Overview | The Math | The Lesson

Shapes without (Sophisticated) Tools

Why would we want to avoid useful tools? There are several good reasons. One is the great educational value of being able to construct rather than to copy. Another is you might not have (or be able to carry) a large enough template! Many real life situations call for the use of geometrical shapes on a large scale. Masons and carpenters use long, taut string to mark straight lines, gardeners use stakes and rope to create flower beds in the shape of an ellipse. Imagine a "Math Field Trip" to the beach where children draw huge circles on firm sand. A more practical example would be drawing a chalk mural in the school yard.

The Basic Constructions

1. The Straight Line

   Make a string taut to get a straight line.




2. The Circle

   A circle is formed by all the points at the same distance (the radius) from a fixed point (the center).

   A circle can be drawn by holding one end of a string (whose length is the radius) at a fixed point (the center), pulling the string taut, and marking the path formed by the other end as you go around the fixed point.
   
   

   • A longer string will produce a bigger circle.
   • In the construction of the circle described above, the string plays the role of a primitive compass.


3. The Ellipse

   An ellipse is made up of all points the sum of whose distances to two fixed points (called the foci) is a constant.

   An ellipse can be drawn by looping a string around two fixed points (the foci). Use a pencil to pull the string taut so that it forms a triangle with corners at the two fixed points and the pencil. As you move the pencil around the fixed points, always keeping the string taut, it will trace an ellipse.




4. The Right Angle

   Imagine dividing a line segment exactly in half with a line that is perpendicular (at a right angle) to it. Each point along the dividing line will have this property: it will be the same distance from one end of the line segment as from the other end. In fact, any point in the plane that is the same distance from the two endpoints must lie on that perpendicular line.
   
   This concept is behind the following technique for drawing a right angle.

   
   Draw a line segment (horizontal in the picture). Draw two circles (as described above) using the same string: one centered at one end of the line segment, the second circle centered at the other end. The circles should be big enough as to cross each other twice. Identify the two points where they meet. Draw a straight line through them. The line segment and the straight line will be at a right angle.




5. Point at Same Distance from two Given Points
   
   This is a by-product of the Right Angle construction above: the point where the circles meet at the top (see figure for Right Angle) is at the same distance from each end of the horizontal line segment. The actual distance is the length of the string used as a radius for the circles. The same is true for the point where the circles meet at the bottom.
   
   This construction is of great help for drawing more complex shapes.

Comments

• It turns out that using string to draw a straight line is not very practical when working on a small scale, as your students will probably do at first. Once the string technique is explained, it might be a good idea to compromise and allow students to use an unmarked, straight edge to draw their lines.
• Even if we cannot measure distances in inches, feet, etc., we can tell whether two distances are equal or not, by matching one to a length of string and comparing to the other.

Download the Lesson Kit (pdf) Continue to the Lesson