Non-unit fractions

Key Stage 2

Maths

Equivalent fractions

﻿Hi, it's Mrs. Lambert here. And quite a little while since we last met, and you have been doing lots of learning about equivalent fractions. And you asked some really tricky math at the end of your last lesson. So I'm just going to go over that together before we move on to our new learning. So your friend was trying to convince you that these were equivalent, because when they looked at the numerator of one, and they said, "Well, you know, if you add three to it equals four." And if I've added three to the numerator, if I add three to this denominator of two equals five, and they were absolutely convinced that they must be equivalent. But you said, "No, no, no, we don't go adding amounts to new races nominators, we need to look at that multiplicative relationship." And what if we scaled one up by we've scaled it up by four, we've multiplied one by four, and one multiplied by four equals four, two multiplied by four equals, come on, join him equals eight. And so four eighths would be equivalent to a half, but not four fifths. We need to look at what we multiply, and we need to look at that multiplicative relationship and not add. So just remember that when you, your friend was wrong. We learn by our mistakes, it's sometimes a good thing when people make mistakes, because it helps us to remember not to make that mistake again and to really understand the mathematics. This one caught me out, this one really caught me out because a half of 40 equals 20. And four fifths of 25 equals 20. And I'm okay with that. But they then said, a half equals four fifths. So I've got to think of a way now to think, are they equivalent, would they be at the same place on a number line? And I know that four fifths is quite close to one and a half is right in the middle between zero and one, so I can kind of picture them in them in a place on a number line. But this might be more convincing and I'm sure that you've used something that's even better than what I've come up with. So they had found fractions of quantities. And we have to be really careful about what we're saying, when we're looking at equivalent fractions, a half of 40 to the top rectangle, the whole bar, the whole rectangle represents 40, a half or 40 equals 20. And if we look at the bottom rectangle, four fifths of 25 equals 20. So we can see that, the shaded part is the same for both shades, but the wholes are different. So we need to be really mindful about what we're actually saying here, what is equivalent, and you can't write that a half equals four fifths, because we look at a half equals four fifths. And the wholes are the same. There's a half, and there's four fifths, and the shaded parts are clearly not the same part of the whole. They are not equivalent. So don't get caught out there, don't find fractions of quantities, and say they're equivalent. When you're looking at equivalent fractions, we're just looking at the fractions. And can we put an equal sign in between two fractures to say that they have the same value on a number line? Tricky stuff, but we're getting our heads around it. Okay, are we ready to move on to today's learning, I hope so, we're going to use the language of non-unit fractions. So we'll just remind ourselves about what a unit fraction is. So I've got some images, and I'd like you to sort in your head 'cause you're not going to be able to do any moving around without actually just looking at the screen and putting them into two different groups, one, where you think I can describe that using a unit fraction, and then those that you'll describe using a non-unit fractions. So have a go pause the video, and press pause now, and then we'll have a look and see how I've sorted them. Okay, let's see if you agree, 'cause I think there's probably one way you might have done it a different way. So let's first of all, and have a look at this egg box. So the egg box, yeah, it doesn't look very much like an egg box, I know, but it's an egg box that could hold 12 eggs. So if I asked you the question, how full is it, what proportion of the egg box is full? We will be able to say that actually, we've got what egg in out of a possible 12, so one twelfth. One twelfth of the egg box has an egg in it. The number line is kind of quite easy, isn't it? Because we can see that it's got one as a numerator and 10, one tenth, the arrow is pointing at one tenth. And how about the shape at the bottom? At the whole has been divided into six equal parts, and one of them has been shaded. So we've got one twelfth, one tenth, one sixth, if it's a unit fraction, the numerator is, yeah, those of you that joined in thank you, one, you'll remember me from all of those lessons ago, where I expect you to just talk at me, even though I can't hear you, I am listening. Okay, let's have a look at this diagram here. Okay, now, there's quite a few options on how you might write your fraction that's been shaded. So I'm going to pick just one of these little shapes here. And I'm going to say that that's one of my parts. So how many of those have I got? Okay, so the whole has been divided into 20 equal parts, and how many have been shaded? Eight, so I could write it as eight twentieth. And I could also, yeah, let's be a bit creative here, I could actually say that that is a part. So how many of those equal parts are there? There are five. It's quite tricky, if you can't see that, I'm just going to circle my part so that you can see where I'm coming from here. I've got five equal parts. And how many of them have been shaded? Two. Hmm, these are actually equivalent fractions, but it's not a unit fraction, because the unit fraction has to have a numerator of one. This one, okay, this one, I think you might have put it over with the unit fractions. It just depends on how you described how the whole has been divided. So if you said that actually, there are 12 spaces, and two of them have been filled with eggs. That's a non-unit fraction. But if you'd said actually, I'm going to describe that as my parts. And there are, how many of those parts are there? Let's have a go at doing this. Okay, I'm going to say that there are six of those. And I've got one sixth, that's a unit fraction. So, it's a bit of a tricky one, you might have put it over the other side, but remember, a non-unit fraction has a numerator that is not one, it is greater than one, okay. Look at this next one, okay. So we will look at non-unit fractions. And we've got a shape here that has been shaded in. And we're going to say, how much of the whole what part of the whole has been shaded? So the whole has been divided into five equal parts, and two of those parts have been shaded. So we can describe this as two fifths. Okay, we're good at this, now, this isn't, this isn't hard, this is easy, okay. And we can put it on a number line because fractions can be describing parts of shapes, but they can also be on a number line. So we need to remind ourselves about that as well, okay. So this looks slightly differently, slightly differently? Slightly different to our two fifths. But can you still see two fifths? Yeah, I can still see two fifths, there's nothing, I haven't shaded anything in, anything more. But I've just drawn this line now. So how many parts are there now? So, there are 10. And how many of those 10 pots have been shaded? Four, okay, so an equivalent fraction to two fifths is four tenths, we can write that equal sign, they're equivalent. And they look a bit different. But on a number line, for tenths is going to be in exactly the same place as two fifths, okay. Still see two fifths? Okay, looks a bit different. What's the denominator going to be? How you're getting good at this fantastic fifteenth 'cause this time the whole has been divided into 15 equal parts and six of them have been shaded. Two fifths have been shaded. But I can also describe it as six fifteenth. Looks a bit different on a number line, we can have them in exactly the same place. Still see two fifths? Looks a bit different. Okay, tell me what fraction you can see that's equivalent to two fifths. I heard loads of you say eight twentieths, you are superstar mathematicians. And there it is on a number line. And why are they there on the number line? Because they have the same value, okay. And you've already got this, you can see we can also say two fifths as ten twenty fifths. We can have actually an endless number of fractions that are equivalent to two fifths. It goes on and on and on. But Mrs. Lambert will not go on and on anymore. Actually a lie. There's one here with 30th. You just never know how many are going... Okay. Equivalent fractions, they can look a little bit different, if you ate a fifth of the pizza, you would cut your pizza into five equal parts, and you'd eat one of those parts. If you cut your pizza into 10 equal parts, the parts would be smaller. So you'd have to eat more of them. So that you ate the same as one fifth, you'd have to eat two of those parts. So one fifth of the pizza, and two tenths of the pizza, they are equivalent. And that's why we've got the scales, they're often our equals symbol. And if we use scales, it can help us realise that they are equivalent. Okay, so here we've got equivalent fractions, two fifths, four tenths, six fifteenths, eight twentieths, ten twenty fifths and twelve thirtieths. And they all have the same value, they would all be placed at the same point on a number line. And you might look and think, what's the relationship between the two and the five? What do I multiply two by so that it equals five? Yeah, it's quite tricky, isn't it? There's not an obvious value here, but I multiply two by two and a half. And it's the same relationship for all of these fractions. If I multiply four by two and a half, equals 10, if I multiply six by two and a half, equals 15, eight multiplied by two and a half equals 20, ten multiply by two and a half, 25, and 12 multiplied by two and a half equals 30. The same relationship between the numerator and the denominator. Okay, so rather than looking at that vertical relationship, let's have a look at the horizontal relationship. What do I scale up two by so that equals four? Yeah, I multiply it by two. So if I multiply the numerator by two, I also need to do the same to the denominator, I need to multiply that by two. So we're looking for that multiplicative relationship between numerators and between denominators. And sometimes it helps to just draw it in that way so that we can see that the relationship is being kept the same so that the fractions are equivalent. So two fifths, can we convince ourselves it does equal six fifteenths. I have scaled two by three, I've scaled it up, I've multiplied it by three, so that it equals six, and therefore when I look at this denominator, if I multiply that by the same scale factor, they will be equivalent, okay. We're going to get good at this because actually, if you're good at your tables, fractions actually are easy, because if you can see that relationship, and you know what you've multiplied a number by, and get going with your practising 'cause it will really really help. Yeah, they'd be multiplied by three, both the numerator and the denominator so that two fifths does equal six fifteenths. And once we multiplied both the numerator and the denominator by here, yeah, we've multiplied it by four, two fifths equals eight twentieths. By five, the scale factor is five, we scaled it up by five. And here we've scaled it up by, yeah, by six. If we've got here 30 parts they're smaller, so we need more of them. It kind of makes sense sometimes fractions are a bit tricky. They represent the same part of the whole, twelve thirtieths represents the same part of the whole as two fifths. Just that we're representing it in a different way, we're using different numerators and different denominators. So if I haven't already convinced you, I'm just going to show you equivalent fractions with some shapes as well, just so that we can make that connection between when we're looking at shapes and when we're looking just at the fractions that have been written. So here, on the left hand side, I've got the whole has been divided into five equal parts, two of them have been shaded. And here the same sized whole has been divided into 10 equal parts, and four of them have been shaded, so they look different, but the same proportion of the shape has been shaded, you try and say that quickly, it's quite tricky. And why is that the case? Is because we have multiplied, we've scaled up both the numerator and the denominator by the same amount, we've multiplied the both by two. And that's really important that we make sure that relationship is the same one, we've multiplied the top and the bottom, sometimes we say the top and the bottom, the top of the bottom numbers, the numerator and the denominator, both by two. And when we look at this one with the shape, the shapes look a little bit different because the parts are smaller. Remember, the bigger the denominator, the smaller, this little part here is and the fifteenths, but we've got six of them. And if we've got six fifteenths, it's equivalent to two fifths, because I've multiplied two by three, and I've multiplied five by three. And looking at the shapes, I can see that the same part of the shape has been shaded, they are equivalent. Two fifths equals eight twentieths. The shapes are the same size, the shaded part is the same proportion of the whole. But in this case, they just look a little bit different because the twentieths are smaller parts, but we've got eight of them. We're multiplying, we're multiplying two by four, so that equals eight, then we're multiplying five by four, so that equals 20. So that the fractions are equivalent. Okay, two fifths is equivalent to ten twenty fifths. So let's have a look at this two. What does that two represent? Well, it's got to be in relationship to this five. So it must be for this shape over here, where I've got 1, 2, 3, 4, 5 equal parts, and two of them have been shaded. In this fraction here, what does the 10 represent? Yeah, it represents the shaded part still, it's just that these parts are smaller. And the 25 represents how many, the whole has been divided into, it's been divided into 25. So even though it looks a bit different, they're equivalent, particularly in this case, when the shapes are the same size, the shapes need to be the same size, if we're going to say that the shaded part is the same, but the same part of the whole has been shaded two fifths and ten twenty fifths are equal. Okay, let's just get rid of this green writing so that you can see that because we've multiplied the numerator and the denominator by five. And then here, what are we multiplied the numerator and the denominator by? Yep, we've multiplied them by six. And when we look at the shapes, it's the same part that's been shaded, it's just that we're describing it in a different way. Okay, I think that you have now got so much information that you're starting to make connections, that you're ready for a challenge. So your task, you're going to find six equivalent fractions for these two fractions. One is a unit fraction, a third, because the numerator is one, and a non-unit fraction two thirds, because the numerator is not one. It's quite a busy slide. So let me just highlight what's going on here, the equal sign, they're in this sequence here, there are several of them. So don't get in a pickle about what all the lines represent. Those are your equal signs and it's the same for the one underneath. And I'm just going to put one of them in for you and equivalent fraction. If I've got a numerator of one, if I mostly supply that by two, I can put in a numerator of two. If I've multiplied the numerator by two, I need to multiply the denominator by two, I scale up by the same amount. So two sixths. Now rather than looking at two sixths and generating my next equivalent fraction, go back to your one third, and see if you can be systematic. And once you've done that, do the same for two thirds. And at the beginning of the next lesson, we'll see what patterns we can spot.

Non-unit fractions

Non-unit fractions

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UnitMaths / Equivalent fractions

Maths