Box 6–13 Expanded Algorithm for Multidigit Multiplication
to drop steps in this algorithm can do so readily, with a result looking something like the common algorithm in Box 6–11, except that it has, in this case, four instead of two partial products to be added. These four can even be collapsed into two for those students who wish to do so. Therefore, the expanded model permits students to function at their own level of competence and is likely to help them understand what they are doing. The key point is that regardless of the algorithm that students use, they should be able to explain what they are doing and why it works.
Multiplying by a three-digit number is an extension of the two-digit version that requires the development of new understanding about multiplying by hundreds. The expanded algorithm for these larger numbers is relatively easy to carry out because the necessary steps are visible, although the number of partial products more than doubles. Given the accessibility of calculators, it might not be wise for students to spend a great deal of valuable school learning time becoming efficient at multiplication with three-digit or larger numbers. There is no research on how much pencil-and-paper computation is necessary or the impact of experiences with calculating with larger numbers on other mathematical understanding. Having some experience working with larger numbers, however, seems essential if students are to extend their conceptual understanding of multiplication and develop their ability to estimate