Properties of Addition (Commutative, Associative, Identity) - Third Grade Mathematics

Addition is one of the most natural operations in mathematics, and it possesses fundamental rules called properties that remain true no matter what numbers you add. Understanding the Commutative, Associative, and Identity properties of addition allows you to rearrange numbers flexibly, make calculations much simpler in your head, and develop deep confidence as a mathematical thinker.

What Are Mathematical Properties?

In mathematics, a property is an unchanging truth or rule that describes how numbers behave. Just like water always freezes at a certain temperature, numbers always follow these mathematical properties. When you learn how these properties work, you can use them as special tools to make hard problems easy.

The Commutative Property of Addition

The Commutative Property of Addition is often called the "Order Property." It states that changing the order of the addends does not change the sum.

Rule: a + b = b + a

Let us look at a visual model using groups of counters:

Group A: 4 stars     Group B: 3 stars
****       +         ***              = 7 stars total

Group B: 3 stars     Group A: 4 stars
***        +         ****             = 7 stars total

Therefore: 4 + 3 = 3 + 4 = 7

Why the Commutative Property Is Helpful

Imagine you need to add 2 + 89. If you start at 2 and count on 89 times, it will take a very long time! But because of the commutative property, you can flip the problem around to 89 + 2. Starting at 89 and counting up 2 (90, 91) takes just one second!

The commutative property works for multi-digit numbers too:
- 350 + 120 = 470

- 120 + 350 = 470


The Associative Property of Addition

The Associative Property of Addition is also known as the "Grouping Property." It states that when adding three or more numbers, changing how you group the numbers using parentheses does not change the final sum.

Rule: (a + b) + c = a + (b + c)

Parentheses tell you which operation to perform first.

Expression 1: (3 + 5) + 2
First do 3 + 5 = 8.
Then add 2: 8 + 2 = 10.

Expression 2: 3 + (5 + 2)
First do 5 + 2 = 7.
Then add 3: 3 + 7 = 10.

Both groupings yield the exact same sum of 10!

Making Friendly Numbers with the Associative Property

The associative property is a superpower for mental math because it lets you search for "friendly numbers" that make 10 or 100.

Suppose you have: 17 + 28 + 3. Without the associative property, you might add 17 + 28 first, which requires regrouping. Instead, use the commutative property to swap 28 and 3: 17 + 3 + 28. Now use the associative property to group (17 + 3) together: (17 + 3) + 28 = 20 + 28 = 48! Adding 20 + 28 in your head is effortless.

The Identity Property of Addition

The Identity Property of Addition is also known as the "Zero Property." It states that the sum of any number and zero is that same number.

Rule: a + 0 = a   and   0 + a = a

Think of a number looking in a mirror. When it adds zero, it keeps its exact identity!

+---------------------+-------------------+
| Problem             | Sum               |
+---------------------+-------------------+
| 9 + 0               | 9                 |
| 0 + 74              | 74                |
| 528 + 0             | 528               |
| 0 + 4,106           | 4,106             |
+---------------------+-------------------+

If you have 15 crayons in a box and add 0 new crayons, you still have 15 crayons. Zero adds no value, leaving the quantity unchanged.

Comparing the Three Properties

+-----------------------+---------------------+-------------------------------------+
| Property Name         | Everyday Nickname   | Example Equation                    |
+-----------------------+---------------------+-------------------------------------+
| Commutative Property  | Order Property      | 14 + 25 = 25 + 14                   |
| Associative Property  | Grouping Property   | (6 + 14) + 9 = 6 + (14 + 9)         |
| Identity Property     | Zero Property       | 38 + 0 = 38                         |
+-----------------------+---------------------+-------------------------------------+

Notice that subtraction does NOT have these properties:
- 8 - 3 is 5, but 3 - 8 is not 5 (subtraction is not commutative).

- (10 - 5) - 2 = 3, but 10 - (5 - 2) = 7 (subtraction is not associative).
These special properties belong uniquely to addition (and multiplication)!

Chapter Practice Exercises

  1. Name the property of addition illustrated by each equation: a. 19 + 42 = 42 + 19 b. 67 + 0 = 67 c. (12 + 8) + 15 = 12 + (8 + 15) d. 0 + 512 = 512 e. 45 + (5 + 30) = (45 + 5) + 30
  2. Fill in the missing number to make each statement true: a. 84 + 16 = ___ + 84 b. (29 + 11) + 7 = 29 + (___ + 7) c. 941 + ___ = 941
  3. Use the commutative and associative properties to solve this problem easily in your head: 26 + 19 + 4. Show the steps you used.
  4. Solve by finding friendly tens first: 15 + 38 + 25 + 12.
  5. True or False: 9 + 0 = 0. Explain your answer.

Solutions and Step-by-Step Answers

  1. Properties: a. 19 + 42 = 42 + 19 illustrates the Commutative Property of Addition (order changed). b. 67 + 0 = 67 illustrates the Identity Property of Addition (adding zero keeps identity). c. (12 + 8) + 15 = 12 + (8 + 15) illustrates the Associative Property of Addition (grouping changed). d. 0 + 512 = 512 illustrates the Identity Property of Addition. e. 45 + (5 + 30) = (45 + 5) + 30 illustrates the Associative Property of Addition.
  2. Missing numbers: a. 84 + 16 = 16 + 84 b. (29 + 11) + 7 = 29 + (11 + 7) c. 941 + 0 = 941
  3. To solve 26 + 19 + 4:
  4. Use the Commutative Property to rearrange: 26 + 4 + 19.
  5. Use the Associative Property to group: (26 + 4) + 19.
  6. Add the friendly ten: 30 + 19 = 49.
  7. For 15 + 38 + 25 + 12:
  8. Group friendly pairs: (15 + 25) + (38 + 12).
  9. 15 + 25 = 40.
  10. 38 + 12 = 50.
  11. 40 + 50 = 90.
  12. False. 9 + 0 = 9, not 0. By the Identity Property of Addition, adding zero leaves the original number unchanged.