Lesson video

Hi, my name's Mr. Peters, and welcome to today's lesson.

In this lesson, we're gonna be thinking about how we can apply our understanding of the properties of numbers, for example, the factors and multiples of these numbers, and how we can use these in number of games and problems as well.

If 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 use knowledge of properties of number to solve problems. Throughout this lesson today, we've got four key words we're gonna be referring to.

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

The first one is factor.

Your turn.

The second one is multiple.

Your turn.

The third one is property.

Your turn.

And the last one is classify.

Your turn.

Let's have a think about what these mean.

Factors are whole numbers that exactly divide another whole number.

A multiple is the result of multiplying a number by an integer, and this doesn't apply to fractions.

A property is a character or a quality that something has.

And when you classify objects, you arrange the objects by some property.

Today's lesson is gonna be broken down into two cycles.

The first one will be thinking about linking factors and multiples, and the second cycle will think about classifying numbers.

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

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

Again, as always, they'll be sharing their thinking, any questions they have along the way.

So our lesson starts here with Izzy and Lucas playing a game.

Hopefully you can see a 100 square on the screen.

The game starts by somebody picking a number.

So Izzy is going to pick a number, and she's chosen number 35.

The next number that she picks has to be either a factor or a multiple of the number that she has chosen.

So, Lucas is saying that Izzy could choose to multiply her number by two to make a multiple of 35.

Lucas knows that 70 is a multiple of 35 because 35 multiplied by 2 is equal to 70.

So there we go.

There would be an example of what Izzy could choose next.

So, now that she's decided to go with 70, she now needs to, again, think about finding a factor or a multiple of 70.

The last link she used was to multiply by two.

And for this game, you cannot use the same link twice throughout your whole chain.

Lucas is suggesting that maybe Izzy should think about finding factors of 70.

We know that the next multiple of 70, if we were to multiply 70 by 2, would take us up to 140.

And that isn't on our grid, is it? So we're going to need to find a factor of 70.

Izzy knows that seven and 10 are factors of 70 because 70 divided by 10 is equal to 7.

Therefore, she's chosen to go for 10 as the next number in her chain.

She's quite pleased with the chain she's got so far, she says, and she's wondering how long she'll be able to make this chain.

Lucas hopes it doesn't go too long because the aim of the game is to create the longest chain you possibly can against your partner.

And Lucas will be wanting to win, so he'll be hoping the chain isn't too long so he can make a chain that's longer than Izzy's.

Okay, time for you to check your understanding now.

The next number that Izzy chose was number one.

What was the link that she made? Take a moment to have a think.

That's right.

Izzy knew that one was a factor of 10 because 1 multiplied by 10 is equal to 10.

So we could say that she's actually divided 10 by 10 and that has given her one as her factor, hasn't it? Hmm, now Izzy's thinking about what she should do next.

She's currently at number one.

Lucas has reminded her and said, "Well, one is a factor of every number, isn't it? So you could choose any number on the board and it would be a multiple of one, wouldn't it? It would be in the one times table." Oh, Izzy's gone for a different choice now, she's gone for 53, and she's locked that in now.

Izzy's now saying, "oh no!" She's made a mistake, but she's already locked her answer in.

Lucas isn't sure.

"What is your mistake, Izzy?" Well, Izzy's realised that 53 is actually a prime number.

It doesn't have any other factors other than one and itself, and therefore, she's already used one so she can't find any other factors of 53.

And Lucas has realised you're right, Izzy, 'cause 53, the next multiple of 53 would be 106, and again, that's not on our grid, is it? So that looks like it's the end of Izzy's turn.

Izzy's created a chain which has five different numbers in it and four different links.

Lucas is up for the challenge though.

He thinks he can beat this.

Let's see how he gets on, shall we? I wonder how he'll get on.

Okay, time to check our understanding again.

Lucas starts on 21 and then looks for a multiple of this.

Take a moment to have a think.

Which one of these numbers could he have chosen? That's right, he could have chosen C or D, 42 or 63.

Why would he have been able to choose those two numbers? Well, that's right.

If you multiply 21 by 2, that would give us 42, and you could multiply 21 by 3, and that would give us 63.

So what other multiples of 21 could he have chosen? That's right, he might have multiplied it by four, which would've given him 84.

Would he be able to multiply it by five? No, that would give us 105, wouldn't it? And 105 wouldn't fit on our grid, would it? Okay, time for you to have a go at our first task then for today.

What I'd like you to do is have a go at the game yourself.

What's the longest chain you can make? If you need to stop and start again, you can do.

Keep going until you can make the longest chain you possibly can.

Izzy's reminding us that once you've used a link once, you can't use it again.

For example, if you multiply a number by two to get another number, you cannot use multiply by two again, you can use divide by two, but you can't use the same number and operation in the same link together again.

You might also like to think about what your strategy is after you've had several goes.

Did you start to realise something which made it easier each time? Good luck with that and I'll see you back here shortly.

Okay, let's have a look at how Lucas got on, shall we? Lucas said his strategy was to avoid using prime numbers.

And he thinks his chain is still going.

Should we have a look at his chain and see if we can identify the links that he made? Well, he started on 21 and then he multiplied that by two to get to 42.

So he found a multiple of 21.

He then found a factor of 42 by dividing that by seven, which left him with six.

He then found a multiple of six by multiplying it by 10.

And then, once he was on 60, he then divided that by six, which gave him a factor of 60 and left him on 10.

He then opted for another factor.

He divided 10 by 2, which left him with five, and then he multiplied his 5 by 20 to get to 100.

So he found a multiple of five and which is, in this case, 100.

He then found a factor of 100.

He knows that four lots of 25 make 100, so he used 25 as a factor.

And he then knows that three lots of 25 make 75, so he found a multiple of 25 and he went to 75.

Once at 75, he then decided to divide that by five, which took him to a factor of 75, which was 15.

And then, once on 15, he made a big leap up to 90.

How did he get there? That's right.

He multiplied it by six, didn't he? 15 Multiplied by 6 is equal to 90.

And then finally, he divided by 10.

He went from 90 down to nine.

He chose nine as a factor of 90, and his chain is still going as well.

And now we can see the links that he made each time as we've just gone through.

And we can just double check that he hasn't repeated himself at all with any of those links.

He hasn't, has he? So, as he's saying he thinks he can keep going.

I wonder how long a chain he'll be able to make by the end of the lesson.

Okay, that's the end of Cycle One.

Let's have a go at Cycle Two now, thinking about classifying numbers.

When we're thinking about factors and multiples of a number, we can say that we're thinking about the properties of these numbers, and we can group numbers by these properties.

Let's use a Venn diagram to do this.

Here, you may have seen one of these before, this is called a Venn diagram.

You can see there are two circles here which overlap in the middle.

Venn diagrams were first invented by an English philosopher known as John Venn in 1880.

He used them as a way to classify items to show their similarities and their differences.

On the left-hand side, we've got two circles.

So he's gonna be comparing two different classes.

And on the right-hand side, you can see he's got three circles, so he's gonna be comparing the similarities and differences of three different objects or classes.

Here's an example of how we could use one.

You can see on the left-hand circle, I've put things that can jump.

And in the right-hand circle, I've suggested we're gonna put things that can swim.

Where they overlap in the middle will be things that can jump and swim.

I wonder if you could think of anything that could fit into these parts of the circles.

Here's an example.

We could say that a grasshopper can jump, but a grasshopper can't swim, can it? So this would go in the left-hand side of the circle.

We could say that a turtle can swim but it can't jump, can it? So that would go in the right-hand side of the circle.

And did you manage to come up with anything that could swim and jump? Of course, a frog.

A frog can swim and a frog can jump.

So a frog would sit in the middle of our two overlapping circles.

We can call the overlapping section of the two circles an intersection.

What do you notice now? We've now got a slug on the right-hand side of our diagram.

Why has it been put on the outside of the diagram? That's right, a slug can't jump or it can't swim either, so it sits on the outside of the diagram.

Okay, so we started looking at different creatures, didn't we? Whereas now we're gonna start thinking about how we can use numbers in a Venn diagram.

You can see here our Venn diagram is separating multiples of seven and multiples of three.

Hmm.

Take a moment to have a think for yourself.

Which numbers do you think would fit into each parts of the circles? Well, let's start with 16.

16 Is not in the seven or the three times table, so therefore, it'll need to stay on the outside of the diagram.

We know that 9 is equal to 3 multiplied by 3, so nine is a multiple of three.

We know that 21 is equal to 3 multiplied by 7, therefore, 21 is a multiple of both three and seven, so that would fit in the intersection of our circles.

12 Is in the three times table, but it's not in the seven times table, so it would fit on the right-hand side of our circles.

14 Is in the seven times table, and we know that because 14 divided by 7 is equal to 2, but it is not in the three times table, so it would fit on the left-hand side of our circles.

And then finally, 42.

Well, we know that 6 multiplied by 7 is equal to 42, so 42 is a multiple of seven.

And also because we know that every multiple of six is actually a multiple of three, we know that 42 would be a common multiple of both three and seven, so it fits, again, in the intersection between the two circles.

Well done if you managed to get those.

Okay, time for you to check your understanding now.

Can you look at the Venn diagram? Which number has been incorrectly placed? Take a moment to have a think.

That's right.

It's the 24, isn't it? At the moment, the 24 is placed in the intersection between the multiples of two and the multiples of five.

However, 24 is not a multiple of five, is it? It is a multiple of two because it's in the two times table, but it's not in the five times table, so it isn't a multiple of five, therefore, we need to take it out and place it in the left-hand side of the circle with the multiples of two.

Well done if you got that.

So far in this lesson, we've already started looking at one type of diagram which would allow us to start classifying numbers by their properties.

We're now gonna look at a different type of diagram.

This is called a Carroll diagram, and it's named after a very famous author named Thomas Carroll.

Do you know any stories that Thomas Carroll may have written? That's right.

Thomas Carroll actually wrote a very famous story called Alice in Wonderland.

You may have read it before.

As well as being an author, Lewis Carroll also regarded himself as a mathematician.

And during his time thinking about mathematics, he came up with this diagram to help classify the properties of numbers.

Let's have a look at an example of one of his diagrams using animals again.

Here you can see, in a very simple version of the diagram, we've got two columns.

We've got things that can jump and things that can't jump.

On the left-hand side, we've got a grasshopper and a lemur, both of those can jump.

And on the right-hand side, we've got an elephant and a snake.

Neither of those can jump.

Carroll diagrams can then be extended to classify objects into two or more groups.

Let's have a look at this example here.

If you look at the diagram, in the rows, we've got square number and not a square number.

And then, in the columns, we've got multiples of nine and not multiples of nine.

So where each row and column meets is where a number has to be placed that fits the criteria of what those rows and columns say.

So let's have a look at these numbers and have a go at classifying them to start off with.

The first number is 16.

Izzy's saying that she knows 16 is a square number because it's 4 multiplied by 4.

Is 16 in the nine times table though? No, it's not, is it? So we know it's a square number, but it's not in the nine times table.

Can you see which part of the diagram we'd have to put 16 into? That's right, it goes in the top right section of the diagram, isn't it? It's in line with the row that says square number and it's also in line with the column that says not a multiple of nine.

Lucas says he knows that 18 is a multiple of nine because 2 multiplied by 9 is equal to 18, but it isn't a square number.

So where would that fit on our diagram? That's right.

It would fit here, wouldn't it? It fits in the row which says it's not a square number, but it also fits in the column which says that it is a multiple of nine.

And the next example, 21.

Izzy's saying that 21 is not a multiple of nine because it's not in the nine times table.

No, it's not.

The next square number is 25 after 16, isn't it? Therefore, where would that fit into our diagram then? That's right.

That'll fit down here, wouldn't it? The row which says not a square number and the column which says not a multiple of nine.

And then, finally, the last one, 81.

Where do you think that's gonna fit in our diagram? Well, we know that 81 is a multiple of nine because 9 multiplied by 9 is equal to 81.

And we also know that 81 is a square.

It's nine squared, isn't it? 9 Times 9 is equal to 81.

So 81 would fit here into our diagram, wouldn't it? Okay, another check for understanding now for you.

Look at the options A, B, C, and D.

Can you tick the correct option, which would fit into the bottom left-hand section of our Carroll diagram? Take a moment to have a think.

That's right.

It's D, isn't it? Actually, there is no number here that would fit into this section here.

I wonder why that is.

Well, we're looking for a number that has a multiple of four and also an odd number.

And we know they don't exist, do we? We know that all multiples of four are actually multiples of two and, therefore, are all even numbers, aren't they? So there is no number that can fit into that section of the Carroll diagram.

So we would just typically leave that blank.

Well done if you managed to get that.

Okay, onto our final tasks for today then.

What I'd like to do here is classify the following numbers into the Venn diagram correctly.

In this diagram we have two circles, whereas in question B, we have three circles.

So we've got three different classes you need to classify the numbers into.

And then, for Question Two, what I'd like to do is organise the numbers into the correct places into the Carroll diagrams as well.

Pause the video now and when you're done, come back, and we'll go through the answers.

Okay, let's see how you got on then.

Have a look at A.

I'll let you go through and see if you managed to tick all of those off correctly.

Did you manage to spot the one that sat outside of the circles? That's right, 14 was neither a multiple of three or a multiple of nine, was it? And then, for Question B, this is a bit trickier 'cause we had the three different circles to use to classify the numbers.

This is where the numbers should have been placed in the diagram.

Did you manage to get all of those? Did you spot that 30 was the only one that would fit for all three circles right in the middle, being a multiple of five, a multiple of three, and a multiple of two? Well done if you got all of those.

Okay, let's run through the answers for these ones then as well.

Question A, we know that 42 and 84 are both multiples of six and multiples of seven.

6 Times 7 is equal to 42.

And if we double that, that will give us 84.

So, therefore, they're both multiples of six and seven.

We know that 21 is not a multiple of six, but it is a multiple of seven because 3 times by 7 is equal to 21.

We know that 12 is not a multiple of seven, but it is a multiple of six because 2 times 6 is equal to 12.

And then, finally, 32 and 15, neither of those are multiples of six or seven as neither of them are in the six and seven times table.

Question B, we're looking for multiples of four or not multiples of four, as well as cube numbers and not cube numbers.

So, this is how they would be laid out.

Eight is a cube number because it's 2 multiplied by 2 multiplied by 2, and it's also a multiple of four.

So that would fit in the top left-hand column.

27 Is not a multiple of four, it's not in the four times table, but it is a cube number.

It's a result of 3 multiplied by 3 multiplied by 3.

Therefore, 27 fits in the top right-hand section of the Carroll diagram.

44, 28 And 92 are actually all multiples of four, but they're not cube numbers.

So they would fit in the bottom left-hand section.

And then finally, 30.

30 Is not a cube number as the previous cube number would be 27 and the next cube number would be 64.

And it's not a multiple of four either 'cause it's not in the four times table.

Well done if you managed to get all of those.

If you've got some extra time on your hands, you might decide to make your own Carroll diagram to test on a friend and see if they can classify the numbers that you decide to put into that diagram as well.

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

To summarise what we've learned about, we can say that the factors and multiples of a number are known as the number's properties.

We know that numbers can be classified into groups by their properties.

And Venn diagrams and Carroll diagrams are often used to help classify numbers.

Well done today.

Hopefully you've enjoyed yourself and you've learned a little bit about how we can classify numbers using those multiple different diagrams. Take care, and I'll see you again soon.