Video Transcript
Multiplying Two-Digit Numbers by
One-Digit Numbers: Area Models
In this video, we’re going to learn
how to use what are called area models to help us multiply two-digit numbers by
one-digit numbers. Now, when we learn times tables
facts, we often memorize up to 10 times the number, maybe even 12 times the
number. But what about larger facts? What about 35 times five or four
multiplied by 26? Are we supposed to memorize these
two? Well, you’d be pleased to hear the
answer to that question is no. There are lots of different,
quicker methods that we can use to help multiply two-digit numbers by one-digit
numbers. And in this video, we’re gonna be
using things called area models. In a moment, we’ll go through how
to use them. But here’s what they look like.
They’re usually made up of a
rectangle split into parts. And in this video, we’re going to
be splitting it into two parts. Then it’s usually labeled with some
numbers around the outside, and these are part of the question, and some numbers on
the inside, and these are part of the answer. So whenever you hear the word area
model, this is what we’re talking about. But how could we use a rectangle
like this to help us multiply two-digit number by a single digit? Let’s go through an example.
Let’s imagine that you’ve been
asked to find the answer to 14 multiplied by three. Now this is the sort of question we
think about in this video. It’s a two-digit number multiplied
by a single digit, and it’s more than those times tables facts we’ve memorized when
we learned our three times tables. So this is a good example of a
question that we might think to ourselves, “I’m gonna use an area model to help me
find the answer.” Now if we were trying to imagine
what 14 times three looks like, what would come into your head? You might think of 14 groups of
three or maybe three groups of 14.
Or if you’re feeling hungry,
donuts. Can you see how these donuts show
14 times three? There are three rows with 14 donuts
in each row, so this amount of donuts that we can see here is actually the answer to
our question, isn’t it? 14 repeated three times, 14 times
three. Now, as we’ve said already, area
models are to do with rectangles. Can you see the rectangle here? It’s our array. Now at the moment, this array shows
14 multiplied by three or three times 14. Now this two-digit number 14 seems
like quite a large number to be multiplying by three, doesn’t it? Perhaps we could split up our array
to help us.
For example, two sevens make
14. And so if we know what seven times
three is, that’s this amount of pink donuts here, we could double it to find the
total number of donuts. In other words, seven times three
plus seven times three will help us find our answer of 14 times three. Now, this is an example where we
split the number 14 in half. But splitting two-digit numbers in
half doesn’t always make calculation easier. There’s another way that we could
split up the number 14 to help us. Can you think what it is? We can partition it into its tens
and ones, and we know that 14 is made up of one 10, which is worth 10, and four
ones, so we could show this in our array.
Let’s draw a line to show where
we’ve partitioned the number. In the first part, we’ve got three
rows of 10. And in the second part, we’ve got
three rows of four. And we know that 10 times three
plus four times three equals 14 times three. And what we’ve done here has a
name. Being able to split up difficult
multiplications like this is called the distributive law of multiplication. We don’t need to learn this phrase
in this video. We just need to know that it
works. We can split our array into 10
times three and four times three, and we’d keep the answer 14 times three.
And hopefully you can see why we
might do this. It’s because the two facts we need
to know, which are 10 times three and four times three, are facts that we already
know. 10 threes are 30, and four threes
are 12. We could even label these numbers
on our array. There are 30 pink donuts and 12
blue donuts. And we know that 30 plus 12 equals
42. And that’s how we know that 14
times three is 42. Now, when we described area models
at the start of this video, one thing we didn’t mention was donuts. So now you understand where this
idea of splitting up a multiplication comes from. Let’s try an example where we use
an area model on its own without any array.
So we’ll begin with a question
where we need to multiply a two-digit number by a one-digit number again. And this time, we want to find the
answer to five multiplied by 32 or 32 times five. So we’re not gonna take any time
making an array now. But we are going to draw a
rectangle, and we’ll label it to show our multiplication. We’ll label the shorter side five
and the longer side 32. And we know that the answer to this
multiplication is going to be the area of this rectangle. If you can imagine this rectangle
was five squares tall and 32 squares along, the number of squares in the middle
would be the answer.
Now, at the moment, we’d have to
work out five times 32 to find the answer. And the whole point of using this
method is that we want to break it up into something a little easier. So we can take our two-digit number
and split it up into its tens and ones. And 32 has three 10s, which are
worth 30, and two ones, which of course are worth two. And so we can draw a line to split
up our rectangle to show these two parts. We’ve partitioned or split up 32
into 30 and two.
Now the area of our first rectangle
is going to be worth five times 30. The area of our second rectangle is
worth five times two. And the distributive law of
multiplication that we’ve talked about already means that if we add these two parts
together, five times 30 plus five times two, we’re going to get the answer to five
times 32.
First of all, what’s five times
30? Well, 30 is another two-digit
number, how are we going to multiply this by five? Well, we could use a fact we
already know to help us here. We know that five times three
equals 15, and so five times three 10s or 30 is worth 15 10s or 150. So let’s fill in the area of our
first rectangle 150. And as we’ve said, the area of our
second rectangle is worth five multiplied by two, which of course we know is 10. Now all we need to do is add our
two areas together: 150 plus 10 equals 160. And that’s how we know that five
times 32, which looked like such a hard calculation to work out, is 160. We’ve split it up into two parts to
make it easier and then combined them.
We’re gonna have a go at answering
some questions now where we have to practice these skills. And the good thing about these
questions is that they’re going to take us through this idea of area models really
slowly. They’re going to help us understand
even more how this method works. So here’s the first one.
James drew this array to help him
calculate four times 15 by splitting 15 into 10 plus five. How many squares are in the orange
part? How many squares are in the green
part? Add these numbers to find four
times 15.
The first thing we can tell from
this question is that James has been asked to find out a multiplication fact that’s
a little bit more than the sorts of facts we memorized when we’re learning at times
tables. Perhaps he’s learned his four times
tables facts up to 10 times four, even 12 times four. But in this question, he needs to
find the answer to 15 times four or four times 15. Now we’re told that James has drawn
an array to help him. We can see that he’s done this on
squared paper. Now James could have drawn that
looked a bit like this, in other words, four rows of 15. Can you see that the area of this
array, in other words, the number of squares that there are altogether, will be the
same as four times 15?
But James has wanted to make this
calculation a little bit easier. And we’re told that he split the
number 15 into its tens and its ones, 10 plus five. And that’s why he split up the
array like this. Can you see how 15 has been split
into 10 and five? Now, in the first part of our
question, we’re asked how many squares are in the orange part. Now we can see that the orange part
is made up of four rows with 10 squares in each. And so to find out the number of
squares in the orange part, we just need to multiply four by 10. And four times 10 is 40. So there are 40 squares in the
orange part.
How many squares are in the green
part? Again, we can see there are four
rows. This time, there are five squares
in each row. And so, the total number of squares
in the green part is going to be the same as four times five, and we know that four
fives are 20. Can you see that the two times
tables fact that James has used here are both ones he’s already learned? 10 fours and five fours. If he puts these two answers
together, he can find the answer to 15 fours. And this is exactly what we’re
asked to do in the last part of the question. We need to add together the numbers
to find four times 15. In other words, we need to add
together 40 and 20, and four 10s plus another two 10s is going to give us six 10s or
60.
We found the answer to four times
15 by splitting 15 into 10 plus five because we know that four times 10 plus four
times five is the same as four times 15. Four times 10 is 40. Four times five equals 20. And that’s how we know four times
15 equals 60.
Jennifer drew this to help her
multiply four and 23. Pick the equation that shows how
she split 23. Does four times 23 equal four times
19 plus two, four times 19 plus four, four times 20 plus three, or four times 20
plus four? She can add the areas of the two
parts to find four times 23. Write the sum she should do. 80 plus eight, 76 plus eight, 80
plus 13, or 80 plus 12. And then finally, what is four
times 23?
We can see from the first sentence
in this problem that Jennifer needs to work out the answer to a multiplication. She wants to multiply four and
23. In other words, she wants to find
the answer to four times 23 or 23 times four. Now, if we look carefully at the
diagram she’s drawn, we can see the numbers four and 23. The rectangle that Jennifer’s drawn
is four squares tall and 23 squares across. But we can see that the way that
Jennifer’s colored her rectangle, she split it into two parts. And it’s this long distance of 23
that she split up. Now, in the first part of the
question, we’re asked to pick the equation that shows how Jennifer split 23.
Well, the smaller rectangle we can
see on the end is perhaps the easiest to count. And we can see that this is three
squares long. Can you see that the area of this
rectangle is worth three multiplied by four? Now, if the whole distance is 23
and we can tell that the smaller rectangle on the end is three, we don’t need to
count every single square for the longer length, do we? We know that this must be 20
squares long because 20 plus three equals 23. And you know, when we’re using an
area model, which is what this is, it often makes sense to split of a two-digit
number into the tens and the ones. And that’s exactly what Jennifer
has done. 23 is two 10s and three ones, or 20
plus three.
And if we look at the different
equations, there’s only one that mentions 20 plus three. And that’s this one. We know that 20 plus three equals
23. And so splitting up 23 in this way
is going to make the calculation a lot easier. And we’re told underneath how
Jennifer can add the areas of the two parts to find the answer to four times 23. We need to write the sum she should
do. And to spot which one of these sums
she should work out, we need to calculate the multiplications ourselves.
Now, as we’ve seen already, the
first part of Jennifer’s diagram, that’s the rectangle that’s lighter green, is 20
squares across and four squares up. And the area of this part, in other
words, the number of squares that it takes up on a page, is worth 20 times four. Now we know that two times four
equals eight, and so two 10s multiplied by four is going to be worth eight 10s or
80. So we know there are 80 squares
hidden under this rectangle. Now, if we come back down and look
at our possible answers. We can see that three of them
involve the number 80. So any one of these three answers
could be right. We know the second one is
incorrect.
But what is Jennifer going to have
to add to 80 to find the overall area? Well, as we’ve said already, the
second part of the diagram, that’s the darker green rectangle, is three squares
across and four squares tall. And three times four, we know, is
12. So we can say that there are 12
squares underneath this part of the rectangle. So to find the overall area, she
just needs to add these two parts together, in other words, 80 plus 12. So what is four times 23? Well, 80 plus 12 equals 92.
So this question has shown us step
by step how Jennifer has used an area model to find the answer to four times 23. First, she drew a rectangle that
was four squares tall and 23 squares across. And then she split up the number 23
into two smaller numbers that were easier to multiply by. The equation that shows how she
split 23 is four times 20 plus three. And Jennifer found the areas of
both parts by working out these easier multiplication. 20 times four is 80, and three
times four is 12. So the sum that Jennifer needs to
work out is 80 plus 12. And that’s how we know the answer
to four times 23 is 92.
This area model can be used to
solve 85 times nine. Find the missing partial
product. Nine times 80 equals what.
This question shows us something we
might do if we wanted to find the answer to 85 times nine. How well do you know your nine
times table? Perhaps you’ve learned facts up to
10 times nine, or maybe even 12 times nine. But I’m guessing you probably
haven’t learned them all the way up to 85 times nine. And that’s why we could use what we
call an area model to help us. Now, one thing we know about the
area of a shape is that it’s the space inside it. And we can find the area of a
rectangle by multiplying the length by the width. So if we were going to draw a
rectangle to show 85 times nine, we might draw the length and label it 85 and the
width and label it nine. The area or the space inside it is
going to be worth 85 times nine.
But as we said already, we probably
don’t know the answer to 85 times nine. We need to make it a lot simpler,
and so we can split up the number 85 into two easier parts. Now one way we could split up 85 is
into its tens and ones. We don’t have to split it up this
way, but probably this is the most common way when we’re using an area model. So 85 is made up of eight 10s,
which are worth 80, and five ones. And so nine times 85 is going to be
the same as nine times 80 plus nine times five. And we could alter our area model
to show this. We could draw a line to split it
into two parts and change our length. Instead of saying 85, we could have
80 and then five.
And the area of each part is going
to be worth nine times 80 add nine times five. And we can see this is now exactly
what we’ve got in front of us. And one of the products has been
calculated. Nine times five equals 45. We just need to solve the first
part, nine times 80. Can you think of a fact that might
help us here? We know that nine times eight is
72, and so nine times eight 10s is going to be worth 72 10s or 720. Although the question doesn’t ask
us to, to find the overall answer to our multiplication, we need to add these two
parts together. But the partial product we were
looking for was 720.
What have we learned in this
video? We’ve learned how to use area
models to multiply two-digit numbers by single digits.
