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数学英语 02 How are Fractions and Division Related?

时间:2010-07-20 01:46来源:互联网 提供网友:ft1186   字体: [ ]
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by Jason Marshall

In the last episode, we talked about exactly what decimal points and decimal numbers are, and why they are so convenient to use. In the next few weeks we’re going to continue talking about decimals and their place in the world of math. Up first today, we’re looking at the relationship between decimal numbers, fractions, and the process of division.
But first, the podcast edition of this tip was sponsored by Go To Meeting. Save time and money by hosting your meetings online. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their web conferencing solution.
How Are Decimals and Fractions Related?
We’ve now talked extensively about fractions and a bit about decimals—both of which are used to represent numbers that are part of a whole. In other words, whereas integers always represent whole numbers like 1, 2, and 3, fractions and decimals represent numbers that have non-whole number parts—like 1/2 and 3.75. But how are all these fractions and decimal numbers really related? Well, the short answer is that they’re all related through the process of division. But my meaning here might be a little fuzzy, so let’s walk through the explanation step by step.
How to Understand Fractions
Let’s start by talking about fractions—specifically, how to understand what fractions mean. For example, take 22/7. What does it mean? Well, one possible answer is right there in how we say the fraction: twenty-two sevenths. In other words, imagine taking 22 sticks that are each 1/7 of a meter long, and then laying them out end-to-end. The unit of measurement here (I chose meters) really doesn’t matter. It could be meters, miles, widths of a hydrogen atom, or anything else—the math is the same. The point is that 22/7 meters is the length you get by lining1 up 22 sticks that are each 1/7 of a meter long.
Okay, now let’s imagine that in addition to sticks that are 1/7 of a meter long, you also have sticks that are 1 meter long. Start setting those meter sticks down end-to-end next to your row of 22 shorter sticks. You’ll find you can lay 3 of these meter sticks down, but that won’t go quite far enough to match the length of the 22 shorter sticks. Putting down another meter stick is too long though, so you know the length 22/7 has to be greater than 3 but less than 4. Being clever, you notice that adding exactly 1 of your 1/7 of a meter long sticks next to the three meter sticks already on the ground will give an exact match. So, 22/7 meters must be the same length as 3 meters plus 1/7 of a meter—better known as 3 1/7 (three and one-seventh) meters.
How are Fractions and Division Related?
Okay, here’s where things get interesting. Have you ever noticed that the way we write fractions is the same as the way we usually write division problems? In other words: two numbers separated by a horizontal or slashed2 line? Well, this notational3 similarity isn’t an accident. Let’s go back to our example fraction 22/7, and think about it in terms of a division problem instead: 22 divided by 7. How can we interpret this problem? Well, imagine you have a single 22 meter long stick that you need to divide into 7 equally sized pieces. How long will each of those pieces be? That question is exactly what the division problem is asking too. So, imagine you go about making marks on your big 22 meter stick and successfully divide it up into 7 pieces. And then imagine using your meter sticks and your 1/7 meter sticks from before to figure out how long one of these divided pieces is. The answer will be 22/7 or 3 1/7—the exact same as before.
This is pretty amazing. In one case—thinking about 22/7 as a fraction—we put down 22 short 1/7 of a meter long stick, and in the other case—thinking about the problem as 22 divided by 7—we divided up one big 22 meter long stick into 7 equally sized parts. Both interpretations4 give the exact same answer because 22/7 and 22 divided by 7 represent the same number—they’re two equivalent ways of thinking about the same problem. Clearly division and fractions are very closely related.
How Fractions, Decimals, and Division are Related
In fact, division (as its name implies) can be thought of as the process of breaking apart whole objects into their fractional or decimal parts. This relationship between fractions and division will be key in our discussions over the next few weeks as we talk about:
how to convert between decimals and fractions,
how to compare the sizes of decimals and fractions, and
how to use decimals and fractions to solve problems in your life.
How to Do Division
One more thing for today: How exactly do you do division? Before we move on to more advanced topics, you certainly should make sure you can successfully work out division problems—there are a couple of practice problems at the end of this article for you to test yourself with. If you find that you do need a refresher, check out this week’s Math Dude “Video Extra!” episode on YouTube where we’ll get you up to speed. In the mean time, here’s my quick and dirty tip on doing division: It’s okay to use a calculator. Seriously. Division is far messier to work out on paper than addition, and there aren’t a lot of easy quick-calculating techniques to speed up the process either. So, if it’s handy, a calculator just might be your best bet.
Wrap Up
Okay, that’s all the math we have time for today. Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
Please email your math questions and comments to。。。。。。。You can get updates about the Math Dude podcast, the YouTube “Video Extra!” episodes, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Also, if you or your company are interested in sponsoring The Math Dude or any of the other Quick and Dirty Tips podcasts, please send an email to。。。。。。for details.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
 


点击收听单词发音收听单词发音  

1 lining kpgzTO     
n.衬里,衬料
参考例句:
  • The lining of my coat is torn.我的外套衬里破了。
  • Moss makes an attractive lining to wire baskets.用苔藓垫在铁丝篮里很漂亮。
2 slashed 8ff3ba5a4258d9c9f9590cbbb804f2db     
v.挥砍( slash的过去式和过去分词 );鞭打;割破;削减
参考例句:
  • Someone had slashed the tyres on my car. 有人把我的汽车轮胎割破了。
  • He slashed the bark off the tree with his knife. 他用刀把树皮从树上砍下。 来自《简明英汉词典》
3 notational 9beeb37cef072d7d56ae97601426a407     
[计] 记数的
参考例句:
  • Notational conventions concerning functions in vector-matrix form are summarized here entirely in terms of vectors. 关系矢量-矩阵形式函数的符号约定在此完全用矢量来描述。
  • A good legendthe notational symbols the name of the symbol, and a description of its usage. 好的图解中包括用到的图符、符号的名字,以及关于符号用法的描述性文字。
4 interpretations a61815f6fe8955c9d235d4082e30896b     
n.解释( interpretation的名词复数 );表演;演绎;理解
参考例句:
  • This passage is open to a variety of interpretations. 这篇文章可以有各种不同的解释。 来自《简明英汉词典》
  • The involved and abstruse passage makes several interpretations possible. 这段艰涩的文字可以作出好几种解释。 来自《现代汉英综合大词典》
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