Properties

Properties  Commutative Property  Associative Property  Distributive Property  Additive Identity  Additive Inverse  Multiplicative Identity  Multiplicative Inverse  Substitution  Symmetric Equality  Transitive Equality  Addition Property of Equality  Multiplication Property of Equality  Zero Product Property

Commutative Property  a + b = b + a  CO is Change order; move; “commute”  Example: 3 + x = ??? 3 + x = ???

Associative Property  (a + b) + c = a + (b + c)  SO is Same Order. Change friends; “associates”.  Numbers stay in the same order, but parenthesis move around.  Example: (6 + x) + 4 = ??? (6 + x) + 4 = ???

Distributive Property  a(b + c) = ab + ac OR (b + c)a = ba + ca  a is multiplied by (distributed) everything inside the parentheses.   Taste the Rainbow (draw arrows on top).  Example: 3(x + 2) = ??? 3(x + 2) = ???

Additive Identity  a + 0 = a  Adds ZERO  stays the same keeps its “identity” keeps its “identity”  Example: = ??? = ???

Additive Inverse  a + (-a) = 0  Opposite (inverse) numbers add to 0  Example: 4 + ??? = ??? = 0

Multiplicative Identity  a * 1 = a  Multiply by 1  stays the same keeps its “identity” keeps its “identity”  Example: 5 * 1 = ??? 5 * 1 = ???

Multiplicative Inverse  2 x 3 =  Opposite (inverse) numbers multiply to 1  Opposite of multiply is divide.  Also called Reciprocals.  Example: 6 * ??? = 1 6 * ??? = 1

Substitution  If a + b = c and b = 2, then a + 2 = c  Substitute given information into equation.  Example: 5 + x = y where x = x = y where x = 4

Symmetric Equality  If a = b, then b = a  Switch sides  Example: If 6 = x, then ??? If 6 = x, then ???

Transitive Equality  If a = b and b = c, then a = c.  “Great cheese comes from happy cows. Happy cows come from California.” Therefore great cheese comes from California. Therefore great cheese comes from California.  Example: If 4 = x and x = 2y, then ??? If 4 = x and x = 2y, then ???

Addition Property of Equality  If a = b, then a + 2 = b + 2  Add the same thing to both sides.  Or Subtract the same thing from both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Multiplication Property of Equality  If a = b, then 2a = 2b.  Multiply both sides by the same thing.  Example: If 4 = x, then 12 = ??? If 4 = x, then 12 = ???

Zero Product Property  If ab = 0, then a = 0 or b = 0.  If the product of 2 numbers is 0, then one of the numbers must be 0 itself.  Example: If 5x = 0, then ??? If 5x = 0, then ???

Properties Examples 2 Block 1 Shown Bl 5 7

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example: 4 * 5 * 7 = 7 * 5 * 4 4 * 5 * 7 = 7 * 5 * 4

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example: (1 * 2) * 3 = 1 * (2 * 3) (1 * 2) * 3 = 1 * (2 * 3)

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: 6(x - 5) = 6x (x - 5) = 6x - 30

Additive Identity  Add ZERO  Identity means stays the same  Example: = = 10

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: = = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: 2 * 1 = 2 2 * 1 = 2 Or Or 1 * 15 = 15 1 * 15 = 15

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: 7 * 1/7 = 1 7 * 1/7 = 1 Or * 2/5 becomes * 5/2 Or * 2/5 becomes * 5/2

Substitution  Replace a letter with a number  Example: 5 + x = y where x = x = y where x = 4 y = 9 y = 9

Symmetric Equality  2 Equations  Switch sides  Example: If = 11, If = 11, then 11 = then 11 = 5 + 6

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = y, then 4 = y If 4 = x and x = y, then 4 = y

Addition Property of Equality  Add Equal things to both sides.  Example: If 6 = x If 6 = x Then 14 = x + 8 (Add 8 to both sides) Then 14 = x + 8 (Add 8 to both sides)

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 10 = x If 10 = x Then 20 = 2x (multiply by 2) Then 20 = 2x (multiply by 2)

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  Example: 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 = -8 So (x + 8) gives x = -8 And (x – 9) gives x = 9 And (x – 9) gives x = 9

Properties Examples 2 Block 5 Shown Bl 1 7

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example: 3 * 5 * 4 * 8 = 8 * 4 * 5 *3 3 * 5 * 4 * 8 = 8 * 4 * 5 *3

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example: ( 6 * 3) * 5 = 6 * (3 * 5) ( 6 * 3) * 5 = 6 * (3 * 5)

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: 3(x - 8) = 3x (x - 8) = 3x - 24

Additive Identity  Add ZERO  Identity means stays the same  Example: = = 5

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: = = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: 72 * 1 = * 1 = 72 Or Or 1 * 72 = 72 1 * 72 = 72

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: 8 * 1/8 = 1 8 * 1/8 = 1 Or Or * 3/4 becomes * 4/3 * 3/4 becomes * 4/3

Substitution  Replace a letter with a number  Example: 5 + x = y where x = x = y where x = 8 y = 13 y = 13

Symmetric Equality  2 Equations  Switch sides  Example: If = 5, If = 5, then 5 = then 5 = 2 + 3

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = y, then 4 = y If 4 = x and x = y, then 4 = y

Addition Property of Equality  Add Equal things to both sides.  Example: If 9 = x If 9 = x then 12 = x + 3 (Add 3 to both sides.) then 12 = x + 3 (Add 3 to both sides.)

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 7 = x If 7 = x Then 28 = 4x (Multiply both sides by 4.) Then 28 = 4x (Multiply both sides by 4.)

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  Example: Question: If (x + 6)(x - 4) = 0, then Question: If (x + 6)(x - 4) = 0, then Answer: (x + 6) = 0 or (x - 4) = 0 Answer: (x + 6) = 0 or (x - 4) = 0 So (x + 6) gives x = -6 So (x + 6) gives x = -6 And (x – 4) gives x = 4 And (x – 4) gives x = 4

Properties Examples 2 Block 7 Shown Bl 1 5

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example: 3 * 2 * 1 = 1 * 2 * 3 3 * 2 * 1 = 1 * 2 * 3

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example: ( 3 * 6 ) * 15 = 3 * ( 6 * 15) ( 3 * 6 ) * 15 = 3 * ( 6 * 15)

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: 4 (x - 7) = 4x (x - 7) = 4x - 28

Additive Identity  Add ZERO  Identity means stays the same  Example: = = 21

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: = = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: 7 * 1 = 7 7 * 1 = 7 Or Or 1 * 8 = 8 1 * 8 = 8

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: 10 * 1/10 = 1 10 * 1/10 = 1 Or * 7/8 becomes * 8/7 Or * 7/8 becomes * 8/7

Substitution  Replace a letter with a number  Example: 5 + x = y where x = x = y where x = 7 y = 12 y = 12

Symmetric Equality  2 Equations  Switch sides  Example: If = 7, If = 7, then 7 = then 7 = 3 + 4

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = y, then 4 = y If 4 = x and x = y, then 4 = y

Addition Property of Equality  Add Equal things to both sides.  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.)

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 3 = x If 3 = x then 21 = 7x (Multiply both sides by 7) then 21 = 7x (Multiply both sides by 7)

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  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 = 7 And (x – 7) gives x = 7

Properties Examples Block 6/7 2011

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example:

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example:

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example:

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example:

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: (x -) = x – (x -) = x – (x + -) = x + - (x + -) = x + -

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: (x -) = x – (x -) = x – (x + -) = x + - (x + -) = x + -

Additive Identity  Add ZERO  Identity means stays the same  Example: + 0 = + 0 =

Additive Identity  Add ZERO  Identity means stays the same  Example: + 0 = + 0 =

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: + = 0 + = 0

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: + = 0 + = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: * 1 = ??? * 1 = ??? Or 1 * = Or 1 * =

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: * 1 = ??? * 1 = ??? Or 1 * = Or 1 * =

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: * 1/ = 1 * 1/ = 1 Or * / becomes * / Or * / becomes * /

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: * 1/ = 1 * 1/ = 1 Or * / becomes * / Or * / becomes * /

Substitution  Replace a letter with a number  Example: 5 + x = y where x = 5 + x = y where x = y = y =

Substitution  Replace a letter with a number  Example: 5 + x = y where x = 5 + x = y where x = y = y =

Symmetric Equality  2 Equations  Switch sides  Example: If + =, If + =, then then

Symmetric Equality  2 Equations  Switch sides  Example: If + =, If + =, then then

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = 2y, then ??? If 4 = x and x = 2y, then ???

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = 2y, then ??? If 4 = x and x = 2y, then ???

Addition Property of Equality  Add Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Addition Property of Equality  Add Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  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 =

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  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 Examples 2 Block X Template

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example:

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example:

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: (x -) = x – (x -) = x – (x + -) = x + - (x + -) = x + -

Additive Identity  Add ZERO  Identity means stays the same  Example: + 0 = + 0 =

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: + = 0 + = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: * 1 = ??? * 1 = ??? Or 1 * = Or 1 * =

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: * 1/ = 1 * 1/ = 1 Or * / becomes * / Or * / becomes * /

Substitution  Replace a letter with a number  Example: 5 + x = y where x = 5 + x = y where x = y = y =

Symmetric Equality  2 Equations  Switch sides  Example: If + =, If + =, then then

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = 2y, then ??? If 4 = x and x = 2y, then ???

Addition Property of Equality  Add Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  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 Examples 2 Block X Template

Commutative Property  1 Equation  CO = Change order;  move numbers; “commute”  Example:

Associative Property  1 Equation  SO = Same Order.  Change groups or ( )  Example:

Distributive Property  1 Equation  Multiply the outside by everything in the inside.  Example: (x -) = x – (x -) = x – (x + -) = x + - (x + -) = x + -

Additive Identity  Add ZERO  Identity means stays the same  Example: + 0 = + 0 =

Additive Inverse  Inverse means Opposite  Add and Subtract the same number or  Positive and Negative  Adds to ZERO.  Example: + = 0 + = 0

Multiplicative Identity  Multiply by 1  Identity means stays the same  Example: * 1 = ??? * 1 = ??? Or 1 * = Or 1 * =

Multiplicative Inverse  Inverse means Opposite  Multiply and Divide the same number are opposites  OR Do reciprocal  numbers multiply to 1  Example: * 1/ = 1 * 1/ = 1 Or * / becomes * / Or * / becomes * /

Substitution  Replace a letter with a number  Example: 5 + x = y where x = 5 + x = y where x = y = y =

Symmetric Equality  2 Equations  Switch sides  Example: If + =, If + =, then then

Transitive Equality  3 equations  The middle of the first two are equal.  The ends create the third.  Example: If 4 = x and x = 2y, then ??? If 4 = x and x = 2y, then ???

Addition Property of Equality  Add Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Multiplication Property of Equality  Multiply Equal things to both sides.  Example: If 5 = x, then 9 = ??? If 5 = x, then 9 = ???

Zero Product Property  Product is multiply  If 2 numbers multiply to 0, then one of the numbers must be 0.  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 =