Presentation of Concepts: Equivalency

Age

9.

Materials

• Metal insets of the square and its subdivisions
• Geometry chart: square 10 (5 and 6 for review)
  

Presentation

1. Isolate the whole and the rectangle halves and triangle halves.
2. Recall the value of each inset.
3. Remove the whole inset from its frame and try to place the two equal rectangles in its frame; these two squares are equal.
4. Repeat the procedure for the triangle pieces.
5. Refer to chart S5.
6. All three squares are equal to one another.
7. Identify 1/2 of the square; a rectangle; also a triangle. Isolate one rectangle and one triangle.
8. Each piece has the value of 1/2 of the same square. Are they congruent? Are they similar?
9. 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).
10. Repeat the experience with the fourths, eighths, and sixteenths.
11. If two figures have the same fractional value of the same whole, then one can be transformed into the other.
12. Use the frame to transform the 1/2 rectangle into a 1/2 triangle.
13. Fill the space vacated by the rectangle with small pieces: 1/4 square, 2/8 triangles.
14. Remove these pieces and arrange them on the table in the shape of the triangle.
15. Change the triangle into the rectangle, using the same smaller pieces and reversing the process.

Purpose

Direct Aim:

•  To furnish the fundamental concepts: congruency, similarity, equivalence.

Indirect Aim:

•  To serve as a base for the following material - constructive triangles.
  

Variation

Exercises:
1. For each of the three concepts, ask the child to identify the insets that bear that relationship.
2. The directress 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.