of whole numbers, but the relationship is hard to see, mostly because there is no “carrying,” from the x to the x2 term, for example. The expanded method below shows the relationship a bit more clearly.

Box 3–10 Examples of Algorithms

The decimal place-value system allows many different algorithms for the four main operations. The following six algorithms for multiplication of two-digit numbers were produced by a class of prospective elementary school teachers. They were asked to show how they were taught to multiply 23 by 15:

In Method 6, sometimes called lattice multiplication,* the factors are written across the top and on the right, the products of the pairs of digits are put into the cells (for example, 15 is written ), and the numbers in the diagonals are added to give the product underneath.

Note that all of these algorithms produce the correct answer. All except Method 4 are simply methods for organizing the four component multiplications and


The method is also called gelosia multiplication and is related to the method of Napier’s rods or bones, named after the Scottish mathematician John Napier (1550–1617).

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