This is a level 2-3 activity from the Figure It Out series.
A PDF of the student activity is included.
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a photocopy of the diagrams in Activity One
string or wool
coloured pencils
a ruler
scissors
FIO, Geometry, Levels 2-3, Follow That Thread, page 19
Activity One
The lines created by these curve-sketching activities create “envelopes” of well-known shapes (loci). A locus is a set of points often governed by a relationship or rule. For example, the lines in the square create the envelope of an ellipse. The more points that are marked on the outside of the square, the closer the envelope would come to approximating an ellipse.
The lines in the angle create the envelope of a parabola, which is the path of a ball when thrown in the air. The lines in the circle create the envelope of another circle. Again, a larger number of matching points on the circumference of the starting circle would make a shape that is closer to a circle.
Ask the students to identify symmetry in the envelope shapes. The ellipse has two lines of reflection symmetry, the parabola has one, and the circle has an infinite number of lines.
The circle has rotational symmetry of infinite order, and the ellipse has half-turn symmetry.
Activity Two
The lines of thread will create the envelope of a hyperbola:
The curves created in this activity are known as the conic sections (see the notes for page 1) as they can be created by cutting a cone with a single cut. Students might enjoy looking up these famous loci in an encyclopedia or on the Internet.
Answers to Activities
Activity One
Answers will vary but could include:
The space left inside the square is an ellipse.
The space left inside the angle is a parabola.
The space left inside the circle is almost another circle.
Activity Two
a. This makes approximately a 1/4 circle (or a hyperbola).
b. To make a complete circle, students could make another 12 slits and create a 1/4 circle in each 1/4 of the square.