Lesson Video: Multiplying Two-Digit Numbers by One-Digit Numbers: Area Models | Nagwa Lesson Video: Multiplying Two-Digit Numbers by One-Digit Numbers: Area Models | Nagwa

Lesson Video: Multiplying Two-Digit Numbers by One-Digit Numbers: Area Models Mathematics • 4th Grade

In this video, we will learn how to use area models to multiply two-digit numbers by one-digit numbers.

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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.

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