Illuminations: Cutting Conics

# Cutting Conics

 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.

### 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

### Materials

 Double-napped cone model (optional) Computers with Internet connections Conic Section Explorer Tool Cutting Conics Activity Sheet

### Questions for Students

 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.] 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.] 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.] 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.] 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).] 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).]

### Assessment Options

 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. Have students submit their activity sheets or answers to the Questions for Students. 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

 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. Students could explore the relationship between circles and ellipses, recognizing that circles are a specific example of an ellipse whose semimajor axis (a) is equal to its semiminor axis (b). 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. 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

### 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? Did the activity enhance their understand of the relationships betweens types of conic sections?

### NCTM Standards and Expectations

 Algebra 9-12Draw reasonable conclusions about a situation being modeled. Geometry 9-12Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools. 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
 This lesson was prepared by Terry Johanson as part of the Illuminations Summer Institute.

1 period

### Activities

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