Lesson video

Hi, thanks for joining me today.

My name is Mr. Peters.

I'm really looking forward to this lesson.

In this lesson we're gonna be thinking about extending on our understanding of factors and multiples to be able to explain how to find common multiples of two or more numbers.

When you're ready to get started, let's get going.

Okay, so by the end of this lesson today, you should be able to say that I can explain how to find a common multiple of two or more numbers.

Throughout this lesson we've got two keywords that we're gonna be referring to.

Let me say them first and then you can repeat them after me.

Are you ready? The first word, multiple.

Your turn.

The second phrase, common multiple.

Your turn.

Let's have a think about what these mean in a bit more detail.

A multiple is a result of multiplying a number by an integer.

This doesn't apply to fractions.

And a common multiple is a multiple that's common for two or more numbers.

Look out for these keywords throughout our lesson today and use them where you can to help develop your reasoning.

Throughout this lesson today we're gonna have two cycles.

The first cycle we're thinking about identifying multiples and the second cycle we're thinking about finding common multiples.

Let's get started with the first cycle.

Throughout this lesson today we've got Laura and Aisha joining us.

They'll be as always sharing their thinking and any questions that they've got along the way.

So let's get started by revisiting our understanding of arrays here.

We can see here that we've got an array which represents 12.

There are two rows and there are six columns.

We know that we can represent an array as a multiplicative equation.

So because we know that 12 is composed of two rows and six columns, we can say that 2 multiplied by 6 is equal to 12.

The 2 represents the two rows.

The 6 represents the six columns, and the 12 represents the total number of tiles altogether.

We can say that 2 is a factor of 12 because 2 multiplied by 6 is equal to 12.

We can also say that 6 is a factor of 12.

Again, because 2 multiplied by 6 is equal to 12.

We can then extend this to say that 12 is a multiple of 2 because 2 multiplied by 6 is equal to 12.

And of course we could say that 12 is a multiple of 6 as well, again, for the same reason that 2 multiplied by 6 is equal to 12.

So drawing back on our language of a multiplication equation, we know that a factor multiplied by another factor is equal to the product.

So we could say that a product represents a multiple of the factors in our equation.

Let's see if we can extend this to our division equations now.

A division equation for this would be 12 divided by 6 is equal to 2.

Take a moment to have a look.

Can you see where the factors and the products are within our division equation now? That's right.

What was the product is now at the beginning of our division equation and we know that we refer to that as the dividend, don't we? You can also see that the factors are represented by the divisor and the quotient in our division equation.

So we can say that 12, the product in the multiplication equation, is actually the dividend in a division equation, and we can say that 6, which was a factor in the multiplication equation, is now represented by the divisor in our division equation.

And 2, which was also a factor in our multiplication equation, is now represented by the quotient in our division equation.

We can say that 2 and 6 are factors of 12 because 12 divided by 6 is equal to 2.

And we can say that 12 is a multiple of 6 and 2 because 12 divided by 6 again is equal to 2.

Let's have a look at another example shall we? Look at our array this time.

What's the product of our array? That's right.

Our multiplication equation would be 3 multiplied by 5 is equal to 15.

We know that 3 is a factor of 15 because 3 multiplied by 5 is equal to 15.

We know that 5 is a factor of 15 because 3 multiplied by 5 is equal to 15.

And we could therefore also say that 15 is a multiple of 5 because 3 multiplied by 5 is equal to 15.

And 15 is a multiple of 3 because 3 multiplied by 5 is also equal to 15.

Again, revisiting our language for a multiplication equation, factor multiplied by factor is equal to product.

We know that the product can represent in multiple of the factors, can't it? Let's extend this to division again now then.

What would the division equation be for this array here? Take a moment to have a think.

That's right.

We could represent this as 15 divided by 3 is equal to 5.

You may have also done 15 divided by 5 is equal to 3 and that would be equally okay.

We know in this example that the dividend is 15, so therefore that would be the multiple, wouldn't it? And our factors would be represented by the divisor and the quotient.

So 3 and 5 would be the factors, wouldn't they? So we can say that 3 and 5 are factors of 15 because 15 divided by 3 is equal to 5.

And we can say that 15 is a multiple of 3 and 5 because 15 divided by 3 again is equal to 5.

So let's go back to Laura and Aisha now.

They are planning to organise a 5-a-side football competition for their friends.

And they're saying in order to be able to run the competition they need to know how many players they need.

And as Aisha's pointed out, that depends on the number of teams that they have, doesn't it? Aisha is suggesting that we skip count in fives now to find out how many players we could have for the number of teams that we have.

We know that for every one team there would be 5 players.

So let's start counting in 5, shall we? Can you count with me? 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55 and 60.

There we go.

We've just skipped counted in our fives and hopefully now we can start to see how many players we would need altogether depending on the number of teams that we have.

Laura is saying that she's noticed something about the total number of players.

She says that the numbers used in the total number of players column are all a part of the five times table.

And that's it, Aisha, exactly that.

All of them are multiples of 5, aren't they? So when looking for multiples, we can use our times tables to help us here.

If you want to find the multiples of 5, we can use our five times table to help us.

So let's have a look here as an example.

Here's a multiplication grid and you can see I've highlighted the 5 and 7 times tables.

And where the 5 and 7 times tables cross, we can find our multiple.

We know that 5 multiplied by 7 is equal to 35, therefore 5 and 7 are factors of 35 or 35 is a multiple of 5 and 7.

Yep, and you're right Aisha, any product of the five times table is a multiple of 5, isn't it? So in order to find out the number of players that we need, all we need to do is multiply the number of teams that we want by 5.

Okay, time for us to check our understanding now.

Can you tick the multiples of 5? Take a moment to have a think.

That's right, it's B, C and D, 15, 40 and 55.

And how did you know that? That's right.

You may have skip-counted in fives, but actually there's a rule isn't there that we know? We know that any multiple of 5 ends in either 0 or 5, doesn't it? So any number in the five times table would end in a 0 or a 5.

And another quick check.

If Laura and Aisha chose to use 12 teams in their competition, how many players would they need altogether? Take a moment to have a think.

Okay, and the answer to that of course is 60.

And how did you know that? Well, as Laura and Aisha mentioned earlier on, we now know that to find out the total number of players, what we needed to do was find out the total number of teams we wanted and multiply that by 5.

So 12 multiplied by 5 is equal to 60, so 60 is a multiple of 12 and 5.

Okay, onto our first tasks for today then.

In task one, what I'd like to do is write in the missing multiple.

For task two, I'd like you to answer these true or false questions and justify your answer each time.

And then for task 3, what I'd like to do is identify the pupil's numbers from the clues that they give you.

Good luck with those tasks.

When you're done, come back and we'll go through the answers together.

See you shortly.

Okay, welcome back.

Let's work through these then.

The first missing multiple was of course 32.

We can see that each multiple is increasing by eight.

So we're actually going up in multiples of eight.

So it's the eight times table, 16, 24, 32 was the missing multiple.

For the second one, it's actually part of the nine times table, isn't it? Therefore the missing multiple here was 54.

And then for the last one, well the multiples are increasing by 15 each time, aren't they? So actually we're going up in the 15 times table here.

15, 30, 45 is the missing multiple for this one.

Okay, true or false then.

Can we justify our answer? 36 is a multiple of 6.

Do you agree with that? It's true, isn't it? We know that 36 is a multiple of 6 because it's in the six times table.

What about the next one? 27 is a multiple of 7.

That's false, isn't it? 27 is not in the seven times table, therefore it is not a multiple of 7.

And finally 84 is a multiple of 4.

What'd you think about that? That's true, isn't it? We can say that 84 is in the extended four times table.

The four times table doesn't just finish at 12 multiplied by 4.

It can continue upwards and actually it would be 21 lots of four that would be equal to 84.

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

Okay, were you able to identify the clues from the pupils? Let's have a look.

Laura is saying, "My number is less than 100.

It's a multiple of 7 and 5 is a factor of the number.

It is also a multiple of 10." Did you manage to get it? Aisha's worked it out.

She said, "Your number must be therefore in the seven and the five times table.

So 7 multiply by 5 is equal to 35.

But hang on, 35 isn't a multiple of 10 is it?" So what can we do with 35 to make it a multiple of 10? Well if we double it, 35 multiplied by 2 would be equal to 70 and now 70 is a multiple of 10.

So we can say that 70 is a multiple of 7.

It's also a multiple of 5, and it is also a multiple of 10.

Well done if you managed to get Laura's number.

Let's have a look at Aisha's number.

Aisha's saying that her number is between 50 and 70.

2 is a factor of the number.

It is a multiple of 9 and it is also a multiple of 6.

So did you manage to get this one? Laura's managed to work it out as well.

She's worked out that the multiples of 9 between 50 and 70 are 54 and 63 so it has to be one of these two numbers.

And we know that also it has to be a multiple of 2 and we know that multiples of 2 are even, don't we? Therefore looking at those two numbers, it must have been 54 'cause 54 is an even number and 63 isn't an even number.

Is 64 also a multiple of 6? Yes, it is.

Nine lots of six is equal to 54.

So 54 is a multiple of 6 as well.

Well done if you've got those.

Perhaps you could create your own set of instructions for a friend to have a go at.

See how tricky you can make it.

See if they can find out using your clues.

Okay, onto cycle two now then, finding common multiples.

Laura is saying how she's always found it interesting how the same number can appear in different times tables.

For example, 20 is in the four times table and the five times table.

If we count up in fours, we could say 4, 8, 12, 16 and 20.

There it is.

And if we count up in fives, we could say 5, 10, 15, 20, 25.

So you can see that the 20 appears in both the four and the five times table.

Aisha's wondering are there any other numbers that also appear in the four and five times table? Well, let's have a look at our multiplication grid here.

We've highlighted the four and the five times table and again where they cross we can find a multiple of both 4 and 5.

And we know that already that 20 is a multiple of both 4 and 5.

Have a look carefully at each of the times tables.

Can you notice any other numbers that are the same? That's right.

40 appears in both the 4 and the five times table.

So we can say that 40 is a multiple of both 4 and 5.

Where a number appears in more than one times table, we can say that these are common multiples of those numbers.

So we can say that 20 is a common multiple of 4 and 5 and we can say that 40 is a common multiple of 4 and 5 as well.

I wonder if you could predict any other common multiples of both the four and five times table.

I'll give you a moment to have a quick think.

Well, you might have noticed that the first number was 20, the second number was 40, therefore maybe the next number could be 60.

If you look carefully, you can see that 60 is in the five times table and we know that because it ends in a 0.

Therefore any number that ends in a 0 or a 5 is in the five times table.

However, what about the fours? Would 60 appear in the four times table? Well, let's extend the four times table out.

There we go.

We can see after 48 it would be 52, then 56 and then 60.

So yes, 60 appears both in the four and the five times table.

So what could we say about 60 then? That's right.

We could say that 60 is a common multiple of both 4 and 5.

I wonder if you could say that with me.

60 is a common multiple of both 4 and 5.

Well done.

Let's have another look at another example then.

Here you can see that we've highlighted the three times table and the nine times table.

Take a moment, what's the first multiple of 3 and 9 that you can think of? That's right, we can always find a multiple really quickly by multiplying those two numbers together.

So 3 multiplied by 9 is 27.

And we can see that, that that's where those two times tables cross in the multiplication grid.

So 27 is a multiple of both 3 and 9.

Are there any others that you can spot? Well, we can see that 9 is a common multiple of both 3 and 9 as it appears in both times tables, doesn't it? We can see that 18 is a common multiple of 3 and 9.

It appears again in both the 3 and the 9 times table.

We know that 27 is a common multiple of 3 and 9 and we can now see that 36 also appears in both the 3 and 9 times table.

So 36 is a common multiple of both 3 and 9.

What'd you notice so far? Aisha thinks she's noticed something.

If we continued, she thinks that every multiple of the nine times table would be in the three times table, and every third multiple of 3 would be in the nine times table.

Hmm, that's an interesting pattern you spotted there so far, Aisha.

I wonder if that's true.

Let's think about extending the three times table outwards.

If we started on 36 and went to 39, 42, 45, oh look, 45 is also in the nine times table as well.

So maybe this is true.

Maybe this would continue just like this all the way through.

If that's the case, then why would that be so? Maybe take a moment to have a think for yourself.

That's right, Laura.

We know that nine consists of three lots of 3, doesn't it? Therefore each time we increase a multiple of nine we are adding on another three lots of 3, aren't we? So that's the reason why every third multiple of the three times table is in the nine times table or why every multiple of the nine times table is in the three times table.

Well done you two, some great thinking there.

Laura's now asking her own question.

She's now beginning to wonder which two times tables would have the most common multiples.

Let's have a look at two and four for example.

We know that 2 multiplied by 4 is equal to 8.

So 8 is a common multiple of 2 and 4.

Have another look.

Can you see any other common multiples? Well we can see that 4 is both a common multiple of 2 and 4.

We can see that 12 is a common multiple of 2 and 4.

We can see that 16 is a common multiple of 2 and 4.

20 is a common multiple of 2 and 4.

24 is also a common multiple of 2 and 4.

What'd you notice this time? Aisha again thinks that every multiple of 4 is also going to be a multiple of 2, and every other multiple of 2 would be a multiple of 4.

Again, why would that be so? Yep, that's right, Laura, because 4 consists of 2 lots of 2, doesn't it? And therefore if we keep increasing by 4 each time going up in fours, that's also the same as saying going up in two lots of two.

Therefore every multiple of four is in the two times table and every other multiple of two is in the four times table.

So Aisha thinks, actually in the previous example we looked at 3 and 9 and 9 was triple 3, wasn't it? Whereas we can see here that 2 and 4 have a lot more common multiples, don't they? 'Cause it's every other multiple that is the same, isn't it? And that would apply to any two numbers that are double one another.

So if we looked at 4 and 8, they would have similar amount of common multiples that 2 and 4 would also have together.

Well done if you managed to share their thinking too.

Okay, time for us to check our understanding now.

Can you tick the numbers that are common multiples of 4 and 7.

Okay, that's right.

28 and 56 are common multiples of 4 and 7.

We know that 4 multiplied by 7 is equal to 28.

And if you double 28, that would make 56.

So therefore 4 and 7 would still be factors of 56 and 56 would be a multiple of both 4 and 7.

56 would also be in the 4 and the 7 times tables.

Okay, true or false? 33 is a common multiple of both 3 and 6.

Take a moment to have a think.

And it's false, isn't it? Look at our justifications.

Which one of these helps you to reason why? And again, it's justification B, isn't it? We know that 33 is not in the six times table.

It is in the three times table, but it's not in the six times table, therefore it is not a multiple of six.

Well done if you managed to get that.

Okay, and onto our final two tasks for today then.

What I'd like you to do for the first one is find the common multiples for A, B and C.

And then for question two, I'd like you to find another, another and then another.

So I'm looking for a common multiple of both 7 and 6.

So could you find a common multiple of both 7 and 6, then find another, then find another, and then find one that nobody else will think of.

Good luck with those two tasks.

I'll see you back here shortly.

Okay, let's go through these together then.

So we're looking for common multiples of 2 and 8 to start off with.

The common multiples of 2 and 8 were 8, 16, and 24.

The common multiples of 3 and 9 were 9, 18 and 27.

And the common multiples of 3, 8 and 12 were 24, 48 and 72.

You may have used your times table grids to help you with that.

Well done if you managed just to know your times tables to help you recognise those though.

And for question two, another, another, and another.

Well, a common multiple of 7 and 6 is 42.

Another common multiple of 7 and 6 could be 84 and I've doubled 42 to get 84 there.

Another one would be to add another group of 42 on top of that, that would give us 126.

So now I've got 3 common multiples of 7 and 6.

And then finally one that nobody else is gonna think of.

Well actually I took 126 and I made that number a thousand times larger.

So 126,000 is also a common multiple of 7 and 6.

Well done if you managed to get those.

I'd be really interested to see which example you came up with and whether anybody else around you also came up with that.

Okay, that's the end of our learning for today.

To summarise what we've learned, we know that a multiple can be represented by the product in a multiplication equation or by the dividend in a division equation.

We know that you can use your multiplication and division facts to help you find multiples of numbers.

And finally, multiples that appear in more than one times table are known as common multiples.

Thanks for joining me today.

Hopefully you're feeling a lot more confident with your understanding around common multiples and what this means and how to find them.

Take care and I'll see you again soon.