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I Must Have Done Something Interesting Over the Last 35 Years! November 22, 2008: Session S114—AMATYC, Hilton Washington Hotel, Washington, D.C. Ned W. Schillow Professor of Mathematics Lehigh Carbon Community College Schnecksville, PA 18078 nschillow@lccc.edu The following are all ideas that have worked well with my students, prompt discussions, and led to real insight or understanding and generated productive discussion to the point that they are worth the time in class! Some of these are the actual handouts I distribute to my students; others are summaries of the actual activity (especially in those settings where handouts are not particularly necessary). Regression Modeling—Knots This activity is fast, it’s hands on, it gets to the point, and the novice should be able to construct a clear sense of much of the crucial nature of linear fitting (and, by extension, other models) with this one example. (Note: This activity also is appropriate in MAT 190, where a review of regression is necessary!) Ultimately, a number of lengths of rope of differing diameters are to be distributed to the students. They are told that eventually an overhand knot will be tied in each, and we want to determine how much rope is needed to form the knot. Ask them how we can make such a determination—with ease they should suggest the strategy of measuring the rope length before and after the knot is tied, the difference being the amount of rope used in forming the knot. It also might be a good idea to have two students (the largest and smallest in the class???) tie a knot in two identical ropes, quickly discussing the types of factors that could have a bearing on the measurements we are about to obtain. Ideally, all the rope should be of the same material: twisted polyethylene or braided nylon are the most readily available, but it might be difficult to get enough of all the same kind of material. At any rate, at least six diameters should be available: 3/16 in., 1/4 in., 3/8 in., 1/2 in., 5/8 in., and 3/4 in. are easiest to obtain (although you might also find 5/16 in. diameter) With five of the six lengths distributed, have before and after measurements made— centimeters might be somewhat more convenient. (For sake of argument, do not distribute the 3/8 in. diameter rope for the time being.) The data is obtained and plotted, calculator capabilities are explored, the regression line is found and plotted over the scatter plot, and a prediction is generated for the amount of rope needed to tie the knot in the 3/8 in. diameter rope (0.95 cm.). Then actually tie the rope and check the result. Handout sheets follow this page! The class should find this to be a meaningful review or introduction to the principles at hand—and for many it might be their first real hands-on check of the type of prediction that regression can lead to. MAT 130 A Knot Tying Investigation Regression Analysis 1. A variety of ropes have been distributed to the class. Their lengths are to be measured before anything is done to them. Then overhand knots will be tied in them, and once again the length of the result will be measured. The difference between the lengths gives us a good estimate of the amount of rope needed to actually create the knot. Enter the results in the table below: Rope Diameter Original Length Knotted Length Rope needed to create the knot 0.48 cm (3/16 in.) 0.64 cm (1/4 in.) 1.27 cm (1/2 in.) 1.59 cm (5/8 in.) 1.91 cm (3/4 in.) 2. You want to create a scatter plot of these results. Plot the diameter along the x- axis and the rope used to create the knot along the y-axis. (Note: The independent variable is along the x-axis as is to be expected.) This can be done using graph paper or by using your calculator: STAT EDIT and enter the data in list L1 and L2 Then set the graphing capabilities using 2 nd STAT PLOT and ZoomSTAT 3. Now we want to find a line that reasonably well fits the data obtained. If you are using graph paper: Draw a line that comes reasonably close to the points Choose two points on the line (which are not necessarily among your data points) Determine the equation of the line based on these two points (regression line) The equation obtained from this approximation is: ___________________ If you are using your calculator (with DIAGNOSTICS ON) Use the keystroke sequence STAT CALC and choose linear regression option 4 for y = ax + b Paste the equation into Y1 by using the keystroke sequence Y= Clear Y 1 if necessary VARS Statistics EQ RegEQ The regression equation obtained by the calculator is: ___________________ 4. How will you use this equation to approximate the amount of rope needed to form a knot using a 3/8 in. (0.95 cm) diameter rope? Actually obtain this approximation: 5. Finally take appropriate measurements of the 0.95 cm (3/8 in.) rope and figure out the knot size. Length of the untied rope: Length of the tied rope: Rope needed to form the knot: 6. Discuss how accurate or inaccurate the prediction from part 4 actually is. 7. If I were to tell you that I used 9.40 cm of rope to form a knot in a piece of rope, how would you obtain an estimate for the diameter of the rope? Explain this, and then actually carry out the computation. Idea adopted from Harrell, Marvin and Linda Fosmaugh, ―Mathematical Modeling and Representation: Wrapping It up with Ropes‖, 2001 Yearbook Pennsylvania Council Teachers of Mathematics, pp. 13-19. Proportions Originally, it was my intention to include the following problem on the day we talked about proportions in Beginning Algebra. However, time ran out, so we didn’t get to it until the next class period, as which point the question took on a life of its own……and frankly, the spirited discussion which followed suggests that I’ll never include it on the day I introduce proportions. Side note: If somebody sees how to do this using proportions right away, it is quite likely the spirited discussion won’t occur (but there certainly is no harm in having the class grappling with this even with a correct approach at the outset)! But if this is not sensed right away, expect some clever alternatives being suggested! Two people pooled their money for lottery tickets. Darrel put in $ 25 while Serena put in $ 20. If they won $ 8.2 million, how much should each person receive? Cabinet Drawers! The following problem is to be discussed together in groups of three or four individuals during class. Each person in the group must understand the solution, so it is everybody's responsibility to make sure that each other member in the group grasps what is going on and to make sure that your own understanding of the problem is solid. ************************************ A cabinet 30 inches high must have a 4-inch thick base and a 1 ½ -inch thick top. Four equal-sized drawers must fit in the remaining space, with ¾ inch between drawers. What is the height of each drawer? Saunders, Hal. When Are We Ever Gonna Have to Use This?, 3rd ed. Palo Alto, California: Dale Seymour Publications, 1988: 3. Building a Cabinet MAT 098/099 The following problem is to be discussed together in groups of three or four individuals during class. Each person in the group must understand the solution, so if is everybody's responsibility to make sure that each other member in the group grasps what is going on and to make sure that your own understanding of the problem is solid. ************************************ A cabinet 10 feet long must have 5 doors. There are 1 ½ inch wide stiles between the doors and at both ends. What is the width of each door? (Round to the nearest 16th of an inch.) 10’ 1½“ 1½“ Saunders, Hal. When Are We Ever Gonna Have to Use This?, 3rd ed. Palo Alto, California: Dale Seymour Publications, 1988: 3. MAT 130—Industrial Mathematics—Radians A very simple activity: Take a piece of string or yarn and approximate the length of the radius. How many of those radius lengths will it take to get all the way around the circle, starting at point P? Actually count this out using the string! P After we discussed things a bit further in class, there are two more things to do: 2. Show an angle in standard position that measures 3.5 radians. 3. a) Is there any way to express 60 in terms of radians? How? b) How can I convert 1.8 radians to degrees? VARIATION Before doing the pendulum exercise on variation in class, we would have already spent some time discussion the idea of variation. For example, the bridge problem on the next page is literally my starting point, which gets them thinking about the nature of such problems…..and provides us with the opportunity to discuss why the multiplicative factor (in direct variation) leads to the notion of “when one value increases, the other also increases” while the dividing factor (in inverse variation) leads to the notion of “when one value increases, the other decreases”. Question 2 also provides the opening for the concept of the coefficient of variation, k. I really don’t expect the students to come up with this on their own, but this bridge exercise gets them to understand the need for it. When I asked them what kind of factors might impact the strength of two bridges with exactly the same dimension, they’ll quickly come up with the idea of it depending also on what the bridge is constructed of. Wonderful! So while the basic rule (the variation equation) holds true for all bridges of this type, the k value will change depending on whether the bridge is made of oak beams, reinforced steel, etc. An Introduction to Variation 1. For a quick discussion in your groups: Which of the ―bridges‖ pictured below is likely to support the greatest weight, and which will support the least? Be able to explain your decision. 2. Which of the following equations would best represent the type of relationship among all three variables, and why? a) W=L+T b) W=TL c) W=TL d) W=L–T Can you support why you think your answer is reasonable? Even if you have the correct answer, can you explain why you think it is not quite correct for all possible bridges of the same dimension? Pendulums and Variation Each group will be provided with the following equipment: --A length of string with a paper clip on one end --Washers --Stopwatch --Ruler or Tape Measure --Protractor 1. You are about to attach some washers to the paper clip on the end of the string provided to your group. One person will pull back the washer and release it, so the string starts to swing back and forth like a pendulum. Someone else will time the pendulum as it swings back and forth. But before you do this, answer the following question: What factor or factors might affect the length of time it takes for the pendulum to swing through each cycle? List those factors below: 2. Now we are ready to experiment, but we only want to change one factor at a time. For right now, attach one washer to the string and choose a certain length that you will keep the same (until later on). The one factor you can control right now is the ___________________________ Keeping this factor fairly small, pull back the pendulum, release it, and use the stopwatch to find out how long it takes to go through 10 cycles. Then calculate how long it takes for just one cycle. Repeat this experiment at least two more times, keeping the weight and length the same, but changing the factor you listed above to larger values. Record your data below. Factor Size Time for 10 cycles Time for one cycle 3. You were probably changing the angle size in part 2. Does there seem to be a relationship between the time for each cycle as the angle size increases? Describe what is happening. 4. Now we want to see if the weight on the end of the pendulum has any control over the length of time each period takes. For this part of the experiment, keep the angle size and the length of string the same, but use different numbers of washers to control the weight. Record your results below. Number of washers Time for 10 cycles Time for one cycle 5. Does there seem to be a trend in what happens to the time for each cycle as the number of washers increases? Describe what is happening. 6. Now we want to see if the length of the pendulum has any control over how much time each period takes. For this part of the experiment, keep the angle size and the weight the same, but use different lengths of the string. Record your results below. String length in Time for 10 cycles Time for one cycle centimeters 7. Does there seem to be a trend in what happens to the time for each period as the length gets longer? Describe what is happening. Pendulums and Variation Part 2 1. From the first part of the experiment you are ready to answer this question: Considering the factors of size of the angle for each cycle, the weight attached, and the length of the string, which factor or factors really appear to influence how long each cycle takes? How do you know you are right? 2. Do you think this factor and the length of time for each cycle are directly or inversely proportional to each other? Why do you think this? 3. Fill in the blank to write a possible proportionality statement of the form P = _______, where P is the period or cycle length of your pendulum. Remember to show a constant of proportionality! 4. Let’s plot a variety of the results that have been gathered by the different groups. 5. In part 3 it is likely that you thought the proportionality statement was P = kL, where L is the length of the string, P is the time period for one swing of the pendulum, and k is the constant of proportionality. Does the graph you formed in part 4 support this? _________ If your answer is yes, explain why you think you are correct. If your answer is no, what kind of formula does the graph suggest for the correct proportionality statement? 6. Once the class agrees on the result from part 5, we want to find an estimate for the value of k. How can each group come up with its own estimate? 7. What is your group’s estimate for k? 8. The laws of physics tell us that if the string length is measured in centimeters, the correct equation relating the length of the string and the period of the pendulum is L T 2 980 Use this to find the true value of the constant, k, of proportionality. 9. Finally, we’ll talk together about how a grandfather clock is adjusted…and decide what this has to do with our experiment. The Volume of a Glass—A Group Project MAT 195 The assignment described below must be done as a group project, with groups of three or four individuals. I leave it up to you to form your own groups, although I might be ―forced‖ to assign a few of you to a group if you fail to do so in timely fashion. The assignment: Using calculus, determine the volume of the glass container that you will choose yourself (subject to my approval). Only one container should be explored by each team. Here are the rules: 1. The groups shall be formed no later than February 19. 2. The glass container should be brought to school and approved by me not later than March 2. 3. The glass container must exhibit rotational symmetry and be more complicated than a straight edged container (that is, it cannot be a standard drinking glass). 4. You will need to find your own mechanisms for splitting up the work fairly. This is not to be a one or two person effort within the group. 5. You are encouraged to use the TI-83, TI-92, and software as an aid in exploring the solution. Printouts will be helpful in defending your findings. 6. During the first two weeks of April, at least two representatives from each team will schedule a meeting with me in my office to discuss your progress and difficulties. In actuality, I would like to sit down with the whole team if possible, but I realize that scheduling might be a problem for some of you. 7. Your solution is due no later than Wednesday, April 29. When the solution is submitted, I expect the following: a) A complete, thorough, professional presentation. b) The container itself must be submitted. c) Each group members must submit his or her own summary of the contri- butions made by himself or herself in reaching the final product submitted. Optional: I will be ruthless with presentations which are not ―complete, thorough, and professional‖ in nature. If you wish, a rough copy of your report can be submitted by April 15, which I will react to over the Spring break. If you choose to do this, please do not turn in the container at that time. Do not use any glass container which will lead to tears, divorce, dismemberment, or disinheritence if it should accidentally be broken. Glass Project Appointments When I first made the assignment to find the volume of a glass, I indicated that at least one representative from each group must schedule an appointment with me to discuss problems, snags, progress, etc. It is time to make these appointments. Keep in mind that at least two persons from each group must come in for this appointment (and I reserve the right to meet with the rest of the team, if such seems necessary). Ideally, I would like to meet with the entire team at the same time, if your schedules allow. Below are the times when I am available for a meeting. Count on approximately 15 minutes for each appointment. A sign-up sheet is on my office door. You must sign up for your appointment, clearly listing exactly who will be coming in! Monday, March 30: 7:30, 7:45, 9:30, 9:45, 10:00, 10:15 10:30, 10:45, 1:00, 1:15, 5:30, 5:45, 6:00 Tuesday, March 31: 9:00, 9:15, 11:00, 11:15, 11:30, 11:45, 12:00, 12:15 (etc.) MAT 195 Glass Project Group members: Grade for Timely submission of glass (5 pts) Timely scheduling of meeting (5 pts) Grade for the project itself (30 pts) Personal involvement grade (10 pts) Total points earned: See the attached sheet for additional comments about your own work and the level of work submitted by your group. Note Cards as Chocolate Chip Cookies Working with sequences and series I hand out two note cards to each group of three students, and make a big deal about the fact that I am giving them luscious chocolate chip cookies. (Good for a laugh…..and besides, this activity cannot be done effectively with real cookies. The rules are simple: They are to find a way to divide the ―cookies‖ evenly between the three of them. Oh……that’s right……I also have to tell them that the only thing they can do is break a ―cookie‖ exactly in half, or break any piece of a cookie exactly in half. Let them work for a short while—I’ll go around the room making sure the interpretation is on target for the most part, and locating a group doing this correctly so I can make use of their pieces later. At first most groups are convinced there is a way to pull this off—but eventually they’ll realize it’s impossible (unless they just throw away a bit of cookie dust!!!) When the timing is right groups can share their (virtually identical approaches) and I will take one share of the cookies (my share, if I had a group of 2 working on this) and tape up the progression on the blackboard. (Alternative, mimic this up front under their instructions using paper, so the pieces are more visible). Line up those pieces—thus providing both a sense of the size of the pieces, the sense of a sequence, and the sense of recursively defined terms. How big are my pieces? The first piece everybody gets is ½ of a cookie. The second is 1/4 of the remaining half, or 1/8 of cookie, with that last 1/8 to share. And on it goes…… 1/2 1/8 1/32 1/128 etc. This can lead to a discussion about geometric sequences and growth. You can develop both to the recursive formula a1 = ½ an = an-1/4, for n > 1 and also the general term formula an= 2 n1 . 1 2 Depending on timing with this, once they agree that with this infinite progression (right down to the chocolate chip cookie atoms, which need to be split---have fund with this!), that each person gets 1/3 of the 2 cookies their group started with…that is, 2/3 of a cookie. Now, go back to the terms, set up the addition of the infinite series, and since it is geometric you are able to sum it easily…..lending real credibility to the notion that a infinite series can really have a finite sum. The Medication Problem This made the rounds at many conferences about 15 years ago and then vanished. It’s a great problem for those of you who never saw it before— and I’ve made use of it in our Allied Health Math class, College Algebra when we talk about sequences, and Calc III with our infinite sequence and series work. The idea is that you’ve been prescribed a certain medication—and you are to take one pill per day. Each day, 25% of the medication in your body is somehow assimilated or eliminated—naturally breaking down, being used, etc. We start exploring this by hand, and I ask them how to determine the amount of medication in the body each day for the next several days. On day 1, you start off with one pill in your body. Day 1 = 1 pill On day 2, 25% has broken down, (so 0.75 pills worth of medication is in your body), but you take another pill, so you have a total of 1 + 0.75. That is, Day 2 = 1.75 pills. On day 3, 25% of the 1.75 pills will be assimilated. Let the students suggest finding 25% of 1.75, getting 0.4375. That way the new balance is 1.75 minus 0.4375 = 1.3125….but you take a new pill, so you are up to Day 3 = 2.3125. If necessary, work out Day 4 as well….and, if necessary, draw out the idea of finding 75% of the previously day’s dosage plus 1, rather than finding 25%, subtracting it off, and then adding 1. (Notice the tie in with a recursive definition of sequences developing here) NOW COMES THE FUN! Take your calculator, type in 1 for the first pill, and press enter. The calculator displays a 1. Now type in .75 + 1 and press enter. The calculator will show this as Ans*.75+1 and will give a result of 1.75 as above. That’s reassuring. Now just press enter again, so the calculator can do its own thing recursively, and you’ll automatically see 2.3125, stress this a day 3’s result. Say day 4 as you press enter, and you’ll see 2.734375 appear. If your students are following you, they’ll probably be pressing enter like crazy at this point to see what happens—be sure that you are stressing that by day 4, there is the equivalent of 2.734375 pills in your bloodstream, coursing through your veins. Now, encourage them to press enter many times and to observe what happens. Two things should come out—that the change in the total gets smaller and smaller, and that the size of the dosage in the body is getting close to 4 pills. When I then do this up front to ―catch up to them‖, I count out days each time I hit enter….. ―Day 5, day 6, 7, 8, 9, ….‖ I go up to at least 21 days, and point out that by this time we are within 1/100 of a dosage of the 4 pill total limit (3.99…… pills). So obviously, you have a real setting in which you are looking meaningfully at the notion of sequences. But we can push this further! Everybody knows that the doctor always tells you to make sure you don’t skip a day….but also, if you aren’t sure if you took a pill earlier, don’t ―play it safe‖ by taking another one. What happens potentially if this occurs? Continue with your calculator from wherever you are, just typing in *.75 and press enter. This gives you the amount in your body after you miss taking a pill. Then press *.75 + 1 to start up the correct process of taking a pill each day. Start counting out the days with each additional push of the enter key… ―1, 2, 3, …‖ Keep counting until you get back to a 3.99….level. This will take over two weeks of pill taking to get back to the ―steady state level.‖ So, stress to the students, missing a pill has impact for over two weeks of depriving your body of the level you need. (I know, I know, there is nothing sacred about the 3.99 level here other than a matter of stressing proximity here, but the idea is important.) Now what about if you think you forget a day and take a second pill? Press +1 (for the second pill of the day) and press enter. Then got back to the *.75+1 sequence, again counting day 1, 2, 3, …. The student reaction is usually an interesting one, as now they observe is shrinking its way back down to the 4.00 dose level. If you again count out the days to fall below the 4.01 level (to be less than 1/100 away), we again see it takes over two weeks of over-dosed levels to return to the safe this-is- what-my-body-needs level. This activity goes very quickly in class, it is effective, it provides a chance to discuss why doctor’s orders are as they are…..and opens up at least one more door: Support L is the desired level of medication the body needs. For this 25% decay rate (and not all meds are assimilated daily at this level, of course), can you lead your students to recognize that the steady state level in this case is: 0.75L + 1 = L Once this makes sense to the students, have them solve the equation for L. This again is a real eye-opener, leads to good discussion, holds their interest, and opens up the mathematics to levels other than a 25% decay rate. Probabilities and the Binomial Distribution (with thanks to Jim Rubillo) On the following page is a listing of 6 multiple choice question, about which your students (and probably you and I as well) know nothing—so answering the questions will be essentially blind guessing. To make use of this activity: administer the ―test‖ (have fun with this—remind them they are not to cheat! That they should have prepared, etc.!) let them correct their own tests, keeping track of how many they get right and wrong, but don’t ask for the results yet figure out (by table or calculator) the probabilities for each of the possible outcomes using the binomial distribution (shown below) ask the students how you can use these probabilities to estimate the number of persons we might expect would get each of the various outcomes correct (i.e., take the number of persons times the corresponding probabilities) now gather the class data, finding out how many had each of the various number of possible correct answers Thanks to the rules of probability, the predicted results and the actual results will almost certainly be very similar—close enough to convince the class about the legitimacy of the binomial distribution! Number correct Probability x P(x) 0 .088 1 .263 2 .329 3 .219 4 .082 5 .016 6 .001 BASIC SKILLS TEST Circle the correct answer for each of the following. (If you don’t have a clue, make a guess—I’m collecting data!) 1. F. Haxo was a) a Belgian impressionistic painter b) an American Civil War financier c) a French military engineer 2. Calluna Vulgaris is the Latin name for a) lilac b) heather c) rose fern 3. Asaias Tegner is a) a Norwegian astronomer b) a Swedish poet c) a Danish inventor 4. Vilnus is located in a) Lithuania b) Latvia c) Estonia 5. Who said, ―To think is to differ‖? a) Ralph Waldo Emerson b) John Stuart Mill c) Clarence Darrow 6. Alofi is in a) the Wallis Islands b) the Futuna Islands c) New Caledonia Thinking About Large Numbers The potato chip discussion on the next page is a good example to provide a class with a real experience involving large numbers. As it turns out, the newspaper numbers had to be in error. This makes for a good problem either as an in class discussion or an out of class assignment. On the other hand, the item two pages hence was widely circulated in late September, 2008, even before the biggest part of the governmental bailout took place. How many persons recognized the computational error of the big numbers? Potato Chips Several years ago, the Philadelphia Inquirer included an article on the potato chip industry. It stated that ―…there are 24 potato-chip manufacturers in Pennsylvania—more than any other state—whose 32 plants produce about 450 billion tons of potato chips each year.‖ Discuss how reasonable this figure is. That is, considering the population of the United States, does it make sense that 450 billion tons of potato chips are produced each year in Pennsylvania? You will probably want to support your discussion with some mathematical computations. 2008 Government Bailout of AIG A widely distributed e-mail To my fellow Americans...... I'm against the $85,000,000,000.00 bailout of AIG. Instead, I'm in favor of giving $85,000,000,000 to America in a We Deserve It Dividend! To make the math simple, let's assume there are 200,000,000 bonafide U.S. Citizens 18+. Our population is about 301,000,000 +/- counting every man, woman and child. So 200,000,000 might be a fair stab at adults 18 and up. So divide 200 million adults 18+ into $85 billon that equals $425,000.00. My plan is to give $425,000 to every person 18+ as a We Deserve It Dividend. Of course, it would NOT be tax free. So let's assume a tax rate of 30%. Every individual 18+ has to pay $127,500.00 in taxes. That sends $25,500,000,000 right back to Uncle Sam. But it means that every adult 18+ has $297,500.00 in their pocket. A husband and wife has $595,000.00. What would you do with $297,500.00 to $595,000.00 in your family? Pay off your mortgage - housing crisis solved. Repay college loans - what a great boost to new grads Put away money for college - it'll be there Save in a bank - create money to loan to entrepreneurs. Buy a new car - create jobs Invest in the market - capital drives growth Pay for your parent's medical insurance - health care improves Enable Dea dbeat Dads to come clean - or else Remember this is for every adult U S Citizen 18+ including the folks who lost their jobs at Lehman Brothers and every other company that is cutting back. And of course, for those serving in our Armed Forces. If we're going to re-distribute wealth let's really do it... If we're going to do an $85 billion bailout, let's bail out every adult U S Citizen 18+! As for AIG - liquidate it. Sell off its parts. Let American General go back to being American General. Sell off the real estate. Let the private sector bargain hunters cut it up and clean it up. Here's my rationale. We deserve it and AIG doesn't. Sure it's a crazy idea that can 'never work.' But can you imagine the Coast-To-Coast Block Party! How do you spell Economic Boom? I trust my fellow adult Americans to know how to use the $85 Billion We Deserve It Dividend more than I do the geniuses at AIG or in Washington DC. And remember, this plan only really costs $59.5 Billion because $25.5 Billion is returned instantly in taxes to Uncle Sam. Ahhh...I feel so much better getting that off my chest.

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