I was teaching (or trying to teach) conic sections. I wanted the students to fully understand the definition of each of these conic sections. The students were having a hard time understanding the idea of the "sum of the distances" for an ellipse, and "absolute value of the differences" for the hyperbola. To understand these phrases from the definitions allows one to take the concept to a deeper depth. So, I decided to design my classroom desks in an ellipse one day, and a hyperbola another. I had a long rope that I had bought from Lowes, and designed the room so that the desks were in a true hyperbola with the "absolute value of the difference from the focal points" being constant. I had a center point, endpoints, vertices, foci so that the shape was true to definition. I randomly had 2 students that were sitting on the "wings" of the hyperbola measure the distance from the foci (another student holding the rope). The students could visually see what was meant by the distance. I did the same with the ellipse. After defining and visually moving thing around, we then came up with the equations using the distance formulas.
The students found this approach more meaningful and more memorable. When a student was stuck at a point in problem solving using hyperbolas and ellipses, the students would say, "Remember when..." and the students would be unstuck.
I am not sure that this would be considered an "off the wall" type of lesson, but it certainly was a lesson that wasn't normally planned.
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