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.
Direct students to the Conic Section Explorer
tool as their primary means of investigation. Before students begin
their own explorations, discuss what 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.
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.
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:
Do not tell students these rules. Use them only as guidelines to help students.]
- 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.
- 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.