Can We Easily Compare Two Fractions With the Same Denominator?

Key Stage 2

Maths

Taking fractions further

﻿Hey and my name is Mrs. Ford. And I'm going to be joining you for your math lesson today. So, first let's have a look at the practise activity you were left with last time. If you've got it with you, then that's great, go through it with me now. But if you don't have it or didn't have a chance to complete it, that's fine, we're going to go through each question now. Okay so question one, you were asked to use the diagrams to fill in the fractions and then put in the inequality symbol. So you're being asked, which fraction is greater and which fraction is smaller. What do we need to look at first? Yes, that's right. You need to look at the total number of equal parts that the whole has been divided into. So let's have a look at this first fraction and we know what we've got, let's have a quick look. One, two, three, four, five, six, seven equal parts. And is the second bar the same? Yes it is. So what does that tell us? That's right. That gives us our denominator because the denominator shows us the total number of equal parts in the whole. Okay, now what do we need to look at? We're going to look at the shaded parts, you're right because that gives us our numerator for each fraction. So for the first one, one, two, three, four equal parts and the second, one, two, three, four, five, six equal parts. Okay, so we've got four sevenths and six sevenths. So now you have to decide, which is greater. Well you already know, that four sevenths is for lots of one seventh and six sevenths is six lots of one seventh. And you know that four is less than six, so four sevenths is less than six sevenths. Brilliant. Okay, let's have a look at the number line. So you've got two eighths and seven eighths and the statement says that two eighths is less than seven eighths. So can we show that on the number line? Let's have a look at the number line here, we've got zero and then we've got one. So what does each one of these parts of the number line represent? Well each step is one eighth, let's count in one eighths, zero, one eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths or one. So where do each of these fractions sit on the number line? Well we've just counted it. One eighth, two eighths, three eighths, four eighths, five eights, six eighths, seven eighths. So which fraction is furthest along the number line and closer to one? That's right, seven eighths. So two eighths is less than seven eighths. And in question three, you were asked to use your verbal reasoning to fill the missing parts of the sentence stems. So we've got four ninths and seven ninths. And your job is to put in the right inequality symbols here. Is four ninths less than seven ninths or is it greater than seven ninths? Well, what do you already know? You know, that four ninths is four lots of one ninth. Are you saying this with me? Let's say that again together. Four ninths is four lots of one ninth. And the next part, seven ninths is seven lots of one ninth. I know that, four is less than seven. So four ninths is less than seven ninths. So you're right, now we need to put in four ninths is less than seven ninths. And again, in question four using your verbal reasoning, the sentence stem is partially completed for you so we know that two fifths is two lots of one fifth, four fifths four lots of one fifth. So what does that show us? Well, we know that two is less than four. So two fifths is less than four fifths. Well done, excellent. Let's move on to today's lesson. The children in my class have been thinking about this question and they've been discussing which method they would use to prove their answer. So the question was, which is greater, 18 twenty-fourths or 23 twenty-fourths James said he's going to to have a go with using an area model. So he's going to start by drawing a circle. Now, what does he need to do? Yes, you're right. He needs to consider the denominator, the total number of equal parts that he needs to divide the whole into. Okay, so he's going to to have to divide his circle into 24 equal parts. Okay. So that's for looking at 18 twenty-fourths but he also has to compare that with 23 twenty-fourths, so what must you do? He's going to have to draw another circle and divide that circle into 24 equal parts as well. Now is he finished? Can you see the comparison between these two fractions? No, not yet, you're right. He has to show us the number of equal parts that we're looking at in each fraction. So, we're looking at 18 out of the 24 equal parts here, so he's going to have to shade in 18 parts and he's looking at 23 equal parts here, so, he's going to have to shade in 23 of the 24 equal parts. So now we know, in the first circle we've got 18 lots of one twenty-fourth shaded and in the second circle we've got 23 lots one twenty-fourth shades. Can you say which fraction is greater? Yes you can. It does work. He can see he compared quite easily and we can see that 23 twenty-fourths is greater than 18 twenty-fourths. Okay, brilliant. Well done James. Farah likes James' method, but she thinks it would take too long. She says she's not very good at drawing circles and she finds dividing that into 24 equal parts really, really tricky. So, she's going to use a number line. So let's have a look at Farah's method. She's got the number line and we've got zero and one. How equal parts will she need between zero and one to show twenty-fourths? Yes you're right, she's going to need to divide this number line into 24 equal parts between zero and one. Okay. Farah is going to need to be really careful that she divides those parts equally. So let's have a look, it might look something like this, 24 equal parts between zero and one. Now where does she need to plot 18 twenty-fourths on the number line? Let's have a think. Is it going to be closer to zero or one? Is it going to be less than halfway or more than half way? What do we know? We know that 12 is half of 24, so it's going to be more than half way, isn't it? So let's have a quick look. Count in one twenty-fourths, one twenty-fourth, two twenty-fourths, three twenty-fourths, four twenty-fourths, five twenty-fourths, six twenty-fourths, seven twenty-fourths, eight twenty-fourths, nine twenty-fourths, 10 twenty-fourths, 11 twenty-fourths, 12 twenty-fourths, half way, 13 twenty-fourths, 14 twenty-fourths, 15 twenty-fourths, 16 twenty-fourths, 17 twenty-fourths, 18 twenty-fourths, do we think it's going to be here? Let's have a look. Yes, 18 twenty-fourths is here. Now, Farah needs to plot 23 twenty-fourths, so we're going to start counting again? Oh no, Farah's got another idea. What do you think her idea might be? What does she know about one? Yes in this case one is 24 twenty-fourths So she knows that 23 twenty-fourths is just one step back from one So let's see if she's right. Yes, 23 twenty-fourths is here. Farah didn't have to do all that counting coz she used what she knows. Well done Farah that was really good. Now, has she shown that 23 twenty-fourths is greater than 18 twenty-fourths? Yes, she has. Okay. Let's move on. Ava disagrees with James and Farah, she knows their methods work but she thinks drawing a model and divided it into 24 equal parts can take a long time and it's a bit easy to lose count or to not divide your parts equally, so she thinks she's going to use a quantity model to show which fraction is greater. She's going to use her favourite sweets. So how many sweets does deserve Ava need? Yes, that's right. She needs 24 sweets Can you explain why? You're right, the whole has 24 equal parts. So each of the sweets represents one twenty-fourth of the whole. So let's have a look, she's got six sweets, 12 sweets, 18 sweets, 24 sweets. Okay, has she shown us 18 twenty-fourths? No. Each one of these sweet is one twenty-fourth of the whole, so we need 18 of them to show 18 twenty-fourths Each one is one twenty-fourth. So now what does she need? She needs another 24 sweet, your right so that she can show 23 twenty-fourths and compare the two fractions. How many sweets will she need to circle this time? 23 sweets. Okay. So this is 18 twenty-fourths and this is 23 twenty-fourths. Each pair is one twenty-fourth of the whole. Which fraction is greater? Has she shown you? Well which fraction would you rather have, would you rather have 23 twenty-fourths of the whole or I think 18 twenty-fourths? Yes. I'm sure you would, you'd rather have 23 twenty-fourths of the whole because that means you'd have more sweets Okay. Well done Ava. Majid said, that he doesn't need to draw any models because he remembers something that we were doing in the last lesson, that will help him reason about his answer. Can you remember what it was? What also do you think Majid has noticed? He's noticed something about these fractions. Yeah, that's right. He's noticed that they've got the same denominator. So that tells him something about the size of the fractions. He knows, you can do this bit with me that 18 twenty-fourths is 18 lots of one twenty-fourth and 23 twenty-fourths is 23 lots of one twenty-fourth. I know that 23 is greater than 18. So, 23 twenty-fourths is greater than 18 twenty-fourths. Do you agree? All of the children in the class have been looking at the same problem and they've all solved in slightly different ways. James used his area model, Farah use the number line, Ava used her sweeties and the quantity model and Majid use his verbal reasoning. Which one do you prefer? Do they work? Yes they do, they all show us the 18 twenty-fourths is less than 23 twenty-fourths so that 23 twenty-fourths is greater than 18 twenty-fourths coz that was our question. Okay. But which one do you think is the most efficient? Majid said, "24 is quite a large denominator, it's quite hard to draw a remodel with 24 equal parts." And he said, "What about if the denominator was an even bigger number? What if the denominator was 75?" Okay, so Majid set this question for his friend, James. He said, "Well which is smaller, 32 seventy-fifths or 23 seventy-fifths?" And Majid had been thinking about this. And he said, "When we compare fractions with the same denominator, the greater the numerator the greater the fraction." James is a little bit stuck because if he's going to use this area model, can he draw a circle and divide it into 75 equal parts? That will be super tricky, wouldn't it? So Majid just reminded him about what we've been doing with the verbal reasoning and the sentence stem. Can you talk it through with me? We've got two fractions with the same denominator, so what do we know? We know that 32 seventy-fifths is 32 lots of one seventy-fifth and 23 seventy-fifths is 23 lots of one seventy-fifth. What else do we know? We know that 23 is smaller than 32. So 23 seventy-fifths is less than 32 seventy-fifths. Do you agree? So James is beginning to understand this really well now. So he's designed some missing symbol problems for his friend Majid to solve. And I'd quite like you to have a go at these too. So make sure you've got pen and paper and you can jot some down or you could even just talk this through in your head. Okay. Now what do we need to remember? Well, we need to remember that when we compare fractions with the same denominator, the greater the numerator, the greater the fraction, and we can use our sentence stem to talk us through. So let's have a look, we've got five twelfths and 10 twelfths. What do we notice? Yes. We noticed that they've got the same denominator and we understand that we can then look at the numerator to compare the size of the fractions. Okay. Pause the video and have a go. Use the sentence stem that we've been using to help you think out loud. Okay. Are you back and ready to have a look? Okay. Look at this pair of fractions first, we've got five twelfths and 10 twelfths. So what do we know? I know that five twelfths is five lots of one twelfth I know that 10 twelfths is 10 lots of one twelfth and I know that five is less than 10. So five twelfths is less than 10 twelfths. Brilliant. And let's have a look at the next one, we've got nine sixteenths and seven sixteenths. What do I notice? They have the same denominator. So I can compare by looking at the numerator. Nine is greater than seven, so nine sixteenths is greater than seven sixteenths. Lets look at this last one, 10 twelfths and 10 twelfths. Okay. Do they have the same denominator? Yes they do. So I understand that I can look at the numerator. So I've got 10 lots of one twelfth and 10 lots of one twelfth Well which is great? Well I know that 10 is equal to 10, so 10 twelfths is equal to 10 twelfths. Excellent. Well done. Ava has been watching carefully and she's designed another problem for her classmates to solve. She asked them this question, "Which symbol should always go in the circle?" Okay. This looks a little bit different from our other problems that we were looking at. What's different? Okay. Can you answer her question, which symbol should always go in the circle? Can you explain why? I wonder if you could give me a couple of examples to prove your answer. Pause the video and have a go. Okay, you're back and ready so answer the question, I wonder what you thought, what did you notice? What did we notice about the fractions? We've got Four is the numerator here and six is the numerator here. What about denominators? Triangles? Well, what does the triangle represent? Yes, you're right. It could be any number but what's the most important thing to notice. Did you say that the most important thing to notice is that the shapes are the same? Yes. That's right, that means that the denominator in each fraction is the same. What if the triangle represented eight? So now, our fractions are four eighths and six eighths. What do we already know, so that we can complete this statement? Yes. When we compare fractions with the same denominator, the greater the numerator, the greater the fraction. So four eighths is less than six eights because four is less than six. Okay. Did you come up with some examples of your own? That's fantastic. Let's try a couple more here. What if the denominator was 10, if the triangle represented 10, is four tenths less than six tenths? Yes, it is. When we compare fractions with the same denominator, we're going to look at the numerator and four is less than six. So four tenths is less than six tenths. Let's do one more just to check. Well if the numerator was six. Okay. We've got four sixths and six sixths. Oh, what do we know about six sixths? Six sixths is one. Okay. So let's look at the numerator, is four less than six? Yes, it is. So four sixths is less than six sixths. Four sixths is less than one. That's fantastic, well done. Now let's have a look at this problem. Okay. What are we being asked to do here? Yes.we're being asked to fill in the missing number here. So, what do you notice? Yeah. You notice that both denominators are the same. So what could you put in this box to make this statement true? Is there only one answer? Could you think of more than one answer? And if you can think of more than one answer, can you explain why? Okay. Pause the video and have a go. Are you back? Okay. What did you think? Yes. You noticed that the denominators are the same, this numerator is five. This symbol is, greater than. So what does that tell us about the number that needs to go in this box? Did you say that it has to be bigger than five? Your right. Okay. Any number greater than five could go in this box? Couldn't it? You could have six, or nine, or 15, it doesn't matter. As long as that number is greater than five. The denominators are the same, so this numerator needs to be greater than five. Well done. Here's part one of your practise activity that we'd like you to have a go of before the next lesson. Have a look and see if you can fill in the missing boxes to make the comparison statements true. They must match the representation however, so you need to look very carefully and think about the factions that are being represented. And in this practise activity you're going to look at using what you know to complete these missing symbol problems. So in each case you have to put in, less than, greater than or equal to and think about all that lovely verbal reasoning that you've been doing and everything that you know to be able to complete these statements. You can use the sentence stems to help you. And in these ones, you needs to complete the boxes to make the statements true. So five ninths is greater than, how many ninths? Might be more than one answer in some of those Okay. Good luck and we'll see you in the next lesson.

Can We Easily Compare Two Fractions With the Same Denominator?

Can We Easily Compare Two Fractions With the Same Denominator?

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UnitMaths / Taking fractions further

Maths