Representing recurring decimals

Key Stage 3

Maths

FDP review (9.1a)

Put these three in order of size from smallest to largest:

Which fraction (in it's simplest form) fits the box on the number line?

﻿Hello, my name's Mr. Langton and today we're going to look at how we represent recurring decimals. You're going to need something to write with, something to write on, and ideally a quite space to work where you won't be distracted. When you're ready, we'll begin. We'll start off with a "try this" activity. Look at Binh and Zaki's calculations. What's the same? And what's different? Pause the video and have a little bit of a think, make some notes, and when you're ready, unpause it, we'll go through it together. So pausing in three, two, one. So what ideas did you come up with? I noticed a couple of interesting things. For a start, they are the same. And I know this is going to be a bit weird, it's a little bit confusing, but we know that three is obviously the same as three. And one-third, right, written as a decimal, is not point three recurring. So they're the same. Three lots of the third is the same as three lots of not point three recurring. But this weird bit here, is that that one equals one, and that one equals not point nine recurring, and it's a really really weird part- one of my favourite parts of Mathematics, That one is equivalent to not point nine, nine, nine, nine, nine, nine, nine, nine, nine, nine, nine, Now if I were to stop it there, not equivalent. But if I go on forever and ever and ever and ever and ever and ever, If I keep going forever, then yes they are equivalent. One is the same as not point nine recurring. And that's something we'll look at in a lot more detail in a few years time. But I just thought it's a rather interesting little bit to start off with for this activity. Right, let's move on. Using what we already know about recurring decimals, we can convert each of these into percentages or decimals. We're going to start off with the first fraction, one-third, because we did that on the last slide. One-third, as a decimal, is not point three recurring. Can put that dot above there, so we make sure that it's recurring. As a percentage, we multiply the decimal by a hundred to get a percentage, so thirty-three point three percent. And that point three is recurring, make sure you say that's recurring. Now a sixth is quite a tricky one. If I were to draw a quick diagram here, to represent thirds, that's one-third like that. If I split it in half, I get a sixth, so a half of third, is a sixth. So if I can do one-sixth, divided by two, so one third, divided by two, I'll get a sixth. So let's draw that now, let's draw a bus stop. We know that one-third is not point three recurring three, three, three, I'm just going to write a few threes. We're going to divide that by two, we might need to put some more threes on, we'll see in a minute. Now, twos into zero, don't go. Make sure you line with your decimal point. The next one. How many times does two go into three, it goes once, with one remainder. How many times does two go into thirteen, that's going to go six times, with one remainder. We've now got the same question, how many times does two go into thirteen? So once again it's six with one remainder, and we can see that's going to keep repeating now. So one-sixth, is zero point one six six six... And the sixes are going to carry on. So I'm just going to put a dot above there. The six is recurring, the one isn't just the six. And the percentage, if I were to multiple that by a hundred, I would get sixteen point six recurring, that six is going to keep recurring, and that's my percentage. Now a ninth, is a third of a third. And that's actually going to be an easier calculation to do. So if I have a third, as a decimal, not point three recurring, and I divide that by three, threes into zero don't go, threes into threes go once, threes into threes go once, and so on. So one-ninth, is the same as not point one, recurring, and as a percentage, that's going to be eleven point one percent, with that one recurring like that. Now just tidied up the slides a little bit to make it a little bit easier to write on. Let's look at seven-thirds now. We know that we could write seven-thirds as two whole ones, and a third. And now, we can turn it into a decimal we've got two whole ones, and we have a third, it's point three recurring. So seven-thirds is the same as two point three recurring. As a percentage, we multiple that by a hundred, we're going to get two hundred and thirty three point three recurring percent. Now seven over thirteen might look a little bit tricky, but we can actually link that in with the one that we did before. Just like with the thirds and the sixths, we said that a sixth is half of a third, seven thirtieths, is a tenth of seven over three. So if we divide our decimals and our percentages by ten we'll get our answers. So two point three recurring divided by ten, is going to be zero point two three recurring and it was just the three that repeats, put the dot there, and as a percentage it's going to be twenty three point three recurring percent. Finally, seventy seven over thirty. Little bit sneaky, if you find an equivalent fraction for seven thirds, seven over thirty, sorry seventy over thirty, we can now add together these two, seventy over thirty and seven over thirty will give us seventy seven over thirty. So we need to do, two point three recurring, added to not point two three recurring. Now because that's repeating, I'm going to put that as a three there and we know that there are more of them but they'll just keep matching up now. So if we add those together, we're going to get two whole ones there, the three plus two is five and six and six and six and so on. And I know I started counting on the left hand side which is normally not the way to do it, but it just stands out a little bit easier with these recurring decimals when you do it that way. So that's giving us two point five six recurring as a decimal and that's going to give us, as a percentage, two hundred and fifty six point six percent. And don't forget that six is recurring. Okay, now it's your turn. Your task is to turn as many of these fractions as possible into decimals. I've given you a couple of reminders, I've put a few percentages up there as a start, but your job is to turn them into decimals. Let's see how many you can do. When you move on to the next slide, it's going to give you the chance to pause the video, if you're not sure, don't pause it straight away, keep it going a little bit longer, and I'll give you a few hints. Good luck. Okay, I'm just going to use this slide, just to give you some hints. Now we know what one sixth is, as a percentage, so as a decimal, it would be not point one six recurring. Two sixth, is the same as two lots of one sixth. So you might want to use your answer from the first part to where we're at now. Five-sixths would be the same as five lots, over sixth. There are two ways you could do seven-sixths, you could do seven lots of one sixths, you could multiply it by seven, or you could do one whole one, and add on one sixth, that might be a bit easier. Seven over sixty, so once we've got that answer from our previous one, we can take our answer from before, seven over six, and we can divide it by ten, and seventy over six, if we take our answer from seven sixths, and we multiple it by ten, that'll let us find it up more. Now let's look over here at the ninths. Two-ninths is, same sort of pattern again, it's going to be two lots of one-ninth. Twenty over nine, is going to be two-ninths multiplied by ten, two over ninety, is going to be two ninths, divided by ten, twenty two over nine is going to be made up of twenty-ninths, add it to two ninths. And one seventh, we've got no methods that are going to help from anything we've got there, what you're going to have to do instead is draw a bus stop and you've got to do one, divided by seven. And I'm going to give you a hint, you're going to need a lot of zeros. I would start with seven. And see if you can get on with it. Okay, right, pause it and finish your activity, when you're ready, unpause it and let's get you onto the answers. So how did you do? I think hopefully, you've had enough of those explanations to be able to get most of those. But what I am going to do, is I'm going to do that one-seventh for you again. Just so you can see how it works. So one divided by seven, and I told you didn't I, put a lot of zeros there. We're not changing the value of one, that's still equivalent to one isn't it. We'll just see where we go. Seven's into one, don't go, carry the one, don't forget that decimal point, it's really important. Seven's into ten, they're going to go once with three left over. Seven's into thirty, seven, fourteen, twenty one, twenty eight. That goes four times, remainder two. Seven's into twenty, almost twenty one, not enough, so it's only going to go twice, with six left over. Seven's into sixty is going to go eight times, with four left over. Seven's into forty go five times, with five left over. Seven's into fifty, I got seven times, with one left over. Seven's into ten, you may know if you've already done that one earlier, seven's into ten go once, and there's three left over isn't there. So you can see that it's now starting to repeat. So this bit here, not point one four two eight five seven, is a unique part, and then it starts repeating with the one before. So that's why here, if you look at my answer, I put a dot above the one, I put a dot above the seven, and that tells you that all the numbers, between the one and the seven, including the one and the seven, they repeat. So it's not point one four two eight five seven, one four two eight five seven, one four two eight five seven. And those six digests keep repeating, over and over again. If you want to have some real fun with this, you could then have a look at two-sevenths or three-sevenths and see if you can spot any patterns. I really like that that's one of my favourite things to do with recurring decimals. I want to leave that one for you to try on your own. So how could you convert these fractions into decimals without using a calculator? Bihn thinks it'll look quite tricky. She says that these fractions look difficult to convert through division. Not sure I entirely agree about the middle one, I think I'd give that a go with a bus stop method. But she's perfectly right that we can do this without doing any division. Zaki thinks that we could use our understanding of fractions to help us, and he's right. Pause the video and just have a think about how you could do it. You don't need to get the answers, but how could you convert them, what methods might he use? Pause the video and have a go. When you're done, unpause it, and we'll do it together. Pausing in three, two, one. Okay, let's have a look. So, we already know how to find one third. And I know one-third is equivalent to ten over thirty. And that's pretty close, I've not really had to do much, but we know that thirty's not going to be recurring so I'm already on my way to getting there. I'm going to need one-thirtieth. And one-thirtieth, if I find a third, and divide that by ten, that gives me one thirtieth, and combined, if I add these together, I'm going to get eleven-thirtieths. Now the second one, like I said, I'd be tempted to use a bus stop method, but we said no division, let's use what we know about fractions. We know that we can find one-ninth, so we know that if we multiply that by two, we get two-ninths. And then if we want twenty-ninths, we're going to need- I should probably put an equals there. So if we take out two-ninths, and we multiply that by ten, we'll get twenty-ninths. If you're feeling really clever, you could've taken one-ninth, and multiplied it by twenty. Sometimes I like to break it down and do it in steps it can be a little bit easier. Twenty three over ninety, right, let me have a think. So, let's start with what we know, we know one-ninth, we know that if we multiple it by two we'll get two-ninths. We know that two-ninths is equivalents to twenty over ninety. So we're getting closer. We know that if we've got one-ninth, and if we divide that by ten, we get one-ninetieth. We need three of those, so if we take out one ninetieth, and we multiple it by three, we get three ninetieths, and if we do twenty ninetieths, I don't know, three ninetieths, and we get twenty three ninetieths. Okay? Did you get those? Well done.

Representing recurring decimals

Representing recurring decimals

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UnitMaths / FDP review (9.1a)

Maths