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.

 

AGE:

4 to 5 years.

CONTROL OF ERROR:

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). 

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)<sup>3

    a + b
x a + b</sup>
    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

      

 

 

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