Lesson video

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

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

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

The lesson is called add multiples of powers of 10, crossing the millions boundaries.

As we move through the learning today, we are going to deepen our understanding of how we can add multiples of powers of 10, using a mental strategy such as a number line, even when we cross the millions boundary.

Now sometimes, new learning can be a little bit tricky.

But I know if we work really hard together and I'm here to guide you, then we can be successful in our learning today.

Let's get started then, shall we? How do we add multiples of powers of 10 and cross that millions boundary? These are the keywords that we will be using throughout our learning today.

We have million and unitize.

It's always good to practise saying these words aloud.

My turn, million, your turn.

Nice, my turn, unitize, your turn.

Fantastic! So 1 million is composed of 1,000 thousands and we write it as a one followed by six zeros.

Then 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 a key to understanding place value and we are going to be unitizing a lot throughout this lesson today to help us.

So let's get started, shall we? We're going to start by thinking about how we can bridge across 1 million.

And in the lesson today, we have Aisha and Lucas to help us.

Aisha and Lucas are playing a game.

Aisha has got 700,000 points and Lucas has 600,000 points.

Wow, that's a lot of points, isn't it? I wonder what game they are playing.

And Aisha wonders what the sum of the points is.

To find the sum, we need to add the scores together.

Hmm, what do you think? Do you agree with Aisha? She says, "These numbers are just too large to add mentally.

We can't do this in our heads.

We need to use a written column method, or what about a calculator?" Ah, Lucas is respectfully challenging her.

Why might he do that? That's right, "We can use our number-sense superpower," instead, can't we, for sure.

We can use unitizing and our known facts to help.

And let's start by looking then at a simpler case and then we can build up to solving this one.

Let's look at 700 add 600.

And Aisha knows that the answer to this will bridge 1,000.

Why does she know that? Could you tell? Why should it be greater than 1,000? That's right, it's because 1,000 is composed of two equal parts of 500 and both of these addends are greater than 500.

So the answer will be greater than 1,000.

"We can use unitizing and known facts to help find this sum." Six added to seven is 13 and that means then that 600 added to 700 is 13 hundred or 1,300.

So that then means if we think about our numbers, 600,000 added to 700,000 must be equal to 1300 thousand.

And we write that as 1,300,000.

So can you see here how using our known facts, seven add six and unitizing 700 add 600 is equal to 1300, so 700,000 and 600,000 is equal to 1,300,000.

See how unitizing and known facts helped us to find the sum of these two numbers? We did not need to use a column or a calculator, did we? So we can represent this unitizing, using part-part-whole models.

We've got seven added to six is 13.

So 700 added to 600 is 1300 or 1,300.

That means 700,000 added to 600,000 is 1,300,000 and we can write that in digits.

We've got 700,000 added to 600,000 is 1300 thousand, or 1,300,000.

So we've used known facts and unitizing.

But actually we could also represent this on a number line.

If we start with 700,000, we know seven and three make 10.

So 700,000 and 300,000 would make 1,000,000.

We can add the 300,000 from the 600,000 and that leaves us another 300,000 to add on.

So we know the sum must be 1,300,000.

And Aisha is saying, she can think of another way, "We could calculate this." Can you? "We could use our near double facts.

We know double six is 12, so double 600,000 must be equal to 1,200,000 and we need to add the extra 100,000," don't we, which would give us the 1300 thousand or 1,300,000.

We could also use our double seven facts and subtracted 100,000.

Sometimes there's lots of different ways to calculate in maths and it's always worth stopping and thinking about how could I do this without a calculator? How could I do this without a column method? If we use the double sevens, then double seven is 14, we would need to subtract that 100,000.

We would still get 1,300,000.

So whichever strategy we use, we used known facts and unitizing, we used a number line and we've used doubling a known fact and unitizing and they've all given us the same sum, haven't they? Let's check your understanding with this.

Could you calculate 800,000 and 700,000 and then check your answer by calculating using a different strategy.

Pause the video while you do this.

You may like to talk to a friend and compare strategies.

Press play when you are ready to go through the answers.

How did you get on? Did you start by maybe using unitizing and known facts? So we know 800 and 700 is 1500, so 800,000 and 700,000 must be 1500 thousand, or 1,500,000.

You might have used partitioning and a number line.

So we start with 800,000.

We know we need to add 200,000 to make the nearest multiple of 1 million.

And then we've still got 500,000 to add on.

So the sum would be 1,500,000.

You might have used near doubles, you might have partitioned the 800,000 into 100,000 and 700,000.

Double 700,000, add that 100,000 is 1,500,000, or you might have done your near doubles with the 800,000, which is 1,600,000 and subtracted 100,000 to get 1,500,000.

I wonder which strategy you used.

As you can see, all the strategies gave the same sum and each of them were more efficient than using a written column algorithm.

It's your turn to practise now.

For question one, could you complete these equations? And then do you spot a pattern? Can you tell me which pair of equations would come next? For question two, could you complete the equations, using the inequality symbols, less than, more than or equals to? And for question three, could you solve this problem? Can you check your answer using a different strategy? So Aisha has 800,000 points and Lucas has 900,000 points.

How many points do they have in total? So you should be able to tell me without me going through the answers that you are correct, because you will check your answer, using a different strategy.

And if you get the same answer, you are likely to be correct.

If you don't, it's worth going back and trying another strategy, just to double-check.

Pause the video while you have a go, 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.

Question one, you asked to complete the equations and see if you spotted a pattern and what would come next.

So we know 500 and 300 is 800.

So 500,000 add 300,000 must be 800,000.

500 add 400 is 900.

So 500,000 add 400,000 must be equal to 900,000.

500 add 500 is 1000.

So 500,000 add 500,000 must be equal to 1000 thousand or 1 million.

500 plus 600 is equal to 1100.

So 500,000 plus 600,000 is equal to 1100 thousand or 1,100,000.

500 plus 700 is equal to 1,200 or 1200.

500,000 add 700,000 is equal to 1200 thousand, which is equal to 1,200,000.

And did you spot what might come next? So you might have spotted that one addend was increasing by either a hundred or by 100,000 and the other addend stayed the same.

This means that the sum would increase by either 100 or 100,000.

And the next pair of equations, then would be 500 add 800 is equal to 1300.

So 500,000 add 800,000 is equal to 1,300,000.

For question two, you needed to complete the equations, using the inequality symbols.

Well, 900,000 add 300,000 was got to be less than 900,000 add 500,000.

Because the 900,000 addend is the same in both pairs, but actually in one, you're only adding 300,000, the other, 500,000.

So I didn't have to work these out.

I could use my number-sense superpower.

300,000 add 800,000, well, that's got to be equal to 800,000 add 300,000.

Commutative law, I've just changed the order of the addends, but the sum will remain the same.

600,000 add 800,000, well, that's got to be greater than 700,000 add 400,000.

Because the 700,000 is 100,000 more.

But actually the 400,000 is 400,000 less than the 800,000.

500,000 add 700,000, well, that's less than 600,000 add 800,000.

You might have reasoned about the addends, or you might have calculated them to be 1,200,000 and 1,400,000.

For the next question, 600,000 and 800,000 is 1,400,000.

Well, that's equal to 700,000 and 700,000.

Seven and seven is 14, so 700,000 and 700,000 will be 1,400,000.

So those are equivalent.

Did you realise that you could use your number-sense superpower to compare these addends? This one here, as we talked about, the 900,000 is the same, isn't it? So the first expression must be less than the second.

And you could use your number-sense superpower for each of those.

But if you calculated it, that's okay.

But it's always worth being aware in the future, stopping and thinking.

For question three, you need to solve a problem.

You might have used unitizing and known facts.

We know that 800 add 900 is 1700.

So 800,000 and 900,000 must be 1700 thousand, or 1,700,000.

You might have used partitioning and a number line.

You might have partitioned the 900,000 into 200,000, because you know that is a complement to 1 million and you had 700,000 left over, which means the sum would be 1,700,000, or you might have used near doubles.

We know that 900,000 is composed of 800,000 and 100,000.

Double eight is 16, so double 800,000 is 1,600,000.

Add the extra 100,000, 1700 thousand or 1,700,000.

You might have worked with the other double and done double nine is 18, so double 900,000 is 1,800,000.

Subtract that 100,000 either way, you would get 1700 thousand or 1,700,000.

How did you get on with those questions? Brilliant! Fantastic learning so far.

Really impressed with how hard you tried with your learning on bridging across 1 million.

We're now going to take it a step further and look at how we can bridge across other millions boundaries and not just 1 million.

So let's look at these calculations.

What do you notice about them? What's the same about them, what's different about them? Aisha has noticed that one addend, that second addend 700,000 stays the same.

Did you notice that? Lucas noticed that the other addend increases by 1 million each time.

We had 500,000, then we had 1,500,000, 2,500,000, 3,500,000.

Did you spot that? But what does that mean for us? Well, if one addend remains the same and the other addend increases by 1 million, then the sum will increase by 1 million.

So if we just determine the sum for the first one, we know the other sums will all just increase by 1 million.

So we can use a number line and partitioning to support us to determine the sum and to make connections.

So let's partition 700,000 into its parts, 500,000 and 200,000.

Why 500,000? Because it's a complement to a million with the other part, with the other 500,000.

And then we know we've still got 200,000 to add on, which gives us 1,200,000.

Let's look at this now with our second calculation, 1,500,000 plus 700,000.

Again, we can partition the 700,000 in the same way.

But this time, it's a complement to 2,00,000.

And we still need to add that extra 200,000, 2,200,000.

We look at our next calculation.

This time, we can still partition the 700,000 into its constituent parts, 500,000 and 200,000.

This time, it's a complement of 3,00,000.

And then we've still got the additional 200,000 to add, 3,200,000.

And then 3,500,000 add 700,000, same strategy.

We can partition 700,000 into 500,000 and 200,000.

This time, it's a complement of 4,00,000.

And we still need to add the extra 200,000, so it's 4,200,000.

Let's compare those number lines.

What do you notice about them? We've said that the first addend is increasing by 1 million.

But we're always adding 700,000.

So the sum will increase by 1 million.

And Aisha has noticed that each time, the sum is 1 million more than the previous sum.

And that proves what we said earlier.

And she's noticed that the 700,000 is partitioned in the same way each time.

We talked about that, didn't we? Aisha also noticed that we bridged through the next multiple of 1 million each time.

First, we bridged 1 million, then 2 million, then 3 million, then 4 million.

Lucas noticed that the sum always had the digit two in the 100,000s place, 1,200,000, 2,200,000, 3,200,000, 4,200,000.

I wonder if you notice any of those things.

And Lucas is saying then that "Using this pattern, we can calculate the sum of 6,500,000 and 700,000." Do you think you could have a go at doing that? That's right, we just need to partition the 700,000, like we've done before into 500,000 and 200,000.

And then we get that complement to the next multiple of 1 million, which would be 7 million and then we can add the remaining 200,000.

So the sum would be 7,200,000.

Let's check your understanding with this.

Could you calculate 8,500,000 added to 700,000? 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 say that we need to partition the 700,000 into 500,000 and 200,000? Then we can add the 500,000, which gives us the next multiple of 1 million, which is 9 million and then we add the remaining 200,000.

And the sum is 9,200,000.

How did you get on with that? Well done.

Let's look at this new equation, 4,900,000 add 700,000.

How would you solve it? Lucas is saying, "We could partition and use a number line." We could, couldn't we? Is there another way though? That's right, have you noticed? 4,900,000, it's really close to the next multiple of 1 million, isn't it? It's really close to 5 million.

So we could just adjust that first addend and then it would be easier to add.

If we adjust it by 100,000, we get 5 million.

5 million and 700,000 is 5,700,000.

Hmm, but is that the answer to the original calculation? Can't be, can it? What we need to do, is remember to adjust the sum or the other addend, so that we get the answer to the original equation.

So we need to subtract that 100,000 that we added, 5,600,000.

Let's check your understanding of this.

Could you solve this equation, using that compensation strategy? 6,900,000 add 400,000.

So think about which multiple of 1 million, is that 6,900,000 close to? How could you adjust that addend? 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 you could add 100,000 to make the next multiple of 1 million, 7 million? Then we can add that quite easily, 7,400,000.

But then we need to remember to adjust that sum by subtracting that 100,000, 7,300,000.

Your turn to practise now.

For question one, could you complete these equations? Remember to use a mental strategy, think about what you notice about the numbers.

Could you use that number line? For question two, could you complete the equations, using the inequality symbols, less than, more than or equals to? For question three, could you solve this problem? "Aisha is thinking of a number.

Her number is 1,700,000 more than 4,900,000.

What number is she thinking of?" 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? Let's have a look.

For question one, you had to complete the equations.

1,800,000 add 600,000, well, 1,800,000, we need the 200,000 from the 600,000, which would make 2 million and 400,000 left over, 2,400,000.

And then here, did you spot something for the next few equations? There was a pattern.

One of the addends was increasing by 100,000.

So the sum increased by 100,000, 2,500,000, 2,600,000.

And the last one, you had to find a missing part, but you could follow the pattern.

Because you could see that the sum increased by that 100,000 and the other addend stayed the same.

So that missing addend must be 900,000.

And then you had to find some missing parts.

Here, we can think about what do we add to 3,700,000 to give us 4,300,000? Well, think about complements to 1 million.

We needed to add 300,000, which would give us to 4 million, but then another 300,000.

So that missing part must be 600,000.

And here, what do we need to add to 800,000 to get 4,400,000? So we know here that we needed to add 3,600,000.

5,400,000 added to 800,000 is equal to 6,200,000.

We can take 600,000 from the 800 and we'd have 200,000 remaining.

6,700,000 add 900,000, well, that's equal to 7,600,000.

2,900,000, well, we need to add 600,000 and that would give us 3,500,000.

Because 100,000 would take us to that 3 million, then we need another 500,000.

So that's 600,000 in total that we need to be adding.

2,900,000, oh, did you spot something here? One addend stayed the same, the other addend increased by 100,000.

So that sum would increase by 100,000, 3,600,000.

Here then, same pattern, did you spot? The sum increased by 100,000, an addend increased by 100,000, so the other addend must've stayed the same, 2,900,000.

Did you unitize? Did you partition or did you use compensation? They're all different strategies you might have used for different questions and they are all equally acceptable to use and far more efficient than doing a column algorithm.

For question two, you needed to complete the equations, using the symbols, less than, more than or equals to.

So here, you might have stopped and looked at these numbers and used your number sense rather than working them out.

And you might have noticed that one part was the same and the other part, well, on the right-hand side, it was 500,000.

On the left-hand side, 300,000.

So 2,900,000 add 300,000 has got to be less than 2,900,000 add 500,000.

On the second question, you might have noticed something very similar here.

We've got 3 million, we've got 400,000, 800,000, just written slightly differently, aren't they, the addends? So they are equal to each other.

600,000 add 8,700,000, well, that's got to be less than 600,000 add 8,800,000.

One addend is the same and one addend is a 100,000 more.

So that must be larger.

500,000 add 7,500,000, that's got to be less than 500,000 add 7,600,000.

6,900,000 add 900,000, well, that's got to be greater than 6,900,000 add 700,000.

Because the first addend is the same and the second addend is smaller on the right-hand side.

Did you realise that when we are comparing, we don't always have to calculate? So in none of those, did I calculate.

I was always just comparing the equivalent addends.

For question three, you had to solve a problem where Aisha was thinking of a number.

So we could represent this in a bar model, showing the two parts.

And to find the whole, we need to find the sum of the known parts.

If we add those together, well, how could we do that efficiently? I was going to try partitioning the 1,700,000.

We can partition the 1,700,000 into 1,100,000 and 600,000.

We can add the 1 million to make 5,900,000.

And we can add the 100,000 to make the next multiple of 1 million, which is 6 million and then add the remaining 600,000.

So the sum would be 6,600,000.

But maybe you thought to use the compensation strategy.

You might have adjusted the first addend to 5 million, but then you needed to remember to adjust the sum by subtracting 100,000 to give 6,600,000.

If we add that 100,000 to one addend, remember, we need to subtract 100,000 from the other sum to compensate, or 100,000 could have been subtracted from the other addend at the same time.

And that would've kept the sum the same as well.

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

Fantastic learning today.

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

We know powers of 10 and their multiples can be added, using the language of unitizing.

We know known facts can be used alongside unitizing to add powers of 10 and their multiples.

And we know addition of powers of 10 and their multiples can be represented on a number line.

Fantastic learning today.

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

I'm really proud of you.

And I look forward to learning with you again soon.