Numbers and numerals

Axioms and arrays

Factors and multiples

Order of operations

Positive and negative numbers

Expressions, equations and inequalities

Angles

Classifying 2-D shapes

Constructing triangles and quadrilaterals

Coordinates

Area of 2-D shapes

Transforming 2-D figures

Prime factor decomposition

Conceptualising and comparing fractions

Manipulating and calculating with fractions

Ratio

Percentages

Different number systems

Sequences

Forming and solving equations

Forming and solving inequalities

Linear graphs

Accuracy and estimation

Algebra and problem solving

Ratio (8.6a)

Ratio, real life graphs, and rates of change

Direct and indirect proportion

Univariate data

Bivariate data

Famous maths problems

Angles and parallel lines

Angles in polygons

Bearings

Volume and surface area of prisms

Area review

Circles

FDP review (9.1a)

Probability

Sets and Venn Diagrams

Solving linear simultaneous equations algebraically

Solving linear simultaneous equations graphically

Angle review (9.5a)

Constructions, congruence, and loci

Pythagoras's theorem

Ratio review (9.7a)

Similarity and enlargement

Surds and trigonometry

Quadratic expressions (9.9)

Quadratic equations (9.10)

Indices and standard form

Growth and decay

Finance

Maths

Y9 Maths

Key Stage 3

Year 9

Statistics and probability

THEME

UNIT

In this lesson, we will recap converting fractions, decimals, and percentages, and use the conversions to make comparisons.

Representing fractions, decimals, and percentages

﻿Hello, I'm Mr. Langton, and today we're going to do a recap on fractions, decimals, and percentages. All you're going to need is something to write with, something to write on, and try and find a quiet space where there's no distractions. When you're ready, we'll begin. Okay. So we'll start with the try this activity. When it splits into two parts, I'm going to use the number card to try and complete the number line. And for that first part, you can use the cards more than once if you want to. Just try and find a way that you can complete the number line. That second part, if you're feeling really, really confident, can you find a way to complete the number line using each card once? So pause the video and have a go. When you're ready, unpause it, and we'll have a go together. You're going to pause it now. Go on. Three, two, one. Right then. I'm going to presume that you've had to go, and now I'm going to start off and show you what I would do as well. So looking at the number line, it starts at zero and goes all the way along to two. So bang in the middle, I'm going to label that. That's one in the middle. So that's my starting point. I'm then going to look at its number keys, okay? So halfway between zero and one, that'll be here. So that's going to be a half or 0.5. And halfway between one and two, that would be 1 1/2, wouldn't it? Or oh, oh 1.5, right? Okay, so that's a good one because that one there is halfway between it. So that must be 1.5 halfway between one and two is 1.5. I'm going to cross five off. So I know I've already used it. Now let's have a look. Let's look at this fraction down here. Let's point out to it there. So it's smaller than one. And it's been split into thirds, which means it's either going to be 1/3, or it's going to be 2/3. It can't be 3/3 because that'd be a whole one. That's too big. Now it's bigger than 1/2. So it can't be 1/3. So that one there must be 2/3. So I'm now going to cross off two because I've used it there. Let's have a look. So this bit here, that would be a 3/4 value or a 75% value. I've been given that both of the ones above it are percentages, I'm going to stick with our 75% of the percentage. So the next 10 up from 75 would be 80. And there's another one there as well, 90. So that works. That's going to be 80, and that's going to be 90. So I can cross those off. I'm starting to make some progress. I'm quite happy with this. And what I've got here is a percentage that ends in a five. It's less than 1/2, which means that this one here is either going to be 15%, 25%, 35%, or 45%. Now I know I've already used two. And actually 25% would be there, wouldn't it? So it can't be 15% either. So this one here is either a three or four. I'm not quite sure yet. So I'm just going to wait. Let's just make a little note there. This one is either three or four. Now, once we get bigger than one, once we go on this side, we're looking at greater than one. We said that's 1.5. Now this, this is a percentage is greater than one. So it must be a hundred and something. So that bit there, must be a one. We know bigger than 1.5. So that could be 160%. It could be 170%, and we've used the eight and nine already. So I'm just going to wait there. If I'm using my one, I want to cross that off. Now let's have a look. So that's either six or seven. What about this fraction here? Is in quarters. Now actually, yes, the whole number lines are split into quarters, isn't it? That's 1/4. That's 2/4, which is 1/2. 3/4. 4/4 is a whole one. 5/4. 6/4 is 1 1/2. So that there is 7/4. I mean that's 7/4. That means that that one there must be 160%. So I've used up my six and not seven. Now, that means that I've only gotten that percentage at the start, which you said is either 35% or 45%. And if I'm honest, it looks closer to the half, to the half. I think it's probably 45%, but we'll just have to see what's going to work over here at the other end. At the reference side, we've got two whole ones, two whole ones as a fraction. Now, normally as a fraction, that'd be two halves of one is a whole one. Now our fraction is out of 15. So it's converting to fifteenths and multiplying by 15. So I'm going to multiply that by 15. So I guess, a fraction of 30/15. So that's a three, and that's a zero. And that's good because I've not used those yet, three and zero, which means that that one there is 45%. So for this one, we've got to write each value as a fraction, a decimal, and a percentage. Let's start with 40%. Now, the important thing about percentage is to remember what the word actually means per cent. It means per 100 or for every 100. So 40% as a fraction, it must be 40/100. Now we can simplify that fraction. Both of those numbers go into 10. Divide the number by 10. I'm going to get the 4/10. And both of those also go into two. So it's 2/5. Now as a decimal. Now, I've got this number line down at the bottom, to help me out just a little bit with my hundreds, my tens, my ones and after the decimal point with my tenths and my hundredths. Now we said that 40% is 40 hundreds or 4/10. Now 4/10 is there, which means that it's going to be 0.4. Okay. Now what about 3/8? I'm going to show you a different method for 3/8. Do you remember that every single fraction can be represented as a division? It actually means three divided by eight. That's quite a tricky thing to do. So I'm going to draw a little bus stop. How many times does eight go into three? Well eight doesn't go into three. So I'm just going to put a zero there. I need to carry my three. I've got a remainder of three. Now, three is the same as 3.0 So now that I put that decimal point in that decimal place, I can carry that three there. That'll make 30. Make sure that you line up your decimal point at the top as well. I can put as many zeros as I want here. And that's still equivalent to three. I don't know how many I'll need yet, but it's just worth bearing that in mind. So let's carry on. Eight into 30. Anything to 30, you're going to go three times, and it's going to be six leftover. Eight into 60. I'm going to go seven times with four left over. Eight into 40 are going to go exactly five times. So as a decimal, 3/8 is 0.375. And if I look at my number line, I'll tell you, just let me, just move it out of the way, if I look at my number line, 0.375. I've got three tenths. I've got seven hundredths. I don't know. It's not there. I've also got five thousandths. So altogether, I've got 375 thousandths. We know that's going to cancel down to 3/8, but we now need to turn it into a percentage, don't we? We said percentages are out of 100. So we said that are hundredths. There are 37 of them with that 0.5. Don't forget the 5 thousandths as well. And for that, it means that 3/8 as a percentage is 37.5%. Finally, we've got 0.64. That's going to be an easy one to add to the percentage because as we've seen, let's just clear some space. 0.64 is going to be six tenths and four hundredths. And percentages are out of 100. So we've got 64%. And a fraction as we said is out of 100, and then we need to simplify it. We want to find the highest common factor of 64 and 100, the largest number that goes into both. Well, they're both even. So I can divide them both by two. They're still both even so I should have divided by four in the first place, shouldn't I? That would've got me 16/25. Is there anything that goes into both of them? No, no, that's good. So lots of different methods there. I hope you were making some notes. If not, by all means, just rewind the video a little bit to look through those methods again. Lots of different ways to go from percentages to fractions and decimals, right? So when you're ready, let's move on to the next part. Okay. So now it's your turn to have a go. What I want you to do is pause the video and have a go with the questions on the worksheet. When you're ready, unpause it, and we'll go through it. Good luck. So how did you get on? Some of the answers are on the screen now. And I thought it might be good if we went through some of them together. I'm going to start off down at the bottom. Which one's largest, 87% or 7/8? Now we're going to turn them both into the same fractions or decimals or percentages. So to do that, I'm going to start with 7/8. It's the trickiest one. It's going to be harder to convert, and I'm going to do that bus stop method that I showed you before. How many times does eight go into seven? Now, stick a few zeros on the end. And we don't know how many we'll need. And I keep my decimal point there now, just so I don't get messed up. Eight into seven don't go. You need to carry that seven to make 70. Eight into 70. You're going to go eight times. Eight and eight is 64. That means I've got six leftover. Eight into 60. They're going to go seven times with four left over. Eight into 40 go five times. So 87% is being compared to 0.875, which as a percentage is 87.5%. So it's very, very close, but the largest one is 7/8 by .5%. Okay. Let's pick a different colour. That's work in red this time. Write down a decimal, which is between 1/3 and 1/2. There are loads of answers you could give here, literally infinite number of answers. But any number that is larger, 1/3 is 0.333333333, 0.3 recurring and 1/2 is 0.5. So if you've got a decimal, so we need something that's going to go between those, don't we? You need a decimal that either begins with 0.4, and you could have some more numbers after it, and that would work. If it begins with 0.3, then your next number, ideally, needs to be a four or a three and a four or a three and a three and a three and a three and then a four or something a little bit higher than that. So it needs to be a little bit bigger than 0.3 recurring. For example, 0.34, 0.35. Anything up to 0.49999999999999999999 And I'll stop there 'cause it's just going to be silly now. Okay. So let's have a different colour so we can do the last problem. Let's just draw a little wavy line there so I don't get confused. This is the question that I'm going to finish with. So complete the number line below. We've got a value of 0.7 here. I'm about 90% there. We've been given those. Now 90% is the same as 0.9. So that's going to be useful. That's going to help me work out what it is that I'm trying to get to. So 0.7. All those gaps that are there must be 0.8 'cause that's evenly spaced between them. And 0.8 is 80%. And I need a fraction that's halfway between 0.7 and 0.8. As a decimal, that would be 0.75, which is 3/4. And I'm going to need, if we know what's going on by 0.5 each time, that's going to be 0.95 or 95%, which means as a fraction that's 95/100. Is that okay? Well, it's almost okay. What you should be saying is the number line there, Mr. Langton. You've not simplified that fraction, and you'd be right. So the highest common factor of 95 and 100 is five. So I'm going to divide each of those numbers by five. So you get 19/20, and that's what this fraction here is. Okay, one last activity. How many ways can you complete the fraction cards to form numbers that lie between those shown on the number line? Pause the video and have a go. See what you can work out. And when you're done, unpause it, and we'll go through it together. So you can start in three, two, one. Okay. So how did you get on? Let's see what we can do. I'm going to start off with this fraction's card on the right hand side, working with something over 1,000. And the reason I'm going to do that is that I can write both of these decimals as fractions out of 1,000. 0.332 is the same as 332/1000 0.334 is the same as 334/1000. And we need to find a fraction over 1,000 that will sit somewhere between 332/1000 and 334/1000. So it must be 333/1000. I want to look at now is the card on the left hand side, the fraction out of 10,000. They've already said that 0.332 can be written as 332/1000. Now I actually want to find a fraction that's out of 10,000. So to do that, I need to multiply my denominator by 10, which means I need to multiply my numerator by 10. So 0.332 is also equivalent to 3,320/10,000. I can do the same thing over here. And you might be able to see where I'm going with that. I'm going to get 3,340/10,000. So I need a fraction after 10,000, that lies between 3,320 and 3,340. That's what my numerator needs to be. So anything as low as 3,321, all the way through to 3,339. Now to get an equivalent fraction for the five thousandths is quite trickier. We said it earlier, but 0.332 is equivalent to 332/1000. Now, I want to try and make that out of 5,000. So to do that, I'm going to need to multiply my denominator by five, which means having to multiply my numerator by five. And 332 times five is 1,660. Don't worry. I don't expect you to do the bit of working out for that. Over on the other side, 0.334 is the same as 334/1000. And if we want it out to 5,000, once again, we have to multiply the numerator and the denominator by five. And in this case, it's 1,670. So any number between 1,660 and 1,670 can be our numerator. So the lowest it could be is 1,661 all the way through to 1,669. And finally, I'll put the rest of the answers up there for you. So you can check your working out. Now you've got the method. You should be able to do all of those yourself. I hope you got on well with this. I'll see next lecture.

In this lesson, we will convert fractions into recurring decimals and percentages, and look at how we can use the links between them to solve problems quickly.

Representing recurring decimals

﻿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.

In this lesson, we will review how to add and subtract fractions, including different denominators and mixed number fractions.

Reviewing addition and subtraction

﻿Hello, I'm Mr. Langton and in this lesson we're going to review adding and subtracting fractions. You're going to need something to write with and something to write on. Then I want you to find a quiet space with no distractions and when you're ready, we'll start. We'll begin with the try this activity. Use the number cards to try and complete the addition frame in as many ways as you can. If you're feeling really confident, then see if you can do this without using the same card more than once. I'm going to pause the video and let you have a go. When you're ready, unpause it and we'll go through it. You can start in three, two, one. So how did you get on? I've come up with a few answers that we can go through now. Firstly, if both the denominators are 12, there's quite a few options that you can go with. I went with five twelfths and two twelfths, that makes seven twelfths. Now instead of five and two, I could've had six and one, or I could've had one and six, I could've had two and five, or I could've had three and four and four and three, so quite a few options there. Now in both cases, I've got the same denominator so it's a good answer, but it's not the best one that I could find. Now I did come up with one for different denominators if I work with thirds and quarters, I found that one third add one quarter will be seven twelfths and the reason for that, if I turn my thirds into twelfths. One third is equivalent to four twelfths and one quarter is equivalent to three twelfths so when I add those together I get seven twelfths. I was quite proud of myself but once again, this time my numerators are the same so I've not used four different cards. I then started thinking, if I compared what I'd got here, and I thought hang on a minute, two twelfths is the same as one sixth, isn't it? And so I can have five twelfths add one sixth. That gives me seven twelfths and I've used four different cards, really pleased with myself there. I've just got to thinking, could I do something similar with this last one? Is there anything here that I could do? So I started thinking and said well, one third is the same as two sixths and I hadn't used two or six in the other one, so I can keep that as a quarter. Two sixths add a quarter, that'll work. There are other ones as well, maybe you got them, but there's some answers you can go with. I want to review adding and subtracting fractions now just to make sure that you're okay with it. I know you've done it before, but it's always good to do a review. Looking at the first question, seven eighths add three quarters. If we're going to add or subtract fractions, we always need to make sure we start off with a common denominator. So I'm looking at my eight and my four and the lowest common denominator would be eight so I'm going to rewrite both fractions so that they're eighths. Now seven eighths is already in eighths, that's useful, I don't need to change that one. Three quarters, if I want to change quarters into eighths, I need to multiply the denominator and the numerator by two which gives me six eighths. Now if I add those together now, seven eighths and six eighths is 13 eighths. And there we go, I've added them together. Now we don't always like having top heavy fractions so sometimes it's nice to change them into mixed numbers. The way that I like to do that is treat it as a division, fractions are divisions. We're going to do 13 divided by eight. How many times does eight go into 13? It goes in once with a remainder of five so 13 over eight is one whole one with five left over and because we're dividing by eight, we're working in eighths, it'll be one and five eighths. Let's look at the second one. Seven eighths take away three fifths. We going to need a common denominator and it's got to be a number that's in the eight times table and the five times table. And the lowest common multiple of eight and five, the lowest number in both those times tables is going to be 40. Now to turn eighths into fortieths, I need to multiply by five so I need to do the same to my numerator which is going to give me 35 over 40. To turn my fifths into fortieths I need to multiply by eight. Multiply that one by eight, three eighths, and 24. And now I can subtract them. 35 take away 24 is 11 over 40. I can't simplify it anymore, there's no number that goes into 11 and 40 so I'm done. Now let's look at the last one. Starting with a mixed number can be tricky. Sometimes you can ignore it and just work out the fractions separately and then put them into number in the end, but it doesn't always work out, you need to be quite careful when you do that. So I like to start off by making my mixed number into an improper fraction. So two and one third, I've got two whole ones and a third. Now if I'm working in thirds, let's draw a diagram over here, if I work in thirds, that's one whole one isn't it? So if I've got two whole ones. That's six thirds. And of course I've got two and a third so altogether two and a third is seven thirds and I'm adding on two fifths. So now I need a common denominator. The lowest common multiple of three and five, so the smallest number that's in both those times tables is going to be 15. So let's change each of these fractions into fifteenths. So 15 goes into three five times, I need to multiply my numerator and my denominator by five which give me 35 over 15. And to get from five to 15, going in three times, I need to multiply by three, it gives me six fifteenths, which altogether, if I add them together, is 41 fifteenths. And once again, let's change it back into a mixed number. So I'm going to do my division. 41 divided by 15. How many does 41, sorry, how many times does 15 go into 41? 15, 30, it's going to go twice which is 30 with 11 left over, so two remainder 11. So that's going to be two whole ones and 11 fifteenths leftover, don't forget we're working in fifteenths. Okay, it's your turn to have a go now. Pause the video and have a go at the questions. When you're done, unpause it and we'll go through the answers together. Good luck. So how did you get on? Here are the answers, have a look and compare them with yours. Some of my fractions are simplified so you might have got slightly different answers so do just check that they're equivalent and of course I've made them into mixed numbers where you might have left them top heavy. The magic square was a little bit sneaky because I didn't think to add up the three fractions first, I was presumed that they were going to add up to make a whole one and so I started subtracting from one and that didn't work out for me so I had to have another look. Finally, we've got the explore activity. Use the number cards to complete the addition and subtraction frames in as many ways as you can. If you're feeling really confident, see if you can do it without using the same number more than once. I'm going to be honest with you, it's really hard and if you're going to be able to manage that, you might want to put on your thinking cap first. Pause it and have a go. Three, two, one. Okay, let's discuss a couple of ideas. I'm going to keep my thinking hat on, it's quite tough. If we use the same denominators, I'm sure you can find lots of numbers that'll add up to make 13 so we can make it out of 12 for each one and let's just see, just for example, what if we make that one three and let's make that one four, that makes seven so we need another six don't we, so that works. Three twelfths add four twelfths add six twelfths. That's going to make 13 twelfths so we've got an answer that we can use, that's great. Over here to make one twelfth, so let's make them twelfths again. And some numbers we're going to subtract to make one twelfth so we could have, well four take away three can't we? So that works. And with my thinking hat on, I was wondering, is there a way to do it without? So I was having a look and I thought, well actually, four twelfths is a third. I'll write that in a different colour, that's one third. Now we've used the three already there but can we change three twelfths? Three twelfths would be a quarter, rub that out let's go for blue this time, one quarter, so I'm almost there but I've used one twice so I need to think of something different there. One third is the same as two sixths so that's no good. I've got it, I've got it, are you ready? Let's, so it's going to be, let's change that quarter to two eighths. One quarter is the same as two eighths. Sorry, I need my pen. Two eighths and there we have it, six completely different digits. There's the three fractions that add up to make 13 over 12. Finally, can we do the same thing over here on the right? We've used 12 in both of them, can one of them simplify? We could make, I don't think that's going to work is it but I do know that that's the same as a third and that's the same as a quarter so I'm not there yet but I know that a quarter is the same as two eighths and that is a possible solution, yep. So once again, we got there. How did you get on? Did you manage to get any of these?

In this lesson, we will review how to multiply and divide fractions using geometrical examples.

Reviewing multiplication and division

﻿Hello, I'm Mr. Langton. In this lesson, we're going to review multiplying and dividing fractions. You're going to need something to write with and something to write on, and try and find a quiet spot where you won't be interrupted, and when you're ready, we'll begin. We'll start with the Try This activity. The diagrams show different lengths marked on the unit square, and the area of the bottom left rectangle is 0.075 square units. Work out the areas of the remaining shaded triangles. And if you're confident, you can pause it and have a go straight away. If you want a hit, just leave it going a little bit longer and I'll give you a little bit of a starting point. So if you ready, pause in three, two, one. Okay, let's go for a hint. So if we're going to look for the area of this, we've got this area here, 0.075 units, let's try and find this length. We know that 1/4 times something is going to be equal to 0.075. I'm going to write that decimal as a fraction, because that would be 75/1000. And at the moment, it's not very easy to see what the numerator and the denominator of this fraction must be. But I know that a quarter is 25%, or 25/100. And we're still trying to make 75/1000 but I can now see what fraction I'm going to need. Can you see it? How I'm going to get from 25 to 75? I could multiply by three and from 100 to 1,000, I'll multiply by 10. So this length here must be 3/10, or if you multiply to the decimal, 0.3. Okay, that's the first hint, now, pause it and carry on and let's see how many of those green areas you can work out. Pause in three, two, one. So what I'm going to do now is see if we can find the other two missing lengths. This one here, and this one here. Let's starts off with the one on the right hand side. So we know the whole length from top to bottom is one, which means 4/10 plus 1/4 plus that third one is going to make a whole one. Now the fact that there are different denominators I could work with, but actually I know that 4/10 is the same as 40%. And I know that 1/4 is 25%. What do I need to add to that to get 100%? So 40 and 25 is 60 that means that that'd be there it must be 35%. Now 35% is equivalent to 0.35 or 35/100. Now actually we've got that as a decimal, and let's do the same there, 0.35. Now I can work out the area of this square. I'm going to be doing 0.35, multiply by 0.35. And 0.35 is 35/100, and that's going to cancel down they both went to five, so it's going to be 7/20. So I need to do 7/20 multiplied by 7/20. That's going to give me 49/400. Now this length in the bottom right, if these two here add up to 0.65, then once again, this is going to be 0.35. So this time, let's do 0.35 multiplied by 1/4. We've said not 0.35 is 7/20, so 7/20 multiplied by 1/4 is going to be 7/80. And now I just need the green rectangle in the top left, which we know that one side is 4/10 and the other side is 0.3, which is 3/10, which means that I've got 12/100. And if I want to show off a little bit, I can cancel it out to 3/25. How many of those did you get first time? Let's go through these four examples together. For the first two, we need to calculate the area of the rectangles. So the area of this first one will be 3/4 multiplied by 7/8. Which is going to be 21/32. And it won't simplify, now don't forget your units, let's put it on there, 21/32 metres squared. Okay, now to the one on the right. I'm going to multiply length by the width again. So we're doing 1.2 multiplied by 0.6. There're few different ways that we can do this. But given that they're decimals, and they're only one decimal place, it doesn't hurt to turn them into fractions into tenths. We've got 12/10 multiplied by 6/10, 12-sixes are 72/100. And we can simplify that fraction, but if we're going to give our answer through the decimal, we're better off leaving it like that because we've got 72/100, 0.72 metres squared. Now for the next two, we need to calculate one of the missing lengths for each rectangle. And that means that this time we need to do a division. So for the first one, we're doing 2/3 divide by 8/9. So what I like to do, I like get a common denominator. The common denominator of three and nine would be nine. So an equivalent fraction will be 6/9, which I'll divide by 8/9. Which is going to give me 6/8 or 3/4. So this length here is 3/4 of a metre. And then finally, we've got 0.063, we're going to have to divide by 0.9, 0.063 divided by 0.9. Now there's lots of ways that we can do this. We'll start off, let's have a look at the numerator of these fractions. So 0.063 is 63/1000. And 0.9 is 9/10. So a common denominator would be 1,000 63/1000 divide by 900/1000, that's going to give us 63/900. Let's cancel that down, they both come into nine, which is going to give me 7/100. And since we're working in decimals, that is 0.07. So this means the length here is 0.07 centimetres. So now we're going to move on to the independent task. This is where you get to show what you can really do. Pause the video, and have a go at the questions. When you're finished, unpause it and we'll go through it together, good luck. So how did you get on? I've simplified some of the fractions here so perhaps you might have slightly different answers to me. Just double check them to see if you've made any mistakes. This is the last activity. The areas of the shaded rectangles in the unit square have been marked. What could the marked lengths be? Can you find more than one set of lengths? It's a really difficult problem. I'm not going to give you any answers, I'm going to leave it with you. And the reason for that is if you can work this out, I want you to share this with us so that we can see just how good you really are. Ask your parents or caregivers if you'll share your work on Twitter, tagging @OakNational and #LearnwithOak. Good luck.

Unit Overview: FDP review (9.1a)

Lessons:

Representing fractions, decimals, and percentages

Representing recurring decimals

Reviewing addition and subtraction

Reviewing multiplication and division