Problem Solving using Maple Animations

Abstract

In our specialist mathematics degree, computer laboratory sessions with Maple are being used to fundamentally change the way we teach. Visualisation and animation are important issues with some modern applications of software. As an example, we mention Finite Element Method calculations of flow such as plastic filling a mould. Some modern FEM packages include a facility to animate the flow to assist with the visualisation of the flow process. Sophisticated computer algebra systems such as Maple (and Mathematica) have the facility to generate animations and this can be exploited in first year mathematics courses.

Since our students have already done some calculus at high school, we have run, since 1998, a new subject, nonlinear mathematics, that is taken concurrently with a fairly standard type of calculus subject. The nonlinear mathematics subject introduces some modern ideas, namely Phase Plane Methods and Iteration, and the use of Maple is integrated throughout this subject. Animation is introduced in this subject. The animations are mostly of parameterised families of curves and details are reported in "Animations using Maple in First Year" by Blyth (to appear).

Students enjoy the animation activities and so we have been encouraged to develop some more advanced animation materials which are being incorporated into our first year program. One topic which students have traditionally found difficult is the standard type of minimise or maximise problem of first year calculus course. To address this we have written our version of the Polya type of problem solving approach using Maple on the standard problem of a farmer fencing a plot of land with one boundary being a river. Students are asked (as an assignment) to use this approach to maximise the area in the Norman window problem.

Although not assessed, students were shown how to use animation of geometrical figures (generated with the Maple plottools package). The example used was a rectangle inscribed in a triangle, where animation was used to find the dimensions of the rectangle with the greatest area. To achieve this, a dynamical display of the dimensions and area of the rectangle is required. It is also instructive to superimpose an animated area versus rectangle width plot on the animated diagram. The maximum is very clearly illustrated and found with this approach.

Exercises (not for credit) are provided. These are to maximise the area of a trapezium inscribed in a semi-circle and the Norman window problem. These animations are very striking and are strong visually. However it is a lot easier to just use calculus, but not visually appealing. We demonstrate both and encourage students to do both.