Problem Solving Skills: Model Algebra

1. SECTION 3-3
Problem Solving Skills:
Model Algebra

2. ESSENTIAL QUESTION
How do you create a mathematical model to describe a
situation?
Where you’ll see this:
Almost everywhere

3. VOCABULARY
1. Model:
2. Mathematical Model:
3. Algebraic Model:
4. Rule:

4. VOCABULARY
1. Model: A physical or numerical representation of a real world
situation
2. Mathematical Model:
3. Algebraic Model:
4. Rule:

5. VOCABULARY
1. Model: A physical or numerical representation of a real world
situation
2. Mathematical Model: Using numbers in a table or list to
describe a situation
3. Algebraic Model:
4. Rule:

6. VOCABULARY
1. Model: A physical or numerical representation of a real world
situation
2. Mathematical Model: Using numbers in a table or list to
describe a situation
3. Algebraic Model: A model that includes one or more variables,
often written as an expression
4. Rule:

7. VOCABULARY
1. Model: A physical or numerical representation of a real world
situation
2. Mathematical Model: Using numbers in a table or list to
describe a situation
3. Algebraic Model: A model that includes one or more variables,
often written as an expression
4. Rule: Another name for a formula or equation

8. THE FIVE-STEP PLAN

9. THE FIVE-STEP PLAN
1. Read:

10. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece

11. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan:

12. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done

13. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve:

14. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve: Set up and solve the problem

15. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve: Set up and solve the problem
4. Answer:

16. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve: Set up and solve the problem
4. Answer: Make sure you have answered each question that is
being asked

17. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve: Set up and solve the problem
4. Answer: Make sure you have answered each question that is
being asked
5. Check:

18. THE FIVE-STEP PLAN
1. Read: Read through the problem, then go back through it piece
by piece
2. Plan: Using the parts of the problem, figure out what needs to
be done
3. Solve: Set up and solve the problem
4. Answer: Make sure you have answered each question that is
being asked
5. Check: Make sure your answers work and make sense

19. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
b. Write an algebraic model of the fare system.

20. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

21. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

22. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

23. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

24. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

25. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

26. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

27. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

28. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

29. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

30. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

31. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

32. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

33. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

34. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

35. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

36. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

37. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

38. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

39. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.

40. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.
Let q equal quarter miles traveled

41. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
a. Make a table that expresses the fare for trips of that range
from 1 to 3 miles, including every quarter-mile.
# of
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Miles
Fare $6.45 $7.25 $8.05 $8.85 $9.65 $10.45 $11.25 $12.05 $12.85
b. Write an algebraic model of the fare system.
Let q equal quarter miles traveled 3.25 + .8q

42. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
c. Write a rule for the fare system.

43. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
c. Write a rule for the fare system.
Let f be the fare and q be a quarter-mile

44. EXAMPLE 1
A taxi charges $3.25 plus $.80 for each quarter-mile.
c. Write a rule for the fare system.
Let f be the fare and q be a quarter-mile
F = 3.25 + .8q

45. EXAMPLE 2
Compare vacation packages offered at two beach resorts. The
Matt Mitarnowski Surf Resort offers a family package of $750
for five days with a charge of $75 for each additional day.
Fuzzy Jeff’s Sun and Sand Resort offers a similar plan for
$350 for three days with a charge of $100 each additional day.
Which resort offers a better deal for a stay of four days? What
about nine days? How about fourteen days?

46. EXAMPLE 2
Compare vacation packages offered at two beach resorts. The
Matt Mitarnowski Surf Resort offers a family package of $750
for five days with a charge of $75 for each additional day.
Fuzzy Jeff’s Sun and Sand Resort offers a similar plan for
$350 for three days with a charge of $100 each additional day.
Which resort offers a better deal for a stay of four days? What
about nine days? How about fourteen days?
Follow The Five-Step Plan

47. EXAMPLE 2

48. EXAMPLE 2
1. Read the problem.

49. EXAMPLE 2
1. Read the problem.
2. Plan.

50. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days

51. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount

52. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.

53. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.
MMSR

54. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.
MMSR
c = 750 +

55. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.
MMSR
c = 750 + 75(d − 5)

56. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.
MMSR FJSSR
c = 750 + 75(d − 5)

57. EXAMPLE 2
1. Read the problem.
2. Plan.
Let c be the cost and d the number of days
Be aware that there is a charge for days after a certain amount
3. Set up and solve.
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)

58. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:

59. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay

60. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay c = 350 + 100(4 − 3)

61. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay c = 350 + 100(4 − 3)
c = 350 + 100(1)

62. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay c = 350 + 100(4 − 3)
c = 350 + 100(1)
c = 350 + 100

63. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay c = 350 + 100(4 − 3)
c = 350 + 100(1)
c = 350 + 100
c = 450

64. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
4 days:
Charges $750 for a five-day stay c = 350 + 100(4 − 3)
c = 350 + 100(1)
c = 350 + 100
c = 450
Charges $450 for a five-day stay

65. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:

66. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5)

67. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5)
c = 750 + 75(4)

68. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5)
c = 750 + 75(4)
c = 750 + 300

69. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5)
c = 750 + 75(4)
c = 750 + 300
c = 1050

70. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5)
c = 750 + 75(4)
c = 750 + 300
c = 1050
Charges $1050 for a nine-
day stay

71. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5) c = 350 + 100(9 − 3)
c = 750 + 75(4)
c = 750 + 300
c = 1050
Charges $1050 for a nine-
day stay

72. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5) c = 350 + 100(9 − 3)
c = 750 + 75(4) c = 350 + 100(6)
c = 750 + 300
c = 1050
Charges $1050 for a nine-
day stay

73. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5) c = 350 + 100(9 − 3)
c = 750 + 75(4) c = 350 + 100(6)
c = 750 + 300 c = 350 + 600
c = 1050
Charges $1050 for a nine-
day stay

74. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5) c = 350 + 100(9 − 3)
c = 750 + 75(4) c = 350 + 100(6)
c = 750 + 300 c = 350 + 600
c = 1050 c = 950
Charges $1050 for a nine-
day stay

75. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
9 days:
c = 750 + 75(9 − 5) c = 350 + 100(9 − 3)
c = 750 + 75(4) c = 350 + 100(6)
c = 750 + 300 c = 350 + 600
c = 1050 c = 950
Charges $1050 for a nine- Charges $950 for a nine-day stay
day stay

76. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:

77. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5)

78. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5)
c = 750 + 75(9)

79. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5)
c = 750 + 75(9)
c = 750 + 675

80. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5)
c = 750 + 75(9)
c = 750 + 675
c = 1425

81. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5)
c = 750 + 75(9)
c = 750 + 675
c = 1425
Charges $1425 for a
fourteen-day stay

82. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5) c = 350 + 100(14 − 3)
c = 750 + 75(9)
c = 750 + 675
c = 1425
Charges $1425 for a
fourteen-day stay

83. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5) c = 350 + 100(14 − 3)
c = 750 + 75(9) c = 350 + 100(11)
c = 750 + 675
c = 1425
Charges $1425 for a
fourteen-day stay

84. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5) c = 350 + 100(14 − 3)
c = 750 + 75(9) c = 350 + 100(11)
c = 750 + 675 c = 350 + 1100
c = 1425
Charges $1425 for a
fourteen-day stay

85. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5) c = 350 + 100(14 − 3)
c = 750 + 75(9) c = 350 + 100(11)
c = 750 + 675 c = 350 + 1100
c = 1425 c = 1450
Charges $1425 for a
fourteen-day stay

86. EXAMPLE 2
MMSR FJSSR
c = 750 + 75(d − 5) c = 350 + 100(d − 3)
14 days:
c = 750 + 75(14 − 5) c = 350 + 100(14 − 3)
c = 750 + 75(9) c = 350 + 100(11)
c = 750 + 675 c = 350 + 1100
c = 1425 c = 1450
Charges $1425 for a Charges $1450 for a
fourteen-day stay fourteen-day stay

87. EXAMPLE 2
4. Answer the questions.

88. EXAMPLE 2
4. Answer the questions.
For a four-day or nine-day stay, Fuzzy Jeff’s Sun and Sand
Resorts offers a better deal, but for a fourteen-day stay, Matt
Mitarnowski’s Surf Resort is better.

89. EXAMPLE 2
4. Answer the questions.
For a four-day or nine-day stay, Fuzzy Jeff’s Sun and Sand
Resorts offers a better deal, but for a fourteen-day stay, Matt
Mitarnowski’s Surf Resort is better.
5. Check to make sure the answers make sense.

90. HOMEWORK

91. HOMEWORK
p. 114 #5-20
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