Rationale
Over the years our students have been conditioned to look for short-cuts when trying to find a solution or solutions to math problems. There was one “shortcut” that my students attempted to teach me when they had to multiply multi-digit numbers. This shortcut was in fact the lattice method, sometimes known as the gelosia method of multiplication or the box method. Originating in India, the gelosia strategy appeared in various Hindu works, where they called the method quadrilateral. As this method became popular in other places, it’s taken on other names such as: the method of the sieve, method of the net, shabakh, and/or venetian squares.
The name was given in Italy as it resembled a grid or lattices that were added to the windows of popular women to protect them from public view. By the time Fibonacci and Europeans were introduced and familiar with the method, it was not practiced as much due its difficulty in being printed. 3
The diagonals in the lattice method represent the various digit place values. Unlike the lattice method the box method offers a very streamlined way of identifying the place value of each unique part. In fact, the lattice method comes from the box method but is very rarely explained.
In the calculation illustrated below, you will see the common schemes in both methods.
Figure 1: The Lattice (Gelosia Method) Box of Multiplication
Here is the basic part of the lattice method for 345´12. The antidiagonals, from the lower right to the upper left, represent the place values, from ones to ten thousand. To complete the calculation, sum along each antidiagonal, and carry the tens part of the sum to the next place.
Figure 2: The Traditional Box Method
Here you can see that each factor was decomposed into its place value parts (300, 40 and 5 for 345, and 10 and 2 for 12), and each place value part of one factor was multiplied by each place value part of the other factor. This is justified by way of an extended version of the distributive rule of multiplication. I will discuss this more general version of the distributive rule with my students, as it forms a key part of the justification of the box method. It is a natural consequence of the relationship between multiplication and area.
When I asked my students what the lattice method was, they could only show me an example of how it was used. The funny thing is, they could use the strategy to apply the rules of multi-digit multiplication; however, they could not explain the make-up or characteristics of the box used or answer why the method worked. “When we do not ask students to think visually about the growth of the shape, they do not have access to important understandings about functional growth.”4 Having seen my kids using the lattice method and almost always multiplying multi digits with accuracy, I decided to capitalize on my opportunity to use similar methods with multiplying polynomials; this time justifying the procedures using basic area models.
Comments: