Lesson Objectives
• Learn how to label the parts of an addition problem as an "addend" or the "sum"
• Learn how to rewrite an addition problem using the commutative property of addition
• Learn how to rewrite an addition problem using the associative property of addition

## What are the Properties of Addition?

What is addition? Addition is the operation that allows us to group items together. It is one of the first operations we learn in elementary school. Usually, our first experience with addition involves a visual aid. Let's suppose we started out with 3 boxes: Now, let's say we came across another 2 boxes and we wanted to determine the total number of boxes. We can count the total number of boxes and we end up with 5. This can be translated into the addition fact: 3 + 2 = 5. After learning via visual cues and object counting, we eventually memorize all of the possible single-digit addition combinations. We can use these addition facts along with a process known as vertical addition to add numbers as large as we would like. When we work with an addition problem, we refer to the numbers being grouped together or added as "addends". The result of the addition operation is referred to as the "sum". In our example featuring the boxes, we added 3 boxes with 2 boxes and got a result of 5 boxes. In other words, the numbers 3 and 2 are being grouped together or added. These numbers (3, 2) are the addends. The result of the addition operation is 5, this is the sum. Example 1: Label the parts of the addition problem 5 + 7 = 12 The 5 and 7 are being added together, these are the addends. The 12 is the sum or result from the addition operation.
• 12 - sum
Let's take a look at a few properties of addition. These will be useful throughout our study of Algebra.

### Identity Property of Zero

The Identity property of zero tells us that adding zero to any number leaves the number unchanged.
Example 2: 17,355 + 0 = ?
The answer here is simple, the number 17,355 will not be changed by adding 0.
17,355 + 0 = 17,355

### The Commutative Property of Addition

The commutative property allows us to reorder the addends in an addition problem. The commutative property of addition tells us that we can add numbers in any order and not change the sum.
Example 3: Rewrite 13 + 5 using the commutative property of addition
To complete this problem, we simply change the order of the addends, 5 will be first and 13 will be last.
13 + 5 = 5 + 13
In each case, the result of the addition is 18.
13 + 5 = 18
5 + 13 = 18
Changing the order of the addends (5, 13) did not change the sum (18).

### The Associative Property of Addition

The associative property of addition allows us to regroup three or more addends in an addition problem. The associative property of addition tells us that the addition of three or more numbers can be grouped in any order, without changing the sum.
Example 4: Rewrite (9 + 1) + 5 using the associative property of addition
To complete this problem, we simply change the grouping. Notice how parentheses are around the 9 + 1 portion of the problem. In order to complete the problem, we can regroup and place parentheses around 1 + 5:
(9 + 1) + 5 = 9 + (1 + 5)
From the order of operations, we work inside of parentheses first:
(9 + 1) + 5 = 10 + 5 = 15
9 + (1 + 5) = 9 + 6 = 15
Changing the grouping, did not change the sum.
In each case, the result of the addition is 15.