## Saturday, April 3, 2010

### Murder Mystery: A Project for Logarithms

[Note added on 2-7-14: This is one of my most-read posts. If it helps you design a good project for your class, that's great. Please do not use it as is. Make adjustments for the age of your students (mine are in college), and fill in the details where I've put notes in brackets. Make sure you try each part out yourself before using this with students! To do otherwise is to ask for trouble.]

There've been some great discussions on teaching logarithms recently at JD2718 and f(t). At JD's blog, I mentioned a project I do in my classes, and nyates asked for the details. Here it is - just enough hook to get students working pretty well in groups. (One class decided we were CSI at CCC.)  :^)   But in my opinion, they're still mostly following the formula. If you have any ideas for improving this, I'd love to hear them.

Years ago, I shared my office with a chemistry teacher. When she complained that we math teachers often did a bad job in Intermediate Algebra courses with the (vital to chemistry) topic of logarithms, I decided to try to do it better. It was right at the end of the course, which always means neglect, so I moved it up a bit somehow. I also wanted to pull the students in more, so I made this project up.

This project uses Newton's Law of Cooling, though I don't mention that. (I often get students coming in the second day with that information. They've searched online, and found it. Good and bad - I like the research skill and initiative, but they want to use the formula they've found, instead of reasoning it out.) I work with them to figure out:  body.temp  = air.temp + excess.temp.at.time0 * b ^ t, where body.temp is a function of time, air.temp is constant, excess.temp.at.time0 is how much hotter the body was than the air when first measured, b < 1 (exponential decay), and t is measured in hours or minutes. The only log work necessary is: $log(x^{a})=alogx$, so this doesn't get them practicing the other log properties.

To help them remember when to use logs, I tell them that historically logs helped people multiply, divide, and find roots, but that now calculators do all that, so "the purpose of logarithms is to get the variable out of the exponent." (What do you think? Is that problematic?)

I have them get in groups, and give them maybe half an hour each day to work on the assignments. They've worked with exponential functions, and I start this project as we begin working on logarithms.

I play the theme music from Gilligan's Island to start class on the day we begin, and I tell them...
"Our whole class has gone on a cruise together and been shipwrecked. There's plenty to eat and drink, so no one's too stressed. But then, our classmate John Doe is murdered! (He was so quiet, you may not remember him.)"

Then I give them this:

Shipwreck and Murder

You’ve all been shipwrecked on a tropical island - a wonderful place, with bananas, coconuts, fish, and a pretty constant temperature of 80°. Your classmate, John Doe, has been murdered. You know there’s no one else on the island - it was one of your classmates that did it! None of you will sleep peacefully in your flimsy grass huts until the murderer is discovered.

You have watches, thermometers, and other simple tools, but no experts on murder investigations. The day John was murdered, everyone was walking around the island by two’s, noting its features, in hopes that would help you all figure out where you are. It turns out that everyone walked by the spot where John's body was found, and recorded the time when they were at a nearby spot (where they could see a volcano on the next island). Figuring out the time of death would likely narrow down the suspects to four or less.

Angel had a hunch that knowing the body temperature would help determine the time of death. So, at 1 pm, she checked the temperature of John’s corpse. It was 96.1°. Then at 2 pm, it was 91.7°.

Finding the murderer will be our goal. It might take us a few days. (I hope you don’t mind some sleepless nights...)

One more clue… this comes from Daniel, “I read a lot of murder mysteries. In one of them, this detective says, ‘A dead body cools off just like a hot cup of joe.’ I don’t know if that helps or not…”

~  ~  ~  ~  ~  ~  ~  ~

[Change Angel and Daniel to the names of students in your class.] After they've read this, and asked me questions about fingerprints, how he was killed ("looks like a coconut to the head"), and a few other distractions, I ask them to do the following assignment.

Assignment 1

[Do each of these assignments in groups of 3 or 4. If you want, you can turn in one copy per group. It will be important later to be able to describe how your understanding of the problem changed over time, so each person should keep neat notes on the case.]

We want to think about how a hot cup of coffee cools off.

1. What would be a reasonable starting temperature?

2. After about how long would it be cold?

3. About what temperature is it when it’s cold? (Why?)

4. Now let time be the x-axis (t-axis) and temperature (T) be the y-axis, and (on graph paper) graph temperature versus time for a cop of coffee, using what you know from common sense. Does a straight line graph make sense for this?

~  ~  ~  ~  ~  ~  ~  ~

I get a few volunteers to promise to actually measure the temperature of a cooling cup of hot water, and point out that old fashioned mercury thermometers will break and other body temp thermometers might break - they'll need a lab thermometer or a cooking thermometer. No matter how many people promise to do this, I know I might have no data by the next day, so I've saved data from an old class.

Assignment 2

Sarah says “We need some numbers here.” And she boils some coffee up over a campfire and measures its temperature with a thermometer Jessica provides.

Here’s what she gets:

The coffee starts out at 176 degrees, and cools off like this…

Min Degrees
1       169
2       162
3       156
5       146
10     125
15     111
20     101
30      90
60      81

1. If he measured the coffee at 2 hours and 3 hours, what temperature would it be?

2. Graph this data, and connect the points with a smooth curve.

3. So we can conclude that this graph has what line as an asymptote?

4. Give an example of a function with this asymptote.

~  ~  ~  ~  ~  ~  ~  ~

Assignment 3

John Doe’s dead body was lying near the viewing spot for the volcano, and it turns out that there was only one path going by that spot. So, after checking with each other, and remembering who passed whom, you all agree that the murderer was most likely one of the people at that spot right before or after the time of death.

Below are the times that each pair walked by the volcano viewing spot:

Colleen & Mouang 11:15am
Tiana & Armoriana 11:38

etc... (all class members listed)

Paolo & Vithaya 1:24pm

When you figure out the time of death, you’ll know the 4 most likely suspects.

Time of Death:
Suspects:

[Note: Before typing up this list in assignment 3, I've figured out the time of death, checked with the class to find out whether anyone objects to playing the killer, and made sure no one who'd be uncomfortable with it will be a suspect. The idea is that the time of death will be between two of the times listed, and the 4 people listed for those 2 times are the suspects. To find the time of death, they'll solve the equation body.temp  = air.temp + excess.temp.at.time0 * b ^ t for the time that the body was 98.6 degrees, which will be the time of death. If we've let the first temperature measured be at time 0, then the time we get from this equation will be negative. That's a nice switch, since some of them think story problems can never have negative answers.]

[Note added on 2-7-14: If you are a teacher planning to use this, the idea is to use your own students' names here. Make sure none is walking by at the exact time of death, so that the pair just before and the pair just after are both suspect, and have pairs walk by every 5 to 15 minutes.]

~  ~  ~  ~  ~  ~  ~  ~

Assignment 4: The Next Day

Of course, all 4 suspects swear they’re innocent. The next day, Rasha finds me dead, left lying right in the clearing.

My body temperature is 93.1°, and it’s 9:46am. You check at 10:16am, and it’s 87.2°.

Here’s everyone’s alibis:
• Colleen, Tiana, Robyn, Pardeep, Maureen, Jianfei, and Tayyaba were all swimming together from 8am to 9:30am.
• Brandon, Adeyinka, Cristina, Ash, Cookie, and Paolo were all looking for clams together from 8:30am to 9:30am.
• Mouang, Armoriana, JoAnn, Adam, Denisha, Edith, and Angel were all gathering coconuts together from 9am until they heard Kevin’s screams.
• Daniel, Josue, Natalie, Danielle, Dwight, and Vithaya were hiking from 9:30 until they heard the screams.

Please find the killer before someone else is murdered!

[Note: Students often like to accuse me of being the killer, so I get killed next. I tell them I think it's because I knew too much. The list of names puts each of the 4 suspects in a different group. The one without an alibi is the serial killer.]

I do this project in Intermediate Algebra (the community college equivalent of a high school Algebra II course, done in one semester) and in Pre-Calc. In Pre-Calc, I often end with an assignment to each write up their closing arguments as prosecuting attorney, explaining to the jury how we know the time of death, and why that means the prime suspect is the killer. A lot of them have fun with that assignment.

I give them a problem like this on their next test, and it's not great. I wish I had collected data on success rate on that question. It's probably less than half of the students. Worse than other questions...

1. Well, I'm not sure when it shifted, but they do better on this test problem these days.
2. If you plan to use this, you will need time to fix up the two lists of names. Work the problems out ahead of time, so you know how it will play out in class.]

Please comment, critique, and suggest improvements.

1. John Doe's initials make me awfully suspicious...

I'm going to play with this a bit. I want to see if I can modify it so that I can squeeze it in over a series of several regular lessons (with the abundance of extra time that I always have...???)

But, at first blush, this is great. I am just frustrated trying to figure how to work it in.

Jonathan

2. Thanks for sharing! We just finished a quick unit on logs in precalc - I'm going to try and find a way to use this when we get back from spring break.

3. I'm glad you both like it. Tell me how it goes. I'm thinking it will be easy nowadays to get that theme music. When I first did this, it was pre-internet. My mom found a tape at the library of TV theme songs.

4. This is a great lesson. I just wrote about it on my blog (mathforlove.com). Thanks for the great blog!

5. Thanks for this post--I just linked to it on my blog (it's called "math for love" if you'd ever like to check it out). I've really enjoying reading your blog since I discovered it recently. I'm very curious though: are you suggesting that people didn't do well at actually absorbing the mathematics of logarithms after this lesson? I would imagine it would work really effectively.

6. I'm impressed you can work with them to make that formula make sense, without needing calculus. I guess I can kind of imagine how it would go, but maybe you can say something, or give a link?

7. That's a nice activity; I'll give it a try with my class when we get to logarithms next month. Thanks for sharing this, Sue!

8. Hey Sue - I just wrote an incredibly long comment that got eaten by the computer. I'll try again, but quickly:

1) This lesson seems really fun!

2) In assignment 3 they have to solve the equation bodytemp = airtemp + (tempattimeofdeath-airtemp)*b^t for t. When and how do they learn how to solve this equation? Was that taught previous to the whole project, or during the project, and is the body-temp scenario used to explain the way the solution works or just to motivate the question?

3) Since you asked, I do think "the purpose of logs is to get the variable out of the exponent" is problematic. Logs have lots of other uses too (in stats, to re-express skewed data to increase symmetry, for example; in all branches of math, to turn problems about multiplication into problems about addition; etc). More importantly, I think it sends the wrong message to treat a fundamental object like it has a single purpose. Ideas come into being to solve specific problems, but once they exist we can use them for whatever uses our imagination and resourcefulness can put them to. Students should get to see this in order to understand math as a creative and evolving field. I think it's empowering for them to see it this way because it invites them into the history of math rather than being mere consumers of the products of that history. Am I taking your question too seriously?

9. Thanks, Ben. You are definitely not taking my question too seriously.

I don't have a good sense of when folks need to use logarithms for other purposes besides getting the variable out of the exponent. (Except, uh yeah, the chemistry work my former colleague pointed to, that got me started on this whole project.)

I think I have a problem with trying to please the students. They want easy answers, easy procedures, and I think I've provided lots of things like that in the past.

[For example, when we're working on factoring problems, I'd say "I think I can..." while writing the two sets of parentheses, and putting in the x's that multiply to x^2. I was just silly enough about it to be memorable, and at least one student uses it as a cue to remind her how to start. That one's probably not a problem, but if it seems so to you, please do say so.]

>When and how do they learn how to solve this equation?

Some before, and some during. It's definitely taught more than discovered. I have them take the log of both sides, once they get it down to c=b^t. The ones who've looked up Newton's law of cooling often do it differently. No problem when they get it right, but harder to figure the partial credit on wrong answers.

>is the body-temp scenario used to explain the way the solution works or just to motivate the question?

Not sure what you're asking here. We definitely talk about how temperature change works. I'm thinking of the Eric Mazur post now, and temperature change does not seem like one of the topics students would have misconceptions about. But maybe they do...

I do talk about how we say our coffee got cold, but it's no colder than the air.

Sorry about the system eating your comment. That's the second one recently. I wonder what the deal is. I have no idea what the character limit is for comments.

Dan, I do think they pretty much try to memorize the procedure. It might be something that will take a long time to sink in, no matter what.

David, good question. I try to get them to experiment with hot water and thermometers. (In one class early on, I felt responsible for a number of mercury spills, when a bunch of students did this with fever thermometers.)

Even if they don't, it seems reasonable that the liquid won't get colder than the air, and that the air temp will be an asymptote. That naturally leads to this equation, more or less.

Does that make sense? (If it doesn't, let's talk, and I can write another post about the formula part. My email is suevanhattum on hotmail.)

11. >temperature change does not seem like one of the topics students would have misconceptions about

Re-reading this months later, and I can't believe I said that!

The students do have some big misconceptions. They want the temperature change to be linear - straight line down to air temp, and then zap, no more change. I should ask them on a test after the project to show a graph of temperature change (no numbers, just shape).

12. So glad to find this while searching today! I have a pre-cal class that I try to do projects for every unit we cover, and I was stymied for our exponential/logarithmic unit. Thanks!! I'm really looking forward to seeing how this goes.

13. Hi Dawn, I'd love to hear how it goes.

14. Thanks so much for sharing this! I used parts of your project and other things I've found to create my own murder mystery. I'm excited to use it in my classroom!

-Ashley

15. Ashley, I'd love to see what you've put together. My email is suevanhattum on the hot email system.

1. Sue,

Just found this project and am hoping to use it in Pre-cal. Did Ashley ever email you her variation on a theme? Would love to see what she has done.
Tina

2. I don't think she did. But it's so long ago I wouldn't remember it. (I searched on 'ashley' in my email.)

16. So glad to find this site...I teach GT students and our Alg I we use CMP2 from MSU that is inquiry-based, so the temp cooling they learn in a unit called "Growing, Growing, Growing" where we actually cooled hot chocolate (and cider and tea...and noticed they cooled at different rates but the shape of the curve was the same, and then graphed first and second differences...cool calculus concepts at algebra I!) Anyway, I'm going to try this this coming week to see if it makes more sense of logs for them. As for the "too simple" question, I wanted to also relate some astrophysical data b/c I know that was when I used logs a lot in my own educational past...am looking for a Star Wars kind of project to come to me---anyone seen anything like that?

17. I haven't. Please let me know what you come up with!

18. Just saw this post and I am really looking forward to trying to use it with my class. I plan on trying to extend it a little with my Precalculus class, but haven't figured out quite how to do that yet. If you have any suggestions or if you have made any modifications since you posted it please let me know. Thanks, Courtney.

19. Hi Courtney, I haven't modified it any. If you come up with any extensions, I'm interested!

We're in the beginning stages of it right now.

21. Hi anon, I don't usually post 'answers'. If you'd like help understanding how this works, feel free to email me (suevanhattum on hotmail).

22. What is supposed to be done in Assignment 4? How do we find the time the murder occurred?

23. This project should come after a unit on exponential functions, and using logarithms to solve them.

Then it's helpful to know that dead bodies (and coffee) cool off like an exponential decay function, using the surrounding temperature as the asymptote (so that what decays exponentially is the 'extra heat').

In my class, we use y=a*b^t+c as a template, and find the values for a, b, and c. (There are other ways to work with this...) We let the first given time be t=0, our second data point comes from the other time, and we also know that the body was 98.6 degrees just before dying.

24. So glad I found your post! I plan to start this in 2 days. Very fitting that the class just watched the pilot episode of LOST to parallel Lord of the Flies. Might be kinda cool to link up with an English teacher for this unit too! Since there is a volcano near by.. it might be fun to use the Richter Scale for an earthquake near by.

25. I also wondered if you happen to have a rubric for grading the project?

26. I don't have a rubric. I test them on a similar problem on the test. (And they do better on this type of problem than they do on population growth problems. I think they've learned this type, but not the bigger concept.)

27. Just worked through all of the math. I thought it would be fun to extend the lesson a bit and I am having 2 suspects in the end. One went hunting and killed a boar. The class will find out that the boar was killed in the same minute as the teacher so it rules out that suspect. I wanted it to be more challenging then just looking for the missing person from a list of alibis.

28. Nice!

(I think this post gets more hits than anything else on my blog. Maybe others will use your alternate ending too.)

29. some one should post the answer so i can see if im right.

30. My teacher gave me a similar problem but i dont really understand it how do you do assignment 2?

31. Please ask your teacher for help. To offer the right help involves understanding which parts the student already understands, and which parts she doesn't. I can't help you without seeing what makes sense to you and what doesn't.

32. I am going to try this problem starting next week in my high school pre-calculus classes. I am introducing logarithms and was looking for an experiment to use next week. What I think I will do, though, is microwave cups of water and use the chemistry teacher's thermometers to collect data. I will not give them the scenario yet. We will collect and graph the data as a classroom exercise as preparation for speaking about logarithms. I will combine assignments 1 and 2 due to limited time with an A/B day schedule.

Very exciting!!!

33. Michelle, I'd love to hear how it goes.

1. Sue,

In the equation, bodytemp = airtemp + (tempattimeofdeath-airtemp)*b^t what does the b of the b^t represent?

34. The rate of exponential decay (sort of). If you prefer to use e^rt, that works too. I think this representation is easier. If you lost a quarter of your excess heat each hour (and you measure time in hours), b would equal .75. So it's actually the percentage of excess heat retained.

35. OK...I think I will use e^rt so that the students use "e" to calculate the answer. If I use e^rt, how would we determine rate (r) and when we solve for t, it will be in hours, right?

36. When I wrote "I think this representation is easier" above, I meant my representation. I'm curious why you prefer to use e.

Here's a bit more about how it goes now, cobbled together in part from a comment of mine above:

We use y=a*b^t+c as a template, and find the values for a, b, and c.

We know the temperature will eventually cool of to be the same as the air temp, so that's the asymptote, or the shift, and must = c.

We always start with two time and temp pairs as data. We let the first given time be t=0, our second data point comes from the other time (measured as minutes or hours since the first time). Plugging in these two data points to the template lets us find a and then b. (If we didn't use t=0, it would still be possible to find a and b from the two data points, but much harder.)

We also know that the body was 98.6 degrees just before dying. Now that we have a proper function, we can find the time of death from this.

37. I wanted to use "e" because we had just introduced it and it would be a way to apply "e" but I see your point. I solved the equation and found that t=-.44 which is a fraction of an hour right? When multiplied by 60, the answer is -26 which means 26 minutes before the 1st temp of the corpse was taken. Is that right?

38. I don't know without re-doing it. Try plotting your points as a graph, using desmos, if that will help you make sense of it. It sounds about right...

39. I'm working through this project. Too fun! Next step is to find the Gilligan's Island soundtrack. I wanted to check my result for Assignment 4...the death of the teacher. I got 38.618 minutes. Not feeling like this is correct. Thanks for your help!

40. I just edited the post to point to a soundtrack on youtube. Thanks for pointing that out.

Email me with the steps you took, and I'll point to your mistake. (suevanhattum on hotmail)

41. I found my mistake. Don't you know it was in the arithmetic, not in the mathematics. Thanks for this fun activity!

42. Sue,

I am wondering if you happen to have even more information about this project, or any sort of additional materials you use. This looks like a fun project and I would like to start it next week but have very limited time to put things together this week. My email is aderosa@mancosre6.edu. Thank you!

43. I tried to email you and it bounced. Try emailing me directly (mathanthologyeditor on gmail).

44. Hey Sue,looks like a great project, do you have an answer key for this you can post?

45. Nope. The teachers can figure out the math. And I don't want it searchable for students. You are welcome to email me with particular questions.

46. My daughter got this project for her 8th grade math class. The teacher gave only the 3 pairs that you have mentioned in this for assignment 3. The time of murder she got is not very close to any of the three values and she thinks the in between values for the other pairs should be there to find the true murderer. The time (given), closest to the value she got is about 1 hr before the murder. Are the 3 times given in your problem enough to find the murderer?

47. No, that's why I said I list the rest of the class in between. The way I do it is to put students in pairs, and have one pair walk by every few (say 4 to 15) minutes. I make sure none walks by at exactly the right time, so there are two pairs who are suspects. The second murder splits everyone up, so only one person is a suspect for both murders.

Did the teacher use the names I gave here, instead of the kids in the class?

48. It was exactly what you posted here. The time she got was roughly an hour after the second pair. She said that the 3rd pair went by after the body was discovered and so they can't be suspects. So it was very confusing to find 2 pairs of suspects. They were supposed to make a poster for the puzzle. She did everything in the poster with out writing the suspects or the murderer. Just wrote the titles, suspects and murderer and wanted to ask the teacher about this part before completing it. When she went to ask her, the teacher has asked her to turn it in for a lower grade as she didn't complete it. Never answered her question too. I didn't check her answers, but I am pretty sure she got the answers right.

49. I don't understand. Email me to discuss answer if you want. (mathanthologyeditor on gmail)

50. This comment has been removed by the author.

51. What do you think, Nelson? (And if you're not old enough to drink coffee, ask someone who is. Or google it. Or google the time McDonald's got sued because someone got burned by their coffee. Lots of options. No need to ask here.) You could buy a cup of coffee to see how long it takes to get cold. Or you could heat up some water, and check that.

Sorry if I sound snippy. I am concerned because I've also gotten questions in email that seem like people don't want to think. I'm not sure yet what I want to do about it...

52. Are you telling me you have no access to a way to heat water up? You can at least test the time, can't you?

53. is it kevin

54. What do you think?

55. I need help I'm stuck

56. The problem can't be solved with the information given. The person using the problem with students needs to add more students (and change all the names). If you're a teacher, please let me know what more you would need to use this. If you don't understand exponential functions and how that relates to cooling off, don't use this in a classroom. You can see in the comments above that a very irresponsible teacher used this just as it is, which makes no sense.

57. I want the answers .!

58. Hello Sue!

Quick question!

Granted, I'm reading this at 4 a.m. and might be overlooking something very simply in my exhaustion. However, I am trying to follow the project (which is amazing, by the way) and I get stuck on Assignment 4. If you've narrowed it down to 4 suspects, and then in step 4, split them up into 4 groups (one per group) you would be able to determine the killer if you could determine which group of the four is unaccounted for. Totally get that. But how do you determine which of the four groups is the one without the alibi? The assignment just seems to stop there.

Thanks!

59. By the time of my death. (I am imagining you awake now, and seeing it yourself without this comment...)

60. Hi Sue,

I am working through this project to do with my students and have a question that I can't seem to resolve: In the first set of data, the body temperature decreases at a rate of 4.4 degrees in one hour. In the second set of data (assignment 4), the body temperature decreases at a rate of 5.9 degrees in 30 minutes or 11.8 degrees per hour. Since it has a lower starting body temp when found, shouldn't it be decreasing more slowly than the first body found?

61. None of my students has paid enough attention to notice this.I would ask: Do different cups of coffee cool at different rates? What does it depend on?

I'm guessing the first person had more insulating body fat (thicker coffee cup), and the second person was smaller overall.

But that difference does seem big. I have always just made up the temps, and then found the time and made sure it worked with the clues. Maybe I should keep them closer to one another. I just looked up body cooling (again), and you must tell student this is an oversimplification of real life.

I re-read what you wrote, and it's important to say "in the first hour" and never say "per hour". That would indicate a constant rate of change. Every time a student says a thing like that I ask what they think will happen in the second hour, and we return to the graph. Students are so used to straight lines, they want to treat everything the way they do straight lines.

62. I would figure that the temperature would bottom out at room temperature or 80 degrees on the island. Also, I would figure 98.6 being the temperature just prior to death. The data follows an exponential graph but takes the data down to 0. So when the cup of coffee gets to 2 or 3 hours the temp should be room temp but using the model my regression takes it down to zero.
I have tried working some transformations using the equation I find but am at a loss as to how to fix the issue.

63. >The data follows an exponential graph but takes the data down to 0.

It sounds like you mean that your regression takes the graph down to 0. If you want to use regression, instead of doing the algebra, you would need to use excess heat (how high above room temp) as your y values.

64. Yes my regression does take the data down to 0. What algebra steps are used to solve this. I know that at whatever time zero the body is 98.6. ...
I got it now. Thanks