Lesson video

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

And it comes from the unit all about the 7 times table, odd and even patterns, square numbers, and tests of divisibility.

Lots to be thinking about.

So if you're ready, let's make a start on today's lesson.

In this lesson we're going to be using divisibility rules for the 2, 3, 4, 5, 6, 8, and 10 times tables to solve problems. Wow, look at all those divisibility rules we know now.

That's really exciting.

I hope you're familiar with them and ready to use them to solve some problems. Let's make a start.

We've got two key words.

Well, a word and a phrase.

We've got "divisible" and "divisibility rules".

Let's just rehearse saying them and then we'll check what they mean.

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

Are you ready? My turn, divisible.

Your turn.

My turn, divisibility rules.

Your turn.

Well done.

They may well be words you're quite familiar with, but let's just check what they mean because understanding them's gonna be really useful to us in our lesson today.

Divisibility is a number's ability to be exactly divided by another number leaving no remainder.

And divisibility rules let you test if one number is divisible by another without having to do too much calculation.

So let's see how those words are going to be used in our lesson today.

There are two parts to our lesson.

In the first part we're going to recap those divisibility rules and just remind ourselves of how they work.

And in the second part we're going to be using the divisibility rules.

So let's make a start on part one.

And we've got Jacob and Sophia helping us in our lesson today.

So what does divisible mean again? Can you remember? So if a number is divisible by another number, it means it can be divided by that number and the division leaves no remainder.

And Sophia says, "Let's imagine numbers as a big family, and in this family there are certain rules that help us know which numbers are related in certain ways".

That's a really nice way of thinking about it, Sophia.

It helps us to find those family groups of numbers.

And the rules are called divisibility rules and they help us quickly see if one number can be divided by another without any leftovers or remainders.

So it's useful to know divisibility rules as they can help us to solve problems. So let's remind ourselves of some.

So you may remember these.

And these are linked, aren't they? The divisibility rule for 2.

"A number is divisible by 2 if the ones digit is even".

So if it's an even number, it will be divisible by 2.

The divisibility rule for 4 says that "If we halve a number and we get an even value, then the number is divisible by 4".

Four is two times two, isn't it? So if we can divide a number by 2 twice, then we've divided it by 4 and we know that if we can divide a number by 2 it's even.

So if we can do that twice so we can divide, get an even number, we know we'd be able to divide by 2 again, it's divisible by 4.

And then what about 8? Well, eight is two times two, which is four, times two, again, which is eight.

So we need to be able to divide by 2 three times.

So if we can halve a number twice and it's even, we'd know we'd be able to do it again.

So the test of divisibility for 8 is if we can halve a number twice and get an even answer, then we know that number is divisible by 8.

So what do you notice about these three rules? "The divisibility rule for 2: A number is divisible by 2 if the ones digit is even".

So if it's an even number.

"For 4: If halving a number gives an even value, then it's divisible by 4".

And for 8: "If halving a number twice gives an even value, the number is divisible by 8".

What do you notice there? You might want to have a little think before we share our thoughts.

Well, you might have noticed that knowing the rule for 2 helps you to check for 4 and 8, because each rule builds on the idea of dividing by 2.

And this makes it easier to remember and to understand the relationships between these rules.

So if we remember that all even numbers are divisible by 2, that four is two times two, so we have to be able to divide by 2 twice, so halve it twice, and eight is two times two times two.

So we have to be able to halve it three times.

We don't have to do the halving three times, we just have to know that we could halve it again.

So if we halve and halve and we do that twice and we get an even answer, we know we could halve it again and that would make it divisible by 8.

So choose, divisible by 2, divisible by 4, or divisible by 8, and explain that divisibility rule to someone near you.

And can you use an example to prove it? You might want to choose each one once.

Pause the video, have a go.

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

Which one did you choose first? We chose 8 first.

So the divisibility rule says that if you can halve a number twice and the answer is even then the number is divisible by 8.

So we chose 32.

So half of 32 is 16, and half of 16 is 8.

So halving 32 twice gives us 8.

8 is an even number.

So 32 is divisible by 8.

And you might know that 32 is in the 8 times table.

So it's a multiple of eight, so it must be divisible by 8.

Jacob and Sophia now explore divisibility rules for 5 and 10.

Can you remember those ones? Think about the 5 times table.

What do you know about multiples of five? That will tell us the divisibility rule.

So "A number is divisible by 5", says Jacob, "if the ones digit is a 5 or a 0".

It's an odd number times table, isn't it? Five is an odd number.

So the products will alternate between being odd and even.

The odd ones have a 5 in the ones and the even ones have a 0 in the ones.

And what about the divisibility rule for 10? Well, 10 is an even number.

So all multiples of 10, all numbers divisible by 10 will be even.

And more than that, all numbers divisible by 10 have a ones digit of 0.

So, "A number is divisible by 10, if the ones digit is a 0".

'Cause that shows us we've got a whole number of 10s.

So "Both of these rules focus on the ones digit of the number and that makes them easy to apply and remember".

So just rehearse those again.

Choose 5 or 10 and explain the divisibility rule to your partner or to someone around you.

And can you use an example to prove it? Pause the video, have a go.

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

How did you get on? We chose 10.

So numbers divisible by 10 have a 0 in the ones column because it shows we've got a whole number of 10s.

So we've chosen 110.

The ones digit is a 0 so 110 is divisible by 10.

It's a multiple of 10.

So now Jacob and Sophia are going to recap or remind themselves of the divisibility rules for 3, 6 and 9.

Can you remember those ones? What have they all got in common? So Jacob says, the divisibility rule for 3, "For a number to be divisible by 3, the sum of its digits must be a multiple of 3".

Do you remember the digit sum means adding together the digits in the number.

So if we had 54 for example, we'd ignore place value and we'd just add the 5 and the 4.

Five plus four is nine, and that is a number that is divisible by 3.

So 54 is divisible by 3.

What about the divisibility rule for 6? Well, six is three times two, isn't it? And six is an even number.

And we know that multiples of an even number or the products in an even times table are all even.

So for a number to be divisible by six, it must be an even number and the sum of its digits must be a multiple of 3.

So it's got to be an even multiple of 3.

Ah, and Sophia's reminding us about that digit sum.

"To find the sum of the digits you add the digits of the number".

So for example, 18, the sum of the digits is 1 plus 8, which is equal to 9.

We know that 18 is one 10 and an 8, but when we find the digit sum, we ignore the place value.

1 plus 8 is equal to 9, that is divisible by 3.

It's a multiple of 3.

So 18 is a multiple of 3, and it's even, so it's also a multiple of 6.

What about the divisibility rule for 9? Hmm.

Well this is very similar to the divisibility rule for 3.

For a number to be divisible by 9, the sum of its digit must be divisible by 9.

So we could look at 18 again, couldn't we? One plus eight is equal to 9.

9 is divisible by 9.

So 18 is divisible by 9.

So let's look at the divisibility rules for 3, 6, and 9, and think about what's the same and what's different.

You might want to pause the video and have a think before we share our thoughts.

What did you spot? "Both 3 and 9 use the sum of the digits to determine divisibility".

So that's a clear link between the two rules, isn't it? "If you know the rule for 3, you can easily remember the rule for 9".

Numbers that are divisible by 3 have a digit sum that is a multiple of 3 or divisible by 3, and the digit sum of numbers divisible by 9 will be a multiple of 9.

And actually even more than that, if you get a two digit answer to your digit sum, if you add those digits, you will get down to a single digit answer of 9 every time.

So if you think about 99, 11 times 9.

99, 9 plus 9 is equal to 18 and 1 plus 8 is equal to 9.

You'll always get a digit sum of 9 if you go down to one digit.

So what's different about these rules then? We've got the 3 and the 6 highlighted there.

"Well the rule for 6 combines the rules for 2 and 3 since 6 is a multiple of both 2 and 3.

A number must be divisible by both to be divisible by 6.

Understanding the rules for 2 and 3 makes the rule for 6 straightforward".

We know that all multiples of 6 or all numbers divisible by 6 must be even.

So they must be even numbers but also divisible by 3 as well.

It's really useful to remember all these divisibility rules.

So you might want to take a little bit of time thinking about them before we move on with our lesson.

And a chance to think about the 3, 6 and 9.

I would recommend you choose each one once or make sure in the group that you are working with or the people you are working with, you've all had a go at all of them.

So pick a number, explain the divisibility rule that works for it, and maybe think of an example to prove it as well.

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

Which one did you choose first? We chose 3.

So for a number to be divisible by 3, the sum of its digits must be divisible by 3.

So we've chosen 42.

4 plus 2 is equal to 6.

We can ignore place value here and just think about the digits themselves.

4 plus 2 is equal to 6.

6 is divisible by 3.

It's a multiple of three.

So 42 is divisible by 3 as well.

It would also be divisible by 6 because it's an even number.

Would it be divisible by 9? Well, no it wouldn't, would it? Four plus two is equal to 6.

And for a number to be divisible by 9, the digit sum must be 9 or a multiple of 9.

So we can say that 42 would be both divisible by 3 and by 6.

Time for you to do some practise.

It's a real chance here to focus in and think about these divisibility rules.

So we're thinking about the number, we're thinking about the divisibility rule and we're giving an example, and you've got some blanks.

So to start off with, we've got the number 2.

What's the divisibility rule for 2? We've got an example here that 10 is an even number.

So can you fill in the divisibility rule given the number and the example? For the next one we've just got the divisibility rule.

If halving a number gives an even value, then the number is divisible by, hmm? So what's the number and can you give an example? And so on, to fill in the gaps in the chart.

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

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

So for the first one, we knew we were looking at 2 and we had an example that 10 is an even number.

So what's the divisibility rule for 2? A number is divisible by 2 if the ones digit is even.

So if it's an even number, it's divisible by 2.

So the next one said, "If halving a value gives an even number, then the number is divisible by 4".

That's right.

We'd need to be able to divide by 2 twice.

So if we divide by 2 once and it's even, we know we'd be able to divide by 2 again.

So that's the divisibility rule for the number 4.

And an example, "Half of 16 is 8, 8 is an even number so 16 is divisible by 4.

So the next one we just said we were looking at the number 8.

So what's the divisibility rule? Well, eight is two times two times two.

So we've got to be able to halve it three times.

So that means that if we halve it twice and we get an even value, then the number is divisible by 8.

So let's look at 32.

Halving 32 twice gives us 8, half of 32 is 16, half of 16 is 8.

8 is an even number.

We could divide it by 2 again.

So 32 is divisible by 8.

So then we had a rule that says, "For a number to be divisible by, hmm, the sum of the digits of the number must be divisible by hmm".

And we had an example of 27, 2 plus 7 is equal to 9.

9 is a multiple of 9.

So that's a test of divisibility for 9.

For a number to be divisible by 9, the sum of the digits of the number must be divisible by 9.

So next we had an example of 90, the ones digit is a 0.

Hmm.

So I wonder what this could have been? Well this could have been 5 or 10, but we already knew that the next one was 5.

So this was the divisibility rule for 10.

A number is divisible by 10 if the ones digit is a 0, because if the ones digit is a 0, we know we've got a whole number of 10s.

And the next one we knew was 5.

So what's the test of divisibility or the divisibility rule for 5? The number is divisible by 5 if the ones digit is a 5 or a 0.

And an example might be 35.

The ones digit is a 5.

For the next one we had 3.

Ah, so that told us that the previous one had to be 9, didn't it? So what is the divisibility rule for 3? "For a number to be divisible by 3, the sum of the digits must be divisible by 3".

So let's have a look at 36.

3 plus 6 is equal to 9.

9 is divisible by 3.

Ooh, what else would we know about 36 then? That's right, it must be divisible by 9 because the sum of the digits is a multiple of 9 as well, or divisible by 9.

And finally, we knew this was 6.

And this is an interesting rule because 6 is an even number, but it is also follows the rule of multiple of 3 as well or divisible by 3 because three times two is equal to 6.

So for a number to be divisible by 6, the number must be both divisible by 2 and 3.

So it must be divisible by 3 and an even number.

So an example here could be 18.

18 is an even number and it's divisible by 3 because 1 plus 8 is equal to 9, and 9 is divisible by 3 as well.

Well done if you've got all of those right and gave your examples.

Those are really useful divisibility rules to know and we're going to go on and use them in the second part of our lesson.

So let's go.

Let's use some divisibility rules.

"Jacob's class is collecting 416 books for the library.

They need to be organised onto shelves equally.

How many shelves could hold all the books equally?" So we need the same number of books on each shelf.

"Can you circle the correct answers?" So what's divisibility got to do with this? Oh, "Sophia says this seems difficult.

I can't divide a three digit number yet".

"Don't worry", says Jacob, "we can use the divisibility rules to help us".

Oh yes! So if we want to put the books onto three shelves, we need to know if 416 is divisible by 3.

Or onto four shelves, is it divisible by 4? Or onto eight shelves, is it divisible by 8? Well done, Jacob.

That's a really good idea.

So Jacob says, "Let's check divisibility by 3".

So we need to find the digit sum.

The sum of the digits in the number.

4 plus 1 plus 6 is equal to 11.

Hmm.

11's not divisible by 3.

It's not in the three times table.

So we couldn't organise those books onto three shelves and have the same number of books on each shelf.

Ah yes.

Sophia's saying yes.

"11 is not a multiple of 3".

So it won't be divisible by three.

No, you're right Jacob.

"We couldn't have three equal shelves of books because 416 is not divisible by 3".

Let's think about 4 then.

"To check divisibility by 4, well, if halving a number gives an even value, then the number is divisible by 4".

We've halved it.

So we've divided by 2.

And if we get an even number, we know we could be divided by 2 again and that would be the same as dividing by 4.

So, let's halve 416 and see if we get an even number.

All right, Sophia says, "Half of 416 is equal to 208, and 208 is an even number".

So yes, we could put all the books onto four shelves and have the same number on each shelf.

What about 8? So to check divisibility by 8, we need to halve it and halve it again.

"So half of 416 was 208, and half of that is 104".

Ah.

And that's an even answer.

So we could divide by two again and then we'd divide it by two and two and two again, two times two times two is equal to eight.

We can divide it by 8.

So yes, we've halved twice and we've got an even number as our answer.

So 416 is divisible by 8.

So we could have eight shelves in the library and the same number of books on each shelf.

As Jacob's confirming.

Well done.

Time to check your understanding.

True or false? 100 is divisible by 8.

And can you explain why? Pause the video.

Have a go.

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

What did you think? Is 100 divisible by 8? It isn't, is it? How do we know? Well, half of 100 is 50 and half of 50 is 25 and 25 is an odd number.

So 100 is not divisible by 8.

And actually if we think about the 8 times table, 10 times 8 is 80, 11 times 8 is 88, 12 times 8 is 96.

Ah, that's only 4 away from 100.

So we wouldn't have another whole 8 to get us to 100.

So no, 100 is not divisible by 8.

Another problem for us to solve.

"For a maths competition, Sophia has 264 stickers.

Can she divide the stickers between the 6 classes taking part with none left over?" Hmm.

I wonder? What do you think? Ah, Sophia's reminding us.

"If a number is divisible by 6, it will be even and the sum of its digits will be a multiple of 3".

Okay, let's check then.

Two plus six plus four.

Well, 6 plus 4 is equal to 10 and another 2 is equal to 12.

12 is a multiple of 3 and 264 is even.

So she can divide the stickers without any leftover.

So she'll be able to share those stickers out between the six classes, divide them up, and there won't be any stickers left over.

That's good news, isn't it, Sophia? Time to check your understanding.

"Jacob and Sophia are helping with the school fair.

They need to put an equal number of cakes into each box.

The families have donated 256 cakes.

Could 3 cakes go in each box?" So if they had lots of boxes, could they fill up boxes by putting 3 cakes in each box and have no cakes left over? Pause the video, have a go.

See if you can remember that divisibility rule for 3.

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

How did you get on? Did you remember the divisibility rule for 3? Jacob says, "To check divisibility by 3 we need to find the sum of the digits of the number".

We can ignore the place value.

We're just going to add 2 and 5 and 6, and 2 plus 5 plus 6, well let's think.

6 plus 5 is 11 plus 2 is 13.

Oh, 13 is not divisible by 3.

It's not a multiple of 3.

So 256 is not divisible by 3.

So we could put the cakes into boxes, three in a box, but we'd have cakes left over at the end.

Well done if you got that right and if you used your divisibility rule.

"Jacob and Sophia are exploring multiples of 100.

They note down their findings into a table".

Can you see? They're exploring dividing 100 by 4 and by 8.

Thinking of multiples of 100 that are divisible by 4 and divisible by 8 what do you notice? Well, you might have said, "All multiples of 100 are divisible by 4 because halving a multiple of 100 will always leave me with an even number".

That's right, isn't it? Half of 100 is 50, half of 200 is 100, half of 300 is 150.

They're all even, aren't they? So that means all multiples of 100 will be divisible by 4.

Sophia says, "So that means all multiples of 100 will also be divisible by 8!" Is that what the table showed us? "Oh, not quite", says Jacob.

Let's have a look.

In your check you told us that 100 is not divisible by 8.

200.

Well let's halve and halve again.

Is it even? Half of 200 is 100, half of 100 is 50.

That's an even number so that is divisible by eight.

300 isn't.

400, 200, 100.

Yes, that is.

500 it isn't.

600, 300, 150, that is.

Oh, can you see what's happening there? Jacob says, "You know half of 100 is 50, which is even, and half of 50 is 25, which is odd.

This means that 100 is not divisible by 8.

Some multiples of 100 are divisible by 8".

Can you see a pattern? Oh, Sophia says she thinks she can see a pattern.

"Me too!" says Jacob.

"If the hundreds digit in the multiple of 100 is even, it is divisible by 8.

If it's odd, it's not divisible by 8".

If it is an odd number, we're always going to have 100 left over when we do some division, aren't we? And we know that 100 is not divisible by 8.

It's a really useful pattern to spot and something to remember.

Jacob notices something else as well.

He says, "I can also tell if a number will be divisible by 8".

He says, "I can see it without using the test sometimes".

"How?" says Sophia.

Well, let's look at 864.

He says, "I know that 800 is divisible by 8 because I know that 8 times 100 is equal to 800.

And 64 is divisible by 8 because I know eight times eight is equal to 64".

So he can see two multiples of 8 in that number.

800 and 64.

So he doesn't need to use the test of divisibility there, or the rule, he can see in the number itself.

"Oh yes!" says Sophia.

"We could have used that thinking to know that 416 was divisible 4 as well".

Oh, we could couldn't we? 400 is four lots of 100, and 16 is four times four.

So we know that 400 and 16 are both multiples of 4.

So therefore 416 must be a multiple of 4 so it'll be divisible by 4.

So can you have a look? Which of these numbers are divisible by 8? Pause the video, have a go.

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

Which ones? Oh, it's just 400, isn't it? Well, we might have seen that that's 40 lots of 10.

And we know that 40 is a multiple of 8 so we know that that's a multiple of 8.

Or we might have known that pattern that even numbers of hundreds are divisible by 8.

Or we could use the divisibility rule, halving the number and halving it again to see if it gives us an even answer.

400 halved is 200, and 200 halved is 100.

So yes it is.

So lots of different ways that you might have been able to reason that 400 is divisible by 8 but 300 and 500 are not.

Well done if you got that whatever thinking you used.

Maybe you did it in different ways.

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

"Using the divisibility rules, circle the correct answers to solve the following questions".

So in 1A, Miss Coe is organising felt tip pens into equal boxes.

There are 432 felt tip pens.

How many pens can be in each box? So could she put them into boxes with 3 in a box, 4 in a box, 6 in a box? And B, oh, this is my class! Mrs. Hopper's class.

Oh my goodness, we've got 864 pencils and we got to put them in equal boxes again.

How many pencils could go in each box? Could we make boxes of 3 pencils, 4 pencils, 8 pencils? Maybe we could do more than one? And again, this time I'm packaging pears into bags.

Each bag must contain the same number of pears.

How many pears could there be in each bag? 2, 4, 5, 10, or maybe more than one option? Pause the video, have a go at those three questions.

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

How did you get on? So for 1A, Miss Coe was organising felt tip pens into equal boxes.

There were 432.

Could she put them in boxes of 3, boxes of 4, boxes of 6? Well, she could do all of them actually couldn't she? So how do we know that? So to check divisibility by 3, we're going to add the digits.

We're gonna forget their value, we're just going to add them up.

3 plus 4 plus 2.

4 plus 3 is 7.

7 plus 2 is equal to 9.

Oh, that's divisible by 3.

So it would be divisible by 3.

It's also an even number, so it must be divisible by 6 as well.

Is it divisible by 4 though? Well, we'd need to halve it and get an even value.

Half of 432 is 216.

And that's an even number.

So it is divisible by 4.

So Miss Coe could have put her felt tip pens into groups of 3, groups of 4 or groups of 6 and there'd have been none left over.

How about my pencils? There are a lot of pencils there, aren't there? Is 864 divisible by 3? Well, let's add up the digits.

8 plus 6 plus 4.

Well, I can see 6 plus 4 is 10, and 10 plus 8 is 18, and 18 is divisible by 3.

So yes, that's right.

We haven't got 6 this time, we've got 4 and 8.

What's half of 864? It's 432.

That's an even number.

So it is divisible by 4.

And to check divisibility by 8 we can use Jacob's thinking here about 864.

It is divisible by 8.

It's 800 and 64 and both of those are divisible by 8.

Or we could halve it twice.

432.

Half of that is 216, that's an even number.

So different ways we could prove that it is divisible by 8.

So again, I could package my pencils into groups of 3, groups of 4 and groups of 8, and there'd be none left over.

And finally, the pears.

2, 4, 5, and 10.

I know that the pears can be put into pairs because 250 is an even number.

I also know that they can be put into bags of 5 and 10 because 250 has a 0 in the ones.

So it's a multiple of 5 and a multiple of 10.

Half of 250 is 125 and that's an odd number.

So I know that I can't put my pears into bags of 4.

Wow, what a lot of divisibility rules you've applied there.

Well done.

I really like your thinking.

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

This time, we've been using divisibility rules for all sorts of numbers.

2, 3, 4, 5, 6, 8, and 10.

So what have we learned? Well, we've learned that they kind of group together, don't they? So divisibility rules for 2, 4, and 8 involve halving being even numbers, don't they? If it's an even number, it's a multiple of 2 or it's divisible by 2.

If we can halve it and halve it again, it's a multiple of 4 or divisible by 4.

And if we can halve it, halve it, and halve it again, three times, then it's divisible by 8.

But for 4 and 8, we can kind of miss off the final halving and just know that if we've landed on an even number we know it's going to be divisible.

So if we can halve it and it's even it's divisible by 4, and if we can halve it twice and it's even it's divisible by 8, and if it's an even number, it's divisible by 2.

We know that multiples of 10 have a 0 in the ones and multiples of 5 have a 5 or a 0 in the ones.

We know that multiples of 3 have a digit sum of a multiple of 3 and that multiples of 6 are the even numbered multiples of 3.

We also know that there's a digit sum rule for 9 as well.

Multiples of 9 have a digit sum that's a multiple of 9.

And we can use all of those divisibility rules to solve problems and sometimes it makes it a lot simpler.

We don't have to do the actual division and we can work with numbers that are perhaps bigger than we know how to do with our written methods for division.

Thank you very much for all your hard work and your mathematical thinking.

I hope you've enjoyed the lesson, I certainly have, and I hope I get to work with you again soon.

Bye-Bye.