Properties of Rational Numbers

1
Properties of Rational Numbers

• Algebra and Functions 1.3
• Simplify Numerical expressions by applying
  properties of rational numbers (e.g. identity,
  inverse, distributive, associative, commutative)

2
Math ObjectiveUnderstand and distinguish
between the commutative and associative properties
3
Five Properties of Rational Numbers

1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

4
The 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.

5
Here are two families of commuters.
Commuter B
Commuter A
Commuter A Commuter B changed lanes.
Remember commute means to change.
Commuter A
Commuter B
6
Home
School
Would the distance from Home to School and then
from school to home change?
Home School School Home
H S S H
A B B A
7
3 groups of 5
5 groups of 3
3 x 5
5 x 3



15 kids
15 kids
8
The Commutative Property

• A B B A

A x B B x A
9
The Commutative Property
You can add or multiply numbers in any order.
Numbers Algebra
4 6 6 4 a b b a
It is called the commutative property of addition
when we add, and the commutative property of
multiplication when we multiply.
10
Five Properties of Rational Numbers

1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

11
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 do
  talk to one another.

12
Lets 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?

13
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!
14
(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.
15
The Associative Property
The parentheses identify which two associates
talked first.
(A B) C A (B C)
THEN
THEN
16
)
(
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?
)
(

17
The Associative Property
When 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)
It is called the associative property of addition
when we add, and the associative property of
multiplication when we multiply.
18
Lets practice !
Look at the problem. Identify which property it
represents.
19
(4 3) 2 4 (3 2)
The Associative Property of Addition
It has parentheses!
20
6 11 11 6
The Commutative Property of Multiplication

• Same 2 numbers
• Numbers switched places

21
(1 2) 3 1 (2 3)
The Associative Property of Multiplication

• Same 3 numbers in the same order
• 2 sets of parentheses

22
a b b a
The Commutative Property of Multiplication
23
(a b) c a (b c)
The Associative Property of Multiplication
24
4 6 6 4
The Commutative Property of Addition
Numbers change places.
25
(a b) c a (b c)
The Associative Property of Addition
Parentheses!
26
a b b a
The Commutative Property of Addition
Moving numbers!
27
Five Properties of Rational Numbers

1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

28
The Identity Property
I am me! You cannot change My identity!
29
Identity Property of Addition
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.
30
Identity Property of Addition
0

• A 0 A

31
Identity Property of Multiplication
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.
32
Identity Property of Multiplication
1

• A 1 A

33
Five Properties of Rational Numbers

1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

34
Inverse Property
Inverse means opposite.
35
Inverse Property
The opposite of addition is
subtraction.
So, when I use inverse operations, I can undo
the original number.
Example 3 (-3) 0
36
Inverse Property
The opposite of division is
multiplication.
So, when I use inverse operations, I can undo
the original number.
Example
37
Lets practice !
Look at the problem. Identify which property it
represents.
38
a 1 a
The Identity Property of Multiplication
39
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.
40
987 1 987
The Identity Property of Multiplication

• Times 1

41
7 (- 7) 0
The Inverse Property

• Undo the operation by using the opposite operation

42
9 1 9
The Identity Property of Multiplication

• Times 1

43
6
1
6
The Inverse Property

• Undo the operation by using the inverse operation

44
3 0 3
The Identity Property of Addition
See the zero?
45
a 0 a
The Identity Property of Addition
Zero!
46
Five Properties of Rational Numbers

1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

47
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

48
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
Lets 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)
49
You have sold many items for the BMMS fundraiser!
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?
5(2 3) (5 2) (5 3) Think Five groups
of (23) or (23) (23) (23) (23) (23)
How many red gifts were distributed? How many
green gifts were distributed?
50
You will be distributing 5 items to each house.
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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
52
The Distributive Property
4( 3n 6)
3( 5 2)
3n
6
5
2


3
4
12n
24
15
6
15 6 21
12n 24
-7( 4 6)
9( -3 - 8)
4
6
-3
- 8

-7

-28
-42
-27
9
-72
-28 - 42 -70
-27 - 72 -99
53
The Distributive Property
-4( 8x 3)
6( 4x - 2)
8x
-3
4x
-2


6
-4
-32x
12
24x
-12
24x - 12
-32x 12
-6n( 2 - 6)
5( -6n 2)
2
-6
-6n
2


-12n
-6n
36n
-30n
5
10
-12n 36n 24n
-30n 10