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Hi, welcome to today's lesson.

My name is Mr. Peters and in this lesson we are gonna be thinking about how different lengths can be recorded, either additively, so using addition, subtraction, or multiplicatively, so using multiplication or division.

If you're ready, let's get started.

By end of this lesson today, you should be able to say that different lengths can be represented as addition and multiplication.

Throughout this lesson today, we're going to need a number of keywords.

The keywords we're gonna be using are these three here.

I'll have a go at saying them first and then you can afterwards.

Are you ready? Common, compose, partition.

Let's think about what these mean in a bit more detail.

Something is common when it is frequently used or it occurs regularly.

When we think about the word composed, we could say that larger numbers are composed of smaller numbers, they're made up of smaller numbers.

And finally, partition.

We partition something when we break it down into smaller parts.

Look out for these words today throughout our lesson and use them where you can as well to help explain and reason your thinking.

Today's lesson, we've broken down into three cycles.

The first cycle, we'll think about common partitions of one.

The second cycle, we'll think of the composition of those common partitions to one.

And then finally, our third cycle, we'll think about the composition of common partitions beyond one.

Let's get going with the first cycle.

So let's take a moment to have a think to start us off with.

Here, I've got a square.

This square represents 100.

There are 100 squares we can see here on the right-hand side of the screen.

If 100 is the whole, how many equal parts do you think we could partition 100 into? Take a moment for you to have a think yourself.

Jacob thinks we could partition 100 into 100 equal parts.

We can see here that each part individually represents one of those 100 parts.

So we've got 100 equal parts here altogether.

We can say that 100 is composed of 100 ones, and we can represent this in a bar model as well underneath.

Here, the whole is 100, and underneath that whole, we've got 100 equal parts.

Each one of those would represent one.

Jacob also thinks we could partition 100 into 10 equal parts.

Let's have a look.

Here's one part, two parts, three parts, four parts, five parts, six parts, seven parts, eight parts, nine parts, and 10 parts.

So again, 100 can be composed of 10 lots of 10, 'cause each one of those parts is made up of one 10.

We can record this again as a bar model.

The whole is 100 and underneath that, we've got 10 equal parts and each one of those parts has a value of 10.

We could partition 100 into five equal parts as well.

Let's have a look here.

Here's one part, two parts, three parts, four parts, five parts.

Take a moment to have a think.

What is the value of each one of those parts? That's right.

100 is composed of five lots of 20.

Each part has a value of 20.

And again, we can record this as a bar model.

100 is the whole, and then we've got five equal parts underneath that, and each one of those parts has a value of 20.

Now we can also extend this into partitioning into four equal parts.

Let's have a look.

One part, two parts, three parts, and four parts.

Hmm, it looks like our 100, which was in a square, wasn't it, has been partitioned into four equal squares.

And each one of those squares has 25 parts.

So we could say that 100 is composed of four lots of 25.

And again as a bar model, 100 as the whole divided into four equal parts and each part has a value of 25.

And finally, if we can partition a number into four equal parts, that means we can also partition the number into two equal parts.

So 100 has been divided into two equal parts here.

Each one of these parts has a value of 50.

So 100 is composed of two lots of 50, and here, as a bar model again, 100 is a whole, and then we've got two 50s underneath that.

We did have a whole which was 100, didn't we? And we broke that 100 into 100 equal parts and each part had a value of one.

But what happens if we change the value of the whole? In the previous examples, the value of the whole was 100, but what if we changed the value of the whole to be one? What would happen to each one of the parts? That's right, each part would have a value of one hundredth or 0.

01.

The whole has been divided into 100 equal parts and each part has a value of one hundredth.

Hmm, so we can say that one is composed of 100 hundredths.

Again, we can represent that as a bar model, can't we? Notice this time that the whole has changed, it's no longer 100, the whole is now one, but we still got 100 equal parts underneath and each one of those parts would represent one hundredth.

Now we can start thinking about how we can compose one of different equal parts.

Let's have a look and see what would happen if we had one whole and we divided that into 10 equal parts.

Here's one part.

Hmm, how many hundredths are there? That's right, there are 10 hundredths.

So one is composed of 10 lots of 10 hundredths.

Here's one lot of 10 hundredths.

Here's two lots of 10 hundredths.

Here's three lots of 10 hundredths.

Here's four lots of 10 hundredths, and so on all the way up to 10 lots of 10 hundredths.

We also know that 10 hundredths is equivalent to one tenth.

So we could say that one is composed of 10 tenths as well.

We can show this in a bar model underneath.

The whole is one and there are 10 equal parts and each part has a value of one tenth.

We can then extend this onto our number line as well.

Notice here how we've got zero at one end and one at the other end.

And our whole has been divided again into 10 equal parts.

And at each point on the interval markers represents one tenth.

Let's count up together, are you ready? Zero, 0.

1, 0.

2, 0.

3, 0.

4, 0.

5, 0.

6, 0.

7, 0.

8, 0.

9, and one whole.

We can also extend this into five equal parts now.

One whole can be partitioned into five equal parts.

Let's have a look.

Very similar to what we had before.

Have a moment to think as well.

What's the value of each one of those parts? That's right.

The value would be 20 hundredths.

You can see that each colour has been shaded.

There's been 20 hundredths shaded each time, so that is equivalent to two tenths.

So in our bar model, we've got one whole divided into five equal parts.

Each part has a value of 0.

2 or two tenths or 20 hundredths.

Again, we can then link this to our number line.

We've got five intervals here and we can count up in those intervals of 20 hundredths or two tenths, ready? This time, we'll say it as tenth, shall we? So zero, two tenths, four tenths, six tenths, eight tenths, and one whole or 10 tenths.

We could then partition our one into four equal parts as well.

You might be starting to think about that in your head already and starting thinking about where those colours might be represented in our whole.

One is composed of four lots of 25 hundredths, 'cause we know that four lots of 25 make 100.

So four lots of 25 hundredths would make one whole.

Again here, one is the whole and we've got four lots of 25 hundredths or four lots of 0.

25.

On our number line, we can see how we'd record this.

Zero, 0.

25, 0.

5, 0.

75, and then one.

And finally, we could also record this into two equal parts, couldn't we? Here, one whole has been divided into two equal parts.

What's the value of each one of those parts? That's right, it would be 50 hundredths, wouldn't it? So one is composed of two lots of 50 hundredths.

We can record that in our bar model.

One is divided into two parts.

Each part has a value of 0.

5 or 50 hundredths.

And once again, we can link that onto our number line.

A whole has been divided into two parts.

It starts at zero.

The interval marker in the middle would be 0.

5 and then at the other end of our number line, it would be one.

Okay, have a look here then.

On the left-hand side, we've got the bar models where we had the whole which was 100.

And on the right-hand side, we've got the bar models where the whole is now one.

Take a moment to have a look.

What do you notice? What's the same about them and what's different about them? Well, Jacob's pointing out to us that the numbers in the bar model on the right-hand side are 100 times smaller than the bar model on the left-hand side.

So the whole has changed from 100 to one and the value of the parts has changed from 20s to two tenths.

So Andeep's saying that if the whole has been made 100 times smaller, then each of the parts also need to be made one hundredth the size.

You may have noticed that the two on the left-hand side bar model was representing two tens each time, wasn't it? And then the two on the right-hand side bar model is now representing two tenths.

So the two itself as a digit stayed the same, didn't it, but the value of that two changed.

It changed from two tens to two tenths and that's because it was made 100 times smaller or one hundredth the size.

Now that we've thought about that, we can then start thinking about applying that to our understanding of a metre.

Here I have one metre and that metre has been divided into 10 equal parts.

And you'll see that I've divided one of those parts, one of those tenths of a metre into another 10 equal parts.

So each one of those little parts represents one hundredth of a metre or one centimetre.

Let's have a look at how we've divided up our one metre to start off with then.

Well, here are my two parts.

I've got the blue part and the green part.

The blue part represents 50 centimetres, so that also means that the green part represents 50 centimetres, 'cause they're the same size.

So we could say that one metre has been divided into two equal parts and each part has a value of 50 centimetres or 0.

5 metres.

We could divide our metre into five equal parts.

Let's have a look.

Here's one part, two parts, three parts, four parts, and five parts.

All of those parts have a value of 20 centimetres or we could say 0.

2 metres.

So again, in our bar model, we've got one metre as the whole this time and each one of those parts has a value of 0.

2 metres.

We could also then partition our metre into four equal parts.

Have a look carefully at this one.

Here's one part, that has a value of 25 centimetres.

Here's our second part, which also has a value of 25 centimetres.

Here's our third part and here is our fourth part.

So again, one metre or our whole has been divided into four equal parts and each one of those parts has a value of 25 centimetres or 0.

25 metres.

Okay, now it's time for us to check our understanding.

Have a look here.

Can you describe the relationship between these two bar models? That's right.

As Jacob has pointed out, the bar model on the right-hand side is 100 times smaller than the bar model on the left-hand side.

We could also say that is one hundredth the size.

Well done if you got that.

Have a look at this example.

Can you spot the incorrect counting sequence? You might need to look very carefully at whether the numbers are increasing by the same amount each time.

That's right, B is the incorrect counting sequence.

Did you spot why? The sequence is increasing in lots of 25 centimetres.

So it would go zero metres, 0.

25 metres, 0.

5 metres, but actually it's got 0.

55 metres here so we know that is incorrect.

So B is the incorrect one.

Okay, onto our first task for today then.

What I'd like you to do is use your knowledge of our common partitions to one to complete these bar models below.

So you need to be thinking about what's the size of each one of the parts and writing in the correct value.

Once you've done that, I'd like you to translate that understanding of common partitions onto our number line and then think about what would be the missing numbers each time on the number lines.

Good luck with those and I'll see you again shortly.

Okay, welcome back.

Here, I've put the answers in for you, have a look at them, and give them a tick if you've got them correct.

On the left-hand side, they were in the standard partitions that we've seen so far, weren't they? Was there anything you noticed on the right-hand side? Yeah, we started combining some of them, didn't we together.

So we partitioned it into not necessarily equal parts, but we started to use those common partitions to recombine them together to make one whole, didn't we? So let's take a look at the first example.

We realised that on the right-hand side at the top, that one metre could be made up of five lots of 0.

1 metres or one tenth of a metre and also one lot of 0.

5 metres.

Well done if you managed to get all of those.

And here are the missing numbers for the number line.

Let's go through them together.

3, 0.

6, and 0.

9 were the missing numbers from the first number line.

5 was the missing number from the second number line.

25 and 0.

75 were the missing numbers from the third number line.

And finally on our vertical number line here, 0.

2 and 0.

6 were the missing parts.

Did you notice here as well that all the number lines were different sizes? It doesn't matter the size of your number line.

Each one of these still represented one whole, because it started at zero and finished at one.

So actually each one of these number lines still represented the distance between zero and one even though some were larger than others.

Right, moving on to cycle two of our lesson then.

Composition of common partitions to one.

We can record our common partitions to one with addition equations.

Let's have a look here.

We've got our number line, starts at zero and ends in one.

The number line has been divided into 10 equal parts and each part has a value of 0.

1 or one tenth.

And we can record that as an addition equation.

One is equal to 0.

1 plus 0.

1 and that continues until there is 10 lots of 0.

We can also turn that equation around, can't we, and write it like this.

So we've got the tenths on the left-hand side of the equal sign and the whole, the one whole on the right-hand side of the equal sign.

Now from previous learning, we should be aware that writing equations like that can take quite a long time and we don't need to record it like that.

We could actually record it multiplicatively if we wanted to.

So how might we do that then instead? So again, we divide it into 10 equal parts, aren't we? But we could say that we've got 10 lots of 0.

1, can't we? So we could say that we've got one is equal to 0.

1 or one tenth multiplied by 10, 10 lots of 0.

Or we could also write it as a division equation.

One has been divided into 10 equal parts and each one of those parts represents one tenth.

Have a look here how we can record this one as an addition equation.

The whole has been divided into five equal parts here.

So each part has a value of 0.

So one is equal to 0.

2 plus 0.

And we know that we can swap the number and the expression to either side of the equal sign to represent the same thing.

And then again, we can then think about, well, that's still quite a long equation, so we can represent it as a multiplication equation as well.

Here we've got five lots of 0.

2, haven't we? So we could say that one is equal to 0.

2 multiplied by five.

That's the same as saying two tenths five times or five lots of 0.

And we could also write it as a division.

One whole has been divided into five equal parts and each part has a value of 0.

Now have a look here, we've got four equal parts this time.

Maybe have a quick think for yourself.

What would the equations be? Well, as an addition equation, we know each part has a value of 0.

25.

So we'd write it as one as equal to 0.

25 plus 0.

25.

And again we could rotate that around or we could write it multiplicatively.

We've got four lots of 0.

25 or 0.

25 four times.

So we can write that as one is equal to 0.

25 multiplied by four or we could record it as one being divided into four equal parts and each part has a value of 0.

25.

And then finally this one here, again, have a think, what might the equations be? Well, as an addition equation, we know we've got two parts, haven't we? Each one of those parts is 0.

5, so we'd write that as one equal to 0.

5 plus 0.

5 or rotate that round.

And then finally multiplicatively, we can describe this as one is equal to 0.

5 multiplied by two or one divided into two equal parts, each part as a value of 0.

Let's extend this a little bit further now.

Look at the equations I've now written.

All of these equations represent the number line above.

What do you notice? That's right, on the left-hand side, our factors have been swapped over, haven't they? And we know that can be applied in multiplication, can't it? We can write this as one is equal to 0.

5 times two or 0.

5 two times, or we could write it as one is equal to two lots of 0.

5, two multiplied by 0.

We know that because the commutative law allows us to swap our factors around and the product would remain the same.

What about on the right-hand side? Well, the dividend, one, stays the same, that stays at the front of our equation each time, isn't it, but we've actually swapped around the divisor and the quotient, haven't we, each time? So one divided by two, the two here is acting as the divisor and the 0.

5 is acting as the quotient here.

Well, underneath that, we then make the 0.

5 the divisor and we make the two the quotient.

The first one would say that our one whole has been divided into two parts and each one of those parts has a value of 0.

5 or our second one would say one has been divided into groups with a value of 0.

5 and in total, there are two groups.

So one divided by 0.

5 is equal to two.

So whilst we've been thinking about that out of context, we can now think about applying that to a context, can't we? So we've now changed our number line from zero to one to zero metres to one metre.

And again our number line here has been divided into four equal parts.

So our one metre has been divided into four equal parts and each one of those parts represents 25 centimetres.

We can record that as one metre is equal to 0.

25 metres multiplied by four.

That's four lots of 0.

25 metres or 0.

25 metres four times.

And we could also write it as a division.

One metre divided into four equal parts and each part has a value of 0.

25 metres.

Jacob's reminding us that 0.

25 metres can also be written as 25 centimetres, can't it? So we could also again rewrite these equations.

We could swap out the 0.

25 metres to 25 centimetres and these would still represent exactly the same thing.

Okay, time for you to check your understanding again.

Tick the equations that represent this bar model.

That's right.

B and C both represent the bar model, don't they? One metre is equal to 0.

25 metres multiplied by four or one metre can be divided into four equal parts and each part represents 25 centimetres.

Okay, onto our task for today then.

What I'd like you to do is look at the equations underneath the squares.

Each large square represents one whole, therefore each small square within that whole will represent one hundredth, 'cause there's 100 small squares within the whole.

What I'd like you to do using the equation underneath is I'd like you to shade in each one of those squares to represent the equation and then have at least three equations to represent each one of the images that you've shaded.

These equations might be addition equations or they might be multiplicative equations.

Good luck and I'll see you again shortly.

Okay, welcome back.

Here's some examples that we've done.

So we can see that we've shaded each one of our wholes now to represent these equations.

And here are three examples of equations that we've written for each one of these images.

Did you get something similar to these or something slightly different? If you can, you might like to check with someone nearby to see what they've recorded as their equations.

Well done if you managed to get all of those.

Finally, onto our last cycle today, composition of common partitions beyond one whole.

So let's have a look here.

You can see I've got my number line and on top of my number line I've put a bar model to represent the value of each one of the intervals.

And our bar model is going up in lots of 0.

2 here, aren't they? Let's count up, zero, 0.

2, 0.

4, 0.

6, 0.

8, and then one.

But what happens if we move our bar model along to here now and we start to partition the whole between one and two on our number line? What would be the value of each one of our interval markers? That's right.

Well, our number line is increasing in 0.

2 each time, isn't it? So from one, it would become one, 1.

2, 1.

4, 1.

6, 1.

8, and then two.

So here, our whole has been divided into two equal parts and each part as a value of 0.

Therefore the missing interval marker in the middle between zero and one would be 0.

But again, we can move our bar model up to the next whole now, so the whole is between one and two this time.

So what would be the missing interval marker this time between one and two? Take a moment to have a think.

That's right.

It would be 1.

5, wouldn't it? Because our numbers are increasing with a value of 0.

5 each time.

Here's another example.

What'd you notice this time? Well, that's right.

Our whole this time has been divided into four equal parts, isn't it? And also our whole has started on four and finished at five this time.

So again, if we shift this up between five and six this time, what's gonna be the missing numbers on each one of those interval markers? That's right, it would be 5.

25, 5.

5, 5.

75, and then six.

Well done if you managed to spot that.

Jacob's pointing out, "The decimal values remain the same no matter what consecutive whole numbers they sit between." So we can see that the whole numbers here were four and five on the left-hand side, whereas now on the right-hand side is between five and six, but the decimal values have stayed the same.

They've either been 0.

25, 0.

5, or 0.

75, haven't they? We can then move this on to expressing this as an equation.

Have a look here.

How many lots of 0.

5 have we got shaded? So we've got five lots of 0.

5 shaded, haven't we? And we can write this as a repeated addition.

We could record this as 0.

5 plus 0.

5, and that would be equal to 2.

5, wouldn't it? Have a look at this example, how would we find the missing value this time? Well again, we've got five lots of 0.

25, but we could record this as four lots of 0.

25, which makes one whole, doesn't it? And an additional lot of 0.

25.

So we can record this as you can see here, as four multiplied by 0.

25 plus one multiplied by 0.

25, as four lots of 0.

25 and an additional one lot of 0.

25, which would be equal to 1.

25.

That's often a really helpful way of thinking about it.

And I often think in a similar way.

If I know that four lots of 0.

25 make the whole, then I'm just thinking about how many more lots do I need after the whole, aren't I, each time.

So the missing value would be 1.

25 here.

Here's one more example, look at this one.

Well, here we can see that our wholes have been divided into five equal parts, haven't they? And this time we've got nine lots of 0.

Hmm.

But I think it might be easy to think about this slightly differently.

If we think about this as 10 lots of 0.

2, that would be two wholes, wouldn't it? So and we know that 10 lots of 0.

2 is two, but I've got one lot of 0.

2 less, so I can record this as an equation.

10 lots of 0.

2 minus one lot of 0.

2 is equal to nine lots of 0.

2 or in this case 1.

And of course, all of this can be applied to our context of centimetres and metres that we've been working with, haven't we? So here, Jacob is saying that, "I'm making a steep slope from my remote control car to drive down.

I've used seven pieces of wood," and each one of those pieces has a value of 25 centimetres.

They're 25 centimetres long.

So how long is the slope altogether? Hmm.

Have a think about how you might tackle and find out how long this slope is.

Well, the way I was thinking about it was that I know that eight lots of 0.

25 would be two metres and I need to find out seven lots of 0.

25.

So what I'm going to do is work out eight lots of 0.

25 metres and I'm gonna minus one lot of 0.

25 metres from that and that would give me a total of 1.

75 metres.

Did you do something similar or did you do it slightly differently? That's okay, it's good to have a range of strategies in order to tackle this kind of problem.

Right, our final checks for understanding today then.

2 is made up of A, B, or C? Take a moment to think.

That's right, it's A, isn't it? 1.

2 is made up of five lots of 0.

2 and one lot of 0.

Five lots of 0.

2 would be one whole and an additional lot of 0.

2 would be 1.

Here is our second check for understanding.

Give the missing number in the sequence.

That's right, the missing number is 3.

75 and we can see that because our numbers are increasing by 0.

25 each time.

So 3.

25, 3.

5, and 3.

75, then four.

Well done if you spotted that.

Okay, onto our final task for today.

What I'd like you to do is fill in the missing numbers here for the examples for A, B, and C here.

And what I'd like to think about here for question two is how many different equations can you write for using common partitions of one where 2.

75 metres is the whole.

And I've given you an example here, I've written that 2.

75 metres could be equal to 11 lots of 0.

25 metres.

Good luck with that and I'll see you back here shortly.

Okay, welcome back.

Let's fly through these answers then, shall we? The first missing number is one.

The second number is 0.

25.

The fourth number is four.

Four lots of 0.

25 is equal to one.

The next number is 1.

25, because four lots of 0.

25 is one and an additional one lot of 0.

25 would be 1.

25.

And then the next missing number is four, because six lots of 0.

25 is equal to four lots of 0.

25 plus two lots of 0.

25.

The missing numbers were three.

5 minus 0.

5 would be equal to four and 4.

5 minus 0.

25 minus 0.

25 would be also equal to four.

What did you notice there? That's right.

Minusing 0.

5 is the same as minusing two lots of 0.

25, isn't it? Okay, onto our last task here as well then, how many different equations could we write where 2.

75 metres was the whole? And only using our common partitions.

Well, here are some examples that Andeep has come up with.

Andeep has said that 2.

75 metres is equal to 12 lots of 0.

25 minus one lot of 0.

25 metres.

And then finally, 2.

75 metres divided by 0.

25 metres.

So 2.

75 metres divided into groups of 0.

25 metres would be equal to 11.

That would be 11 groups.

Well done if you managed to come up with those examples or some of your own as well.

Okay, that's the end of our lesson for today.

Hopefully you're feeling a lot more confident about how different lengths can be composed, both additively and multiplicatively.

Let's run through some of the key points we've been thinking about in our lesson today.

One can be composed of two, four, five, 10, and 100 equal parts.

Common partitions to one can be recorded additively, and there are some examples below there.

Common partitions to one can be recorded multiplicatively.

And again, there's another two examples there for you.

Common partitions to one can be extended beyond one whole, can't they? So it doesn't just work between zero and one.

It could work between any two consecutive numbers.

And common partitions to one can be used within measures of length as well.

So the whole time today, we've been applying our learning back to our unit of measures for length, so centimetres and metres.

Thanks for learning with me today.

I've enjoyed myself.

Hopefully you have too and I hope to see you again soon.

Take care.

I've finished the video