 Hello, Guest!
 Instructional Focus DocumentAlgebra II
 TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities SUGGESTED DURATION : 14 days

Unit Overview

Introduction
This unit bundles student expectations that address writing equations of parabolas given various characteristics; writing quadratic functions given three points;  formulating, solving, and determining the reasonableness of solutions to a system of equations consisting of a linear equation and a quadratic equation; and  solving quadratic 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 Algebra I Units 07 and 08, students investigated quadratic functions and equations. Students also formulated quadratic models to represent problem situations and applied various methods to solve quadratic equations. In Algebra I Unit 05 and Algebra II Unit 03, students investigated systems of linear equations.

During this Unit
Students use a system of three equations in three variables to write quadratic functions given three specified points in a plane and justify the quadratic function using the graphing calculator. Students transform quadratic functions from standard form, f(x) = ax2 + bx + c, to vertex form, f(x) = a(xh)2 + k, and identify attributes of f(x), including vertex, symmetries, y-intercept, maximum, and minimum. Students write equations of parabolas from attributes including vertex, focus, directrix, axis of symmetry, and direction of opening. Students define the complex number system and its subsets as well as perform operations (addition, subtraction, multiplication) with complex numbers. Students solve quadratic equations using various methods, including graphing, factoring, completing the square, and the quadratic formula, and verify solutions by graphing and multiplying factors created by roots. Students solve quadratic inequalities graphically and algebraically. Students formulate quadratic equations from tables of data and real-world problem situations, solve the quadratic equations by a method of choice, and justify the solution in terms of the problem situation. Students formulate systems of equations consisting of two equations, the first linear and the second quadratic, solve the system algebraically, and determine the reasonableness of the solution in terms of the problem situation.

After this Unit
In Units 06, 07, 08, and 11, students will continue to apply the concepts of quadratic functions, equations, and inequalities. In subsequent mathematics courses, students will also continue to apply these concepts when quadratic functions, equations, and inequalities arise in problem situations.

Research
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions, Grades 9-12, understanding of the function concept is essential to describing and analyzing quantities which vary with respect to one another. According to research from the National Council of Teachers of Mathematics (2000), high school algebra should provide students with insights into mathematical abstraction and structure. High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with a higher degree of sophistication. Students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities.

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. (2011). Developing essential understanding of expressions, equations, and functions, grades 6-8. 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 judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Quadratic functions are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can quadratic functions model?
• What graphs, key attributes, and characteristics are unique to quadratic functions?
• What pattern of covariation is associated with quadratic functions?
• How can the key attributes of quadratic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a quadratic function?
• What are the real-world meanings of the key attributes of a quadratic function model?
• How can the key attributes of a quadratic function be used to make predictions and critical judgments?
• What relationships exist between the algebraic forms of a quadratic function and the graph and key attributes of the function?
• 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 structure of the equation.
• How can equations be used to represent relationships between quantities?
• What methods can be used to write quadratic equations?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations?
• How are properties and operational understandings used to transform quadratic?
• 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?
• 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?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Axis of symmetry
• Symmetries
• Focus
• Directrix
• Functions
• 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 judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Different families of functions, including their related relations, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Quadratic relations are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can quadratic relations model?
• What graphs, key attributes, and characteristics are unique to quadratic relations?
• What patterns of covariation are associated with quadratic relations?
• How can the key attributes of quadratic relations be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a quadratic relation?
• How can the key attributes of a quadratic relation be used to make predictions and critical judgments?
• What relationships exist between the algebraic forms of a quadratic relation and the graph and key attributes of the relation?
• 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 structure of the equation.
• How can equations be used to represent relationships between quantities?
• What methods can be used to write quadratic equations?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations?
• How are properties and operational understandings used to transform quadratic?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Vertex
• Axis of symmetry
• Focus
• Directrix
• Functions and Equations
• Patterns, Operations, and Properties
• Relations and Generalizations
• 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.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place. How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy? Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Complex numbers create a more sophisticated number system where new relationships exist within and between sets and subsets of numbers.
• What representations can be used to visually demonstrate relationships between sets and subsets of numbers?
• How does organizing numbers in sets and subsets aid in understanding the relationships within and between complex numbers?
• What relationships exist between real and complex numbers?
• The ability to represent quantities in various forms develops the understanding of equivalence and allows for working flexibly with algebraic expressions in order to communicate and reason about quantities.
• How can expressions be used to represent situations?
• What mathematical conventions are used when representing expressions? Why?
• How can it be determined if two expressions are equivalent?
• How are properties and operational understandings used to generate equivalent expressions?
• Why can it be useful to simplify expressions?
• 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 quadratic equations and quadratic inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations and quadratic inequalities?
• What methods can be used to solve quadratic equations and quadratic inequalities?
• How does the structure of the equation influence the selection of an efficient method for solving quadratic equations?
• How can the solutions to quadratic equations and quadratic inequalities be determined and represented?
• How are properties and operational understandings used to transform quadratic equations and quadratic inequalities?
• Functions, Equations, and Inequalities
• Equations and Inequalities
• Patterns, Operations, and Properties
• Number and Algebraic Methods
• Expressions
• Complex numbers
• Patterns, Operations, and Properties
• 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 judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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.
• Quadratic functions are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about situations.
• What kinds of mathematical and real-world situations can quadratic functions model?
• What graphs, key attributes, and characteristics are unique to quadratic functions?
• What pattern of covariation is associated with quadratic functions?
• How can the key attributes of quadratic functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a quadratic function?
• What are the real-world meanings of the key attributes of a quadratic function model?
• How can the key attributes of a quadratic 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 quadratic equations and quadratic inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing quadratic equations and quadratic inequalities?
• What methods can be used to solve quadratic equations and quadratic inequalities?
• How does the structure of the equation influence the selection of an efficient method for solving quadratic equations?
• How can the solutions to quadratic equations and quadratic inequalities be determined and represented?
• How are properties and operational understandings used to transform quadratic equations and quadratic inequalities?
• Functions, Equations, and Inequalities
• Functions, Equations, and Inequalities
• Patterns, Operations, and Properties
• Relations and Generalizations
• Statistical Relationships
• 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.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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?
• How does the solution for a system of a linear equation and a quadratic equation in two variables differ from the solution to a system of two linear equations in two variables?
• Functions, Equations, and Inequalities
• Functions
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Equations
• Linear and quadratic
• 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 terms zeros, x-intercepts, roots, and solutions are all the same and can be used interchangeably rather than understanding that when an equation is set equal to zero, these will be equivalent, but not at other times. Roots and solutions pertain to equations, while x-intercepts and zeros pertain to functions.
• Some students may think that in order to be a complex number, the number must contain an imaginary part rather than that all numbers can be written in complex form, e.g., 25 can be written as 25 + 0i, and its conjugate is 25 – 0i.
• 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.

Unit Vocabulary

• Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
• Directrix – horizontal or vertical line not passing through the focus whose distance from the vertex is |p| and is perpendicular to the axis of symmetry
• Focus – point not on the directrix whose distance from the vertex is |p| and lies on the axis of symmetry
• Imaginary number – number in the form of bi where b is a real number and i = • |p| – distance from vertex to directrix or distance from vertex to focus
• Parabola – the locus of points, P, such that the distance from P to a point F (the focus) is equal to the distance from P to a line q (the directrix)
• x-intercept(s)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; the x-intercepts

Related Vocabulary:

 Axis of symmetry Completing the square Discriminant Factoring Fundamental Theorem of Algebra Gaussian method Horizontal shift Inverse matrix Locus of points Maximum Minimum Operations of complex numbers Quadratic equation Quadratic formula Quadratic function Quadratic inequality Quadratic regression rref Real numbers Roots Solutions Standard form, f(x)= ax2 + bx + c Substitution method Symmetric point Transformation Vertex Vertex form, f(x) = a(x – h)2 + k Vertical compression Vertical shift Vertical stretch
Unit Assessment Items System Resources Other Resources

Show this message:

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

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 II Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

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.
• 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.

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
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 AND SYSTEMS CONSISTING OF TWO EQUATIONS, THE FIRST LINEAR AND THE SECOND QUADRATIC

Including, but not limited to:

• Systems of linear equations
• Two equations in two variables
• Three equations in three variables
• Systems of one linear equation and one quadratic equation in two variables

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.3C Solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation.
Supporting Standard

Solve

SYSTEMS OF TWO EQUATIONS IN TWO VARIABLES CONSISTING OF A LINEAR EQUATION AND A QUADRATIC EQUATION, ALGEBRAICALLY

Including, but not limited to:

• Two equations in two variables
• One linear equation
• One quadratic equation
• Methods for solving systems of equations consisting of one linear equation and one quadratic equation
• Tables
• Common points on tables
• Graphs
• Identification of possible solutions in terms of points of intersection
• Algebraic methods
• Substitution of linear equation into quadratic
• Solve by factoring
• Solve by quadratic formula
• Solve by completing the square

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.3D Determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables.
Supporting Standard

Determine

THE REASONABLENESS OF SOLUTIONS TO SYSTEMS OF A LINEAR EQUATION AND A QUADRATIC EQUATION IN TWO VARIABLES

Including, but not limited to:

• Types of equations in system
• Two equations in two variables
• One linear equation
• One quadratic equation
• Justification of reasonableness of solutions to systems of equations
• Tables
• Graphs
• Substitution of solutions into original functions
• Restriction of solutions in terms of real-world problem situations
• Verbal description in terms of 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
• 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.4 Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:
2A.4A Write the quadratic function given three specified points in the plane.
Supporting Standard

Write

THE QUADRATIC FUNCTION GIVEN THREE SPECIFIED POINTS IN THE PLANE

Including, but not limited to:

• 3 × 3 system of three linear equations in three variables
• Determination of a linear system of three equations in three variables using the three points and the standard form of the quadratic function, ax2 + bx + c = y
• Methods for solving the linear system of three equations in three variables
• Substitution
• Gaussian elimination
• Graphing calculator technology
• Inverse matrix
• rref
• Quadratic regression using the graphing calculator
• Three points required
• Correlation of determination, or r2 value, closer to ±1, the better the fit of the regression equation

Note(s):

• Algebra I wrote quadractic functions given real solutions and graphs of their related equations.
• Algebra I formulated a quadratic function using technology.
• Algebra II solves systems of three linear equations in three variables using various methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• A2 – Recognize and distinguish between different types of 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.4B Write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening.

Write

THE EQUATION OF A PARABOLA USING GIVEN ATTRIBUTES, INCLUDING VERTEX, FOCUS, DIRECTRIX, AXIS OF SYMMETRY, AND DIRECTION OF OPENING

Including, but not limited to:

• Parabola – the locus of points, P, such that the distance from P to a point F (the focus) is equal to the distance from P to a line q(the directrix)
• Quadratic equation representations
• Standard form
• Vertical axis of symmetry: y = ax2 + bx + c
• Horizontal axis of symmetry: x = ay2 + by + c
• Vertex form
• Vertical axis of symmetry: y = a(xh)2 + k
• Horizontal axis of symmetry: x = a(yk)2 + h
• Parabola (conic form)
• Vertical axis of symmetry: (xh)2 = 4p(yk)
• Horizontal axis of symmetry: (yk)2 = 4p(xh)
• Connection between a and p in the vertex form and parabola (conic form)
• a = • Attributes of a parabola
• Vertex: (h, k)
• Axis of symmetry
• Vertical axis of symmetry for a parabola that opens up or down: x = h
• Horizontal axis of symmetry for a parabola that opens to the right or to the left: y = k
• Positive value of a or p, the parabola opens up or to the right
• Negative value of a or p, the parabola opens down or to the left
• |p| – distance from vertex to directrix or distance from vertex to focus
• Directrix – horizontal or vertical line not passing through the focus whose distance from the vertex is |p| and is perpendicular to the axis of symmetry
• Focus – point not on the directrix whose distance from the vertex is |p| and lies on the axis of symmetry

Note(s):

• Algebra I wrote quadratic equations in vertex form (f(x) = a(xh)2 + k), and rewrote from vertex form to standard form (f(x) = ax2 + bx + c).
• Precalculus will address parabolas as conic sections.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B2 – Identify the symmetries of a plane figure.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4D Transform a quadratic function f(x) = ax2 + bx + c to the form f(x) = a(x - h)2 + k to identify the different attributes of f(x).
Supporting Standard

Transform

A QUADRATIC FUNCTION f(x)= ax2 + bx + c TO THE FORM f(x)= a(xh)2 + k

Including, but not limited to:

• Forms of quadratic functions
• Standard form: f(x) = ax2 + bx + c
• Vertex form: f(x) = a(xh)2 + k
• Completing the square to transform from the standard form f(x) = ax2 + bx + c to vertex form f(x) = a(xh)2 + k

To Identify

THE DIFFERENT ATTRIBUTES OF f(x)

Including, but not limited to:

• Attributes from the vertex form, f(x) = a(xh)2 + k
• Vertex of the function, (h, k)
• Minimum point of function if a > 0
• Maximum point of function if a < 0
• y-intercept, ah2 + k
• Axis of symmetry, x = h
• Attributes from the standard form, f(x) = ax2 + bx + c
• Vertex of the function, • Minimum point of function if a > 0
• Maximum point of function if a < 0
• y-intercept, c
• Axis of symmetry, x = Note(s):

• Algebra I wrote quadratic equations in vertex form (f(x) = a(xh)2 + k), and rewrote from vertex form to standard form (f(x) = ax2 + bx + c).
• Algebra I solved quadratic equations having real solutions by completeing the square.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B2 – Identify the symmetries of a plane figure.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4E

Formulate quadratic and square root equations using technology given a table of data.

Supporting Standard

Formulate

QUADRATIC EQUATIONS USING TECHNOLOGY GIVEN A TABLE OF DATA

Including, but not limited to:

• Data collection activities with and without technology
• Data modeled by quadratic functions
• Real-world problem situations
• Real-world problem situations modeled by quadratic functions
• Data tables with at least three data points
• Technology methods
• Transformations of f(x) = x2
• Solving three by three matrix to determine ab, and c for f(x) = ax2 + bx + c

Note(s):

• Algebra I solved quadratic equations having real solutions using tables, graphs, factoring, completing the square, quadratic formula, and technology.
• Algebra I wrote, using technology, quadratic functions that provide a reasonable fit to date to estimate solutions and make predictions for real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4F

Solve quadratic and square root equations.

Solve

Including, but not limited to:

• Methods for solving quadratic equations with and without technology
• Tables
• Zeros – the value(s) of x such that the y value of the relation equals zero
• Domain values with equal range values
• Graphs
• x-intercept(s) – 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
• Algebraic methods
• Factoring
• Solving equations by taking square roots
• Solving quadratic equations using absolute value
• x2 = 25,  therefore, x = ±5
• Completing the square
• Quadratic formula, x • The discriminant, b2 – 4ac, can be used to analyze types of solutions for quadratic equations.
• b2 – 4ac = 0, one rational double root
• b2 – 4ac > 0 and perfect square, two rational roots
• b2 – 4ac > 0 and not perfect square, two irrational roots (conjugates)
• b2 – 4ac < 0, two imaginary roots (conjugates)
• Connections between solutions and roots of quadratic equations to the zeros and x-intercepts of the related function
• Complex number system
• Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
• Imaginary number – a number in the form of bi where b is a real number and i = • 2 = –1
• i = • Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• a + bi and a – bi
• Operations with complex numbers, with and without technology
• Complex solutions for quadratic equations
• One real solution
• One rational double root
•  Two real solutions
• Two rational roots
• Two irrational root conjugates
• Two complex solutions
• Two complex root conjugates
• Reasonableness of solutions

Note(s):

• Algebra I solved quadratic equations having real solutions using tables, graphs, factoring, completing the square, and the quadratic formula.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• A2 – Define and give examples of complex numbers.
• B1 – Perform computations with real and complex numbers.
• 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.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.4H Solve quadratic inequalities.
Supporting Standard

Solve

Including, but not limited to:

• Methods for solving quadratic inequalities with and without technology
• Graphs
• Tables
• Algebraic methods
• Factoring
• Solving inequalities by taking square roots
• Solving quadratic inequalities using absolute value
• x2 ≤ 25, |x| ≤ 5; therefore, –5 ≤ x ≤ 5
• Completing the square
• Testing and identifying acceptable regions on a number line
• Graphical analysis of solution sets for quadratic inequalities
• One-dimensional on a number line
• Two-dimensional on a coordinate plane
• Comparison of solution sets of equations and inequalities
• Comparison of one-dimensional solutions and two-dimensional solutions, e.g. intervals versus points
• Reasonableness of solutions

Note(s):

• Algebra I solved quadratic equations.
• Algebra II introduces quadratic inequalities.
• 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.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.7 Number and algebraic methods. The student applies mathematical processes to simplify and perform operations on expressions and to solve equations. The student is expected to:
2A.7A Add, subtract, and multiply complex numbers.
Supporting Standard

COMPLEX NUMBERS

Including, but not limited to:

• Complex number system
• The complex number system, C, is composed of real and imaginary numbers.
• Real numbers, ℜ, are composed of rational numbers, Q, and irrational numbers, ℜ – Q.
• Rational numbers, Q, are composed of integers, Ζ, whole numbers, N ∪ 0, and natural numbers, N.
• Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
• Real part of a complex number, a
• Imaginary part of a complex number, b
• Imaginary number – a number in the form of bi where b is a real number and i = • Imaginary number unit, i, is a number whose square equals –1; therefore, the = i.
• If x is a non-negative, real number .
• Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• When complex conjugates are added or multiplied the imaginary part equals 0.
• Operations with complex numbers
• Addition/subtraction of complex numbers
• Real parts combine with real parts and imaginary parts combine with imaginary parts.
• Multiplication of complex numbers
• Distribute and collect like terms.
• The imaginary unit, i, can only have a power of 1.
• Any i² units must be converted to –1.

Note(s):

• Algebra II introduces the system of complex numbers and operations with complex numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• D1 – Interpret multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 