properties (answers) commutative property associative property distributive property additive...

Properties (Answers)Properties (Answers) Commutative PropertyCommutative Property Associative PropertyAssociative Property Distributive PropertyDistributive Property Additive IdentityAdditive Identity Additive InverseAdditive Inverse Multiplicative IdentityMultiplicative Identity Multiplicative InverseMultiplicative Inverse

SubstitutionSubstitution Symmetric EqualitySymmetric Equality Transitive EqualityTransitive Equality Addition Property of Addition Property of

EqualityEquality Multiplication Property Multiplication Property

of Equalityof Equality Zero Product PropertyZero Product Property

Game 2

Definitions:Definitions:

Choose the property Choose the property that matches the that matches the

definitiondefinition

Two numbers that add Two numbers that add to 0to 0

Switch sides Switch sides

with 2 equationswith 2 equations

Two numbers that Two numbers that multiply to 1multiply to 1

““Great cheese comes from happy Great cheese comes from happy cows. Happy cows come from cows. Happy cows come from California.”California.”

Therefore great cheese comes Therefore great cheese comes from California.from California.

Multiply equal amounts Multiply equal amounts to both sides.to both sides.

Replace a variable with Replace a variable with a number.a number.

Change order of the Change order of the numbers from left to numbers from left to

right side.right side.

Add equal amounts to Add equal amounts to both sides.both sides.

If the product of 2 If the product of 2 numbers is 0, numbers is 0,

then one of the numbers then one of the numbers being multiplied must being multiplied must

equal 0.equal 0.

Multiply by 1Multiply by 1

Multiply a parenthesis Multiply a parenthesis by an outside numberby an outside number

Add and subtract the Add and subtract the same number.same number.

Add zeroAdd zero

ReciprocalReciprocal

Positive and Negative Positive and Negative of the same number.of the same number.

5B - B = 4B5B - B = 4B

Replace B with Bananas, Replace B with Bananas, so this means: Eat 1 so this means: Eat 1

Banana out of a bunch of Banana out of a bunch of 5 Bananas and 4 are left5 Bananas and 4 are left

Properties with VariablesProperties with VariablesExamplesExamples

a + b + c = b + a + ca + b + c = b + a + c

If a = b, then 4a = 4bIf a = b, then 4a = 4b

If a = b and b = c, If a = b and b = c, then a = c.then a = c.

(a + b) + c = a + (b + c)(a + b) + c = a + (b + c)

If ab = 0, If ab = 0,

then a = 0 or b = 0then a = 0 or b = 0

a(b + c) = ab + ac a(b + c) = ab + ac

a + 0 = aa + 0 = a

a + (-a) = 0a + (-a) = 0

a * 1 = aa * 1 = a

a x 1/a = 1a x 1/a = 1

If a = b, then a + 2 = b + 2If a = b, then a + 2 = b + 2

If a + b = c and b = 2, If a + b = c and b = 2, then a + 2 = cthen a + 2 = c

If 12 = b, then b = 12If 12 = b, then b = 12

Examples fromExamples fromBlock 1 Block 1

If x = 6If x = 6

Then x + 8 = 14Then x + 8 = 14

10 + 0 = 1010 + 0 = 10

3 equations. The middle of 3 equations. The middle of the first two are equal. The the first two are equal. The

ends create the third.ends create the third.Example:Example:

If 4 = x and x = y, then 4 = yIf 4 = x and x = y, then 4 = y

x 2/5 becomes x 5/2x 2/5 becomes x 5/2

1 * 15 = 151 * 15 = 15

If (x + 8)(x - 9) = 0, then If (x + 8)(x - 9) = 0, then

(x + 8) = 0 or (x - 9) = 0(x + 8) = 0 or (x - 9) = 0

So (x + 8) gives x = -8So (x + 8) gives x = -8

And (x – 9) gives x = 9And (x – 9) gives x = 9

-57 + 57 = 0-57 + 57 = 0

6(x - 5) = 6x - 306(x - 5) = 6x - 30

If 5 + 6 = 11, If 5 + 6 = 11,

then 11 = 5 + 6then 11 = 5 + 6

y = 5 + xy = 5 + x

when x = 4, then y = 9when x = 4, then y = 9

2 * 1 = 22 * 1 = 2

4 * 5 * 7 = 7 * 5 * 44 * 5 * 7 = 7 * 5 * 4

If 10 = xIf 10 = x

Then 20 = 2xThen 20 = 2x

7 * 1/7 = 17 * 1/7 = 1

(1 * 2) * 3 = 1 * (2 * 3)(1 * 2) * 3 = 1 * (2 * 3)

When you volunteer to When you volunteer to wash Mr. Burkholder’s wash Mr. Burkholder’s

car but send your car but send your younger sibling to do younger sibling to do

it instead.it instead.

PropertiesPropertiesExamples 2Examples 2

Block 5Block 5Shown Bl 1 7Shown Bl 1 7

3 * 5 * 4 * 8 = 8 * 4 * 5 *33 * 5 * 4 * 8 = 8 * 4 * 5 *3

( 6 * 3) * 5 = 6 * (3 * 5)( 6 * 3) * 5 = 6 * (3 * 5)

3(x - 8) = 3x - 243(x - 8) = 3x - 24

5 + 0 = 55 + 0 = 5

-12 + 12 = 0-12 + 12 = 0

72 * 1 = 7272 * 1 = 72

Or Or

1 * 72 = 721 * 72 = 72

Multiplicative InverseMultiplicative Inverse

Inverse means Opposite Inverse means Opposite Multiply and Divide the same number are Multiply and Divide the same number are

oppositesopposites OR Do reciprocalOR Do reciprocal numbers multiply to 1numbers multiply to 1 Example:Example:

8 * 1/8 = 18 * 1/8 = 1 Or Or * 3/4 becomes * 4/3* 3/4 becomes * 4/3

SubstitutionSubstitution

Replace a letter with a numberReplace a letter with a number

Example:Example: 5 + x = y where x = 85 + x = y where x = 8 y = 13y = 13

Symmetric EqualitySymmetric Equality

2 Equations2 Equations

Switch sidesSwitch sides

Example:Example: If 2 + 3 = 5, If 2 + 3 = 5, then 5 = 2 + 3then 5 = 2 + 3

Transitive EqualityTransitive Equality

3 equations3 equations

The middle of the first two are equal.The middle of the first two are equal. The ends create the third.The ends create the third. Example:Example:


Addition Property of EqualityAddition Property of Equality

Add Equal things to both sides.Add Equal things to both sides.

Example:Example: If 9 = xIf 9 = x then 12 = x + 3 (Add 3 to both sides.)then 12 = x + 3 (Add 3 to both sides.)

Multiplication Property of EqualityMultiplication Property of Equality

Multiply Equal things to both sides.Multiply Equal things to both sides.

Example:Example: If 7 = xIf 7 = x Then 28 = 4x (Multiply both sides by 4.)Then 28 = 4x (Multiply both sides by 4.)

Zero Product PropertyZero Product Property

Product is multiplyProduct is multiply If 2 numbers multiply to 0, then one of the If 2 numbers multiply to 0, then one of the

numbers must be 0.numbers must be 0. Example:Example:

Question: If (x + 6)(x - 4) = 0, then Question: If (x + 6)(x - 4) = 0, then Answer: (x + 6) = 0 or (x - 4) = 0Answer: (x + 6) = 0 or (x - 4) = 0 So (x + 6) gives x = -6So (x + 6) gives x = -6 And (x – 4) gives x = 4 And (x – 4) gives x = 4


Block 7Block 7Shown Bl 1 5Shown Bl 1 5

Commutative PropertyCommutative Property

1 Equation1 Equation

CO = Change order; CO = Change order; move numbers; “commute”move numbers; “commute”

Example:Example: 3 * 2 * 1 = 1 * 2 * 33 * 2 * 1 = 1 * 2 * 3

Associative PropertyAssociative Property


SO = Same Order. SO = Same Order. Change groups or ( )Change groups or ( )

Example:Example: ( 3 * 6 ) * 15 = 3 * ( 6 * 15)( 3 * 6 ) * 15 = 3 * ( 6 * 15)

Distributive PropertyDistributive Property


Multiply the outside by everything in the Multiply the outside by everything in the inside.inside.

Example:Example: 4 (x - 7) = 4x - 28 4 (x - 7) = 4x - 28

Additive IdentityAdditive Identity

Add ZEROAdd ZERO Identity means stays the sameIdentity means stays the same

Example:Example: 21 + 0 = 2121 + 0 = 21

Additive InverseAdditive Inverse

Inverse means Opposite Inverse means Opposite Add and Subtract the same number orAdd and Subtract the same number or Positive and NegativePositive and Negative Adds to ZERO.Adds to ZERO.

Example:Example: +8 - 8 = 0+8 - 8 = 0

Multiplicative IdentityMultiplicative Identity

Multiply by 1Multiply by 1 Identity means stays the sameIdentity means stays the same

Example:Example: 7 * 1 = 77 * 1 = 7 Or Or 1 * 8 = 81 * 8 = 8




10 * 1/10 = 110 * 1/10 = 1 Or * 7/8 becomes * 8/7Or * 7/8 becomes * 8/7



Example:Example: 5 + x = y where x = 75 + x = y where x = 7 y = 12y = 12




Example:Example: If 3 + 4 = 7, If 3 + 4 = 7, then 7 = 3 + 4then 7 = 3 + 4



Example:Example: If 8 = x If 8 = x then 15 = x + 7 (Add 7 to both sides.)then 15 = x + 7 (Add 7 to both sides.)



Example:Example: If 3 = xIf 3 = x then 21 = 7x (Multiply both sides by 7)then 21 = 7x (Multiply both sides by 7)



numbers must be 0.numbers must be 0.

Example:Example: If (x + 2)(x - 7) = 0, then If (x + 2)(x - 7) = 0, then (x + 2) = 0 or (x - 7) = 0(x + 2) = 0 or (x - 7) = 0 So (x + 2) gives x = -2 So (x + 2) gives x = -2 And (x – 7) gives x = 7And (x – 7) gives x = 7


Block X TemplateBlock X Template

Commutative PropertyCommutative Property


CO = Change order; CO = Change order; move numbers; “commute”move numbers; “commute”

Example:Example:

Associative PropertyAssociative Property


SO = Same Order. SO = Same Order. Change groups or ( )Change groups or ( )

Example:Example:

Distributive PropertyDistributive Property


Multiply the outside by everything in the Multiply the outside by everything in the inside.inside.

Example:Example: (x -) = x –(x -) = x – (x + -) = x + - (x + -) = x + -

Additive IdentityAdditive Identity

Add ZEROAdd ZERO Identity means stays the sameIdentity means stays the same

Example:Example: + 0 = + 0 =

Additive InverseAdditive Inverse

Inverse means Opposite Inverse means Opposite Add and Subtract the same number orAdd and Subtract the same number or Positive and NegativePositive and Negative Adds to ZERO.Adds to ZERO.

Example:Example: + = 0+ = 0

Multiplicative IdentityMultiplicative Identity

Multiply by 1Multiply by 1 Identity means stays the sameIdentity means stays the same

Example:Example: * 1 = ???* 1 = ??? Or 1 * =Or 1 * =




* 1/ = 1* 1/ = 1 Or * / becomes * /Or * / becomes * /



Example:Example: 5 + x = y where x = 5 + x = y where x = y =y =




Example:Example: If + = , If + = , then then




If 4 = x and x = 2y, then ???If 4 = x and x = 2y, then ???



Example:Example: If 5 = x, then 9 = ???If 5 = x, then 9 = ???



numbers must be 0.numbers must be 0.

Example:Example: If (x + )(x - ) = 0, then If (x + )(x - ) = 0, then (x + ) = 0 or (x - ) = 0(x + ) = 0 or (x - ) = 0 So (x + ) gives x = So (x + ) gives x = And (x – ) gives x = And (x – ) gives x =

properties (answers) commutative property associative property distributive property additive...

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