Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 1 Real Numbers and Introduction to Algebra

22 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6

33 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7

44 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7 (-8)(-5) + (-1) = 40 + (-1) = 39

55 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7 (-8)(-5) + (-1) = 40 + (-1) = / -20 = + 2/5

66 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7 (-8)(-5) + (-1) = 40 + (-1) = 39 2( ) = 2(-7) = / -20 = + 2/5

77 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7 (-8)(-5) + (-1) = 40 + (-1) = / -20 = + 2/5 2( ) = 2(-7) = / -30 = 3/10

88 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Bellwork: 1. 7 subtracted from the quotient of 0 and added to the product of -8 and the quotient of -8 and twice the sum of -3 and the quotient of -9 and the difference of -9 and the product of -4 and -6 0/5 – 7 = -7 (-8)(-5) + (-1) = 40 + (-1) = / -20 = + 2/5 2( ) = 2(-7) = / -30 = 3/10 -9 – (-4)(-6) = -9 – (+24) = -33

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. 1.7 Properties of Real Numbers

10 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Objectives:  Define and use properties of real numbers

11 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Commutative Property Addition: a + b = b + a Multiplication: a · b = b · a

12 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Commutative Property Addition: a + b = b + a Multiplication: a · b = b · a “reorder”

13 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Activity  Can you illustrate the commutative property using a group of people? Show what the commutative property means using a 2 or 4 member group.

14 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Associative Property Addition:(a + b) + c = a + (b + c) Multiplication: (a · b) · c = a · (b · c)

15 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Associative Property Addition:(a + b) + c = a + (b + c) Multiplication: (a · b) · c = a · (b · c) “regroup”

16 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Activity  Can you illustrate the associative property using a group of people? Show what the associative property means using a 3 member group.

17 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

18 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

19 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

20 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

21 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

22 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 1 Use the commutative or associative property to complete. a. x + 8 = ______ 8 + x b. 7 · x = ______ x · 7 c. 3 + (8 + 1) = _________ (3 + 8) + 1 d. ( ‒ 5 ·4) · 2 = _________ ‒ 5(4 · 2) e. (xy) ·18 = ___________ x · (y ·18)

23 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. a(b + c) = ab + ac Distributive Property

24 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. a(b + c) = ab + ac “Multiplication Over Addition” Distributive Property

25 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.  Multiplication 1 is the identity for multiplication a · 1 = a and 1 · a = a Identity Properties  Addition 0 is the identity for addition a + 0 = a and 0 + a = a

26 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.  Multiplication 1 is the identity for multiplication a · 1 = a and 1 · a = a Identity Properties  Addition 0 is the identity for addition a + 0 = a and 0 + a = a “doesn’t change it”

27 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.  Multiplication 1 is the identity for multiplication a · 1 = a and 1 · a = a Identity Properties  Addition 0 is the identity for addition a + 0 = a and 0 + a = a “same as what you started with”

28 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

29 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

30 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

31 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

32 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

33 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 2 Use the distributive property to write each expression without parentheses. Then simplify the result. a. 7(x + 4y) = 7x + 28y b. 3( ‒ 5 + 9z) = 3( ‒ 5) + (3)(9z)= ‒ z c. ‒ (8 + x ‒ w) = ( ‒ 1)(8) + ( ‒ 1)(x) ‒ ( ‒ 1)(w) = ‒ 8 ‒ x + w

34 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Additive Inverse The numbers a and –a are additive inverses or opposites of each other because their sum is 0; that is a + ( – a) = 0. Multiplicative Inverse The numbers b and (for b ≠0) are reciprocals or multiplicative inverses of each other because their product is 1; that is Inverse Properties

35 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Additive Inverse The numbers a and –a are additive inverses or opposites of each other because their sum is 0; that is a + ( – a) = 0. Multiplicative Inverse The numbers b and (for b ≠0) are reciprocals or multiplicative inverses of each other because their product is 1; that is Inverse Properties “opposites”

36 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Additive Inverse The numbers a and –a are additive inverses or opposites of each other because their sum is 0; that is a + ( – a) = 0. Multiplicative Inverse The numbers b and (for b ≠0) are reciprocals or multiplicative inverses of each other because their product is 1; that is Inverse Properties “opposites” “reciprocals”

37 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Additive Inverse The numbers a and –a are additive inverses or opposites of each other because their sum is 0; that is a + ( – a) = 0. Multiplicative Inverse The numbers b and (for b ≠0) are reciprocals or multiplicative inverses of each other because their product is 1; that is Inverse Properties “opposites” “flip it”

38 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. a. 5(x + y) = 5 · x + 5 ·y Distributive property b. (n + 0) + 9 = n + 9 Identity element for addition c. ‒ 5 · (x · 11) = ( ‒ 5 ·11) · x Commutative and associative properties of multiplication

39 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. a. 5(x + y) = 5 · x + 5 ·y Distributive property b. (n + 0) + 9 = n + 9 Identity element for addition c. ‒ 5 · (x · 11) = ( ‒ 5 ·11) · x Commutative and associative properties of multiplication

40 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. a. 5(x + y) = 5 · x + 5 ·y Distributive property b. (n + 0) + 9 = n + 9 Identity element for addition c. ‒ 5 · (x · 11) = ( ‒ 5 ·11) · x Commutative and associative properties of multiplication

41 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. a. 5(x + y) = 5 · x + 5 ·y Distributive property b. (n + 0) + 9 = n + 9 Identity element for addition c. ‒ 5 · (x · 11) = ( ‒ 5 ·11) · x Commutative and associative properties of multiplication

42 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. d. 5 + (x + y) = (5 + x) + y Associative property of addition e. Multiplicative inverse property f. ‒ = 0 Additive inverse property

43 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. d. 5 + (x + y) = (5 + x) + y Associative property of addition e. Multiplicative inverse property f. ‒ = 0 Additive inverse property

44 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. d. 5 + (x + y) = (5 + x) + y Associative property of addition e. Multiplicative inverse property f. ‒ = 0 Additive inverse property

45 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Example 3 Name the property illustrated by each true statement. d. 5 + (x + y) = (5 + x) + y Associative property of addition e. Multiplicative inverse property f. ‒ = 0 Additive inverse property

46 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say commutative, you say _______.

47 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say commutative, you say _______. “reorder”

48 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say associative, you say _______.

49 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say associative, you say _______. “regroup”

50 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say distributive, you say _______.

51 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say distributive, you say _______. “multi. over add”

52 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say identity, you say _______.

53 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say identity, you say _______. “doesn’t change it”

54 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say identity, you say _______. “same as what you started with”

55 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say additive inverse, you say _______.

56 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say additive inverse, you say _______. “opposite”

57 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say multiplicative inverse, you say _______.

58 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say multiplicative inverse, you say _______. “reciprocal”

59 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure: I say multiplicative inverse, you say _______. “flip it”

60 Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Closure:  I say commutative, you say _______.  I say associative, you say _______.  I say distributive, you say _______.  I say identity, you say _______.  I say additive inverse, you say _______.  I say multiplicative inverse, you say _______.