Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson.

Are you ready to do some maths? This lesson comes from the unit Division with Remainders.

So we're going to be thinking a bit more about division, how we can represent it, how we can record it, and what it means to have remainders.

So if you're ready to make a start, let's get going.

In this lesson, we're going to be thinking about how the remainder in a division relates to the divisors in a division equation.

So remember what the divisor is, that's the number we are dividing by and the remainder is what's left.

So we're going to think about how we can relate the remainder to the divisor in a division equation.

Let's have a look at what's going to be in our lesson.

We've got 3 key words in our lesson today: Dividend, divisor, and remainder.

Hmm, two of those were in our outcome, weren't they? So let's have a practise saying them and then we'll look at what they mean.

So I'll take my turn, then it'll be your turn.

My turn, dividend.

Your turn.

My turn, divisor.

Your turn.

My turn, remainder.

Your turn.

Well done.

I'm sure they're words you are quite familiar with.

Maybe you've used them before quite recently.

Let's remind ourselves what they mean.

They're gonna be really useful to us in our lesson today.

The dividend is the whole amount to be divided into groups or divided into equal parts.

It's what we are dividing.

The divisor is the number in each group or the number of equal parts that the whole is divided into or between.

It's what we are dividing by.

And the remainder is the amount left over after division when the dividend does not exactly divide by the divisor.

So let's have a look at those words in action as we go through our lesson.

In part one, we're going to explore the remainder when the dividend changes, and in part 2, we're going to be spotting errors in remainders.

So let's get started on part one.

And we've got Jun and Sofia helping us with our learning today.

Sofia's mom has a fruit shop.

The children are helping her to put the fruit out.

Sofia counts 28 apples.

She needs to pack 4 apples in each tray.

How many trays of apples will she put out? She says, "I am dividing the apples into equal groups.

I will write a division equation." 28 divided into groups of 4, 28 divided by 4.

She says, "There are 28 apples.

This is the whole, or the dividend when we're talking about division." They are divided into groups of 4, so 4 is the divisor, it's the number we are dividing by.

"I wonder if there'll be any apples left over," says Sofia.

What do you think? Ah, 28 is a multiple of 4 so there will not be a remainder in the equation.

Let's solve this.

She says, "We can use counters to represent the apples." So there are 28 counters representing the 28 apples.

"There are 28 apples and I must subtract groups of 4 to put in each tray," she says.

So here's her number line.

1 group of 4, 2 groups of 4, 3 groups of 4, 4 groups of 4, 5 groups of 4, 6 groups of 4, 7 groups of 4.

So she's subtracted 7 groups of 4, she's got no apples left, she's reached 0 on her number line.

"I subtract 7 groups of 4 and there are no apples left over," she says.

"I can fill 7 trays with apples and there is no remainder." So 28 divided into groups of 4 is equal to 7.

28 divided by 4 is equal to 7.

Jun says, "I think it would've been easier to use your multiplication facts to solve this." We can skip count up in 4s: 4, 8, 12, 16, 20, 24, 28.

7 times 4 is equal to 28.

So we can make 7 groups of 4, and we know there will be 7 groups of apples.

"I can fill 7 trays of apples and there is no remainder," says Jun.

28 divided by 4 is equal to 7.

Jun finds another apple on the shelf.

Now, he has 29 apples.

He wonders how many trays of apples he can fill now.

What do you think? There's his extra apple represented by one more counter.

He says, "The equation will change because the dividend is now 29." Our whole group is 29 and not 28, so we need to change our equation.

29 divided by 4 divided into groups of 4.

He says, "Let's find out how many groups of 4 are in 29." Do you think you know? Let's use our knowledge of the 4 times table.

1 group of 4, 2, 3, 4, 5, 6, 7 groups of 4 is equal to 28 and there's 1 apple remaining.

So we haven't changed the number of groups we can make, we've just added in 1 more apple, which is a remainder.

There are still 7 trays of apples, but this time there is a remainder of 1 because 29 is 1 more than 28.

So 29 divided by 4 is equal to 7 remainder 1.

Is this statement true or false? "It would be easier to solve this equation by subtracting groups of 6." Sometimes we can solve a division by counting on or by subtracting groups of the divisor.

Would it be easier to do that? What do you think, true or false? Pause the video, have a think, and when you're ready for some feedback, press Play.

What did you think? Well, you might be able to do it.

It's not that it's the wrong way to solve it, but it might not be the easier way to do it.

Why is that? Well, 29 is not a multiple of 6.

So in counting backwards, we would not use our multiples of 6 to help us to subtract 6 each time.

Counting forwards would be easier because we could count in multiples of 6 or use a fact from the 6 times table.

The difference is that if we count up, our remainder, if there is one, is the difference between that largest multiple and our dividend.

When we count backwards, when we get to the point where we can find no more groups of 6, the number we've landed on is our remainder.

So there are good things about counting on, we can use our knowledge of multiples, and there are some good things about counting back because we land on the remainder, but it is harder to count backwards if we're not counting in our known multiples.

So there we go.

If we count forwards, we can use our multiples of 6 and we can land on 24 and know that we have a remainder of 5.

Jun finds another apple on the shelf.

Remember we started with 28, then he found another one for 29, now he's got another one, he has 30 apples.

He wonders how many trays of apples he can fill now.

Remember, he has to put 4 apples in each tray.

There's his 30th apple, has he made another group of 4 yet? Hmm.

He says, "The dividend is now 30.

That's my whole," so he's changed his division, 30 divided by 4 divided into groups of 4.

He says, "I know that 28 is the largest multiple of 4 that is less than or equal to 30 because 7 times 4 is equal to 28," that was his 7 trays of apples he'd filled already.

So can he make another tray of apples? He can't, can he? He says "30 is 2 more than 28, so there must be a remainder of 2.

There are still 7 trays of apples, but there is now a remainder of 2." So 30 divided by 4 will be equal to 7 remainder 2.

Sofia finds another apple, so she now has 31 apples.

There's her extra apple.

She thinks she can predict how many trays she can fill and how many apples there will be left over.

Hmm, I wonder if you can.

You might want to have a go before Sofia shares her thinking.

Go on then Sofia, go for it.

She says "31 divided by 4," and she's got 31 on her number line.

She says, "I've noticed a pattern.

Each time I add 1 more, the remainder increases by 1.

I know that 28 is the largest multiple of 4 that is less than or equal to 31 because 7 times 4 is still equal to 28." She hasn't got another group of 4 yet.

"31 is 3 more than 28," 28 add 2 is 30 and another 1 is 31, "so there must be a remainder of 3.

There are still 7 trays of apples, but this time, there is a remainder of 3." 31 divided by 4 is equal to 7 remainder 3 Sofia's mom gives Jun another apple.

Now, they have 32 apples.

What's this going to look like? Aha, what do you notice now? We've changed our division.

Our dividend is now 32.

32 divided by 4.

Have you spotted something? Jun says, "I know that each time I add 1 more, the remainder increases by 1, so there will be 7 trays of apples with a remainder of 4." 7 times 4 with a remainder of 4.

Hmm, are you thinking what I'm thinking? Do you agree with Jun? Oh, what does Sofia think? She says, "I respectfully challenge you!" That's a very nice way of doing it, Sofia.

She's got a different idea, she wants to challenge his thinking.

"I've noticed that there is a remainder of 4, so I could fill another tray of 4 apples." Oh, well done, Sofia.

Had you spotted that as well? There's another group of 4.

So we've got 7 groups of 4 and another group of 4.

The largest multiple of 4 that is less than or equal to 32 is 32 because 8 times 4 is equal to 32.

So this time, 8 groups of 4 apples is our 32 apples.

We've used all the apples and we've got no remainder again.

Let's look at the equations we wrote.

28 divided into groups of 4 was equal to 7.

So that was our first number of apples.

7 trays, no remaining apples.

Then we had 29 was our dividend and we divided by 4, then 30, then 31, and then 32.

What do you notice about the divisors and the remainder? So the divisors is the number we are dividing by, which is 4 in each case, and we had a remainder of nothing then 1, 2, 3, and then nothing.

What do you notice? There's our divisor, and there's the remainder.

Jun says, "I notice that the remainder is always less than the divisor." We had a remainder of 1, a remainder of 2, and a remainder of 3.

And then when that last apple went in, we were able to make another group of 4, weren't we? We didn't have a remainder of 4.

If the remainder is greater than or equal to the divisor, another group can be made.

And that was Jun's learning point, wasn't it? When we had 32 apples, the remainder didn't increase to 4, it meant we could make another group of 4 apples.

So our quotient, our answer, increased from 7 and possibly some remainders to 8.

With all that thinking in mind, can you solve these equations? You could draw a number line or use your multiplication facts and find the largest multiple of the divisor to help you.

Can you predict and solve the next equation in the sequence? Remember what we've just found out on that slide before, the remainder will always be less than the divisor.

If it's the same as or greater than, we can make another group.

Have a go at these equations, see if you can predict the next one in the sequence, and when you're ready for some feedback, press Play.

How did you get on? Okay, so here was A, we were dividing by 7 so we were looking for numbers that were multiples of 7.

So 35 divided by 7.

Well, we know that the highest multiple of 7 that is equal to or less than 35 is 35, 5 times 7.

So 35 divided by 7 will be equal to 5.

What about 36 divided by 7? Our dividend or whole has increased by 1.

We can't make another group of 7 out of that, so we must have a remainder of 1, 5 remainder 1.

What about 37? Yep, we've added another 1 so we've got a remainder of 2.

For 38, we'll have a remainder of 3.

For 39, we'll have a remainder of 4.

What would the next equation be? Well, it would be 40 divided by 7, which would be 5 remainder 5.

The dividend increased by 1, but the remainder was still less than the divisors, so that also increased by 1.

40 divided by 7 is equal to 5 remainder 5.

And what about in B? Let's have a look.

30 divided by 6.

So our divisor is 6, so we're thinking of multiples of 6.

So you may have used your multiplication facts and knowledge of the largest multiple of 6 to help you.

5 times 6 is equal to 30, so 30 is the largest multiple of 6 that is less than or equal to 30.

Well, it's equal to, isn't it? So 30 divided into groups of 6 or divided between 6 will be 5.

We can make 5 groups of 6.

Or if we divided between 6, there'd be 5 in each group.

What about 32 divided by 6.

Oh, what have we increased our dividend by this time? We've gone up by 2, haven't we? So 30 is still the largest multiple of 6 that is less than or equal to 32.

5 times 6 is equal to 30.

32 is 2 more than 30 so our answer this time is going to be 5 remainder 2.

What about the next one? Can you see we've gone up to 34 again? Again, 30 is still the largest multiple of 6, but this time we've got 4 more so it must be 5 remainder 4.

36 divided by 6, what do you notice now? Ah, 36 is the largest multiple now, 6 times 6 is 36.

We've got another group of 6.

We've increased our dividend by 6, so 36 divided by 6 is equal to 6.

And what about 38 divided by 6? Well again, we've increased our dividend by 2.

So our largest multiple is still 36, but now we've got 2 more so we've got a remainder of 2.

So our answer, the result of our division will be 6 remainder 2.

What about the next equation in the sequence? This would be going up in 2s, 40 divided by 6.

So we've got 2 more again.

We haven't got enough to make another group of 6, so our answer, the result of our division will be 6 remainder 4.

The dividend increase by 2, but the remainder was still less than the divisor, so that also increased by 2.

Well done if you spotted all of those.

And now we're going to be doing some more spotting, spotting errors in remainders.

So the children are still helping Sofia's mum, this time they're sorting bananas.

There are 34 and they're put into boxes of 7.

Jun writes this equation to work out how many boxes of bananas he can fill and how many will be left over.

Is he right? 34 divided by 7.

Sofia says, "Well, there are 34 bananas and we must subtract groups of 7 to put in each tray.

I will use the largest multiple of 7 that is less than or equal to 34 to help me." 4 times 7 is equal to 28.

"4 times 7 is equal to 28, so 28 is the largest multiple of 7.

34 is 6 more than 28, so there will be 6 bananas left over.

So there'll be 4 full boxes of bananas and a remainder of 6 bananas." So 34 divided into groups of 7 gives us 4 groups of 7 and a remainder of 6.

Jun is right, he wrote the correct equation and Sofia solved it very well.

Well done, Sofia.

Sofia finds some different boxes and decides to put the 34 bananas into boxes of 8.

Instead she writes this equation.

34 divided by 8 is equal to 3 remainder 10.

Hmm, you might want to have a think about that.

Let's find out if she's right.

"There are 34 bananas and we must subtract groups of 8 to put in each tray," she says, "3 times 8 is equal to 24 and 34 is 10 more than 24 so there will be 3 boxes of bananas and a remainder of 10 bananas." Hmm, you spotted something? 3 times 8 is 24, and 24 plus 10 is equal to 34.

The number line is correct, but what about that division? Jun says, "That can't be right! 10 is greater than 8, so you can make another box of bananas." If she only had 3 boxes, then she would only be able to put 24 bananas into the boxes.

But I don't think there's a limit to the number of boxes she's got.

If we add 1 more lot of 8, we get to 32, don't we? And then we've got 2 more because 4 times 8 is equal to 32, and then we'd have 2 bananas left over.

"Remember," says Jun, "the remainder is always less than the divisor unless there's a special case." Sofia says, "I will correct my mistake.

There will be 4 boxes of 8 bananas and a remainder of 2 bananas." 4 times 8 is 32, and the 2 bananas gives us 34.

Now the children have learned more about remainders, they decide to check for mistakes in some earlier equations that they've solved.

So we've got 25 divided by 6 is equal to 3 remainder 7.

Sofia says, "I think this is incorrect because 7 is greater than 6." And they've been thinking that if you have a remainder that is greater than your divisor, you can make another group of the divisor.

3 times 6 is equal to 18, but 18 add 7 is equal to 25.

That's right, but that means that our remainder is greater than our divisor.

And in our lesson today, the remainder is always less than the divisor, so we can make another group of 6.

So 3 groups of 6 plus another group of 6.

4 times 6 is equal to 24, and 25 is 1 more than 24 so it should be 4 groups of 6 with a remainder of 1.

So we can change our equation.

25 divided into groups of 6 gives us 4 groups of 6 and a remainder of 1.

Now let's check this equation.

25 divided by 9.

So 25 divided into groups of 9 is equal to 2 remainder 7.

Ooh, that's a big remainder.

Is that remainder okay? Ah, Jun says, "I think this is correct because 7 is less than 9.

The remainder is less than the divisor so we can't make another group of 9." Well spotted, Jun, 7 is quite a large remainder, but our divisor is 9, it's not enough to make another group of 9.

2 times 9 is equal to 18, and 7 more is equal to 25.

So this division is correct, the divisor is greater than the remainder, we can't make another group.

Let's check your understanding.

Can you sort these equations according to whether they are correct or not, thinking about the divisors and the remainder? Pause the video, have a go, and when you're ready for the answers and some feedback, press Play.

What did you think? Let's look at the first one.

49 divided by 6 is equal to 7 remainder 7.

If we're following the rule that the remainder must be less than the divisor, then this is incorrect, isn't it? We could make another group of 6 out of that remainder, so we're going to put that one into this side of the table.

So what would it look like if it was correct? Well, 49 divided by 6 is equal to 8 remainder 1.

6 times 8 is equal to 48 and 1 more is equal to 49, so 8 lots of 6 and 1 remaining.

What about the next one? 49 divided by 8 is equal to 5 remainder 9.

Hmm, no, our remainder is greater than our divisor, isn't it? We can make another group of 8.

So 48 divided by 8 is equal to 6 remainder 1.

6 times 8 is equal to 48.

So if we divided 49 into groups of 8, we could make 6, which would be 48 and there'd be 1 remaining.

What about 49 divided by 9 is equal to 5 remainder 4? Well, this is correct 'cause our remainder of 4 is less than our divisor of 9.

So yes, that is correct.

9 times 5 is equal to 45 and 4 more is equal to 49.

Well done if you spotted those and if you were able to correct them as well.

And it's time for you to do some practise.

Use a tick or cross to show whether each equation is correct or not, correct in that the remainder is less than the divisor.

And if you need to, can you correct the equation? For question 2, choose a times table and create some equations of your own which have a remainder that is less than the divisor.

So for example, Jun says he's going to choose the 9 times-table and he's going to list some multiples of 9.

"These will not have a remainder, so I will pick a number between the two multiples as the dividend," so he's going to pick 21.

"I know 9 is the divisor, so I'll write my equation." 21 divided by 9 is equal to (pauses).

Okay, so you are going to create some of those and ask a friend to solve your equations.

Pause the video, have a go at questions 1 and 2, and when you're ready for some feedback, press Play.

How did you get on? So let's have a look.

18 divided by 4.

So 18 divided into groups of 4, and we've got 3 groups remainder 6.

That's not right, is it? Because our remainder is 6 and our dividend is 4, so we can make another group of 4.

4 times 4 is equal to 16 and 2 more is equal to 18.

So 18 divided into groups of 4 will be 4 groups of 4, and a remainder of 2.

20 divided by 5 is equal to 3 remainder 5.

Oh, well, we've got another group of 5 there, haven't we? And we know that 5 times 4 or 4 times 5 is equal to 20, so there's no remainder.

20 divided into groups of 5 gives us 4 groups.

22 divided by 6 is equal to 3 remainder 4.

That's correct, the remainder is less than the divisor.

3 times 6 is 18, and 4 more is equal to 22.

23 divided by 7 is equal to 3 remainder 2.

Yes, that's correct as well.

3 groups of 7 is equal to 21 and a remainder of 2, 2 more to equal 23.

And 31 divided by 8 is equal to 3 remainder 7.

That's correct as well.

3 times 8 is equal to 24.

So if we divide 31 into groups of 8, we can make 3 groups to give us 24 and there'll be 7 remaining.

So the first two were incorrect and the remaining three were correct.

So for question 2, you were making up your own divisions and asking a friend to solve them.

So you might have chosen the 7 times table and then looked for numbers that were between the multiples.

So 26 divided into groups of 7.

26 divided by 7.

Oh, now Sofia says, "I will solve it.

3 times 7 is equal to 21, so 21," she says, "is the largest multiple of 7 that is less than or equal to 26.

26 is 5 more than 21, so there will be 3 sevens with a remainder of 5." So the answer would be 3 remainder 5.

I hope you had fun creating and solving each other's equations.

And we've come to the end of our lesson.

We've been explaining how the remainder relates to the divisor in a division equation.

In a division equation or problem, the remainder is always less than the divisors.

We can decide if a division equation is incorrect by comparing the size of the remainder to the size of the divisor.

Thank you for all your hard work and your mathematical thinking in this lesson, and I hope I get to work with you again soon.

Bye-bye.