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```© 21st Century Math Projects
Project Title: Anesthesiologist
Standard Focus: Algebra & Functions
Time Range: 2-3 Days
Supplies: Basic Stuff
Topics of Focus:
-
Solving and Graphing Rational Functions
Benchmarks:
1. Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
Creating Equations
A-CED
Creating Equations
A-CED
Reasoning with
Equations and
Inequalities
A-REI
Interpreting Functions
F-IF
7. Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.★
Interpreting Functions
F-IF
9. Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions).
2. Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
Procedures:
A.) Students will complete “Sedation Class”. In this assignment, students will use rational models to create a
table and construct a graph of the data.
B.) Students will complete “Medicine”. In this assignment, students will investigate 5 different rational models
of medicine to determine which has the longest lasting effects.
C.) Students will complete “Anesthesiology Crisis”. In this assignment, students will wear their
anesthesiologist hat and have found a paperwork disaster. There are 6 patients and the levels and types of
sedation were mixed up. Using graphs or tables students must determine which rational model belongs to
which patient. They must determine when the patients will wake from anesthesia and they have to figure out
if patients will need more anesthesia.
Sedation Class
For surgeries and serious injuries anesthesiologists are responsible for administering
anesthesia. If correct dosages are not given and monitored it is possible that patients
may awaken in the middle of a procedure. Avoiding this disaster takes analysis of an
anesthesia’s concentration in the bloodstream. When a concentration of an anesthesia is below
0.1 mg/L it is no longer effective. In these two cases, you must use rational functions to model the
anesthesia’s concentration in the blood stream and determine when the patient may wake up.
1.
Given a model for this patient’s concentration over time complete the table with useful times
or amounts. Use the model to determine when they may wake up.
Model
y=
2.8
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
0.4 2 +1.3
10
20
30
60
0.4
0.2
0.1
0.01
Conclusion
Based on the model, when will the patient wake up?
If the procedure lasts approximately 1 hour and thirty minutes, will additional anesthesia be
required?
Can you determine the maximum amount of concentration that the patient had in their
bloodstream? If it is over 2 mg/L then it could be dangerous.
2.
Given this patient’s concentration log, plot the points as coordinate pairs and sketch a rational
model to attempt to complete the rest of the log.
Chart
Concentration Log
Minutes
Since
Dose
(x)
1
Concentration
in mg/L
(y)
.83
2
1.27
3
1.37
4
1.32
5
1.21
10
20
30
40
50
60
When will the
concentration be… ?
1
0.5
0.2
0.1
Conclusion
Based on your model, when will the patient wake up?
If the patient was put to sleep at 11:12 AM and the procedure lasts approximately 1 hours and 15 minutes,
If additional anesthesia needs to be administered 30 minutes before the previous anesthesia would wear
off, at what time does it need to be administered?
Medication
In the world of medicine, every minute of relief can count for something.
Once medicine enters the blood stream it will soon have its most powerful
effect because it has its highest concentration. Over time the concentration
reduces and once it reaches a certain level, the medicine will no longer be
effective. The concentration level of medicines can be most accurately
modeled by rational functions.
In this assignment, there are models for 5 different prescription medications. It is generally accepted
that when a concentration of a medicine is below 0.15 mg/L is no longer effective. It is your job to
determine how long each medicine is above the 0.15 mg/L level and rank them from longest relief to
shortest relief.
Model
Anvil
(t is in minutes)
5.6
A(t) =
0.2 2 +2.3
Bozak
B(t) =
Crylenol
C(t) =
Depto
D(t) =
Equal
E(t) =
Length of Effectiveness
(round to the nearest minute)
4.85
0.1 2 +3.2
8.2
0.3 2 +4.6
10.2
0.3 2 +3.1
7.135
0.4 2 +5.2
--Longest acting to shortest acting—
Suppose each of the medicines were administered at the times below. At what time would they no
longer be effective?
Anvil
Bozak
Crylenol
Depto
Equal
No longer effective at…
3:16 PM
7:05 AM
6:51 PM
11:41 PM
10:28 AM
Anethesiology Crisis
Just as an anesthesiologist begins their shift, a crisis scenario occurs. A
patient has unexpectedly awaken from sedation in the middle of a
procedure. The information from the previous shift is no longer reliable
and they must quickly figure out the concentration of anesthesia in each
patient’s blood stream.
When a concentration of sedation is below 0.1 mg/L a patient is liable to wake up. You discover there
has been a serious paperwork mishap and models were written for the incorrect patients. Currently
there are six patients under sedation. You must use what you know about rational functions and
modeling to determine which model belongs to which patient and determine when they will wake up.
Patient 1 38/F
Chart
Sedated at:
3:15PM
Model
Conclusion
Based on the chart, when will the patient wake up?
At what approximately time will the patient awaken from the anesthesia?
If the patient is scheduled to be in surgery until 4:45, will more anesthesia need to be given?
Additional anesthesia must be administered 30 minutes before they would awaken. At what time
will more anesthesia be needed?
Patient 2 14/F
Model
Sedated at: 2:48PM
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
1.99
10
1.11
20
0.57
30
0.38
60
0.19
When will the concentration be ___ ?
0.4
0.2
0.1
Conclusion
Based on the model, at what time will the patient wake up?
If the patient is scheduled to have a procedure until 4:25, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed?
Model
Patient 3 73/F
Chart
Sedated at: 11:56 AM
Conclusion
Based on the model, at what time will the patient wake up?
If the patient is scheduled to have a procedure until 4:45, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed?
Patient 4 27/M
Model
Sedated at: 10:15 AM
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
4.02
30
1.27
60
0.65
90
0.43
120
0.32
When will the concentration be ___ ?
0.4
0.2
0.1
Conclusion
Based on the model, at what time will the patient wake up?
If the patient is scheduled to have a procedure until 5:15, will he need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed?
Model
Patient 5 51/F
Chart
Sedated at: 12:55 PM
Conclusion
Based on the model, at what time will the patient wake up?
If the patient is scheduled to have a procedure until 5:00, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed?
Patient 6 7/M
Model
Sedated at: 2:55 PM
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
2.39
15
0.95
25
0.58
50
0.29
75
0.19
When will the concentration be ___ ?
0.4
0.2
0.1
Conclusion
Based on the model, at what time will the patient wake up?
If the patient is scheduled to have a procedure until 5:50, will he need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed?
Summary
Which patients will need additional anesthesia? Create a schedule for your day to ensure every patient is
properly cared for.
Models
A(x) =
D(x) =
7.8
0.20 2 +4.7
4.8
0.32 2 +6.3
B(x) =
E(x) =
5.1
0.35 2 +1.9
3.5
0.11 2 +2.6
C(x) =
F(x) =
2.2
0.28 2 +1.1
6.1
0.53 2 +2.1
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Boomdiggy.
Sedation Class
For surgeries and serious injuries anesthesiologists are responsible for administering
anesthesia. If correct dosages are not given and monitored it is possible that patients
may awaken in the middle of a procedure. Avoiding this disaster takes analysis of an
anesthesia’s concentration in the bloodstream. When a concentration of an anesthesia is below
0.1 mg/L it is no longer effective. In these two cases, you must use rational functions to model the
anesthesia’s concentration in the blood stream and determine when the patient may wake up.
1.
Given a model for this patient’s concentration over time complete the table with useful times
or amounts. Use the model to determine when they may wake up.
Model
y=
2.8
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
1.24
10
0.68
20
0.35
30
0.23
60
0.12
0.19 and 17.3
0.4
0.09 and 34.9
0.2
0.05 and 70.0
0.1
0.005 and 700 min
0.01
0.4 2 +1.3
Conclusion
Based on the model, when will the patient wake up?
The patient would wake up in 70 minutes
If the procedure lasts approximately 1 hour and thirty minutes, will additional anesthesia be
required?
Yes, the patient will need more because there are at least 20 minutes left in the procedure.
Can you determine the maximum amount of concentration that the patient had in their
bloodstream? If it is over 2 mg/L then it could be dangerous.
At 1.8 minutes it is 1.94 mg/L so it is close to dangerous.
2.
Given this patient’s concentration log, plot the points as coordinate pairs and sketch a rational
model to attempt to complete the rest of the log.
Chart
Concentration Log
Minutes
Concentration
Since
in mg/L
Dose
(y)
(x)
1
.83
2
1.27
3
1.37
4
1.32
5
1.21
10
0.75
20
0.40
30
0.27
40
0.20
50
0.16
60
0.14
When will… ?
~6.9
1
min
~15.8
0.5
~40.6
0.2
~81.6
0.1
Conclusion
Based on your model, when will the patient wake up?
The patient would wake up in approximately 81.6 minutes.
If the patient was put to sleep at 11:12 AM and the procedure lasts approximately 1 hours and 15 minutes,
This patient should be find and will get an extra 5 minutes of sleep!
If additional anesthesia needs to be administered 30 minutes before the previous anesthesia would wear
off, at what time does it need to be administered?
51.6 minutes after 11:12 is 12:03
Medication
In the world of medicine, every minute of relief can count for something.
Once medicine enters the blood stream it will soon have its most powerful
effect because it has its highest concentration. Over time the concentration
reduces and once it reaches a certain level, the medicine will no longer be
effective. The concentration level of medicines can be most accurately
modeled by rational functions.
In this assignment, there are models for 5 different prescription medications. It is generally accepted
that when a concentration of a medicine is below 0.15 mg/L is no longer effective. It is your job to
determine how long each medicine is above the 0.15 mg/L level and rank them from longest relief to
shortest relief.
Model
Anvil
(t is in minutes)
5.6
A(t) =
0.2 2 +2.3
Bozak
B(t) =
Crylenol
C(t) =
Depto
D(t) =
Equal
E(t) =
4.85
0.1 2 +3.2
8.2
0.3 2 +4.6
10.2
0.3 2 +3.1
7.135
0.4 2 +5.2
Length of Effectiveness
(round to the nearest minute)
From 0.04 minutes to 279.96 minutes ~
280 minutes – 4 hours and 40 min.
From 0.10 minutes to 323.23 minutes ~
485 minutes – 5 h and 23 min
From 0.08 minutes to 182.14 minutes ~
273 minutes – 3 hr and 3 min
From 0.05 minutes to 226.62 minutes ~
340 minutes – 3h and 47 min
From 0.11 minutes to 118.8 minutes ~
178 min – 1 h and 59 min
--Longest acting to shortest acting—
Bozak, Anvil, Depto, Crylenol and Equal
Suppose each of the medicines were administered at the times below. At what time would they no
longer be effective?
Anvil
Bozak
Crylenol
Depto
Equal
No longer effective at…
3:16 PM
~7:56PM
7:05 AM
~12:28PM
6:51 PM
~9:53PM
11:41 PM
~3:28 AM
10:28 AM
~12:27 PM
Anethesiology Crisis
Just as an anesthesiologist begins their shift, a crisis scenario occurs. A
patient has unexpectedly awaken from sedation in the middle of a
procedure. The information from the previous is no longer reliable and
they must quickly figure out the concentration of anesthesia in each
patient’s blood stream.
When a concentration of sedation is below 0.1 mg/L a patient is liable to wake up. You discover there
has been a serious paperwork mishap and models were written for the incorrect patients. Currently
there are six patients under sedation. You must use what you know about rational functions and
modeling to determine which model belongs to which patient and determine when they will wake up.
Patient 1 38/F
Chart
Sedated at:
3:15PM
Model
C(x) =
2.2
0.28 2 +1.1
Conclusion
Based on the chart, when will the patient wake up?
78.5 minutes or 1 hour and 18 minutes
At what approximately time will the patient awaken from the anesthesia? 4:33
If the patient is scheduled to be in surgery until 4:45, will more anesthesia need to be given? Yes.
Additional anesthesia must be administered 30 minutes before they would awaken. At what time
will more anesthesia be needed? 4:03.
Patient 2 14/F
Model
F(x) =
Sedated at: 2:48PM
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
1.99
10
1.11
20
0.57
30
0.38
60
0.19
When will the concentration be ___ ?
28.6
0.4
57.5
0.2
115
0.1
6.1
0.53 2 +2.1
Conclusion
Based on the model, at what time will the patient wake up?
The patient would wake up in 115 minutes. 1 hour and 55 minutes. So 4:43 PM
If the patient is scheduled to have a procedure until 4:25, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed? No, she will not need more.
Model
Patient 3 73/F
Chart
Sedated at: 11:56 AM
E(x) =
3.5
0.11 2 +2.6
Conclusion
Based on the model, at what time will the patient wake up? 318.1 minutes or 5 hours and 18 minutes.
5:14 PM
If the patient is scheduled to have a procedure until 4:45, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed? No, that anesthesia will be sufficient.
Sedated at: 10:15 AM
Patient 4 27/M
Model
A(x) =
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
4.02
30
1.27
60
0.65
90
0.43
120
0.32
When will the concentration be ___ ?
97.3
0.4
198.9
0.2
389.9
0.1
7.8
0.20 2 +4.7
Conclusion
Based on the model, at what time will the patient wake up?
The patient would wake up in 389.9 minutes or 6 hours and 29 minutes. 4:44.
If the patient is scheduled to have a procedure until 5:15, will he need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed? Yes, more will be needed. So it will be needed at 4:14.
Model
Patient 5 51/F
Chart
Sedated at: 12:55 PM
D(x) =
4.8
0.32 2 +6.3
Conclusion
Based on the model, at what time will the patient wake up? 149.9 MINUTES or 2 hours and 30 min. So
3:25PM
If the patient is scheduled to have a procedure until 5:00, will she need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed? Yes at 2:55
Patient 6 7/M
Model
B(x) =
5.1
0.35 2 +1.9
Sedated at: 2:55 PM
Concentration Log
Minutes Since Dose(x)
Concentration in mg/L (y)
5
2.39
15
0.95
25
0.58
50
0.29
75
0.19
When will the concentration be ___ ?
36.2
0.4
72.8
0.2
145.7
0.1
Conclusion
Based on the model, at what time will the patient wake up?
The patient would wake up in 145 minutes. 2 hours and 25 minutes. 5:20PM
If the patient is scheduled to have a procedure until 5:50, will he need more anesthesia? Additional
anesthesia must be administered 30 minutes before they would awaken. At what time will more
anesthesia be needed? It would be needed at 4:50.
Summary
Which patients will need additional anesthesia? Create a schedule for your day to ensure every patient is
properly cared for.
Patient 5 by 2:55, Patient 1 by 4:03, Patient 4 by 4:14, Patient 6 by 4:50