Addition may be a pretty simple idea. You take two or more numbers, combine their value, and get a sum, or total. But addition is so much more interesting and exciting than that! Addition has properties, or special math rules, that make adding easy and fun. One of these properties is called the associative property.
The associative property is a rule that says, in any addition problem, you can add the numbers together in whatever order you want. And guess what? The sum of the numbers doesn’t change! That means that you can add the numbers together in all kinds of different ways, but the answer will always be the same.
Even if the addition problem has three basic digits…
3 + 5 + 3
Or too many digits to count…
4 + 2 + 9 + 10 + 5 + 7 + 7 + 1 + 6 + 11 + 3 + 9 + 8 + 15
Or big numbers that make you think really hard…
435 + 7,290 + 560
The associative property of addition tells us that we can group those numbers in any order we want.
Associative Property of Addition
Addition has many rules, or properties, that are true for all addition problems. There is the commutative property, the distributive property, and, of course, the associative property. Although the vocabulary, or words, in these properties is difficult to understand at first, once you break down the real meaning, it becomes quite simple and clear.
The root word of associative is associate, which means to connect or group. For example, you might associate a rainbow with rain, a beach with swimming, or football with your favorite team.
When we think of the word associate in terms of addition, it has a similar meaning. The associative property tells us that we can associate, or group, any numbers together in an addition problem without changing the answer- the sum remains the same!
Let’s look at a basic example.
5 + 6 + 7
Thanks to the associative property of addition, there are multiple ways to solve this problem.
You can simply add the addends in order of how they appear.
However, because the associative property of addition is true, you can start with any of the three numbers and still get the correct answer- 18.
Typically, math problems involving the associative property of addition use parentheses to show where the addends can be, or should be, grouped.
Parentheses act like a flag that shows us where to begin solving a math problem. In most math problems, the parentheses are really important; in order to get the right answer, you must start with the numbers and the operation given inside the parentheses. However, because of the associative property, the parentheses can be moved around in an addition problem without changing the sum.
In our example above, the parentheses can be added to show which pair of numbers were added first.
The associative property and parentheses can come in handy when you have to complete problems that you want to solve in your own, unique way. Most math problems are written a certain way and can’t be changed. However, the associative property makes addition problems a little more flexible.
When you are given a set of numbers to add, sometimes it makes sense to add the addends out of order. You might want to add a pair of addends first because they are easier, because they add up to a nice round sum (like 10 or 100), or because you already know that sum off the top of your head.
Let’s look at the following example:
620 + 63 + 20 + 44
In this problem, it might be a breeze for you to add the addends in order- 620 to 63, and then 20, and then 44. However, using the associative property and grouping the addends in a different order might make the addition problem even easier!
It is much easier to add 20 and 44 before any of the other numbers. We can represent that thinking with parentheses grouping 20 and 44 together.
Then, because of the associative property, you can add another set of numbers, 620 + 63 or 63 + 64.
Next, you find the sum of the last two addends.
683 + 64 = 747 620 + 127 = 747
The associative property says that either method of solving the problem works and delivers the same sum.
The properties of addition are helpful and make a simple idea, like addition, even easier. The associative property of addition becomes more and more important as math gets more and more complex. Hopefully, one day, you can associate complex math problems with the ease of the associative property.
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