Metric System and Fractions ```Name: Serena Status: student Grade: 6-8 Location: TX Country: USA Date: November 2008 ``` Question: I am a math educator and just learned from a science educator that students are not allowed to represent the metric system numbers as fractions. They are only allowed to represent them as decimals. Is it wrong to allow them to represent 3.2 as possibly 3 1/5? In math we teach them number sense and being able to represent the different forms of numbers as percents, fractions, decimals, etc. Replies: Having been a teacher of both Math and Science I appreciate your concern. As a mathematician, all forms of a fraction are viable, and the ability to convert from one form to another is a very useful mathematical skill. It is scientific convention, however, that ALL fractions are represented as decimals, and in the case of large factors, then Exponential notation is used - e.g. Avogadro's Number = 6.02 x 10E23 Is it wrong - No. It allows a free exchange of scientific discoveries by the use of a uniform "language" of science. By convention even American Scientists use the metric system of units. No one else in the USA would use millimetres, but its use by scientists is not wrong. (In Scientific circles by the way the system of measurements is called SI - System International ) Nigel Skelton Tennant Creek High School AUSTRALIA Hi Serena Being also an educator and living in a country( Brasil) where the metric system is the legal and used one, maybe I can be of some help. The metric(decimal system) is based on the decimal number system, base 10 of course and all multiples and sub multiples are base 10 determined, at all type of measurements (length, surface, volume). Then it is more practical and I would say logic to use the decimal fractions. To remember also it is easier since the prefixes are all similar and taken from the Latin language: meter, decimeter, centimeter, millimeter, kilometer; and so on: liter, deciliter, centiliter, milliliter, kiloliter... Maybe it is not "wrong" to use other kind of fractions but it breaks the logic behind the decimal system. Thanks for asking NEWTON! Mabel (Dr. Mabel Rodrigues) You sound like a very thoughtful teacher. I think the answer to your dilemma lies in your goals and objectives for the lesson. It sounds as if you are using metric system numbers in a lesson to teach number sense. The numbers without the units are just numbers. If you want to be a purist about the metric system, drop the units to teach the number sense and relationship between decimals, fractions, etc. The concept of units can be added later. OR keep doing what you are doing, and let the students know that when they do science, they use decimals only. The science teacher can reinforce this (perhaps you can take the opportunity to develop an interdisciplinary unit so students are doing similar work in math and science--- perhaps a measurement lab or similar measurement-intensive work with the same sorts of math operations required in both classes). Thank you for your efforts! Patricia Rowe Hi Serena One of the reasons might be that in science we quantify things based on observations and measurements. A fraction is a very precise thing - an object can be split mathematically into exactly three one-third pieces. However, when we measure something physically, we can never achieve the precision that a fraction represents. The degree of precision depends on the instrument used for the measurement. For example, if I give you a ruler and ask you to measure the cover of a textbook, you could probably measure it within a sixteenth of an inch or so - a small percentage of the size of the cover. If I then give you a dime and ask you to measure the thickness with the same ruler, you will have difficulty achieving the same degree of accuracy, as the marks on the ruler are much too coarse compared to the thickness of the dime. So we need a better ruler. Think now about the ruler required to measure precisely one-third of something. How can we ever resolve 0.3333333... ? Hope this helps. Bob Froehlich I am not sure who claims that 3.2 does not equal to 3 1/5 or 16/5 for that matter. However, I can think of reasons why decimals are more convenient, but not "wrong". First, in addition and subtraction there is no need to find a "common denominator". Second, in multiplication things get even more messy. Example: (3 1/5)^2 = (3 + 1/5)x(3 + 1/5)= 3x3 + 2x(3x 1/5) + (1/5)^2 = 9 + 6/5 + 1/25 = 9 + 30/25 +1/25 = 9 + 31/25 = 9 + 1 + 6/25 = 10 6/25. You have to admit that is messy and prone to numerical mistakes. Vince Calder In general life, it is good to know both fractions and decimals. However, decimals are the vast, vast preference for science. There are many reasons why fractions are inappropriate in science, especially when used with the metric system. Perhaps the easiest reason has to do with comparing and ordering numbers. With fractions, there are many different options for what denominator to use -- and it becomes much more difficult to compare two values if they are not using the same denominator (such as if you reduced them to their simplest form). It is quick and easy to know that 10.4 is smaller than 10.42, but it takes a split second to think about if 10 2/5 is smaller than 10 3/7. Second is the prevalence of computational equipment (such as calculators). It is much easier to perform calculations using decimals as opposed to fractions, and most computers and calculators default to decimal output. With how important computational equipment is in science, decimals are the overwhelming preference. Third, the metric system is designed to work with base-10 numbering system to not only be easily scaled (from very large to very small). It is much easier to start with 42cm and move the decimal place to 0.42m instead of converting from 42cm to 3/7m (or 21/50m). Last, and perhaps the most important, there is the concept of precision. In science, we always try to measure values precisely. However, we should never represent a measurement as having more precision than it actually has. With fractions, there is no way to represent precision. You would say 10 2/5 for either 10.4cm or 10.400000 -- there is no way to distinguish the two. Let me illustrate this with an example: if I use a ruler to measure the size of a, say, turtle, it would not be appropriate to say it was 10.43593 cm long because the tool (a ruler) is not appropriate for that kind of precision. It is more likely that I measured only 10.4cm, or even just 10cm if it was squirming around. Suppose you used a laser to measure a distance very, very precisely, and suppose it ended up exactly 10.40000cm. You really measured all those zeroes, and it is important to represent them. With fractions, you would just end up using 40000/100000, at which point you may as well use decimals. English/standard system measurements are not in base-10 units (e.g. pints-quarts-gallons, inches-feet-yards-miles). For this system, in every-day use (e.g. not science), fractions are very important to know. People are used to thinking of things in terms of quarter-inches and half-pints, so a student should be able to speak in those terms as well. Hope this helps, Burr Zimmerman As one who instructs both math and science in the same semester, I appreciate your question. In science, we are concerned with the accuracy of real world measurements. For example 3.2 is a far less accurate measurement than 3.2000. A science book that addresses "significant digits" will give you more information than I can fit in here. In math, a fraction such as 1/5 is a precise, rational number. It would be inappropriate to list it as a fraction in the science lab because it would not give us any idea of the accuracy of the measurement. In math, 1/2 + 1/2 = 1 is an exact problem, and one assumes infinite accuracy if the topic ever comes up. But in science, we ask, "How sure are you of the measurements of those halves?". Are they measured to the nearest tenth (0.5), the nearest thousandth (0.500) or nearest hundred thousandths (0.50000) of a unit? My advice? Just keep on what you are doing in your math classes and let science teachers explain why they want only decimals. (It all becomes pretty apparent to the students when they hit high school science.) Hope this helps. Bob Avakian Oklahoma State University Institute of Technology Strictly speaking, there is nothing stopping you from saying 3 1/5 concerning metric measurements, but as the system is based on 10 as opposed to doubling (2, 4, 8, 16, etc) as the Imperial pounds and ounces are, then decimals are far better in the metric system. e.g 2 miles equal 3.2 kilometres. Howard Barnes Serena, As a retired math and science teacher, I, like you, wrestled with this issue many times. I often questioned teaching the algorithms of adding, subtracting, dividing and multiplying fractions is a waste of time within a math curricula. I also insisted in fractions not being used in science class. I have been reminded by my other jobs (teachers never made much money) that USA construction projects is still stuck with manipulating fractions. Working with tile installations, I was amazed at the amount of fraction calculations that were necessary to plan the tile placement and cutting determination. It was done with fractions. I brought up that there was an easier method, but contruction workers find the SI system too difficult. It is a sad case, but points out that in our society we still use fractions because we have failed to completely adopt the SI system. But I think the key to understanding fractions and their manipulations comes more focused with dealing with algebra and calculus functions. Here in the abstract students are placed in the position of manipulating fractions; so the skill can be of benefit with higher mathematics. Now with this said, one of my colleagues insists that one of the prime reasons students do not like working math and have trouble with algebra and calculus is the confusion they develop with the time they frustrated themselves in grade school with fraction manipulations. Her opinion may or may not be justified, but it shows that the US student is confronted with a series of algorithums that may contradict or confuse mathematical understandings. A great deal of this is caused by our duel system of measurement. Science does not represent data as fractions, but science uses fraction all the same. Students are often asked to cut quantities in half or quarters in experiments, and oral explainings often use fractions instead of 0.125. Fraction have their use. Ok, let's portion our pie into 0.25. Steve Sample In my class, I am delighted if my students exhibit number sense. If they describe the numbers using fractions, decimals, percentiles, or appropriate descriptors like "greater than" or "less than," it's fine with me. I will say "half a meter" WITH no guilt feelings. Richard Barrans Department of Physics and Astronomy University of Wyoming Serena, The biggest reason that comes to mind about decimals vs. fractions in science is the idea of significant figures. Using fractions you cannot easily express the significant figures of a number, which can be critical when performing math related to measurement and testing. Ian Farrell Hi Serena, I am not sure about "allowed", but to a scientist the number 3 1/5 has a very different meaning, at least in our context, than the number 3.2. The number 3 1/5 is an exact number, whereas in science most things are not exact. Virtually all measured quantities have uncertainty in them. Some have very little uncertainty and some have a lot of uncertainty, and so we write numbers down in a way which explicitly shows this fact of life. In that sense the number 3.2 (which experimentally might be 3.1 or 3.3 depending on who does the measuring) is not the same as the number 3.200 (which might be 3.202 or 3.198 if someone else measures it). However, the number 3 1/5 is an exact number representing the fraction 16/5, or 3.20000000000000... However, there are exact numbers in a few experimental contexts and it would undesirable, sometimes even incorrect, for a scientist to use decimals to represent them. As a simple example: if I have 20 eggs, how many dozens of eggs do I have? Answer: 20/12 = 1 2/3 dozen, exactly. Not 1.7, or 1.67, or 1.667. I suppose 1.666666... would be okay, but it would never be used in a journal article by a scientist, since 1 2/3 gives the exact answer in a more compact form. All of this, by the way, has nothing to do with the metric system. The issue would still be there whether we measured length in meters, yards or ells. It simply has to do with scientists' preference for numeric representations which reflect the confidence that other scientists can have in getting the same result under the same conditions. In the end, students need to be able to represent numbers as fractions, percentages, decimals...and science students also need to learn scientific notation, which we use a lot (2344.55 = 2.34455 x 10^3) Hope this is helpful. Dr. Topper Hello, I imagine that the confusion comes from a valid point, though it may not appear at first to be one. When I first saw your question my reaction (probably typical) was to think "of course you can write them as fractions." However, I think there is a reason you are being asked to have students write out in decimals even if it has not been made clear. The reason to write decimals instead of fractions has to do with measurement. So long as you are dealing with pure mathematics, it does not matter if you write 3.2 or 3 1/5. However, once you are talking about a measurement, the number can tell you more than just the value. One of the most common ways to convey both the magnitude and precision of a measured quantity is to use "Significant Figures." Suppose we are talking about length. If I say an object is 3.2m (meters), that implies the precision in my measurement of the object is no better than 0.1m (which is pretty poor!). If on the other hand I say the object is 3.200 meters, then you know I measured it down to 0.001m precision. The reason for not using fractions is now clear, if I say the object is 3 1/5 meters long, then you will not know how precise that value is. It could be 3.2, 3.20000, or anything else. What has perhaps not been made clear by the science educator is that this is not a question of using the metric system, but its applicable to any system of units/measurements. So if this person maintains that it is OK to say 3 1/5 inches, then they are wrong. Another way to say it is that you could in fact use fractions, but specify both the value and the precision. So the length of the object could be written as "3 1/5 meters, to the nearest 1/1000th meter." That answer is then entirely equivalent to the decimal answer. cheers, PS I had almost decided to neglect this as it is one of my personal beafs, but on second thought I will go ahead and stick my neck out. While the basic idea of "significant figures" is useful (more so than not having any measure of precision), it is not the best thing in practice. The motivation for using it is good and in the right place, but in practice it is better (I think) to just go with the "full" answer. My problem with significant figures is that if I write 3.20 meters it states a precision to 0.01 meters while 3.200 implies 0.001 meters, but there is nothing in between (which is nonsense). As a practicing scientist (and I would suspect in any endeavor where precision and accuracy are important), there are many cases where the precision is in-between and it is important to know just how precise the value is. And thankfully there is a much better way to express it. I can write the number as measured and then using either Â± or parentheses writing the precision. So, 3.20 Â± 0.02 meters, or equivalently (should the Â± plus/minus symbol not work) 3.20(2) meters. Even when it comes to doing arithmetic with measured quantities, I will still say it is better (and not much more complicated) to teach students the correct way than the "significant figures" and "significant arithmetic" way. Michael S. Pierce "Fractions" are a very inefficient way of expressing numbers. This would not be so confusing if the "denominator" was always "100" in which case the fraction and decimal notation would be the same. Confusion results because even simple computations, for example:3/7 + 1/19 = ??? The underlying mathematical issue is that while rational numbers can always be expressed as a repeating decimal, the length of the repeating string may be very long. The important concept is that every rational number can be expressed as a repeating decimal, and every repeating decimal number is rational. Physical measurements -- lengths, widths, areas, volumes, etc. -- are almost always expressed in "base 10". It is a sad commentary that not only has the U.S. (and a few other countries) clung to the "English" units of "inches, feet, yards, etc., but that conversions even within of these units of measure are a different conversion factor. What sense does that make??? Vince Calder Click here to return to the General Topics Archives

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