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 x^{2} + y^{2} = 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).
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. 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. 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 |