Try some software to help pupils approach proof via explanation, says Michael De Villiers.
Which mathematics teacher has not been faced with students demanding: "Why do we have to prove this result? It's so obvious." Indeed, one of the biggest challenges facing maths teachers is convincing their students of the usefulness and value of proof.
Teachers may find it more meaningful to introduce proof as a means of explanation rather than as a means of verification. I have developed some activities to do this with Geometer's Sketchpad software.
This dynamic tool makes it easy to explore triangles, quadrilaterals, circles, and other geometric figures without the mechanical restraints of pencil and paper, compass, and straight edge. Students can transform their figures with the mouse, while preserving the geometric relationships of their constructions. They can examine an entire set of similar cases in a matter of seconds, leading them naturally to generalisations. This encourages a process of discovery where pupils or students first visualise and analyse a problem, and make conjectures before attempting a proof.
Working with 14- to 17-year-olds, often a difficult age to engage with maths, teachers can often get through to them by using unusual conjectures to elicit surprise and create a need for further understanding. The word "proof" is initially avoided. To start off with, the word "explanation" is used instead. Students appear to find it more meaningful to see the exercise not as an attempt to verify, but rather to understand, why a conjecture is true.
Sum of distances A beautiful piece of mountainous land in the Drakensberg in South Africa approximates the shape of an equilateral pentagon (ie, has equal sides) and is bordered by a stream along each side. Since trout and other fishing are quite excellent in these streams, the owner wants to build a small holiday cabin. She anticipates that visitors would visit each stream with the same frequency and decides to build the cabin so that the sum of the distances from the cabin to all five streams is as small as possible. Where should she build the cabin so that this is possible?
Guess 1. Make a guess where you think she should build the cabin. Mark a point in the equilateral pentagon corresponding to your guess.
2. The next step is to use a dynamic geometry program to investigate the problem, and for further explorations. A free trial period demonstration version of Sketchpad can be downloaded from www.keypress.comsketchpadsketchdemo.html and used to manipulate the dynamic sketches downloadable from http:mzone.mweb.co.zaresidentsprofmddist.zip (WinZip is required to unzip the dynamic sketches).
Conjecture1. Open the sketch dist4.gsp supplied by the teacher. Drag point P to experiment with your sketch.
2. Double-click the Distances Sum button to show the distance sum. Drag point P around the interior of the equilateral pentagon. What do you notice about the total sum of the distances?
3. Drag a vertex of the equilateral pentagon to change its size or shape. Then again drag point P around the interior of the pentagon. What do you notice now?
4. What happens if you drag P outside the equilateral pentagon?
5. Organise your observations from steps 2-4 into a conjecture. Write your conjecture using complete sentences.
Explaining Conjecture: As long as P is an interior point, the total sum of the distances from point P to all five sides of a given equilateral pentagon is always constant.
But can you explain why this is true? Although further exploration with Sketchpad may convince you of the truth of your conjecture, this does not amount to an explanation. The regular observation that the sun rises every morning is not an explanation; it only reconfirms the validity of the observation. To explain something, we have to try to explain it in terms of something else - for example, we explain sunrises by discussing the rotation of the earth around the polar axis.
It will help to use sketch dist4.gsp and work though the following to develop a logical explanation for your conjecture above.
6. Double-click the Small Triangles button.
7. Drag a vertex of the equilateral pentagon. Why are the five different sides labelled a?
8. Write an expression for the area of each small triangle using the variables h1 h2, h3, h4, h5 and a.
9. Add the five areas and simplify your expression by taking out common factors.
Answer:Fah1 + Fah2 + Fah3 +Fah4 + Fah5 = Fa(h1 + h2 +h3 + h4+ h5) 10. How does the sum in step 9 relate to the total area of the equilateral pentagon? Write an equation to show this relationship. Use A for the area of the equilateral pentagon.
Answer: A = Fa(h1 + h2 + h3 +h4, + h5) 11. Use your equation from 10 to explain why the sum of the distances to all five sides of an equilateral pentagon is always constant.
Answer: Since h1 + h2 + h3 + h4 + h5 = a and A and a are constant for a fixed equilateral pentagon, it follows that h1 + h2 + h3 + h4 + h5 is constant.
12. Explain why yourexplanation in step 11 would not work if the pentagon was not equilateral.
Further exploration 13a. Where should you locate P to minimise the sum of the distances to the three sides for an equilateral triangle? Open sketch dist1.gsp to investigate the problem.
b. Explain your observation in step 13a, and generalise to polygons with a similarproperty.
14a. Where should you locate P to minimise the sum of the distances to the three sides for an arbitrary triangle? Open sketch dist1b.gsp to investigate the problem.
b. Explain your observation in step 14a.
15 a. Where should you locate P to minimise the sum of the distances to the four sides of a parallelogram? Open sketch dist3.gsp to investigate the problem.
b. Explain your observation in step 15a, and generalise to polygons with a similar property.
* Michael De Villiers's book, Rethinking Proof withGeometer's Sketchpad, and the software, are available from QED Books, Pentagon Place, 195 Berkhamsted Road, Chesham, Bucks. HP5 3AP. Tel: 08457 402275 Michael De Villiers is lecturer in mathematics education at the University Durban-Westville, South Africa. E-mail:email@example.com