Decimals Versus Fractions: Pluses and Minuses
Date: 08/19/2011 at 12:22:34 From: Terri Subject: Why do we sometimes use decimals instead of fractions? I have just started teaching middle school math. My students' questions routinely take the form of, "Why do they write it that way"? My answer is usually to equate "math shorthand" with the texting lingo of kids -- just a quicker way to write something. When asked what they thought the hardest part of math is, many answered fractions and decimals. So when we talked about how a decimal represents the same thing as a fraction, they asked, "Why use decimals?" I really don't know that answer, unless it too is simply another shorter, quicker way to express quantities. My thought is that .023 is easier, quicker, and neater than writing out 23 over 1,000. However, I don't want to pass that along if there is more to it than that. Is that why, or is there another reason? I guess I'm searching for a deeper meaning of decimals and their use.
Date: 08/19/2011 at 14:47:52 From: Doctor Ian Subject: Re: Why do we sometimes use decimals instead of fractions? Hi Terri, > I have just started teaching middle school math. Congratulations! > My students' questions routinely take the form of, "Why do they write > it that way"? My answer is usually to equate "math shorthand" with the > texting lingo of kids -- just a quicker way to write something. That's a nice way to tie it to something they know. But there's also more to it than just writing things more quickly. And sometimes the fraction is actually quicker to write. For example, I'd rather write this ... 1/7 ... than this ... ______ 0.142857 ... wouldn't you? > When asked what they thought the hardest part of math is, many answered > fractions and decimals. So when we talked about how a decimal > represents the same thing as a fraction, they asked, "Why use > decimals?" I really don't know that answer, unless it too is simply > another shorter, quicker way to express quantities. One way to get them to answer this for themselves is to ask them to do some basic operations using the two different notations. What they will find, I think, is that for the most part, decimals make it easier to add things: 1/4 + 1/5 = 5/20 + 4/20 = 9/20 0.25 + 0.20 = 0.45 But in some sense, that's just because decimals already have common denominators (or are nearly there if you just tack some zeros on the end): 0.12 + 0.3456 = 0.1200 + 0.3456 = 0.4656 So if you have fractions with the same denominator, there's really no advantage. In fact, sometimes the fractions are easier, e.g., 1/7 + 2/7 = 3/7 0.142857 + 0.285714 = ... I don't even want to add those! Using decimals can also make comparisons easier: 3/5 < 4/7 ? Uh, maybe. 0.6 < 0.57 ? Clearly not! On the other hand, with fractions, multiplication is easy, while with decimals it's more work: 3/4 * 2/3 = 6/12 = 1/2 0.75 * 0.667 = 0.50025 And fractions have the nice property that you can often cancel numerators and denominators, so you don't even have to multiply: / / / 2 3 4 5 2 1 - * - * - * - = - = - 3 4 5 6 6 3 / / / Compare that to the decimal alternative, 0.667 * 0.75 * 0.8 * 0.833 = 0.3333666 Note, too, that in all these cases, the result from the fraction is exact, while the result with the decimal has to be rounded off somewhere... which may actually have adverse consequences in a later calculation. The invention of calculators skewed things in favor of decimals, since they're usually easier to enter and to show on a simple display. But the tide is turning back as systems like Mathematica, which can do exact calculations, become more common. So in some sense, the reason we have both fractions and decimals is the same reason we have different kinds of hammers, and different kinds of saws, and different kinds of screwdrivers.... Any notation is a kind of tool, and not every tool is the best for every job. And that's a good lesson for your students to learn, because it's really quite rare in mathematics for there to be one best way to solve a problem. Part of the art of mathematics is in deciding which approach is likely to be best, before jumping in with the first one that occurs to you, or the one you habitually use. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 08/19/2011 at 14:59:06 From: Terri Subject: Thank you (Why do we sometimes use decimals instead of fractions?) Yes, that helped tremendously! (That's why I love this site!) I really like comparing the various representations to saws, hammers, and drills.... different tools for different jobs. That will help them so much. My goal is to help these kiddos truly understand numbers, not just memorize steps and procedures. I think your analogy and examples will help with that a lot.
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