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:

2A.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:

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.
Readiness Standard

Formulate
SYSTEMS OF EQUATIONS, INCLUDING SYSTEMS CONSISTING OF THREE LINEAR EQUATIONS IN THREE VARIABLES AND SYSTEMS CONSISTING OF TWO EQUATIONS, THE FIRST LINEAR AND THE SECOND QUADRATIC
Including, but not limited to:
 Systems of linear equations
 Two equations in two variables
 Three equations in three variables
 Systems of one linear equation and one quadratic equation in two variables
Note(s):
 Grade Level(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.
Readiness Standard

Note(s):
 Grade Level(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
 One quadratic equation
 Methods for solving systems of equations consisting of one linear equation and one quadratic equation
 Tables
 Graphs
 Identification of possible solutions in terms of points of intersection
 Algebraic methods
 Substitution of linear equation into quadratic
 Solve by factoring
 Solve by quadratic formula
 Solve by completing the square
Note(s):
 Grade Level(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
 Realworld problem situations represented by systems of inequalities
 Two linear inequalities
 Linear programming problem situations
Note(s):
 Grade Level(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 realworld problem situations
Note(s):
 Grade Level(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
 Realworld problem situations
 Realworld problem situations modeled by quadratic functions
 Data tables with at least three data points
 Technology methods
 Transformations of f(x) = x^{2} and f(x) =
 Solving three by three matrix to determine a, b, and c for f(x) = ax^{2} + bx + c
 Quadratic regression
Note(s):
 Grade Level(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 realworld 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.
Readiness Standard

Solve
QUADRATIC EQUATIONS
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
 xintercept(s) – x coordinate of a point at which the relation crosses the xaxis, 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
 Quadratic formula, x =
 The discriminant, b^{2} – 4ac, can be used to analyze types of solutions for quadratic equations.
 b^{2} – 4ac = 0, one rational double root
 b^{2} – 4ac > 0 and perfect square, two rational roots
 b^{2} – 4ac > 0 and not perfect square, two irrational roots (conjugates)
 Connections between solutions and roots of quadratic equations to the zeros and xintercepts of the related function
 Reasonableness of solutions
Note(s):
 Grade Level(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 realworld situations, including exponential relationships written in recursive notation.
Supporting Standard

Formulate
EXPONENTIAL EQUATIONS THAT MODEL REALWORLD 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
 Realworld problem situations
 Realworld problem situations modeled by exponential functions
 Exponential growth
 Exponential decay
 Representations of exponential equations
 Tables/graphs
 Verbal descriptions
 Recursive notation – notation which defines how each term of an iterative rule or formula is related to one or more preceding terms
 Recursive formula: starting value and recursion equation
 Exponential recursive formulas
 Exponential growth: a_{1} = 5, a_{n} = 4a_{n – 1}
 Exponential decay: a_{1} = 64, a_{n} = a_{n – 1}
 Technology methods
 Transformations of f(x) = b^{x} and y = log_{b}x
 Exponential regression
Note(s):
 Grade Level(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 = ab^{x} 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.
Readiness Standard

Solve
EXPONENTIAL EQUATIONS OF THE FORM y = ab^{x} 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 = ab^{x}
 a – initial value at x = 0
 b – common ratio
 Solving exponential equations
 Application of laws (properties) of exponents
 Realworld problem situations modeled by exponential functions
 Exponential growth
 f(x) = ab^{x}, where b > 1
 f(x) = ae^{k}^{x}, where k > 0
 Exponential decay
 f(x) = ab^{x}, where 0 < b < 1
 f(x) = ae^{k}^{x}, where k < 0
Note(s):
 Grade Level(s):
 Algebra I applied exponential functions to problem situations using tables, graphs, and the algebraic generalization, f(x) = a • b^{x}.
 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.
Readiness Standard

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
 Realworld problem situations involving variation
 Reasonableness of solutions mathematically and in context of realworld 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
 Realworld problem situations modeled by rational functions
 Justification of reasonableness of solutions in terms of realworld problem situations or data collections
Note(s):
 Grade Level(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.
Readiness Standard

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 realworld 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 realworld problem situations
 Mathematical justification
 Substitution in original problem
 Justification for predictions using the coefficient of determination, r^{2}
Note(s):
 Grade Level(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 realworld 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
