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 Instructional Focus DocumentAlgebra II
 TITLE : Unit 13: Exploring a Business Venture SUGGESTED DURATION : 10 days

Unit Overview

This is a project-based unit where students apply bundled student expectations that address collecting, representing, and analyzing data to develop a platform for a business venture in order to make predictions and defend decisions in terms of the business situation. 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 quadratic, rational, and exponential functions and equations. 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 equation and quadratic equation.

During this project-based unit, students apply prior knowledge to investigate and plan business ventures. Student groups explore possible scenarios for business ventures and each group selects a business venture to develop. Students represent at least two different data sets involved in the business venture that can be modeled by linear, quadratic, exponential, inverse variation, or other appropriate functions. Students identify characteristics of the functions, and analyze various aspects involving their business venture in terms of the representative functions, equations, inequalities, and systems of equations and inequalities. Students create a final written summary of their business venture, including appropriate diagrams, displays, algebraic generalizations, and calculations, justifying all solutions and conclusions in terms of their business venture. Students then analyze and critique other business venture reports.

After this unit, in subsequent courses in mathematics and later careers, these concepts will continue to be applied to problem situations involving functions, equations, inequalities, and systems of equations and inequalities.

In Algebra II, applying functions, equations, inequalities, and systems of equations and inequalities to model real-world situations are identified in STAAR Readiness Standards 2A.3A, 2A.3B, 2A.4F, 2A.5D, 2A.6L, and 2A.8C and STAAR Supporting Standards 2A.3C, 2A.3E, 2A.3G, 2A.4E, and 2A.5B.  These standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Functions and Their Inverses, 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 A2, B1; II. Algebraic Reasoning A1, B1, C1, D1, D2; III. Geometric Reasoning C1; VII. Functions A2, B1, B2, C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), Principles and Standards for School Mathematics, students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities. According to Navigating through Algebra in Grades 9 – 12, “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 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,” (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, 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.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Equations and inequalities can model problem situations and be solved using various methods.

• Why are equations and inequalities used to model problem situations?
• How are equations and inequalities used to model problem situations?
• What methods can be used to solve equations and inequalities?
• Why is it essential to solve equations and inequalities using various methods?
• How can solutions to equations and inequalities be represented?

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

• Why are systems of equations and inequalities used to model problem situations?
• How are systems of equations and inequalities used to model problem situations?
• What methods can be used to solve systems of equations and inequalities?
• Why is it essential to solve systems of equations and inequalities using various methods?
• How can solutions to systems of equations and inequalities be represented?

Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.

• Why are functions classified into families of functions?
• How are functions classified as a family of functions?
• What graphs, key attributes, and characteristics are unique to each family of functions?
• What patterns of covariation are associated with the different families of functions?
• How are the parent functions and their families used to model real-world situations?

Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical judgments in terms of the problem situation.

• Why is it important to determine and apply function models for problem situations?
• What representations can be used to analyze collected data, and how are the representations interrelated?
• Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
• How can function models be used to evaluate one or more elements in their domains?
• How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
• How does technology aid in the analysis and application of modeling and solving problem situations?

Statistical data are collected, analyzed graphically and numerically, and interpreted to determine the reliability of the data, make predictions, and draw conclusions.

• What is the purpose of analyzing statistical data?
• Why is it important to understand the analysis and interpretation of statistical data?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

• Equations
• Estimation
• Expressions
• Inequalities
• Multiple Representations
• Solve
• Systems of Equations
• Systems of Inequalities

Functions

• Attributes of Functions
• Inverse Variation
• Linear Functions
• Non-Linear Functions

Statistical Reasoning

• Conclusions/Predictions
• Data
• Regression
• Statistical Representations
• Summary Statistics

Associated Mathematical Processes

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

Equations and inequalities can be used to model and solve mathematical and real-world problem situations.

• How are equations and inequalities used to model problem situations?
• What methods can be used to solve equations and inequalities?
• What are the advantages and disadvantages of various methods used to solve equations and inequalities?
• What methods can be used to justify the reasonableness of solutions to equations and inequalities?

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

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

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

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

Systems of equations in two variables consisting of a linear equation and a quadratic equation can be used to model real-world problem situations by analyzing the problem situation and various representations in order to interpret and make predictions and critical judgments.

• What representations can be used to display the system of equations?
• How can the representations of the system of equations be used to interpret and make predictions and critical judgments in terms of the problem situation?
• How can solutions be justified for reasonableness in terms of the problem situation?

Each family of functions has unique graphs and attributes.

• What representations can be used to represent functions?
• What are the key attributes of each family of functions and how can they be determined from the various representations?

Data collected from real-world situations can be matched with an appropriate function family comparing key attributes from the graph of the data with key attributes from the graph of the parent function.

• How can the function family be determined for a set of data?
• How are the key attributes of data analyzed to select an appropriate function model for a problem situation?

Unique characteristics of data, scatterplots of data, calculation of regression equations, and comparison of correlation coefficients and coefficients of determination can be used to select an appropriate model from among linear, quadratic, or exponential functions.

• What characteristics of sequential data can be used to determine if the data is linear, quadratic, or exponential?
• How does the rate of change compare for linear, quadratic, and exponential data?
• How are scatterplots used to determine if data is best represented by a linear, quadratic, or exponential function?
• How can a regression equation be determined using a graphing calculator?
• How does the correlation coefficient help determine whether to represent the data by a linear or exponential function?
• How does the coefficient of determination help determine whether to represent the data by a linear, quadratic, or exponential function?

Predictions and critical judgments can be made from models selected to represent linear, quadratic, or exponential data.

• How can representations of the data be used to make predictions and critical judgments?
• How can the coefficient of determination help determine the validity of predictions using the model regression equation?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• None Identified

Unit Vocabulary

Related Vocabulary:

 Correlation value Direct variation Domain/range Equations Exponential function Inequalities Inverse variation Linear function Parent functions Quadratic function Rational function Representations Reasonableness of solutions Scatterplot Systems of equations Systems of inequalities x-intercepts Zeros
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity

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

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
2A.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
2A.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• VIII. Problem Solving and Reasoning
2A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• X. Connections
2A.1G Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxCCRS:
• IX. Communication and Representation
2A.3 Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:
2A.3A Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

Formulate

SYSTEMS OF EQUATIONS, INCLUDING SYSTEMS CONSISTING OF THREE LINEAR EQUATIONS IN THREE VARIABLES AND SYSTEMS CONSISTING OF TWO EQUATIONS, THE FIRST LINEAR AND THE SECOND QUADRATIC

Including, but not limited to:

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

Note(s):

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

Solve

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

Including, but not limited to:

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

Note(s):

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

Formulate

SYSTEMS OF AT LEAST TWO LINEAR INEQUALITIES IN TWO VARIABLES

Including, but not limited to:

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

Note(s):

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

Determine

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

Including, but not limited to:

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

Note(s):

• Algebra I solved systems of two linear inequalities in two variables using graphs, tables, and algebraic methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• C1 – Apply known function models.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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.4E

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

Supporting Standard

Formulate

QUADRATIC EQUATIONS USING TECHNOLOGY GIVEN A TABLE OF DATA

Including, but not limited to:

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

Note(s):

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

Solve quadratic and square root equations.

Solve

Including, but not limited to:

• Methods for solving quadratic equations with and without technology
• Tables
• Zeros – the value(s) of x such that the y value of the relation equals zero
• Domain values with equal range values
• Graphs
• x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• Zeros – the value(s) of x such that the y value of the relation equals zero
• Algebraic methods
• Factoring
• Solving equations by taking square roots
• Solving quadratic equations using absolute value
• Completing the square
• 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)
• Connections between solutions and roots of quadratic equations to the zeros and x-intercepts of the related function
• Reasonableness of solutions

Note(s):

• Algebra I solved quadratic equations having real solutions using tables, graphs, factoring, completing the square, and the quadratic formula.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• A2 – Define and give examples of complex numbers.
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.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.5B

Formulate exponential and logarithmic equations that model real-world situations, including exponential relationships written in recursive notation.

Supporting Standard

Formulate

EXPONENTIAL EQUATIONS THAT MODEL REAL-WORLD SITUATIONS, INCLUDING EXPONENTIAL RELATIONSHIPS WRITTEN IN RECURSIVE NOTATION

Including, but not limited to:

• Data collection activities with and without technology
• Data modeled by exponential functions
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Real-world problem situations
• Real-world problem situations modeled by exponential functions
• Exponential growth
• Exponential decay
• Representations of exponential equations
• Tables/graphs
• Verbal descriptions
• Recursive notation for exponential relationships
• Recursive formula: starting value and recursion equation
• Exponential recursive formulas
• Exponential growth: a1 = 5, an = 4an – 1
• Exponential decay: a1 = 64, an = an – 1
• Technology methods
• Transformations of f(x) = bx and y = logbx
• Exponential regression

Note(s):

• Algebra I determined formulas and terms for geometric sequences given in recursive and function notation.
• Algebra II introduces formulating exponential equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.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
• 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

Note(s):

• Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = a • bx.
• Algebra II solves exponential equations algebraically.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.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.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:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
2A.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:
• I. Numeric Reasoning
• B1 – Perform computations with real and complex numbers.
• II. Algebraic Reasoning
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• D1 – Interpret multiple representations of equations and relationships.
• D2 – Translate among multiple representations of equations and relationships.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections