
 Bold black text in italics: Knowledge and Skills Statement (TEKS)
 Bold black text: Student Expectation (TEKS)
 Strikethrough: Indicates portions of the Student Expectation that are not included in this unit but are taught in previous or future unit(s)

 Blue text: Supporting information / Clarifications from TCMPC (Specificity)
 Blue text in italics: Unitspecific clarification
 Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards (TxCCRS)

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:

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:
 VIII. Problem Solving and Reasoning

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:
 VIII. Problem Solving and Reasoning

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:
 IX. Communication and Representation

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:
 IX. Communication and Representation

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:

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:
 IX. Communication and Representation

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.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 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 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°
 Interval for angle A in standard position, [0°, 360°]
 If angle A is in 1st quadrant, (0°, 90°), then the reference angle is A.
 If angle A is in 2nd quadrant, (90°, 180°), then the reference angle is 180° – A
 If angle A is in 3rd quadrant, (180°, 270°), then the reference angle is A – 180°
 If angle A is in 4th quadrant, (270°, 360°), then the reference angle is 360° – A
 Given an angle not in standard position, use addition or subtraction of multiples of 360° in order to determine an angle with the same terminal side in standard position, [0°, 360°].
 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:
 III. Geometric Reasoning
 A1 – Identify and represent the features of plane and space features.
 C1 – Make connections between geometry and algebra.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

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 of the unit circle
 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
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. Geometric Reasoning
 A3 – Recognize and apply right triangle relationships including basic trigonometry.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.4I 
Use vectors to model situations involving magnitude and direction.

Use
VECTORS
Including, but not limited to:
 Describing vectors with words and symbols
 Vector vocabulary
 Vector – a directed distance
 Magnitude of a vector – the length of a vector
 Direction of a vector – the standard position rotation angle associated with a vector
 Components of a vector – the horizontal (xcomponent) and vertical (ycomponent) measures of a vector, written as an ordered pair
 Vector symbols
 Vector representation:
 v: bolded, nonitalicized lowercase letter
 : halfarrow above italicized, nonbolded letter
 Magnitude: v
 Direction:
 Components: = v = x, y
 Vector formulas
 If a vector v has components given as v = x, y = 6.84, 18.79, the magnitude and direction can be found using:
 Magnitude: v =
 Direction: = tan^{1}, when x > 0, or = tan^{1} + 180°, when x < 0
 If a vector v has a magnitude of v and a direction of , the components can be found using:
 x = vcos
 y = vsin
To Model
SITUATIONS INVOLVING MAGNITUDE AND DIRECTION WITH VECTORS
Including, but not limited to:
 Describing situations with vectors
 Measures of magnitude
 Distance
 Speed (velocity)
 Force
 Measures of direction
 Angle
 Compass direction
 Navigational bearing
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 uses the trigonometric ratios in order to measure and describe different attributes of vectors.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 III. Geometric Reasoning
 A3 – Recognize and apply right triangle relationships including basic trigonometry.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.4J 
Represent the addition of vectors and the multiplication of a vector by a scalar geometrically and symbolically.

Represent
THE ADDITION OF VECTORS GEOMETRICALLY AND SYMBOLICALLY
Including, but not limited to:
 Geometrically
 Headtotail (or tiptotail) method for adding a + b = x_{1}, y_{1} + x_{2}, y_{2}
 Draw vector a in standard position, with the tail at (0, 0).
 Draw vector b starting at the head of vector a, with the tail at (x_{1}, y_{1}).
 Draw vector a + b by connecting the tail of vector a to the head of vector b.
 Parallelogram method for adding a + b = x_{1}, y_{1} + x_{2}, y_{2}
 Draw vectors a and b in standard position, with the tails at (0, 0).
 Form a parallelogram using the vectors as adjacent sides.
 Draw vector a + b by drawing the diagonal of the parallelogram from the origin.
 Vector addition represented in different frames of reference
 Coordinate plane with at least one vector in standard position
 Coordinate plane with neither vector in standard position
 Diagram using horizontal and vertical components
 Symbolically
 If a = x_{1}, y_{1} and b = x_{2}, y_{2}, then a + b = x_{1}, y_{1} + x_{2}, y_{2} = x_{1} + x_{2}, y_{1} + y_{2}
Represent
THE MULTIPLICATION OF A VECTOR BY A SCALAR GEOMETRICALLY AND SYMBOLICALLY
Including, but not limited to:
 Geometrically
 Headtotail (or tiptotail) method for finding k•a, where a = x_{1}, y_{1}
 Draw vector a in standard position, with the tail at (0, 0) and the tip at (x_{1}, y_{1}).
 Copy vector a again, starting with the tail at (x_{1}, y_{1}) and with the tip at (2x_{1}, 2y_{1}).
 Copy vector a again, starting with the tail at (2x_{1}, 2y_{1}) and with the tip at (3x_{1}, 3y_{1}).
 Repeat until k vectors are drawn endtoend.
 Multiplication of a vector by a scalar represented in different frames of reference
 Coordinate plane with vector in standard position
 Coordinate plane with vector not in standard position
 Diagram using horizontal and vertical components
 Symbolically
 If a = x_{1}, y_{1}, then k•a = k • x_{1}, y_{1} = kx_{1}, ky_{1}
Note(s):
 Grade Level(s):
 Geometry analyzed lines and segments on the coordinate grid to include distance, midpoint, and slope.
 Precalculus introduces the operations of addition and scalar multiplication of vectors.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II. Algebraic Reasoning
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 III. Geometric Reasoning
 C1 – Make connections between geometry and algebra.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections

P.4K 
Apply vector addition and multiplication of a vector by a scalar in mathematical and realworld problems.

Apply
VECTOR ADDITION AND MULTIPICATION OF A VECTOR BY A SCALAR IN MATHEMATICAL AND REALWORLD PROBLEMS
Including, but not limited to:
 Mathematical problem situations
 Combining vectors through addition and scalar multiplication
 Realworld problem situations
 Combining two or more directed distances
 Combining two or more forces
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 uses the trigonometric ratios in order to measure and describe different attributes of vectors.
 Precalculus applies the operations of addition and scalar multiplication of vectors in realworld problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxCCRS:
 II. Algebraic Reasoning
 B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).
 III. Geometric Reasoning
 A3 – Recognize and apply right triangle relationships, including basic trigonometry.
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
