G Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

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:
• IX. Communication and Representation
2A.3 Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:
2A.3A

Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

Formulate

SYSTEMS OF EQUATIONS, INCLUDING SYSTEMS CONSISTING OF THREE LINEAR EQUATIONS IN THREE VARIABLES

Including, but not limited to:

• Systems of linear equations
• Two equations in two variables
• Three equations in three variables
• Real-world problem situations

Note(s):

• Algebra I solved systems of two linear equations in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.3B Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

Solve

SYSTEMS OF THREE LINEAR EQUATIONS IN THREE VARIABLES BY USING GAUSSIAN ELIMINATION, TECHNOLOGY WITH MATRICES, AND SUBSTITUTION

Including, but not limited to:

• 3 x 3 system of linear equations
• Three variables or unknowns
• Three equations
• Standard form for systems of equations – variables on left side of the equal sign in alphabetical order with constant on the right side of the equal sign
• Ex:
2x + y – 4z = 7
2x + 4y + 2z = 40
6x – 2y + 4z = 44
• Methods for solving systems of three linear equations in three variables
• Gaussian elimination – sequence of elementary row operations on a matrix of coefficients and answers to transform the matrix into row echelon form (ref)
• Ex:
• Elementary row operations
• Row switching
• Multiplication of a row by a non-zero number
• Addition of a multiple of one row with another row
• Technology with matrices
• Standard form for systems of equations – variables on left side of the equal sign in alphabetical order with constant on the right side of the equal sign
• Ex:
3x + 4y – 3z = 5
x + 6y + 2z = 3
6x + 2y + 3z = 4
• Inverse matrices
• Matrix form for inverse matrices
• Ex:
• Solution matrix form for inverse matrices
• Ex:
• Augmented matrices
• Matrix form for augmented matrices
• Ex:
• Substitution
• Elimination
• Special cases
• All variables are eliminated
• Infinite number of solutions – remaining constants yield a true statement
• No solutions – remaining constants yield a false statement
• Calculator gives an error message
• Infinite number of solutions – last row is all zeros and yields 0 = 0, which is a true statement.
• Ex:
• No solutions – last row is not all zeros and ends up 0 = 1, which is not a true statement.
• Ex:

Note(s):

• Algebra I solved systems of two linear equations in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.3E Formulate systems of at least two linear inequalities in two variables.
Supporting Standard

Formulate

SYSTEMS OF AT LEAST TWO LINEAR INEQUALITIES IN TWO VARIABLES

Including, but not limited to:

• Systems of linear inequalities in two variables
• Two variables or unknowns
• Two or more inequalities
• Mathematical problem situations
• Graphical interpretation
• Verbal interpretation
• Real-world problem situations represented by systems of inequalities
• Two linear inequalities
• Linear programming problem situations

Note(s):

• Algebra I wrote linear inequalities in two variables given a table of values, a graph, and a verbal description.
• Algebra I solved systems of two linear inequalities in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.3F Solve systems of two or more linear inequalities in two variables.
Supporting Standard

Solve

SYSTEMS OF TWO OR MORE LINEAR INEQUALITIES IN TWO VARIABLES

Including, but not limited to:

• Systems of linear inequalities in two variables
• Two variables or unknowns
• Two or more inequalities
• Method for solving system of inequalities
• Graphical analysis of system
• Graphing of each function
• Solid line
• Dashed line
• Shading of inequality region for each function
• Representation of the solution as points in the solution region

Note(s):

• Algebra I solved systems of two linear inequalities in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.3G Determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.
Supporting Standard

Determine

POSSIBLE SOLUTIONS IN THE SOLUTION SET OF SYSTEMS OF TWO OR MORE LINEAR INEQUALITIES IN TWO VARIABLES

Including, but not limited to:

• Method for solving system of inequalities
• Graphical analysis of system
• Graphing of each function
• Solid line for ≤ or ≥
• Dashed line for < or >
• Shading of inequality region for each function
• Methods for solving linear programming problem situations
• Graphical analysis of system
• Graphing of each function
• Shading of inequality region for each function
• Identification of common or feasible region of intersection
• Determination of points of intersection by solving system of equations
• Testing the points of intersection that create the vertices of the feasible region by substituting them into the objective function and determining the appropriate outcome
• Conclusion in terms of the linear programming problem situation
• Representation of the solution as points in the solution region
• Justification of solutions to system of inequalities
• Verbal description
• Tables
• Graphs
• Substitution of solutions into original functions
• Justification of reasonableness of solution in terms of real-world problem situations

Note(s):

• Algebra I solved systems of two linear inequalities in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections