In the past, you may have learned particular algorithms for the multiplication and division of fractions. We are now going to use some of the visual models we've employed earlier in this course to better understand what is actually happening when we perform these operations. Note 2

First we'll use an area model -- one that superimposes squares that are partitioned into the appropriate number of regions, and shaded as needed -- to clarify what happens when you multiply fractions. For example, here's how we would use the area model to demonstrate the problem 3/8 • 2/3:

Shade one square, partitioned vertically, to represent 3/8 (shown below in pink):		Shade another square, partitioned horizontally, to represent 2/3 (shown below in blue):		Superimpose the two squares. The product is the area that is double-shaded (shown below in purple):

What is the value of this purple area? There are 3 • 2, or 6, purple parts out of 8 • 3, or 24, parts in all, so the value of the purple area is 6/24.

This model visually demonstrates the familiar algorithm: To multiply two fractions, multiply the numerators and then multiply the denominators. This algorithm "counts" both the purple parts (the product of the two numerators) and the total number of parts (the product of the two denominators).

We can also use this model to "reduce" the fraction. First we swap the positions of some of the purple parts. Two of the purple parts can be moved to the top, and thus, two of the eighths are now shaded. These two eighths are the same area as one quarter:

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