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Hello, how are you today? My name is Dr.
Shorrock.
I am really excited 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 unit: Calculating with decimal fractions.
The lesson is called: Explain the effect of multiplying and dividing a number by 10, 100 and 1,000.
Throughout the learning today, we will review our understanding of what tenths, hundreds and thousandths are.
And we will also deepen our understanding of multiplying and dividing by 10 and 100 and use this to determine the effect of multiplying and dividing by 1,000.
Throughout the lesson, we are going to use the place value chart and the Gattegno chart to support us to make connections in our learning.
Sometimes new learning can be a little bit tricky, but don't worry, I am here to guide you and I know if we work really hard together, then we will be successful.
Let's get started, shall we? So how do we explain the effect of multiplying and dividing by 10, 100 and 1,000? These are the keywords that we will be using in our learning today.
We've got a thousandth and equivalent.
You may have heard those words before, but it's always good to practise.
So let's practise together.
My turn: thousandth.
Your turn: Nice.
My turn: equivalent.
Your turn: Fantastic.
Well done.
So when we talk about one-thousandth, we mean one part in one-thousand equal parts.
And if we say something is equivalent, well, it has the same value.
For example, one pound is equivalent to 100 pence.
They have the same value.
We will start our learning today by reviewing our understanding of tenths, hundredths and thousandths.
We have Lucas and Sofia who will help us throughout our learning today.
Let's have a look at this.
So Lucas and Sofia find some base 10 blocks.
I wonder if you've seen any like that before.
Ah, Lucas definitely remembers using these before.
What do we know about these? Hmm? Is there anything you can remember about base 10 blocks like this? Ah, thank you Sofia.
Sofia has remembered that each block is 10 times smaller than the previous block.
Let's use that information to help us in our learning today.
Sometimes you may have used this block and given it a value of 1,000, but I'm going to challenge you today to think about what if this block is our whole but today it's worth one, one whole.
So one is our whole.
And we can represent one on a place value chart, one in the one's place.
So we know the larger cube has a value of one.
So how can we describe this flatter block in relation to our whole? That's right, the whole has been divided into 10 equal parts.
Each part is one-tenth of the whole and we can form an equation to represent this.
Our whole was one and we are dividing it into 10 equal parts.
So each part is worth one-tenth and 10 tenths make up the whole.
So this part here, this block is one-tenth of the whole, because 10 of them would make that one cube, that one whole.
And we can represent one-tenth on a place value chart.
We've got one in our tenth position, we've got no ones because our one block has been divided into ten equal parts and we've got one of those parts.
So we can say 1 divided by 10 = one-tenth and we can represent that as 0.
1.
Now I wonder if you knew this, dividing by 10 is equivalent to, that's that key word, it means the same as, X one-tenth or 0.
1.
So we can say 1 divided by 10 = 0.
1, but it's also equal to 1 X 0.
1.
And we know that when we multiply a number by one, the number does not change.
So 1 X 0.
1 = 0.
1.
Let's look at this other block now.
How can we describe this block in relation to our whole? Well the whole has been divided into 100 equal parts.
Each part is one hundredth of the whole.
And again, we can form an equation to represent this.
The whole was one and we have divided the one whole into 100 equal parts.
Each part is worth one-hundredth, and there must be 100 hundredths in the whole.
So this block is one-hundredth of our whole, because 100 of them would make that one whole block.
And yes, we can represent one-hundredth on a place value chart.
We've got one in the hundredth position.
1 divided by 100 = 0.
01.
And dividing by 100 is equivalent to.
So it's the same as multiplying by one-hundredth or 0.
01.
So we can rewrite our equation as 1 divided by 100 = 1 X 0.
01.
And we know if we're multiplying by one, the number does not change.
So 1 X 0.
01 = 0.
01.
But is there another way that we can describe this one-hundredth block? We know we've said it's one-hundredth or 0.
01 of the whole.
Is there another way we can describe it? Ah, thank you Sofia.
We could think about it in relation to the one-tenth block.
When one-tenth of a whole is divided into 10 equal parts, each part is one-hundredth of the whole.
So we can see that one-tenth block.
If we divide that to 10 equal parts, we would get our one-hundredth block and there are 10 hundredths in one-tenth.
We can form equations to represent this.
We've got one-tenth, it's divided into 10 equal parts.
One-tenth is equivalent to 0.
1 and we're dividing into 10 equal parts and we would get one-hundredth or 0.
01.
Dividing by 10 is equivalent to multiplying by 0.
1.
So we can write a second equation.
0.
1 times 0.
1 = 0.
01.
One-tenth of one-tenth is one-hundredth is another way we could say that equation.
So we can see that each part is worth one-hundredth or 0.
01.
Let's look at this last little block then, shall we? How can we describe this block in relation to our whole? The whole has been divided into 1,000 equal parts.
Each part is one-thousandth of the whole and we can form an equation to represent this: We've got one whole is divided into 1,000 equal parts.
Each part must be worth one-thousandth.
And there are 1,000 thousandth in the whole.
So our block, the small block is one-thousandth of the whole.
We could have 1,000 of them, it would make that one whole block.
And we can represent one-thousandth on the place value chart.
You can see we've got one in the thousandth place and we can then form an equation.
Our whole divided into 1,000 equal parts is worth one-thousandth.
And we know dividing by 1,000 is equivalent to multiplying by one-thousandth or 0.
01.
So we can write our equation, our division equation with an equivalent multiplication equation, 1 divided by 1,000.
Well that's the same as 1 X one-thousandth or 0.
001.
And we know when we multiply by one, the number remains the same.
So 1 X 0.
001 = 0.
001.
So is there another way then that we can describe the one-thousandth block? What do you think? Thank you Sofia.
When one-hundredth of a whole is divided into 10 equal parts, each part is worth one-thousandth of the whole, and we can represent that in an equation.
One-hundredth is 0.
01.
If we divide that to 10 equal parts, each part is worth one-thousandth or 0.
001.
There are 10 of our small one-thousandth blocks in one-hundredth.
And dividing by 10 is equivalent to multiplying by 0.
1.
So we can form a second equation, a multiplication equation where we've got one-hundredth and we're multiplying by one-tenth or 0.
1 and that would give us one-thousandth because one-tenth of one-hundredth is one-thousandth.
Is there another way that we can describe the one-thousandth block? Hmm? What do you think? Thank you Sofia.
When one-tenth of a whole is divided into 100 equal parts, each part is worth one-thousandth of the whole.
I wonder if you can think about what that equation might be.
We've got one-tenth and we're dividing into 100 equal parts.
Hmm.
Ah, that's right.
One-tenth is the same as 0.
1 and 0.
1 divided by 100.
Well that gives us our one-thousandth and there are 100-thousandths in one-tenth and we've learned that when we divide by 100 it is equivalent to multiplying by one-hundredth.
So I wonder if you can think, what's our next equation going to be? The multiplication equation that is equivalent to 0.
1 divided by 100.
That's right.
0.
1 X 0.
01, that's also equal to one-thousandth or 0.
001.
Let's check your understanding.
Have a look at these equations and let me know which equation or equations, so there might be more than one, match this statement: One-hundredth has been divided into 10 equal parts.
Each part is one-thousandth of the whole.
Pause the video while you have a look at the equations and when you are ready to go through the answers, press play.
How did you get on? Did you work out that equation A was correct? Because we've got one-hundredth or 0.
01 and we are dividing into 10 equal parts and each part is one-thousandth or 0.
001 of the whole.
What about B? Can't be B, can it? Because B is one-tenth divided by 10 and we wanted one-hundredth divided by 10.
What about C? Oh yes, C is correct.
We've got one-hundredth and we are dividing into 10 equal parts, but we know dividing by 10 is the same as multiplying by 0.
1.
What about D? Hmm, not D.
D, we've got one-hundredth and it's divided into 100 equal parts.
A hundredth of a hundredth.
So that can't be correct.
What about E? Well E we've got one-hundredth and it's divided into 10 equal parts.
Hmm but then the answer has stayed the same so that can't be correct.
The answer is not one-thousandth.
So it is key to remember that when we divide by 10 it is the same as multiplying by 0.
1.
How did you get on with those? Well done.
Now let's have a look at this Gattegno chart and see what relationships we can spot between the whole, tenths, hundreds and thousandths.
I wonder if you've seen something like this before.
You might have done.
In particular, let's have a look at this first column, and we can see that the numbers underneath each other are 10 times smaller.
So Lucas has given us an example.
We could say 0.
1 is 10 times smaller than one, or we could say 0.
001, which is one-thousandth, is 10 times smaller than 0.
01 or one-hundredth.
I wonder if you can see any more relationships on that column.
And then let's have a look at it the other way around.
The numbers above each other, well they are all 10 times larger.
So Lucas is telling us that he can see from this that 0.
01, or one-hundredth, is 10 times larger than one-thousandth, or 0.
001.
He can also see that 0.
1 is 10 times larger than 0.
01.
I wonder if you can see any other relationships in that column? Ah, good question, Sofia.
What about numbers that are not directly above or below each other? Ah yes, we can use the relationship that each row is 10 times larger or smaller than the other one and if we've got two rows it must be 10 tens larger or 100 times larger.
So Lucas can tell us that a 0.
1 is 100 times larger than 0.
001.
One-tenth is 100 times larger than one-thousandth.
We can look at it the other way round.
This time you can see we've started with one and we have moved down three rows.
So 10 X 10 X 10.
10 tens are 100, 100 tens are 1,000.
So we can see that one-thousandth is 1,000 times smaller than one.
Let's check your understanding on that.
Could you use this Gattegno chart to help you complete the sentence? One is mm times larger than 0.
01.
Pause the video while you work it out and when you are ready to go through the answers, press play.
How did you get on? Did you work out that one is 1,000 times larger than 0.
001.
One is three rows above the one-thousandth, so it must be 1,000 times larger.
Well done.
Your turn to practise now.
Question one: Could you match the division equation to its equivalent multiplication equation? For question two, could you use a Gattegno chart to complete the sentences? This is a copy of a Gattegno chart that you can use.
For question three: Could you look at the equation? Is it true or is it false? And whatever you decide, could you give me some reasons for your answer? One divided by 1,000 = 0.
01 divided by 10.
Pause the video while you have a go at all three questions and when you are ready to go through the answers, press play.
How did you get on? With question one, you had to match the division equation to its equivalent multiplication equation.
When we start with one-hundredth and we divide by 10, well, divided by 10 is the same as multiplying by 0.
1.
The second equation, we start with one-tenth and we divide by 100, divided by 100 is the same as multiplying by 0.
01.
We started with one for the next equation, we divided by 100, the same as multiplying by 0.
01.
Then we started with one and we divided by 10, which is the same as multiplying by 0.
1.
Then we started with 0.
1 and we divided by 10, which is the same as multiplying by 0.
1.
And lastly, we started with one and we divided by 1,000, which is the same as multiplying by 0.
001.
For question two you were asked to use a Gattegno chart to complete these sentences.
So one is 1,000 times larger than 0.
001 and that's the same as a thousandth, isn't it? So one has to be 1,000 times larger than one-thousandth.
One-tenth is 100 times larger than 0.
001.
One-thousandth is 1,000 times smaller than one.
One-hundredth is 10 times larger than one-thousandth.
One-thousandth is 100 times smaller than one-tenth, and one-tenth is 10 times larger than one-hundredth.
For question three, you were asked to say whether this equation was true or false and give a reason for your answer.
So you might have said that this is true and have given a reason like this: So this is true because when one whole is divided into 1,000 equal parts, each part is worth one-thousandth and when you divide one-hundredth, so 0.
01 into 10 equal parts, each part is also worth one-thousandth.
So those equations were equivalent.
They are true.
How did you get on with all three questions? Brilliant.
Fantastic learning so far.
I am really impressed with how hard you are trying, well done.
We are now going to move on to look at multiplying and dividing by 1,000.
We're going to start by looking at the effect of multiplying and dividing by 10 and 100, which is something that you might remember, and we're then going to use this to determine the effect of multiplying and dividing by 1,000.
What does happen then when we multiply it by 10? You can see we've got six ones.
Ah yes, that's right.
When a number is multiplied by 10, the digits move one place, value place to the left making the number 10 times the size, one place.
And we can write an equation.
Six tens are 60.
That six has moved one place to the left, it was six ones, we make it 10 times larger and it becomes six tens or 60.
What happens then when we divide by 10? Ah, dividing by 10, we've seen from the first part of the lesson, is equivalent to multiplying by one-tenth or 0.
1.
And when a number is divided by 10 or multiplied 0.
1, the digits move one place to the right, making the number one-tenth times the size.
So 60 divided by 10 is 6.
We had six tens, we divide by 10 or multiplied by 0.
1 and we end up with six ones.
What happens then when we multiply by 100? Any thoughts? What do you think? Ah, that's right.
When a number is multiplied by 100, the digits move two place value places to the left making the number 100 times the size.
One, two.
We had six ones, we now have six hundreds and we can form an equation.
Six one hundreds are 600.
What about when we divide by 100? Hmm? What happens? Well we know dividing by 100 is equivalent to multiplying by one-hundredth or 0.
01.
And when the number is divided by 100 or multiplied by 0.
01 the digits move two places to the right, making the number one-hundredth times the size.
One, two.
We had six hundreds, we divide by 100 and we end up with six ones.
And this is the same as dividing by 100 is the same as multiplying by 0.
01.
Let's revisit these equations.
What do you notice about them? Is there something that is the same? Is there something that is different? Hmm? Ah.
Lucas noticed in the first two equations where we are multiplying, we are increasing the size of six 10 times then 100 times.
And Lucas has also spotted that the equations given are inverses, 6 X 10 = 60.
And the inverse of that is the division 60 divided by 10 = 6.
What might the next equations be then? Any thoughts? If we follow the pattern? That's right, the next one would be six still, but this time we would be multiplying by 1000, Six one-thousandths are 6,000.
And the inverse equation? Well we would take the whole, 6,000 and divide it into 1,000 equal parts.
Each part would be worth six.
So let's have a look at this on a place value chart.
What happens when we multiply by 1,000? Hmm? We know when we multiply by 10, that number six, that would move one place to the left and when we multiply by 100, six would move two places to the left.
What do you think will happen when we multiply by 1,000? That's right.
When a number is multiplied by 1,000, the digits move three place value places to the left making the number 1,000 times the size.
One, two, three.
We had six ones.
And when we make six 1,000 times greater, we get six thousands and we can represent that in an equation.
Six one thousands are 6,000.
But what about if we divide by 1,000? Well when we divide by 10, the digits move one place to the right.
When we divide by 100, the digits move two places to the right.
What about 1,000, do you think? And remember, dividing by 1,000 is equivalent to multiplying by one-thousandth or 0.
01.
That's right.
So then when a number is divided by 1,000 or multiplied by 0.
001, the digits move three places to the right, making the number one-thousandth times the size.
We had six thousands, we divided by 1,000 and we get six ones.
The digit 6 has moved three places to the right.
Remember we can also represent that as a multiplication equation.
6,000 X 0.
001 = 6.
Let's check your understanding on this.
Look at these equations and complete the sentences.
2 X 1,000 = 2,000.
2000 divided by 1,000 = 2.
2000 X 0.
001 = 2.
And the sentences to complete are: when a number is multiplied by 1,000, the digits move mm place value places to the left.
When a number is divided by 1,000 or multiplied by 0.
01 the digits move mm places to the right.
Pause the video while you do that and when you are ready for the answers, press play.
How did you get on? Did you realise that the first number sentence, we needed to replace the blank with the word three.
When a number is multiplied by one-thousandth, the digits move three place value places to the left.
And in the second question, when a number is divided by 1,000 or multiplied by 0.
001, the digits move three places to the right.
How did you get on? Well done.
Let's explore this relationship then on a Gattegno chart.
Let's have a look at the 6,000 column.
Remember, numbers below each other are 10 times smaller and those above each other are 10 times larger.
So if we're going to look at six tens are 60, we can see that 60 is 10 times larger than 6, and we can see the relationship, the inverse relationship that 60 divided by 10 is 6.
So 6 is one-tenth the size of 60.
We can also use the Gattegno chart to look at the relationship between 6 X 100.
We can see 600 is 100 times larger than 6 'cause it's two rows above 10 tens.
And 10 tens are 100.
So 600 is 100 times larger than 6.
We can look at the inverse relationship.
600 divided by 100 is 6.
600 multiplied by 0.
01 or one-hundredth is 6.
And we can see 6 is 100 times smaller than 600.
Let's look at the relationship when we do 6 X 1,000.
We can see that 6,000 must be 1,000 times larger than 6.
It is three rows above six, 10 X 10 X 10, 10 tens are 100, times another 10 is 1,000.
And we look at the inverse relationship.
6,000 divided by 1,000 is 6.
6 is 1,000 times smaller than 6,000.
Let's check your understanding with that.
Use this part of a Gattegno chart to complete the equations.
3 X 1,000 is mm, 3,000 divided by 1,000 is mm.
And 3,000 multiplied by mm = 3.
Pause the video while you do that, maybe find someone to have a chat to about it.
When you are ready for the answers, press play.
How did you get on? Did you notice that 3 X 1,000 = 3000? If we multiply by 1,000, we have to go up three rows on the Gattegno chart.
Start with 3, 30, 300, 3,000.
And if we're going to divide 3,000 by 1,000, we must get 3 and then 3,000 multiplied by mm.
What's the same as dividing by 1,000? That's right.
Multiplying by 0.
001.
How did you get on? Nice! Your turn to practise now.
For question one, could you look at the equation? Do you agree? Could you give reasons for your answer? Is 8,000 X 0.
001 equal to 80? For question two, you've got some equations to complete.
For question three, could you complete these two sequences? For each sequence, could you write three equations? So one multiplication equation and then a division equation, and then a second multiplication equation that matches the division equation.
And those three equations connect the two missing numbers for each sequence.
Pause the video while you have a go at all three 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 you had to say whether you agreed or not with my equation, and you might have said you disagree and given a reason like this: you disagree because multiplying by 0.
001 is equivalent to dividing by 1,000.
There are eight one thousands in 8,000, so the product should be eight, not 80.
The equation should have been 8,000 X 0.
001 = 8.
For question two, you had to complete the equations.
4 X 1,000 = 4,000.
4,000 divided by 1,000 must therefore be equal to 4, 4,000 multiplied by 0.
001.
Well that's equivalent to dividing by 1,000, so that must also be 4.
Nine 1,000 are 9,000.
So 9,000 divided by 1,000 must be 9.
And we know dividing by 1,000 is the same as multiplying by 0.
001.
So that must also be 9.
5,000.
Well that's the same as five one-thousands 5 then must be equal to 5,000 divided by 1,000.
And we know dividing by 1,000 is the same as multiplying by 0.
001.
7,000, well that's equal to seven one-thousands.
So 7 must be equal to 7,000 divided by 1,000.
And we know that dividing by 1,000 is the same as multiplying by 0.
001.
For question three you have some sequences to complete.
For the first sequence, the missing number was 4,000 and 4.
And then you had to write three equations that connect the missing numbers.
So 4 X 1,000 = 4,000, and then we've got 4,000 divided by 1,000 = 4 or 4,000 X 0.
001, or one-thousandth = 4.
For the second equation, the missing numbers were 2000 and 2, and the equation, two one-thousands are equal to 2,000.
2,000 divided by 1,000 = 2 and 2,000 X 0.
001 or one-thousandth is also equal to two.
How did you get on with those questions? Brilliant, well done.
Fantastic learning today, everybody.
I am really proud of how far you have come in your learning and how hard you have tried.
We have learned to multiply a number by 10, 100 or 1,000.
We need to move the digits one, two, or three places to the left and that makes the numbers 10, 100 or 1,000 times the size.
We also know dividing by 10, 100 or 1,000 is equivalent to multiplying by 0.
1, 0.
01 or 0.
001 respectively.
I have had great fun learning with you today, well done and I look forward to learning with you again soon.
