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
Commutative
Associative
Identity
Inverse
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 = School + Home
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
Commutative
Associative
Identity
Inverse
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. 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!

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) B B A C
THEN A
THEN C

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

19. The Associative Property of Addition
(4 + 3) + 2 = 4 + (3 + 2)
The Associative Property of Addition
It has parentheses!

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

21. The Associative Property of Multiplication
(1 • 2) • 3 = 1 • (2 • 3)
The Associative Property of Multiplication
Same 3 numbers in the same order
2 sets of parentheses

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

23. The Associative Property
(a • b) • c = a • (b • c)
The Associative Property
of Multiplication B A C

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

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

26. The Commutative Property of Addition
a + b = b + a
The Commutative Property of Addition
Moving numbers!

27. Five Properties of Rational Numbers
Commutative
Associative
Identity
Inverse
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
Commutative
Associative
Identity
Inverse
Distributive

34. Inverse means “opposite”.
Inverse Property
Inverse means “opposite”.

35. The opposite of addition is… subtraction.
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. So, when I use inverse operations, I can “undo” the original number.
Inverse Property
The opposite of division is…
multiplication.
So, when I use inverse operations, I can “undo” the original number.
Example:

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

38. a • 1 = a
The Identity Property
of Multiplication

39. The Identity Property of Addition
= 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. The Identity Property of Addition
3 + 0 = 3
The Identity Property of Addition
See the zero?

45. The Identity Property of Addition
a + 0 = a
The Identity Property of Addition
Zero!

46. Five Properties of Rational Numbers
Commutative
Associative
Identity
Inverse
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. 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 (2+3) or
(2+3) + (2+3) + (2+3) + (2+3) + (2+3)
How many red gifts were distributed? How many green gifts were distributed?

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
4( 3n + 6)
3( 5 + 2) 3n 6 5 2 3 4
12n 24 15 6
= 21
12n + 24
-7( 4 + 6)
9( ) 4 6 -3
- 8 -7
-28
-42 9
-27
-72
= -70
= -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
-6n
-12n
36n 5
-30n 10
-12n + 36n = 24n
-30n + 10