1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 Complete each equation. 1.1 week = days2.1 foot...

1-1

Variables and ExpressionsVariables and ExpressionsPRE-ALGEBRA LESSON 1-1PRE-ALGEBRA LESSON 1-1

Complete each equation.

1. 1 week = days 2. 1 foot = inches

3. 1 nickel = cents 4. 1 gallon = quarts

5. 1 yard = feet

Solutions

1. 1 week = days 2. 1 foot = inches

3. 1 nickel = cents 4. 1 gallon = quarts

5. 1 yard = feet


1-1

7 12

5 4

3

Identify each expression as a numerical expression or a

variable expression. For a variable expression, name the variable.


numerical expression

Variable expression; t is the variable.

a. 7 3

b. 4t

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.

Expression 29 p•

1-1


Pages 6–7 Exercises

16. z – 8

17. 2 • 12

18. d • 12

19. 5 • 7

20. 5n

21. 4 • 3

22. 4g

23. Variable expression; d is the variable.

1. Variable expression; b is the variable.

2. numerical expression

3. Variable expression; n is the variable.


5. Variable expression; x is the variable.


7. m + 16

8.

9. 3c

10. p – 2

11. 3b

12. j – 4

13.

14.

15. 2 – x

6z

3n

n3

1-1



25. Variable expression; g is the variable.


27. 7 • 4

28. 7w

29.

30.

31.

32.

33. d – 20

34. 110e + 55

35. 70a + 100b

36. B

37. C

38. D

39. A

40. Both use numbers and operation symbols; only variable expressions use variables.

41. Answers may vary. Sample: The student confused the positions of n and 5.

42. B

43. I

44. C

160 16

z16

100 12

i12

1-1


45. H

46. 225

47. 72

48. 8

49. 75

50. 13

51. 9,563

52. $2.95

53. a. 500

b. 400

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 three 4. a divided by 7

p – 3

h + 56 d – 3

a ÷ 7

1-1

1-2

The Order of OperationsThe Order of OperationsPRE-ALGEBRA LESSON 1-2PRE-ALGEBRA LESSON 1-2

(For help, go to Skills Handbook p. 760.)

Find each quotient.

1. 164 ÷ 2 2. 344 ÷ 8 3. 284 ÷ 4

4. 133 ÷ 7 5. 182 ÷ 13 6. 650 ÷ 25


Solutions

1. Estimate: 2. Estimate: 3. Estimate:160 ÷ 2 = 80 360 ÷ 8 = 45 280 ÷ 4 = 70 82 43 712 164 8 344 4 284 –16 –32 –28 4 24 4 – 4 – 24 – 4 0 0 0

4. Estimate: 5. Estimate: 6. Estimate:140 ÷ 7 = 20 195 ÷ 13 = 15 625 ÷ 25 = 25 19 14 267 133 13 182 25 650 –7 –13 –50 63 52 150 –63 – 52 –150 0 0 0

1-2


Simplify 8 – 2 • 2.

8 – 2 • 2

8 – 4 First multiply.

Then subtract.4

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.

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

1-2



17. 33

18. 13

19. 8

20. 4

21. 3

22. The student subtracted 1 from 6 before dividing, instead of dividing and then subtracting.

1. 27

2. 3

3. 2

4. 4

5. 16

6. 16

7. 11

8. 31

9. 4

10. 6

11. 14

12. 4

13. 13

14. 108

15. 49

16. 17

1-2


23. We must agree on an order of operations to ensure that everyone gets the same value for an expression.

24. 3

25. 24

26. 12

27. 16

28. 33

29. 25

30. 24

31. 22

32. 4

33. >

34. >

35. >

36. <

37. >

38. <

39. (7 + 4) • 6 = 66

40. [7 • (8 – 6)] + 3 = 17

41. (3 + 8 – 2) • 5 = 45

42. (6 • 3) + (9 – 4) = 23

43. 4 • 9 + 5; 41

44. 21 – (15 + 5); 1

45. 17 – (25 ÷ 5); 12

46. 4 + 7 • 3; 25 hours

1-2


47–49. Answers may vary. Samples are given.

47. (6 5) – (4 2) and

(6 3) + (2 2); 22 in.2

48. (6 6) – (2 2) and

(2 6) + (2 6) +

(2 2) + (2 2);32 m2

49. (5 4) – (3 3) and

(4 1) + (4 1) +

(3 1); 11 ft2

50. Answers may vary. Sample: Payton bought 4 pairs of blue socks and 3 pairs of red socks at $3 a pair. She also bought a hat for $2. What was the total cost of her purchases? $23

51. Answers may vary. Sample: 1 + 2 + 3 + 4 + 5 + 6 + 7 + (8 • 9)

k20

52. D

53. G

54. D

55. F

56. 8n

57.

58. h – 6

59. 10d

60. 15c

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

13 8

9

1-2

1-3

Evaluating ExpressionsEvaluating ExpressionsPRE-ALGEBRA LESSON 1-3PRE-ALGEBRA LESSON 1-3

(For help, go to Lesson 1-2.)


1. 6(9 + 1) 2. 17 – 2 + 3

3. 9 + 8 • 2 + 4 4. [3(5) + 1] • 2


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


Evaluate 18 + 2g for g = 3.

18 + 2g = 18 + 2(3) Replace g with 3.

= 18 + 6 Multiply.

= 24 Add.

1-3


Evaluate 2ab – for a = 3, b = 4, and c = 9.

= 2 • 3 • 4 – 3 Work within grouping symbols.

= 6 • 4 – 3 Multiply from left to right.

= 24 – 3 Multiply.

= 21 Subtract.

c3

2ab – = 2 • 3 • 4 – Replace the variables.c3

93

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

x36 = Evaluate for x = 180.180

36

x36

= 5 Divide.

The Omelet Café should buy 5 cartons to get 180 eggs.

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

pizza plus

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 p• 2+

1-3



16. a. 24d + 4

b. $76

17. 15

18. 735

19. 11

20. 15

21. 91

22. 16

23. 99

1. 35

2. 2

3. 5

4. 23

5. 1

6. 107

7. 60

8. 12

9. 4

10. 10

11. 26

12. 6

13. 14

14. 65

15. a. 55m

b. 1,100 words

1-3


24. 125

25. 2

26. 10

27. 8d; 24 km

28. a. 153w

b. 306 calories

29. Answers may vary. Sample: You did not work within the grouping symbols first.

30. 10

31. 100 + 25n; $400

32. a. 5 + 2r

b. $17

c. 5 rides

33. Answers may vary. Sample: Liam has 5 fewer than 3 times the number of cards that Jamie has. Jamie has 5 cards. How many cards does Liam have?

34. D

35. H

36. D

37. F

38. 6

39. 53

40. 0

41. 19 – t

42.

43. 8 + n

d20

1-3


44. No; Valerie’s average grade should be in the 80s. Valerie evaluated 96 + 82 + 78 + 76 ÷ 4 on her calculator instead of

.96 + 82 + 78 + 76 4

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 = 2 4. fg – g for f = 5, g = 7

26

17 7

28

1-3

1-4

Integers and Absolute ValueIntegers and Absolute ValuePRE-ALGEBRA LESSON 1-4PRE-ALGEBRA LESSON 1-4


Write an integer for each situation.

1. lose $7 2. find $9

3. 8 steps forward 4. 3 yards gained

5. 5 floors down


Solutions

1. lose $7 2. find $9

–7 9

3. 8 steps forward 4. 3 yards gained

8 3

5. 5 floors down

–5

1-4


Write a number to represent the temperature shown by the thermometer.

The thermometer shows 2 degrees Celsius below zero, or –2°C.

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


Use a number line to find |–5| and |5|.

|–5| = 5 |5| = 5

1-4



16. 1, 1

17. 2, 2

18. 8, 8

19. 7, 7

20. 6, 6

21. 4, 4

1. 250

2. –18

3. –45

4. 110

5. –50

6. 7

7. –300

8. –8

9. 3,400

10. 2

11. 5

12. –4

13. –9, –2, 8

14. –12, –9, –3

15. –6, 0, 6

1-4


22. 18

23. 9

24. 3

25. 6

26. 7

27. 2

28. Answers may vary. Sample: loss of 1,000 points in a board game

29. Answers may vary. Sample: 28 golf strokes over par

30. Answers may vary. Sample: checkbook balance for checks totalling $126 more than is in the account

31. 6

32. –2

33. 2

34. –8

35. 0

36. 1,000

37. –13

38. 56

39. –23

40. –12

41. <

42. >

43. <

1-4


44. <

45. <

46. =

47. 10h

48. – d

49. r + n

50. –12,500; –15,617

51. Answers may vary. Sample: My friend did not take into account

51. (continued)the signs of the numbers.

52. zero

53. negative

54. positive

55. negative


56. –3, –2

57. –2, –1

58. –11, –10

59. a.

b. Nevada

60. Answers may vary. Sample: If sea level is zero, then water levels above sea level can be

13

1-4


60. (continued)described by positive integers, and water levels below sea level can be described by negative integers.

61. No. Explanations may vary. Sample: If x and y have opposite signs,| x + y | | x | + | y |. For x = –1 and y = 3,| –1 + 3 | = 2 while| –1 | + | 3 | = 4.

62. B

63. I

64. B

65. F

66. 14

67. 24

68. 440

69. >

70. <

71. c + 6

=/

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

1-5

Adding IntegersAdding IntegersPRE-ALGEBRA LESSON 1-5PRE-ALGEBRA LESSON 1-5


Compare. Use >, <, or = to complete each statement.

1. –6 –3 2. 2 –15

3. –5 | 5 | 4. | 10 | | –10 |

5. | 9 | | –2 | 6. | –8 | | 0 |


Solutions

1. –6 < –3 2. 2 > –15

3. –5 < | 5 | 4. | 10 | = | –10 |

5. | 9 | > | –2 | 6. | –8 | > | 0 |

1-5

(–7) + 3 Model the sum.

– 4 Group and remove zero pairs. There are four negative tiles left.


Use tiles to find (–7) + 3.

(–7) + 3 = – 4

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

1-5

Start at 0. To represent –20, move left 20 units. To add positive 4, move right 4 units to –16.


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


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


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



17. 53

18. –65

19. 100

20. 0

21. –185

22. 2,620 m

23. 23

24. 12

1. –4 + 7; 3

2. 5 + 0; 5

3. –4 + (–2); –6

4. 3 + (–8); –5

5. –3

6. –3

7. 3

8. –7

9. –5

10. –9

11. –1

12. 1

13. 3

14. –9

15. –13

16. –14

1-5


25. –61

26. 12

27. Negative; both numbers are negative.

28. Positive; the number with greater absolute value is positive.

29. Zero; the numbers are opposites.

30. 15

31. 0

32. –8

33. 7

34. 16

35. –40

36. <

37. >

38. =

39. >

40. >

41. >

42. –7

43. –8

44. –45

45. –20 + 18; –2

46. 200 + (–75); 125

47. 120 + (–25); 95

48. –35 + 10; –25

49. 1

1-5


50. –3

51. –60

52. 4 yd loss

53. $158

54. Answers may vary. Sample: My friend subtracted 5 instead of adding 5.

55. Answers may vary. Sample: First find the difference of the absolute values of the

55. (continued)two numbers. Then give the answer the sign of the number with the greater absolute value.

56. negative

57. positive

58. negative

59. positive

60. B

61. H

62. B

63. G

64. <

65. >

66. >

67. >

68. <

69. <

1-5


70. (116 – 8) + 130, or 130 + (116 – 8); 238

71. 25 + 10n; $55

1-5

Find each sum.

1. –37 + (–5) 2. 14 + (–4)

3. –100 + 5 + (–3) 4. Evaluate 33 + t for t = –11.


–98

–42 10

22

1-5

1-6

Subtracting IntegersSubtracting IntegersPRE-ALGEBRA LESSON 1-6PRE-ALGEBRA LESSON 1-6


Find each sum.

1. 8 + (–9) 2. –11 + (–18) 3. –4 + (–6)

4. 14 + (–3) 5. 6 + (–6) 6. –13 + (–10)


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

Start with 7 negative tiles.

Take away 5 negative tiles. There are 2 negative tiles left.


Find –7 – (–5).

–7 – (–5) = –2

1-6

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.


Find 2 – 8.

2 – 8 = –6

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)

= –8 Simplify.

1-6



17. –12

18. 6 + (–2); 4

19. 6 + 2; 8

20. –6 + (–2); –8

21. 2 + (–6); –4

22. 2 + 6; 8

23. –2 + (–6); –8

24. 5 + (–11); –6

1. –9 – (–2) = –7

2. 3 + (–8) = –5 or3 – 8 = –5

3. –4

4. –8

5. 1

6. –2

7. 2

8. –7

9. –5

10. 2

11. 15

12. 17

13. –6

14. 0

15. –1

16. –5

1-6


25. 75 + 25; 100

26. 22 + 7; 29

27. 87 + 9; 96

28. 35 + (–15); 20

29. 100 + 91; 191

30. –$19

31. –15

32. –124

33. –80

34. 19

35. 170

36. 913

37. –60

38. –83

39. –68

40. –52

41. –30

42. 850

43. 10

44. 66


45. 3 – 3 = 0; (–4) – (–4) = 0

46. 15 – 5 = 10; –5 – (–15) = 10

47. 1 – 7 = –6; –10 – (–4) = –6

1-6


48. 5 – 20 = –15; –20 – (–5) = –15

49. 7 – 4 = | –3 |;–2 – (–5) = | –3 |

50. 12 – 1 = | 11 |;–5 – (–16) = | 11 |

51. It decreases.

52. –24°C

53. –8°C

54. 2

55. –60

56. –7

57. 150

58. 10

59. 66

60. 12

61. –40

62. 0

63. 18

64. –30

65. –20

66. –2 – 4; –6; $6

67. 3,000 – 600; 2,400; 2,400 ft

68. 0 + 15 – 25; –10; –10°F


69. 50

1-6


70. –280

71. –470

72. –110

73. 230

74. 80

75. a. Answers may vary.

Sample: The temperature at 5:00 P.M. was 5°C, but it dropped 7 degrees after sundown. What

is the temperature

75. (continued) after sundown?

b. 5 + (–7); –2°C

76. a. 16, 20, 24, 28

b. 28

c. 24

77. a. a 5 Example: When a = 6, | 6 – 5 | = | 6 | – 5.

b. a < 5 Example: When a = 4, | 4 – 5 | > | 4 | – 5.

c. none

78. –3;

79. –15;

80. 6;>–

1-6


81. –6;

82. D

83. H

84. D

85. G

86. –5

87. 7

88. –6


89. –7

90. 3

91. 0

92. –6

93. 100 + 6 • 9; 154

1-6

Find each difference.

1. –24 – (–5) 2. 19 – (–4) 3. –33 – 11

4. 14 – 46 5. –200 – 50 – (–10)


–44–19 23

–32 –240

1-6

1-7

Inductive ReasoningInductive ReasoningPRE-ALGEBRA LESSON 1-7PRE-ALGEBRA LESSON 1-7

(For help, go to Skills Handbook.)

Find each difference.

1. –3 – 4 2. –7 – 4

3. –11 – 4 4. –15 – 4


Solutions

1. –3 – 4 2. –7 – 4

–3 – 4 = –3 + (–4) –7 – 4 = –7 + (–4)

= –7 = –11

3. –11 – 4 4. –15 – 4

–11 – 4 = –11 + (–4) –15 – 4 = –15 + (–4)

= –15 = –19

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.

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


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 – 10 – 10

The first number is 110. 110, 100, 90, 80

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


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

(continued)


c. The next figure in the pattern below has 16 dots.

The conjecture is correct. The diagram belowshows the next figure in the pattern.

1-7



8. Start with 1 and alternately add 1 and 3; 10, 13

9. Answers may vary. Sample: No. Mario’s catching a cold could be due to many different reasons.

10. Incorrect; an ostrich cannot fly.

11. correct

1.

a square with four corners shaded

2.

a six-sided figure with a six-sided figure inside it

3. Start with 100 and subtract 15 repeatedly; 40, 25

4. Start with 5 and multiply by 4 repeatedly; 1,280, 5,120

5. Start with 2 and add 5 repeatedly; 22, 27

6. Start with –10 and add 6 repeatedly; 14, 20

7. Start with 1 and add 3 repeatedly; 13, 16

1-7


12. Incorrect; • = ,

which is less than .

13.

an eight-sided figure with bottom right eighth shaded

14.

a square inside a circle inside a

14. (continued)triangle inside a diamond

15. Start with 1 and add 0.5 repeatedly; 3.5, 4, 4.5

16. Start with –1 and alternate between finding the opposite and the next negative integer. –4, 4, –5

17. Start with 6 and add –2 repeatedly; –2, –4, –6

18. Incorrect; some have 4.

19. Incorrect; 8 + (–6) is 2,

2 < 8.

20. correct

21. Start with 1, and then double and add 2, repeatedly; 190, 382, 766

22. Start with 1 and alternately subtract and add consecutive multiples of 3; 10, –11, 13

12

12

14

12

1-7


23. a. Answers may vary. Sample: The unemployment rate will decrease; it has decreased every year shown in the graph.

b. Answers may vary. Sample: Look up unemployment records for 2001.

24. C

25. G

26. C

27. –7

28. 5

29. 103

30. –4

31. –2

32. 20

33. 1,500n; 36,000 gallons

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, –4 63, 127, 255

1-7

1-8

Look for a PatternLook for a PatternPRE-ALGEBRA LESSON 1-8PRE-ALGEBRA LESSON 1-8


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, . . .


Solutions

1. 8 11 14 17 20 23 26

+3 +3 +3 +3 +3 +3Start with 8 and add 3 repeatedly.

2. 1 5 4 8 7 11 10 14

+4 –1 +4 –1 +4 –1 +4Start with 1. Alternate adding 4 and subtracting 1.

3. 3 5 10 12 24 26 52 54

+2 2 +2 2 +2 2 +2Start with 3. Alternate adding 2 and multiplying by 2.

4. 1 4 7 10 13 16 19

+3 +3 +3 +3 +3 +3Start with 1 and add 3 repeatedly.

1-8

Each student on a committee of five students

shakes hands with every other committee member. How

many handshakes will there be in all?


The pattern is to add the number of new handshakes to the number of handshakes already made.

4 the number of handshakes by 1 student

4 + 3 = 7 the number of handshakes by 2 students

1-8


(continued)

There will be 10 handshakes in all.

Make a table to extend the pattern to 5 students.

Student

Number of originalhandshakes

Total number ofhandshakes

1

4

4

2

3

4 + 3

3

2

7 + 2

4

1

9 + 1

5

0

10 + 0= 7 = 9 = 10 = 10

1-8



12. (continued)

a 4 4 square

13.

a circle divided into 5 pieces

14. 9°F

15. 0

16. 17

17. 25

1. 36 laps/day

2. 78 students

3. $10.23

4. 365 min, or 6 h and 5 min; 10 min

5. 11 pieces; 16 pieces

6. a. 4, 3, 1; 9, 8, 1; 16, 15, 1; 25, 24, 1. The differences are all 1.

b. 11 • 11; 1

c. 2,208

6. (continued)d. 4,225

7. a. $59; $21

b. 10 people

8. 3 mi

9. C

10. I

11. B

12.

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

1-9

Multiplying and Dividing IntegersMultiplying and Dividing IntegersPRE-ALGEBRA LESSON 1-9PRE-ALGEBRA LESSON 1-9



1. 5 4 2. 3 8 3. 5 5

4. 14 2 5. 6 5 6. 20 7


Solutions

1. 5 2. 3 3. 5 4 8 5 20 24 25

4. 14 5. 6 6. 20 2 5 7 28 30 140

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.

1-9


a. –2(7)

Use a pattern to find each product.

2(7) = 14 Start with products you know.

1(7) = 7

0(7) = 0

–1(7) = –7 Continue the pattern.

–2(7) = –14

1-9


b. –2(–7)

(continued)

2(–7) = –14 Start with products you know.

1(–7) = –7

0(–7) = 0

–1(–7) = 7 Continue the pattern.

–2(–7) = 14

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.

= 36 Multiply. The product of two negative integers is positive.

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


(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.

–1084=

1-9



16. 0

17. –360

18. 0

19. –96

20. –1

21. –18

22. –10

23. –7

1. 5 • (–2) = –10

2. 4(–9); –36

3. 5(–5); –25

4. –35

5. –9

6. –44

7. –24

8. –50

9. –18

10. –30

11. –81

12. –72

13. –48

14. 15

15. –60

1-9


24. –6

25. 10

26. 19

27. –12

28. –3

29. –2°C

30. 2 yd

31. 0

32. $92

33. Positive; the integers have the same sign.

34. Negative; the integers have opposite signs.

35. Negative; the integers have opposite signs.

36. Positive; the first product is negative, so the second is a product of integers with the same sign.

37. A

38. D

39. C

40. B

41. –15

42. –14

43. 4,661

44. –21,384

45. 8

46. 216

1-9


47. –76

48. –20

49. 12,288

50. a. –3

b. $40 per share

51. >

52. <

53. >

54. <

55. =

56. <

57. 12

58. –15

59. –27

60. –15, 5


61. 4 and 8; 7

62. 3 and –2; –1

63. 2 and –4; 1

64. –11 and 0; –6

65. a. negative; positive; negative

b. If there are an even number of negative integers, the sign of the product will be positive; otherwise, the sign will be negative.

1-9


66. Negative; the numerator is positive, and the denominator is negative, so the quotient is negative.

67. $.60; $520

68. C

69. F

70. C

71. G

72. 37

73. <

74. >

75. <

76. 50 – n

77. 60y

78. x + y

79. d5

1-9

Find each product or quotient.

1. –7(–3) 2. –36 ÷ (–9)

3. –12 • 2 4. 7(–3)

5. –6 • (–2) • (–1)


–24

21 4

–21

–12

1-9

1-10

The Coordinate PlaneThe Coordinate PlanePRE-ALGEBRA LESSON 1-10PRE-ALGEBRA LESSON 1-10


Graph numbers on a number line.

1. –2, 1, –5 2. 0, 2, –4

3. –3, 3, –2 4. –1, –5, –8


Solutions

1. –2, 1, –5 2. 0, 2, –4

3. –3, 3, –2 4. –1, –5, –8

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.

1-10

The coordinates of point G are (–2, –3). Point G is located in Quadrant III.


Graph point M(–3, 3).

1-10


1. III

2. II

3. IV

4. I

5. II

6. III

7. (0, –7)

8. (3, 0)

9. (–2, 4)

10. (–4, –2)

11. (–8, 3)

12. (1, –4)

13–28.

29. (0, 0)

30. Q

31. F

32. M

33. P

34. (2, –3)

35. (–2, 3)

36. (–5, 0)

37. (6, 6)

1-10



38. (–5, –2)

39. (0, –4)

40. (3, 0)

41. IV

42. I

43. II

44. III

45. I

46. y-axis

47. III

48. IV

49. y-axis

50. II

51. triangle

52. rectangle

53. parallelogram

1-10


54. square 59. Frankfort, Kentucky

60. Santa Fe, New Mexico

61.

62. (continued)b.

55. (0, –5)

56. (–1, –2)

57. about 90° W, 32° N

58. about 96° W, 39° N

62. a.

c.

1-10


62. (continued)d.

64. Check students’ work.

65. Explanations may vary. Sample: No; since a and b describe positions on twonumber lines, (a, b) and (b, a) describe different points (unless a = b).

66. A(3, 0), B(0, 3), C(–3, 0), D(0, –3)

67. D

68. I

69. A

70. H

71. A

72. –121

73. –9

74. 0

75. –925 ft

63. Answers may vary. Sample: 62a flips the figure across the y-axis. 62b flips the figure across the x-axis. 62c flips the figure across one axis and then the other. 62d doubles the lengths of the sides.

1-10


76. 8

77. –95

78. –12

79. 20

80. 4

81. 3

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

1-A

Algebraic Expressions and IntegersAlgebraic Expressions and IntegersPRE-ALGEBRA CHAPTER 1PRE-ALGEBRA CHAPTER 1

1. n + 19

2. –3 –10

3. –5x

4. –y + 5

5. –10

6. 56

7. 22

8. 18

9. 5

10. 9

11. –7

12. 15

13. <

14. >

15. <

16. >

17. 8

18. 4

19. 63

20. –9

21. –288

22. 8

23. 143

24. –68

25. –66

26. –180

27. 46

Algebraic Expressions and IntegersAlgebraic Expressions and IntegersPRE-ALGEBRA CHAPTER 1PRE-ALGEBRA CHAPTER 1

28. 16

29. 26

30. 0

31. II

32. y-axis

33. III

34. (–3, 2)

35. (1, –2)

36. (2, 0)

37. (–2, –2)

38. a. 25j + 15s

b. $150

c. 2

39. D

40. –175

41. Start with 100 and alternately subtract 10 and 5; 55, 45, 40.

42. 15th floor

43. Place them on a number line and write them as they appear from left to right.

1-A

1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 Complete each equation. 1.1 week = days2.1 foot...

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Transcript of 1-1 Variables and Expressions PRE-ALGEBRA LESSON 1-1 Complete each equation. 1.1 week = days2.1 foot...