Properties of Rational Numbers

Properties of Rational Numbers

Algebra and Functions 1.3Simplify Numerical expressions by

applying properties of rational numbers (e.g. identity, inverse, distributive,

associative, commutative)

Algebra and Functions 1.3Simplify Numerical expressions by

applying properties of rational numbers (e.g. identity, inverse, distributive,

associative, commutative)

Math Objective:Understand and distinguish between the commutative and associative properties

Math Objective:Understand and distinguish between the commutative and associative properties

Five Properties of Rational Numbers

Five Properties of Rational Numbers

1. Commutative

2. Associative

3. Identity

4. Inverse

5. Distributive

The Commutative PropertyThe Commutative Property

• Background– The word commutative comes from the

verb “to commute.”– Definition on dictionary.com

• Commuting means changing, replacing, or exchanging

– People who travel back and forth to work are called commuters.• Traffic Reports given during rush hours are

also called commuter reports.

Here are two families of commuters.Here are two families of commuters.

Commuter A

Commuter B

Commuter A

Commuter B

Commuter A & Commuter B changed lanes.

Remember… commute means to change.

Home School

Would the distance from Home to School and then from school to home change?

Home + School = School + HomeHome + School = School + Home

H + S = S + HH + S = S + H

A + B = B + AA + B = B + A

3 groups of 5 =

=

15 kids =15 kids

3 x 5 5 x 3=5 groups of 3

The Commutative PropertyThe Commutative Property

A + B = B + AA + B = B + A

A x B = B x AA x B = B x A

The Commutative PropertyThe Commutative PropertyYou can add or multiply numbers in any order.

Numbers Algebra

4 + 6 = 6 + 4 a + b = b + a

3663 abba

It is called the commutative property of addition when we add, and the commutative property of multiplication when we multiply.

Five Properties of Rational Numbers

Five Properties of Rational Numbers

1. Commutative

2. Associative

3. Identity

4. Inverse

5. Distributive

The Associative PropertyThe 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 do talk to one another.

Let’s look at another hypothetical situation

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?

A C

B

Here are three associates. Here are three associates.

B calls A first He calls C last

If he called C first, then called A, would

it have made a difference?

NO!

(The Role of Parentheses)(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….So….

The Associative PropertyThe Associative Property

(A + B) + C = A + (B + C)(A + B) + C = A + (B + C)

A C

B

A C

B

THEN THEN

The parentheses identify which two associates talked first.

Notice the first two students are associating with each other in the first situation. In the second situation, the same girl is associating with a different student. Have the students changed? Have the students moved places?

=

( )

( )

The Associative PropertyThe Associative PropertyWhen adding or multiplying, you can change the grouping of numbers without changing the sum or product. The order of the terms DOES NOT change.

Numbers Algebra

(3 + 9) + 2 = 3 + (9 + 2) (a + b) + c = a + (b + c)

2)(4324)(3 c)(bacb)(a

It is called the associative property of addition when we add, and the associative property of multiplication when we multiply.

Let’s practice !Let’s practice !

Look at the problem.

Identify which property it represents.

(4 + 3) + 2 = 4 + (3 + 2)

The Associative Property of Addition

It has parentheses!

6 • 11 = 11 • 6

The Commutative Property

of Multiplication

•Same 2 numbers

•Numbers switched places

(1 • 2) • 3 = 1 • (2 • 3)

The Associative Property of Multiplication

•Same 3 numbers in the same order

•2 sets of parentheses

a • b = b • aThe Commutative Property

of Multiplication

A

The Associative Property

of Multiplication

C

B

(a • b) • c = a • (b • c)

4 + 6 = 6 + 4The Commutative Property of Addition

Numbers change places.

A

(a + b) + c = a + (b + c) The Associative Property of Addition

Parentheses!

C

B

a + b = b + aThe Commutative Property of Addition

Moving numbers!

Five Properties of Rational Numbers

Five Properties of Rational Numbers

1. Commutative

2. Associative

3. Identity

4. Inverse

5. Distributive

The Identity PropertyThe Identity PropertyI am me!

You cannot changeMy identity!

I am me!You cannot change

My identity!

Zero is the only number you can add to something

and see no change.

This property is also sometimes called the

Identity Property of Zero.

Identity Property of AdditionIdentity Property of Addition

Identity Property of AdditionIdentity Property of Addition

A + 0 = A

+ 0 =

One is the only number you can multiply by something

and see no change.

This property is also sometimes called the

Identity Property of One.

Identity Property of MultiplicationIdentity Property of Multiplication

Identity Property of MultiplicationIdentity Property of Multiplication

A • 1 = A

• 1 =

Five Properties of Rational Numbers

Five Properties of Rational Numbers

1. Commutative

2. Associative

3. Identity

4. Inverse

5. Distributive

Inverse means “opposite”.

Inverse PropertyInverse Property

Inverse Property Inverse Property

The opposite of addition is…subtraction.

So, when I use inverse operations, I can “undo” the

original number.

Example: 3 + (-3)= 0

Inverse Property Inverse Property

The opposite of division is…

multiplication.

So, when I use inverse operations, I can “undo” the

original number.

Example: 113

31

Let’s practice !Let’s practice !

Look at the problem.

Identify which property it represents.

a • 1 = a The Identity Property

of Multiplication

12 + 0 = 12

The Identity Property of Addition

It is the only addition property that has two addends and one of them is a zero.

987 • 1 = 987

The Identity Property

of Multiplication

•Times 1

7 + (- 7) = 0

The Inverse Property

•Undo the operation by using the opposite operation

9 • 1 = 9

The Identity Property

of Multiplication

•Times 1

The Inverse Property

•Undo the operation by using the inverse operation

66 = 1

3 + 0 = 3 The Identity Property of Addition

See the zero?

a + 0 = a The Identity Property of Addition

Zero!

Five Properties of Rational Numbers

Five Properties of Rational Numbers

1. Commutative

2. Associative

3. Identity

4. Inverse

5. Distributive

The Distributive PropertyThe 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)

5(2 + 3) = (5 • 2) + (5 • 3)Think: Five groups of (2+3) or(2+3) + (2+3) + (2+3) + (2+3) + (2+3)

You went to five houses. Every family bought 5 items total, 2 red gifts and three

green gifts! How many gifts did you deliver all together?

How many red gifts were distributed? How many green gifts

were distributed?

You have sold many items for the BMMS fundraiser!

You will be distributing 5 items to each house.

5(2 + 3) = (5 • 2) + (5 • 3)

You distributed (delivered) these all in one trip.

You need to deliver 5 gifts to each house.

To each house, you will deliver 2 red gifts and 3 green gifts.

How many red gifts?

How many green gifts?

5 houses x 2 red gifts and 5 houses x 3 green gifts = (5x2) + (5x3) = 25 items all together

The Distributive Property The Distributive Property

3( 5 + 2)

3

5 2

15 6

15 + 6 = 21

4( 3n + 6)

4

3n 6

12n 24

12n + 24

-7( 4 + 6)

-28-74 6

-42

-28 - 42 = -70

9( -3 - 8)

-279

-3 - 8-72

-27 - 72 = -99

The Distributive Property The Distributive Property

6( 4x - 2)

6

4x -2

24x -12

24x - 12

-4( 8x – 3)

-4

8x -3

-32x 12

-32x + 12

-6n( 2 - 6)

-12n-6n

2 -636n

-12n + 36n = 24n

5( -6n + 2)

-30n5

-6n 210

-30n + 10