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

Commutative Properties Changing the order of the numbers in addition or multiplication will not change the result. Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a. Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

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

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)

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

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.

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.

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

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.

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

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

Verbal Hints for Multiplicative Inverse Flip the number Reciprocal

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

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

Distributive Property Multiplication distributes over addition.

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

Let’s play “Name that property!”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Properties Using Negatives

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

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

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

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

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

Identity Property of Addition 18. Which Property? -8 + 0 = -8 Identity Property of Addition

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

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

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

Distributive Property 22. Which Property? (5 + 4)9 = 45 + 36 Distributive Property

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

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

Properties Using Fractions

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

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

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

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

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

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

Properties Using Variables

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

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

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

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

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

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

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

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

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

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

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