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The Lattice Form of Multiplication
The following came from Dr math@ www.mathforum.com

Problem Statement
How to use the lattice method of solving a multiplication problems


Problem setup

 

Let's multiply 469 x 37.

 

Plans to Solve/Investigate the Problem

 

Multiplication really takes three steps: 

1. Multiply

2. Carry

3. Add


The method we typically use does the multiply and carry steps together.

 

The lattice method does all three steps separately.

Investigation/Exploration of the Problem

It gets its name from the fact that to do the
multiplication you fill in a grid which resembles a lattice one
might find ivy growing on.  Let me see if I can explain it with an
example.

 

Let's multiply 469 x 37.

First write the 469 across the top, and the 37 down the right side of
a 3x2 rectangle.  (It's 3x2 because the factors have three and two
digits respectively.)

     
Now fill in the lattice by multiplying the two digits found at
the head of the column and to the right of the row. When the partial
product is two digits, the first (10's) digit goes above the diagonal
and the second (1's) digit goes on the lower right of the diagonal. 
If the partial product is only one digit, a zero is placed in the
triangle above the diagonal in the square.

    

At this point, we have the multiplication done.  Now we add along the
diagonals beginning in the lower right to get the final product.  Any
"carries" when adding are illustrated outside the rectangle.

 
Extensions of the Problem

469
     x  37
     -----

The basis of any method of multiplying is the distributive property:

     a x (b + c) = a x b + a x c

In this case,

     469 x 37 = (400 + 60 + 9) x (30 + 7)
              = (400 + 60 + 9) x 30 + (400 + 60 + 9) x 7
              = 400 x 30 + 60 x 30 + 9 x 30 + 400 x 7 + 60 x 7 + 9 x 7

In other words, we can break each number up into a sum of terms, one
term for each digit, and the product will be the sum of all possible
products of a term from one number and a term from the other. How can
we do that easily? A multiplication table does the same thing - a
table of all products. So let's make a multiplication table for these
terms:

                      400     60     9
                   +-------+------+-----+
                   |       |      |     |
                   | 12000 | 1800 | 270 | 30
                   |       |      |     |
                   +-------+------+-----+
                   |       |      |     |
                   |  2800 |  420 |  63 | 7
                   |       |      |     |
                   +-------+------+-----+

(I've written the labels for the rows on the right rather than the
left, just because I know that they would be in the way on the left.)

Notice that the products along a diagonal from top right to bottom
left have the same number of zeroes (the same exponent of 10). That's
because when you go left a column and down a row, you add a zero and
take it back off. So we can add the numbers along each diagonal line,
then sum the results:

                      400     60     9
                   +-------+------+-----+
                   |       |      |     |
                   | 12000 | 1800 | 270 | 30
                   |       |      |     |
                   +-------+------+-----+
                 / |       |      |     |
               /   | 2800  |  420 |  63 | 7
             /     |       |      |     |
           /       +-------+------+-----+
         /       /       /      /
     12000 +  4600  +  690  +  63 = 17353

We can drop the zeroes and just write the product of the non-zero
digits, and put the zeroes back in when we add:

                     4      6      9
                 +------+------+------+
                 |      |      |      |
                 |  12  |  18  |  27  | 3
                 |      |      |      |
                 +------+------+------+
               / |      |      |      |
             /   |  28  |  42  |  63  | 7
           /     |      |      |      |
         /       +------+------+------+
       /       /      /      /
     12      46     69     63

     12
      46
       69
        63
     -----
     17353

But we can make it a little easier by noticing that the first digit of
each product will add into the second digit of the diagonal to its
left. So we can split each product into two digits and draw diagonals
to show this:

                       4      6      9
                   +------+------+------+
                   | 1   /| 1   /| 2   /|
                   |   /  |   /  |   /  | 3
                   | /  2 | /  8 | /  7 |
                   +------+------+------+
                 / | 2   /| 4   /| 6  / |
               /   |   /  |   /  |   /  | 7
             /     | /  8 | /  2 | /  3 |
           /       +------+------+------+
         /       /      /      /      /
        1      5      22     15      3

       1
        5
        22
         15
           3
       -----
       17353

Now if you just do the carries as you write the sum of each diagonal,
you'll have the lattice method, with as little writing as possible:

                       4      6      9
                   +------+------+------+
                   | 1   /| 1   /| 2   /|
                   |   /  |   /  |   /  | 3
                   | /  2 | /  8 | /  7 |
                   +------+------+------+
                 / | 2   /| 4   /| 6  / |
               /   |   /  |   /  |   /  | 7
             /     | /  8 | /  2 | /  3 |
           /       +------+------+------+
         /       /      /      /      /
     1      7       3      5      3

You can see the same process at work in our normal method, which we
can spread out as:

       469
     x  37
     -----
        63  <-- 7x9
       42   <-- 7x60
      28    <-- 7x400
       27   <-- 30x9
      18    <-- 30x60
     12     <-- 30x400
     -----
     17353  <-- (30+7) x (400+60+9)

(notice the same six products we had before), or write more compactly
as:

       469
     x  37
     -----
      3283  <-- 7x(400+60+9)
     1407   <-- 30x(400+60+9)
     -----
     17353  <-- (30+7)x(400+60+9)

Have fun using the lattice!

 


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