Family Literacy. readers. Easy Algebra Reading Practice

Transcription

1

2 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.0 A Review of Unit 1

3 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.0 A Review of Unit 1

4 contents Topics Sets 1 6 The Order of Operations 7 Definitions 8 Properties 9 Like Terms 11 Products and Factors Expanded and Power Forms Formulas in Algebra 18 Perimeters Areas Formulas 27 Page UNIT 2 Integers and Expressions 2.0 A Review of Unit 1

5 Algebra Words and Phrases sets finite sets infinite sets subsets empty sets null sets disjoint sets sets of numbers The Order of Operations variable term coefficient algebraic expression Commutative Property Associative Property Distributive Property like terms products factors powers expanded form perimeter of a rectangle perimeter of a square perimeter of a triangle circumference of a circle area of a square area of a rectangle area of a triangle A Review of Unit 1 READER 2.0

6 Defining Sets A set is a group of things (objects or ideas). A set may include the players on the football or soccer teams. There are two sets pictured here: one is a set of 16 boys on the soccer team; the other is the set of 20 boys on the football team. We often identify sets with letters. These two sets could be called sets A and B Set A Set B READER 2.0 A Review of Unit 1 1

7 Finite and Infinite Sets Those things which belong to the set are called elements or members of the set. Each boy on the soccer team is a member of that set. Similarly, each boy on the football team is a member of that set. If a set has a limited number of members or elements, it is a finite set. If its members or elements are unlimited, it is an infinite set. These teams are finite sets because they have a limited number of members Members of a Set Members of a Set Members of finite sets (a limited number they can be counted) 2 A Review of Unit 1 READER 2.0

8 Subsets Look at the set of soccer players again. We ll call this team set A Three of the players are subs on the team. We ll choose these three players from set A and call them set B. They are also still in set A. Since they are members of both sets, set B is a subset of set A Set A Set B READER 2.0 A Review of Unit 1

9 The Null or Empty Set Suppose we define Set E as the set of Natural numbers between 4 and 5: Set E = {Natural numbers between 4 and 5} Because there are no Natural numbers between 4 and 5, set E has no members. This finite set is called an empty set, or null set. It is written as E = { } or E = Ø. (The symbol for the null set is Ø.) A Review of Unit 1 READER 2.0

10 Disjoint Sets Disjoint sets have no elements in common. Let s define set A as the members on the soccer team and set B as members on the football team. Let s say that no one plays on both teams, so they have no members in common. They are two disjoint sets. No member or element of set A belongs to set B and no element or member of set B belongs to set A. Set A Set B READER 2.0 A Review of Unit 1

11 Sets of Numbers There are many sets of numbers that will be used in algebra. You don t need to understand all of them now. They are included to show how sets and subsets work with sets of numbers. Natural Numbers: N = {1, 2, 3, 4,...} Whole Numbers: W = {0, 1, 2, 3, 4,...} Integers: Z = {...-3, -2, -1, 0, 1, 2, 3,...} Rational Numbers: Q = { a, a and b integers, b 0} b Irrational Numbers: Real Numbers: I = {Non-terminating, nonrepeating decimals} R = {All rational or irrational numbers} Counting numbers Natural numbers, plus 0 Whole numbers, plus an opposite for each natural number. (-1 is the opposite of 1, -2 is the opposite of 2, etc.) Common fractions: (repeating or terminating decimals). Examples: π, N, W, Z, Q, and I are all subsets of R A Review of Unit 1 READER 2.0

12 The Order of Operations Calculate operations within any grouping symbols, starting with the innermost symbols first. Calculate multiplication and/or division in order from left to right. Calculate addition and/or subtraction in order from left to right. Simplify by following the Order of Operations. There are grouping symbols ( ) inside other grouping symbols [ ]. Do the operation in the innermost symbols first: 10 [15 2(1 + 4)] + 5 = 10 [15 2(5)] + 5 = 10 [15 10] + 5 = = = 7 READER 2.0 A Review of Unit 1

13 Definitions A variable is a symbol (usually a letter) in an expression that can be replaced by a number or other symbols. Can you identify the variable in this expression? 2x + 3 = 11 The variable is x. A term is a number by itself, a product with a letter and a number (from a multiplication problem), or a quotient (from a division problem). Can you identify the terms in this expression? 2x + 3 = 11 The terms are 2x, 3, and 11. In any term, the numerical factor is called the coefficient of the variable factor(s). Can you identify the coefficient in this expression? 2x + 3 = 11 The coefficient is 2. An algebraic expression is a single term, or several terms linked by addition or subtraction operations. Is this an algebraic expression? 2x + 3 Yes, this is an algebraic expression. An algebraic expression is not always an equation. A Review of Unit 1 READER 2.0

14 Properties The commutative, associative, and distributive properties help to combine like terms and simplify expressions. Commutative Property: For Addition: a + b = b + a = For Multiplication: ab = ba 2 3 = 3 2 Associative Property: For Addition: (a + b) + c = a + (b + c) (2 + 3) + 4 = 2 + (3 + 4) For Multiplication: (ab)(c) = (a)(bc) (2 3)(4) = (2)(3 4) READER 2.0 A Review of Unit 1

15 Distributive Property for Multiplication Over Addition: Both the b and c are multiplied by the a. Remember the chocolate factory from Unit 1? (a)(b + c) = ab + ac (2)(3 + 4) = A Review of Unit 1 READER 2.0

16 Like Terms Like terms are terms in an expression that have exactly the same variables (or letters) in them or that have no variables (or letters) at all. Can you find the like terms in the following algebraic expressions? 2x + 3y + 5x 5r x + 7xy + 2z + y + 1 The terms 2x and 5x are like terms because in these terms, one factor is a number and the other factor is x. The terms 32, 75, and 5 are like terms because none of these terms have variables in them. There are no like terms here. We combine like terms by adding the coefficients like this: 2x + 3y + 5x = 7x + 3y = READER 2.0 A Review of Unit 1 11

17 Products When two or more numbers or variables are multiplied together, the result is the product. 5 5 = 25 Product 12 A Review of Unit 1 READER 2.0

18 Factors The numbers or variables which are multiplied are called factors. 5 a = 25 Factors factors product READER 2.0 A Review of Unit 1 13

19 Powers and Factors a n is a product called the n th power of a. This means a is used as a factor n times. For example, a 3 = a a a. The general form looks like this: a n = a a a... a n factors{ 14 A Review of Unit 1 READER 2.0

20 Expanded and Power Forms When factors of a product are written out, as seen below, it is called the Expanded Form. y 4 = y y y y 3 3 = When exponents are used in the product, it is the Power Form. y 4 = y y y y 3 3 = READER 2.0 A Review of Unit 1 15

21 Powers For the power, b n, b is the base, and n is the exponent. b n = b b b b... n factors 2 5 = = 32 5 factors 16 A Review of Unit 1 READER 2.0

22 Study these examples: Power Form Base Exponent Expanded Form x 5 x 5 x x x x x p 2 r 3 p, r 2, 3 p p r r r READER 2.0 A Review of Unit 1 17

23 Formulas in Algebra Formulas are used in Algebra to express ways to solve problems using algebraic expressions. Instead of writing long verbal expressions, we write a shortened version using an algebraic expression. For example: Instead of saying, To determine the distance around a rectangle, we add the two lengths and the two widths of the rectangle together, we would write the formula for the distance around a rectangle as this: 2L + 2W (two lengths) (two widths) where L is a measure of the length and W is a measure of the width. 18 A Review of Unit 1 READER 2.0

24 Perimeter of a Rectangle The formula for the perimeter (or distance around) of a rectangle is: L + L + W + W or 2L + 2W or 2(L + W) 1L + 1L = (1+1)L = 2L 1W + 1W = (1+1)W = 2W LL W READER 2.0 A Review of Unit 1 19

25 Perimeter of a Square Here's the algebraic formula for the perimeter of a square, where s is equal to the length of one side. P = 4s s 1(s) + 1(s) + 1(s) + 1(s) = ( )s = 4s s s s 20 A Review of Unit 1 READER 2.0

26 Perimeter of a Triangle The perimeter of a triangle with sides a, b, and c is equal to a + b + c. P = a + b + c a b c READER 2.0 A Review of Unit 1 21

27 Circumference of a Circle The perimeter of a circle is called the circumference, where π (pi) is equal to 3.14 and r is equal to the radius of the circle. C = 2πr circumference r radius 22 A Review of Unit 1 READER 2.0

28 Area of a Rectangle The formula for the area of a rectangle or square is length x width. L 50' 140' W A = L W 50' 140' = 7000 ft 2 Note: Since area deals with square units, remember to add the squared exponent to your final answer. READER 2.0 A Review of Unit 1 23

29 Area of a Square The formula for the area of a rectangle or square is length x width. Because we know that all sides of a square are equal, the area of a square is: A = L W A = s s A = s 2 S S S S 24 A Review of Unit 1 READER 2.0

30 The area of a triangle is 1 2 bh or 1 2 bh 2 Area of a Triangle the length of the base times the height or height base READER 2.0 A Review of Unit 1 25

31 Here's how to find the area of this triangle: 1 A = 2 bh A = 1 2 (7yd)(4yd) 1 A = 2 (28yd) 2 A = 14yd 2 height 4 yd. 7 yd. base Remember that 14 yd 2 means 14 square yards, not linear yards. 26 A Review of Unit 1 READER 2.0

32 Formulas Using Algebraic Expressions Figure Dimensions Perimeter Area Rectangle width = w P = 2L + 2w A = LW length = L P= 2(L + w) Square sides = s P = 4s A = s 2 1 Triangle sides = a, b, c P = a + b + c A = bh 2 height = h Circle radius = r C = 2πr π = 3.14 where C is circumference READER 2.0 A Review of Unit 1 27

33 28 A Review of Unit 1 READER 2.0

34 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.1 Absolute Value of Integers

35 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.1 Absolute Value of Integers

36 contents Topics Page Taking a Vacation 1 Number Lines 4 Word Phrases 11 Just for Fun 1 14 Absolute Value 15 Working with Absolute Values Just for Fun 2 19 Answers 20 UNIT 2 Integers and Expressions 2.1 Absolute Value of Integers

37 Algebra Words and Phrases number lines origin positive numbers negative numbers set of Integers set of Natural numbers set of Whole numbers set of Real numbers graphic representation word phrases absolute value Absolute Value of Integers READER 2.0

38 Taking a Vacation Paul is taking a trip from El Paso, Texas to San Diego, California. The trip is 724 miles. San Diego El Paso READER 2.1 Absolute Value of Integers 1

39 It is the same distance between San Diego and El Paso no matter which way Paul travels, 724 miles. San Diego El Paso 2 Absolute Value of Integers READER 2.1

40 Paul is also taking a trip to Padre Island. Notice the distance from El Paso to Padre Island is the same as the distance from El Paso to San Diego (only in the opposite direction.) San Diego El Paso Padre Island READER 2.1 Absolute Value of Integers

41 Number Lines A number line is much like the path Paul traveled. The origin is the point assigned to zero on a number line and is the place where Paul starts El Paso. The destinations of San Diego and Padre Island can also be shown on the number line. (Notice they are equal lengths apart in opposite directions.) San Diego El Paso Padre Island Absolute Value of Integers READER 2.1

42 The positive numbers are to the right of the origin and represent his trip to Padre Island. San Diego El Paso Padre Island READER 2.1 Absolute Value of Integers

43 The negative numbers are to the left of the origin and represent his trip to San Diego. San Diego El Paso Padre Island Absolute Value of Integers READER 2.1

44 The arrows mean the line continues in both directions The integers include the set of all natural numbers, their opposites, and 0: {... 3, 2, 1, 0, 1, 2, 3...}. We can show them in red on a number line. Integers READER 2.1 Absolute Value of Integers

45 Remember some sets of numbers are: Natural Numbers: N = {1, 2, 3, 4, 5...} (counting numbers) Whole Numbers: W = {0, 1, 2, 3, 4...} (natural numbers plus zero) Integers: Z = {... 2, 1, 0, 1, 2...} (whole numbers and their opposites) Real Numbers: R = The set of all repeating and non-repeating decimals (We will explain these later.) Absolute Value of Integers READER 2.1

46 A number line is a graphic representation of all real numbers. Real Numbers Natural Numbers Whole Numbers READER 2.1 Absolute Value of Integers

47 Here is a graphic representation of Integers, Rational, and Irrational numbers. Notice that the Rational and Irrational number line includes all the numbers between the Integers (represented by the red line). The Integers are the natural numbers, their opposites, and zero, but nothing in-between. Integers Rational and Irrational Numbers The Rational and Irrational Numbers make up the set of Real Numbers. 10 Absolute Value of Integers READER 2.1

48 Word Phrases We can use Integers to describe word phrases like this: A gain of $ feet below sea level. 7 READER 2.1 Absolute Value of Integers 11

49 Remember Paul s trips? The distance we drive is always positive, no matter which direction we go. Movement in either direction is progress toward a destination whether it be up or down, forward or backward. San Diego El Paso Padre Island Absolute Value of Integers READER 2.1

50 This is true of number lines as well as roads. San Diego El Paso Padre Island So if we count from 0 to 3, we are counting 3 units without regard to the direction. We are just progressing a certain distance. READER 2.1 Absolute Value of Integers 13

51 Just for Fun 1 Patty needs to go the grocery store for food and to the mall for shoes. She starts at home and travels 4 miles to the mall. Then she goes to the grocery store, which is 3 miles past her house in the opposite direction. Use the number line below to figure how many miles Patty drove. Mall Patty s House Grocery Absolute Value of Integers READER 2.1

52 Absolute Value The absolute value of any given number is the distance from 0 to that number on a number line. The absolute value of a positive number or zero is that same number. The absolute value of a negative number is its opposite. Absolute value is expressed by the grouping symbol. Here are examples of absolute values: 7 = 7 7 = 7 5 = 5 6 = 6 0 = 0 1 = 1 READER 2.1 Absolute Value of Integers 15

53 Working with Absolute Values We can simplify numeric expressions with absolute value like this: = = Absolute Value of Integers READER 2.1

54 Here is another example of how we can simplify a numeric expression with absolute values in it: = = = = 8 READER 2.1 Absolute Value of Integers 17

55 And one more example with absolute values: 2 (4 1) 8 2 = 2 (3) 8 2 = = 2 2 = 2 2 = 4 18 Absolute Value of Integers READER 2.1

56 Just for Fun 2 Use what you know about absolute values to find the answer to this problem: 5 (7 3) 4 3 = READER 2.1 Absolute Value of Integers 19

57 Answers Page 14: 4 miles (to mall) + 4 miles (back home) + 3 miles (to grocery) + 3 miles (home again) = = 14 Page 19: 5 (7 3) 10 2 = 5 (4) 10 2 = = 10 2 = Absolute Value of Integers READER 2.1

58 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.2 Addition of Integers

59 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.2 Addition of Integers

60 contents Topics Party Balloons and Algebra 1 Adding Two Negatives 9 Just for Fun 1 10 Adding Two Positives 14 Just for Fun 2 16 Juan at the Store 17 Adding Integers with Unlike Signs 22 Just for Fun 3 25 Solving Equations Containing Like and Unlike Signs 26 Just for Fun 4 31 Answers 32 Page UNIT 2 Integers and Expressions 2.2 Addition of Integers

61 Algebra Words and Phrases positive negative adding two negatives adding two positives like signs unlike signs larger absolute value positive sign negative sign Addition of Integers READER 2.2

62 Party Balloons and Algebra Maria has 3 balloons. Rita has 4 balloons. 3 Maria 4 Rita READER 2.2 Addition of Integers

63 Since Maria has three balloons, we will call this number positive 3 or +3. Maria 3 Rita 2 Addition of Integers READER 2.2

64 Since Rita has four balloons, we will call this number positive 4 or +4. Maria 3 Rita 4 READER 2.2 Addition of Integers

65 Maria and Rita together have seven balloons. 3 AND 4 = 7 + Addition of Integers READER 2.2

66 Since Maria and Rita actually have seven balloons, we will consider what they have to be a positive number AND 4 = 7 + We label positive integers with the symbol (+) in front of the number or no symbol at all. READER 2.2 Addition of Integers

67 If Maria were to lose her three balloons, we could consider the lost number to be negative. 3 Addition of Integers READER 2.2

68 If Rita also lost her balloons, we could consider that lost number to be negative too. 4-4 READER 2.2 Addition of Integers

69 Maria lost three balloons and Rita lost four balloons. Together they lost seven balloons. We can represent the total number of balloons lost as a negative number. 7 7 = 0 Addition of Integers READER 2.2

70 Adding Two Negatives A negative plus a negative equal a negative = 7 READER 2.2 Addition of Integers

71 Just for Fun 1 How many glasses are on both tables together? Table 1 Table Addition of Integers READER 2.2

72 Now suppose the table legs broke and the glasses spilled. Table 1 Table 2 READER 2.2 Addition of Integers 11

73 Since Table 1 lost all 5 glasses, we can represent the number of glasses it doesn t have as negative 5. Likewise, Table 2 lost all of its glasses. We can represent the number of glasses it doesn t have as negative 6. Both tables have zero glasses. Table 1 Table 2 12 Addition of Integers READER 2.2

74 Since we lost 5 glasses on Table 1 and 6 glasses on Table 2, we have negative 5 and negative 6 glasses altogether. 5 + ( 6) = 11 READER 2.2 Addition of Integers 13

75 Adding Two Positives To add integers with the same sign, just add the numbers and keep the sign. For example: positive + positive = positive negative + negative = negative (+8) + (+4) = +12 ( 6) + ( 4) = Addition of Integers READER 2.2

76 The rule for adding two integers with the same sign is shown below. 2 + ( 6) = Same sign 2 + ( 6) = 8 Add the numbers (2 + 6) = ( 6) = 8 Keep their sign READER 2.2 Addition of Integers 15

77 1. Add these integers. 2. Add these integers. 3. Add these integers. Just for Fun ( 7) = ( 9) = = Addition of Integers READER 2.2

78 Juan at the Store Juan saved 10 dollars and is on his way to the store to spend it. SALE READER 2.2 Addition of Integers 17

79 Juan has $10. He wants to buy something. Can he buy this cap? Of course he can! Since Juan has more money than the cap costs, he is able to buy the cap and receive change back. 18 Addition of Integers READER 2.2

80 Juan has $10. The cap costs $5. He gets $5 change back and the cap. $10 + ( $5) = $5 and the cap or just $10 $5 = $5 and the cap READER 2.2 Addition of Integers 19

81 Now let s say instead of the cap, Juan would like the CD. Can Juan buy the CD? Of course he can t. He has $10 and the CD cost $15. Juan is missing $5. So Juan is minus $5 or negative $5 ( 5). 20 Addition of Integers READER 2.2

82 Juan has $10 and the CD costs $15. Juan needs (or is missing) $5. $10 + ( $15) = $5 READER 2.2 Addition of Integers 21

83 Adding Integers with Unlike Signs When adding integers, if the signs are different (like the 7 and the +3 below), subtract the smaller number from the larger number and keep the sign of the larger absolute value. IMPORTANT: Notice that sometimes when we add integers, we end up subtracting to get the answer = 4 (7 3) = 4 7 is greater than 3, so we keep the negative sign ( ). 22 Addition of Integers READER 2.2

84 To add integers with unlike signs, = These integers have unlike signs = Keep the sign of the larger absolute value ( ) = 16 Subtract the smaller number from larger number smaller number the larger one. READER 2.2 Addition of Integers 23

85 Another example showing how to add integers with unlike signs: = These integers have unlike signs = + Keep the sign of the larger absolute value = 3 Subtract the smaller number from the larger one. 24 Addition of Integers READER 2.2

86 Just for Fun 3 1. Add these integers = Add these integers. 2 + ( 9) = READER 2.2 Addition of Integers 25

87 Solving Equations Containing Like and Unlike Signs Following the Order of Operations (going from left to right), we will add the first two numbers, then we will add the next number to that sum. 7 + ( 3) + 2 = = 6 26 Addition of Integers READER 2.2

88 Let s take a look at another problem. Look at the sign in front of the first number. It is negative. 6 + ( 4) = READER 2.2 Addition of Integers 27

89 Think of what you do when the signs are the same. 6 + ( 4) = 10 Keep the sign ( ) and add the numbers (6 + 4). The answer is Addition of Integers READER 2.2

90 Remember: Positive integers are numbers or quantities that we actually have on hand. We label positive integers with the positive or plus symbol (+), or no symbol at all in front of the number. Example: +4 = 4 Notice no sign is in front of the number 4. When no sign is given, it is understood to be positive. READER 2.2 Addition of Integers 29

91 Remember: Negative numbers denote numbers or quantities we do not have or are missing. We label negative numbers with the negative or minus symbol ( ). Negative numbers must have the symbol ( ) before the number. Example: 7 = 7 The negative sign is necessary. 30 Addition of Integers READER 2.2

92 Just for Fun 4 1. Solve these equations ( 4) = Solve these equations. 8 + ( 6) + 2 = READER 2.2 Addition of Integers 31

93 Answers Page 10: = 11 Page 16: ( 7) = ( 9) = = 12 Page 25: = ( 9) = 7 Page 28: If signs are the same, add the numbers and keep the sign. Page 31: ( 4) = 2 + ( 4) = ( 16) + 2 = = Addition of Integers READER 2.2

94 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.3 Properties of Addition Applied to Integers

95 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.3 Properties of Addition Applied to Integers

96 contents Topics Page Val and Sal s Graduation 1 The Commutative Property of Addition 4 The Associative Property of Addition 7 Additive Identity Property 10 Additive Inverse Property 14 UNIT 2 Integers and Expressions 2.3 Properties of Addition Applied to Integers

97 Algebra Words and Phrases The Commutative Property of Addition The Associative Property of Addition identity The Additive Identity The Additive Identity Property Additive Inverses Properties of Addition Applied to Integers READER 2.3

98 Val and Sal s Graduation In March it looked like the valedictorian of the senior class was going to be Val and the salutatorian was going to be Sal. The valedictorian is the person who is number one in the graduating class and the salutatorian is the number two person. Usually both of them give speeches at graduation time. #1 Val I m first, you re second. #2 Sal Just wait. In May I ll be first. READER 2.3 Properties of Addition Applied to Integers

99 They both continued to study until the end of the year. Val Sal 2 Properties of Addition Applied to Integers READER 2.3

100 Sal worked a little harder than Val and got better grades. By May graduation, the valedictorian was Sal and the salutatorian was Val. The order had changed. #2 Sal #1 Val READER 2.3 Properties of Addition Applied to Integers 3

101 The Commutative Property of Addition This lesson shows how properties of addition, such as the Commutative and Associative Properties, work with integers. Integers include negative numbers as well positive numbers. Here is our first one. For all numbers x and y: x + y = y + x This is called the Commutative Property of Addition. It says that it doesn t matter which order the two numbers are added, the answer will be the same. This property is always true, even when adding negative integers. The Commutative Property of Addition can look like this: 5 + ( 3) = ( 3) = 2 Properties of Addition Applied to Integers READER 2.3

102 A real life example of the Commutative Property of Addition might look like this: I don t like this grouping. We must change it. READER 2.3 Properties of Addition Applied to Integers

103 Even though the order is changed, there are still the same number of water fowls. Properties of Addition Applied to Integers READER 2.3

104 The Associative Property of Addition For all numbers x, y, and z : (x + y) + z = x + (y + z) This is called the Associative Property of Addition. The Commutative Property of Addition said it didn t matter in which order two numbers were added; the Associative Property says when there are more than two numbers, it doesn t matter which two are added first. (x + y ) + z = ( + ) + = x + ( y + z ) + ( + ) READER 2.3 Properties of Addition Applied to Integers

105 We can also use the Associative Property of Addition to work with negative numbers. Watch how the groupings can change because of this property: [3 + ( 7)] + 5 = 3 + [( 7) + 5] = 3 + ( 2) 1 = 1 Properties of Addition Applied to Integers READER 2.3

106 If you have $35 and spend $4, you will have $31 left. $35 + ( $4) = $31 $35 $4 = $31 READER 2.3 Properties of Addition Applied to Integers

107 Additive Identity Property If you have $35 and spend no money you will still have $35. $35 + $0 = $35 When you add 0 to a number, the value of the number is kept. We say zero keeps the identity. This is the Additive Identity Property. 10 Properties of Addition Applied to Integers READER 2.3

108 It is the same as when you look in the mirror. You see yourself nothing added or subtracted. Your identity is the same. READER 2.3 Properties of Addition Applied to Integers 11

109 Does = 2? Does = 4? Yes! Zero added to any number is the number itself. This is called the Additive Identity. The Additive Identity Property states, for any number x: x + 0 = x and 0 + x = x 12 Properties of Addition Applied to Integers READER 2.3

110 For example: = = ( 7) = 7 READER 2.3 Properties of Addition Applied to Integers 13

111 Additive Inverse Property If you start with $35 and you buy a T-shirt for $35, the result is that you will have no money left. This is called the Additive Inverse Property. 14 Properties of Addition Applied to Integers READER 2.3

112 For any number, x the symbol x means the opposite of x, or the additive inverse of x. READER 2.3 Properties of Addition Applied to Integers 15

113 The Additive Inverse Property states that the sum of any number and its opposite is 0. For any number x: x + ( x) = 0 $35 + ( $35) = 0 16 Properties of Addition Applied to Integers READER 2.3

114 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.4 Subtraction of Integers

115 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.4 Subtraction of Integers

116 contents Topics Page Sophia s Shoe Story 1 Subtracting Integers 12 Just for Fun 1 16 Just for Fun 2 17 Just for Fun 3 18 Just for Fun 4 19 Answers 20 UNIT 2 Integers and Expressions 2.4 Subtraction of Integers

117 Algebra Words and Phrases subtract subtracting integers opposite change the sign compute Subtraction of Integers READER 2.4

118 Sofia is on her way to buy shoes. Sofia s Shoe Story READER 2.4 Subtraction of Integers

119 She wants to purchase shoes that cost $26 plus the tax. 2 Subtraction of Integers READER 2.4

120 Sofia brought $30 to buy her shoes. READER 2.4 Subtraction of Integers

121 What if Sofia had only taken $20 with her? Could Sofia still purchase the shoes she chose for $26? 4 Subtraction of Integers READER 2.4

122 The answer is no, because the shoes cost more than the money she has. $20 $26 READER 2.4 Subtraction of Integers

123 Let s examine the two situations? we have just seen. Subtraction of Integers READER 2.4

124 Situation one. Sofia buys the shoes when she has $30. $30 $26 Situation two. Sofia can t buy the shoes when she has $20. $20 $26 READER 2.4 Subtraction of Integers

125 In situation one, Sofia has 4 dollars extra. $30 $26 = $4 +4 In situation two, Sofia needs 6 more dollars. $26 $30 = $6 6 Subtraction of Integers READER 2.4

126 For Sofia to know if she had enough money to purchase the shoes, she had to subtract the cost of the shoes from the money she had. Money she had minus the cost of the shoes. READER 2.4 Subtraction of Integers

127 First, situation one: money she brought to the store cost of the shoes = $30 $26 = $4 = enough money Now, situation two: money she brought to the store cost of the shoes = $20 $26 = $4 = not enough money 10 Subtraction of Integers READER 2.4

128 When Sofia subtracted the cost of the shoes from the money she had, she discovered whether she could buy the shoes. money she brought to the store cost of the shoes = enough money (+) or not enough money ( ) READER 2.4 Subtraction of Integers 11

129 Subtracting Integers Based on mathematics which we learned in earlier grades, we might think we cannot subtract a larger number from a smaller one. Algebra allows us to work with positive and negative numbers. To subtract an integer, we simply add its opposite. Adding the opposite looks like this: x y = x + ( y) = 20 + ( 26) 6 = 6 12 Subtraction of Integers READER 2.4

130 When working with integers, to find an opposite all one does is change the sign of the number in question. The opposite of a positive is a negative. The opposite of +3 is 3. and... The opposite of a negative is a positive. The opposite of 3 is +3. READER 2.4 Subtraction of Integers 13

131 Consider this example: 4 2 = 4 + ( 2) = [4 2 means 4 plus the opposite of 2] ( 7) = = [6 ( 7) means 6 plus the opposite of 7] Subtraction of Integers READER 2.4

132 Let s look at another example: 4 ( 5) = = [ 4 ( 5) means 4 plus the opposite of 5] 2 READER 2.4 Subtraction of Integers 15

133 Just for Fun 1 Change the subtraction to an equivalent addition problem, then compute. Fill in the parentheses and then compute the answer. 9 3 = 9 + ( ) Point to the box with the number that should go inside the parentheses Point to the box with the right answer Subtraction of Integers READER 2.4

134 Just for Fun 2 Change the subtraction to an equivalent addition problem, then compute. Fill in the parentheses with the correct next step and then compute the answer. 7 ( 4) = 7 + ( ) Point to the box with the number that should go inside the parentheses Point to the box with the right answer READER 2.4 Subtraction of Integers 17

135 Just for Fun 3 Change the subtraction to an equivalent addition problem, then compute = Subtraction of Integers READER 2.4

136 Just for Fun 4 Change the subtraction to an equivalent addition problem, then compute. 13 ( 19) = READER 2.4 Subtraction of Integers 19

137 Answers Page 16: 9 3 = 9 + ( 3) = 6 Page 17: 7 ( 4) = 7 + (4) = 3 Page 18: 13 5 = 13 + ( 5) = 8 Page 19: 13 ( 19) = 13 + (19) = Subtraction of Integers READER 2.4

138 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.5 Adding Integers: A Simplified Approach

139 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.5 Adding Integers: A Simplified Approach

140 contents Topics Page A Football Game 1 Just for Fun 1 4 Grouping Like Signs 5 Omitting Signs 8 Just for Fun 2 14 Just for Fun 3 15 Answers 16 UNIT 2 Integers and Expressions 2.5 Adding Integers: A Simplified Approach

141 Algebra Words and Phrases gain loss like signs simplify omit symbols group mentally regroup mentally Adding Integers: A Simplified Approach READER 2.5

142 A Football Game The quarterback releases the ball. He has a man open. He s on the 40 yard line. 40 yard line READER 2.5 Adding Integers: A Simplified Approach

143 The football was caught at the 40 yard line on the other side of the 50 yard line. The receiver gained 20 yards. A gain of 20 yards! What a play! 2 Adding Integers: A Simplified Approach READER 2.5

144 On the next play, the running back was tackled behind the line of scrimmage for a loss of 15 yards. A loss of 15 yards! READER 2.5 Adding Integers: A Simplified Approach

145 Just for Fun 1 At this point in the game, we have a gain of 20 yards and a loss of 15 yards. What would be the total yardage at this point? 20 + ( 15) = 5 yards 35 yards 45 yards Adding Integers: A Simplified Approach READER 2.5

146 Grouping Like Signs Integers may be added in any order. It is sometimes easier if like signs are grouped together. READER 2.5 Adding Integers: A Simplified Approach 5

147 Here is an example of how grouping like signs works. Notice that brackets [ ] are used here to group like signs. To simplify this numeric expression we do the following: ( 13) = [ 2 + ( 13)] + 9 = Group like signs = Add like signs 6 Simplify Adding Integers: A Simplified Approach READER 2.5

148 Here is another example of how grouping like signs works. To simplify this numeric expression we do the following: 24 + ( 7) ( 28) = (24 + 9) + [( 7) + ( 28)] = Group like signs 33 + ( 35) = Add like signs 2 = Simplify READER 2.5 Adding Integers: A Simplified Approach

149 Omitting Signs When adding negative numbers, the + and ( ) symbols may be omitted (or taken out) to simplify writing the addition problem. Adding Integers: A Simplified Approach READER 2.5

150 Here is an example of how omitting the symbols help make our work easier and simpler. To simplify this numeric expression we do the following: 5 + ( 7) + ( 11) = = Omit symbols 2 11 = Simplify 13 = Simplify READER 2.5 Adding Integers: A Simplified Approach

151 The following is another example of how omitting the + and ( ) symbols help make our work easier and simpler. To simplify this numeric expression we do the following: 9 + ( 3) ( 13) = = Omit appropriate symbols = Simplify 4 13 = Simplify 17 Simplify 10 Adding Integers: A Simplified Approach READER 2.5

152 Grouping like signs can make adding integers easier = ( 35 20) + ( ) READER 2.5 Adding Integers: A Simplified Approach 11

153 The following shows the example on the previous page. Grouping like signs and omitting the + and ( ) symbols make our work easier and simpler. To simplify this numeric expression we would do the following: = ( 35 20) + ( ) = Think of this as grouping = Add each step 10 Simplify 12 Adding Integers: A Simplified Approach READER 2.5

154 Here is how we can use the same rules to simplify a fraction: = = = = Regroup Add like signs Simplify READER 2.5 Adding Integers: A Simplified Approach 13

155 Just for Fun 2 Joe and Thomas decide to set up a lemonade stand. They spend $2.00 on lemonade, paper cups and materials to make a sign. On their first day of business, they sell 20 cups of lemonade at 25 each, collecting a total of $5.00. The next day, the buy more lemonade mix for $1.50 and sell 36 cups of lemonade, collecting $9.00. An equation can be written to describe their sales and expenses: $ $5.00 $ $9.00 After subtracting their expanses, how much money did Joe and Thomas make in the two days their lemonade stand was open? 14 Adding Integers: A Simplified Approach READER 2.5

156 Just for Fun 3 Simplify: READER 2.5 Adding Integers: A Simplified Approach 15

157 Answers Page 4: 20 + ( 15) = 5 yards Page 14: $ $5.00 $ $9.00 $2.00 $ $ $9.00 $ $14.00 $10.50 Page 15: = = = Adding Integers: A Simplified Approach READER 2.5

158 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.6 Multiplication of Integers

159 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.6 Multiplication of Integers

160 contents Topics Juan the Wrestler 1 Multiplying Integers 1 3 Multiplying Integers 2 6 Just for Fun 1 8 Just for Fun 2 9 Multiplying Integers with Like Signs 10 Multiplying Integers with Unlike Signs 11 Just for Fun 3 13 Doing Mental Multiplication 15 Powers with Integers 17 Multiplying with Exponents 19 Just for Fun 4 23 Answers 24 UNIT 2 Integers and Expressions Page 2.6 Multiplication of Integers

161 Algebra Words and Phrases gain loss like signs simplify omit symbols group mentally regroup mentally Multiplication of Integers READER 2.6

162 Juan the Wrestler Juan is a wrestler. If he gains 3 pounds per month, how much weight will he gain in 4 months? READER 2.6 Multiplication of Integers

163 If he gains 3 pounds per month, Juan the wrestler will gain 12 pounds in 4 months: +3 Pounds per month (+4) (+3) Gain in 4 months (+4) (+3) = +12 Gained a total of 12 pounds 2 Multiplication of Integers READER 2.6

164 Multiplying Integers 1 When multiplying two positive integers, you get a positive product: Positive Positive = Positive Example: Multiply: 6 8 = 48 positive 7 11 = 77 positive 7 3 = 21 positive READER 2.6 Multiplication of Integers

165 Now suppose Juan needs to lose weight before the wrestling season? If Juan can lose 3 pounds per month, ( 3 pounds per month) how much will he lose in 4 months? Multiplication of Integers READER 2.6

166 To figure out how much weight he will lose, Juan multiplies: 4 months 3 pounds per month = 12 pounds Since there is a loss, the answer is negative. READER 2.6 Multiplication of Integers

167 Multiplying Integers 2 We can also change the order of the multipliers using the Commutative Property: 3 4 = 12 When a positive and a negative integer are multiplied in any order, the product is always a negative integer. Positive Negative = Negative Negative Positive = Negative 6 Multiplication of Integers READER 2.6

168 Multiply: 6 3 = = = = = = 20 READER 2.6 Multiplication of Integers

169 Just for Fun 1 Multiply: 12 5 = ( 3) = Multiplication of Integers READER 2.6

170 Just for Fun 2 Multiply: 7 4 = ( 8) = READER 2.6 Multiplication of Integers

171 Multiplying Integers with Like Signs When multiplying two integers with like signs (the same signs), the product is always positive: negative negative = positive 2 3 = +6 positive positive = positive = Multiplication of Integers READER 2.6

172 Multiplying Integers with Unlike Signs When multiplying two integers with the same sign, the product is positive and when multiplying two integers with different signs, the product is negative: negative negative = positive 2 3 = +6 positive positive = positive = +6 negative positive = negative 2 +3 = 6 positive negative = negative +2 3 = 6 READER 2.6 Multiplication of Integers 11

173 More examples: ( 3) ( 12) = +36 Like signs (two negatives) Negative negative = positive 8 ( 4) = 32 Different signs (negative and positive) Positive negative = negative 12 Multiplication of Integers READER 2.6

174 Just for Fun 3 Multiply: (12) ( 8) = ( 7) ( 8) = READER 2.6 Multiplication of Integers 13

175 Note: Use the commutative and associative properties to do quick mental multiplication. 14 Multiplication of Integers READER 2.6

176 Doing Mental Multiplication Here is an example of how to use the commutative and associative properties to do quick mental multiplication ( 5) 4 = ( 3 5) (25 4) = Regroup (put like signs together) (15) (100) = Multiply 1,500 READER 2.6 Multiplication of Integers 15

177 The following is another example of a problem you can easily figure out in your head by regrouping: ( 4) (6) ( 5) = ( 4 5) (6) = Regroup (20) (6) = Multiply 16 Multiplication of Integers READER 2.6

178 Powers with Integers A negative integer raised to an even power results in a positive integer. ( 3) 2 = 9 Multiply: ( 3) 2 = 3 3 = 9 READER 2.6 Multiplication of Integers 17

179 Multiply: ( 2) 4 = = 4 4 = Multiplication of Integers READER 2.6

180 Multiplying with Exponents Now for an explanation of how we work with exponents like this: ( 2)( 4) 2 READER 2.6 Multiplication of Integers 19

181 This is how we multiply with exponents. ( 2)( 4) 2 = ( 2)( 4)( 4) = ( 2)(16) = Multiplication of Integers READER 2.6

182 Remember that a positive number times a negative number is always negative. A negative integer raised to an odd power results in a negative integer. Expand: ( 2) 3 = ( 2 2) 2 = 4 2 = 2 READER 2.6 Multiplication of Integers 21

183 Expand: 2 3 = (2) 3 = (2 2) (2) = (4 2) = (8) = 8 22 Multiplication of Integers READER 2.6

184 Just for Fun 4 Expand: ( 3) 4 = Expand: ( 3)(6) READER 2.6 Multiplication of Integers 23

185 Answers Page 8: 12 5 = 60 6 ( 3) = 18 Page 9: 7 4 = 28 2 ( 8) = 16 Page 13: (12) ( 8) = 96 ( 7) ( 8) = 56 Page 23: 3 4 = ( ) = 81 ( 3)(6) 3 = ( 3)( 6)( 6)( 6) = ( 3)( 60) = Multiplication of Integers READER 2.6

186 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.7 Division of Integers

187 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.7 Division of Integers

188 contents Topics Algebra and the Weather 1 Division and Multiplication 4 Division with Like Signs 5 Just for Fun 1 7 Division with Unlike Signs 8 Just for Fun 2 10 Division of Fractions 11 Just for Fun 3 13 Just for Fun 4 14 Summary 15 Answers 16 Page UNIT 2 Integers and Expressions 2.7 Division of Integers

189 Algebra Words and Phrases consecutive mean average Celsius opposites expressed positive quotient negative quotient reciprocal Law of Exponents Division of Integers READER 2.7

190 Algebra and the Weather On four consecutive days, the temperature in a city was 5, 7, 2, and 6 degrees Celsius. What was the mean (average) temperature for the four days? January 15 January 16 January 17 January 18 5º C 7º C 2º C 6º C READER 2.7 Division of Integers

191 The mean (or average) temperature is found by adding the four temperatures and then dividing by 4 because there are four temperatures being averaged. 5 + ( 7) + ( 2) + ( 6) = = 5 The mean, or average, temperature is 5 degrees Celsius. 2 Division of Integers READER 2.7

192 This problem shows one of the rules for division with integers (whole numbers and their opposites). Notice that when dividing a positive number by a negative number the answer is a negative number. 5 + ( 7) + ( 2) + ( 6) 4 = 20 4 = 5 The mean, or average, temperature is 5 degrees Celsius. READER 2.7 Division of Integers

193 Division and Multiplication Division is based on the same rules and facts as multiplication. Division can be expressed in terms of multiplication: 8 2 means 8 1/2 The sign rules for multiplication also apply to division. Division of Integers READER 2.7

194 Division with Like Signs Division with like signs results in a positive quotient. 6 2 = = = 3 READER 2.7 Division of Integers

195 These problems illustrate that dividing integers with like signs results in a positive quotient = 3 16 ( 8) = 2 Division of Integers READER 2.7

196 Just for Fun 1 Divide: 50 2 = READER 2.7 Division of Integers 7

197 Division with Unlike Signs Division with unlike signs results in a negative quotient. 6 3 = = 2 Division of Integers READER 2.7

198 Note these examples: 8 ( 2) = = 2 The signs are different (one positive and one negative) so the quotient is negative. READER 2.7 Division of Integers

199 Just for Fun 2 Divide: = Division of Integers READER 2.7

200 These rules also apply to fractions. Division of Fractions = = = 2 Unlike Signs Multiply by the reciprocal Result is negative (A negative positive = negative) READER 2.7 Division of Integers 11

201 Use the Order of Operations to compute the equation below. The numbers in red give a hint as to what operations to compute first. [ 64 ( 2) 2 ] [ 8 ( 4)] = [ 64 4] [ 2] = Law of Exponents [ 16] [ 2] = Like signs 8 12 Division of Integers READER 2.7

202 Just for Fun 3 Use the Order of Operations to compute the equations below. 1. [32 ( 2) 2 ] [ 4 2] = [80 ( 2) 3 ] [ 8 4] = READER 2.7 Division of Integers 13

203 Just for Fun 4 Solve = = Division of Integers READER 2.7

204 Summary To divide integers: Like signs: positive positive = positive negative negative = positive Unlike signs: positive negative = negative negative positive = negative Can you restate this in your own words? READER 2.7 Division of Integers 15

205 Answers Page 7: 50 2 = 25 Page 10: 30 = 3 Page 13: 1. [32 ( 2) 2 ] [4 2] 2. [80 ( 2) 3 ] [8 4] 32 4 [4 2] 80 ( 8) [8 4] 8 2 = = 5 Page 14: = = = = = = Division of Integers READER 2.7

206 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.8 Evaluating Variable Expressions

207 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.8 Evaluating Variable Expressions

208 contents Topics Shopping at Quick Mart 1 Replacing Letters with Numbers 5 Just for Fun 1 6 Adding and Subtracting Integers 7 Just for Fun 2 9 Multiplying Integers 10 Just for Fun 3 12 Rewriting Subtraction Problems with Addition 13 Multiplying with Powers 15 Using the Distributive Property with Integers 18 Just for Fun 4 19 Answers 20 UNIT 2 Integers and Expressions Page 2.8 Evaluating Variable Expressions

209 Algebra Words and Phrases replace letters with numbers replace variables with numbers larger absolute value unlike signs negative product positive product rewriting subtraction problems with addition rewrite as a sum substitute The Distributive Property exponents factor Evaluating Variable Expressions READER 2.8

210 Shopping at Quick Mart Let's stop and get a soda! READER 2.8 Evaluating Variable Expressions

211 If the price of one six-pack of soda is $2.00 we can show it like this: = $2.00 Then the price of two six-packs of soda is 2 $2.00 and we can show it like this: = 2 $2.00 = $4.00 Then the price of three six-packs of soda is 3 $2.00 and we can show it like this: = 3 $2.00 = $ Evaluating Variable Expressions READER 2.8

212 If we let n = the number of six-packs, then n $2.00 = the cost of n six-packs To find out the cost of 5 six-packs, we evaluate. That means we replace the letter with a number (or value). In this example, the n is replaced with a 5. n $2.00 with n = 5 READER 2.8 Evaluating Variable Expressions

213 To find out the cost of 5 six-packs, we evaluate: Replace n with 5. $10.00 is the cost of 5 six-packs. n $2.00 with n = 5 5 $2.00 = $10.00 Evaluating Variable Expressions READER 2.8

214 Replacing Letters with Numbers As the previous example showed, when we evaluate variable expressions we replace a letter with a number. In other words... we replace a variable with a number. For example, if the question asks you to evaluate x = 3 in the expression 2 + x, we substitute 3 for x in the equation. 2 + x = = 5 READER 2.8 Evaluating Variable Expressions

215 Try one. Evaluate x = 4 for 7 x: Just for Fun 1 7 x = 3 11 Evaluating Variable Expressions READER 2.8

216 Adding and Subtracting Integers When you re adding or subtracting two integers, remember the following rules: For numbers with unlike signs, subtract and keep the sign of the larger number (larger absolute value) = 8 For numbers with like signs, simply add and keep the sign = ( 11) + ( 3) = 14 READER 2.8 Evaluating Variable Expressions

217 Here s how to evaluate x = 15 for the expression x 5: x 5 = ( 15) 5 = 20 Here s how to evaluate x = 8 for 2 + x: 2 + x = 2 + ( 8) = 6 8 Evaluating Variable Expressions READER 2.8

218 Just for Fun 2 Evaluate x = 3 for 5 + x. 5 + x = 8 2 READER 2.8 Evaluating Variable Expressions

219 Multiplying Integers When multiplying two integers, remember the following rules: Unlike signs result in a negative product. Example 1: 2 2 = 4 Example 2: (5) ( 10) = 50 Like signs result in a positive product. Example 1: 2 2 = 4 Example 2: (5) (10) = Evaluating Variable Expressions READER 2.8

220 Evaluate x = 2 for 4x. 4x = 4 ( 2) = 8 Evaluate x = 1 for 7 3x. 7 3x = 7 3 ( 1) = 7 + ( 3 1) = = 10 READER 2.8 Evaluating Variable Expressions 11

221 Just for Fun 3 Evaluate x = 2 for 15 2x. 15 2x = Evaluating Variable Expressions READER 2.8

222 Rewriting Subtraction Problems with Addition Some questions will ask you to evaluate a subtraction problem with negative variables. Sometimes it s easier to rewrite the problem as an addition problem before evaluating an expression for a negative variable. Here is an example. Evaluate for x = x = 3 + ( 12x) = Rewrite as a sum. 3 + [ 12 ( 3)] = Substitute. 3 + [36] = Multiply. 33 Answer READER 2.8 Evaluating Variable Expressions 13

223 Here is another example. Evaluate for a = 2, b = 1, c = 5. 5a 2b + 6c = 5a + ( 2b) + 6c = Rewrite as a sum. 5 ( 2) + [ 2 ( 1)] + 6 ( 5) = Substitute ( 30) = Multiply = Subtract 18 Answer 14 Evaluating Variable Expressions READER 2.8

224 Multiplying with Powers Evaluate for x = 3. 5x 4 = 5( 3) 4 = Substitute for x. 5( 3)( 3)( 3)( 3) = 5(81) = Multiply 405 Answer READER 2.8 Evaluating Variable Expressions 15

225 When you have variables with exponents, the negative or positive sign in front of the variable stays as a factor. Example: Evaluate for x = 1. x 3 = x x x = ( 1) ( 1) ( 1) = 1 16 Evaluating Variable Expressions READER 2.8

226 Evaluate for x = 2. 3 x 4 = 3 ( 2) 4 = 3 [( 2)( 2)( 2)( 2)] = 3 [16] = 13 READER 2.8 Evaluating Variable Expressions 17

227 Using The Distributive Property with Integers Use the distributive property to simplify an expression before evaluating. Evaluate for x = 1, y = 2. 3(4x + 3y) + 2x = (3)(4x) + (3)(3y) + 2x = Distribute the 3 first. 12x + 9y + 2x = Multiply. (12 + 2)x + 9y = Combine like terms. 14x + 9y = Combine like terms. 14( 1) + 9( 2) = Substitute for x and y ( 18) = Multiply. 32 = Answer. 18 Evaluating Variable Expressions READER 2.8

228 1. Evaluate for x = 2, y = 5. Just for Fun 4 3x 2y = Evaluate for x = 2, y = 1. 5(3x + y) y = READER 2.8 Evaluating Variable Expressions 19

229 Answers Page 6: 7 x = 7 4 = 3 Page 9: 5 + x = 5 + ( 3) = 2 Page 12: 15 2x = 15 2( 2) = = 19 Page 19: 1. 3x 2y = 3( 2) 2( 5) = = (3x + y) y = 5[3 2 + ( 1)] ( 1) = 5[6 1] + 1 = 5[5] + 1 = = Evaluating Variable Expressions READER 2.8

230 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.9 Simplifying Expressions By Combining Like Terms

231 preface About Family Literacy Algebra Prep Readers These readers are not intended to be a complete course in algebra, but rather a simple introduction to the vocabulary used in a traditional, more advanced math class, thus offering an advantage for students who will take a course in algebra later on. They are intended to provide practice for anyone wishing to: (1) sharpen reading skills, (2) sharpen concentration and analysis skills, and (3) build a vocabulary in mathematical terms commonly used in more advanced math classes. The stories in each book relate the unknown and unfamiliar to the known and the familiar. Real life experiences are depicted where math can help and concepts can be made clearer. There are multiple books in this series with the concepts and vocabulary of one leading to the next. The books labeled 2.0, 3.0, and so on, are review storybooks which summarize and very quickly put previous unit story concepts and vocabulary together. Lesson 1.0, however, is an introductory lesson on "sets," because there are no previous lessons to review. Students and tutors who become proficient in the vocabulary introduced in these storybooks may want to experience the more involved Algebra course offered by Family Literacy Centers, Inc. which can be accessed online and in newsprint text books. Contact Family Literacy Centers, Inc. for more information at flcinc.org. UNIT 2 Integers and Expressions 2.9 Simplifying Expressions By Combining Like Terms

232 contents Topics Page Algebra and Trading Cards 1 Combining Like Terms with Integers 8 Using The Distributive Property 9 Just for Fun 1 10 Applying the Distributive Property Mentally 11 Just for Fun 2 13 Simplifying with Integer Coefficients 14 Multiplication Properties of 1 and Just for Fun 3 19 Answers 20 UNIT 2 Integers and Expressions 2.9 Simplifying Expressions By Combining Like Terms

233 Algebra Words and Phrases grouped separated sorted organized integer coefficients The Distributive Property combine like terms simplify evaluate Multiplication Properties of 1 and 1 Simplifying Expressions By Combining Like Terms READER 2.9

234 Algebra and Trading Cards Joe went to a yard sale and bought a box of trading cards. READER 2.9 Simplifying Expressions By Combining Like Terms

235 He brought the cards home and found that all of them were from the National Football League, National Basketball Association, National Hockey League, and Major League Baseball seasons. 2 Simplifying Expressions By Combining Like Terms READER 2.9

236 Joe has many options for organizing and displaying his new card collection. READER 2.9 Simplifying Expressions By Combining Like Terms

237 He could separate the cards by brand (the company who made the cards.) Simplifying Expressions By Combining Like Terms READER 2.9

238 Joe could also sort them according to year READER 2.9 Simplifying Expressions By Combining Like Terms

239 The cards could also be grouped by sport. Football Hockey Baseball Basketball Simplifying Expressions By Combining Like Terms READER 2.9

240 There are many other ways Joe can organize his trading cards. Each method of sorting serves a specific purpose. Remember, however, that whether the cards are grouped, separated, sorted, or organized, the actual cards remain the same. READER 2.9 Simplifying Expressions By Combining Like Terms

241 Combining Like Terms with Integers We combine like terms with integer coefficients in the same way that we combine like terms with whole number coefficients. Simplify the following expression using the Distributive Property. 2x + 7x = ( 2 + 7)x = We group like terms using the Distributive Property. 5x The coefficients are combined. Simplifying Expressions By Combining Like Terms READER 2.9

242 Using The Distributive Property Here is how we simplify an algebraic expression using the distributive property: 3y 7y = We group like terms using the Distributive Property. (3 7)y = The coefficients are combined. 4y READER 2.9 Simplifying Expressions By Combining Like Terms 9

243 Just for Fun 1 Use the Distributive Property to combine like terms. 11m + 2m = 13m 9m 9m 9 10 Simplifying Expressions By Combining Like Terms READER 2.9

244 Applying the Distributive Property Mentally When combining like terms, do the distributive step mentally as soon as it can be done without making errors. Watch as the distributive property is done mentally. 5x 4x = The Distributive Property is followed mentally. ( 5 4)x = 9x READER 2.9 Simplifying Expressions By Combining Like Terms 11

245 Here is another example of the distributive property used mentally. 2a 7a + 3a = The Distributive Property is followed mentally. ( )a = Combine like terms. 2a Answer 12 Simplifying Expressions By Combining Like Terms READER 2.9

246 Just for Fun 2 Combine like terms. 13a + 6a 7a 2 7a 7a 19a 5m 9m 4m m 4m READER 2.9 Simplifying Expressions By Combining Like Terms 13

247 Simplifying with Integer Coefficients To simplify an algebraic expression when the terms have integer coefficients: First group like terms; then combine coefficients. Simplify by grouping like terms. 3x + 2y x 8y = Group like terms ( 3 + 5)(x) + (2 8)(y) + 4 = Use the Distributive Property. (+2x) + ( 6y) + 4 = Combine the coefficients. 2x 6y Simplifying Expressions By Combining Like Terms READER 2.9

248 Simplify; then evaluate for a = 2, b = 3. Remember to group like terms before replacing the variable with the number. 2a 8 + 5a 12b + 2 6a 2b = 2a + 5a 6a 12b 2b = a 14b 6 = 2 14(3) 6 = = 38 READER 2.9 Simplifying Expressions By Combining Like Terms 15

249 Multiplication Property of 1: Multiplication Properties of 1 and 1 For each integer x, 1 x = x and x 1 = x Multiplication Property of 1: For each integer x, 1 x = x and x 1 = x 16 Simplifying Expressions By Combining Like Terms READER 2.9

250 Simplify: 4x 2y 3x + y = 4x 3x 2y + y = 1x 1y = x y 1x = x, 1y = y READER 2.9 Simplifying Expressions By Combining Like Terms 17

251 Simplify; then evaluate for x = 3, y = 4. 2x y + 8x = 2x + 8x y = 6x 1y = 6( 3) 1( 4) = = Simplifying Expressions By Combining Like Terms READER 2.9

252 Simplify; then evaluate for j = 8, k = -8. Just for Fun 3 3j 2k + 2j + k READER 2.9 Simplifying Expressions By Combining Like Terms 19

253 Answers Page 10: 11m + 2m = ( )m = 9m Page 13: 13a + 6a = ( )a = 7a 5m 9m = (5 9)m = 4m Page 19: 3j 2k + 2j + k = 3j + 2j 2k + k = 5j k = 5(8) ( 8) = = Simplifying Expressions By Combining Like Terms READER 2.9

254 Family Literacy readers Easy Algebra Reading Practice UNIT 2 Integers and Expressions 2.10 Simplifying Expressions Containing Parentheses