Lesson Video: Comparing Multiplication and Division Expressions Mathematics • 3rd Grade

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.