What does it mean to multiply fractions? When we multiply whole numbers, we can think of it as rows and columns, or groups and amounts. For example, 3 X 5 can be modeled as 3 groups of 5 (or 5 columns and 3 rows).
But what about 2/3 X 4/5? Can we have 2/3 group of 4/5? It is easier to use an area model.
AREA MODEL (room and rug)
On graph paper, draw a rectangle with the dimensions of your denominators. For this example, your rectangle will be 3 by 5. This is the size of your "room".
Now use the numerators of each fraction to draw another rectangle in the corner of your room. Make sure you choose the numerator to be on the same side as its denominator. That is, make the rug as wide as 2/3 or two out of three squares (in this picture, up and down). The rug will be as long as 4/5, or four out of five (in this picture, across).
Now set up a proportion for the rug and room -- how many squares does the rug cover? (8) How many squares does the room cover? (15) So the rug covers 8/15 of the room. You may simplify the fraction answer if needed.
Now think about the traditional multiplication algorithm for fractions. In our picture, we simply put the dimensions of the rug (2x4) over the dimensions of the room (3x5). By this model we see that we need to multiply across the top and multiply across the bottom.
DISCUSSION: Do we need a common denominator for fraction multiplication?