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CC Algebra I 3- Properties of Real Numbers Name: __________________________ Number_____ Date: ______________________ Do Now: Suzy draws a picture of squares to represent the product 3 × 4. Ben moves to the end of the table and says, “From my new seat, your picture looks like the product 4 × 3.” What picture might Suzy have drawn? Why would Ben see it differently from his viewpoint? Properties of Addition Property (a, b, and c are real numbers, variables, or algebraic expressions) Examples 3 4 43 ab ba Meaning Verbal Hints The order of the “Commute – to get up numbers change and move to a new but the sum is the location: switch same places” 3 (4 5) (3 4) 5 a (b c) (a b) c The grouping of numbers change but the sum is the same “regroup – elements do not physically move, they simply group with a new friend” a0 a 40 4 **the identity element for addition is ZERO** When zero is added to any number, the result is the number itself “the value that returns the input unchanged’ When you add a number and its opposite, the result is zero “the value that brings you back to the identity element under addition a (a) 0 4 ( 4) 0 **the result is the identity element** Properties of Multiplication Property (a, b, and c are real numbers, variables, or algebraic expressions) Examples Meaning Verbal Hints 3 4 43 ab ba The order of the numbers change but the product is the same “Commute – to get up and move to a new location: switch places” 3 (4 5) (3 4) 5 a (b c) (a b) c The grouping of numbers change but the product is the same “regroup – elements do not physically move, they simply group with a new friend” a 1 a 4 1 4 **the identity element for multiplication is ONE** When ONE is multiplied to any number, the result is the number itself “the value that returns the input unchanged’ When you multiply a number and its reciprocal, the result is ONE “the value that brings you back to the identity element under multiplication 5(4 x) 20 5 x 3( y 5) 3 y 15 The number outside the parentheses is multiplied with each number inside. “multiplication distributes across addition and subtraction” ab a b c c c ab a b c c c Each term in the numerator is divided by the term in the denominator “dividing both sides of an equation by the same non-zero value will not change the truth value of the equation” 1 a ( ) 1 a 1 4( ) 1 4 **the result is the identity element** Match each example to its corresponding property. _____ 1) (2 b) 9 2 (b 9) A. Multiplicative Identity _____ 2) r (1) r B. Distributive Property _____ 3) 5𝑥+2𝑥 3 = 5𝑥 3 + 2𝑥 3 C. Additive Identity _____ 4) 3 3 0 D. Associative Property of Multiplication _____ 5) 6 (4 x) 6 ( x 4) E. Commutative Property of Multiplication _____ 6) (ab)c a(bc) F. Distributive Property _____ 7) 3 3 x0 x 2 2 G. Commutative Property of Addition 1 _____ 8) a 1 a H. Associative Property of Addition _____9) 3(4 x) (4 x)3 I. Multiplicative Inverse _____10) 7(c d ) 7c 7d J. Additive Inverse 11) When solving the equation 4(3x 2 2) 9 8x 2 7 Emily wrote 4(3x 2 2) 8x 2 16 as her first step. Which property justifies Emily’s first step? (1) addition property of equality (2) commutative property of addition (3) multiplication property of equality (4) distributive property of multiplication over addition 12) Use these abbreviations for the properties of real numbers and complete the flow diagram. C for the commutative property of addition C x for the commutative property of multiplication A for the associative property of addition Ax for the associative property of multiplication 13) Draw a flow diagram, and use both the commutative and associative properties to change the outcome. You must use one of each property for a minimum of two changes. a) (xy)z b) (x + y) + z Each expression has been simplified or solved one step at a time. Next to each step, identify the property, transformation or simplification used in the step. 14) 4 5( x 7) _________________________ 4 (5 x 35) _________________________ (4 35) 5 x _________________________ 39 5x _________________________ 15) x 19 23 _________________________ x 19 (19) 23 (19) _________________________ x 0 23 (19) _________________________ x 4 _________________________ 16) Fill in the blanks of this proof. Write either “Commutative Property”, “Associative Property”, or “Distributive Property” in each blank. ( w 5)( w 2) = ( w 5) w ( w 5) 2 _________________________ w( w 5) ( w 5) 2 _________________________ w( w 5) 2( w 5) _________________________ w2 w 5 2(w 5) _________________________ w 2 5w 2(w 5) _________________________ w 2 5w 2w 10 _________________________ w 2 (5w 2w) 10 _________________________ w 2 7 w 10 CC Algebra I Period ____ Name:__________________________ Date:___________________________ Homework -Number Properties 1) The following portion of a flow diagram shows that the expression ab + cd is equivalent to the expression dc + ba. Fill in each circle with the appropriate symbol: C for the commutative property of addition, C x for the commutative property of multiplication, A for the associative property of addition, Ax for the associative property of multiplication 2) Fill in the blanks of this proof. Write either “Commutative Property”, “Associative Property”, or “Distributive Property” in each blank. ( x 4)( x 3) = ( x 4) x ( x 4) 3 __________________________ x( x 4) ( x 4) 3 _________________________ x( x 4) 3( x 4) ____________________________ x 2 4 x 3x 12 _____________________________ x 2 (4 x 3x) 12 ____________________________ x 2 1x 12 _________________________________ Properties of Real Number Word Bank Multiplicative Identity Distributive Property Additive Identity Associative Property of Multiplication Commutative Property of Multiplication Distributive Property Commutative Property of Addition Associative Property of Addition Multiplicative Inverse Additive Inverse Properties of Real Number Word Bank Multiplicative Identity Distributive Property Additive Identity Associative Property of Multiplication Commutative Property of Multiplication Distributive Property Commutative Property of Addition Associative Property of Addition Multiplicative Inverse Additive Inverse Properties of Real Number Word Bank Multiplicative Identity Distributive Property Additive Identity Associative Property of Multiplication Commutative Property of Multiplication Distributive Property Commutative Property of Addition Associative Property of Addition Multiplicative Inverse Additive Inverse