© Mathalicious 2014 lesson guide POUNDING HEADACHE How much Tylenol can you safely take? Acetaminophen is one of the most popular over-­‐the-­‐counter pain relievers in the country, but it’s also one of the most common causes of liver failure. There isn’t a big difference between helpful and harmful dosages, and sometimes even following manufacturer recommendations isn’t enough to keep people out of harm’s way. In this lesson students use exponential functions and logarithms to explore the risks of acetaminophen toxicity, and discuss what they think drug manufacturers should do to make sure people use their products safely. Primary Objectives Given acetaminophen’s half-­‐life, calculate its hourly rate of elimination from the body Write an exponential rule for determining how much of the drug remains in a person’s body after a given time Use logarithms to determine the required time until a person’s body contains a given amount of the drug Describe how the amount of medication changes over time during a period of continued use Discuss some of the dangers of exceeding the recommended dosage guidelines Discuss what drug manufacturers should do to help prevent accidental acetaminophen overdoses •
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Content Standards (CCSS) Algebra Functions SSE.3c BF.1a, BF.5, IF.8b, LE.5 Mathematical Practices (CCMP) Materials MP.3, MP.4 •
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Student handout LCD projector Computer speakers Before Beginning… Students should be familiar with writing rules describing exponential decay. They will need to use logarithms to solve for variable exponents, and they should be able to apply basic rules of exponents and logarithms, for instance that (a)bc = (ab)c and log(ab) = b log(a). Lesson Guide: POUNDING HEADACHE 2 Preview & Guiding Questions Students watch a commercial from 1990 touting the benefits of Tylenol over other pain relievers. In the commercial, a woman tells the viewer that, “Tylenol won’t irritate your stomach, the way ibuprofen sometimes can. That’s a medical fact.” The ad closes with the tag line “Tylenol: the pain reliever hospitals use most.” After watching the ad, ask students what sorts of pain relievers their family uses around the house. Any student who’s had a moderate headache will probably be familiar with a few of the more popular options: Tylenol, Advil, maybe Aleve. With so many pain relievers on the market, it’s natural to ask why there’s so much choice: there must be something that people prefer about one option over the other. The ad suggests that Tylenol has an advantage in safety; it’s easier on your stomach than something like Advil – which contains ibuprofen – might be. Based on the commercial, some students might even think that Tylenol is the safest option around! Once students have talked about their experiences with these pain relievers, ask them if any of them know how they work. Students may have a general idea that these pain relievers are absorbed by your body and, in turn, are able to dull pain. In Act One, they’ll come up with a more robust quantitative model to build on this intuition. •
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Does your family keep medicine around the house for headaches or pain? If so, what kinds? Why do you think people might prefer one type of pain relief over another? Do you think Tylenol is the safest pain reliever? Why or why not? Does anyone know how pain relievers like Tylenol work? Act One In Act One students examine how acetaminophen is eliminated from a person’s body over time. They’re told that the drug’s half-­‐life (the time it takes for one half of the drug to be eliminated) is approximately three hours, and they use that information to calculate the percent that’s removed from someone’s system each hour. Then students examine how the level of drug changes over time for someone following the recommended dosage guidelines for three different Tylenol products. Act Two In Act Two students look at changing acetaminophen levels for someone who continues to use the drug over the course of a 24-­‐hour period, and predict what would happen if he or she continued to take it for an entire week. It seems that the peak levels never get very far above the recommended one-­‐time dose, but then students listen to a story from This American Life about the dangers of even slightly exceeding the daily limit. Finally, they discuss what they think drug manufacturers should do to help prevent accidental acetaminophen overdoses. Lesson Guide: POUNDING HEADACHE 3 Act One: Take Two of These 1 The active ingredient in Tylenol is acetaminophen, which has a half-­‐life of 3 hours. If you take a single 325 mg tablet now, you’ll have 162.5 mg of the drug in your system in three hours, 81.25 mg in six hours, etc. Write a formula to model the amount of acetaminophen, A, in your system after t hours. Based on this, what percent of the drug is eliminated from your body every hour? After 3 hours you’ll have 162.5 mg of acetaminophen in your system because 325(1/2) = 162.5. After 6 hours you’ll have 81.25 mg of acetaminophen in your system because 325(1/2)2 = 81.25. After t hours, the amount of acetaminophen in your system will be A(t) = 325(1/2)t/3 = 325(1/2
Since 1/2
1/3
1/3 t
) . ≈ 0.794, each hour you retain about 79.4% of the drug in your system and eliminate about 20.6%. Explanation & Guiding Questions Although this is just an application of exponential decay, there are at least three main challenges for students trying to find the hourly rate. First, coming up with the form of the exponent takes a bit of reasoning. It may help to have students begin by thinking of the exponents in terms of half-­‐lives. The amount of drug in your system is multiplied by 1/2 after each half-­‐life, So the amount of drug in your system after h half-­‐lives have passed should equal the initial amount times (1/2)h. But since we want the rate in terms of hours (t), we need students to consider the relationship between h and t. Since one half-­‐life is three hours long, the number of half-­‐lives that have passed (h) is equal to one-­‐third the number of hours (t). In other words, h = t/3. If students have difficulty seeing this relationship, consider having them make a small table of values. Next, students have a formula written in terms of hours rather than half-­‐lives, but in order to find the hourly rate of decay, they need to rewrite it so that the exponent contains only the number of hours, t. To do that they need to recall that (a)bc = (ab)c, and thus (1/2)t/3 = (1/21/3)t. Taking the cube-­‐root of 1/2 yields a rate of about 0.794. Third, students should realize that 0.794 represents the fraction of drug remaining in your system after an hour, not what has been eliminated. The fraction eliminated is 1 – 0.794, or about 20.6%. (If students are familiar with writing exponential functions in terms of base e, that’s another route to the solution. The interactive includes equations written in both forms, so you can toggle back and forth if need be. For instance, 325(1/2)t/3 = 325et ln(1/2)/3 ≈ 325e-­‐0.231t.) •
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How many times have you multiplied 325 mg by 1/2 after one half-­‐life? Two half-­‐lives? Three half-­‐lives? What is the rule for how much of a drug will be in your system after h half-­‐lives have passed? For each half-­‐life that has passed, how many hours have passed? How can you replace h in your rule with an expression involving t? How can you rewrite your rule so that t is isolated in the exponent? By what percent are you multiplying the amount of drug in your system each hour? What percent of the drug remains after each hour, and what percent is eliminated? Deeper Understanding •
At this rate, how much time will it take for the drug to be completely eliminated? Does that make sense? (This model seems to imply that the drug will never be completely gone, which clearly isn’t the case. At some point, the last molecule of acetaminophen can’t be halved, and it’s eliminated as well.) Lesson Guide: POUNDING HEADACHE 4 2 Tylenol comes in different strengths, including Regular Strength, Extra Strength, and Children’s. If you take a single recommended dosage, how much acetaminophen will you have in your system after 0, 2, 8, and 24 hours? Milligrams per Pill 325 mg 500 mg 80 mg 2 tablets (adult) 2 caplets (adult) 3 tablets (4-­‐5 years) 0 hours 650 mg 1000 mg 240 mg 2 hours 410 mg 630 mg 151 mg 8 hours 103 mg 158 mg 38 mg 24 hours 2.6 mg 3.9 mg 0.9 mg Recommended Dose Acetaminophen in system (mg) Explanation & Guiding Questions In the previous question students developed a formula for the amount of drug remaining in your system after t hours: A(t) ≈ 325 × 0.794t. Now students will generalize the rule slightly to see how various starting dosages – corresponding to the recommended dosages for some different Tylenol products – affect the amount of acetaminophen in your system over time. For example, if you take two regular strength Tylenol, the amount of acetaminophen in your system will start out at 650 mg. After two hours, it will be around 650 × 0.7942, or 410 mg. And so on. Note that it would be slightly more accurate to calculate A(t) using (1/2)1/3 as the base and then round at the end, but in this case the difference is very small, less than 1 mg for any of the values in the table. •
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Where did the number 325 in your rule come from? If you take two regular Tylenol tablets, how much acetaminophen is in your system after 0 hours? 2 hours? What about the other types of Tylenol? Deeper Understanding •
Write a general form of your rule giving the amount of acetaminophen in your system after t hours for any starting amount. (A(t) ≈ A(0) × 0.794t ) Lesson Guide: POUNDING HEADACHE 5 3 Everyone responds differently to acetaminophen. However, let’s assume that for the average person the drug stops being effective below 200 mg. For Regular Strength Tylenol, the manufacturer suggests that adults take 2 tablets every 4-­‐6 hours. For Extra Strength: 2 caplets every 6 hours. If an adult takes the suggested initial dose, how many hours will it take for the amount to reach 200 mg, and is this consistent with the manufacturer’s recommendations? Regular Strength t
200 = 650 × 0.794 Extra Strength 200 = 1000 × 0.794t 200/650 ≈ 0.794t 200/1000 ≈ 0.794t ln(200/650) ≈ ln(0.794t) ln(200/1000) ≈ ln(0.794t) ln(200/650) ≈ t ln(0.794) ln(200/1000) ≈ t ln(0.794) ln(200/650) / ln(0.794) ≈ t ln(200/1000) / ln(0.794) ≈ t 5.11 ≈ t 6.98 ≈ t For Regular Strength, a person will drop below 200 mg after about five hours, which is right in the middle of the window of 4 – 6 hours. For Extra Strength, it will take about seven hours, which is slightly longer than the recommended time between doses. Explanation & Guiding Questions In the previous question students calculated the amount of drug left in a person’s body after t hours. Now they’re simply working in reverse: given a particular amount of drug, how long will it take to get down to that level? There are two basic approaches here for students. In either case, they need to realize that they will need to use logarithms because the unknown is in the exponent. Maybe the simpler of the two approaches is to isolate the expression containing the exponent on one side of the equation, and then take logarithms of both sides. Then, students can make use of the fact that ln(ab) = b ln(a), leaving them with only a simple division to complete the solution. Alternatively, students could take logarithms first, and then use log properties to rearrange terms. For instance, in the case of Regular Strength Tylenol, students could end up with ln(200) = ln(650 ⋅ 0.794t), and rewrite the right-­‐
hand side of the equation as ln(650) + t ln(0.794). Students can use logarithms in any base they choose. They could use the fact that the inverse operation of raising a base to a power is taking a logarithm in that base (if they have a calculator with a logb button), but here our base isn’t particularly convenient, so using either the base-­‐10 or natural logarithm to convert the expression to multiplication (and possibly addition) might be more straightforward. •
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Which values in your rule are known, and which are unknown? How can you rewrite the expression to move t out of the exponent? How can you get t by itself on one side of the equation? How do the times line up with the manufacturer recommendations? Deeper Understanding •
How long should you wait to take more Tylenol if you take an initial dose d? ( ln(200/d)/ln(0.794) ) Lesson Guide: POUNDING HEADACHE 6 Act Two: Call Me in the Mourning 4 Imagine an adult hurts his back and takes Extra Strength Tylenol. Instead of taking the recommended two pills every six hours, he takes three. Graph how much acetaminophen will be in his system over the first day. Then, if he continues at this rate, how much of the drug do you estimate there will be in his system after a week? Initial 1 hour 2 hours 3 hours 4 hours 5 hours 6 hours (a) 1,500 mg 1,191 mg 946 mg 751 mg 596 mg 473 mg 376 mg 6 hours (b) 7 hours 8 hours 9 hours 10 hours 11 hours 12 hours (a) 1,876 mg 1,489 mg 1,183 mg 939 mg 768 mg 592 mg 470 mg 12 hours (b) 13 hours 14 hours 15 hours 16 hours 17 hours 18 hours (a) 1,970 mg 1,564 mg 1,242 mg 986 mg 783 mg 622 mg 494 mg 18 hours (b) 19 hours 20 hours 21 hours 22 hours 23 hours 24 hours (a) 1,994 mg 1,583 mg 1,257 mg 998 mg 792 mg 629 mg 500 mg mg 2100 1800 1500 1200 900 600 300 0 Answers will vary. It looks as though the “peak” amount of Tylenol in his system is increasing with each dose, but at a decreasing rate. It seems to level off around 2000 mg for the day. Maybe by the end of a week he’ll be somewhere around 2500 m g or so. 0 2 4 6 8 10 12 14 16 18 20 22 24 hours Explanation & Guiding Questions Filling out the table should be fairly straightforward; once they know the initial dose, students need to apply the decay rule from Act One. The biggest difference here is that we need to account for additional doses every 6 hours. However, the rate of decay is always exponential, regardless of how much Tylenol this person has taken. Because someone following the recommended guidelines will have some of the drug in his system when he takes the next dose, at first it seems as though the amount in his body will continue to increase. But because the decay is exponential, the larger the amount of drug in his system, the larger the amount that gets eliminated. The total amount of acetaminophen seems to be leveling out at a peak amount around 2000 mg for the day. •
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How much acetaminophen is in his system after one hour? Two hours? Why does the amount of acetaminophen jump after six hours? How much is there after seven hours? How did you get that? What do you think will happen as he continues to take Tylenol over the course of a week? Deeper Understanding •
Does the graph you just plotted above represent a function? Why or why not? (The graph itself does not represent a function, because every six hours there are two outputs corresponding to a single input. In reality, of course, the amount of acetaminophen doesn’t change instantly when you take a new dose.) Lesson Guide: POUNDING HEADACHE 7 5 Listen to the clip from This American Life. If having more than 5000 mg of acetaminophen in your system can cause liver damage, do you think the man above who takes three pills every six hours needs to worry? Explain. Answers will vary. Even though the man is taking more of the drug than he should, it seems the amount of acetaminophen in his system at any one point is never going to get near 5000 mg. Or at least it would take a very, very long time to get up to that level. Explanation & Guiding Questions From the previous question, students have some idea that the man would have to take 1500-­‐mg doses of Tylenol for a really long time to get anywhere near 5000 mg. (In fact, at this rate he’ll never get much above 2000 mg. See the Deeper Understanding Question below). Given that information, it seems as though taking an extra pill doesn’t present that much danger. •
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What was the most acetaminophen the man ever had in his system over the course of a day? How long do you think he would have to continue taking Tylenol to get over 5000 mg? Does it seem like taking an extra pill per dose would be particularly dangerous? Deeper Understanding •
What is the most Tylenol that will ever be in this person’s body at one time? (Just after a new dose, the amount of Tylenol in his system equals the amount in his system at the last dose, times six hours of exponential decay, plus the new dose. In other words: A(t + 6) = A(t) ⋅ 0.7946 + 1500. If the peak amount indeed levels off, then eventually A(t + 6) = A(t). Substitute A(t) for A(t + 6) in the equation about, and solve. The maximum amount turns out to be approximately 2002 mg. When he has 2002 mg of acetaminophen in his system, the amount that he eliminates over the next six hours exactly equals the new dose, leading to a steady cycle that repeats indefinitely.) Lesson Guide: POUNDING HEADACHE 8 6 In reality, it’s not how much acetaminophen you have in your system at any given moment that matters; it’s how much you take over a 24-­‐hour period. If this danger zone starts at 5000 mg, do you think the man with back pain needs to worry now? If so, what do you think could be done to prevent such accidental overdoses? Answers will vary. Explanation & Guiding Questions Now students learn why it could be relatively easy to mistakenly overdose on acetaminophen. It’s not so much that taking more than recommended puts you at unsafe levels at any one point; it’s that, over time, your body can’t handle much more than the recommended dose without putting strain on your liver. So even though our man from above is only taking one extra pill per dose, over the course of a day – or days – he’s definitely crossing the 5000 mg per day threshold and putting himself at risk. Another problem with acetaminophen is that it’s so common. It’s not only a standalone pain reliever, it’s also an ingredient in lots of other medications. So if you’re already taking the daily recommended dose of Tylenol because you’re sick and have a headache, there’s a good chance that the cough syrup you’re also taking could put you into dangerous territory. And you might not even realize it. One thing that could be done – and in fact the FDA has already recommended it – is to reduce the amount of acetaminophen in products. Another is to make sure that products with acetaminophen have conspicuous labeling so that people realize when they’re putting more into their bodies. Manufacturers could also lower the recommended dose on bottles. In any case, students should come away from this lesson knowing that, even though it seems harmless, acetaminophen can actually be extremely dangerous if it’s not used properly. •
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What do you think now? Should the man be concerned that he’s taking more than the recommended dose? Can you name any other products (cold medicines, allergy medicines, cough syrups) that might also contain acetaminophen? Do you think people are always aware that the medicines they take contain acetaminophen? What do you think should be done to protect people from overdosing accidentally?