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property is frequently employed when terms within parentheses or other grouping symbols cannot be combined. This allows for each term to interact with the multiplier.

      

Q.

      A. Distribute the 3 over the by multiplying each term by 3: . Then simplify the terms: .

      Q.

      A.

      7

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      9

10

      11

      12

      Making Associations Work

      The associative property has to do with grouping — that is, how you deal with two or more terms when you perform operations on them. Think about what the word associate means. When you associate with someone, you’re close to the person, or you’re in the same group with them. Say that Anika, Becky, and Cora associate. Whether Anika drives over to pick up Becky and the two of them go to Cora’s house and pick her up, or Cora is at Becky’s house and Anika picks up both of them at the same time, the same result occurs: all three are in the car at the end.

      

The associative property means that even if a particular grouping of the operation changes, the result remains the same. (If you need a reminder about grouping, refer to the section, “Getting a Grip on Grouping Symbols,” earlier in this chapter.) Addition and multiplication are associative operations. Subtraction and division are not associative. So,

      You can always find a few cases where the associative property works even though it isn’t supposed to. For example, in the subtraction problem , the property seems to work. Also, in the division problem , it seems to work. I just picked numbers very carefully that would make it seem like you could associate using subtraction and division. Although there are exceptions, a rule must work all the time, not just in special cases.

      Here’s how the associative property works:

      

This rule is special to addition and multiplication. It doesn’t work for subtraction or division. You’re probably wondering why you would even use this rule. That’s because it can sometimes make the computation easier.

      

Q. Use the associative property to create an easier problem:

      A. With the current grouping, you need to add to 111 and then add the result to 14. Instead, take advantage of the associative property and regroup: . Now you have a resulting problem of — much easier!

      Q. Use the associate property to create an easier problem:

      A. Regroup, putting the first two numbers together:

      13

      14

      15

      16

      Computing by Commuting

      Before discussing the commutative property, take a look at the word commute. You probably commute to work or school and know that whether you’re traveling from home to work or from work to home, the distance is the same: The distance doesn’t change because you change directions (although getting home during rush hour may make that distance seem longer).

      The same principle is true of some algebraic operations: It doesn’t matter whether you add or , the answer is still 3. Likewise, multiplying or yields 6.

      

The commutative property means that you can change the order of the numbers in an operation without affecting the result. Addition and multiplication are commutative. Subtraction and division are not. So,

      In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For example, if a and b are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer. In the case of division, if a and b are opposites, then you get –1 no matter which order you divide them in. By the way, this is why, in mathematics, big deals are made about proofs. A few special cases of something may work, but a real rule or theorem has to work all the time.

      You can use this rule to your

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