Properties of Numbers 10/20/2016

Do you remember these Properties of Addition? Commutative Property of Addition The numbers move around a + b = b + a Associative Property of Addition Grouping with parentheses (a + b) + c = a + (b + c) Identity Property of Addition The identity of the problem does not change a + 0 = a

In multiplication, you will see these same properties, plus 2 more…

Five Properties of Multiplication These are the basically the same as addition Commutative Associative Identity These belong to multiplication only Zero Distributive

Multiplicative (Do you see most of the word “multiply” in this word? Let’s review the addition properties— from the multiplicative perspective… Multiplicative (Do you see most of the word “multiply” in this word?

The Commutative Property of Multiplication

The Commutative Property Background The word commutative comes from the verb “to commute.” Definition on dictionary.com Commuting means changing, replacing, exchanging, switching places, trading places People who travel back and forth to work are called commuters.

Here are two families of commuters. Hi! Remember us? Commuter B Commuter A Commuter A & Commuter B changed lanes. Remember… commute means to switch places. Commuter A Commuter B

The Commutative Property A • B = B • A

Here is another example…

3 groups of 5 = 5 groups of 3 3 x 5 = 5 x 3 = = 15 kids 15 kids

What commutative means to multiplication… Remember… in Lesson 1-11 we said that the word “of” means multiply 3 groups of 5 = 5 groups of 3 3 • 5 = 5 • 3 a • b = b • a

The Associative Property of Multiplication

The Associative Property Background The word associative comes from the verb “to associate.” Definition on dictionary.com Associate means connected, joined, or related People who work together are called associates. They are joined together by business, and they have to talk to one another.

Let’s look at another hypothetical situation Three people work together. Associate B needs to call Associates A and C to share some news. Does it matter who he calls first?

Here are three associates. B B calls A first He calls C last A C If he called C first, then called A, would it have made a difference? NO!

(The Role of Parentheses) In math, we use parentheses to show groups. In the order of operations, the numbers and operations in parentheses are done first. (PEMDAS) So….

The Associative Property The parentheses identify which two associates talked first. (A  B)  C = A  (B  C) B B A C THEN A THEN C

Also known as the zero property of the addition Property #3a The Identity Property of Addition Also known as the zero property of the addition Make foldable

The Identity Property of Addition I am me! You cannot change My identity!

0 is the only number you can add to something and see no change.

Identity Property of Addition + 0 = A + 0 = A

The Identity Property of Multiplication Property #3b The Identity Property of Multiplication Make foldable

The Identity Property I am me! You cannot change My identity!

1 is the only number you can multiply something by and see no change.

Identity Property of Multiplication x 1 = a x 1 = a

Identity Property of Multiplication x 1 = x 1 = x 1 = a x 1 = a

These are 3 of the Properties of Multiplication Commutative Property of Multiplication The numbers move around a • b = b • a Associative Property of Multiplication Grouping with parentheses (a • b) • c = a • (b • c) Identity Property of Multiplication The identity of the problem does not change a • 1 = a

There are two more properties which are unique to multiplication The Zero Property The Distributive Property

The Zero Property of Multiplication Make foldable

The Zero Property of Multiplication This looks like a mixture of the identity property of addition and the identity property of multiplication… Be careful not to mix them up!

If I have 2 pockets with NO money in them, then I have NO money! The Zero Property Any time you multiply a number by zero, your answer is zero! If I have 2 pockets with NO money in them, then I have NO money! 2 • 0 = 0 The End

The Distributive Property of Multiplication

The Distributive Property Background The word distributive comes from the verb “to distribute.” Definition on dictionary.com Distributing refers to passing things out or delivering things to people

The Distributive Property a(b + c) = (a • b) + (a • c) A times the sum of b and c = a times b plus a times c Let’s plug in some numbers first. Remember that to distribute means delivering items, or handing them out. Here is how this property works: 5(2 + 3) = (5 • 2) + (5 • 3)

You have sold many items for the RCMS fundraiser! You went to two houses on one street and three houses on a different street. Every family bought 5 items! 5(2 + 3) = (5 • 2) + (5 • 3) You went to two houses on one street and three houses on a different street. Every family bought 5 items!

You will be distributing 5 items to each house. 2 3 5 4 1

5(2 + 3) = (5 • 2) + (5 • 3) You distributed (delivered) these all in one trip. There are (2+3) five houses all together. You need to deliver 5 gifts to each house. You need to put 25 items on your wagon at one time. 5 items x 5 houses = 25 items all together

5(2 + 3) = (5 • 2) + (5 • 3) and 10 You distributed your items in two trips (+). On the first trip you distributed 5 items to each of 2 houses (5 x 2 = 10). On the second trip you distributed 5 items to each of 3 houses (5 x 3 = 15). That means you distributed (delivered) 10 items plus 15 items. That makes 25 items altogether. + 15 25

The Distributive Property Make 1 trip. You have 5 houses. You need to bring 5 items to each house. You need 25 items on your wagon. DISTRIBUTION CENTER 5(2 + 3)

The Distributive Property Make 2 trips. You have 2 houses for your first trip and you need to bring 5 items to each house. You have 3 houses on your second trip and need to bring 5 items to each house. When your second trip is over, you will have distributed 25 items. DISTRIBUTION CENTER (5 • 2) + (5 • 3)

How do I tell the properties apart? Commutative Numbers switch places Associative Parentheses on both sides Only multiplication on each side Identity Multiply by 1 Zero Property Multiply by zero Distributive Parentheses on each side One side has a multiplication sign AND a plus sign

Let’s practice ! Look at the problem. Identify which property it represents.

The Distributive Property 4(5 + 6) = (4 • 5) + (4 • 6) The Distributive Property of Multiplication 3 numbers on one side—4 on the other Multiplication AND addition 3 sets of parentheses

987 • 1 = 987 The Identity Property of Multiplication Times 1

Zero Property of Multiplication 3 • 0 = 0 Zero Property of Multiplication Times zero

The Associative Property of Multiplication (1 • 2) • 3 = 1 • (2 • 3) The Associative Property of Multiplication Same 3 numbers Multiplication only 2 sets of parentheses

The Commutative Property 6 • 11 = 11 • 6 The Commutative Property of Multiplication Same 2 numbers Numbers switched places

The Commutative Property 9 • 7 = 7 • 9 The Commutative Property of Multiplication Same 2 numbers Numbers switched places

Zero Property of Multiplication 12 • 0 = 0 Zero Property of Multiplication Times zero

The Associative Property of Multiplication (9 • 8) • 7 = 9 • (8 • 7) The Associative Property of Multiplication Same 3 numbers Multiplication only 2 sets of parentheses

The Distributive Property 9(8 + 7) = (9 • 8) + (9 • 7) The Distributive Property of Multiplication 3 numbers on one side—4 on the other Multiplication AND addition 3 sets of parentheses

9 • 1 = 9 The Identity Property of Multiplication Times 1

a • 1 = a The Identity Property of Multiplication

The Commutative Property a • b = b • a The Commutative Property of Multiplication

The Associative Property (a + b) + c = a + (b + c) The Associative Property of Multiplication B A C

a • 0 = 0 The Zero Property of Multiplication

The Distributive Property a(b • c) = (a • b) + (a • c) The Distributive Property of Multiplication

Ticket Out the Door http://www.softschools.com/quizzes/math/algebra_properties/quiz1038.html