Lesson video

Hiya, my name Miss Lambell.

Really pleased that you've decided to pop along and do some maths with me today.

Welcome to today's lesson.

The title of today's lesson is checking understanding of prime numbers.

And that features in our unit on properties of numbers.

By the end of today's lesson, you will be able to: state the properties of prime numbers and also identify prime numbers.

There are some really, really cool facts about prime numbers.

You might decide you want to go away and have a look at what some of them are.

But I'm just going to share this one with you.

Now, I'm not sure how I would say this number as an actual integer, so I'm just going to read out the digits.

So it's 20 digits long, and it's actually a prime number.

See if you can spot why it's interesting.

One, two, three, four, five, six, seven, eight, nine, 10, nine, eight, seven, six, five, four, three, two, one.

Did you spot why it's quite an interesting number? So 20 digits and it's prime.

I'll just repeat it again.

One, two, three, four, five, six, seven, eight, nine, 10, nine, eight, seven, six, five, four, three, two, one.

That 20-digit number was the numbers one to 10 and then repeating those in descending order.

And that's a prime number.

Pretty special, I think you'll agree.

So keywords for today's lesson.

We've only got one keyword, which may be new to you, and that is prime.

Pause the video and write down what you think prime means.

When you're done, unpause the video and come back and see.

So a prime number is an integer greater than one with exactly two factors.

And we're going to be exploring that throughout today's lesson.

There are two other words that we are going to be using in today's lesson, which you are really familiar with, but it might be worth just us checking in on the definition of these two words.

And they are multiple and factor.

So a multiple is the product of a number and an integer.

So remember product is multiplication.

And a numerical factor is a factor that is an integer.

We've done lots and lots of work recently on factors, so you are expert at those now.

We're going to divide today's lesson into two separate learning cycles.

In the first one, we'll be looking at the properties of prime numbers.

So what is special about them? We've already mentioned something there.

And then we'll be moving on to identifying prime numbers.

So let's start with properties of prime numbers.

Jacob is thinking about the numbers one to 10 and their factors.

He likes to draw arrays.

Well in fact, he loves to draw arrays, and he noticed something when he was drawing them for different integers.

He says, "I've noticed that some integers only have one array." Can you think of any integers which only have one array? So remember, an array is an arrangement of dots in a rectangle or a square.

Pause the video and see if you can find any integers between one and 10 that only have one array.

We'll take a look at some of those integers.

We're going to take a look at five, six, seven, and nine.

So arrays for five.

I don't know if you may have drawn this one out, but actually, there's only one array for five, which is one by five dots.

Let's now look at number six.

The arrays for six.

So we've got one by six dots, but we could also have two by three dots.

And remember, we could have it as three by two dots, but then, that is the same array.

Arrays for seven.

Did you do seven? Well, seven only has one array, one dot by seven dots.

What about nine? Nine, we could do nine as one row of nine dots, or we could do three rows of three dots.

You'll notice that seven and five, we were only able to draw one array for.

Integers with exactly one factor pair are prime numbers.

We can also find prime numbers by listing their factors.

We'll look at the number 11.

So remember our first factor pair, the one that people often miss, is one and the number itself.

So our first factor pair is one multiplied by 11.

Is two a factor of 11? I'm sure you said no, because two, to be a factor of two, it needs to be even, and 11 is not even.

Three, is three a factor of 11? Again, I'm sure you all said no.

So here, we've got a three, the divisibility check for three is the sum of the digits.

The sum of the digits here for 11 is two, and that's not divisible by three.

So we'll move on to four.

But why don't we need to check four? There's a reason why, actually, we don't need to check four.

And that's because, if it's not a multiple of two, then it isn't a multiple of any other even number.

Five.

Is five a factor of 11? Well, we know there's something special about multiples of five.

In the ones column, there's either a zero or a five.

And for 11 in the ones column, it's a one, so no, the digits are neither zero nor five.

Is seven a factor of 11? No, seven does not divide into 11 without leaving a remainder.

Eight, nine, and 10 will not be factors due to the other tests of divisibility we have considered.

So the number 11 actually only has one factor pair.

Sofia and Andeep are talking about prime numbers.

Sofia says, "One is prime because it can only be divided by itself and one." Andeep says, "One is not prime, as it only has one factor.

A prime has exactly one factor pair." Who do you agree with? Pause the video and have a think about who you agree with, and then come back.

Hopefully you said you agreed with Andeep.

Andeep was right.

Prime numbers have exactly two factors.

It's a common mistake for people to think that one is a prime number because it's only divisible by itself and one.

The definition of a prime number is that it has exactly one factor pair.

It has two factors.

Now, I've got another check just to make sure that you've really understood what the difference is between a prime and a non-prime.

15 is a prime number.

Is that true or false? And don't forget, we always want to be able to justify our answer.

So is the justification of it, 15 is only divisible by one and 15 or b, 15 has more than two factors? Pause the video, have a think about your answer.

So true or false and then your justification, and then you can come on back.

Great work.

Now let's have a look then.

So 15 is a prime number.

I hope you all said false.

False, it is not a prime number.

But what was the justification for that? So a says 15 is only divisible by one and 15.

But actually, if we were to have drawn an array, I know I could draw an array of three dots by five dots.

So actually, it has more than two factors.

And those are one and 15 and three and five.

You're now ready to have a go at working out which of these are prime numbers.

And we're going to use the listing of factors.

So we've done this previously.

We know that we're going to use a systematic approach, and we are going to list them in pairs.

So list the factors of the following, and then decide which of these numbers are prime numbers.

Pause the video, have a go, and come back when you're ready.

Great work.

Now, the second part to this task.

There's quite a lot of information on the screen.

But let's go through it together.

And then when I've gone through it, you can then pause the video and you can have a go.

But remember, you've got all of the skills to be successful at this.

Alex writes out the first seven prime numbers.

And they're two, three, five, seven, 11, 13, and 17.

And he uses these numbers to make the following statements.

So he's just used those seven numbers.

What I want you to do is to decide whether you agree with the statements that Alex has made.

And remember to give an example to show support of your answer.

So don't, we're not just going to say agree/disagree.

We need that justification, because it shows we really understand what we're talking about.

So the statements are: most prime numbers are odd, all numbers ending in a three are prime, prime numbers don't end in nine, numbers ending in five can be prime, seven and 17 are prime, so 27 will be too.

So you need to decide which ones you agree with, which ones you disagree with, give an example to support your answer, and then also the challenging part of this task is to create your own statements about prime numbers.

And see if you can find one that is always true, one that is sometimes true, and one that is never true.

So you're ready now.

It's a lot of information there.

Pause the video, take as long as you need to work through each of these statements.

Good luck.

All done working through that? Now we can check our answers.

So number one, part a, the factor pairs of 27 were one and 27 and three and nine, so it's not prime because it hasn't got just one factor pair.

B is one and 37, so it is prime.

It's only got one factor pair.

It's got exactly two factors.

C, 87, sorry, one and 87 and three and 29, so not prime.

And then d, 57 is one and 57 and three and 19, so again, not prime because it has more than two factors.

Now let's look at this next task, which like I said, it was challenging, but I'm sure you've done fine.

So first one: most prime numbers are odd.

So this is an example.

You may have written something similar to this.

I agree, all prime numbers are odd except two.

All numbers ending in three are prime.

I don't agree.

33 ends in a three, but it is not prime because it has two factor pairs, one and 33 and three and 11.

Numbers ending in five can be prime.

I agree.

Five ends in five and is prime.

However, this is the only prime ending in five, as all other numbers ending in five are multiples of five.

Prime numbers don't end in nine.

I don't agree.

29 is a prime number.

Seven and 17 are prime so 27 will be too.

I don't agree.

27 is not prime because it has two factor pairs, one and 27 and three and nine.

And here are some examples of my always, sometimes, never true.

Prime numbers always have two factors.

That's true always.

Prime numbers end in nine.

Well, that's sometimes true, 'cause if we look at the top of this page, we've got 29 is prime, but nine is not prime.

Numbers ending in zero are prime.

That's never true, because any number ending in zero is a multiple of 10.

Well done on that first learning cycle.

We're now ready to move on.

So we have already identified some prime numbers, but we are going to look beyond the ones that we already know and look at some methods there are for finding those prime numbers.

In the last learning cycle, we looked at using factors to help us identify prime numbers.

But we can also identify prime numbers using multiples.

A mathematician, Eratosthenes, who was born in modern-day Libya, developed an algorithm to identify prime numbers using multiples.

So an algorithm is a process, a set of rules.

Now, if you look him up online, it will say that he was a ancient Greek.

So when he was born, which was BC, he actually lived in Greece, but it's now part of modern-day Libya, which is in Africa.

This is called the sieve of Eratosthenes and is one of the most efficient ways of identifying prime numbers.

We're actually gonna have a go at that ourselves now.

And it works on this fact.

So this is the algorithm.

This is the process, a set of rules, that the multiples of a prime number are themselves not prime.

Because they would have more than two factors.

It would be useful now to have a 100-square grid.

But don't worry if you haven't got one.

All you need to do is to write out in some form on a piece of paper the numbers from one to 100.

So pause the video, and when you're done, come back and we're going to perform the sieve of Eratosthenes ourselves.

So remember, not the prime itself.

So we are going to cross out all of the multiples of the prime numbers that we know, but not the multiple itself.

So we're going to start with the smallest prime number that we know, which is two.

So we're not gonna cross out two, but we are gonna cross out the multiples of two.

So we're gonna start with four.

So you're gonna pause the video, and you're gonna cross out all of the multiples of two.

Remember, all the multiples of two are even.

When you've crossed them out, then come back.

Super work.

So this is what your grid should now look like.

This is nice easy one to check.

So we can see that it's the second, fourth, sixth, eighth, and 10th column.

But remember, we didn't cross out two itself.

So our next prime number is three.

We're gonna cross out the multiples of three, but not three itself.

So again, pause the video, and then cross out the multiples of three, remembering not to include three.

When you've crossed out all of your numbers, then you can come back.

Super, so we should've crossed out nine and then those numbers there.

You might want to pause the video and check that you've crossed off the right ones.

Or you might choose to wait until the end and check your whole grid.

What's our next prime number? That's right, it's five.

So we are going to cross out all of the multiples of five, but we are not going to cross out five itself.

This one shouldn't take so long.

So pause the video, come back when you're done.

So we should've crossed out 25 and then the rest of that column.

So five at the top, not crossed out, but all of the ones below it crossed out.

Next prime number is seven.

So pause the video, cross out any multiples of seven that have not already been crossed out.

And then when you're done, you can come back.

So 49 and then there are two more.

The numbers we have left are all prime numbers.

They have exactly one factor pair, exactly two factors.

Is that true or false? I'm hoping you recognised there that actually, one is not a prime number, so let's cross that off in our grid.

But the other numbers that are left are all prime numbers.

Now we're going to have a quick check for understanding.

Why is two the only even prime number? Is it a, because it is even, b, because it only has one factor, or c, because all other even numbers are multiples of two? Well done, I'm sure you said c, because all other even numbers are multiples of two.

Well done.

We're ready now to have a go at a task.

Now, I'm going to challenge you here.

What I'd like you to do is to do this task, but without looking at that grid of prime numbers we've just created.

You can if you want to double check something, but you could also try and do this task without and see whether you can really remember those prime numbers.

But remember the grid's there for a reference if you need it.

So you're going to start on the number two in the bottom left-hand corner.

You can move vertically and horizontally.

So left and right or up and down, not diagonally.

And you are only allowed to move onto boxes that contain a prime number.

So starting at two in the bottom left-hand corner, find your way through the grid horizontally and vertically onto prime numbers.

And what number on the right-hand side do you finish on? Pause the video and have a go at this task.

Well done, hopefully you enjoyed that.

We've got a second question now.

Use the numbers on the cards to complete the sentences.

We must use each card once only.

So our numbers are the numbers nine through to 16.

You need to choose a number to answer each of those questions, and try your hardest to use each number only once.

And just as we had in the last learning cycle, you can challenge yourself then to write a sentence that's true for all of the numbers, true for some numbers, and not true for any numbers.

Good luck with that task, and come back when you're ready.

Pause the video now.

Great work.

Let's check.

So this should've been your path through the maze.

So two, 13, 19, seven, 31, 61, 43, three, 79, 53, 29, 97, five, 67, 83, 11, 71, 89, 23, and then finishing up on 17.

Hopefully you found your way through that grid correctly.

Question number two.

A: you could've chosen nine or 16.

B: 11 or 13.

C: 10, 14 or 15.

D: 10 or 14.

E: 11 or 13.

F: nine or 16.

G: 10, 14 or 15.

H: 12.

And here are some statements I have chosen.

Remember these are just examples.

All of the numbers have two or more factors.

Some of the numbers are prime numbers.

None of the numbers have more than six factors.

Well done if you got all of those right.

Let's now summarise what we've looked at during today's lesson.

You've done fantastically well, and I'm really pleased that we've got to the end.

Prime numbers have exactly two factors.

So remember that really, really important word there, exactly two factors.

Prime numbers can be identified using arrays.

So that's what we started looking at.

We drew the array of dots, so you can see there, for our example, we've got one row of seven dots.

So seven is prime.

We then looked at the fact we could use factor pairs.

So if I do my factor pairs for seven, I've just got one and seven.

And then we moved on to using multiples and the sieve of Eratosthenes.

Thank you so much for joining me for your maths learning today.

Well done.