Area Models
In this lesson, we build on the ideas developed in the counting Piece Arrays lesson to develop multiplication and division models for signed numbers. The use of minimal arrays and the three generalized models developed in the last lesson make these operations with integers easy without having to memorize rules.
Here are some examples of multiplication:
The procedure is the same for any multiplication problem; if both edges are black (positive), the array does not flip at all, and therefore remains black. If one edge is red, either the rows or columns only turn to red. The students are quick to discover the generalizations:
We can use logical deduction to solve division problems using the same rectangle model. In an earlier lesson we learned that in a division problem, the dividend (the number being divided) can be represented by the area of a rectangle, and the divisor by one dimension. The answer is seen in the other dimension of the model. Here are a couple of examples with signed numbers:
We do lots of exploration of various multiplication and division concepts in class, including division with zero, but these examples demonstrate the basic ideas. Out of these explorations comes a 'mega' generalization about multiplying and dividing signed numbers. Ask your child about it; it's very cool!