In algebra, the distributive property is required for the multiplication of even a monomial by a binomial. In arithmetic, the distributive property is used but not required.

      To set the stage for OPERATIONS (addition, multiplication, division, factoring, etc.) work with REPESENTATION (represent, write, show) is very important. If you have not worked with representing and writing expressions, please do so before continuing with multiplication.

      Example 1, below, illustrates how the distributive property develops naturally from work with representing and writing algebraic expressions. Consider how representing and writing an expression for "double the sum of a number and three" evolves into "multiply 2 times x+3."

      Example 3 is even more complicated. It illustrates that distributing a binomial over another binomial is what is usually called multiplying two binomials.

Example 3 edited in red at the right, displays the traditional "first, outer, inner, last" format for binomial multiplication called FOIL.

      This author teaches FOIL, prefers FLOI for its use in multiplication of complex numbers, but, emphasizes most that as long as 4 multiplications are completed, it doesn’t matter what order the multiplications are performed.

      Example 4 illustrates that removing zeros might be required.

      Example 5 "looks funny" and is NOT USUALLY DISCUSSED MANIPULATIVELY. It emphasizes the fact that the displayed product is really the sum of the areas no matter what shape that area takes.

Example 6 can't be discussed manipulatively with area TILES. The product involves a cube, a volume. The x³ tile or token represents a 3 dimensional idea in a 2 dimensional way. A volume TOKEN is required.
The symbol on the cube says: This is a cube with a volume of x³. Its base has an area of x² and its height is x.

      Using the cube token, products not normally considered with manipulatives may be examined. Read "Special Products".