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Precalculus
TITLE : Unit 10: Vectors SUGGESTED DURATION : 8 days

Unit Overview

This unit bundles student expectations that address representing, using, and applying vectors and vector operations, including vector addition and multiplication of a vector by a scalar. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this unit, in Geometry Unit 05, students determined the lengths of sides and the measures of angles in a right triangle by applying special right triangle relationships, the Pythagorean theorem, Pythagorean triples, and the trigonometric ratios, sine, cosine, and tangent. In Precalculus Unit 07, students extended the use of the trigonometric ratios to solve real-world problems, including those involving navigational bearings. In Geometry Units 02 and 07, students analyzed lines and segments on the coordinate grid to derive and use formulas for distance, midpoint, and slope. In Precalculus Unit 08, students represented rotation and reference angles on a coordinate grid and described the relationship between the two verbally and symbolically.

During this unit, students model situations with vectors graphically using the concepts of rotation angles in mathematical situations. Students describe vectors using proper notation for magnitude, direction, and vector components and use trigonometry to determine the magnitude and direction of a vector, given its component form. Students use trigonometry to determine the components of a vector when given its magnitude and direction. Students use vectors to model situations involving magnitude and direction, using appropriate formulas and notation for magnitude, direction, and components. Students model real-world problem situations using vectors, including problems that incorporate directional bearing. Students geometrically represent multiplication of a vector by a scalar in order to determine a symbolic representation for vector multiplication by a scalar. Students apply vector multiplication by a scalar in both mathematical and real-world situations. Students geometrically represent vector addition in order to determine a symbolic representation for vector addition. Students apply vector addition in both mathematical and real-world situations, with a continued focus on describing vectors using magnitude, direction, and components and converting between these different representations. Students apply vector addition and multiplication of a scalar to real-world problem situations in order to combine two or more directed distances or forces.

After this unit, in Precalculus Unit 11, students will continue to use trigonometry and vectors to model and solve mathematical and real-world problems using parametric equations. In subsequent mathematics courses, students will continue to apply these concepts as they arise in problem situations.

Geometric reasoning serves an integral role in college readiness. Identifying and using geometric representations and making connections between algebraic and geometric representations are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning B1; III. Geometric Reasoning A1, A3, C1; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

According to the National Council of Teachers of Mathematics (2000), students in grades 9-12 should use their understanding of integer operations to explore number systems in which new patterns and properties emerge, such as with vectors and vector combinations. Students should understand vectors as systems that share certain properties with the real-number system and should develop an understanding of the properties of and representations for vector addition and multiplication (NCTM, 2000). Furthermore, students in grades 9-12 should develop fluency with vector operations and basic proficiency with using vectors to solve problems, including the use of mental computation, paper-and-pencil calculations, and technology as appropriate (NCTM, 2000). Additionally, dynamic geometry software can be used to explore and illustrate vector operations (NCTM, 2000). Vectors are important geometric tools that provide a means for simultaneously representing magnitude and direction (NCTM, 2000). Vectors are useful for modeling a number of real-world phenomena, especially in physics. Vectors also have important connections to transformational geometry and can be used to understand and represent translations in the plane (NCTM, 2000). The study of vectors continues into calculus, where students analyze planar curves (including those given in parametric form, polar form, and vector form) as an important part of functional analysis and derivative studies (College Board, 2012).

College Board.  (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/collegereadiness/crs.pdf.

OVERARCHING UNDERSTANDINGS and QUESTIONS

Geometric and spatial reasoning are necessary to describe and analyze geometric relationships in mathematics and the real-world.

  • Why are geometric and spatial reasoning necessary in the development of an understanding of geometric relationships?
  • Why is it important to visualize and use diagrams to effectively communicate/illustrate geometric relationships?
  • How do geometric and spatial reasoning allow for the understanding of different geometric systems as models for the world?

 

Application of attributes and measures of figures can be generalized to describe geometric relationships which can be used to solve problem situations.

  • Why are attributes and measures of figures used to generalize geometric relationships?
  • How can numeric patterns be used to formulate geometric relationships?
  • Why is it important to distinguish measureable attributes?
  • How do geometric relationships relate to other geometric relationships?
  • Why is it essential to develop generalizations for geometric relationships?
  • How are geometric relationships applied to solve problem situations?
  • How do different systems of measure relate to one another?
Performance Assessment(s) Overarching Concepts
Unit Concepts
Unit Understandings
Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

Algebraic Reasoning

  • Multiple Representations
  • Patterns/Rules
  • Ratios

 

Geometric Reasoning

  • Geometric Relationships
  • Pythagorean Theorem
  • Trigonometric Relationships

 

Measurement Reasoning

  • Angle Measures
  • Formulas
  • Length

 

Numeric Reasoning

  • Addition
  • Multiplication

 

Associated Mathematical Processes

  • Application
  • Tools and Techniques
  • Problem Solving Model
  • Communication
  • Representations
  • Relationships 
  • Justification

Models and diagrams can be used to visualize and illustrate geometric relationships and aid in solving problems.

  • Why are models and diagrams necessary for visualizing the geometric relationships found in the problem situation?
  • How are models and diagrams used to organize information from the problem situation?
  • How do models and diagrams aid in calculations when solving problems?
  • Why is the coordinate plane used to diagram two-dimensional figures?

 

Geometric tools such as degree measures are used to represent angles based on the concept of rotation in mathematical and real-world problems.

  • How can angles be represented in degrees using the concept of rotation?
  • What role do reference angles play in solving problems with rotational angles?
  • What problem situations can be modeled and solved using rotational angles?

 

Geometric tools such as vectors can be used to model situations and solve problems involving magnitude and direction.

  • How can vectors be represented?
  • What relationships exist between the different representations of vectors?
  • What situations can be modeled with vectors?
  • How can vectors be used to model situations involving magnitude and direction?
  • What mathematical and real-world problems can be solved with vectors?
  • How can vectors be used to solve problems involving magnitude and direction?
  • How do vectors, trigonometry, and the Pythagorean theorem relate?

 

Vector addition and scalar multiplication exhibit various geometric and algebraic relationships that can be represented and verified using a variety of methods and can be applied to model and solve problems involving magnitude and direction.

  • How can addition of vectors be represented geometrically?
  • How can addition of vectors be represented symbolically?
  • How can the multiplication of a vector by a scalar be represented geometrically?
  • How can the multiplication of a vector by a scalar be represented symbolically?
  • What geometric and algebraic relationships can be verified by exploring vector addition?
  • What geometric and algebraic relationships can be verified by exploring multiplication of a vector by a scalar?
  • How can vector addition be applied to solve mathematical and real-world problems?
  • How can multiplication of a vector by a scalar be applied to solve mathematical and real-world problems?

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • Some students may think that inverse trigonometric functions are the same as the reciprocalsof the trigonometric functions. For example, tan–1(x) is not the same as .
  • Some students may incorrectly evaluate trigonometric functions or inverse trigonometric functions if they use a calculator in the wrong angle mode.
  • Some students may misinterpret the magnitude of a vector. For example, the magnitude of a velocity vector represents the speed an object is traveling, not the distance it has traveled.
  • Some students may confuse angle measurements for angles in standard position versus angles represented as bearings.
  • Some students may struggle to determine whether a specific vector given in a problem situation represents a vector to be summed or the resultant vector itself.
  • Some students may incorrectly model vector addition by drawing two vectors with the same initial point and then drawing a “resultant” vector from the tip of one vector to the tip of the other. Rather, students can represent vector addition by using the tail-to-tip method or the parallelogram method.
  • Students may incorrectly determine the angle of rotation for vectors in Quadrants II and III. For example, students may incorrectly determine that the angle of rotation for the vector  is tan–1  = –53.13°. However, the angle of rotation for this vector is actually –53.13° + 180° = 126.87°.

 

Underdeveloped Concepts:

  • Some students may struggle with using angle relationships (including supplementary, complementary, alternate interior, corresponding, and exterior angles) to determine angle measures in problem situations involving vector addition and directional bearing.
  • Some students may believe that vectors are tied to a particular location in the coordinate plane (i.e. that the vector  has its initial point at the origin and its terminal point at (–3, 4)). However, a vector can actually be represented anywhere on the coordinate plane , so long as it maintains its magnitude and direction.
  • Some students may believe that multiplying a vector by a scalar always results in a vector that is longer than the original vector. However, this is only true if the absolute value of the scalar is greater than 1.

Unit Vocabulary

Key Content Vocabulary:

  • Components of a vector – the horizontal (x-component) and vertical (y-component) measures of a vector, written as an ordered pair
  • Direction of a vector – the standard position rotation angle associated with a vector
  • Magnitude of a vector – the length of a vector
  • Quadrantal angle – an angle whose terminal side lies on the x-axis or y-axis
  • 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 x-axis
  • Vector – a directed distance

 

Related Vocabulary:

  • Angle of depression
  • Angle of elevation
  • Bearings
  • Conventional bearing
  • Head-to-tail method
  • Navigational bearing
  • Negative rotation
  • Parallelogram method
  • Positive rotation
  • Pythagorean Theorem
  • Quadrant bearing
  • Resultant
  • Rotation angle
  • Scalar
  • Scalar multiplication
  • Standard position
  • Tip-to-tail method
  • Trigonometric ratios
  • True bearing
  • Vector addition

 

Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards (select CCRS from Standard Set dropdown menu)

Texas Instruments – Graphing Calculator Tutorials

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – Mathematics TEKS: Supporting Information

Texas Education Agency – Interactive Mathematics Glossary

TEKS# SE# TEKS Unit Level Specificity
 
  • Bold black text in italics: Knowledge and Skills Statement (TEKS)
  • Bold black text: Student Expectation (TEKS)
  • Strike-through: 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: Unit-specific 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:
    • X. Connections
P.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:

  • Problem-solving model
    • Analyze given information
    • Formulate a plan or strategy
    • Determine a solution
    • Justify the solution
    • Evaluate the problem-solving 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 non-examples, patterns, etc.
      • Current knowledge to new learning

Note(s):    

  • The mathematical process standards may be applied to all content standards as appropriate.
  • TxCCRS:
    • X. Connections
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 real-world 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 x-axis
    • Direction of rotation
      • Positive rotation: counter-clockwise
      • Negative rotation: clockwise
  • Describing rotation of angles in degrees
    • x- and y-axes (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 x-axis.
    • Quadrantal angle – an angle whose terminal side lies on the x-axis or y-axis
      • 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 real-world problems, including directional bearing.

Use

TRIGONOMETRY IN MATHEMATICAL AND REAL-WORLD 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
  • Real-world 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 north-south 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 real-world 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 (x-component) and vertical (y-component) measures of a vector, written as an ordered pair
    • Vector symbols
      • Vector representation:
        • v: bolded, non-italicized lowercase letter
        • : half-arrow above italicized, non-bolded 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 = |v|cos
        • y = |v|sin

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
    • Head-to-tail (or tip-to-tail) method for adding a + b = x1, y1 + x2, y2
      • 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 (x1, y1).
      • Draw vector a + b by connecting the tail of vector a to the head of vector b.
    • Parallelogram method for adding a + b = x1, y1 + x2, y2
      • 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 = x1, y1 and b = x2, y2, then a + b = x1, y1 + x2, y2 = x1 + x2, y1 + y2

Represent

THE MULTIPLICATION OF A VECTOR BY A SCALAR GEOMETRICALLY AND SYMBOLICALLY

Including, but not limited to:

  • Geometrically
    • Head-to-tail (or tip-to-tail) method for finding ka, where a = x1, y1
      • Draw vector a in standard position, with the tail at (0, 0) and the tip at (x1, y1).
      • Copy vector a again, starting with the tail at (x1, y1) and with the tip at (2x1, 2y1).
      • Copy vector a again, starting with the tail at (2x1, 2y1) and with the tip at (3x1, 3y1).
      • Repeat until k vectors are drawn end-to-end.
    • 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 = x1, y1, then ka = kx1, y1 = kx1, ky1

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 real-world problems.

Apply

VECTOR ADDITION AND MULTIPICATION OF A VECTOR BY A SCALAR IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

  • Mathematical problem situations
    • Combining vectors through addition and scalar multiplication
  • Real-world 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 real-world 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 
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/24/2016
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