Lesson 1-6. Algebra Slide Show: Teaching Made Easy As Pi, by James Wenk Identities 5 12 = 12 5

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1 Lesson -6 Objective - To simplify expressions using commutative and associative properties. Commutative - Order doesn t matter! You can flip-flop numbers around an operation. Commutative Property of Addition a + b = b + a = = Commutative Property of Multiplication a b= b a 8 = 8 3 2= 2 3 Write an equivalent expression. ) (6 + ) = ( + 6) 2) 3 8 ) = = 2 5 5) 3 + (+ = = 3 + (8+ ) or ( Addition and multiplication are commutative. Is subtraction commutative? Difficult to prove true! Easy to prove false! Counterexample - An example that proves a statement false. Counterexample: Therefore, subtraction is not commutative. Give a counterexample that shows division is not commutative. Counterexample: Therefore, division is not commutative. State whether each situation below is commutative or not commutative. ) Waking up in the morning and going to school. not commutative 2) Brushing your teeth and combing your hair. commutative Putting on your socks and putting on your shoes. not commutative ) Eating cereal and drinking orange juice. commutative Identities Identity Property of Addition x + 0 = x Zero is sometimes called the Additive Identity. Identity Property of Multiplication x = x One is sometimes called the Multiplicative Identity.

2 Lesson -6 (cont.) Identify the property shown below. ) = ) Comm. Prop. of Add. (0 ) = ( 0) Comm. Prop. of Mult. (2 + 0) + 3 = (0 + 2) + 3 Comm. Prop. of Add. ) 5 ( 7 ) = ( 7 ) 5 Comm. Prop. of Mult. 5) 7 0= 0 Mult. Prop. of Zero 6) = 7 Identity Prop. of Add. 7) 7 = 7 Identity Prop. of Mult. Associative Property Associative - Re-grouping is ok! You can re-group numbers together. Associative Property of Addition (a + b) + c = a + (b + c) ( + 2) + 9 = + (2 + 9) Associative Property of Multiplication (a b) c = a (b c) (3 5) 7 = 3 (5 7) Write an equivalent expression. ) ( + 5) + 9 = + (5 + 9) 2) 3 ( = (3 ) 8 (6 7) = ( 6) 7 ) (6 + 2) + 3 = 6 + (2 + 5) 3 ( = (3 ) 8 Addition and multiplication are associative. Give counterexamples to prove that subtraction and division are not associative. Subtraction Counterexample: (0 2 0 (3 2) Therefore, subtraction is not associative. Division Counterexample: (6 2 6 (8 2) Therefore, division is not associative. Identify each property shown below. ) 5 + = + 5 Comm. Prop. Of Add. Commutative Associative (2 + 7) + 8 = (7 + 2) + 8 (2 + 7) + 8 = 2 + (7 + 2) 3 ( = (3 ) 8 Assoc. Prop. Of Mult. Flip-flop Re-group 7 2= 2 7 Comm. Prop. Of Mult. (2 + 7) + 8 = 8 + (2 + 7) ) (6 + 2) + 3 = (2 + 6) + 3 Comm. Prop. Of Add. Flip-flop ( ) does not imply Associative

3 Lesson -6 (cont.) Identify each property shown below. ) (9 + + = (3 + 9) + 2) 0 (5 = (0 5) 3 ) Comm. Prop. Of Add. Assoc. Prop. Of Mult. + 8= 8+ Comm. Prop. Of Add. (3 2) = (3 2) Comm. Prop. Of Mult. Give the property that justifies each step. Statement Reasons 6 + (27 + Given 6 + (8 + 27) Comm. Prop. of Add. ( (00) + 27 Assoc. Prop. of Add. Simplify 27 Simplify Use the commutative and associative properties to simplify each expression. ) 25 (37 ) 2) 2 + ( ( 37) 2 + (8 + 29) (25 ) 37 ( (00) 37 (20) Distributive Property Distributive Property a(b+ c) = a b+ a c or a(b c) = a b a c Order of Operations 3(8( 2) ) = 3(6) ( ) = 8 3( 3(2) 2 6 It works! 8 Why use the distributive property? 3(x 2) = 3(x) 3(2) = 3x 6 Use the distributive property to simplify. ) 3(x + 7) 6) x(a + m) 3x + 2 ax + mx 2) 2(a + ) 7) (3 + r) 2a r 7(8 + m) 2(x m 2x + 6 ) 3( + a) 9) 7(2m + 3y + ) 2 + 3a m + 2y ) (3 + k)5 0) (6 + 2y + a) k 8 + 6y + 3a Opposite of a Sum or Difference Opposite of a sum -(x + y) = -x + -y -(x + y) = -x + -y Simplify the following. ) (x + 5) x 5 2) (y y + 3 Opposite of a difference -(x - y) = -x + y -(x - y) = -x + y (m + y 2) m y+ 2 ) (a 2b + 7) a + 2b 7

4 Lesson -6 (cont.) ) -(x + 6) -x - 6 2) -(2x - -2x (m m - 3 -m + 7 ) 2(x - 5) - (x - 2x x + 3 x - 7 Subtracting a Quantity 6) -(3a + ) -3a - 7) -(-3x + 2y - 7) +3x - 2y (3y y + 8-3y - 9) (3k - 5) - (2k + 9) 2k k - 9 0k - 29 Use the distributive property to help simplify the following without a calculator. ) 5(9.96) 2) 7(8.2) 5(0 0.0) 7( ) 5(0) 5(0.0) ( + 7(0.2) Use the distributive property to help simplify the following without a calculator. 8($.30) ) ($ + $0.30) 7(6 0.05) 8($) + 8($0.30) $88 + $2.0 $90.0 7(6) 7(0.05) Closure Property A set of numbers is said to be closed if the numbers produced under a given operation are also elements of the set. Tell whether the whole numbers are closed under the given operation. If not, give a counterexample. ) Addition Multiplication 2) Subtraction Not 5-7 = -2 ) Division Not 2 8 = 0.25 Closure Property A set of numbers is said to be closed if the numbers produced under a given operation are also elements of the set. Tell whether the integers are closed under the given operation. If not, give a counterexample. ) Addition 2) Subtraction Multiplication ) Division Not 2 8 = 0.25 Field Properties (Axioms) Used in Proofs The Closure Properties If a and b are rational, then a + b is rational. If a and b are rational, then abis rational. The Commutative Properties a + b = b + a, a b = b a The Associative Properties (a + b) + c = a + (b + c), (ab) c = a (b c) The Identity Properties a + 0 = a, a = a The Inverse Properties a+ (- a) = 0, a = (where a 0) The Distributive Property a ab ( + c) = ab+ ac

5 Lesson -6 (cont.) Additional Properties (Axioms) Used in Proofs Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a - c = b - c. Multiplication Property of Equality If a = b, then a c = b c. Subtraction Property of Equality If a = b, then a c = b c. Reflexive Property a = a Other Properties Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Example of Direct Proof (Deductive) Prove: If a = b, then -a = -b. Statement Reason a = b Given a + (-b) = b + (-b) Addition Property of Equality a + (-b) = 0 Inverse Property (-a) + [a + (-b)] = 0 + (-a) Addition Property of Equality [(-a) + a] + (-b) = 0 + (-a) Associative Prop. of Addition 0 + (-b) = 0 + (-a) Inverse Property -b = -a Identity Property of Addition -a = -b Symmetric Property 3x + 5(2 x) Given 3x x Distributive Property 3x + 5x + 0 Commutative Prop. of Add. (3 + 5)x + 0 Distributive Property 2x + 0 Simplify 5(2x = 35 Given 0x 5 = 35 Distributive Property 0x 5 + 5= Addition Property of Equality 0x = 50 Inverse Property 0x = 50 Multip. Property of Equality 0 0 x = 5 Inverse Property 2a + 3(a ) = 3 Given 2a + (3a 2) = 3 Distributive Property (2a + 3a) 2 = 3 Associative Property of Add. (2 + a 2 = 3 Distributive Property 5a = 3+ 2 Addition Property of Equality 5a = 25 Inverse Property 5a = 25 Multip. Property of Equality 5 5 a = 5 Inverse Property