Multiplication by Method of Square & Diagonals

Aside from the standard multiplication algorithm (algorithm really means steps or methods) that we already know, there is another method to do multiplication. That is, to place the numbers in squares. This method can be used for multiplying 2-digit numbers. We will call this method the Square Diagonal Multiplication Algorithm

Steps in Multiplication Algorithm

1. Create a 2 by 2 square and place the numbers on top and on the right side. Each digit should be assigned to one row and one column. The example is shown below for 63 × 17.

2. In each square, draw a diagonal slanting to the right. 

3. To multiply, the product of the digits is assigned to the square corresponding to the columns and rows where they are placed. For example, the product of 3 which is on the right and 7 which is on the bottom should be assigned to the bottom-right square.

4. Place the digits of each product such that the tens digit is on the triangle on the left and the ones digit is in the triangle on the right.  For example, if we multiply 3 x 7, in the product which is 21, 2 goes to the triangle on the left and 1 goes to the triangle  on the right.

5. If the product is a 1-digit number, put 0 on the left. For example, if we multiply 3 and 1, the product becomes 03.

6. Fill out all the triangles.

6. To get the product, add the digits diagonally to get the product as shown in the color codes shown below.

Ones digit: 1

Tens digit: 2 + 2 + 3 = 7

Hundreds digit: 4 + 6 + 0 = 10.

Thousands digit: 0

Since the hundreds digit is 10, 0 goes to the hundreds digit and 1 is carried over to the thousands digit. So, the product of 63 x 17 = 1071.

The Whole Number Mixed Fraction Multiplication Secret

In the previous post, we have learned a strategy on multiplying whole numbers by mixed fractions. We have learned that instead of converting the mixed fraction part to improper fractions, we split the numbers and the fraction and then multiplied them separately.

One example for this is 8 ¾ × 2. Instead of converting 8 ¾ to the improper fraction 35/4, we then split it to 8 and ¾ and then multiply each by 2. Therefore, 8 × 2 = 16 and ¾ × 2 = 3/2 which means equals to 17 ½.

Now, why does this math trick work?

First we know that 8 ¾ really means 8 and ¾ which is equal to 8 + ¾. Now, if we multiply 8 + ¾ by 2, we have

2(8 + ¾) = 2(8) + 2(¾) = 16 + 6/4 = 16 + 3/2.

Are the expressions and equations familiar? 

The Whole Number Mixed Fraction Multiplication Secret

Yes they are. If you have already discussed the distributive property of multiplication over addition in your class, then you are familiar with the equation

a(b + c) = ab + ac

where a, b, and c are real numbers.

In the expression above, a = 2, b = 8 and c = ¾.

And as mentioned above, the distributive property works for all real numbers, so it will work with 3/4 since 3/4 is a real number.

That’s it. At least now, you know why the math trick works.

Remember, it’s cool to know the trick. But it’s even cooler if you know why the trick works.

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