Here is the task:Student 1 is to draw two points on the board that are on an imaginary horizontal axis. I introduce these as "the foci."
Student 2 is to take the marker and pull the string taught. They are then to go all the way around and draw an oval, which is to be the perfect ellipse. See the video for the demonstration.
It was a bit like twister, which was really fun. But the beauty of it is that kids began to understand, without even talking about ellipses and their equations, why the major axis of an ellipse must contain its foci.
They get the definition of an ellipse, that the sum of the distances from the foci to any point on the ellipse, is a constant. And, that this distance is 2a*
Even better, some could not draw the ellipse because the points were too close together and their taught string would go off the board. So placement of foci was starting to have meaning.
But what was really cool was when students spoke about eccentricity without knowing what it was. The farther apart the two foci, the narrower the ellipse. The closer they were, the closer the shape was to a circle. (see last picture)
|*This is a nice way to show how the string length = 2a...there is overlap from the foci to the left vertex. That part was missing on the right. You can also hold the string from one end to the other.|
|Demonstration of how the location of the foci affects the ellipse...discovered by students and untouched for picture.|