Lesson video

Hello, how are you today? I trust you all well, and on top form ready for your maths lesson.

My name is Dr.

Shorrock, and I look forward to guiding you through the learning today.

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

This lesson is called, "Solving problems using column addition and subtraction." As we move through our learning today, we are going to deepen our understanding of the formal written algorithms for addition and subtraction.

And we will use these to solve problems when the numbers are precise.

So, using a mental strategy would be a little bit trickier.

Now, sometimes new learning can be a bit tricky, but it is okay because I am here to guide you, and I know if we work really hard together, then we can be successful.

Let's get started then, shall we? How can we solve problems using column addition and subtraction? This is our key word for the learning today.

The word is regroup.

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

Let's have a go together.

My turn.

Regroup.

Your turn.

Fantastic.

Now, the process of unitising and exchanging between place values is known as regrouping.

So, for example, 10 tens can be regrouped for one hundred.

One hundred can be regrouped for 10 tens.

Let's get started with our learning today then, shall we? We're going to start by looking at one-step problems. And in our lesson today we've got Aisha, Lucas, and Izzy to help us.

Aisha and Lucas are researching the area of different European countries, and they present their findings in a table.

Have a look at the table.

Can you see it's got two columns.

The first column is the European country.

We have France, Spain, Sweden, Germany, Italy, and the United Kingdom.

And then the second column is their area in kilometres squared.

Now we want to know how much greater is the area of Germany than the area of the the United Kingdom.

Can you see those in the table? So the area of Germany, 357,022, and the area of the United Kingdom, can you read that number? It's 243,610.

And we can see that the area of Germany is greater, but we want to find out how much greater.

So, good idea, Lucas, let's represent this in a bar model.

And we're comparing those two values, aren't we? We want to know which one is greater.

So one of the values is going to be our whole, and the other is going to be a part.

The area of Germany is the larger amount, so this is going to be our whole.

So I've set up my bar model and now I can substitute the country names for their areas.

And then we need to find that missing part, which is the difference between the two.

And when we find the difference, we need to subtract the known part from the whole.

And I formed an equation from my bar model, 357,022 subtract 243,610.

How would you subtract 243,610 from 357,022? Aisha says, "We could use our understanding of place value and use a mental method." What do you think? Is that what you would do? Lucas respectfully challenges her.

I wonder why he does that.

Ah, Lucas is looking at the numbers and he could see that there will be some regrouping.

So using a mental method, although it's possible, it might just be a little bit tricky, there might be a more efficient way.

So he suggests it will be more efficient to use column subtraction algorithm.

So when we use the column subtraction algorithm, we need to lay out the column very neatly.

and the minuend, which is our whole, or the 357,022, is the number at the top of the algorithm.

So we can see it's the whole in our bar model, and it's the first number in our equation, and then it's the top of our written algorithm.

The subtrahend, which is the part, is lined up underneath very carefully, and the large equal sign, well that frames the difference, our answer will be the difference.

It will tell us how much greater the area of Germany is than the area of the United Kingdom.

Let's work through this column algorithm together, shall we? We always start on the right-hand side, so in the ones column.

Two ones minus zero ones, well that's equal to two ones.

Two tens minus one ten is equal to one ten.

Oh, what do you notice next though? There are insufficient hundreds, aren't there? We can't subtract six hundreds from zero hundreds, so we need to regroup from the column to the left.

That would be the thousands column.

We can regroup one thousand for 10 one hundreds.

So we're going to take one of those thousands, so we've only got six thousands left, and we can regroup it for 10 one hundreds.

Now we can do that subtraction, can't we? Ten one hundreds minus six one hundreds is equal to four one hundreds.

Six one thousands minus three one thousands is equal to three one thousands.

Five ten thousands minus four ten thousands is equal to one ten thousand.

And then three 100,000s minus two 100,000s is equal to one 100,000.

The area of Germany is 113,412 kilometres squared greater than the area of the United Kingdom.

Let's check your understanding with this.

Could you tell me the difference in the area of France and the area of Spain? Pause the video while you have a go at that.

And when you are ready to go through the answer, press play.

How did you get on? First of all, we had to identify the numbers that we were going to use.

France and Spain are the top two countries in that table.

So we can form an equation, 646,809, that's the larger number, so that is our minuend, it's the first in our equation, it's the whole.

And we're going to subtract 505,370.

Then we've got nine ones subtract zero ones is nine ones.

Oh, do you notice something here? We have insufficient tens.

So we need to regroup from the column to the left.

So we had eight hundreds, we're going to regroup one hundred, so we're left with seven hundreds and that gives us 10 tens.

So 10 tens subtract seven tens is three tens.

Seven hundreds subtract three hundreds is four hundreds.

Six thousands subtract five thousands is one thousand.

Four ten thousands subtract zero ten thousand leaves us with four ten thousands And six 100,000s subtract five 100,000s is one 100,000.

So we can see the difference in area of France and the area of Spain, is 141,439 kilometres squared.

How did you get on with that? Well done.

Aisha now wants to know how much larger the total area of India is than Nigeria.

How would you work this out? Would you do a mental method or would you use a column algorithm? And what do you notice about the numbers? There's something that's slightly different this time.

Have you noticed it already? And Aisha knows that to find the difference in land area, we need to subtract.

So let's set up this column subtraction algorithm.

Hmm, do you notice something? Yes, "Stop," Lucas is saying, the numbers have different numbers of digits, don't they? So what does that mean? It means that she set up the column algorithm incorrectly.

Because digits in the same place value place should be aligned.

They should be underneath each other.

So now we've set the column algorithm up accurately.

Our ones digits are underneath each other.

Let's have a go working through this one together.

Zero ones subtract zero ones, well that's zero ones.

Nine tens subtract zero tens is nine tens still.

Oh, now what do you notice? I've got five hundreds and I need to subtract eight hundreds.

There's insufficient hundreds, isn't there? So we need to regroup from the column to the left.

I'm going to regroup one thousand for 10 one hundreds, so I've got 15 one hundreds.

Now I can subtract the eight one hundreds, which is seven one hundreds.

Six one thousands subtract three one thousands is three one thousands.

Eight ten thousands subtract two ten thousands is six ten thousands.

Oh, do you notice something here? I've got two 100,000, and I need to subtract nine 100,000, so there's insufficient 100,000.

So I need to exchange from the column to the left, the three millions.

And I can exchange one of those millions for ten 100,000.

So I've got 12 100,000s subtract nine 100,000s is three 100,000s.

And then I can finish off by subtracting two.

Oh, there's nothing to take away.

Ah, so it's the same as subtracting zero.

So, two subtracts zero is two millions.

That means the total area of India is 2,363,790 kilometres squared greater than the total area of Nigeria.

Let's check your understanding with this.

Could you now use the column algorithm to calculate this calculation? Pause the video while you have a go.

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

How did you get on? So we've got our equation and we can set up the column algorithm.

We've noticed that the larger number, the minuend, is on the top of our algorithm, but the part, that subtrahend, has got fewer digits.

Three ones subtract zero ones, well that's three ones.

Five tens subtract zero tens, well that's five tens.

Four hundreds subtract one hundred, we can do that, so that's three hundreds.

Five thousands subtract three thousands, we can do that, that's two thousands.

Oh, now we've got four ten thousands and we need to subtract six ten thousands.

There's insufficient ten thousands, so we need to exchange from the column to the left.

I'm going to exchange one of the 100,000, that gives me 10 ten thousands, so in total now I've got 14 ten thousands, so I can now subtract those six ten thousands, which is eight ten thousands.

Eight 100,000s subtract seven 100,000s is one 100,000.

And then I've got five and I'm subtracting nothing, I'm subtracting zero, so I've got five millions remaining.

So the answer is 5,182,353.

Let's compare now the area of the countries Myanmar, which is roughly 680,378 kilometres squared and Cameroon, which is roughly 475,442 kilometres squared.

How would you work this out? Mental method or column algorithm? Well, Aisha was saying she can see Myanmar has the greater land area.

But how much greater? That's what we're trying to work out, isn't it? So let's have a go at a column algorithm.

Eight ones subtract two ones is six ones.

Seven tens subtract four tens is three tens.

Oh, we can't do the next one, can we? We've got three one hundreds and we need to subtract four one hundreds, so we need to regroup from the column to the left.

Oh, but the column to the left, that's a zero.

Oh, we can't, can we? There's nothing there to regroup from.

What do we need to do? That's right, we need to look to the column to the left of that.

So to the ten thousands column with the eight, and we actually need to regroup twice.

We regroup from the eight, which leaves us with seven ten thousands, and then that means we've got 10 one thousands, but that's no good.

We need to regroup again.

That leaves us with nine one thousands, and then we can have our 10 hundreds added to the three hundreds that we've already got, we've got 13 one hundreds.

Now we can subtract the four one hundreds, which leaves us with nine one hundreds.

Then we can look at the thousands column.

Nine one thousands subtract five one thousands is four one thousands.

Seven ten thousands subtract seven ten thousands, well that's zero ten thousands.

And then six 100,000s subtract four 100,000s is two 100,000s.

So the area of Myanmar is 204,936 kilometres squared greater than the area of Cameroon.

Did you notice this time that there was a zero in the column that we needed to regroup from? So we needed to regroup from the next column to the left as well.

So we had to regroup twice.

So it was a little bit trickier.

Let's check your understanding with that.

Could you use the column algorithm to calculate 940,587 subtract 203,602? Pause the video while you have a go.

And when you're ready for the answer, press play.

How did you get on? Seven ones subtract two ones, we can do that, that's five ones.

Eight tens subtract zero tens, we can do that, that's eight tens.

Five hundreds subtract six hundreds.

Oh, there's insufficient hundreds isn't there? So we need to regroup.

Oh, there's a zero.

We can't regroup from that zero, so we need to look to the column to the left of the zero.

It's a four ten thousands.

We can regroup from that.

So we can regroup one of the ten thousands for 10 one thousands.

Then we need to regroup one of the thousands for 10 one hundreds.

Now I have 15 one hundreds, so I can subtract the six one hundreds.

That's nine one hundreds.

Now I can continue with the subtraction.

Nine one thousands subtract three one thousands is six one thousands.

Three ten thousands subtract zero ten thousands is three ten thousands.

And then nine 100,000s subtract two 100,000s is seven 100,000s.

So the difference, the answer, 736,985.

How did you get on with that? Well done.

Let's revisit this table.

And this time, we want to know what is the combined area of France and Sweden.

What do we mean by combined? That means we have to put together.

So this is going to be an addition.

Let's start by representing this as a bar model.

We're finding a combined area, so we must have two parts.

The parts are France and Sweden.

Then I can substitute the country names for their areas.

To find the combined area, we need to find the sum of the parts so I can form an equation from my bar model.

646,809 add 450,185.

How would you add these numbers? Would you use a mental strategy or would you use a column algorithm? Good idea.

I think we'll do a column algorithm.

The numbers are really quite precise, aren't they? And I can see there will be some regrouping, so it's probably more efficient to use a column algorithm.

Let's work through this.

Nine ones added to five ones is equal to 14 ones.

Oh, but the ones column sum is greater than 10, we've got 14, so we need to regroup.

14 ones is equal to one tens and four ones.

Now we can work on our tens.

We've got zero tens added to eight tens is eight tens.

But we need to add that regrouped tens so we've actually got nine tens.

Eight one hundreds added to one one hundreds is nine one hundreds.

Six one thousands added to zero one thousands is six one thousands.

Four ten thousands added to five ten thousands is nine ten thousands.

Six 100,000s added to four 100,000s is ten 100,000s.

Oh, what do you notice? That 100,000s column sum is greater than 10, so we need to regroup.

And then we have one one million.

So the combined area of France and Sweden is 1,096,994.

Let's work together to find the sum of 435,798 and 1,928,791.

Do you notice something about these numbers? It's always worth stopping and thinking what do I notice.

I've noticed something.

I've noticed that the numbers have a different amount of digits, so we need to make sure they're properly aligned.

One ones add eight ones is nine ones.

Nine tens add nine tens is 18 tens, so we need to regroup.

Seven one hundreds add seven one hundreds is 14 one hundreds, but we've got to add that regrouped one, so that's 15 one hundreds.

We're going to need to regroup again.

We've got five one hundreds and one one thousand.

Eight one thousands and five one thousands, well that's 13 one thousands.

I've got to add the regrouped one, so that's 14 one thousands.

So, again, I need to regroup.

Two ten thousands and three ten thousands is five ten thousands.

I've got to add that regrouped one, so that's six ten thousands.

Nine 100,000s add four 100,000s, well that's 13 100,000s.

So again, I'm going to need to regroup and then I can add the millions.

One million add that regrouped million is two millions.

So the sum is 2,364,589.

If the numbers that we are adding have a different amount of digits, then the digits with the same value place should be aligned.

Remember that, key learning.

Could you find the sum of 609,251 and 2,135,814? Pause the video while you have a go at this.

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

How did you get on? Did you set up your column algorithm very carefully to make sure that digits of the same place value are aligned? And then let's work through this.

Four ones and one ones, that's five ones.

One tens and five tens, well that's six tens.

Eight one hundreds and two one hundreds, well that's 10 one hundreds.

That digit sum is 10, so we're going to need to regroup.

But five one thousands and nine one thousands is 14 one thousands, add that regrouped one, is 15 one thousands.

So again, the digit sum is 10 or greater, so I need to regroup.

Three ten thousands added to zero is three ten thousands.

But I've got to add that regrouped one, so that's four ten thousands.

One 100,000s added to six 100,000s is seven 100,000s.

And then two million added to nothing, well that's just two million.

So the sum is 2,745,065.

How did you get on with that? Well done.

Time for you to practise now.

For question 1, could you calculate these using the appropriate column algorithm? For question 2, got a problem to solve.

Lucas and Aisha play a game and they both win points.

For Part A, could you tell me what their combined points total is.

For part B, how many more points did Lucas score than Aisha? 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.

For question 1, you were asked to calculate these using the appropriate column algorithm.

The first was an addition.

I lined up both add-ins, and when I added the parts together, the answer was 4,268,158.

For Part B, you might have noticed this was a column subtraction.

So I lined the parts up and I performed the subtraction, being very careful to keep an eye out for when there was insufficient of any digit that I would need then to regroup.

And the difference was 1,865,954.

For Part C, we were adding, this time, I noticed there were a different number of digits, so I was very careful when I lined my parts up with each other.

And the sum was 2,629,201.

Part D was a subtraction.

Again, the numbers had different amounts of digits, so I was very careful when I lined the digits up, and I was very careful to look out for any places where I might need to regroup.

And the difference was 1,992,230.

For Part E, it was a subtraction.

Again, very carefully lining my column algorithm up and looking out for places where I might need to regroup, especially if I needed to regroup from a zero when I had to regroup from the digit to the left of that.

And the difference was 1,087,122.

For part F, again, lining up the part and the whole very carefully, and looking out for places where I might need to regroup because it was insufficient to any digit, and the difference was 4,804,348.

For question 2, Lucas and Aisha were playing a game and they both won point.

First all, you had to work at their combined points total.

When we combine something, we add the parts.

So, using a column algorithm, their combined score points total is 2,384,589.

For Part B, you had to work at how many more points Lucas scored.

So here, you can see we've got the whole and a part, and we're needing to find the difference, so it's a subtraction.

And then lining my digits up very carefully and looking out for places where I might need to regroup, I could find the difference.

And Lucas scored 1,512,993 points more than Aisha.

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

Fantastic learning so far.

I'm really impressed with how hard you are trying.

Let's move on and look at multi-step problems. Aisha and Lucas invite Izzy to play their game.

They record their scores in a table.

And the question is, what is their combined points total? Remember, combined means to put together.

So let's start by representing this in a bar model.

And we must have three add-ins.

As our bar model, we've got three parts.

What strategy would you use to combine these three add-ins? What would you do? Would you use a mental strategy? Lucas is saying, "We could add two of the add-ins using the column addition algorithm, and then add the other one in another column addition.

That's not efficient.

Is there a more efficient way? That's right.

Let's just put all three in one column algorithm.

Let's have a look.

See what it looks like.

The numbers have different amounts of digits, so we do have to be really careful and check that the appropriate digits align.

There's our column algorithm, and this time we can work through this together.

We've got nine ones add one ones add two ones is 12 ones.

We need to regroup.

Two tens add seven tens add one tens is 10 tens, add the additional regrouped tens, is 11 tens, we need to regroup.

Eight one hundreds add two one hundreds add six one hundreds, add that regrouped one, we've got 17 one hundreds, so we need to regroup.

Five one thousands, add three one thousands, well that's eight one thousands, add another eight, we know double eight is 16, add the regrouped one, is 17 one thousands.

We need to regroup again.

Four ten thousands, add eight ten thousands, add nine ten thousands, add that one ten thousand, is 22 ten thousands.

We need to regroup.

Two 100,000s, add five 100,000s add those two regrouped ones is nine 100,000s.

We've got one million added to three million is four million.

So their combined points total is 4,927,712.

So you can put more than two add-ins in a column addition.

Lucas decides to practise the game at home so he can become even better at it.

First, he wins 102,341 points.

Then, he loses 1,310,230 points.

Oops.

Finally, he then wins 5,620,000 points.

Brilliant.

How many points does he have at the end of his three practises? Hmm.

Well, let's form an equation to represent this good idea.

So his total points, well, first he wins 102,341.

Then he loses, so that's a subtraction.

But then he wins some more, so it's an addition.

So in this one equation, we've got two operations.

And how would we solve this equation then? Lucas notices that 1,310,230 is greater than now a starting number of 102,341.

What can we do? We can represent this in a bar model.

These are the points that Lucas won.

This is the whole amount, and these are the points that Lucas lost.

And actually, we need to find that missing part.

How would you calculate that unknown part? Well, first we can find the whole by adding, then we can subtract the part.

So, all we're going to do is rearrange that equation.

We change the order of the equation.

We decided to add the parts first, the parts that he won first before we then do the subtraction.

You can see how we have reordered that equation.

So now we can add first before we then subtract.

So this is the whole amount of points that he won.

We can then subtract the points he lost.

So Lucas scored 4,412,111 points in total.

Let's check your understanding with that.

The order of the equation should be changed to support the calculation.

So 257,109, subtract 1,046,245 add 3,890,000.

Should we change the order of the operation? Pause the video while you think about that.

When you're ready for the answer, press play.

How did you get on? Did you say, "Yes it's true"? But why is it true? Is it because we should change the order of the equation so we are completing the addition first? Or is it B, we should change the order of the operation to start with a negative number? Which of those do you think? Pause video while you think about it, maybe chat to somebody else about this.

When you're ready for the answer, press play.

How did you get on? Did you say it was A, we should change the order of the equation so we can complete the addition first? Aisha also practises the game.

She wins 8,399,768 points.

Then she loses 1,306,008 points.

Finally, she loses another 74,641 points.

Oh dear.

How many points does she have at the end of the game, and how would you calculate her points? Good idea.

Let's represent this in a bar model.

How would you find that missing part? So the total point she has is her whole subtract the two parts.

But we need to find that missing part.

First, that's right, we can add the known parts, then we subtract from the whole.

So we have an answer of 1,380,649.

We can then find that missing part by subtracting that from the whole.

So Aisha scored 7,019,119 points in total.

Let's check your understanding with that.

Could you look at the bar model and choose the most efficient way to find the missing part? Pause the video while you have a look at the options.

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

How did you get on? Did you say that "Yes, ideally, to find the missing part, first we add the known parts, then we subtract from the whole." Well done.

Your turn to practise now.

Using these clues, could you order the final points that the children scored from least points to most points? Pause the video while you look at each of the clues, and when you're ready to go through the answers, press play.

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

We can represent this in a bar model.

We can see we're looking to find a missing part, and we can form an equation from that.

And first we're gonna add the known parts then we're going to subtract from the whole.

So I can add the known parts, so the total parts are 616,806.

Then we can subtract this from the whole.

So, the total points that Lucas has 1,781,250.

Our next clue about Aisha, we can form an equation to represent this.

She wins 98,034, then loses 1,093,004, then wins 4,390,062.

Here, we're going to reorder the equation.

So we can do the additions part first.

Then we need to subtract the part that she lost.

So the total points Aisha wins is 3,395,092.

Our last clue.

We can represent this in a bar model and we can see she wins.

So we're going to combine three parts in a column algorithm.

So the total points Izzy wins is 1,438,482.

So we can then order the children's points.

Izzy won the least, then Lucas, then Aisha.

How did you get on with those? Well done.

Fantastic learning today.

I'm really impressed with the progress you have made at solving problems using column addition and subtraction.

We know that sometimes those formal written methods are more efficient than mental methods.

And we know that using algorithms, if the numbers have different amounts of digits, we need to really be careful that the digits are aligned in their appropriate place value columns.

Really well done today.

I'm proud of how hard you have tried.

I've loved learning with you, and I look forward to learning with you again soon.