Lesson video

Hello, and welcome back to lesson 11 on fractions.

You've learned quite a lot since I saw you, and I've been really looking forward to developing your thinking even further today.

In the previous lesson, you revised multiples and factors, and your teacher left you these activities.

So let's have a look at the first one.

Multiple of six will also be a multiple of three and two.

Is that sometimes, always, or never true? What did did put? Did you say that that was always true? Fantastic! And shall we have a look at the reasons why? So if we look at our multiples of six here, why would they be a multiple of two as well? Straightaway, we can see they're all even numbers, aren't they? So we can see that they're multiples of two.

And if we look at our threes, can we find the multiples of six? Six, 12, 18, 24? Yes, we can.

And what can we say why does that happen? Ah yes, so our six times tables are double our three times tables.

So our multiples of three are always also multiples of six.

Excellent! Let's look at the second example.

Two multiples of five added together will equal a multiple of 10.

Is this sometimes, always, or never true? What did you say? Did you say that hat was sometimes true? Fantastic! Why is it sometimes true? Let's have a look at them.

Well, we know that multiples of five end in a zero or a five, don't we? So if we have two of the multiples that end in zero, so that would be for example, 10 and 20, then we know we'd get a multiple of 10 then.

How else might we get a multiple of 10? Yes, indeed, we could add a multiple ending in five, and another multiple ending in five.

So for example, 15, and five.

Because the five ones, in both of those numbers, would make one 10.

So that would make a multiple of 10.

And what instance wouldn't we make a multiple of 10? Yes, you are spot-on.

So if we were to add a multiple of 10, which is 20, and a multiple of five ending in five which is 25, then we would get a multiple of five, but doesn't make a multiple of 10.

Excellent! And our final challenge here was to look at what Captain Conjecture had said.

And he said, "Factors come in pairs.

"So all numbers have an even number of factors.

"Do you agree? And can you explain your reasoning?" So this you really, had to think really deeply for this.

Did you find a set of numbers that don't have an even number of factor pairs, or did you find that all numbers have an even number a factor pairs? Ah, fantastic! So you didn't agree? Brilliant! So most numbers do, don't they? But not all? And can you tell me which ones don't have an even number of factors? Did you find it was square numbers? Fantastic, yes.

And I've explained my reasoning here, because I thought, if you've explained your reasoning so succinctly, I would too.

So most numbers have factors which come in pairs.

So factors of 12, for example, we got one and 12, two and six, three and four.

And if we put them in a list, then we get an even number of factors.

But if we say, "Look at 36," which is a square number, yes, we can generate our factor pairs.

But because it's a square number, it has a repeated factor of six.

So when we write those in an ordered list, we only write the repeated factor once, and so we end up with an odd number of factors.

So you can say we've got nine factors out, which is an odd number.

So all square numbers have an odd number of factors, although most numbers have factors which come in pairs.

Super, really well done.

And I hope you really enjoyed that task.

It was a really interesting one.

Let's get started on today's learning.

So what can you tell me about this family of equivalent fractions? We've said, haven't we, in previous sessions, that a family of equivalent fractions may be expressed differently, but they have the same value, which is why we call them equivalent.

So what is the same and what is different about this family of equivalent fractions? Just to give you a moment to think about that, but we've spoken about it in quite a lot of detail.

Yes, what can you tell me? Oh, fantastic, something that's the same.

So each of these fractions is the same proportion of the whole.

Fantastic! I'm glad you remember that.

And is there anything that's different? Ah, so the numerators and denominators appear different.

Why is that? Because as we divide the whole into more parts, each of those parts get smaller, and we need more of those parts to make the same proportion of the whole, really well done.

So for example there, 4/16 we need four of those equal parts to make the same proportion of the whole as one quarter or 1/4, if a whole is divided into four equal parts.

Fantastic.

And we can also say that the simplest form in this family of fractions is 1/4.

Have you heard that language before, simplest form? You might have heard that phrase before.

Let's say this together.

"The simplest form in this family of fractions is 1/4." And again, "The simplest form "in this family of fractions is 1/4." And we're really going to have a little thing throughout this lesson, about what that language means.

I've put up another area diagram here, and I'd like you to write a family of four equivalent fractions that are represented by the green part of each whole.

Suppose we know, and if you could write the family of fractions this time.

How did you get on? Did you find that 1/2 was equivalent to 2/4, and that was equivalent to 4/8, equivalent to 8/16? Really well done.

And has what we said earlier, does that apply in this family of fractions? So yes, the proportional relationship is the same.

So 8/16 is the same proportion of the whole as 2/4.

And can you tell me why? Can you tell me because of what the relationship is, between the numerator the denominator? So the denominator is twice the numerator, and the numerator is 1/2 of the denominator, which represents the whole.

Super.

Now you might already be thinking you can identify which is the simplest form here.

Let's have a look.

The simplest form in this family of fractions is 1/2.

Shall we say that together? "The simplest form in this family of fractions is 1/2." And again, "The simplest form "in this family of fractions is 1/2." Really well done.

You might be beginning to identify why you think that is.

Let's have a look at another example.

Here's another way of representing equivalent fractions on a multiplication table, which you saw in a previous lesson.

So putting out right, the family of seven equivalent fractions that are represented by those shaded bars.

So if pause me now, I'll give you a moment to do that.

Did you come back, what did you find? This list of equivalent fractions? Super.

What can you tell me about them? So once again, we can seek out whether that proportional relationship between the numerator and denominator is the same.

And the denominator is seven times the numerator, and the numerator is 1/7 of the denominator, which represents the whole.

Could you identified the simplest form? The simplest form was 1/7, well done.

Let's say this together.

"The simplest form in this family of fractions is 1/7.

"The simplest form, in this family of fractions is 1/7." So again, I think you might be making some generalisations about what you think the simplest form is now.

Let's have a see if you're right.

If we look at the three examples that we had, in each example, the ringed fraction is the simplest form.

So we said 1/4 was the simplest form of those fractions, 1/2 is the simplest form of those, and 1/7 is the simplest form of that group.

So what can you conclude about the simplest form? What do you think the simplest form of the fraction means? And I'd like you to pause me again now, and have a little thing.

And can you describe it in different ways? How might you describe that? So have a little thing.

What do you think the simplest form of the fraction means, considering those examples that we've seen? See you in a moment? Okay, what did you find? Let's have a look.

Well, I've been keeping my ears out at some of the things that I've heard you saying and writing down.

And I spotted that some of you had set this.

"In the simplest form, "we have scaled the numerator and denominator "to make them as small as we can." So let's have a look at one of those.

And we used that language as well, during the enterprise session, when we were comparing the numerators and comparing the denominators.

We said that they had been scaled.

So have they been scaled to make them as small as they can? So yes, there.

The 1/2, the numerator is as small as it can possibly be, and the denominator is small.

How were the numerators been scaled? They've been scaled by a factor of two.

Haven't they? So eight divided by two is four, four divided by two is two, two divided by two is one.

Excellent! And is that the same for the denominator? Have a little look at that.

So have the denominators been scaled by the same factor? Yes.

So let's scale the opposite way this time.

So two times two is four, four times two is eight, and eight times two is 16.

So really well done for all those people that said, "Oh, actually, I can see here, "that the numerator and the denominator "have been made as small as we can possibly can make them "by scaling those numerators and denominators." So that's really good.

Now I heard some children saying, that in the simplest form, the numerator is one.

So I'm really interested to hear that, which is why I picked that example out, from what I've heard you saying.

So in the simplest form, we made the numerator one.

Do you agree, is that correct? Okay, yes.

So in this case, if you have written that, in this case, in the simplest form we have made the numerator one, and in all the examples that we looked at the numerator has been one, hasn't it? 1/2, 1/7, and 1/4 we looked at.

Didn't we? Brilliant! Though, we'll find out as we continue looking at the simplest form, if that's always true.

Because we don't know yet if that's always true.

It's just true at the moment with the examples we've looked at.

And then if we look at this diagram again, and we can see why that happens.

And again, proportional relationship there, which means that we have the same proportion of the whole.

What's about these two fractions? So 1/3 and 3/9.

What can you tell me about those? What's the same, and what's different.

And can you identify the simplest form? and why do you think it's the simplest form? I give you a moment to think about that.

Okay, what you said if the same? Yes, we getting so good at that now.

It's the same proportion of the whole, isn't it? We can see that so clearly in these area models.

And what's different? The numerator and denominators are different.

So in 3/9, the whole has been divided into nine equal parts, and three of those parts is the same proportion of the whole as 1/3.

Fantastic! But actually, when we do look at that what can we say about the proportional relationship between the numerator and the denominator? So the denominator is three times the numerator, and the numerator is 1/3 of the denominator.

Excellent! And did you identify the simplest form? So you said it was 1/3.

Brilliant! And actually does that agree with some of the statements we looked at earlier? So yes, the numerators have been scaled by that factor of three, haven't they? And the denominators have also been scaled by a factor of three, so scaled down, divided by three, and we've made the numerator and denominator as small as they possibly can be.

And in this example, again, it is a unit fraction.

So in the simplest form, we're making, oh, actually, yes.

In the simplest form, we're making the fractions as small as possible.

So this is also something that I heard some of you say.

Let's read that again.

"In the simplest form, "we are making the fractions as small as possible." Do you agree? I'm going to ask you to pause again now, and think about how we described the simplest form in the previous slides.

You can go back and have a look at that if you like.

And then look at that statement.

Do you think that we're making the fractions as small as possible? Pause me now.

Okay, what did you think? I'm hoping, yes, what did you say? No, no, we're not making the fractions as small as possible.

Why not? The fractions are the same size, aren't they? And we've said that over and over again in this lesson, and in other lessons.

So equivalent fractions are the same proportion of the whole.

So they're not changing in size at all.

They are the same size.

So we're not making the fractions as small as possible.

We're actually making the numerator and denominator as small as possible, whilst keeping the proportion the same.

Shall we say that together? "The numerator and denominator "are made as small as possible, "while keeping the proportion the same." And in this case, what is the proportional relationship between the numerator and the denominator? The denominator is three times the numerator, and the numerator is 1/3 of the whole." Superb! So let's say that again.

"The numerator and denominator "are made as small as possible, "while keeping the proportion the same." So we haven't made the fractions any smaller, we have made the numerator and denominator as small as possible.

And in this example, yes, we have made a unit fraction to do that.

Have a look at it now.

I've extended the family of fractions here.

So we've seen two of those in the previous slide.

This is what we've just talked about, is that correct for these? Is the proportional relationship the same? Can you see the simplest form? Yes, well done.

It's 1/3.

So the numerator and denominator are made as small as possible, while keeping the proportion the same.

And we can see that yes, we have scaled down the numerators as we compare them, scaled down the denominators, while that proportional relationship where the denominator is three times the numerator is absolutely intact.

I would really like you now to apply what we found out about the simplest form to these fractions.

On the top row, we've got a list of fractions, and on the second row, each of those has an equivalent part.

So I'd like you to match up the fractions and then ring the one that you feel is the simplest form.

Let's do the first one together.

So the equivalent fraction to 1/5.

And you should know this straightaway, because we looked at this in intense detail in a previous lesson.

It is 2/10.

Superb! Which one is the simplest form? The simplest form is 1/5.

And why? Can you tell me why? The simplest form is 1/5 because, the numerator and denominator have been made as small as possible, whilst keeping the proportion the same.

And what is that proportional relationship? The denominator is five times the numerator, and the numerator is 1/5 the denominator.

Excellent! So just apply that across the rest of those fractions, match them up, and tell me which is the simplest form and why.

Pause now.

How did you get on? Shall we have a look at your answers? So 4/8 is equivalent to 1/2.

Well done if you got that.

Which one is the simplest form? Did you identify that 1/2 is the simplest form? Excellent! And can you tell me why? It's because the numerator and denominator have been made as small as possible.

Is the proportion the same? Yes, so the denominator is two times the numerator, and the numerator is 1/2 of the denominator.

Brilliant! What about 1/3? This is equivalent to 5/15.

Correct.

What can you tell me about the simplest form in that pair? The simplest form of that equivalent fraction pair is 1/3.

Brilliant! Why? Because the proportional relationship has been kept the same, whilst making the numerator and denominator as small as possible.

And that proportional relationship is, can you tell me? The denominator is three times the numerator, and the numerator is 1/3 of the denominator.

Cool.

And finally, did you find that 2/3 was the same as 4/6? You did! What can you tell me that's different about this example? Fantastic! So this one hasn't made a unit fraction, has it? And all the other examples we looked at were unit fractions, although we have said, haven't we? They don't always have to be.

And we're going to look at that in more detail in the next lesson.

Okay, it's time for me to leave you to it, and to stop waffling that too.

So here's your practise activity.

And you can see in front of you there, you've got nine fractions, and I want you to sort them again, into pairs of equivalent fractions, like we did in the last example.

And some of the very observant children out there, which is all of you will say to me, "Oh, there's one leftover," and you're quite right.

So for the equivalent fraction pairs, decide which one's the simplest form, and then you can use the sorting diagram to write those in the correct area.

And then you'll see that one of the fractions is missing its equivalent pair partner.

And so can you write an equivalent fraction part of it, and then decide, which is the simplest form out of those.

And we'll come back and look at those in the next lesson.

And I've also just put on a little challenge for you.

And that's my adorable puppy at the top of the screen there, he really is a love book.

So I was planning a walk with my adorable puppy today.

But I only had a limited time to walk.

'cause I've been really busy planning this lesson and getting it ready for you all.

So I asked him.

Now he's really good at whole numbers as my pop, but he's not so good at fractions.

So I asked him if he rather go for 1/2 an hour or for 30/60 of an hour, quite tricky to say in that one.

Now, which one do you think he preferred.

I'll tell you next time which one he barked out at me.

And what would you say to him, about those fractions? What would you say to him about those fractions, thinking about everything that we've discussed today? What we now know about equivalent fractions? Super! Well, that's everything for today, I've really enjoyed working with you again.

I hope you enjoy your independent tasks, bye-bye.