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 Instructional Focus DocumentPrecalculus
 TITLE : Unit 12: Polar Equations SUGGESTED DURATION : 6 days

#### Unit Overview

This unit bundles student expectations that address the polar coordinate system, including graphing points in polar coordinates, converting between polar and rectangular coordinates, and graphing polar equations by plotting points and using technology. 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 Grade 6 Unit 08, students plotted ordered pairs of rational numbers on a coordinate grid.  In Geometry Unit 05, students used trigonometric ratios (including sine, cosine, and tangent) 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 08, students represented rotation and reference angles on a coordinate grid. In Precalculus Unit 11, students graphed parametric equations and converted between parametric equations and rectangular relations. In Algebra I, Units 02 – 04, 08, and 09, students analyzed the graphs of linear, quadratic, and exponential functions. In Algebra II, Units 01, 02, and 05 – 11, students analyzed linear, quadratic, and exponential functions as well as absolute value, square root, cubic, cube root, rational, logarithmic functions. In Precalculus Units 01, 03 – 05, and 08, students analyzed the graphs of exponential, logarithmic, rational, polynomial, power, trigonometric, inverse trigonometric, and piecewise defined functions, including step functions.

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 analyzed the graphs of linear, quadratic, exponential, absolute value, square root, cubic, cube root, rational, logarithmic, polynomial, power, trigonometric, and inverse trigonometric functions.

During this unit, students graph points in the polar coordinate system using both radians and degrees. Students determine the polar coordinates for points and discover multiple ways to identify the same point in polar coordinates. Students convert polar coordinates into their rectangular coordinates and convert rectangular coordinates into their polar coordinates, while continuing to graph points in the polar coordinate system. Students graph polar equations by creating a table of values and plotting the polar points. Students graph polar equations with technology and analyze these graphs to explore various types of polar equations, including circles, cardioids, limacons, roses, and lemniscates.

After this unit, in Precalculus Unit 13, students will continue to study complex curves and relations that are not defined in terms of f(x). Specifically, students will explore the conic sections and their equations in rectangular coordinates. In subsequent mathematics courses, students will continue to apply polar coordinates and equations as they arise in problem situations.

Algebraic and geometric reasoning serve an integral role in college readiness. Translating among multiple representations of equations and relationships, analyzing the features of functions, and making connections between geometry and algebra are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning D2; III. Geometric Reasoning C1; IV. Measurement Reasoning B1; VII. Functions A2, B1; 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 “specify locations and describe spatial relationships using coordinate geometry and other representational systems” (p. 308). Specifically, students should use Cartesian coordinates and other coordinate systems, including polar coordinates, to analyze geometric situations (NCTM, 2000). To this end, students should explore problem situations where using a different coordinate system (such as the polar coordinate system) provides a simpler representation of the geometric relation, while also being able to explain how both a polar representation and a rectangular representation describe the same geometric relation (NCTM, 2000). Cooney, Beckmann, & Lloyd (2010) contend that polar coordinates provide a criticism of the vertical line test, which is often used to determine the functionality of a relation graphed using Cartesian coordinates, but may deter students from exploring important aspects of graphical representations of other functional relationships. Specifically, the circle represented by the rectangular relation x2 + y2 = 16 does not represent a function for y in terms of x when using Cartesian coordinates; however, when this relation is graphed using the polar equation r(θ) = 4, then the set of ordered pairs of polar coordinates for this equation does represent a function of r in terms of θ (Cooney, Beckmann, & Lloyd, 2010). Lawes (2013) offers an alternative method for students to graph polar equations in addition to the traditional methods of plotting polar coordinates and using graphing technology. Specifically, Lawes recommends having students graph the polar function on the rectangular plane (where the horizontal axis represents θ and the vertical axis represents r) before graphing it on the polar plane. This method reinforces previous concepts related to graphing transformations of trigonometric functions, while also allowing students to visually determine the critical characteristics of polar curves and quickly sketch the graph of nearly any polar equation (Lawes, 2013). The study of polar 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 area of a region, including a region bounded by polar curves (College Board, 2012).

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.
Lawes, J. F. (2013). Graphing polar curves. Mathematics Teacher, 106(9), 660-667.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

#### OVERARCHING UNDERSTANDINGS and QUESTIONS

Relations are algebraic models that describe how two quantities relate to one another. Functions are a subset of relations.

• What are types of relations?
• How can relations be represented?
• Why do some relations not define a function?
• Why do some relations define a function?
• Why can function models describe how two variable quantities change in relation 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

• Equations
• Multiple Representations
• Relations

Associated Mathematical Processes

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

Polar equations have unique characteristics and can be represented in multiple ways, including as graphs in the polar coordinate system.

• What is the polar coordinate system and why is it used?
• How are points graphed in the polar coordinate system?
• How can rectangular coordinates be converted into polar coordinates?
• How can polar coordinates be converted into rectangular coordinates?
• What relationships exist between the rectangular coordinates and polar coordinates for a point?
• Does each point in the xy-plane have a unique set of rectangular coordinates? A unique set of polar coordinates? Explain.
• What are the characteristics of polar equations?
• How do the characteristics of polar equations and their corresponding rectangular relations compare?
• How can polar equations be used to represent functions and relations?
• How can polar equations be represented using tables?
• How can polar equations be represented using graphs?
• How can polar equations be represented algebraically?

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that inverse trigonometric functions are the same as the reciprocals of the trigonometric functions. For example, tan–1 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 incorrectly graph polar equations if they use a calculator in the wrong angle mode.
• Some students may believe that a point expressed in rectangular coordinates has a unique set of corresponding polar coordinates. Instead, each set of rectangular coordinates can be represented by an infinite set of polar coordinates.
• Some students may believe that polar equations are not functions since they do not pass the “vertical line test.” However, polar equations are functions where r is a function of θ.
• Some students may struggle to graph points in the polar coordinate system with negative r-values.

Underdeveloped Concepts:

• Some students may not understand how the θ-Step value affects the graph of parametric equations created using graphing technology.
• Some students may only graph polar equations for 0 ≤ θ < 2. However, it may be necessary to extend these values in order to obtain a complete graph of a polar equation.

#### Unit Vocabulary

Related Vocabulary:

 Cardioid   Circle   Clockwise   Convert   Convex   Cosine   Counterclockwise   Degrees   Function Inverse functions   Lemniscate   Limaçon   Polar coordinate system   Polar coordinates   Polar equation   Pole   Quadrant   Radians Rectangular coordinates   Relation   Rose   Rotation angle   Sine   Standard position   Tangent   xy-plane
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 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

TEKS# SE# TEKS Unit Level Specificity

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.3D Graph points in the polar coordinate system and convert between rectangular coordinates and polar coordinates.

Graph

POINTS IN THE POLAR COORDINATE SYSTEM

Including, but not limited to:

• Recognizing the meaning of the polar coordinates (r)
• The “r” coordinate is the distance from the origin (or the “pole”).
• When r > 0, start at the origin (or pole) and move to the right a distance of r units.
• When r < 0, start at the origin (or pole) and move to the left a distance of r units.
• The “” coordinate is a rotation angle in standard position (in degrees or in radians).
• When  > 0, rotate counter-clockwise.
• When  < 0, rotate clockwise.
• Plotting points on a set of polar axes
• Naming coordinates for points in the polar coordinate system
• Multiple ways to name the same point
• If a point has coordinates (r), then it can also be named in the following ways:
• In degrees
• (r ± 360°)
• (–r ± 180°)
• (r ± 2π)
• (–r ± π)

Convert

BETWEEN RECTANGULAR COORDINATES AND POLAR COORDINATES

Including, but not limited to:

• Using formulas to change polar coordinates (r) into rectangular coordinates (xy)
• x = rcos
• y = rsin
• Using formulas to change rectangular coordinates (xy) into polar coordinates (r)
• r =
•  = tan–1 , Quadrant I
• = tan–1  + π, Quadrant II
•  = tan–1  + π, Quadrant III
•  = tan–1  + 2π, Quadrant IV

Note(s):

• 6th Grade Mathematics plotted ordered pairs of rational numbers on a coordinate grid.
• Geometry applied the trigonometric ratios (sine, cosine, tangent) to determine side lengths and angle measures in triangles.
• Precalculus combines these skills to plot points in the polar coordinate system and to convert between polar and rectangular coordinates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• IV. Measurement Reasoning
• B1 – Convert from one measurement system to another.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
P.3E Graph polar equations by plotting points and using technology.

Graph

POLAR EQUATIONS

Including, but not limited to:

• Determine which graphing technique is most appropriate for a given situation
• Graphing calculator MODE to POLAR when applicable
• Plotting points
• Making a table
• Evaluating the polar equation
• Recognizing that the input (independent) variable is , on the right side of the table
• Plotting points on a polar graph
• Using both angle measurements
• Degrees
• Recognizing that different sets of coordinates may graph to the same location
• Using technology
• Mode settings
• Changing from “function” mode to “polar” mode
• Entering equations where r is a function of  (or in the form r())
• Window settings
• Minimum and maximum values of
• Interval between -values (“-Step”)
• Types of polar equations
• Circles
• r = a
• Center at the pole (or origin)
• r = a cos , or r = a sin
• Center on one of the coordinate axes
• Diameter = a
• Cardioids
• r = a + b cos , or r = a + b sin , where |a| = |b|
• Limacons
• r = a + b cos , or r = a + b sin , where |a|  |b|
• If |a| < |b|, the limacon has an inner loop.
• If |b| < |a| < 2|b|, the limacon is called “dimpled.”
• If |a| ≥ 2|b|, the limacon is considered to be convex.
• Roses
• r = a cos (n)
• r = a sin (n)
• If n is odd, the rose has n petals.
• If n is even, the rose has 2n petals.
• The length of each petal is a (or a is the “radius” of the rose).
• Lemniscates
• r2 = a2 sin (2)
• r2 = a2 cos (2)
• The length of each loop is a (or a is the "radius" of the lemniscate).

Note(s):

• Algebra II analyzed the graphs of quadratic, square root, exponential, logarithmic, cubic, cube root, absolute value, and rational functions.
• Precalculus extends these skills to include the graphing of polar 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.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections