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Let us start multiplication procedure by seeing how children can multiply two single digit numbers. Taking the example of 3 X 4, they have 3 visual ways to get the product, as the result of a multiplication is called.
3 baskets with 4 fruits in each basket I.e 4 + 4 + 4 = 12 (4 added three times)
4 baskets with 3 fruits in each basket I.e 3 + 3 + 3 + 3 = 12 (3 added 4 times)
Students standing in an "array" with 4 rows & 3 columns
(1) & (2) are "repeated additions" of the same number. But using the above examples exposes children to the 2 "different" ways in which 3 X 4 can be visualised.
To perform these operations, they would require cups & pebbles, to act as baskets & fruits.
To perform the array method, they can use broom sticks which they can keep in an array of 3 sticks horizontally or vertically and another array of 4 sticks perpendicular to the 3 sticks. The number of “junctions” gives the product.
Advantages of the Array Method
The array method is simpler in terms of the material required and the procedure to be adopted. In our crowded classrooms, they can be done on the desk easily. All multiplication facts up to 9 X 9 can be got by having just 18 broomsticks!
It is also more powerful conceptually, since it is very obvious that 3 X 4 has the same product as 4 X 3. It is just a question of viewing the array from 2 different directions. In the example with fruits & baskets, this concept is not "visible". The total number of fruits in each case has to be worked out to find the relation.
Multiplication by Zero
This also can be visualised through the Array method. Let us take 4 X 0.
Let us start with 4 X 3 and visualise it as 4 sticks vertically crossed by 3 sticks horizontally. There are 12 junctions and the product is 12.
Now remove one of the horizontal sticks. We have 4 sticks crossed by 2 sticks giving 8 junctions. Hence 4 X 2 = 8. Proceeding further, we can get 4 X 1 =4.
Now 4 X 0 means there should be 0 sticks (or No sticks) in the horizontal direction. How many junctions are there now? There are no junctions or 0 junctions. Hence 4 X 0 = 0.
Extending the same logic, The product of any number multiplied by 0 will be 0.
Multiplication by One
Using the visual of the array method it is obvious that the product of any number when multiplied by 1 will be the same number. There would only be one horizontal stick and the number of intersections would be equal to the number of sticks.
Multiplication by Ten
When multiplying a number (say 2) by ten why do we just add a 0 to the right and write it as 20?
The explanation is easy if we visualise 2 in terms of 2 sticks. When you multiply by ten it is same as having ten sets of 2 sticks, which can also be seen as 2 bundles of ten sticks. This is written as 20, the 0 implying that there are no sticks!
Multiplication by Five
The table of 5 can be visualised as the division of the minutes in a clock which is 5, 10, 15, 20 .....
Greek Method of Multiplying
Greeks developed a way to multiply any number with a 2nd number by writing down repeatedly the double of the number and noting down the doubling factor as 1, 2, 4, 8 etc.
It uses the same principle as a binary representation of the number by which we multiply.
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