Historically, the lattice method of multiplication appears in the first printed arithmetic book, printed in Treviso (Italy) in 1478. It shows this exact method and a variation as well as some variations of the long multiplication algorithm commonly taught today.
These methods were introduced into Europe by Fibonacci. He was an Italian who learnt the use of Arabic numerals from a Moorish teacher in North Africa. Before the Hindu-Arabic system was used in Europe, multiplication was often done with counters because Roman numerals were ill-suited to calculation and very few people knew how to multiply. The Hindu-Arabic system has made calculation fairly simple.
The lattice method of multiplication is illustrated by the following examples, taken from the Teaching Mental and Written Computation. See References.
There is an animated description of the lattice method in the National Library of Virtual Manipulatives. See References.
In lattice multiplication, the partial products are laid out in a lattice and adding along the diagonals gives the answer to the multiplication.
Example 1: 28 × 57 = 1596
As 28 and 57 have two digits each, a lattice is set out with two columns and two rows. The diagonals are drawn in each cell as shown below. 28 is written above the lattice with 2 above the first column and 8 above the second. 57 is written to the right of the lattice with 5 along the first row and 7 along the second.
The partial products of these digits taken two at a time is set out in the corresponding cells with the tens above the diagonal and ones below. For example, the partial products in this case are 5 × 8 (= 40), 5 × 2 (= 10), 7 × 8 (= 56) and 7 × 2 (= 14).These are set out as shown below.
The sum along each diagonal is then recorded as shown below and these digits 1, 5, 9 and 6 form the answer to the multiplication. As usual, start adding at the ones (in this case ‘6’ which comes from multiplying 8 ones by 7 ones), proceeding from right to left around the lattice.
Thus 28 × 57 = 1596
Example 2: 183 × 49 = 8967
The lattice set out for this multiplication will have 3 columns and two rows as 183 has 3 digits (it could also be done as 2 columns and 3 rows as 49 × 183). As before the numbers are set out as shown below and the partial products are written down in their respective positions. The numbers along the diagonals are added to give the answer.
Note that in this example adding along the third diagonal gives 19 which needs 1 to be carried to the diagonal to the left, in other words, 19 hundreds is 10 hundreds + 9 hundreds, then the 10 hundreds is renamed as 1 thousand and the 1 is then written in the thousands column. Therefore the addition should begin with the lowest diagonal on the right hand side (the product of the ones from the two numbers).
183 × 49 = 8967
National Library of Virtual Manipulatives. Utah State University - http://nlvm.usu.edu/en/nav/vlibrary.html
Teaching Mental and Written Computation. (Version 1.1) University of Melbourne (CD-ROM). Stacey, K., Varughese, N., Marston, K. (2003) - http://extranet.edfac.unimelb.edu.au/DSME/tmwc/