Karen Kinel PT3 Project Proposal Introduction Every year the students in my sophomore, curriculum 1 math class struggle with conic sections. They not only have trouble differentiating between them, but they also struggle with the skills needed to find the important characteristics. With parabolas, circles, ellipses, and hyperbolas they have difficulty putting the equations into the modified forms for graphing. They also have trouble making the connections with the foci points and how that affects the shape of the graph; the fact that a parabola has a directrix only further confuses matters. I would eventually like to create a power point lesson that helps to explain this information. I would like to incorporate dynamic models and Quick Time movies into the lessons for circles, parabolas, ellipses and hyperbolas. I would eventually like to design a companion web site that supports and parallels the lessons through the utilization of teacher created material and links to related websites. Graphing calculators and other mathematics programs such as Geometer’s Sketchpad will also be utilized. This lesson is to be incorporated into an integrated math class that covers topics in Geometry and Algebra II. The students have been only studying functions in algebra up to this section. This will be their first time working algebraically with graphs that are not functions. This material is an entire chapter, so for the purpose of this class I would like to design the introduction and some basic work with circles. This could then be expanded upon for the rest of the unit. Newton South High School is undergoing a major construction project. As part of the construction, new technology is being integrated into the school. Every teacher has received a laptop computer with PowerPoint. Every classroom has been wired with access to the Intranet and has the ability to project from the computers onto a screen at the front of the room. It is a goal of the school to integrate computers, and technology in general, into the classroom. Additionally, the mathematics department is undertaking an examination of how computers and information technology will impact teaching and benefit the students’ comprehension of the material. Problem Statement The students started the year with a section on Quadratic Functions. They are familiar with circles from work in Geometry (they know the geometric definition), and from “real world” examples. The students are familiar with the terminology of circle (i.e. center and radius). They are familiar with calculating both circumference and area. They are not familiar with the “unit circle”. They do not know how the distance formula relates to both the radius and the algebraic formula for a circle. As a result of the lesson, students should become familiar with the basic formula for a quadratic relation. They should understand how conic sections were discovered. They should know how to find the center and radius of any circle. This lesson should address several learning styles. Some of the students have IEPs that require explicit procedures for all complex mathematics along with modified instructions. Students will investigate some properties in groups, and some information will be shown in a presentation format. Instructional Goals Students will understand that a quadratic relation (conic section) comes from slicing one or two cones. Students will be able to describe where a quadratic relation comes from. Students will be able to identify a circle given an equation in standard form Students will discover that the center of a circle is moved when you add, subtract and/or modify the x and y terms. Students will be able to describe how the center of a circle moves when you add, subtract and/or modify the x and y terms. Students will understand the Geometrical definition of a circle. Students will understand the distance formula and its relation to a center of a circle. Students will be able to quickly sketch the graph of a circle given its equation or inequality. Students will be able to write the equation of a circle. Instructional activities: The students will use real models and see computer-generated models to look at how the conic shapes are formed. Have students work in groups to plot one of four relations. They should select values of x and calculate the corresponding values of y. They should continue until they have enough points for a smooth curve. They will be using graphing calculators and approximating any radicals. We will then look at their graphs and at computer generated graphs of the same equations in order to make conclusions about the shape, size, and location of each graph. The four equations are: x2 + y2 = 25 x2 + y2 + 6x = 16 x2 + y2 – 4y = 21 x2 + y2 + 6x – 4y = 12 A circle is the set of all coplanar points that are equidistant (the length of the radius) from a given fixed point called the center. See a power point presentation on how this works. Review of how the distance formula relates to the Pythagorean theorem. The equation of a circle is derived from the distance formula. The presentation should also include what the graph of an inequality for a circle will look like. The students will have to determine when to shade the interior and when to shade the exterior of the circle. Go through power point presentations explaining how to graph a circle of the form x2 + y2 = r2 and (x – h)2 + (y – k)2 = r2. Assessment The assessment of the group work will be done through observation. I will also have a discussion with the class about circles before starting the lesson; here they can explain what they already know. At the end of the lesson, there will be a quiz to evaluate their knowledge. The students will also have to do a project. They will also have to videotape themselves explaining how to graph a circle to a student that has not yet studied this topic. This will allow them to demonstrate their understanding. Resources I will use computers, graphing calculators and the students’ textbooks. I will use power point and computer programs such as Power Point, Geometer’s Sketchpad, Mathepedia and other programs in order to achieve my objectives. TI – 83 graphing calculators will be utilized as well. Duration This lesson will last one or two classes. Eventually it will be expanded to a month long unit, with every conic section looked at dynamically.
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