Lesson video

Hi, thanks for joining me today.

My name is Mr. Peters and I'm really excited for this lesson today.

In this lesson, today we're gonna be thinking about something that's really important for the creation of positive whole numbers or integers themselves.

Hopefully we can have quite a nice investigation and think about how numbers are created in a slightly different way.

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

So by the end of this lesson today, you should be able to say that you can identify a prime factor of a number.

In this lesson, we've got two key words we're gonna be thinking about.

First one is factor, and the second one is prime factor.

I'll have a go at saying it first and then you can repeat it after me.

Are you ready? Factor, your turn, prime factor, your turn.

Okay, thinking about what these mean then, a factor is a whole number that exactly divides another whole number and a prime factor is a factor of a number that is a prime number.

Today's lesson we've broken down to two cycles.

The first cycle we'll think about identifying prime factors, and then the second cycle, we'll think about decomposing numbers multiplicatively.

Let's get started with the first cycle.

Throughout our lesson today, we've got Jun and Jacob joining us.

They will as always be sharing their thinking, any questions they've got as we go.

So Jun and Jacob are starting their day thinking about exploring the factors of number 12.

They've said that 12 has six factors, and we can see that here on our factor bug, can't we? We've got the 12 in the head of the factor bug and we know 12 has factors of one and itself, so one and 12.

12 can also be divided into an array which represents two rows and six columns, or it could be represented as three rows and four columns.

Therefore two and six and three and four are also factors of 12.

Jun's saying that, "Some of these factors are either composite, square or prime numbers." Looking at these factors then, which ones would be composite numbers? That's right, six and 12 are composite numbers because they have more than two factors themselves.

Two and three are prime numbers, aren't they? And we know they would be prime numbers because they only have one and themselves as a factor.

And finally number 12 has one and four as square numbers.

They've then gone to look at another example.

Let's have a look at 16.

16 has five factors, and then here's our table again.

Take a moment to have a think for yourself.

Which ones of these factors of 16 are composite, square or prime? That's right, number eight is the only composite number that 16 has.

Again, it has more than two factors.

Number two is the only prime number that 16 has as a factor.

And actually 16 has got three square numbers as factors.

It's got one, four and itself, 16.

That's an interesting number, isn't it? Jun's got a great question.

He's noticed already that both 12 and 16 so far have actually got a square number as a factor, don't they? He's now wondering, do all numbers have a square number as a factor? Let's have a look at one more example.

This time we're gonna look at 17.

17 has two factors, one and itself.

Let's see how we can record these as either composite, prime or square.

We know that actually 17 doesn't have any factors that are composite numbers.

It has one factor that's a prime number, which is 17, and it also has one factor that is a square number, again, number one.

So let's look at the numbers that we've investigated so far.

What do we begin to notice? Jacob's saying that not every number has a composite number as a factor, and he's saying that we already knew that every number has number one as a factor.

Therefore every number does have a square number as a factor because number one is a square number.

But Jacob's noticed something else.

He said he didn't realise that every number has a prime number as a factor.

And that's right, we have a special name for these.

We call these prime factors.

Let's revisit number 12.

We can see the factors of 12 here on our bug.

And we know that two and three are prime numbers, therefore we can call these prime factors.

Two and three are prime factors of 12.

I wonder if you could say that again for yourself.

Let's have a look at another example then.

Here's 15 this time.

Were you able to spot the prime factors? That's right, three and five are prime numbers, aren't they? Therefore we can say that three and five are prime factors of 15.

Three and five are factors of 15, and they're both prime numbers, so they are known as prime factors of 15.

Could you say that for yourself as well? Well done, okay, time for you to check your understanding now.

Can you identify the prime factors of 20? Have a quick look for yourself.

That's right, two and five are prime numbers, aren't they? So we can say that two and five are prime factors of 20, can't we? Well done if you got that.

Okay, time for you to do some practise now.

What I'd like you to do is find all of the factors for each of these numbers here and record them on our factor bugs.

And once you've done that, I'd like you to circle each of the prime factors that you found for each number.

Good luck with that and I'll see you again shortly.

Okay, let's go through these then.

So the factors of 18 are one and itself, one and 18, two and nine and three and six.

And we know that the prime numbers here were two and three.

So two and three are prime factors of 18.

The factors of 36 were one and 36, two and 18, three and 12 and four and nine.

Again, two and three were the prime numbers here.

Therefore two and three are prime factors of 36.

And finally for 29, 29 actually only has two factors, one and itself, so it's a prime number.

And which one of these therefore is the prime factor? That's right, 29 is the prime factor, isn't it? 29 is the prime factor of 29.

Okay, that's the end of our first cycle then.

Let's have a look at our second cycle now, decomposing numbers multiplicatively.

Here, I've recorded numbers one up to 50 and we've got some circles here representing each one of those.

What I'd like you to do is take some time for yourself now to have a think.

What do you notice when you look at this image here? Take a good few moments and we'll come back and talk about it shortly.

Okay, let's see what the boys noticed.

Jun is saying that he noticed that any prime number greater than 10 had a red circle around it.

Hmm, that's interesting, Jun.

And Jacob's saying that he noticed that any number that was in the two times table had a yellow segment on it.

And we can see these here.

Number two, number six, number 14 and number 20 are all examples of numbers that would be in the two times table.

And each one of them has at least one yellow segment on it, doesn't it? Jun's asking, "Why do you think number six was represented like this?" Well, Jacob said as he is realised that all of the numbers in the two times table had a yellow segment on it, he realises that the yellow must represent number two.

And so he is wondering, well, what would work with two to make six? So he said two times three is equal to six.

So maybe the yellow segment represents two and the green segment might represent a three, mightn't it? Hmm, that's some great detective work, you two.

I wonder what else we can find out.

Ah, here's a good question.

Let's think about number eight then.

Well, "Why is number eight represented like this?" Jun's saying.

He's saying two times four is equal to eight, so surely it would have a yellow segment for the two and then we'd have a different colour to represent the four.

Ah, Jacob spotted something though.

He said, well, actually there's three segments here that are yellow, aren't there? So that would be two, two and two.

Let's multiply those together.

Two multiplied by two multiplied by two.

Well, we know that two times two is equal to four and then times that by two again is equal to eight.

Ah, so they are multiplying each other, aren't they, by looks of it.

And we can see that it can't be addition because two plus two plus two would be equal to six, therefore it has to be multiplication that we're looking at here, doesn't it? Jun feels as it's starting to make a bit more sense to him now and wants to look at a few more numbers.

Jun's spotted those numbers that have some purple in them.

Let's have a look at these.

What do you notice about all of these? That's right, Jacob said they're all in the seven times table, aren't they? So maybe the purple represents seven each time.

Let's check that for example with number 14.

We know that the yellow represents two and we know that if the purple represents seven, then two times seven is equal to 14.

Yes, that works, doesn't it? We also know that the green could represent three.

So three times seven is equal to 21, and that's why 21 has a green segment.

Ah, this is all starting to come together now, isn't it? So we think that the yellow segment is worth two, we think that the green segment is worth three, and so far we know that the purple segment we think represents seven, don't we? Is there anything else that we can work out, I wonder, let's go back and have a look.

Jun has spotted something with the blue segments.

What do you notice? Yeah, good spot, actually they look like numbers that are in the five times table, don't they? So can we say that each blue segment represents five, can we? Let's check it.

We know again that the green segment represents three.

So three multiplied by five would be 15, and therefore 15 has a green segment and a blue segment.

We spotted that, didn't we? And what about 25 then? Well, 25 has got two blue segments, hasn't it? So that must represent five times five.

Ah, that's really interesting, isn't it? Let's have a look at 20 in a bit more detail.

20's got three segments, it's got two yellows and a blue.

And we know that the yellow represents two, don't we? So that must mean it's two multiplied by two, multiplied by five.

We know that two times two is equal to four.

And then multiply that by five, that'll make 20, won't it? So we've got some interesting ways of making these numbers here, haven't we? And Jun's starting to ask, well, Why are we representing numbers in this way? Let's have a look again at what we know so far.

We know that the yellow is two, the green is three, the light blue is five, the purple is seven.

And any number that has a prime number greater than 10 has red, doesn't it? What'd you notice about each of these numbers and what they're representing? Jun thinks he's got it, he's right.

Each of the colours so far are representing a different prime number, aren't they? So each circle is showing the prime factors that make up each one of those numbers, isn't it? Jacob's relieved, he feels a lot more confident now.

He started to have an understanding for the patterns that we've been looking at so far.

Let's look at a few more in detail again.

We know that every number is made up of prime factors.

So let's look at 18 for example.

We know that 18 can be made up of either one or 18 or two or nine or three and six.

For the moment, we're gonna look at two and nine.

We know that two is a prime factor, don't we? And we know the other factor that works with two is nine.

However, if we found the factors of nine and decomposed nine into its prime factors, then actually we know three and three multiply together to make nine, don't they? So three and three would also act as prime factors.

So when we look at the circle then, we can see we've got one yellow segment, that segment represents two as a prime factor, and then we've got two green segments, both of those segments represent the threes.

So we can say that two multiplied by three multiplied by three would be equal to 18.

And we know that because three multiplied by three is equal to nine, and then two multiplied by nine is equal to 18, don't we? So we've decomposed both factors as far as we can into their prime factors.

If we look at 18 being made up as three and six as well, we know three is a prime factor, don't we? And when we think about decomposing six into its prime factors, we know that two multiplied by three is equal to six and two and three are both prime factors as well, aren't they? They're both prime numbers, so they are both prime factors.

They are prime factors of six, and they're also prime factors of 18.

So again, looking at the 18 circle, we can see that it has one yellow segment.

Again that's represented by the two.

And we've got two green segments represented by both of the threes that you can see on our bug.

We can represent that as two multiplied by three, multiplied by three, and that is equal to 18.

So we can multiply these prime factors together to make 18, can't we? We can decompose the factors of a number into their prime factors and that can be multiplied back together to make the initial number that we had originally.

Jacob's really enjoying this.

He says he's never really thought of it like this before.

It's a really mega discovery for him.

He wants to have a look at another number.

Let's have a look at 32, shall we? Well, again, we know there's lots of different factors that could make up 32.

One and itself is definitely one way, but we could also represent it as eight and four.

Eight times four is 32.

So let's have a think about these then.

At the moment, we don't have any prime factors, do we? Neither eight or four are prime factors and decompose that into two factors.

So we know now we've got two multiplied by four is equal to eight, well, two is a prime factor, we know that because it's a prime number, but four still isn't a prime number yet, is it? It's a square number.

So whilst we've now got one prime factor, we're gonna have to decompose the four again into its prime factors.

And we know that two multiplied by two is equal to four.

Therefore we've got another two prime factors here.

So far we've got three lots of two, haven't we? Let's think about the four on the other side of our bug now.

Again, we already know that four can be decomposed into two and two for its prime factors.

So we've got two multiplied by two would be equal to four.

And so, we can now see we've got another two sets of two as a prime factor.

Have a look at the image now, what do you notice about the image for 32? That's right, it's made up of five orange segments and each one of those represents a two.

And when we multiply each of those twos by each other, so it becomes two multiplied by two, multiplied by two, multiplied by two multiplied by two, that means all of that would be equal to 32.

Let's check it, shall we? Two multiplied by two is four, then multiply it by another two is eight, then multiply that by two, gives us 16, and then multiply that by two, the last two, that gives us 32.

Wow, that's a really interesting way of representing 32, isn't it? So that's why we have five yellow segments to represent the 32.

Okay, time for you to check your understanding now.

I've given you a key to show you what each number represents.

Which image represents 16 as prime factors? Take a moment to have a think.

That's right, it's A, isn't it? And why is it A, I wonder? Well, for example, we know four multiplied by four is equal to 16 and four is not a prime factor, is it? So we need to decompose those fours and both of those fours would be decomposed into two twos, wouldn't they? So it'd become two multiplied by two and then multiplied by two and multiplied by two again.

So that means we're gonna need four orange segments to represent the four twos that multiply with one another.

Okay, another check for understanding now.

True or false, every number can be decomposed to just prime factors? Take a moment to have a think.

Okay, and that's true, isn't it? Which one of these justifications helps you to reason that? That's right, it's A, isn't it? Every number has its own unique set of prime factors.

And you saw that with the initial image we had so far.

We represented one to 50, didn't we? With all of those circles, every single one representing the number as prime factors.

Okay, and onto our final tasks for today then.

What I'd like to do here is find the missing prime factors and record these in the equations on the left hand side.

And then for task two, what I'd like you to do is create your own prime factor circles for each of the following numbers.

Good luck with those tasks and when you're done, come back and we'll have a look at them together.

Okay, so I've put the answers in here.

We can see that 36 is made up of two multiplied by two, multiplied by three, multiplied by three.

Two times two is equal to four, multiply that by three gives us 12.

And then 12 multiplied by three is equal to 36.

30 would be equal to two multiplied by three, multiplied by five, two times three is six, and six times five is 30.

49 is equal to seven multiplied by seven.

Seven is a prime number.

Therefore seven is a prime factor of 49.

So seven multiplied by seven gives us the 49.

38 is equal to two multiplied by 19, isn't it? 19, again, is also a prime number, therefore it acts as a prime factor of 38.

Two multiplied by 19 is equal to 38.

And then finally for 40, 40 can be made up of two, multiplied by two, multiplied by two, multiplied by five.

Two multiplied by two is four, then multiply that by two, gives us eight, and then multiply that by five, gives us 40.

Well done if you've got all of those.

And then here is each circle for each of the numbers.

28 can be represented as two multiplied by two, multiplied by seven.

So we've got two yellow segments and one purple segment.

39 can be represented as three multiplied by 13.

So remember, any prime number greater than 10 was represented with a red.

We could write number 13 in that.

Just to be really clear what the red segment represented on the number.

And we've got one green segment there, don't we? To represent the three.

56 can be represented as two multiplied by two, multiplied by two, multiplied by seven.

So we've got three segments representing the twos and one segment representing the seven.

63 is the product of two lots of three, two lots of green segments and one purple segment for the seven.

And then finally, 100 at the bottom.

100 can be composed of two multiplied by two, multiplied by five, multiplied by five.

Two times two is equal to four.

Multiply that by five is equal to 20, and then multiply 20 by five is equal to 100.

So as you can see, you'd represent it as two yellow segments and two blue segments.

Jacob has got a great question.

He's wondering, "I wonder what one million would look like." Let's have a look.

Wow, there we go.

One million is made up of six yellow segments and six blue segments.

So each yellow segment represents a two, and each blue segment represents a five, doesn't it? So two multiplied by two, multiplied by two, multiplied by two, multiplied by two, multiplied by two, that's all of the yellow segments, and then multiplied by five, multiplied by five, multiplied by five, and then again, multiply that by five, multiply that by five, and then multiply that by five.

That would make one million.

And there we go, there's a representation of that for you.

That's a lot of multiplying, isn't it (laughs)? Maybe you could take some time to explore a number that you'd like to look into and find the prime factors of that.

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

Thanks for joining me.

I really enjoyed that lesson thinking about the prime factors.

And as I said, prime factors are the building block of all positive whole numbers.

So they're really interesting for us to understand how these numbers can be composed and decomposed.

So to summarise our learning today, we know that if a number has a factor that has a prime number, then this factor is known as a prime factor.

All positive integers or whole numbers can be decomposed into prime factors.

And again, all positive integers have their own unique set of prime factors, which compose that number.

Thanks for joining me again today.

Go and investigate those prime factors and hopefully I'll see you again soon.