Distributive Property

In class today we began discovering the distributive property.  They have several notes pages that will help you with their homework.  We also determined the definition of even and odd numbers, and found some things to be true:

The sum of 2 even numbers will always be EVEN
The sum of 2 odd numbers will always be EVEN
The sum of an even and odd number will always be ODD

The product of 2 even numbers will always be EVEN
The product of 2 odd numbers will always be ODD
The product of an even and odd number will always be EVEN.

We also learned two different ways to right a multiplication problem.  Lets look at 16.

8 x 2 = 16
8 · 2 = 16
8(2) = 16

From here on out, we will use the parentheses version to multiply.

Before we continue with the distributive property, lets review order of operations.


Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

When doing math problems, we have to look at this to know what order to perform the operations.

In terms of distributive property, we typically do the parentheses in different ways.  Lets look at the number 24.

24 can be seen at 3 (8) = 24.  However, it can always be seen as

 3 ( 2 + 6) =24  (we can read this at 3 times the sum of 2 and 6)

In the second case, we can "distribute" the 3 to both numbers inside the ( ).  This would look like

3 (2) + 3 (6) = 24, or

6 + 18 = 24.

Notice we had to multiply AFTER distributing the 6 through the parentheses.  Otherwise it would have looked like this:

3 (2) is 6, so 6 + 3 (5), then 6 + 3 is 9, so 9 (5) = 45.  That entire process is wrong according to order of operations.

Let's look at one in terms of area of a rectangle.

The area of the entire rectangle can be found by added together the areas of the two smaller rectangles.  Remember, area is equal to length times width ( a = l (w) )

Using the distributive property we can "distribute" the 5 to what is inside the parentheses.

 5 (10 + 4) = 5 (10) + 5 (4) = 50 + 20 = 70

We can actually do this another way.  We can add the lengths on the bottom (10 + 4) to get 14 then multiply by the width, 5.

5 (14) = 70.