1. 1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 How many weeks in 363 days? 51 weeks 6767

2. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Complete each equation. 1.1 week = days2.1 foot = inches 3.1 nickel = cents4.1 gallon = quarts 5.1 yard = feet Check Skills You’ll Need 1-1

3. Solutions 1.1 week = days2.1 foot = inches 3.1 nickel = cents4.1 gallon = quarts 5.1 yard = feet Variables and Expressions PRE-ALGEBRA LESSON 1-1 712 54 3 1-1

4. Identify each expression as a numerical expression or a variable expression. For a variable expression, name the variable. Variables and Expressions PRE-ALGEBRA LESSON 1-1 numerical expression variable expression; t is the variable. a. 7  3 b. 4t Quick Check 1-1

5. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Write a variable expression for the cost of p pens priced at 29¢ each. = number of pens. Words times Let 29¢ number of pens p The variable expression 29 p, or 29p, describes the cost of p pens. Expression29 p Quick Check 1-1

6. Variables and Expressions PRE-ALGEBRA LESSON 1-1 Write a variable expression for each word phrase. 1.the total of h and 56 2.three less than d 3.p decreased by three4.a divided by 7 p – 3 h + 56d – 3 a ÷ 7 1-1

7. 1-2 The Order of Operations PRE-ALGEBRA LESSON 1-2 Divide. Round answers to the nearest tenth. a. 75 ÷ 8 b. 740 ÷ 7 9.4 105.7

8. The Order of Operations PRE-ALGEBRA LESSON 1-2 (For help, go to Skills Handbook p. 760.) Find each quotient. 1.164 ÷ 22.344 ÷ 83.284 ÷ 4 4.133 ÷ 75.182 ÷ 136.650 ÷ 25 Check Skills You’ll Need 1-2

9. The Order of Operations PRE-ALGEBRA LESSON 1-2 Solutions 1.Estimate:2.Estimate:3.Estimate: 160 ÷ 2 = 80360 ÷ 8 = 45280 ÷ 4 = 70 82 43 71 2 1648 3444 284 –16 –32 –28 4 24 4 – 4 – 24 – 4 0 0 0 4.Estimate:5.Estimate:6.Estimate: 140 ÷ 7 = 20195 ÷ 13 = 15625 ÷ 25 = 25 19 14 26 7 13313 18225 650 –7 –13 –50 63 52 150 –63 – 52 –150 0 0 0 1-2

10. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 8 – 2 2. 8 – 2 2 8 – 4 First multiply. Then subtract. 4 Quick Check 1-2

11. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 12 ÷ 3 – 1 2 + 1. 12 ÷ 3 – 1 2 + 1 Add and subtract from left to right. 2 + 1 4 – 2 + 1 Multiply and divide from left to right. 3 Add. Quick Check 1-2

12. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify 20 – 3[(5 + 2) – 1]. 20 – 3[(5 + 2) – 1] 20 – 3[ 7 – 1] Add within parentheses. Subtract within brackets. 20 – 3 [6] 20 – 18 Multiply from left to right. Subtract. 2 Quick Check 1-2

13. The Order of Operations PRE-ALGEBRA LESSON 1-2 Simplify each expression. 1.7(3) – 2 4 2.6 ÷ 2 + 1 5 3.10 ÷ (4 + 1)4.3[9 2 ÷ (10 – 4)] 2 138 9 1-2

14. 1-3 Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate each expression. a. 8 + 6 – 2 b. 16 – 8 + 2 c. 12 + 9 2 12 10 30

15. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 (For help, go to Lesson 1-2.) Simplify each expression. 1.6(9 + 1)2.17 – 2 + 3 3.9 + 8 2 + 44.[3(5) + 1] 2 Check Skills You’ll Need 1-3

16. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 Solutions 1. 6(9 + 1) 2.17 – 2 + 3 6(10) 15 + 3 60 18 3.9 + 8 2 + 4 4.[3(5) + 1] 2 9 + 16 + 4 [15 + 1] 2 29 16 2 32 1-3

17. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate 18 + 2g for g = 3. 18 + 2g = 18 + 2(3)Replace g with 3. = 18 + 6Multiply. = 24Add. Quick Check 1-3

18. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate 2ab – for a = 3, b = 4, and c = 9. = 2 3 4 – 3Work within grouping symbols. = 6 4 – 3Multiply from left to right. = 24 – 3Multiply. = 21 Subtract. c3c3 2ab – = 2 3 4 –Replace the variables. c3c3 9393 1-3 Quick Check

19. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 The Omelet Café buys cartons of 36 eggs. a.Write a variable expression for the number of cartons the Café should buy for x eggs. b.Evaluate the expression for 180 eggs. a. x eggs b. 180 eggs x 36 = Evaluate for x = 180. 180 36 x 36 = 5Divide. The Omelet Café should buy 5 cartons to get 180 eggs. 1-3 Quick Check

20. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 The One Pizza restaurant makes only one kind of pizza, which costs $16. The delivery charge is $2. Write a variable expression for the cost of having pizzas delivered. Evaluate the expression to find the cost of having two pizzas delivered. $2 delivery charge = number of pizzas. Words for each Let $16 p pizzaplus Evaluate the expression for p = 2. 16 p + 2 = 16 2 + 2 = 32 + 2 = 34 It costs $34 to have two pizzas delivered. Expression 16 p2+ 1-3 Quick Check

21. Writing and Evaluating Expressions PRE-ALGEBRA LESSON 1-3 Evaluate. 1.7(b) – 4 for b = 3 2.h ÷ 2 + 1 for h = 12 3.3c + 4 ÷ d for c = 8, d = 24.fg – g for f = 5, g = 7 26 177 28 1-3

22. 1-4 Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Order from least to greatest:, –1, 0.5, – 0.2, 2323 1818 – 1, – 0.2,, 0.5, 1818 2323

23. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 (For help, go to Skills Handbook p. 775.) Write an integer for each situation. 1.lose $72.find $9 3.8 steps forward4.3 yards gained 5.5 floors down Check Skills You’ll Need 1-4

24. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Solutions 1.lose $72.find $9 –79 3.8 steps forward4.3 yards gained 83 5.5 floors down –5 1-4

25. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Write a number to represent the temperature shown by the thermometer. The thermometer shows 2 degrees Celsius below zero, or –2°C. Quick Check 1-4

26. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Graph 2, –2, and –3 on a number line. Order the numbers from least to greatest. The numbers from least to greatest are –3, –2, and 2. 1-4 Quick Check

27. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Use a number line to find |–5| and |5|. |–5| = 5 |5| = 5 1-4 Quick Check

28. Integers and Absolute Value PRE-ALGEBRA LESSON 1-4 Write an integer to represent each situation. 1.A debt of $50 2.A dive of 23 feet below the surface Simplify. 3.| –12 |4.| 8 | 12 –50 –23 8 1-4

29. 1-5 Adding Integers PRE-ALGEBRA LESSON 1-5 List each set of fractions in order from greatest to least. a.,,b.,,c.,, 5656 9 10 3535 1616 2323 2727 1414 1313 1515 2727 1313 1414 1515 2727 2323 1616, 9 10 5656 3535,,

30. Adding Integers PRE-ALGEBRA LESSON 1-5 (For help, go to Lesson 1-4.) Compare. Use >, <, or = to complete each statement. 1.–6 –32.2 –15 3.–5 | 5 |4.| 10 | | –10 | 5.| 9 | | –2 |6.| –8 | | 0 | Check Skills You’ll Need 1-5

31. PRE-ALGEBRA LESSON 1-5 Solutions 1.–6 –15 3.–5 < | 5 |4.| 10 | = | –10 | 5.| 9 | > | –2 |6.| –8 | > | 0 | 1-5 Adding Integers

32. (–7) + 3Model the sum. – 4Group and remove zero pairs. There are four negative tiles left. Adding Integers PRE-ALGEBRA LESSON 1-5 Use tiles to find (–7) + 3. (–7) + 3 = – 4 Quick Check 1-5

33. Adding Integers PRE-ALGEBRA LESSON 1-5 From the surface, a diver goes down 20 feet and then comes back up 4 feet. Find –20 + 4 to find where the diver is. The diver is 16 feet below the surface. –20 + 4 = –16 Start at 0. To represent –20, move left 20 units. To add positive 4, move right 4 units to –16. 1-5 Quick Check

34. Adding Integers PRE-ALGEBRA LESSON 1-5 Find each sum. b. 13 + (–17) a. –20 + (–15) Since –17 has the greater absolute value, the sum is negative. 13 + (–17) = – 4 Since both integers are negative, the sum is negative. –20 + (–15) = –35 Find the difference of the absolute values. |–17| – |13| = 17 – 13= 4 Simplify. 1-5 Quick Check

35. Adding Integers PRE-ALGEBRA LESSON 1-5 A player scores 22 points. He then gets a penalty of 30 points. What is the player’s score after the penalty? Write an expression. 22 + (–30) Find the difference of the absolute values. |–30| – |22| = 30 – 22 Since –30 has the greater absolute value, the sum is negative. 22 + (–30) = – 8 The player’s score is – 8. = 8 Simplify. 1-5 Quick Check

36. Adding Integers PRE-ALGEBRA LESSON 1-5 Find –7 + (– 4) + 13 + (–5). Add from left to right. –7 + (– 4) + 13 + (–5) –7 + (– 4) + 13 + (–5) = –3 –3 |5| – |2| = 3. Since –5 has the greater absolute value, the sum is negative. The sum of the two negative integers is negative. –11 + 13 + (–5) |13| – |11| = 2. Since 13 has the greater absolute value, the sum is positive. 2 + (–5) 1-5 Quick Check

37. Find each sum. 1.–37 + (–5)2.14 + (–4) 3.–100 + 5 + (–3)4.Evaluate 33 + t for t = –11. Adding Integers PRE-ALGEBRA LESSON 1-5 –98 –4210 22 1-5

38. 1-6 Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find all possible number pairs for which the sum of two whole numbers is 63 and the difference between the numbers is less than 10. 31 and 32; 30 and 33; 29 and 34; 28 and 35; 27 and 36

39. Subtracting Integers PRE-ALGEBRA LESSON 1-6 (For help, go to Lesson 1-5.) Find each sum. 1.8 + (–9)2.–11 + (–18)3.–4 + (–6) 4.14 + (–3)5.6 + (–6)6.–13 + (–10) Check Skills You’ll Need 1-6

40. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Solutions 1.8 + (–9)4.14 + (–3) | –9 | – | 8 | = 9 – 8| 14 | – | –3 | = 14 – 3 = 1 = 11 8 + (–9) = –1 14 + (–3) = 11 2.–11 + (–18)5.6 + (–6) –11 + (–18) = –29| –6 | – | 6 | = 6 – 6 = 0 6 + (–6) = 0 3.–4 + (–6)6.–13 + (–10) –4 + (–6) = –10–13 + (–10) = –23 1-6

41. Start with 7 negative tiles. Take away 5 negative tiles. There are 2 negative tiles left. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find –7 – (–5). –7 – (–5) = –2 Quick Check 1-6

42. Start with 2 positive tiles. There are not enough positive tiles to take away 8. Add 6 zero pairs. Take away 8 positive tiles. There are 6 negative tiles left. Subtracting Integers PRE-ALGEBRA LESSON 1-6 Find 2 – 8. 2 – 8 = –6 1-6 Quick Check

43. Subtracting Integers PRE-ALGEBRA LESSON 1-6 An airplane left Houston, Texas, where the temperature was 42°F. When the airplane landed in Anchorage, Alaska, the temperature was 50°F lower. What was the temperature in Anchorage? The temperature in Anchorage was –8°F. Write an expression. 42 – 50 To subtract 50, add its opposite. 42 – 50= 42 + (–50) = –8Simplify. 1-6 Quick Check

44. Find each difference. 1.–24 – (–5) 2.19 – (–4)3.–33 – 11 4.14 – 465.–200 – 50 – (–10) Subtracting Integers PRE-ALGEBRA LESSON 1-6 –44–1923 –32–240 1-6

45. 1-7 Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Find the difference between and. 3 12 1818 3 24 1818 =

46. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 (For help, go to Skills Handbook.) Find each difference. 1.–3 – 42.–7 – 4 3.–11 – 44.–15 – 4 Check Skills You’ll Need 1-7

47. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Solutions 1.–3 – 42.–7 – 4 –3 – 4 = –3 + (–4)–7 – 4 = –7 + (–4) = –7 = –11 3.–11 – 44.–15 – 4 –11 – 4 = –11 + (–4)–15 – 4 = –15 + (–4) = –15 = –19 1-7

48. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Use inductive reasoning. Make a conjecture about the next figure in the pattern. Then draw the figure. Observation: The circles are rotating counterclockwise within the square. Conjecture: The next figure will have a shaded circle at the top right. Quick Check 1-7

49. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Write a rule for each number pattern. a. 0, – 4, – 8, –12,... b. 4, – 4, 4, – 4,... c. 1, 2, 4, 8, 10,... Start with 0 and subtract 4 repeatedly. Alternate 4 and its opposite. Start with 1. Alternate multiplying by 2 and adding 2. 1-7 Quick Check

50. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Write a rule for the number pattern 110, 100, 90, 80,... Find the next two numbers in the pattern. The rule is Start with 110 and subtract 10 repeatedly. The next two numbers in the pattern are 80 – 10 = 70 and 70 – 10 = 60. The next numbers are found by subtracting 10. – 10 The first number is 110. 110,100,90,80 1-7 Quick Check

51. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 A child grows an inch a year for three years in a row. Is it a reasonable conjecture that this child will grow an inch in the year 2015? No; children grow at an uneven rate, and eventually they stop growing. 1-7 Quick Check

52. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Is each conjecture correct or incorrect? If it is incorrect, give a counterexample. a. Every triangle has three sides of equal length. b. The opposite of a number is negative. The conjecture is incorrect. The figure below is a triangle, but it does not have three equal sides. The conjecture is incorrect. The opposite of –2 is 2. 1-7

53. (continued) Inductive Reasoning PRE-ALGEBRA LESSON 1-7 c. The next figure in the pattern below has 16 dots. The conjecture is correct. The diagram below shows the next figure in the pattern. 1-7 Quick Check

54. Inductive Reasoning PRE-ALGEBRA LESSON 1-7 Find the next three numbers in each pattern. 1.1, –1, 2, –2, 3,... 2.1, 3, 7, 15, 31,... 3.–11, –8, –5, –2,... 1, 4, 7 –3, 4, –463, 127, 255 1-7

55. 1-8 Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 Describe the pattern in this sequence of numbers and find the next two numbers. 5, 8, 4, 9, 3,... description: + 3, – 4, + 5, – 6; next two numbers: 10, 2

56. Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 (For help, go to Lesson 1-7.) Write a rule for each pattern. Find the next three numbers. 1.8, 11, 14, 17,...2.1, 5, 4, 8, 7,... 3.3, 5, 10, 12, 24,...4.1, 4, 7, 10,... Check Skills You’ll Need 1-8

57. Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 Solutions 1.8 11 14 17 20 23 26 +3 +3 +3 +3 +3 +3 Start with 8 and add 3 repeatedly. 2.1 5 4 8 7 11 10 14 +4 –1 +4 –1 +4 –1 +4 Start with 1. Alternate adding 4 and subtracting 1. 3.3 5 10 12 24 26 52 54 +2  2 +2  2 +2  2 +2 Start with 3. Alternate adding 2 and multiplying by 2. 4.1 4 7 10 13 16 19 +3 +3 +3 +3 +3 +3 Start with 1 and add 3 repeatedly. 1-8

58. Each student on a committee of five students shakes hands with every other committee member. How many handshakes will there be in all? Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 The pattern is to add the number of new handshakes to the number of handshakes already made. 4the number of handshakes by 1 student 4 + 3 = 7the number of handshakes by 2 students 1-8

59. Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 (continued) There will be 10 handshakes in all. Make a table to extend the pattern to 5 students. Student Number of original handshakes Total number of handshakes 1 4 4 2 3 4 + 3 3 2 7 + 2 4 1 9 + 1 5 0 10 + 0 = 7= 9= 10 Quick Check 1-8

60. Problem Solving Strategy: Look for a Pattern PRE-ALGEBRA LESSON 1-8 Solve using any strategy. 1.You have a penny, a nickel, a dime, and a quarter. You give away three coins. How many different amounts of money can you give away? Name the values. 4; 16¢, 31¢, 36¢, 40¢ 1-8

61. 1-9 Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 For 3 weeks, Jessica worked 2 hours per day, 4 days per week baby sitting. If she earned $3.50 per hour, how much did she earn in all? $84.00

62. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 (For help, go to Skills Handbook p. 759.) Simplify each expression. 1.5  42.3  83.5  5 4.14  25.6  56.20  7 Check Skills You’ll Need 1-9

63. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 Solutions 1. 52. 33. 5  4  8  5 20 24 25 4. 145. 66. 20  2  5  7 28 30 140 1-9

64. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 A diver is descending from the surface of the water at a rate of 5 ft/s. Write an expression with repeated addition to show how far the diver is from the surface of the water after four seconds. 4 (–5) = (–5) + (–5) + (–5) + (–5) = –20 The diver is 20 feet below the surface of the water. Use a number line to show repeated addition. Quick Check 1-9

65. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 a. –2(7) Use a pattern to find each product. 2(7) = 14Start with products you know. 1(7) = 7 0(7) = 0 –1(7) = –7Continue the pattern. –2(7) = –14 1-9

66. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 b. –2(–7) (continued) 2(–7) = –14Start with products you know. 1(–7) = –7 0(–7) = 0 –1(–7) = 7Continue the pattern. –2(–7) = 14 1-9 Quick Check

67. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 Multiply 6(–2)(–3). 6(–2)(–3) = (–12)(–3)Multiply from left to right. The product of a positive integer and a negative integer is negative. = 36Multiply. The product of two negative integers is positive. 1-9 Quick Check

68. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 Use the table to find the average of the differences in the values of a Canadian dollar and a U.S. dollar for 1994–1997. Write an expression for the average. –27 + (–27) + (–26) + (–28) 4 1-9

69. Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 (continued) The quotient of a negative integer and a positive integer is negative. = –27 For 1994–1997, the average difference was –27¢. Use the order of operations. The fraction bar acts as a grouping symbol. –108 4 = 1-9 Quick Check

70. Find each product or quotient. 1.–7(–3)2.–36 ÷ (–9) 3.–12 24.7(–3) 5.–6 (–2) (–1) Multiplying and Dividing Integers PRE-ALGEBRA LESSON 1-9 –24 214 –21 –12 1-9

71. 1-10 The Coordinate Plane PRE-ALGEBRA LESSON 1-10 Order from greatest to least: –1, 3, 0, – 2, – 4 3, 0, – 1, – 2, – 4

72. The Coordinate Plane PRE-ALGEBRA LESSON 1-10 (For help, go to Lesson 1-4.) Graph numbers on a number line. 1.–2, 1, –52.0, 2, –4 3.–3, 3, –24.–1, –5, –8 Check Skills You’ll Need 1-10

73. The Coordinate Plane PRE-ALGEBRA LESSON 1-10 Solutions 1.–2, 1, –5 2.0, 2, –4 3. –3, 3, –24.–1, –5, –8 1-10

74. The Coordinate Plane PRE-ALGEBRA LESSON 1-10 Write the coordinates of point G. In which quadrant is point G located? Point G is located 2 units to the left of the y-axis. So the x-coordinate is –2. The point is 3 units below the x-axis. So the y-coordinate is –3. The coordinates of point G are (–2, –3). Point G is located in Quadrant III. Quick Check 1-10

75. The Coordinate Plane PRE-ALGEBRA LESSON 1-10 Graph point M(–3, 3). 1-10 Quick Check

76. The Coordinate Plane PRE-ALGEBRA LESSON 1-10 Draw a coordinate grid. Graph each point. 1.S(2, 3) 2. T(2, –3) 3.U(–2, 3)4. K(0, –3) 5. L(3, 0) 1-10