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 – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 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:
 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 problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 nonexamples, 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)=x^{3}, f(x)=, f(x)=b^{x}, f(x)=x, and f(x)=log_{b} (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.
Readiness Standard

Graph
THE FUNCTIONS f(x) = , f(x) = , f(x) = x^{3}, f(x) = , f(x) = b^{x}, f(x)= x, AND f(x) = log_{b}(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) = x^{3}
 Cube root, f(x) =
 Exponential, f(x) = b^{x}, where b is 2, 10, and e
 Absolute value, f(x) = x
 Logarithmic, f(x) = log_{b}(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, 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
 Ex: x
 Ex: –3 < y ≤ 6, y
 Ex: y ≥ 0, y
 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: {xx , x < 5}
 Ex: {xx }
 Ex: {yy , –3 < y ≤ 6}
 Ex: {yy , 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
 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
 yintercept(s) – y coordinate of a point at which the relation crosses the yaxis, 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.
 Use key attributes to recognize and sketch graphs
 Application of key attributes to realworld problem situations
Note(s):
 Grade Level(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) = x^{2}, and f(x) = b^{x} 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.2B 
Graph and write the inverse of a function using notation such as f ^{1}(x).
Supporting Standard

Graph, Write
THE INVERSE OF A FUNCTION USING NOTATION SUCH AS f ^{–1} (x)
Including, but not limited to:
 Inverse of a function – function that undoes the original function. When composed f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x.
 Inverse functions
 Quadratic and square root
 Exponential and logarithmic
 Cubic and cube root
 Inverses of functions on graphs
 Inverses of functions in tables
 Interchange independent (x) and dependent (y) coordinates in ordered pairs
 Inverses of functions in equation notation
 Interchange independent (x) and dependent (y) variables in the equation, then solve for y
 Inverses of functions in function notation
 f ^{–1}(x) represents the inverse of the function f(x).
Note(s):
 Grade Level(s):
 Algebra II introduces inverse of a function.
 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.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.
 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.
Readiness Standard

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 onetoone functions are functions.
 Inverses of functions that are not onetoone 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) = x^{2} 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) = x^{3} and g(x) =
 Exponential function and logarithmic function, f(x) = b^{x} and g(x) = log_{b}(x) where b is 2, 10, and e
Note(s):
 Grade Level(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.2D 
Use the composition of two functions, including the necessary restrictions on the domain, to determine if the functions are inverses of each other.
Supporting Standard

Use
THE COMPOSITION OF TWO FUNCTIONS
Including, but not limited to:
 Composition of functions – process of substituting one function into another function to create a new function such that the range of one function becomes the domain of the other function
 Composition notation given f(x) and g(x)
 Verbal
 f composed with g of x; g composed with f of x
 f of g of x; g of f of x
 Symbolic
 f(g(x)); g(f(x))
 (f ○ g)(x); (g ○ f)(x)
To Determine
IF TWO FUNCTIONS ARE INVERSES OF EACH OTHER, INCLUDING THE NECESSARY RESTRICTIONS ON THE DOMAIN
Including, but not limited to:
 Characteristics of inverse relations
 Interchange of independent (x) and dependent (y) coordinates in ordered pairs
 Interchange of independent (x) and dependent (y) coordinates in an equation and resolving for y
 Reflection over the f(x) = x line
 Domain of the function becomes an appropriate range of the inverse function
 Range of the function becomes an appropriate domain of the inverse function
 Composed as f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x
 Domain and range of the function versus domain and range of the inverse of the given function
Note(s):
 Grade Level(s):
 Algebra II introduces inverse of a function.
 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).
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.5. Evaluate the problemsolving process.

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.7I 
Write the domain and range of a function in interval notation, inequalities, and set notation.
Supporting Standard

Write
THE DOMAIN AND RANGE OF A FUNCTION IN INTERVAL NOTATION, INEQUALITIES, AND SET NOTATION
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
 Reasonable domain and range values in problem situations
 Domain and range of the function versus domain and range of the contextual situation
 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
 Ex: x ∈ ℜ
 Ex: –3 < y ≤ 6
 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: {xx ∈ ℜ, x < 5}
 Ex: {xx ∈ ℜ}
 Ex: {yy ∈ ℜ, –3 < y ≤ 6}
 Ex: {yy ∈ 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, ∞)
Note(s):
 Grade Level(s):
 The notation represents the set of real numbers, and the notation represents the set of integers.
 Algebra I introduced domain and range written in inequality notation.
 Algebra II introduces domain and range written in set and interval notation.
 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.
