PPT on 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 Objective:
Understand 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
A
Commuter
B
Commuter
A
Commuter
B
Commuter A & Commuter B
changed lanes.
Remember… commute
means to change.

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

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
3
6
6
3 

 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. 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?

13. A C
B
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
(A + B) + C = A + (B + C)
A C
B
A C
B
THEN THEN
The parentheses identify which two associates talked first.

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)
2)
(4
3
2
4)
(3 



 c)
(b
a
c
b)
(a 




It is called the associative property of addition
when we add, and the associative property of
multiplication when we multiply.

18. Let’s 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
The Associative Property
of Multiplication
C
B
(a • b) • c = a • (b • c)

24. 4 + 6 = 6 + 4
The Commutative Property of Addition
Numbers change places.

25. A
(a + b) + c = a + (b + c)
The Associative Property of Addition
Parentheses!
C
B

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. 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 Addition

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

31. 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 Multiplication

32. Identity Property of Multiplication
A • 1 = A
• 1 =

33. Five Properties of Rational
Numbers
1. Commutative
2. Associative
3. Identity
4. Inverse
5. Distributive

34. Inverse means “opposite”.
Inverse Property

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: 1
1
3
3
1



37. Let’s 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. The Inverse Property
•Undo the operation by using the inverse
operation
6
6 = 1

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
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)

49. 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!

50. You will be distributing 5 items to each house.

51. 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
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
-7
4 6
-42
-28 - 42 = -70
9( -3 - 8)
-27
9
-3 - 8
-72
-27 - 72 = -99

53. 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 -6
36n
-12n + 36n = 24n
5( -6n + 2)
-30n
5
-6n 2
10
-30n + 10