The Effect of
Student-generated Application Problems
Jean
Foley
Math
Back
to Scholarship of Teaching: 2003-04 Participants and Their
Projects
Project Title: Student-generated Application Problems as an Aid to Understanding Practical Uses
of Logarithmic, Exponential and Trigonometric Functions in College Math II
Investigator: Dr. Jeanne M. Foley , Department of Mathematics, Statistics and Computer Science
Course Context:
Math 121 (College Math II) introduces the topics of conic sections and logarithmic, exponential and trigonometric functions. The majority of the students enrolled in this course are not math majors; most come from engineering, construction and applied sciences majors. The course is required for these majors. For some it is the terminal math course, but for many it is a prerequisite to a calculus program requirement.
Project Summary:
The course material in College Math II naturally lends itself to applications within the major fields of the typical enrollee, but in my experience most students still struggle to understand the practical application of the theory to problems and to see the specific relevance of the material to applications in their own fields.
This project evaluated the impact on learning (and motivation to learn) of a series of required and extra-credit assignments in which students used Internet resources to develop applications of course topics to problems in their own fields of interest. For each assignment, students were required to electronically submit a written report including a description of the general problem, with web links for illustration and explanation, plus a detailed problem of their own design, along with a step-by-step solution. These reports were posted on the E-Scholar course web page, and some of these student-generated problems were used on subsequent in-class tests and quizzes.
End-of-term evaluations indicated a significant positive impact of the assignments on studentsí attitude toward taking this course. Seeing the relevance to problems in their own fields, combined with the active nature of digging up their own applications and seeing their efforts posted for class perusal and included in class-wide assessments resulted in a gradual, if sometimes grudging evolution from dread of the course to appreciation of its utility, and even to enjoyment of the challenge for some students. Analysis of final exam scores suggests a link between degree of participation on the project assignments and mastery of the course fundamentals, although the study design did not allow for controlled comparisons.
Key Learning Activity:
Three application project opportunities were provided to students. The first assignment, developing an application of logarithmic or exponential functions, was a required project worth 30 quiz points out of a total of 200 quiz points, with the total quiz points counting for 20% of the final course grade. (The instructions given to students are reproduced on the following page.)The second assignment was identical in format to the first, but covered applications of trigonometric functions to periodic or cyclical phenomena. This project was presented as an optional, extra credit assignment worth 15 points; however, the five students who failed to complete the first assignment were given the chance to substitute this assignment for the first 30-point project. A third and final project on conic sections was offered for 15 additional extra credit points two weeks prior to the final exam. An initial brief assignment introduced students to the electronic drop box option in the course delivery software. A summary of this introductory assignment is given in Appendix 1.
Instructions to students for first outside project assignment for MATH 121 S04:
This required project is worth 30 quiz points and is due by 5 p.m. Friday, March 19.
The assignment must be submitted as a Word file via the E-Scholar Course Work Assignment drop box, just as you did for the 5-point assignment at the beginning of the semester.
The purpose of this assignment is to have you search out and describe an application of logarithms or exponential growth or decay thatís related to your major or intended career or some area of interest to you. Sources of leads on what kinds of applications might be used in your major would include textbooks or teachers of your major classes. Search engines on the web are also a great source (e.g. Google). NOTE: If you do an ìadvancedî search and limit your search to web sites ending in .edu, youíll get a lot less junk coming up in your searches. If you canít come up with anything, check with me and Iíll see if I can get you started.
Once you find an appropriate application, your job will be to write up an explanation of the application as well as to create an application problem with a detailed solution (see sample writeup at end for an example of what this should look like.) These projects will be posted in the ìResourcesî section of the E-Scholar web site so your classmates can view them, and some of the submitted problems will be used on a quiz towards the end of the semester when we review this material in preparation for the final exam.
Your report should contain the following sections: (Instructions are given in italics.)
1. NAME: List your name here, and if you have created a personal web page, you can include a hyperlink to the page. (If you need help figuring out how to insert hyperlinks into a Word document, check with me.)
2. DATE: Put the submission date here.
3. TYPE OF APPLICATION: Put an ìXî by the type of application problem you chose.
(check one) ___ Logarithms __ Exponential Growth __ Exponential Decay
4. AREA OF INTEREST: List your major or career or interest area here.
5. E-MAIL QUESTIONS OR COMMENTS TO: Insert your email address here, with a ìmailtoî hyperlink.
6. OVERVIEW: Write a brief paragraph giving an overview of your application.
7. DETAILS OF THE APPLICATION: Provide a clear and detailed explanation of the problem and how logs or exponents are used in approaching the problem. This will probably be your longest section, from a few paragraphs up to one page in length. (See sample provided. ) NOTE: Sample is appended as Appendix 2.
8. THE PROBLEM: Create an application problem that can be solved given information and numbers you provide in this section and using your explanation from part 7. This section will probably be pretty short, kind of like a homework problem or a test question.
9. THE SOLUTION: Provide a step-by-step demonstration of exactly how you would solve the problem you described in part 8. This section should carefully lay out ALL of the steps youíd need to go through to solve the problem, including the results of any intermediate calculations.
10. OTHER RESOURCES: Provide links to at least one web site that gives further information on this topic.
NOTE: Additional hyperlinks can be embedded in other sections (see sample), but AT LEAST ONE link must be included in the ìOTHER RESOURCESî section. These are links to outside web pages that you found in the web, not web pages that you have to design yourself.
Key Findings:
The two key research questions examined in this study were:
1. Can studentsí understanding of how to apply abstract mathematical concepts to practical problems be enhanced by having students develop their own application problems within their fields of interest?
2. Does encouragement of active exploration of the relationship of course material to studentsí own interest areas enhance their engagement in learning these mathematical concepts?
Answering the first question rigorously would have required a research design randomly assigning students to control (no project assignments) vs. treatment groups, a path I chose not to take. However, I did observe that of the 39 students enrolled in the course, the 16 students who completed at least one optional assignment in addition to the required assignment averaged 76.5% on the final exam (grade of C), whereas those 18 students who completed only the one required assignment averaged 71.1% (grade of C-). Of the five students who completed no projects, three failed to take the final and received F grades for the class; the remaining two averaged 65% (grade of D). The significance of these observations is of course open to question, since it is likely that the students who came into the course with better skills and higher motivation would also be the ones more likely to completed assigned and optional coursework.
Some insight into the second question may be gleaned from student responses to an end-of-term survey of students perceptions of the application project assignments. (26 of 39 students completed the surveys; a blank form is attached as Appendix 3.) Two thirds of respondents indicated that the assignments helped their understanding ìa fair amountî or ìa lotî, and the vast majority (85%) felt that the points assigned to the projects were worth the time spent working on them. A tally of responses is provided below, and excepts from open-ended comments are given in the next section.
QUESTION: How would you rate the contribution of doing the project(s) towards helping you understand the given topic(s)? (check one)
|
RESPONSES: |
Percent of total responses |
|
None |
4% |
|
Helped a little |
27% |
|
Helped a fair amount |
50% |
|
Helped a lot |
19% |
QUESTION: Was the time you invested in the project(s) worth the points you got for it?
|
RESPONSES: |
Percent of total responses |
|
Yes |
85% |
|
No |
0% |
|
Sort of |
15% |
Another subjective indicator of an increase in studentsí level of engagement in the learning process was the increasing creativity of the students in formulating applications for the mathematical concepts covered in class. Whereas most of the first-round projects reported relatively straightforward applications to studentsí majors, the second and third submissions often ranged farther afield into potential career applications or even into the realm of whimsy. This progression is illustrated by the sample projects included in the appendices. Appendix 4 is a typical round-one application of exponential decay to a nuclear engineering problem. Appendix 5 is a second-round report by a student who realized while sitting in the barn on a milking stool that the sound of the milking machine was a perfect example of rhythmic changes that could be modeled with a vertically-shifted sine function. Appendix 6 is another studentís humorous description of how to use elliptical conic sections to figure out where to sit in a famous cathedral in order to eavesdrop on two elderly women gossiping about the participants in a wedding ceremony.
Evidence of Student Learning:
n Relationship between project participation and exam scores:
The following table presents the relationship between number of application projects submitted and the average final exam score for students in each category.
31 of 39 students completed the first (required) project assignment. Three of the eight individuals who failed to submit the first application report took advantage of the opportunity to make up the points by completing Project 2, while 12 of the students who turned in the first assignment also completed the second one as an optional extra credit assignment. Only nine students completed the third project report, which was offered exclusively as an extra credit assignment. The low participation on the final application project may have been due at least in part to the time demands on students as final exams approached.
n Impact of application project participation on student attitudes -- Excepts from student comments from end-of-term survey:
… I think the application projects were very helpful in understanding the section because they give or made you find real world application helping you understand the topic and how the topic applies to the real world. It gave a reason to learn the topic and why the topic is needed in the real world instead of just do this problem and you will never see it again. If itís not too much of a time constraint it would be nice to have a couple more.
… I think that projects better represent a studentís ability to do real life problems than tests do. I know that when I can relate a problem to a real life solution I learn better and can understand the material far better than when all I am learning is the equation and practicing multiple problems in the homework.
… I liked the write up assignments a lot. Not only was it an opportunity for me to get more points, but it actually helped me be able to apply it to real live situations. I am normally one to hate word problems but I didnít mind these as much because the way you had the steps set up was easy for me to follow. I normally give up on math problems when I think that I will not need them in everyday life but this helped me understand that even what I think I wouldnít use in my everyday life I actually do.
… I liked how in the first project you wanted us to relate it to our major. I thought this brought meaning to what we were doing in class.
… I think that the projects were a really good idea. It really helped me understand logs a lot better.
… I think that doing these problems helps with seeing how this kind of math can be applied to what you will be doing later on in life. Most of us go through math class saying that we never see the actual point in it since it will never apply to what we are doing. However it does apply to just about everything we do, it's just that we don't see it.
… I thought the project was a really good idea. It wasnít very hard to find ideas for it, and making up the problems myself helped a lot to understand what we were doing. It was also a really great way to make up for some of the points that were lost on quizzes.
… I liked the layout of the project it was a good opportunity to use our imagination and apply it to math and it was easy to do online. It was also very helpful when it was offered for extra credit.
… I thought the project assignment was a great idea. This is the first math class that I have had that made me understand how math can apply to real life situations.
… I thought the projects were worthwhile and I learned a lot. They helped me learn more about the material.
… I believe each project was worth while, although I only attempted the first one on Logs/Exponents. Nevertheless, they will help other kids and sections. Sometimes, people need a little project to stimulate their initiative, so continue with the projects.
… I thought this was a good project, especially considering that two of them were for extra credit and they really werenít all that difficult. The hardest part about them was trying to find a topic and finding good research. The format you had laid out for the project was easy to follow step by step which gave me an idea of what you were looking for.
List of Appendices:
1: (page 7) Procedures used to introduce students to the use of the E-Scholar portal and the electronic drop box function in the E-Scholar Course Delivery Tool.
2: (page 10) Sample project report provided to students for use as a model.
3: (page 12) Survey form for student responses to the application project assignments
4: (page 13) Sample student project report from Assignment 1
5: (page 15) Sample student project report from Assignment 2
6: (page 19) Sample student project report from Assignment 3
APPENDIX 1: Procedures used to introduce students to the use of the E-Scholar portal and the electronic drop box function in the E-Scholar Course Delivery Tool.
1. Introductory e-mail and hot link to the E-Scholar portal:
The week before classes start, I send an e-mail message containing a hot link to the E-Scholar portal and instructions for logging on to the site to all students on the E-Scholar roster for each of my courses. This message is brief, outlining the directions for logging on, explaining that the site is set up to be viewed using Internet Explorer rather than other web browsers, and giving directions on how to click on the course web link once logged on. This message also informs students that the site contains a copy of the course syllabus with a day-by-day schedule of lecture topics, assignments and tests for the entire semester to give them an incentive to log on. Within this message I also note the availability and location on the web page of the ìCourse Delivery Helpî button and suggest that students nose around the site and see how it works before classes get underway.
(Outcome note: This spring semester I was pleased to note that many of the students had actually viewed the page prior to attending the first class meeting, as evidenced by the fact that a number of them came with hardcopies of the syllabus already in hand. Furthermore, I had included in the ìAnnouncementsî section of the Course Home page a note that there would be an open-notes quiz at the end of the first lecture, which encouraged not only first-day attendance, but also rewarded those who actually took the time to look at the page with some advance information other students didnít have.)
2. First assignment using the ES-CDT:
On the first day of class, I announce a brief take-home assignment to be done for the following class day and requiring logging on to the E-Scholar course web site. To complete the assignment, students must log on the site, then follow directions in the ìAnnouncementsî on the Course Home page to reach the ìAssignmentsî page. Here the students are directed to click on first assignment and follow directions on how to open and print a one-page profile form asking for background information (previous related coursework, next planned math or stats course, major, intended career plans, type of calculator the student has, and any concerns they have about the course.) Printing this form requires following the on-screen instructions, and/or using the Course Delivery Help module. The student is then directed to fill out the form and turn it in at class the next day for an automatic 5 points of homework credit.
(Outcome note: In addition to ensuring prompt return of the student profile sheets, this exercise also immediately identifies any students with log-on problems that I can then get cleared up before any ìrealî assignments are posted. This exercise also further encourages the habit of frequent checking of the course web page, because I also post an announcement giving advance warning of the first ìpopî quiz, usually given on the second or third day of class.)
3. Second assignment using the ES-CDT:
The second week of class, after all students have successfully logged on to the E-Scholar site at least once (and late-add students have access), I post a formula sheet in the ìLecture Notesî section of the ES-CDT that students can print off and use on upcoming quizzes and the first exam.
(Outcome note: If the previous assignment hasnít succeeded in getting students to log on, this one always does, since nobody want to be without the legal ìcheat sheetî for quizzes and tests...)
4. Third assignment using the ES-CDT:
A new item I added this semester with the advent of the new ìCourseworkî module is a short assignment introducing the methods for copying a posted file onto the studentís desktop, renaming the file with their Stout username, and then using the ìbrowseî feature of the drop box to submit the assignment to me for another 5 points of homework credit.
(Outcome note: I have only used this assignment once so far, in a pilot run in my Spring í04 MATH 121 class, as a preparatory assignment to a class project that will be part of my Scholarship of Teaching and Learning project. The assignment proved to be a quick, easy exercise [most students reported that it took them no more than 5 minutes] that reinforced the ease of use of the CDT. This writeup of this assignment is what has been recently used by the ES folks as a ìbest practiceî in their training sessions. A copy of the instructions for this assignment is attached to this report.)
SUMMARY COMMENTS:
When I first began using the E-Scholar course delivery tool in the spring semester of 2003, there was some resistance to its use, particularly among the non-laptop (then non first-year) students. This resistance has diminished considerably in each of the two successive semesters Iíve used the tool, most likely as a consequence of its growing use among instructors as well as the doubling of the number of students involved in the laptop initiative. Iíve also definitely seen a turnaround in attitude among students over the course of any given semester as they begin to appreciate the easy availability of resources such as practice tests, answer keys, links to calculator help websites, tutor information, etc. Several students whoíve previously used Blackboard commented about the generally faster response time afforded by the E-Scholar portal. I have also noticed an increasing number of students writing favorable comments on their course evaluations about the extent and easy availability of extra course materials that I post on this website.
MATH 121 (Foley) -- Instructions for Quiz 3 (take-home, 5 points)
This assignment is due by class time on Monday 2/2/04.
THIS IS A QUICK EXERCISE THAT WILL EARN YOU 5 EASY QUIZ POINTS AND PREPARE YOU FOR A MORE COMPLEX E-SCHOLAR BASED ASSIGNMENT THAT WILL BE DUE LATER IN FEBRUARY.
1. Log into the E-Scholar course website (https://portal.uwstout.edu/index.cfm ) using your Stout email username and password.
NOTE: The E-Scholar portal is set up to run using Internet Explorer. Things probably wonít work properly if you use another browser such as Netscape Navigator.
2. Click on the MATH 121 listing in your schedule at the upper right.
3. Click on the ìCourse Workî button at the left-hand side of the screen.
4. Click on the ìAssignmentsî tab within the Course Work module.
5. From the assignment entitled ìQuiz 3 Take-home Assignmentî, copy the file highlighted as Q3_tkhom.doc to your disk or desktop, open it in Word, then save it as a file named with your Stout email username followed by the numeral 1. (For example, I would save my file under the name foleyj1). You donít need to make any changes to the document, just save it as is, but with the new file name.
To save a file to your disk or desktop, right-click on the file and chose the ìSave Target Asî menu item. If you have any trouble saving/copying/printing things from the E-Scholar web site, look at the information in the ìCourse Delivery Helpî tab at the upper left of your screen.
6. Use the ìBrowseî function in the Drop Box in the E-Scholar Coursework Assignments section for this assignment to locate and add the file you just created into the drop box.
Again, if you have any trouble with this step, look at the information in the ìCourse Delivery Helpî tab at the upper left of your screen.
7. Collect 5 points.
APPENDIX 2: Sample of completed assignment for first outside project for MATH 121 S04:
1. NAME: Jeanne Foley
2. DATE: 2/29/04
3. TYPE OF APPLICATION:
(check one) _X_ Logarithms __ Exponential Growth __ Exponential Decay
4. AREA OF INTEREST: Physiology/biochemistry of exercise and sports
5. E-MAIL QUESTIONS OR COMMENTS TO: foleyj@uwstout.edu
6. OVERVIEW: Because of my background in college basketball coaching and my research in exercise physiology, I have developed an interest in the physiology and biochemistry of muscular contraction. Intense muscular exercise, such as performed in sprint-type activities like a fast break in basketball, produces lactic acid. The acidifying effect of this compound is measured on a logarithmic scale called the ìpHî scale.
7. DETAILS OF THE APPLICATION: As lactic acid builds up in muscle, it acidifies the cytosol or fluid component of the muscle cell. This compound also leaks out into the vascular system to acidify the blood. The level of acidification is measured on a logarithmic scale called the ìpH scaleî. (Follow this link for further explanation of pH: http://www.sp.uconn.edu/~geo101vc/Lecture12/sld006.htm.)
The mathematical definition of pH is:
pH = -log [H+]
where [H+] represents the concentration of free hydrogen ion in units of moles per liter (M/L).
Concentrations of the acid ion H+ are typically extremely low, with the middle or neutral value of the scale being a concentration of 0.0000001 M/L. In scientific notation, this concentration is expressed as 10 -7, a number whose base 10 logarithm is simply the power of 10, or -7 in this case. (In fact, the ìpî in pH stands for ìpowerî, i.e. the ìpowerî or exponent above the base 10.) The negative sign in the formula ensures that values for pH will be positive, since H+ concentrations measured in nature vary from extremes of 0.1 M/L (pH = 1) on the high end of the scale to 1 x 10-14 (pH = 14) on the low end.
A logarithmic scale is needed because these extremes of concentration vary over a range that is 100,000,000,000,000 (1 x 1014) times larger at the high end than at the low end. This huge range would make it practically impossible to graph pH changes without using logarithms to condense the range to a manageable scale which ranges from 1 to 14. Because of the negative sign in the pH equation, lower pH numbers actually represent higher H+ concentrations, or more acidic conditions, whereas higher pH numbers indicate lower H+ concentrations, referred to as either basic or alkaline conditions. A pH of 7 is considered neutral, neither basic nor acidic.
The typical pH inside a resting muscle cell is about 7.4, indicating a slightly basic or alkaline environment. Shortly after beginning a bout of intense, anaerobic exercise, the lactic acid produced during muscle energy production begins to lower the muscle pH. Even under the most intense exercise conditions, however, muscle pH almost never reaches a value lower than 6.0 because protective regulatory systems intervene to shut down the muscle contractive apparatus before it can produce conditions that could permanently damage or even kill the cell.
8. THE PROBLEM:
a). If hydrogen ion concentration doubles during a bout of intense exercise, what will the new pH value be? (Assume an initial or resting pH of 7.4).
b). Determine the factor by which muscle cell hydrogen ion increases during exercise that decreases the pH from a resting level of 7.4 to a value of 6.7 at the end of the exercise bout.
9. THE SOLUTION:
(See the following web site for a brief review of the rules of logarithms, including a sample pH problem at the end: http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html)
.
Part a). At a resting pH of 7.4, the muscle cellís hydrogen ion concentration satisfies the equation:
7.4 = -log [H+]
The unknown quantity [H+] is inside the ìlogî or common (base 10) logarithm in this equation, so we must use the base 10 exponential function to get it out. First we have to get the log part of the equation isolated by itself, which means dividing both sides of the equation by -1. This gives:
-7.4 = log [H+]
Next, we use the base 10 exponential function on both sides of the equation to get:
10-7.4 = 10log [H+]
Since the base 10 exponential function and base 10 log function are inverses of each other, they will cancel each other out on the right side of this equation, giving the new equation:
10-7.4 = [H+]
This equation gives the resting hydrogen ion concentration of the muscle cell in units of M/L.
If this concentration is doubled due to exercise-induced lactic acid production, the new hydrogen ion concentration will be 2 x 10-7.4 . We must now convert this number into pH units using the definition of pH and the laws of logarithms:
pH = -log [H+] = -log(2 x 10-7.4 ) = -(log2 + log 10-7.4 ) = - (0.301 + -7.4) = - (-6.99) = 7.099
FINAL ANSWER: A 2-fold increase in hydrogen ion concentration only decreases pH by
about 0.3 pH units, from 7.4 to approximately 7.1 if we round our answer off to one decimal place.
Part b). Using the same method outlined above to calculate that resting [H+] = 10-7.4 , we convert
the post-exercise pH of 6.7 into H+ concentration as follows:
6.7 = -log [H+] ---> -6.7 = log [H+] ---> 10-6.7 = 10log [H+] ---> 10-6.7 = [H+]
Next, we calculate the ratio of post-exercise to pre-exercise hydrogen ion concentrations:
post-exercise [H+] = 10-6.7 = 10-6.7 - (-7.4) = 10-6.7 + 7.4 = 100.7 = 5.0118723 (value from calculator)
pre-exercise [H+] 10-7.4
FINAL ANSWER: We see that a relatively small decrease in pH, from 7.4 to 6.7 or a change of only
-.7 pH units, increases the acid ion concentration by a factor of slightly more than 5-fold.
10. OTHER RESOURCES: The following web site gives a somewhat more technical explanation of the mathematics and chemistry of pH, and also gives some more sample pH problems:
http://scidiv.bcc.ctc.edu/wv/acid_base/definitions-ph.html
APPENDIX 3: Survey of student responses to the application project assignments
MATH 121 Spring 2004 NAME: ___________________________
(5 points extra credit)
NOTE: There are no right or wrong answers here. Simply answering the questions guarantees you the five points of extra credit.
1. I did (or plan to do) the application project(s) on: ___ logs/exponents ___ trig functions ___ conic sections
(check all that apply)
2. Please indicate whether or not you would be willing to allow your project writeup to be used as an example or test problem in future versions of this class.
____ Yes, with attribution (i.e. with you identified as the creator of the problem)
____ Yes, without my name attached
____ No
3. Please indicate whether or not you would be willing to allow your project writeup to be used as an example in written or oral presentations given to other teachers on this topic.
____ Yes, with attribution (i.e. with you identified as the creator of the problem)
____ Yes, without my name attached
____ No
4. How would you rate the contribution of doing the project(s) towards helping you understand the given topic(s)? (check one)
___None ___Helped a little ___Helped a fair amount ___Helped a lot
5. Was the time you invested in the project(s) worth the points you got for it? (check one)
___Yes ___No ___Sort of
6. Do you have any other comments about the project assignment and/or suggestions for how to improve the assignment for future sections of this class? (Write comments/suggestions in the space below.)
APPENDIX 4: Sample of student report for first outside project for MATH 121 S04
(used by permission of student)
1. NAME: Ryan Finnessy
2. DATE: March 19 2004
3. TYPE OF APPLICATION: Put an ìXî by the type of application problem you chose.
(check one) ___ Logarithms __ Exponential Growth _X_ Exponential Decay
4. AREA OF INTEREST: Engineering Technology
5. E-MAIL QUESTIONS OR COMMENTS TO: Finnessyr@uwstout.edu
6. OVERVIEW: I am designing a problem on exponential Decay. This problem deals with an engineer that works at a nuclear waste disposal site called Yucca Mountain located in Nevada. The engineer has a couple of barrels of Radium-226 that he has to test. The Radium in the barrels is 500 years old and has 80.4% of the radioactive element remaining. As standard procedure the engineer has to calculate the half-life of the Radium for company records to help them understand the material better.
7. DETAILS OF THE APPLICATION: Radium-226 is a radioactive material. It must be kept in a safe area and away from the public. To do this it has to be taken to a nuclear waste disposal site. An example of this would be the Yucca Mountain project in Nevada. The Yucca Mountain project is a repository. The purpose of a repository is to safely isolate highly radioactive nuclear waste for at least 10,000 years.
Most elements found in nature are stable. They do not change over time. Some elements, however, are unstableóthat is, they change into a different element over time. Radium-226 is an example of this. Elements that go through this process of change are called radioactive, and the process of trans-formation is called radioactive decay.
A half-life is the time that it takes for half a certain amount of a radioactive material to decay, and it can range from less than a second to billions of years.
Radioactive decay happens very steadily, engineers can use radioactive elements like clocks to measure the passage of time. By looking at how much of a certain element remains in an object and how much of it has decayed, Engineers can determine an approximate age for the object. However in this problem 80.4% of the element remains in the object and the engineer is going to try and find out how many years its going to take for the object to have 50% of the element remaining.
This engineer at the Yucca Plant has to calculate the half-life of the radioactive material Radium-226. This is an exponential decay problem.
The general equation for exponential decay is:
A=A0ekt
A-The original amount
K-The constant
T-Time in years
8. THE PROBLEM: When will the half-life of the Radium-226 occur after being put in the Yucca Mountain Project when the Radium is already 500 years old?
9. THE SOLUTION:
The general equation for exponential decay is:
A=A0ekt
A- The original amount
K- the constant
T- Time in years
With information obtained on the internet I found out that after 500 years the Radium-226 had decayed by 80.4%. This information can be used to help us find the constant K. The engineer now plugs these values into the equation to find K.
0.804 A0=A0ek(500).
The starting rates cancel out giving the engineer the equation below:
0.804 = e500k
At this next part of the problem the engineer uses Natural logs to cancel out e and get 500k from being a power.
ln ( 0.804) = ln (e500k) ýln ( 0.804) = 500k
Next the Engineer divides ln (0.804)/500 to get K alone.
k= (ln 0.804)/500
When the Engineer calculated this he found out that k = -0.000436
The Engineer now has the decay model for Radium-226ý A=A0 e-0.000436 t
From this information the engineer can determine the half-life of the Radium-226.
Set A to ‡ to get the half-life.
(1/2)A0=A0 e-0.000436 t
Cancel out the A0 on both sides of the equation.
1/2 = e-0.000436 t
They engineer now solves for t by using natural logs on both sides.
ln(1/2) = ln[e-0.000436 t].
The Natural log cancels out e on the right hand side of the equation.
ln (1/2) = -0.000436 t
To get the time the engineer divides by -.000436
t= ln(1/2) /(-0.000436)
With his calculator the Engineer finds out that the Radium will reach its half-life in 1590 years!
The engineer now subtracts 500 from 1590 and records that the radium will reach its half life in 1090 years.
Looks like someone is going to get a raise! J
10. OTHER RESOURCES: http://www.ocrwm.doe.gov/
http://www.nrc.gov/reading-rm/basic-ref/students/radiation.html
APPENDIX 5: Sample of student report for second outside project for MATH 121 S04
(used by permission of student)
1. NAME: (name withheld by request)
2. DATE: 4/06/04
3. TYPE OF APPLICATION:
_X_ Cosine function
4. AREA OF INTEREST: employment (dairy farming-pulsator)
5. E-MAIL QUESTIONS OR COMMENTS TO: (email address withheld by request)
6. OVERVIEW: All milker units operate in basically the same way and consist of the following components:
As the pulsator operates, it causes the chamber between the shell and regularly from vacuum to air source. Keep in mind that the inside of the teat-cup liner is under a milking vacuum at all times. Thus when air is admitted between the shell and liner (Figure 1A) the line collapses around the cow's teat. The pressure of the collapse liner is applied to the teat giving a massaging action. This is called the rest or massage phase. Milk does NOT flow from the teat during this phase.
Figure 1A. Massage Phase
A cycle refers to the total time in seconds that a pulsator takes to complete one milk phase and one massage phase.
The pulsation rate refers to the number of cycles that the pulsator makes in one minute. Pulsators on the market have pulsation rates ranging from 40 to 60 cycles per minute.
The pulsation ratio is the length of time in each cycle that the pulsator is in its milk phase compared to its massage phase. The pulsation ratio may be expressed as a simple ratio or it can be expressed as a percentage. Examples of pulsation ratios are as follows:
1:1 or 50:50
1 1/2:1 or 60:40
2 1/2:1 or 70:30
Therefore, a 60:40 pulsator means that within any given cycle the teat-cup liner will be open and milking 60% of the time and closed or massaging the teat 40% of the time.
The pulsation phase refers to the method of pulsation known as simultaneous (4 x 0) or alternating pulsation (2 x 2).
Some milking machine units are designed to operate with an alternating action; that is, while two teat-cup liners are milking the other two liners are massaging. Depending on the manufacturer, the alternating action may be from the left side to the right side or it can be from front quarters to back on an individual cow.
Pulsators can either be vacuum or electrically operated. The vacuum-operated pulsator uses air to move the plunger or slide valve which covers or uncovers the air passages to produce the pulsating action. The plunger or slide valve may be housed in oil for smoother action. The rate of pulsation is controlled by a needle valve which may be factory set or may be manually adjustable. Temperature changes tend to affect the pulsation rate of vacuum-operated pulsators; so be conscious of this factor and maintain the pulsator at normal operating temperatures to help reduce rate variations.
The electric pulsator may be operated by a master control which sends, via an electric current, the proper command to the pulsator to perform a preset pulsation rate and ratio. The electric pulsator is unaffected by temperature and therefore, has the advantage of producing a constant pulsation rate.
Some electronic pulsators have a computer chip internally mounted. These pulsators function to a preset rate and ratio once they are inserted into a stall cock electrical-vacuum source.
Some pulsators have variable pulsation rates and ratios. This feature allows the individual farmer to better choose the pulsation rate and ratio that suits the dairy herd's needs. However, a word of caution: DO NOT experiment unless you fully understand the technical aspects of pulsation rates and ratios and know how they influence the cow's milking; otherwise, severe injury could result.
Information from: http://www.gov.on.ca/OMAFRA/english/livestock/dairy/facts/89-103.htm
7. DETAILS OF THE APPLICATION: By looking at the control panel in the milk house I was able to determine that the pulsator pulses at a rate of sixty times in one minute or one pulse every second, I also notice that the ratio was set at 1:1. I also took a volt meter and tested the pulsator to find the high and low voltage being sent to the pulsator. When I touched the ends of the volt meter to the pulsator I was able to observe the readings cycle from 0volts to 24volts. With this information I was able to get start on putting a function together that fit this formula: y=Acos(Bx-C)+D
First we can figure D by adding the minimum value and the maximum value divide by two. (max + min/2) So, (24 + 0/2) = 12. 12 would be my baseline or vertical shift. D = 12 With this we can now figure out our amplitude. Our amplitude A is our maximum value subtract our baseline. (max ñ baseline) = A
So, (24 ñ 12) = 12. 12 would also be our amplitude for this function.
Our period is 60sec/1 min. or just 1. Since the period = 2Π/B, or B = 2Π/period then B would equal 2Π divided by 1. (B = (2Π/1)
We have no C or phase shift
Now we can fill out our function formula with this information:
Y = 12cos(2Π/1X) + 12
8. THE PROBLEM:
A) Graph this function on your graphing calculator for two periods using the proper window. (NOTE: Make sure you are in radian mode.)
B) Now say we are technically inclined on the aspects of ratios and rates for pulsators that we can ignore the above warning and we want to change our rate from sixty pulses a minute to forty pulses a minute. Rewrite the formula using the new period?
C) Now graph this new function on your calculator and the old one for sixty pulses a minute. Tell me at what times do the two graphs meet?
9. THE SOLUTION:
A) These are the settings you should have entered in your calculator to see two periods.
Xmin= 0 Ymin= 0
Xmax= 2 Ymax= 24
Xscl=1 Yscl= 1
Xres= 1
B) The only thing the changes is our period
Our period is 40sec/1 min. or 40divided by 60 (40/60) or just 2/3. Since the period = 2Π/B, or B = 2Π/period then B would equal 2Π divided by 2/3. (B = (2Π/(2/2))
New function:
Y = 12cos(2Π/(2/3)X) + 12
C) By putting both the functions into your calculator and setting the window to:
Xmin= 0 Ymin= 0
Xmax= 8 Ymax= 24
Xscl=1 Yscl= 1
Xres= 1
We can see that the two meet up a 2sec. , 4sec. , 8sec. and so on.
10. OTHER RESOURCES: The following websites can give you more detail on the operation of a whole milking machine or manufacturer homepages.
http://www.gov.on.ca/OMAFRA/english/livestock/dairy/facts/89-103.htm
Dave Pechmiller (farmer)
APPENDIX 6: Sample of student report for third outside project for MATH 121 S04
(used by permission of student)
1. NAME: (name withheld by request)
2. DATE: 5/11/04
3. TYPE OF APPLICATION: Ellipse
4. AREA OF INTEREST: Applied Science / Concentration in Scientific Laboratory Management
5. E-MAIL QUESTIONS OR COMMENTS TO: (email address withheld by request)
6. OVERVIEW:
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The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it. |
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