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

Determine
THE VALUES OF THE TRIGONOMETRIC FUNCTIONS AT THE SPECIAL ANGLES
Including, but not limited to:
 Trigonometric functions – a function of an angle expressed as the ratio of two of the sides of a right triangle that contain that angle
 Trigonometric functions are called circular functions because they are based on the unit circle (circle with radius of 1 unit and center at the origin) and right triangle relationships within the circle.
 Sine (sin x or sin θ)
 Cosine (cos x or cos θ)
 Tangent (tan x or tan θ)
 Cotangent (cot x or cot θ)
 Secant (sec x or sec θ)
 Cosecant (csc x or csc θ)
 Special angles
 Degrees
 Multiples of 30°
 Multiples of 60°
 Multiples of 45°
 Special right triangles – right triangles which have angles that measure 30°– 60°– 90° or 45°– 45°– 90°
 In a 45°– 45°– 90° triangle, the proportions of the sides are 1x:1x:x.
 In a 30°– 60°– 90° triangle, the proportions of the sides are 1x:x:2x.
 Special angles as reference angles
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.4E 
Determine the value of trigonometric ratios of angles and solve problems involving trigonometric ratios in mathematical and realworld problems.

Solve
PROBLEMS INVOLVING TRIGONOMETRIC RATIOS
Including, but not limited to:
 Mathematical problem situations involving right triangles
 Finding side lengths and angle measures of right triangles
 Determining trigonometric values
 Applying trigonometric values
 Applying the Pythagorean Theorem
 Determining measures of right triangles on a coordinate grid for any radius value
 Realworld problem situations
 Problems involving right triangles
 Problems involving distances and angles
 Problems involving angles of elevation and depression
Note(s):
 Grade Level(s):
 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 real angle measure.
 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.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
 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.4F 
Use trigonometry in mathematical and realworld problems, including directional bearing.

Use
TRIGONOMETRY IN MATHEMATICAL AND REALWORLD PROBLEMS, INCLUDING DIRECTIONAL BEARING
Including, but not limited to:
 Mathematical problem situations involving right triangles
 Finding side lengths and angle measures of right triangles
 Determining and applying trigonometric values
 Determining measures of right triangles on a coordinate grid for any radius value
 Determining measures of rotation angles
 Realworld problem situations
 Problems involving right triangles
 Problems involving distances and angles
 Problems involving angles of elevation and depression
 Problems involving directional bearing
 True bearing (navigational bearing)
 Angle measure in degrees between 0° and 360°, determined by a clockwise rotation from the north line
 Conventional bearing (quadrant bearing)
 Angle measure in degrees between 0° and 90°, determined by a rotation east or west of the northsouth line
 Relationship between standard position angles, true (navigational) bearings, and conventional (quadrant) bearings
 Tools to use when solving trigonometric problems
 Pythagorean Theorem
 Pythagorean triples
 Trigonometric ratios
 Right triangle relationships
 Special right triangle relationships
 Law of Sines and/or the Law of Cosines
 Formula for Law of Sines
 Formulas for Law of Cosines
 a^{2} = b^{2} + c^{2} – 2bc cos A
 b^{2} = a^{2} + c^{2} – 2ac cos B
 c^{2} = a^{2} + b^{2} – 2ab cos C
Note(s):
 Grade Level(s):
 Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine side lengths and angle measures in right triangles.
 Precalculus extends the use of the trigonometric ratios to solve realworld problems, including those involving navigational bearings.
 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.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
 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.4G 
Use the Law of Sines in mathematical and realworld problems.

Use
THE LAW OF SINES IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Law of Sines
 Recognizing when the Law of Sines can be applied to solve for side lengths and angle measures in a triangle
 Given two angles and the included side of a triangle (or “ASA”), determine one of the remaining side lengths.
 Given two angles and a nonincluded side of a triangle (or “AAS”), determine one of the remaining side lengths.
 Given two sides and a nonincluded angle of a triangle (or “SSA”), determine one of the remaining angle measures.
 Analyzing both solutions to triangles where the Law of Sines produces an ambiguous case
 Solving various types of mathematical and realworld problem situations
 Mathematical problems
 Realworld problems
Note(s):
 Grade Level(s):
 Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine side lengths and angle measures in right triangles.
 Precalculus extends the use of the trigonometric ratios to solve for side lengths and angle measures in certain triangles using the Law of Sines.
 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.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
 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.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.4H 
Use the Law of Cosines in mathematical and realworld problems.

Use
THE LAW OF COSINES IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Law of Cosines
 a^{2} = b^{2} + c^{2} – 2bc cos A
 b^{2} = a^{2} + c^{2} – 2ac cos B
 c^{2} = a^{2} + b^{2} – 2ab cos C
 Recognizing when the Law of Cosines can be applied to solve for side lengths and angle measures in a triangle
 Given three sides of a triangle (or “SSS”), determine one of the angle measures.
 Given two sides and the included angle of a triangle (or “SAS”), determine the length of the remaining side.
 Solving various types of mathematical and realworld problem situations
Note(s):
 Grade Level(s):
 Geometry used the trigonometric ratios (sine, cosine, and tangent) to determine side lengths and angle measures in right triangles.
 Precalculus extends the use of the trigonometric ratios to solve for side lengths and angle measures in any triangle using the Law of Cosines.
 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.
 III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
 III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
 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.
 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.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.
