Concepts in Action: Smaller hexagonal box
- At first the large yellow equilateral triangle and the six obtuse-angled isosceles triangles are removed from the box.
- The remaining triangles are all small equilateral triangles: six grey, three green, two red.
- The child begins as usual.
- He names the figures he has formed: hexagon, trapezoid, rhombus.
- Then values are assigned to the three figures.
- Superimpose all of the triangles to show congruency.
- Reconstruct each figure and count the number of congruent triangles in each.
- The trapezoid has 1/2 as many pieces as the hexagon.
- Separate the hexagon to show two trapezoids.
- The rhombus has 1/3 as many pieces as the hexagon.
- Separate the hexagon into three rhombi.
- 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.
- Examine the relationship between the lines of the three figures.
- Note this time that the diagonals of the hexagon connect opposite vertices. classify the trapezoid.
- It is an isosceles trapezoid, but it is more than isosceles.
- Since it is made up of three equilateral triangles, it has an extraordinary characteristic.
- The longer base is equal to the equal legs.
- Present the inset and the label and add it to the other insets.
- Just as the equilateral triangle is an isosceles triangle plus, the equilateral trapezoid is an isosceles trapezoid plus.
- Show also that bilateral symmetry exists in the hexagon and rhombus.
- At this point the large yellow equilateral triangle and the six red obtuse-angled triangles are returned.
- When the child joins the triangles, three rhombi are formed.
- The triangles are stacked up to prove that they are congruent.
- Thus the three rhombi are congruent.
- Put the three rhombi together to form a hexagon, thus the hexagon is formed of six equal triangles.
- Open the hatch of the hexagon and take out the three red triangles, replacing them with the yellow equilateral triangle.
- Observe that the equilateral triangle is inscribed in the hexagon.
- 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.
- 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.