Concepts in Action: Smaller hexagonal box

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  1. At first the large yellow equilateral triangle and the six obtuse-angled isosceles triangles are removed from the box.
  2. The remaining triangles are all small equilateral triangles: six grey, three green, two red.
  3. The child begins as usual.
  4. He names the figures he has formed: hexagon, trapezoid, rhombus.
  5. Then values are assigned to the three figures.
  6. Superimpose all of the triangles to show congruency.
  7. Reconstruct each figure and count the number of congruent triangles in each.
  8. The trapezoid has 1/2 as many pieces as the hexagon.
  9. Separate the hexagon to show two trapezoids.
  10. The rhombus has 1/3 as many pieces as the hexagon.
  11. Separate the hexagon into three rhombi.
  12. Comparing the trapezoid to the rhombus, we see that the rhombus is 2/3 of the trapezoid, or the trapezoid is 3/2 of the rhombus.
  13. Examine the relationship between the lines of the three figures.
  14. Note this time that the diagonals of the hexagon connect opposite vertices. classify the trapezoid.
  15. It is an isosceles trapezoid, but it is more than isosceles.
  16. Since it is made up of three equilateral triangles, it has an extraordinary characteristic.
  17. The longer base is equal to the equal legs.
  18. Present the inset and the label and add it to the other insets.
  19. Just as the equilateral triangle is an isosceles triangle plus, the equilateral trapezoid is an isosceles trapezoid plus.
  20. Show also that bilateral symmetry exists in the hexagon and rhombus.
  21. At this point the large yellow equilateral triangle and the six red obtuse-angled triangles are returned.
  22. When the child joins the triangles, three rhombi are formed.
  23. The triangles are stacked up to prove that they are congruent.
  24. Thus the three rhombi are congruent.
  25. Put the three rhombi together to form a hexagon, thus the hexagon is formed of six equal triangles.
  26. Open the hatch of the hexagon and take out the three red triangles, replacing them with the yellow equilateral triangle.
  27. Observe that the equilateral triangle is inscribed in the hexagon.
  28. Superimpose the red triangles (that were just removed) on the yellow triangle to show that the equilateral triangle is made up of three red triangles.
  29. Since the hexagon is made up of six red triangles, the triangle is 1/2 of the hexagon.

Control Of Error

Points Of Interest

This hexagon will be called H2 and the large yellow equilateral triangle is called T2. Therefore T2 = 1/2 H2 and T2 is inscribed in H2. The smaller equilateral triangles which are congruent to the small equilateral triangles of the first box, and therefore have the value of 1/4 T1, will be called T3.