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**Properties of Real Numbers**

Commutative Associative Identity + × Inverse + × Zero Property Reflexive Distributive

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**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.

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**Verbal Hints for Commutative Property**

Commute Switch Places Move to a new location

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**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)

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**Verbal Hints for Associative Property**

Regroup They simply group with a new friend ( ) change places but numbers stay the same

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**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.

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**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.

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**Verbal Hints for Identity Property**

The value that returns the input unchanged Remember “I” in the word identity

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**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.

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**Verbal Hints for Additive Inverse**

Opposite numbers Same number but different sign Zero Pairs

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**Multiplicative Inverse Property**

For each real number a there exists a unique real number such that their product is 1.

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**Verbal Hints for Multiplicative Inverse**

Flip the number Reciprocal

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**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

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**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

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**Distributive Property**

Multiplication distributes over addition.

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**Verbal Hints for Distributive Property**

Multiplication distributes across addition or subtraction. Multiply across the parentheses!!!!!

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**Let’s play “Name that property!”**

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**State the property or properties that justify the following.**

3 + 2 = 2 + 3 Commutative Property

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**State the property or properties that justify the following.**

10(1/10) = 1 Multiplicative Inverse Property

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**State the property or properties that justify the following.**

3(x – 10) = 3x – 30 Distributive Property

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**State the property or properties that justify the following.**

3 + (4 + 5) = (3 + 4) + 5 Associative Property

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**State the property or properties that justify the following.**

(5 + 2) + 9 = (2 + 5) + 9 Commutative Property

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**Commutative Property of Addition**

2. Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

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**Identity Property of Addition**

3. Which Property? 8 + 0 = 8 Identity Property of Addition

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**Commutative Property of Multiplication**

5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

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**Inverse Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Associative Property of Addition**

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

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

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**Distributive Property**

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

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**Associative Property of Multiplication**

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

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**Identity Property of Multiplication**

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

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Properties Using Negatives

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**Distributive Property**

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

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**Identity Property of Multiplication**

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

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**Inverse Property of Addition**

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

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**Associative Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Identity Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Addition**

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

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**Commutative Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Multiplication**

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

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**Commutative Property of Multiplication**

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

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Properties Using Fractions

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**Identity Property of Addition**

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

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**Commutative Property of Addition**

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

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**Identity Property of Multiplication**

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

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**(fraction)(fraction) = fraction**

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

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**Identity Property of Addition**

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

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**Associative Property of Multiplication**

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

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Properties Using Variables

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**Distributive Property**

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

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**Commutative Property of Addition**

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

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**Commutative Property of Multiplication**

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

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**Identity Property of Addition**

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

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**Associative Property of Multiplication**

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

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**Identity Property of Multiplication**

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

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**Commutative Property of Addition**

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

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**Distributive Property**

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

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**Associative Property of Addition**

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

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**Inverse Property of Addition**

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

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**Properties of Real Numbers**

Commutative Associative Distributive Identity + × Inverse + ×

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