Lesson video

Hello and welcome to this lesson on square numbers.

I am Mr. Maseko.

Before you get on with today's lesson, make sure you have a pen, a pencil and something to write on.

Okay, now that you have those things, let's get on with today's lesson.

Try this activity.

A square number is the result of multiplying an integer by itself.

So pause the video here and give this activity a go.

Okay, now that you've done this, let's see what you come up with.

Well, you were given an example of 25, which is the square number we get by multiplying five by itself.

And we can draw an array of 25 square tiles that represents the square number 25.

So list three more square numbers.

Well, what could you have had? Well, you could have had the square number if you did three multiplied by three, that is nine.

And you could have done the same thing.

So there you go, there is a square that represents nine.

Or you could have done something like two multiplied by two.

And that is four.

And again, you could have done the same thing.

Draw yourself an array that represents four.

So these are squares.

So what happens when we multiply a square number by four? Well, let's explore this.

So if you look at the square number 25 and we multiply that by four, or 25 is five multiplied by five, and four is two multiplied by two.

Because in multiplication we can rewrite it in a different way, that still means the same thing, that is five multiplied by two, multiplied by five multiplied by two, which is the same as saying 10 multiplied by 10.

And that gives us 100.

So when we take a square number, and multiply it by four, we get another square number.

Now 100 is 10 times 10.

So that's a 10th square number.

Okay, let's see if this works for other square numbers.

Now let's look at nine multiplied by four.

Well, nine is three multiplied by three, and that's multiplied by two times two.

Now we can rearrange this to make three multiplied by two, multiplied by three multiplied by two, which gives us six times six, which is 36.

Again, a square number multiplied by four gave us another square number.

Is this is a coincidence? Why is this happening? Does this only apply to four? Does it apply to other numbers other than four? Can you multiply by any number at all? Well, let's see.

First, let's try this independent task.

Pause the video here and give this a go.

Okay, let's see what you come up with.

Well, the first 10 Square numbers that is 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Ten squared take away four squared.

That is 100 take away 16 and that would give you 84.

Write 64 as a product of two square numbers, both greater than one.

So we can't use one.

So we've got 64.

What are the factors of 64? Well, the factors of 64, you could do two times 32 and do four times 16.

Oh, look, four and 16 are both square numbers.

So 64 can be written as four times 16.

It's that property again, when we multiply a square number by four, it gets us another square number.

Now these numbers have been written so that each pair of adjacent numbers sums to a square number.

So three and six make nine, six and 19 make 25.

19 and 30 makes 49.

30 and six makes 36.

Six and 58 makes 64.

So, the line of numbers you should have made was so that any pair of adjacent numbers, those added together to give you a square number.

Now let's look at this explore task.

Now, Anton says, "Nine times four is 36, "which is a square number.

"If I multiply a square number by another square number, "the answer will always be a square number." Now Anton is no longer saying if I multiply a square number by four, I get a square number.

He's saying if I multiply a square number by another square number, the answer will always be a square number.

Can you derive calculations and draw diagrams to show that Anton is correct? Is there a way we can show this is true every time we multiply a square number by another square number? Pause the video here and give this a go.

Okay, let's see what you have come up with.

Well, let's do 25 multiplied by nine.

Well, if we do 25 multiplied by nine, well that is five multiplied by five, multiplied by three, multiplied by three.

That's five, times three, multiplied by five times three.

That's 15 times 15, which gives us 225, which is a square number.

Could we obtain it by multiplying 15 by itself? Looks like Anton is correct.

Will this work with another example? Well, let's see, let's try another example.

Well, let's do, surely let's try.

Let's do 100 times 36.

Well, that is 10 times 10, multiplied by six times six, which is 10 times six, multiplied by 10 times six.

So that's 60 times 60, which is a square number, because you're multiplying 60 by itself, which gives you 3600.

So it's not just four.

Anytime you multiply a square number by another square number, it looks like we're always getting another square number.

Can you see why this is working? Think back, how do we get square number? We multiply an integer by itself.

Well, if you look at this, we take a square number and write it as an integer, we multiply it by itself.

And we have two square numbers.

We can make another calculation, because we have a pair.

So we have two pairs, so we can combine them to make two different pairs that are identical.

And because they're identical, that multiplication will always give us a square number.

Now, is there a way we can show that this is correct every time? Because these are examples, and this is what we call conjectures.

But these are not proofs, because we'd have to try for every single number.

And we don't have time for that.

So is there a way that can show that this is true every single time? Can you figure out a way? Well, if you can, try it.

Well, that's it for today's lesson.

I hope you've learned something about the properties of square numbers, especially what happens when you multiply square numbers with other square numbers.

I will see you again next time.

Bye for now.