Dimensions and Net Values
Dimensions and net values of counting piece arrays form the basic models for integer multiplication and division. In this lesson, we develop the model using counting pieces and edge pieces.
Here is a simple array whose net value will change when rows and columns are flipped over (we always start with a black array before flipping a row or column)...
Did you notice what happened to the tile where the flipped row and column intersect? Since it flipped twice, it turned to red and back to black again.
Edge pieces are used like instructions to turn over a row or column. Red edge pieces tell us to turn over the row or column; black pieces mean to leave it unturned. For example, what would the array inside these edge pieces look like?
Click here to see the array and its net value.
In order to develop models that are easy to use and remember, we use the idea of a minimal array. A minimal array has edges of one color only, and the array counting pieces are all one color as well. We can think of starting with an array and removing 'zero pair' rows or columns until all such combinations are removed. Here's an example...
From exploration and discussion of arrays, three general models for arrays are developed. Although the models use one edge piece, it could be considered a generalized edge, having any given positive or negative net value. Here are the models:
