### Video Transcript

Relating Multiplication and
Division

In this video, we’re going to learn
how to model the relationship between multiplication and division. We’re also going to learn how to
write a set of facts that are related to each other. Let’s start with a really difficult
maths test. Don’t worry, it’s not for you. It’s for this group of 10
children. Before they start, let’s arrange
them into equal rows. There we are. We’ve arranged the children in two
rows with five children in each row. Or if we look at the children a
different way, we could say we have five columns or groups of two. You know, we could use number
sentences to show the total number of children, which we know is 10, in two
different ways. Two lots of five make 10, and five
lots of two make 10.

And this just highlights something
we already know about multiplication facts. We can switch the number that we’re
multiplying around, and they’ll still make the same answer. So two times five equals 10, and
five times two equals 10. Would you say these were different
number facts or the same fact? Let’s describe them as being
different number facts but related. They’re in the same family, aren’t
they? They contain the same numbers. Both multiplications contain two,
five, and 10. Are there any other number facts
that we can make using these numbers? What if we think backwards?

To help us understand what this
means, let’s think about addition and subtraction for a moment. We know that addition means putting
numbers together. But subtraction is all about the
opposite, breaking numbers apart and taking them away. Addition and subtraction are what
we call the inverse of each other; they’re opposite operations. And so if we have a number fact
like two plus three equals five, we can think backwards and use the inverse
operation to change this fact into something related. If we start with five and take away
three, we’ll be left with two. And how do we know this? We know this because two plus three
equals five. And in the same way that addition
and subtraction are inverse operations, so multiplication and division are
opposites, too.

So when we talk about looking at
these multiplications and starting to think backwards, what we’re really saying is
“Let’s start with the answer to our multiplications, the whole amount, which is 10
children, and let’s use the inverse.” Let’s divide 10 in different ways,
using our multiplication facts to help. If you remember, the first
multiplication fact we noticed was to do with the number of rows. We said there were two rows with
five children in each row. So if we start with a group of 10
children, we can ask ourselves what happens if we take those 10 children and split
them up or divide them into two equal parts? There’ll be five children in each
part, won’t there, just like there are five children in each row.

And there’s another division fact
in this family, too. Just like we know that five lots of
two make 10, we can divide 10 into five equal groups. And we know there’ll be two
children in each group. 10 divided by five equals two, so
the numbers two, five, and 10 are part of a family. We can use them to write related
multiplication and division facts. Two times five equals 10, and five
times two equals 10. 10 divided by two equals five, and
10 divided by five equals two.

Here’s a new multiplication
fact. Five times four equals 20. How many other facts can we write
that are part of the same family? Well, as well as this
multiplication fact, remember, we can switch the two numbers that we’re
multiplying. So four times five equals 20 as
well. And then we can start with 20 and
think of the inverse, too.

If five lots of four equals 20,
then we could divide 20 into groups of four, and there’ll be five of them. And if we know four lots of five
are 20, we could split up 20 into groups of five, and there’ll be four of them. We found two multiplication facts
and then looked at the inverse and found two division facts. They each contain the numbers five,
four, and 20.

Do you think you got hang of this,
now? Here’s another times table
fact. How many related facts can we think
of? Well, this is a bit of a trick
question; I wonder if you can see why. The two numbers that we’re
multiplying together are the same. So if we swap them around, they’re
just gonna make the same fact. And if we think of the inverse, we
can write 36 divided by six equals six. But again, the other division we
can write is exactly the same, 36 divided by six equals six. That was a bit of a trick question
to end with. There are only two facts in this
family. Let’s have a go at answering some
questions now, and there won’t be a trick question among them. We just need to try to put into
practice everything that we’ve learned about these families of multiplication and
division facts.

Find the missing numbers by making
six equal groups from these 54 books. Six times what equals 54. 54 divided by six equals what. And 54 divided by nine equals
what.

In the picture, we can see 54
books. And they’ve been arranged in such a
way that they can help us answer this question. We’re given three number sentences,
a multiplication and two divisions. And in each number sentence,
there’s a missing number. We’re told that we need to find
these missing numbers by making six equal groups from the 54 books. Can you see how we can draw some
lines on our diagram to do this? The way that the books have been
arranged, they’re in six equal rows. This is a quick way we could find
our six equal groups. There we go, six equal groups. And there are one, two, three,
four, five, six, seven, eight, nine books in each group.

Let’s use what we can see then to
help us fill in these related facts. Six multiplied by what equals
54. In this sentence, we can think of
the number six as representing our six rows. And the number 54, that’s the
answer to our multiplication, is the total number of books that we have. So six lots of what equal 54. As we’ve already seen, there are
nine books in each row. So we can say that six times nine
equals 54. Our next number sentence is the
inverse of multiplication. We need to divide. 54 divided by six equals what. In other words, if we start with
the whole number of books, that’s 54, and we split it into six equal rows, how many
will there be in each row? The answer is going to be nine. If six lots of nine make 54, then
we can take 54 and split it up into six equal groups; there’ll be nine in each
group.

Look at how both of our number
sentences contain the numbers six, nine, and 54. Do you think our final number
sentence is part of the same family of facts? It certainly contains a 54 and a
nine, so we might expect that our missing number is going to be a six. Let’s see whether we’re right. 54 divided by nine equals what. Now, as well as this array of books
showing six rows of nine, we could also look at the books a different way and think
of them as nine columns of six. So if we divide 54 into nine equal
groups, just as we guessed, there’ll be six in each group. We’ve used this array of 54 books
to help us find three related multiplication and division facts. We know they’re in a family because
they contain the same three numbers: six times nine equals 54, 54 divided by six
equals nine, and 54 divided by nine equals six. The missing numbers are nine, nine,
and six.

Find all the missing numbers in
these related facts. Six times three equals what, three
times what equals 18, 18 divided by six equals what, and what divided by three
equals six.

In this question, we’re given four
number facts. And we’re told these aren’t just
any number facts. These are related. They’re part of a family. What’s the same and what’s
different about these facts? Well, firstly, as we’ve heard
already, each of the facts has a missing number. They’re all in different places,
though, so we need to think carefully about what numbers are missing. Two of the calculations are
multiplications, and two are divisions. And although these may seem like
different sorts of number sentences, they are related because we know that division
is the inverse or the opposite of multiplication.

You know, there’s one other
interesting thing about these number facts. Can you spot it? The same numbers keep cropping up
again and again. We can see the numbers six, three,
and also 18. These are the only numbers that are
being used in our number sentences. I wonder why this is. Well, the key to the whole thing
really is our first number fact. Six times three equals what. Let’s model this calculation using
counters. We can think of six times three as
being worth six lots of three. And this is the same as three, six,
nine, 12, 15, 18 altogether. Our missing number is the third of
our three numbers. Six times three equals 18.

Do you think maybe all of our
number facts have the same three numbers in them? Let’s have a look at the second
one. Three times what equals 18. Well, we know that we’ve got 18 in
this group, but 18 is worth three lots of what? To help us find the missing number,
we’re going to have to use some division, even though this is a multiplication
question. We’re going to need to start with
18 and split it into three equal groups. There are six counters in each
group. As though if we know six lots of
three are 18, we also know three lots of six equal 18, too. And of course, this is something we
know about multiplications, isn’t it? We can swap the numbers that we’re
multiplying around and they’ll still make the same product.

Our next fact is a division. And because the missing number is
the answer, the number that comes after the equal sign, we can just work out the
division. 18 divided by six equals what. Now, we could do one of two things
here. We could find out how many sixes
there are in 18, or we could split 18 into six equal groups and see how many there
are in each group. Let’s do the second one. One, two, three, four, five,
six. Now, I’m guessing you probably knew
what the answer was going to be before we started spitting up this number. If we know that six times three
equals 18, then we also know the inverse. 18 divided by six equals three.

And what number do we divide by
three to get six? The missing number in our last
division is 18. We know that multiplication and
division are inverse operations; they’re opposites. And so, with the number six times
three equals 18, we can find four related number facts. Six times three equals 18, three
times six equals 18, 18 divided by six equals three, and 18 divided by three equals
six. Our missing numbers are 18, six,
three, and 18 again.

If four times seven equals 28, then
28 divided by four equals what.

Lots of what we learn in maths is
to do with building on knowledge we already have. And in this question, we’re given
an if-then statement. If we know one thing, then we can
use it to help know something else. And the building block that we’re
supposed to know, the thing that we’re supposed to use to help us find the answer,
is a multiplication fact: Four times seven equals 28. We could show this number fact as a
bar model. Four lots of seven are the same as
28. Or if we want to think of it a
different way, four repeated seven times equals 28, too. Let’s use these number facts and
bar models to help us find the missing number, because we’re told if we know that
four lots of seven are 28, then we should be able to work out what 28 divided by
four is.

Let’s think about what 28 divided
by four might look like. We could think of it as 28 split
into four equal groups, and the answer will be the number that’s in each group. Or we could think of it as 28 split
into groups of four, where the answer is the number of groups that there are. Can you see that both of these
ideas match with our bar models? If we know that four lots of seven
make 28, then we also know if we start with 28 and we split it into four equal
groups, there’ll be seven in each group. And we could also say that if we
know that we could multiply the number four seven times to make 28, then we could
start with 28 and split it up into groups of four, we know there are going to be
seven of them. We’ve used two different ways to
help show how to use a multiplication fact to find an inverse or opposite fact. If four times seven is 28, then 28
divided by four equals seven. The missing number is seven.

What have we learned in this
video? We’ve learned how to model the
relationship between multiplication and division and write a set of related
facts.