Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson.

The lesson comes from our unit on multiples of a thousand.

So we're gonna be looking at those bigger numbers.

How are they put together? How are they composed? Where do we find them in real life? And how can we learn about them so that we can work with them when we're solving problems? So if you're ready to make a start, let's get going.

So in this lesson, we're going to be reading and writing numbers up to one million using a place value chart.

It's really useful to be able to think of those place value headings when we're thinking about really big numbers, like numbers up to one million.

So let's make a start.

We've got three keywords in our lesson today.

We've got comma, factor and multiple.

So I'll take my turn to say them and then it'll be your turn.

So my turn comma, your turn.

My turn factor, your turn.

My turn multiple, your turn.

Now, I'm sure you are familiar with those words and perhaps one of them from your English, maybe rather than from Math but let's have a look.

Just remind ourselves what they mean.

They're going to be useful to us today.

So a comma is a punctuation mark and it's used to indicate a break in a sentence and you can see what it looks like there.

But we are going to use it to indicate a break in our numbers, in our place value system.

So we're going to use commas within our numbers today.

You may have done it before.

Factors are two or more numbers that we can multiply together to create a product.

And a multiple is the result of multiplying a number by another whole number.

So watch out for those words as we go through our lesson today.

There are three parts to our lesson.

In the first part, we're going to be looking at powers of 10.

In the second part, we're going to be looking at multiples of larger powers of 10.

And in the third part, we're going to be constructing larger powers of 10.

So let's make a start with part one.

And we've got Lucas and Alex helping us with our learning today.

So what number is this? What do you think? Alex says, "I think it's 1 million?" Do you agree with Alex? Lucas says, "It's not that big.

Maybe a thousand?" Hmm, what do you reckon? Well, we can use a place value chart to help read numbers.

"Sometimes," Alex says, "it's easier to write the zeros first." So let's start from the ones.

We definitely had a zero in our ones, so let's put the zeros in.

And then a one.

And Lucas says, "Oh, so it was 100,000." Ad we can see we've got a one in the hundred thousands and nothing else.

We've got zeros in all the other columns so it must be the number 100,000.

Lucas says, "But we can't draw a place value chart each time we have a big number." Can we? We need another way to think about this.

So we've taken away the place value chart.

Alex says, "How else could we read the numbers more easily then?" Well, let's go back to the place value chart for a bit.

We can't draw one every time, but maybe we can use it to help us to work out what these larger numbers are.

What do you notice about the place value chart? You might want to pause and have a think.

Well, Alex says he spotted that the colour changes after every three columns.

"But why?" Says Lucas.

Alex says, "I think I've spotted it." Have you spotted it too? Have you noticed something about those sets of three in different colours? Alex says, "The counting unit changes every three columns." Let's see what he means by that.

He says the red column, that's the three on the left.

Tell us how many millions there are altogether.

And you can see we've got a hundred millions, 10 millions and one millions.

The yellow column, that's the middle three, tell us how many thousands there are.

Hundred thousands, ten thousands, one thousands.

Ah, can you see, we've got hundreds, tens, and ones of millions and then hundreds, tens and ones of thousands.

And Lucas says, "So the blue columns must tell us how many ones we have." Hundreds, tens and ones.

So our unit of count in each set of three changes from ones, to thousands, to millions.

"Clever," says Lucas.

Alex says, "Let's put our number back in and then remove the place value chart." So there was our number, which was 100,000.

And we'll remove the place value chart.

So Alex says, "We just need a way to separate the thousands and the ones now." So we've got our 100,000, with no 10 thousands, and no one thousands.

And then no one hundreds, no one tens and no ones.

So what could we do? Ah, that's it.

Alex says, "I think we can use a comma to separate them." You may well have seen numbers written like that before.

So we're going to put our comma in to separate our number of thousands from our number of ones.

Now we've got 100 thousands.

100 thousands and zero ones.

"And if there are zero ones," Lucas says, "we don't need to say this, so we can just say that the number is 100,000." Usually, we'd read how many millions there are, how many thousands there are, and how many ones there are.

But because there aren't any ones, we can just say there is 100,000.

Time to check your understanding.

Can you tick the number with the comma in the correct place? Think about what the place value chart looked like.

Think about what the comma was doing.

Pause the video, have a think, and then we'll talk about the answer.

Which one did you think? That's right.

It was C, wasn't it? It was our a hundred thousand again.

And Lucas says, "Starting from the ones, the comma should be placed after every three digits." Okay, so let's have a look at this.

Now, you might have used a Gattegno charts perhaps in the past, and this is sort of one column from a Gattegno charts, isn't it? If that's what you've seen before.

Otherwise, let's just have a look at this and think, what operation could you use to move up each row? So look at the link between one row and the row above it.

Well, Alex is focused in on the one and 10.

And he says, "Well, we could add nine." Well, we could.

One add nine is equal to 10.

Lucas says, "Does that work for every row though? 10 plus nine is not equal to 100." So it's not going to be the same for each row, is it? "Oh, I see," said Alex, "You multiply by 10 as you go up each row." So 0.

001 times 10 is equal to 0.

01, and so on all the way up there.

And we can see where Alex started one times 10 is 10 and 10 times 10 is equal to 100.

So we are multiplying by 10 every time we move up a row.

Lucas says, "That means to go down each row, you would divide by 10." 100,000 divided by 10 is 10,000 and so on.

And if you think about these numbers on a place value chart, you might be able to see them moving one column to the left as we multiply by 10 or one column to the right as we divide by 10.

So if you took a digit one and moved it around the place value chart, you would create these numbers and each time you moved it to the right, you'd be dividing by 10.

Each time you moved it to the left, you'd be multiplying by 10.

And you can make sure that we know exactly what the value of that one is by the number of zeros that you've got to fill in the columns to the ones.

So one times 10 is equal to 10.

And we can write this as 1 x 10.

Then if you want to go up another row, we can write this as 1 x 10 x 10.

And that's the same as saying 1 x 10 squared.

"Yes," says Lucas.

And if we went up another row, it would be 1 x 10 x 10 x 10, which is 1 x 10 with a little three.

So what do you notice here? Well, we've got some equations written out there.

And Alex says, "In these examples, we multiply by 10 more than once.

So we multiply by the same factor several times." So 10 is a factor and we keep multiplying by 10 so we multiply by 10 more than once.

And Lucas says, "Instead of writing times 10 lots of times, we can write a smaller digit next to it.

And this is called a power." So 1 x 10 x 10 is the same as saying 1 x 10 to the power of two.

So which expressions are equal to 1 x 10 to the power four? Pause the video and have a look.

What did you think? Well, the first two are 1 x 10 to the power four.

Multiplying a number by one doesn't change the product, does it? So we can write that without the one.

We are leaving it there because it's useful to know that all of this means 1 x 10 to the power of.

So the first expression says 1 x 10 x 10 x 10 x 10, which is 1 x 10 to the power of four.

And the second one has 10 multiplied by itself four times.

10 x 10 x 10 x 10, which is 10 to the power of four.

And if we multiplied that by one, it would still be equal to 10 to the power four.

So those are equivalent expressions and they both represent 1 x 10 to the power four.

So what is 1 x 10 to the power of four equal to? Is it A, B, C, or D? Pause the video and have a think, and we'll look at the answer together.

It's C, isn't it? 1 x 10 to the power four is equal to 1 x 10 x 10 x 10 x 10, which is equal to 10,000.

10 times 10 is equal to 100, 100 x 10 is equal to 1,000, and 1,000 times 10 is equal to 10,000.

Time for you to do some practise now.

Write down the numbers given in words in digits.

And remember to use commas where appropriate.

And question two says, write down the equivalent number in digits.

And you've got a few to look at there.

Pause the video, have a go at your tasks, and we'll come back to together for some feedback.

How did you get on? A, asked you to write the number 100,000 in digits using a comma.

And there we have it 100,000.

And our comma shows us where the number of thousands stop and the number of hundreds, tens and ones begins.

1,000 has the comma to separate the thousand from the hundreds, tens and ones.

What about 10,000? Yeah, 10,000 is a five-digit number.

one ten thousand and then zeros to show us that that one is in the ten thousands column.

And our comma there between the thousands and the hundreds.

Ooh, now we have to think.

Ten, ten thousands.

So we've got 10,000 there, we want 10 of them.

So what's that equivalent to? Well, it's 100,000 again, isn't it? 10 x 10 is equal to 100.

So 10 x 10,000 will be equal to 100,000.

What about ten, one thousands? There's a clue there.

It's 10,000, isn't it? So it's going to be written the same way as we wrote C, 10,000.

What about one thousand tens? Hmm.

Well, that's 10,000 as well, isn't it? So 1,000 x 10 is equal to 10,000.

So one thousand tens will be 10,000.

And for question two, you were writing down the equivalent number in digits.

So A said, what number is 100 times the size of 10? So we could think about using the place value chart to help us, or we could remember that multiplication is commutative.

So 100 x 10 is the same as 10 x 100, which is equal to 1,000.

Ah, and in fact, that's B, isn't it? What number is 10 times the size of 100? It's that commutativity, it's 1,000 again.

C asks what do you get if you divide 100,000 by 10? And you might be able to see things moving on the place value chart to help you there.

It's 10,000.

Or you might have thought of that column that we'd taken from the Gattegno chart and thinking about what the number underneath 100,000 on the Gattegno chart looks like.

D asks what do you get if you multiply one by 10 to the power of three? Now remember, 10 to the power of three was 10 multiplied by itself three times.

So 1 x 10 x 10 x 10.

10 x 10 is 100, 100 x 10 is 1,000.

So our answer is 1,000.

And E, what do you get if you divide 10,000 by 10 and then another 10? So 10,000 divided by 10 and then divided by another 10.

We could do them separately, or we know that 10 tens are 100, so we'd be dividing by a hundred.

So our answer is 100.

10,000 divided by 10, and then another 10 is equal to 100.

And you can imagine that column in the Gattegno chart coming down two columns, we've divided by 10, divided by 10 again and we've got to 100.

And F says if I multiply a number by 10 and then 10 again, I will end up with 10.

What was my original number? Oh, so let's think about that chart.

So we've multiplied by 10 and multiplied by 10.

So to get back to the original number, we've got to divide by 10 and then divide by 10 again.

10 divided by 10 is equal to one, and one divided by 10 is equal to one 10th, which is 0.

1.

So our original number was 0.

1.

Well done if you've got those right.

Lucas says, "Can you create your own question to challenge a friend with?" Hmm, I wonder.

And on into part two of our lesson, where we're going to look at multiples of larger powers of 10.

So here we've got more of Gattegno chart.

Can you spot the multiples of 10,000? The left hand column is the column that we were looking at in the first part of our lesson, but now we're trying to look for the multiples of 10,000.

So we're probably looking across a row this time.

Can you spot them? There they are.

Can you imagine them with commas in as well? We can see the 10,000 in our question has a comma, but in the Gattegno chart, we haven't got the commas.

But those are all the multiples of 10,000.

So let's count in multiples of 10,000.

10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000, 90,000, 100,000.

Can you spot the multiples of 100,000? There they are, we just landed on a hundred thousand when we counted up past 90,000, didn't we? Alex says, "Let's count in multiples of 100,000." Let's go.

Are you ready? 100,000, 200,000, 300,000, 400,000, 500,000, 600,000, 700,000, 800,000, 900,000.

Where do we go after that? Lucas says, "Let's play a game.

I'll say a number and you can point to it on your chart." Okay, Alex says he's ready.

You might want to play along with this as well.

500,000.

Can you spot it? There we go.

Alex has found it for us.

50,000.

It's the row below.

800,000.

We're back in our hundred thousands, this time we've got eight of them.

Alex says, "Okay, my game now." He says, "I'll say a number and you can write it on your whiteboard." I'm sure you could join in with this, couldn't you? 70,000.

And we go, well done, Lucas.

That is 70,000.

And he's used the comma to show where the thousands are separated from the hundreds, tens and ones.

200,000.

Got it.

There we go.

200,000.

And the comma's there again to separate our thousands from our hundreds, tens and ones.

What about thirty thousands, says Alex.

That's right.

Three tens of thousands this time.

Time to check your understanding.

You're going to see circles appear around some of the numbers on this Gattegno chart and you are going to say the number out loud.

So are you ready? Let's go.

40,000.

700,000.

90,000.

Well done if you've got those ones.

You might want to carry on playing the game.

Perhaps someone could point to the numbers and you could say them out loud.

You can now apply powers of 10 to any number.

So we are going to start with three and we're going to say 3 x 10 x 10.

3 x 10 is 30, times another 10 is 300.

So three times 10 times 10 is equal to three times 10 to the power two.

Can you see now why we have the 1 x 10 x 10 in part one? Because sometimes, we are not just multiplying the tens together, we're applying them to another number that we're starting with.

And 3 x 10 to the power two is equal to 300.

What's about 50 then? If we start with 50? So not a single digit number this time.

So we're going to do 50 x 10 x 10 x 10.

So let's use the chart to help us.

We're starting with 50, times 10 is 500, times another 10 is 5,000, times another 10 is 50,000.

How would we write this down? That's right.

It's equal to 50 x 10 to the power three.

We've got three tens multiplied together there.

And it's equal to 50 multiplied by 1,000.

10 x 10 x 10 is equal to 1,000.

Time to check your understanding.

Which expression is equal to 800 x 10 to the power four? Pause the video, have a look, and we'll come back for the answer.

How did you get on? Did you remember what 10 to the power four looks like? It's 10 x 10 x 10 x 10, four tens multiplied together.

So B was 800 x 10 to the power four.

800 multiplied by ten, four times.

In the other ones, well, we had four tens in A, but we were only multiplying eight and we wanted 800 multiplied by 10 to the power four.

And then in the other two, we didn't have the right number of tens to represent 10 to the power four.

Another check.

What is 70 x 10 to the power of two? So which number is equivalent to 70 x 10 to the power of two? Pause the video, have a look, and we'll come back for the answer.

How did you get on? What did 10 to the power two mean? Well, it meant 10 x 10, which is 100.

So we're looking for the number equivalent to 70 multiplied by 100.

And that number is 7,000.

And the comma there to show us that we have seven thousands, no hundreds, tens and ones.

Time for you to do some practise again.

Can you spot the odd ones out in this set of equations? So you're going to answer the equations and see if you can spot the odd ones out.

So pause the video, have a go, and we'll come back for some feedback.

How did you get on? Did you spot the odd ones out? So in the first one we were looking at, 8 x 10 x 10 x 10 x 10.

And if we multiplied 8 by 10 all those times, we'd get the answer 80,000.

What about the second one? 8 x 10 to the power four.

Have you spotted something? Yeah, it's a different way of writing out the first equation, isn't it? 10 x 10 x 10 x 10 is equivalent to 10 to the power four.

So our answer again was 80,000.

What about 8 x 100 x 10? Can we think about how many tens we've got there? 100 is 10 times 10, so we've got 8 x 10 x 10 x 10.

Ah, we've only got three tens there, so that's 8 x 10 to the power of three.

So the answer to that one is 8,000, not 80,000.

What about the next one? 8 x 10 to the power of three times another 10? Well, we've got 10 to the power four again there, haven't we? So the answer must be 80,000.

And 10 to the power two x 10 to the power two.

So again, we've got 10 x 10 x 10 x 10, we've got those four tens back again so we've got 10 to the power four.

So that one also is going to be 80,000.

Oh, and 10 to the power four x 8, is just the same as the second one, but with our factors written in a different order, so that also is equal to 80,000.

What about the last one? 10 to the power two x 8, and then times 10 to the power three.

Ah, so we've got 10 x 10, and then 10 x 10 x 10, so we've got five times that we're multiplying by 10.

So that must be 8 x 10 to the power five.

So that's going to be 10 times bigger so that one must be 800,000.

So which were our odd ones out? Well, there were two weren't there? All of the equations were equal to 80,000 except for two.

So those ones are the odd ones out.

And on into the final part of our lesson, we're going to be constructing larger powers of 10.

So what do you notice here? We've got three numbers circled on our Gattegno chart.

We've got 300, and 70 and five.

300 + 70 + 5 = 375.

What do you notice now? Hmm, it's similar, but it's a bit different, isn't it? This time we've got 300,000, 70,000 and 5,000.

So we had 300, 70, and five.

So now we've got 300,000, 70,000 and 5,000 and that's equal to 375,000.

So if you think about the place value chart, our first set of numbers were all sitting in the hundreds, tens and ones.

And the second set were sitting in the hundred thousands, ten thousands and one thousands.

So we can read it as 375, but our unit this time was thousands, 375,000.

And now, we've got our addition to show it with the numbers written out in full.

What do you notice this time? What have we got? We've got 800 and 60 and three.

And 800 + 60 + 3 = 863.

What can you see now? Hmm, so our 800 is now 800,000.

Our 60 is now 60,000 and our three is now 3,000.

So we're now counting in thousands.

So our sum now is 863,000.

And we can write that out with the digits in full as well, using the commas to show us where our thousands are separated from our hundreds, tens and ones.

What about this one? We've got 800 and three.

So we've got eight one hundreds and three ones so we've got no tens, so we need that zero there to show us that there are no tens and that our eight is in the hundreds and our threes in the ones.

And what's happened now? Well, we've got 800,000 and 3,000 as well.

So again, that looks like 803,000.

We still need that zero there, to show us that there are no ten thousands, just as we needed it in 803 to show that there were no tens.

And if we write that out with the digits in full, we can see 803,000.

The comma there showing us that the 803 are thousands and that we don't have any extra hundreds, tens and ones.

What about this one? We've got 860.

This time, we've got no ones, so we need zero there to show us there are no ones.

And now we have 800,000 and 60,000, we have 860,000.

And you can see that number repeated again, can't you? And when we write it out in full, we've got 860 in the thousands and then the zeros to show that we've got no additional hundreds, tens and ones.

Time to check your understanding.

Can you fill in the missing numbers? We've got 700 + 50 + four = 754.

So what would that be if we had those numbers in thousands? And then what would that look like as an addition calculation with the numbers written in full? So pause the video, have a go, and we'll come back together for some feedback.

How did you get on? So we had 700,000 + 50,000 + 4,000, 754,000.

And written out as the full numbers, we can see 754, 000.

The zeros to tell us that those are thousands numbers and that we don't have any additional hundreds, tens and ones.

Time for you to do some practise.

You've got some of these to fill out for yourself.

So here we've given you the calculation in hundreds, tens and ones for both A and B.

And you are going to fill in the number of thousands, and you are going to fill in what that would be if it was in thousands and then write the calculation.

And the same for C and D.

And then for E and F, we haven't given you the sum, so you need to work the sum out.

Think carefully about zeros in this one.

And for G and H, we've given you the thousands part of our calculation.

And so you've got to go back and think, "What would that look like if we were thinking about hundreds, tens and ones." Pause the video, have a go, and we'll come back for the answers.

How did you get on? So in A, we had 500 thousand, 30 thousand and 1 thousand, giving us 531,000.

And they're written out in digits with the commas used to show us where our thousands break from our hundreds, tens and ones.

B, was 400 thousand + 70 thousand + 4 thousand, 474,000.

And again, written out in digits.

Here, we had to think about zeros, didn't we? 809,000.

So we had 800 thousand and 9 thousand giving us 809,000.

So we've got a zero in our ten thousands there.

Really important because we still need to know that that eight is 800,000 and that nine is 9,000.

So the zero is really important.

230,000 this time the zero to show us that we had no 1,000.

So 230,000.

We've swapped the order around here, but we know that tradition is commutative.

So that's 349.

So 349,000.

And there written out in full with the commas to separate the thousands from the hundreds, tens and ones.

And here we just had 49.

So this time we had 9 thousand and 40 thousand.

We have no hundred thousands this time, so we don't have a six-digit number this time.

We just have a five-digit number.

So for G and H, we were given the thousands and we had to work backwards.

So 400,000 and 3,000 we'd have had 400 and 3.

So 403,000 with a zero in the ten thousands 'cause we don't have any extra ten thousands.

And what about the last one? We were only even the 164,000.

So we must have been thinking about 164, which is 100 and 60 and 4.

100,000 and 60,000 and 4,000.

And there it's written out in the digits.

Well done.

It's really useful to be able to read and write large numbers accurately and especially useful to think about the zeros.

Where are they needed and why are they needed? And we've come to the end of our lesson.

So we've been reading and writing numbers up to 1 million.

We almost got to a million, but not quite.

And we've been using a place value chart.

So what have we learned about today? We've learned that you can use commas to separate the ones, thousands and millions when reading larger numbers and when writing them.

When you multiply a number by the same factor several times, you can say that you are multiplying that number by that factor to the power of the number of times it's multiplied.

So remember that 3 x 10 x 10 x 10, was three multiplied by 10 to the power of three.

And additive equations of adding hundreds, tens and ones can be applied to adding multiples of a thousand.

And that's what you were doing in that last practise.

Well done for all your thinking around big numbers today.

I hope you've enjoyed it as much as I have and I hope I get to work with you again soon.

Bye bye.