**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.

**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.

**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.

**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.

**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.

**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.

**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.

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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