The basic operations have some properties that are interesting:
Their names are sort of intimidating, but they really are quite simple if we use models to understand just what they mean.
The distributive property means that we can break up one factor of a multiplication problem into its addends, multiply them by the other factor, and then add the products together to get the whole answer. let's take a look at an example to show what this means.
Suppose we consider the multiplication problem 5 x 12:
The idea is to make the problem easier to solve in our heads, with numbers that are easier to use. We could just as easily have done it this way:
This model can work for harder problems, and in fact models another important cousin of the distributive property - partial products. Let's see how it would look for 23 x 35:
In the above example, 600, 90, 100 and 15 are the partial products of the 23 x 35 multiplication problem. The partial products add up to the total product.
This same model can be used to ease mental computation by using the idea of subtraction, too. Here's a simple example for 12 x 19:
These properties seem almost self-evident, but that does not lessen their fundamental importance. They are basic to mental computation strategies and most useful in algebraic computation. We will see examples of these in future lessons.