What do these represent?
First, let's consider the lid of the binomial:
Look at each side and label each section a or b - so the length of each side is a + b
To find the area of the whole thing, we would multiply the length times the height (a + b)*(a+b)
We can also write that as (a+b)squared
Ok, so let's multiply that out - we end up with these pieces:
a-squared + ab + ab + b-squared
And that is what we have: a red square, 2 rectangles representing a times b, and a blue square.
But we have a CUBE.
So we multiply length by height by depth - (a + b)*(a+b)*(a+b)
Multiplying this out, we end up with a component for each prism within the binomial cube box.
Each piece represents a mathematical principle.
The same concept applies for the trinomial cube, except there are 3 factors: a, b, c
As the children proceed through other mathematical experiences, they will begin seeing these same images in SO many other places. Because these principles surround us.
And most of you only remember those formulas from high school algebra, right!? ;)