Lesson Video: Relating Multiplication and Division | Nagwa Lesson Video: Relating Multiplication and Division | Nagwa

Lesson Video: Relating Multiplication and Division Mathematics • 3rd Grade

In this video, we will learn how to model the relationship between multiplication and division and write a set of related facts.

15:39

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.

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