數(shù)學(xué)英語 02 How are Fractions and Division Related?
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 lining 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 slashed line? Well, this notational 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 interpretations 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!
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 lining 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 slashed line? Well, this notational 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 interpretations 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!
