Properties ofAddition &

Multiplication

Before We Begin…Before We Begin…• What is a VARIABLE?A variable is an unknown amount in a number sentence represented as a letter: 5 + n 8x 6(g) t + d = s

Before We Begin…Before We Begin…• What do these symbols mean? ( ) = multiply: 6(a) or group: (6 + a) * = multiply · = multiply

÷ = divide / = divide

Commutative PropertyCommutative Property• To COMMUTECOMMUTE something is to change it• The COMMUTATIVECOMMUTATIVE property says that the order of numbers in a number sentence can be changed• Addition & multiplication have COMMUTATIVECOMMUTATIVE properties

Commutative PropertyCommutative Property

Examples: (a + b = b + a)

7 + 5 = 5 + 79 x 3 = 3 x 9

Note: subtraction & division DO NOT have commutative

properties!

Commutative PropertyCommutative PropertyPractice: Show the

commutative property of each number sentence.

1. 13 + 18 =2. 42 x 77 =3. 5 + y = 4. 7(b) =

Commutative PropertyCommutative PropertyPractice: Show the

commutative property of each number sentence.

1. 13 + 18 = 18 + 132. 42 x 77 = 77 x 423. 5 + y = y + 54. 7(b) = b(7) or (b)7

Associative PropertyAssociative Property• To ASSOCIATEASSOCIATE something is join it, partner it, or combine it• The ASSOCIATIVEASSOCIATIVE property says that the way we group numbers in a number sentence can be changed• Addition & multiplication have ASSOCIATIVEASSOCIATIVE properties

Associative PropertyAssociative PropertyExamples: (a + b) + c = a + (b + c)

2 + (3 + 4) = (2 + 3) + 4

5 x (3 x 7) = (5 x 3) x 7Note: subtraction & division DO NOT

have associative properties!

Associative PropertyAssociative PropertyPractice: Show the associative

property of each number sentence.

1. (7 + 2) + 5 =2. 4 x (8 x 3) =3. 5 + (y + 2) = 4. 7(b x 4) =

Associative PropertyAssociative PropertyPractice: Show the associative

property of each number sentence.

1. (7 + 2) + 5 = 7 + (2 + 5)2. 4 x (8 x 3) = (4 x 8) x 33. 5 + (y + 2) = (5 + y) + 24. 7(b x 4) = (7b) x 4 or (7 x

b)4

Identity PropertiesIdentity Properties• An IDENTITYIDENTITY is the state of being one’s self or staying the same• The IDENTITYIDENTITY properties says that with certain operations, a number can stay the same• Addition & multiplication have ASSOCIATIVEASSOCIATIVE properties

Identity PropertiesIdentity PropertiesExamples: a + 0 = a b x 1 = b Additive Identity: 7 + 0 = 7(when you add 0 to a number, it stays the same)

Multiplicative Identity: 7 x 1 = 7

(when you multiply a number by 1, it stays the same)

Identity PropertiesIdentity PropertiesPractice: Show the ADDITIVE and MULTIPLICATIVE identity properties of each number.

1. 9 =2. 17 = 3. 8m =4. 5 + (6 x 9) =

Identity PropertiesIdentity PropertiesPractice: Show the ADDITIVE and

MULTIPLICATIVE identity properties of each number.

1. 9 = 9 + 0 and 9 x 12. 17 = 17 + 0 and 17 x 1 3. 8m = 8m + 0 and (8m) x 1 (why

parentheses?)

4. 5 + (6 x 9) = 5 + (6 x 9) + 0 and 5 + (6 x 9) x 1

Distributive PropertyDistributive Property• To DISTRIBUTEDISTRIBUTE something is give it out or share it.• The DISTRIBUTIVEDISTRIBUTIVE property says that we can distribute a multiplier out to each number in a group to make it easier to solve• The DISTRIBUTIVEDISTRIBUTIVE property uses MULTIPLICATIONMULTIPLICATION and ADDITIONADDITION!

Distributive PropertyDistributive PropertyExamples: a(b + c) = a(b) + a(c)

2 x (3 + 4) = (2 x 3) + (2 x 4)

5(3 + 7) = 5(3) + 5(7)Note: Do you see that the 2 and the 5 were

shared (distributed) with the other numbers in the group?

Distributive PropertyDistributive PropertyPractice: Show the distributive

property of each number sentence.

1. 8 x (5 + 6) =2. 4(8 + 3) =3. 5 x (y + 2) = 4. 7(4 + b) =

Distributive PropertyDistributive PropertyPractice: Show the distributive

property of each number sentence.

1. 8 x (5 + 6) = (8 x 5) + (8 x 6)

2. 4(8 + 3) = 4(8) + 4(3)3. 5 x (y + 2) = (5y) + (5 x 2)4. 7(4 + b) = 7(4) + 7b

Zero PropertyZero Property• Only MULTIPLICATIONMULTIPLICATION has a ZEROZERO property.• The ZEROZERO property of multiplication says that when we multiply any number by ZEROZERO, the answer is always ZEROZERO.

Zero PropertyZero PropertyExamples: a(0) = 0

2 x 0 = 05(3 + 7) x 0 = 0

Zero PropertyZero PropertyPractice: Show the zero

property of multiplication for each number or number sentence.

1. 52. 4k3. 9 + (d + 6)

Zero PropertyZero PropertyPractice: Show the zero

property of multiplication for each number or number sentence.

1. 5: 5 x 0 = 02. 4k: 4k (0) = 03. 9 + (d + 6): 9 + (d + 6) x 0

= 0

POP QUIZ!POP QUIZ!Tell which property each

sentence represents:1. 5 + 0 = 52. 9 x 8 = 8 x 9 3. (7c + 1) x 0 = 04. (8 + 4) + 7 = 8 + (4 + 7)5. 9a + (5 x 4) x 1 = 9a + (5 x 4)6. 3(4 + 5) = 3(4) + 3(5)

POP QUIZ POP QUIZ ANSWERSANSWERS!!Tell which property each

sentence represents:1. 5 + 0 = 5 (add. identity)2. 9 x 8 = 8 x 9 (commutative) 3. (7c + 1) x 0 = 0 (zero)4. (8 + 4) + 7 = 8 + (4 + 7) (associative)5. 9a + (5 x 4) x 1 = 9a + (5 x 4) (mult.

identity)6. 3(4 + 5) = 3(4) + 3(5) (distributive)