Concepts in Action: Difference and ratios between similar figures

6-9.

Materials

• From the triangular box : T1-grey equilateral, and T3- red equilateral
• From the large hexagonal box: T1-yellow equilateral, three yellow obtuse-angled triangles having the black line on the hypotenuse
• From the small hexagonal box: T2-yellow equilateral, six small grey equilaterals

Presentation

1. Identify the two triangles by the symbol names T1 and T2.
2. Construct the hexagons and identify them H1 and H2.
3. If the child has not already discovered it, lead him to the conclusion that T1 - T2 = 1/4 T1.
4. Since T1 has the value of 4/4ths, and T2 has the value of 3/4ths, T1 - T2 is the same as saying 4/4 - 3/4 which equals 1/4.
5. Set up the equation using the materials and card for the signs. T2 = T1 - T3.
6. This means that the small red equilateral triangle has the same value as the grey portion that is left showing when T2 is superimposed on T1 concentrically, or so that one vertex coincides.
7. Knowing that H1 is the double of T1 and that H2 is the double of T2, we can conclude that their difference would be the double of 1/4 of T1, that is 2/4.
8. 2T1 - 2T2 = 2 (1/4 T1) = 2/4
2(4/4) - 2 (3/4) = 8/4 - 6/4 = 2/4
9. Using T1 as the unit, assign relative values to the hexagons: H1 = 8/4 and H2 = 6/4 (T1 = 4/4).
10. The large hexagon is 8/4 of the grey triangle; the small hexagon is 6/4 of the grey triangle.
11. Therefore, the small hexagon is 3/4 of the large hexagon.
12. Examine also the inverse of each of these statements as they are also true.
13. Since the ratio between the small triangle and the large triangle is 3:4, then the same relationship exists between their doubles, the small hexagon and the large hexagon ... 3:4.
14. Superimpose H2 on H1 concentrically with the sides parallel.
15.  The frame is the difference between the two hexagons, and, therefore, has the value of 2 1/4 equilateral triangles.
16. Thus the difference between the hexagons is a rhombus.
17. This is the arithmetical way of showing their difference.