Concepts in Action: Triangular Box

Age

6-9.

Presentation

1. The child empties the box, sorts the pieces and unites them as usual.
2. They all make equilateral triangles.
3. Superimpose each equilateral triangle formed of several pieces on the grey unit triangle.
4. Determine that the unit has been divided into 2, 3, 4 equal pieces and assign the fraction values to each piece.
5. Determine how the unit triangle was divided in each case.
6. Isolate the two right-angled scalene triangles.
7. As the child did with the blue triangles of the first series, he tries to form all of the figures possible with these two triangles.
8. Excluding the original equilateral triangle, there are five: rectangle, two different parallelograms, obtuse-angled isosceles triangle, and a kite.
9. The child names the figures as he makes them.
10. Determine equivalence between each figure and the unit triangle.
11. Superimpose the two triangles on the unit triangle to show congruency, and to recall their value.
12. Any other figure made with these two halves has the same value as the unit, and is equivalent.
13. Show that the figures are equivalent among themselves.
14. rectangle = equilateral triangle
15. > therefore: rectangle = kite
16. kite = equilateral triangle
17. As a parallel exercise, the children may trace the triangles onto colored paper, cut them out, construct the figures using a compass and ruler.
18. Next, we draw the child's attention to the relationships that exist between the lines of the equilateral triangle and those of the five figures.
19. Then we examine the relationship between the lines of the five figures.
20. Recall the nomenclature of the equilateral triangle: side, base, height (shown by superimposing one-half) and semi-base.
21. Be careful to position the parallelograms so that the height is represented by a side of the triangles which form it.
22. Identify the properties of the kite.

Rectangle: base = 1/2 the base of the equilateral triangle
height = the height of the triangle
Parallelogram: base = the height of the triangle
height = 1/2 the base
side = side
Kite: longer side = height
shorter side = 1/2 the base
long diagonal = side

  23.   Note: In identifying the nomenclature of this figure, the child showed where the shorter diagonal would be

  24.   The characteristic of this figure is that one diagonal is the perpendicular bisector of the other.

  25.   However, there is no relationship to be made with the short diagonal.

  26.   Later the child may discover that this short diagonal is equal to the side of T2.

Obtuse angled isosceles triangle: base = twice the height
height = 1/2 the base
equal sides = sides

  27.  In examining the relationship between the lines of the various figures, paper figures must be constructed for each figure in order to leave the triangle free for use as a measuring instrument.

  28.  Be sure that the child understands that the number of comparisons to be made will decrease with each figure: the first is compared with four others, the second with three, the third with two and the fourth with only   

        one, the last figure.

  29.  Arrange the figures so that the key figure is isolated above the row of other figures.

  30.  Place the triangle first on the key figure and name a line, then find the corresponding line on the figure below.

  31.  Identify all of the relationships between the key figure and the first figure, then go on to do the same with the other figures in the row.

  32.  After the possibilities of the first key figure have been exhausted, one of the figures below becomes the key figure, and so on.