Math Options Be Bold! Inc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 4, Number 5 September 2002 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This newsletter is best viewed with a Courier font, size 10. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note - Chemistry is an Experimental Science Schedule of Public Classes Math in Industry - Why Design an Experiment? Software Review - UCon -- Convert Units for Practically Anything! Family Math - Playing Cards and Learning Math Revisited Guest Author - Gary Foreman: Mortgages, Taxes, and Bigger Homes Ask Statman - Median vs. Mean ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "In spite of its many successful theories, chemistry remains, and probably always will remain, an experimental science." E. J. Slowinski ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor's Note ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Reader, When I left college for the working world I thought the proper application of the chemistry theories I had learned would allow me to solve nearly any problem I encountered. I was mistaken. I discovered that the theories taught in college contain many simplifying assumptions that just don't ever seem to apply in the real world. While these theories used properly can sometimes guide us in the right direction, they very frequently lead us far astray. Only experiments can actually help us understand how nature behaves in any particular situation. Learning to perform objective, efficient experiments is the key to success in industrial chemistry. Design of Experiments, or DOE, is an extremely valuable tool for chemists because it helps us to study nature experimentally. Good technique plus good experimental design equal successful experimentation. If you want to know how you can learn objective, efficient experimentation, please stop by http://www.MathOptions.com. Take care, Bill Kappele. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Oct. 17-18 Creating Custom Experiment Designs Bellingham, WA Learn to create experiment designs to fit YOUR experimental needs. You will no longer have to change your experiment to fit available designs. Nov. 4 - 6, 2002 Performing Objective Experiments Bellingham, WA Learn to use the power of DOE in your work. When you leave this workshop you will know how to identify the problem to be solved how interactions among factors affect your results how to optimize your product or process how to make contour plots to show to customers and your management how to measure and report precision in your results how to find the most important factors when experimenting on a tight budget how and when to use different types of designs for efficient, cost-effective, yet sufficiently thorough experiments Nov. 7, 8 2002 Objective Experiments for Mixtures Bellingham, WA and Discrete Factors When you leave this workshop you will know how to design and perform an experiment for mixtures how to design and perform an experiment for discrete factors how to design and perform an experiment for mixtures and process factors how to design and perform an experiment for discrete and process factors how to optimize your product or process how to find the most important mixture components when experimenting on a tight budget how and when to use different types of designs for efficient, cost-effective, yet sufficiently thorough experiments You can learn more about these classes and register to attend at http://www.mathoptions.com/public.htm You can learn about hosting these classes at your company at http://www.mathoptions.com/training1.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Why Design an Experiment? Designing an experiment helps you get as much information from your data as possible with as little experimentation as possible. Let's put this in familiar terms to make this a little more clear. To do this, consider running errands as an analogy to experimtnetaion. Each errand is a step toward your goal of accomplishing your tasks, as each experiment you perform is a step toward accomplishing your goal of understanding. When you set out to run errands you could use the "one-errand- at-a-time" approach. You could go to the grocery store because you need groceries. When you finish shopping you could consider what else needs to be done. You need to go to the Post Office, but the package you need to mail is at home. So you head home and get the package you need to mail. When you are finished mailing your package you consider what else needs to be done. You need to take your suit to the dry cleaner. So you head home again, get your suit, and head to the dry cleaner. When you finish with the dry cleaner you realize that you need to return a video you had rented. So its off for home again to get the movie to return. After returning the movie, you realize that you need to put away the groceries you bought earlier. Oh dear -- the ice cream is ruined! Not many people find this a good approach to running errands. It requires many unnecessary trips home and consumes a lot of time. It would be much beter to plan everything you needed to do in advance and to plan the best order in which to run your errands. For example, you could plan to take your suit to the dry cleaner on the way to return the video you rented, then swing by the post office and mail your package before grocery shopping. You could put everything you need in the car and avoid multiple trips home. Planning ahead also lets you pick the most effiecient order for your errands. You can plan to grocery shop last so no food has to stay a long time in a hot car. How does this analogy apply to experimentation? Many people believe they should perform one experiment, consider the outcome, then decide which experiment to do next. This can lead to a very helter-skelter path of experiments leading to your goal, just as one-errand-at-a-time can lead to a lot of wasted trips home. For example, you could try putting 3 grams of dye in an ink and see what it looks like. A little light, so you try 6 grams. This makes the ink too thick to print, so you try 4.5 grams. This looks good, but maybe its still a little too light. 5 grams does the trick. Now you need to decide how much anti-bacterial agent to add. You try 0.3 grams, but it doesn't kill the bacteria. So you try 0.5 grams. This kills the bacteria, but it makes the color look too light. So you return to adjusting the amount of dye. You get the picture? Designing your experiment is like planning ahead for your errands. You decide what the minimum set of experiments is that will provide the information you are seeking and the best order in which to run them. As with planning your errands, you will generally be much more efficient and accomplish your goal more quickly. You will also understand how the different factors -- such as dye and anti-bacterial agent -- influence each other. Unfortunatley, designing an experiment is not as obvious as planning your errands. That's why DOE was invented -- it provides you with a simplified, systematic way to plan your experiments. So why design your experiments? So you can have time for a life! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You can learn Design of Experiments in a practical, hands-on workshop at your company. Let Bill Kappele show you how to USE DOE in your work - not just talk about it. Please visit http://www.mathoptions.com/training1.htm for details. Have you taken "Performing Objective Experiments" but are feeling pretty rusty? You can repeat the workshop for $495. Please call Bill Kappele for details - (888) 764-3958. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Software Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ UCon UCon is an amazing program! You can convert practically anything in any units to different units at a click of a button. Do you want to know what an Acre is in square yards? No problem. How about converting it to Hectares? Just click a button. You can make conversions for acceleration, density, viscosity, force, power, surface tension, temperature, and volumetric flow to name just a few. You can literally convert units for almost anything. And that's not all. UCon also includes a scientific calculator to assist you in calculations using your converted units. This wonderful tool is also very inexpensive -- only $16.95 at its regular price. Please give UCon a try. You'll be glad you did. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ HERE's A TIP FOR E-MATH NEWS READERS: you can download a free 3-day trial version of UCon at http://www.Qivx.com. This trial will provide you with a discount coupon so the price to you is only $10.17. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Playing Cards and Learning Math Revisited by Neil Flatter This is in response to Playing Cards and Learning Math by Beth Heffernan in Emath News Vol. 4 No. 4, July 2002. It seems to me you missed one of the more obvious possibilities to incorporate math and cards with children. Windows ships with a game called Solitaire. My seven year old can quickly recite from 2 through ace, but isn't nearly as fast going backwards. The standard game adds red/black patterning. King's Corners and 7-Up work along the same vein and involve the whole family. Other versions use various combinations of thirteen, and that drills basic math skills. Elimination uses the spotted cards in combinations to match a dice roll. There must be thousands of variations on solitaire! One of the first card games we played with our children was Old Maid. From the one card one match, we went to Go Fish where there are four of each card available for pairing. Our next game was probably Crazy Eight's. We play Uno now. Both prompt the kids to think not simply about matching colors, but recognizing the value of numbers to change the suit [to say nothing about the added joy of being able to make Dad draw more cards]. Don't forget to let your kids keep score. Adults can check the math, but mistakes are golden opportunities to show checksum methods. There's also the rummy family. Numbers are presented both as matches and sequences. We play with deuces wild which opens up more learning experiences. After both a run and a set of three are laid down, a player can build on other runs and matches from their set or from other players. Wild cards can be swapped for the face value of the card to add interest. First one out of cards wins. Any cards held when a player goes out are counted, regardless if they would have played. Spotted cards count face value and face cards are all worth ten. The whole game moves pretty quickly and doesn't require the concentration of gin rummy. As children, my brother and I progressed through Rook, Euchre, Spades, and Hearts. My wife's family added Pinochle and Bridge. All require players to keep track of trump. It's hard to be good at them if you don't count the cards played. If these suggestions didn t spark a few ideas, pick up a copy of Hoyle's book. The many games and variations will keep your family entertained while sharpening their math skills, many times without any one noticing. Except for the savvy parents who read this newsletter, that is. While I'm standing on this soapbox, let me compliment you on the practical applications featured in your newsletter. As a parent as well as a tutor for math and physical science, the object is to be able to take advantage of a teachable moment. A flat helium balloon from the fair becomes a discussion on buoyancy. A sale on half gallons of milk is an opportunity to teach unit cost. Counting change, travel times, and sales tax all open the door for learning if we will only recognize them. Thanks for the continued ideas to use with my children. Note from Beth Heffernan: Thank you so much, Neil, for your input and kind words. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Neil Flatter is an E-Math News reader. You can reach him at mailto:Neil.Flatter@Juno.Com Beth Heffernan is Vice President of Math Options. You can reach her at mailto:Beth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Guest Author: Gary Foreman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Mortgages, Taxes, and Bigger Homes Dear Gary, We have very nearly paid off our mortgage! We put a lot of spare money into it because the mortgage had a higher interest rate than any safe investment we could find. But for some personal reasons, we would like to have a different house, probably one that is nicer than our current one. My husband says that since interest is tax-deductible getting a new house makes financial sense especially with today's fairly low interest rates. So he's all for it. To me, as much as I'd like to have a new house, it feels as if we have finally "caught up with our tails" only to begin chasing them again. Can you give us some perspective? Thank you, Rebecca Congratulations, Rebecca! It sure does feel good to own a home without a mortgage. Financial life is much easier without a mortgage payment. On the other hand, she and her husband have a lot of company in wanting a bigger and better home. According to the National Association of Home Builders, the average home has increased in size from 1,500 square feet in 1970 to 2,265 square feet in 2000. That's a 50% increase in just 30 years. Rebecca's husband isn't the only one to think that the deductibility of mortgage interest makes a more expensive home a good deal financially. But sometimes the 'conventional wisdom' isn't really wise. So let's pull out our calculators and take a look at mortgages, taxes, and housing prices. We'll assume that Rebecca is in the highest tax bracket. That would mean she gets the biggest possible benefit from the deductibility of mortgage interest. In 2002, the top bracket is 38.6%. So for every dollar of interest that Rebecca pays the mortgage company, her tax bill would be reduced by 38.6 cents. Not such a good deal. In fact, she could cut out the middleman and just give a buck to a friend. I'm sure that the friend would be willing to give her 40 cents in return! Is it really that simple? Probably not. There are other factors to consider. Some people would argue that it's still a good deal because of the benefits of using OPM (other people's money). That's an old idea. And one that does indeed work well when prices are increasing. Let's see how it works. Suppose Rebecca buys a house and she's paying a mortgage at 8% per year. But with the tax deduction, the true cost of the mortgage is really 4.9%. How did we get the 4.9% figure? To calculate the true cost of your mortgage, first you'll need to know how much your deduction will be worth. To get that multiply the interest rate on the mortgage (in this case 8%) by your tax bracket (38.6%). That works out to 3.1%. Next you'll subtract the deduction rate from the mortgage interest rate to get your true cost to borrow (8.0% minus 3.1% = 4.9%). Now back to OPM. For Rebecca to benefit from the money she borrowed, the house would need to appreciate by more than 4.9%. Is that possible? The Office of Federal Housing Enterprise Oversight publishes an index that compares housing prices going back to 1980. For the first quarter of 2002, housing prices across the U.S. had increased by 171% compared to 1980. That works out to about a 4.4% annual increase in price. So it would be close for Rebecca. There were some regional differences. Some areas did quite well for awhile. But others did not. For instance in the Northeast, prices dropped after 1989. Prices didn't return to 1989 levels until 1998. So all housing markets aren't created equal. Even though you can't predict the future, studying the history of your community should give you an idea of how lively the housing market is. As Rebecca has pointed out, there are also personal reasons to want a nicer home. And only she can put a value on what a nicer home would mean to her family. Should Rebecca go ahead and buy the bigger house? That's up to her. But if they are going to do it, her husband is right. Low mortgage rates does make it easier. Whatever they decide I hope that they enjoy their home and it's never a financial burden to them. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Gary Foreman is a former Certified Financial Planner who currently edits The Dollar Stretcher website http://www.stretcher.com. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Median vs. Mean Editor's Note: This article includes graphics that are not easily represented in text. You can find a Microsoft Word document at http://www.MathOptions.com/medianvsmean.doc with the full text and graphics. Dear Statman- For some time I have been using the average of my data to describe the "center" of its distribution. However, some people tell me it is better to use the median since it is more robust. I don't know what "robust" means or when to use the average or the median. Can you help? Signed, Above Average Dear AA- Ah, yes, the eternal question. Which is better, the median or the mean? As usual, I will begin with some background and then explain when one estimate might be preferred. Most people are familiar with the average, or mean. You find it by adding up all the observations and then dividing by the number of observations. The mean is usually written x with a line over it, (read "x-bar"). Here are a few numbers for use in an example. 2.1, 3.6, 5, 6.2, 8, 10.5, 15, 21, 30.2 The sum of the above nine data points is 101.6. The average is found by dividing the sum by 9, since there are 9 observations. So, x-bar= 101.6/9 = 11.29. That's easy enough. Now, how do we calculate the median? The median splits the data in half, so that half the observations are above it, and the other half are below it. You find it by ranking the data from smallest to largest and then picking off the middle value. For the above data, the median is 8. The median is usually written as an x with a tilde over it (read "x-tilde"). I think you can see that both the mean and the median can be used as estimates for the location of the center of the data. The median has an interesting property. Suppose we added a new observation to the above list. Let the next observation be 62. What is the new average? In that case, the average would change to 16.36, up from 11.29. What about the median? Well, it increases from 8 to 9.25. (When there is an even number of data points, the median is taken as the average of the two middle observations. In this case, the two middle values are 8 and 10.5 and their average is 9.25.) So, the mean moved up a bit more than the median. Supposed the next observation we added was very big, like 350. Now the average is 46.69, but the median is only 10.5. So, the median is stable in the sense that it doesn't change much, even if you add some extreme observations. In contrast, the average can be greatly affected by extreme observations as the above demonstrates. It can be helpful to have an estimate that is stable since extreme values might be suspect. Chalk one up for the median. Now as an experienced person, you might realize that you can't get something for nothing. If you were to use the median all the time, what would you give up? The answer is you might be giving up accuracy in knowing where the true mean of your distribution lies. Let me explain what that means. Consider data from a normal distribution (bell curve). Figure 1 shows a pile of data (or histogram) with a smooth bell curve superimposed over the top. The pile of data is for individual observations. It has a mean (often called m) and a standard deviation (often called s). The value of s is a measure of spread in the data. The bell curve is symmetrical about the mean, since I could flip one half of the curve over about the mean and have it line up with the curve on the other side. For symmetrical distributions like the normal, the mean and the median are the same. Suppose I wanted to take averages from the pile in Figure 1. For example, I could take 5 observations at random and find the average. Then, take another 5 observations and find that average. Suppose I repeated this hundreds of times and then asked the question, "How do these different averages pile up"? Figure 2 is a graph showing the resulting histogram. Note that the histogram for the averages is less spread out than the histogram for the individual observations, whereas the means are the same. It turns out that the standard deviation for the bell curve of the averages is equal to the standard deviation for the individual observations divided by the square root of 5 since we were taking averages of 5 data points. If I had used 10 data points, then the standard deviation of the averages would have been equal to the standard deviation for the individual observations divided by the square root of 10, etc. If you have been reading these articles for a while, I am sure by now you are a budding statistician. As such, you realize that estimation of the average is only half the story. Due to the random nature of data, the average you calculate will probably not be the same as the true average, but it will be close. That implies that you need to acknowledge the uncertainty in the calculated average by using a confidence interval. The above discussion is useful to know when you want to find confidence intervals for the average. A confidence interval is a range where we expect to find the true average (m) a high percentage of the time, like 95% or 99%. For data that follow a normal distribution, 95% of the observations will lie within about 2 standard deviations of the mean, and 99% of the data will fall within 2.6 standard deviations. The bigger the standard deviation is, the wider the confidence interval will be. If the confidence interval is very wide, then we aren't very sure where the true mean lies. If the confidence interval is very narrow, then we have a pretty accurate idea about the actual value for the mean. Let's do a quick example to see how this is used. Suppose we have tons of data from previous work and so we have a very good idea what s is. (Incidentally, if you aren't comfortable using a historical value of s and want to use the observed value from the 5 data points, you would need to increase the number you multiply s by in order to get the same confidence levels. You would need to use numbers from a table of the t-distribution. This subject has been covered in previous Statman articles.) Then we get 5 new data points and want to find a confidence interval for the mean. We do our calculation on the data and find that x-bar = 25.2. Since the data are coming from a well-known process, we feel comfortable in using an historical value of s = 2.1. What is a 95% confidence interval for the mean of our distribution? Well, we take 2 x 2.1 / square root(5) = 1.88 and add and subtract it to 25.2. The resulting 95% confidence interval for the mean is then [25.2 - 1.88, 25.2 + 1.88] or [23.32, 27.08]. Thus, if we did hundreds of experiments and each time observed a slightly different mean and formed the confidence interval the same way, then we would expect that 95% of the time, the computed range would contain the true mean, m. What if we wanted to calculate a confidence interval for m, but use the median instead of the average? First we need to know the spread in the pile of medians. If you were to take a sample of 5 from our normal data in Figure 1 and calculate the median, and repeated this hundreds of times, the medians would pile up like those in Figure 3. It looks very similar to the pile of averages in Figure 2, with one exception. It is a little wider. In fact, the standard deviation of the pile of medians is 1.25 x s / square root(5) where s is the standard deviation from the original individual observations in Figure 1. Remember the standard deviation of the pile of averages was only s / square root(5). This demonstrates that if we were to form a confidence interval using the median instead of the mean, it would be 1.25 times larger. If we increased the sample size from 5 to 8 and then constructed the histogram of medians, it would have the same spread as the pile of means using samples of 5. Said another way, we would need to increase our sample size for the median in order to get the same confidence interval width as that of the mean. We say that the median is less efficient than the mean, because it requires more data for the same accuracy. So, if your data come from a normal distribution, you are better off using the mean over the median. Chalk one up for the mean. Now, on to robustness. To understand that, we first need to understand the concept of efficiency. The efficiency of an estimate is usually discussed relative to a different estimate. For example, the mean and median both estimate the center of the distribution (pile of data). The relative efficiency of the median to the mean is the ratio of the variances of the estimates. The variance is just the square of the standard deviation. In the above example, the standard deviation of the pile of means was just s / square root(5). So, the variance of the mean is s x s/5 and similarly, the variance of the median would be (1.25 x s) x (1.25 x s)/5 = 1.57 x s x s/5. Then the relative efficiency of the median to the mean is 1/1.57 = 0.64. A robust estimate is one where the relative efficiency is high (rarely less than 1) over a wide range of distributions. By different distributions I mean the way in which the data pile up. For example, your data may not follow a bell curve. It might pile up so that it's lop-sided. Examples of distributions that are lop-sided are the chi-squared, the lognormal and the Weibull. It turns out that in some cases for these and other distributions, the relative efficiency of the median is greater than 1. Determining when this is the case is a little complicated. You might want to ask your in-house statistician. So, in some special cases you can have the best of both worlds when using the median since it won't change with the addition of extreme observations, and it will provide narrow confidence intervals. In many cases, though, the median will give wider confidence intervals than the mean, which is one reason the mean might be preferred. I hope this helps! Thanks, Statman. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you have a question for Statman, please send it to mailto:Statman@MathOptions.com. Statman will answer questions about basic statistics that are of general interest to people working in industry. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2002 by William D. Kappele, Neil Flatter, Gary Foreman, and Charles S. Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you like E-Math News, please forward it to a friend. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A free newsletter published every other month by Math Options Inc. http://www.MathOptions.com 814 Lakeway Drive #179 FAX (503) 218-6587 Bellingham, WA, 98229 Toll Free (888) 764-3958 William D. Kappele, Editor Bill@MathOptions.com To subscribe to or unsubscribe from E-Math News please visit http://www.mathoptions.com/e-math.htm. If you don't have access to the World Wide Web, please send E-Mail to mailto:EMathNews-request@listdelivery.com with either "subscribe" or "unsubscribe" in the message body.