Multiplication 2

Now let us look at the logic underlying the procedures for Multiplication with 2 digit numbers. 

The procedure involves “joining” several groups, each having the same number of items. Commonly it is called “repeated addition”.

Visualising Multiplication

Since the result is also likely to be a 3 digit number, let us use the visual imagery of Sheets (hundreds), Strips (tens) and Pieces (Units) instead of bundles & sticks.

Let us visualise 35 as 3 Strips and 5 Pieces. Joining 3 such collections is equal to joining 3 collections of 3 Strips and 3 collections of 5 Pieces. Multiplication tables are used for quickly doing these “repeated additions”.

In the above case we are left with 9 Strips and 15 Pieces. The 15 sticks can be regrouped into 1 Strip and 5 Pieces. Hence the total collection becomes 10 Strips and 5 Pieces. The 10 Pieces can be made into a Sheet. Hence we are left with 1 Sheet, 0 Strips and 5 Pieces i.e 105.

The Multiplication Algorithm

The multiplication algorithm depends on the distributive law of multiplication of (a + b ) X (c + d) which resolves into ac + ad + bc + bd. This process ultimately breaks the multiplication into multiplication facts with single digit numbers.

Let us take an example of 32 X 45. This can be written as (30 + 2) X (40 + 5).

This can be written as 30 X40 + 30X5 + 2X40 + 2X5

This essentially boils down to finding the following multiplication facts – 3X4, 3X5, 2X4, 2X5. Multiplying by 30, because of the place value system, reduces to multiplying by 3 and adding a 0 at the right end to the product.

So any multiplication problem can be reduced to a sum of the product of multiplication of 2 single digit numbers!

30 X 40 = 1200, 30 X 5 = 150, 2 X 40 = 80, 2 X 5 = 10 and their sum is 1440!

If we look carefully at a standard multiplication procedure, we can see it as an addition of the above products or a combination of them! Let us see it through actually doing it.