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Cutting Conics

  • Lesson
9-12
1
AlgebraGeometry
Terry Johanson
Langham, SK, Canada

Students explore and discover conic sections by cutting a cone with a plane. Circles, ellipses, parabolas, and hyperbolas are examined using the Conic Section Explorer tool. Physical manipulatives such as dough can optionally be used as well.

Students may have heard the term conic section before, or at least heard of the shapes circle, ellipse, hyperbola, and parabola. To begin the day's exploration, ask them how the shapes might be related. Most students who have not explored cutting a double-napped cone will be unfamiliar with the relationship. You might consider listing students' hypotheses on the board and leaving them up during the exploration. In this way, students can revisit their earlier ideas as they investigate the conic sections.

In this activity, students explore how a double-napped cone can be cut to create each of the 4 conic sections, as shown below. If students are unfamiliar with double-napped cones, you might want to have a model to show them. Inform students that all the conic sections can be created by passing a plane through a double-napped cone and looking at the cross section. However, do not tell them how to do this or show them the image below at this point.

2907 conics 

Direct students to the Conic Section Explorer Tool as their primary means of investigation. Before students begin their own explorations, discuss the various parameters of the tool control and how they change the graph. If time permits or if you would like to allow an additional class day to complete the lesson, you might also consider building physical models out of a malleable material such as dough, which will give students a different perspective. See the Extensions for ideas on how to achieve this.

appicon Conic Section Explorer Tool 

Hand out the Cutting Conics Activity Sheet. Read through the introductory text on the activity sheet as a class. You may wish to reference the hypotheses made earlier, the double-napped cone model, and the Conic Section Explorer tool as you do this. However, your goal should be to provide as little information as possible to the students, allowing them to discover the answers on their own. Simply clarify any questions on the meanings of new terms or the functionality of the online tool.

pdficon Cutting Conics Activity Sheet 

While students are experimenting with the online tool parameters to create conic sections, listen to group discussions regarding how they are deciding to cut the cone to create each cross section. What processes do they go through to discover the slope and position of the cut to create the required shape?

If students are struggling to create a given shape, you can direct their exploration with questions like:

  • What happens if you change the slope of the cut?
    [There is a continuum related to the angle of the cut:
    • If the slope is 0, the cross section is a circle.
    • If the slope is between 0 and the slant of the cone, the cross section is an ellipse.
    • If the slope is greater than the slant of the cone, the cross section is a hyperbola.
    • If the slope is equal to the slant of the cone, the cross section is a parabola.
    Do not tell students these rules. Use them only as guidelines to help students.]
  • What happens if you change the location of the cut and place it closer to the origin? What happens if you change the value of b
    [As the cut gets closer to the origin:
    • The size of circles and ellipses are decreased.
    • Parabolas become less rounded at their vertex.
    • The distance between vertices in hyperbolas decreases, and the vertices become less rounded.]
  • What happens if you try to cut the cone through the origin?
    [This creates "degenerate cases" where the cross sections are not unique to the double-napped cone. If the cut goes through the origin, the following occurs:
    • If the slope is equal to the slant of the cone, a line is created.
    • If the slope is less than the slant of the cone, a point created.
    • If the slope is greater than the slant of the cone, an X is created.]

If students use both the Conic Section Explorer Tool and physical models to explore the conic sections, emphasize that they only need to write additional ideas for the second method used. They do not need to repeat observations they have already written down.

When students have completed their explorations, bring them together as a large group to discuss the answers to their answers. This is a good time to reread the hypotheses written on the board at the beginning of the class. Were they correct? If they were incorrect, can students now explain why things behaved differently? Let the students lead the discussion by having them share their answers.

In the ClassroomSuccessStoryThumb

"This is one of my favorite lessons; it allows students to ask, 'What happens if ...?' and actually see the result of the changes. I think it really helps kids connect why they are called conic sections." -Carol McDonald, HS math teacher (Jasonville, IN)

 
 

Assessment Options

  1. In whole class discussions, ask students to describe how they discovered how to cut their cones to create each conic section. Encourage and validate a variety of responses.
  2. Have students submit their activity sheets or answers to the Questions for Students.
  3. Separate the class into 6 groups (or a multiple of 6 if your class is large). Assign two conic sections to each group. There are 6 different ways to do this: circle/ellipse, circle/hyperbola, circle/parabola, ellipse/hyperbola, ellipse/parabola, and hyperbola/parabola. Each group should create a poster summarizing what they've learned about their two conic sections and comparing and contrasting them.

Extensions 

  1. You may wish to allow students to explore conic sections with physical models as well. Dough is a malleable substance that lends itself well to this exploration. Prepare dough in advance, or have students create their own. A possible recipe is:
    • 21/2 to 3 cups flour
    • 2 cups boiling water with 1 package Kool-Aid (any flavor)
    • 3 tablespoons corn oil
    • 1/2 cup salt
    • 1 tablespoon alum (can usually be found in the spices aisle of a grocery store)

    Have students press the play dough into cone-shaped funnels or paper cups. This will create solid cones. They will have to do this 8 times to create 4 double-napped cones. Create the double-napped cones by connecting two separate cones at their vertices using toothpicks. To create the cross sections, cut the double-napped cones carefully (so the dough does not compress and change shape) with dental floss.

    Before you allow students to begin this exploration, be certain to discuss the limitations of these models. Specifically, mathematical double-napped cones extend infinitely, so students will need to remember that their cross sections will likely extend past the cross sections they create.

  2. 2907 ellipseStudents could explore the relationship between circles and ellipses, recognizing that circles are a specific example of an ellipse whose semi-major axis (a) is equal to its semi-minor axis (b).
  3. Have students explore foci and their function in conic sections. How are the various foci related between different types of conic sections? The Cutting Conics Activity Sheet shows the foci for the same graphs at the top of the activity sheet. This might be a good place to start the discussion.
  4. Parabolas are often described using the standard equation:

    y = 1/4p (x – h)2 + k 

    Show how this can be expanded to create the general equation:

    y = ax2 + bx + c 

    Have students relate the values of a, b, and c to the distance between the vertex and the focus (p) and to the location of the vertex (h,k). They should be able to find these equations:

    a = (1/4·p)

    b = (1/2·p)×h

    c = (1/4·p)×h2 + k  

Questions for Students 

1. How can a cone be cut to create a circle?

[To create a circle, the slope needs to be 0 so that the plane is parallel to the ends of the cone. The farther the cut is from the origin, the larger the circle will be.]

2. How can a cone be cut to create a ellipse?

[To create an ellipse, the slope needs to be greater than 0 but less than the slant of the cone. The farther the cut is from the origin, the larger the ellipse will be. Many students will need to be reminded that the cone does not end, so even if it looks like the ellipse does not fully close, they need to consider a larger cone. To help them do this, suggest they increase the cone height.]

3. How can a cone be cut to create a hyperbola?

[To create a hyperbola, the slope must be greater than the slant of the cone. The farther the cut is from the origin, the greater the distance between vertices and the more rounded the vertices of the hyperbola.]

4. How can a cone be cut to create a parabola?

[To create a parabola, the slope must be parallel to the slant of the cone. This cut cannot go through the origin. The farther the cut is from the origin, the more rounded the vertex of the parabola. May students will have trouble distinguishing between an ellipse and a parabola when the slope is nearly equal to the slant of the cone. For these students, suggest they increase the height. The ends of the parabola are always moving farther away from each other, so if the cross section seems to be closing, it is an ellipse.]

5. How are circles and ellipses related?

[Both shapes are closed and do not extend past the end of the cone. Also, both shapes get larger as the cut moves away from the origin (i.e., As the value of b increases).]

6. How are hyperbolas and parabolas related?

[Both shapes are open and always extend past the end of the cone. Also, both shapes have vertices that get rounder as the cut moves away from the origin (i.e., As the value of b increases).]

Teacher Reflection 

  • How did your lesson address auditory, tactile and visual learning styles?
  • Did students prefer using the online tool? Dough or another physical model? Why did they enjoy one more than the other?
  • What were some of the ways that the students illustrated that they were actively engaged in the learning process?
  • How did the activity enhance their understanding of the relationships between types of conic sections?
 

Learning Objectives

Students will:

  • Explore the conic sections: circles, ellipses, hyperbolas, and parabolas.
  • Describe how to cut a double-napped cone to create the various conic sections.

NCTM Standards and Expectations

  • Draw reasonable conclusions about a situation being modeled.
  • Analyze properties and determine attributes of two- and three-dimensional objects.
  • Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.
  • Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP7
    Look for and make use of structure.