Explore the commutative, associative, and identity properties of addition.
In this article, we'll learn the three main properties of addition. Here's a quick summary of these properties:
Commutative property of addition: Changing the order of addends does not change the sum. For example, $4 + 2 = 2 + 4$.
Associative property of addition: Changing the grouping of addends does not change the sum. For example, $(2 + 3) + 4 = 2 + (3 + 4)$.
Identity property of addition: The sum of $0$ and any number is that number. For example, $0 + 4 = 4$.

The commutative property of addition says that changing the order of addends does not change the sum. Here's an example:
$4 + 2 = 2 + 4$
Notice how both sums are $6$ even though the the ordering is reversed.
If Jill starts with $4$ bananas
and then gets another $2$ bananas
she ends up with the same amount as if she started with $2$ bananas
and got $4$ more bananas.
Here's another example with more addends:
$1 + 2 + 3 + 4 = 4 + 3 + 2 + 1$
Which of these is an example of the commutative property of addition?

The associative property of addition says that changing the grouping of the addends does not change the sum. Here's an example:
$\blueD{(2 + 3) + 4} = \goldD{2 + (3 + 4)}$
Remember that parentheses tell us to do something first. So here's how we evaluate the left-hand side:
$\phantom{=}\blueD{(2 + 3) + 4}$
$= 5 + 4$
$=9$
And here's how we evaluate the right-hand side:
$\phantom{=}\goldD{2 + (3 + 4)}$
$= 2 + 7$
$=9$
Notice that both sides sum to $9$ even though we added the $2$ and the $3$ first on the left-hand side, and we added the $3$ and the $4$ first on the right-hand side.
Which of these is an example of the associative property of addition?
The identity property of addition says that the sum of $0$ and any number is that number. Here's an example:
$0 + 4 = 4$
This is true because the definition of $0$ is "no quantity", so when we add $0$ to $4$, the quantity of $4$ doesn't change!
The commutative property of addition tells us that it doesn't matter if the $0$ comes before or after the number. Here's an example of the identity property of addition with the $0$ after the number:
$6 + 0 =6$