Conic Sections

Angela Wood, NSF Scholar, 2002-03
Mathematics & Science Center & Virginia Commonwealth University

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:
Conic Sections worksheet
Folding Conic Sections Worksheets
4 pieces of 11 x 14 in. wax paper
4 pieces of construction paper
Compass
Ruler
Permanent Marker
Computer with Internet access
4 Worksheets from http://www.exploremath.com
Parabolas
Circles
Ellipses
Hyperbolas


For each group:
Slicing a Cone Worksheet
Styrofoam Cone
Plastic knife or wire


For the class:
Flashlight
Power point on conics

 

 

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 5-minute mini-presentation 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.

 

 

 

Objective
Paper-Pencil Test
Product/
Performance

Recognize and describe the real world applications of conics.

Part 1

Real world applications

 

Define and identify the four different conics sections.

Part 2

Definitions

 

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

Part 3

Equations

 

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 the center, vertices, and other variables.

Part 4

Manipulating graphs and writing equations

 

Major Understanding: Conic sections are apparent all around us.  For example, they can be found in structures, mirror and satellite dishes.  Conic sections also represent orbit paths.  It is very important to understand the four different conic sections; parabolas, circles, ellipses and hyperbolas.  It is also necessary to understand the different components of the equations that form the four different conic sections.

  Assign each student a conic section and have them create a visual aid and a written document.

Record and analyze the pulse rate of an organism as the temperature of the water increases and discuss how this relates to thermal pollution.

4  
 


Teaching Tips

 


For additional information on teaching this lesson, go to the following links:

Answer Keys:

Parabolas Key

Circles Key

Ellipses Key

Hyperbolas Key

Conic Practice Key

Paper-Pencil Test: Conic Sections Assessment

Frequently Asked Questions

 

 

References

 

Drawing the Paths of the Planets
http://www.galashiels.org.uk

Historical View of Conic Sections
http://www.krellinst.org/UCES/archive/resources/conics/node5.html

Information for Power Point
http://mathworld.wolfram.com/ConicSection.html

Mathematics & Science Center
Information on educational programs available to students, teachers and school divisions and procedures for registering for programs.
http://mathsciencecenter.info

Mathematics & Science Center:  On-Line Educational Programs
Learn through on-line virtual classrooms, web-based lessons and on-line courses. Access proven lesson plans and instructional modules.
http://MathInScience.info

Newtonian Gravitation and the Laws of Keplar
http://csep10.phys.utk.edu/astr161/lect/history/newtonkepler.html

Occurrence of the Conics
http://britton.disted.camosun.bc.ca/jbconics.htm

Orbits
http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/ BrombacherAarnout/OrbitsLesson/orbits_lesson_series.html

Parabola, Ellipse and Hyperbola Lesson Plan and Interactive Activity
http://www.exploremath.com