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Hello, how are you today? My name is Dr.

Sharrock.

and I'm really excited to be learning with you today.

We're gonna have great fun as we move through the learning together.

Today's lesson is from our unit "Calculating with Decimal Fractions." The lesson today is all about using known multiplication facts and unitizing to multiply tenths with whole numbers.

As we move through the learning today, we will deepen our understanding of what tenths are, and then we will use this knowledge to help us multiply tenths with whole numbers.

Sometimes new learning can be a little bit tricky, but I know if we work really hard together, then we can be successful.

And I am here to help you if you need it.

So, should we get started then? How do we use our known multiplication facts and unitizing to multiply tenths with whole numbers? This is the key word that we will use throughout our learning today.

The key word is tenth, and I'm sure you've heard it before, but let's practise saying it anyway.

My turn, tenth.

Your turn.

Lovely.

And a tenth is when we have one part in 10 equal parts.

We will start today's lesson thinking about tenths and really deepening our understanding of what they are.

We have Lucas and Sofia to help us today.

Let's have a look at this.

Lucas's mum has made a cake for Lucas and his friends to share during his birthday party.

That looks a tasty cake, doesn't it? "There are 10 of us," Lucas is saying, "so the cake will need to be cut into 10 equal slices." And that's right Sofia, each slice will be 1/10 of the whole because 1/10 is one part in 10 equal parts.

10 of the slices would make the whole cake, so 10 one-tenths must be equal to one whole.

Let's practise counting in tenths using our number line.

And Sofia is reminding us that we can count up in different ways.

So first, let's count up using the fraction vocabulary of tenths.

And we're gonna have a practise counting up and counting down.

So keep an eye out what happens with that number line.

Let's start.

We start with zero, 1/10, 2/10.

Can you continue? What would be next? That's right, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10.

Oh, what would be next? What would come after 9/10? That's right, it would be 10/10, and 10/10 is the same as one whole.

Let's keep going, 11/10, 12/10, 13/10, 14/10, 13/10, 12/10.

Ooh, did you spot that? I switched to counting down now, counting backwards.

11/10, 10/10, 9/10.

Let's swap back again, shall we, and keep counting up.

10/10, 11/10, 12/10, 13/10, 14/10, 15/10, 16/10, 17/10, 18/10, 19/10.

Ooh, what would be next, you think, after 19/10? That's right, it would be 20/10, and 20/10 is the same as two wholes.

So after 20/10, it would be 21/10.

I wonder if you can keep going.

26/10, 27/10.

Let's count backwards again.

26/10, 25/10, 24/10, 23/10.

And forward, 24/10, 25/10, 26/10, 27/10, 28/10, 29/10.

And what would come next? That's right, 30/10, and 30/10 is the same as three wholes.

But we can also count up using decimal fraction vocabulary.

We start with zero, and we've got 0.

1, 0.

2.

So 0.

2 is the same as 2/10.

It's just a different way of saying the number.

So 0.

2, let's keep going.

Can you continue this? 0.

3, 0.

4, 0.

5, 0.

6, 0.

7, 0.

8, 0.

9.

Ooh, what would be next? Would it be 0.

10? No, 0.

10 is the same, or we don't really say 0.

10.

It would be 0.

10, which would be the same as 1/10.

After 0.

9, we get to one.

Let's keep going, 1.

1, 1.

2, 1.

3.

Let's switch it and go backwards, 1.

2, 1.

1, one, 0.

9, 0.

8.

And forwards again, 0.

9, one, 1.

1, 1.

2, 1.

3.

Can you keep going? 1.

9.

What would be next? That's right, not 1.

10.

It would be two.

And keep going, 2.

1.

2.

2.

Keep going.

2.

7, 2.

8, 2.

9, and three.

Fantastic counting.

We could also practise counting up in tenths using a Gattegno chart.

And again, we could count up in decimal fraction vocabulary, 0.

1, 0.

2, or we could count up in fraction vocabulary, 1/10, 2/10.

What should we do? Let's count up in tenths, shall we? Are you ready to count with me? We'll start with 1/10, 2/10, 3/10.

Can you continue? 8/10, 9/10.

Ooh, what would come next after 9/10? That's right, it would be 10/10, which is equivalent to one.

Let's keep going.

1 1/10, 1 2/10.

Keep going.

1 7/10, 1 8/10, 1 9/10.

What would be next? That's right, it would be two.

Are we ready to keep going? 2 1/10, 2 2/10, 2 3/10.

Keep going.

2 9/10, and then three.

Well done.

Lovely counting.

Let's check your understanding.

Could you use the number line to complete the sequences? Maybe talk to somebody about this.

Pause the video while you do this.

And when you're ready to go through the answers, press play.

How did you get on? For sequence a, we had 1.

7, 1.

8, 1.

9.

So we were counting up in one-tenths.

So after 1.

9 would be two, or 2.

0, and then 2.

1.

For sequence B, this time we were counting down in tenths, weren't we? 12/10, 11/10, 10/10, and 9/10.

How did you get on? Well done.

What then do we know about the composition of tenths? So what are tenths made from? Let's look at 0.

3.

We could say 3/10.

So 0.

3 or 3/10 is composed of three one-tenths, and we can write that as an addition equation.

1/10 add 1/10 add 1/10 is equal to 3/10, or 0.

1 added to 0.

1 added to 0.

1 is equal to 0.

3.

We could say there are three 0.

1s There are three one-tenths, which are equal to 3/10 or 0.

3.

What about 1.

8? What's 1.

8 composed of? Ah, thank you Sofia.

1.

8 is composed of eight one-tenths.

What do you think? Do you agree with Sofia? Ah, Lucas doesn't.

"I respectfully challenge you," he says.

Why might he be challenging her? There is an eight in the tenth column, but 1.

8 is actually composed of 18/10.

Hm.

Ah, Sofia understands now.

One whole, we must remember, one whole is composed of 10/10.

And there's another 8/10, so that is 18/10 in total.

And we can see my bar model.

1.

8 is composed of 10/10 and another 8/10, which is 18/10 in total, not just 8/10.

Let's check your understanding on that.

2.

3 is composed of 3/10.

Is that true or false? Pause the video while you have a think about that, and then when you are ready, press play.

How did you get on? Did you say that was false? It's not composed of 3/10, but why not? Is it because, a, the two is a whole number, which are ones and not tenths? Or is it b, one whole is composed of 10/10, so two wholes will be composed of 20/10, and we have another three more tenths.

This is 23/10.

Pause the video.

Maybe find someone to chat to about this.

See if you can agree.

And when you are ready for the answers, press play.

How did you get on? Did you say it must be b? 2.

3 is not composed of 3/10.

That's because one whole is composed of 10/10, and two wholes will then be 20/10 and another three.

So we've got 23/10 in total.

2.

3 is composed of 23/10.

Your turn to practise now.

For question one, could you complete the equations? And for question two, Sofia says that 30/10 is written as 0.

3.

Do you agree or disagree with her? And could you convince me that you are correct? Pause the video while you have a go at both questions, and when you are ready for the answers, press play.

Let's see how you got on.

You were asked to complete the equations, and you might have noticed that you could use your known facts to help you.

For example, we know 3/1 and 7/1 are equal to 10/1, so 3/10 and 7/10 must be equal to 10/10, which is equal to one.

7/1 and 13/1 is equal to 20.

So 7/10 and 13/10 would be equal to 20/10, which is the same as two.

And similarly, for the second set of equations, you could use known facts.

And we know 4/1 and 6/1 are equal to 10/1, so 4/10 and 6/10 must be equal to 10/10, which is equal to one.

And similarly, 4/1 and 16/1 are equal to 20/1.

So 4/10 and 16/10 are equal to 20/10, which is equal to two.

And then we know one is composed of 0.

9 and 0.

1, 9/10 and 1/10.

And two must be composed of 9/10 and 11/10, or 0.

9 add 1.

1.

And the same for subtracting.

We can use our known facts.

We know 10/1 subtract 4/1 is equal to 6/1, so 10/10 subtract 4/10 must be equal to 6/10.

And then the opposite to that, one subtract 0.

6 must be equal to 0.

4.

Two subtract 0.

6 would be 1.

4.

And two subtract 1.

6 must be equal to 0.

4.

And 1.

1, well, we've got 11/10 there.

If we subtract another tenth, we get left with 10/10, or one.

11/10 subtract 3/10 would be 8/10.

And then 0.

8, well, that is equal to one, or 10/10, subtract 2/10.

And 0.

5 would be equal to one subtract 5/10, or 10/10 subtract 5/10.

Question two.

Sofia says that 30/10 is written as 0.

3.

Did you agree or disagree? You might have said you disagree with Sofia and given a reason like there are 10/10 in one whole, and we have 30/10.

30/10 are equivalent to three wholes.

So 30/10 should be written as three, not 0.

3.

How did you get on with both those questions? Well done.

Fantastic learning so far.

I can see how hard you are trying, and you have really made good progress on deepening your understanding of what tenths are.

We're going to move on now and look at multiplying with tenths, so using all the knowledge we know about tenths to help us.

Let's look at this.

Sofia is building a tower, and each block that she uses is four centimetres tall.

Can you visualise that? What does that look like? Can you see a tower with a block that is four centimetres tall? That's what it looks like to me in my head.

What might the question be then, do you think? Ah, that's right, how tall could her tower become? So not how tall is her tower because we don't know how many blocks she's used, but how tall could it become? So let's start by representing this on a number line, and we can write some equations to match.

So if she has no blocks, well, no fours are equal to zero.

If she has one block, one four is equal to four.

If she had two blocks, which is four add another four, two fours, that would be eight.

If she had three blocks, we would add another four.

Three fours are equal to 12.

And we could continue adding blocks.

So this is what we could have written.

What do you notice? That's right, the height of Sofia's tower could be any multiple of four 'cause we didn't say how many blocks, so we just kept adding a block each time.

And each block is worth four centimetres.

And we know that multiplication is commutative, so we could also write the equations like this.

Instead of zero times four equals zero, we could write four times zero equals zero.

Lucas is also building a tower.

Each block that he uses is 0.

4 centimetres tall.

Hm, do you notice what's the same and what's different there between his block and Sofia's block? That's right, his block is 0.

4 centimetres tall, 4/10.

Hers was four whole centimetres.

So what about his tower then? How tall could his tower become? Can you visualise that? That's right, he's using different blocks, isn't he? His blocks are smaller.

They are only 0.

4 or 4/10 of a centimetre tall.

Let's start by representing this on a number line and writing the related equations.

If he has no blocks, well, his tower would be zero centimetres tall.

If he has one block of 0.

4, it would be 0.

4 centimetres tall.

If we had two blocks, we would add another 0.

4.

Two 0.

4s are equal to 0.

8.

If he has three blocks, we would add another 0.

4.

Three multiplied by 0.

4 is equal to 1.

2.

Three lots of 4/10 are 12/10, and 12/10 are the same as 1.

2.

And we could keep adding blocks of 0.

4 centimetres each time.

What do we notice then about how tall Lucas's tower could be? That's right, it could be any multiple of 4/10 'cause each block is 4/10, or 0.

4, centimetres.

We know multiplication is commutative, so we could write the equations this way round as well.

Let's compare the multiples of four and 0.

4.

The multiples of four are how tall Sofia's tower could be.

And the multiples of 0.

4 are how tall Lucas's tower could be.

What do you notice about the multiples? Is there something that is the same? Is there something that is different? That's right, let's have a look at this example.

We can see three fours are equal to 12.

So three four-tenths are equal to 12/10.

Lucas and Sofia look at this equation, seven times 0.

5.

And we can use our stem sentence.

Seven times 5/1 is equal to 35/1, so seven times 5/10 is equal to 35/10.

35/10 is equivalent to 0.

35.

Ooh, why is that Sofia? Sofia is respectfully challenging Lucas.

Sofia is explaining that there are 10/10 in one whole, so 35/10 is equal to three wholes and 5/10.

So we need to write it as 3.

5, not 0.

35.

Let's check your understanding.

Could you complete the equations? Use the stem sentence to help.

Pause the video while you have a go, and when you're ready to go through the answers, press play.

How did you get on? Did you say six times 4/1 is equal to 24/1? So six times 4/10 is equal to 24/10, and 24/10 is equal to 2.

4 because 10/10 is equal to one, so 20/10 must be equal to two, and we have another 4/10.

How did you get on with that? Well done.

Let's look at this using arrays.

You can see here a representation of three groups of four, and we know three multiplied by four is equal to 12.

And we can say that three multiplied by four ones is equal to 12 ones.

But what if each counter has the value of 0.

1? We can see there are three groups of 0.

4, which is equal to 1.

2.

Three multiplied by 4/10 is equal to 12/10.

And we can say our stem sentence to help us.

Three times 4/1 is equal to 12/1, so three times 4/10 is equal to 12/10.

Let's check your understanding on this.

Could you use the arrays to help complete the equations? Pause the video while you have a go, and when you are ready to go through the answers, press play.

How did you get on? Did you work out that four times seven is equal to 28? So four times 7/1 must be equal to 28/1.

Then we must know that four times 7/10 is equal to 28/10, and four multiplied by 0.

7 is equal to 2.

8.

How did you get on with those? Well done.

Your turn to practise now.

For question 1a, could you fill in the missing numbers? And for part b, could you fill in the missing symbols less than, greater than, or equal to? For question two, could you solve these problems? Part a, Lucas is making a necklace of beads.

Each bead is 0.

6 centimetres long.

How long could it be? Tick the possible lengths and convince me that you are correct.

For part b, it takes 0.

3 kilogrammes of flour to make one cake.

How much flour is needed to make seven cakes? And for part c, Sofia has a blue ribbon that is five metres long.

She also has a red ribbon that is 3/10 times the length of the blue ribbon.

How long is the red ribbon? And for question three, could you look at this statement that Lucas has made? Do you agree or disagree? And convince me that you are correct.

So Lucas's statement is, "I know seven sevens are equal to 49, so 0.

7 times seven must be equal to 0.

49.

Pause the video while you have a go to all those questions, and when you are ready to go through the answers, press play.

How did you get on? Let's have a look.

First for question 1a, you are asked to fill in the missing numbers.

Four eights are 32, so four eight-tenths must be 32/10, or 3.

2.

Three sixes are 18, so three six-tenths must be equal to 18/10, which is 1.

8.

Five multiplied by nine is equal to 45.

Five multiplied by 9/10 would be 45/10, which is 4.

5.

Again, five multiplied by nine is equal to 45, and 0.

5 multiplied by nine, or 5/10 multiplied by nine, will be 45/10, which is equal to 4.

5.

Then we have 11 multiplied by 4/10 is equal to 44/10, or 4.

4.

5/10 multiplied by seven is 35/10, which is equal to 3.

5.

Zero multiplied by 0.

9, well, that's equal to zero, isn't it? Remember anytime we multiply by zero, the product is equal to zero.

Then 4/10 multiplied by 10 is 40/10, which is equal to four.

For part b, you were asked to fill in the missing symbols.

Five multiplied by 4/10 must be less than six multiplied by 4/10.

We've only got five four-tenths rather than six four-tenths, so it's one four-tenth less, isn't it? Eight multiplied by 9/10 must be less than nine multiplied by 9/10.

Seven multiplied by 2/10 is greater than five multiplied by 2/10.

Six multiplied by 8/10 is equal to eight multiplied by 6/10.

They are the same because multiplication is commutative.

Four multiplied by 3/10, if we add another 3/10, well, that would be equal to five multiplied by 3/10.

Did you notice that you did not need to actually calculate any of these? You could have filled in the missing symbols using your number sense and reasoning skills.

If the multipliers are the same, we can just compare the multiplicand.

An understanding of the commutative property of multiplication and the distributive law would help as well.

For question two, we had to solve some problems. The first problem was about how long a necklace could be.

Each bead is 0.

6 centimetres long.

We can use our six times-table to help us here to help us identify multiples of 0.

6.

That would be 3.

6 centimetres, six centimetres, and 4.

8 centimetres.

You might have said that the length of the necklace would have to be a multiple of 0.

6 because the necklace is composed of equal parts of 0.

6.

To identify multiples of 6/10, we can use the six times-table.

Six sixes are equal to 36, so six times 6/10 must be 36/10, which is 3.

6.

10 multiplied by six is equal to 60.

10 multiplied by 6/10 is equal to 60/10, which is equal to six.

And then eight sixes are 48, so eight six-tenths are 48/10, which is equal to 4.

8.

For part b, we had a problem about some flour, and we needed to make seven cakes, so I'm gonna have to form an equation.

Three multiplied by seven is equal to 21, and I can use that to help me.

3/10 multiplied by seven is equal to 21/10, which is equal to 2.

1.

We could also have said that seven multiplied by three is equal to 21, and seven multiplied by 0.

3 is equal to 2.

1.

2.

1 kilogrammes of flour is needed to make seven cakes.

Question c was a question about ribbons.

Sofia has a blue ribbon that is five metres long and a red ribbon that is 3/10 times the length of the blue ribbon.

We can use that information to form an equation.

We know five times three is equal to 15, so five times 3/10 is equal to 15/10, or 1.

5.

The red ribbon is 1.

5 metres long.

For question three, you are asked to say if you agree or disagree with Lucas's statement.

You might have said that you agree with Lucas that seven sevens are 49.

However, you might have disagreed that 0.

7, or 7/10, times seven was equal to 0.

49.

You might have reasoned that if seven sevens are equal to 49, then 7/10 times seven is equal to 49/10, and that is 4.

9, not 0.

49.

How did you get on with those questions? Well done.

Fantastic learning today, everybody.

You should be really proud of how hard you have tried and how much progress you have made with multiplying tenths with whole numbers.

We know that there are 10/10 in one whole, and we can use a stem sentence, mm times mm ones is equal to mm ones, so mm times mm tenths is equal to mm tenths.

And that supports us to multiplying tenths.

Really well done today.

I have had a lot of fun learning with you, and I look forward to learning with you again soon.