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 =