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Of course, there is no "right" way to do multiplication as long as you come up with the right answer, but there is a way that minimizes the opportunities to make an error and optimizes the chances of getting a correct answer. That way is called "lattice multiplication". And it's easier than traditional multiplication.

It's called "lattice" because you begin by drawing a square and dividing it into rows and columns, one row for each digit of one number, and one column for each digit of the second number. The set up is pictured below. A diagonal is then drawn through each cell from top right to lower left. Then the nmbers are placed at the top and to the right of the lattice.

To multiply, each digit at the top is multiplied by each digit at the right. The product is placed in the cell in the row and column of the digits multiplied. If there are two digits in the result, the tens digit is place above the diagonal and the ones digit is placed below. It doesn't matter in what order you multiply the digits.

Then you add across the diagonals. Finally any tens place digits in the sums are carried to the sum to the left and added.

The figure to the right should be clear.

It's called "lattice" because you begin by drawing a square and dividing it into rows and columns, one row for each digit of one number, and one column for each digit of the second number. The set up is pictured below. A diagonal is then drawn through each cell from top right to lower left. Then the nmbers are placed at the top and to the right of the lattice.

To multiply, each digit at the top is multiplied by each digit at the right. The product is placed in the cell in the row and column of the digits multiplied. If there are two digits in the result, the tens digit is place above the diagonal and the ones digit is placed below. It doesn't matter in what order you multiply the digits.

Then you add across the diagonals. Finally any tens place digits in the sums are carried to the sum to the left and added.

The figure to the right should be clear.

Now, how much better is lattice multiplication? Below, I list each step of each, the ways people commonly make mistakes at each step, and the number of likely errors.

Traditional method

1) 6 x 3 Multiplication rules, carry, placement (3)

2) 6 x 5 Multiplication rules, carry, placement (3)

3) 6 x 2 Multiplication rules, placement (2)

4) 9 x 3 Multiplication rules, carry, placement (3)

5) 9 x 5 Multiplication rules, carry, placement (3)

6) 9 x 2 Multiplication rules, placement (2)

7) 8 Bring down 8 (1)

8) 7 + 1 Addition rules (1)

9) 7 + 5 Addition rules, carry (2)

10) 2 + 1 Addition rules, carry (2)

11) 2 Bring down 2 (1)

Lattice method

1) Organize numbers around lattice (1)

2) Draw diagonals (4)

3) 9 x 3 Multiplication rules (1)

4) 9 x 5 Multiplication rules (1)

5) 9 x 2 Multiplication rules (1)

6) 6 x 3 Multiplication rules (1)

7) 6 x 5 Multiplication rules (1)

8) 6 x 2 Multiplication rules (1)

9) 8 Add first diagonal (1)

10) 7 + 1 Add 2nd diagonal (1)

11) 2 + 5 + 3 + 2 Add 3rd diagonal, carry (2)

12) 4 + 8 + 1 + 1 Add 4th diagonal, carry (2)

13) 1 + 1 Add 5th diagonal (1)

For the traditional method, there are 23 chances to make an error; for the lattice method, there are only 18 chances.

The two most common kinds of error are carry errors and multiplication rule errors. Both methods come out the same for the number of times you have to multiply.You always have to multiply each digit in one number by every digit in the other number regardless of what method you use. But notice that there are 6 carries in the traditional method and only two in the lattice method.

Also notice that you have to worry about where 8 digits of partial products go, where, in lattice multiplication, it's quite clear that you place the result in the cell below one digit and to the left of the other. The major difficulty in the lattice method is making sure that you add all the digits along each diagonal.

I think you'll find that the lattice method of multiplication is both easier than the traditional method and much less likely to allow errors. I especially recommend it to people who have learning disabilities in mathematics.

Find Another Way

I've said elsewhere that there is no substitute for long division – that isn't precisely true. There's this thing called Vedic mathematics that I'm just beginning to be exposed to. It shows some promise but I don't know enough about it yet to recommend it.

Otherwise, although I don't know an easier form of division which will always lead to an exact (to a given decimal point) answer, there are two useful procedures that will give approximate answers and that's usually all that's needed.

The first takes a division problem as a fraction and simplifies it in the hopes of ending up with a short division problem. For example, let's divide 252 by 105. In terms of fractions, that would be 252/105. We can simplify this fraction by ferreting out the factors of both the denominator and the numerator and then cancel the common factors.

We know that 3 is a factor of the denominator because the digits sum to a number divisible by 3 (1 + 0 + 5 = 6). We also know that 5 is a factor since any number ending in 5 or 0 is divisible by 5. The only factor left is 7 which is a prime, so the denominator is 7 • 5 • 3.

Now the numerator. It's an even number so 2 is a factor:

2 • 126

When that's factored out, another even number is left so:

2 • 2 • 63

63 is obviously divisible by 3 to give 21 which is also divisible by 3 (2 + 1 = 3) so:

2 • 2 • 3 • 3 • 7

Now we can cancel out the common factors in the top and bottom of the fraction and end up with :

= 12/5

That's a short division problem that you can do in your head. 12/5 = 2.4.

It doesn't always work out so nicely. Let's try 35 into 274. That can't be simplified by factoring but we can still get a single digit denominator by multiplying the top and bottom by 2 to get 548/70, and dividing the top and bottom by 10 to get 54.8/7. That's a short division problem that yields (to one decimal point) 7.8. Checking the answer by multiplying the answer by 35 gives 273, which is very close to 274.

The other way to do it is to search for the answer or 35 ÷ 274. We know that the correct answer times 35 will be 274, so we can make calculated guesses that will bring us closer and closer to the correct number. Since we know the 30 goes into 300 10 times (30 is close to 35 and 300 is close to 274) we might start at 10 for the first guess. 10 x 35 is 350, which is too large so we try a smaller number, say 5. 5 X 35 = 175, which is too small so we try 7 and the progression is like:

7 x 35 = 245 – too small

7.5 x 35 = 262 – still to small but closer

7.6 x 35 = 266 – closer still

7.7 x 35 = 269.5

7.8 x 35 = 272 – and that's close enough for government work.

So there's 3 ways to do division. Choose the easiest way.

People process information differently. Traditional teaching is designed to teach the average learner. The more the way you process information diverges from the way the average learner processes information, the more difficulty you will have in a traditional learning situation. Your best strategy, then, is to find a nontraditional method that works for you. If long division doesn't work for you, try fraction simplification. If traditional multiplication doesn't work, try lattice multiplication. If traditional, linear reading doesn't work, try scanning text.

A good place to try out a variety of alternative methods for processing different kinds of information is the Study Guides and Strategies website (http://www.studygs.net/).

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