Lesson video

Hi, my name is Mr. Peters.

Thanks for joining me today for this lesson.

Today we're gonna be thinking about how we can use a complete list of factors to help us identify when a number is a square number.

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

So by the end of this lesson today, you should be able to say that I can use complete list of factors to help identify when a number is a square number.

In this lesson today we're gonna have two key words we're gonna be referring to.

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

The first one is factor, your term.

The second one is square number, your term.

Let's think about what these mean then.

Factors are whole numbers that exactly divide another whole number.

And a square number is a number that is the result of multiplying an integer, a whole number by itself.

This doesn't apply to fractions.

So for this lesson today we've got two cycles.

The first cycle is gonna be thinking about finding the factors of square numbers, and then the second cycle is gonna be investigating square numbers even further.

If you're ready, let's look at the first cycle.

Throughout this session today we're gonna have both Laura and Alex joining us.

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

Okay, Alex is investigating the factors for the number nine.

He's drawn them out on his factor bug and he is realised that nine has factors of one and itself just like every other number.

And he also knows that three multiply by three also makes nine.

So three and three would be factors of nine.

Alice is saying nine only actually really has three factors then.

It has one, nine and three 'cause we don't need to say the three twice as a factor.

Laura's realised that numbers that she's looked at so far have have all gotten even number of factors 'cause they're all in pairs.

So he's wondering are there any more numbers that are like this one? Alex starts to think, should we investigate some of these numbers then.

Starting with number one, we know that one has factors of one in itself.

Well actually that's just one.

So factors of one are one and that isn't odd number of factors, isn't it? Let's have a look at number two.

Two has factors of one and itself again.

So it has two factors, one and two, therefore it has an even number of factors.

Three has factors of one and itself again.

So this is just two factors.

So that would be an even number of factors once more.

And then let's have a look at number four.

Four has one and itself, but it can also have two rows and two columns as an arrangement.

So two can also be a factor.

So four actually has three factors as well.

It one, two, and four.

Therefore has an odd number of factors.

Alex is asking, "So what's special about these numbers Then?" Let's look at them more carefully.

Well, Laura thinks she spotted something.

They all have factors of one and it itself, but these tiles representing these numbers could be rearranged, couldn't they? Let's have a look.

If we rearrange them to look just like this, we realise that one has factors of one and one, that hasn't changed.

We realise that four now has a factor of two 'cause it's got two rows and two columns and we realise that nine has a factor of three because it's got three rows and three columns.

What do you notice now? That's right Alex.

They're all squares, aren't they? So all of these numbers have the same length and they have the same width.

When the lengths and the widths are the same, we can represent it like this.

The little two that's used in the air represents that the number has been multiplied by itself.

And when a number has been multiplied by itself, we can say that it has been squared.

So one squared is equal to one, two squared is equal to four, and three squared is equal to nine.

And Laura's making a valid point here.

When listing these as factors on a factor bug, for example, we don't need to write the number twice.

So for the factors of four, we wouldn't record two twice.

We just record it once and say that two is a factor of four.

Alex has got a great question now.

He wonders if this is only applicable to square numbers.

I wonder if you'll be able to help answer that for him as we go forward.

Okay, time for you to check your understanding now.

Look at the lists of factors below.

Each of these lists of factors represent a number.

Which one of these lists would represent a square number? Take a moment to have a think.

That's right, it'd be a and c, wouldn't it? And why is that? That's right, we know that a square number has an odd number of factors, doesn't it? Because one of those factor pairs would technically be multiplying a number by itself and we only record that factor once.

So for the first number, for example, 25 is a square number because five squared or five multiplied by five would be equal to 25.

And therefore we just record the five once and the third number would represent 36.

We know that because the largest factor is 36 and 36 is the product of six multiplied by six.

So we only record the six once in our list of factors.

And that's the reason why square numbers have an odd number of factors.

Okay, another check for understanding now.

True or false.

All square numbers have the same number of factors.

Take a moment to have a think.

That's right, it's false, isn't it? Which justification helps to reason that.

That's right, it's a, isn't it? Some square numbers have more factors than others.

For example, four only has three factors, whereas 36 has nine factors.

Well done if you've got that.

Okay, time for you to practise now.

What I'd like to do is investigate the number of factors that each of these following numbers have here.

You can record whether it's an odd or an even number of factors as well, and you can let us know whether it's a square number by giving a tick or a cross in that box at the end.

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

You can see here that we've listed all of the factors below for each number and we're just gonna run through whether they have an even or an odd number of factors.

So 15 has an even number of factors, doesn't it? 16 has an odd number of factors and every other number also has an even number of factors except for 25 which has an odd number of factors.

And if you look carefully, we've realised that both 16 and 25 are square numbers and they are the only two numbers there that have an odd number of factors.

Therefore, that helps to support our thinking.

That square numbers appear to only have an odd number of factors.

Alex is wondering whether we could go on and find out what other numbers have an odd number of factors.

Let's have a look at that in cycle two.

So cycle two is about investigating square numbers.

Let's have a look at this together.

So Alex is saying that we know that square numbers have an odd number of factors and here's an example of the first five square numbers.

One has one factor, four has three factors, nine has three factors, 16 has five factors, and 25 has also three factors.

Continuing this we can look a bit more closely.

36, which is the product of six multiplied by six has seven factors.

49, which is the product of seven multiplied by seven has three factors.

And 64, which is the product of eight multiplied by eight, has seven factors.

Alex has recorded this in a table below.

In each column he's recorded the number multiplied by itself, which enables us to make the square number.

And he's recorded the number of factors so far that number has.

He says it feels like there's a bit of a pattern building.

Can you notice anything yourself? Let's have a quick check for understanding now.

Find the factors of nine squared.

Take a moment to have a think.

Okay, well 81 is the product of nine squared.

So we know that one in 81 would be factors.

81 is actually a multiple of three.

So we can say that three multiplied by 27 would be equal to 81.

So three and 27 are factors of 81.

And we also know that nine multiplied by nine would be 81, so nine is a factor of 81.

Well done, have you got that? Okay, time for you to practise now.

Can you continue investigating the number of factors that each square number has? You can finish recording the table to the right hand side and then ask yourself what is it that you notice as you go through.

And then after that I've got an always, sometimes, and never question for you.

The factors of a square number are also the same factors as the previous square number.

And then the last task.

Have a look at these square numbers here.

What'd you notice about the difference between each successive square number? Good luck with those tasks and I'll see you back here shortly.

Okay, here's the rest of the table.

I wonder if you were able to fill out exactly what we managed to do as well.

Hopefully you notice that nine squared or 81 has five factors.

10 squared or a 100 has nine factors.

11 squared or 121 has three factors, and 12 squared or 144 has 15 factors.

Wow, that's a lot more factors isn't it? Was there anything you were able to notice? Looking at the number of factors each square number has? Alex says he's realised that the larger the number gets, it doesn't necessarily mean that it's gonna have more factors.

And Laura's noticed most odd square numbers only really have three factors.

That's a good spot, Laura.

Okay, always, sometimes, never.

The factors of a square number are also the same factors as the previous square number.

So Alex is looking at an example of this.

He says the factors of 25 are one, five and 25, and the factors of 16 are one, two, four, eight, and 16.

The only factor that both of these numbers share is number one.

So it doesn't seem to be true so far, does it? And Laura's confirming that, she thinks it's definitely never.

And the reason for that is that some previous square numbers may share some of the same factors.

However, any new square number would have to include itself as a factor and therefore 25, for example, if that was the new square number, that could not be a factor of 16, which was the previous square number because 25 is obviously too big a number because we know that the largest possible factor and number can have is itself.

Well done if you managed to come up with any reasoning like that for yourself.

Okay, and then what did you notice about the difference between these squares here? Well, Alex was saying that each square number can be found in the next square number, and hopefully you can see that's been represented by the green in itself.

For example, if we look at 16, we can see that there's actually a square of nine in there, so nine would fit into 16.

However, each one of these squares almost has a reverse element, doesn't it? And that's because each square has been given an extra row and an extra column, and where those extra rows and columns overlap, which you can see here.

That means by adding on the extra row and column together, we're always gonna be adding an odd number of tiles onto the square because of the way that the tiles will overlap with each other.

So we can say that the difference between any two consecutive square numbers will always be an odd number.

We can see that the difference between one and four would be three.

The difference between four and nine would be five.

The difference between nine and 16 would be five, and the difference between 16 and 25 would be nine.

Well done if you managed to get that.

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

Hopefully you've enjoyed yourself and again, you're feeling a lot more confident thinking about square numbers and their factors as well.

To summarise what we've learned today, we know that square numbers have an odd number of factors.

This is because a square number is created when an integer or a whole number is multiplied by itself.

When recording factors of square numbers, we do not need to record the same factor twice.

Thanks for joining me today, I really enjoyed that lesson.

Hopefully you have too.

Take care and I'll see you again soon.