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1 Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes STRAND 7: CONIC SECTIONS TOPIC 7.4: APPLICATIONS OF CONIC SECTIONS Topic Notes Mathematical focus The mathematical focus of this topic is to use conic sections, their geometric properties, and algebraic representations to model physical situations and solve application problems. Topic focus This topic contains six tasks: Task 7.4.1: Radio Astronomy Task 7.4.2: Earthquake Energy Task 7.4.3: An Arching Dilemma Task 7.4.4: LORAN Task 7.4.5: Rocket Engine Design Task 7.4.6:Communication Satellite Mission Design The goal of this topic is for participants and students to solve application problems involving physical representations of conics sections. They must be able to recognize the conic that will model a situation, write its equation, and solve it either algebraically or graphically for needed information. Finding and using the appropriate conic model is often a matter of recognizing the locus definition in an applied context. In some situations, the solution to the equation must them be interpreted in the context of the problem situation. TExES standards focus TExES Standard III.014 Geometry and Measurement. The teacher understands coordinate, transformational, and vector geometry and their connections. The beginning teacher: (F) Uses coordinate geometry to derive and explore the equations, properties, and applications of conic sections (i.e., lines, circles, hyperbolas, ellipses, parabolas). TExES Standard IV.019 Mathematical Processes and Perspectives. The teacher understands mathematical connections both within and outside of mathematics and how to communicate mathematical ideas and concepts. The beginning teacher: (B) Applies correct mathematical reasoning to derive valid conclusions from a set of premises. (C) Uses inductive reasoning to make conjectures and uses deductive methods to evaluate the validity of conjectures. TEKS/TAKS focus TEKS 2A.3 Foundations for functions. The student formulates systems of equations December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 2 Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to: (A) analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems. TEKS 2A.5 Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to: (B) sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph. Materials Materials needed Graphing Calculator Conics APP for TI-83 Plus Chart Paper Poster Markers Blank Transparencies Transparency Markers Task 7.4.1 * Task 7.4.2 * Task 7.4.3 * * * * * Task 7.4.4 * * * Task 7.4.5 * * * Task 7.4.6 * * * * * * Procedure Facilitators may choose to do any of the Tasks. Task 7.4.1 – Parabola Task 7.4.2 – Circles Task 7.4.3 – Ellipse Task 7.4.4 – System of Hyperbolas Task 7.4.5 – Parabola Task 7.4.6 – System of 2 Circles and an Ellipse Task 7.4.1 should probably be done together in class to illustrate how the locus definition of a conic will be useful in determining the type of conic that can be used to model a physical situation. Participants may need help to understand that some decisions in the solution of the problem such as placement of the axes are left to their discretion and that the answer to the equation or system of equations used must be interpreted in the context of the problem situation. Divide the participants into groups and assign one or two groups to do the remaining tasks. They should record their work on a poster or transparency and be prepared to explain to the group the decisions they made in the solution of their problem. Summary The ability to do these problems easily will be ample evidence that the participants or students understand both the geometric and algebraic representations of the conic sections. The connection between the locus definition of a conic, its equation, and an application problem is very important if conic sections are going to be useful mathematical tools. December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board. 3 Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Topic Notes Assessments These tasks are assessments for the entire strand. Participants should have reviewed the skills necessary to solve the problems by completing the tasks in Topics 7.2 and 7.3. Teacher use only Task 7.4.1 Task 7.4.2 Task 7.4.3 Task 7.4.4 Task 7.4.5 Task 7.4.6 Modify for students Ready for students * * * * * * December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.