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.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) = x^{3}, f(x) =
Including, but not limited to:
 Representations of functions, including graphs, tables, and algebraic generalizations
 Cubic, f(x) = x^{3}
 Cube root, f(x) =
 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, AND MAXIMUM AND MINIMUM GIVEN AN INTERVAL, WHEN APPLICABLE
Including, but not limited to:
 Domain and range of the function
 Domain – set of input values for the independent variable over which the function is defined
 Continuous function – function whose values are continuous or unbroken over the specified domain
 Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
 Range – set of output values for the dependent variable over which the function is defined
 Representation for domain and range
 Verbal description
 Ex: x is all real numbers less than five.
 Ex: x is all real numbers.
 Ex: y is all real numbers greater than –3 and less than or equal to 6.
 Ex: y is all integers greater than or equal to zero.
 Inequality notation – notation in which the solution is represented by an inequality statement
 Ex: x < 5, x ∈ ℜ
 Ex: x ∈ ℜ
 Ex: –3 < y ≤ 6, y ∈ ℜ
 Ex: y ≥ 0, y ∈ Ζ
 Set notation – notation in which the solution is represented by a set of values
 Braces are used to enclose the set.
 Solution is read as “The set of x such that x is an element of …”
 Ex: {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, ∞)
 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.
 Maximum and minimum (extrema)
 Relative maximum – largest ycoordinate, or value, a function takes over a given interval of the curve
 Relative minimum – smallest ycoordinate, or value, a function takes over a given interval of the curve
 Use key attributes to recognize and sketch graphs
 Application of key attributes to 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:
 III. Geometric Reasoning
 B1 – Identify and apply transformations to figures.
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A1 – Recognize whether a relation is a function.
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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
 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:
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A1 – Recognize whether a relation is a function.
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.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
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
 Cubic function and cube root function, f(x) = x^{3} and g(x) = ^{ }
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:
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VII. Functions
 A1 – Recognize whether a relation is a function.
 A2 – Recognize and distinguish between different types of functions.
 B1 – Understand and analyze features of a function.
 B2 – Algebraically construct and analyze new functions.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.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.6A 
Analyze the effect on the graphs of f(x) = x^{3} and f(x) = when f(x) is replaced by af(x), f(bx), f(x – c), and f(x) + d for specific positive and negative real values of a, b, c, and d.
Supporting Standard

Analyze
THE EFFECT ON THE GRAPHS OF f(x) = x^{3} AND f(x) = WHEN f(x) IS REPLACED BY af(x), f(bx), f(x – c), AND f(x) + d FOR SPECIFIC POSITIVE AND NEGATIVE REAL VALUES OF a, b, c, AND d
Including, but not limited to:
 General form of the cubic and cube root functions
 Cubic
 Cube root
 f(x) =
 Representations with and without technology
 Graphs
 Tables
 Verbal descriptions
 Algebraic generalizations
 Effects on the graphs of f(x) = x^{3} and f(x) = when parameters a, b, c, and d are changed in f(x) = a(b(x – c))^{3} + d and f(x) =
 Effects on the graphs of f(x) = x^{3} and f(x) = , when f(x) is replaced by af(x) with and without technology
 a ≠ 0
 a > 1, the graph stretches vertically
 0 < a < 1, the graph compresses vertically
 Opposite of a reflects vertically over the xaxis
 Effects on the graphs of f(x) = x^{3} and f(x) = , when f(x)is replaced by f(bx) with and without technology
 b ≠ 0
 b > 1, the graph compresses horizontally
 0 < b < 1, the graph stretches horizontally
 Opposite of b reflects horizontally over the yaxis
 Effects on the graphs of f(x) = x^{3} and f(x) = , when f(x) is replaced by f(x – c) with and without technology
 c = 0, no horizontal shift
 Horizontal shift left for values of c < 0 by c units
 For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left two units.
 Horizontal shift right for values of c > 0 by c units
 For f(x – 2), c = 2, and the function moves to the right two units
 Effects on the graphs of f(x) = x^{3}, and f(x) = , when f(x) is replaced by f(x) + d with and without technology
 d = 0, no vertical shift
 Vertical shift down for values of d < 0 by d units
 Vertical shift up for values of d > 0 by d units
 Connections between the critical attributes of transformed function and f(x) = x^{3} and f(x) =
 Determination of parameter changes given a graphical or algebraic representation
 Determination of a graphical representation given the algebraic representation or parameter changes
 Determination of an algebraic representation given the graphical representation or parameter changes
 Descriptions of the effects on the domain and range by the parameter changes
 Effects of multiple parameter changes
 Mathematical problem situation
 Realworld problem situation
Note(s):
 Grade Level(s):
 Algebra I determined effects on the graphs of the parent functions, f(x) = x and f(x) = x^{2} when f(x) is replaced by af(x),
f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d.
 Algebra II introduces the cubic and cube root functions and their transformations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 B1 – Identify and apply transformations to figures.
 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.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

2A.6B 
Solve cube root equations that have real roots.
Supporting Standard

Solve
CUBE ROOT EQUATIONS THAT HAVE REAL ROOTS
Including, but not limited to:
 Application of laws (properties) of exponents
 Application of cube roots to solve cubic equations
 Applications of cubics to solve cube root equations
 Reasonableness of solutions
 Substitution of solutions into original problem
 Graphical analysis
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra II introduces cubic and cube root functions and solving cube root equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.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.7H 
Solve equations involving rational exponents.
Readiness Standard

Solve
EQUATIONS INVOLVING RATIONAL EXPONENTS
Including, but not limited to:
 Laws (properties) of exponents
 Product of powers (multiplication when bases are the same): a^{m} • a^{n} = a^{m+n}
 Quotient of powers (division when bases are the same): = a^{mn}
 Power to a power: (a^{m})^{n} = a^{mn}
 Negative exponent: a^{n} =
 Rational exponent:
 Equations when bases are the same: a^{m} = a^{n} → m = n
 Solving equations with rational exponents
 Isolation of base and power using properties of algebra
 Exponentiation of both sides by reciprocal of power of base
 If the denominator of the reciprocal power is even, then the variable must be represented using absolute value.
 Simplification to obtain solution
 Verification of solution
 Realworld problem situations modeled by equations involving rational exponents
 Justification of reasonableness of solutions in terms of realworld problem situations
Note(s):
 Grade Level(s):
 Prior grade levels simplified numeric expressions, including integral and rational exponents.
 Algebra II introduces equations involving rational exponents.
 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.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
