Lesson video

Hi, thanks for joining me today.

My name is Mr. Peters.

I'm really looking forward to this lesson today.

Today, we're gonna be thinking about how we can identify either a prime number or composite number.

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

By the end of this lesson today, you should able to explain how to identify either a prime number or a composite number.

Throughout this lesson, we've got a couple of key words we're gonna be referring to.

I'll say them first, and you can have a go afterwards.

The first word is factor.

Your turn.

The second word is prime number.

Your turn.

The third one is composite number.

Your turn.

And finally, the last one is square number.

Your turn.

So let's think about what these mean in a bit more detail.

A factor is a whole number that exactly divides another whole number.

Prime number is a whole number greater than one, which has exactly two factors.

A composite number is a whole number that has more than two factors.

And finally, a square number is a result of multiplying an in integer by itself.

And this doesn't include fractions.

Okay, this lesson is broken down to two cycles.

The first cycle, we'll think about identifying prime numbers and the second cycle, we'll think about identifying composite numbers.

Let's get going.

Throughout this lesson today, we're gonna be joined by Lucas and Andeep.

They'll be sharing their thinking, as well as any questions that they have as we go throughout the lesson.

Okay, so we start our lesson today with Lucas and Andeep who are exploring the number of factors different numbers have.

Lucas says he's noticed something.

He's realised that some of these numbers only have two factors.

Can you spot which ones they are? That's right.

In the numbers they've investigated so far, numbers two, three, and five only have two factors, don't they? Lucas is wondering why that is.

Well, let's have a look in a little bit more detail.

At the moment, you can see numbers one to five represented as an array.

At the moment, these arrays only have one column.

They could also be represented as an array, where there is only one row.

Lucas is saying, "That's because every number has one and itself as a factor." And as we know, we can help find our factors by understanding number of rows and columns that an array has.

For example, let's look at number four.

Number four has one row and four columns.

So we can say that one and four are factors of four.

So that's right.

Andeep noticed that every number here has an even number of factors, isn't it? It has one and itself as factors except for number one.

Well, why does number one only have one factor? Well, as we know, all numbers have one and itself as a factor.

But number one, when you say itself, number one is itself.

So it has one row and one column, and therefore, we only need to record number one as the factor once.

We don't need to write it down twice.

That's right.

So we can't repeat a number as a factor, can we? Hmm, but Andeep's thinking that some of these numbers can be rearranged into a slightly different arrangement.

Let's have a look.

We know that four can be made into a square, and we know that six can be represented as two rows of three.

While some of these numbers can be rearranged into a different array, numbers two, three, and five can't be rearranged into a different array to represent two different factors.

Let's have a look more closely at these numbers.

There is no way that we can rearrange these rows and columns.

So there's an equal number of items in each row and in each column.

So Lucas is asking, "Well, these numbers surely have a special name, do they?" And that's right, they do, Lucas.

These numbers are called prime numbers.

Prime numbers only have two factors, one and itself.

Hmm, that's a great question.

I wonder what other prime numbers there are, as well.

Should we keep investigating this? Let's have a think about whether we can find prime numbers on a multiplication grid.

Here, you can see this multiplication grid represents each of the times tables from 0 up to 12.

Well, because each number in the one times table has one and itself as a factor.

Therefore, it has two factors.

That must mean every number in the one times table is a prime number.

Hmm, Andeep disagrees.

He says, "Not quite." You're right in saying that every number has one and itself as a factor.

However, lots of numbers appear more than once in other times tables.

For example, let's have a look here.

Number eight, for example, appears in the two and four times table.

So number eight cannot be a prime number.

Can you take a moment to spot any other numbers that also appear in the one times table, but also appear in our multiplication grid? I'm sure you spotted a few more.

So we can't just say that every number in the one times table is a prime number.

Of course, it's not.

So can you spot any other numbers that don't appear in any other times table other than the one times table and its own times table? The numbers that only appear in the one times table and its own times table on this grid are numbers 3, 5, 7, and 11.

Well done if you managed to spot them.

So we can say that every other number on this grid has three or more factors.

And that's right, Lucas.

Every number on this grid, except for a few are in the one times table would be a composite number.

And we're gonna look at those in a little bit more detail later on.

Okay, time to check your understanding now.

True or false, one is a prime number.

Take a moment to have a think.

That's right, it's false, isn't it? And which one of these justifications helps you to reason that? That's right, it's A, isn't it? Prime numbers have exactly two factors, one and itself, and we know that number one is itself, so it only has one factor.

We don't repeat it.

Okay, time for you to practise now.

What I'd like to do is fill in the remainder of this table here to help identify whether the number is a prime number or a square number.

You might want to use some tiles to represent the numbers as you go, which will help you identify the factors that they have.

And then for question two, what I'd like to think about is this.

Always, sometimes, or never.

Adding two prime numbers together always makes a square number.

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

Okay, let's go through the answers then.

You'll have seen that I've recorded all the factors ready for you to have a look at.

You might want to go through those now, and tick them off and check that you've got them.

Now, let's see how we can class these different numbers.

Well, number seven is a prime number.

Number eight is neither prime or square.

Number nine is a square number.

Number 10 is neither prime or square.

Number 11 is, in fact, a prime number.

Number 12 is neither prime or square.

Number 13 is also a prime number, and our next square number is not until number 16.

Then our next prime number is actually number 17.

And then we've got one more prime number in this collection numbers here, and that, in fact, is number 19.

Well done if you've got all of those.

Okay, let's have a think about this one then.

Always, sometimes, or never.

Adding two prime numbers together always mix a square number.

Andeep is saying he doesn't think that's true.

He knows that two and three are prime numbers, and if you add those together, that makes five.

And we know that five isn't a square number.

Hmm.

Here's an example of that.

Here's our number two and here's our number three.

And as we can see, it doesn't make a square, does it? So, so far, it sounds like it might be never.

But hang on, Lucas has identified, if you add two to itself, so two and two, we know two is a prime number.

So if we add two and two to itself, actually, it makes four, and four is a square number.

So the answer is sometimes.

Did you manage to find any other pairs of prime numbers that added together to make a square? It'd be really interesting to investigate how many of those you found, and whether there are any patterns at all.

Okay, moving on to the second phase of our lesson then, identifying composite numbers.

So going back to our investigation so far, we know that some numbers can be recorded as prime numbers or square numbers.

However, there's a couple of numbers here that are neither prime nor square.

And Lucas is right.

If we were to make these numbers with tiles, they could all make rectangular arrangements.

So Lucas is asking, "Well, do we call them rectangular numbers?" Whilst rectangular numbers would be a great name for them, we actually call them composite numbers.

Could you have a go at saying that? Let's have a look at some of them in a bit more detail.

So the numbers we recognise as composite numbers are 8 and 10 so far.

Let's have a look at number 8 and number 10.

They can both be recorded in one row with either 8 or 10 columns.

In this arrangement, we could say that both one and itself are factors, aren't they? We could, however, rearrange it to find different factors.

Let's have a look at number eight now.

So we can see that eight has now been rearranged.

It's now got two rows and four columns.

So two and four are factors of eight.

And let's have a look at number 10 now.

Number 10 has been rearranged into 2 rows and 5 columns.

So we can say that 2 and 5 are factors of 10.

So so far, we can say that number eight has factors of one, two, four, and eight.

And number 5 has factors of 1, 2, 5, and 10.

8 and 10 are composite numbers.

And the reason we call them composite numbers is because they have more than two factors.

Andeep's already investigated number 12, and realised how much of a good example number 12 is as a composite number.

Here again, you can see it has one in itself as factors.

We have 1 row and 12 columns.

We could now represent it again to show different factors.

So let's have a look.

We've now got two rows and six columns.

And if we rearrange it again, we've now got three rows and four columns.

So 12 has 6 factors altogether.

It has 1, 2, 3, 4, 6, and 12 as factors.

Okay, time for you to check your understanding again.

Tick the composite numbers, I'll give you a moment to have a think.

That's right, C and D are the composite numbers here, and the numbers are 18 and 20, aren't they? You may have used tiles to help you rearrange them, but I also noticed that both of these numbers are even numbers, aren't they? And we know that any even number can be divided by two, so therefore, they could always be represented as two rows.

That's a really good indicator of whether a number is composite or not.

Have another look here then.

Each of these answers here are showing the factors of a number.

Can you tick the ones that would represent a composite number? Take a moment to think.

That's right, it would be B, C, and D.

And how did we know that? Well, we know that a composite number has more than two factors, and we can see here that B, C, and D all have more than two factors.

However, you may have wondered about C a little bit, weren't you? C is 16, and it has 3 factors.

It has 1, 4, and 16.

And when you're creating 16 earlier on, you may have made it into a square shape.

So 16 is, in fact, a square number, but a square number is actually a special type of composite number.

So whilst we do call it a square number, it could also be classed as a composite number, because it makes a rectangular shape, in this case, a square.

Okay, and onto our final task for today then.

What I'd like to do is find the composite number less than 50, which has the largest amount of factors.

Once you've tackled that, I'd like to have a go at filling in this Venn diagram here.

You have to place the numbers in the correct places on the diagram to represent whether they're composite, square, or prime.

Or, in fact, could be classed as more than one of them.

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

Okay, let's go through the answers then.

So the composite number less than 50 with the large amount of factors is, in fact, 48.

48 has 10 factors.

These factors are 1 and 48, 2 and 24, 3 and 16, 4 and 12, and 6 and 8.

Well done if you managed to find that.

And then here's the arrangement that each of the numbers should have been placed into on our Venn diagram.

We can see that there is only one number that is a square number by itself, and that's number one, and that's because we know that number one only has one factor.

It doesn't have more than two factors.

We can see that any other square number is also placed as a composite number as well, because they can be classed as both composite and square numbers.

We can see that 11 and 7 are classed as prime numbers.

And also, you may have noticed zero doesn't fit into any of these categories since zero multiplied by any number is equal to zero.

Zero actually has an infinite number of factors, and a composite number actually only has a set amount of factors.

For example, every number that we've identified so far as a composite number has, for example, 4 factors, or 8 factors, or 12 factors, and has an exact amount of factors, where a 0 has an infinite number of factors.

Therefore, we cannot class zero as a composite, square, or prime number.

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

Hopefully, you've enjoyed that, and you're a bit more secure thinking about what prime numbers and composite numbers are.

To summarise our learning today, we can say that a prime number has only two factors, one and itself.

A composite number has more than two factors.

And one is neither a composite number or a prime number as it only has one factor, which is itself.

However, we can class number one as a square number, as if we use tiles, for example, to represent it, it can indeed make a square shape.

Thank you very much for joining me today.

Hopefully, you've enjoyed yourself.

Take care, and I'll see you again soon.