# Introduction to Hierarchical Material

INTRODUCTION

These materials are the geometric representation of the quantities from one unit to one million-the powers of ten; 100 to 106 Having reached one million the child will easily imagine the succeeding hierarchies.

Age: 7 years

Materials:
…the wooden materials made of light wood to facilitate movement, in relative proportions:
…1 – green cube – .5cm
…10 – blue rod with green lines .5 x .5 x 5cm
…100 – red square with blue lines .5 x 5 x 5cm
…1000 – green cube with red lines 5cm
…10,000 – blue rod with green lines 5 x 5 x 50cm
…100,000 – red square with blue lines 5 x 50 x 50cm
…1,000,000 – green cube with red lines 50cm
…numeral cards 1; 10; 100; 1,000; 10,000; 100,000; 1,000,000 all on white backing with numerals printed in black
…a ruler or stick or an expensive laser pen

Presentation:
The materials should be laid out in a row as they are presented, from right to left. Isolate the unit cube and identify it. This is one. If I had ten of these little cubes and placed them end to end, I would have this rod. This is ten. Place the cube along the side of the rod to count the ten sections. If I had 10 of these tens, I would get one hundred-this square. Count the sections of the square using the rod. Place the three pieces on the table in a row, and place the ruler on top. These all have the same height. Identify the three pieces again – 1, 10, 100, unit, ten, hundred. They are numbers of the simple class. Set the stick aside.
Isolate the thousand cube. This is still a unit, but it is a unit of the thousands. Compare its color and shape to the unit cube. Present the ten and hundred as before. Place the stick on top to see that they are all the same height. They are 1, 10, 100 but of thousands.
Present the million cube. This is still a unit, but it is a unit of millions. Imagine the ten of millions. It would be as long as ten of these side by side. Imagine also the hundred, which would be made of ten of these tens. These would make up the class of millions.
Review the first period giving the names of the classes and the names 1, 10, 100 for the cube, rod, square, to show how these three orders are repeated in each class. The dominant figure of each class is the cube for it gives us the name of the class. Compare these materials to the concepts of point, line, surface and solid-which is only a point of the next line. The point is represented bigger each time. Even the Earth, as big as it is, is just a tiny point in space. Three period lesson

Give the child the symbols to match by placing on top of the material. Identify for the child the symbols of 10,000 · 100,000 · and 1,000,000. Notice that the comma corresponds to a change in hierarchical class.

Games:
Distribute the cards to a group of children and they place the cards on the appropriate material. Give each child a piece of material and he finds the right card.
Ask the child to identify a piece of material, the class to which it belongs, the reason for the color of the lines, of what it is composed. In this way the child will be able to form definitions in his own words. Emphasize that the superior hierarchy is always formed of 10 of the preceding hierarchy.
The child draws the material in his notebook or cuts and pastes the pieces using a different scale of measurement. The cube is drawn as a three-dimensional image.