If Pythagoras had a geoboard…

Ciak PA Em ann A IF PYTHAGORAS HAD A GEOBOARD …. By WILLIAM A. EWBANK Toylor University Upland, Indiana use a model at least 10 x 10 (i.c., 100 nails). Or one could use four smaller boards together, provided the borders are one-half the standard nail spacing (fig. 1). THE theorem of Pythagoras is an old . . . . . . favorite of junior and senior high mathematics teachers, and it lends itself to much interesting discovery. This article dezeribes an investigation into the famous theorem, using geoboards. “\ mathematician who developed the Cuizenrire-Gattegno system of colored rods. The geoboard (or nailboard) is a flat board . . . . . . . into which nails or panel-pins have been driven in à regular pattern. Geometric . . fgures can then be formed, using rubber hd , . e bands. There are three types of geoboards in use: (11 the square grid geoboard, (2) the eıreular geoboard, and (3) the isometric eri geoboard. This investigation used (1) wid 13), which are analogs of squared paper and isometric paper, respectively. However, unlike paper, the geoboard is tin a manner of speaking) a dynamic device, and this I have found to be a . . . » . First, a word about the geoboard. The invention of the geoboard has been credited to Dr. Caleb Gattegno, the British . « . . . . . . . e . . . . . . . . . . . . . . . . . . . à ° . . . . . . . . . . ° . . . 4 Fig. 1 On this type of geohoard the arca of any simple closed curve can easily be found (that of a nonsimple closed curve is a much harder task). It is best done on the subtractive prineiple: by surroundobtainable from several educational suppliers (see listing at the end of this article) or can easily be homemade. Howing the figure with a rect rete (nr a polygonal figure with cach angle one or three right angles), as shown in fig. 1. The area of quadrilateral (a) is given by subtracting from the area of the square surrounding it (4 x 4, or 16), the sum of the areas of triangles (b), (cj), (dj, and (e). These are 15(3 x 2), M(2 x 1), MIZ x 1), and 12(3 x 1), respectively, or a total of 7 units. Thus the quadrilateral (a) has area 16 — 7, or 9 units. A facility with calculations like this is essential for a satisfactory investigation into the theorem of Pythagoras. Using this method, the student can set ‘ver. for this investigation, it is best to up arrangements Jike those on the other strong point in its favor in the classroom. I do not know of any other teaching li| «vite at once so simple and so versatile. At one end of the scale, it is attractive to kindergarten children (Liedtke and Kieren 1970), and at the other, it can challenge the senior mathematics major in “lege (Buckeye, Ewbank, and Ginther 1971), sy far the best-known geoboard is the “are grid model. It is inexpensive and Matbemalics lacher March 1978

IF PYTHAGORAS HAD A GEOBOARD... Author(s): WILLIAM A. EWBANK Source: The Mathematics Teacher, Vol. 66, No. 3 (MARCH 1973), pp. 215-221 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/27959243 . Accessed: 19/09/2013 10:32 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Mathematics Teacher. http://www.jstor.org This content downloaded from 192.87.31.20 on Thu, 19 Sep 2013 10:32:11 AM All use subject to JSTOR Terms and Conditions

IF PYTHAGORAS HAD A GEOBOARD . By WILLIAM A. EWBANK Taylor University . use a model at least 10 x 10 (i.e., 100 nails). four Or one could use smaller boards together, provided the borders are Upland, Indiana one-half the standard nail spacing (fig. 1). THE theorem of Pythagoras is an old favorite of junior and senior high mathematics teachers, and it lends itself to much interesting discovery. This article describes an investigation into the famous theorem, using geoboards. First, a word about the geoboard. The invention of the geoboard has been credited to Dr. Caleb Gattegno, the British mathematician who developed the Cuisenaire-Gattegno system of colored rods. The geoboard (or nailboard) is a flat board into which nails or panel-pins have been driven in a regular pattern. Geometric figures can then be formed, using rubber bands. There are three types of geoboards in use: (1) the square grid geoboard, (2) the Fig. 1 circular geoboard, and (3) the isometric grid geoboard. This investigation used (1) and (8), which are analogs of squared paper and isometric paper, respectively. On this type of geoboard the area of any simple closed curve can easily be found (that of a nonsimple closed curve However, unlike paper, the geoboard is is a much harder task). It is best done (in a manner of speaking) on the subtractive principle: by surrounda dynamic device, and this I have found to be a ing the strong point in its favor in the classroom. polygonal figure with each angle one or I do not know of any other teaching three right angles), as shown in fig. 1. The figure with a rectangle (or a device at once so simple and so versatile. area of quadrilateral (a) is given by sub- At one end of the scale, it is attractive tracting from the area of the square surto rounding it (4 X 4, or 16), the sum of the kindergarten children (Liedtke and Kieren 1970), and at the other, it can areas of triangles (b), (c), (d), and (e). challenge the senior mathematics major in These are 12(3 x 2), %(2 x 1),12(3 x college (Buckeye, Ewbank, and Ginther 1971). 1), and 44(3 x 1), respectively, or a total of 7 units. Thus the quadrilateral (a) By far the best-known geoboard is the has aréa 16 — 7, or 9 units. A facility with square grid model. It is inexpensive and calculations like this is essential for a obtainable from several educational supsatisfactory investigation into the theorem pliers of Pythagoras. (see listing at the end of this article) or can easily be homemade. How- Using this method, the student can set ever, for this investigation, it is best to up arrangements like those on the other March 1973

three boards (fig. 1) to illustrate, with the right triangles shown: top right, 1? + 1? = 2; lower left, 12 + 22 = 5; and lower right, 2 + 2 = 2?. An arrangement with a “skewed” right triangle on a 10 X 10 geoboard is shown in figure 2. The areas of the squares can be calculated to be 2, 8, and 10. Fig. 3 avoid the labor of marking out the lines, it is recommended that a piece of isometric graph paper be taped to the board. With this, the nails can be driven through the intersections at desired spacing (%” is convenient), and the paper then removed. With the board in hand, a snag arises Fig. 2 which, at first sight, seems rather formidable: You cannot make a square on If doubt arises as to whether a certain triangle is, in fact, a right triangle, and if the isometric geoboard! This makes it impossible to check on the area of the the students have had enough experiences square on the side of any triangle, and establishing the principles as illustrated difficult to work out any area in square above, they can use the converse of the units. theorem of Pythagoras to determine what Now the basic unit of shape on the sort of triangle they have made. Thus, isometric board is the equilateral triangle. if the area of the square on the longest So why not use the area of the smallest side of a triangle is equal to the sum of equilateral triangle as our unit of area? the areas of the squares on the two shorter Figure 4 shows, with triangles of area 1, sides, the triangle must be a right triangle. 4, and 9 and a hexagon of area 24, that Figure 3 illustrates how a situation of this system works well with simple figures. inequality In figure 5, the isosceles obtuse triangle can be determined, showing that because 1 + 2 < 5, the first triangle has area 1, because it is half of a rhombus is obtuse, and because 2 + 5 > 5, the of area 2. In figure 6 the obtuse triangle second triangle is acute. has area 2 because it is half of a paral- The investigation now moves to the lelogram of area 4. In this way the area isometric geoboard. The isometric board of any simple closed curve on the isois not nearly so well known. At the time metric geoboard can be found. For exof writing I know of only one supplier in ample, the area of quadrilateral the United States (Math Media), but it figure 7 can be found by (a) in subtracting is fairly easy to make a board for yourfrom the area of the outside equilateral self. As it is important that the nails be triangle (side 6, area 36), the sum of the as accurately lined up as possible and to areas of triangles (b), (c), (d), (e), and Mathematics Teacher

March 1973

(f). This gives 36 — (3+5+3+44+1), a geoboard. So we must resort to paper and pencil. From the numerical relationor 20. To check this result, the quadrilateral can ship be decomposed into the triangles (p), (q), (r), and (s), which have area 9 + 4 + 4 + 3, or 20. Although squares cannot be made on the isometric geoboard, rectangles can be made, also right triangles (fig. 8). The three rectangles in this diagram are congruent, and each is of area 15. The diagonals divide them into congruent right triangles. Whenever right angles are formed, one arm must lie on one of the three main lines of nails, i.e., one of the axes of symmetry of the system. The reader is invited to attempt to form right angles in other positions, and to satisfy himself that it is not possible. We are now ready to investigate Pythagoras further. Since it is possible to make right triangles, but impossible to make squares, why not try equilateral triangles on the sides of a right triangle? Figure 9 shows such an arrangement, and calculation gives the areas of the equilateral triangles to be 3, 9, and 12. So far Pythagoras, duly adapted, still holds we can derive: ka” = kb” + ke’, where k is a constant. We shall now establish a formula for the area of a regular n-gon of side a. Using the regular pentagon as model (fig. 12), we find: 180 a? 180 aa area AABC = 2 9 cot 5 “4 cot 5 a? 180 area pentagon = 5| — cot —— 4 5 na? . area regular n-gon of side a = a 180 © t ra = Ka? where K is constant for the type of polygon under consideration. Since Ka? = Kb’ + Kc the theorem of Pythagoras must be true for any given regular n-gon, and not just for squares. good! What about figure 10? Does it work here? Since equilateral triangles will tessellate to form regular hexagons, perhaps Pythagoras can be adapted as follows: The area of the regular hexagon on the hypotenuse of a right triangle 1s equal to the area of the regular hexa- / ou! ra == — |A eee ; + (Py À Lis a4 gons on the other two sides. Figure 11 shows that this does appear to be true, though of course one illustration does not prove the rule. Let us assume that Pythagoras holds good for equilateral triangles, squares, and regular hexagons. Does it hold good for any regular n-gon? Due to the nature of the geoboard, this cannot be checked, as only the above three regular polygons can be formed on 218 Mathematics Teacher j j B O a Q 2 Fig. 12 What about nonpolygonal figures, e.g., semicircles? Is the following statement true?

The area of the semicircle on the An example of similar rectangles simhypotenuse of a right triangle 1s equal to the sum of the areas of the semicircles on the other two sides. ilarly placed is given in figure 14, and of Just as above we found Pythagoras was true for any given regular n-gon, since the areas of regular n-gons are proportional to the squares of their sides, so here, all we scalene triangles in figure 15. a e e a d a > . e re ra 4 > have to do is to show that the areas of semicircles are proportional to the squares of their diameters. This is clearly so, since the area of a semicircle of diameter a 2 would be rà Thus we have stg 8 So far the adaptations of the Pythagorean theorem have been true only for figures that are similar. Erect nonsimilar rectangles on the sides of a right triangle, and obviously there will be no area relationship, except by chance. Even if the rectangles chosen were similar, e.g., with ratio of sides 2:1, their areas may not satisfy the Pythagorean relationship. This can easily be shown on a geoboard It is not easy to make a series of similar (fig. 13). This situation arises when the Pythagorean theorem is adapted to figures which, although similar, are not “similarly nonregular polygons on a geoboard of placed” or “positioned” on the triangle. However, Euclid stated in Proposition 31 of Book VI of The Elements that “in right-angled triangles, the rectilinear figure described on the side opposite the right angle is equal to the similar and similarly described figures upon the sides containing the right angle.” on the sides of a right triangle on an limited size. In figure 16 an attempt has been made to construct similar trapezoids isometric board, but since 16 + 9% 24, one of the trapezoids must be dissimilar. It is fairly easy to see which one it is. In figure 17 parallelograms have been similarly placed on the sides of a right triangle, again on an isometric board. This time we can see that 36 + 10 # 44 and attempts to alter the largest parallelogram in shape only result in areas of 48 (fig. 18) or 52 (fig. 19). It is probably easier to pursue investigations such as these on isometric dot paper, and those with expansive ideas can Fig. 13 stick two or more sheets together. Isometric dot paper is quite easily made by March 1973

Mathematics Teacher

placing a piece of isometric graph paper on top of a ditto mat and pricking through vertices at regular intervals. Square grid dot paper can be made, with tolerable accuracy, with a typewriter and a ditto mat. The writer feels that with these simple materials an interesting field of discovery is open and that great value is to be found in evolving simple rules to cope with strange and novel situations. REFERENCES Buckeye, D. A, W. A. Ewbank, and J. L. Ginther. A Cloudburst of Math Lab Ezxperiments. Vols. 3 and 4. Troy, Mich.: Midwest Mathematics textbooks that stimulate learning Publications, 1971. Liedtke, Warner, and Thomas E. Kieren. “Geoboard Geometry for Preschool Children.” ARITHMETIC TEACHER 17 (February 1970): 123-27. GEOMETRY A Transformation Approach Pa avin { 02, VY 4 0 L An exciting new textbook for high school students of average ability which simplifies the presentation and facilitates understanding of SOME GEOBOARD SUPPLIERS basic Euclidean geometry by the use of transformations. Creative Publications P.0. Box 328 Palo Alto, Calif. 94302 Cuisenaire Co. of America INTRODUCTION TO HIGH SCHOOL MATHEMATICS HIGH SCHOOL MATHEMATICS APPLYING 12 Church St. New Rochelle, N.Y. 10805 Ideal School Supply Co. Oak Lawn, Ill. 60453 Math Media P.O. Box 345 Danbury, Conn. 06810 Selective Educational Equipment 3 Bridge St. Newton, Mass. 02195 Walker Educational Book Corp. 720 Fifth Ave. New York, N.Y. 10019 ALGEBRA 1— Theory and Application ALGEBRA 2 AND TRIGONOMETRY— Theory and Application GEOMETRY— Theory and Application [en PL THE SPECTRUM MATHEMATICS SERIES France » Clarke A series of six, non-graded, consumable mathematics texts for pupils who need assistance with the basic concepts and skills of computation and reasoning. FOR FURTHER WRITE TO INFORMATION LAIDLAW OR CONTACT YOUR BROTHERS LAIDLAW REPRESENTATIVE à LAIDLAW NEW MATH TEACHERS. Volunteer PEACE CORPS. Two years in local school systems of developing countries overseas. Develop curriculum/teaching aids, train teachers, participate in team teaching workshops. Information: Bruce Mazzie, ACTION, OCP Box 158, Washington, D.C. BROTHERS A Division of Doubleday Thatcher and Madison River Forest, Illinois 60305