Subtraction

In the previous chapter, we summarised the concepts underlying the procedures used in the addition operation. Let us now look at subtraction.

Basic Module for Subtraction

Using the same logic as in addition, the computation skills & the required algorithm for subtraction can be mastered with just the algorithm for two 2 digit numbers. Extension of the idea to multi-digit numbers is easy. These in turn can be reduced to the following skills & concepts

Skill of subtracting a single digit numbers from another single digit number or a double digit number less than 20.
Concept of “borrowing” when the number “to be subtracted” is more than the number “subtracted from”

The skill of subtracting needed above can be practiced on fingers and remembered by repeated use. (We will see how in chapter 11.7)

For understanding the concept of “borrowing” let us now revisit the Addition of two 2 digit numbers.

Let us visualise, 43 as 4 bundles and 3 sticks and 29 as 2 bundles and 9 sticks. We are required to Take Way 2 bundles and 9 sticks from a collection of 4 bundles and 3 sticks. Normally we can take away the bundles from bundles and sticks from sticks.

But we have a problem that 9 sticks have to be taken from 3 sticks. There are not enough sticks to carry out this operation.

Hence from the 4 bundles in the collection, we break one bundle into “ten” sticks. Hence now we have “thirteen sticks” and 3 bundles.

Now 9 sticks can be taken from 13 sticks, leaving 4 sticks. From 3 bundles we can take away 2 bundles, leaving 1 bundle. Hence what is left over is 1 bundle and 4 sticks. Hence the answer is 14.

This process of removing 1 (bundle) from the ten’s place and writing it (adding ten) in the Unit’s place along with 3 as 13, is called “borrowing”. There could be 2 confusions in the minds of students.

They think of it as borrowing 1 whereas it is actually borrowing a ten (or a bundle)
Hence, they are not clear as to how 3 becomes 13

If students are helped to visualise 2-digit numbers as bundles and sticks, the above confusions will just disappear.

“Borrowing” should be understood as “regrouping” i.e converting one of the bundle to sticks in order to have enough sticks to complete the take away process. In general, any bigger bundle can be changed into smaller bundles of ten or its multiples.

The one bundle when broken into sticks yields ten sticks and together with the already existing 3, become thirteen sticks.

Some bright students may even wonder as why the borrowed number is never returned. Since there is no actual borrowing, the question of returning does not arise.

We can see in the above example use of non-standard representation of numbers when 4 3 is written as 3 13. This is similar to paying Rs 43, either with 4 ten rupees & 3 One-rupee notes OR paying it as 3 ten rupee notes and 13 one rupee notes.

It would also be clear that in any single step only one bundle needs to be broken up.

Direction of Subtraction

Subtraction of sticks may require a bundle to be broken up. Hence, we realise that sticks have to be subtracted before the bundles. I.e the subtraction also has to start from the Unit’s Place and then proceed to higher value places.

Apart from the above “standard” procedure, there are other methods to do additions and subtractions. We will see some of them in the next chapter.

It is a good principle to allow students to explore different ways of doing a given problem, before showing them the standard method. These different methods will deepen their understanding of the operation

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