Lesson video

Hello, my name's Mrs. Hopper and I'm really looking forward to working with you in this lesson.

And it comes from the unit all about the 7 times table, odd and even patterns, square numbers, and tests of divisibility.

Lots to be thinking about.

So if you're ready, let's make a start on today's lesson.

In this lesson, we're going to be representing square numbers.

So by the end of it, you should be able to represent a square number and explain what one is.

Let's make a start.

We've got two key sort of phrases in our lesson today.

We've got "square number" and "square array," so let's practise saying those and then we'll look at what they mean.

I'll take my turn, then it'll be your turn.

So my turn.

Square number.

Your turn.

My turn.

Square array.

Your turn.

Well done.

They might be new phrases for you.

We're going to explore what they mean in today's lesson.

Let's look at their meanings.

So a square number is the result of multiplying a whole number, not a fraction, by itself.

So when we take a whole number and multiply it by itself, we get a square number.

We're going to be exploring that today.

And a square array is a group of objects arranged in rows and columns, where the number of rows is the same as the number of columns, making a square.

They're going to be really useful in exploring our square numbers in this lesson.

There are two parts to our lesson.

In the first part, we're going to be representing square numbers, and in the second part we're going to be thinking about the squared symbol.

Hmm, I wonder what that's going to be.

So let's make a start on part one.

And we've got Jacob and Sofia helping us in our lesson today.

Jacob is celebrating his birthday.

Happy birthday, Jacob.

He's watching a rugby tournament with Sofia and his mum.

Rugby Sevens tournaments have rugby teams made of 7 players.

Jacob says, "I can see 7 teams." And Sofia says, "Each team has 7 players." How many players are there all together? 7 teams and 7 players in each team.

Sofia represents this with an array.

There are 7 teams with 7 players in each team.

So she's got 7 counters there representing one team of 7 players, and then she knows that there are 7 teams. 1, 2, 3, 4, 5, 6, 7.

So what do you notice about the array? "Ah," says Sofia, "it's a square.

All the sides are the same length.

So you can call this a square array." One of our keywords or key phrases for today.

It's a square, isn't it? There are 7 rows and 7 columns.

There are 7 counters in each row and 7 counters in each column.

The array represents the 7 groups of 7.

And from our times table knowledge, we probably know that 7 times 7 is equal to 49.

So there are 49 players playing in the Rugby Sevens tournament.

What do you notice about the factors? Remember, the factors are the numbers that we multiply together to get our product.

So what do you notice about our factors? "Oh," says Sofia, "they're both the same! The factors are 7 and 7." So our factor is the same number repeated.

It's one number multiplied by itself.

7 times 7 is equal to 49.

When both factors have the same value, the product is called a square number.

So 49 is a square number.

And Sofia says, "49 counters can make a square array." So if we have a number of counters that we can create a square from by having the same number of rows and columns, then we know that we have a square number of counters.

Let's explore a bit further.

Time to check your understanding.

Can you use the array to complete the equation and the sentence? So hmm times hmm is equal to 16.

When both factors have the same hmm, the product is called a hmm.

So have a look at the array that's there and see if you can fill in the equation and the gaps in the sentences.

Pause the video, have a go, and when you're ready for some feedback, press Play.

How did you get on? So, when both factors have the same value, the product is called a square number.

So what's our square number this time? Well, our square number is 16 and our factors are 4 times 4, the same number repeated, 4 multiplied by itself.

4 times 4 is equal to 16, and we can see in that array we've got 4 rows of 4 counters, or 4 columns with 4 counters.

4 multiplied by 4 is equal to 16.

16 is our square number.

Let's look at another scenario.

Basketball teams have 5 players.

There are 5 teams taking part in a basketball tournament.

So Jacob says, "I can see 5 teams" and Sofia says, "Each team has 5 players." So how many players are there all together? Can you think what our array would look like this time? Well, there we are, an array showing 5 teams with 5 players in each team.

So we can think about 5 rows of 5 counters or 5 columns of 5 counters.

But however we look at it, it's 5 times 5.

5 lots of 5.

So what do you notice about the array? That's right, it's a square.

The sides are the same length and you can call this a square array.

It's got 5 rows of 5 or 5 columns of 5.

There are 5 counters along each side.

The array shows 5 times 5 is equal to 25.

There are 25 counters representing the 25 players in the tournament.

What do you notice about the factors there in that equation? 5 times 5 is equal to 25.

That's right, they're both the same.

The factors are 5 and then 5 again.

5 times 5 is equal to 25.

When both factors have the same value, the product is called a square number.

So 25 is a square number.

And 25 counters can make a square array, as we can see there on the screen.

Time to check your understanding again.

In the Olympic games, a special type of basketball is played with 3 players in each team.

How many players are there if we have 3 of these basketball teams? So 3 players in a team and 3 teams. Can you represent this using counters? If you haven't got counters, you could probably draw them.

Is the total number of players a square number? How would you know? Pause the video, have a think, and when you're ready for some feedback, press Play.

What did you think? Maybe you've watched the Olympics and you've seen that 3 against 3 basketball.

It's quite exciting.

So here's an array.

3 lots of 3.

3 teams with 3 players in each team.

So we can think about 3 rows of 3 or 3 columns of 3.

But can you see that we formed a square with our counters? Each side is the same length.

And that square array represents 3 times 3, which is equal to 9.

And we know that when the two factors are the same in our multiplication, then the product is a square number.

There are 9 players altogether.

9 is a square number.

9 counters can be arranged as a square array.

Now Sofia represents square numbers onto squared paper.

Can you see one corner looks quite interesting, doesn't it? I wonder what she's going to do.

So she says 1 times 1 is equal to 1.

We know that a square number is when the factors are the same.

So even 1 times 1 is equal to 1.

We've got 1 lot of 1, it's a square number.

Now we've got 2 times 2.

So 2 multiplied by itself will give us a product which is a square number.

2 times 2 is equal to 4.

Can you see that we've represented? We can still see our 1 times 1 is 1, and then we've got some extra squares to give us a square that has 4 squares in it.

What's going to be next? That's right, 3 times 3 is equal to 9.

Can you see how these squares are growing on the paper? So now we've got 3 rows of 3 squares or 3 columns of 3 squares.

3 multiplied by 3, the factors are the same, so our product is a square number, 9.

4 times 4 is equal to 16.

4 rows of 4 squares or 4 columns with 4 squares.

The sides of our square are the same length, the factors in our equation are the same, so the product must be a square number.

4 times 4 is equal to 16.

16 is a square number.

What do you notice? Well, each squared number forms a perfect square shape on the paper.

So if you imagine our 1, and then you can imagine our 4, which was our next square number, if we coloured them all in the same colour, it's a perfect square.

And so on for our others.

Each squared number forms a perfect square shape on the paper.

As you go from one squared number to the next, as the numbers increase, you add more rows and more columns.

Each time you form a bigger square, you add 1 more row and 1 more column, and that means that your square goes from being 1 times 1 to 2 times 2, to 3 times 3, to 4 times 4, and onwards.

If you plot all the squares on the paper, they fall along the diagonal line starting from the top left corner to the bottom right.

Can you see that? That we're sort of creating an arrow with our extra squares, pointing down towards that bottom right corner starting from the top left? So what will the next square number be? Can you predict? That's right, it's 5 times 5.

We're adding one to the side length each time.

So we've added one to the number in the row and one to the number in each column.

Now we've got 5 times 5.

The factor's the same, so our product is a square number.

So 25 is our product, and it's a square number.

25 squares coloured in from a perfect square on our square paper.

Each new square can be seen as adding a new layer around the previous square.

Time to check your understanding.

What will the next square number be? Can you picture it on the paper? And how would you represent it as an equation? Pause the video, have a go, and when you're ready for some feedback, press Play.

What did you reckon? I'm sure you can see the pattern there.

It's going to be 6 times 6, isn't it? And 6 multiplied by 6 is equal to 36.

The factors are the same, so the product is a square number.

And you can see we've added an extra layer onto a column and onto a row to create our square that is 6 along and 6 down.

Each new square number can be seen as adding a new layer around the previous square, just around those two sides that fit on the squared paper.

And again, we're heading down the diagonal towards that bottom right hand corner.

Time for you to do some practise.

So for question one, you're going to need a graph paper or squared paper.

Any paper that's been divided into perfect squares.

And you are going to create a raise using counters first and then record this onto squared paper.

If you haven't got counters, you might be able to draw them.

And you're going to identify if the following numbers are square numbers or not.

So it would be good if you've got some counters to be able to experiment with first.

So you're going to see whether 4, 9, 12, 16, 20, 25, and 30 are square numbers.

Can they make a square array? And for question two, you're going to colour in the arrays on the graph paper to visually represent the square numbers.

So you're going to experiment with the counters and then you're going to create your arrays on the squared paper.

So pause the video, have a go at questions one and two, and when you're ready for some feedback, press Play.

How did you get on? So for question one, you are creating a raise using the counters and then recording it onto the squared paper.

And you were going to try and find out whether those numbers were square numbers or not.

So we started with 4 here.

4 is a square number, we can arrange those counters into a perfect square.

2 rows and 2 columns.

So 2 rows of 2, and 2 columns of 2.

2 times 2 is equal to 4.

We can do the same with 9.

9 is equal to 3 times 3.

And 16.

16 is equal to 4 times 4.

So 4, 9, and 16 are square numbers.

What about the others? Well, let's have a look.

We've got 20 here, but 20 can't be made into a square.

We can make 4 rows of 5 counters or 5 columns of 4 counters, but we can't make a perfect square.

20 is equal to 4 times 5.

Our factors are not the same, so 20 is not a square number.

25 though we can make into a square array.

5 rows of 5 counters, or 5 columns of 5 counters.

25 is equal to 5 times 5, so it is a square number.

And if we experimented with 12 and 30 as well, we would find that they are not square numbers.

So 4, 9, 16, and 25 are square numbers, but 12, 20, and 30 are not square numbers.

And for question two, you are going to represent them by colouring in the perfect squares on your squared paper.

So 4 is equal to 2 times 2, 9 is equal to 3 times 3, 16 is equal to 4 times 4.

They're all square numbers and they fill a perfect square on the paper.

20 is not, but 25 is.

25 is equal to 5 times 5.

So 25 is a square number, the factors are the same.

And let's move on to part two of our lesson, thinking about the squared symbol.

So Sofia and Jacob are continuing to explore square numbers.

Sofia says, "We can represent squared numbers in a different way." Here we've got an array that shows that 7 times 7 is equal to 49.

It's a square array.

7 rows of 7 counters or 7 columns with 7 counters.

The lengths of our square are the same size, our factors are the same.

7 times 7 is equal to 49, but there's another way we can record this and we've got some squared paper to record it on.

Are you ready? We can use a little 2 above and to the right of our 7, and we read that as 7 squared.

And 7 squared is equal to 49.

So you can use the squared symbol to show 7 times 7.

The little 2 written above the number is said as "squared," and what it represents is that 7 has been multiplied by itself.

So that little 7 squared indicates 7 times 7.

When you see it after a number, it means you multiply that number by itself.

So for example, 7 with the little 2 written means 7 multiplied by 7.

Time to check your understanding.

Can you write 4 times 4 is equal to 16 using that squared notation? So can you rewrite the equation? Pause the video, and when you're ready for the answer and some feedback, press Play.

How did you get on? That's right, it's 4 with that little 2 written above, and that indicates 4 multiplied by itself, 4 times 4.

And 4 times 4 is equal to 16.

So 4 squared is equal to 16, and that's how we read it.

So let's have a look at another one.

What does our square array show this time? That's right, it's 5 times 5 is equal to 25.

25 is a square number.

So how are we going to write that using the squared notation? We're gonna put that little 2 there to show 5 squared, and that's equal to 5 times 5.

So it's equal to 25.

You can use the squared symbol to show 5 times 5.

The little 2 written above the number is said as "squared." 5 squared is equal to 25, and that's the same as 5 times 5.

Remember, when you see it after a number, it means that you multiply the number by itself.

So 5 with a little 2 means 5 multiplied by 5.

Time to check your understanding.

Jacob has represented 6 squared.

Is he correct? Explain your thinking to your partner.

He's said that 6 squared is equal to 12.

Is that correct? Pause the video, have a think and explain, and then when you're ready for some feedback, press Play.

What did you think? Is he correct? He's not correct, is he? What did the little 2 mean? Ah, that's right, thank you, Sofia.

The squared symbol, that little 2, means to multiply the number by itself.

I think Jacob's multiplied by 2 here, hasn't he? So we need to remember that it means multiplied by itself.

So it's representing 6 times 6, which is equal to 36, not 6 times 2, which is equal to 12.

Jacob and Sofia are sorting cards into the correct category.

So they're thinking about square numbers, but they're thinking about whether they are even numbers or odd numbers, and they're wondering if they can think about this without calculating the value of the square.

So here they've got 2 squared, and we know that 2 squared represents 2 times 2.

And Sofia says, "Remember, even numbers end in 0, 2, 4, 6, and 8.

Odd numbers end in 1, 3, 5, 7, and 9." You know that 2 squared is equal to 2 times 2.

Ah, both factors are even, so the product will be even! We don't even need to work out what the product is.

We know it will be even because both of the factors are even.

So 2 squared will be an even number.

What about 5 squared? "So remember," says Sofia, "even numbers end in 0, 2, 4, 6, and 8.

Odd numbers end in 1, 3, 5, 7, and 9." "Ends in" means that they have that digit in the 1's position, in the 1's place.

Well, we know that 5 squared is equal to 5 times 5, and both factors are odd, so the product will be odd.

You might have been learning about this recently.

The only way you can get an odd product is if both of the factors are odd.

And in 5 times 5, both of those factors are odd.

So 5 squared will be an odd number.

You might know that 5 squared is equal to 25, but we don't need to know the product, we don't need to know the square.

We just need to know that this is representing 5 times 5, so therefore the product will be odd.

So what's the equation you're calculating here to find the total number of sides? We've got 3 triangles with 3 sides each.

So what's the equation? How could we represent this? 3 triangles with 3 sides each is the same as 3 squared, 3 multiplied by 3.

So we can represent 3 multiplied by 3 as being 3 squared using our little 2, our squared notation, and we know that that is equal to 9.

So there'd be 9 sides on those 3 triangles.

But even without calculating that, we'd know there'd be an odd number of sides because we know that we are multiplying 3 by itself.

So we have two odd factors.

So 3 triangles with 3 sides each will give us an odd number of sides in total.

Jacob and Sofia noticed something.

Jacob says, "I know when a square number product will be odd or even by just looking at the digits that is to be squared." Ooh, go on, Jacob, tell us more.

He says, "When we square a number, we multiply the number by itself." So, "Ah," says Sofia, "so if that number is odd, the product of the squared number will be odd, because odd multiplied by odd equals odd product." If we have two odd factors, the product will be odd.

"If the number is even, the product of the squared number will be even, because two even factors will give us an even product.

Even multiplied by even is equal to an even product." So can you put that to the test and check your understanding? Can you fill in the blanks? We've got 7 jars of 7 sweets here.

So can you represent that as an equation, and using the squared notation? And will the product be odd or even? Pause the video, have a think, and when you're ready for some feedback, press Play.

How did you get on? Well, 7 jars of 7 sweets is 7 multiplied by 7, which we can say is 7 squared, and that's equal to 49.

And because we were squaring 7, we knew that the product was going to be an odd number because an odd number multiplied by an odd number will give an odd product.

Well done if you got that right.

And it's time for you to do some practise.

So use your knowledge of squared numbers, can you match the equations to the correct squared notation and to the correct product? So you're going to match 1 times 1 to how that would look using the squared notation and to its product.

So lots of matching up to do there.

And for question two, you're going to sort the following cards that come up on the next slide by representing them in square number notation in order to complete the table.

And then you're going to sort those square numbers into whether they will give an even product or an odd product as their square number.

And there are the cards you're going to be looking at.

So pause the video, have a go at questions one and two, and when you're ready for the answers and some feedback, press Play.

How did you get on? Let's look at question one first.

So you were using your knowledge of squared numbers and matching the equations to the correct product and to that squared notation.

So 1 times 1 is represented by 1 squared, right at the bottom, and it's equal to 1, right back up at the top again.

Then we have 3 times 3.

Well, 3 times 3 is equal to 3 squared, 3 with that little 2.

And 3 times 3, or 3 squared, is equal to 9.

5 times 5 we can write as 5 squared with that little 2 as our squared notation.

And 5 times 5, or 5 squared, is equal to 25.

7 times 7 is equal to 7 squared, and that's equal to 49.

And 9 times 9 is equal to 9 squared, and that's equal to 81.

Did you notice that all the numbers we were squaring were odd, and then all the products were odd as well? So when we square an odd number, we get an odd product.

And for question two, you were sorting those cards and you were representing them in square number notation and then sorting them into this table to see whether the square number they produce would be odd or even.

So you can see there that if your card represented an even number squared, then it's going to give an even square.

And if your card was representing an odd number squared, it's going to give an odd number as its square number.

Well done if you got all of those right.

And we've come to the end of our lesson.

We've been representing a square number.

So we understand that a square number of objects can be arranged into a square array.

A square number can be represented as the product of two factors which are the same.

And we also understand that when both factors have the same value, the product is called a square number.

And a square number can be represented with a small 2 written above the number to show that it has been multiplied by itself.

For example, here we've got 5 squared is equal to 5 times 5.

Thank you for all your hard work.

I hope you've enjoyed exploring squared numbers, and I hope I get the chance to work with you again soon.

Bye-bye.