Mental Addition of Numbers up to
In this video, we’re going to learn
how to use different mental strategies to efficiently add numbers with up to four
digits. To begin with, it might help us to
start thinking about a toolkit. It contains lots of different tools
suitable for doing lots of different jobs. And when we have a toolkit like
this, it’s best to choose the right tool for the right job.
For example, if a tiny little screw
falls out the bottom of our laptop, although we could probably fix it by charging a
power electric screwdriver for about an hour and then screwing it in, it would
actually be a lot quicker for us to take a normal screwdriver and just screw it in
by hand. Would only take a couple of
seconds. We could say using a normal
screwdriver is a lot more efficient. And in this video, we’re going to
use the word efficient to describe something that does the job quickly. It’s quick and accurate. Cutting your lawn with a lawnmower
is much more efficient than using a pair of scissors.
Now, this video isn’t about the
sorts of tools that we can see in this toolkit. It’s about the sort of tools that
we have in our heads. In particular, we’re going to be
thinking about strategies we can use to add numbers together. Now we know if we want to add
together two four-digit numbers together, we can always rely on the standard written
method. But in a way, it’s like using that
electric screwdriver just to fix that tiny little screw. Using a written method like this
might be accurate, but it’s not always very quick. Working out the answer in our heads
is often a lot more efficient.
So let’s dip into our addition
toolkit. What methods could we use to add
numbers in our heads? And as we think about each new
strategy, let’s ask ourselves a question, when might I want to use this? What sort of numbers would I be
The first strategy we could look at
is counting on. This strategy is useful if we’re
being asked to add on a number of tens or maybe hundreds or even thousands. And to use this strategy, we don’t
need to draw a number line, but we can imagine one in our heads. And we start with the more
complicated number in the addition and then just count on how many tens or hundreds
or thousands we need to. In this case, we’re going to add
two 1,000s, so we could think 4,632, 5,632, 6,632. Because we’ve counted on two
1,000s, the thousands digit has gone from 4,000 to 6,000. None of the other digits have
changed. So you can see why counting on can
be a really quick method to use in our heads. Let’s try an example.
4,667 plus 3,000 equals what.
In this question, we’ve got two
four-digit numbers to add together. Now, one way to find the answer
might be to get a pencil and paper to write both numbers neatly on top of each other
and then go through and add the ones, the tens, and so on. And although this would be a way to
find the correct answer, with an addition like this, it’s not the quickest method we
could use. It’s much more efficient for us to
work out the answer in our heads. Can you spot the reason why? It’s because the second number in
our addition is quite a simple number. It’s a multiple of 1,000. We don’t need to worry about any
hundreds, tens, or ones. All we have to do to find the
answer is to add on three 1,000s.
It makes sense to use a strategy
like counting on here. We could start with 4,667 and count
on three 1,000s. Now, we should be able to do this
just by thinking through each number in our heads. But let’s draw a number line here
to show us what we’re thinking about. Let’s begin with a more complicated
number, 4,667. And we know we need to add 3,000,
which is the same as counting on 1,000 three times. And because we’re counting on in
thousands, we’d expect the thousands digit to change. So we could start by saying 4,667
and then 5,667, 6,667, and finally 7,667.
We started with a number that had
four 1,000s. And by adding 3,000, we’ve ended
with a number that has seven 1,000s. Because we were simply adding a
multiple of 1,000 here, we chose to use the counting on strategy. And we found the answer in our
heads. 4,667 plus 3,000 equals 7,667.
So the first strategy we’ve
practiced then is counting on. And you know, we could even use
this to add a number like 2,030. All we’d have to do is to count on
two 1,000s and then three 10s. But what if the numbers that we’re
adding don’t really lend themselves to counting on? What other strategies could we
use? We could try breaking numbers
apart, in other words, finding the total of their thousands, their hundreds, their
tens, and their ones and then combining all the parts back together again to find
the answer. Let’s look at a question that’s a
really good example of this strategy.
Pick the sum equal to 3,324 plus
5,564. Is it 9,000 plus 800 plus 80 plus
eight, 8,000 plus 900 plus 80 plus eight, 8,000 plus 800 plus 80 plus nine, 8,000
plus 800 plus 80 plus eight, or 8,000 plus 800 plus 90 plus eight?
In this question, we’re given a
pair of four-digit numbers to add together. But instead of finding a number
that’s equal to their total, we’re asked to pick a sum. Our five possible answers are all
additions, and only one of them makes a total that’s exactly the same as the total
of the two numbers in the question. One way to find the answer could be
to start off by adding our two numbers together and then go through each of our five
possible answers adding them to see which one has the same total. But that’s a lot of addition we’d
need to do. And there’s a quicker way to find
Can you spot anything interesting
about the five additions that are listed? Well, they each show a number of
thousands plus a number of hundreds plus a number of tens plus a number of ones. It’s almost as if our total of
3,324 and 5,564 has been split up into its thousands, hundreds, tens, and ones. Let’s try seeing how many
thousands, hundreds, tens, and ones there’ll be in our answer. Let’s break apart the two
four-digit numbers we’re given. Now, this is the sort of addition
we can do in our heads, but we’re going to make a note of everything that we do so
that it’s clear what strategies we’ve used.
So to begin with, let’s see how
many 1,000s there’re going to be in our answer. Our first number contains three
1,000s and there’s a five in the thousands place of our second number, and 3,000
plus 5,000 equals 8,000. Looks like the number of 1,000s in
our answer is going to be eight. And if we look at each of the
possible additions, we can see that one stands out. The first one has nine 1,000s, not
eight. So let’s cross this one
through. It’s not the right answer. The others though could be. Let’s break apart the hundreds. Our first number contains three
100s and our second number contains five 100s. This is a very similar addition to
the one we’ve just done, isn’t it? 300 plus 500 equals 800.
Now, if we look at our remaining
possible answers, we can see that most of them do have 800 as the hundreds number in
the addition. But can you spot one of them
doesn’t? The first one on our list has
900. So we’re gonna cross this
through. This one is not correct. Now it’s time to look at the
tens. Our first number contains two 10s
or 20 and our second number contains six 10s or 60, and 20 plus 60 equals 80. Now, if we look at our remaining
answers, we can see that two of them do have 80 in them. But if we look at the last answer,
we can see 8,000 plus 800 plus 90.
Let’s cross this one through. We want to be adding 80 not 90. Finally, we can break off the ones
in both of our numbers and add those. Four ones plus another four ones
makes a total of eight ones. We’ve found the total of 3,324 and
5,564 by breaking these two numbers apart, adding the thousands, then the hundreds,
then the tens, then the ones.
Now, if we were to find the actual
total, we’d combine all our answers together again. But of course, our question doesn’t
ask us to do this. We’re just looking for a sum that
represents 8,000 plus 800 plus 80 plus eight, and here it is. It would make an interesting
answer, wouldn’t it? 8,888. The sum that’s equal to 3,324 plus
5,564 is 8,000 plus 800 plus 80 plus eight.
So breaking apart can be a really
useful tool in our mental toolkit, especially if we’re able to hold all those
numbers in our heads.
A third tool for our addition
toolkit is compensation. Compensation is useful when one of
our numbers would be really simple to add if only we could adjust it a little
bit. For example, in the calculation
4,021 plus 2,434, we might look at the first number and say to ourselves, “if only
it didn’t end in 21, then we could just add 4,000 to the second number.” So compensation is all about making
one of our numbers easier, but adjusting our second number to make up for it. In this example, we’ve moved the 21
from our first number to our second number which is pretty easy to do. And now the calculation looks a lot
easier. Let’s try an example.
Calculate 7,452 plus 2,011.
In this question, we’ve got two
four-digit numbers to add together. And if we look at our second number
in particular, we could think to ourselves, “if only it didn’t end in 11, then we
could just add 2,000.” And that’s really quick to do. Well, we can use the mental
strategy of compensation to help us here. Instead of adding 2,011, we could
just add 2,000 like we want to. But we know if we do this, our
total is going to be 11 less than it should be. We could solve this problem by
adjusting our first number and making it 11 more than it should be. This means adding one 10 and one
one. So instead of five 10s, we now have
six 10s. And instead of two ones, we now
have three ones.
Now look at our two additions. The second one is a lot easier to
work out, isn’t it? 7,463 plus 2,000. The hundreds, tens, and ones in our
number are going to stay exactly the same. But instead of our number having
seven 1,000s, we’re adding another two 1,000s. So that’s nine 1,000s
altogether. We found the answer to 7,452 plus
2,011 by adjusting both numbers by 11. We subtracted 11 from the second
number to make it a lot easier to add. But then we compensated for this by
adding 11 to our first number. 7,452 plus 2,011 is 9,463.
The final strategy we’re going to
look at in this video is very similar to compensation. Again, it’s all about looking at
the numbers we’re adding and thinking that one of them could be made a lot
easier. For example, in the addition we can
see here, 3,999 might look like a complicated number. But if we can recognize that it’s
only one away from 4,000, this will make our calculation a lot easier. 3,999 is what we call a near
multiple. It’s very nearly a multiple of
4,000. So in our heads, we could round it
up to 4,000 and then add 4,000 to 1,526. But then, the important thing is
that we need to remember we’ve added one too many, so we can then adjust our answer
by subtracting one from it. Adding 4,000 and taking away one is
exactly the same as adding 3,999. It’s a lot more efficient. Let’s have a go at practicing this
Find 4,762 plus 2,999.
In this question, we’ve got two
four-digit numbers to add together. Now, one way we could find the
answer would be to get a pen and paper to write our two numbers as a column addition
and work it out that way. But we’d need to do a lot of
regrouping if we worked it out this way. Most of our columns would add to a
two-digit number. Instead, there’s a much quicker way
to find the answer. We could actually find the total
really quickly in our heads. To spot how to do this, we need to
look closely at our second number. What’s special about it? Well, although 2,999 looks like a
complicated number to add with all those nines in it, it’s only one away from a very
simple number to add. We could call 2,999 a near multiple
because it’s only one away from a multiple of 1,000.
So to find our answer really
quickly, we could round this second number up to 3,000. So in our heads, we could begin
with 4,762 and then add 3,000 instead. So 4,762 is going to become
7,762. But although that was really easy
to add, it’s not our final answer because if you remember, we rounded 2,999 up to
3,000. In other words, we’ve added one
more than we needed to. So we need to adjust our answer by
taking away one. We didn’t need to use the standard
written method to add these two numbers together. We did it really quickly in our
heads, and this was because we spotted that the second number was a near multiple of
We found our answer by rounding
2,999 up to 3,000 and adding that which was really quick to do and then adjusting
our answer by one because we’d added one too many. 4,762 plus 2,999 equals 7,761.
So what’ve we learned in this
video? We’ve learned how to use different
mental strategies to add numbers with up to four digits efficiently.