Lesson video

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Hello, I'm Mr Langton and today we're going to look at multiplying a fraction by an integer.

All you're going to need is something to write with and something to write on.

Try and find a quiet spot where you can work without any distractions.

When you're ready, we'll begin.

We'll start off with a try this activity.

Each rectangle has an area of 120 centimetres squared.

Find the area of each shaded section.

If you're feeling confident, pause the video and have a go.

If you want a hint, just leave the a little bit longer and I'll go through a little bit with you.

So if you're feeling confident, get ready.

Starting in three, two, one, go.

So a little hint for you, if you're not quite sure what to do.

This one on the left has been split down here like that.

It's been split in half.

So if the whole area is 120, a half of that would be 60.

So try and see what fractions the rest of it's missing to as well.

Looking at the one on the right, it's been split horizontally into three equal sections there.

So each of those represents a third of the shape.

So if you find a third of the area, that should get you off on a starting point.

Okay, so, pause it, off you go.

Three, two, one.

And here are the answers, how'd you get on? Now Antoni and Cala are working out the product of 12 and a third.

Antoni has done 12 lots of a third.

Can you see each of these shapes has been split into thirds? And 12 lots of a third, one, two, three, four, five six, seven, eight, nine, ten, eleven, twelve lots of a third give us four whole ones.

Cala has done a third of 12.

She's taken 12, split it up into three equal pieces and she's also got four whole ones as well.

So we can see that multiplying by a third gives us the same result as dividing by three.

Now what I'd like you to do is draw two diagrams that represent a half times five.

Pause the video and have a go.

When you're done, unpause it and we'll compare answers.

So I want you to pause in three, two, one, go.

Okay so one answer you could have got, if you did it like Antoni did, we could have done five lots of a half.

So I'm going to draw a diagram there.

That one's split into half so that will be one half there.

And I've got five of them so I need to draw identical models and we said five halves.

So if that was one half, that's the second half, a third half, a fourth half and a fifth half.

So altogether, I've got two whole ones and another half.

So a half times five is two and a half.

You could've done it Cala's way.

Cala would've done this into a half of five.

So if I draw a bar model to represent five and if I find half of it, if I cut it down the middle there, each of those parts is 2.

5 or two and a half.

So we can see that multiplying by a half gives the same result as dividing by two.

Now it's your turn to have a go.

Pause the video and access the worksheets.

When you're ready, unpause it and we'll go through it together.

Good luck.

How did you get on? I put some of the answers on the screen now and I'll go through some more with you.

Hopefully, you've done pretty well so far.

Now this one here, let's draw an arrow to it, which is a unit fraction, one over something, multiplied by 12 gives us something else.

The reason that I've not filled this one in is because there are actually lots of different answers you could give.

You could say one over six multiplied by 12 gives you two.

You could reverse those numbers and say a half of 12 is six.

You could instead have said a quarter multiplied by 12 gives you three.

And there are other ones as well so hopefully you got one of those or something else that works.

Underneath that, which is largest? A third of 21, or 21 multiplied by a third or 21 multiplied by three.

Now if you're really on the ball and if you've spotted it, I was trying to trick you a little bit there 'cause actually, they're all equal.

None of them is larger than the other.

They're all three different ways of calculating the same thing in each case.

A third of 21 is seven, 21 multiplied by a third is seven, 21 divided by three is seven.

And finally, this calculate question.

Again, if you're really on the ball, remember what we said earlier? Dividing by three is the same as multiplying by a third.

So we've got four multiplied by a third multiplied by nine and dividing by two is the same as multiplying by a half.

Now, multiplication is commutative.

We can move around the order of them and still get the same answers.

So we could do, four lots of a half multiplied by nine lots of a third.

Now four lots of a half, a half of four is two and nine lots of a third, or a third of nine, is three and three times two is six.

Did you get that? This last activity follows on from that last question that we just went through.

How many different ways can you solve each of these problems? Pause the video and have a go and we'll discuss it after you're done.

Pause in three, two, one, go.

Okay, so one idea that I mentioned is to replace these divisions with multiplying by a fraction.

So we could do 12 multiplied by a fifth multiplied by 10 multiplied by a third.

And we need to look for what goes well together.

Which ones can we multiply and end up with whole number answers? We could do 12 times a third, which is four, and we can do 10 times a fifth, which is going to be two.

So that's eight.

That's one way you could do it, there are lots of others.

This next one, should we do it a different way? Let's try and do it without using fractions this time.

We could do our 14 divided by seven.

We can multiply that by 27 divided by nine.

You might find that easier.

That's going to be two multiplied by three to get six and remember, it'd be just the same.

Dividing by nine is the same as multiplying by a ninth.

So you could do it that way as well.

This last one is a really long one but once again, once you look for links, and there are different sorts of links you could look for, you might, for example, notice that here, we're dividing by two.

That's the same as multiplying by a half.

Now all the way here at the end, we're multiplying by two.

And if we multiply by a half and we multiply by two, then together, a half of two is just one.

So that bit there is equivalent to multiplying by one.

And you might find, if you look closely, you might spot a few more of those pairs as well.

I'm going to leave that there so you can have another look at it.

And also, see if you can write a similar question of your own.