TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Prerequisites

Three dimensional plotting of parametric plots, equations of surfaces, trigonometry, integration for surface area in three dimensions and for area in two dimensions .

Time allotment - time management

IF students have to spend all their time constructing the various surfaces mathematically and/or visualizing these by hand then this could take hours and some students might never get it. If reasonable diagrams such as the surface diagram offered in the statement and the line diagram with suggestive lines and curves in the solution are offered then the students' time and frustration can be shortened considerably.

We would imagine that spending half a class on it with a chance to answer questions, consolidate gains, discuss pictures, etc. at the end of the hour and at the first one quarter of the next hour may be enough for students to tghen go out on their own and do the problem.

Expectations

Students will either see the "flash" solution or will make diagrams (in itself visualization practice) to gain understainding of the surfaces involved.

Future payoffs

Students will be able to visualize in three dimensions and see the shape "rolled out" in two dimensions.

Extensions

One could ask what other figures "drawn" on a cylinder, say a surface of an airliner, would have to look like in their planar version as they are prepared for application to the side of the cylindrical airliner tube.

References and Sources