2. Welcome to Interactive Chalkboard Mathematics: Applications and Concepts, Course 3 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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4. Contents Lesson 1-1A Plan for Problem Solving Lesson 1-2Variables, Expressions, and Properties Lesson 1-3Integers and Absolute Value Lesson 1-4Adding Integers Lesson 1-5Subtracting Integers Lesson 1-6Multiplying and Dividing Integers Lesson 1-7Writing Expressions and Equations Lesson 1-8Solving Addition and Subtraction Equations Lesson 1-9Solving Multiplication and Division Equations

5. Lesson 1 Contents Example 1Use the Four-Step Plan Example 2Use the Four-Step Plan

6. Example 1-1a HOME IMPROVEMENT The Vorhees family plans to paint the wall in their family room. They need to cover 512 square feet with two coats of paint. If a one-gallon can of paint covers 220 square feet, how many one- gallon cans of paint should they purchase? Explore Since they will be using two coats of paint, we must double the area to be painted. Plan They will be covering square feet or 1,024 square feet. Next, divide 1,024 by 220 to determine how many cans of paint are needed.

7. Example 1-1a Solve Examine Since they will purchase a whole number of cans of paint, round 4.7 to 5. Answer: They will need to purchase 5 cans of paint.

8. Example 1-1b HOME IMPROVEMENT Jocelyn plans to paint her bedroom. She needs to cover 400 square feet with three coats of paint. If a one-gallon can of paint covers 350 square feet, how many one-gallon cans of paint should she purchase? Answer: 4

9. Example 1-2a GEOGRAPHY Study the table. The five largest states in total area, which includes land and water, are shown. Of the five states shown, which one has the smallest area of water? Largest States in Area StateLand Area (mi 2 ) Total Area (mi 2 ) Alaska570,374615,230 Texas261,914267,277 California155,973158,869 Montana145,556147,046 New Mexico121,364121,598 Source: U.S. Census Bureau

10. Example 1-2a Explore What do you know? You are given the total area and the land area for five states. What are you trying to find? You need to find the water area. Plan To determine the water area, subtract the land area from the total area for each state.

11. Example 1-2a Solve Examine Compare the water area for each state to determine which state has the least water area. Answer: New Mexico has the least water area with 234 square miles.

12. Example 1-2b GEOGRAPHY Study the table. The five smallest states in total area, which includes land and water, are shown. Of the five states shown, which one has the smallest area of water? Smallest States in Area StateLand Area (mi 2 ) Total Area (mi 2 ) Rhode Island1,0451,212 Delaware1,9451,983 Connecticut4,8454,872 Hawaii6,4236,471 New Jersey7,4177,787 Answer: Connecticut

13. End of Lesson 1

14. Lesson 2 Contents Example 1Evaluate a Numerical Expression Example 2Evaluate Algebraic Expressions Example 3Evaluate Algebraic Expressions Example 4Identify Properties Example 5Find a Counterexample

15. Example 2-1a Evaluate Divide inside parentheses first. Multiply next. Add and subtract in order from left to right. Answer: 4

16. Example 2-1b Evaluate Answer: 2

17. Example 2-2a Replace r with 6 and s with 3. Do all multiplications first. Add and subtract in order from left to right. Answer: 20 Evaluate the expression

18. Example 2-2b Answer: 24 Evaluate the expression

19. Example 2-3a The fraction bar is a grouping symbol. Evaluate the expressions in the numerator and denominator separately before dividing. Replace q with 5 and r with 6. Do all multiplications first. Subtract in the denominator. Then divide. Answer: 2 Evaluate the expression

20. Example 2-3b Answer: 2 Evaluate the expression

21. Example 2-4a Name the property shown by Multiplying by 1 does not change the number. Answer: This is the Multiplicative Identity.

22. Example 2-4b Name the property shown by the statement Answer: Commutative Property of Multiplication

23. Example 2-5a State whether the following conjecture is true or false. If false, provide a counterexample. The sum of an odd number and an even number is always odd. Answer: This conjecture is true.

24. Example 2-5b State whether the following conjecture is true or false. If false, provide a counterexample. Division of whole numbers is associative. Answer: false;

25. End of Lesson 2

26. Lesson 3 Contents Example 1Write Integers for Real-Life Situations Example 2Write Integers for Real-Life Situations Example 3Compare Two Integers Example 4Compare Two Integers Example 5Order Integers Example 6Expressions with Absolute Value Example 7Expressions with Absolute Value Example 8Expressions with Absolute Value

27. Example 3-1a Write an integer to represent a discount. Answer: The integer is –35.

28. Example 3-1b Write an integer to represent a tax. Answer: 25

29. Example 3-2a Write an integer for the expression a fever of 4 degrees above normal. Answer: The integer is 4.

30. Example 3-2b Write an integer for the expression a loss of 10 yards in football. Answer: –10

31. Example 3-3a Replace  with, or = to make –3  3 a true sentence. Use the integers graphed on the number line below. –3 is less than 3, since it lies to the left of 3. Answer: –3 < 3

32. Example 3-3b Answer: –2 < 2 Replace  with, or to make –2  2 a true sentence. Use the integers graphed on the number line below.

33. Example 3-4a –2 is less than –1 since it lies to the left of –1. Answer: –2 < –1 Replace  with, or to make –2  –1 a true sentence. Use the integers graphed on the number line below.

34. Example 3-4b Answer: –4 > –6 Replace  with, or to make –4  –6 a true sentence. Use the integers graphed on the number line below.

35. Example 3-5a FOOTBALL The table below shows the number of yards rushing for several players on a football team during one game. Order these statistics from least to greatest. Player Yards Rushing Marty 8 Autry 4 Shane 5 Brad –10 Glyn 3 Jim –19

36. Example 3-5a Write the numbers as they appear from left to right. Answer: The number of yards rushing are –19, –10, 3, 4, 5, and 8 from least to greatest. Graph each integer on a number line. –19, –10, 3, 4, 5, 8

37. Example 3-5b WEATHER The table below shows the temperatures for several cities on January 31, 2000. Order these statistics from least to greatest. CityTemp. on Jan. 31, 2000 (°F) Kahului, HI83 Homestead, FL81 Buckland, AK–54 W. Yellowstone, MT–30 Source: www.usatoday.com/weather/wext0001 Answer:

38. Example 3-6a Evaluate. Answer: The graph of 5 is 5 units from 0 on the number line.

39. Example 3-6b Answer: 3 Evaluate.

40. Example 3-7a Evaluate. The absolute value of 6 is 6. The absolute value of –5 is 5. Simplify. Answer: 1

41. Example 3-7b Evaluate. Answer: 3

42. Example 3-8a Evaluate Answer: 17 Replace x with –4. Simplify.

43. Example 3-8b Evaluate Answer: 9

44. End of Lesson 3

45. Lesson 4 Contents Example 1Add Integers with the Same Sign Example 2Add Integers with the Same Sign Example 3Add Integers with Different Signs Example 4Add Integers with Different Signs Example 5Add Integers with Different Signs Example 6Add Three or More Integers Example 7Add Three or More Integers Example 8Use Integers to Solve a Problem

46. Example 4-1a Find Method 1 Use a number line. Move 8 units to the left. From there, move 4 units left. Start at zero.

47. Example 4-1a Method 2 Use counters. Answer: So,

48. Example 4-1b Find Answer: –9

49. Example 4-2a Add and Find Both numbers are negative, so the sum in negative. Answer: –26

50. Example 4-2b Find Answer: –25

51. Example 4-3a Find Method 1 Use a number line. Move 4 units right. From there, move 6 units left. Start at zero.

52. Example 4-3a Method 2 Use counters. Answer: So,

53. Example 4-3b Find Answer: –2

54. Example 4-4a Find Method 1 Use a number line. Move 5 units left. From there, move 9 units right. Start at zero.

55. Example 4-4a Method 2 Use counters. Answer: So,

56. Example 4-4b Find Answer: 2

57. Example 4-5a To find subtract from Find The sum is positive because > Answer: 17

58. Example 4-5b Find Answer: 10

59. Example 4-6a Find Associative Property Simplify. Answer: –6

60. Example 4-6b Find Answer: –5

61. Example 4-7a Find Commutative Property Associative Property Simplify. Answer: 26

62. Example 4-7b Find Answer: 12

63. Example 4-8a STOCKS An investor owns 50 shares in a video game manufacturer. A broker purchases 30 shares more for the client on Tuesday. On Friday, the investor asks the broker to sell 65 shares. How many shares of this stock will the client own after these trades are completed? Selling a stock decreases the number of shares, so the integer for selling is –65. Purchasing new stock increases the number of shares, so the integer for buying is +30. Add these integers to the starting number of shares to find the new number of shares.

64. Example 4-8a Associative Property Simplify. Answer: The number of shares is 15.

65. Example 4-8b MONEY Jaime gets an allowance of $5. She spends $2 on video games and $1 on lunch. Her best friend repays a $2 loan and she buys a $3 pair of socks. How much money does Jaime have left? Answer: $1

66. End of Lesson 4

67. Lesson 5 Contents Example 1Subtract a Positive Integer Example 2Subtract a Positive Integer Example 3Subtract a Negative Integer Example 4Subtract a Negative Integer Example 5Evaluate Algebraic Expressions Example 6Evaluate Algebraic Expressions

68. Example 5-1a Find To subtract 6, add Add. Answer: –4

69. Example 5-1b Find Answer: –4

70. Example 5-2a Find Add. Answer: –12 To subtract 5, add

71. Example 5-2b Find Answer: –8

72. Example 5-3a Find Add. Answer: 19 To subtract –8, add 8.

73. Example 5-3b Find Answer: 18

74. Example 5-4a Find Add. Answer: 3 To subtract –9, add 9.

75. Example 5-4b Answer: 4 Find

76. Example 5-5a To subtract –7 add 7. Answer: 19 Replace r with –7. Add. Evaluate

77. Example 5-5b Answer: 12 Evaluate

78. Example 5-6a Answer: –9 Replace q with –3 and p with 6. Add. To subtract 6, add Evaluate

79. Example 5-6b Answer: –9 Evaluate

80. End of Lesson 5

81. Lesson 6 Contents Example 1Multiply Integers with Different Signs Example 2Multiply Integers with Different Signs Example 3Multiply Integers with the Same Sign Example 4Multiply More than Two Integers Example 5Divide Integers Example 6Divide Integers Example 7Evaluate Algebraic Expressions Example 8Find the Mean of a Set of Integers

82. Example 6-1a Find Answer: –32 The factors have different signs. The product is negative.

83. Example 6-1b Find Answer: –18

84. Example 6-2a Find Answer: –35 The factors have different signs. The product is negative.

85. Example 6-2b Find Answer: –12

86. Example 6-3a Find Answer: 144 The factors have the same sign. The product is positive.

87. Example 6-3b Find Answer: 64

88. Example 6-4a Find Answer: 48 Associative Property

89. Example 6-4b Find Answer: 30

90. Example 6-5a Find Answer: –6 The dividend and the divisor have different signs. The quotient is negative.

91. Example 6-5b Find Answer: –6

92. Example 6-6a Find Answer: –3 The dividend and the divisor have different signs. The quotient is negative.

93. Example 6-6b Find Answer: 6

94. Example 6-7a Evaluate The product of 3 and –10 is negative and the product of –4 and –4 is positive. Answer: –46 Replace x with –10 and y with –4. To subtract 16, add –16. Add.

95. Example 6-7b Evaluate Answer: –24

96. Example 6-8a WEATHER The table shows the low temperature for each month in McGrath, Alaska. Find the mean (average) of all 12 temperatures. Average Low Temperatures MonthTemperature (°C) January–27 February–26 March–19 April–9 May1 June7 July9 August7 September2 October–8 November–19 December–26 Source: weather.com

97. Example 6-8a To find the mean of a set of numbers, find the sum of the numbers. Then divide the result by how many numbers there are in the set. Answer: McGrath, Alaska has an average low temperature of for the year.

98. Example 6-8b WEATHER The table shows the record low temperature for each month in Brook Park, Ohio. Find the mean (average) of all 12 temperatures. Record Low Temperatures MonthTemperature (°F) January–20 February–15 March–5 April10 May25 June31 July41 August38 September34 October19 November3 December–15 Source: weather.com Answer: about

99. End of Lesson 6

100. Lesson 7 Contents Example 1Write an Algebraic Expression Example 2Write an Algebraic Expression Example 3Write an Algebraic Equation Example 4Write an Equation to Solve a Problem

101. Example 7-1a Words Variable Expression Write double the price of a loaf of bread as an algebraic expression. double the price of a loaf of bread Let p represent the price of a loaf of bread. double the price of a loaf of bread 2  p Answer: The expression is 2p.

102. Example 7-1b Write triple the cost of a cup of coffee as an algebraic expression. Answer: 3c

103. Example 7-2a Words Variable Write stalks of celery are divided into 4 portions as an algebraic expression. Stalks of celery are divided into 4 portions. Let c represent the stalks of celery. Answer: The expression is Stalks of celery are divided into 4 portions. c 4 Equation

104. Example 7-2b Write pies are divided into 8 portions as an algebraic expression. Answer:

105. Example 7-3a Words Variable Write the price of a book plus $5 shipping is $29 as an algebraic equation. The price of a book plus $5 shipping is $29. Let p represent the price of the book. Equation Answer: The equation is $29. 29 is The price of a book p 5 plus $5 shipping

106. Example 7-3b Write the price of a toy plus $6 shipping is $35 as an algebraic equation. Answer:

107. Example 7-4a NUTRITION A particular box of oatmeal contains 10 individual packages. If the box contains 30 grams of fiber, write an equation to find the amount of fiber in one package of oatmeal. Words Variable Ten packages of oatmeal contain 30 grams of fiber. Let f represent the grams of fiber per package. Answer: The equation is Ten packages of oatmeal Equation contain 30 grams of fiber. 3010f

108. Example 7-4b NUTRITION A particular box of cookies contains 10 servings. If the box contains 1,200 Calories, write an equation to find the number of Calories in one serving of cookies. Answer:

109. End of Lesson 7

110. Lesson 8 Contents Example 1Solve an Addition Equation Example 2Solve a Subtraction Equation Example 3Write and Solve an Equation

111. Method 1 Vertical Method Example 8-1a Solve Check your solution. Write the equation. c is by itself. Subtract 15 from each side.

112. Example 8-1a Write the equation. Subtract 15 from each side. Method 2 Horizontal Method c is by itself.

113. Example 8-1a Write the original equation. Replace c with –8. Is this sentence true? Check The sentence is true. Answer: The solution is –8.

114. Example 8-1b Solve Check your solution. Answer: –5

115. Example 8-2a Method 1 Vertical Method Write the equation. Add 16 to each side. z is by itself. Solve

116. Example 8-2a Method 2 Horizontal Method Write the equation. Add 16 to each side. z is by itself. Answer: The solution is 11.

117. Example 8-2b Answer: 6 Solve

118. Example 8-3a MULTIPLE- CHOICE TEST ITEM What value of y makes the difference of 30 and y equal to 12? A –42 B –18 C 18 D 28 Read the Test Item To find the value of y, write and solve an equation.

119. Example 8-3a Solve the Test Item The difference of 30 and y equals12. 30 – y12 Answer: C Write the equation. Add y to each side. Subtract 12 from each side.

120. Example 8-3b MULTIPLE- CHOICE TEST ITEM What value of x makes the difference of 30 and x equal to 5? A 15 B –15 C 25 D 10 Answer: C

121. End of Lesson 8

122. Lesson 9 Contents Example 1Solve a Multiplication Equation Example 2Solve a Division Equation Example 3Use an Equation to Solve a Problem

123. Example 9-1a Solve Write the equation. Divide each side of the equation by 7. and Identity Property; Answer: The solution is –7.

124. Example 9-1b Answer: –8 Solve

125. Example 9-2a Write the equation. Answer: The solution is –54. Solve Multiply each side by 9 to undo the division in −6(9) = 54

126. Example 9-2b Solve Answer: –50

127. Example 9-3a SURVEYING English mathematician Edmund Gunter lived around 1600. He invented the chain, which was used to measure land for maps and deeds. One chain equals 66 feet. If the south side of a property measures 330 feet, how many chains long is it? Words Variable Equation One chain equals 66 feet. Let the number of chains in 330 feet. Measurement of property 330 = 66c 66 times the number of chains. is

128. Example 9-3a Write the equation. Divide each side by 66. Answer: The number of chains in 330 feet is 5.

129. Example 9-3b HORSES Most horses are measured in hands. One hand equals 4 inches. If a horse measures 60 inches, how many hands is it? Answer: 15 hands

130. End of Lesson 9

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