Vertical and horizontal relationships

Key Stage 2

Maths

Equivalent fractions

﻿Hello everybody, welcome back to Lesson 5, where we are going to carry on finding equivalent fractions, and we're going to explore the relationships between the equivalent fractions that we find, using the accurate mathematical language we used yesterday. So I sent you these tasks, let's have a look at those individually. So this first one here, you started with 4/8, and did you find that was equivalent 2/4? Which is equivalent to 1/2. And I asked you to use the STEM sentences to find the scale factor so, you should have said that the numerator is scaled up by a factor of two. And that a 1/2, is the same proportion of the shape as 2/4 and 4/8. Well done. So 4/16 is equivalent to 2/8, which is equivalent to 1/4 or a fourth. And 1/4 is the same proportion of the whole shape as 4/16 and 2/8. Has the scale factor changed this time? The scale factor is now four, isn't it? So the numerator, in each of those fractions has been scaled by the same factor four, well done. And then move them up to an equivalent fractions here. So if we start with that top one So 1/5, the scale factor is five, so if we know the scale factor is five, we're looking for another fraction to match it with the same scale factor. And we can see count whether that will be for 4/20. So 1/5 and 4/20 are equivalent, because four times five is 20. Let's look at a quarter. So the scale factor is four and which of a fraction has a scale factor four, it is indeed 3/12, well done. So we should have matched a quarter with 3/12. And finally, if we look at a third, we can say that that would have matched with 3/9. So 3/9 takes up the same proportion of the shape as a third, and the numerator has been scaled up by factor of three has been multiplied by three. So that's been a really good recap on our language there, I'm thinking about those STEM sentences we met and we're going to need those again this session. Let's have a quick look at a challenge. Did you ask some of the adults in your house how much pizza they'd like and try and cut them out? I was thinking that I go to one of my babe and said someday would you like a half of a pizza? Or would you like 50 or hundreds? because he sounds like he's getting nosy if you're having 50 slices. Well, actually we both know, don't we? That, that's smallest slices. And so I hope you managed to catch some of your adults though. And I'm looking at the four children in the playground. 2/8 of them are wearing the hat. How many of them are wearing the hats? Now I found it was easier, if I found an equivalent fraction to help to think about this. 'Cause I knew I have four children and I had four children, actually 2/8 didn't help me to solve it very well. What if I found the equivalent fraction of a quarter? Then I could really quick to see that those four children, out of the four children, one of those children was wearing a hat. So I hope you managed to see that too. But we visit some of the language that we used yesterday. Looking at a vertical relationship between the numerator and denominator. Yesterday we saw, that we could stay low the numerator by the same scale factor, and that would show us that fractions were equivalent. Or trying to just take a look at the numerators in both of those fractions. And can you see a relationship between them? And have a look at the denominators as well, can you see a relationship between those? Now I know yesterday we spoke, didn't we? about additive relationships and that may be, we can look at those new numerators and say, well, if I add three to one I get four and then I add if I add 15 to five, I get twenty. But we said that we weren't looking at additive relationships when we were thinking about comparing the equivalent fraction. So we need to look at multiplicative relationships. Can you find a relationship between the way the numerator and the denominator have been scaled up? Your moment to look at those. I would say, that they've both been scheduled up a factor four. So one multiply by four is four, and five multiplied by four is 20. So we can see that actually, the relationship we were looking at yesterday vertically, we've got the same thing happening horizontally. And it's that scaling up and scaling down of the numerators and denominators. Have a look at this one. You might want to pause me and write this one down and then have a say, is the scale factor the same vertically for both of these equivalent fractions, and then is it the same horizontally? Can you see a relationship horizontally. Now have paused me and have a look at that. Okay, if you come back. What have you found? Did you find that the scale factor was three? So the numerator has been scaled up by a factor three. Yep, so that's what we looked at yesterday, isn't it? And we can see that, that helps us understand that there are equivalent fractions. What about that horizontal relationship? Did you see a relationship between the numerator and denominator? Yes, did you see that, that both multiplied by six? So that's scaled up by a factor of six. So we can use this language again, can't we? The new numerator has been scaled up or down by a specific factor. The denominator has been scaled up or down by a specific factor. And then we can say the fractions are or are not equivalent. So having a look at those two examples, you might want to pause me again and really, really try and identify those multiplicative relationships again both horizontally and vertically this time. And you might want to draw those arrows off and just really check that they're the same. And how many of those fractions are or are not equivalent. Just pause me there. Okay, if you come back. So did you use the STEM sentences to help you? So, for the first example, the numerators have been scaled up by a factor of four. The denominators have been scaled up by a factor four, horizontally. But then vertically, the numerator has been scaled up by a factor of five. One times five is five and four times five is 20. Did you find that in the second example as well, fantastic. Now for the same example can we then now showing the vertical and the horizontal multiplicative relationships. And I've got this generalisation that says, "A fraction preserves its value only when both the numerator and denominator are scaled by the same factor." So, really need to think about what that language means. What does it mean to preserve something? So it means to keep it the same, doesn't it? So, a fraction keeps the equivalent value, only when both the numerator and denominator are scaled by the same factor. Yesterday, we had a look at that vertical relationship anyway. And so we could see that the numerator was scaled up by a factor of five. But now we're looking at that horizontal relationship. And we can see that, that's been scaled the same for the numerators and the denominators by four. Can you draw anything to help you understand that? Could you maybe draw a bar to help you visualise that? Pause me now. And see if you can draw the bar that helps you show those equivalent fractions. Did you get something like this? So on the bottom bar, the hole has been divided into five equal parts, and I could see one of those parts. And the top bar, the hole has been divided into 20 equal parts. And four of those parts is the same proportion of the hole. And we can see that those multiplicative relationships are the same. So horizontally, they've both be scaled by factor four and vertically, they've both been scaled by factor five. So it proves that in order to find an equivalent fraction in a more abstract manner, and as long as we preserve the scale factor, we will preserve the value of that equivalent fraction. Now, I think it might be helpful here for us to think about if two fractions aren't equal. What does that do to the scale factor? So I want you to have a look at this one and you're going to have to pause me again for this one I think. I want you to think about the scale factors. So you are going to have to write this one out and then draw the arrows from the numerator to the denominator to see what the vertical relationship is. and then draw the arrows horizontally, between the numerators and between the denominators and see if all of those scale factors are the same. So you should have the same scale factors going vertically, and the same scale factors going horizontally. So I'll let you have a little go with that. And then we'll maybe have a look at that pictorial as well. So pause me now and have a go with that. Okay, how did you get on? It's interesting, isn't it? When we looked at this one. Because when we look at the scale factor vertically, one multiplied by five is five. Now is five multiply by five. Is that 20? So we can see, can't we? That the vertical scale factor isn't correct. They're not the same. Were the horizontal scale factors accurate? So one multiplied by five is five, five multiplied by five. So no, they weren't. They weren't the same either, were they? And we can see the image shows that, doesn't it? Because the bar has been split into five equal parts. But actually, if we do the same sized hole into 20 equal parts, then five of those parts is greater than 1/5. So it proves to us that, if we want to preserve the value of the equivalent fraction the scale factors have to be the same. Have a look at this. Again you might want to pause me now. I'd like you to find the vertical scale factor and the horizontal scale factor to the numerator and the denominator. Use the STEM sentences and tell me if these are equivalent and why. All right, have you come back? Did you find they were equivalent? Yes. So they have the same scale factor of seven, particular. So the numerator can be scaled up by seven, And the denominator can be scaled down by seven. And then horizontally, what did you find? They were both scaled by a factor three. So the numerators have been multiplied by three. And the denominators have been multiplied by three. So we had equivalent fractions there and that generalisation we used before about the equivalence being preserved, has been proved. Because the scale factors are the same. Here's another one. Pause me again, have a go with that. How did you get them? Have you used the STEM sentences? So the numerator has been scaled up by eight, well done. The numerators have been scaled up by eight. What about the numerator and the denominator horizontally? were they scaled up all down? They were scaled down, weren't they? So they were scaled down by a factor of five, fantastic. So we had to divide by five, didn't we? When we spoke about that inverse relationship yesterday. So this is your independent tasks now. So we're drawing on everything that we looked at yesterday. And we're thinking about those multiplicative relationships and how we can prove if fractions are equivalent or not, not only by pictorially representing them, but by looking mathematically at those multiplicative relationships. So I've got four examples here for you. I want you to use the STEM sentences and explain what is happening in each time. So you can do this on your own. We'll go through them at the next session, but really have a think about, draw the arrows on, think about those relationships to see whether they are equivalent or not. Your second task, we've got some problems for you to solve. I'm going to let you read through those on your own. So you've got a problem there, looking at equivalent fractions, and then if you're ready for a challenge, have a look at that one and see if you agree or disagree with both statements. And that's the end of today's session. So good luck with your independent work. And I look forward to seeing you again. Bye.

Vertical and horizontal relationships

Vertical and horizontal relationships

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UnitMaths / Equivalent fractions

Maths