### Video Transcript

Division Rules for Zero and One

In this video, we’re going to learn
how to model what happens when we divide using the numbers one and zero. You’ll be pleased to know that
dividing by zero isn’t really going to cause any black holes. But it is one of those ideas in
maths that can be quite difficult to get our heads around. So, to begin with, let’s start with
the easier of our two numbers.

What happens if we divide a number
by one? Well, if you remember, there are
two ways we can think about division, as grouping and sharing. Let’s use nine divided by one as an
example. Now if we’re thinking about this
division as grouping, what we’re saying is if we start with nine and we split this
number up into groups of one, how many groups will there be? Of course, we’ll have nine groups,
won’t we? This is the same sort of idea as if
there were nine of you in your class. And your teacher said, “Right,
you’ve got to work individually this afternoon.” You’re all going to work at a desk
on your own. You’d need to have nine desks,
wouldn’t you? Nine divided by one equals
nine.

We can find the same answer by
thinking of dividing by one as sharing. In other words, what happens if we
start with nine and we share it with, well, one? Imagine you had nine sweets and you
shared them just with yourself. In a way, it’s not really sharing,
is it? There’s only one of you. You’ll get all nine sweets. Nine shared into one group equals
nine. Now can you spot anything
interesting about this division? We started with nine. And by the time we divided it by
one, it hadn’t changed at all. Our answer is the same number we
began the division with.

And this brings us to a rule that
we can remember when we’re dividing by one, no matter what the number. And that’s that any number divided
by one equals itself. And so, 14 split into one group
equals 14. 35 divided by one is 35. And 296,352 divided by one equals
296,3 — well, you get the picture. Something else we could use to help
us understand what happens when we divide by one is multiplication. So, for example, if we want to find
out what 12 divided by one is, we could think of the opposite or inverse operation,
which is multiplication, and ask ourselves, “What will be multiplied by one to give
the answer 12?” We know that 12 times one equals
12. And if we know that 12 lots of one
make 12, we also know if we divide 12 by one, we get the answer 12.

Now, so far, we’ve just thought
about dividing by one. And we’ve seen that any number
divided by one equals itself. But you know, we can switch this
statement around to make another rule. Any number divided by itself equals
one. If we take six divided by six as an
example, if we have six counters and split them into groups of six, we’ll only be
able to make one group. Six divided by itself equals
one. And so, 18 divided by 18 equals
one. It doesn’t matter what number we
work with. If we divide it by itself, the
answer will always be one. So we’ve learned two different
facts or rules here for when a division involves the number one. Any number divided by one stays the
same. And any number divided by itself
equals one. Let’s have a go at answering a
couple of questions now where we can put into practice these rules.

Complete: Four divided by what
equals four.

In this question, we’re given a
division where the divisor, that’s the number we’re dividing by, is missing. The number that we’re dividing is
four. And then we divide it by something
and the answer is also four. This is interesting. What number could we divide by so
that the number we start with doesn’t change? Let’s sketch a bar model to help
us. If we divide four into groups of a
certain amount and we find that we can make four groups, what will each group be
worth? Each group is going to have a value
of one. Four split into equal groups of one
equals four. And we know this is correct because
we’re reminded of an important fact to do with dividing by one. And that’s that any number divided
by one equals itself. So as soon as we saw that our
starting number didn’t change in this division, we realized we must have divided it
by one. Four divided by one equals
four. Our missing number is one.

Complete the following: Two divided
by what equals one.

In this question, we’re given a
division with a missing number. What do we divide two by to give us
the answer one? Well, we know that division and
multiplication are inverse operations; they’re opposites. So to help find a missing number in
a division like this, we can think about the multiplication fact that goes with it,
the opposite of it. So instead of asking ourselves,
“What do we divide two by to give us the answer one?”, we can work backward and ask
ourselves, “What do we multiply one by to give us the answer two?” Well, this is quite a simple
multiplication fact, isn’t it? One times two equals two. And if we know there’s one lot of
two in two, if we divide to by itself, we’re going to get the answer one.

Another way we know this is true is
because we know a rule that applies every time a number is divided by itself. Any number divided by itself equals
one. And so, because our division showed
the answer one, we knew that the first number must be divided by itself. Two divided by two equals one. Our missing number is two.

Now, if you remember, the title of
this video was “Division Rules for Zero and One.” And although we’ve looked at the
rules for dividing when there’s a number one involved, it’s now time to think about
zero. What happens when we divide using a
zero? Do you remember that cartoon we
used of the two characters falling down a black hole because one of them had just
tried to divide by zero. Now this was just a little bit of
fun. But you know, dividing by zero can
actually make your head spin a little bit. So what we’ll do is take this
really slowly to try to understand it. Perhaps this time around, it might
make sense to start with the rule.

This rule is really easy to
remember, perhaps not so easy to understand, but it’s definitely easy to
remember. And that’s that we can’t divide by
zero. Let’s have a go at explaining why
not, and we’ll try 100 divided by zero as an example. So here are 100 counters. And if we split them into groups of
100, there’d only be enough groups for one person. If we divide 100 into groups of 50,
then two people would get a group of 50 each. And then just as one more example,
if we split our 100 counters into groups of one, 100 people would all get a counter
each. Now looking at these divisions can
help us when it comes to thinking about dividing by zero.

Notice how as the number we’re
dividing by goes down, the answer to each division goes up. And so we’d expect the answer to
100 divided by zero to increase again. But if we stop and think about it,
100 counters split into groups of zero means that one person could come up and then
we could share out to them zero counters, but we’d still have just as many counters
as we had to start with. And this would still be the case if
someone else came along or even another 100 people. Hundreds and hundreds, even
thousands of people, could knock at our door asking for zero counters and we’d still
be able to do this because zero is nothing. It doesn’t affect our 100 counters
at all. There just isn’t a way we can split
up 100 into zero.

Some other things that we can use
to help us understand that we can’t divide by zero are multiplication facts. For example, if we’re asked to do
the impossible and to divide 15 by zero, we’d think about the inverse operation and
we’d ask ourselves, “What number do I multiply by zero to give the answer 15?” And then hopefully we’d start to
frown and say to ourselves, “This isn’t possible.” If each plate that we have has zero
bananas on it, how many of these empty plates would we need to have 15 bananas? Well, it makes no sense, does
it? We just can’t answer this. We could have half a million empty
bites of bananas and we’d still wouldn’t have 15 to eat. We might as well cross out
calculations like this. They just don’t make sense.

But what if we have a division that
starts with zero? Is there a rule for this? What about zero divided by 10 for
example? Here’s a picture of zero sheep. Some of you might think it looks
more like zero elephants, but It’s definitely zero sheep. You need to look more closely. Now what if we take our zero sheep
and divide them into 10 equal groups? Here we are. What do you mean you can’t see
them? Of course there aren’t going to be
any sheep in these groups are there because we had zero sheep to begin with. Zero divided by 10 equals zero. In fact, we could have shared zero
into any number of groups. The answer is always going to be
the same.

And this brings us on to our second
rule to do with dividing by zero. And that’s that if we start with
zero and we divide it by any number at all, it always results in zero. Let’s practice what we’ve learned
about dividing with zero then.

Complete the following: What
divided by 12 equals zero.

This might seem like an unusual
division because it ends with the answer zero. And it’s not often that we see a
division that ends in zero. At the start of this number
sentence, we can see we’ve got a missing number that we need to complete. What number if we divide it by 12
would give an answer zero? To help us find this missing
number, we can start at the end of this division and work backwards, because we know
that the inverse or the opposite of division is multiplication. If we divide a number by 12 and we
have zero lots of 12, then to find that missing number, we just need to find zero
lots of 12.

We can use our knowledge of
multiplying by zero to help us here because we know that any number multiplied by
zero equals zero. And that’s how we know our missing
number must be zero. This reminds us of a fact we know
to do with working with zero in divisions. Dividing zero by any number at all
always results in zero. And as we know our answer is zero,
we know the number we divided by 12 to get there must also be zero. Zero divided by 12 equals zero. Our missing number is zero.

So what have we learned in this
video? We’ve learned how to model what
happens when we divide one or zero and also when we divide a number by one or
zero. We’ve learned these facts. Any number divided by one equals
itself, and any number divided by itself equals one. Also, zero divided by any number is
zero. And dividing by zero isn’t
possible.