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Hi, my name's Mr. Peters.

Thanks for joining me today.

I'm really looking forward to starting our learning.

Today, we're gonna be thinking about decimal numbers with tenths and how these can be formed using our multiplicative understanding.

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

Hopefully, by the end of today, you'll feel confident in explaining that decimal numbers can be represented as multiplication.

For this lesson, we've got some key words that we're gonna be keeping an eye on and thinking about throughout.

The first one is decimal number.

The second one is equation.

The third one is factor, and the fourth one is product.

So a decimal number is a number that has a decimal point in it.

And any numbers that are placed after decimal point are representing the fractional parts of a number.

An equation is used to show that a number, a calculation or an expression is equal to one another.

A factor is a word that you may have heard before and can be used to describe numbers which divide exactly into another number.

And finally, product is also another word that you may have heard before.

The product is the result of two numbers that multiply together.

When we write a multiplication equation, the parts of that equation we can represent are as follows.

Factor multiplied by a factor is equal to the product.

So this lesson today is broken down into two parts.

The first part, we'll think about decimal numbers less than one as multiplication.

And the second part of our lesson, we'll think about decimal numbers greater than one as multiplication.

Let's get going with the first part.

Throughout this lesson, you might see Alex and Andeep pop up to help us with our thinking as we go.

So let's get started.

Here, we have one whole, which we've now divided into 10 equal parts, which Alex has just pointed out.

He says that the whole has been divided into 10 equal parts and each part has a value of 1/10.

Let's look at our whole now.

Ah, well now we've got three of those parts that have been shaded.

So if three parts have been shaded, that means three tenths of the whole have been shaded or we have three tenths.

This can be written as 0.

3 = 0.

1 + 0.

Hmm, look at the whole this time.

What do you notice? Ah, actually most of the whole is shaded.

There's only one part that isn't shaded.

So that must mean we have nine tenths shaded this time.

I wonder what that would look like as an addition equation.

Well, we could write it as 0.

9 = 0.

1 + 0.

I think you can get the picture.

We're gonna have nine 0.

1s there, aren't we? And as Alex has just pointed out, he says, "That's gonna take us a long time to write it down.

There must be a quicker way of doing that." Thankfully, as Andeep just pointed out, there is a quicker way we can write that down.

"We can write it as a multiplication equation.

We can write nine groups of 0.

1." So here, you can see our equation now is 0.

9 = 9 x 0.

We could also write it as 0.

1 nine times.

So again, 0.

9 = 0.

1 x 9.

Let's have a look at these equations a little bit more carefully now.

What do the nines in our equations represent? Have a little think.

The nine represents a number of parts that have been shaded or the number of groups that have been shaded.

We have nine groups and this is a factor in our equation.

What does a 0.

1 represent? Well, the 0.

1 represents the value of each one of those parts or each one of those groups, and this is also a factor in our equation.

Finally, what does a 0.

9 represent? The 0.

9 represents a total amount.

In this case, a total amount that has been shaded.

9 in our equation is the product.

Now we're more familiar with how we can write multiple lots of one tenth as a multiplication equation.

Let's have a little bit more a think about it in a different context.

Here, we've got a jug with some water in it.

If you look at the scale of our jug, when the water reaches the top of the scale, that would be one litre which has been filled into the jug.

Therefore, each one of those parts represents 1/10 of the litre.

Let's have a look at how much of water is in our jug.

Yeah, I can see four tenths of a litre of water have been poured into our jug.

We can represent this as 0.

4 = 4 x 0.

1 of a litre or we can write this as 0.

4 = 0.

1 litres x 4.

So again, what does the four represent this time in our equation? Well, the four represents a number of groups again and this is a factor in our equation.

What does the 0.

1 litre represent? Well, the 0.

1 litre represents the value of each group and this is also a factor again in our equation.

4 litres is the product of our equation.

Okay, time for us to check your understanding now.

I'm gonna have a go first and then you're gonna have a go afterwards.

Have a look at our number line on the left-hand side, it starts at zero metres and it finishes at one metre and there are 10 equal parts throughout.

So each part is 1/10.

How many of those parts has been shaded? That's right.

Three of those parts has been shaded.

So that would represent three tenths.

How do we write this as an equation then? Well, I could write this as 0.

3 is equal to three groups of 0.

1 metre.

So we could write this as 0.

3 = 3 x 0.

1 metres.

I could also write this as 0.

3 = 0.

1 metres x 3.

And we can do that with multiplication, can't we? Because we know that multiplication is commutative.

Now it's your turn.

Have a look at number line I've got for you on the right-hand side.

Have a go at writing those equations.

Good luck and I'll see you in a second.

Welcome back.

How did you get on? Hopefully, you realise that actually there were five parts shaded, so we could write this as 0.

5 = 5 x 0.

1 metres or we could write this as 0.

5 = 0.

1 metres x 5.

You may have written it where the expressions or the number on either side of the equal sign were rotated to the other side and that is also fine.

Okay, onto our first task today then.

The first task is asking you to match the pictures to the correct equation.

Down the left-hand side you've got four pictures, and down the right-hand side, we've got four equations.

You need to draw a line from each one of those pictures to the equation which you think represents that picture.

Your second task is to complete the table by writing in either the additive or multiplicative equation that's missing.

Down the left-hand side of the table, you'll see additive equations, and down the right-hand side of the table, you'll see it's corresponding multiplicative equations.

You need to fill in the boxes and write the missing additive or multiplicative equation.

Have a good go at that and I'll see you again shortly.

Okay, welcome back.

Let's see how you got on compared to our answers.

Let's have a look at the top picture, first of all.

It's a circle that's been divided into 10 equal parts and two of those parts has been shaded.

So that would match to this equation here, 0.

1 x 2 = 0.

The second picture is one whole, and that again has been divided into 10 equal parts and there are eight parts that have been shaded.

So I'm looking for an equation which has a number eight in it to represent the number of groups.

And I'm looking for an equation that has 0.

1 as a factor and 0.

8 then as the product.

Have you managed to spot that one? Yeah, that's correct.

It was the bottom one.

8 x 0.

1 = 0.

The third picture is a number line representing one metre and that metre has been split into 10 equal parts and three of them are shaded.

And that picture represents the equation at the top, 0.

3 = to 3 x 0.

1 metres.

And finally, the last image at the bottom, we've got some place value counters.

There are three lots of one tenth.

Hmm, which equation's that? It's very similar to the last equation.

Although, the last equation was in metres and this one isn't.

So this picture matches to this equation here, which is 0.

3 = 0.

1 x 3.

Well done if you've got those.

Task two then was asking us to either convert from an additive equation to a multiplicative equation or from a multiplicative equation to an additive equation.

And the first two were done for you.

Let's have a look at the second row, 0.

2 = 0.

1 + 0.

That's two groups of 0.

So we need to represent that as 0.

2 = 2 x 0.

In the third row, we're given the multiplicative equation, 0.

3 = 3 x 0.

That's three groups of 0.

So we could record that as 0.

3 = 0.

1 + 0.

In the fourth row, we've got 0.

4 = 4 x 0.

While in the previous example, we had three 0.

1s.

This time, we're gonna have four 0.

1s.

So we've record this as 0.

4 = 0.

1 + 0.

We've got five groups of 0.

1 here, so we could record this as 0.

5 = 5 x 0.

Fantastic, let's move on to the second phase of our learning now where we're gonna start thinking about decimal numbers greater than one and how they can be written multiplicatively.

Have a look at our bar model this time.

Well, as Andeep was pointing out, that's a lot of tenths again, isn't it? How many tenths are there altogether? Well, there are actually 18 tenths altogether.

And if we were to write that as an addition equation again, that would be a long equation to write, wouldn't it? So it would be entirely appropriate for us to write it as a multiplication equation.

Have a little think for yourself.

How do you think you could record this as a multiplication equation? Maybe write it down if you can.

Andeep's got some ideas for how we could have recorded this.

Andeep is saying that 1.

8 can be divided into 18 groups of 0.

So we could record that as 1.

8 = 18 x 0.

We could also record this as 18 groups of 0.

1 make 1.

So as you can see, we've got 18 x 0.

1 = 1.

Let's look at our equations again more carefully.

What does the 18 represent in our equations? Well, again, the 18 represents the number of groups, doesn't it? In our bar model, that is also a factor.

What does the 0.

1 in our equations represent? Well, the 0.

1 represents the value of each one of those groups.

And again, the 0.

1 is a factor in our equation.

And finally, what does the 1.

8 represent in our equation? We know that the 1.

8 is the product of our equation and the product here is representing the total amount.

Alex is starting to think about how we could take this a little bit further.

Alex knows that 1.

8 could be written as 18/10.

And you can see this here in our bar model now where the one in 1.

8 represents 10/10 or one and the eight in 1.

8 represents 8/10 or 0.

So as Alex is quite rightly pointing out, 10/10 + 8/10 = 1.

And so in our equations now, we've changed it slightly so that 1.

8 can be represented with a fraction of 18/10 and the 0.

1 can be represented with a fraction of 1/10.

So so far, we've been writing our equations using multiplication only and now, we can start to think about actually how we could write these equations as multiplication and addition.

Have a look at our bar model.

We still have 1.

8 as the whole and we also have 18 lots of 0.

What'd you notice has happened now? Well, we've actually got 10 tenths that have been shaded and we know that 10/10 is equal to one and we've got a remaining eight tenths that haven't been shaded and we know that is equal to eight lots of 0.

1 or 8 x by 0.

So we can write this as an equation, 1.

8 = 1 + 8 x 0.

1 or we could write it as 1 + 8 x 0.

1 = 1.

What do each of these numbers in the equations represent? Let's start with the one.

Well, we know in the equation that we've just written that the one represents the one whole or the 10/10 in our bar model at the top.

And to that one whole, we can add eight lots of 0.

So we can add 8 x 0.

1, which therefore means that 1.

8 is equal to 1 plus 8 lots of 0.

1 or how we'd write that in the equation as 8 x 0.

Alex is now wondering if we could write it a different way this time.

Have a look carefully.

How many have been shaded this time? Well, this time, we've shaded eight tenths and we know that we can represent 8/10 as 0.

That also means, we've got 10 tenths remaining.

And instead of using a one this time, we can represent that as 10 x 0.

And if we combine the 0.

8 and the 10 groups of 0.

1 together in an equation, we can write 1.

8 = 0.

8 + 10 x 0.

We could also write it as 0.

8 + 10 x 0.

1 = 1.

What do each of the numbers again represent in our equation? Well, the 0.

8 represents the eight tenths, doesn't it? And the 10 represents the 10 groups that haven't been shaded and each one of those groups is 0.

So the 10 multiplied by 0.

1 represents those 10 tenths that have not been shaded in our bar model.

Okay, time for us to have another go.

I'll have one more go and then you can have a practise as well.

Let's look at our bar model this time.

The whole is 1.

2 for my example, that means there are 12 lots of one tenth.

I can write this as 1.

2 = 12 x 0.

1 or I can write it as 1.

2 = 0.

1 x 12.

I could then extend this and swap the positions of the expressions and numbers on either side of the equal sign.

And that's what we've done underneath here.

I've now rotated it so that 0.

1 x 12 is on the left-hand side of the equal sign and the 1.

2 is on the right-hand side of the equal sign.

And again, I've done that underneath as well.

So I've got 12 x 0.

1 = 1.

Have a look at your example now on the right-hand side.

Can you ever go at writing four corresponding equations like I have done? Good luck.

Okay, welcome back.

Hopefully, you got an okay with that and realise that you could represent it as 1.

4 = 14 x 0.

If you then took both of those equations and rotated the expressions and numbers to either side of equal sign, you could write it as this, 0.

1 x 14 = 1.

4 and 14 x 0.

1 = 1.

Well done if you've got that.

Now I've got one more task for you.

We've got the same numbers in our bar models again.

What do you notice has changed this time? Ah, in the last bar models, they were all shaded whereas now, we've shaded some of them.

So we're gonna have a go at writing an equation using both multiplication and addition again.

I'll have a go first on the left-hand side.

As you can see, my whole is still 1.

2 and I've got 10 tenths that are shaded in green and two tenths that are not shaded in green.

So I could represent this as 1.

2 = 10 x 0.

1, which is same as saying 10 groups of 0.

Plus the remaining 0.

2 or those two tenths that haven't been shaded.

I could also write it as 1.

2 = 0.

2, that's those two tenths that aren't shaded, + 0.

1 x 10.

We know that in multiplication, you can rotate the factors due to the commutative law and it doesn't matter which order they are in.

So this time, I've got the 0.

1 coming first and then I'm multiplying that by the 10.

And the 0.

1 multiplied by 10 is representing those 10 green shaded tenths in my bar model.

Instead this time, I could actually choose to take the 10 tenths and write that as a one instead.

We know that 10/10 is equal to one whole.

So here, I could write it as 1 whole plus 2 multiplied by 0.

1 and the 2 multiplied by the 0.

1 represents those two unshaded tenths at the end of our bar model.

All of that is equal to 1.

Also, I could represent it as 2 x 0.

1, which again represents the white parts, plus one whole, which represents the green parts, the 10 green shaded parts.

And all of that again is equal to 1.

Now finally, it's your turn.

You have a look at your bar model again.

See if you can have a go at writing equations which combine both addition and multiplication.

Okay, well done.

You may have written 1.

4 = 10 x 0.

1 + 0.

4 or you may have written 1.

4 = 0.

4 + 0.

1 x 10.

Underneath that, you may have written 1, which again represents those 10 green shaded tenths, + 4 x 0.

And the 4 x 0.

1 represents the white parts that have not been shaded.

All of that is equal to 1.

4 or you could have written as 4 x 0.

1 + 1 whole to represent those 10 tenths.

And that, again, is equal to 1.

Well done if you managed to get some of those.

Okay, into our second task for today now.

I want you to have a look at these images here on the left-hand side.

We've got A, B, and C.

And I think you can record these in two different ways, at least for each equation using multiplication.

So on the right-hand side, I'd like you to write two multiplicative equations to represent each image of tenths on the left-hand side.

Task two gives you another bar model here similar to what you've seen already.

And I'm wondering how many different equations do you think you could write to represent this bar model? There's a challenge for you.

It'll be really interesting to see what you can come up with.

Good luck and I'll see you again shortly.

Okay, let's see how you got on.

On the left-hand side at the top, we have 1.

3 and that has been broken down into 13 parts and each one of those parts represents 1/10.

You may have recorded that as 1.

3 = 0.

1 x 13 or you may have written it as 1.

3 = 13 x 0.

1 using the commutative law to find two different ways of writing that.

The second image has one tenths frame full of tenths and an additional five tenths.

So that means, we have 15 tenths altogether.

We can record that as 15 x 0.

1 = 1.

5 or you could record that as 0.

1 x 15 = 1.

And finally the bottom one, it's not so much of a picture, more a set of numbering words written as 24 tenths.

And we can record this as 24 x 0.

1 = 2.

4 or we could record it as 2.

4 = 24 x 0.

Well done if you managed to get all of those.

Okay, and for task two, I wonder how many different ways you were able to represent this bar model.

Here, I've represented it as 1.

7 = 0.

1 x 17 or I've represented it as 1.

7 = 17 x 0.

I then went on to consider the different shadings of the parts.

And this time, I realised that 10 white parts had been unshaded, so I could represent that with a one.

And then the seven remaining green parts that have been shaded could be represented as 7 x 0.

So my equation says, 1.

7 = 1 + 7 x 0.

And finally, I could have gone on to write it like this.

Instead this time, I combined the seven one tenths and wrote that as 0.

So here, I've got 1.

7 is equal to 0.

7 plus 10 lots of 0.

So I've written that as 10 x 0.

Well done if you managed to get any of those.

Andeep is just finally adding that we could have written it using our fractional notation as well instead of our decimal numbers.

So again here, Andeep has written an equation as 1 and 7/10 = to 1/10 x 17.

Gosh, that's a different way of thinking about it again, isn't it? And that means we could have written lots more equations.

I wonder how many equations could you potentially write now? Maybe go off if you get an extra chance to and write a few more equations to represent this bar model.

Okay, that's it for our learning today.

Well done for sticking along and hopefully, you feel a bit more confident about representing decimal numbers using multiplication equations.

So let's just recap over some of the key points from our learning today.

Firstly, we thought about decimal numbers and how they could be represented as an equation using either repeated addition or multiplication.

And we'd often use multiplication to make things easier for us 'cause writing some repeated addition equations could take a long time.

When writing 'em as a multiplication equation, we then knew that the 0.

1 would act as a factor in our equation.

There's an example there for you, 1.

2 = 12 x 0.

2 is the product and 12 and 0.

1 are factors.

When working with decimals that were greater than one, we could do the same thing and we could also extend that thinking about how we could record it as an improper fraction rather than a decimal number.

So here, instead of saying 1.

2, we could say 12/10 = 12 x 0.

1 or 1/10.

And finally, we were able to represent some of our decimal numbers using both multiplication and addition within an equation, weren't we? And there's an example again at the bottom where we had 1.

2 was equal to one whole or 10/10 plus 2 lots of 1/10, 2 x 0.

And 0.

1 here would act as a factor again.

Thank you for learning with me today.

I've really enjoyed myself and we had to do a lot of hard thinking around how we can represent decimal numbers, didn't we? Well done and I'll see you again soon.

I've finished the video