Alternative Math Methods

A lot of math is broken down into processes. Start with a problem, do this, do that and, voila, you have the answer. Most people in America learn the same processes for doing most basic arithmetic, and it’s easy to forget there might be other ways to solve these problems. The positional decimal system developed by the Indians (of India) is the standard taught in every school I’ve ever been in, but it’s not the only way to multiply two numbers1. The following is a description of another method I like for its enforced structure and visual guides.

The Lattice method of mulitiplication2 was published in Europe by Fibonacci in the early 13th century. The lattice, though time-consuming to create for variable lengths of multiplicands, brings a helpful bit of structure when teaching multi-digit multiplication to young students. Use the animation below as a guide when reading the instructions, and I think you’ll find this method very intuitive and interesting.

• Draw the lattice with the same number of squares per side as the number of digits in each multiplicand. Add a diagonal line from upper-right to lower-left for each square, extending the line out the bottom-left of the left-most and bottom-most squares. For students, I’d recommend pre-creating the lattice for them and photocopying.
• Place the digits of the multiplicands across the top and down the right side of the lattice.
• In each of the squares, multiply the multiplicand digits at the top and far right of the square, placing the ones-digit of the product in the bottom-right of the divided square, and the tens digit (0 if none) in the upper-left portion of the square. Repeat for all squares in the lattice.
• Note how the triangular sections of adjacent squares form diagonal “rows”. Starting with the lower-right triangle, sum these rows, placing the ones digits of the sums in the spaces formed by the extension lines outside the lattice. If a tens digit exists, add it to the next row’s total. Note the color-coding of the rows in the example image, which would help a new user of this method. Perhaps shading every other row would be appropriate when pre-making the lattice for your students?
• When completed, the final product can be read down the left side and across the bottom of the lattice.

How does it work? Well, a little study will show that this is really no different than the long multiplication we all know, only some of the operations have been re-ordered; essentially, all of the carrying has been moved to the end of the process. What I find so elegant about this method is the complete lack of “spacing”, padding, or “shifting” that we use in long multiplication which is so difficult to get students to understand and remember. Long multiplication can also be confusing when attempting to place carry values from two different operations in the same space, which can’t happen using the lattice.

Now, I’m not saying that this lattice method should be the new, one-and-only method of multiplication you should use. The lattice is obviously a little cumbersome to draw, especially for larger numbers, though it doesn’t take much more room than long multiplication would. I wouldn’t necessarily expect students to perform multiplication faster using the lattice, but they shouldn’t perform slower, either. As I said, these methods are really very similar when broken down to their elements. While you could certainly use this as a primary teaching method for long multiplication, you could also just present it as an example of the fact that, while math is universal, the methods for performing math are not.

The Count

1. Multiplication on Wikipedia
2. Lattice method on Wikipedia