## 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.

T2 + 2TU + U2 = (T + U)2   or
a2 + 2ab    + b2 = (a  + b)2
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 asked 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:

The teacher takes the binomial cube to a table, sits next to the
child, and takes the cube out of the box.  The teacher then invites the child to view the cube from all sides.

The teacher removes the top layer of the cube.  She lets the child see that the two layers are different in height.  Then, starting with the red cube (a3) from the tallest layer, the teacher takes the layer apart and arranges the pieces in front of the child in order (setting the pieces according to the formula).

The cube is taken apart piece by piece beginning with "a3."  The first row is set out at height "a," and the second row is set out at height "b" according to the formaula.

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 the child and work slowly.
And so, 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 the child how to lay out the pieces.

When the formula has been set out on the table the teacher and
child view it for a minute or so.

The teacher then shows the child how
to rebuild the cube, starting with the "a3," 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 he or she is ready to do so. When the child has finished, the
teacher shows the child how to replace the cube in the box.

### EXERCISE:

Cube 1: Colored Binomial

The child takes the cube apart beginning with a3 and lays out the
pieces as shown, according to the formula. The child 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. The child must build the
cube in the same way as before, but matching faces by size instead of color.
This, leads towards the mathematical understanding of the cube.
Formula for the cube (a+b)3

a + b
x a + b

a2 + ab
ab + b2

a2 +2ab + b2
x           a  + b
a3+2a2b +ab2          First layer         (Multiplied by a)
_____a2b+2ab2+b3    Second layer     (Multiplied by b)
a3+3a2b+3ab2+b3

### Note:

One way that humans attempt to survive is by understanding the world around them.  The human brain is a pattern seeking organism.  So, by nature, children are interested in finding patterns, relationships, and order.  If children have worked their way through the materials for dimension, color, and shape, they will have found order, patterns, and relationships in those materials, and will have developed the ability to discriminate attributes to a point where they will enjoy the challenge of exploring the order inherent in the binomial and trinomial cubes.  For this age, 4 to 6, the purpose of the material is not to teach math, but instead, to provide a challenge for a child's ability to find patterns and relationships. Therefore, the material is presented as a sensorial activity.  It is presented like a three dimensional puzzle.  Anyone who likes to do puzzles knows that in order to master a puzzle, you have to pay attention to the relationship between the pieces.  People who are masters at puzzles will tell you that they take out, and organize, puzzle pieces very carefully.  This is what is modeled for the child in this activity.

The math presented above and below is provided for the teacher and is not to be presented or discussed with the child of this age.  The math is presented to the children when they are older and are ready for it.

As mentioned above, the binomial represents two numbers represented symbolically as (a + b).  We could represent the numbers with (T + TU) for Tens plus Tens times Units.  The pattern for the binomial squared is apparent on each of the faces of the binomial cube.  It is represented below:

This pattern for the binomial squared can also be seen when building a square of the number with the golden beads.  For example, for the number 13, which is (10 + 3), the pattern for the square of the number looks like:

(10 + 3)2

a + b
x  a + b

a2 + ab
ab + b2

a2 +2ab + b2

or

10 + 3
x   10 +3
102+(10x3)
_          (10x3) + 32
102+2(10x3)+32