Division Metaphors 1

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There are 4 metaphors for Division.

Equal Sharing

Children are familiar with sharing eatables or tokens with their friends. This is one of their earliest introductions in life to division.

8 &divide; 2 can be interpreted as sharing 8 chocolates, equally among 2 persons, the result 4 being the number of chocolates each gets.

In actual practice, the chocolates are handed over one by one in turns to each person, thus ensuring that at the end of each round, both persons have the same number of chocolates. What is not known at the beginning is the number of chocolates each would get.

Equal Grouping

Here again, it is a sharing activity but with a difference. Let us take an example.

Ram has 12 chocolates. His mother tells him to share with his friends but ensure that each friend gets 3 chocolates. The operation is 12&divide;3 and the result 4 is the number of friends who receive chocolates.

In actual practice, chocolates are handed over to friends in &lsquo;groups of 3&rsquo;. What is not known is the number of friends who would receive chocolates.

In both &lsquo;Equal Sharing&rsquo; and &lsquo;Equal Grouping&rsquo;, the share of each person is equal. The difference is in the process by which the distribution is done. This difference has to be demonstrated to students many times before they can internalise it.

Equal Grouping is also called &ldquo;Repeated Subtraction&rdquo; as at every step a certain quantity is subtracted from the total. But &ldquo;repeated subtraction&rdquo; is a process and not a concept. It is the process by which any division problem can be solved. I feel calling it &ldquo;equal grouping&rdquo; would give the sense of the concept in a much better way.

&ldquo;Equal Grouping&rdquo; can also be imagined as the reverse of &ldquo;Join Several Equal Quantities&rdquo; multiplication situation.

Equal Grouping – More Powerful Idea

Though &ldquo;equal sharing&rdquo; is an earlier experience for children compared to &ldquo;equal grouping&rdquo;, the latter is a more powerful arithmetic idea.

For example a problem like 8 &divide;⅓ cannot be visualised as an &ldquo;equal sharing&rdquo; problem. How do you share 8 chocolates among ⅓ friends? But it can be visualised as an &ldquo;equal grouping&rdquo; problem of sharing 8 chocolates among friends such that each gets⅓ of a chocolate. The answer which naturally emerges (without the rule of invert and multiply) is that 24 friends can get chocolates.

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