Addition

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In the previous chapter, we summarised the concepts underlying the procedures used for performing the 4 basic arithmetic operations.

Let us look at the addition operation closely.

The skills required in the adding several multi-digit numbers can be reduced in principle to the addition of 2 multi-digit numbers. These in turn can be reduced to the following skills &amp; concepts

Hence, students need to master only the algorithm of adding just 2 multi-digit numbers. The idea can easily be extended to multi-digit numbers.
 * 1) Skill of adding 2 single digit numbers &amp;
 * 2) Concept of &ldquo;regrouping&rdquo; and &ldquo;carry over&rdquo; when the total of an addition is greater than 9

The skill of adding 2 single digit numbers can be practiced on fingers and remembered by repeated use. (We will see how in chapter 11.7)

For understanding the concept of &ldquo;carry over&rdquo; let us now revisit the Addition of two 2 digit numbers.

Let us visualise, 23 as 2 bundles and 3 sticks and 39 as 3 bundles and 9 sticks. We can join the bundles together and the sticks together. If we join the sticks together (3 and 9) we get &ldquo;twelve&rdquo; sticks which can be "regrouped" as 1 bundle and 2 sticks.

Obviously, the bundle should go with the other bundles. Hence this bundle is "carried over" and placed along with bundles. Hence we get 5 bundles (from the original) and 1 bundle "carried over", resulting in 6 bundles. Hence the total is 6 bundles and 2 sticks, which is written as 62.

Though this procedure is called "carry over", it is preceded by the procedure of "regrouping".

While adding just with numbers, we write 3 + 9 as 12. This hides the important concept of getting twelve (or a dozen) sticks which are "regrouped" into one bundle & two sticks which is written as 12. So "regrouping" has almost been forgotten.

Let us see the steps which cause confusion to the students.

The addition of the sticks has resulted in a mix of bundles &amp; sticks. It is visually obvious that though bundles &amp; sticks are represented by numbers, they cannot be added together. 1 (bundle) and 2 (sticks) do not become 3! The resulting bundle has to be &ldquo;carried over&rdquo; to the bundles.
 * 1) Why carry over?

While doing the addition on paper and pencil, 3 + 9 yields 12. Students are confused whether 1 should be &lsquo;carried over&rsquo; or 2. This happens because in its form as 12, it can represent either &ldquo;twelve&rdquo; sticks or &ldquo;one bundle and two sticks&rdquo;. The student is expected to mentally shift from &ldquo;twelve&rdquo; sticks or &ldquo;one bundle and two sticks&rdquo;.
 * 1) Which number to carry over?



If students are helped to visualise 2-digit numbers as bundles and sticks, and regroup twelve sticks into 1 bundle and 2 sticks, the above confusion will disappear.

It is also obvious that the addition of any 2 numbers less than 9 can only result in a number with 1 bundle or a number less than twenty. Hence the number to be carried over, from the Unit&rsquo;s Place cannot be more than &ldquo;1&rdquo;.

Direction of Addition

One question which arises in the minds of students is the reason from proceeding from the unit&rsquo;s place to the ten&rsquo;s place and so on.

Addition of bundles would result in bundles. Addition of sticks may yield sticks and, in some cases, also yield some bundles&amp; sticks. If we start adding the bundles then what do we do if the addition of sticks also yields some bundles. Hence it is better if sticks are added before the bundles. I.e the addition should start from the Unit&rsquo;s Place and then proceed to higher value places.

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