Lesson video

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

Shark.

I am so happy to be learning with you today.

You have made a great choice to learn maths with me and I'm here to guide you through the learning.

Today's lesson is from our units calculating with decimal fractions.

The lesson is called explain the relationship between multiplying by 0.

1 and dividing by 10.

As we move through the learning today, we are going to deepen our understanding of this connection between multiplying by 0.

1 or one-tenth and dividing by 10.

Sometimes new learning can be challenging, but that's okay.

I am here to guide you and I know if we work really hard together then we can be successful.

So let's find out, shall we? What is this relationship between multiplying by 0.

1 and dividing by 10? These are the key words that we will use throughout our learning today.

We have got unit fraction and tenth, let's practise those words together.

My turn, unit fraction.

Your turn.

Nice.

My turn, tenth.

Your turn.

Fantastic.

And when we talk about a unit fraction, we mean any fraction that has a numerator of one and a tent, well, that's one part in 10 equal parts.

So one-tenth would actually be a unit fraction because its numerate is one.

Let's start our learning today thinking about how we find one-tenth, and we have Lucas and Sofia to guide us through the learning.

Let's look at this problem.

Lucas has a one metre length of string.

Okay, can you imagine what would that look like? Sofia's string is one-tenth times the size of Lucas's string.

Can you visualise that? Does that mean it's longer or shorter if it's one-tenth times the size? Well, we've got my metre ruler here and we can see Lucas's string is one metre long and Sofia's string is one-tenth times the size.

So it must be shorter.

What might our question be though? We don't have a question yet.

That's right.

How long is Sofia's string? So we know Lucas's string is one metre long and we know Sofia string is one-tenth times the length of Lucas's string.

So we need to find one-tenth of one metre and that will tell us how long Sofia's string is.

One-tenth is written as 0.

1.

We need to multiply one metre by 0.

1 that way we would find one-tenth times metre.

When we multiply by one-tenth, the digits move one place to the right.

So one metre multiplied by 0.

1 or one-tenth is equal to 0.

1 metres.

One-tenth of one metre is 0.

1 metres.

The length of Sofia string is 0.

1 metres or one-tenth of a metre.

Lucas now has a two metre length of string.

Sofia has a new piece of string that is one-tenth times the size of Lucas's string.

Can you visualise that? I've got Lucas's string is two metres and Sofia string is shorter than that, is one-tenth times the size of Lucas's.

So how long is Sofia string this time? Well, we need to find one-tenth of two metres this time.

So we need to multiply the two metres by one-tenth, which remember is written as 0.

1 and when we multiply by one-tenth the digits move one place to the right.

Two metres multiplied by one-tenth is equal to 0.

2 metres.

One-tenth of two metres is 0.

2 metres.

One-tenth of two metres is two-tenths of a metre.

So the length of Sofia's string this time is 0.

2 metres or two-tenths of a metre.

And the children then investigate other lengths of string and they work systematically, so in an order, and present their results in a table.

We can see the length of Lucas' string and the length of Sofia's string.

So they've looked at pieces of string that are one metre, two metre, three metre, four metres and five metres.

And we can say one-tenth of three metres is three-tenths of a metre or 0.

3.

One-tenth of four metres is four-tenths of a metre or 0.

4.

One-tenth of five metres is five tenths of a metre or 0.

5.

Can you spot a pattern? Hmm, what might come next? Well we've got one, two, three, four, five.

So what might come next? Got 0.

1, 0.

2, 0.

3, 0.

4, 0.

5 or one-tenth, two-tenths, three-tenths, four-tenths, five-tenths.

What might come next? That's right, we've got six metres next, haven't we? And we can use the stem centres to help us with any of these lengths.

We can say that one-tenth of mm metres is mm 10th of a metre or zero point mm.

So let's say.

We can say that one-tenth of six metres is six-tenths of a metre or 0.

6.

One-tenth of seven metres is seven-tenths of a metre or 0.

7.

But what if the original length of string is 10 metres or more than 10 metres? Well, we can still say the same stem sentence.

One-tenth of 10 metres is ten-tenths of a metre or one.

One-tenth of 11 metres is eleven-tenths of a metre or 1.

1.

It just means that when our length of string is 10 metres or greater, that we get one whole now as well as an amount of tenths.

So one-tenth of 11 metres is eleven-tenths of a metre or 1.

1.

I wonder if you could continue saying that stem sentence for the other length.

So one-tenth of 12 metres is 12 tenths of a metre or 1.

2.

Maybe have a go at saying the rest of those.

To summarise so far, to find one-tenth we can multiply by 0.

1.

To multiply by 0.

1, we move the digits one place to the right.

We can use the stem sentences to support us.

One-tenth of mm metres is mm 10th of a metre or mm.

Let's check your understanding on that.

Lucas has an 18 metre length of string.

Sofia's string is one-tenth times the length.

Complete the stem sentence to determine the length of Sofia's string.

One-tenth of mm metres is mm tenths of a metre or mm.

Maybe talk to somebody about this, compare your answers, pause the video while you do that.

When you are ready for the answers, press play.

How did you get on? Did you say that one-tenth of 18 metres is 18 tenths of a metre or 1.

8? Well done.

Let's have a look at this now.

Lucas has a 10 metre length of string.

Is there another way to find one-tenth? We know we could multiply by 0.

1, but is there another way? To find one-tenth of a number we need to multiply by one-tenth and that's what we've just learned.

10 metres multiplied by one-tenth.

But we can also write one-tenth as a fraction instead of a decimal.

One metre multiplied by one-tenth.

That's our unit fraction, isn't it? It has a numerator of one.

When we multiply by a unit fraction, it is the same as dividing by the denominator of that unit fraction.

What's the denominator? That's right, it's 10, isn't it? So when we multiply by this unit fraction one-tenth, it must be the same as dividing by 10.

10 metres divided by 10.

When we divide by 10, the digits move one place to the right.

10 metres divided by 10 is equal to one metre.

So we have one 10, we now have one one.

One one-tenth of 10 metres is 10 tenths of a metre or one.

So each of these expressions represent the same relationship.

10 metres multiplied by 0.

1 or multiplied by one-tenth or divided by 10, they represent the same relationship.

They all represent ways that we can find one-tenth of 10 metres.

So we could write them all in one long equation.

10 metres multiplied by 0.

1, well, that's the same as or equal to 10 metres multiplied by one-tenth, which is equal to 10 metres divided by 10.

Let's check your understanding with this.

Which equation or equations, so there might be more than one, represent finding the length of a piece of string that is one-tenth times the length of a 20 metre length of string? You've got options A, B, C, and D.

Pause the video, maybe talk to somebody about this and when you are ready to go through the answers, press play.

How did you get on? Did you realise it must be A because 20 metres multiplied by 0.

1.

Well, 0.

1 is the same as one-tenth.

So that's the the same as finding one-tenth times and when we find one-tenth of a number, the digits move one place to the right.

So the answer would be two.

What about B? Well, B is the same starting equation but the answer is 0.

2.

So that can't be correct because there the digits have moved two places not one.

C is correct because we've just learned that multiplying by 0.

1 is the same as dividing by 10 and that is the same as multiplying by one-tenth.

So equations A, C, and D were correct.

How did you get on with those? Well done.

Let's look at this concept now though on a place value grid.

So if we start with 20, we know if we multiply by 0.

1 it's the same as dividing by 10 and what do you notice? That's right, when we divide by 10 or multiply by one-tenth the digits move one place to the right.

We had two tens, we now have two ones.

Let's look at this different number.

We've got 12 on our place value grid, when we divide by 10 or multiply by one-tenth the digits move one place to the right.

One-tenth of 12 metres is 12 tenths of a metre or 1.

2.

Let's look at four.

When we divide by 10 or multiply by one-tenth, the digits move one place to the right.

We had four ones, we now have four tenths.

Four multiplied by 0.

1 is the same as multiply by one-tenth, which is the same as divided by 10.

We had four ones, we now have four tenths.

We need to remember though to write the placeholder in the ones, we can't just write the decimal point and a four, we need that zero in the ones place.

Let's look at this number.

We've got 0.

9.

And if we multiply it by one-tenth, which is the same as multiplying by 0.

1 or di divided by 10, well, the digits move one place to the right.

We had nine tenths, we now have nine hundredths.

What do you notice? That's right.

We need to remember to write those placeholders in the ones and in the tenths.

We can't just write a point and then leave a gap and have a nine.

We need the zero in the ones and the zero in the tenths so we make sure our number looks the correct size.

So let's summarise this.

When a number is multiplied by 0.

1, which is the same as one-tenth, the digits move one place to the right and when a number is divided by 10, the digits move one place to the right.

So they have to be equivalent actions.

When we multiply by 0.

1 and divide by 10, it does the same thing to the number, so they are equivalent.

Let's check your understanding on this.

Could you complete these two related equations that represent finding one-tenth of 67 metres? Pause a video while you do that.

When you are ready to go through the answers, press play.

How did you get on? Did you remember that to find one-tenth we can multiply by 0.

1? The digits would move one place to the right so we would have 6.

7.

And multiplying by one-tenth is the same as dividing by 10.

Again, the digits move one place to the right So the answer would be 6.

7.

When we divide by 10 or multiply by one-tenth, the digits move one place to the right.

It's your turn to practise now.

For question one, could you fill in the missing numbers and could you explain any pattern that you spot? And then if you finish, have a go at making up your own group of equations that have eight similar connection.

For question two, could you complete these equations by filling in the blanks? Pause the video while you have a go at both questions and when you are ready for the answers, press play.

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

So for question one, you had to fill in the missing numbers and explain any pattern that you spot.

Well, 150 divided by 10 is 15.

So 15 divided by 10 must be 1.

5, must be 10 times smaller.

And then we had 700 multiplied by 0.

1 or one-tenth is 70 because the digits move one place to the right.

17 multiplied by one-tenth will be seven.

Seven multiplied by one-tenth will be 0.

7.

And 0.

7 multiplied by one-tenth will be 0.

07 or seven hundredths.

You might have explained that when the multiplicand can becomes 10 times smaller then the product also becomes 10 times smaller if the multiplier remains the same.

You were then asked to make up your own group of equations that had a similar connection and you might have made up a group of equations where the multiplicand can became 10 times smaller like this.

I did mine with 920 and then my multiplicand became 10 times smaller, 92, 9.

2, 0.

92.

I kept the multiplier the same and so that meant that my product would become 10 times smaller.

For question two, you are asked to complete these equations, 50 multiplied by 0.

1 or one-tenth would be five.

So 50 divided by 10 must also be five because they are equivalent actions.

So 3.

1 multiplied by one-tenth would be 0.

31 because those digits will move one place to the right and that is the same as dividing by ten.

Seven multiplied by one-tenth would be 0.

7, and seven divided by 10 would also be 0.

7.

0.

24, well, that would be equal to 2.

4 multiplied by one-tenth.

Then the digits can move one place to the right and you would get 0.

24.

And 0.

32, well, that is equal to 3.

2 divided by 10.

Then 60 multiplied by one-tenth would be six.

And here the multiplicand has got 10 times smaller but the multiplier has stayed the same.

So our product will become 10 times smaller, 0.

6.

750 divided by 10 would be 75, and 75 divided by 10 would be 7.

5 80 divided by 10, well, that would be equal to 80 times 0.

1 because both of those are ways of finding one-tenth.

So 63 divided by 10 would be equal to 63 times 0.

1 or one-tenth.

72 divided by 10 is equal to 72 multiplied by 0.

1 or one-tenth.

And nine divided by 10 is equal to nine multiplied by 0.

1.

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

Fantastic learning so far.

I am really impressed with how hard you are trying and when we try hard then we can be successful.

We're going to move on now and have a little bit of a look at problem solving with this concept.

So let's look at this problem.

Sofia has saved 87 pounds.

Lucas has saved one-tenth of that amount.

Can you visualise that? Has he saved more or less money than Sofia? What might the question be? Well, we want to find out how much money Lucas has saved and actually it will be less money because he has saved one-tenth of that amount.

So let's represent this in a bar model.

Sofia has saved 87 pounds and that is our whole, and Lucas has saved one-tenth of that amount.

To find one-tenth times the size, we need to multiply by one-tenth.

And we know when we multiply by one-tenth it is the same as dividing by 10.

And you can see in the bar model I have divided my whole 87 into 10 equal parts.

When we divide a number by 10, the digits move one place to the right.

So 87 divided by 10 would be 8.

7.

So that means Lucas has saved 8.

70 pounds.

And remember, because it's in an amount of money, we need to have that placeholder in the hundredth place.

Let's look at a different problem.

Sofia has made a cake that has a mass of 0.

6 kilogrammes.

Lucas has made a cake with a mass that is one-tenth of that amount.

Can you visualise that? I think there's my cakes that I'm seeing in my head.

Really nice yummy cake.

That's 0.

6 kilogrammes.

That's the larger cake.

And a smaller cupcake that Lucas has made that has got a mass that is one-tenth of that amount of the whole larger cake.

What might the question be here then, do you think? Hmm.

What is the mass of Lucas's cake? That's right, we don't know that yet, do we? We just know it's one-tenth that of Sofia's cake.

Let's represent this in a bar model.

We can see the whole amount 0.

6 kilogrammes and we know that Lucas's is one-tenth times that amount.

To find one-tenth times the size we need to multiply by one-tenth or 0.

1.

And we know that multiplying by one-tenth is the same as divided by 10.

And so I've divided my bar model into 10 equal parts.

And when we divide by 10, the digits move one place to the right.

We had six tenths, we've now got six hundredths.

Lucas's cake has a mass of 0.

06 kilogrammes or six hundredths of a kilogramme.

We can convert the mass given in kilogrammes to grammes by multiplying by 1000 because there are 1000 grammes in one kilogramme.

So that means Lucas's cake has a mass of 0.

06 kilogrammes and where we've multiplied that by 1000, the digits will move three places to the left.

So that would be 60 grammes.

Let's check your understanding with this.

Could you use the bar model to 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 realise that we were multiplying by one-tenth because there were 10 equal parts and that's the same as dividing by 10? And when we divide by 10, all the digits move one place to the right.

So 0.

7 litres divided by 10 is equal to 0.

07 litres or seven hundredths of a litre.

Your time to practise now.

For question one, could you solve these problems? You could represent them as a bar model to support you to form an equation.

Part a, Lucas measures the length of a leaf to be 30 millimetres.

What is the length of the leaf in centimetres? The height of a daffodil is 76 centimetres.

The height of a daisy is one-tenth times this height.

What is the height of the daisy? And part C, the mass of a banana is 125 grammes.

The mass of a strawberry is one-tenth times this mass.

What is the mass of the strawberry? For question two, is this true or false? And could you give reasons for your choice? Lucas has 30.

50 pound.

Sofia has 3.

50 pound.

Sofia has one-tenth the amount of money that Lucas has.

Is that true or false? Let me know why.

Pause the video while you have a go at both questions and when you are ready for the answers, press play.

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

So the first question about Lucas measuring the length of a leaf to be 30 millimetres.

Well, we know there are 10 millimetres in one centimetre, to convert between millimetres and centimetres, we need to divide by 10.

30 divided by 10 is three.

The length of the leaf is three centimetres and that is equivalent to 30 millimetres.

For part B, the height of a daffodil is 76 centimetres and we can represent this in a bar model.

And we are told the height of a daisy one-tenth times this height.

So we know we need to multiply by one-tenth and we know that is the same as dividing by 10.

So the bar model is divided into 10 equal parts.

When we divide by 10, the digits move one place to the right.

So 76 divided by 10 is 7.

6.

So the height of the daisy is 7.

6 centimetres.

For part C, the mass of the banana is 125 grammes and the mass of the strawberries one-tenth times this mass.

There's my bar model.

The whole is 125 and I need to find one-tenth times this mass.

So I'm multiplying by 0.

1, and we know multiplying by 0.

1 is the same as divided by 10.

And when we divide by 10, all the digits move one place to the right, which gives us 12.

5.

So the mass of a strawberry is 12.

5 grammes.

Our second question, was this true or false? Is 3.

5 pound one-tenth times the amount of money that Lucas has? So 30.

50 pound, what did you think? You might have said it's false and reasoned that to find one-tenth we need to divide by 10.

When we divide by 10, all the digits move one place to the right.

We can't just remove a zero.

So 30.

50 pound divided by 10 is equal to 3.

050.

So if Sofia had one-tenth the amount of money that Lucas had, she would've had three pounds and five pent, not 3.

50 pound.

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

Fantastic learning today, you have really deepened your understanding of that connection between multiplying by 0.

1 and dividing by 10.

We know 0.

1 can be represented as the fraction one-tenth, and we know multiplying by 0.

1 is the same as multiplying by one-tenth, and we know multiplying by one-tenth is the same as dividing by 10.

So multiplying by 0.

1 must also be the same as dividing by 10.

They are all equivalent actions, and we can use the stem sentence to support us.

One-tenth of mm metres is mm tenth of a metre or mm.

You should be really proud of how hard you have tried today.

When we try really hard, that is when we can be most successful.

I have had great pleasure learning with you today and I look forward to learning with you again soon.