Cabri-Géomètre, Euclid's Revenge

Cabri-Géomètre, Euclid's RevengeAuthor(s): David GreenSource: Mathematics in School, Vol. 21, No. 2 (Mar., 1992), pp. 46-50Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214862 .

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CABR-

GEOME by David Green. Loughborough University

o~ee o~ee

Fig. 1

The cry of the modern mathematics movement of the early sixties was "Euclid must go!". A remarkable new piece of software called Cabri-geom~tre might well reverse the demise of Euclidean geometry in schools. It has been likened to LOGO. This is an accurate comparison not so much because of the mathematical content as in the approach. Cabri-geometre, like LOGO, has a "low thres- hold, high ceiling", being quite easy to begin to use but capable of considerable development and sophistication. It shares with LOGO the facility to build up more complex structures from simple basic 'objects' which then them- selves become 'objects' of the enriched system. LOGO does this with procedures, Cabri-geometre with macro- constructions. As with LOGO, it is perhaps unwise to ask (or at least to try to answer) the question "What can it do?". The response must be "It can do a great deal, and no one knows its limitations".

What is Cabri-g The manual says it is "An interactive notebook for learning and teaching geometry". It takes the familiar point, line and circle as most basic objects and allows the user to draw geometrical shapes on the screen and to specify relationships between them (e.g. a point on a line, or a line perpendicular to another line). The drawn shapes can be moved and any defined relationships will be pre- served. By this means the invariant features can be investi-

gated, which is the essence of geometry. Hypotheses can be formed and tested visually and numerically and deduct- ively and theorems proved. Cabri-geometre has potential from primary school through to university level.

What computer does it run on? The software is available for the Macintosh family of computers and an MS-DOS version has recently been released. It does not seem likely that the French originators will make a version for (say) the Acorn A3000 but it is so impressive that teachers should not dismiss it for that reason. This article describes the latest version for the Macintosh (version 2.1), which has some definite enhance- ments over version 2.0.

Getting started (Creation) Cabri-geometre takes as its fundamental objects Basic point, Basic line, Basic circle. These objects may be drawn on the screen with simple commands from a Cre- ation menu. The drawn figures can be re-positioned quite simply using the mouse. There are four other Creation commands. Two are Line by 2 points and Circle by centre and radius point. The other two - Line- segment and Triangle - are available more for con- venience than necessity. Figure 1 shows examples of simple figures drawn using the seven commands from the Cre- ation menu.

46 Mathematics in School, March 1992



Drawn objects can be moved about the screen (by "dragging" with the mouse). The three basic shapes move as entities, but the four other shapes (defined in terms of points) are moved and deformed by the movement of each point. For example a circle defined by centre and radius point may be displaced and enlarged by moving one or other of its defining points, and similarly a line segment may be lengthened and rotated. This much is not particularly impressive, of course, being the standard fare of drawing packages.

Defining Geometrical Relationships (Construction Commands) A very powerful feature of Cabri-geometre is the ability to define relationships between objects and to explore graphi- cally the implications. This is fundamental to Euclidean geometry and is likely to be the most used feature of this software at the school level.

The Construction menu has a number of commands which allow relationships between objects to be established. The following simple example illustrates the kind of activity which is possible. It uses commands from both Creation and Construction menus.

Description Result Command [menu]

1. Draw a circle

2. Draw a diametric line

3. Determine the unknown endpoint of the diameter

5. Define a free point on the circle

6. Draw a triangle with base the diameter and apex on the circumference

Circle by centre and radius point [Creation] Position mouse at centre and click, then at a point on the circumference and click.

Line by 2 points [Creation]

- Position mouse and click on centre and then on other point.

Intersection of 2 objects [Construction] Point and click on the line and on the circle.

Point on an object [Construction] Point to circle and click

Triangle [Creation] Point and click at the three points.

7. Look for anything interesting in the figure. 8. Form an hypothesis

(e.g. the angle at the circumference formed by a diameter is always rightangled).

9. Check the hypothesis visually by moving the free point around the circumference.

Such work could be carefully structured (contrived, some might say) or could be the outcome of very free exploration. As with LOGO various modes and styles of working are possible, each with its own merits.

There are further more powerful ways to test an hypothesis which will be described later.

Labelling and Measuring It can be helpful to label points and lines on diagrams and this is easily achieved using the Label command.

Although measurement of lengths and angles in standard units is not fundamental to this approach to geometry there is a clear use for this, and the Measure command allows this. As figures are moved around the screen the labels move too and the values of angles and lengths are moved and updated. Further calculation facilities are available in a temporary window. These include Coordinates of a point, Area enclosed by a polygon, and Slope of a line. Unfortu- nately the Calculations window disappears as soon as adjustments are made to the drawing. A good use of this is to demonstrate Pythagoras' Theorem (Fig. 2).

T

S B M

c A N

P Q

Fig. 2

Towards Proof The ability to calculate and display measures of angles, lengths and areas can corroborate numerically a visual impression (such as an angle being constant at 90', or one length always being twice another). For the triangle-in- the-circle example above the next step might be:

Description 10.Check the hypothesis

numerically by measuring the angle and moving the free point. Also, check further by altering the circle itself by moving the points defining the centre and radius.

Command [Menu] Measure

-angle [Miscellaneous]

Click in order on the three points to specify the angle.

The result is shown in Fig. 3. There is a further and most significant step towards a

complete confirmation of the validity of an hypothesis, and

Mathematics in School, March 1992 47



P 900

A O B

Fig. 3

confirmation that it is a theorem. This uses the Check property command:

11.Prove or refute the hypothesis by asking Cabri-geometre if it is generally true.

Check property -perpendicular lines

[Miscellaneous] Click on the two lines which are thought to be perpendicular.

The software responds with:

This property is true in the general case.

How does it know? Presumably, not by having theorems stored. It seems to have an intelligent knowledge base which is consulted but more likely it is essentially numerically driven. The basic facts and rules of Euclidean geometry are consulted and a test is made to see if the conclusion logically follows from the given relationships of the diagram. If the property is not generally true then that is reported. If the property is true (or nearly so) in the particular diagram but not generally so then the diagram is modified (e.g. a point or line is moved) to provide a counter-example. For example, a general triangle ABC is drawn such that it appears isosceles (Fig. 4).

B C

Fig. 4

Then an enquiry is made as to whether the two sides AB, AC, are equal (which they are in the particular figure). The response is:

This property seems true in this case ... but is false in a general position.

Show a counter-example?

If a counter-example is requested Figure 5 is obtained. The position of B is moved and the old position of AB is shown as a dotted line and the two lengths AB, AC (no longer equal) flash.

48

A

B

C

Fig. 5

In principle, any theorems of two dimensional Euclidean geometry can be investigated, demonstrated or validated. This of course raises difficult questions about what is proof and about how it should be treated in the mathematics classroom. It may be helpful to think of Cabri-geometre as confirming whether or not it is worth trying to prove a conjecture, rather than considering that the software actu- ally proves anything.

Loci The locus command enables paths of moving points to be traced on the screen (and there are printing facilities too, of course).

An example is given here to find the path of a point equidistant from a fixed point and from a point constrained to move along a fixed line. The sequence of steps is given below. All have corresponding commands in the Cabri- geome'tre menus; points and lines are positioned by simple mouse actions.

1. Draw the fixed point, P. 2. Draw the fixed line, m. 3. Place the free point, Q, on the line m. 4. Find the midpoint between the two points P and Q. 5. Ask for the locus of the midpoint. 6. Move the free point and so obtain the locus. 7. Print the resulting diagram.

The result is shown in Fig. 6.

Fig. 6

Mathematics in School, March 1992



The reader might like to consider how to modify the sequence to obtain the locus of a point moving equidistant from a fixed point and a fixed line, which gives the result in Fig. 7.

a

P

Fig. 7

A limitation of the software - hardly a surprising one - is that the drawn locus is no more than a trace on the screen. The trace itself cannot be manipulated or tested in any way. Nevertheless the idea of locus is a powerful one and Cabri-geome"tre complements other software already available for such explorations.

Dependence and independence Ideas of dependence and independence can be explored by establishing relationships among points on figures. A simple example is for one person unobserved by the rest of the group to place three points on the screen, A, B, C. The midpoint of B and C is found and labelled D (see Fig. 8). The question is then asked "What is the relation- ship between the points?", and this can be deduced by moving the points. Using Cabri-geometre the points A, B and C can all be moved independently but D will move if and only if B or C is moved. D is dependent on B and C and independent of A.

A B

D

C

Fig. 8

The nature of the dependence is simple in this case but can be made more subtle, of course. For example, in the example above the point B could be hidden or the midpoint of AD could be found and labelled E and D hidden. Figures involving points of intersection of lines or of lines and circles (with the lines and circles hidden) provide interesting challenges.

Making it Simpler The supplied system has seven Creation menu commands (described already) and ten Construction menu commands.

Of those ten, we have already met:

Locus of point Point on object Intersection of two objects Midpoint Centre of circle

The others available are:

Perpendicular bisector Parallel line Perpendicular line Symmetrical point Bisector (of angle)

This may all seem too much! Fortunately, there is a facility to remove items from the menus to create a simpler system, and so tailor it to the user's needs. For example the two main menus might be restricted to:

Creation Basic point Line-segment

Construction Locus of point Point on object Intersection Midpoint

This would limit investigation to polygonal shapes, without parallelism and perpendicularity being available for construction.

Building Up The Creation and Construction menus provide the basic building blocks for geometrical investigation. As has been mentioned, Triangle is provided as a Creation command. This is not strictly necessary as a triangle can easily be constructed from three line segments. Triangle is then a convenient extra. How about an equilateral triangle or square or general rectangle or ...? Indeed it would be very convenient to have other shapes available, particularly for specific topic work. This can be achieved by means of macro-constructions. A macro-construction is essentially a prepared procedure written once (e.g. by the teacher) and added, perhaps permanently, to the Construction menu. Creating a macro-construction is not a trivial exercise but it can be accomplished with a little forethought, care and practice.

We see, then, that the menus can be customised to suit the situation and this can greatly help beginners to make progress without the distractions of many commands on the menus or without the need to construct a shape fundamental to a particular exploration. Also, the ability to work with different subsets of commands (equivalent to having different restrictions on drawing apparatus) is an attractive feature.

And yet more ... There is not space to give full justice to the richness of the Cabri-geometre microworld, and its potential remains

Mathematics in School, March 1992 49



a matter for speculation. Brief mention here of two other features will underline the similarity in philosophy to LOGO - the Exposition command displays in a window, in natural language, a list of the objects shown on the screen, and the History command allows the user to reconstruct a completed figure step by step.

The Cabri-geome"tre microworld allows exploration of any aspects of mathematics capable of a geometric interpretation. This not only includes such obvious ideas as the properties of circles and triangles and their inter- relationships and associated loci, but also symmetry, trans- formations, tessellations, projections, location problems, trigonometry, tangents and gradients, and even some aspects of arithmetic and the number line, algebra and statistics. (However, it should be noted that this geometrical microworld only allows "ruler and compasses" constructions, which is a severe restriction). Emphasis on the process of doing mathematics, and on the exploration of the nature of mathematical proof are further features. If past experience is anything to go by this software will be used in ways as yet unimagined.

All-in-all Cabri-geometre can give us the best of both mathematical worlds - 'traditional' and 'modern'. What would the members of the A.I.G.T. - the Association for the Improvement of Geometrical Teaching- have thought of Cabri-geometre, I wonder?

What is going on in the UK? The NCET have produced an excellent nine page booklet containing seventeen "Starters for Cabri-geometre". There are already a number of individuals and small groups in the UK exploring Cabri-geometre, including one at Lough- borough University. A more elementary package working on similar principles is currently under development by Richard Bridges. It is written in BBC BASIC for the RML Nimbus. Leo Rogers of Roehampton Institute, with the

assistance of the NCET, is drawing up a list of people with some experience of Cabri-geometre and those who wish to be kept informed of developments, and some coordinated classroom reseach is underway. Those inter- ested are welcome to make contact with the appropriate person or group (see below).

Further Information Cabri-g om tre is available for 490 French Francs from:

Laboratoire LSD2 IMAG, Tour Irma BP S3X, 38041 Grenoble cedex, France.

Leo Rogers' address is: Mathematics Education, Digby Stuart College, Roehampton Institute, London, SW15 5PH. [Leo has made good contact with Grenoble and invites people to channel their orders through him. Past experience has shown that Grenoble can be very slow to respond (correctly) to individual orders. It is important to specify whether the Macintosh or the MS-DOS version is required and to say whether the English or French version of the software and manual is required. If you write to Grenoble it might be better to do so in French.]

The NCET contact is: Ronnie Goldstein, NCET, Sir William Lyons Road, Science Park, University of Warwick, Coventry, CV4 7EZ (Tel 0203-416994). [If you want the "Starter" booklet send you request to Nina Stone, at NCET, together with a stamped addressed A4 envelope.] Richard Bridges, King Edward School, Birmingham B15 2UA.

The author's address is: David Green, Department of Mathematical Sciences, University of Technology, Loughborough, Leicestershire, LE11 3TU (Tel 0509- 222864).

This article is one of a series prepared by members of the Mathematical Associations working group on Computers in Mathematics Teaching, co- chaired by David Green.

Book Reviews

Teaching Mathematics and Art Edited by Lesley Jones Stanley Thornes, 0748704566, e1 5.99 This is an interest book for teachers which examines the mathematics of some aspects of art. It is also a resource book since it provides practical examples and explanations of links between the two areas and thereby presents suggestions for cross curricular initiatives.

A first flip through the pages revealed many diagrams and pictures which were familiar and so encouraged me to believe that I might find ideas which I could develop in the classroom. On closer inspection I was disappointed to find an almost exclusive emphasis on two- dimensional art with no examination of three- dimensional models.

The book contains nine chapters each written by a different author. The wide range of expertise of the authors covering art and mathematics, and its teaching in schools is illus- trated in the short bibliographies in the introduction. The topics covered include, Mathematics and Aesthetic Perception, How Children Map 3D Volumes and Scenes on to 2D Surfaces, Proportion, Perspective Draw- ings, Geometry and Art, Tessellations, Islamic Design, Fractals and Plotting Spheres.

The chapters follow a similar format, each containing historical and factual information

about the topic, together with exercises to complete, so as to enhance the teacher's knowledge and understanding rather than provide classroom ready material for teaching. Much of this interest work is centred either on the diagrams and pictures provided or those produced as a result of the exercises. The mathematical knowledge and skills required to study a chapter varied but none of it would prove too difficult for an able 16 year old.

Having worked through the book, for the purpose of writing this review, without com- pleting all the exercises, I'm not sure that this is how the book should be used. The chapters are not inter-dependent but contain informative insights into topics which would be best studied separately in depth. My advice is therefore to start reading at the chapter which offers most initial intrigue and study it in depth. The ideas studied may then inspire and suggest classroom activities.

PETER PARKER

Mathematics Meets Technology by Brian Bolt Cambridge, 0521376920, e12.95 Technology is now a core subject to be studied alongside mathematics within the National Curriculum. Many teachers will therefore be examining the possibilities of cross-curricular

links between the two areas. This book seeks to promote such links by providing teachers with a range of activities to be adapted to school schemes of work.

The content is presented in a style similar to that of the author's previous books such as Mathematical Activities, that is as sets of activities followed by hints and solutions. The twelve chapters provide activities on pulleys and chains, gears, linkages, winding mechanisms, rollers and wheels, cams and ratchets, levers and hydraulic rams and robot designs. The activities concentrate on the making of models and a study of how and why they work, thereby developing the geometry of moving shapes in a practical manner.

The objective of the book is to provide the teacher with an insight into the design of mechanisms as seen through the eye of a mathematician. It is essentially a resource book, allowing the teacher to study particular ideas and select and adapt activities for classroom use. It certainly provided me with new interesting historical and scientific knowledge about mechanisms in addition to presenting me with a challenge in constructing some of the models.

A typical chapter is that entitled, "From Rocking Horses to Steam Engines", which examines the application of the isosceles tra- pezium linkage. Here we are shown how this linkage idea was developed from a mechanism for a rocking horse, via the horse drawn coach

50 Mathematics in School, March 1992



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Cabri-Géomètre, Euclid's Revenge