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xGeometry :: 6-9 :: Chapter One
xTable of Contents:

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xIntroduction to Geometry


Maria Montessori described it "psycho-geometry" and defined it so the "measurement of the earth together with the consciousness of the reciprocal relationship between Man and the objects of the environment, and between the objects themselves."

Maria Montessori extends this definition beyond the etymology (geometry: Greek ge the earth, Mother Earth from mythology, and metron measure, measurement of a dimension that is so vital to our lives.) Her definition especially this consciousness of geometry is so practical and so attached to reality for we are all a part of this.

Many refer to geometry as abstract, thus giving it to the child at a much later age and with the attitude that it is far removed from us and from reality.

Geometry, as it will be dealt with here, are the fundamental concepts of Euclidean geometry.

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xThe Geometry Cabinet


INTRODUCTION

With relation to the senses, Maria Montessori has extended the number of senses from five to seven. To the senses of smell, taste, sight, hearing and touch, she added the stereognostic sense, (the knowledge of 3-dimensionality) and the basic sense (the sense of mass, that is, of heaviness or lightness). The visual and stereognostic senses are directly related to the following work in geometry.
Maria Montessori has also identified three different aspects of education of the visual sense: according to size, form and color. In geometry we will deal with visual education according to size and form, thus eliminating color. If the child did not have previous Children's House training, this visual education must be offered differently, because it is really only pertinent to a younger age.

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PLANE INSETS

Materials:
...Geometry cabinet
...Additional insets, including pictures of the figures
...2 of the 3 boxes of pictures of the figures:
...entire figure "surface" shaded; the fine "contour" margin of the figure
...Reading labels
...box of command cards

Description of Materials:
geometry cabinet- The presentation of this material follows the order in which the drawers are arranged. Since the presentations differ from Children's House to the elementary school, so the order of the drawers and the arrangement of the contents of each drawer differs from Children's House to the Elementary school.

Order:
Presentation tray - 0 comes first at both levels.

The names of the drawers in the Children's House and their order is:

1- circles; 2 - rectangles; 3 - triangles; 4 - polygons; and 5 - different figures.

At the elementary level the names and order are:

1 - triangles; 2 - rectangles; 3 - regular polygons; 4 - circles; and 5 - other figures.

At the Children's House level, the children worked directly for the education of the visual sense, and only indirectly to learn the geometric figures. In the elementary school what was a sensorial exploration becomes a linguistic exploration via etymology. What was an indirect approach to geometry becomes an actual study of geometry.

Therefore in elementary the drawer of triangles comes first because the triangle is the first polygon we can construct in reality, having the least number of sides. The second drawer logically follows as the quadrilaterals, specifically rectangles. Regular polygons follow beginning with the five-sided figure progressing to ten sides. circles follow, because a circle is the limit of a regular polygon having an infinite number of sides.

From Children's House to elementary the order has changed: from easiest to most difficult, to: from threes sides to an infinite number of sides. This correlates with the change from seeing, touching, and naming to a focus on etymology and reasoning.

Arrangement:
The presentation tray contains the three fundamental figures of geometry, that is the only regular figures. The equilateral triangle is the only regular triangle. The square is the only regular quadrilateral. The circle is the limit of all regular polygons having an infinite number of sides. the triangle is "the constructor of reality". For every plane figure can be decomposed into triangles, just as all solids can be decomposed into tetrahedrons. The square is the "measurer of surfaces" just as the cube is the measurer of solids. The circle is the measurer of angles. In Children's House. In the Children's House the arrangement is square (left), circle (top), triangle (right). In elementary the arrangement is triangle (left), square (top), circle (right).

The triangle tray examines triangles according to their sides on top; the bottom three examine triangles according to their angles, at both levels. In the Children's House the order is (top - from left to right): equilateral, isosceles, scalene (bottom - from left to right), acute-angled, right-angled, obtuse-angled. In elementary (top - from left to right): scalene, isosceles, equilateral (bottom - from left to right), right-angled, obtuse-angled, acute- angled.

In the rectangle tray, the base of the smallest figure is 5 cm. which is 1/2 the base of the largest which is a square. In Children's House the order is largest to smallest, elementary the reverse.

The regular polygon tray is ordered identically at both levels, progressing from five to ten sides. It is understood that these are the regular polygons having more than four sides, since the equilateral triangle and the square (first tray) are also regular polygons.
In the circle tray, the diameter of the smallest is 5 cm.; the diameter of the largest is 10 cm. It is ordered from largest to smallest in the Children's House and the reverse in elementary.

The arrangement in the other figure drawer is the same for both levels: trapezoid, rhombus, quatrefoil, oval, ellipse, and curvilinear triangle (Reuleaux triangle).

Additional insets for the geometry cabinet:

Two triangles: acute-angled scalene triangle, obtuse-angled scalene triangle

Eight quadrilaterals:
common quadrilateral (four different sides and four different angles)
common parallelogram (opposite sides are parallel and equal)
four trapezoids
equilateral trapezoid
(constructed from three equilateral triangles)
scalene trapezoid
right-angled trapezoid
obtuse-angle trapezoid (two obtuse angles opposite)
Two deltoids or kites: one with unequal diagonals
one with equal diagonals
Two quatrefoils: quadrilobed
epi-cycloid

Including surface cards for each.

Note: Ten dominates all of the plane insets:

Presentation tray: triangle sides - 10cm.; square sides - 10 cm.; circle diameter - 10 cm.

Triangles: Hypotenuse of the obtuse-angled triangle - 10 cm.

Rectangles: Height of each - 10 cm.

Regular polygons: All can be inscribed in a 10 cm. diameter circle

Circles: Diameter of largest - 10 cm.

Other figures: Trapezoid base, short diagonal in rhombus, distance between opposite lobes in quatrefoil, distance between two opposite cusps in oval and ellipse, base of triangle used to construct curvilinear triangle, all - 10 cm.

Extra figures: Triangles, diagonal of parallelogram, equilateral trapezoid base all 10 cm.
Distance between points on adjacent lobes of quadrilobed quatrefoil, and between opposite lobes of epi-cycloid - 10 cm.
No 10 cm. exists in the common quadrilateral, deltoids and the last three trapezoids.

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THE GEOMETRY CABINET

Introduction:
In this second presentation of the geometry cabinet (first being in CH) the visual memory is aided by etymology, and no longer by the tactile sense. therefore the emphasis on that element is eliminated. Instead the emphasis is placed on etymology - the heart of our language.

Presentation tray

Materials:
...Appropriate drawer
...Three reading labels - "triangle", "square/quadrangle", and "circle"

Presentation: With only the tray on the table, the teacher takes out the triangle and identifies it.. this is a triangle. The child is asked to identify the angles and count them (triangle: Latin tres, tria - three and angulus - an angle; thus triangulum - triangle). Triangle means three angles. Place the inset in its frame in the drawer.

The teacher isolates the square and identifies it (square: Old French esquarre, esquerre <Latin ex - out, and squadra - square; thus to make square>). It is such an old word that the etymology doesn't help us as much. Put the square back. Isolate the circle and identify it (circle: Latin circulus - a diminutive of circus - a circle). Again the etymology doesn't help us because this shape has been called a circle as far back in time as we know.

As all three inset are placed on the table, review the first period. Rearrange the order and continue with the second and third periods. Invite the child to place the insets in their frames.

Exercise: Give the child the reading labels to place on the insets in their frames: triangle, circle, square/quadrangle. Note: The word quadrangle is not used at this point.

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TRIANGLES

Materials: Reading labels - "scalene triangle", "isosceles triangle", "equilateral triangle", "right-angled triangle", "obtuse- angled triangle", "acute-angled triangle"

Presentation: Take out the first triangle in the first row. Invite the child to identify the three sides and observe whether the sides are alike or different. all three sides are different, this is a scalene triangle. Relate the story of the farmer and the ladder he used to pick fruit from his trees. Unlike the ladders we use today, the rungs of this ladder were all different lengths. These ladders are still used today in lesser developed countries. Just as all the rungs are different lengths, the sides of this triangle are all different lengths (scalene: Latin scala, usually plural scalae - ladder, flight of steps or Greek: skalenas - limping, uneven).

Isolate the second triangle in the first row. Invite the child to carefully observe its sides - two are alike. This is an isosceles triangle (isosceles: Greek isos - equal, and sceles - legs; thus having equal legs). Here it means two equal legs, or sides.

Isolate the third triangle. By observing and turning the inset in its frame, the child sees that all of the sides are the same. This is an equilateral triangle (equilateral: Latin aequus - equal, and latus, lateris - a side; thus having equal sides). Place the three insets on the table and do a three period lesson.

Isolate the first triangle in the second row. Identify the right angle. This is a right angle, it is erect. This is a right-angled triangle. How many right angles does it have? Only one.

Isolate the second triangle. Identify the obtuse angle. Obtuse means dull. This is an obtuse-angled triangle. Count the obtuse angles... only one.

Isolate the third triangle. All of these angles are smaller than the right angle. They are acute angles. Acute means sharp, pointed. (feel how it is sharper than the right or obtuse angles). This is an acute-angled triangle. How many acute angles does it have? Three.
Bring out the three triangles and review the first period. The triangle must have one right angle to be a right-angled triangle... and so on. Second and third periods follow. Give the child the reading labels.

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RECTANGLES

Materials: Reading labels: five "rectangle" and one "rectangle/square"

Presentation: Isolate the first inset. Identify it and give etymology (rectangle: Latin rectus- right, and angulus - an angle; thus having all right angles. Invite the child to identify the other rectangles as they are isolated.

Isolate the last inset. This is also a rectangles because it has all right angles, but it is also a square. Do a three period lesson and give the child the reading labels.

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REGULAR POLYGONS

Materials
: Reading labels: "pentagon", "hexagon", "octagon", "nonagon", "decagon", and a series of ten cards: "</angulus", "3/tria-", "4/quatuor-", "5/pente", "6/hex-", "7/hepta-", "8/okto-", "9/nonus or ennea", "10/deca-", "n/polys-"
Drawer 3 and the frame and inset of triangle and square from the presentation tray.

Presentation: Position the two extra insets to the left of the drawer in line with the top row. Isolate the triangle. Invite the child to identify an angle. Identify one on the square also. Isolate the decagon and invite the child to identify an angle. Feel it and compare it to the triangle and square. This angle is less sharp than the angles f the triangle.

Present the symbol card which represents angle (<). Identify the angles on the triangle and count them. Place the 3 card and the angle card side by side over the inset frame. Continue with each of the other figures, counting the angles, and placing the corresponding numeral card with the angle card. Since there is only one angle card, it floats from one inset to the next as needed.

Isolate the triangle inset and the two cards 3 <. The child identifies the figure and gives the meaning of its name. Then turn over the cards reading the Latin words which were made into a compound word to get triangle. Return the inset to its frame with its number card.
Isolate the square inset and cards: 4 <. Turn over the cards to find that 4 angles was quatuor angulus, from which our word quadrangle was derived.

Go on naming the other figures in this way using the Greek word for angle - gonia. Note: nonus - ninth, and ennea - nine.

After ten we have no more figures in our materials. Imagine a figure with any number of sides... 15, 20, 100, any figure with more than three sides. We can indicate this number by n. Bring out the card and place next to it the angle sign. turn over the cards: polys - many, and gonia - angle. Any figure that has more than three sides is a polygon. All of these figures we've examined up to now are polygons.
Beginning with the triangle turn all of the figures in their frames to show that the sides and angles are equal. All of these are "regular polygons". Name each figure: regular triangle is an equilateral triangle; a regular quadrangle is a square; a regular pentagon; a regular hexagon... and so on. Do a three-period lesson and give the reading labels.

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CIRCLES

Materials: Reading labels: 4 "circle", 1 "circle (smallest)", 1 "circle (largest)"

Presentation: The child identifies all as circles and puts out the reading labels.

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OTHER FIGURES

Materials: Reading labels: "trapezoid", "rhombus", "quatrefoil", "oval", "ellipse", "curvilinear triangle" or "Reuleaux triangle"
Frame of the circle inset (for presentation of ellipse)

Presentation: Isolate the trapezoid and identify it (trapezoid: Greek: trapezion - a little table). In order to understand why this figure has its name we must go back in time to see what a table of the Greeks looked like. Nowadays our tables don't look trapezoidal. Some Spanish tables have two legs but still not trapezoidal. The Greek table was like a Spanish table because it had two legs, yet it was more stable because the legs were inclined.

Isolate the rhombus and identify it. This is a rhombus (rhombus: Greek: rhombos - magic wheel, top) In ancient Greece, in the city of Athens, during a religious procession through the streets, a priest walked along with a cane (rod) raised over his head. At the end of the cane there was a cord attached, and at the end of the cord there was a rhombus-shaped figure attached. He rotated the cane in the air as he walked causing this figure to spin around like a top, making a characteristic sound. This was part of a religious ritual.
Isolate the quatrefoil and identify it (quatrefoil: Old French quatre - four, and foil - leaf). This figure has the shape of a four-leaf clover, considered a sign of good luck.

Isolate the oval and identify it (oval: French ovale <Latin ovum> - egg). This figure has the shape of an egg.

Isolate the ellipse and identify it (ellipse: Greek elleipsis - an omission or defect <elleipo - to leave out>). What has been left out? Think of the ideal figure, the circle. Place the inset of the ellipse in the circle frame and it is easy to what is missing. This is also the shape of the path that the earth follows around the sun.

Isolate the Reuleaux triangle and identify it (curvilinear: Latin curvus - curved, and linear - a line). This triangle has three sides which are curved lines. It is named after a man name Reuleaux who discovered the properties of this shape. He found that a drill bit made in this shape will make square holes.

Give three-period lesson and give the reading labels.

Age: 6 years and on

Aim: Knowledge of the geometric figures and their relative exact nomenclature.

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xConstructive Triangles - First Series


Note
: The triangle, the smallest figure in reality is the constructor of all other figures in reality. The tetrahedron, the smallest solid in reality constructs all of the other solids in space.

Materials:
Box 1-
Two yellow equilateral triangles
Two yellow, two green right-angled isosceles triangles
Two yellow, two green, two gray right-angled scalene triangles
One red smaller right-angled scalene triangle
On red obtuse-angled scalene triangle
(Each triangle has a black line along one side)

Box 2 -
Two blue equilateral triangles
Two blue right-angled isosceles triangles
Two blue right-angled scalene triangles
One blue obtuse-angled scalene triangle
One blue right-angled scalene triangle (corresponds to the red triangle from Box 1)

Box 3 -
Twelve blue right-angled scalene triangles with no black lines. The angles measure 30, 60, and 90 degrees.

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FIRST BOX

Presentation: Invite the child to remove the triangles from the box, and then sort the triangles according to shape. Having done this, ask the child to separate each pile according to color, resulting in various piles of triangles having both shape and color in common. Isolate the two red ones to be used later.

The teacher takes the pile of two equilaterals and separates them in such a way that the two black lines are facing each other. Watch, these black lines are like a magnet. Slide the two triangles together so that the black lines meet. Invite the child to do the same, leaving the joined triangles in place.

Identify the figures that have been constructed: a yellow rhombus, a green square, and a gray rectangle. The teacher identifies the other three figures as common parallelograms (parallelogram: Greek parallelogramium <parallelos, parallel, and grame, figure>). therefore a parallelogram is a plane figure having parallel sides. By simultaneously running two fingers along two parallel sides, the teacher gives a sensorial impression of parallel. We also call them common parallelograms to differentiate them from the square, rectangle, and rhombus which could also be considered parallelograms. The child names each figure as they are indicated by the teacher.

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SECOND BOX - CONSTRUCTION OF QUADRILATERALS

Presentation: Invite the child to sort the triangles by shape. As before, set aside the two small triangles which correspond to the red ones of the first box.

Isolate the two equilateral triangles and invite the child to form all of the possible quadrilaterals. try as he might, he can only form one. The child identifies it as the rhombus.

Leaving the rhombus intact, the teacher takes the two right-angled isosceles triangles and forms the possible figures. The child identifies the figures as they are made. there are two: the square and the common parallelogram.

The child may see two different parallelograms. Trace one on a sheet of paper. Form the other and superimpose it. The second parallelogram doesn't fit inside the contours of the first. Trace the second parallelogram and cut out the two figures. By placing the cut-outs back to back, we can see that one is the mirror image of he other, therefore they are the same parallelogram.

The child is invited to form the possible quadrilaterals with the two right-angled scalene triangles. The child identifies the three:

rectangle, common parallelogram, a different parallelogram.

One by one, isolate each type of triangle, ask the child to classify the triangle according to its sides, and ask, "Of how many different lengths are the sides of this triangle"? Conclude that with a triangle whose sides have all one measure, we can form only one figure - the rhombus. With a triangle whose sides have two different measures, we can form to figures - square and common parallelogram. With a triangle whose sides have three different measures, we can form three figures - rectangle and two parallelograms.

Direct Aim: To give the relationship between the number of different lengths of the sides and the number of figures which can be possibly constructed.

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SECOND BOX - HEADS & TAILS

Presentation: Every plane figure, like a coin, has two sides. One side is called obverse or heads; it is the side which has a face or the principle design. When you turn it over, you have the reverse side or tails. All of these figures have an obverse (blue) and a reverse (natural wood or white) side.

Isolate the two equilateral triangles. Invite the child to form as many figures as possible. As before, he can make only one. Suggest that he tries with one obverse and one reverse side. It won't help. There is only one figure he can make.

With the two isosceles triangles he makes the two possible quadrilaterals with the obverse sides. Invite the child to make a triangle. The child classifies the triangle: isosceles. By turning one triangle to its reverse side, the child can make no new figures.

With the two scalene triangles, the child tries to form all possible quadrilaterals first with the two obverse sides (yielding the same three figures as before) and then with one reverse side. The child is able to form a new figure: a kite (or a deltoid, having the form of the Greek capital letter Delta ).

Invite the child to make triangles, first with obverse sides (yielding none) and then with one reverse side. The child classifies the triangles he makes: acute-angled isosceles and obtuse-angled isosceles triangles.

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TRAPEZOID

Presentation: With the two small red triangles, invite the child to unite the triangles along the black lines and identify the figure obtained - a trapezoid.

Using the two corresponding blue triangles, invite the child to form the figure which he already knows and to identify it - a trapezoid. Continue making other quadrilaterals using only the obverse sides. The teacher identifies the figure obtained. since it has four sides we can call it a quadrilateral. It is a concave quadrilateral - a boomerang (it may also be called a re-entrant).

Invite the child to turn over one triangle to form any other figures, quadrilaterals or triangles. The quadrilateral is called a common quadrilateral. The triangle is an obtuse-angled isosceles triangle. (Note: This triangle has great importance in the later study of the area of a trapezoid.) Recall the figures formed by these triangles; there are four.

Age: After 6 years

Aim: Exploration of the triangle as the constructor of triangles and quadrilaterals.

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THIRD BOX

Presentation: Isolate one triangle. Ask the child to identify each of the angles, and the biggest and smallest angles. This angle which is neither smallest nor biggest we can call the medium angle. The child names each angle: smallest angle, medium angle, biggest angle.

First star: Let's unite all the triangles by their smallest angle. The teacher positions a few and allows the child to continue. How many points does this star have? Twelve. with all of the triangles at our disposal, we can make only one star with twelve points.

Second star: Let's unite all the triangles by the medium angles. How many points does this star have? Six. Try to make another star with the triangles that are left. With all the triangles at our disposal, we can make two stars with six points.

Third star: Let's unite all of the triangles by the largest angle. How many points does this star have? Four. This symbol is very famous; it is the star of Saint Brigid, the patron saint of Ireland. Try to make another star like this. With all the triangles at our disposal, we can make three stars with four points.

Aim: Use of the triangle as a constructor to indirectly demonstrate the following:
30o x 12 triangles = 360o 60o x 6 triangles = 360o 90o x 4 = 360o
360o / 30o = 12 tris. 360o / 60o = 6 tris. 360o / 90o = 4 tris.
360o / 12 triangles 360o / 6 = 60o 360o / 4 = 90o

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Related activities
:

1. Construct the first star. Notice that the triangles meet at a point in the center. We must divide this star into two equal parts, leaving six triangles on one side and six on the other. Many possibilities exist; simply choose on and slide the triangles away to leave a gap.
We want to make the point at the top of one side meet the point on the top of the other. Slide one half along and then towards the other to make the two points meet at the top. We see that they have met at the bottom also, and where there was a point in the center there is now a line segment.

Again divide the figure in half, this time along the other side of the triangle which was displaced before. Separate the two halves to leave a gap. Identify the two points at the top and bottom which should meet. Slide one half into position. We see that a quadrilateral (a rhombus) has been created at the center.

Continue in the same manner, identifying the figure formed at the center each time: equilateral hexagon, equilateral octagon, equilateral decagon, equilateral and equiangular, therefore regular dodecagon. This is the first diaphragm. It is like the diaphragm of a camera. Bring one in to demonstrate.

2. Construct the second star. As before, divide into two equal parts. Slide one side so that the vertices of the extreme angles meet. Note the change from a point to a line segment. Continue naming each of the figures made, ending with the equilateral and equiangular, therefore - regular hexagon.

3. Construct the third star. Divide as before and slide one half. In this only the point, line segment and square are formed in the center. This is the third diaphragm.

Note: This third diaphragm will serve as a point of reference for two algebraic demonstrations of the Pythagorean theorem.

4. The children draw, cut, and paste the stars and diaphragms.

5. Older children may solve for the areas of the diaphragms and their internal figures, and find the relationship between them.

6. Constructing the second or third star, the child forms other figures by fitting in the angles.

7. Encourage further explorations using these triangles.

Direct Aim: Exploration of shapes using triangles.

Indirect Aim: Preparation for the sum of exterior and interior angles.

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