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+44 + 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)(2 + 3) + 4 = 2 + (3 + 4).
Identity property of addition: The sum of 00 and any number is that number. For example, 0+4=40 + 4 = 4.

Commutative property of addition

The commutative property of addition says that changing the order of addends does not change the sum. Here's an example:
4+2=2+44 + 2 = 2 + 4
Notice how both sums are 66 even though the the ordering is reversed.
If Jill starts with 44 bananas
and then gets another 22 bananas
she ends up with the same amount as if she started with 22 bananas
and got 44 more bananas.
Here's another example with more addends:
1+2+3+4=4+3+2+11 + 2 + 3 + 4 = 4 + 3 + 2 + 1
Which of these is an example of the commutative property of addition?
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Associative 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:
(2+3)+4=2+(3+4)\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:
=(2+3)+4\phantom{=}\blueD{(2 + 3) + 4}
=5+4= 5 + 4
=9=9
And here's how we evaluate the right-hand side:
=2+(3+4)\phantom{=}\goldD{2 + (3 + 4)}
=2+7= 2 + 7
=9=9
Notice that both sides sum to 99 even though we added the 22 and the 33 first on the left-hand side, and we added the 33 and the 44 first on the right-hand side.
Which of these is an example of the associative property of addition?
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Identity property of addition

The identity property of addition says that the sum of 00 and any number is that number. Here's an example:
0+4=40 + 4 = 4
This is true because the definition of 00 is "no quantity", so when we add 00 to 44, the quantity of 44 doesn't change!
The commutative property of addition tells us that it doesn't matter if the 00 comes before or after the number. Here's an example of the identity property of addition with the 00 after the number:
6+0=66 + 0 =6
Which of these is an example of the identity property of addition?
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