Montessori Elementary Materials - English Restoration --- ## Chapter 16 - Geometry ## PART IV ## I **PLANE GEOMETRY** The geometric insets used for sensorial exercises in the "Children's House" made it possible for the child to become familiar with many figures of plane geometry: the square, rectangle, triangle, polygon, circle, ellipse, etc. By means of the third series of corresponding cards, where the figures are merely outlined, he formed the habit of recognizing a geometric figure represented merely by a line. Furthermore, he has used a series of iron insets reproducing some of the geometric figures which he previously had learned through the use of wooden geometric insets. He used these iron insets to draw the outline of a figure, which he then filled in with parallel lines by means of colored pencils (an exercise in handling the instruments of writing). The geometric material here presented to the elementary classes supplements that used in the "Children's House." It is similar to the iron insets; but in this material each frame is fastened to an iron foundation of exactly the same size as the frame. Since each piece is complete in itself, no rack is needed to hold them. The frame of the inset is green, the foundation is white, and the inset itself—the movable portion—is red. When the inset is in the frame, the red surface and the green frame are in the same plane. This material further differs from the other in that each\[260\] inset is composed not of a single piece, as in the first material, but of many pieces which, when put together on the white foundation, exactly reproduce the geometric figure there designated. The use to which these modified insets may be put is most varied. The main purpose is to facilitate the child's auto-education through exercises in geometry and often through the solution of real problems. The fact of being able actually to "handle geometric figures," to arrange them in different ways, and to judge of the relations between them, commands the child's absorbed attention. The putting together of the insets, which deal with equivalent figures, reminds one of the "games of patience"—picture puzzles—which have been invented for children but which, while amusing them, have no definite educational aim. Here, however, the child leaves the exercises with "clear concepts" and not merely with general "notions" of the principles of geometry, a thing which is very hard to accomplish by the methods common to the older schools. The difference between like figures, similar figures, and equivalent figures, the possibility of reducing every regular plane figure to an equivalent rectangle, and finally the solution of the theorem of Pythagoras—all these are acquired eagerly and spontaneously by the child. The same may be said about work in fractions, which is made most interesting by the exercises with the circular insets. The real meaning of the word *fraction*, operations in fractions, the reduction of common fractions to decimal fractions—all of this is mastered and becomes perfectly clear in the child's mind. These are formative conquests and at the same time a dynamic part of the child's intellectual activity. A child who works spontaneously and for a long period of time\[261\] with this material not only strengthens his reasoning powers and his character but acquires higher and clearer cognitions, which increase his mental capacity. In his succeeding spontaneous flights into the abstract he will show ability for surprising progress. While a high school child is still wasting his mental effort in trying to understand the relation between geometrical figures, which it seems impossible for him to comprehend, our child in the primary grades is "finding it out for himself" and is so elated by his discovery that he immediately begins the search for other geometrical relations. Our children gallop freely along over a smooth road, urged on by the inner energy of their growing psychic organism, while many other children plod on barefooted and in shackles over stony paths. Every positive conquest gained through objects with our method of freedom—allowing the child to exercise himself at the time when he is most ready for the exercise and permitting him to complete this exercise—results in spontaneous abstractions. How is it possible to lead a child to perform abstractions if his mind is not sufficiently mature and he is without adequate information? These two points of support are, as it were, the feet of the psychic man who is traveling toward his highest mental activities. We shall always see the repetition of this phenomenon. Every ulterior exercise of inner development, every ulterior cognition, will lead the child to new and ever higher flights into the realm of the abstract. It is well, however, to emphasize this principle: that the mind, in order to fly, must leave from some point of contact, just as the aeroplane starts from its hangar, and that it must have reached a certain degree of maturity, as is the case with the small bird when it tries its wings\[262\] and starts on its first flight from the nest where it was born and gained his strength. An aeroplane of perpetual flight without a means of replenishing its supplies, and a bird with only an "instinct of flight" without the process of development that takes place from the egg to the first flight, are things that do not exist. A machine flying perpetually without need of replenishing the fuel for its propelling energy, and an instinct without a corresponding organism, are pure fancies. The same is true of the flight of man's imagination, which soars through space and creates. Though this is the mind's "manner of being," its "highest instinct," yet it also needs to find support in reality, to organize its inner forces from time to time. The longer a material can claim and hold a child's attention, the greater promise it gives that an "abstract process," an "imaginative creation" will follow as the result of a developed potentiality. This creative imagination, which is ever returning to reality to gain inspiration and to acquire new energies, will not be a vain, exhaustible, and fickle thing, like the so-called imagination which our ordinary schools are trying to develop. Without positive replenishment in reality there never will be a spontaneous flight of the mind; this is the unsurmountable difficulty of the common schools in their attempt to "develop the imagination" and to "lead to education." The child who without any impelling force from within is artificially "borne aloft" by the teacher, who forces him into the "abstract," can at most learn only how to descend slowly like a parachute. He can never learn to "lift himself energetically to dizzy heights." This is the difference; hence the necessity for considering the positive basis which holds the mind of the child to\[263\] systematic auto-exercises of preparation. After this it suffices merely to grant freedom to the child's genius in order that it may take its own flight. I need not repeat that even in the period of replenishing, freedom is the guide in finding the "particular moment" and the "necessary time"; for I already have spoken insistently and at length concerning this. It is well, however, to reaffirm here even more clearly that a material for development predetermined by experimental research and put into relation with the child (through lessons) accomplishes so complete a work by the psychic reactions which it is capable of stimulating that marvelous phenomena of intellectual development may be obtained. These geometric insets furnish rich materials for the application of this principle and respond wonderfully to the "instinct for work" in the child mind. The exercises with this material not only are exercises of composition with the pieces of an inset or of the substitution of them into their relative metal plates; they are also exercises in drawing which, because of the labor they require, allow the child to take cognizance of every detail and to meditate upon it. The designing done with these geometric insets, as will be explained, is of two kinds: geometric and artistic (mechanical and decorative). And the union of the two kinds of drawings gives new ways of applying the material. The geometric design consists in reproducing the figure outlined by the corresponding insets. In this way the child learns to use the different instruments of drawing—the square, the ruler, the compass, and the protractor. In these exercises he acquires, with the aid of the special portfolio which comes with the material, actual and real cognitions in geometry. \[264\] Artistic designs are made by combining the small pieces of the various geometric insets. The resulting figures are then outlined and filled in with colored pencils or watercolors. Such combinations on the part of the child are real esthetic creations. The insets are of such reciprocal proportions that their combination results in an artistic harmony which facilitates the development of the child's esthetic sense. With our insets we were able to reproduce some of the classic decorations found in our masterpieces of art, such as decorations by Giotto. A combination of geometric design and artistic design is formed by decorating the different parts of the geometric figure—as the center, the sides, the angles, the circumference, etc.; or by elaborating with free-hand details the decorations which have resulted from the combination of the insets. But a far better concept of all this will be gained as we pass on to explain our didactic material. --- \[265\] ## II **THE DIDACTIC MATERIAL USED FOR GEOMETRY** ### Equivalent, Identical and Similar Figures First 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: | A | square | divided | into | two equal rectangles | | - | ------ | ------- | ---- | -------------------- | | " | " | " | " | four equal squares | | " | " | " | " | eight equal rectangles | | " | " | " | " | sixteen equal squares | | " | " | " | " | two equal triangles | | " | " | " | " | four equal triangles | | " | " | " | " | eight equal triangles | | " | " | " | " | sixteen equal triangles | 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\[266\] relation between the triangles and the rectangles; indeed, they are each half of the same square; yet they differ greatly in form. Inductively the child gains an idea of equivalent figures. The two triangles are identical; the two rectangles also are identical; whereas the triangle and the rectangle are equivalents. The child soon makes comparisons by placing the triangle on the rectangle, and he notices at once that the small triangle which is left over on the rectangle equals the small triangle which remains uncovered on the larger triangle, and therefore that the triangle and the rectangle, though they do not have the same form, have the same area.  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\[267\] square by a medial line and are identical; the two triangles are formed by dividing the original square by a diagonal line, etc. *Similar* figures, on the other hand, are those which have the same form but differ in dimension. For example, the rectangle which is half of the original square and the one which is half of the smaller square—that is, an eighth of the original square—are neither identical nor equivalent but they are *similar* figures. The same may be said of the large square and of the smaller ones which represent a fourth, a sixteenth, etc. 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.  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\[268\] from 0° to 180°. Each sector is then subdivided into ten parts or degrees.  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\[269\] circle is divided into from two to ten parts. The protractor shows approximately that: | 1/3 | circle | = | 120° | and | 360° | ÷ | 3 | = | 120° | | --- | ------ | - | ---- | --- | ---- | - | - | - | ---- | | 1/4 | " | = | 90° | " | 360° | ÷ | 4 | = | 90° | | 1/5 | " | = | 72° | " | 360° | ÷ | 5 | = | 72° | | 1/6 | " | = | 60° | " | 360° | ÷ | 6 | = | 60° | | 1/7 | " | = | 51° | " | 360° | ÷ | 7 | = | 51° | | 1/8 | " | = | 45° | " | 360° | ÷ | 8 | = | 45° | | 1/9 | " | = | 40° | " | 360° | ÷ | 9 | = | 40° | | 1/10 | " | = | 36° | " | 360° | ÷ | 10 | = | 36° | 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\[270\] pieces. From this he can draw the following conclusion: 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.  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: \[271\] 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\[272\] equal to them. In the space occupied by the five tenths may be placed one half, and in that occupied by the two eighths, one fourth; then the circle is filled thus: 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: \[273\] \[8 + (7/7 + 18/9 + 24/2) + 1\] =\ 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\[274\] which marks the five is distinguished from the others by its greater length. Each of the larger divisions is marked respectively with the numbers, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 0. The 0 is at the top and there is a raised radius against which are placed the sectors to be measured.  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](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#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\[275\]\ \[276\] 1/3, 1/4, and 1/8 of a circle are placed within the circle divided into hundredths. Result: 1/3 + 1/4 + 1/8 = 0.70.  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\[277\] the circle, and the reduction is made. All the parts which result are so many hundredths. Hence: 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\[278\] triangle. This demonstrates that the rectangle and the triangle are equivalent. The triangular space is filled by two pieces formed by a horizontal line drawn through the triangle parallel to the base and crossing at half the altitude. Taking the two pieces out and putting them one on top of the other the identity of the height may be verified.  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\[279\] there are also an entire rhombus and an entire rectangle. If they are placed one on top of the other they will be found to have the same height. As the equivalence of the two figures is demonstrated by these pieces of the rectangle which may be used to fill in the two figures, it is easily seen that the area of a rhombus is found by multiplying the side or base by the height.     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\[280\] the bases. For the latter it is only necessary to cut the long rectangle in half and superimpose the two halves. 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.  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.  > **The analysis of the decagon.** In the material there are two decagon insets, one consisting\[281\]\ \[282\] of a whole decagon and the other of a decagon divided into ten triangles. [Page 281](https://www.gutenberg.org/cache/epub/42869/pg42869-images.html#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.  > **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 FIGURES *A.* 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\[283\] rectangle are each divided into two triangles. The triangles of the rhombus are different, for they are divided by opposite diagonal lines. The three different triangles resulting from these divisions have the same base (this can be actually verified by measuring the bases of the different pieces) and fit into the same long rectangle which is found below the first three figures. Therefore, it is demonstrated that the three triangles have the same altitude. They are equivalent because each one is the half of an equivalent figure.  > **The decagon and the rectangle can be composed of the same triangular insets.**  > **The triangular insets fitted into their metal plates.**  > *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. \[284\] 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.  > **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,\[285\]\ \[286\] so that they are both of the same color; while the square formed on the hypotenuse can be filled with varied designs by various combinations of the small squares of the sides which are in two different colors. *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\[287\] movable pieces actually fit into this white background which gives the child the opportunity for reasoning out the theorem, and not the abstract idea of the corresponding relations between the dimensions of the sides and the different heights of the figures. Reduced to these terms the exercise is easily performed and proves very interesting. 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. \[288\]  > **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.  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\[289\]\ \[290\] triangles of the plane insets used in the "Children's House." He measures the degrees of the angles and learns to distinguish a right angle (90°) from an acute angle (<90°) and from an obtuse angle (>90°). 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\[291\] square in the 5 centimeters square; and finally the tiny square, which is 1/16 part of the large square, in the 3.5 centimeters square. 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). --- \[292\] ## III **SOLID GEOMETRY** Since the children already know how to find the area of ordinary geometric forms it is very easy, with the knowledge of the arithmetic they have acquired through work with the beads (the square and cube of numbers), to initiate them into the manner of finding the volume of solids. After having studied the cube of numbers by the aid of the cube of beads it is easy to recognize the fact that the volume of a prism is found by multiplying the area by the altitude. In our didactic material we have three objects for solid geometry: a prism, a pyramid having the same base and altitude, and a prism with the same base but with only one-third the altitude. They are all empty. The two prisms have a cover and are really boxes; the uncovered pyramid can be filled with different substances and then emptied, serving as a sort of scoop. These solids may be filled with wheat or sand. Thus we put into practise the same technique as is used to calculate capacity, as in anthropology, for instance, when we wish to measure the capacity of a cranium. It is difficult to fill a receptacle completely in such a way that the measured result does not vary; so we usually put in a scarce measure, which therefore does not correspond to the exact volume but to a smaller volume. One must know how to fill a receptacle, just as one\[293\] must know how to do up a bundle, so that the various objects may take up the least possible space. The children like this exercise of shaking the receptacle and getting in as great a quantity as possible; and they like to level it off when it is entirely filled. The receptacles may be filled also with liquids. In this case the child must be careful to pour out the contents without losing a single drop. This technical drill serves as a preparation for using metric measures. By these experiments the child finds that the pyramid has the same volume as the small prism (which is one-third of the large prism); hence the volume of the pyramid is found by multiplying the area of the base by one-third the altitude. The small prism may be filled with clay and the same piece of clay will be found to fill the pyramid. The two solids of equal volume may be made of clay. All three solids can be made by taking five times as much clay as is needed to fill the same prism. . . . . . . . Having mastered these fundamental ideas, it is easy to study the rest, and few explanations will be needed. In many cases the incentive to do original problems may be developed by giving the children definite examples: as, how can the area of a circle be found? the volume of a cylinder? of a cone? Problems on the total area of some solids also may be suggested. Many times the children will risk spontaneous inductions and often of their own accord proceed to measure the total surface area of all the solids at their disposal, even going back to the materials used in the "Children's House." The material includes a series of wooden solids with a base measurement of 10 cm.: \[294\] A quadrangular parallelopiped (10 X 10 X 20 cm.)\ A quadrangular parallelopiped equal to 1/3 of above\ A quadrangular pyramid (10 X 10 X 20 cm.)\ A triangular prism (10 X 20 cm.)\ A triangular prism equal to 1/3 of above\ The corresponding pyramid (10 X 20 cm.)\ A cylinder (10 cm. diameter, 20 altitude)\ A cylinder equal to 1/3 of above\ A cone (10 cm. diameter, 20 altitude)\ A sphere (10 cm. diameter)\ An ovoid (maximum diameter 10 cm.)\ An ellipsoid (maximum diameter 10 cm.)\ Regular Polyhedrons\ Tetrahedron\ Hexahedron (cube)\ Octahedron\ Dodecahedron\ Icosahedron (The faces of these polyhedrons are in different colors.) ### Applications: *The Powers of Numbers*. Material: Two equal cubes of 2 cm. on a side; a prism twice the size of the cubes; a prism double this preceding prism; seven cubes 4 cm. on a side. The following combinations are made: > The two smaller cubes are placed side by side = 2. > > In front of these is placed the prism which is twice as large as the cube = 22. > > On top of these is placed the double prism, making a cube with 4 cm. on a side = 23. > > One of the seven cubes is put beside this = 24. \[295\] In front are placed two more of the seven cubes = 25. On top are put the remaining four equal cubes = 26. In this way we have made a cube measuring 8 cm. on a side. From this we see that: 23, 26 have the form of a cube.\ 22, 25 have the form of a square.\ 2, 24 have a linear form. *The Cube of a Binomial:* (a + b)3 = a3 + b3 + 3a2b + 3ba. Material: A cube with a 6 cm. edge, a cube with a 4 cm. edge; three prisms with a square base of 4 cm. on a side and 6 cm. high; three prisms with a square base of 6 cm. to a side and 4 cm. high. The 10 cm. cube can be made with these. These two combinations are in special cube-shaped boxes into which the 10 cm. cube fits exactly. . . . . . . . *Weights and Measures:* All that refers to weights and measures is merely an application of similar operations and reasonings. The children have at their disposal and learn to handle many of the objects which are used for measuring both in commerce and in every-day life. In the "Children's House" days they had the long stair rods which contain the meter and its decimeter subdivisions. Here they have a tape-measure with which they measure floors, etc., and find the area. They have the meter in many forms: in the anthropometer, in the ruler. Then, too,\[296\] they use the metal tape, the dressmaker's tape measure, and the meterstick used by merchants.  > **Hollow geometric solids, used for determining equivalence by measuring sand, sugar, etc.** The twenty centimeter ruler divided into millimeters they use constantly in design; and they love to calculate the area of the geometric figures they have designed or of the metal insets. Often they calculate the surface of the white background of an inset and that of the different pieces which exactly fit this opening, so as to verify the former. As they already have some preparation in decimals it is no task for them to recognize and to remember that the measures increase by tens and take on new names each time. The exercises in grammar have greatly facilitated the increase in their vocabulary. They calculate the reciprocal relations between length, surface, and volume by going back to the three sets which first represented "long," "thick," and "large." The objects which differ in length vary by 10's; those differing in areas vary by 100's; and those which differ in volume vary by 1000's. The comparison between the bead material and the cubes of the pink tower (one of the first things they built) encourages a more profound study of the sensory objects which were once the subject of assiduous application. By the aid of the double decimeter the children make the calculations for finding the volume of all the different objects graded by tens, such as the rods, the prisms of the broad stair, the cubes of the pink tower. By taking the extremes in each case they learn the relations between objects which differ in one dimension, in two dimensions, and in three dimensions. Besides, they already know that the square of 10 is 100, and the cube of 10 is 1000. . . . . . . . \[297\]  > **Designs formed by arranging sections of the insets within the frames.** The children make use of various scientific instruments: thermometers, distillers, scales, and, as previously stated, the principal measures commonly used. By filling an empty metal cubical decimeter, which like the geometric solids is used for the calculation of volume, they have a liter measure of water, which may be poured into a glass liter bottle. All the decimal multiples and subdivisions of the liter are easily understood. Our children spent much time pouring liquids into all the small measures used in commerce for measuring wine and oil. They distil water with the distiller. They use the thermometer to measure the temperature of water in ebullition and the temperature of the freezing mixture. They take the water which is used to determine the weight of the kilogram, keeping it at the temperature of 4°C. The objects which serve to measure capacity also are at the disposal of the children. There is no need to go into more details upon the multitudinous consequences resulting from both a methodical preparation of the intellect and the possibility of actually being in contact with real objects. A great number of problems given by us, as well as problems originated by the children themselves, bear witness to the ease with which external effects may he spontaneously produced when once the inner *causes* have been adequately stimulated.