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# Subtracting different ways

Sal subtracts numbers using models and number lines. Created by Sal Khan.

Video transcript

Let's review a little bit
about what we know so far about subtraction. So if I say 5 minus 3,
what does that mean? Well, there's a couple of
ways to think about it. I could have 5-- let's
say I had to 5 berries. So 1, 2, 3, 4, 5. So I could have 5 berries, and
when I say minus 3 you're subtracting 3 from it. I can view that as saying
I'm going to take away 3 of these berries. So if I take away that berry,
that berry, and that berry. So I took away 1, 2, 3 berries. How many berries
do I have left? Well the only berries I have
left are right here-- 1, 2. So I have 2 berries
left just like that. Now the other way, the other
way that I could visualize or think about 5 minus 3,
I'll do it over here. 5 minus 3-- is to think
about what the difference between 5 and 3 is. So let me draw this. So let's say I have 5 berries. 1, 2, 3, 4, 5. And let's say that
you have 3 berries. Here's a slightly
different color. You have 3 berries. So another way to think about
5 minus 3 is how many more berries do I have
than you have? And if you look right here,
well, you see, this berry is another-- you have
also one berry there. We both have one berry there,
we both have one berry there. But I've got 1, 2 berries
that you don't have. So once again, I have 2 more
berries than you have. Now we can also think of
this from the number line point of view. So let me draw a number
line just like that. It's my number line. We've learned on the
addition videos we can keep going off forever. And actually, we could even go
to the left of 0 and go into negative numbers, which
we'll see in future videos. But I'll start at 0. 0, 1, 2, 3, 4, 5--
I'll just go up to 7. So if we do 5 minus 3, if we
view 3 as being taken away from 5, 5 minus 3 means start at 5. If I did 5 plus 3 I would jump
3 spots to the right because that's increasing the
number of things I have. But since I'm subtracting 3,
I want to decrease by 3. So I decrease by 1, 2, 3 and
I get to 2 just like that. Now if we visualize it from
this point of view, let me draw another number line. I want to show you. I mean this is, I'm taking away
3 and here I'm saying, how many more is 5 than 3? Even though they're the exact
same answer, but there are two different ways to
think about it. Let me draw a number
line here again. Let me draw the
same number line. I have 0, 1, 2, 3, 4, 5, 6, 7. So if I were to plot where 5
is on this number line, so this is the 5 right there. I'll put a little pink
square around it. 5 is right there. Now 3, let me do 3 in
this yellow color. 3 is right here on
the number line. So in this way of thinking
about 5 minus 3 you're saying, what is the difference--
let me write that down. Here we're saying, what is the
difference between 5 and 3? And to figure out the
difference you actually have to say, how much do you have to
add to 3 to get to 5? So the difference here, how
different is 5 than 3? Well you have to go up 1
and then up 2 to get to 5. So the difference between 5,
which is all the way over here, and 3, which is just that
far, is 2, just like that. That right there is 2. Let me draw that
in another box. So that's 2 right here. I want to make this difference
between subtraction and difference-- I want to make it
at least reasonably clear to you because these are two
different ways of viewing subtraction, but it ends up
being the exact same operation. You're going to get the same
answer regardless of which way you think about it. Now, I could view-- let me
do different numbers now. Let me do 7 minus 4. So I could view this as,
maybe I have a 7 foot long piece of wood. It's 7 feet long. If I put a ruler up against
it I would have 0, 1, 2, 3, 4, 5, 6, 7. So I have a 7 foot
long piece of wood. And then I could saw
off 4 of those feet. So if I were to saw off
4 of these feet-- so I saw off 1, 2, 3, 4. How much wood do I have left? So all of this stuff right
here, I'm eliminating. I'm sawing it off. I'm sawing it off of the wood. Maybe I should do that in
a darker color to show that I'm sawing it off. So all of this stuff is
going to disappear. I'm grinding it away. I'm sawing that off. So I'm just left with-- after I
saw the 4 inches or feet or whatever of the wood, I'm left
with 1, 2, 3 inches of wood. So this is 3. So 7 minus 4 is equal to 3. This is viewing subtraction
as literally taking away. I sawed off the wood,
so I took away wood. Now I could think of it in a
slightly different way of thinking about it, but give
you the exact same answer. We could say 7 minus 4. So once again, I could have
the 7 inch long piece of wood like that. So if I put a ruler here
that's 1, 2, 3, 4, 5, 6, 7. So once again, a 7 inch
long piece of wood. And now instead of taking 4
away of it I'm comparing it-- so that's a 7-- I'm comparing
it to a 4 inch long piece of wood. So I have another 4 inch long
piece of wood right there. That's my 4 inch long piece of
wood. that's 7, this is 4. You could view 7 minus 4 as
taking 4 inches away from the long piece of wood. Or you could view seven minus 4
as the difference between the 4 inch piece of wood and the
7 inch piece of wood. So in this case, what's
the difference? To go from the 4 inch piece of
wood to the 7 inch piece of wood I would have to grow by 3
inches, or I would have to add a 3 inch piece of wood somehow. Or the wood would somehow
have to grow by 3 inches in order to become 7 inches. So these are 2 completely
equivalent ways to view subtraction. That's all a little bit of a
review from the last video. Now what I also want to do in
this video is start tackling slightly larger problems. But you'll see that really,
the number line applies just equally as well as to kind
of the simpler problems that we've done before. Let's do 17 minus 9. So just like everything
else, there's two ways we could've done it. You know, the more slow way is
you could draw 17 objects. Let's say I have 17 chips. 1, 2, 3, 4, 5, 6, 7, 8, 9 10,
11, 12, 13, 14, 15, 16, 17. And I'm going to take
away 9 of them. So I'm going to take away
1, 2, 3, 4, 5, 6, 7, 8, 9. How many am I left with? I'm left with 1, 2,
3, 4 5, 6, 7, 8. So 17 minus 9 is equal to 8. But that took a long time and
you could imagine, if this number was a lot bigger it
would've taken me forever to draw all of these circles and
then scratch out things. And it would've wasted
paper and time. And we have other things to do. So another way you could do it,
and maybe this would be easier for you to visualize, is
to draw the number line. You always don't
have to start at 0. So if we draw the number line,
if we say that's 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8,
7-- you could imagine, I could keep going to the left
all the way to 0. But I start at 17. I could start at 17 and
take away 9 from it. So I go 1, 2, 3,
4, 5, 6, 7, 8, 9. And once again, we
are left at 8. Now this was, at least in my
head, a little bit cleaner and faster than this one. But in either case, you don't
want to do this every time you have to subtract 9 from 17 or
want to find the difference between 17 and 9. And then to realize that's 8. So this is something that
eventually you want to internalize. You'll want to know by heart
that, oh, 17 minus 9? I know that is 8. And by the way, 17 minus 8? What's 17 minus 8? Well, that is 9. And now why does all
of this make sense? Because 8 plus 9
is equal to 17. So 17 minus 9 is 8. Or 17 minus 8 is 9. When I say 17 minus 8, I'm
essentially saying that is equal to some number that if I
were to add to 8 will equal 17. Well, that's 9. When I say 17 minus 9 that's
saying, there's some number, that if I were to add
it to 9, I'll get 17. And that's 8. So all of these, all of these
statements, are kind of saying the same thing. That 8 plus 9 are 17. Or the difference
between 17 and 9 is 8. Or the difference
between 17 and 8 is 9. Hopefully I'm not
confusing you. So for most of these
subtraction problems where the answer is a one-digit answer,
you should eventually have them memorized, but in your head
it's good to be imagining this number line. Let's do a couple
more of these. And then, once we have these
memorized or at least be able to do a number line if we
forget, I'll show you have to do any subtraction problem,
arbitrarily for super large numbers. So let's say we're going
to do 13 minus 5. So once again, I'm not going
to do the whole circles or the berries this time. I'm just going to draw
the number line. Just draw the number
line like that. Let's start at 14, 13, 12, 11,
10, 9, 8, 7, 6, 5-- and you just can keep going
lower and lower. You can go to 0 or you
can even go past 0. We'll talk about
that in the future. But we start at 13. We're starting at 13. And we're going to
take 5 away from it. So this is the subtraction
view of subtraction; we're taking away. 1, 2, 3, 4, 5 and we land at 8. So 13 minus 5-- let me
do this in a new color. 13 minus 5 is equal to 8. Now another way we could
have thought about that, I plotted where 13 is. I can plot where 5 is. I could say look, this is 5. 5 is right here on
my number line. What do I have to add
to 5 to get to 13? So let's see. I would have to go 1,
2, 3, 4, 5, 6, 7, 8. I have to add 8 to
5 to get to 13. 5 plus 8 is equal to 13. So that tells me that 13
minus 5 is equal to 8. This also tells me that 13
minus 8 is equal to 5. All of these, are on some
level, telling me the exact same thing. But the difference
between 13 and 5 is 8. The difference between
13 and 8 is 5. 5 plus 8 is 13. So hopefully you have the hang
of that and if you haven't done so already, it'll be good
to practice all of these. Taking a teen number and then
subtracting any of the one-digit numbers from
those teen numbers. That's in general, very,
very good practice for you.