Properties

Most of these will seem like common sense to the average middle school student, because you've been using them for several years. You may or may not know the "fancy-schmancy" name for them. So here's a quick rundown.

Definition Associative Commutative Identity Zero Distributive

A PROPERTY is a rule, or a special feature of certain operations that can be used to make solving problems easier.

The associative property allows you to change the grouping of numbers. Think that your parents don't want you to ASSOCIATE (hang out with) the wrong GROUP of kids.

The associative property applies in both addition and multiplication, but NOT subtraction and division.

The associative property can be spotted easily because of the parentheses ( ) ... but don't be fooled. Just because parentheses are present, doesn't mean the associative property is being used.

The associative property will have at least three numbers/letters (addends or factors).

EXAMPLES:

(3 + 4) + 6 = 3 + (4 + 6)	< Notice that on the left, 3 + 4 was grouped, and on the right, 4 + 6 was grouped.

x (y z) = (x y) z	< Notice that on the left, y times z was grouped, and on the right, x times y was grouped.

The commutative property allows you to change the order of (or move) numbers. Think that many parents commute to work when they drive to Philadelphia (they have to move around to get to work).

The commutative property applies in both addition and multiplication, but NOT subtraction and division.

The commutative property will have at least two numbers/letters (addends or factors).

EXAMPLES:

4 + 5 = 5 + 4	< Notice that the numbers "flip-flopped" on either side of the equals sign.

26 x y = 26 y x	< Notice that the numbers "flip-flopped" on either side of the equals sign.

The identity property allows you to perform an operation without changing the value of the original number. Think that your identity is who you are. If we change your hair or clothes or polish your nails, it doesn't change who you are, what you think and believe.

The identity property applies in both addition and multiplication, but each has its own special number.

The Additive Identity is the number zero.

The Multiplicative Identity is the number one.

EXAMPLES:

25 + 0 = 25	< Notice that the original value was not changed when zero was added.

(36) (1) = 36	< Notice that the original value did not change when multiplied by one.

The multiplication property of zero states that any number times zero is zero.

EXAMPLES:

2 · 0 = 0

0 · 4,693 = 0

The distributive property is used with multiplication, over addition and subtraction.

Remember that to distribute means to give something out, fairly, to each.

EXAMPLES:

4(x + 3)	< 4(x) + 4(3) = 4x + 12

12 ( 6 - y)	< 12(6) - 12(y) = 72 - 12y

But just as you can give out a number when you distribute, you can take away and "undistribute" (called factoring). This can help with mental math.

EXAMPLES:

5(23) - 5(3)
5(23 - 3)
5(20)
100 < What's the same in both parts? The 5. So take that out of each and rewrite what's left.
< Now you can simplify inside the parantheses.
< Then multiply.
< Your answer, no calculator needed!

8(207)
8(200 + 7)
1600 + 56
1656 < Think of another way to represent 197.
< Use the distributive property.
< Your answer, no calculator needed!

6(894)
6( 900 - 6)
5400 - 36
5364 < Think of another way to represent 894.
< You could use addition, but subtraction is bit quicker.
< Distribute.
< Your answer, no calculator needed!

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5(23) - 5(3) 5(23 - 3) 5(20) 100	< What's the same in both parts? The 5. So take that out of each and rewrite what's left. < Now you can simplify inside the parantheses. < Then multiply. < Your answer, no calculator needed!

8(207) 8(200 + 7) 1600 + 56 1656	< Think of another way to represent 197. < Use the distributive property. < Your answer, no calculator needed!

6(894) 6( 900 - 6) 5400 - 36 5364	< Think of another way to represent 894. < You could use addition, but subtraction is bit quicker. < Distribute. < Your answer, no calculator needed!