What is this fraction in its simplest form?

Ronaldo simplifies the fraction 12/18 so that the denominator is 9. What is the numerator?

Understanding Equivalence

Key Stage 2

Maths

Fractions

Where would four tenths go on the number line?

If the value of the yellow rod is 1, what is the value of the red rod?

Which symbol should be entered to make the number sentence correct: <, > or =?

Read the statement below. Do you agree or disagree? Explain your answer.

﻿Welcome to our second lesson on Fractions. Today, we'll be learning to understand and describe equivalence. First of all, make sure that you've got everything you'll need for today's lesson. All you need is a pencil and a piece of paper or an exercise book. Pause the video now and get your equipment ready. In today's lesson, we're going to learn to understand and describe equivalence. You'll start the lesson with a quiz to test your knowledge from your previous lesson then we'll work on identifying equivalent fractions pictorially then with numerical representations followed by reasoning about equipment, simplifying, finishing off with an independent test to practise what we've learned today, and then a quiz to test all of your knowledge from today's lesson. So, let's start with the Knowledge Quiz. Pause the video and take the quiz before we start the lesson. Now for our Do Now. Look at the three measurements. What can you notice? What's the same and what is different about them? Pause the video and make some notes. The key concept here is same value different appearance. Equivalent means equal in value. Equivalence means when numbers of that value is the same. In the example given, note 0.1 metres is equal to 10 centimetres, which is also equal to a 100 millimetres. What has changed is their units. Equivalence can also be seen in fractions where they have different numerators and denominators, but fractions are actually equal in value, same value, different appearance. We'll be exploring this further today. Now, looking at these fractions, I need to work out what fraction of each rectangle has been shaded. In my first one, it has been divided into one, two, three, equal parts. That over four is my denominator. One of those four parts has been shaded so one out to four parts is shaded that is equal to one-quarter. In the second rectangle, one, two, three, four, five, six, seven, eight equal parts so that's the total number of parts. Two of those have been shaded. Two-eighths of the rectangle has been shaded. In my final one, let me be more efficient and counting groups of four. Four, eight, 12, 16 equal parts of which four have been shaded. Four-sixteenths have been shaded. Now what is important here is for us to ask ourselves what patterns do we notice in the numbers in the fractions? Why did the numbers change when the shaded part has stayed the same? Now we need to look at the relationship between the two numerators and the two denominators. So let's take the first and second fractions. Let me look at the two numerators first. I can see that in the second fraction, the numerator is exactly two times greater than the numerator in the first fraction. One times two is equal to two. In the denominators, I can see that the denominator of the second fraction is exactly two times greater than the denominator in the first fraction. Four times two is equal to eight. Therefore, these fractions are equivalent. One-quarter is equal to two-eighths because the relationship between the two numerators is the same as the relationship between the two denominators. Let's have a look this time at the first fraction and the third fraction. One-quarter and four-sixteenths. So, numerators. The numerator in the second fraction is four times greater than the numerator in the first fraction. One times four is equal to four. The denominator in the second fraction is also four times greater than the denominator in the first fraction. Four times four is equal to 16. Therefore, one-quarter is equal to four-sixteenths. Let's check one more together. Two-eighths and four-sixteenths. So this is the second and the third fraction. I can see that the numerator in the second fraction is two times greater than the numerator in the first fraction. Two times two is four. In order for these fractions to be equivalent, the relationship between the denominators must also be times by two. Eight times by two is equal to 16. Therefore, these two fractions are equivalent. Now it's your turn. Pause the video and work out what fraction of each rectangle has been shaded. Are the fraction is equivalent? How do you know? This is how you should have described those relationships. In the first rectangle, two thirds has been shaded. In the second rectangle, six-ninths and then the third rectangle twelve-eighteenths. Now, they appear pictorially to be equivalent but let's check using the numerical representations. The first fraction two-thirds and the second fraction, six-ninths. I can see, looking at the numerators that the numerator in the second fraction is three times greater than the numerator in the first fraction. Two multiply by three is equal to six. Is this the same for the denominators? The denominator in the second fraction is three times greater than the denomination in the first fraction. Three times three is equal to nine. Therefore, these two are equivalent. Let's check one more together. We'll look at the first fraction and the third fraction, two-thirds and twelve-eighteenths. Remember that the relationship between the two numerators must be the same as the relationship between the two denominators. I can see that the numerator in the second fraction is six times greater than the numerator in the first fraction. Two times six is equal to 12. Is this the same for our denominators? The denominator in the second fraction is six times greater than the denominator in the first fraction. Three times six is equal to 18. Therefore, because these relationships are the same, these fractions are equivalent. Let's move on to reasoning about equivalence. I've given two fractions and I need to prove whether they are equivalent or not. Three-quarters and eleven-sixteenths. I meant to use two approaches. My first approach will be to draw a diagram to compare the fractions pictorially. So I will need two rectangles of the same length. The first one, I'll divide into four equal parts. The second one, I'll divide into 16 equal parts. First I'll divide it into four and then each of those four I will divide into further four parts. Now you can see from my messy diagram that this is not a particularly accurate way of representing fractions, but it gives us a good estimate so that we know where we're working from. So three-quarters will be represented in my first rectangle. Three out of the four boxes are shaded. My second rectangle, I will shade 11 out to the 16 parts three, four, five, six, seven, eight, nine, 10, 11. Okay, so at first glance, it looks like they're not equivalent. It looks like eleven-sixteenths is less than three-quarters, but look at the relationships between the two numerators and the two denominators. So looking at the numerators and using my knowledge of times tables, I can see that three is not a factor of 11. I can't divide the 11 by any whole number to get three. Equally, 11 is not multiple of three. I can't multiply three by any whole number to get to 11. Let's check the denominator, sorry. So, I can straightly see that four is a factor 16, 16 divided by four is equal to four. I can also that foresee that 16, four multiplied four is equal to 16. Now, I can see that the relationship between the denominators and the numerators are not equal. Therefore, these two fractions are not equivalent. Let's see if I did have the same relationship. So, if the denominator is four times greater than the numerator in the first one, then the denominator should be four times greater, I mean, sorry, the numerator should be four times greater than the numerator in the first one so that will twelve-sixteenths. These two fractions are equivalent. I'm going to explain that one more time because I stumbled over my words. In the first fraction, the denominator is multiplied by four to get 16 in the second fraction. In the first fraction, the numerator is also multiplied by four, which gives us 12 in the second fraction. Because these have been treated exactly the same in the second fraction, the denominators and numerator are four times greater than in the first fraction. These fractions are equivalent. Let's move back to the diagram and this time I should have 12 parts shaded. Now I can see, even though this is a very rough diagram, that those two fractions are equivalent. Now it's you turn. Pause the video and prove whether these fractions are equivalent or not. Remember to use two approaches to prove your answer. First, show pictorially, followed by looking at the relationship between the two numerators and denominators. This is how the relationship should have been reasoned. We can see from the pictorial representation that the fractions don't seem to be equivalent, but let's have a look at the relationship between two numerators and the two denominators. Looking at the numerators and using my knowledge of times tables, I can see that three is not factor eight. Eight cannot be divided by any whole number to get the answer three. Equally, eight is not a multiple of three. Three cannot be multiplied by any whole number to get eight. Looking at the denominators, I can see a relationship straight away. The denominator in the second fraction 12 is two times greater than the denominator in the first fraction six. But because the relationship between the two denominators and the two numerators this is not to say, these fractions are not. Time to deepen our understanding. Look at these number line representations. Now we can see that the fractions seem to be equivalent because they come up to the same point on each number line but when we start to look at the relationship between them, we can see that they're not factors or multiples of each other. So if I take the first two, two-eighths and three-twelfths, I can immediately see a relationship between the two numerators and the two denominators. Now, that just because the initial fraction, two-eighths is not actually in its simplest form. The numerator, two and the denominator, eight are both multiples of two. Therefore they can both be divided by two in order to get the fraction one quarter. Because I've done the same thing to both numerators and denominators, these fractions are equivalent. I can show this pictorially by drawing a fourth number line this time, dividing it into four equal parts, dividing it into two quarters, and I can place the one quarter here and I can see that that is equivalent. Let's have a quick look at the relationship with the other fractions. So I'll just look at one-quarter and three-twelfths. Are they equivalent? Let's look at the two numerators. I can see that the numerator in the second fraction is three times greater than the numerator in the first fraction. One times three is three. The numerator in the the second fraction, the denominator, sorry, in the second fraction is also three times greater than the denominator in the first fraction four times three is equal to 12. Because these relationships are the same for both numerators and denominators, these fractions are equivalent. Now it's your turn to simplify each fraction so that they are in their simplest form. The first one has been done for you. Pause the video and simplify the fractions. Here we go, solutions. The first one is done for you. In the second one, I know that two is a factor of six and eight so when I divided both by two, my answer was three-quarters. Six-eighths is equivalent to three-quarters. For the third one, common factor was two again so I divided both numerator and denominator by two which gave me five-sixths, therefore ten-twelfths is equivalent to five-sixths. In the fourth one, the common factor was three. So I divided both and denominator by three, which gave me two-thirds. Six-ninths is equivalent to two-thirds. Now it's time Independent Task. Pause the video and complete the Independent Task. Come back here when you have finished so that we can go through the answers. The question one of your Independent Task, you were asked to simplify the fraction and then find an equivalent to complete the missing fractions. So I can simplify six-ninths. I know that three is a factor of six and nine so if I divide both by three, then I get two-thirds. Going the other way, I can't go through all of the possibilities here because you might have multiplied it by 10 to get six-ninths, you might multiply that by 20. I'm going to give the example of multiplying by three and do it outside of the box. You would have got eighteen twenty-sevenths. For the second one, I know that 16 is a factor of both 16 and 48. So I could have divided both numerator and denominator by 16 to get the answer one-third. Now, you may not know your 16 times tables so you could have done this in stages. Two is a factor of both 16 and 48 so if I divide both the denominator and numerator by two, that gives me eight twenty-fourths. Now I can use my times tables and see that eight is a factor of both of these numbers. So, eight twenty-fourths is equivalent to one-third. Again, going the other way, there's too many possibilities for me to cover, but I multiplied mine by two and I got the answer thirty two ninety-sixths. For question two, you're asked to reason your answer. eighteen twenty-sevenths is equivalent two-thirds. Prove it. I'm not going to draw a pictorial representation, because that would mean dividing a bar model into 27 equal parts, which I know I can't do accurately. So I need to look at the relationship here between the two numerators, two denominators. I can see that nine is a factor both 18 and 27 so if I divide 18 by nine, that gives me two. If I divide 27 by nine, that gives me three. Therefore, these fractions are equivalent because the relationship between the two numerators and the relationship between the two denominators are the same questions. Question three, Elizabeth knows that one equivalent fraction of thirty three fifty-fifths has a numerator three. What is the fraction? What is the relationship between 33 and three? Well, I know that 33 is 11 times greater than three so I divide 33 by 11 to get three. I have to treat the denominator exactly the same way. 55 divided by 11 is five. Therefore three-fifths is equivalent to thirty three fifty-fifths. Question four, Youcef wants to share his cake equally. He gives one-sixth to Sarah and four eighteenths to Metila. Did he share it equally? Did Sarah and Metila get the same amount each? We're being asked here, is one-sixth equivalent four eighteenths? So, I look at the relationship between the two numerators and I can see that the numerator in the second fraction is four times greater than the numerator in the first fraction. Is that the same in second fraction? Sorry, is that the same for the denominators? The denominator in the second fraction is actually only three times greater than the denominator in the first fraction. Therefore, he did not share the cake equally. They are not equivalent fractions because these two relationships are not the same. Finally, how many different ways can you correctly complete these two fractions so that they are equivalent? Now, the way I approach this is to look at the factors of 24. So I start off with two, I know 24 can be divided by two to give me 12. So to go the other way, I need to use the inverse of division. So I know that three multiplied by two gives me six. Six twenty-fourths is equal to three-twelfths. Let's look at another option. I know that three is a factor 24, 24 divided by three is eight so to go the other way I use the inverse of division three multiplied by three is equal to nine. A third option here. I know that four is a factor 24. 24 divided by four is equal to six. Using the inverse, three multiplied by four is equal to 12, twelve twenty-fourths is equivalent to three-sixths. And then my final option. I know that six is is a factor of 24. 24 divided by six is equal to four. Use the inverse, three multiplied by six is equal to 18. Now it's your time for your Final Knowledge Quiz. Pause the video and complete the quiz to see what you've remembered. Fantastic work today! I'm looking forward to meeting you back here when we'll continue to work on finding equivalent fractions. See you then!

Understanding Equivalence

Understanding Equivalence

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