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 Precalculus 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 Precalculus 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.
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.