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Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
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Modeling the Meaning of
Multiplication As with whole numbers and integers, multiplication may be represented as repeated addition. This approach works well when one of the factors is a whole number. For instance, the product 4 x 1/3 may be thought of as 1/3 + 1/3 + 1/3 + 1/3. Products such as this may be easily modeled using fraction bars (See Figure 5.1) and number lines (See Figure 5.2).
The meaning of a product such as 1/2 x 1/3 is less obvious. One of the most useful models for explaining fraction multiplication is the area model. In Figure 5.3, a unit square is partitioned two different ways. On the left of Figure 5.3, the shaded portion represents 1/2 the area of the unit square. In the middle of Figure 5.3, the shaded portion represents 1/3 the area of the unit square. On the right of Figure 5.3, the two partitions are superimposed on one another. The cell in the upper-left hand corner is shaded darker than the rest because two shaded cells overlap there. The region of overlap represents 1/6 of the area of the unit square. In the area model for fraction multiplication, this area of overlap is the only portion of the unit square associated with the answer, 1/6, modeled as the quantity “one-half of one-third”. The other shaded squares are only useful to the extent that they help to identify the area of overlap.
Figure 5.3: An Area Model for Fractions This area of overlap may be thought of as representing … · “one-third of one-half” or “one-half divided by three”, since the horizontal lines divide the shaded region representing 1/2 the area of the unit square into 3 equal parts; or as · “one-half of one-third” or “one-third divided by two”, since the vertical line divides the shaded region representing 1/3 of the area of the unit square into 2 equal parts. In both of these verbal representations, use of the preposition of provides insight into the meaning of the multiplication symbol in the expression 1/2 x 1/3 and justification for the expectation that fraction multiplication may lead to products that are smaller than either factor. Figure 5.4 relates the concept models, written notation, and spoken language associated with these representations.
Figure 5.4: Multiple Representations An alternative shading strategy called cross-hatching is used in Figure 5.5. In this representation, the overlap is seen as four cells with diagonal lines running through them both directions. Each of these cells represents 1/15 of the area of the unit square. These four cells model the quantity 2/3 of 2/5 and collectively constitute the answer, 4/15.
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