Lesson video

Hello, my name is Mrs. Behan, and in this lesson I will be your teacher.

In this lesson, we are going to be thinking about calculating efficiently.

If something is done efficiently, it means its done well and quickly.

Instead of going the long way round, it's the shortest, most direct route to it.

So we're going to be using doubling and halving to help us efficiently calculate.

Let's begin by looking at the lesson agenda.

We will explore doubling first.

Then, we will explore halving.

After that, you will have a chance to practise doubling and halving.

And then there will be an independent task, and I will go through the answers with you, because I know you'll be wanting to know how you got on.

There are just a couple of things you will need to take part in this lesson: something to write with, like a pencil or a pen, and something to write on.

If you don't have those things to hand now, just pause the video here whilst you go and get them.

Remember to try and work somewhere quiet where you won't be disturbed.

So let's have a look at doubling first.

What does doubling actually mean? Well, doubling is the same as multiplying by 2.

So, we're firstly going to transport ourselves to Junaid's school.

Here is Junaid's teacher, and he's showing these two calculations.

2 multiplied by 13 multiplied by 2, and 13 multiplied by 4.

The teacher says, "Junaid, will these calculations give different answers or the same answer? And Junaid's reply is, "They must give different answers, because one has three numbers, and the other only has two." What do you think? Is Junaid right? Or has Junaid made a mistake? Well, let's take a look.

To multiply by 4, we can double, and then double again.

So, to multiple by 4, we multiply by 2, and then multiply by 2 again.

And if we use our Law of Associativity, we know that actually, multiplying by 4 is the same as multiplying by 2 and by 2 again.

We have those factors in this calculation up here.

So, let's calculate.

32 multiplied by 4, and the way we're going to do it is by doubling it, and doubling it again.

What do you notice has happened here? Well, we've used our Law of Commutativity to just change the order of the factors.

And that's fine.

The product will still remain the same.

So let's double then double again.

Double 32.

Let's use a partitioning strategy here.

So, let's partition 32 into 30 and 2, and let's double them.

We then make 60 plus 4.

A total of 64.

Here we have Dienes to represent what we've just shown.

32 doubled is 64.

But we don't just want to stop there, because that would just be doubling once.

We're going to double again, to multiply by 4.

Double 64 is the same as 120 plus 8.

Just tell me what's happened here.

That's right, we partitioned 64 into 60 and 4, and then we doubled each part.

We now have a total of 128.

The Dienes will show us what that looks like.

So, 64 doubled is 128.

And what do we notice about the number of tens we have? That's right.

We can do some regrouping.

So here I regrouped ten tens for 100.

I'm going to give you something to have a think about now.

If I know that 32 multiplied by 4 is equal to 128, how could this help me calculate 32 multiplied by 8? Just pause the video here whilst you have a think.

Okay, are you ready? So, did you have think then? How can 32 multiplied by 4 equals 128 help us to calculate 32 multiplied by 8? Well, I know that 32 multiplied by 8 will be double 32 times 4 because 8 is double 4.

So, we can make a relationship here, or a connection between the 4 and the 8.

These factors are double, so the 4 has been doubled to make 8, so we know that 32 multiplied by 8 will be double 32 multiplied by 4.

So let's test out that generalisation that we have just made.

So we're going to practise multiplying by 2, by 4, and 8 using doubling.

So, I've got an example for you.

I wanted to know what double 14 was, so I partitioned 14 into the tens and the ones, and doubled it, so 20 plus 8.

And you can see I've represented this with base ten, or Dienes to the right hand side.

So I now know that 14 multiplied by 2 is equal to 28.

So, I'm not going to stop there.

I want to look at 14 multiplied by 4.

Well I'm going to use double 28, that I found out here, to help me.

So I'm going to partition 28 into 20 and 8, and then double both parts.

As again you can see in my pictures over here, at the side.

So double 20 is equal to 40, and double 8 is equal to 16.

Now you can see I've drawn this purple little wiggly line around ten ones there, because I know that I will need to regroup.

So 14 multiplied by 4 is equal to 5 tens, which I have here the 4 and this one here, that I've not regrouped yet, and 6 ones.

So, 14 multiplied by 4 is equal to 56.

Well, how will I work out 14 multiplied by 8? I'm going to double again.

So we're going to use double 56 to help us.

So let's start by partitioning 56 into 50 and 6, and then doubling each part.

You can see 56 here, and I have doubled it.

We now have 100 plus 12 ones.

And I've done a little bit of regrouping because I've noticed that I have ten tens here.

14 multiplied by 8 is equal to 112.

I could also have drawn a line around ten of these ones, couldn't I? Just to make it really, really clear.

So 14 multiplied by 8 is equal to 112.

Do you think you could have a go? Can you multiply 12 by 2, then 4, and then 8? Pause the video here whilst you have a go.

When you're ready, come back to me, and I'll show you what I did.

Okay then, welcome back.

Let's see how you got on multiplying by 2, 4, and 8.

So, I hope you have something that looks like this.

We started with double 12, and our first job is to partition 12 into 10 and 1.

Into 10 and 2 ones, even.

And I do find it useful to draw the pictures at the side.

So these sticks and the ones represent the tens and the ones.

My lines and the dots.

So, I now have 2 tens and 4 ones.

So 12 multiplied by 2 is equal to 24.

I then worked out that 12 multiplied by 4 is equal to 48, and I'm sure you've got this too.

You'll also know your tables facts to know that 12 multiplied by 4 is equal to 48.

But, let's double 24 anyway.

So let's partition 24 into 20 and 4, and then we'll double each part.

So 20 doubled is equal to 40, and we can see that here in these representations.

And then 4 doubled is 8.

Let's combine it all together, and we understand that 12 multiplied by 4 is equal to 48.

So to multiply by 8, what do we have to do? Well let's double again.

So 12 multiplied by 8 is equal to 96.

Using the drawings really does help.

Partition 48 into 40 and 8 and then double each part.

So 40 doubled becomes 80, and 8 doubled becomes 16.

So 12 multiplied by 8 is equal to 96.

So what is the inverse of doubling? Well, doubling means multiply by 2.

Halving is the inverse.

Halving means divide by 2.

So now let's work backwards.

96 divided by 4.

If you're up for a challenge, pause the video here and have a go at it first.

See if you can do the inverse of our doubling procedure.

When you're ready, come back and we'll go through the next part together.

So we're going to halve and halve again.

If we can multiply by 4, and double and double again, we can halve by 4, by halving and halving again.

The Dienes and blocks representations come in real handy here.

So we're going to think about half of 96.

Now, straight away, I can see there is an odd number in here: 90.

90 is a little bit trickier to divide by 2.

That's what we're doing, we're halving, or we're dividing by 2 first.

So I need to find some way of partitioning 96 where it's going to give me two even numbers.

So I can represent 96 as 80 plus 16.

So this is what I've chosen to do: 80 in tens and 16 ones.

So I need to find what half of 80 plus 16 is.

Can you help me? What is half of 80 and 16? Well half of 80 and 16 is 40 and 8.

And you can see here, in the abstract version, 40 and 8.

We're not going to stop there, though.

That's half of 96.

But what do we want to do? We actually want to divide by 4.

So we're going to halve again.

So we've halved our blocks again, so we now have how many tens? 2 tens and 4 ones.

Let's record that over here.

We've now halved it, so we have 20 and 4 ones.

So 96 divided by 4 is equal to 24.

Pause the video here, and using the strategy that we've just done, please can you have a go at dividing 72 by 4.

Remember, halve and halve again.

Okay then, let's see how you got on.

So these are my drawings here that I used to show my working out, and it really did help.

Half of 72.

These are the numbers that I decided to partition 72 into, because again, I had an odd digit.

I had an odd number of tens.

So I'll go with 60 and 12.

I halved it and I ended up with 3 tens and 6 ones.

So half of 72 is 36.

But we don't stop there.

We want to divide by 4.

So we need to find half of 36.

So, 36 is equal to 20 plus 16.

I had to do the same thing and choose two numbers that could be partitioned easily, because I had another 3, an odd number.

They don't divide easily by 2.

So, we then halved 36.

Sorry, we halved 20, and it made 10, and half of 16 is equal to 8.

So, oh, my screen's hiding it.

Let me just hide my face.

There we go.

So, 72 divided by 4 is equal to 18.

So you have all of the knowledge you need to complete the independent task.

For each calculation, use one of the mental strategies you have learned in this lesson.

Write down any jottings that help you to remember your calculations.

But if you don't need to write anything down, that's fine.

Complete the sentence to explain which strategy you have used.

So on the first question, 27 multiplied by 4, you will write the product here.

And I want to know.

Did you use a doubling strategy, or a halving strategy? How many times did you have to double or halve? Pause the video here whilst you complete your task.

When you're ready, come back to me, and we will go through the answers together.

Okay, let's go through the answers together.

You can see all of the answers on screen, so you can pause the video here whilst you mark your work.

I'm going to go through each answer.

27 multiplied by 4 is equal to 108.

I used a doubling strategy two times.

Remember to multiply by 4, you can double, and then double again.

33 multiplied by 4 is equal to 132.

I used a doubling strategy two times.

22 multiplied by 8 is equal to 176.

I used a doubling strategy 3 times, and that's because to multiply by 8, we double, double again, and double a third time.

76 divided by 4 is equal to 19.

I used a halving strategy two times.

Remember halving is the inverse of doubling.

So to multiply by 4 we double and then double again.

So to halve by 4, we halve, and then halve again.

19 multiplied by 8 is equal to 152.

I used a doubling strategy for this one, three times.

92 divided by 4 is equal to 23.

I used a halving strategy two times.

If you'd like to, please ask your parents or carer to share your work on Instagram, Facebook, or Twitter, tagging @OakNational, @LauraBehan21, and #LearnwithOak.

So in this lesson we have learned that doubling and halving really does help us to calculate efficiently.

Remember, to multiply by 4, you double, and double again.

Divide by 4, you halve, and halve again.

Don't forget to take the quiz to test out your new learning.

See you again soon.

Bubbye!.