# Properties by ewghwehws

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```									Properties
Properties
   Commutative Property         Substitution
   Associative Property         Symmetric Equality
   Distributive Property        Transitive Equality
   Multiplicative Identity      Multiplication Property
   Multiplicative Inverse        of Equality
   Zero Product Property
Commutative Property
a    +b=b+a

 CO   is Change order; move; “commute”

 Example:
   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 = ???
Distributive Property
 a(b    + c) = ab + ac OR (b + c)a = ba + ca

a    is multiplied by (distributed) everything
inside the parentheses.

the Rainbow (draw arrows on top).
 Taste
 Example:
   3(x + 2) = ???
a    +0=a

 stays the same
   keeps its “identity”

 Example:
   3 + 0 = ???
a    + (-a) = 0

 Opposite    (inverse) numbers add to 0

 Example:
   4 + ??? = 0
Multiplicative Identity
a    *1=a

 Multiplyby 1
 stays the same
   keeps its “identity”

 Example:
   5 * 1 = ???
Multiplicative Inverse

  1                2 x 3 =1
a  1
a •              3   2

 Opposite  (inverse) numbers multiply to 1
 Opposite of multiply is divide.
 Also called Reciprocals.
 Example:
   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 = 4
Symmetric Equality
 If   a = b, then b = a

 Switch     sides

 Example:
   If 6 = x, then ???
Transitive Equality
 If   a = b and b = c, then a = c.

 “Greatcheese comes from happy cows.
Happy cows come from California.”
   Therefore great cheese comes from
California.

 Example:
   If 4 = x and x = 2y, then ???
 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 = ???
Multiplication Property of Equality
 If   a = b, then 2a = 2b.

 Multiply    both sides by the same thing.

 Example:
   If 4 = x, then 12 = ???
Zero Product Property
 If   ab = 0, then a = 0 or b = 0.

 Ifthe product of 2 numbers is 0, then one
of the numbers must be 0 itself.

 Example:
   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
Associative Property
1    Equation

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

 Example:
   (1 * 2) * 3 = 1 * (2 * 3)
Distributive Property
1    Equation

 Multiply   the outside by everything in the
inside.

 Example:
   6(x - 5) = 6x - 30
 Identity means stays the same

 Example:
   10 + 0 = 10
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   -57 + 57 = 0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

 Example:
   2*1=2
   Or
   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
   Or * 2/5 becomes * 5/2
Substitution
 Replace    a letter with a number

 Example:
   5 + x = y where x = 4
   y=9
Symmetric Equality
2    Equations

 Switch   sides

 Example:
   If 5 + 6 = 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
 Add   Equal things to both sides.

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

 Example:
   If   10 = x
   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
   (x + 8) = 0 or (x - 9) = 0
   So (x + 8) gives x = -8
   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
Associative Property
1    Equation

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

 Example:
   ( 6 * 3) * 5 = 6 * (3 * 5)
Distributive Property
1    Equation

 Multiply   the outside by everything in the
inside.

 Example:
   3(x - 8) = 3x - 24
 Identity means stays the same

 Example:
   5+0=5
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   -12 + 12 = 0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

 Example:
   72 * 1 = 72
   Or
   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
   Or
   * 3/4 becomes * 4/3
Substitution
 Replace    a letter with a number

 Example:
   5 + x = y where x = 8
   y = 13
Symmetric Equality
2    Equations

 Switch   sides

 Example:
   If 2 + 3 = 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
 Add   Equal things to both sides.

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

 Example:
   If     7 =x
   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
   Answer: (x + 6) = 0 or (x - 4) = 0
   So (x + 6) gives x = -6
   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
Associative Property
1    Equation

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

 Example:
   ( 3 * 6 ) * 15 = 3 * ( 6 * 15)
Distributive Property
1    Equation

 Multiply   the outside by everything in the
inside.

 Example:
   4 (x - 7) = 4x - 28
 Identity means stays the same

 Example:
   21 + 0 = 21
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   +8 - 8 = 0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

 Example:
   7*1=7
   Or
   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
   Or * 7/8 becomes * 8/7
Substitution
 Replace    a letter with a number

 Example:
   5 + x = y where x = 7
   y = 12
Symmetric Equality
2    Equations

 Switch   sides

 Example:
   If 3 + 4 = 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
 Add   Equal things to both sides.

 Example:
   If   8 =x
   then 15 = x + 7 (Add 7 to both sides.)
Multiplication Property of Equality
 Multiply   Equal things to both sides.

 Example:
   If    3= x
   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
   (x + 2) = 0 or (x - 7) = 0
   So (x + 2) gives x = -2
   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 + -
Distributive Property
1    Equation

 Multiply   the outside by everything in the
inside.

 Example:
    (x -) = x –
   (x + -) = x + -
 Identity means stays the same

 Example:
   +0=
 Identity means stays the same

 Example:
   +0=
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   + =0
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   + =0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

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

 Example:
   * 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
   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
   Or * / becomes * /
Substitution
 Replace    a letter with a number

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

 Example:
   5 + x = y where x =
   y=
Symmetric Equality
2    Equations

 Switch   sides

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

 Switch   sides

 Example:
   If   +=,
   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 ???
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 ???
 Add   Equal things to both sides.

 Example:
   If 5 = x, then 9 = ???
 Add   Equal things to both sides.

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

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

 Example:
   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
   (x + ) = 0 or (x - ) = 0
   So (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
   (x + ) = 0 or (x - ) = 0
   So (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 + -
 Identity means stays the same

 Example:
   +0=
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   + =0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

 Example:
   * 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
   Or * / becomes * /
Substitution
 Replace    a letter with a number

 Example:
   5 + x = y where x =
   y=
Symmetric Equality
2    Equations

 Switch   sides

 Example:
   If   +=,
   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 ???
 Add   Equal things to both sides.

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

 Example:
   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
   (x + ) = 0 or (x - ) = 0
   So (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 + -
 Identity means stays the same

 Example:
   +0=
 Inverse means Opposite
 Add and Subtract the same number or
 Positive and Negative

 Example:
   + =0
Multiplicative Identity
 Multiply by 1
 Identity means stays the same

 Example:
   * 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
   Or * / becomes * /
Substitution
 Replace    a letter with a number

 Example:
   5 + x = y where x =
   y=
Symmetric Equality
2    Equations

 Switch   sides

 Example:
   If   +=,
   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 ???
 Add   Equal things to both sides.

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

 Example:
   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
   (x + ) = 0 or (x - ) = 0
   So (x + ) gives x =
   And (x – ) gives x =

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