
The squares of binomials are sometimes addressed manipulatively. The cubes of binomials are almost never addressed manipulatively or in this depth. Term Tiles and, in particular, Tokens permit this presentation. There is insight to be gained through these tokens, even for the teacher.
A binomial squared is a prefect trinomial square. Any binomial may be squared, but the square of the sum of two numbers and the square of the difference of two numbers are important for the patterns they display.
A binomial cubed takes a binomial squared and multiplies it by the binomial. Above, the tiles and tokens have been oriented and labeled to facilitate verification of the multiplication. Notice that there are two groups of three token which have equivalent volumes and shapes, but, which appear very different. One more orientation is important to examine. The cube of the sum is really a cube with each side the length of the sum. The following examines the 2 layers of prisms required by the multiplication which creates the cube of the sum. The cube of the difference of two numbers is very similar. Take the square of the difference and multiply it by the difference. Shown below, the cube of the difference of two numbers is displayed in layers and as a cube of "negative and positive volumes."

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