Chapter 2: Arithmetic 19DEFINITION
The associative property is a property of addition or multiplication that says that
when you must add or multiply more than two numbers, you may group them in
different ways without changing the result.If a problem has lots of numbers, you’ll sometimes see parentheses around some of the numbers.
This is a way of saying “do this part first.” It might have 3 + (7 + 4) + 9 to tell you to add the
7 and the 4 first. That can be helpful, and sometimes absolutely necessary, but the associative
property says that if the problem is all addition or all multiplication, you can move those
parentheses. It’s telling you that you’ll get the same answer no matter which part you do first, as
long as the problem has just one operation, addition or multiplication. 3 + (7 + 4) + 9 =
(3 + 7) + 4 + 9. You can regroup.
When you let yourself use both the commutative and the associative properties, you realize that
as long as addition is the only thing going on, you can tackle those numbers in any order. That’s
one step toward speedier work. If you can add any two numbers at a time, do the simple ones
first. Adding 8 + 5 + 5 + 2 is easier if you rearrange it to 5 + 5 + 8 + 2. Add the 5s to get 10, add
the 8 + 2 to get another 10, and the two 10s give you 20.
MATH TRAP
Like the commutative property, the associative property is for addition (or multiplica-
tion), not for subtraction (or division) and not for combinations of operations. (8 + 7) + 3
is the same as 8 + (7 + 3), but (8 + 7) v 3 is not the same as 8 + (7 v 3).Grouping the numbers in this way is easier because 5 and 5 are compatible numbers, and so are 8
and 2. Compatible numbers are pairs of numbers that add to ten. Why ten? Because our decimal
system is based on tens. You might have your own version of compatible numbers, which add to
something else.
For example, if you were a shepherd and had to keep track of a f lock of seven sheep, you’d spend
a lot of your time counting to seven to be sure you had them all. You’d quickly get to know that
if you saw 5 of them in the pasture and 2 on the hill, all was well. If 3 were by the stream and 4
under the tree, you were good. You’d know all the pairs of numbers that added to 7: 1 + 6, 2 + 5,
and 3 + 4. For general addition purposes, however, compatible numbers are pairs that add to 10:
1 + 9, 2 + 8, 3 + 7, 4 + 6, and 5 + 5.