The associative property of addition states that how the numbers in an addition problem are grouped doesn't change the sum. Given that a, b, and c are addends in an addition problem, it doesnt matter whether a and b are grouped together and added first, or if b and c are grouped together and added first. The result will be the same. This can be written in the form of an addition sentence as:

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

One way to visualize the associative property of addition is to use a set of objects. The figure below shows 3 sets of stars with different colors. The objects being added are identical, and the different colors represent the addends.

In both of the cases above, the result is still 12, regardless whether we combine the blue and yellow or yellow and green stars first. Since they are all the same object being added together, as long as each part is counted in the total, how each part is counted does not affect the sum.

The associative property holds true for any number of addends, as long as there are at least 3 (you can't group 2 addends in different ways).

Examples

4 + 7 + 12 + 15 + 48 = 86

Using parentheses to indicate order of operation, we can group the above addition problem in a number of ways:

Grouping 1:

(4 + 7) + (12 + 15 + 48) = 86

(11) + (75) = 86

Grouping 2:

(4 + 7 + 12) + 15 + 48 = 86

(23) + 15 + 48 = 86

Grouping 3:

(4 + 7) + 12 + (15 + 48) = 86

(11) + 12 + (63) = 86

There are more ways to group the above numbers than what is shown, but in all cases, the sum will still be 86. This is the nature of the associative property of addition, and it is just one of the many properties of addition.

Learning the various properties of addition is important because it helps to form a foundation for learning more complex mathematical concepts in the future. For example, when solving an algebraic equation, it can often be helpful to group numbers in a certain way, or even rearrange them (using the commutative property of addition). Even for relatively simple addition problems, grouping the addends as compatible numbers can simplify the problem.