Video: Factors and multiples

﻿Hi, Mrs. Amber here. At the end of the last lesson, Mrs. Holmes asked you to have a go at making your own number cards. And well, I had to go and I used one, three, five, and 15. You kind of had to have knowledge of how the cards were connected, and you might have had a go at creating some, and they didn't make any sense at all. So this lesson is going to build on how you might go back and recreate that activity with a little bit more knowledge as a result of some language that you might have. This was one where you kind of had to look at the relationship. So I kind of thought, okay, if you've got 1/3 and then I multiply the numerator by eight, that will equal eight. And yeah, if I multiply the denominator three by eight I'm equal 24. It was a bit kind of more like pot luck than anything else. So let's hang onto that and think about how some further understanding of language will actually help us to be better at creating that number card game. So this generalisation, we used a number of lessons ago, and we're going to say it together. And you'll know me by now that when I'm sat here, and I'm reading things out, I do want you to join in. So let's say this together. When the numerator and denominator are multiplied or divided by the same number, the value of the fraction remains the same. Okay, so let's have a think about this fraction here. This is one quarter, brilliant. And I'm going to look at this denominator, and I need to think about what have I multiplied or divided? Well, I know that I'm going to multiply this by two. So if I multiply the denominator by two, I need to multiply the numerator by two. Yep, so I can create a fraction that has the same value. It's an equivalent fraction. So let's have a look at this fraction, 6/12. The numerator here, well, the numerator has been divided by, it's been divided by six. So I need to divide the denominator of 12 by six. I need to divide it by the same number. So six divided by six equals one, 12 divided by six equals two. So I kind of have already been doing this a little bit, but I'm going to build up my understanding to think about what can I multiply or divide fractions by so that the fractions remained the same. Here, I haven't been given a numerator or denominator. So I could pick anything here. I could multiply this numerator by three. So six multiplied by three would equal 18. If I multiply the numerator, I need to multiply the denominator by the same number. So if I've said six multiplied by three equals 18, 12 multiplied by three would equal 36. And I could divide it by a number and if I divide the numerator by a number, I need to divide the denominator by the same number. We're going to come back to that, because this is how are we going to build up our understanding of generating equivalent fractions? So I apologise if you are already familiar with the language of factor, but this lesson is very much a revision. 'cause we need it when we are simplifying fractions. So when we were introduced to multiplication, we looked at the language of factor and product. Factor times factor equals product. We knew also that we could start with product, and say that the product equals factor times factor. So that's not new. That's something that we've been using since we were in year two, but this is going to come in really handy for fractions. Bear with me, 'cause you'll see how useful it will be. Okay, so when we were looking at factors, we also thought about how we could make a rectangular arrays and a factor can be the length or the width of an array. So we've got here 12, I'm going to call them tiles just because these little squares look a little bit like tiles. And I'm going to think about how many different ways I can make arrays. So there were 12 tiles, and here there's one row and 12 columns. So 12 and one are factors of 12. And I could also look at it in that orientation. And I could say here that there are 12 rows and one column. So 12 and one are factors of 12. It's the same rectangle, it's just been twisted round. Same 12 tiles, but I've now made a different array. This time, there are six rows and two columns. So six and two are factors of 12. Okay. I've realised, I say, okay, quite a lot. I need to stop doing that, that's a about habit. And here I've got two rows and six columns. So six and two are factors of 12, right? You'll go four rows, three columns, so four and three are factors of 12. Now what you don't want to have to keep doing is making rectangular arrays. But if you remember, we use this to help us define factors. When we were looking at arrays, probably back in years, I don't know, two, three, and four. There's a good way, but it's not the only way. And when you're looking at fractions, we're going to look at how we can use other things to help us to find factors of numbers. Let's have a look at using a multiplication square. It will really help us practise using that language of factor. So I'm going to use seven. I'm going to start with seven, and I'm going to say seven is a factor of 42, because 42 is in the seven times table. Okay, let's try it with another set of values. Six is a factor of 48, because 48 is in the six times table. Okay, I'm going to use my laser, and see if you can use that sentence and put the numbers in. Yeah, some of you joined in, some of you didn't. Nine is a factor of 81, because 81 is in the nine times table. Have a go at just creating your own now without me using the laser. And I want to hear you. Thank you to everybody that joined in. And just one last one. Let's go for 11 is a factor of 44, because 44 is yeah, in the 11 times table, really helpful. We don't have to keep drawing arrays. And let's just think about that one. So we said seven is a factor of 42, it's in the seven times table. 42 divided by seven equals six. So I could make a rectangular array. That was a six by seven, six and seven are factors of 42. And you might want to just go back and think about how that might look as a rectangle to make those connections. Let's have a look at this in a different way. If you divide a number and the quotient is a whole number, then the divisor and the quotient of factors of the starting number. Now that's quite hard to remember. So let's just use an example. So, okay. Let's just think of a positive whole number. We'll not include zero because when we write fractions, we just use positive whole numbers. So if you divide a number and the quotient is a whole number, okay, let's think of 12. Okay. So is two a factor of 12, can I divide 12 by two? Yeah, I can. I can divide 12 by two, and the quotient and the answer is six. So two and six are factors of 12. So we could write it like this. Remember we had factor, factor product. So two and six are factors of 12. So another way to talk about what a factor is, factors are positive integers, positive integers or positive whole numbers that you could multiply them together. And they give you they're equal to a given number. So two and six, because you could multiply them together and they equal 12. They are factors of 12. Okay, let's do it with another example. See this is easy now. That sentence is quite difficult to begin with, but hopefully it's starting to make sense. So the next one, let's think is three a factor of 24? Okay, so I'm going to think. Okay, 24 divided by three equals eight. So the quotient, the answer to that calculation is eight. So yes, three is a factor of 24, three 24 divided by three equals eight, three multiplied by eight equals 24 factor factor product. Three and eight of factors of 24. Is five a factor of 24? Well, it's not because 24 divided by five will not give me a whole number answer. So if you're really good at using your division knowledge and your multiplication knowledge, this will be really, really useful when we are simplifying fractions. So we're going to look now at the term common factor. A common factor is a factor that's shared by two or more numbers. So I've listed my factors of six. I know that six divided by one equals six. So one and six are factors. Six divided by two equals three. So two and three are factors. Two multiplied by three equals six. And that's all of them. I haven't got any other factors of six. And then I've listed my factors of 12. One, two, three, four, six, and 12. And a common factor needs to be in both lists. If it's a common factor of six and 12, it will be in both of those lists. So what are they? One, two, three, and six. It's not four because six divided by four does not give me a whole number answer. And it's not 12 because six divided by 12 equals a half, and it has to be a whole number answer. So it's just one, two, three, and six. And this is really, really important that we understand this term, particularly when we're looking at fractions. So we're going to go back to this generalisation that you'll be fed up of by the end of this lesson. When the numerator and denominator are multiplied or divided by the same number, the value of the fraction remains the same. So I need a common factor of six and 12. Well, I know that six is divisible by six and 12 is divisible by six. So I'm going to divide numerator and denominator by six. So six divided by six equals one, and 12 divided by six equals two. So 6/12 is equivalent to the 1/2. I can generate equivalent fractions. When I know that common factors can be used, okay. Multiple, this is the other bit of language that we need to get our heads around. This revision, I know you already know it, but I'm just reminding you. So a multiple of a number is the product of that number and an integer. So for example, two multiplied by five equals 10. So let's just put that there. 10 is a multiple of two, because two multiplied by five equals 10. 20 is a multiple of 10, because 10 multiplied by two equals 20. You can have a go at putting in some values there, so that you've really understood what multiple means. Pause the video now and put in some of your own numbers. Okay. I'm going to do some skip counting. I'm not even going to show you anything. Let's just skip counting threes. 3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39, 42. And I'll stop. All of those numbers are multiples of three. Okay, let's do some more. Let's do some skip counting and sixes. All the numbers that we say when we skip count will be multiples of six. Six, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66. Brilliant. So if you're good at skip counting, your next bit is to be able to just say those multiplication facts and link them in to relate to division facts. So if you were a bit hesitant, joining with me, counting in those multiples, let's have a look at using our multiplication grid. So let's count in multiples of six. Six, we'll miss out zero, because we know that when we write fractions, we don't ever use zero as a numerator or denominator. So let's start at the six and counts our multiples. Six, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72. They're all multiples of six. Multiples of six, they're also across there. Okay, let's have a look at some other multiples. That's how about look at our multiples of 12? So we're going to say them together. 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144. And you might've used this row to help you. So this grid is really, really helpful to see the multiples, because when you skip count, you say the multiples. Let's see what it would look like, now, on a hundred grid. It would look different to what it looks like on a multiplication square. So let's have a listen to someone who's used this hundred grid to look at multiples of three and six.

So, if I wanted to skip count in threes and then highlight all my multiples of three, that's what it was looked like on a hundred grid. And then if I skip counted in multiples of six and went to six, 12, 18, and carried on, that's what it would look like when I highlight them on my hundred grade. And I could actually have both of those shown. And you can see that some of the numbers I'm going to say when I skipped count in threes and I skipped count in sixes.

I'm going to ask you some questions now. What can you tell me about the number 15? Yeah, 15 is a multiple of three, but it's not a multiple of six. What can you say about six? And 12, 36, and 72? Yeah, they're multiples of six. They're also multiples of three. How about the number 58? 58 is not a multiple of three, and it's not a multiple of six. So a hundred grid is really useful to see patterns. And you might want to have a go at highlighting some of those multiples and asking your friends about what they notice when they see those patterns. So, we're going to come back to this generalisation again, that you're probably fed up of, but it's so helpful when you are simplifying fractions. When the numerator, yeah, I'm saying it. You need to say it as well. When the numerator and denominator are multiplied or divided by the same number, the value of the fraction remains the same. So I'm going to use multiplied by this time. And I'm going to say six multiplied by three. So six multiplied by three equals 18. If I multiply the numerator by three, I now need to multiply the denominator by three. So 12 multiplied by three equals 36. So I can start to look at these relationships. And I can say that 18 is a multiple of six, because six multiplied by three equals 18. And I can say that 36 is a multiple of two, because 12 multiplied by three equals 36. So that knowledge of how I can use the language of multiple will also help me when I'm writing equivalent fractions. So the activity that I reviewed right at the beginning of this lesson looks at generating number cards. And I bet you, you'd be better now at creating a game, because you've got that knowledge of how you need to have an understanding of factors and multiples. Remember I picked one, three, five, and 15, and it wasn't just by accident. It was because I'd understood the relationship between those numbers. So you might want to go back and actually create that game again with this understanding of factors, common factors and multiples of you might not want to. It's absolutely fine if you don't want to have a go at doing the game again, but if you do want to have a go, that would be great. The teacher who will teach you in the next lesson will have expected you to have done this practise activity, so they will review it. So you'd be better be ready with some really good examples of why they might be sometimes always or never true. So a multiple of six will also be a multiple of three and two, and you can read, but I'll just read it out for you. Two multiples of five added together will equal a multiple of 10. So just be ready with sometimes always or never true and keep remembering what those terms mean. Factor, common factor, and multiple, because when you are simplifying fractions, you'll need to really understand what those terms mean.

Lesson summary: Factors and multiples

In this lesson, we revise the language of 'factor', 'multiple' and 'common factor. '

Maths

Factors and multiples

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Lesson summary: Factors and multiples

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