Danyel Graves C&I 303, B. Arvold October 7, 2003 Conic Sections – Parabolas Goals Students need an introduction to conic sections. Ideally, with parabolas, they should already have a basic knowledge through graphing quadratic functions. It is important in this section that students learn the different parts of the parabola and how they are found from the formula. Prerequisite Knowledge Students should know how to graph quadratic functions. They should also have a knowledge of how to use a dynamic geometry program. Actions for Special Needs Students Groups should be specially chosen so that students can participate to their fullest and not hindered by other members. Computer projects can be done in pairs if needed (i.e. if computer space is limited or if some students need extra help). Materials Computers that have dynamic geometry software loaded on them are needed for Day 2’s activities. Day 1 To introduce the idea of conic sections to the students, it would be beneficial to ask the students what they know (if anything) about conic sections. Ideally, students should have at least seen a parabola, ellipse, hyperbola, or a circle. An introduction (or demonstration) of how to construct conic sections with a piece of paper rolled into a cone should provide adequate understanding for students to know how are formed. Sample questions: Tell me what you know about cones. What kinds of shapes can you make when you intersect a cone with a plane? Misconceptions: Students may not be aware the cones are infinite. Beginning with the parabola, students should make observations about it (within small groups of about 3 students) and by the end of the period, the class as a whole should begin to develop pieces of the definition of a parabola. They should observe various aspects and characteristics of the parabola and begin to form ideas about how to graph them. Day 2 A great way to get a handle on parabolas would be to utilize computers. Using a program (such as Mathmatica, Geometer’s Sketchpad, or other dynamic geometry programs) students can get a feel of how parabolas are formed from the three-dimensional cone. Ideally, they should be able to construct their own and begin to see patterns and similarities between the parabolas they have created. This day may need more direction for the students so that they are able to construct the parabolas. Certain codes may need to be given so that students do not end up frustrated and overwhelmed with this exploration of parabolas. HW: The students should construct a few parabolas on a sheet of paper and do their best to find the equations for them so that they can be used for the next lecture. Day 3 This day should begin with an informal assessment of what the students learned the day before. The teacher should ask questions concerning what they noticed about the parabolas they created. Next, specific parts of the parabola should be explored in depth (the focus, axis of symmetry, vertex, directrix, etc.) and explained. The students should then work in groups to draw more parabolas and find the equations to these parabolas. After four or five are drawn, they should be able to generalize the formula and come up with a formula that works with all parabolas. Sample questions: What happens when you put an integer coefficient on the front of the equation? What about a fraction? A negative sign? What if you add or subtract a number to the end of the equation? Can you make the parabola slide left or right on the axis? Misconceptions: Students might think that the focus is the same as the vertex. They may also be unaware of the reason they need to know the directrix. HW: There will be a worksheet with exercises and essay questions concerning the day’s activities (explanation of what certain parts of the general formula stands for). Day 4 This day will be a quiz so that students can test their knowledge and apply what they learned in the previous days. This quiz should cover the general form of the parabola and a basic sketch of a parabola should be drawn from a given equation. It should also have a graph of a parabola where they should be able to give the equation. Day 5 First off, the quizzes should be handed back and any questions the students have on specific problems should be addressed. Next, it is time to move on to the next conic section, ellipses. In the same fashion as Day 1, ellipses should be explored.
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