  THE DIDACTIC MATERIAL USED FOR GEOMETRY Equivalent, Identical and Similar FiguresFirst Series of Insets: Squares and Divided Figures. This is a series of nine square insets, ten by ten centimeters, each of which has a white foundation of the same size as the inset. One inset consists of an entire square; the others are made up in the following manner:
The child can take the square divided into two rectangles and the one divided into two triangles and interchange them: that is, he can build the first square with triangles and the second with rectangles. The two triangles can be superimposed by placing them in contact at the under side where there is no knob, and the same can be done with the rectangles, thus showing their equivalence by placing one on the other. But there also is a certain drawing This exercise in observation is repeated in a like manner with all the other insets, which are divided successively into four, eight, and sixteen parts. The small square which is a fourth of the original square, resulting from the division of this latter by two medial lines, is equivalent to the triangle which was formed by dividing this same original square into four triangles by two diagonal lines. And so on. By comparing the different figures the child learns the difference between equivalent figures and identical figures. The two rectangles are the result of dividing the large Through these divisions of the square an idea of fractions is gained intuitively. However, this is not the material used for the study of fractions. For this purpose there is another series of insets. Second Series of Insets: Fractions. There are ten metal plates, each of which has a circular opening ten centimeters in diameter. One inset is a complete circle; the other circular insets are divided respectively into 2, 3, 4, 5, 6, 7, 8, 9, and 10 equal parts. drawing The children learn to measure the angles of each piece, and so to count the degrees. For this work there is a circular piece of white card-board, on which is drawn in black a semicircle with a radius of the same length as that of the circular insets. This semicircle is divided into 18 sectors by radii which extend beyond the circumference on to the background; and these radii are numbered by tens drawing The diameter from 0° to 180° is outlined heavily and extends beyond the circumference, in order to facilitate the adjustment of the angle to be measured and to give a strict exactness of position. This is done also with the radius which marks 90°. The child places a piece of an inset in such a way that the vertex of the angle touches the middle of the diameter and one of its sides rests on the radius marked 0°. At the other end of the arc of the inset he can read the degrees of the angle. After these exercises, the children are able to measure any angle with a common protractor. Furthermore, they learn that a circle measures 360°, half a circle 180°, and a right angle 90°. Once having learned that a circumference measures 360° they can find the number of degrees in any angle; for example, in the angle of an inset representing the seventh of the circle, they know that 360° ÷ 7 = (approximately) 51°. This they can easily verify with their instruments by placing the sector on the graduated circle. These calculations and measurements are repeated with all the different sectors of this series of insets where the
In this way the child learns to write fractions: 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 He has concrete impressions of them as well as an intuition of their arithmetical relationships. The material lends itself to an infinite number of combinations, all of which are real arithmetical exercises in fractions. For example, the child can take from the circle the two half circles and replace them by four sectors of 90°, filling the same circular opening with entirely different 1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4. He also may say that two halves are equal to four fourths, and write accordingly: 2/2 = 4/4. drawing This is merely the expression of the same thing. Seeing the pieces, he has done an example mentally and then has written it out. Let us write it according to the first form, which is, in reality, an analysis of this example: 1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4. When the denominator is the same, the sum of the fractions is found by adding the numerators: 1/2 + 1/2 = 2/2; 1/4 + 1/4 + 1/4 + 1/4 = 4/4. The two halves make an entire circle, as do the four fourths. Now let us fill a circle with different pieces: for example, with a half circle and two quarter circles. The result is 1 = 1/2 + 2/4. And in the inset itself it is shown that 1/2 = 2/4. If we should wish to fill the circle with the largest piece (1/2) combined with the fewest number of pieces possible, it would be necessary to withdraw the two quarter sectors and replace them by another half circle; result: 1 = 1/2 + 1/2 = 2/2 = 1. Let us fill a circle with three 1/5 sectors and four 1/10 sectors: 1 = 3/5 + 4/10. If the larger pieces are left in and the circle is then filled with the fewest number of pieces possible, it would necessitate replacing the four tenths by two fifths. Result: 1 = 3/5 + 2/5 = 5/5 = 1. Let us fill the circle thus: 5/10 + 1/4 + 2/8 = 1. Now try to put in the largest pieces possible by substituting for several small pieces a large piece which is 1 = 1/2 + 1/4 + 1/4 = 1/2 + 2/4. We can continue to do the same thing, that is to replace the smaller pieces by as large a sector as possible, and the two fourths can be replaced by another half circle. Result: 1 = 1/2 + 1/2 = 2/2 = 1. All these substitutions may be expressed in figures thus: 5/10 + 1/4 + 2/8 = 1/2 + 1/4 + 1/4 = 1/2 + 2/4 = 1/2 + 1/2 = 2/2 = 1. This is one means of initiating a child intuitively into the operations used for the reduction of fractions to their lowest terms. Improper fractions also interest them very much. They come to these by adding a number of sectors which fill two, three, or four circles. To find the whole numbers which exist under the guise of fractions is a little like putting away in their proper places the circular insets which have been all mixed up. The children manifest a desire to learn the real operations of fractions. With improper fractions they originate most unusual sums, like the following: [8 + (7/7 + 18/9 + 24/2) + 1] = 8 [8 + (1 + 2 + 12) + 1] = 8 8 + 15 + 1 = 24/8 = 3. 8 We have a series of commands which may be used as a guide for the child's work. Here are some examples: —Take 1/5 of 25 beads —Take 1/4 " 36 counters —Take 1/6 " 24 beans —Take 1/3 " 27 beans —Take 1/10 " 40 beans —Take 2/5 " 60 counters In this last there are two operations: 60 ÷ 5 = 12; 12 X 2 = 24; or 2 X 60 = 120; 120 ÷ 5 = 24, etc. Reduction of Common Fractions to Decimal Fractions: The material for this purpose is similar to that of the circular insets, except that the frame is white and is marked into ten equal parts, and each part is then subdivided into ten. In these subdivisions the little line drawing To reduce a common fraction to a decimal fraction the sector is placed carefully against the raised radius, with the arc touching the circumference of the inset. Where the arc ends there is a number which represents the hundredths corresponding to the sector. For example, if the 1/4 sector is used its arc ends at 25; hence 1/4 equals 0.25. Page 275 shows in detail the practical method of using our material to reduce common fractions to decimal fractions. In the upper figure the segments correspond to 1/3 + 1/4 + 1/8 = 0.70. drawing The lower figure shows how the 1/3 sector is placed: 1/3 = 0.33. If instead we use the 1/5 sector we have: 1/5 = 0.20, etc. Numerous sectors may be placed within the circle; for example: 1/4 + 1/7 + 1/9 + 1/10. In order to find the sum of the fraction reduced to decimals, it is necessary to read only the number at the outer edge of the last sector. Using this as a basis, it is very easy to develop an arithmetical idea. Instead of 1, which represents the whole circle, let us write 100, which represents its subdivisions when used for decimals, and let us divide the 100 into as many parts of a circle as there are sectors in 1/4 = 100 ÷ 4 = 25 hundredths: that is, 25/100 or 0.25. The division is performed by dividing the numerator by the demoninator: 1 ÷ 4 = 0.25. Third Series of Insets: Equivalent Figures. Two concepts were given by the squares divided into rectangles and triangles: that of fractions and that of equivalent figures. There is a special material for the concept of fractions which, besides developing the intuitive notion of fractions, has permitted the solution of examples in fractions and of reducing fractions to decimals; and it has furthermore brought cognizance of other things, such as the measuring of angles in terms of degrees. For the concept of equivalent figures there is still another material. This will lead to finding the area of different geometric forms and also to an intuition of some theorems which heretofore have been foreign to elementary schools, being considered beyond the understanding of a child. Material: Showing that a triangle is equal to a rectangle which has one side equal to the base of the triangle, the other side equal to half of the altitude of the triangle. In a large rectangular metal frame there are two white openings: the triangle and the equivalent rectangle. The pieces which compose the rectangle are such that they may fit into the openings of either the rectangle or the drawing Already the work with the beads and the squaring of numbers has led to finding the area of a square by multiplying one side by the other; and in like manner the area of a rectangle is found by multiplying the base by half other. Since a triangle may be reduced to a rectangle, it is easy to find its area by multiplying the base by half the height. Material: Showing that a rhombus is equal to a rectangle which has one side equal to one side of the rhombus and the other equal to the height of the rhombus. The frame contains a rhombus divided by a diagonal line into two triangles and a rectangle filled with pieces which can be put into the rhombus when the triangles have been removed, and will fill it completely. In the material Material: To show the equivalence of a trapezoid and a rectangle having one side equal to the sum of the two bases and the other equal to half the height. The child himself can make the other comparison: that is, a trapezoid equals a rectangle having one side equal to the height and the other equal to one-half the sum of The large rectangular frame contains three openings: two equal trapezoids and the equivalent rectangle having one side equal to the sum of the two bases and the other side equal to half the height. One trapezoid is made of two pieces, being cut in half horizontally at the height of half its altitude; the identity in height may be proved by placing one piece on top of the other. The second trapezoid is composed of pieces which can be placed in the rectangle, filling it completely. Thus the equivalence is proved and also the fact that the area of a trapezoid is found by multiplying the sum of the bases by half the height, or half the sum of the bases by the height. drawing With a ruler the children themselves actually calculate the area of the geometrical figures, and later calculate the area of their little tables, etc. Material: To show the equivalence between a regular polygon and a rectangle having one side equal to the perimeter and the other equal to half of the hypotenuse. drawing The analysis of the decagon. In the material there are two decagon insets, one consisting Page 281 shows a table taken from our geometry portfolio, representing the equivalence of a decagon to a rectangle having one side equal to the perimeter and the other equal to half the hypotenuse. photograph The bead number cubes built into a tower. The photograph shows the pieces of the insets—the decagon and the equivalent rectangle—and beneath each one there are the small equal triangles into which it can be subdivided. Here it is demonstrated that a rectangle equivalent to a decagon may have one side equal to the whole hypotenuse and the other equal to half of the perimeter. Another inset shows the equivalence of the decagon and a rectangle which has one side equal to the perimeter of the decagon and the other equal to half of the altitude of each triangle composing the decagon. Small triangles divided horizontally in half can be fitted into this figure, with one of the upper triangles divided in half lengthwise. Thus we demonstrate that the surface of a regular polygon may be found by multiplying the perimeter by half the hypotenuse. SOME THEOREMS BASED ON EQUIVALENT FIGURESA. All triangles having the same base and altitude are equal. This is easily understood from the fact that the area of a triangle is found by multiplying the base by half the altitude; therefore triangles having the same base and the same altitude must be equal. For the inductive demonstration of this theorem we have the following material: The rhombus and the equivalent photograph The decagon and the rectangle can be composed of the same triangular insets. photograph The triangular insets fitted into their metal plates. drawing B. The Theorem of Pythagoras: In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the two sides. Material: The material illustrates three different cases: First case: In which the two sides of the triangle are equal. Second case: In which the two sides are in the proportion of 3:4. Third case: General. First case: The demonstration of this first case affords an impressive induction. In the frame for this, shown below, the squares of the two sides are divided in half by a diagonal line so as to form two triangles and the square of the hypotenuse is divided by two diagonal lines into four triangles. The eight resulting triangles are all identical; hence the triangles of the squares of the two sides will fill the square of the hypotenuse; and, vice versa, the four triangles of the square of the hypotenuse may be used to fill the two squares of the sides. The substitution of these different pieces is very interesting, and all the more because the triangles of the squares of the sides are all of the same color, whereas the triangles formed in the square of the hypotenuse are of a different color. Second case: Where the sides are as the proportion of 3:4. In this figure the three squares are filled with small squares of three different colors, arranged as follows: in the square on the shorter side, 32 = 9; in that on the larger side, 42 = 16; in that on the hypotenuse, 52 = 25. drawing Second Case The substitution game suggests itself. The two squares formed on the sides can be entirely filled by the small squares composing the square on the hypotenuse, Third case: This is the general case. The large frame is somewhat complicated and difficult to describe. It develops a considerable intellectual exercise. The entire frame measures 44 × 24 cm. and may be likened to a chess-board, where the movable pieces are susceptible of various combinations. The principles already proved or inductively suggested which lead to the demonstration of the theorem are: (1) That two quadrilaterals having an equal base and equal altitude are equivalent. (2) That two figures equivalent to a third figure are equivalent to each other. In this figure the square formed on the hypotenuse is divided into two rectangles. The additional side is determined by the division made in the hypotenuse by dropping a perpendicular line from the apex of the triangle to the hypotenuse. There are also two rhomboids in this frame, each of which has one side equal respectively to the large and to the small square of the sides of the triangle and the other side equal to the hypotenuse. The shorter altitude of the two rhomboids, as may be seen in the figure itself, corresponds to the respective altitudes, or shorter sides, of the rectangles. But the longer side corresponds respectively to the side of the larger and of the smaller squares of the sides of the triangle. It is not necessary that these corresponding dimensions be known by the child. He sees red and yellow pieces of an inset and simply moves them about, placing them in the indentures of the frame. It is the fact that these This material may be used for other demonstrations: Demonstration A: The substitution of the pieces. Let us start with the frame as it should be filled originally. First take out the two rectangles formed on the hypotenuse; place them in the two lateral grooves, and lower the triangle. Fill the remaining empty space with the two rhomboids. The same space is filled in both cases with: A triangle plus two rectangles, and then A triangle plus two rhomboids. Hence the sum of the two rectangles (which form the square of the hypotenuse) is equal to the sum of the two rhomboids. In a later substitution we consider the rhomboids instead of the rectangles in order to demonstrate their respective equivalence to the two squares formed on the sides of the triangle. Beginning, for example with the larger square, we start with the insets in the original position and consider the space occupied by the triangle and the larger square. To analyze this space the pieces are all taken out and then it is filled successively by: The triangle and the large square in their original positions. The triangle and the large rhomboid. photograph Showing that the two rhomboids are equal to the two rectangles. Demonstration B: Based on Equivalence. In this second demonstration the relative equivalence of the rhomboid, the rectangles, and the squares is shown outside the figure by means of the parallel indentures which are on both sides of the frame. These indentures, when the pieces are placed in them, show that the pieces have the same altitude. This is the manner of procedure: Starting again with the original position, take out the two rectangles and place them in the parallel indentures to the left, the larger in the wider indenture and the smaller in the narrower indenture. The different figures in the same indenture have the same altitude; therefore the pieces need only to be placed together at the base to prove that they are equal—hence the figures are equal in pairs: the smaller rectangle equals the smaller rhomboid and the larger rectangle equals the larger rhomboid. Starting again from the original position you proceed analogously with the squares. In the parallel indentures to the right the large square may be placed in the same indenture with the large rhomboid, which, however, must be turned in the opposite direction (in the direction of its greatest length); and the smaller square and the smaller rhomboid fit into the narrower indenture. They have the same altitude; and that the bases are equal is easily verified by putting them together; therefore here is proof that the squares and the rhomboids are respectively equivalent. Rectangles and squares which are equivalent to the same rhomboids are equivalent to each other. Hence the theorem is proved. . . . . . . . Showing that the two rhomboids are equal to the two squares. This series of geometric material is used for other purposes, but they are of minor importance. Fourth Series of Insets: Division of a Triangle. This material made up of four frames of equal size, each containing an equilateral triangle measuring ten centimeters to a side. The different pieces should fill the triangular spaces exactly. One is filled by an entire equilateral triangle. One is filled by two rectangular scalene triangles, each equal to half of the original equilateral triangle, which is bisected by dropping a line perpendicularly to the base. drawing The third is filled by three obtuse isosceles triangles, formed by lines bisecting the three angles of the original triangle. The fourth is divided into four equilateral triangles which are similar in shape to the original triangle. With these triangles a child can make a more exact analytical study than he made when he was observing the Furthermore he finds in measuring the angles of any triangle that their sum is always equal to 180° or to two right angles. He can observe that in equilateral triangles all the angles are equal (60°); that in the isosceles triangle the two angles at the opposite ends of the unequal side are equal; while in the scalene triangle no two angles are alike. In the right-angled triangle the sum of the two acute angles is equal to a right angle. A general definition is that those triangles are similar in which the corresponding angles are equal. Material for Inscribed and Concentric Figures: In this material, which for the most part is made up of that already described, and which is therefore merely an application of it, inscribed or concentric figures may be placed in the white background of the different inset frames. For example, on the white background of the large equilateral triangle the small red equilateral triangle, which is a fourth of it, may be placed in such a way that each vertex is tangent to the middle of each side of the larger triangle. There are also two squares, one of 7 centimeters on a side and the other 3.5. They have their respective frames with white backgrounds. The 7 centimeters square may be placed on the background of the 10 centimeters square in such a way that each corner touches the middle of each side of the frame. In like manner the 5 centimeters square, which is a fourth of the large square, may be put in the 7 centimeters square; the 3.5 centimeters There is also a circle which is tangent to the edges of the large equilateral triangle. This circle may be placed on the background of the 10 centimeters circle, and in that case a white circular strip remains all the way round (concentric circles). Within this circle the smaller equilateral triangle (1/4 of the large triangle) is perfectly inscribed. Then there is a small circle which is tangent to the smallest equilateral triangle. Besides these circles which are used with the triangles there are two others tangent to the squares: one to the 7 centimeters square and the other to the 3.5 centimeters square. The large circle, 10 centimeters in diameter, fits exactly into the 10 centimeters square; and the other circles are concentric to it. These corresponding relations make the figures easily adaptable to our artistic composition of decorative design (see following chapter). Finally, together with the other material, there are two stars which are also used for decorative design. The two stars, or "flowers," are based on the 3.5 centimeters square. In one the circle rests on the side as a semi-circle (simple flower); and in the other the same circle goes around the vertex and beyond the semi-circle until it meets the reciprocal of four circles (flower and foliage). |
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