## Add Whole Numbers

**Addition Properties**

Mathematical properties are often used to simplify computation. Below are three addition properties stated in words, shown with a numeric example, and shown with an algebraic example. The Zero Property of Addition is also called the Identity Property of Addition.

**Associative Property of Addition**

When numbers or variables are added, the addends can be grouped in different ways without changing the result.

(a + b) + c = a + (b + c)

**Commutative Property of Addition**

When numbers or variables are added, the order of theaddends can be changed without changing the result.

a + b = b + a

**Zero Property of Addition**

When 0 is added to a number or variable, the result is the same number or variable.

a + 0 = a

**The Addition Algorithm**

In its basic form, the addition problem 8 + 5 calls for five successive additions of 1 to 8. Students who have not mastered the addition table are likely to solve this problem by using their fingers to add on from 8. Of course, such a system is not practical for evaluating a sum such as 3,768 + 2,597.

To find 3,768 + 2,597, the expanded form can be used to write this sum as

(3,000 + 700 + 60 + 8) + (2,000 + 500 + 90 + 7).

The Associative and Commutative Properties of Addition allow the parentheses to be removed and the addends to be reordered. The properties can be used repeatedly to find the sum as follows.

(3,000 + 2,000) + (700 + 500) + (60 + 90) + (8 + 7) =

5,000 + 1,200 + 150 + 15 = 5,000 + (1,000 + 200) + (100 + 50) + (10 + 5) =

(5,000 + 1,000) + (200 + 100) + (50 + 10) + 5 = 6,000 + 300 + 60 + 5 = 6,365

By carrying out simpler addition problems in both expanded and algorithmic forms, students can be made aware of the fact that the addition algorithm corresponds to a regrouping and reordering of terms in accordance with the laws of arithmetic.

In the addition and subtraction algorithms, digits are aligned according to place value and the computation is completed from right to left.

Add the ones. |
Add the tens |
Add the hundreds. |
Add the thousands. |

15 ones = 1 ten + 5 ones |
16 tens = 1 hundred + 6 tens |
13 hundreds = 1 thousand + 3 hundreds |

**Teaching Model 4.1:** Addition Properties