
Conic SectionsAngela Wood, NSF Scholar, 200203 Developed with funding from the National Science Foundation (DGE0086320) 
Major Understanding 
Conics are apparent all around us. For example, they can be found in structures, mirrors, and satellite dishes. Conics also represent orbital paths. It is very important to understand the four different conics: parabolas, circles, ellipses and hyperbolas. It is also necessary to understand the different components of the equations that form the four different conic sections.

Grade/Subject 
Algebra II

Objectives 
Recognize and describe the real world applications


Define and identify the four different conic sections.


Recognize the four different standard equations and compare and contrast them.


Relate the different components of the various equations for each conic to how each component affects the shape and placement of the graph. (i.e.: a, b, h, k, etc…) Be able to write equations for conics, given center, vertices, and other variables.

Time 
Anticipatory Set 
5 min  
Power Point Presentation: Real World Applications and Definitions (tie to the history of astronomy) 
20 min  
Folding Conic Sections worksheet 
60 min  
Students explore and graph four types of equations in order to determine each conics general equation and how each component of the equation affects the graphs (and complete worksheets). 
45 min  
Discuss student findings and conclusion 
10 min  
Wrap up with flashlight demonstration 
5 min 
Materials 
For each student:

State and National Correlations 
Virginia Standards of Learning: Algebra II (AII.18) National Science Education Standards: Historical perspectives including the history and nature of science; origin and evolution of the universe; Earth in the solar system; science and technology NCTM Standards: Interpret representations of functions of two variables. Understand relations and functions and select, convert flexibly among, and use various representations for them. Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model relationships. Interpret representations of functions of two variables.

Instructional Strategies 
1. Anticipatory Set: In groups, have students slice a cone into the four different conic sections. 2. Power Point presentation: Discuss include real life applications of conics, pictures, and definitions. 3. Folding Conic Sections activity: Have students read directions to create four conic sections by folding wax paper. 4. Casio Graphing Calculator Exploration: Give students the four worksheets (Parabolas, Circles, Ellipses, and Hyperbolas) and a graphing calculator to guide them through their exploration. First, give students a brief overview of the graphing calculator. Then have students complete the four worksheets. 5. Discussion: Ask students what they found about the different components of the equations and how each component affected the graphs. Show students one example of each from the Power Point and have students compare their answers. 6. Conclusions: Review the four conics, their graphs and equations. 7. Flashlight Demonstration: Use a flashlight to create the four different conics by changing the position and direction of the flashlight.

Practice 
Guided Practice: Give each student a conic section. Have them describe and list all of their knowledge about that conic. Have students exchange their list with a neighbor and modify or add to that list. Repeat the process as necessary. Independent Practice: Conic Practice: Students have to identify and describe conics. Students may use their Internet tool to complete the worksheet.

Closure 
Review objectives: Hold up flash cards of various graphs and equations and ask students to identify them. Encourage students to further their exploration about conics.

Extensions 
1. Given a graph, have students write the equations of the graph. Then have students create their own graphs and write the equations. 2. Give each student a particular conic and ask him/her to create a visual aid, and present a 5minute minipresentation to the class. 3. Ask students to research comets and astronomy and relate to conic sections. Ask students to also research other real world applications such as planetary motion. 4. Use the conic applets in http://www.exploremath.com for another way of discussing the various components of the equations. 
Assessment 
Three sample items are provided for use in assessing students’ understanding:
The following table shows how the assessment items are related to specific objectives.



Answer Keys: PaperPencil Test: Conic Sections Assessment

References 
Drawing
the Paths of the Planets Historical View of Conic Sections
Information for Power Point
Mathematics & Science Center Mathematics & Science Center:
OnLine Educational Programs Newtonian Gravitation and
the Laws of Keplar Occurrence of the Conics Parabola, Ellipse and Hyperbola
Lesson Plan and Interactive Activity 