Decomposing an Array

One of the ways to view multiplication is to consider the area of an array. The dimensions of the array are the factors and the area is the product. Using array cards, students work on their basic facts. Array cards not only help with fluency, but students also begin to understand that the shape of the array is an indicator of how far the factors are "apart". A square array indicates a square number (both factors are the same)and a long and skinny array indicates factors that are very different in magnitude.
One of the strategies we use to solve multiplication problems with larger factors is decomposing. To model this strategy, we use generic arrays. An example of a generic array is pictured below.


A generic array can be decomposed by drawing either a horizontal or vertical line through the array. In doing so, one of the two factors is broken into two smaller numbers which add to the original dimension. In the first example, 12 is decomposed into 10 and 2. The idea is to create two smaller problems that are much easier to solve. 15 x 10 and 15 x 2 are much easier to solve than 15 x 12. Decomposing the large array into multiples of 10, 5, and 2 seem to be the easiest. Once the two smaller areas (products)are found, they are recomposed to find the area (product) of the large array. In our example: 15 x 10 = 150 and 15 x 2 = 30, so 15 x 12 = 150 + 30 = 180.