The Properties of Addition & Multiplication As shown in different types of algebraic equations. Yes yes.

Addition Properties Closure Property Commutative Property Identity Property Associative Property Inverse Property

What is a property? What do we mean by property? Something that a person owns Ex. This is Chuck’s cow. It is his property.

Boiling Point = 100°C Freezing Point = 0°C What is a property? The other definition of property A distinctive attribute or quality of something Ex. What are the properties of water? Boiling Point = 100°C Freezing Point = 0°C

The Properties of Addition An addition property is “A distinctive quality unique to problems involving addition.”

Keyword = ORDER Commutative Property Commutative Property The order in which two numbers are added does not change the sum Ex. 5 + 6 = 6 + 5 Keyword = ORDER

Commutative Property How will I remember? Commute = move from one place to another (like when you commute to school every morning) With addition, numbers can commute (move around), and we still get the same answer.

Keyword = GROUPED Associative Property Associative Property The way three numbers are grouped when adding does not change the sum Ex. (4 + 3) + 9 = 4 + (3 + 9) Keyword = GROUPED

Associative Property Ex. (5+12)+13 = 5 + (12+13) How will I remember? Associate = to hang out with someone (like when you associate with your homies) With addition, numbers can associate (be grouped with other numbers), and we still get the same answer. Ex. (5+12)+13 = 5 + (12+13)

Identity Property Identity Property The sum of a number and zero (0) is the number. Ex. 14 + 0 = 14 How will I remember? 0 has no identity (poor guy). So there is no way that this nothing can change a number. He is worthless.

Inverse = Opposite -3.25 = 5 = 987 = -3.14… = Inverse Property Inverse Property The sum of a number and its opposite is zero (0). Ex. 24 + -24 = 0 How will I remember? Inverse = Opposite What’s the opposite of… -3.25 = 5 = 987 = -3.14… =

Let’s Practice

Which property is represented in the equation below? 8 + (-8) = 0 Inverse Property

Which property is represented in the equation below? (12 + 4) +13 = 12 + (4 +13) Associative Property

Which property is represented in the equation below? 1234 + 4321 = 4321 + 1234 Commutative Property

Which property is represented in the equation below? 98765 + 0 = 98765 Identity Property

This addition stuff is boring. Can’t we do multiplication now? Yes yes young grasshopper. We can embark on the wonderful journey of multiplication properties!!!

Multiplication Properties Distributive Property Closure Property Commutative Property Multiplication Properties Identity Property Associative Property Inverse Property

Keyword = ORDER Commutative Property Commutative Property The order in which two numbers are multiplied does not change the product Ex. 5 x 6 = 6 x 5 Keyword = ORDER

5 x 6 = 6 x 5 Commutative Property How will I remember? Commute = move from one place to another (like when you commute to work) With addition, numbers can commute (move around), and we still get the same answer. 5 x 6 = 6 x 5

Keyword = GROUPED Associative Property Associative Property The way three numbers are grouped when multiplying does not change the product Ex. (4 x 3) x 9 = 4 x (3 x 9) Ex. (ab)c = a(bc) Keyword = GROUPED

Associative Property Ex. (5x12)x13 = 5 x (12x13) How will I remember? Associate = to hang out with someone (like when you associate with your homies) With mutliplication, numbers can associate (be grouped with other numbers), and we still get the same product. Ex. (5x12)x13 = 5 x (12x13)

1 Identity Property Identity Property I am worthless The product of a number and 1 is the number. Ex. How will I remember? Just like 0 in addition, 1 really serves no purpose in multiplication. It is worthless and has no effect on the number. It is the 0 of multiplication. I am worthless 1

Inverse Property Inverse Property The product of a number and its reciprocal is 1. Ex. How will I remember? When you are inverted, you are flipped over.

Let’s Practice

Which property is represented in the equation below? 54 x 11 = 594 Closure Property

Which property is represented in the equation below? 31(5) = 5(31) Commutative Property

Which property is represented in the equation below? Inverse Property

Which property is represented in the equation below? 56342 x 1 = 56342 Identity Property

Which property is represented in the equation below? (983 x 3)17 = 983(3 x 17) Associative Property

LET’S RECAP

Associative Property States that changing the grouping in an addition or multiplication expression does not change the sum or product.

Associative Property ONLY works with addition and multiplication! a + (b + c) is the same as (a + b) + c a x (b x c) is the same as (a x b) x c

Associative Property – Example 1 a = 4, b = 8 , c = 3 4 + (8 + 3) = ???

Show each side is equal by simplifying and solving Associative Example 2 Show each side is equal by simplifying and solving 4 x (8 x 3) = (4 x 8) x 3 Simplify: Solve:

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

Apply the Associative Property (12 + 9) +15 = 12 + (9 + 15)

Apply the Associative Property (48 + 7) + 32 = 48 + (7 + 32)

Prove the Associative Property Simplify each side to show equivalency (12 + 4) + 2 = 12 + (4 + 2) Simplify: Solve: 16 + 2 = 12 + 6 18 = 18

Prove the Associative Property Simplify each side to show equivalency 30 + (5 + 15) = (30 + 5) + 15 Simplify: Solve: 30 + 20 = 35 + 15 50 = 50

Check Yo’ Self 1. 2. 3. b. (8 + 1) + 2 c. k + (g + m) Simplify: 13 + 5 = 2 + 16 Solve: 18 = 18 3. a. 3 b. 4

Commutative Property States that two or more terms can be added in any order without changing the sum, and two more terms can be multiplied in any order without changing the product.

Commutative Property ONLY works with addition and multiplication! a + b = b + a a x b = b x a

Commutative Property – Example 3 5 + 9 = ???

Commutative Property- Example 4 Use the commutative property to rewrite each number sentence in two different ways. 9 + 6 + 3 = __________________ _________________

Apply the Commutative Property 17 + 19 = 19 + 17

Apply the Commutative Property 32 + 12 + 9 = 32 + 9 + 12 12 + 9 + 32 12 + 32 + 9 9 + 32 + 12 9 + 12 +32

Apply the Commutative Property 12.5 + 7 + 19 = 12.5 + 19 + 7 7 + 12.5 + 19 7 + 19 + 12.5 19 + 12.5 + 7 19 + 7 + 12.5

Commutative Challenge: Associative or Commutative Property? (3 + 5) + 7 = 7 + (3 + 5) Commutative

Check Yo’ Self 1. b. 3 + 8 c. d + m 4. B

Distributive Property A number outside parenthesis must be multiplied by everything inside Ex. 3(5 + 2) = 3x5 + 3x2 Keyword = SHARED

Distributive Property A number outside parenthesis must be multiplied by everything inside Ex. How will I remember? Distribute means “to share.” The outside number must be distributed (shared) with every other number. 3(5 + 2) = 3x5 + 3x2

Distributive Property Think of it like this… Ex. 3(5 + 2) = ??? ** 3 is the candy. We must share the candy with everyone. That is, we must share 3 with 5 & 2. 3 (5 + 2) 3x5 + 3x2

Distributive Property Think of it like this… Ex. 6(2 + 7) = ??? 6 (2 + 7) There is too much of a spotlight on 6. The spotlight must be shared with 2 and 7 6(2) + 6(7) 12 + 42 54

Distributive Property Think of it like this… Ex. -10(3 + 9) = ??? (3 + 9) Uh oh! Don’t lose the negative sign as you distribute the spotlighted number!! -10 -10(3) + -10(9) -30 + -90 -120

Distributive Property Think of it like this… Ex. -5(4 – 7) = ??? -5 (4 – 7) Make sure you know where your negative signs are in this problem!!!!! -5(4) - -5(7) -20 – (-35) 15

And now it’s time for … No!! It can’t be!! Say it ain’t so!! Yes class. It’s time for variables!!! Mwahahaha!!!

Distributive Property Think of it like this… Ex. 3(x + 4) = ??? 3 (x + 4) There is too much of a spotlight on 3. The spotlight must be shared with x and 4 3(x) + 3(4) 3x + 12

Distributive Property Think of it like this… Ex. -11(x + 5) = ??? (x + 5) Uh oh! Don’t lose the negative sign as you distribute the spotlighted number!! -11 -11(x) + -11(5) -11x + -55

Distributive Property Think of it like this… Ex. -5(2x – y) = ??? -5 (2x – y) Make sure you know where your negative signs are in this problem!!!!! -5(2x) - -5(y) -10x – (-5y)

Let’s Practice

Distributive Property Share, share, share! 1. 4(6 + 5) 2. 6(3 + 8) 3. 15(12 + t) 4. -6(x + 12) 5. -12(3 + r) 6. -15(4 + t) 7. -9(6 + p) 8. -5(7 + t)