Conic Cuts

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Purpose

This is a level 5 geometry strand activity from the Figure It Out series.

A PDF of the student activity is included.

Achievement Objectives
GM5-7: Construct and describe simple loci.
Student Activity

  

Click on the image to enlarge it. Click again to close. Download PDF (367 KB)

Specific Learning Outcomes

explore the properties of conic sections

Required Resource Materials

sticky tape

cardboard

torch

greaseproof paper

FIO, Level 4+, Geometry, Book Two, Conic Cuts, page 18

Activity

These activities investigate three of the four conic sections, that is, the different cross-sections you can get by cutting a cone. The first of these, the circle, appears in the level 4 Measurement and Geometry achievement objectives. The others do not appear in the Algebra achievement objectives until level 6 and in the Geometry
ones until level 8, but they are in the suggested learning experiences for levels 5 and 6. This illustrates the fact that students can enjoy exploring these curves and their applications without getting too far into the mathematical theory.


The cone is a familiar shape. Your students should be able to think of examples of how the cone shape is found in nature and how it is used in product design (for example, road cones, ice cream cones, funnels for pouring, or the nose cone on rockets). In each of these uses, the shape has been chosen for good, practical
reasons. Can the students suggest what these might be?


The following diagram shows that, mathematically speaking, a cone actually has two parts that touch at a point. If it is cut:

  • parallel to the base, we get a circle
  • parallel to the sloping side, we get a parabola
  • at an angle between the above two cases, we get an ellipse
  • at an angle greater than the angle of the sloping side, we get an hyperbola.

diagram.

Only the circle and the ellipse are closed curves. If a closed curve is not a circle, it's an ellipse. The parabola and hyperbola are open and extend into infinity. The hyperbola has two completely separate parts to it and is avoided in this activity because it is conceptually more difficult than the other three sections.


Activity One

This is a simple experiment in which light is used to define the shape of the three sections.
 

Activity Two
 

In this activity, the students use paper folding to define the outline of a parabola and an ellipse. Your students should:

  • use greaseproof paper or tracing paper so that they can easily see the marked point when they fold the top layer of paper onto it
  • mark the point towards the edge of the long side of the sheet of paper
  • make many creases so that the shape of the curve (which is formed by the edge of the unfolded part of the paper) is clearly defined
  • use a pencil to draw a smooth line around the curve they have made.
     

Activity Three

Students should see that in Game One, to be fair, all players must stand the same distance from the rubbish bin. This means that they should stand on the circumference of a circle, with the bin in the centre.

Game Two is trickier. One approach is for the students to mark any point that is the same distance from the cone and the wall and then to mark other such points. These points will lie on the locus of the line that defines all suitable starting points. The students may need prompting to see that Activity Two question a is relevant here. The cone in the game is equivalent to the dot, the wall to the line, and the locus of points equidistant from both is a parabola. Folding the paper was a means of finding midpoints.
One approach to Game Three is to use a closed loop of string to represent the total distance travelled by each player and two drawing pins stuck into a piece of card to represent the cones. The player can be represented by a pencil. The pencil can move anywhere as long as it keeps the loop of string taut at all times. As it moves
to all possible positions, the pencil draws an ellipse. If the students consider the 3 legs of the journey each player makes, they will see that, although lengths a and b change depending on the starting point, the total a + b is constant (as is the total a + b + c). This means that the game will be fair as long as the two players start anywhere on the ellipse.
In Activity Two, question b, the students create an ellipse by finding points
equidistant from a fixed point and a circle. Although Game Three also involves an ellipse, its shape is derived in a completely different way (using two fixed points).
 

ellipse.



As an extension to these activities, the students  could create a parabola on square grid paper, using square numbers, as in the diagram, and joining the
marked points with a smooth curve. (It's over to you whether you introduce the squares of negative numbers at this point.)
Another extension activity would be for the students to investigate the use of the parabola in technology, for example, in a torch or a solar cooker. They could
also try to discover why the parabola is the most suitable shape for such purposes.

diagram.
For a short history of the conic sections and a visual demonstration of them, see this University of Georgia site: http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Erbas/emat6690/Insunit/conicsunit.html

Answers to Activities

Activity One

a. The conic section is a circle only if the torch is held at right angles to the surface. If you alter this even slightly, the section becomes an ellipse.
b. The conic section becomes a parabola when the outer edge of the beam is parallel to the surface.

Activity Two
a. A parabola

answer.

b.

diagram.


Activity Three
Game One

The fairest way is to draw a circle around the rubbish bin. If the
players shoot from any point on this circle, the throwing distance
is the same for all of them.

bin.

Game Two
Draw a parabola, as in the diagram. Every point on this curve is exactly the same distance from the cone and the wall, so X can be anywhere on it.
parabola.
Game Three

The fairest way is to draw an ellipse, as in the diagram. If the two players start from anywhere on this curve, they will run the same distance.

ellipse.

Attachments
ConicCuts.pdf163.22 KB
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Level Five