Lesson video

Hello, I'm Miss Miah and I'm so excited to be a part of your learning journey today.

I hope you enjoy this lesson as much as I do.

In this lesson, you'll be able to solve problems using divisibility rules for divisors of 3, 6, and 9.

And your key words are on the screen.

Now I'd like you to repeat them after me.

Digit sum, divisible, divisibility, multiple.

Well done.

Let's find out what these words mean.

Now the sum of the digits of a number is called its digit sum.

So for example, if I have the number 17, the digit sum for 17 would be 8, and that's because the digits in 17 are one and 7.

And if we sum those 2 numbers together, we get 8.

More on that later.

Divisibility.

Now divisibility is when division of a number results in another whole number.

A number is divisible by another if it can be shared exactly with no remainder.

A multiple is the result of multiplying a number by another whole number.

Now this lesson is all to do with our 3 times tables, 6, and 9 times tables.

Now we've got 2 lesson cycles here.

Our first lesson cycle is to do with the divisibility rule for multiples of 9, and then we're going to move on to solving problems. A very exciting lesson ahead.

So let's get started.

To help us in our journey today, we've got Andeep and Izzy who will be helping us with our mathematical thinking.

Now, divisibility rules can help you to figure out if a number can be divided by another number without having to do too much calculation.

Now this is super important because as you progress in your mathematical journey, you are to find that these rules will help you to solve problems efficiently.

Now they are super helpful to test whether a number is a multiple of another.

So today we will be looking at the divisibility rules for 9, 3, and 6.

And we're going to start off with looking at the divisibility rules for 9.

A number is divisible by 3 if it is a multiple of 3, it can be divided into groups of 3 exactly with nothing left over.

So Izzy says 36 is a multiple of 3 because it is 12 groups of 3.

Andeep's got the number 90, he says he can go bigger.

90 is a multiple of 3 because it is 30 groups of 3.

If its digit sum is a multiple of 3, it is divisible by 3.

Now let's look at our 6s.

A number is divisible by 6 if it is a multiple of 6, which means it can be divided into groups of 6 exactly with nothing left over, For example, 36.

Now 36 is a multiple of 6 because it is 6 groups of 6.

Andeep challenges Izzy and says he can go bigger.

So he's got the number 360.

Now 360 is a multiple of 6 because it is 60 groups of 6.

If a number is even and the digit sum is a multiple of 3, then the number is also a multiple of 6.

So that is a key point there.

Multiples of 3 are different to multiples of 6 because with multiples of 3, the number has to be a multiple of 3 and that's it.

That's the only condition.

Whereas with multiples of 6, it has to be a multiple of 3 and also be an even number.

So remember that.

Now, multiples of 9 can be shown on a number line.

Let's chart the multiples.

Are you ready? Let's begin.

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108.

Have a look at the screen, what multiple is missing? And I'd like you to explain how you know.

You can pause the video here and click play when you're ready to rejoin us.

So what did you get? 81 was the missing multiple, and that's because we can add 9 to 72 to calculate what the adjacent multiple is.

So 72, add 9, is equal to 81.

So 81 was a missing multiple.

Well done if you got that correct.

Now you can find out if a number is a multiple of 9 by using the digit sum of the number.

So here we go.

You've got the number 90 on the screen.

To find a digit sum, you need to add together the digits.

So the digits here are 9 and 0.

So now if we were to add those together, 9, add 0 is 9.

The number is 90 and its digit sum is 9.

But what now? Well, Andeep and Izzy are looking at the multiples of 9.

They spot a pattern.

Andeep says I can see a pattern in the sum of the digits for the multiples of 9.

What do you mean? Well, let's have a look at the table, and I'm sure you spotted it as well.

All of the digit sums are 9 for the multiples of 9.

For example, the digit sum of 18 is 1, add 8, which is 9.

Over to you.

Is this true? Prove it.

So the next multiple of 9 has a digit sum of 9 too.

You can pause the video here and click play when you're ready to rejoin us.

So how did you do? Well, the next multiple of 9 is 81.

In order to calculate the digit sum of 81, we need to look at the digits within 81, and that's 8 and 1.

So 8 plus 1 is equal to 9.

So it's true.

Well done if you've got that correct.

Izzy and Andeep look at larger multiples of 9.

If 36 is a multiple of 9, then 360 is a multiple of 9.

I agree.

360 is 36 tens.

And 36 tens can be divided into 4 groups of 9 tens.

9 is also a multiple of 3.

So 360 must also be a multiple of 3.

So we've got the number 360 there and here this is how I would record the digit sum.

So I take apart the digits and add them together.

So Andeep says great, the digits are 3, 6, and 0.

So we can add them together.

And the digit sum is 9.

9 is a multiple of 9.

So we now know that 360 is divisible by 9.

Over to you.

Explain why 342 is a multiple of 9.

Can you use the digit sum? You could pause the video here and click play when you're ready to rejoin us.

So how did you do? Well, if I were you, I would look at the number 342 and I would identify the digits.

So the digits here are 3, 4, and 2.

I would then add the digits separately.

So 3, add 4 is 7, add 2 is 9.

So the digit sum is 9, which is a multiple of 9, meaning 342 is a multiple of 9.

Well done if you got that correct.

Now there is another way that we could have solved this as well.

Let's see Izzy's reasoning for this.

So 300 is a multiple of 3.

We know that because 3 100s are 300.

And 42 is also a multiple of 3, which means 342 is also a multiple of 3.

It's an even number.

So it's also a multiple of 6.

Let's move on.

So the divisibility rule for multiples of 9 works for all multiples of 9.

And we're going to prove this using a 4 digit number.

So Izzy says I'm going to use digit cards to make a 4 digit number.

Oh, she's made a 4 digit number here.

What number have I made? Can you say it? If you said 9,036, you are correct.

Andeep, is this a multiple of 9? Hmm.

Well, how can Andeep check? Do you think the rule that we applied before would work now? Let's have a look.

Well, Andeep does suggest that.

He says, let's find the digit sum.

One digit is 0, so you can ignore that.

The digit sum for 9,036 is 18.

Hmm, 18.

Now we know so far that multiples of 9 must have a digit sum of 9.

But this time because we've got 4 digits which have added together to to give us 18, wonder what we'll do next.

Now, Izzy says yes, 9,036 is exactly divisible by 9.

But coincidentally enough, she also knows that it is a multiple of 9 because its parts are divisible by 9.

Let's have a look.

9 1,000s can be divided into 9 groups of 1,000.

36 is a multiple of 9 because it is 4 groups of 9, which means if we were to partition 9,036 into our highest multiples of 9 that we know, we would've ended up with 9,036 as our parts and then by dividing both by 9.

And those parts are both divisible by 9, making 9,036 divisible by 9.

Now there's also one more thing that Andeep points out.

Let's see what he says.

He's looking at the number 18.

He says that 1 and 8 is equal to 9.

So 9,036 is a multiple of 9.

Let's move on to another example.

Izzy says she's going to make another 4 digit number.

So what number has she made? Can you say it? So this time she's made the number 4,545.

Andeep, is this a multiple of 9? What do you think? How can Andeep check? So let's use the divisibility rule for 9 for as well.

Andeep suggests to find the digit sum.

So 4, add 5 is 9 and that's repeated.

The digit sum is 18.

18 is a multiple of 9.

So yes, 4,545 is also a multiple of 9.

Yes, 4,545 is exactly divisible by 9.

I also know it is a multiple of 9 because its parts are divisible by 9.

45 100s can be divided into 9 groups of 5 100s.

And then we also know that 45 can be divisible by 9 because it gives us 5.

Now, Andeep has also noticed that the digit sum 18 can be summed together to make 9, which is also a multiple of 9.

Which of these 4 digit numbers is a multiple of 9? I'd like you to explain how you know.

You can pause the video here and click play when you're ready to rejoin us.

So how did you do? Well, 4,599 is a multiple of 9, and that's because the digit sum is 27, which is a multiple of 9.

6,331 is not a multiple of 9 because the digit sum is 13, which is not a multiple of 9.

Well done if you got that correct.

And you remembered to use the divisibility rule for 9 to help you.

Onto the main task for this lesson cycle.

For question 1, you are going to use the digit sum divisibility test to circle the multiples of 9.

On the screen you have the numbers 3,636, 6,613, 8,295, 2,412, 2,045, and 4,185.

You could pause the video here and click play when you're ready to rejoin us.

So how did you do? If you got this, you are correct.

So let's look at 3,636.

The digit sum for this number is 18, and we know that 18 is a multiple of 9.

Now, 6,613 is not a multiple of 9 'cause the digit sum is 6, add 6 which is 12, 12 add one is 13, 13 add 3 is 16.

16 is not a multiple of 9.

8,295.

So the digit sum for this was 24.

24 is not a multiple of 9.

For 2,045, well, we know that the digit sum is 11, so that is not a multiple of 9 either.

Now for 2,412, the digit sum is 9.

So 2,412 is divisible by 9.

And lastly, the digit sum for 4,185 is 18.

So 4,185 is divisible by 9.

Now let's move on to our second lesson cycle, which is to solve problems. Izzy tells Andeep that 135 is divisible by 9, so it must be divisible by 3.

Do you agree? I'd like you to justify your thinking to your partner.

Now, what did you guys discuss? You may have said something like this.

Divisible usually means to divide, but Izzy says that she can't divide a large number.

Andeep says you can use your divisibility rules to help you instead.

For example, the digit sum of the number 135 is 1 add 3 add 5, which is equal to 9.

And we know that 9 is a multiple of 9.

We also know it is a multiple of 3 because 3 times 3 is equal to 9.

That means 135 is divisible by 3 and 9.

So you were able to calculate that by just using the divisibility rules.

There was no division required.

Wow, you worked that out so quickly.

Now Izzy tells Andeep that 234 is divisible by 6, so it must be divisible by 3 and 9.

Do you agree? I'd like you to justify your thinking to your partner.

Have a think about the divisibility rules that you already know about your 3s and 6s and 9s now.

Izzy reminds Andeep that divisible usually means to divide, but again, she can't divide a large number.

So Andeep says you can use your divisibility rules to help you instead.

Now the digit sum for 234 is equal to 9.

And that's because 2 add 3 add 4 is 9, 9 is a multiple of 9.

We also know it is a multiple of 3 because 3 times 3 is 9.

It's an even number with a digit sum that's divisible by 3.

So it's divisible by 6 as well.

And Izzy says, wow, Andeep, you have worked that out so quickly yet again.

And it's true.

That's because he knows the divisibility rules for 3, 6, and 9.

Over to you.

So using your knowledge of the divisibility rules for 3, 6, and 9, decide whether 303 is a multiple of 3, 6, 9, or a combination.

You could pause the video here and click play when you're ready to rejoin us.

So what did you get? Well, let's have a look.

303 is a multiple of 3 because the digit sum is 6, and we know that because 3 add 0 add 3 is 6.

Now 6 is a multiple of 3, which means it is divisible by 3.

Now, 303 is not a multiple of 6 because it is not an even number.

303 is not a multiple of 9 because its digit sum is not a multiple of 9.

Well done if you got that correct.

Now Andeep tests Izzy with larger numbers.

He says that 1,212 is divisible by 6, so it must be divisible by 3 and 9.

Have a think.

Do you agree? I'd like you to justify your thinking to your partner.

So Izzy reminds Andeep again that divisible usually means to divide, but she can't divide an even larger number.

So here's where the divisibility rules help us.

The digit sum of the number 1,212 is 1, add 2, add 1, add 2, which is equal to 6.

6 is a multiple of 6.

We also know it is a multiple of 3 because 3 times 2 is 6.

The digit sum is not a multiple of 9.

So 1,212 is not divisible by 9.

And Izzy says yet again, wow, you worked that out so quickly.

Back to you.

Using your knowledge of divisibility rules for 3, 6, and 9, justify whether the number 3,126 is a multiple of 3, 6 or 9.

You could pause the video here and click play when you're ready to rejoin us.

So how did you do? Now, 3,126 is a multiple of 3 because the digit sum is 3, add 1, add 2, add 6, which is equal to 12.

Now we know that 12 is a multiple of 3 because 3 times 4 is equal to 12.

This ultimately means it is divisible by 3.

It is a multiple of 6 because it is a multiple of 3 and it is an even number.

And lastly, it is not divisible by 9 because 12 is not a multiple of 9.

Goodness me, it's like the number has to go through quite a few checks just so we can find out whether it is divisible by 3, 6 or 9.

But it's well worth it if you know these rules because they will help us solve trickier problems in the future.

Now, Andeep and Izzy makes some other numbers using the same digit cards we've got on the screen here, 9,675 and 5,679.

Izzy says that both of these are divisible by 3 too.

The digit sum is still 27.

So all of these are multiples of 9 as well.

This is exciting.

Now they're not multiples of 6 because they are not even numbers, and we can see that because the last digit for the first number is 5.

That's not an even number.

And the last digit for the second number is 9, which is also not an even number, which makes 5,679 not an even number.

Use the digit cards to make a different 4 digit number that is a multiple of 6.

Explain how you know it is a multiple of 6.

So you've got the digits 5, 7, 6, 9.

You can pause the video here and click play when you're ready to rejoin us.

So what 4 digit number did you make? So you may have made 7,556 and 7,596.

Now because the digit sum is always 27, we know that's a multiple of 3, and those numbers are even numbers.

So they will be multiples of 6.

Well done if you've got that correct.

Let's move on.

Izzy has another idea about multiples of 3, 6, and 9.

She's got the number 4,986.

Now she says 6 and 4 in the number sum to 10.

I wonder if I can change those.

So she has, she says that's cool.

I can swap the digit 6 and 4 too.

6,984 is also a multiple of 3.

Andeep says yes, and that's because the digit sum is still 27.

So our new number is still divisible by 3 and 9.

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

He then says that he agrees and that there are so many different combinations.

Over to you.

Now I'd like you to try Izzy's idea.

Explain how you know your new number is still a multiple of 3, 6, and 9.

So you've got the number 3,798 on the screen.

Izzy says that she knows the last digit has to be an even number.

She also knows that 8 and 3 sum to 11.

Have a think.

Can you use this to think of more combinations? You could pause the video here and click play when you're ready to rejoin us.

So how did you do? You may have got these numbers.

9,738, 9,378 and 7,398.

As long as the digit sum remained the same and was a multiple of 3 so that it is a multiple of 6 as well.

Well done if you've got those correct.

Now onto the main task for your lesson cycle.

For question 1, you're going to be answering the questions that you see on the screen.

So Jun tells Andeep that 504 is divisible by 9.

So it must be divisible by 3.

Aisha said that 504 must also be divisible by 6.

Are they right? Why or why not? Question 2, Jun tells Andeep that 333 is divisible by 9, so it must be divisible by 3.

Aisha said that 333 must also be divisible by 6.

Is Aisha correct? Question 3, a baker baked 396 cakes.

Can he split them into 4 packs of 9? Why or why not? Can he then split them into 4 packs of 6 and then 3? And I'd like you to reason why or why not.

Question 4, you are going to use 4 digit cards each time to make 4 digit multiples of 3, 6, and 9.

You are going to make the largest and smallest multiples of 3, 6, and 9 that you can.

So the numbers that you've got on the screen are 2, 2, 4, 6, 8, 9.

So how did you do? This is what you should have got for question 1.

Now we're looking at 504.

So 504 is divisible by 3 because the digit sum is 9, which is a multiple of 3.

It is also divisible by 6 because it is an even number.

Now for question 2, we're working with the number 333.

Now, 333 is divisible by 3 because each digit is a multiple of 3, and the digit sum is 9, which is a multiple of 3.

Now, Aisha said that 333 must also be divisible by 3.

Was Aisha correct? Well, Aisha is incorrect because even though 333 is a multiple of 3, it is not an even number.

So it cannot be divisible by 6.

Question 3, we've got 396 cakes and we are trying to figure out if the number is divisible by 9, 3 and 6.

So let's find out if they are divisible by 9 first.

Well, yes, the digit sum is 18, and that's because 3, add 9, add 6 is 18, which is a multiple of 9.

Now, can he split them into full packs of 6? The answer is yes again, and that's because the digit sum is a multiple of 3 and it's an even number.

So 396 is divisible by 6.

And lastly, can he split them into full packs of 3? Yes, because the digit sum is a multiple of 3.

Well done if you've got all of those answers correct so far.

Let's move on.

Now for question 4, this is what you may have got.

So 9,864 is the largest multiple of 9, 3 and 6, and 2,682 would've been the smallest, and if you've got any more combinations of numbers which are multiples of 9, 3, and 6, well done, as long as it was an even number and is a multiple of 3, and the digit sum was also a multiple of 9.

Well done.

We've made it to the end of this lesson.

I'm super proud of you.

Let's summarise our learning.

You now understand that for a number to be divisible by 3, the digit sum of the number must be divisible by 3.

You also understand that for a number to be divisible by 6, it must be divisible by both 2 and 3.

In other words, it needs to be an even number and divisible by 3.

You should also understand that you can use this knowledge now to solve problems more efficiently.

Well done for making it to the end of this lesson.

I look forward to seeing you in the next one.

Bye.