 Hello, Guest!
 Instructional Focus DocumentPrecalculus
 TITLE : Unit 11: Parametric Equations SUGGESTED DURATION : 7 days

Unit Overview

Introduction
This unit bundles student expectations that address graphing parametric equations, using parametric equations to model and solve problem situations, and converting between parametric equations and rectangular relations. 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 Algebra I Units 02 – 04, 08, and 09, Algebra II Units 01, 02, and 05 – 11, and Precalculus Units 01, 03 – 05, and 08, students graphed various types of functions and their transformations, including linear, quadratic, exponential, absolute value, square root, cubic, cube root, rational, logarithmic, polynomial, power, trigonometric, and inverse trigonometric functions. Additionally, students analyzed the features of these functions, including domain and range. In Algebra I Units 01 and 07, students solved literal equations for specified variables. In Algebra II Unit 01 and Precalculus Unit 02, students used function composition to model and solve problems. In Geometry Unit 05, students used trigonometric ratios (including sine and cosine) to determine the lengths of sides and the measures of angles in right triangles. In Precalculus Unit 07, students extended the use of the trigonometric ratios to solve real-world problems, including those involving navigational bearings. In Precalculus Unit 10, students represented, used, and applied vectors and vector operations to model and solve problem situations.

During this Unit
Students graph parametric equations by hand using tables and explore the characteristics of these equations, including the effect of the parameter t and the direction of the graph over time. Students graph two sets of parametric equations on the same graph (in mathematical and real-world problem situations) and explore whether the two paths meet at the same time. Students compare parametric equations and their corresponding rectangular relations to determine what additional information is provided by parametric equations. Student graph parametric equations using graphing technology and analyze these graphs to model and solve problem situations. Students explore the effect of the t-step value on the graph of a parametric equation created using graphing technology. Students convert parametric equations into rectangular relations, and convert rectangular relations into parametric equations. Students model linear situations with parametric equations, including modeling linear motion and vector situations. Students use these parametric models to solve real-world problems. Students model projectile motion with parametric equations and then use these models to solve real-world problems.

After this Unit
In Unit 12, students will continue to study complex curves and relations that are not defined in terms of f(x). Specifically, students will use trigonometry to graph polar equations and convert between polar and rectangular coordinates. In Unit 13, students will convert between parametric equations and rectangular relations for conic sections and use these representations to model and solve problems. In subsequent mathematics courses, students will continue to apply parametric equations as they arise in problem situations.

Algebraic reasoning serves an integral role in college readiness. Translating among multiple representations of equations and relationships and analyzing the features of functions are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning C1, D2; VII. Functions B1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (2000), students in grades 9-12 should use algebraic symbols to represent and analyze mathematical situations. Specifically, students should represent functions and relations using a variety of symbolic representations, including parametric equations (NCTM, 2000). Parametric equations are useful for representing graphs of curves that cannot be represented as functions where y is defined in terms of x and for modeling situations involving motion along a path where position coordinates (x(t), y(t)) can be determined over time (Herman, 2006). Cooney, Beckmann, & Lloyd (2010) contend that parametric equations provide a way to create functions that map real numbers (for the parameter t) to specific points (x(t), y(t)) for rectangular relations where y is not a function of x. In this way, parametric equations provide an example of functions where the domain and range of the function do not have to be numbers (Cooney, Beckmann, & Lloyd, 2010). The study of parametric equations 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). These planar curves can represent a number of real-world applications, including problems incorporating velocity and acceleration. Additionally, students in calculus will determine the length of a curve, including a curve represented with parametric equations (College Board, 2012).

College Board.(2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf.
Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
Herman, M. (2006). Introducing parametric equations through graphing calculator explorations. Mathematics Teacher, 99(9), 637-640.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Parametric equations express a set of quantities as explicit functions of an independent variable (a parameter), connect algebraic and geometric relations, and can be represented, transformed, and analyzed to model problem situations to make predictions and critical judgments.
• What kinds of mathematical and real-world situations can be modeled by parametric equations?
• How can parametric equations be used to represent problem situations?
• How can parametric equations be represented?
• What connections can be made between parametric equations and their corresponding rectangular relations?
• How are properties and operational understandings used to transform …
• parametric equations into rectangular relations?
• rectangular relations into parametric equations?
• Relations and Geometric Reasoning
• Algebraic and Geometric Relations
• Parametric equations
• Rectangular relations
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 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.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may incorrectly evaluate trigonometric functions if they use a calculator in the wrong angle mode.
• Some students may incorrectly graph parametric equations if they use a calculator in the wrong angle mode.
• When graphing parametric equations, some students may only graph the path of the parametric equations without noting specific t values along this path.
• Some students may believe that two sets of parametric equations that trace out the same path are equivalent. However, students most also consider the role of the parameter t in determining the “speed” with which each set of parametric equations traces out the path.
• Some students may believe that each rectangular relation has a unique set of corresponding parametric equations. Instead, each rectangular relation can be represented by an infinite set of parametric equations.

Underdeveloped Concepts:

• Some students may struggle with writing multiple sets of parametric equations to represent a rectangular relation.
• Some students may use the incorrect value for the acceleration due to gravity (either 32 ft/s2 or 9.8 m/s2) when formulating parametric equations to model projectile motion.
• Some students may not understand how the t-step value affects the graph of parametric equations created using graphing technology.

Unit Vocabulary

Related Vocabulary:

 Acceleration due to gravity Convert Direction Function Graph Initial height Initial velocity Motion Parameter Parametric curve Parametric equations Path Projectile motion Rate of change Rectangular relation Relation Starting point Vectors
Unit Assessment Items System Resources Other Resources

Show this message:

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

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Precalculus Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

The phase 2 College Readiness English Language Arts and Reading vertical alignment team found that the College Readiness Standards in English Language Arts and Reading are well aligned with the Texas Essential Knowledge and Skills.
TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
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.3 Relations and geometric reasoning. The student uses the process standards in mathematics to model and make connections between algebraic and geometric relations. The student is expected to:
P.3A Graph a set of parametric equations.

Graph

A SET OF PARAMETRIC EQUATIONS

Including, but not limited to:

• Characteristics of parametric graphs
• Parameter (t)
• Direction
• Methods for graphing parametric equations
• Constructing tables and plotting points
• Using graphing calculator technology
• Mode settings
• Changing from “function” mode to “parametric” mode
• Entering two equations for one graph (x(t) and y(t))
• Window setting
• Minimum and maximum values of t
• Interval between t-values (“t-Step”)
• Convert parametric equations to rectangular relations to graph
• Algebraic methods
• Solving one equation in a set of parametric equations (either x(t) or y(t)) for the parameter t
• Substituting the expression for t into the other equation (either y(t) or x(t))
• Graphing locus of points for the rectangular relation

Note(s):

• Algebra II graphed various types of functions, including their transformations and compositions.
• Algebra II also analyzed the domains and ranges of functions, written in inequality and set notation.
• Precalculus extends these ideas through the graphing of sets of parametric equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.3B Convert parametric equations into rectangular relations and convert rectangular relations into parametric equations.

Convert

PARAMETRIC EQUATIONS INTO RECTANGULAR RELATIONS

Including, but not limited to:

• Algebraic methods
• Solving one equation in a set of parametric equations (either x(t) or y(t)) for the parameter t
• Substituting the expression for t into the other equation (either y(t) or x(t))

Convert

RECTANGULAR RELATIONS INTO PARAMETRIC EQUATIONS

Including, but not limited to:

• Rectangular relations into general parametric equations
• Rectangular functions of the form y = f(x)
• Letting x = t
• Writing y as y = f(t)
• Rectangular relations of the form x = f(y)
• Letting y = t
• Writing x as x = f(t)
• Rectangular relations, given with specific information about x(t) or y(t)
• Write an expression to describe x(t) or y(t)
• Substitute the expression for x or y into the given rectangular relation

Note(s):

• Algebra I solved literal equations for a specified variable.
• Algebra II used the composition of two functions.
• Precalculus extends these skills to convert between parametric and rectangular equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.3C Use parametric equations to model and solve mathematical and real-world problems.

Use

PARAMETRIC EQUATIONS

Model, Solve

MATHEMATICAL AND REAL-WORLD PROBLEMS USING PARAMETRIC EQUATIONS

Including, but not limited to:

• Linear motion
• General form: • Variables
• t (time)
• (xy) (location at time t)
• (x0y0) (starting point)
• a (rate of change in the horizontal direction)
• b (rate of change in the vertical direction)
• (slope)
• Vector equations
• General form: • Variables and constants
• t (time in seconds)
• (xy) (location at time t)
• (x0y0) (starting point)
• v (constant velocity)
• θ (direction, measured as a rotation angle in standard position)
• Applications
• Projectile motion
• General form: • Variables and constants
• t (time in seconds)
• (xy) (location at time t)
• v (initial velocity)
• θ (angle at which the projectile is launched or released, measured from the horizontal)
• h0 (initial height)
• g (acceleration due to gravity, 32 ft/s2 or 9.8 m/s2)
• Applications
• Curve sketching
• General form: • Parametric curve is determined by the points (x, y) = (f(t), g(t)) as t varies
• Applications

Note(s):

• Geometry applied the trigonometric ratios (sine, cosine) to determine side lengths and angle measures in triangles.
• Algebra I analyzed the features of linear and quadratic functions.
• Precalculus combines these skills to develop parametric equations that model real-world situations such as linear and projectile motion.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• II. Algebraic Reasoning
• D2 – Translate among multiple representations of equations and relationships.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 