Introduction This unit bundles student expectations that address determining, analyzing, and writing equations for the conic sections formed when a plane intersects a double-napped cone. Connections are made between the locus definitions of the conic sections and their equations in rectangular coordinates. Additionally, converting between parametric equations and rectangular relations for conic sections and using these representations to model and solve problems are also included. 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 07, students identified the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres. In Geometry Units 02 and 07, students derived and used the distance formula to verify geometric relationships. In Geometry Unit 06, students showed that the equation of a circle with center at the origin and radius r is x^{2} + y^{2 }= r^{2} and wrote the equation for a circle centered at (h, k) with radius r using (x – h)^{2} + (y – k) = r^{2}. In Algebra II Unit 05, students wrote the equation of a parabola using given attributes, including vertex, focus, directrix, axis of symmetry, and direction of opening. In Precalculus Unit 11, students graphed parametric equations, used parametric equations to model and solve problem situations, and converted between parametric equations and rectangular relations for various linear and parabolic situations.
During this Unit Students explore the cross-sections of a double-napped cone in order to determine the four conic sections formed when a plane intersects a double-napped cone. Additionally, students explore the special cases of the intersection of a plane and a double-napped cone yielding the degenerate conic sections. Students determine the general characteristics of the four types of conic sections (circles, parabolas, ellipses, and hyperbolas), including shape, cross-section conditions, key attributes (vertex/vertices, asymptotes, etc.), equations (for circles and parabolas), and real-world applications. Students explore the locus definition of a circle and make connections to its equation in rectangular coordinates. Specifically, students connect the distance formula to the equation of a circle. Students write equations of circles in both mathematical and real-world problem situations. Students explore the locus definition of an ellipse and make connections to its equation in rectangular coordinates. Specifically, students use the distance formula to determine the general equation of an ellipse. Students write rectangular equations of ellipses using their various characteristics, including major axis, semi-major axis, minor axis, semi-minor axis, focal length, coordinates for the center, coordinates for the foci, and coordinates for the vertices. Students write equations of ellipses in both mathematical and real-world problem situations. Students explore the locus definition of a hyperbola and make connections to its equation in rectangular coordinates. Specifically, students explore the conditions that give rise to a hyperbola and then use the distance formula to determine the general equation of a hyperbola. Students write rectangular equations of hyperbolas using their various characteristics, including focal length, coordinates for the center, coordinates for the foci, coordinates for the vertices, distance from the center to the vertices, slopes of the linear asymptotes, and equations for the linear asymptotes. Students write equations of hyperbolas in both mathematical and real-world problem situations. Students explore the locus definition of a parabola and make connections to its equation in rectangular coordinates. Specifically, students explore how the distance formula yields the equation of a parabola. Students write equations of parabolas in both mathematical and real-world problem situations. Students convert rectangular relations for circles, parabolas, ellipses, and hyperbolas into their parametric equations, then convert parametric equations for these conics into their rectangular relations. Students model real-world situations using rectangular and parametric equations of ellipses and hyperbolas and use these equations to solve problems.
After this Unit In subsequent mathematics courses, students will continue to apply conic sections as they arise in problem situations.
Additional Notes Algebraic and geometric reasoning serve an integral role in college readiness. Translating among multiple representations of equations and making connections between geometry and algebra are emphasized in the Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning B1, C1, D2; III. Geometric Reasoning C1; 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 “understand relations and functions and select, convert flexibly among, and use various representations for them” (p. 296). Additionally students should use visualization, spatial reasoning, and geometric modeling to solve problems including analyzing the cross-sections of three-dimensional objects (NCTM, 2000). Students should be provided with rich mathematical settings in which they can hone their visualization skills in order to use visualization as a problem-solving tool. Visualizing the shape of a conic section formed by the intersection of a plane and a cone can be facilitated by the use of physical models, drawings, and software capable of manipulating three-dimensional objects (NCTM, 2000). Herman (2012) describes how the modern study of conics is considered analytic geometry where the conic sections represent graphs of certain types of equations that are defined in terms of distance relationships between specific points and/or lines. Brown and Jones (2005) advocate for the use of two different coordinate systems which are well suited for graphing conic sections and exploring these distance relationships: a focus-directrix coordinate system and a focus-focus coordinate system. Using these alternative coordinate systems allows students to explore the definitions of the conic sections in a way that may be masked by using equations or technology to plot points alone (Brown & Jones, 2005). The study of conic sections continues into calculus, where students analyze conics and other implicitly defined curves using implicit differentiation (College Board, 2012).
Brown, E. M., & Jones, E. (2005). Understanding conic sections using alternate graph paper. Mathematics Teacher, 99(5), 322-327. College Board. (2012). AP calculus course description. Retrieved from http://media.collegeboard.com/digitalServices/pdf/ap/ap-calculus-course-description.pdf. Herman, M. (2012). Exploring conics: Why does b^{2} – 4ac matter? Mathematics Teacher, 105(7), 526-532. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc. |