Multiplication 2

< 12.1 Multiplication 1 | Topic Index | 12.3 Multiplication Tables >

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

The procedure involves &ldquo;joining&rdquo; several groups, each having the same number of items. Commonly it is called &ldquo;repeated addition&rdquo;.

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 &amp; 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 &ldquo;repeated additions&rdquo;.

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.

 

Standard Algorithm (Procedure)

In the traditional algorithm, steps 1 &amp; 2 and then steps 3 &amp; 4 are combined and the total is written directly.

Since all teachers are familiar with the algorithm, I am not describing it. The idea is to bring out that the algorithm is actually (a + b ) X (c + d)!

The standard algorithm also requires the student to write &ldquo;X&rdquo; in the place of the additional 0s which are the result of multiplying by 40 instead of 4. This is how it looks like. But frankly we should not be using it in schools where the idea is to &ldquo;enlighten&rdquo; and not to unnecessarily confuse students!

Visual understanding of Multiplication

Students, who find the algorithm difficult, can start with this visual way. The numbers in brackets are only for understanding. They need not actually be written.

This idea can be extended even to bigger numbers! It is also an introduction to algebraic thinking!

< 12.1 Multiplication 1 | Topic Index | 12.3 Multiplication Tables >