Video: Multiplying a Three-Digit Number by a One-Digit Number: Area Models

In this video, we will learn how to multiply a two- or three-digit number by a one-digit number by writing the number in expanded form and applying the distributive property.

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Video Transcript

Multiplying a Three-Digit Number by a One-Digit Number: Area Models

In this video, we’re going to learn how to multiply a two- or a three-digit number by a one-digit number. And we’re going to use area models to help. This involves writing the larger number in expanded form and then applying what’s called the distributive property.

Let’s imagine that we want to multiply together 16 and four. One way of modeling this would be as an array. Here, we’ve got four rows, and there are 16 squares in each row. And we know that four lots of 16 is the same as 16 lots of four. We also know that if we want to make this calculation easier, we can split up our array. There are lots of ways we could partition or decompose the number 16, but one ways to split it up into 10 and six. Let’s model this on our array by using different-colored squares.

We can see now that 16 times four is the same as calculating 10 times four, six times four, and then adding the two parts together. Can you see why we might want to do this? We might not know our four times table up to 16 times four. But we do know what 10 times four and six times four are. We’ve split a fact that we don’t know into two facts that we do know. The way that we can split up a multiplication like this is called the distributive property of multiplication.

Now notice that, so far, we haven’t actually counted those squares in our array, have we? Although we’ve taken the time to lay them out neatly in rows, we could actually find out the answer without counting them at all. So we could just represent what we’ve got here using two rectangles or rather one rectangle that we then split into two. Because we know that the area of a rectangle, that’s the space inside it, is found by multiplying the length by the width, we can see that our first rectangle represents four times 10 or 10 times four and our second rectangle has an area worth four times six or six times four. And because of everything we know about the distributive property of multiplication, we know that the area of the whole rectangle is worth 10 times four plus six times four or 16 times four all together.

Let’s quickly go through and find the answer then. We know that 10 times four equals 40, six times four equals 24, and if we add 40 and 24 together, we get 64. So we can say 16 times four equals 64. Now how could we apply this sort of area model to help us multiply not just with two-digit numbers like this, but with three-digit numbers?

What’s 257 multiplied by five? This time, we’re multiplying a three-digit number by a single digit, and we can use the same method as before. So to begin with, we can think about how to decompose or partition the number 257 to make the whole thing a lot easier. There are lots of ways we could split up the number 257 but with a three-digit number like this. Perhaps the quickest way is to think about it in expanded form, in other words, split it up into its hundreds, tens, and ones. In the hundreds place, we have the digit two, so this has a value of 200. In the tens place, we have the digit five; this is worth 50, five 10s. And there’s a seven in the ones place. 257 is the same as 200 plus 50 plus seven. And we can use this to help us solve the multiplication.

Now, if we were to represent 257 multiplied by five or five times 257 as an area model, it might look like this. But once we’ve partitioned our number, we can split up our area model too because the total of 200 times five, 50 times five, and seven times five will be the same as 257 times five. First, we multiply the hundreds. What’s 200 times five? Well, we know that two times five equals 10. So two 100s multiplied by five must equal 10 100s, which we know is the same as one 1000.

Now, we need to find the area of the next part of our model. We need to multiply the tens. What is 50 times five? Well, again, we can use a fact we already know to help us. Five times five equals 25. And we know that 50 is the same as five 10s. So we know five 10s times five must equal 25 10s, and 25 10s is 250. And we should know the fact already for the last part of our area model; seven fives are 35. Now, all we need to do is to find the total, and you might be able to add together 1000, 250, and 35 in your head. But if not, we could just use a standard written method. We have a total of five ones. There are eight 10s altogether, two 100s, and one 1000.

So, although 257 multiplied by five seemed like a really difficult calculation at the start, by splitting up this three-digit number into its expanded form — in other words, its hundreds, its tens, and its ones — we could sketch an area model to help us because the distributive property of multiplication tells us that we can split up multiplication is like this to make them easier. 257 times five equals 1285. Let’s have a go at solving some problems now where we can use area models to help.

Pick the multiplication expression that matches this area model. Seven times 406, seven times 164, 146 times seven, 700 times 641, or 641 times seven.

In the picture, we can see an area model, and we know that these are a way of helping us to visualize multiplications. They don’t tell us what the answer is, but they do give us an idea of how to get there. We’re asked to pick the multiplication expression that matches this model. And underneath, we’ve got five to choose from. So what’s this area model showing us? Firstly, we can see that this side of the rectangle is labeled seven. This tells us that one of the numbers we’re going to be multiplying by is going to be seven. Now, if we look at our possible answers, we can see that most of them do have seven as a factor. However, there’s one that doesn’t.

So perhaps, the first thing we can do is to cross off 700 multiplied by 641. This isn’t the right answer at all. But what are we multiplying seven by? This is where our area model becomes interesting because we can see that it’s been split up into three parts. The length of the first part is labeled 100. And because we know that the area of a rectangle is worth its length multiplied by its width, we know that the area of this first part is 100 multiplied by seven. The length of our second part is labeled 40. So we know that the area of this part is 40 times seven. And we can see that the area of our last part is six times seven.

So our area model seems to represent three multiplications, not one. But we know if we add these three partial products together, we can find the answer to an overall multiplication. 100 plus 40 plus six equals 146. And so, by adding together 100 times seven, 40 times seven, and six times seven, we can find the answer to 146 times seven. This area model is exactly the sort of thing we could sketch if we wanted to find the answer to 146 times seven. It shows us how we could partition the three-digit number to make it easier to multiply. The multiplication expression that matches this area model is 146 times seven.

Calculate 211 multiplied by three.

In this question, we’re asked to multiply a three-digit number by a single digit. Now, when we learned our three times tables fact, we probably only learned them up to 10 times three or maybe 12 times three. How are we going to find out what 211 times three is? To help us, we could split 211 into smaller parts. Perhaps the quickest way to do this is to think about 211 in expanded form. In other words, what the hundreds, the tens, and the ones are worth in this number. We know that 211 has a two in the hundreds place, so that’s worth 200. The one in the tens place has a value of 10. And the one in the ones place, of course, is just worth a single one.

So we can partition or decompose 211 into 200 plus 10 plus one. Now, how can we use this new split-up version of 211 to help us with this multiplication? We could show how using an area model. We know that we can find the area of a rectangle by multiplying its length by its width. And so this area model represents the calculation we want to find out, 211 multiplied by three. Now, what’s it going to look like if we split it up? Something like this. Notice how it’s still the same size. We’ve just split it up a little. The area of our first part is worth 200 times three. Our second part has an area of 10 times three. And our last part is worth one times three.

And if we add these three partial products together, we can find the overall area, which is going to be the answer to our multiplication. First, let’s multiply the hundreds. We know that two times three is six, so 200 multiplied by three is going to be 600. Now, we multiply the tens and 10 times three equals 30. And finally, the ones. One times three equals three. And to find the overall answer, we just need to add together 600 plus 30 plus three. This is the sort of calculation we can do in our heads; it’s 633.

We’ve multiplied a three-digit number by a single-digit number here by first partitioning our number into its hundreds, tens, and ones and then by using this to sketch an area model to help us see what we needed to do. We multiplied each part out separately and then added all the products together. 211 multiplied by three equals 633.

Calculate 145 multiplied by six.

Now finding out what 145 sixes are isn’t the sort of fact that we already know in our heads. How are we going to multiply this three-digit number by six? We could represent this calculation by sketching an area model with a length worth 145, a width of six, and an area that represents the multiplication we need to solve. Now this is a bit tricky to try to work out all in one go. But is there a way that we could split up the number 145 to help us? We could write the number 145 in expanded form, showing what the hundreds, the tens, and the ones are worth.

145 equals 100 plus 40 plus five. And we can split up our area models to show this. The area of our first part is worth 100 times six. Our second part is worth 40 times six. And the area of our last part is five times six. So if we want to solve 145 times six, we just need to add together 100 times six, 40 times six, and five times six. First, we can multiply the hundreds. 100 multiplied by six — it’s quite quick to do this one. It’s 600.

Next, we can multiply the tens. What’s 40 times six? Well, we can use a six times tables fact we already know to help us here. We know that four times six is 24, so four 10s times six equals 24 10s or 240. And finally, we multiply the ones. Five sixes are 30. So to find the overall answer, we just need to combine our three parts back together. The answer to 600 plus 240 plus 30 is going to have zero ones, seven 10s, and eight 100s. It’s 870. To find the answer to 145 times six, we split up 145 into its hundreds, tens, and ones. We multiplied each part by six separately and then combined them all back together again. 145 times six equals 870.

So what have we learned in this video? We’ve learned how to multiply a two- or three-digit number by a single digit by first writing the larger number in expanded form and then using the distributive property to help. We used area models to help us see what to do.

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