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.

Japanese Multiplication Method

In the Japanese Multiplication Method, a student is able to complete a multiplication problem of two large numbers by merely drawing a few lines and counting the points of intersections.  For example, to multiply 21 x 23, the student first represents 21 by 2 diagonals lines that are drawn up and to the right followed by 1 diagonal line that is drawn in the same direction just underneath this, and then 23 by 2 diagonal lines that are drawn down and to the right followed by 3 diagonals lines that are drawn in the same direction just above this, so that the four groups of lines form a diamond shape. 

Japanese Multiplication of 21 x 23

The lines intersect in a total of 4 points on the left side of the diamond, a total of 8 points in the top and bottom sides of the diamond, and a total of 3 times on the right side of the diamond for a final and correct answer of 483.

The Japanese Multiplication Method can be extended to handle more digits by creating a larger diamond, and handle a digit of zero by drawing a different colored line.  In some cases, carrying is required in the final addition steps.

Japanese Multiplication Method Tutorial Video

Despite its name, the origin of the Japanese Multiplication Method is unknown.  The method is also known as Indian Multiplication and Chinese Stick Multiplication, but it is not known if it actually did originate from Japan, India, China, or elsewhere.

Validation

When it comes to multiplying two numbers of multiple digits, most people use the traditional vertical method that is taught in most elementary schools.  For example, to multiply 21 x 23, 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 1 x 3 = 3 (Step 1). 

Step 1	Step 2	Step 3	Step 4	Step 5
21         x 23               3	21         x 23            63	21         x 23            63            20	21         x 23            63          420	21         x 23            63      + 420          483

Vertical Multiplication of 21 x 23

Then the first digit of the top number 2 is multiplied by the last digit of the second number 3 to obtain 2 x 3 = 6, and so the first product is 63 (Step 2).  Then the last digit of the first number 1 is multiplied by the first digit of the second number 2 to obtain 1 x 2 = 2 and placed beside a place holder of 0 (Step 3).  Then the first digit of the first number 2 is multiplied by the first digit of the second number 2 to obtain 2 x 2 = 4, so the second product is 420 (Step 4).  The two products 63 and 420 are then added together to obtain the final answer of 63 + 420 = 483 (Step 5).

In summary, in the traditional vertical method a student first multiplies 1 x 3, then 20 x 3, then 1 x 20, then 20 x 20, and then adds all the results together.  In other words, when multiplying numbers with multiple digits, each digit of the first number is multiplied by each digit of the second number (with appropriate positioning or zeroes to preserve place value) and then added all together.  Algebraically, this can be expressed as (20 + 1)(20 + 3) = 3·1 + 20·3 + 1·20 + 20·20 = 3 + 60 + 20 + 400 = 483.  Tweaking the order a little bit gives the algebraic equivalent of (20 + 1)(20 + 3) = 20·20 + 20·3 + 1·20 + 1·3 = 400 + 60 + 20 + 3 = 483, which is the same order as the popular FOIL method (first, outer, inner, last) taught in most algebra classes. 

The same multiplication problem can be visualized with a table using Base Ten Blocks.  The 21 can be represented as 2 rod blocks and 1 unit block as row headers, and the 23 can be represented as 2 rod blocks and 3 unit blocks as column headers. 

Base Ten Block Multiplication of 21 x 23

The table would then be filled in by 4 square blocks (hundreds), a group of 6 rod blocks and another group of 2 rod blocks for a total of 8 rod blocks (tens), and 3 unit blocks (ones) for a final answer of 483.  Once again, each digit of the first number is multiplied by each digit of the second number (this time with appropriate shapes to preserve place value) and then added all together.

The Japanese Multiplication Method is simply a transformation of the Base Ten Block Table.  Instead of 4 square blocks there are 4 points of intersection on the left side of the diamond of lines, instead of a group of 6 rod blocks and a group of 2 rod blocks there are 6 points of intersection on the top side of the diamond of lines and 2 points of intersection on the bottom side of the diamond of lines, and instead of 3 unit blocks there are 3 points of intersection on the right side of the diamond of lines.  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.

Because of its similarities with the FOIL method, it should be noted that the Japanese Multiplication Method can also be used to multiply polynomials.  The above Japanese Multiplication diagram that shows 21 x 23 = 483 can also be used to multiply (2x + 1)(2x + 3) and obtain the result 4x² + 8x + 3.  (The equation 21 x 23 = 483 is a specific example of (2x + 1)(2x + 3) = 4x² + 8x + 3 when x = 10.)

Evaluation

After watching the video and examining the above example, it is tempting to conclude that the Japanese Multiplication Method is the superior algorithm for multiplying two numbers.  After all, it just requires drawing a few lines and counting its intersections.  Unfortunately, the above example is a bit misleading because all of the numbers used have small digits (3 and under).  Here is an example of multiplying some numbers with larger digits, 69 x 78, using the Japanese Multiplication Method:

Japanese Multiplication of 69 x 78

In this example, a lot more lines have to be drawn, and a lot more points of intersection are formed.  Counting the points of intersection becomes time-consuming and carrying is required.  In this example, the Japanese Multiplication Method takes longer than the traditional vertical method of multiplication.

Still, the Japanese Multiplication Method is a great way to visualize the multiplication process, especially for numbers with smaller digits.

Conclusion

The Japanese Multiplication Method is an algorithm for multiplying two large numbers by representing both numbers by a group of lines that form a diamond pattern.  The number of points of intersection near each vertex of the diamond are then counted in a certain order 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.  Unfortunately, the Japanese Multiplication Method is too time-consuming for multiplying numbers with larger digits, but remains a great visual aid for the multiplication process.