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 using knowledge of division equations and remainders to solve some problems. Have you been doing lots of division recently? I hope so.

Let's see if we can put some of what we've learned into practise and solve some problems. We've got three key words in our lesson today, dividend, divisor, and remainder.

I'll take my turn to say them, and then it'll be your turn.

Are you ready? My turn.

Dividend.

Your turn.

My turn.

Divisor.

Your turn.

My turn.

Remainder.

Your turn.

Well done.

I hope you're quite familiar with those words now, but let's just check what they mean.

They're going to be really useful to us 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 a remainder is the amount left over after division, when the dividend does not divide exactly by the divisor.

There's lots of words in there, aren't there? And we will make sure that they're really clear as we go through the lesson.

In the first part of our lesson, we're going to be exploring equal groups with remainders, and in the second part we're going to be solving a range of division problems. So let's make a start on part one.

And we've got Jun and Sofia with us in our lesson today.

The children are making party hats for Sofia's party.

Sofia has a piece of string that is 30 centimetres long.

She wants to cut it into equal lengths to decorate her party hat.

She says, "I wonder how many equal length pieces I could make, and whether I would have any string left over." What do you think? Jun says, "It would depend on the length of each piece.

Let's try some different options," he says.

We can record the different options in a table.

So the total length of our string is 30 centimetres.

We can decide though, the length of each piece of string, and then we can work out the number of equal length pieces we can get from our 30 centimetres, how much string is left over, and we can record that all as an equation.

So Sofia says, "Well, let's start with 30 centimetres, and let's cut lengths of three centimetres." She says, "10 groups of three is equal to 30, so I could cut 10 lengths of three centimetres with no string left over." So she'd have 10 equal length pieces, no string left over, and her equation would be 30 divided into groups of three.

30 divided by three.

Jun says, "Seven times four is equal to 28.

So I could cut seven lengths of four centimetres and there would be two centimetres left over." 28 add two is equal to 30.

So if he had four centimetre length pieces, he'd have seven pieces and two centimetres of string left over.

So what would his equation be? Well he's taken his 30, divided it into groups of four, and he's got seven remainder two.

What about five centimetre pieces? Sofia says, "Six times five is equal to 30." She says, "I could cut six lengths of five centimetres and there would be no string left over." Six groups of five is equal to 30.

So there'd be six pieces of string, there'd be no string left over, and her equation would be 30 divided into groups of five is equal to six.

What else could they do? "Ah," she says, "Well five times six is equal to 30.

So I could cut five lengths of six centimetres with no string left over." If she can cut six lengths of five centimetres, she can cut five lengths of six centimetres, can't she? And there'll be no string left over.

This time, 30 has been divided into groups of six.

30 divided into groups of six is equal to five.

What about seven centimetre pieces? Oh, well we knew that we could get seven lots of four centimetres, so we must be able to get four lots of seven centimetres.

Four times seven is equal to 28, and there'll be two centimetres remaining.

So four lengths of string that are seven centimetres long, four times seven, plus two, because four times seven is equal to 28 and we've got 30 centimetres of string.

So 30 divided into groups of seven gives us four pieces with two centimetres remaining.

And what about eight centimetre pieces then? Well, Sofia says, "Three lots of eight is equal to 24, so I could cut three lengths of eight centimetres and there would be six centimetres left over." So she can cut three lengths of eight centimetres, three times eight is equal to 24, and she's got six centimetres remaining.

So 30 divided into groups of eight is equal to three, remainder six.

Can you use your times table knowledge to complete the rest of the table? Can you divide 30 centimetres of string into nine centimetre pieces? How many pieces will you get, and how much string will be left over, and write an equation.

And then the same for 10 centimetre pieces.

Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? Well, we've got nine centimetre pieces.

So how many groups of nine can we make before we get to 30? Well, three groups of nine is equal to 27.

So I could cut three lengths of nine centimetres and there would be three centimetres left over.

30 divided into groups of nine gives me three groups of nine with three remaining.

And what about 10 centimetres? Well, three times 10 is equal to 30.

So I could cut three lengths of 10 centimetres and there'd be no string left over.

Three equal length pieces with no string remaining.

30 divided into groups of 10 is equal to three.

Do you see? We got three equal length pieces when we used nine centimetres and 10 centimetres, but we had three centimetres left over from the nine centimetre pieces, and if we added an extra one centimetre onto each, we'd get three lots of 10 centimetres.

And that's effectively what we do when we divide by 10.

We get three equal pieces, and no remainders.

Jun has a box of stars.

He wants to divide them equally between the six children who are decorating their hats.

So we've got a few stars there.

I wonder how many stars each child will receive, and whether there will be any left over.

This time, we know there are six children, but do we know how many stars there are? Jun says, "It will depend upon how many stars are shared out." Ah, so this time we know that we are dividing the stars by six.

Six is our divisor, but we don't know how many stars we're starting with.

We don't know our dividend.

Again, Sofia says, "Let's record some of the options in a table." So this time we need to decide our total number of stars.

We know there are six children, and we're going to fill in the number of stars each child receives, the number of stars left over, and how we'd record that as an equation.

So let's start with 12 stars.

The dividend, the number of stars we're starting with, this is 12, is a multiple of the divisor.

We're dividing by six.

12 is equal to six times two.

So each child will receive two stars, and there'll be no stars left over.

And our equation will be 12 divided between six children is equal to two each.

12 divided by six is equal to two.

What if we started with 15 stars? Well, this time, the dividend is not a multiple of the divisor.

15 is not a multiple of six.

Two times six is still equal to 12.

But there'll be three stars left over.

We've got three more stars.

So each child will receive two stars.

We can make two groups of six, and each group of six is one star each, but there'll be three stars remaining.

So 15 divided between six is equal to two remainder three.

They'll get two stars each, and there'll be three stars left over.

What could we start with this time? Oh, we're starting with 18 stars this time.

What do you spot? That's right.

Our dividend, 18 stars in total, is a multiple of our divisor, which is the six children.

So this time there'll be no stars left over, because 18 is equal to six times three.

So we'll be able to make three groups of six.

So that's three stars for each child, and none left over.

18 divided between six is three each.

What if we started with 20 stars? Ah, well 20 is not a multiple of six.

Our dividend is not a multiple of our divisor, six.

We've got two more stars this time, haven't we? So each child will receive three stars, and there'll be two left over.

We know that six times three is equal to 18, and 18 plus two is equal to 20.

And our division equation is 20 divided between six is equal to three, remainder two.

We can make three groups of six, so each child will get three stars.

What about 25 stars and six children? Well, 24 is a multiple of six, but 25 is one more, isn't it? Our dividend is not a multiple of our divisor.

25 is equal to six times four, plus one.

24 plus one.

So each child will receive four stars, and there'll be one left over.

25 divided between six is equal to four, remainder one.

We can make four groups of six, and in each of those groups, each child gets one star.

So four stars each, remainder one.

And finally, 30 divided by six.

Ah, well we know a multiplication fact here, don't we? 30 is equal to six times five.

So we can make five groups of six out of our 30, and there'll be none left over.

The dividend is a multiple of the divisor.

And our division says that 30 divided between six or shared between six is equal to five each.

Time to check your understanding.

Can you complete the different parts of this table for a total number of stars of 50, and a total number of 60 stars? Pause the video, have a go, when you're ready for some feedback, press play.

How did you get on? Did you spot whether our dividend, our total number of stars, was a multiple of the divisor? Well, for 50 stars, the dividend is not a multiple of the divisor.

Six times eight is equal to 48, and we've got two more stars than that.

So six times eight plus two is equal to 50.

So we know that each child will get eight stars, and there'll be two stars left over.

50 divided between six is equal to eight, remainder two.

And what about 60 stars as our dividend? Oh, well this time the dividend is a multiple of the divisor, isn't it? We know that 60 is equal to six times 10.

So each child will receive 10 stars, and there'll be no remainders.

60 divided by six is equal to 10.

Jun sets Sofia a challenge.

Jun says, "Find as many ways as you can to divide 25 into equal groups.

Predict whether there will be a remainder each time, and record your findings in a table like this." So this time, 25 is our total value, our dividend.

We've got to decide on our group size, our divisor, work out the number of groups, if there's a remainder, and then record a multiplication equation and a division equation.

Gosh, lots to fill in there.

Let's have a look.

Jun says, "You could use a number line to help you, if you want to." Okay Sofia, off you go.

Sofia says, "I will make groups of one.

25 groups of one is equal to 25, so there'll be 25 groups of one and no remainder." So our group size is one, and we can make 25 groups of one.

There'll be no remainder.

25 times one is equal to 25.

25 divided into groups of one gives us 25 groups.

A good start there, Sofia.

Well done! What could we do next, do you think? Ah, Sofia's going to be systematic.

She started with a group size of one.

Now she's going to have a group size of two.

So her divisor is two.

Will there be a remainder? There will, won't there, because 25 is an odd number.

So what is a multiple of two that is as close to but less than 25? Well it's 24, isn't it? 12 times two is equal to 24.

So there'll be 12 groups of two, with a remainder of one.

12 times two is equal to 24, and one more gives us our 25.

So there'll be 12 groups and a remainder of one.

25 is equal to 12 times two, plus one.

So our division says 25 divided by two will be equal to 12, remainder one.

Aha, and your task is to continue Sofia's investigation.

Use a number line or your times table facts to complete the table.

Continue until you have found all the possibilities.

So can you keep going all the way up to 12? Pause the video, have a go, and when you're ready for some feedback, press play.

So we had a group size of three next, didn't we? 25 divided into groups of three, or shared between three.

So 25 is not a multiple of three, so we will have a remainder, but we know that eight times three is equal to 24, and one more is 25.

So we can make eight groups of three and have a remainder of one.

25 is equal to eight times three plus one.

25 divided by three is equal to eight, remainder one.

What about four? Well, 25 is not a multiple of four either, is it? But we know that six times four is equal to 24, plus one.

So we can make six groups of four with one remainder.

25 is equal to six times four plus one.

25 divided by four is equal to six, remainder one.

What about a group size of five? Oh, hooray! Yes.

Our dividend is a multiple of our divisor.

Five times five is equal to 25.

So we can make five groups with no remainders.

25 divided by five is equal to five.

What about six? Well, I think I can see a pattern emerging here.

Can you? We can use what we knew about a group size of four to help us here.

When we had a group size of four, we made six groups and there was one remainder.

So when we've got a group size of six, we must be able to make four groups with one remainder.

25 is equal to four times six, plus one.

So 25 divided by six is equal to four, remainder one.

What about seven? No, 25 is not a multiple of seven, is it? But seven times three is equal to 21, plus four to equal 25.

So 25 divided by seven will give us three groups, with a remainder of four.

Three, remainder four.

What about eight? Well do you remember, we had eight threes were equal to 24, and then another one.

So three eights are equal to 24 and another one.

25 is equal to three times eight, plus one.

25 divided by eight is equal to three, remainder one.

Nine.

Hmm.

Well, two times nine is equal to 18, plus seven this time.

So 25 divided by nine is equal to two, remainder seven.

Big remainder, but it's still smaller than the divisor, which is nine.

25 divided by 10.

Well we know 25 isn't a multiple of 10, but we can make two groups of 10 and we've got five remaining.

25 divided by 10 is equal to two, remainder five.

11.

Ooh, no.

I don't think that's going to work, is it? But two groups of 11 is equal to 22, and we've got three remaining.

So 25 divided by 11 is equal to two, remainder three, because two times 11 plus three is equal to 25.

And finally, 25 divided by 12.

25 isn't a multiple of 12, but two times 12 is equal to 24.

So plus one for our remainder.

25 divided by 12 is equal to two, remainder one.

Two groups of 12, and one remainder.

Whew.

Well done.

I hope you were able to use your times table knowledge to help you there.

And on into the second part of our lesson, we're going to solve a range of division problems. The children set each other some more challenges.

Sofia writes some division equations for Jun to solve.

So we're dividing by four this time.

Our divisor is four, and we're dividing 20 by four, 21 by four, 22 by four, 23 by four, and 24 by four.

Hmm.

I wonder if you can think what's gonna happen here.

Let's start with looking at 20 divided by four.

The dividend, 20, is a multiple of our divisor, four.

Four times five is equal to 20.

So there'll be no remainder.

So 20 divided into groups of four, or divided by four, is equal to five, and there's no remainder.

What about 21 divided by four? Can we use our last answer to help us here? Well, Jun says the dividend, that's 21, has increased by one, but the divisor has stayed the same.

I can still make five groups of four, but this time the remainder will increase by one.

So we will have a remainder of one.

So 21 divided by four is going to be five, remainder one.

We can make five groups of four out of 20, but we'll have one left over.

Can we use that thinking to help us with the next one? 22 divided by four.

Well again, the dividend has increased by one.

21 is now 22, but the divisor has stayed the same.

So the remainder will increase again by one.

Five, remainder two.

What about 23 divided by four? Again, our dividend's increased by one, but our divisor is the same.

So four times five is still equal to 20, but this time we've got a remainder of three.

What about 24 divided by four? Ah! Jun says he's used the pattern.

24 divided by four, he says, we've increased our dividend by one.

So it must be five remainder four.

Hmm.

Can you spot something there? You might want to have a think before Jun has a look carefully.

Have you spotted something, Jun? Well, he says, "Let's use a number line to explore the pattern." He's not quite sure yet.

20 divided by four.

Five groups of four is equal to 20.

No remainders.

21, well we've got our five groups of four, and one remainder.

22.

Five groups of four, and an extra two.

23.

Five groups of four, and an extra three.

24.

Five groups of four, and another four.

What can we do here? "Oh," he says, "I remember that the remainder is always less than the divisor, but our remainder is equal to our divisor here.

We could make another group of four." 24 divided by four is going to be equal to six, remainder zero.

So we can make six groups of four, and there's no remainder.

So it's always worth checking to see if you can make another group.

So check your remainder carefully.

So have a look at this, time to check your understanding.

Which of the following equations could be answered in a different way? So which of them have been maybe recorded incorrectly? And explain how you know.

Pause the video, have a go, and when you're ready for some feedback, press play.

So let's have a look.

46 divided by nine.

Well, nine times five is 45, remainder one.

Yeah, so five, remainder one.

That's okay.

46 divided by eight.

Well, eight times five is equal to 40.

And that's a remainder of six.

So five, remainder six.

That's okay.

46 divided by five.

Well, eight times five is equal to 40, plus six, but oh, hang on a minute.

Hmm.

The remainder should be less than the divisor.

The divisor is five.

So there shouldn't be a remainder of six.

We can make an extra group of five, can't we? 46 divided by five is equal to nine, remainder one.

Nine times five is 45, plus one is 46.

Now, Jun sets Sofia a challenge.

Jun says, "I have fewer than 45 straws.

If I shared them equally between eight people, there would be no remainder.

If I shared them equally between three people, there would be no remainder either." He says, "I must have 24 straws.

Is this true or false?" You might want to have a think before Sofia has a go.

Sofia says, "When the straws are divided between eight, there is no remainder.

So the dividend must be a multiple of eight that is less than 45." Good thinking, Sofia.

Can we list some? What else did Jun say? He said, when they are divided between three, there is no remainder.

So the dividend must also be a multiple of three.

Can you remember the rule of divisibility for three? That's right.

The sum of the digits must be a multiple of three, or divisible by three.

Well, eight isn't.

One plus six is seven.

Two plus four is equal to six.

So yes, 24 is a multiple of three.

32 isn't, and 40 isn't.

So 24 is the only multiple of both three and eight, below 45.

So it's true! There must be 24 straws.

24 divided by eight is equal to three, and 24 divided by three is equal to eight.

And there's no remainder each time.

Well done, Jun.

You were correct.

Time to check your understanding.

True or false? There are fewer than 30 sweets in a bag.

When they are divided between eight children, there is no remainder.

And when they are divided between six children, there is no remainder.

There must be 24 sweets.

Is that true or false? Pause the video.

Have a go.

And when you're ready for some feedback, press play.

How did you get on? It's true! Let's have a look at that number, 24.

So 24 is a multiple of eight, and 24 is a multiple of six.

It's in the six and the eight times tables.

There is no remainder, so the dividend must be a multiple of the divisor.

So it's true.

The only number below 30 that is a multiple of both eight and six is 24.

Well done if you were able to explain that.

Sofia sets a missing number problem for Jun.

Something divided by something is equal to four remainder three.

Hmm.

And Sofia says, "Circle all the possible values of the star." So that's our divisor, isn't it? The number we're dividing by.

Which of these numbers could it be? One, two, three, four, or five? I wonder what Jun's going to think about this.

You might want to pause and have a little think before he has a go.

Go on then, Jun.

Take it away.

He says, "We know that all numbers are divisible bye one, so if the divisor is one, there will be no remainder." So it can't be one.

He says, "The remainder is always less than the divisor, and the remainder is three, so the divisor must be more than three." Oh, well done, Jun.

I really like your thinking there.

So we can't be dividing by two or three, because if we had a remainder of three with a divisor of two or three, we'd be able to make another group, wouldn't we? So the possible values for our divisor are four and five.

"Now," says Sofia, "Can you complete the equation?" If our divisor was four, what would our dividend be? Well, Jun says, "There are four groups of four, because when we've divided into groups of four, we've got four groups of four, and three remainder.

So the dividend would be 19." Four groups of four is equal to 16, and an extra three gives us our 19.

Excellent thinking there, Jun.

Well done.

What about if our divisor was five? Well, this time, we've divided into groups of five, and we've made four groups of five, with three remaining.

So five times four, or five four times is equal to 20, plus another three is equal to 23.

So our dividend must have been 23.

Really good thinking there, Jun.

Excellent problem, as well there, Sofia.

I like that.

The children work together to solve this problem.

Let's help them.

Can we write down all the possible values of the rectangle? And this time, the rectangle is our remainder.

So what could our remainder be? Sofia says, "The divisor is five, and we know that the remainder must be less than the divisor.

So the possible values of our remainder are one, two, three, or four." If we had a remainder of five or more, we'd be able to make another group of five.

And what's its largest possible value? Well the largest possible value is four, because the remainder cannot be equal to the divisor, because we'd be able to make another group.

Great thinking and reasoning there, Jun and Sofia.

And I hope you were able to follow their reasoning as well.

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

There's a couple of problems here for you.

Use what you've learned and solve the problems. Remember, you could draw a number line to help you if you want.

So in A, you're going to circle all the possible values of the star, our divisor.

Something divided by something is equal to six, remainder four.

And in B, you're going to find all the possible values for the rectangle, or our remainder.

Something divided by eight is equal to six, remainder something.

And then for both of them, can you find all the possible solutions to the equation? And in question two, you're going to decide whether each of the following are true or false, and explain how you know.

Pause the video, have a go at your questions, and when you're ready for the answers and some feedback, press play.

How did you get on? So here we were looking for the divisor.

Possible values of the divisor.

The remainder is four, so the divisor must be more than four.

So our divisor must be five or six.

Something divided by five is equal to six, remainder four.

Well, we've made six groups of five, which is 30, plus four.

So our dividend must be 34.

And if it was six, we've made six groups of six, with four remaining.

Six sixes are 36, plus four is equal to 40.

So the dividend could be 40.

In B, we had to find all the possible values for the remainder.

So if we've got a divisor of eight, then the remainder must be less than the divisor.

So it must be less than eight.

So our remainder could be one, two, three, four, five, six, or seven.

So if it was a remainder of one, we're dividing something by eight and we've made six groups of eight.

So six eights are 48, plus one is 49.

If we had a remainder of two, our dividend would be 50, for three, it would be 51, for four, it would be 52, for five, it would be 53, and for six, it would be 54, and for seven, it would be 55.

Each time we're adding one more on, aren't we? We've still got six groups of eight being 48.

And then we've added on one, two, three, four, five, six, and seven, to work out what our dividend was.

In question two, there are fewer than 45 children in the playground, and when they're divided into teams of 10, there is no remainder.

And when they're divided into teams of five, there is no remainder.

It must be 35 children.

Well, there are no remainders, so the dividend must be a multiple of the divisors in each case.

So it must be a multiple of 10 and five that is less than 45.

So the multiples of 10 that are less than 45 are 10, 20, 30, and 40.

All of the numbers shown are also multiples of five.

So any of these numbers are possible answers.

35 is not a possible answer, because it's not a multiple of 10.

So A was false.

B, there were fewer than 56 pencils in a box, and when they're divided between nine children, there is no remainder, and when they're divided between four children there is no remainder.

There must be 36 pencils.

There are no remainders, so the dividend must be a multiple of the divisors in each case.

So it must be a multiple of nine and four, and less than 56.

So the multiples of nine that are less than 56 are nine, 18, 27, 36, 45, and 54.

36 is the only multiple here that is both a multiple of four and nine.

So it must be true.

I hope you were able to reason that and work it out using your knowledge of multiples.

Great thinking.

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

We've been using knowledge of division equations and remainders to solve problems. What have we thought about today? Well, if the dividend is a multiple of the divisor, there will be no remainder.

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

If we know the remainder, we can identify the possibilities for a missing divisor.

And if we know the divisor, we can identify the possibilities for a missing remainder.

And this knowledge can help us to solve problems. Thank you for all your hard work and your mathematical thinking, and I hope I get to work with you again soon.

Bye bye!.