Lattice Multiplication Method

In the Lattice Multiplication Method, a student is able to complete a multiplication problem of two large numbers by arranging numbers in a lattice grid.  For example, to multiply 48 x 12, the student would start out by creating a 2 x 2 lattice grid with the digits of 48 along the top as column headers, and the digits of 12 along the right side as row headers (Step 1).   The student would then multiply the digits in each column and row header and place each product in its corresponding position (using leading zeroes for single digit products), so in this case 4 x 1 = 04 (Step 2), 8 x 1 = 08 (Step 3), 4 x 2 = 08 (Step 4), and 8 x 2 = 16 (Step 5). 

Lattice Multiplication of 48 x 12

Finally, the student would add the digits along each diagonal, from the rightmost diagonal to the leftmost diagonal, and place each sum at the bottom left of each diagonal (carrying when necessary), so in this case 6 = 6 (Step 6), 8 + 1 + 8 = 17 which breaks down to 7 and a carried 1 (Step 7), 1 + 0 + 4 + 0 = 5 (Step 8), and 0 = 0 (Step 9).  The final answer can be determined by the right side column digits and the bottom row digits, in this case a final and correct answer of 576.  The Lattice Multiplication Method can also be extended to handle more digits by creating a larger lattice.

Lattice Multiplication Tutorial Video

The Lattice Multiplication Method is also known as Gelusia Multiplication, Sieve Multiplication, Shabakh, Venetian Squares, and the Chinese Lattice.  It is not known whether the method originated in Europe, the Middle East, or China, but it has been known since at least the 13^th century.

Validation

When it comes to multiplying two numbers of multiple digits, most people use the traditional Vertical Multiplication Method that is taught in most elementary schools.  For example, to multiply 48 x 12, the numbers are placed vertically, and the last digit of the top number is multiplied by the last digit of the second number to obtain 8 x 2 = 16, which is breaks down as 6 and a carried 1 (Step 1). 

Step 1	Step 2	Step 3	Step 4	Step 5
148         x 12               6	148         x 12            96	48         x 12            96            80	48         x 12            96          480	48         x 12            96     + 1480          576

Vertical Multiplication of 48 x 12

Then the first digit of the top number 4 is multiplied by the last digit of the second number 2 to obtain 4 x 2 = 8, plus the carried 1 makes 8 + 1 = 9, and so the first product is 96 (Step 2).  Then the last digit of the first number 8 is multiplied by the first digit of the second number 1 to obtain 8 x 1 = 8 and placed beside a place holder of 0 (Step 3).  Then the first digit of the first number 4 is multiplied by the first digit of the second number 1 to obtain 4 x 1 = 4, so the second product is 480 (Step 4).  The two products 96 and 480 are then added together to obtain the final answer of 96 + 480 = 576 (Step 5).

The Lattice Multiplication Method is simply a rearrangement of the same numbers in the Vertical Multiplication Method.  In the Lattice Multiplication Method, the digits of 48 and 12 are found in the top row and right column, and the digits in 96 and 480 are found inside the lattice (recall that the 9 was actually 8 and 1), and the digits of 576 are found in the left column and bottom row.

The Lattice Multiplication Method and the Vertical Multiplication Method both work for the same reason – the distributive property.  In both methods, the student multiplies 40 x 10, 40 x 2, 8 x 10, and 8 x 2 in some order and adds all the results together.  This is the algebraic equivalent of (40 + 8)(10 + 2) = 40·10 + 40·2 + 8·10 + 8·2 = 400 + 80 + 80 + 16 = 576, which is the same order as the popular FOIL method (first, outer, inner, last) taught in most algebra classes to show the distributive property.  In other words, the integrity of the multiplication is maintained because each digit of the first number is multiplied by each digit of the second number (with appropriate positioning to preserve place value) and then added all together.

Conclusion

The Lattice Multiplication Method is an algorithm for multiplying two large numbers by arranging numbers in a lattice grid.  Each digit of the two numbers are separated and placed as column and row headers, then the product of each column and row header is found and positioned inside the grid, and then the sum of each diagonal is found placed at the bottom left of each diagonal, and finally these sums can be read to obtain the solution.  Because each digit of the first number is multiplied by each digit of the second number (with appropriate positioning to preserve place value) and then added all together, it is a valid algorithm for multiplying two numbers.  The Lattice Multiplication Method is both organized and visually appealing, making it an ideal algorithm for multiplying two large numbers.