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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 12: Making Connections SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address a review of families of functions, their attributes, transformations, and applications to problem situations. Student expectations also address a review of formulating, solving, and justifying solutions to 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, students investigated linear functions, equations, and inequalities and systems of two linear equations in two variables. Students also investigated quadratic functions and equations and exponential functions. In Algebra II, students investigated absolute value, quadratic, square root, cubic, cube root, rational, exponential, and logarithmic functions and equations. Students investigated absolute value and quadratic inequalities. Students solved systems of three linear equations in three variables, systems of at least two linear inequalities in two variables, and a system involving a linear and quadratic equation.

During this Unit
Students review graphing functions and analyzing the key attributes such as domain and range (representing domain and range using interval notation, inequalities, and set notation), intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval as covered previously in Algebra II. Students review analyzing inverse functions as covered previously in Algebra II. Students review analyzing the effect on the graph of parent functions, f(x), when f(x) is replaced by af(x), f(bx), f(xc), and f(x) + d for specific positive and negative real values of a, b, c, and d, as covered previously in Algebra II. Students review investigating parameter changes and key attributes in terms of real-world problems, including comparisons of linear, quadratic, and exponential functions as covered previously in Algebra II. Students review solving systems of three equations in three variables, systems involving one linear and one quadratic equation, and systems of at least two linear inequalities in two variables as covered previously in Algebra II. Students review solving equations, including square root, exponential, logarithmic, cube root, absolute value, and rational equations as covered previously in Algebra II. Students review solving absolute value and quadratic inequalities and determining factors and roots for cubic and quartic equations as covered previously in Algebra II. Students review application of equations to model real-world problem situations as covered previously in Algebra II.

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

After this Unit
In Unit 13, students will apply linear, quadratic, and exponential functions, equations, and systems of equations and inequalities to model problems in a business venture. In subsequent courses in mathematics, these concepts will continue to be applied to problem situations involving functions, equations, inequalities, and systems of equations and inequalities.

In Algebra II, graphing and analyzing key attributes of functions and using functions to model problem situations are identified in STAAR Readiness Standards 2A.2A, 2A.2C, 2A.4C, 2A.5A, and 2A.8C and STAAR Supporting Standard 2A.6K. These standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses, STAAR Reporting Category 4: Quadratic and Square Root Functions, Equations, and Inequalities, STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations, and STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. Solving equations, inequalities, and systems of equations and inequalities are identified in STAAR Readiness Standards 2A.3B, 2A.4B, 2A.4F, 2A.5D, 2A.6E, 2A.6I, 2A.6L, and 2A.7E and STAAR Supporting Standard 2A.3C. 2A.3F, 2A.6B, 2A.6F, and 2A.7D. These standards are subsumed under STAAR Reporting Category 1: Number and Algebraic Methods, STAAR Reporting Category 3: Writing and Solving Systems of Equations and Inequalities, STAAR Reporting Category 4: Quadratic and Square Root Functions, Equations, and Inequalities, STAAR Reporting Category 5: Exponential and Logarithmic Functions and Equations, and STAAR Reporting Category 6: Other Functions, Equations, and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1; II. Algebraic Reasoning A1, B1, C2, C3, D1, D2; III. Geometric and Spatial Reasoning C2; V. Statistical Reasoning A1, C2; VI. Functions A2, B1, B2, C1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics (2000), students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions,” (NCTM, 2002, p. 3). Research found in National Council of Teachers of Mathematics (NCTM) also states, “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,” (as cited by NCTM, 2009, p. 41). This unit places particular emphasis on multiple representations. State and national mathematics standards support such an approach. The Texas Essential Knowledge and Skills repeatedly require students to relate representations of functions, such as algebraic, tabular, graphical, and verbal descriptions. This skill is mirrored in the Principles and Standards for School Mathematics (NCTM, 1989). Specifically, this work calls for instructional programs that enable all students to understand relations and functions and select, convert flexibly among, and use various representations for them. More recently, the importance of multiple representations has been highlighted in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics (NCTM, 2007). According to this resource, students should be able to translate among verbal, tabular, graphical, and algebraic representations of functions and describe how aspects of a function appear in different representations as early as Grade 8. Also, in research summaries such as Classroom Instruction That Works: Research-Based Strategies for Increasing Student Achievement, such concept development is even cited among strategies that increase student achievement. Specifically, classroom use of multiple representations, referred to as nonlinguistic representations, and identifying similarities and differences has been statistically shown to improve student performance on standardized measures of progress (Marzano, Pickering & Pollock, 2001).

Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
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.
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? Statistical displays often reveal patterns within data that can be analyzed to interpret information, inform understanding, make predictions, influence decisions, and solve problems in everyday life with degrees of confidence. How does society use or make sense of the enormous amount of data in our world available at our fingertips? How can data and data displays be purposeful and powerful? Why is it important to be aware of factors that may influence conclusions, predictions, and/or decisions derived from data?
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 relationship?
• How can the most appropriate function model be determined for a set of data?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by …
• linear functions?
• square root functions?
• exponential functions?
• logarithmic functions?
• cube functions?
• cube root functions?
• absolute value functions?
• rational functions?
• What graphs, key attributes, and characteristics are unique to different families of functions?
• What patterns of covariation are associated with different families of functions?
• How can the key attributes of functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of functions?
• What relationships exist between the mathematical and real-world meanings of the key attributes of function models?
• How can key attributes be used to make predictions and critical judgments about the problem situation?
• 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 can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• What relationships exist between a function and its inverse?
• How are the key attributes of a function related to the key attributes of its inverse?
• How can the inverse of a function be determined and represented?
• How can function composition be used to analyze relationships between functions?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Asymptotes
• Axis of symmetry
• Symmetries
• Functions and Equations
• Square root
• Exponential
• Logarithmic
• Cube
• Cube root
• Absolute value
• Rational
• Inverse
• Patterns, Operations, and Properties
• Relations and Generalizations
• Transformations
• Parent functions
• Transformation effects
• Data
• Data and Statistics
• Data
• Models
• Conclusions and predictions
• 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.

 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)
• 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 factor expressions?
• How does the structure of the expression influence the selection of an efficient method for factoring polynomial expressions?
• What relationships exist between the factors of polynomial expressions and the zeros of their associated polynomial functions?
• Different families of functions, 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 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 types of equations produce extraneous solutions, and why are the solutions considered extraneous?
• What methods can be used to write …
• square root equations?
• exponential equations?
• logarithmic equations?
• cubic equations?
• cube root equations?
• absolute value equations?
• absolute value inequalities?
• rational equations?
• equations involving inverse variation?
• How does the given information and/or representation influence the selection of an efficient method for writing equations and inequalities?
• What methods can be used to solve …
• square root equations?
• exponential equations?
• logarithmic equations?
• cubic equations?
• cube root equations?
• absolute value equations?
• absolute value inequalities?
• rational equations?
• equations involving inverse variation?
• How does the structure of the equation influence the selection of an efficient method for solving the equation?
• How can the solutions to equations and inequalities be determined and represented?
• How are properties and operational understandings used to transform equations and inequalities?
• Systems of equations and 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 equations or inequalities in the system.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can systems of equations and 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 equations?
• systems of inequalities?
• What methods can be used to solve …
• systems of equations?
• systems of inequalities?
• How does the structure of the system influence the selection of an efficient method for solving the …
• system of equations?
• system of inequalities?
• How can the solutions to systems of equations and systems of inequalities be determined and represented?
• How are properties and operational understandings used to transform systems of equations and systems of inequalities?
• 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 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?
• What kinds of algebraic and graphical relationships exist between inequalities in a system with …
• no solutions?
• infinitely many solutions?
• What relationships exist between the solution set of each inequality in a system and the solution set of the system of inequalities?
• How can the boundaries and the vertices of the solution set of a system of inequalities be used to make predictions and critical judgments in problem situations?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Vertex
• Axis of symmetry
• Focus
• Directrix
• Equations and Inequalities
• Square root
• Exponential
• Logarithmic
• Cube
• Cube root
• Absolute value
• Rational
• Inverse variation
• Patterns, Operations, and Properties
• Systems of Equations and Inequalities
• Linear
• Number and Algebraic Methods
• Relations and Functions
• Expression
• Polynomial
• Patterns, Operations, and Properties
• 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 domain and range of a function and the domain and range of the problem situation are always equivalent rather than that the domain and range of the problem situation may be restricted by constraints in the problem.
• Some students may think that the inverse of a function means to take the opposite sign or reciprocal of the function rather than switching the independent and dependent variables.
• Some students may think that |x| can equal a negative number such as |x| = –1 rather than the fact that it is the x inside the absolute that can equal –1.
• Some students may think that |x| = 2 only gives x = 2 rather than x = 2 and x = –2.
• 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 the cube root of a negative number is imaginary rather than understanding that the cube root of a negative number is just the negative number.
• Some students may think an asymptote is a line that can never be crossed rather than a line that is approached. Although a vertical asymptote cannot be crossed, a horizontal asymptote can be crossed and approached in another section of the graph.
• Some students may think that when zeros of an expression occur in the denominator of the function, it always produces a vertical asymptote rather than understanding that if an x-value makes both the numerator and the denominator equal to zero, it indicates a removable discontinuity, not a vertical asymptote.
• Some students may have misconceptions about the nature of the graph of a function when using a graphing calculator, rather than understanding that sometimes the calculator obscures the details or hidden behavior of a function.

#### Unit Vocabulary

• Asymptote – a line that is approached and may or may not be crossed
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• 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
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Direct variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• 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
• Discontinuity – characteristic of a function where it is not continuous at some point along its graph; place where there is a break in the continuous curve of the function
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Extraneous solution – solution derived by solving the equation algebraically that is not a true solution of the equation and will not be valid when substituted back into the original equation
• Focus – point not on the directrix whose distance from the vertex is |p| and lies on the axis of symmetry
• Gaussian elimination – sequence of elementary row operations on a matrix of coefficients and answers to transform the matrix into row echelon form (ref)
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve. A horizontal asymptote describes the long run behavior of the rational function.
• Imaginary number – a number in the form of bi where b is a real number and i • Inequality notation – notation in which the solution is represented by an inequality statement
• Interval notation – notation in which the solution is represented by a continuous interval
• Inverse variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = • 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)
• Point (removable) discontinuity – hole or discontinuity in the graph of a rational function generated when a factor in the denominator, that should create a vertical asymptote, reduces out with an equivalent factor in the numerator
• Range – set of output values for the dependent variable over which the function is defined
• Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
• Relative maximum – largest y-coordinate, or value, a function takes over a given interval of the curve
• Relative minimum – smallest y-coordinate, or value, a function takes over a given interval of the curve
• Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
• Set notation – notation in which the solution is represented by a set of values
• 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
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Absolute value function Correlation value Cubic function Cube root function Equations/inequalities Exponential function Factor theorem Independent/dependent variables Inverse functions Linear function Logarithmic function Quadratic function Remainder theorem Rational function Reasonableness of solutions Representations Square root function Systems of equations Systems of inequalities
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 II 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.
• 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
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
2A.2 Attributes of functions and their inverses. The student applies mathematical processes to understand that functions have distinct key attributes and understand the relationship between a function and its inverse. The student is expected to:
2A.2A Graph the functions f(x)= , f(x)=1/x, f(x)=x3, f(x)= , f(x)=bx, f(x)=|x|, and f(x)=logb (x) where b is 2, 10, and e, and, when applicable, analyze the key attributes such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum given an interval.

Graph

THE FUNCTIONS f(x) = , f(x) = , f(x) = x3, f(x) = , f(x) = bx, f(x)= |x|, AND f(x) = logb(x) WHERE b IS 2, 10, AND e

Including, but not limited to:

• Representations of functions, including graphs, tables, and algebraic generalizations
• Square root, f(x) = • Rational (reciprocal of x), f(x) = • Cubic, f(x) = x3
• Cube root, f(x) = • Exponential, f(x) = bx, where b is 2, 10, and e
• Absolute value, f(x) = |x|
• Logarithmic, f(x) = logb(x), where b is 2, 10, and e
• Connections between representations of families of functions
• Comparison of similarities and differences of families of functions

Analyze

THE KEY ATTRIBUTES OF THE FUNCTIONS SUCH AS DOMAIN, RANGE, INTERCEPTS, SYMMETRIES, ASYMPTOTIC BEHAVIOR, AND MAXIMUM AND MINIMUM GIVEN AN INTERVAL, WHEN APPLICABLE

Including, but not limited to:

• Domain and range of the function
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Representation for domain and range
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, y ∈ Ζ
• Set notation – notation in which the solution is represented by a set of values
• Braces are used to enclose the set.
• Solution is read as “The set of x such that x is an element of …”
• Ex: {x|x ∈ ℜ, x < 5}
• Ex: {x|x ∈ ℜ}
• Ex: {y|y ∈ ℜ, –3 < y ≤ 6}
• Ex: {y|y ∈ Ζ, y ≥ 0}
• Interval notation – notation in which the solution is represented by a continuous interval
• Parentheses indicate that the endpoints are open, meaning the endpoints are excluded from the interval.
• Brackets indicate that the endpoints are closed, meaning the endpoints are included in the interval.
• Ex: (–∞, 5)
• Ex: (–∞, ∞)
• Ex: (–3, 6]
• Ex: [0, ∞)
• Domain and range of the function versus domain and range of the contextual situation
• Key attributes of functions
• Intercepts/Zeros
• 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
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Symmetries
• Reflectional symmetry – symmetry in which one half of the image is a mirror image of the other over a line of reflection
• Rotational symmetry – symmetry that occurs if a figure can be rotated less than 360° around a central point and still looks the same as the original. The number of times a figure fits into itself in one complete rotation is called the order of rotational symmetry.
• Asymptotic behavior – behavior such that as x approaches infinity, f(x) approaches a given value
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Maximum and minimum (extrema)
• Relative maximum – largest y-coordinate, or value, a function takes over a given interval of the curve
• Relative minimum – smallest y-coordinate, or value, a function takes over a given interval of the curve
• Use key attributes to recognize and sketch graphs
• Application of key attributes to real-world problem situations

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Algebra I studied parent functions f(x) = x, f(x) = x2, and f(x) = bx and their key attributes.
• Precalculus will study polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts 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.
2A.2C Describe and analyze the relationship between a function and its inverse (quadratic and square root, logarithmic and exponential), including the restriction(s) on domain, which will restrict its range.

Describe, Analyze

THE RELATIONSHIP BETWEEN A FUNCTION AND ITS INVERSE (QUADRATIC AND SQUARE ROOT, LOGARITHMIC AND EXPONENTIAL), INCLUDING THE RESTRICTION(S) ON DOMAIN, WHICH WILL RESTRICT ITS RANGE

Including, but not limited to:

• Relationships between functions and their inverses
• All inverses of functions are relations.
• Inverses of one-to-one functions are functions.
• Inverses of functions that are not one-to-one can be made functions by restricting the domain of the original function, f(x).
• Characteristics of inverse relations
• Interchange of independent (x) and dependent (y) coordinates in ordered pairs
• Reflection over y = x
• Domain and range of the function versus domain and range of the inverse of the given function
• Functionality of the inverse of the given function
• Quadratic function and square root function, f(x) = x2 and f(x) = • Restrictions on domain when using positive square root • Restrictions on domain when using negative square root • Cubic function and cube root function, f(x) = x3 and g(x) = • Exponential function and logarithmic function, f(x) = bx and g(x) = logb(x) where b is 2, 10, and e

Note(s):

• Algebra I determined if relations represented a function.
• Algebra II introduces inverse of a function and restricting domain to maintain functionality.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
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.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.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.
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
• 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 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.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.
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.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.
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.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)
• 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:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VIII.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.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
2A.4C Determine the effect on the graph of f(x) = when f(x) is replaced by af(x), f(x) + d, f(bx), and f(x - c) for specific positive and negative values of a, b, c, and d.

Determine

THE EFFECT ON THE GRAPH OF f(x) = WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(bx), AND f(xc) FOR SPECIFIC POSITIVE AND NEGATIVE VALUES OF a, b, c, AND d

Including, but not limited to:

• General form of the square root function
• f(x) = • Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Effects on the graph of f(x) = , when parameters a, b, c, and d are changed in f(x) = • Effects on the graph of f(x) = , when f(x) is replaced by af(x) with and without technology
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the x-axis
• Effects on the graph of f(x) = , when f(x) is replaced by f(bx) with and without technology
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the y-axis
• Effects on the graph of f(x) = , when f(x) is replaced by f(xc) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left or right by |c| units
• Left shift when c < 0
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
• Right shift when c > 0
• For f(x – 2), c = 2, and the function moves to the right two units
• Effects on the graph of f(x) = , when f(x) is replaced by f(x) + d with and without technology
• d = 0, no vertical shift
• Vertical shift up or down by |d| units
• Down shift when d < 0
• Up shift when d > 0
• Connections between the critical attributes of transformed function and f(x) = • Determination of parameter changes given a graphical or algebraic representation
• Determination of a graphical representation given the algebraic representation or parameter changes
• Determination of an algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situation
• Real-world problem situations

Note(s):

• Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d.
• Algebra II introduces the square root parent function and its transformations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
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, |x| = 5; 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 = • i2 = –1
• i = • Complex conjugates – complex numbers having the same real part but an opposite imaginary part
• a + bi and abi
• 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
• Methods for solving square root equations with and without technology
• Tables
• Zeros – the values of x such that the y value of the relation equals zero
• Domain values with equal range values
• Graphs
• x-intercept(s) – xcoordinate 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
• Identification of extraneous solutions
• 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:
• 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.
2A.5 Exponential and logarithmic functions and equations. The student applies mathematical processes to understand that exponential and logarithmic functions can be used to model situations and solve problems. The student is expected to:
2A.5A Determine the effects on the key attributes on the graphs of f(x) = bx and f(x) = logb(x) where b is 2, 10, and e when f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive and negative real values of a, c, and d.

Determine

THE EFFECTS ON THE KEY ATTRIBUTES ON THE GRAPHS OF f(x) = bx AND f(x) = logb(x) WHERE b IS 2, 10, AND e WHEN f(x) IS REPLACED BY af(x), f(x) + d, and f(xc) FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, c, AND d

Including, but not limited to:

• General form of the power function
• Exponential functions, f(x) = bx, where b is 2, 10, and e
• f(x) = 2x; f(x) = 10x; f(x) = ex
• Logarithmic functions, y = logb(x), where b is 2, 10, and e
• f(x) = log2(x); f(x) = log10(x) or f(x) = log(x); f(x) = loge(x) or f(x) = ln(x)
• Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Key attributes
• Intercepts
• Asymptotes
• Effects on the graphs of f(x) = bx and y = logb(x) when parameters a, b, c, and d are changed in f(x) = ab(xc) + d and f(x) = a • logb(xc) + d
• Effects on the graphs of f(x) = 2x and f(x) = log2(x) when f(x) is replaced by af(x) with and without technology
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the x-axis
• Effects on the graphs of f(x) = 10x and f(x) = log(x), when f(x) is replaced by f(xc) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
• Horizontal shift right for values of c > 0 by |c| units
• For f(x – 2), c = 2, and the function moves to the right two units.
• Effects on the graphs of f(x) = ex and f(x) = ln(x), when f(x) is replaced by f(x) + d with and without technology
• d = 0, no vertical shift
• Vertical shift down for values of d < 0 by |d| units
• Vertical shift up for values of d > 0 by |d| units
• Connections between the critical attributes of transformed functions and f(x) = bx and y = logb(x)
• Determination of parameter changes given a graphical or algebraic representation
• Determination of a graphical representation given the algebraic representation or parameter changes
• Determination of an algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situation
• Effects of parameter changes in real-world problem situations

Note(s):

• Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2 when f(x) is replaced by af(x),
f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d.
• Algebra II continues to investigate the exponential parent function and introduces logarithmic parent function and transformations of both functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
2A.5D Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions.

Solve

EXPONENTIAL EQUATIONS OF THE FORM y = abx WHERE a IS A NONZERO REAL NUMBER AND b IS GREATER THAN ZERO AND NOT EQUAL TO ONE

Including, but not limited to:

• Exponential equation, y = abx
• a – initial value at x = 0
• b – common ratio
• Solving exponential equations
• Application of laws (properties) of exponents
• Application of logarithms as necessary
• Real-world problem situations modeled by exponential functions
• Exponential growth
• f(x) = abx, where b > 1
• f(x) = aekx, where k > 0
• Exponential decay
• f(x) = abx, where 0 < b < 1
• f(x) = aekx, where k < 0

Solve

SINGLE LOGARITHMIC EQUATIONS HAVING REAL SOLUTIONS

Including, but not limited to:

• Single logarithmic equation, y = logb(x)
• x – argument
• b – base
• y – exponent
• Solving logarithmic equations
• Transformation to exponential form as necessary
• Real-world problem situations modeled by logarithmic functions

Note(s):

• Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = abx.
• Algebra II solves exponential equations algebraically.
• Algebra II introduces logarithms and solving logarithmic equations.
• Precalculus will use properties of logarithms to solve equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• 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.
2A.6 Cubic, cube root, absolute value and rational functions, equations, and inequalities. The student applies mathematical processes to understand that cubic, cube root, absolute value and rational functions, equations, and inequalities can be used to model situations, solve problems, and make predictions. The student is expected to:
2A.6B Solve cube root equations that have real roots.
Supporting Standard

Solve

CUBE ROOT EQUATIONS THAT HAVE REAL ROOTS

Including, but not limited to:

• Application of laws (properties) of exponents
• Application of cube roots to solve cubic equations
• Applications of cubics to solve cube root equations
• Reasonableness of solutions
• Substitution of solutions into original problem
• Graphical analysis
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra II introduces cubic and cube root functions and solving cube root equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• 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.
2A.6E Solve absolute value linear equations.

Solve

ABSOLUTE VALUE LINEAR EQUATIONS

Including, but not limited to:

• Methods for solving absolute value linear equations with and without technology
• Graphs
• Algebraic methods
• Solving process
• Transform the equation so that the absolute value expression is on one side of the equation and all other variable terms and constants are on the other side of the equation.
• Separate the equation into two parts divided by “or”:
• Expression inside the absolute value equal to the other side of the equation
• Expression inside the absolute value equal to the opposite of the other side of the equation
• |x| = 5 → x = 5 or x = –5
• Extraneous solution – solution derived by solving the equation algebraically that is not a true solution of the equation and will not be valid when substituted back into the original equation
• Solving absolute value equations involves separating the absolute value into both the possible positive value inside the absolute and the possible negative value inside the absolute. In the case of |x| = 2, The x value can be either positive or negative 2. However, this is not a reversible situation, |x| = 2 but |x| ≠ –2.
• Justification of solutions with and without technology
• Graphs
• Substitution of solutions into original functions
• Extraneous solutions
• Real-world problem situations modeled by absolute value functions
• Justification of reasonableness of solutions in terms of the real-world problem situations or data collections

Note(s):

• Grade 6 defined absolute value and identified the absolute value of a number.
• Algebra II introduces the absolute value equation and its applications.
• 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.
2A.6F Solve absolute value linear inequalities.
Supporting Standard

Solve

ABSOLUTE VALUE LINEAR INEQUALITIES

Including, but not limited to:

• Methods for solving absolute value linear inequalities with and without technology
• Graphs
• Algebraic methods
• Solving process
• Isolation of absolute expression on one side of the inequality
• Separation of the inequality into two parts
• Greater than (>) or greater than or equal to (≥)
• First part: expression inside the absolute value set greater than or greater than or equal to other side of the inequality
• Second part: expression inside the absolute value set less than or less than or equal to the opposite of the other side of the inequality
• Parts separated by “ or ”
• Representation of solutions
• Symbolic notation
• Interval notation
• Graphical notation
• Less than (<) or less than or equal to (≤)
• First part: expression inside the absolute value set less than or less than or equal to other side of the inequality
• Second part: expression inside the absolute value set greater than or greater than or equal to the opposite of the other side of the inequality
• Parts separated by “ and ”
• Representation of solutions
• Symbolic notation
• Interval notation
• Graphical notation
• Justification of solutions of absolute value inequalities with and without technology
• Graphs
• Substitution of solutions into original functions
• Removal of extraneous solutions

Note(s):

• Grade 6 defined absolute value and identified the absolute value of a number.
• Algebra II introduces absolute value inequalities.
• 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.
2A.6G Analyze the effect on the graphs of f(x) = 1/x when f(x) is replaced by af(x), f(bx), f(- c), and f(x) + d for specific positive and negative real values of a, b, c, and d.
Supporting Standard

Analyze

THE EFFECT ON THE GRAPHS OF f(x) = WHEN f(x) IS REPLACED BY af(x), f(bx), f(xc), AND f(x) + d FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, b, c, AND d

Including, but not limited to:

• General form of the rational function
• Rational function
• f(x) = • Representations with and without technology
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations
• Effects on the graph of f(x) = , when parameters a, b, c, and d are changed in or • Effects on the graph of f(x) = , when f(x) is replaced by af(x) with and without technology
• a ≠ 0
• |a| > 1, the graph stretches vertically
• 0 < |a| < 1, the graph compresses vertically
• Opposite of a reflects vertically over the x-axis
• Effects on the graph of f(x) = , when f(x) is replaced by f(bx) with and without technology
• b ≠ 0
• |b| > 1, the graph compresses horizontally
• 0 < |b| < 1, the graph stretches horizontally
• Opposite of b reflects horizontally over the y-axis
• Effects on the graph of f(x) = , when f(x) is replaced by f(xc) with and without technology
• c = 0, no horizontal shift
• Horizontal shift left for values of c < 0 by |c| units
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
• Horizontal shift right for values of c > 0 by |c| units
• For f(x – 2), c = 2, and the function moves to the right two units
• Effects on the graph of f(x) = , when f(x) is replaced by f(x) + d with and without technology
• d = 0, no vertical shift
• Vertical shift down for values of d < 0 by |d| units
• Vertical shift up for values of d > 0 by |d| units
• Connections between the critical attributes of transformed function and f(x) = • Determination of parameter changes given a graphical or algebraic representation
• Determination of a graphical representation given the algebraic representation or parameter changes
• Determination of an algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Descriptions of the effects on the asymptotes by the parameter changes
• Effects of multiple parameter changes
• Mathematical problem situations
• Real-world problem situation

Note(s):

• Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(xc), f(bx) for specific values of a, b, c, and d.
• Algebra II introduces the rational function and its transformations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
2A.6I Solve rational equations that have real solutions.

Solve

RATIONAL EQUATIONS THAT HAVE REAL SOLUTIONS

Including, but not limited to:

• Rational equations composed of linear or quadratic functions
• Limited to real solutions
• Methods for solving rational equations with and without technology
• Graphs
• Algebraic methods
• Solving processes
• Identification of domain restrictions; denominator ≠ 0
• Methods to solve
• Application of cross products for proportional problems
• Multiplication by least common denominator
• Determination of least common denominator
• Multiplication of least common denominator to eliminate fractions
• Transformation of equation to solve for unknown
• Justifications of solutions with and without technology
• Graphs
• Substitution of solutions into original functions
• Removal of extraneous solutions
• Real-world problem situations modeled by rational functions
• Justification of reasonableness of solutions in terms of real-world problem situations or data collections

Note(s):

• Algebra II introduces the rational equation and its applications.
• 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.
2A.6K Determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation.
Supporting Standard

Determine

THE ASYMPTOTIC RESTRICTIONS ON THE DOMAIN OF A RATIONAL FUNCTION

Including, but not limited to:

• Discontinuity in rational functions
• Discontinuity – characteristic of a function where it is not continuous at some point along its graph; place where there is a break in the continuous curve of the function
• Asymptote – a line that is approached and may or may not be crossed
• Vertical asymptote – vertical line approached by the curve as the function approaches positive or negative infinity. Vertical asymptotes are never crossed by the curve.
• Discontinuity where the denominator cannot equal zero
• Determination of vertical asymptotes by setting the denominator ≠ 0
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve. A horizontal asymptote describes the long run behavior of the rational function.
• If the degree of the numerator is less than the degree of denominator, the horizontal asymptote is f(x) = 0.
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is f(x) = where p is the coefficient of the highest degreed term of the numerator and q is the coefficient of the highest degreed term of the denominator.
• Oblique (slant) asymptote – non-vertical and non-horizontal line approached by the curve as the function approaches positive or negative infinity. Oblique (slant) asymptotes may be crossed by the curve.
• If the degree of the numerator is one more than the degree of the denominator, then the oblique asymptote is of the form y = mx + b determined by the quotient of the numerator and denominator through long division.
• Point (removable) discontinuity – hole or discontinuity in the graph of a rational function generated when a factor in the denominator, that should create a vertical asymptote, reduces out with an equivalent factor in the numerator
• Determination of canceled factor
• Determination of x-value in canceled factor that would create a zero in the denominator
• Calculation of the corresponding y-value of the point discontinuity using the reduced rational function
• Graphical analysis using discontinuity
• Domain and range
• Limitations from discontinuities
• Vertical asymptote(s) restrictions on domain
• Horizontal asymptote restrictions on range
• Point(s) of discontinuity restrictions on domain and range
• End behavior
• Single and compound inequality statements to identify domain and range
• Analyzing graph of function in regions formed on graph
• Point tested in regions
• Symmetry
• Intercepts
• Appropriate curve sketched in each region

Represent

DOMAIN AND RANGE USING INTERVAL NOTATION, INEQUALITIES, AND SET NOTATION

Including, but not limited to:

• Inequality notation
• Ex: x < 5 or x > 8?
• Ex: –3 < y < 6
• Ex: x < –3 or 0 < x< 2 or x > 4
• Set notation
• Ex: {x|x ∈, ℜ, x < 5 or x > 8}
• Ex: {y|y ∈, ℜ, –3 < y < 6}
• Ex: {x|x ∈, ℜ, x < –3 or 0 < x < 2 or x > 4}
• Interval notation
• Ex: (–∞,5) ∪ (8,∞)
• Ex: (–3, 6)
• Ex: (–∞,–3) ∪ (0,2) ∪ (4,∞)

Note(s):

• Algebra II introduces the rational function and its attributes.
• Precalculus will continue to investigate rational functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
2A.6L Formulate and solve equations involving inverse variation.

Formulate

EQUATIONS INVOLVING INVERSE VARIATION

Including, but not limited to:

• Characteristics of variation
• Constant of variation
• Particular equation to represent variation
• Types of variation
• Direct variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• y varies directly as x
• General equation: y = kx
• Connection of direct variation to linear functions
• Inverse variation – a relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = • y varies inversely as x
• General equation: y = • Connection of inverse variation to rational functions
• Real-world problem situations involving variation
• Reasonableness of solutions mathematically and in context of real-world problem situations

Solve

EQUATIONS INVOLVING INVERSE VARIATION

Including, but not limited to:

• Methods for solving variation equations with and without technology
• Graphs
• Algebraic methods
• Solving processes
• Determination of a particular equation to represent the problem
• Direct variation, y = kx
• Inverse variation, y = • Transformation of equation to solve for unknown
• Justification of solutions with and without technology
• Substitution of solutions into original functions
• Real-world problem situations modeled by rational functions
• Justification of reasonableness of solutions in terms of real-world problem situations or data collections

Note(s):

• Prior grade levels studied direct variation and proportionality.
• Algebra II introduces inverse variation and its applications in problem situations.
• Precalculus will continue to investigate rational functions.
• 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.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VI.B. Functions – Analysis of functions
• VI.B.2. Algebraically construct and analyze new functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VI.C.2. Develop a function to model a situation.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• 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.
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.7D Determine the linear factors of a polynomial function of degree three and of degree four using algebraic methods.
Supporting Standard

Determine

THE LINEAR FACTORS OF A POLYNOMIAL FUNCTION OF DEGREE THREE AND OF DEGREE FOUR USING ALGEBRAIC METHODS

Including, but not limited to:

• Connections between roots and factors
• If x = c is a root of a polynomial, then (xc) is a factor of the polynomial.
• Determination of linear and quadratic factors from tables
• Identification of roots from a table, x values where y values equal zero
• Writing roots as factors
• Determination of linear and quadratic factors from graphs

• Identification of roots from a graph, x-intercepts or zeros
• Writing roots as factors
• Determination of linear and quadratic factors by depressing polynomials
• Rational root theorem to determine possible roots
• For polynomial equation anxn + an–1xn–1 + ... + a1x1 + a0 = 0 with integral coefficients of degree n in which an is the coefficient of xn, and a0 is the constant term, then possible rational roots are where q is a factor of the leading coefficient, an, and p is a factor of the constant term, a0.
• Analysis of possible rational roots by synthetic division
• Remainder Theorem
• If the remainder is zero, x = c is an actual root of the polynomial.
• When the polynomial is depressed to a quadratic expression, remaining roots can be determined by factoring or solving using the quadratic formula.
• The calculated rational roots must be a part of the set of possible rational roots, .

Note(s):

• Algebra I introduced factorization of polynomials of degree one and degree two.
• Algebra II introduces synthetic division of degree three and four polynomials by degree one polynomials.
• Algebra II introduces depression of polynomials to determine roots and factors of the polynomial.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
2A.7E Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping.

Determine

LINEAR AND QUADRATIC FACTORS OF A POLYNOMIAL EXPRESSION OF DEGREE THREE AND OF DEGREE FOUR, INCLUDING FACTORING THE SUM AND DIFFERENCE OF TWO CUBES AND FACTORING BY GROUPING

Including, but not limited to:

• Determination of linear and quadratic factors by factorization
• Greatest common factor
• Difference of squares: a2b2 = (a + b)(ab)
• Trinomials
• Sum of cubes: a3 + b3 = (a + b)(a2ab + b2)
• Difference of cubes: a3b3 = (ab)(a2 + ab + b2)
• Grouping methods
• Verify factorization by re-multiplying the factors.
• Factor using non-algebraic techinques to determine rational roots
• Tables
• Graphs

Note(s):

• Algebra I introduced factorization of polynomials of degree one and degree two.
• Algebra II introduces factorization of polynomials of degree three and degree four.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II.A. Algebraic Reasoning – Identifying expressions and equations
• II.A.1. Explain the difference between expressions and equations.
• II.B. Algebraic Reasoning – Manipulating expressions
• II.B.1. Recognize and use algebraic properties, concepts, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
2A.8 Data. The student applies mathematical processes to analyze data, select appropriate models, write corresponding functions, and make predictions. The student is expected to:
2A.8C Predict and make decisions and critical judgments from a given set of data using linear, quadratic, and exponential models.

Predict, Make

DECISIONS AND CRITICAL JUDGMENTS FROM A GIVEN SET OF DATA USING LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

Including, but not limited to:

• Mathematical and real-world problem situations modeled by linear, quadratic, and exponential functions and equations
• Predictions, decisions, and critical judgments from function models
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations
• Mathematical justification
• Substitution in original problem
• Justification for predictions using the coefficient of determination, r2

Note(s):

• Algebra I introduced the linear, quadratic, and exponential functions.
• Algebra I introduced the correlation coefficient as a measure of the strength of linear association.
• Algebra I applied linear, quadratic, and exponential functions to model and make predictions in real-world problem situations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.2. Make connections between geometry, statistics, and probability.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations. 