Video: Column Addition of Numbers up to 10,000: Regrouping

In this video, we will learn how to use the standard algorithm to add numbers with up to four digits when we have to regroup.

15:08

Video Transcript

Column Addition of Numbers up to 10,000: Regrouping

In this video, we’re going to learn how to use the standard written method to add numbers with up to four digits. And the examples we’re going to look at in this video are those that we’re going to have to regroup somewhere along the way.

Let’s imagine that we need to find the sum of 5,629 and 3,690. And we know, of course, that finding the sum of two numbers means we need to add them together. Now, although we can add two four-digit numbers together in our heads, particularly for simple ones, we can see that there are a lot of larger digits in these numbers. And so looking at our calculation horizontally like this isn’t really going to help us. What would be much more helpful is if we could put the digits from our numbers into columns so that we could add the ones, the tens, the hundreds, and the thousands separately, in other words, use what we call the standard written method.

As well as using this written method, let’s use place value counters too so that we can model what we’re doing. 5,629 is made up of five 1,000s, six 100s, two 10s, and nine ones. And the number that we are adding, 3,690, is made up of three 1,000s, six 100s, nine 10s, and zero ones. Now, as I’m sure you remember, whenever we use the standard written method, we start by adding the ones digits first, and then we move from right to left. So if we look at our ones, we can see that there are nine ones in the first number, but there aren’t any ones in the second number. And nine plus zero equals nine.

Onto the tens column, there are two 10s in our first number, which are worth 20, and nine 10s in our second number, worth 90. Now we know that two plus nine or nine plus two is equal to 11. So two 10s plus nine 10s must equal 11 10s. But wait a moment; there’s something not quite right about this. We know that each place value column only contains one digit. And so the largest value of a place value column can only be nine. Now, in this case, we found two more 10s than nine 10s, and we’ve come up with a total of 11 10s. We need to include these 11 10s in our answer. But we can’t write two digits in the tens place. Do you remember what we have to do?

We know that 10 10s are the same as one 100. And so we can take 10 of our 11 10s and regroup them. Can you see what we’ve done with our place value counters? We’ve regrouped 10 10s for one 100, and so we can represent our 11 10s as one 100 and one 10. Now, watch how we do this using the standard written method. We take 10 of our tens and we exchange them for one 100. We need to include this 100 in the hundreds column. So we can write a little one underneath like this. And then we’ve also got one 10. So we write out one in the tens place.

Now let’s find the total of our hundreds. Both of our numbers contain six 100s. Now we know that six plus six is 12. So six 100s plus six 100s must equal 12 100s. But don’t forget we also need to include the 100 that we just regrouped. So that’s actually a total of 13 100s we’ve got. Once again, we can’t represent a two-digit number like this, so we’re going to have to regroup. Now, in the past, when we’ve been working with this method, perhaps we’ve learned that 10 ones are the same as one 10. Or as we’ve just seen, 10 10s are the same as one 100.

But now that we’re working with four-digit numbers, we need to think about what 10 100s might be worth. We can regroup 10 of our hundreds and exchange them for one of the next column along. That’s one 1,000. So if we now record what we’ve done using our standard written method, we’re going to say that 600 plus 600 plus the 100 we’ve regrouped is 13 100s, and we’re going to exchange 10 of those 13 100s for one 1,000. We’re going to write a little one underneath in the thousands column. And we’re going to write the remaining three 100s in the hundreds place. 13 100s are the same as 1,300.

Now, finally, we need to add our thousands digits. And we can’t forget that extra 1,000 that we’ve just regrouped. So we have 5,000 plus 3,000, which equals 8,000. And then adding the extra one underneath, we have a total of 9,000. And so, by using the standard written method, we’ve found that the sum of 5,629 and 3,690 is 9,319. And as long as we know how to use this standard written method and, in particular, how to regroup when the total of a column has more than one digit, then we’ve got a way of adding four-digit numbers together that’s going to work every single time.

Let’s have a go at putting into practice what we’ve learned then. We’ll try a question where we use the standard written method on its own. But to begin with, let’s also use place value counters so that we can model what we’re doing.

Fill in the blank: 1,366 plus 353 equals what.

In this question, we’ve got two large numbers to add together. We’ve got a four-digit number and a three-digit number. Now, if these were simple numbers like perhaps 2,000 and 400, we could look at a calculation written horizontally like this and just work out the answer in our heads. I don’t think we could describe these numbers as particularly simple, could we? We can’t just look at them and see what the answer is. We’re going to have to use a strategy to help us. And when we’re adding two numbers like this, it’s going to be important that we keep track of the ones, the tens, the hundreds, and the thousands. To do this, we can make sure that that digits in both of these numbers are written in columns. We call this the standard written method.

Our first number, which is 1,366, is made up of one 1,000, three 100s, six 10s, and six ones. Can you see by writing it in a place value grid like this, we’ve put each digit in the correct column? Now we’re adding 353. Is this right? It’s not, is it? 353 is a three-digit number, and we need to make sure those three digits are in the correct place. They’re not at the moment. Let’s try again. 353 is made up of three 100s, five 10s, and three ones. That’s better. So although we’ve set out our numbers in a place value grid, this is really what the standard written method looks like: numbers written on top of each other so that the ones digits, the tens digits, the hundreds digits, and the thousands digits are all separate.

Now, before we use this written method to find the answer, let’s model our two numbers using place value counters just so that we can show what we’re doing. 1,366 plus 353. Let’s start by adding the ones digits. 1,366 has six ones, and 353 has three ones. And six plus three equals nine. So we know our answer is going to end in a nine. Now we need to add our tens digits. 1,366 has six 10s, and 353 has five 10s. By the way, can you see how it’s much harder to spot those two tens digits if we were reading our calculation across the page? It’s so much easier when we write it vertically, isn’t it?

But we’ve got a bit of a problem here because we know that six plus five is 11, so six 10s plus five 10s equals 11 10s. And we know that each place in a number can only have one digit in it. We can’t write 11 10s. We’re going to have to show our 11 10s a different way. We’re going to have to regroup them. We can take 10 of our 11 10s and exchange them for one 100. It’s important to see we haven’t changed our total at all here; one 100 and one 10 is exactly the same as 11 10s. We’ve just shown it a different way.

Onto the hundreds digits, 1,366 contains three 100s, and so does 353. Now we know 300 plus 300 equals 600. But we need to remember to include the one 100 we’ve regrouped too. It’s often easy to forget this. The total number of hundreds that we have is seven 100s. And a nice easy one to end with: our second number doesn’t have any thousands, so all we have is the one 1,000 in our first number. We found the total of these two numbers by using the standard written method. And our place value counters helped us to see what to do. When we made a total that was more than nine in one of the columns. We regrouped and we exchanged 10 10s for one 100. 1,366 plus 353 equals 1,719.

2,897 plus 5,497 equals what.

In this question, we need to find the total of two four-digit numbers. Now, the way that this calculation is being written, horizontally or across the page, might make us think that that’s how we need to try to find the answer. We’ve just got to look at these two four-digit numbers as they are and try to add them together. But never forget just because we’re shown a calculation like this doesn’t mean we can’t rewrite it in a different way. And when we’re adding two four-digit numbers like this, a really helpful way to rewrite the calculation is vertically, in other words, writing both numbers so that the thousands, the hundreds, the tens, and the ones digits are on top of each other in separate columns.

Now we can add each pair of digits separately. And we always start by adding the ones. Do you know why we always start by adding the ones by the way? Before we begin, look at the two digits in the thousands column and make a prediction. How many thousands do you think are going to be in the answer? We’ve got 2,000 in our first number and 5,000 in our second number. Well, we know two plus five equals seven, so we might predict that our answer is going to contain 7,000. Well, we’d come back to that prediction in a moment, but to begin with, let’s find the total of our ones.

Both numbers contain seven ones. Seven plus seven equals 14 ones. Now we know we can’t show 14 ones in the ones place because we can only show one digit. So we need to take 10 of our 14 ones and exchange them for one 10. We’ll write the little one underneath like this. So we’re still showing 14 ones, but we’re writing it as one 10 and four ones. If we look at our tens digits, we can see that they’re both the same too. Both numbers have a nine in the tens place. Nine plus nine is 18, so nine 10s plus another nine 10s is 18 10s. We mustn’t forget the 10 that we got when we exchanged either, so that’s 19 10s altogether. Again, this is a two-digit number, so we’re going to need to exchange. We can take 10 of our 19 10s and exchange them for one 100. So we can express 19 10s as one 100 and nine 10s.

In our hundreds column, we have eight 100s plus another four 100s. This gives us a total of 12 100s, but we can’t forget the one 100 that we’ve exchanged underneath. So that makes 13 100s. We’re going to need to exchange again. A lot of exchanging in this calculation, isn’t there? We need to take 10 of our 13 100s and exchange them for one 1,000 because one 1,000 and three 100s is the same as 13 100s.

Finally, let’s add our two 1,000s and five 1,000s that we talked about at the start. 2,000 plus 5,000 equals 7,000, but we’ve got one extra 1,000 that we’ve exchanged. Instead of seven 1,000s, our answer has eight 1,000s. Good job we didn’t start by adding the thousands, isn’t it? This is why we always start on the right and work to the left. It’s because if we have to regroup in a column and exchange, the next column to the left is going to be affected. We found the total of these two four-digit numbers by using the standard written method. 2,897 plus 5,497 equals 8,394.

What have we learned in this video? We’ve learned how to use the standard written method or column method to add numbers with up to four digits when we have to regroup.

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