## What is Multiplication – part VI

In our series of posts looking at multiplicative structures I wouldnt want to leave out what is often called the area model, though I think array model is a better fit.   As I suggested in a prior post, the notion that inches times inches give you square inches isn’t as obvious as it first sounds.

After all, three oranges times two oranges don’t give you six square oranges.

Yet what is nice about the arrangement in the picture is that you can identify rows and columns.  The arrangement is in an array.  We could say the oranges are arrayed as three rows of two oranges.  This is different from the situation we encountered with rolls of Mentos.  If there are three rolls of Mentos, and each roll has 14 Mentos candies in it, the rolls are carefully wrapped to keep the 14 candies together.  Not only are the 14 candies kept together, they are also seen as separate from all the other Mentos candies.  When we look at the three rows of two oranges each, the rows are visually separate, we can see them as separate.  “Row” is something that we bring to the story.  We can just as easily see the oranges as arrayed in two columns of three oranges each.  We can see the “dotted lines” horizontally or vertically, whichever we please.  This is an important point.  The oranges, through their arrangement, can at once be seen as three rows of two oranges each, as two columns of three oranges each, or as 6 individual oranges.

There is a certain symmetry between rows and columns in that there are no wrappers or identifying marks that make seeing rows somehow more natural than seeing columns.  This is the beauty of the array model.

In contrast,  in other situations the grouping is “forced” on us from the outside, as it were.  In the picture of birds below, it is much easier to see it as three couples than as two triplets.  The bird picture resembles the 3 rolls of Mentos more.

The array, as an arrangement of things in either rows or in columns, exhibits multiplicative structure.

$\begin{matrix} 3 & rows & of\ 2\ oranges/row & = & 6 & oranges \\ 2 & columns & of\ 3\ oranges/column & = & 6 & oranges \\ 1 & arrangement & of\ 6\ oranges/arrangement & = & 6 & oranges \\ 6 & groups & of\ 1\ orange/group & = & 6 & oranges \end{matrix}$

So the array model really has nothing to do with square oranges nor oranges squared.  All we are ever counting is oranges.

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