Lesson video

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

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

We are gonna have a lot of fun as we move through the learning together.

Today's lesson is from our unit, order, compare and calculate with numbers with up to eight digits.

This lesson is called subtract multiples of powers of 10 crossing the millions boundary.

As we move through the learning today, we are going to deepen our understanding about which mental strategies we can use to subtract across the millions boundary, such as using our known facts and unitizing or number lines.

Now, sometimes new learning can be difficult, but it's okay because I know if we work really hard together and try our absolute best and I'm here to guide you, then I know that we will be successful.

So let's get started, shall we? How do we subtract multiples of powers of 10 crossing that millions boundary? These are the keywords that we will be using in our lesson today.

We have million, unitize and power of 10.

Now when we have keywords, it's always good to say them aloud, so let's have a go at doing that together.

My turn, million, your turn.

Nice, my turn.

Unitize.

Your turn.

Well done.

And my turn, power of 10, your turn.

Fantastic.

One million is composed of 1,000 thousands and we write one million as a one followed by six zeros.

When we unitize, we treat groups that contain or represent the same numbers of things as units or ones.

And being able to unitize is key in understanding place value and we'll be using unitizing a lot as we move through the learning today.

Now, a power of 10 is when 10 is multiplied by itself a certain number of times and there are lots of powers of 10.

Some powers of 10 are 0.

01, 10, 1,000, 1 million, 10 million.

And we're going to start our learning today thinking about how we can subtract crossing the millions boundaries.

And in our lesson, we've got Aisha and Lucas to help us today.

Aisha and Lucas are playing a game.

Aisha says she that she has got 1,400,000 points and Lucas has 600,000 points, and Aisha wonders how many more points that she's got.

So how many more points is she winning by? And to find how many more, well, we need to find a difference.

And when we find the difference, we need to subtract.

And Aisha is saying, well, these numbers are a bit too large to subtract mentally, aren't they? 1,400,000 and 600,000.

They are large numbers, aren't they? So she recommends we use a column method or a calculator.

Do you agree? Lucas doesn't.

He's respectfully challenging her.

Lucas thinks we can use our number sense superpower and we can use unitizing and known facts to help.

We can start by looking at a simpler case.

Let's look at 1,400 subtract 600.

Now, Aisha is saying she knows the answer will bridge 1,000.

Do you know why the answer will bridge 1,000? That's because 1,400, 1,400 is only 400 more than 1,000 and we're actually subtracting 600.

So it will bridge 1,000.

We can use unitizing and known facts to help us find the difference here.

We know 14 subtract six is eight.

So 1,400 subtract 600 is 800.

And how can we use that then to solve our original equation? Well, in which case, we can say 14 hundred thousand subtract 600,000 must be equal to 800,000.

So you can see here how we used unitizing and known facts to help us.

We didn't need a calculator or a column algorithm, did we? And yes, we can represent what we have just done using a part-part-whole model.

14 subtract six is equal to eight.

And so we know 1,400 subtract 600 must be equal to 800 and 14 hundred thousand subtract 600,000 is equal to 800,000.

And we can represent that in numerals as well.

Now, we could also have represented this on a number line as a different strategy.

We know 14 subtract four equals 10.

So 14 hundred thousand subtract 400,000 would equal 1,000,000 or 1 million.

So we can subtract the 400,000 from 600,000 and that leaves us another 200,000 to subtract.

So if we subtract that 400,000, we know it takes us to that million and then we're bridging that million by subtracting another 200,000, which leaves us 800,000.

Let's check your understanding on this so far.

Can you calculate 1,300,000 subtract 700,000? And then check your answer by using a different strategy so that you know whether you are correct or not.

Pause the video while you do that.

You might want to talk to someone about this.

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

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

You might have used unitizing and known facts.

You might have known that 1,300 subtract 700 would be 600.

So 13 hundred thousand subtract 700,000 would be 600,000.

You might have used partitioning and a number line.

You might have partitioned the 700,000 into 300,000 or 400,000.

And then you can subtract, that's 300,000 to take you to that millions number.

And then you can subtract the remaining 400,000 to take you to 600,000.

Either way, either strategy, you should have come up with the same answer, 600,000.

Let's look at these calculations.

What do you notice? Is there something that is the same? Is there something that is different? Hmm, Aisha noticed that that part, 700,000, it stays the same in each calculation, doesn't it? And Lucas noticed that the whole is increasing by one million each time.

We've got 1,200,000, 2,200,000, 3,200,000.

What does that mean for us? It means if the part remains the same but the whole increases by 1 million, then the difference will increase by 1 million and we can use a number line on partitioning to support us to determine the difference and to make connections.

Let's have a look at this calculation.

We know we can partition the 700,000 into 200,000 and 500,000.

Why are we partitioning it like that? Well, that's because we've noticed from the whole we've got 1,200,000.

So we know if we subtract that 200,000, it will take us to that 1 million boundary.

So we can subtract 200,000.

Then we know we've still got another 500,000 to subtract, which leaves us with 500,000.

And let's have a look at this calculation.

This is the same part, 700,000.

So we can still partition it in the same way because our whole has just increased by one million.

So we've still got that 200,000, which we can subtract first.

This time it's taking us to the two millions boundary.

Then we still need to subtract that 500,000, which leaves us with 1,500,000.

And again, we can have a look at this.

What do you notice? Well, the whole is increased by 1 million but the part is remaining the same.

We can partition that part as we've done before because it would be really useful to subtract 200,000 first.

That would take us to three million.

We then need to subtract the remaining 500,000, which leaves us at 2,500,000.

So let's compare these number lines that we've just looked at.

What do you notice? Did you notice that the number that we land on is the difference each time and the difference is one million more than the previous one and each time that 700,000 is partitioned in the same way.

Why is that? That's right.

That's because the number we are starting with, our whole always has 200,000.

So it's really useful to subtract that 200,000 first.

Aisha has also noticed that we bridged through the previous multiple of 1 million each time.

So first we bridged through 1 million, then 2 million, then 3 million.

And Lucas notices that the difference always had the digit five in the 100,000 place.

We had 500,000, 1,500,000, 2,500,000.

It's always worth stopping and taking a moment and thinking about what do you notice, what is the same and what is different when you're looking at calculations.

So Lucas wins the next game with 4,200,000 points.

Wow, that's a lot.

Aisha only scores 700,000 points.

So how many points does Lucas win by? So we formed our equation, we know we have to find the difference and we know that's a subtraction and we can use what we've noticed previously to calculate this.

Do you see something that is the same or something that is different? We can partition that 700,000 into 200,000, and 500,000 still, can't we? Because our whole has got 200,000 and it would be really useful to subtract that first and then that would leave us at the 400 millions boundary.

So we need to subtract that 200,000 and we get to the previous multiple of 1 million, which is 4 million, and then we're gonna subtract that remaining 500,000.

So Lucas is saying he won by 3,500,000 points.

Well done, Lucas.

Let's check your understanding with that.

Could you calculate 6,200,000 subtract 700,000? Remember to use what we have noticed.

Pause the video, have a go at this question and when you are ready to go through the answer, press play.

How did you get on? Did you notice that the part was the same and in the whole, we still had 200,000? So it's useful to partition the 700,000 part into 200,000 and 500,000.

We can then subtract the 200,000 to get to the previous multiple of 1 million, which is 6 million.

And then we subtract the remaining 500,000.

And the difference here is 5 million, 500,000.

How did you get on with that? Well done.

It's your turn to practise now.

For question one, could you complete these equations? So a handy hint here.

Stop.

Think.

What do you notice about these sets of equations? For question two, could you complete the equations.

This time, I'd like you to use the inequality symbols less than, more than or equals to.

Again, stop and think.

What do you notice? Question three, solve this problem.

Aisha's thinking of a number.

Her number is 800,000 less than 2,300,000.

What number is she thinking of? Pause the video while you have a go at those questions and when you are ready to go through the answers, press play.

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

Question one, you had to complete these equations.

1,300,000 subtract 300,000.

Well, we can see that that would leave us with 1 million.

Then did you notice a pattern? Here we were subtracting one extra 100,000.

So our difference would be 100,000 less.

So 900,000 and then 800,000, 700,000 and 600,000.

For our second set of equations, 5,600,000 subtract 800,000.

Well, we could partition here, couldn't we? We know we need to subtract 600,000 to take us to the previous multiple of 1 million, which would be 5 million.

We'd then still have to subtract another 200,000.

4,800,000.

For the second equation, did you notice you had to find the part, but did you notice something that would help us? The whole stayed the same, the other part increased by 100,000.

So the part must have decreased by 100,000.

So 700,000.

Then you were asked to find the whole.

Maybe you spotted a pattern with the previous two equations.

Our difference is 4,700,000 and we needed a whole that when we subtracted 900,000, it would equal that difference.

So the whole must have been the same as the previous wholes, 5,600,000.

Then our next set of equations, 9,200,000 subtract 500,000.

Well, there's that 200,000 again.

So we need to partition that 500,000 into 200,000 to take us to the previous multiple of 1 million, which is 9 million.

But then we would still have 300,000 to subtract.

8,700,000.

And then spot the pattern here.

The whole is the same and the difference is 100,000 more.

So the part that we're looking for should be 100,000 less 400,000.

And then we had to find the whole amount here.

And we needed a whole amount that when we subtracted 600,000 from, it would equal 8,800,000.

So I can see that it's 200,000 less than the nearest multiple of 1 million, which would be 9 million.

And then we've still got another 400,000.

So it's 9,400,000.

For question two, you had to complete these equations using the inequality symbols or the equal sign.

Here you could have hopefully just compared the different parts of the expressions.

So the whole is the same but on the left-hand side, the part is 500,000.

And on the right-hand side, it's 600,000.

So if we're subtracting 500,000, the difference must be greater than or more than.

For the second equation, one of our wholes is 100,000 less but both parts are the same.

So the left-hand side, 1,400,000 subtract 500,000 must be more than.

2,100,000 subtract 600,000 is equal to 2,200,000 subtract 700,000.

The differences remained the same there.

The whole and the part are both bigger by 100,000.

4,300,000 subtract 400,000.

That's got to be greater than 4,100,000 subtract 300,000 because the whole on the left-hand side, that 4,300,000 is greater.

8,200,000 subtract 400,000, well, that's going to be equal to 8,400,000 subtract 600,000.

Now I'm just comparing the different parts of these expressions, but it might be that you work them out and that is absolutely okay to do as well.

So here I maybe I work these out.

So I would've determined that 8,200,000 subtract 400,000 is 7,800,000 and that's equal to 8,400,000 subtract 600,000, which is also 7,800,000.

7,300,000.

If I'm going to subtract 800,000, well, I'm gonna subtract that 300,000 first and then subtract the remaining 500,000.

So that takes me to 6,500,000.

And then 7,500,000 subtract 700,000.

Well, that's got to be greater than 'cause I'm starting with a whole that's greater and I'm subtracting less.

For question three, you had to solve this problem.

Aisha is thinking of a number and her number is 800,000 less than 2,300,000.

So we could represent this in a bar model to show that we're looking to find a part and to do that we need to subtract.

We can partition that 800,000 into 300,000 and 500,000.

We can then subtract the 300,000 to take us to 2 million and then subtract the remaining 500,000, 1,500,000.

So Aisha is thinking of the number 1,500,000.

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

Fantastic learning so far, everyone.

Really impressed with how you have worked so hard at learning how to subtract across millions boundaries.

We're going to move on now and look at how we subtract from multiples of 1 million.

Let's look at these calculations.

What do you notice about them? Anything, anything that's the same, anything that's different? Could you spot a pattern or could you tell me what might come next? That's right, it would be 1 million subtract 100 and that would be equal to 999,900.

And then 1 million subtract 10, which would be equal to 999,990.

Then we could do 1 million, subtract one, which would be equal to 999,999.

Can you explain this pattern though? Why is that happening? Our whole is always 1 million and we are subtracting powers of 10.

Let's look at this first one.

Let's represent this with place value counters on a place value chart.

We've got 1 million and we're subtracting 100,000 and that is equal to 900,000.

And this is because each power of 10 is composed of 10 of the previous power of 10 and 1 million is equal to ten 100,000s.

That means if we subtract 100,000, we'll have nine hundred thousands left.

Let's look at our next calculation and represent this on a place value chart.

Remember, each power of 10 is composed of 10 of the previous power of 10.

So that means 1 million is equal to 100 ten thousands.

So if 1 million is composed of 100 ten thousands and we subtract one 10,000, we'll have 99 ten thousands left.

And our next calculation, again if we represent this, remember each power of 10 is composed of 10 of the previous power of 10.

So that means that one million is equal to 1,000 one thousands.

And if 1 million is composed of one thousand thousands, if we subtract one of those thousands, we will have 999 thousands left.

And our next calculation where we are subtracting 100 from one million, again, we can represent this on a place value chart.

And we know each power of 10 is composed of 10 of the previous power of 10.

So that means 1 million is equal to 10,000 hundred.

And then if we subtract 100, we will have 9,999 hundreds left.

And our next calculation, remember, each power of 10 is composed of 10 of the previous power of 10.

So 1 million must be equal to 100,000 tens.

If we subtract one 10, we will be left with 99,999 tens.

And our last calculation, again, we can represent one million in ones.

one million is equal to one million ones.

And if we subtract one one, we must have 999,999 ones left.

Let's look at these different calculations.

What do you notice about them? Can you spot a pattern with them? What might come next? Do you spot something? Do you spot that that whole is the same and that we are subtracting? The part we are subtracting are the powers of 10.

So next will be 5 million subtract 1,000, which would be 4,999,000.

Then we might subtract 100.

4,999,900.

And subtract 10, 4,999,990 and subtract one, 4,999,999.

I wonder if you spotted that pattern with the differences as well.

Let's check your understanding with that.

Could you complete these equations? Remember, stop and think.

What do you notice? Is there something that is the same? Is there something that is different? And can you use a pattern to help you complete them? Pause the video while you have a go.

Maybe compare your answers with somebody else.

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

How did you get on? Did you determine that 3 million subtract 1 million is 2 million and then we subtract 100,000? Well, we must have 2,900,000 because the extra 100,000 would take us to that three millions number.

Subtracting 10,000 would be 2,990,000, subtracting 1,000 would be 2,999,000.

Subtracting 100 would be 2,999,900, subtracting ten, 2,999,990.

And then if we just subtracted that one, it would be 2,999,999.

How did you get on with those? Well done.

Lucas and Aisha finished playing that game.

Lucas has 300,000 points.

Aisha has four million points and wants to calculate how many more points she has.

So how many points has she won by? So we know it's going to be a subtraction 'cause we are finding the difference.

And Lucas is saying, well, actually, "I know 1 million is composed of ten one hundred thousands because each power of 10 is composed of 10 of the previous power.

So 4 million then, that must be composed of 40 one hundred thousands 'cause there are four times as many.

And we can now use our known facts.

40 subtract three is 37.

That means 40 hundred thousand subtract 300,000 is 37 hundred thousand.

And we can represent that in numerals as well.

Let's check your understanding with that.

Could you calculate 5 million subtract 6,000? Think about the learning that we've done.

Make a connection.

Try and use your unitizing and your known facts to help you.

Pause the video while you have a go.

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

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

Did you work out that you could do 5,000 and subtract six? That would leave us with 4,994 and they're both thousand, so it'd be 4,994,000.

Well done if you spotted you could use known facts and unitizing like that.

Well done.

It's your term to practise now.

For question one, could you complete the equations? Remember, stop and think.

What do you notice? Is there a pattern that would help you? For question two, could you complete these equations? And for question three, could you solve this problem? Lucas has scored 6 million points during his computer game.

He then gets captured and loses 40,000 points.

How many points does he have now? Pause the video while you have a go at those three questions and when you are ready to go through the answers, press play.

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

For question one, you were asked to complete these equations.

Well, 7 million subtract 1 million, well, we know seven subtract one is six, so 7 million subtract 1 million is 6 million.

And then we can use the pattern that we've learned about before.

If we're going to subtract 100,000, the difference would be 6,900,000.

If we're subtracting 10,000, the difference would be 6,990,000.

If we're subtracting 1,000, the difference would be 6,999,000.

If we're subtracting 100, the difference would be 6,999,900.

If we're subtracting 10, the difference would be 6,999,990.

And if we're subtracting one, the difference would be 6,999,999.

For question two, you had some more equations to complete.

Here we could use unitizing and known facts to help us.

9 million subtract 2 million, well, nine subtract two is seven.

So 9 million subtract 2 million is 7 million, 9 million subtract 400,000.

Well, I know that 9 million is the same as 90 hundred thousand.

90 hundred thousand subtract 400,000 would be 86 hundred thousand or 8,600,000.

9 million subtract 60,000 would be 8,940,000.

9 million subtract 8,000 would be 8,992 thousand.

9,000,000 subtract 300 would be equal to 8,999,700.

5 million subtract 30,000 would be equal to 4,970,000.

3 million subtract 3,000 is equal to 2,997,000.

For question three, you had a problem to solve.

We could represent this in a bar model and we would see that we've got the whole and a known part.

So we know we need to subtract.

We've got 600 ten thousands subtract four ten thousands, which is equal to 596 ten thousands.

So 6 million subtract 40,000 must be equal to 5,960,000.

So that's how many points Lucas has at the end of the game.

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

Fantastic learning today.

I'm really impressed with how hard you have tried.

You have really deepened your understanding on how we can subtract multiples of powers of 10, crossing those millions boundaries.

We know powers of 10 and then multiples can be subtracted using the language of unitizing.

And we know known facts can be used alongside unitizing to subtract powers of 10 and their multiples.

Subtraction of powers of 10, and their multiples can be represented on a number line as well.

So lots of strategies that you can use there before you rush into using a calculator or a column algorithm.

I've had great fun learning with you today and I look forward to learning with you again soon.