**Presentation of the Concepts**

**1. Congruency**

**Materials**: Metal insets of the square and its subdivisions

**Presentation**: Invite the child to bring out the material and arrange it properly. Take any two inset pieces from the same frame.

Note: Montessori suggests the fourths to be taken from the frame as pictured. .

Superimpose the two pieces. Indicating the surface, tell the child that every point of one figure corresponds to a point on the other figure, that is, they can be superimposed perfectly. (The Norwegians say that one “answers” the other, rather than on “corresponds” to the other) This is a special quality that these two figures have. We can say that one is congruent to the other.

Repeat the experience with the other fractional divisions.

A. Similarity

**Materials**:

Metal insets of the square and its subdivisions

A red cardboard rectangle 20 x 2.5 cm

Metal insets of the triangles

Geometry charts: S9, T5

**Presentation**: Discuss the meaning of the word similar and give examples of things resembling one another, having the same traits, but different in other ways (similar).

Isolate the whole inset and the inset of the square 1/4. Ask the child to identify their shapes. They have the same name. are they congruent? Superimpose to find out. Hold up the small square while the child walks away with the large square to see that they look alike.

These two figures are similar. All squares are similar simply by their definition.

Isolate the 1/2 and 1/8 pieces and identify their shapes as rectangles. They have the same name. Recall the nomenclature – sides, base, height, angles. superimpose the angles to show that the angles are equal. Place two small rectangles adjacent to the larger rectangle to show that the base of the larger is twice the base of the smaller. Repeat the experiences in reference to the height. Thus, these two rectangles are similar to each other because their angles are respectively equal and the sides are in proportion to one another: the base and height of the larger are double those of the smaller.

Here the name was not sufficient to determine similarity, nor are equal angles sufficient; the sides must be proportionate.

Repeat the experience with two triangular fractional pieces: 1/2, 1/8. Classify them. Superimpose the angles. Use an extra 1/8 to demonstrate that the sides are proportionate.

Note: All square by their definition are similar to each other. Rectangles however must have proportionate sides. Compare the cardboard rectangle – 20 x 2.5cm to the 1/2 rectangle inset to see the extreme case of two non-similar rectangles. The base of the metal rectangle is twice the base of the other, while the height of the metal rectangle is half the height of the other.

(Triangle) Isolate the 1/2 triangle inset and classify it. Draw another right-angled scalene triangle which is not similar to show that the classification is not enough to render them similar. bring out a 1/3 inset and classify it. These two triangles (1/2 and 1/3) have nothing in common.

Show that the small triangle (1/4) which has the same name as the whole triangle (and thus, has equal angles) also has proportionate sides. Like the square which is the perfect quadrilateral, the equilateral triangle, which is the perfect triangle, is similar to all other equilateral triangles by its definition.

B. Equivalence

**Materials**:

Metal insets of the square and its subdivisions

Geometry chart: square 10 (5 and 6 for review)

**Presentation**: Isolate the whole and the rectangle halves and triangle halves. Recall the value of each inset. Remove the whole inset from its frame and try to place the two equal rectangles in its frame; these two squares are equal. Repeat the procedure for the triangle pieces. Refer to chart S5. All three squares are equal to one another.

Identify 1/2 of the square; a rectangle; also a triangle. Isolate one rectangle and one triangle. Each piece has the value of 1/2 of the same square. Are they congruent? Are they similar?

When two figures do not have the same shape, but have the same fractional value; they are equivalent (equivalent: Latin *aequus* – equal; *valere*, to be worth). Repeat the experience with the fourths, eighths, and sixteenths.

If two figures have the same fractional value of the same whole, then one can be transformed into the other. Use the frame to transform the 1/2 rectangle into a 1/2 triangle. fill the space vacated by the rectangle with small pieces: 1/4 square, 2/8 triangles. Remove these pieces and arrange them on the table in the shape of the triangle.

Change the triangle into the rectangle, using the same smaller pieces and reversing the process.

Exercises:

1. For each of the three concepts, ask the child to identify the insets that bear that relationship.

2. The teacher chooses an inset piece and asks the child to find a piece which is congruent (similar or equivalent) to it.

3. The child constructs his own geometry charts.

4. The child makes different shapes (). To find the value of the pine tree, place the pieces in an empty square frame. 1/2 + 1/4 + 1/8 + 1/16 = What will make the square complete? 1/16. Thus this shape is equivalent to 1/16 less than the whole: 15/16. Later, when the child has done addition with fractions having unlike denominators, the calculation can be done arithmetically.

Note: The village is formed with triangles and squares. Therefore the value of the village is 15/16 + 15/16 = 30/16 = 1 7/8. Each roof is equivalent to one whole.

**Age**: Nine years

**Direct Aim**: To furnish the fundamental concepts: congruency, similarity, equivalence.

**Indirect Aim**: To serve as a base for the following material – constructive triangles.

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