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.
