To appear in** : Mathematical Digest, **No. 118, Jan 2000

Sketchpad Solutions and Results '99

Michael de Villiers, Dynamic Math Learning

In the July 1999 issue, a copy of the software package *Sketchpad* was offered as a prize to Grade 9 learners or below for the most elegant set of solutions for the three problems given below. Submissions were received from Harry Wiggens (Graad 9, Worcester), Theo Mokgatlhe (Grade 9, St Dominic College, Welkom) and Adrian Frith (Grade 6, St George’s Grammar School, Mowbray). The prize was awarded to Adrian who gave the best set of solutions to the three problems. The small number of responses was rather disappointing as the first and last problems could very easily be solved by scale drawings, and a trail and error approach. This either implies that scale drawings are not dealt with at school, or if dealt with, clearly under-emphasized as an extremely useful tool for solving many real world problems. Although paper and pencil scale drawings and measurement could be tedious and relatively inaccurate, scale drawings on computer by means of a programme like *Sketchpad* is quick and easy, and very accurate.

- A rider is traveling from point D to point E between a river and a pasture. Before she gets to E, she wants to stop at the pasture to feed her horse, and again at the river to water him. If angle BAC = 45°, EJ = 2 km, AJ = 5 km, DK = 7 km and JK = 10 km, what path should she take to travel the shortest possible distance?

Geometrically this problem can only be solved with the aid of an accurate scale drawing on which one can either try many different positions for L and M, or use reflections. In general, the principle of reflection is a useful one for finding minimum paths. For example, by reflecting D in AB and E in AC (and since reflections preserve distance), the problem is now simply reduced to finding the shortest distance from D’ to E’, which is clearly the straight line D’E’. The shortest distance would therefore be found when D’, L, M and E’ all lie in a straight line. This occurs (from measurement on a scale drawing) when AL = approx 8.4 km and AM = approx 5.3 km.

Although a trail and error approach by hand with an accurate scale drawing is tedious, using Sketchpad simplifies the process somewhat (one needs to first drag one of L or M and then the other, etc.) Although this problem could also be solved numerically by trial and error, and using the theorem of Pythagoras, and some trigonometry, this becomes even more cumbersome and tedious.

2. *Determine the sum of the angles at the labelled vertices of an arbitrary septagon of the type shown below.*

The sum of the angles is 180° and follows from a repetitive application of the exterior angle theorem of a triangle. For example, Angle 1 + Angle 2 = ext. Angle 3, Angle 4 + Angle 5 = ext. Angle 6, Angle 6 + Angle 7 = ext. Angle 8, and Angle 9 + Angle 10 = ext. Angle 11. But angles 3, 8 and 11 form a triangle. Therefore their sum is equal to 180° , which is also the sum of the angles at the seven vertices.

Theo Mokgatlhe generalized the result to any "star" polygon (*n* > 6) where each vertex is joined to the third vertex to its immediate left and right, with the interior angle sum given by 180° (*n* — 6).

3. *A railroad has to be built between two towns A and B as shown below. The ground to the North of line PQ is very rocky and as a result it costs R20000 per km to build there, as opposed to the R14000 per km to the South of this line. If PQ is 200 km, where should X be chosen so that construction of the railroad is as cheap as possible?*

Geometrically this problem can only be solved with the aid of an accurate scale drawing and using a trail and error approach. For example, by choosing several different positions of X, measuring the corresponding distances, and calculating the costs, one can find the minimum position. Although this is tedious by hand, one can set up a scale drawing on Sketchpad, do the necessary measurement and calculation, easily finding the minimum by dragging X along PQ (see above). This gives a solution of XQ = approx 43.7 km.

The problem can also be approached numerically by choosing different positions of X, and using the theorem of Pythagoras to calculate the distances, one continues the process until a minimum is found. If we let XQ = *x*, we can algebraically specify the problem as one of finding the minimum of the following function (in thousands of rand):

.

To find the minimum of this function requires knowledge of advanced calculus (e.g. the chain rule), but one can draw a graph of the function with a graphics calculator or computer programme, and zooming in to its minimum value as shown above, giving a solution of XQ = approx 43.8 km.

A demo version of *Sketchpad* can be downloaded at www.keypress.com or provided on request. Excellent mathematics books such as *One equals Zero* by Nitza Movshovitz-Hadar & John Webb, and others on fractals, problem solving, geometry, etc., are also available. For ordering information about *Sketchpad*, and other specialized mathematical publications, please contact Michael de Villiers at

Tel: 031-2044252(w); 031-7083709(h); fax: 031-2044866(w); e-mail: dynamiclearn@mweb.co.za OR Pearl de Villiers at Tel: 031-2504431(w). Dynamic Learning, 8 Cameron Rd, Sarnia (Pinetown) 3615.

http://mzone.mweb.co.za/residents/profmd/homepage.html