Lesson video

Hello everyone.

Welcome back to another maths lesson with me, Mrs. Pochciol.

I can't wait for us to have lots of fun together and hopefully learn lots of new things.

So let's get started.

This lesson is called "Explain how a dividend of zero affects the quotient," and it comes from the unit doubling, halving, quotative and partitive division.

By the end of this lesson, you should be able to explain how a dividend of zero affects the quotient.

Let's have a look at our keywords for this lesson.

Dividend and quotient.

Let's practise.

My turn.

Dividend.

Your turn.

My turn.

Quotient.

Your turn.

Fantastic.

Let's have a look at our lesson outline for today.

In the first part of our learning, we are going to be looking at when zero is the number of groups.

And in the second part of our learning, we're going to be looking at when zero is the number in each group.

So let's get started with that first learning cycle, when zero is a number of groups.

Jacob and Sofia are back to help us with our learning again.

Are you ready, guys? Let's get started.

Jacob and Sofia have a look at this problem.

Zero times by five is equal to something.

Hmm.

What are you thinking guys? Sofia notices that she can see that one of the factors there is zero.

So we can see this as zero groups of five.

If we have no groups of five, we have nothing.

So zero times five is equal to zero.

Well done, guys.

I really love how you used your prior learning there.

Jacob and Sofia now explore this as a division problem.

I am putting cookies into bags of five.

I have no cookies.

How many bags of five can I make? We know that we are putting cookies into bags of five, so this must be our divisor because that's the number in each group.

We don't have any cookies that are being put into the bags, so our dividend or the whole is zero.

Zero divided by five.

When the dividend is zero, we have nothing.

So that must mean that our quotient must also be zero.

Because if we don't have any cookies, they can't make any bags of five cookies at all.

Zero divided into groups of five is equal to zero.

Jacob and Sofia noticed something else.

Zero divided by five is the related division problem to zero times by five, which is equal to zero.

Yes, the product of zero and five, which is zero, becomes our dividend, and then when we divide by five, it is equal to zero, the other factor.

So that's our related division problem to the multiplication that we solved right at the beginning.

Over to you then.

Solve this problem and record this as an equation.

I am putting cookies into bags of 10.

I have zero cookies.

How many bags of 10 can I make? Use the stem sentence to help you to explain your equation.

Mm, divided into groups of, mm, is equal to, mm.

Pause this video, have an explore of that word problem, and come on back when you've written the equation and can explain it using the stem sentence.

Welcome back.

Let's have a look then at how we got on.

In this problem, we have zero cookies being put into bags of 10, so that's going to be zero divided by 10.

When the dividend is zero, the quotient is also going to be zero.

So zero divided into groups of 10 is equal to zero.

Zero divided into groups of 10 is equal to zero.

That's because if we don't have any cookies, we can't make any groups of 10.

Well done to you if you said this.

Now let's use this knowledge to fill in the missing numbers and to complete these equations.

Zero times 10 is equal to, mm, so zero divided by 10 is equal to, mm.

Pause this video, have a think, use what we've learned so far in our learning to complete the equations, then come on back when you are ready to see how you've got on.

Welcome back.

Let's have a look then.

One of our factors is zero in the multiplication, so we can see this as zero groups of 10.

If we have no groups of 10, we have nothing.

So zero times 10 must be equal to zero.

Then, let's have a look at the division.

If zero times by 10 is equal to zero, then zero divided by 10 is also equal to zero.

Just like the equation that we've just solved in our first check.

Well done to you if you said that zero was the answer to both of those equations.

Jacob and Sofia explore the equations that they've just solved, and they begin to notice something.

We know that when one of the factors is equal to zero, the product is also equal to zero.

And similarly, when the dividend is equal to zero, the quotient is also equal to zero.

Wow.

Did you notice that? So when we have no groups of something, our product is zero.

And when we have no whole or no dividend, that means that the quotient will also be zero.

Wow.

Sofia decides to explore this pattern in the equations that they've solved so far and use this to try and solve one more equation.

What is the same and what is the difference between our division equations here? What pattern do you think she might have noticed? Have a think.

What is the same and what is different in the equations that they've solved so far? Sofia has noticed that in both of the equations that we've solved so far, zero is our dividend.

And Jacob notices that in both of the grouping problems that they've just explored, the quotient was also zero.

When thinking about what is different, Sofia notices that in both of the equations so far, the divisors were different in both problems. Jacob also noticed this, but the quotient was still zero.

So what does that tell us? That must mean that we can see that no matter what the divisor, if the dividend is zero, the quotient will also be zero.

Wow.

What a great thing to notice there, Jacob and Sofia.

Shall we use this knowledge to help us solve that last equation then, Jacob? Jacob knows that zero divided by two will be equal to zero because the dividend is zero, the quotient will also be zero.

What a great thing to notice there.

That's really going to help us to efficiently solve lots of division problems, because as soon as we see that zero is our dividend, we know that the quotient will also be zero.

Over to you then.

Help Jacob to complete some more equations.

Zero divided by eight is equal to something.

Zero divided by four is equal to something.

And something is equal to zero divided by nine.

Have a think of what we've covered so far in our learning.

What are those missing quotients going to be? Pause this video and come on back once you've had a go at answering all three problems. Welcome back.

Let's have a look then.

What did you notice, Jacob? Even though the divisor is different in each of these problems, the dividend is still zero.

So that means that the quotients will also be zero.

In the third problem, we've noticed that the equal symbol has changed positions, but the problem is still zero divided by nine, which we know is going to be equal to zero.

So the quotient will still be zero.

Well done to you if you manage to solve all three of those problems. Let's continue to practise our learning with task A.

In task A, fill in the missing numbers to complete the equations.

So you have some multiplication and you have some division.

So make sure you look carefully as to what equation you are solving.

So pause this video, and once you've completed all of those equations, come on back to see how you've got on.

Welcome back.

Let's have a look then.

How did we get on with these? We can see that in the multiplication equations, all of the products are zero, and when the product is zero, that tells us that one of the factors must be zero.

The factors that we can see are not zero.

We have times by two, times by four, and times by six.

So that tells me that all the other factors must be zero.

Zero times by two is equal to zero.

Zero times by four is equal to zero.

And zero times by six will be equal to zero.

Well done if you said that.

Now let's have a look at those divisions then.

Zero divided by two, zero divided by four, and zero divided by six.

In all of those problems, we can see that our dividend is zero.

We know that when the dividend is zero, the quotient will also be zero because we haven't got anything to divide.

So that means that the answer is going to be nothing or zero.

Well done if you manage to complete task A.

Let's have a look then at the next part of our learning.

We are now going to be looking at when zero is the number in each group.

Jacob and Sofia decide to explore if what they have found works in every situation.

Zero times by two is equal to something.

We know that one of the factors here is zero, so we can see this as zero groups of two.

If we have no groups of two, that means we have nothing.

So zero times by two is equal to zero.

We know that zero times by two is equal to zero, so we can also write the related division fact.

If zero times by two is equal to zero, then zero divided by two will be equal to zero, the other factor.

Well done, guys.

Jacob has remembered that division can be seen as grouping or sharing.

We've already explored some problems where the dividend was zero and we couldn't make any groups, but Sofia wonders if this is still going to be the same in a sharing division problem.

Should we have a look? Jacob and Sofia now explore this problem as a sharing worded problem.

We are sharing apples between two people.

If we have zero apples, how many apples will each person get? I can see that keyword there of sharing.

So we know that we are sharing our whole between two equal groups.

Our dividend is zero, so we will have zero apples, and the divisor is two, so we are sharing them between two people.

Here we have zero apples that are being shared between two people.

If we don't have any apples, then neither person will get any apples at all.

So zero divided between two groups is equal to zero.

That's the same as our other problems. So that means that even when we are sharing between equal groups, if the dividend is zero, the share in each group will be zero.

It doesn't matter whether we are grouping or sharing.

Let's have a look at another problem to see if this is still the case.

Solve this problem and record it as an equation.

We are sharing apples between five people.

If we have zero apples, how many apples will each person get? So record the equation and use that stem sentence to explain what you found.

Pause this video and come on back when you are ready to see how you've got on.

Welcome back.

Let's have a look then at how you got on.

Here we have zero apples being shared between five people, so we can see this as zero divided by five.

If we don't have any apples, we can't share them between five people.

So that means that zero divided between five groups must be equal to zero.

Zero divided between five groups is equal to zero.

Well done if you recorded that equation and said that each person would get no apples.

Let's have a look then at the two equations we've solved so far.

What is the same and what is different? Again, we can see that in all of these equations, our dividend is zero, and in both of our worded sharing problems, the quotient was also zero.

Again, Sofia notices that the divisors were different in both problems. But again, just like before, the quotient was still zero.

So what does this tell us? This tells us that no matter whether we are sharing or grouping, if the dividend is equal to zero, the quotient, either the number of groups or the share in each group will also be zero.

What a great thing that you've noticed there, guys.

Well done.

Over to you then.

Let's use what we've learned to see if we can create our own sharing problems to represent these equations.

Here is a stem sentence that you might like to use.

I'm sharing, mm, between, mm, people.

If I have zero, mm, each person will get, mm, mm.

So use the equations on your left to create your own sharing problem.

You don't have to use that stem sentence to help you.

You might be able to create your own, but just make sure that you check carefully that it matches exactly what the equation is telling us.

Pause this video, have a go at creating your own word problem and share it with the people around you, then come on back to continue the learning.

Welcome back.

Let's have a look then.

I wish I could hear all of your great problems. Jacob decided to choose zero is equal to zero divided by 10, and his problem sounds a little bit like this.

The farmer is sharing her chickens between 10 pens.

If she has zero chickens, then she will have zero chickens in each pen.

Of course.

Because if she hasn't got any chickens, she can't put any in any of those pens, even though I'm sure those 10 pens are very lovely.

Well done, Jacob.

That's a great problem there.

I hope you managed to share your problem with some people around you, and I bet they were just as wonderful as Jacob's problem.

Let's have a look then at the next part of our lesson.

Jacob and Sofia now explore a range of equations, and Jacob thinks he's finished them all already.

Oh.

Hmm.

Yeah, Sofia, I don't think Jacob's read those problems properly either.

Yes, I know that this lesson is all about dividends and quotient being zero, but that doesn't mean that zero is just the answer to every box, Jacob.

I don't think you've read those problems at all.

You've just assumed and put zero in every box.

I think we might need to have another try.

Come on then, Sofia, let's have a look at each of those problems carefully and see if Jacob was correct to put zero in every box.

Hmm.

Let's have a look at the first one then.

Yes, Jacob.

The product is 15, so the missing factor cannot be zero at all for this first problem, and that's the first one.

You didn't even check the first one, nevermind the rest of them.

Let's have a look then.

Something times by five is equal to 15.

Again, the product is 15, so that means that zero cannot be the missing factor.

So what's the missing factor going to be? After looking again, Jacob thinks that three is the missing factor.

How do you know that then, Jacob? He knows that three times by five is equal to 15.

So 15 divided by five will be equal to three.

Well done.

See, Jacob, you did know the answer.

You just didn't read the questions properly.

Let's have a look at the next one then.

We know that two times by five is equal to 10, so 10 divided by five will be equal to two.

Not a zero in sight.

And finally, we can see that the product is zero this time, so that means that the missing factor will be zero.

And here, we can see that the dividend is zero.

So that means that the quotient will also be zero.

You are right, Sofia.

It is a good job that we check through those problems. Well done for spotting that.

And make sure, Jacob, in future, you are looking at all of the problems carefully, not just assuming what the answer is going to be.

I think we need to explore one more problem, don't you think? Sofia and Jacob now use what they know to complete this equation.

Zero divided by something is equal to zero.

Sofia thinks that she could put 10 in here because when the dividend is zero, the quotient will also be zero.

So zero divided by 10 will be equal to zero.

Come on then, Jacob, what number do you think you could put in there? Oh.

175? 175, that number is way too big, Jacob.

Don't be silly.

Oh.

You're quite right, Jacob.

If the dividend is zero, the divisor could be any number because we know that the quotient will still be zero.

That is a really good point there, Jacob.

So zero divided by 175 will be still equal to zero.

So it wasn't a silly answer.

Remember, zero divided by any number will be equal to zero.

Well done to Sofia, and well done to Jacob.

Let's move on then to task B.

In task B, fill in the missing numbers to complete the equations.

Don't fall into the same trap as Jacob and just assume that every answer is zero.

Make sure you look at each problem carefully and use the knowledge that you have to solve them.

And part two is to add your own advisor to complete this equation and explain your thinking.

So pause this video, have a go at part one and part two, and then come on back when you are ready to continue with the lesson.

Welcome back.

Let's have a look then.

I hope you read those problems carefully and didn't just put zero in all the boxes like Jacob did.

Let's have a look.

We can see that something times by 10 is equal to 20.

We know that two times by 10 is equal to 20, so 20 divided by 10 will be equal to two because that's our related division fact.

Something times by 10 is equal to 30.

I know that 30 is three 10s, so three times by 10 is equal to 30.

So 30 divided by 10 will be equal to three.

And finally, we can see here that the product is zero.

So one of our factors must be zero.

The factor that we have already is 10, so that means the other one must be zero.

And here, we can see that the dividend is zero.

So as soon as we see that that dividend is zero, we know that the quotient will also be zero.

It doesn't matter what the divisor is.

Well done if you managed to complete part one.

Let's have a look then at part two.

When the dividend is zero, we can divide by any number and the quotient will still be zero.

So for this problem, you could have put any number.

You might have put 10, 20, 30, 752.

I hope you managed to create lots of different options there for that problem and managed to share them with your friends around you.

Unfortunately, that is the end of our lesson today, so let's have a look at what we've covered.

If the dividend is zero, then the quotient will be zero.

Zero divided by any number is equal to zero.

Thank you for all of your hard work today.

As always, I can't wait to see you all again soon for some more maths learning.

See you soon.