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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. © 21st Century Math Projects 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. © 21st Century Math Projects 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, will additional anesthesia be required? 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? © 21st Century Math Projects 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 Administered at… No longer effective at… 3:16 PM 7:05 AM 6:51 PM 11:41 PM 10:28 AM © 21st Century Math Projects 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? © 21st Century Math Projects 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? © 21st Century Math Projects 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? © 21st Century Math Projects 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. © 21st Century Math Projects 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 © 21st Century Math Projects Thank you for being my Math Friend! st If you liked this 21 Century Math Project You might like others. (Click the logo) Math it Up. Boomdiggy. © 21st Century Math Projects 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. © 21st Century Math Projects 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, will additional anesthesia be required? 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 © 21st Century Math Projects 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 Administered at… 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 © 21st Century Math Projects 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. © 21st Century Math Projects 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. © 21st Century Math Projects 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 © 21st Century Math Projects 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 © 21st Century Math Projects