1. Properties of Real Numbers
Commutative
Associative
Identity + ×
Inverse + ×
Zero Property
Reflexive
Distributive

2. Commutative Properties
Changing the order of the numbers in addition or multiplication will not change the result.
Commutative Property of Addition states: = or a + b = b + a.
Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

3. Verbal Hints for Commutative Property
Commute
Switch Places
Move to a new location

4. Associative Properties
Changing the grouping of the numbers in addition or multiplication will not change the result.
Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c
Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

5. Verbal Hints for Associative Property
Regroup
They simply group with a new friend
( ) change places but numbers stay the same

6. Additive Identity Property
There exists a unique number 0 such that zero preserves identities under addition.
a + 0 = a and 0 + a = a
In other words adding zero to a number does not change its value.

7. Multiplicative Identity Property
There exists a unique number 1 such that the number 1 preserves identities under multiplication.
a ∙ 1 = a and 1 ∙ a = a
In other words multiplying a number by 1 does not change the value of the number.

8. Verbal Hints for Identity Property
The value that returns the input unchanged
Remember “I” in the word identity

9. Additive Inverse Property
For each real number a there exists a unique real number –a such that their sum is zero.
a + (-a) = 0
In other words opposites add to zero.

10. Verbal Hints for Additive Inverse
Opposite numbers
Same number but different sign
Zero Pairs

11. Multiplicative Inverse Property
For each real number a there exists a unique real number such that their product is 1.

12. Verbal Hints for Multiplicative Inverse
Flip the number
Reciprocal

13. Zero Property of Multiplication
Any number multiplied by 0 is equal to 0
A ● 0 = 0
-23 ● 0 = 0
½ ● 0 = 0
0.25 ● 0 = 0
“zero times any value is 0

14. A real number is always equal to itself
Reflexive Property
A number is always equal to itself
A = A
-2 = -2
5 = 5
A real number is always equal to itself

15. Distributive Property
Multiplication distributes over addition.

16. Verbal Hints for Distributive Property
Multiplication distributes across addition or subtraction.
Multiply across the parentheses!!!!!

17. Let’s play “Name that property!”

18. State the property or properties that justify the following.
3 + 2 = 2 + 3
Commutative Property

19. State the property or properties that justify the following.
10(1/10) = 1
Multiplicative Inverse Property

20. State the property or properties that justify the following.
3(x – 10) = 3x – 30
Distributive Property

21. State the property or properties that justify the following.
3 + (4 + 5) = (3 + 4) + 5
Associative Property

22. State the property or properties that justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property

23. Commutative Property of Addition2.
Which Property?
3 + 7 = 7 + 3
Commutative Property of Addition

24. Identity Property of Addition3.
Which Property?
8 + 0 = 8
Identity Property of Addition

25. Commutative Property of Multiplication5.
Which Property?
6 • 4 = 4 • 6
Commutative Property of Multiplication

26. Inverse Property of Addition6.
Which Property?
17 + (-17) = 0
Inverse Property of Addition

27. Commutative Property of Multiplication7.
Which Property?
2(5) = 5(2)
Commutative Property of Multiplication

28. Associative Property of Addition1.
Which Property?
(2 + 1) + 4 = 2 + (1 + 4)
Associative Property of Addition

29. 8.
Which Property?
even + even = even
Closure Property

30. Distributive Property9.
Which Property?
3(2 + 5) = 3•2 + 3•5
Distributive Property

31. Associative Property of Multiplication
10.
Which Property?
6(7•8) = (6•7)8
Associative Property of Multiplication

32. Identity Property of Multiplication
11.
Which Property?
5 • 1 = 5
Identity Property of Multiplication

33. Properties Using
Negatives

34. Distributive Property
13.
Which Property?
(6 – 3)4 = 6•4 – 3•4
Distributive Property

35. Identity Property of Multiplication
14.
Which Property?
1(-9) = -9
Identity Property of Multiplication

36. Inverse Property of Addition
15.
Which Property?
3 + (-3) = 0
Inverse Property of Addition

37. Associative Property of Addition
16.
Which Property?
1 + [-9 + 3]
= [1 + (-9)] + 3
Associative Property of Addition

38. Commutative Property of Multiplication
17.
Which Property?
-3(6) = 6(-3)
Commutative Property of Multiplication

39. Identity Property of Addition
18.
Which Property?
= -8
Identity Property of Addition

40. Distributive Property
19.
Which Property?
3•7 – 3•4 = 3(7 – 4)
Distributive Property

41. Associative Property of Addition
20.
Which Property?
6 + [(3 + (-2)]
= (6 + 3) + (- 2)
Associative Property of Addition

42. Commutative Property of Addition
21.
Which Property?
7 + (-5) =
Commutative Property of Addition

43. Distributive Property
22.
Which Property?
(5 + 4)9 =
Distributive Property

44. Associative Property of Multiplication
23.
Which Property?
-3(5 • 4) = (-3 • 5)4
Associative Property of Multiplication

45. Commutative Property of Multiplication
24.
Which Property?
-8(4) = 4(-8)
Commutative Property of Multiplication

46. Properties Using
Fractions

47. Identity Property of Addition
25.
Which Property?
51/7 + 0 = 51/7
Identity Property of Addition

48. Commutative Property of Addition
26.
Which Property?
3/4 – 6/7 = – 6/7 + 3/4
Commutative Property of Addition

49. Identity Property of Multiplication
27.
Which Property?
12/5 • 1 = 12/5
Identity Property of Multiplication

50. (fraction)(fraction) = fraction
28.
Which Property?
(fraction)(fraction)
= fraction
Closure Property

51. Identity Property of Addition
29.
Which Property?
-8 2/5 + 0 = -8 2/5
Identity Property of Addition

52. Associative Property of Multiplication
30.
Which Property?
[(-2/3)(5)]9 = -2/3[(5)(9)]
Associative Property of Multiplication

53. Properties Using
Variables

54. Distributive Property
31.
Which Property?
6(3 – 2n) = 18 – 12n
Distributive Property

55. Commutative Property of Addition
32.
Which Property?
2x + 3 = 3 + 2x
Commutative Property of Addition

56. Commutative Property of Multiplication
33.
Which Property?
ab = ba
Commutative Property of Multiplication

57. Identity Property of Addition
34.
Which Property?
a + 0 = a
Identity Property of Addition

58. Associative Property of Multiplication
35.
Which Property?
a(bc) = (ab)c
Associative Property of Multiplication

59. Identity Property of Multiplication
36.
Which Property?
a•1 = a
Identity Property of Multiplication

60. Commutative Property of Addition
37.
Which Property?
a +b = b + a
Commutative Property of Addition

61. Distributive Property
38.
Which Property?
a(b + c) = ab + ac
Distributive Property

62. Associative Property of Addition
39.
Which Property?
a + (b + c) = (a +b) + c
Associative Property of Addition

63. Inverse Property of Addition
40.
Which Property?
a + (-a) = 0
Inverse Property of Addition

64. Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×