Number & Operations for Teachers
Copyright David & Cynthia Thomas, 2009
Modeling Multiplication with Regrouping
In addition to understanding the meaning of multiplication and division and their properties, students must also become fluent in performing complex arithmetic computations involving these operations. For many students, the standard algorithms associated with these operations are a mystery and a source of considerable frustration.
Figure 3.11: Modeling Multiplication
Figure 3.11 presents a concept model and expanded algorithm that emphasizes the meaning of multiplication and explains why the standard algorithm works. In this model, the number 17 is represented as the sum 10 + 7. The ten is modeled as a strip and the seven as a string of ones. This quantity is then multiplied by two, seen as two identical rows, each containing a strip and seven ones. The expanded algorithm is an interpretation of the distributive law: 2(17) = 2(10 + 7) = 2(10) + 2(7) = 20 + 14 = 34.
A more complex example is seen in Figure 3.12. This figure shows an area model for the product 43 x 25 that could be assembled on a table top using base ten blocks or drawn on a piece of graph paper. Note that each large square is 10 x 10 = 100 square units and that the long, narrow rectangles are 1 x 10 = 10 square units. Beside the concept model is the standard algorithm for performing such calculations. For many students, the rules associated with this algorithm appear arbitrary and are easily confused.
The purpose of the expanded algorithm is to add meaning to the computational process and to associate each partial product in that process with a shaded portion of the Concept Model. The first step in constructing this algorithm is to rewrite the factors as 40 + 3 and 20 + 5. The arrows in the expanded algorithm indicate the four partial products formed. Each partial product is written down in a manner that displays its true value, not in the abbreviated form seen in the standard algorithm. As seen in the section labeled Partial Products as Areas, each partial product listed in the Expanded Algorithm is seen to represent a different area of the shaded Concept Model.
Further comparison of the Standard Algorithm and the Expanded Algorithm may be used to justify why the Standard Algorithm works, if that is important. But students who prefer the openness of the Expanded Algorithm should feel free to adopt that procedure as their favorite, if they so desire.
Figure 3.12: Concept Model and Expanded Algorithm for Multiplication of Whole Numbers
Sketch a concept model for the indicated product 11 x 32