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 use our knowledge of times tables and divisibility rules to identify when there will be a remainder in a division equation.

So have your times tables knowledge at your fingertips, and we're gonna have a look at some divisibility rules.

Can you remember those? Here are some keywords to help us in our lesson.

We've got dividend, divisors and remainder.

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

So my turn, dividend, your turn.

My turn, divisors, your turn.

My turn, remainder, your turn.

Well done.

I hope they're words that you are quite familiar with and you've maybe been using quite a lot recently, but let's just remind ourselves of 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 being 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.

Look out for those words as we go through today's lesson.

They're gonna be really useful.

There are two parts in our lesson.

In the first part, we're going to be using multiplication facts and in the second part we're going to be using the rules of divisibility.

So let's make a start on part one.

And in this lesson, we've got Jun and Sophia helping us with our learning.

The children are tidying the classroom.

Oh, well done guys.

It's really helpful when you tidy the classroom.

I hope you are helpful in tidying your classroom as well.

There are 48 pencils.

Each packet holds eight pencils.

Jun wonders if there will be any pencils left over after he's filled each packet.

He's got a stem sentence here to help him.

48.

Hmm.

A multiple of eight.

So when it is divided into groups of hmm, there are hmm leftover; there is hmm remainder.

So we are thinking about multiples here.

So are things a multiple of the divisor or not? And if they are or they aren't when they're divided into groups, will there be anything left over? So let's have a think.

48 pencils and they're going into packets of eight.

Let's complete the stem sentence to help us find out.

June says, I'm dividing the pencils into groups of eight.

I must think about whether 48 is a multiple of eight.

Because we are dividing 48 by 8, 48 divided into groups of eight.

Ah, well if we think about our times tables, six times eight is equal to 48.

So yes, 48 is a multiple.

So 48 is a multiple of eight.

So when it is divided into groups of eight, there are none left over.

There is no remainder.

So thinking about whether a number is the multiple of the divisors helps us to work out if there will be a remainder or not.

So Jun can put his pencils into packs of eight.

One pack, 2, 3, 4, 5, 6 groups of eight is equal to 48.

So 48 divided by eight is equal to six and there's no remainder.

Sophia collects 49 felt tips.

Hmm.

These must also be tidied into packets of eight.

Have you spotted something? Let's think about whether there will be any felt tips left over once the packets are filled.

Sophia says, when I divided 48 into eights, there was no remainder.

So will there be a remainder when I divide 49 into eights? Hmm, I wonder.

49 is one more than 48.

So I predict there will be one felt tip left over.

So let's have a look.

49 divided by 8.

49 divided into groups of eight.

Oh six times eight is 48, but there's one leftover isn't there? 49 is not a multiple of eight.

So when it is divided into groups of eight, there are some leftover, there is a remainder and we can see on the number line what our remainder will be.

Let's put the 49 pens away in packs of eight.

1, 2, 3, 4, 5, 6 groups of eight and one remaining.

So 49 divided into groups of eight is six remainder one, six groups of eight, and one remaining.

Time to check your understanding.

The children have 50 pieces of card and they put eight pieces on each table.

How many tables do they give card to? And do they have any pieces left over? Write the equation to represent the problem and predict whether it will have a remainder and then solve it.

Maybe you could think about the stem sentence.

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

What did you think? This time it was 50 divided into groups of eight.

50 divided by eight.

Well we know that 48 is a multiple of eight and 50 is two more than 48.

So 50 is not a multiple of eight.

We can predict that when it is divided into groups of eight, there will be some leftover, there will be a remainder.

So how many groups of eight can we make? 1, 2, 3, 4, 5, 6 groups of eight and two pieces of card remaining.

Six groups of eight is equal to 48 and we need two more to reach 50.

So 50 divided by eight is equal to six remainder two.

50 divided into groups of eight gives us six groups and two left over.

There are 48 paint brushes.

Sophia divides them equally between the seven tables where children are painting.

Hmm, have you spotted something? She says ,"When I divided the 48 pencils into groups of eight, there were none left over.

So I predict that when I divide the 48 paint brushes, there will be none leftover either." Hmm, do you agree with her? What do you think? Jun says let's write the equation.

48 paintbrushes dividing them equally between the seven tables.

Ah.

So 48 shared between seven tables.

Hmm, what have you spotted? Hang on, Sophia, the divisors is seven, not eight.

I must consider whether 48 is a multiple of seven this time.

Ah, well spotted, Sophia.

We have to be careful.

What is it we are dividing by? It's a multiple of the divisors we are looking for.

Six times seven is equal to 42 and seven times seven is equal to 49.

So 48 is not a multiple of seven and we are looking for the multiple of seven that is less than or equal to our dividend, which is 48.

So six groups of seven is equal to 42 and then there'll be six remaining.

So 48 is not a multiple of seven.

So when this is divided into groups of seven, there are some leftover.

Each time we take a group of seven, there'll be one for each table won't there? So our answer to our division is that there will be six paintbrushes on table and six leftover because there are six groups of seven in 42 and then there will be six leftover.

Jun finds another paintbrush.

So now there are 49 paintbrushes to divide equally between the seven tables.

Will there be a remainder now? What do you think? Jun says, "There was a remainder when I divided 49 by eight, but this time I'm dividing by seven." Hmm.

Yes, do you remember there were 49 felt tips to put into groups of eight.

I think it was Sophia that did that bit of tidying, but Jun was helping.

He says, "I must consider whether 49 is a multiple of seven this time." Is it? What do you think? 49 divided by seven.

Yes, seven times seven is equal to 49.

So 49 is a multiple of seven.

Seven groups of seven are equal to 49.

So when the paint brushes are divided between seven tables, there are none left over.

There is no remainder.

Each table gets seven paintbrushes.

Time to check your understanding.

Which of the following will not have a remainder? So think, will it have a remainder or will it not have a remainder? Pause the video and have a go.

When you're ready for some feedback, press play.

What did you think? Did you look carefully at the divisors? 24 is a multiple of six.

So when it is divided by six, there will be none leftover, there will not be a remainder, and 24 is a multiple of eight.

So when it is divided by eight, there will be none leftover, there will not be a remainder.

So in A and C there will be no remainder because 24 is a multiple of six and 24 is a multiple of eight, six times four is equal to 24 and three times eight is equals 24.

What about B? Well, 24 is not a multiple of seven is it? So there will be a remainder if we divide 24 by seven, three times seven is 21.

So there'd be a remainder of three.

The children have 56 water pots, when they divide them between some tables, there is no remainder.

And Jun says, "There are more than five tables in the class, but fewer than 10." How many tables could they have divided them between? Hmm, you might want to pause and have a think before they share their thinking.

Sophia says, "The divisors must be between five and 10." So it must be 6, 7, 8, or nine.

And remember, there are 56 water pods and there's no remainder.

"56 must be a multiple of the divisor," says Sophia.

Then there'll be no remainder.

So 56 divided by something.

Let's have a think.

Well, six times nine, nine times six is equal to 54.

Oh no, so that's not going to work, is it? Another group of six would take us to 60.

So 56 is not a multiple of six.

Well, eight times seven is equal to 56.

So 56 is a multiple of seven.

So there could be seven tables.

56 divided by seven is equal to eight.

Ah, what about eight? Well seven times eight is equal to 56.

So 56 is a multiple of 8.

56 divided by eight will be equal to seven.

What about nine? Well, no we already know nine times six is 54, so six times nine must be equal to 54.

So 56 is not a multiple of nine.

So they could have divided the water pots between seven or eight tables in the classroom.

Did you notice anything about the two equations we found that worked? 56 divided by seven is equal to eight and 56 divided by eight is equal to seven.

Jun says, "Once I remember that seven times eight is equal to 56, I knew that if I could make eight groups of seven from 56, I could also make seven groups of eight from 56." Let's have a look.

I can share out eight groups of seven and seven groups of eight.

And again, it's my favourite times table fact, seven times eight is equal to 56.

Jun wants to write some equations where he divides by seven.

He has a method for deciding if his equation will have a remainder or not.

Let's find out what it is.

He says, "I will write down the multiples of seven.

So there we go, from one time seven up to 10 times seven.

My divisors is seven.

So all of these are multiples of the divisors.

So if we are dividing by seven, all of these would give us no remainder.

He says, "I know if I choose any one of these numbers as the dividend, there will not be a remainder because they're all multiples of seven." Seven divided by seven, 14 divided by seven, 21 divided by seven, 35 divided by seven.

Those would not give us a remainder because the dividend is a multiple of the divisors.

Sophia wonders which numbers she should choose as the dividend for the equation to have a remainder.

She says, "You can choose any number that is not a multiple of seven if you are dividing by seven." So she's going for a number between 42 and 49, 45.

45 divided by seven or 62 divided by seven or 29 divided by seven.

Now the dividends, the numbers we are dividing are not multiples of the divisors, they're not multiples of seven.

So there will be a remainder.

Time for you to do some practise now.

Can you write and solve three equations with a divisors of nine that will not have a remainder.

And then write and solve three equations with a divisors of nine that will have a remainder.

Remember, just like Jun did, you can write out your multiples of nine to help you and then choose multiples where you need to and choose numbers or dividends that are not multiples when you need to as well.

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

How did you get on? There were lots of equations that you could have chosen, but here are some examples.

To write an equation with no remainder, you could have chosen any multiple of nine as the dividend, the number you were dividing, because when the dividend is a multiple of the divisor, there is no remainder.

So in this case, if you chose a dividend, a whole that's a multiple of nine, our divisor, there'll be no remainder.

So you could have had 45 divided by nine is equal to 5, 54 divided by nine is equal to six, or 63 divided by nine is equal to seven.

What about for the other set where we needed a remainder? To write an equation with a remainder, you could have chosen any number that is not a multiple of nine because when the dividend is not a multiple of the divisor, there is a remainder.

So when our whole is not in the nine times table or a multiple of nine, there will be a remainder.

So we could have had 47 divided by nine is equal to five, remainder two.

56 divided by nine is equal to six, remainder two.

And 65 divided by nine is equal to seven, remainder two.

Can you see what we did? We added two to the dividend each time so that meant there would be two remaining.

And on into part two where we're going to think about using the rules of divisibility.

Sophia thinks she can use the rules of divisibility to predict whether any division equation will have a remainder.

Do tell Sophia I'm interested.

Let's recap some of the rules first.

Sophia says multiples of five have a one digit of five or zero.

So here are some multiples of 5.

45, 50, 55, 60, 65, 70, 75.

Going a bit beyond our times table knowledge there.

But we know that they are multiples of five.

Multiples of two are always even.

When you halve a multiple of two, you will get a whole number.

So all of those numbers are even, we can look at the ones digit.

If it's a ones digit of 0, 2, 4, 6 or eight, then we know that we have an even number, a multiple of two.

Multiples of four are always even as well.

But to have a multiple of four, you've got to be able to divide it by two twice because two times two is equal to four.

So if you halve a multiple of four, you will get an even number or to check that a number is a multiple of four, it must be even to start with.

And then when you halve it, it must also be even.

And for a multiple of eight, you've got to be able to do that again.

Eight is two times two times two.

So you'd have to be able to divide by two three times.

So if we have a number and have it again and the answer is still even, then we will be able to divide it again and we'll have a multiple of eight.

So let's use these rules of divisibility to see if these equations will have a remainder.

What about 75 divided by five? Well Sophia says we know that multiples of five have a ones digit of five or zero.

So 75 is a multiple of five.

The dividend 75 is a multiple of our divisor five, so there'll be no remainder.

What about 75 divided by two.

Well, multiples of two are always even and 75 is odd.

So the dividend is not a multiple of the divisor.

75 is not a multiple of two.

So there will be a remainder this time.

Time to check your understanding.

Which of the following will have a remainder and explain how you know.

Pause the video, have a go.

When you're ready for some feedback, press play.

How did you get on? Let's look at A.

37 divided by two.

Well, that will have a remainder, won't it? Multiples of two are always even and 37 is odd.

So the dividend, 37 is not a multiple of the divisors of two.

So there will be a remainder.

What about 65 divided by five? Nope, there'll be no remainder.

Multiples of five have a one digit of five or zero.

So 65 is a multiple of five.

The dividend is a multiple of the divisor, so there will be no remainder.

What about 41 divided by two? Well again, 41 is an odd number isn't it? And multiples of two are always even.

So the dividend 41 is not a multiple of the divisor two.

So there will be a remainder.

Well done if you've got all of those.

Let's use the rules of divisibility for dividing by four and eight to decide if these equations will have a remainder.

Remember, if we can halve a number and it's even, it's a multiple of four.

And if we can halve a number twice and the answer's even, it's a multiple of eight.

Jun says, "We know multiples of four, if halved, will reach an even number.

Half of 30 is 60, which is even, so 60 is a multiple of four.

The dividend is a multiple of the divisor so there will be no remainder when we divide 60 by four." What about 60 divided by eight? What about multiples of eight? Well, if we halve a number twice and it's even, then it's a multiple of eight.

Half of 60 is 30, half of 30 is 15.

Oh, that's an odd number.

60 is not a multiple of eight so the dividend is not a multiple of the divisor so there must be a remainder when this division happens.

Time to check your understanding.

Which of the following will have a remainder when we do the division and explain how you know.

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

How did you get on? So in A, we've got 67 divided by four.

Oh, well, 67 is an odd number, isn't it? Multiples of four are always even, 67 is odd.

So the dividend is not a multiple of the divisor and there will be a remainder.

What about 108 divided by eight? Well, 108 is an even number.

We have to halve it and halve it again and the answer be an even number.

Well half of 108 is 54 and half of 54 is 27, which is odd.

So the dividend again is not a multiple of the divisor.

So there will be a remainder.

And 99 divided by eight, well, we we know 99 is an odd number.

Multiples of eight are always even, and 99 is odd so it's not a multiple of eight.

The dividend is not a multiple of the divisor so there will be a remainder again.

Well done if you spotted that all of those answers would have remainders.

Let's remind ourselves of the rules of divisibility for dividing by three, six, and nine.

Multiples of three have digits that sum to a number that is divisible by three.

So for example, if we look at 24, the sum of 2 and 4 in 24 is six, which is divisible by three.

When we do a digit sum as we call it, we ignore the place value.

We know that 2 in 24 is actually worth 22 10s.

But when we add the sum of the digits, we literally just take the value of the digit and add it together.

And we know that six is an even number.

So multiples of six must also be even.

Their digits sum to a number that is divisible by three and they are even, they have to be divisible by two and three.

And multiples of nine have a digit sum that is divisible by nine.

So here are the multiples of nine.

So 45, 4 plus 5 is equal to nine, and that is a digit sum that is divisible by nine.

So let's use those rules of divisibility to decide if these equations will have a remainder.

58 divided by three.

Hmm.

So we've got to add the digits together.

Multiples of three have a digit sum, which is divisible by three.

So the digit sum of 58, five plus eight is 13, which is not divisible by three.

So 58 is not a multiple of three.

There will be a remainder.

We know that multiples of six must be even and their digit sum must be divisible by three.

Well, we know that 58 does not have a digit sum divisible by three, so therefore the dividend is not a multiple of the divisor and there will be a remainder.

So what about 130 divided by nine? Well, multiples of nine have a digit sum that is divisible by nine.

So the digit sum of 130, one plus three plus zero is four.

That's not divisible by nine, is it? So the dividend is not a multiple of the divisor.

It will have a remainder as well.

So all of those divisions would have a remainder.

Time to check your understanding.

Which of the following will have a remainder and explain how you know? Pause the video, have a go.

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

How did you get on? Let's look at A.

66 divided by three.

Multiples of three have a digit sum that is a multiple of three.

The digit sum of 66 is 12, which is a multiple of three.

So the dividend is a multiple of the divisors so there'll be no remainder.

What about 85 divided by six? Oh, well we know this isn't, it's an odd number, isn't it? Multiples of six are even, so 85 is not a multiple of six because it's an odd number, it's not even.

So this will have a remainder.

The dividend is not a multiple of the divisor.

And what about 124 divided by nine? Multiples of nine have a digit sum that is divisible by nine.

The digit sum of a 124, one plus two plus for is seven.

That's not a multiple of nine.

The dividend is not a multiple of the divisor.

So there will be a remainder.

I hope you are able to use your rules of divisibility to work those out.

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

Use the dividend and the divisor, so that's the whole number that we're starting with and the number we are dividing with to decide whether the equation will have a remainder.

Use your rules of divisibility to help you.

And then write it in the correct part of the table, you'll see the table in a moment and then solve the equation.

So here's our table.

You're going to write the equation into remainder side or no remainder side and then solve the equation.

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

How did you get on? That was your table, let's have a look.

So 56 divided by two.

Well, that is an even number, isn't it? So there won't be a remainder.

56 divided by two is like saying half of 56.

Half of 50 is 25, half of six is 3, 25, add three is 28.

So 56 divided by two or halved is 28.

54 divided by five.

All multiples of five have a five or zero in the ones so the dividend here is not a multiple of the divisor.

So there will be a remainder.

But what do we know here? Well, 10 groups of five is 50 and four remaining.

So our answer will be 10, remainder four.

10 times five is equal to 50, and four more is 54.

62 divided by three.

So we are looking for a digit sum here.

Six plus two is equal to eight.

That's not a multiple of three is it? So the dividend is not a multiple of the divisor.

We've used our rule of divisibility for three.

So there will be a remainder.

But let's have a look at this.

This might be outside our times table knowledge, but let's have a think.

Well, 60, 62 times three so 60 must be 20 times three and a remainder of two.

So 20, remainder two.

And 75 divided by four.

Well, 75 is an odd number, isn't it? Multiples of four are always even.

So the dividend is not a multiple of the divisor, that is a remainder.

Ooh, 75.

Can we think about this? Well, I could partition 75 into 40 and 35.

40 is 10 lots of four and 35 is eight lots of four and three remainder.

So 18 remainder three.

And what about the next lot? 72 divided by eight.

If 72 is a multiple of eight, I should be able to halve it, halve it again and get an even number.

Half of 72 is 36 and half of 36 is 18, which is even.

So yes, it is a multiple of eight.

In fact, it's nine times eight, isn't it? The dividend is a multiple of the divisor.

So there's no remainder.

What about 82 divided by nine? Well, I can do a digit sum.

Eight plus two is 10, that's not a multiple of nine, not divisible by nine.

So there will be a remainder.

But I do know that nine times nine is 81, so it'll be nine, remainder one.

25 divided by eight.

Well, it's an odd number, isn't it? So it there will be a remainder because multiples of eight are always even.

But I know that three times eight is 24, so it'll be three remainder one.

And finally, 72 divided by six.

So to be a multiple of six, the digit sum must be a multiple of three and the number must be even.

72 is even, seven plus two is equal to nine.

So yes, multiples of six are even, and the digit sum is divisible by three.

So there will be no remainder here.

And I know that 72 divided by six is equal to 12.

Well done if you used your rules of divisibility and then used some different division strategies to be able to work out those answers.

Great thinking.

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

We've been identifying when there will be a remainder when dividing.

What have we thought about? Well, we know that if the dividend is a multiple of the divisors, there will be no remainder.

The dividend is our whole, the divisors is the number we're dividing by.

And we also know that if the dividend is not a multiple of the divisor, there will be a remainder.

We can use multiplication facts to decide whether there will be a remainder using our times table knowledge and we can also use the rules of divisibility to decide if there will be a remainder.

You've done lots of fantastic mathematical thinking today.

Thank you so much for your hard work and I hope I get to work with you again soon.

Bye-bye.