Lesson video

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

And it comes from the unit all about the "Seven 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 the divisibility rules for the 3, 4, 6, and 8 times tables to solve problems. Have you come across divisibility rules before? I wonder if you have.

Well, in this lesson we're going to be recapping some and seeing how we can use them to solve problems. So let's make a start.

There are two key words in our lesson.

Well, one word and one phrase.

One is divisible and one is divisibility rules.

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

So are you ready? My turn, divisible, your turn.

My turn, divisibility rules, your turn.

Well done.

So divisible and divisibility rules are obviously linked, aren't they? Let's check out what they really mean.

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

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

So when we do a division and there's no remainder, the number we've divided by.

We can say that our number that we started with is divisible by the number we've divided by.

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

There are some tricks and some that we can understand why they work.

So let's have a look at some as we make a start in this lesson.

There are two parts.

In the first part, we're going to be recapping those divisibility rules, reminding ourselves of them.

And in the second part, we're going to be using them.

So let's make a start with some recaps.

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

So Jacob's just reminding us what does divisible mean.

And he says, "It means 'can be divided by.

'" And it can be divided by without leaving anything left over.

No remainders.

So it's useful to know divisibility rules as they can help to solve problems. They can also help you to check whether an answer is correct or not.

Let's remind ourselves of a couple.

Jacob's reminding us of the divisibility rule for 3.

He says, "For a number to be divisible by three, the sum of the digits of the number must be a multiple of three." Oh, there's a lot in there.

I think he's gonna show us an example a bit later on.

The divisibility rule for 3 is all about the sum of the digits of a number.

Sofia is reminding us about the divisibility rule for 4.

She says, "If halving a number gives an even value, then the number is divisible by 4." Hmm, what else do we know about halving? If we can halve a number, then it's a multiple of two.

It's divisible by two.

And if we halve it and we get an even number, it means we can divide by two again.

And dividing by two and dividing by two again is the as dividing by four.

2 times 2 is equal to 4.

So if we divide by two once, and divide by two again, we've divided it by four.

So if we can halve a number and the answer is an even number, then we know we can halve it again, so it must be divisible by 4.

What about the divisibility rule for 6 then? Hmm.

Well, six is an even number, isn't it? We may know, you may have had a lesson recently where you've thought about the patterns in the times tables, and we know that all multiples of six must be even numbers.

If one of our factors is even, then the product will be even as well.

So a multiples of six, or numbers that are divisible by six, must be even.

So Jacob says, "For a number to be divisible by 6, the number must be divisible by both 2 and 3, because 2 times 3 is equal to 6." So it must be an even number and it must be divisible by 3.

And we know for it to be divisible by 3 that the sum of its digits will be a multiple of 3.

So if the sum of the digits is a multiple of 3 and the number is even, then we know it is divisible by 6.

And what about 8? Well, this links to our 2s and our 4s.

Remember, if we can halve a number, it's divisible by 2.

If we can halve it and it gives an even answer, then we can halve again, it's divisible by 4.

And if when we've halved it twice, it's still even, then we can halve it again, and that means it's divisible by 8.

2 times 2 is equal to 4 and 4 times 2 is equal to 8.

So if we can halve a number once, twice, and three times, then it will be divisible by 8.

So we know that if we halve a number twice and it gives an even number, then it is divisible by 8.

Ooh.

Let's just have a pause and check our understanding of that.

Have a look at these four statements and select the ones that are true.

So A says, "For a number to be divisible by 3, the sum of the digits must be divisible by 3." B says, "If halving a number gives an odd value, then the number is divisible by 4." C says, "For a number to be divisible by 6, the number must be divisible by both 2 and 3." And D says, "If halving a number twice gives an odd value, the number is divisible by 8." Pause the video, have a look at those statements, decide which are true, and when you're ready for some feedback, press play.

So which ones were true? That's right, A and C were true.

For a number to be divisible by 3, the sum of the digits must be divisible by 3.

B is not true.

If halving a number gives an odd value, then it is not divisible by 4.

If halving it gives an even value, then it will be divisible by 4.

And C, for a number to be divisible by 6, the number must be divisible by both 2 and 3, because 2 times 3 is equal to 6.

Yes, that's true.

And D, if halving a number twice gives an odd value, no, then that number is not divisible by 8.

If halving it twice gives an even value, then it is divisible by 8.

Well done if you've got those ones right.

Ooh, now we've got our cauldrons back to sort our numbers into.

Have you met these ones before? Jacob and Sofia need to sort numbers into the correct cauldrons, and the cauldrons show what numbers are divisible by.

So we've got divisible by 3, by 4, by 6, and by 8 on our cauldrons, and we've got the number 15.

I wonder if you can see something.

I wonder if you can work it out just by looking at the number 15.

What do you know about odd multiples? Well, let's have a think about what Sofia and Jacob are going to say.

Ah, they've said, "Remember, a number is divisible by 3 if the sum of its digits is divisible by 3." So what do we mean by the sum of the digits? Well, to find the sum of the digits, we add them together.

So we forget that 1 is a ten and this 5 ones.

We just look at the digits themselves.

So we've got a 1 and a 5, and 1 plus 5 is equal to 6.

6 is divisible by 3, it's a multiple of 3, so 15 is divisible by 3.

Let's check.

Yes, a puff of purple smoke.

Now, yes, did you remember? Because it's an odd number, it will not be in the 4, 6, or 8 times tables, so it will not be divisible by 4, 6, or 8.

We know that if we have an even times table, then all of the products in the times table will also be even.

So 15 is not divisible by 4, 6, or 8.

It's not in those times tables.

Time to check your understanding.

We've got two cauldrons here, divisible by 3 and divisible by 4.

Which cauldron would the number 18 go into? Pause the video, have a think, and when you're ready for some feedback, press play.

What did you work out? Well, Sofia says, "For a number to be divisible by 3, the sum of the digits must be a number divisible by 3." So what is the sum of our digits here? 1 plus 8 is equal to 9.

9 is divisible by 3.

It's in the 3 times table.

So 18 is divisible by 3.

Yes, well done.

What about 4? So if we halve a number and we get an even number, then it's divisible by 4.

Well, half of 18 is 9, and that's an odd number, so 18 is not going to be divisible by 4.

And you might also be able to work this out by thinking about your 4 times table.

16 is a multiple of 4, and 20 is a multiple of 4, but not 18.

So 18 is not divisible by 4.

So our number would go into the divisible by 3 cauldron.

So Sofia and Jacob are moving on to another number.

This is 36.

So we've got 3, 4, 6, and 8 here.

I wonder which cauldrons it's going to go into.

It's an even number this time, isn't it? So we can't say straight away that it won't go into any of them.

So let's think about that divisibility rule for 3.

We need the sum of the digits, don't we? Oh, 3 plus 6 is equal to 9.

Remember, when we're finding the sum of the digits, we can ignore the fact, and we shouldn't really, but we can ignore the fact that the 3 represents 3 tens.

We're just looking at the 3 in its own right.

3 plus 6 is equal to 9.

9 is divisible by 3.

So that will go into the divisible by 3 cauldron.

Let's have a look.

Yes, puff of purple smoke for us.

So what else do we know? It's divisible by 3 and it's an even number.

Ah, that means 36 is also divisible by 6 because 36 is an even number and it is a multiple of 3.

So yes, it's going to be divisible by 6.

And there goes the puff of purple smoke.

Do you remember the 4? If we can halve the number and it gives us an even value, then it must be divisible by 4 because we could halve it again.

So what's half of 36? Well, half of 36 or 36 divided by 2 is equal to 18, half of 30 is 15, and half of 6 is 3, and 15 plus 3 is equal to 18.

18 is an even number, so we could halve it again.

So that means 36 is divisible by 4.

Yes, correct.

What about 8 then? This is where we have to halve it and halve it again and get an even number.

Hmm.

So is 36 divisible by 8? Well, 36 halved is 18.

We know that 18 halved is 9, and 9 is not an even number, it's an odd number, so 36 is not divisible by 8.

And we could check by thinking about our 8 times table.

We know 32 is a multiple of 8 and 40 is a multiple of 8, but 36 isn't.

Well done if you were able to reason that along with Jacob and Sofia who are speaking to us even though we can't see them on the slide this time.

Time to check your understanding.

Which cauldron would 24 go into? And can you explain why to your partner? We've got divisible by 3, 4, 6, and 8 here.

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

Which cauldrons did you think it would go into? Well, it goes into 3, doesn't it? Because 2 plus 4 is equal to 6, so it is divisible by 3.

What about by 4? Well, we'd have to halve it, and the answer to our halving would have to be even.

Half of 24 is 12.

That's an even number, so yes, it is divisible by 4 as well.

Well, we know it's divisible by 3 if we're thinking about 6 and it's an even number.

So yes, it's divisible by 6 as well.

It's divisible by 3 and it's an even number.

What about 8? Is it divisible by 8? So we'd have to halve it and halve it again.

So half of 24 is 12 and half of 12 is 6.

That's an even number again, isn't it? We could have it again, so it is.

So 24 is divisible by 3, 4, 6, and 8.

Well done if you've got all of those right.

Were you expecting it to be in all of those cauldrons? Time for you to do some practise.

You are going to use the divisibility rules to sort the numbers below into the correct cauldrons.

We'll see the numbers in a moment.

So there's two parts to this question.

For each number, you're going to justify your thinking.

So A says, are there any numbers which do not go in any of the cauldrons? And B, can you think of a number that can go into all of the cauldrons? So you're going to sort the numbers and then spot if there are any that don't go into any cauldrons, and if you can think of a number that would go into all of them.

So here are your cauldrons and your numbers.

Sort them and then think about parts A and B.

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

How did you get on? So here are all the numbers sorted into their correct cauldrons.

So did you see that 60, 12 and 6 were all divisible by 3, but because they're all even numbers, they're also divisible by 6? 60 and 12, also divisible by 4, as were 40, 32, and 28.

We can halve all of those numbers and the answer is even so we know it's a multiple of 4.

And 40 and 32, we could halve and halve again and still get an even number.

So half of 40 is 20, half of 20 is 10, we can half 10 again, so that makes it divisible by 8.

And, in fact, we know that 5 times 8 is equal to 40.

And 32, half of 32 is 16, half of 16 is 8, so we could half that again.

And we know that 8 times 4 is equal to 32.

So well done if you sorted all of those correctly.

Let's think about A and B.

So which of the numbers were not divisible by 3, 4, 6 and 8? Well, 37, 10, and 25.

None of those were divisible by all of those numbers.

We could tell that 37 and 25 weren't going to be divisible by 4, 6, or 8 because they're odd numbers, aren't they? And they weren't divisible by 3 because the sum of their digits was not divisible by 3.

7 plus 3 is equal to 10.

That's not a multiple of 3.

And 2 plus 5 is equal to 7, and that's not a multiple of 3.

Ten is an interesting one, isn't it? You think of 10 as a very friendly number, but it's not a multiple of 3, 4, 6, or 8 so it's not divisible by them either.

And for B, we asked you to think of a number that could go into all the cauldrons.

So two numbers you may have come up with were 48 and 96.

They're both even numbers, so that means they can be divisible by even numbers.

And the sum of the digits is equal to a multiple of 3, so they must be divisible by 6.

And when we halve them and halve them, we get an even number, so they're divisible by both 4 and 8 as well.

I wonder if you came up with any other numbers, but well done for your hard work on that task.

And let's go into the second part of our lesson.

This time, we're going to be using those divisibility rules.

Jacob and Sofia are playing a game called Two Truths and a Lie.

Now, I know lying isn't something we encourage people to do, is it? It's not a very good thing to do, but when we're playing this game, I think we'll let it be.

So they're going to tell you two things about a number that are true and one that's not true.

And they're going to use the divisibility rules for 3, 4, 6, and 8 to create two true statements and one false statement.

So Jacob started.

He says, "36 is: divisible by 3, divisible by 4, and divisible by 8." Two of those are true and one of them is false.

Can you work out which one's false? You might want to pause and have a think before Sofia shares her thinking.

So which statement is false? Sofia says, "I can use my knowledge of the divisibility rules to solve this." She thinks about 36.

So she starts thinking about the statement: is divisible by three.

She says, "Yes, 3 plus 6 is equal to 9, and 9 is divisible by 3." 9 is a multiple of 3, so that statement is true.

36 is divisible by 4.

I can halve it and it gets 18.

That's an even number so I could halve it again, so yes, it is divisible by 4.

Is it divisible by 8? Let's think.

We have to halve it and halve it again and get an even number.

36 halved is 18.

18 halved is 9.

Oh, that's an odd number, isn't it? So no, 36 is not divisible by 8, so that was the false statement.

Sofia says, "If you halve 36, that's 18, half of 18 is 9, and that's not an even number, so 36 is not divisible by 8." So that was Jacob's false statement.

Now, it's Sofia's turn.

She says, "50 is: divisible by 3, is not divisible by 4, and is divisible by 10." Oh, so she's got a not statement in there as well.

So which is the false statement? Jacob says, "I can use my knowledge of divisibility rules to solve this." Could you? Maybe you might have a go before Jacob shares his thinking.

Let's have a look.

We're starting with 50.

It's not divisible by 3 because 5 plus 0 is equal to 5, and 5 is not divisible by 3.

Hmm.

So that's false.

Let's check the others are both true then.

Sofia said it is not divisible by 4.

And Jacob says, "Yes, that's right.

50 is not divisible by 4." Half of 50 is 25, and that's an odd number, so we couldn't have it again.

And it is divisible by 10.

Yes, it is divisible by 10, isn't it? We know that 5 tens are equal to 50.

And we know it's divisible by 10 because the ones digit is a 0.

So that means we have a whole number of tens.

So which was the false statement? 50 is divisible by 3, that was our false statement.

And Jacob carefully checked that the other two were true.

It's time to check your understanding now.

Which statement is false? And can you justify your thinking to your partner? Is it A, 60 is divisible by 3, B, 60 is divisible by 6, or C, 60 is divisible by 8? Which statement is false? Pause the video, have a go, when you're ready for some feedback, press play.

So which statement was it? That's right, it was C.

A and B were true and C was false.

So let's think.

Is it divisible by 3? Well yes, the sum of the digits is 6.

So yes, that is a multiple of 3, or it's divisible by 3, and it's an even number,.

so it must be divisible by 6.

But let's think about the divisibility of 8.

So we have to halve and halve again and get an even number.

Half of 60 is 30, and half of 30 is 15, and 15 is an odd number, so it is not divisible by 8.

So that was our false statement.

Well done if you work that one out.

They're going to go on now and solve a divisibility puzzle.

They have to place the digits below in each space to make the statements true.

And they can only use each digit once.

So they've got to use 0, 1, 2, 3, 4, or 5 as a ones digit to make a number that's divisible by 5.

I wonder what you would use.

Sofia says, "Remember, we can use divisibility rules to help us." "Yes!" says Jacob.

"I know that 20 is divisible by 5 because the ones digit is a 0." And we know that the multiples of 5 alternate between having a ones digit of five and a ones digit of zero.

So we could have 20 is divisible by 5, and that makes our statement true.

And we've used the 0.

This time, we've got to make a 50 something number that is divisible by 3.

And Sofia's reminding us the divisibility rule for 3 is that the sum of the digits of the number must be a multiple of 3.

So 5 plus something has got to equal a multiple of 3.

Jacob says, "Let's try 54.

5 plus 4 is equal to 9.

9 is divisible by 3." So we could use the 4.

54 is divisible by 3.

Excellent.

It's time to check your understanding now.

It's your turn.

We've got 1, 2, 3, and 5 left, and we've got to make a two-digit number with a ones digit of 10 that is divisible 6.

So can you remember the test of divisibility for 6? Pause the video, have a think, and when you're ready for some feedback, press play.

Which number did you use? Sofia's reminding us the divisibility rule for 6 is that the number must be divisible by both 2 and 3.

So it must have a digit sum where the sum of the digits is a multiple of 3, or divisible by 3, and it must be an even number.

Oh, it must be an even number.

So we can't use the 1, 3, or the 5, can we? So it must be the 12.

Because 12 is an even number and it is divisible by 3.

1 plus 2 is equal to 3.

Well done if you've got that right.

And well done if you use some reasoning to get there.

They move on to another puzzle.

Ooh, can you see here, this time, we've got a three-digit number.

We've got to put a ones digit in from 0, 1, 2, 3, 4, 5 to make our three-digit number divisible by 3.

Sofia says, "Oh, this time it's a 3-digit number.

I might need help." But Jacob says, "Remember, we can use the divisibility rules to help us." The divisibility rules help for any size of number, any whole number, no matter how many digits it has.

Sofia says, "Right, I know the divisibility rule for 3.

The sum of the digits must be divisible by 3." So she's got 5 plus 2 is equal to 7.

So 7 plus something has got to equal a number that's divisible by 3.

What could she use? "Ah, yes!" says Jacob.

"Let's try 525." 5 plus 2 plus 5, or 5 plus 5 is equal to 10, and the extra 2 makes 12, doesn't it? So 5 plus 2 plus 5 is equal to 12 and 12 is divisible by 3.

So we can use the 5 and make the number 525, and we know that it's divisible by 3.

"Oh," says Sofia, "that wasn't too bad!" So there's another one here.

This time, it's 620 something and it's got to be divisible by 4.

And Sofia says, "I know the divisibility rule for 4.

If halving a number gives an even value, then the number is divisible by 4." So we've got to find a number to put in the ones, and when we halve the number, it's going to give us an even value.

Hmm, I wonder if you can see which number it's going to be.

"Yes," says Jacob, "let's try 624.

Half of 624 is 312.

312 is an even number so that means it's divisible by 4." Excellent.

Great thinking there, Jacob.

Well done.

So 624 is divisible by 4 and we've used the digit 4.

Your turn again.

We've got 0, 1, 2, and 3 left, and we've got to have 800 and something is divisible by 8.

Can you remember the divisibility rule for 8? Pause the video, have a go, and when you're ready for some feedback, press play.

How did you get on? There was only one possibility here, wasn't there? It couldn't be 1 or 3 because they're odd numbers and we know that all multiples of 8 or numbers divisible by 8 have to be even.

If it was 802, half of 2 is 1, so that would give us an odd number straight away.

So it must be 800.

And Sofia reminds us the divisibility rule for 8 is if halving a number twice gives an even value, then the number is divisible by 8.

And Jacob says, "Yes! Let's try 800.

800 halved is 400, halved again is 200.

200 is an even number so that means it is divisible by 8." And look, we've got 8 hundreds, so we know that's got to be divisible by 8, don't we? Oh, there we go.

Jacob's saying, "We also know that 800 must also be divisible by 8 as it is 8 hundreds.

Right.

Time for you to do some practise.

So you are going carry on helping Jacob and Sofia playing their game two truths and one lie, find the lie.

So you've got to find the statement that is false in A, for 32, and B, for 54.

And then in question two, have a go at creating your own two truths and one lie game with a partner using the divisibility rules for 3, 4, 6, and 8.

And then for question three, you're going to fill in the blanks with the digits 0, 1, 2, 3, 4, and 5 exactly once each to complete the numbers.

And you've got six numbers there to complete using the digits 0 to 5 once each.

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

How did you get on? So did you get which ones were the false statements? So Sofia said, "32 is divisible by 4, 6, and 8." So which of those was wrong? So 32 is divisible by 4.

Yes, that's true.

Half of 32 is 16, that's an even number.

Is it divisible by 6? Well, 3 plus 2 is equal to 5, and that's not a multiple of 3.

So it is an even number, but it's not a multiple of 3.

So that was our false statement.

That's the lie.

And it is divisible by 8.

Half of 32 is 16, half of 16 is 8.

That is an even number, so 32 is divisible by 8.

And Jacob said, "54 is divisible by 3, 6, and 8." Well, let's have a look at that digit sum.

5 plus 4 is equal to 9, so yes, 54 is divisible by 3.

And it will be divisible by 6 as well because it's an even number and it's divisible by 3.

So is it divisible by 8? I think that's going to be our lie, but let's just check.

Half of 54 is 27, and that's an odd number, so no, it's not divisible by 8.

So that was our false statement.

Well done if you've got those ones right.

So question two, you are going to have your own go at creating two truths and a lie.

So you may have written a statement like "80 is divisible by 4, 6, and 8." So which is the false statement there? "Let's check," says Jacob.

80 is divisible by 4, that's the truth, yes.

Half of 80 is 40, and that's an even number.

80 is divisible by 6.

So it's got to be divisible by 3, so the digit sum must be a multiple of 3, or divisible by 3.

8 plus 0 is 8.

That's not divisible by 3, so it's not divisible by 6.

Is it divisible by 8? Well, we see 8 tens, isn't it? So it must be, but we can use our divisibility check.

Half of 80 is 40, half of 40 is 20.

That's an even number, so yes, it is.

So the false statement was that 80 is divisible by 6.

That's not true.

And for question three, you are filling in the blanks.

So 123 is divisible by 3.

1 plus 2 plus 3 is equal to 6.

444 is divisible by 6.

4 plus 4 plus 4 is 12, and it's an even number, so it is divisible by 6.

681 is divisible by 3.

So 8 plus 1 is 9 and 9 plus 6 is 15.

That's a multiple of 3.

400 is divisible by 8.

Half of 400 is 200, half of 200 is 100, and that's an even number.

312 is divisible by 4.

Half of 312 is 156.

That's an even number, so it is divisible by 4.

And 225 is divisible by 3.

5 plus 2 is equal to 7, plus another 2 is equal to 9.

So the sum of the digits is divisible by 3.

Well done if you've got all of those right.

I wonder what reasoning you use to get those in numbers in the right places.

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

We've been using the divisibility rules for the 3, 4, 6, and 8 times tables to solve problems. So what have we learned about today? Well, we can understand that multiples of 3 have a digit sum of a multiple of 3, and that multiples of 6 are even multiples of 3.

We know that if you can halve a number and get an even number, then the original number is a multiple of 4.

And if halving a number twice gives an even number, then the number is divisible by 8.

And we know that we can use the divisibility rules for the 3, 4, 6, and 8 times tables to solve problems. Thank you for all your hard work and your mathematical thinking, and I hope I get to work with you in another lesson soon.

Bye-bye.