P.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


P.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.

P.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.

P.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.

P.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.

P.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.

P.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.

P.1G 
Display, explain, and 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.

P.2 
Functions. The student uses process standards in mathematics to explore, describe, and analyze the attributes of functions. The student makes connections between multiple representations of functions and algebraically constructs new functions. The student analyzes and uses functions to model realworld problems. The student is expected to:


P.2E 
Determine an inverse function, when it exists, for a given function over its domain or a subset of its domain and represent the inverse using multiple representations.

Determine
AN INVERSE FUNCTION, WHEN IT EXISTS, FOR A GIVEN FUNCTION OVER ITS DOMAIN OR A SUBSET OF ITS DOMAIN
Represent
THE INVERSE OF A FUNCTION USING MULTIPLE REPRESENTATIONS
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.
 Characteristics of inverse functions
 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
 Multiple representations
 Inverse function notation
 When a function f(x) has an inverse that is also a function, the inverse can be written with f ^{–1}(x).
 For the function f(x) = x + 4, the inverse function is f ^{–1}(x) = x – 4.
 For the function g(x) = x^{2}:
 If the restricted domain of g(x) is x ≥ 0, then the inverse function is g^{–}^{1}(x) = .
 If the restricted domain of g(x) is x ≤ 0, then the inverse function is g^{–}^{1}(x) = –.
 Algebraic
 The inverse of a function can be found algebraically by:
 Writing the original function in “y = ” form
 Interchanging the x and y variables
 Solving for y
 A function’s inverse can be confirmed algebraically if both of the following are true: f(f ^{–1}(x)) = x and f ^{–1}(f(x)) = x.
 Tabular
 From the table of values for a given function, the tabular values of the inverse function can be found by switching the x and yvalues of each ordered pair.
 Graphical
 The graphs of a function and its inverse are reflections over the line y = x.
 Verbal description of the relationships between the domain and range of a function and its inverse
 Restrictions on the domain of the original function to maintain functionality
 Inverse functions over a subset of the domain of the original function
Note(s):
 Grade Level(s):
 Algebra II analyzed the relationship between functions and inverses, such as quadratic and square root, or logarithmic and exponential, including necessary restrictions on the domain.
 Precalculus extends the analysis of inverses to include other types of functions, such as trigonometric and others.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.

P.2F 
Graph exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

Graph
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS
Including, but not limited to:
 Graphs of the parent functions
 Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
 Various methods for graphing
 Curve sketching
 Plotting points from a table of values
 Transformations of parent functions (parameter changes a, b, c, and d)
 Using graphing technology
Note(s):
 Grade Level(s):
 Algebra II graphed various types of functions, including square root, cube root, absolute value, and rational functions.
 Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewisedefined, and others.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 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.
 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.

P.2G 
Graph functions, including exponential, logarithmic, sine, cosine, rational, polynomial, and power functions and their transformations, including af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and d, in mathematical and realworld problems.

Graph
FUNCTIONS, INCLUDING SINE and COSINE FUNCTIONS AND THEIR TRANSFORMATIONS, INCLUDING af(x), f(x) + d, f(x – c), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d, IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 General form of parent function
 Sine and cosine functions: f(x) = sin x, f(x) = cos x
 Representations with and without technology
 Graphs
 Verbal descriptions
 Algebraic generalizations (including equation and function notation)
 Changes in parameters a, b, c, and d on graphs
 Effects of a on f(x) in af(x)
 a ≠ 0
 a > 1, the graph stretches vertically
 0 < a < 1, the graph compresses vertically
 Opposite of a reflects vertically over the horizontal axis (xaxis)
 Effects of d on f(x) in f(x) + d
 d = 0, no vertical shift
 Translation, vertical shift up or down by d units
 Effects of c on f(x) in f(x – c)
 c = 0, no horizontal shift
 Translation, horizontal shift left or right by c units
 Effects of b on f(x) in f(bx)
 b ≠ 0
 b > 1, the graph compresses horizontally
 0 < b < 1, the graph stretches horizontally
 Opposite of b reflects horizontally over the vertical axis or yaxis
 Combined transformations of parent functions
 Transforming a portion of a graph
 Illustrating the results of transformations of the stated functions in mathematical problems using a variety of representations
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra II graphed transformations of various types of functions, including square root, cube, cube root, absolute value, rational, exponential, and logarithmic functions.
 Precalculus extends the analysis of functions to include other types, such as trigonometric, power, piecewisedefined, and others.
 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.
 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.

P.2H 
Graph arcsin x and arccos x and describe the limitations on the domain.

Graph
arcsin x AND arccos x
Including, but not limited to:
 Graph of sine function,f(x) = sin x, and inverse sine function, f(x) = arcsin x
 Graph of cosine function, f(x) = cos x, and inverse cosine function, f(x) = arccos x
 Inverse relationships
 x and y values of the ordered pairs are switched.
 Graphs are reflections over the line y = x.
 f(x) = sin x and f(x) = arcsin x, which can also be written as f(x) = sin^{–1}x, are inverses.
 f(x) = cos x and f(x) = arccos x, which can also be written as f(x) = cos^{–1}x, are inverses.
 Graphs of both parent functions and other forms of the identified functions from their respective algebraic representations
Describe
THE LIMITATIONS ON THE DOMAIN OF arcsin x AND arccos x
Including, but not limited to:
 Limitations
 Comparison of domain and range of f(x) = sin x and f(x) = arcsin x
 The range of f(x) = sin x is limited so that the graph of f(x) = arcsin x is a function
 Comparison of domain and range of f(x) = cos x and f(x) = arccosx
 The range of f(x) = cos x is limited so that the graph of f(x) = arccos x is a function
 Domain
 f(x) = arcsin x has a domain of –1 ≤ x ≤ 1
 f(x) = arccos x has a domain of –1 ≤ x ≤ 1
 Range
 f(x) = arcsin x has a range of – ≤ f(x) ≤
 f(x) = arccos x has a range of 0 ≤ f(x) ≤
Note(s):
 Grade Level(s):
 Algebra II graphed inverse functions and wrote the inverse of a function using notation such as f^{–1}(x).
 Precalculus extends the analysis of inverses to include trigonometric 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.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of a 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.

P.2I 
Determine and analyze the key features of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions such as domain, range, symmetry, relative maximum, relative minimum, zeros, asymptotes, and intervals over which the function is increasing or decreasing.

Determine, Analyze
THE KEY FEATURES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS, INCLUDING STEP FUNCTIONS SUCH AS DOMAIN, RANGE, SYMMETRY, RELATIVE MAXIMUM, RELATIVE MINIMUM, ZEROS, ASYMPTOTES, AND INTERVALS OVER WHICH THE FUNCTION IS INCREASING OR DECREASING
Including, but not limited to:
 Covariation – pattern of related change between two variables in a function
 Domain and range
 Represented as a set of values
 Represented verbally
 All real numbers greater than or equal to zero
 All real numbers less than one
 Represented with inequality notation
 Represented with set notation
 {xx ∈ ℜ, x ≥ 0}
 {yy ∈ ℜ, y < 1}
 Represented with interval notation
 Symmetry
 Reflectional
 Rotational
 Symmetric with respect to the origin (180° rotational symmetry)
 Relative extrema
 Relative maximum
 Relative minimum
 Zeros
 Roots/solutions
 xintercepts
 Asymptotes
 Vertical asymptotes (x = h)
 Horizontal asymptotes (y = k)
 Slant asymptotes (y = mx + b)
 Intervals where the function is increasing or decreasing
 Represented with inequality notation, –1 < x ≤ 3
 Represented with set notation, {xx ∈ ℜ, –1 < x ≤ 3}
 Represented with interval notation, (–1, 3]
 Connections among multiple representations of key features
 Graphs
 Tables
 Algebraic
 Verbal
Note(s):
 Grade Level(s):
 Algebra II analyzed functions according to key attributes, such as domain, range, intercepts, symmetries, asymptotic behavior, and maximum and minimum values over an interval.
 Precalculus extends the analysis of key attributes of functions to include zeros and intervals where the function is increasing or decreasing.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II.D. Algebraic Reasoning – Representing relationships
 II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
 VI.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.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.

P.2O 
Develop and use a sinusoidal function that models a situation in mathematical and realworld problems.

Develop, Use
A SINUSOIDAL FUNCTION THAT MODELS A SITUATION IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Sinusoidal functions to generate models
 General forms of sinusoidal equations
 Sine: f(x) = a • sin(b(x – c)) + d
 Cosine: f(x) = a • cos(b(x – c)) + d
 Characteristics of sinusoidal functions
 Amplitude, a
 Period, P =
 Horizontal axis (middle axis), y = d
 Maximum value, d + a
 Minimum value, d – a
 Local maxima (high points)
 If a sinusoidal function has a maximum at x = k, then other maxima occur at x = k + nP (where P is the period and n is an integer).
 Local minima (low points)
 If a sinusoidal function has a minimum at x = k, then other minima occur at x = k + nP (where P is the period and n is an integer).
 Sinusoidal model from data
 Analyzing data
 Table
 Graph
 Verbal description
 Determining characteristics (amplitude, period, etc.)
 Developing the model
 General forms of sinusoidal equations
 Using transformations
 Using attributes of functions
 Sinusoidal regression equations (using technology)
Note(s):
 Grade Level(s):
 Algebra I analyzed and investigated quadratic and exponential functions and their applications.
 Algebra II analyzed and investigated logarithmic, exponential, absolute value, rational, square root, cube root, and cubic functions.
 Algebra I and Algebra II analyzed and described the effects of transformations on the parent functions with changes in a, b, c, and d parameters.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
 VI.B. Functions – Analysis of functions
 VI.B.1. Understand and analyze features of functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 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.

P.2P 
Determine the values of the trigonometric functions at the special angles and relate them in mathematical and realworld problems.

Relate
THE VALUES OF TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Geometry determined side lengths of right triangles using the Pythagorean Theorem and the relationships of special right triangles (30°– 60°– 90° and 45°– 45°– 90°).
 Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
 Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure.
 Precalculus generalizes the relationships of special right triangles to determine the values of the trigonometric functions of any multiple of 30° or 45°.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.3. Recognize and apply right triangle relationships including basic trigonometry.
 VI.B. Functions – Analysis of functions
 VI.B.2. Algebraically construct and analyze new functions.
 VI.C. Functions – Model realworld situations with functions
 VI.C.1. Apply known functions to model realworld situations.
 VI.C.2. Develop a function to model a situation.
 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.

P.4 
Number and measure. The student uses process standards in mathematics to apply appropriate techniques, tools, and formulas to calculate measures in mathematical and realworld problems. The student is expected to:


P.4A 
Determine the relationship between the unit circle and the definition of a periodic function to evaluate trigonometric functions in mathematical and realworld problems.

Determine, To Evaluate
THE RELATIONSHIP BETWEEN THE UNIT CIRCLE AND THE DEFINITION OF A PERIODIC FUNCTION IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Periodic functions
 Definition of a periodic function
 Verbal: A function is periodic if it repeats in regular intervals (or periods).
 Symbolic: A function f(x) is periodic if f(x + P) = f(x) for some positive value P. If P is the smallest such value for which this is true, then P is the period.
 Unit circle
 Unit circle on the coordinate grid
 Radius: r = 1
 Center: (0, 0)
 Equation: x^{2} + y^{2} = 1
 Rotation angles in standard position
 Vertex: (0, 0)
 Initial side (ray): the positive xaxis
 Direction of rotation
 Positive rotation: counterclockwise
 Negative rotation: clockwise
 Trigonometric functions of an angle in standard position whose terminal side intersects the unit circle at the point (x, y)
 sinθ = y
 cosθ = x
 tanθ =
 cotθ =
 secθ =
 cscθ =
 Relationship between degree measures in relation to the points on the unit circle
 Periodic functions on the unit circle
 Rotation angles for positive and negative rotation
 If θ is a standard position rotation angle that intersects the unit circle at the point (x, y), then x(θ) and y(θ) are both periodic functions with a period of 360°.
 The relationship between the unit circle and period functions is every 360° the function values repeat themselves after equal intervals.
 Graphs of trigonometric functions and angle measures to determine periodic relationships
 Rotation angles
 If θ is a standard position rotation angle that intersects the unit circle at the point (x, y), then x(θ) and y(θ) are both periodic functions with a period of 360°.
 The relationship between the unit circle and periodic functions is that every 360° the function values repeat themselves on equal intervals.
 Mathematical problem situations
 Realworld problem situations
Note(s):
 Grade Level(s):
 Algebra I and Algebra II analyzed the symmetry of various graphs, including quadratic functions.
 Algebra II analyzed the symmetry of various graphs, including quadratic, absolute value, and rational functions.
 Precalculus uses symmetry to analyze features of periodic functions.
 Geometry determined side lengths in right triangles using the Pythagorean Theorem and the relationships in special right triangles (30° – 60° – 90° and 45° – 45° – 90°).
 Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine triangle side lengths and angle measures.
 Precalculus extends trigonometric ratios from angles in triangles to include any real radian measure or degree measure.
 Precalculus generalizes the relationships in right triangles to determine the values of the trigonometric functions for any angle measure.
 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.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 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.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.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.

P.4B 
Describe the relationship between degree and radian measure on the unit circle.

Describe
THE RELATIONSHIP BETWEEN DEGREE AND RADIAN MEASURE ON THE UNIT CIRCLE
Including, but not limited to:
 Defining the unit circle
 Unit circle on the coordinate grid
 Radius: r = 1
 Center: (0, 0)
 Rotation angles in standard position
 Vertex: (0, 0)
 Initial side (ray): the positive xaxis
 Direction of rotation
 Positive rotation: counterclockwise
 Negative rotation: clockwise
 Labeling the unit circle
 Degree measure – measured as a central angle with respect to the unit circle
 Radian measure – measured as the directed length of an intercepted arc on the unit circle
 Coordinates of the unit circle
 Unit Circle
 Computing common unit conversions
 Relationship between degrees and radians
 180° = π
 Degrees into radians: multiply by
 Radians into degrees: multiply by
Note(s):
 Grade Level(s):
 Students began measuring angles in Grade 4 and continued solving problems with angle measures throughout middle school.
 Grade 6 converted units within a measurement system.
 Geometry used the Cartesian equations for circles and solved problems with central angles and arc lengths.
 Geometry described radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.
 Precalculus combines these skills together to describe the relationship between degree and radian measure.
 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.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 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.
 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.

P.4C 
Represent angles in radians or degrees based on the concept of rotation and find the measure of reference angles and angles in standard position.

Represent
ANGLES IN RADIANS OR DEGREES BASED ON THE CONCEPT OF ROTATION
Including, but not limited to:
 Rotation angles in standard position
 Vertex: (0, 0)
 Initial side (ray): the positive xaxis
 Direction of rotation
 Positive rotation: counterclockwise
 Negative rotation: clockwise
 Describing rotation of angles in radians and degrees
 x and yaxes (quadrants on the coordinate grid)
 Initial and terminal sides (rays)
 Arcs with direction arrows to indicate rotation
Find
THE MEASURE OF REFERENCE ANGLES AND ANGLES IN STANDARD POSITION
Including, but not limited to:
 Reference angle – If A is an angle in standard position, the reference angle to angle A is an acute angle created by the terminal side of angle A and the xaxis.
 Quadrantal angle – an angle whose terminal side lies on the xaxis or yaxis
 Degrees in standard position: 0°, 90°, 180°, 270°, 360°
 Radians in standard position: 0, , π, , 2π
 Interval for angle A in standard position, [0°, 360°] or [0, 2π]
 If angle A is in 1st quadrant, (0°, 90°) or , then the reference angle is A.
 If angle A is in 2nd quadrant, (90°, 180°) or , then the reference angle is 180° – A or π – A
 If angle A is in 3rd quadrant, (180°, 270°) or , then the reference angle is A – 180° or A – π
 If angle A is in 4th quadrant, (270°, 360°) or , then the reference angle is 360° – A or 2π – A
 Given an angle not in standard position, use addition or subtraction of multiples of 360° or 2π in order to determine an angle with the same terminal side in standard position, [0°, 360°] or [0, 2π].
 Mathematical problem situations
Note(s):
 Grade Level(s):
 Students began measuring angles in Grade 4 and continued solving problems with angle measures throughout middle school.
 Grade 7 solved equations based on geometry concepts, including angle relationships.
 Precalculus represents rotation and reference angles on a coordinate grid and describes the relationship between the two verbally and symbolically.
 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.
 III.A. Geometric and Spatial Reasoning – Figures and their properties
 III.A.2. Form and validate conjectures about one, two, and threedimensional figures and their properties.
 III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
 III.C.1. Make connections between geometry and algebraic equations.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.1. Find the perimeter and area of twodimensional figures.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
 IX.A. Connections – Connections among the strands of mathematics
 IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.

P.4D 
Represent angles in radians or degrees based on the concept of rotation in mathematical and realworld problems, including linear and angular velocity.

Represent
ANGLES IN RADIANS OR DEGREES BASED ON THE CONCEPT OF ROTATION IN MATHEMATICAL AND REALWORLD PROBLEMS, INCLUDING LINEAR AND ANGULAR VELOCITY
Including, but not limited to:
 Describing rotation in radians and degrees
 x and yaxes (quadrants on the coordinate grid)
 Initial and terminal sides (rays)
 Arcs with direction arrows to indicate rotation
 Describing rotation in terms of revolutions
 Describing angular velocity
 Angular velocity – rate of change of angular displacement, measured in radians per unit of time
 Formula: angular velocity, ω =
 ω = angular velocity
 θ = angle measure in radians
 t = unit of time
 Writing and simplifying ratios of rotation and time
 Converting between and among various measures of angular velocity
 Computing linear velocity
 Linear velocity – the speed, measured in distance per unit of time, with which an object moves along a circular or rotational path
 Formula: linear velocity,
 v = linear velocity
 r = radius
 = angular velocity (ω), where θ is angle measure in radians and t is unit of time
Note(s):
 Grade Level(s):
 Grade 6 converted units within a measurement system.
 Grade 7 solved problems with circumference of a circle.
 Geometry described radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.
 Precalculus combines these skills together to determine linear and angular velocities in a variety of units of measure.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 VII.A. Problem Solving and Reasoning – Mathematical problem solving
 VII.A.3. Determine a 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.
 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.
 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.
