## Binomial Cube

( a + b)3

### MATERIAL:

Cube 1: A cube composed of 8 wooden blocks which fit together
in a binomial pattern, representing the cube of two numbers, (a + b),
or tens plus units. All the blocks fit into a natural wood box.
Each side of the cube has the same dimensions and pattern, and
represents the square of (a + b) or (t + u). The faces of the small
blocks are color coded: a2 is always red, b2 is always blue, and
"ab" is always black.

Cube 2: This cube contains blocks of the same dimensions as Cube 1,
but it is made from plain unpainted wood.

### PURPOSE:

The child is at the stage of the absorbent mind. She child is
not made to understand the formula, but is using the cube in a
mathematical way. The child will build up a predisposition to
enjoy and understand mathematics later.

4 to 5 years.

Teacher control.

### PRESENTATION:

Take the binomial cube to a table. The teacher sits next to the
child, and invites her to view the cube. The teacher takes the
cube apart piece by piece beginning with a3, and lays it out very
carefully.

The first row is set out at height "a", and the second row is set
out at height "b", according to the formula. Do not explain to the
child why you are setting the cube out in this order, or talk
about the mathematics of the cube. Simply show him. Work slowly.

When the formula has been set out on the table the teacher and
child view it for a minute or so. The teacher shows the child how
to rebuild the cube, starting with the cube of a, taking each
piece in order. She lets the child see that she is matching the
faces according to color. She pauses after finishing the first
layer. Then, taking a2b, she builds the second layer by taking the
pieces in order, matching the colored faces. When the teacher is
finished, she lets the child view the cube from all sides. If
necessary she may lay out the cube again and rebuild it. The child
works alone when she is ready to do so. When she has finished, the
teacher shows the child how to replace the cube in the box, beginning
by placing the smallest cube (b3) in the back of the box.

### EXERCISE:

Cube 1: Colored Binomial

The child takes the cube apart beginning with a 3 and lays out the
pieces as shown, according to the formula. He reconstructs the cube,
matching red faces, black faces, and blue faces, beginning with a3.

Cube 2: Unpainted Binomial

This cube is introduced later. The teacher shows the child how to
handle the cube as cube 1, take it to pieces beginning with a3 as
before, and lay out the pieces according to the formula. The child then
rebuilds the cube. There is no color to help the child. He must build the
cube in the same way as before, but matching faces by size not color.
This, leads towards the mathematical understanding of the cube.