Caleb Gattegno’s classic mathematical invention involves nothing more than brass pins set in a plywood board and everyday elastic bands. In this guide for teachers, Gattegno advocates the use of Geoboards and outlines some basic activities for students.
Geoboard Geometry Caleb Gattegno Educational Solutions Worldwide Inc. First published in 1971. Reprinted in 2009. Copyright ÂŠ 1971-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-020-2 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com Table of Contents Introduction ................................................................ 1 Exploration.................................................................. 5 Description Of The Geoboards..................................... 7 Example Of Use ..........................................................11 The 25-Pin Lattice Geoboard...................................... 19 Parallelograms .......................................................................19 Area ....................................................................................... 26 Pythogorasâ€™ Theorem............................................................ 29 Isosceles and Scalene Triangles ........................................... 38 Lesson With Polygon Boards ..................................... 53 The Regular Pentagon .......................................................... 53 The Regular Hexagon, Octagon and Dodecagon ................. 54 The Circle .............................................................................. 56 More Systematic Work ......................................................... 62 Animated Geometry Films ....................................................67 Introduction There will be no dispute that it is essential for every pupil to gain experience in geometry before any codification takes place. The term ‘experience’ however can be used with a variety of meanings, and most teachers will think of it here as being Stage A Geometry (experimental drawing, measuring of angles, areas, etc.) as advocated by reformers in Great Britain earlier this century. But the position needs further examination for we do not know how much of what is done in this way is really relevant to geometrical thinking. It is readily accepted that if we actually measure the angles of two or three triangles and find sums that are approximately 180°, it follows that the Euclidean proposition that the sum actually is 180° is made more convincing. Nothing of the sort can in fact be concluded from the experiments. It is only because there are other reasons for believing that the sum is 180° that teachers are ready to encourage their pupils to think that it is reasonable to conclude, from the evidence they obtain, as Euclid did. Such a procedure 1 Geoboard Geometry will not bear criticism. Those who know, for example, that sixty years ago the speed of light in a vacuum was given as 3 x 1010 cm/sec. as found from formulae, and that experimental advances have made this simple number much more complicated, must agree that the propositions of geometry cannot be based upon crude experiments and insufficiently accurate measurements. These in fact are the bases of measurement, not the reverse. In order to avoid these pitfalls we shall not attempt to introduce geometry through physical experiments but by giving experience that is akin to the end product, making the pupils move forward towards a freeing of their minds from the conditions put by the restricted field presented. There is of course more than one approach. We need to see that geometry is concerned with statements about space relations, space being at first the vague set of mental structures that make possible and organise our actions, but slowly gaining independence and becoming those same relations abstracted from experience and considered in themselves. Hence there is a two-way movement from mental structures involving feelings of extension and tension to mental structures in which these feelings become potential or virtual. The organisation of oneâ€™s field of experience consists in the recognition of what in the situation is relevant or irrelevant to the relation singled out. This will be clarified when we come to some examples. 2 Introduction For teaching to be in agreement with the true activity of the geometer and all the time at the level of the pupils, it is again enough to use perception and action in the proper direction. 3 Exploration If each pupil is given a few colored elastic bands and a board on which nails have been fixed, it is obvious that various polygons can be formed by simply stretching the bands over the nails. Free manipulation shows that similar gestures produce similar results as far as perception is concerned, and that these actions can be labelled. Pupils can perceive that a band can be extended over 3, 4, 5 or more nails and that sometimes the figures obtained with different sets of 3, 4, . . . nails create the same impression. 5 Geoboard Geometry Fig. 1 This feeling will be the basis of a recognition of structure which will remain as a mental structure that can be used and easily recalled for action. When children are allowed to play with one of these â€˜Geoboardsâ€™ and a few elastic bands, they discover on their own the set of relations existing in a given situation. The method consists in first allowing time for exploring the various Geoboards described below, and then in establishing a graded exploitation of the situations involved, with the production of statements that express the findings. 6 Description Of The Geoboards These are square pieces of plywood, with on the one hand a lattice background, and on the other a circle; small brass pins are placed at the centres of the squares or at particular points on the circumference of the circle and at the centre. A variety of boards sufficient to cover the ordinary school syllabus have so far been used. These are as follows: three rectangular lattices of 9-, 16-, and 25-nails, (boards with 36 and 49 nails, although mathematically most attractive, become too complicated for young children); the regular decagon, the regular hexagon, the regular octagon, the regular dodecagon, the regular pentagon and the double hexagon. These form for the moment what we shall call a set.* * The set of Geoboards available from Educational Solutions Inc. comprises six boards: 9-, 16-, and 25-pin lattices and the regular octagon, decagon and dodecagon. 7 Geoboard Geometry Fig.2 The pentagon, hexagon and double hexagon boards are not illustrated. Although the 25-pin lattice contains several sub-boards of 9 and 16 nails it is nevertheless advisable to have the three boards available. With the 9-pin lattice we have enough, but not too much, variety. While it holds the pupilsâ€™ interest it does not present them with questions that are too difficult, and it allows them to acquire experience of exploring a mathematical situation. The 16-pin board has no centre of symmetry formed by a nail as in the boards with an odd number of nails. Questions that appear only in special forms on the 9-pin board will require extension, and this board would be valuable for that purpose alone. Of course the pupils will always find more interest and fun in using the 25-pin board once they have 8 Description Of The Geoboards experience of it. Similarly, while the regular dodecagon board contains two regular hexagons, for some purposes it is more convenient to have only the one hexagon marked. The regular pentagon is the least productive of all the Geoboards and is of very limited use, but this is in itself of interest in that it will show that certain situations are easily exhausted while others are very fruitful. With the double regular hexagon, formed of two hexagons having one side in common, questions can be put which cannot be put on any other board. The Geoboards should be given to the pupils to handle. They should be used by the teacher for demonstration only, when a question is to be put to the class by means of gestures. 9 Example Of Use 1 There can here be no question of explaining every application of the Geoboards as it would be impossible to exhaust all that could be done; this text will be restricted to indicating how they can be used and why they are advocated. 2 For the purpose of exploration any board and any number of rubber bands can be given to the pupils. After a few hours of pattern making they will be familiar with stretching the bands over the nails, and will have learnt to contemplate the intermingling of lines. A 9-pin board and a single rubber band can then be given to each pupil. 3 On his own board the teacher makes a figure with the band, asking the pupils to make one on theirs. Usually they make a figure congruent to his. He can then rotate his board about its central nail letting the learners see the various positions the figure can take. They do the same with theirs, watching what happens. 11 Geoboard Geometry Fig. 3 The figure might be a triangle. The teacher will by his actions produce a set of triangles differing from the first only by rotation (which is at once recognised by the pupils even if the word is long to remain unknown). If the teacher himself moves holding the board steady in his hands he is submitting the figure to another displacement (called translation), leaving the triangle conspicuously unchanged. Fig. 4 12 Example Of Use These two gestures are important and it is well to be acquainted with them from the start. Many geometrical notions â€” congruency, similarity, symmetry â€” are in some way connected with them. But they are also important because they introduce at once a dynamic dimension into the situation. A triangle on the blackboard is a static figure; on the Geoboard it yields, merely by its displacement, as many triangles as we want. When a triangle has been formed, a single pull on one side will produce a different figure - whether a trapezium, a parallelogram, a different type of triangle, etc. This gesture must be watched when the teacher makes it and be reproduced by the pupils. Fig. 5 They should also experience the return to the original figure so that links between figures are formed in both directions. If a triangle, for instance, is visible, a parallelogram is virtually there. Conversely there is a potential triangle in the parallelogram. 13 Geoboard Geometry 4 As yet there is no need for labels. The elastic bands will generate figure after figure, and the gestures can be so controlled that a figure that is visible may be reproduced at will, or one figure altered to generate another. Some of these transformations can be obtained either by altering the position of the band or by merely rotating the board about its centre. In this way an organic link would be formed between the two transformations, which are apparently so different. When it is that rotation is operative, and when we must act upon the elastic band, can be examined. Those figures which do not require any change of the band will provide the class of congruent figures, whereas the others are not congruent. This will be established as a certainty long before there is any need to use the term or to make any formal statement. 5 It will soon appear that among the figures formed by means of one elastic band, some, such as squares, are totally unaffected by orientation, and others are visibly affected either to the right or to the left. These orientated figures come to light very early. Though they may not be singled out in ordinary textbooks of geometry they worry pupils, and we should be 14 Example Of Use aware of their existence. Two such figures cannot be made to coincide by rotation or translation. * 6 Once the pupils have noticed these obvious properties of the figures formed and are accustomed to considering how figures are affected by rotation of the board or action upon the band, the teacher can begin introducing labels. When he has made a figure on his board and his class has reproduced it, its name can be written on the chalkboard and copied by the learners into their notebooks. When two or three names are on the chalkboard the pupils can be asked to construct on the Geoboards the corresponding figures, in any order, and to show the result. When the number has been increased to contain: square, triangle, quadrilateral, trapezium, parallelogram, right-angled triangle, isosceles triangle, isosceles trapezium, rectangle, right-angled trapezium questions such as these can be put: The two triangles shown here, for example, cannot be made to coincide by displacement or rotation although they can be seen to be inversely congruent. They are distinguished by their orientation. Fig. 6 * 15 Geoboard Geometry 1 Form all the squares possible on your Geoboard. How many can you form? 2 Form all the right-angled triangles. How many are there of these? The same can be done with all the figures noted. These are simple but profound questions because there are various ways of getting the answer - three of them being by rotation, symmetry and similarity - and this the pupils discover. 3 Find the figures congruent with or similar to a given figure. This exercise will clearly show the distinction between congruency and similarity, and these words can then be written down and used. With other Geoboards we can introduce regular polygons and their names, and form figures that were not possible with the rectangular lattices. With the hexagon board, for instance, we can form rhombi and equilateral triangles.* When circles appear on the board new adventures open up and new elements present themselves, resulting from the position of the nails on the circumference. Thus chords, radii, diameters are seen to have obvious characteristics. The centre is also the centre of symmetry and rotation can be used to establish results or to show what figures look like when upside down or on their * This is illustrated here on the dodecagon board. 16 Example Of Use sides. This training will be valuable in the education of geometric imagination, often neglected. Fig. 7 8 The fact that the nails are placed on the circumference of course limits the possibilities of study, but use of the Geoboards does not preclude the introduction of other teaching aids, such as mathematical films or even colored chalk on the blackboard; this matter is treated in greater detail at the end of this booklet. It should be said here that there is a place, after experience has been obtained with the Geoboards, for drawing, and making patterns in onesâ€™s exercise book. Practice with ruler and compasses will be the source of a new study of geometry â€” not the geometry of Euclidean space but of the piece of paper and these instruments. ***** 17 The 25-Pin Lattice Geoboard A few of the more advanced lessons that can develop one from the other on the 25-pin rectangular lattice Geoboard are considered here. Parallelograms 1 Various parallelograms can be formed, some congruent, others not. Starting from two nails on one side at one corner, and using different colored bands, we can form four parallelograms whose other side will be on the next row of the lattice, i.e. parallelograms which are on the same base and between the same parallels. Their differences can be noted: â€˘ angles increase at one vertex and diminish at the other on the same side; 19 Geoboard Geometry â€˘ while two sides remain constant, the other two, which are equal, increase as we pass from one figure to the next on the right. If we start with a square, we can see that it does in some way belong to the series of four parallelograms; but the notion of identifying a square or a rectangle with a parallelogram requires care and attention and may take some time. Fig. 8 This work can be repeated taking as the common side the two nails in the middle of one side of the board. A new feature, symmetrical pairs of parallelograms, results. If we cease to restrict ourselves to the next row of the lattice many more parallelograms can be obtained which could form the basis for another enquiry. This might be a study of diagonals and their property of bisecting each other. To give such a study its full value, it should be included in a general consideration of diagonals of quadrilaterals. It is already known that with one elastic band we can form: 20 The 25-Pin Lattice Geoboard 1 a general quadrilateral 2 a quadrilateral a little more specialised: the trapezium (of which two special cases are the isosceles and the right-angled trapeziums) 3 a quadrilateral twice trapezium: the parallelogram 4 a parallelogram with right angles: the rectangle 5 a parallelogram with equal sides: the rhombus 6 a square Fig. 9 Conversely, 1 by stretching two parallel sides of a square we obtain a rectangle; 2 by slanting this a little we change it into a parallelogram; 3 by stretching one side only of the square we have a trapezium; 21 Geoboard Geometry 4 and by moving one of the vertices of the rectangle out of the row, a general quadrilateral is produced. 3 In each of these cases two rubber bands can be introduced as diagonals joining opposite vertices. The resulting alterations in passing from one figure to the next in the series in specialisation or generality can be examined. Pupils will find no cause for comment in the case of the general quadrilateral nor in those of the trapeziums that are not isosceles for they will see that the diagonals are not always equal for all cases of the figure and positions on the board (and there are several possibilities for position on the 25-pin board). The teacher may now place two bands on his board so that each stretches from one vertex of a lattice square to the vertex opposite. The two lengths can be recognised as equal. Fig. 10a If he then places one length as the continuation of the other (or stretches one to double its length), the nail that was at one end of the first line becomes the centre of the new line produced. 22 The 25-Pin Lattice Geoboard Fig. 10b Putting another band on the board with the same nail as its mid point, he has two lines clearly bisecting each other. Fig. 10c By placing another band on the four end nails of the two lines, he constructs a parallelogram (or a rectangle or a square according to what was done, but nothing else in the 9-pin Geoboard, also a rhombus on the 25-pin one). 23 Geoboard Geometry Fig. 10d The pupils, on constructing a parallelogram on their own boards, will recognise that their sequence of gestures was the inverse of those of the teacher, and will conclude that the diagonals of a parallelogram are unequal but bisect each other. If they are also equal the figure is a rectangle. But if the diagonals of a quadrilateral are equal and do not bisect each other, the figure is in no way special unless it is symmetrical. It will not be easy to show that for a square, in addition to equality and bisection of diagonals there is perpendicularity. If we return to the 9-pin board we find there are only the unit squares and two others that can be constructed: the one formed round the eight external nails, and that round the four that are the mid-points of the sides. In these it is obvious that the diagonals are at right angles. 24 The 25-Pin Lattice Geoboard Fig. 11 If we transfer the figure to the 16-, and 25-pin boards new gestalts are formed and the property is more easily seen. Since we do not use set-squares or protractors, we shall have to make use of rotation to ensure that the perpendicularity of the diagonals of a square to each other is rigorously established. If in fact we place our elastic bands as diagonals of a square on the board, a rotation of a quarter of a revolution round the centre of the square will replace one of the diagonals by the other. This will be our definition of perpendicularity. 4 The hexagon board (or the regular dodecagon) is helpful for finding rhombi and their other properties. * A classification with chalkboard drawings and notes about diagonals will give the following table, which can be considered as our pupils’ acquired geometrical knowledge: * See page 21 25 Geoboard Geometry Area With the material so far obtained the areas of parallelograms can be studied. If the smallest square that can be formed on the lattice is taken as the unit of area, the areas of various easily formed figures can be calculated: 1 right-angled triangles half the square; 2 rectangles composed of two contiguous squares; 3 right angled-trapeziums, etc. These exercises are easily given and solved. All that is required is to construct rectangles with a rubber band, mark the squares contained with bands of different colors, and count them. 26 The 25-Pin Lattice Geoboard In the case of a fraction, (a half, for example) the two triangles form a square, their equality having been recognised by their congruence. Fig. 12 A parallelogram with a side on two consecutive nails and the parallel side on the next row of the lattice, has an area equal to one unit. Proof of this may require several steps, the main point being that in every parallelogram each diagonal bisects the area. Fig. 13 27 Geoboard Geometry Once this is recognised it only remains to see that conversely if we start with a square and halve it by means of a diagonal, twice this half (a triangle) will yield a parallelogram. If we take the other diagonal, we form another triangle whose area is both half that of the square and that of the parallelogram, i.e. half a unit, and its double yields another parallelogram with area one unit. Fig.14 We have here in particular the proof that parallelograms which have the same base and are between the same parallels are equal in area. At the same time, by halving, the proof of a similar proposition for triangles may be obtained.* Many problems on area can be dealt with as corollaries of the previous study. For example, the proposition that if two figures are similar their areas are in a ratio equal to the square of the ratio of similarity (only easy cases naturally being considered). * It may be remarked in passing that the Geoboards can be used for the study of fractions in arithmetic, the intuitive support being, in the present case, areas. 28 The 25-Pin Lattice Geoboard For example, in the illustration the areas of the larger and smaller parallelograms are 4 units and 1 unit respectively. The lengths of two corresponding sides are 2 units and 1 unit (taking unit length as the distance between adjacent nails on the board) thus the square of the ratio of these lengths is 4 : 1. By forming diagonals we obtain similar triangles whose areas are two units and and half a unit respectively, and whose corresponding sides have lengths in the ratio of 2 : 1. The ratio of the areas of these new figures is 4 : 1 and the square of the corresponding sides is also 4 : 1. Fig. 15 Pythogorasâ€™ Theorem 1 We already know from the 9-pin board that the middle square formed round the 4-mid points of the sides is equal to two squares of the lattice unit. (Fig. 16a) If we insert a rightangled triangle such that the unit squares are squares on two sides, the original square lies on the hypotenuse. (Fig. 16b). 29 Geoboard Geometry Fig. 16a Fig. 16b 2 On the 16-pin board we can show, in addition to this figure, the squares on the right-angled triangle of sides one and two unitsâ€™ length. Here again the square on the hypotenuse is easily proved to be equal to five unit squares by finding the sum of the squares on the other two sides, of four units and of one unit respectively (Fig. 17a). It is not easy to describe the proof, but as before the converse is seen at once. Fig. 17a 30 The 25-Pin Lattice Geoboard We form a square round the third pin on each side of the board, always moving in the same direction. If we insert a triangle so that its hypotenuse is on one side of the square, the other sides are one and two unitsâ€™ length respectively, and by forming on them the corresponding squares we get one square of one unit of area and one of four. The original square lies on the hypotenuse of the constructed triangle. Its area can be seen to be composed of one central unit square and four right-angled triangles each equal to half a rectangle composed of two unit squares (Fig. 17b). Hence its area is five units. Fig. 17b 3 On the 25-pin board besides the two cases given above we can consider right-angled triangles whose sides are two and two, or one and three units of length: the same actions will yield similar results. But we can also show that if a triangle has an obtuse angle instead of a right angle then 31 Geoboard Geometry the square on the side opposite the obtuse angle is equal to the sum of the squares on the other two sides plus twice a constructed rectangle with dimensions measuring the length of one of the shorter sides by the length of the projection on that side of the other short side. This is more easily seen through the diagrams that follow than described. ***** In a triangle ABC with an obtuse angle at C, opposite which is the longest side AB, the following relationship between the squares on the sides obtains: a = b + c + twice a rectangle measuring the length of one of the shorter sides by the length of the projection on that side of the other short side. 32 The 25-Pin Lattice Geoboard Fig. 18a Let us project side BC on AC to obtain the length CD1. Fig. 18b Constructing a rectangle measuring AC by CD1, we obtain the rectangle d1. Hence a = b + c + 2d1 (Here a = 5 units of area, b = 2 units, c = 1 unit, d1 = 1 unit). 33 Geoboard Geometry Fig. 18c Now let us instead project the other side AC, on BC, to obtain the length CD2 Fig. 18d We could construct a rectangle, d2, measuring BC by CD2. Hence a = b + c + 2d2. 34 The 25-Pin Lattice Geoboard Fig. 18e It should be noted however that this second construction cannot in fact be executed on the Geoboards owing to the restriction imposed by the placing of the nails on the lattice. Nevertheless d2 can also be recognised as being equal to 1 unit of area. If in the triangle ABC in Fig. 18 a we considered the acute angle at A, a relationship could be established between the square on the side opposite this angle and the squares on the other two sides. In the diagram, b = a + c MINUS twice a rectangle measuring the length of one of the other sides by the length of the projection of the remaining side on it. Thus by projecting AB on AC we produce the length AD1. The constructed rectangle d1 measures AD1 x AC and b= (a + c) - 2d1 where b = 2 units of area, a = 5, c = 1 and d1 = 2. 35 Geoboard Geometry Fig. 18f In the alternative situation the side AC is projected on AB to produce, by construction, the rectangle d2. Once again this particular arrangement cannot be executed on the Geoboards owing to the restriction of the lattice. However the classroom can consider what the area of rectangle d2 would be. Clearly if the other acute angle (at B) is considered, a similar relation obtains between the area of the square on the side opposite to it and the sum of the other squares. 36 The 25-Pin Lattice Geoboard Fig. 18g The theorem of Apollonius can also be examined here, but only with respect to certain triangles owing to the lattice and the limited size of the Geoboards. In the triangle ABD having the line BC as one median bisecting the side AD, Fig. 19 37 Geoboard Geometry the sum of the squares on sides AB and BD can be seen to be equal to twice the square on half the third side AD plus twice the square on the median which bisects the third side. In the illustration the squares with the continuous outline are together equal in area to those with the dotted outline. On this triangle, the cases when AB and BD are the third side respectively can then be examined, followed by a study of entirely new triangles. This relationship between the sum of the squares on two sides of a triangle and the sum of twice the square on half the third side and twice the square on the median bisecting that side will be found to hold for all triangles. ***** Isosceles and Scalene Triangles 1 If we start with the 9-pin board we can form two sets of four isosceles triangles which are not right-angled but which are congruent within the set. One member from each of the two sets is shown here: 38 The 25-Pin Lattice Geoboard Fig. 20a Fig. 20b Considering one of the triangles only, we can insert three rubber bands either as altitudes or medians. In both cases they appear to be concurrent. Fig. 21a Fig. 21b We have first to establish that the lines we called â€˜altitudesâ€™ are truly perpendicular. This is proved by showing that, on reproducing on a similar board one line representing one of the altitudes and another representing the corresponding side of the triangle, and rotating the board a quarter of a revolution round the centre point, one of the lines exactly takes the place of the 39 Geoboard Geometry other. After one quarter of a revolution the altitude lies exactly where the side it intersected was. Fig. 22a Fig. 22b Returning to the first board we can see that in the isosceles triangle the three altitudes all pass through the same point. 2 We can prove that the other lines, joining nails, bisect the corresponding sides. We insert a band to construct a parallelogram on four nails, one being at each end of one side of the triangle; a third at the point of intersection of the medians; and the fourth at a point on an extension of the median such that the distance from the origin of the median to this point is twice that from the origin to the point of intersection of the medians. 40 The 25-Pin Lattice Geoboard Fig. 23 Since we know that the diagonals of a parallelogram bisect each other our proof is obtained. Moreover, we can see that as the diagonal of the parallelogram which is not a side of the triangle is equal to the length of the median from the vertex to the point of intersection of the medians, the portion from this point to the side of the triangle is half the longer portion. It follows that the three medians cut each other in the ratio of one to two with respect to their parts, or of one to three if we compare the smaller portion with the whole. It follows that, by using parallelograms as above, the medians of a triangle are concurrent. 3 On passing to the 16- and 25-pin boards we can find new triangles, not isosceles, which will allow the placing of altitudes. Here again by using rotation, we can prove the perpendicularity 41 Geoboard Geometry of the altitudes to the corresponding sides, and it can also be seen that the three altitudes are concurrent. Fig. 24a Fig. 24b It can further be observed that in acute-angled triangles the altitutes are concurrent inside the triangle; in obtuse-angled triangles the point of concurrence lies outside. **** All this is, of course, only a gathering of experience which will serve for understanding certain propositions. Since we are concerned with watching figures, the facts we want to establish will be seen by contrast with what happens to other lines: for instance, two medians and one altitude in scalene or rightangled triangles do not concur. We must bring into relief the fact that certain things occur when certain lines are considered in certain situations; teachers will find this quite easy when they have themselves played for a while with the Geoboards. 42 The 25-Pin Lattice Geoboard It is remarkable how many figures can be formed with one board. There are about two hundred different propositions of interest on the 25-pin board and more than fifty on the 9-pin board. Not all of them can be taken further or seen in relation to later developments of geometry, but by introducing children to an aid which is eloquent in their language - that of perception and action - we can enable them to see the facts that cluster round the results of certain gestures: parallelograms, concurrency, orientation, etc. **** 4 Intercepts theorems in triangles and trapeziums can follow, as a result of the placing of rubber bands as parallel lines, along the sides of the board or along the slanting rows of nails, and using others as transverse lines. This arrangement provides many questions of interest. A special case is that which appears as either a triangle or a trapezium, with a line joining the midpoints of two sides. Fig. 25a Fig. 25b 43 Geoboard Geometry These are important facts of geometry and are readily proved by the insertion of other rubber bands to form parallelograms from which conclusions already met can be drawn. The mid-point theorem can in its turn be used to prove that in the general case parallel lines cut, on transverse lines, segments that are in the same ratio. 5 We have found with rectangular lattices that the most immediate geometrical facts are concerned with shapes, distances and areas. As we shall also need angles we can introduce the notion in the following way. Starting with a rubber band forming a line along one side of the board: â€˘ we can place another band at one end and stretch it on another nail, â€˘ remove it from this to the next nail above it on the perpendicular to the initial line, â€˘ and so on to all the nails on the perpendicular. Fig. 26 44 The 25-Pin Lattice Geoboard The process can be repeated again and again so that it can be seen that at the common nail something varies, and that the variation is continuous, with halts at the openings marked by the nails. We can then leave different colored bands in position so that the successive angles can be seen. When we remove the bands one at a time, certain angles disappear while others become more obvious. Since we can repeat the process at the other end of the original rubber band, or stretch it to other points on the side, or place it at random on any pair of nails, we discover: â€˘ that we need two lines to form an angle and that these must be so placed that they have one point in common (or vertex); â€˘ that the same vertex and one line give as many angles as we want; â€˘ that the figure so formed is deeply related to rotation around the vertex. 6 Taking advantage of the lattice, we can form equal angles by placing rubber bands in positions that show the degrees of freedom of the initial choice. We see angles with one side common but with different vertices on it, and the other sides 45 Geoboard Geometry parallel, or anti-parallel (meaning symmetrical of the parallel with respect to an appropriate line). Fig. 27 These are, with respect to the board, equivalent in that it is a matter of indifference how we choose the vertex on the line and the direction, right or left, in which the other band is stretched. This is another important experience for our pupils. Fig. 28a 7 From this we can go on to; 46 The 25-Pin Lattice Geoboard 1 corresponding and alternate angles; 2 parallel lines and equality of angles in situations where there are parallel lines and corresponding angles. Fig. 28b Fig. 28c We can also see that: 3 adjacent angles with their exterior sides on a line are supplementary; 47 Geoboard Geometry 4 and by drawing the perpendicular we can introduce right angles and complementary angles. Adjacent or not, as long as they are related properly, angles formed on the board can be complementary or supplementary. Fig. 28d 8 We can now: • form triangles by joining points on the sides of an angle, • show what we mean by ‘common angle’, • form triangles with angles equal, • and, in the case of right-angled triangles, find that the other two angles are complementary. Inserting a band at one of the vertices which is not the right angle, we can use what we have learnt so far to see that the angle at the third vertex is alternate and equal to the one that is complementary to the second angle (Fig. 29a). 48 The 25-Pin Lattice Geoboard Fig. 29a Fig. 29b If we now put the same question, but with respect to a triangle which is not right-angled, we have enough experience to see that the same construction gives a slightly different result, this time the three angles of the triangle being involved (Fig. 29b). Since this can be repeated with any number of different triangles, the fact that the sum of the angles of a triangle (on the boards) is equal to two right angles can be considered as established. 9 Taking any quadrilateral, or indeed any convex polygon and dividing it into triangles, we can, following the same approach, gain experience of the sum of the interior angles of a polygon (Fig. 30a). 49 Geoboard Geometry Fig 30a 1 Any convex pentagon can be divided into three triangles, the sum of each of whose angles is two right angles. Therefore the sum of the interior angles is six right angles Fig. 30b 2 Alternatively in the pentagon, or again in any convex polygon, as many triangles can be formed as there are sides to the figure by utilising a point within the figure. It can be seen that the sum of the 50 The 25-Pin Lattice Geoboard interior angles of the polygon is equal to two right angles for each side (since there are two right angles for each triangle) LESS the sum of the angles around the point forming the third vertex for all the triangles, i.e. 4 right angles. In the case of the pentagon the result is again 6 right angles; but the general rule can also be stated: that the sum of the interior angles of a convex polygon is equal to 2n - 4 right angles, where n is the number of sides of the polygon. For the exterior angles, there is no difficulty in forming at a chosen point a star whose rays are parallel to the sides of the convex polygon, when it will be seen at once that their sum, whatever the number of sides, is always 4 right angles (Fig. 30c). Fig. 30c 3 A convex polygon produced on the board in this manner shows the sides produced, and enables us to examine the sum of the exterior angles of a polygon. Considering the continuous lines only we see there are supplementary adjacent angles at each vertex. The sum of the angles, both exterior and interior, of a polygon is equal to 2n right 51 Geoboard Geometry angles, where n is the number of vertices and hence also of sides. But as the sum of interior angles alone has been shown to be equal to 2n - 4 right angles, the sum of the exterior angles alone must be 4 right angles. Fig. 31 It is also easy to see that in any triangle the angles and the opposite sides (or as they are often called, the â€˜correspondingâ€™ sides) increase or decrease together and this will relate magnitude of sides and angles in triangles. Several corollaries of these facts can also be made evident. In an isosceles triangle angles at the base are equal; in an equilateral triangle (three times isosceles on a hexagon board for example) the three angles are equal, and equal to two thirds of a right angle. 52 Lesson With Polygon Boards The Regular Pentagon The pentagon can provide, when one band is stretched over two nails forming one of its sides and a second band stretched over another pair of nails forming the parallel diagonal, two lengths in the ratio of the golden section (Fig. 32a). Fig. 32a Fig. 32b The examples here are illustrated on the decagon board. 53 Geoboard Geometry Joining each nail to the next, we form the convex regular pentagon. Joining each nail to the next but one we form the stellar regular pentagon (Fig. 32b). The ratio of the sides of these two pentagons is the golden section. The Regular Hexagon, Octagon and Dodecago Fig. 33a Joining first adjacent nails, then every other nail on the hexagon board yields the convex regular hexagon on the one hand and two equilateral triangles on the other (Fig. 33b). Thus a difference from the example provided by the pentagon boards can already be seen. 54 Lesson With Polygon Boards Fig. 33b (Note that this is illustrated here on the dodecagon board.) In the case of the octagon and the dodecagon the situation becomes more complex. For while all convex cases are straightforward, stellar polygons can be formed, in addition to the subdivision into two squares in the one case and into two hexagons in the other. The systematic approach of joining consecutive vertices, then skipping 1, then 2, then 3 etc. nails, will allow the pupils to obtain all the possible figures on each board. The use of bands of different colors makes the pattern obvious. The example of the octagon board is illustrated here. 55 Geoboard Geometry Fig. 34a Fig. 34b Fig. 34c Fig. 34d The Circle Facts about the circle will emerge quite soon. For example, the diameter perpendicular to a chord bisects the chord, the arc, and the angles subtented at the centre or at the circumference. Fig. 35a 56 Lesson With Polygon Boards On the hexagon board we could form an equilateral triangle from two radii and one side. (Once again, the examples are illustrated here on the dodecagon board.) Fig. 35b Any rotation of any multiple of one sixth of a revolution will make this triangle take the place of another such triangle. By placing other colored bands on three nails each time we generate a triangle by rotation, and see that the hexagon is made of six equilateral triangles. The angles at the centre have thus been evaluated. Fig. 35c 57 Geoboard Geometry Two consecutive such triangles form a rhombus, and there are six rhombi which can be made evident by the use of the different colored bands. Hence in each of these rhombi the angle at the centre is one third of a revolution. Fig. 35d The diagonals of the rhombus are perpendicular to each other since one is a chord and the other the bisecting diameter or radius. Fig. 35e By forming a large equilateral triangle, using three vertices of the hexagon as above, we find a figure which is such that if we rotate the board by one third of a revolution the triangle will 58 Lesson With Polygon Boards coincide with itself, the three angles and the three sides thus being respectively equal. (This property could have been one of the first to be discovered through playing with the hexagonal or dodecagonal lattices; and triangles that coincide with themselves by a rotation round their centre of one third of a revolution or its multiples we can call either ‘equilateral’ or ‘equiangular’.) With the small equilateral triangles, we must measure the side of the hexagon with the radius, and this will require a different tool from the Geoboard and rotations. We shall soon see how we can build up knowledge that will allow us to overcome these difficulties. Fig. 35f Starting with a diameter and forming a triangle with one side the diameter and the opposite vertex on the circumference, we see on all circular boards that the angle at the circumference is a right angle. 59 Geoboard Geometry Fig. 36a To prove it we insert on the dodecagon board a band along the radius joining the centre to the vertex opposite the diameter. This line sub-divides the triangle into two isosceles triangles whose angles at the base are equal. (On the 9-, 16-, and 25-pin lattice boards this fact resulted from the symmetry of the lattice and the gestalt formed by the triangle on it.) Hence the angle at the vertex is half the sum of the angles in the triangle. Fig. 36b Mere displacement of the rubber band so that the previous diameter becomes a pair of radii not necessarily on a line, will show that the same argument leads to a more general statement: 60 Lesson With Polygon Boards the angle at the circumference is half the angle at the centre subtending the same arc. Fig. 36c But it is easy to see that, keeping the arc, we can move the point at the circumference to different positions and find the same relation. Hence all angles formed on the same segment are equal; on opposite segments they are supplementary. Fig. 36d The figure formed by two angles on the circumference, on the same chord, but on either side of it, is a quadrilateral, a so-called â€˜cyclic quadrilateralâ€™. Hence the opposite angles of a cyclic quadrilateral are supplementary. 61 Geoboard Geometry Fig. 36e All this has shown how we can relate results by merely inserting various colored bands into the pattern or removing them from it. We have of course been concerned only with these results that are needed by pupils who are studying geometry in the orthodox fashion. Many other facts are to be found in using the Geoboards and we leave the reader to enjoy discovering these for himself. More Systematic Work In what has so far been said, the Geoboards have played an important role, but teachers will probably supplement the findings of their pupils through writing, through colored diagrams on the blackboard, and through geometric drawing and measurements. It is clear that once we are aware of some relation on one of the Geoboards, we shall want to make sure that its truth does not 62 Lesson With Polygon Boards depend on the special choice of distances and angles on the lattice. We can then take a piece of paper and instruments and produce similar situations to those on the Geoboards, but with arbitrary angles and distances. If the arguments follow exactly the same lines and the facts are again established, we can say that they are true for these new conditions - i.e. paper and instruments. The fact will need to be stated in a way comparable to that generally used and we gradually learn so to express them. This is the training in formal proof. Certain facts are obvious and there will be no need to state them. In such cases it will be of value to draw figures in which the fact in question becomes far less obvious and presents a challenge to the mind. Thus the need for a statement of the obvious, which is not always obvious, is created. For example, if lines meet on the Geoboard or the paper, the point of intersection can be seen. But if they are made to meet outside the field then one must imagine that they do so and find reasons which will convince others of the fact. Fig. 38 63 Geoboard Geometry It will be clear that with pupils whose experience is being gathered rather rapidly the teacher will have a different function from that of providing the experience in books and notes. He will need to watch that this experience is co-ordinated, integrated, discussed, expressed and stored. He will not say ‘prove that . . .’, but, ‘does this line or this figure show this or that?’ And then, ‘Why?’, so that instead of there being insistence on the formal expression of knowledge, the dynamic approach to situations becomes second nature, every question being viewed as a challenge, a rider, as it certainly once was in the past for someone. Children will be able to demonstrate their understanding first by inserting rubber bands in the appropriate way to indicate what they mean, then in words, and then by writing in an agreed notation; for example: Fig. 38 • for parallelogram; 64 Lesson With Polygon Boards • • (1, B) (2, A) to indicate the joining of two points on the board — in this case the nail to the right of the origin and the nail above the origin; for hence, etc. As these notational elements can be introduced when the need arises and for the purpose of making communication easier and univocal, each teacher will suggest his own solution, the merits of which can be discussed after experiment. A logical argument will not be achieved at once. It is only when pupils feel the need for it that they make the effort to acquire the habit of disciplining their minds so that they express their awareness in terms that can be generally acceptable. A technique which is fruitful consists in translating the message formulated by a writer into actions different from those he intended, when the words or symbols used cover more classes of objects than he meant to suggest. If for instance a child says ‘polygon’ instead of ‘quadrilateral’, and we draw a hexagon as he goes through his argument, he will see the need to correct his terminology to exclude hexagons. By working patiently on all insufficiently accurate statements related to a given situation, teachers can give their pupils a sense of what constitutes adequate expression of thought, and this is education in rigorous thinking. There is need for understanding in this important matter. Children will certainly acquire through perception and action what Geoboards or other techniques have to offer. If we are careful we can develop in them habits of communication, and 65 Geoboard Geometry give them a progressive sense of how notational communication corresponds to the actual steps of the mind. Since these are within the pupils’ reach, all we have to ensure is that our requirements do not exceed their awareness. In particular, we must refrain from introducing writing too soon or when too much is to be retained. After figures have been recognised and produced with rubber bands on the boards, drawn in pencil and on the blackboard, or in other material (threads in embroidery for instance), the words written and produced as a sound will be identified with a varied and deep experience. When it reappears, it will do so with all its dimensions and will suggest to the mind a number of possible connections. Every time a new situation is mastered, the relevant speech material can be introduced and used as a field for question and answer, first in conjunction with the concrete material and afterwards without it. There is no doubt that children are tremendously uplifted by their success in this field and that their minds are freed and their energy made available for further new adventure.* The reader may be interested to know of other writings by C. Gattegno on the use of his Geoboards. These have been published in various books and at different times and are listed below. A list of Animated Geometry films is also given, for a study of geometry through another medium. 1 For the Teaching of Mathematics, Vol. I * In Gattegno Mathematics Text-book 7 readers will find a more detailed study of the 9-pin Geoboard as well as additional material not developed in this booklet. 66 Lesson With Polygon Boards pp. 65-67. ‘Some problems involved in the teaching of mathematics’ 2 For the Teaching of Mathematics, Vol. II pp. 81 et seq. ‘Three dimensional vision and the use of the senses’ pp. 102 et seq. ‘The idea of dynamic patterns in geometry’ pp. 109 et seq. ‘Teaching mathematical films’ through 3 For the Teaching of Mathematics, Vol. III pp. 91 et seq. ‘Multivalent materials’ Animated Geometry Films All films are silent, 16 mm, black and white, 3 minutes run. * in color by J. L. Nicolet Three points determine one circle * Circles tangent to two concentric circles * Contact point of parallel tangents to circles * Subtended arc A given line seen at a given angle 67 Geoboard Geometry Angles at the circumference Internal bisectors of a triangle External bisectors of a triangle The construction of the regular pentagon The golden section and the regular pentagon Triangle formed from sides of regular polygons Hypocycloid motion with circles in a ratio of 1:2 Two given circles seen under equal angles The strophoid and the golden section Poles and polars in the circle Generation of an ellipse I Locus of vertex of right angles tangent to an ellipse Generation of an ellipse II Generation of a hyperbola Generation of a parabola Another generation of a parabola Common generation of conics by C. Gattegno Extensions of Pythagorasâ€™ Theorem Sections of the cube Generation of some plane curves * Sections of the cone 68