How Does The Algorithm Work?
Most of us, when asked to multiply, say, 35 by 23, write
Why do we place the mark x (or 0, or leave a blank) in the second row of the calculations?
The multiplication algorithm is based on the distributive law that we were just discussing. In this algorithm, when we multiply a single digit number by a 2- digit number the steps are:
i) Break up the 2-digit number into the tens part and the ones part.
ii) Then multiply each of these parts by the single digit number separately.
iii) Finally, add the two products to get the answer.
For example, 13 x 7 = (10 + 3) x 7
= (10 x 7) + (3 x 7), by distributive.
In the standard algorithm, we begin multiplication from the smallest unit of the number that is, the ones digit, and move leftwards. This is because we first add the smaller sized groups, those of the size of the digit in the ones place. This result is then converted into tens and ones. While the ones are retained at the ones place, the tens are came forward and added to the number obtained by multiplying the digit in the tens place of the number by the multiplier. The entire algorithm of multiplications based on this process.
For instance, in 13 x 7, we first multiply 3 by 7 to get 2 1. We retain the 1 of 2 1 in the ones place and carry-over the 20 as 2 tens, to be added to the number obtained by multiplying the 1 ten of 13 with 7, i.e., the 2 tens are added to the 7 tens to get 9 tens.
We need to give children an adequate feel for this process. As you can see, this requires them to understand distributive. Further, we need to encourage the children to use this law for multiplying numbers by breaking them up into ones and tens. We can then lead them to the formal algorithm.
Of course, the children may not immediately understand how the algorithm works. This understanding may require a couple of years. During this time they should come back repeatedly to the working of the algorithm. They could ,also compare its working with other algorithms that they learn during this period.