Contents. How You May Use This Resource Guide
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1 Contents How You May Use This Resource Guide ii 15 An Introduction to Plane Analytic Geometry 1 Worksheet 15.1: Modeling Conics Worksheet 15.2: Program to Graph the Conics Worksheet 15.3: Exploring with Real-World Conics Worksheet 15.4: Write About It Worksheet 15.5: Drawing by Definition Worksheet 15.6: Conic Corporate Logos Answers 10 i
2 How You May Use This Resource Guide This guide is divided into chapters that match the chapters in the third editions of Technical Mathematics and Technical Mathematics with Calculus by John C. Peterson. The guide was originally developed for the second editions of these books by Robert Kimball, Lisa Morgan Hodge, and James A. Martin all of Wake Technical Community College, Raleigh, North Carolina. It has been modified for the third editions by the author. Each chapter in this Resource Guide contains the objectives for that chapter, some teaching hints, guidelines based on NCTM and AMATYC standards, and activities. The teaching hints are often linked to activities in the Resource Guide, but also include comments concerning the appropriate use of technology and options regarding pedagogical strategies that may be implemented. The guidelines provide comments from the Crossroads of the American Mathematical Association of Two-Year Colleges (AMATYC), and the Standards of the National Council of Teachers of Mathematics, as well as other important sources. These guidelines concern both content and pedagogy and are meant to help you consider how you will present the material to your students. The instructor must consider a multitude of factors in devising classroom strategies for a particular group of students. We all know that students learn better when they are actively involved in the learning process and know where what they are learning is used. We all say that less lecture is better than more lecture, but each one of us must decide on what works best for us as well as our students. The activities provided in the resource guide are intended to supplement the excellent problems found in the text. Some activities can be quickly used in class and some may be assigned over an extended period to groups of students. Many of the activities built around spreadsheets can be done just as well with programmable graphing calculators; but we think that students should learn to use the spreadsheet as a mathematical tool. There are obstacles to be overcome if we are to embrace this useful technology for use in our courses, but it is worth the effort to provide meaningful experiences with spreadsheets to people who probably will have to use them on the job. Whether or not you use any of the activities, we hope that this guide provides you with some thought-provoking discussion that will lead to better teaching and quality learning. ii
3 Chapter 15 An Introduction to Plane Analytic Geometry Objectives After completing this chapter, the student will be able to: Calculate the distance between two points; Find the midpoint between two points; Determine the slope of a line given two points on the line; Determine the slope of a line given its angle of inclination; Find the angle of inclination of a line given its slope; Determine the slope of a line perpendicular to a given line; Write the equation of a line using the slope intercept form or the point slope form; Solve applications involving linear relationships; Write the equation of a circle, ellipse, parabola, or hyperbola from given information; Write an equation in standard form given the equation of any of the conic sections; Graph any conic section and determine all features of interest; Identify by inspection whether a given second-degree equation represents a circle, ellipse, parabola, or hyperbola; Write polar equations of conic sections from given information; Solve applied problems involving any of the conic sections. 1
4 Peterson, Technical Mathematics, 3rd edition 2 Teaching Hints 1. Introduce the distance formula through the use of an application. Example 15.1 A straight roadway banked with a slight incline is to be paved. When put on a coordinate grid the beginning of the roadway is placed at the origin and the top of the incline has the coordinates (23, 6) feet. Find the width of the roadway. This could also be used as the occasion to review volume and unit conversions by having the students find the cubic yards of pavement needed if the road is 1 mile long and the pavement needs to be 5 inches deep. 2. Example 2, Midpoint in a Number Line (Formula Development), page 72, in Crossroads, is a hands-on activity to help students develop the concept of midpoint. 3. Students need to understand that the slope of a line will give its orientation as well as its inclination. Have students look at the graphs of several lines and use the slopes to discover what causes lines to be parallel or perpendicular. 4. Show students how the conic sections can be obtained by slicing a cone. Or, better yet, have the students build models. (see Activity 15.1) 5. Introduce each conic section with an application to help students understand the relevance of studying the conic sections. Conic sections alone are a topic that should be given decreased attention, but the applications involving conics should be given increased attention. 6. See teaching hint 12 in Chapter 9 about completing the square in Chapter Demonstrate how the standard equation of a circle can be obtained by using either the distance formula or the Pythagorean Theorem. This will help students to understand the relevance of the center and the radius in the standard equation, and it will help them in understanding the standard equations of the other conic sections. 8. Students should be able to identify conics from the general equation of a conic section. 9. Work with students producing graphs of conics on a graphing utility. Since most conics are not functions they will need to graph the conics in parts to get the entire conic to graph. Students may need a review on the steps needed to solve the standard equations for y. Guidelines One of the pedagogy standards in Crossroads says, Mathematics faculty will actively involve students in meaningful mathematics problems that build upon their experiences, focus on broad mathematical themes, and build connections within branches of mathematics and between mathematics and other disciplines so that students will view mathematics as a connected whole relevant to their lives.
5 Peterson, Technical Mathematics, 3rd edition 3 With the topic of conic sections students will often ask the question, When will I ever use this? Therefore, they must be shown the importance of conic sections through applications. Students need to be aware of the many applications of conics in the real world. Once students are exposed to the wide use of conic sections, the question, When will I ever use this? will answer itself. Crossroads suggests a decrease in attention to the complex algebraic manipulations of conic sections in mathematics-intensive programs. This decreased attention does not imply that conic sections should not be taught, but that they should be approached in such a way as to avoid the complexity of them. Applications should be the underlying reason for studying conic sections. Graphing utilities can be used to eliminate much of the complex algebraic manipulation. Increased Attention Use of statistical software and graphing calculators Problems related to the ordinary lives of students Guidelines for Content Decreased Attention Paper-and-pencil calculations and fourfunction calculators Problems unrelated to the daily lives of most students Guidelines for Pedagogy Increased Attention Decreased Attention Problem solving and multistep problems One-step, single-answer problems Active involvement of students Passive learning Activities 1. Modeling Conics In groups, students will model and research the conic sections. 2. Program to Graph the Conics Students will write a program to graph conic sections on a graphing utility. 3. Exploring with Real-World Conics Involves applications that give students hands-on experience with conic sections. This activity can be done in groups or individually. 4. Write About It Students will write about the uses of conic sections in the real world. 5. Drawing by Definition Students will write a step-by-step plan that uses the definition of each conic to produce a drawing. 6. Conic Corporate Logos Students will use one or more conic sections as they design a corporate logo.
6 Peterson, Technical Mathematics, 3rd edition 4 Student Worksheet 15.1 Modeling Conics Your group is to prepare a presentation for the class on how the (your instructor will assign each group a conic section) can be created by intersecting a plane and a cone. Have illustrations and/or models for your classmates to examine. Also include in your presentation uses for the conic. You may want to bring in objects that rely on your conic. Your presentation should be creative as well as informative.
7 Peterson, Technical Mathematics, 3rd edition 5 Student Worksheet 15.2 Program to Graph the Conics The conic sections can be rather difficult to graph, even on a graphics calculator. They must first be solved for y and then graphed in two separate parts. Write a program for a graphing utility that will graph a conic given the type of conic you have and certain input values. For a circle have the input values be the center and the radius; for a parabola the vertex, whether it opens up or down, and a point the hyperbola will pass through; for the ellipse and hyperbola the center and the values for a and b. Test your program out for several conics to make sure it will graph each type. You will also need to include a feature that will set up the needed range for each graph.
8 Peterson, Technical Mathematics, 3rd edition 6 Student Worksheet 15.3 Exploring with Real-World Conics 1. Find a satellite dish at your school, home, or a place of business that sells them. Get permission to take measurements of the dish. Your measurements should include the location of the receiver from the dish, and enough measurements to calculate where the focus point of the dish should be. Use your measurements to calculate where the focus of the dish is located. How does this compare to the location of the receiver? 2. Do the same for the lamp inside a flashlight. 3. Find an elliptical-shaped roasting pan. Take measurements to find where the focus points should be located; put colored weights in the pan at these points. Now fill the pan with water and tap the top of the water with a pencil at one of the focus points. What do you observe about the water? Is this what you thought would happen? If not, explain why there might be a difference. 4. The light source in a dental lamp shines against a reflector shaped like a portion of an ellipse, in which the light source is one focus of the ellipse. Reflective light enters a patient s mouth at the other focus of the ellipse. Make a visit to a local dental office and take measurements of their light. Find out how far the light source needs to be from a patient s mouth. 5. Research your field of study and find out if conics are involved in any way. Write a paper on the uses of conics in your field of study. If you find that conic sections do not in any way apply to your field of study, pick one of the conics and write a paper on the real-world uses of that conic section.
9 Peterson, Technical Mathematics, 3rd edition 7 Student Worksheet 15.4 Write About It Find examples of conic sections in the real world. Write a report on Conics in our World. Do conic sections play a part in your field of study? If so, in your report explain which conics are involved and how they are used or relate to your field.
10 Peterson, Technical Mathematics, 3rd edition 8 Student Worksheet 15.5 Drawing by Definition Use the definition of each of the conic sections to describe a method to physically draw each conic. You may use string, nails, etc. in your construction. Write a step-by-step plan that someone not familiar with conics could follow to draw each of the conic sections. Include in your plan any needed materials, and the changes that must be made in your plan to create the same conic in a different size.
11 Peterson, Technical Mathematics, 3rd edition 9 Student Worksheet 15.6 Conic Corporate Logos Several companies, such as Toyota and DISH Network, have a logo based on the conic sections. To quote from the Frequently Asked Questions on the Toyota web page, 1 In 1990, Toyota debuted the three overlapping Ellipses logo on American vehicles. The Toyota Ellipses symbolize the unification of the hearts of our customers and the heart of Toyota products. The background space represents Toyota s technological advancement and the boundless opportunities ahead. Exercises 1. Get a copy of the Toyota logo for a newspaper ad, a brochure from a Toyota dealer, or by making a pencil rubbing of the logo on a Toyota vehicle. 2. Each of the three overlapping ellipses in the Toyota logo is actually formed by two ellipses one ellipse forms the outside and the other forms the inside. Write the formulas for the six ellipses used to make the Toyota logo. You may assume that all six ellipses are centered on the y-axis. 3. Think of an existing company or make up one and design a logo for the company. Your logo should include at least one conic section. Write a paragraph explaining what the company does and why your logo is appropriate. Include the formulas for all conic sections that you used in your design. 1 alp.php
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