### Video Transcript

Comparing Multiplication and
Division Expressions

In this video, we’re going to learn
how to compare multiplication and division expressions. Here’s an interesting fact to start
our video with. Did you know we don’t always have
to work out the answer to a calculation to solve a problem? In other words, we don’t always
have to add, subtract, multiply, or divide to find out an answer. Sometimes we can use what we know
about numbers instead, a little bit of common sense really. We could call it number sense. And although we might have to do
some calculating in this video, we’re also going to use our number sense where we
can too.

Ice lollies are sold in boxes of
five. We’ve got some lemon flavor and
some raspberry flavor. Are there more lemon flavor ice
lollies or raspberry flavor? Well, if we count the number of
boxes, we can see that there are six boxes of lemon ice lollies and eight boxes of
lollies with a raspberry flavor. So we know there are more boxes of
raspberry lollies. But does this mean there are more
lollies?

Well, as we said at the start, both
boxes contain the same number of ice lollies, five. And so just like we said at the
start of this video, this is an example of the sort of problem where we don’t need
to work out or calculate the exact answer for. All the boxes have the same number
of ice lollies in them. And because there are more boxes of
raspberry flavor lollies, we know there must be more lollies.

We could use multiplication
expressions to show what we found. For the lemon flavor, there are six
boxes of five lollies. We could write this as six times
five. And then for the raspberry flavor
lollies, there are eight boxes of five. So we could write this as eight
times five. And you can see now why we didn’t
need to work out any calculations, can’t you? We know that six lots of five is
going to be smaller than eight lots of five. Because both multiplications share
a factor that’s the same, that’s the number five, we can just compare them without
working them out. Six times five is less than eight
times five.

Now, tropical flavor ice lollies
are slightly different. They’re sold in boxes of eight. So do you think we have more
raspberry or tropical flavor ice lollies? Let’s count the boxes to begin
with. There are eight boxes of raspberry
flavor. And as we know already, each box
contains five ice lollies. And so we could write eight groups
of five as the expression eight times five.

Now, it looks like we have less
tropical flavor boxes, doesn’t it? There are seven boxes of this
flavor. But of course, this time, we know
that we can’t just compare the number of boxes because there are more ice lollies in
each box for this flavor. There are eight in each box. So seven groups of eight is the
same as writing seven times eight.

Now, if we want to compare these
two multiplication expressions, what do you think we’re gonna have to do? Are we going to have to calculate
each answer? Or can we use our number sense with
these expressions too? Well, if we look closely at these
expressions, we can see that they have something in common again. The number eight is in a different
position in each multiplication, but it’s still there. We could say these multiplications
once again have got a factor in common. It’s the number eight.

Now here’s where we have to start
using our number sense to help. One thing we know about multiplying
numbers, which is the same about adding them, is that we can swap the numbers in a
multiplication around and the answer stays the same. So this means that eight times five
is exactly the same as five times eight. And by thinking of the number of
raspberry ice lollies as five times eight, the whole thing becomes much easier to
compare. We can see the answer now, can’t
we? We don’t need to do any
multiplication because we know that five times eight is going to be less than seven
times eight. So we could use a symbol for less
than in between both multiplication expressions. Eight times five is less than seven
times eight.

Now we worked out the answer
without doing a single multiplication. Now, the title of this video was
comparing multiplication and division expressions. So far, we’ve only compared
multiplication, so let’s have a go at looking at some division expressions.

Choc ices are sold in boxes. We’ve got the milk chocolate flavor
and this white chocolate flavor. Now, although it looks like the boy
has more choc ices than the girl, it turns out both of them have the same
number. If they were to open all their
boxes and tip all their choc ices into a bowl, they’d both have 18 each. Now, this can only mean one
thing. A box of milk chocolate choc ices
must contain a different number than the white chocolate. Which type of box contains the most
choc ices?

We did say that this problem was
going to involve division. So let’s try to write this as a
division expression. We know that the boy has 18 choc
ices altogether. So this is the number that we can
think of dividing or splitting up. Now, we can see that there are six
boxes. We can think of these as six equal
groups. So to find the number in each box
or in each group, we need to split 18 into six equal groups, 18 divided by six.

Now, it could be that we need to
calculate the answer to this. But maybe it’s another one of these
problems where we can just use our number sense. Let’s find out. We’ve been told already that both
children have the same number of choc ices in total. So the girl has 18 too, but her 18
choc ices have been shared equally between less boxes. They’ve been divided into three
equal groups, not six.

Now, we’ve got both divisions in
front of us. We can look at them carefully and
think to ourselves, do we have to work out the answer to compare them, or can we use
number sense? Well, one thing we can notice about
these division expressions is that they both start with the same number. So the only thing that’s different
is the number of groups that we’re splitting it into. Six is more than three. So what happens if we take the same
number and we split it into more parts?

Well, if you can imagine a
delicious chocolate bar that you have all to yourself, and then a friend comes, so
you split it into half, and then more and more friends come, so you have to share it
into more and more parts. Each part is going to end up being
quite small. The more we divide a number by, the
smaller the part. And because both of our division
expressions start with the number 18, we just need to look at the second number. Six is a larger number than
three. And as we’ve just said, when we
divide a number into more and more equal groups, there’ll be less in each group. If we split 18 choc ices into six
equal groups, there’ll be less in each group than if we had 18 choc ices and split
them into three equal groups. 18 divided by six is less than the
same number divided by three.

Let’s have a go at answering some
questions now where we have to compare multiplication and also division
expressions. And where we can, let’s use what we
know about numbers to help us. Let’s use our number sense.

Look at these cards: two times
nine, zero times nine, eight times nine, and nine times four. Which expression has the smallest
product? Which expression has the largest
product?

In this question, we’re given four
cards to look at. And on each one, there’s a
multiplication expression. Now, we’re asked two very similar
questions. We need to look at the expressions
on the cards and decide which one has the smallest product and which one has the
largest product. Remember that the word product is
what we get when we multiply numbers together. It’s the answer to a
multiplication. So really, our questions are asking
us which multiplication has the smallest value or the smallest answer and which one
has the largest.

Now, there are two ways we could
solve this problem. Firstly, we could go through each
card, multiply the numbers together, and then just compare all the answers. This would definitely be a way to
solve the problem. But is there a quicker way to find
the answer? If we look really carefully at
these number cards, do you notice anything? Two times nine, zero times nine,
eight times nine, nine times four. The number nine keeps cropping up a
lot, doesn’t it? In fact, the number nine is a
factor in each of the multiplications. They’ve all got it in. In fact, the first three
multiplications are very easy to compare because the number nine is in the same
position in each of them.

We can see straightaway which is
the smallest out of two times nine, zero times nine, and eight times nine. But if we look at our last
multiplication, the number nine is at the start. Does this make any difference to
us? Not at all, because we know it
doesn’t matter which order we multiply two numbers together. They give the same answer or the
same product. So we know that nine times four is
exactly the same as four times nine. And if it helps us, we could think
of this last card as showing four times nine.

Now, all our cards show something
times nine. So which has the smallest
product? Two times nine, zero times nine,
eight times nine, or four times nine. Because we’re multiplying by nine
each time, we simply need to look for the smallest number that we’re multiplying by
nine. And that’s zero. And the opposite is true. If we want to find the expression
with the largest product, we need to find the one that has the largest number that
we’re multiplying by nine. And that’s eight times nine.

Although we could’ve compared these
multiplication expressions by working each one out individually and comparing all
the answers, we noticed that they had something in common. And we used the fact that they were
all to do with multiplying by nine to help us solve the problem without working out
any of the answers. The expression that has the
smallest product is zero times nine, and the expression that has the largest product
is eight times nine.

Compare the expressions. Which symbol is missing? 16 divided by four, what, 20
divided by four.

In this problem, we are given two
expressions to compare, and they’re both division expressions. We’ve got 16 divided by four and
then 20 divided by four. And in between them, we’ve got a
gap where there’s a missing symbol. And when we’re comparing
expressions or numbers or values like this, do you remember the sorts of symbols
that we use? Is 16 divided by four less than 20
divided by four? Is it greater than 20 divided by
four? Or are the two expressions the
same?

Now, one way we could find the
answer might be to actually work out each value. We could divide 16 by four and then
work out 20 divided by four and compare the two answers together. We could definitely find the answer
this way. But perhaps there’s a quicker way
to find the answer. We don’t wanna do any calculations
unless we have to.

Now, what do we notice about these
division expressions? They both show a number divided by
four. Perhaps we could use this to
help. So we could think of both divisions
as looking for the number of fours in the starting number. In other words, how many fours are
there in 16 and how many fours are there in 20? Well, we know if we just look at
the first number in each division, 16 is less than 20. It’s a smaller number. And because it’s a smaller number,
there must be less fours in 16 than there are in 20. 16 divided by four must be less
than 20 divided by four.

We found the answer without having
to do any division at all. Because we divided 16 and 20 by the
same amount, we know that the smaller number will give the smaller answer. There are less fours in 16 than
20. 16 divided by four is less than 20
divided by four. The symbol that’s missing is the
one that means “is less than.”

Use the symbol for “is less than,”
“is equal to,” or “is greater than” to fill in the blank. Six times seven, what, 42.

In this question, we’re being asked
to compare two values together. On one side of the gap, we’ve got a
multiplication expression, six times seven. And on the other side, we’ve got a
number, 42. Is six times seven less than
42? Are they both worth the same? Or is it greater than 42?

Now, sometimes when we compare
expressions like this, we don’t have to work anything out. Sometimes we can see similar
numbers in the expressions. And then we can think about
properties we know that can help us. But in this particular question,
because we’ve got a multiplication on one side and a number on the other, really the
only way we can find the answer is by finding out what this multiplication is
worth. Then we can compare it with the
number. So what is six times seven?

Let’s skip count in sevens six
times. It could be a good chance to
practice our seven times tables facts. Seven, 14, 21, 28, 35, 42. We skip counted in sevens six times
to find that six times seven equals 42. Six times seven isn’t less than or
greater than 42. It’s exactly the same as 42. And because six times seven is
worth exactly the same as 42, the symbol that we need to use in between to fill in
the blank is the one that means “is equal to.” We need to use the equal sign.

Use the symbol for “is less than,”
“is equal to,” or “is greater than” to fill in the blank. 12 divided by six, what, four
divided by two.

In this question, we need to
compare two division expressions together. And once we’ve compared them, we
need to choose the correct symbol to put in between them. Is 12 divided by six less than four
divided by two? Are they both worth exactly the
same? Or is it greater than four divided
by two?

Now, sometimes we can look at a
couple of expressions like this and we can see something in common between them. Perhaps the starting number is the
same or the number that we’re dividing by is exactly the same. When we spot little patterns like
this, we can use it to help us. We can often find out the actual
answer without needing to calculate anything. We can just use our number sense to
help, our knowledge of how numbers work and how divisions work.

But if we look at the two divisions
in this question, we can see that they both contain different numbers. It looks like perhaps that quickest
way to find out the answer to this question is going to be to solve each
division. If we can find out what each
expression is worth, then we can compare the answers.

Firstly, let’s think about 12
divided by six. This is asking us, how many sixes
are there in 12? We know that two times six equals
12. And so we can say 12 divided by six
equals two. The value of our first expression
is two. Our second expression is four
divided by two. It’s asking us, how many twos are
there in four? Well, we know that two times two
equals four. And so four divided by two must be
equal to two. It looks like the value of both
division expressions are exactly the same. And so we need to use a symbol that
shows that 12 divided by six is the same as four divided by two. The correct symbol to use to fill
in the gap is the equal sign.

What have we learned in this
video? We’ve learned how to compare
multiplication and division expressions. We’ve also learned to look for
opportunities to use what we know about the properties of numbers to help us.