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 seven 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.

So in this lesson, we're going to be identifying and using square numbers to solve problems. Hopefully, by the end of this lesson, you'll be really good at spotting those square numbers and using them to solve some problems. Let's make a start.

We've only got one key, sort of, phrase in our lesson today and that's square number.

So I'll take my turn to say it and then it'll be your turn.

Are you ready, my turn, square number.

Your turn.

Excellent.

You may have done some learning about square numbers very recently.

I hope so.

If not, we're going to learn a bit about them today.

Let's just check what they mean.

Let's just check what square number really is and look at its definition.

So a square number is the result of multiplying a number by itself.

So that's known as a square number.

And it might be that you've explored creating square numbers using different numbers of counters.

If you can arrange a number of counters into a perfect square, then you've got a square number of counters.

There are two parts to our lesson today.

In the first part, we're going to be looking at inequality problems, and in the second part, we're going to be solving some problems. So let's make a start on part one.

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

Jacob and Sofia are solving square number problems. They need to use one of the symbols, less than, greater than, or equal to between those two square numbers.

And you remember, we can record square numbers with that little 2.

That little 2 means a number multiplied by itself.

So we've got 2 squared and 3 squared here.

Are they greater than, less than, or equal to each other? Well, Jacob says, "2 squared is equal to 4, 2 times 2 is equal to 4.

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

So 3 squared is greater than 2 squared." You are right, Jacob.

Well done.

And there's 2 squared represented, 2 lots of 2, and 3 squared, 3 multiplied by 3.

And our arrays there form perfect squares.

They're square arrays.

They represent square numbers.

Sofia says, "Yes, but you also know that the factor 3 is greater than 2 so you can solve this without having to calculate." She's right, isn't she? If we've got 2 times 2 and we're comparing it with 3 times 3, then 3 times 3 is going to be greater.

So 2 squared is less than 3 squared.

They move on to the next question.

Again, they need to use one of those symbols.

This time, we've got 4 squared and 3 squared.

Hmm.

I wonder what Jacob's going to do.

Oh, Jacob's gone for calculating.

He says, "4 squared is equal to 16 because 4 squared, 4 with that little 2 represents 4 times 4.

And 3 squared represents 3 times 3, which is equal to 9.

So 4 squared which is 16 is greater than 3 squared, which is 9." Yes, you're right, Jacob.

And there's 4 squared represented with a squared array of counters and 3 squared.

And we've used more counters to represent 4 squared, haven't we? "Yes," says Sofia, "but you also know that four is greater than three.

So you can solve this without having to calculate.

The factors in 4 square is 4 and 4, 4 times 4.

And in 3 squared, it's 3 times 3.

So we know that 4 times 4 is going to be greater than three times three without having to calculate the value.

So 4 squared is greater than 3 squared." Time to check your understanding.

Without calculating the product, which of the following have a value greater than 4 squared? Is it A, B, or C, or is it more than one of them? Pause the video, have a think.

And when you're ready for some feedback, press Play.

How did you get on? You were going to do this without calculating.

So we were going to think about Sofia's reasoning rather than Jacob's good calculating and his number knowledge.

So we had to find out which ones have a value greater than 4 squared.

Well, 4 squared means 4 times 4, doesn't it? And we've got 5 squared there, which is 5 times 5.

Well, that's got to be greater, hasn't it? And 6 squared, which is 6 times 6, that's got to be greater too, doesn't it? And then 3 squared, which is 3 times 3, I think that'll be less.

Yes, that's right.

You know that the factors of 5 and 6 are greater than 4.

So you can solve this without having to calculate the product.

5 times 5 and 6 times 6 are both going to be greater than 4 times 4 in value.

Well done if you sorted that out without having to calculate.

And there are the calculations just to keep Jacob happy.

5 times 5 is equal to 25, 6 times 6 is equal to 36.

And we know that 4 times 4 is equal to 16, which is less than both of those.

Now, they look at a different question, but they still need to put in the correct symbol.

This time we've got 3 squared and we're comparing it with 3 times 2.

Hmm.

Well, let's have a look.

Jacob says, "3 squared is equal to 9.

And 3 times 2 is equal to 6.

So 3 squared is greater than 3 times 2." And there we can see 3 lots of 2.

We've only got 6 counters there.

But 3 times 3, 3 squared is equal to 9.

"Yes," says Sofia.

"Remember the small 2 in 3 squared means to multiply 3 by itself, which is different from multiplying by 2.

That's doubling, isn't it?" So 3 squared is greater than 3 times 2.

Now, they look at a different one.

They still need to insert that symbol.

This time, we are comparing 3 squared and 3 squared plus 3.

"This is tricky," says Jacob.

"3 squared plus 3 is a mixed operation expression." We've got two different things to do there.

We need to work out 3 squared and then add 3.

Well, Sofia says, "3 squares add 3 means adding another group of 3 to 3 squared." "Oh," says Jacob.

"So 3 squared plus 3 will be greater because we've added another group of 3." Great reasoning there.

We can see it there with the arrays to show us.

We've got 3 squared, and we've then got 3 squared and another 3.

And we can see that that's greater, but really good reasoning to see that we have the same 3 squared on both sides.

But on one side, we were adding another group of 3.

So 3 squared is less than 3 squared plus 3.

"Ooh, true," says Sofia.

What if we had subtracted a group of 3?" Let's have a look.

So here, we've got 3 squared, and we're comparing it to 3 squared subtract 3.

Hmm, now what? "Ah," Jacob says, "3 squared will be greater this time because it has three more." 3 squared is 3 multiplied by 3, isn't it? And this time, we've got 3 multiplied by 3, but we are subtracting 3.

So there'll be three more in 3 squared.

And we can represent that with counters.

3 squared, and then 3 squared subtract 3.

Yes, so 3 squared is greater.

3 squared is greater than 3 squared subtract 3.

"I agree," says Sofia.

I'm glad you do, Sofia.

Do you agree as well? I hope so.

Time to check your understanding.

Which of the two expressions is greater? Can you insert the correct symbol? We've got 5 squared and 5 squared subtract 5.

So you might want to think about arrays here or you might be able to reason through it, thinking about what you know about squared numbers.

Pause the video, have a go.

When you're ready for some feedback, press Play.

What did you think? Well, here, we can see 5 squared, and then 5 squared subtract 5.

So 5 squared is greater than 5 squared subtract 5 because it has five more.

We've taken five away from 5 squared on the right hand expression.

Well done if you got that right.

And it's time for you to put this into practise.

So for question one, using your knowledge of square numbers, fill in the missing symbols between those expressions.

Are they less than, greater than, or equal to each other? And what do you notice as you work through? Pause the video, have a go.

When you're ready for some feedback, press Play.

How did you get on? So we're going to focus on A and B first.

We had 7 squared and then 7 plus 7.

Ah, now this is where we need to be really careful, because 7 squared is 7 multiplied by 7, 7 times 7.

It's nothing to do with adding, or as B says, multiplying by 2.

Do you notice they're the same thing? So 7 squared is going to be greater than 7 plus 7, and 7 squared is greater than 7 times 2.

7 times 2 is the same as 7 plus 7.

It's 7 two times.

But however we record it, 7 squared is greater because that's 7 multiplied by 7.

So onto C, 7 squared is equal to 7 times 7.

That's what the little squared symbol, the little to means.

It means multiplied by itself.

So 7 squared is equal to 7 times 7.

What about D, 7 squared and 2 plus 7? Oh, no, 7 squared is gonna be greater than that.

7 squared means 7 multiplied by 7.

We can see that with C.

What about E then, 7 squared and 8 times 8? Ah, now, we can think about that reasoning they were using earlier.

7 squared means 7 times 7, and seven is smaller than eight.

So 8 times 8 is going to be greater.

So we can say 7 squared is less than 8 times 8.

And I think, we can use similar reasoning for F, can't we? 9 squared is 9 times 9, 7 squared 7 times 7.

So we know that 9 squared is going to be greater, because nine is greater than seven.

So 9 squared is greater than 7 squared.

Well, G, we've got 7 squared and 7 squared, they're equal to each other, aren't they? What about H? 7 squared and 7 squared plus 7? Well, we're adding seven more, so that's going to be greater, isn't it? So 7 squared is less than 7 squared plus 7.

What about I, 7 squared and 7 squared subtract 7? Oh, we're taking seven away from one of those 7 squared, so that's going to be less.

So 7 squared will be greater than 7 squared subtract 7.

And then let's compare 7 squared and 6 squared.

Well, seven is greater than six.

So 7 squared will be greater than 6 squared.

So in G, we can see that 7 squared is equal to 7 squared.

And in I, we've got 7 squared on one side of our symbol.

And then we've got 7 squared subtract 7, so that's going to be less than, isn't it? And those are the arrays to show us the symbols that's correct for G and I.

Well done if you've got all those correct, and well done if you reasoned without calculating the value of those expressions.

And on into part two of our lesson, we're solving problems. Sofia and Jacob are making designs using tiles in their art lessons.

Oh, that sounds like fun.

Come on then, Jacob, shows what you've been doing.

Oh, wow, look at that one.

He says, "The square tiles come in different colours.

Mine looks a bit like a J," he says.

Well done, a J for Jacob there.

Sofia's design is a square, one side is six tiles long.

So how many tiles has she used? Can we tell just from one side? What's the key thing there? Ah, well done, Jacob.

A square has four sides that are the same length.

So if one side is six tiles long, then the other side must be six tiles long as well.

So now, we know we've got a square and we can fill in that square.

So we've got six lots of six tiles.

Jacob says, "You could count each tile one by one, but that's not efficient, is it? We know this is a square.

We can use our knowledge of square numbers to solve this," he says.

6 squared is equal to 36, 6 times 6, 6 lots of 6.

This is a square number because it's formed by multiplying the side length 6 by itself, 6 times 6, and that's equal to 36.

So Sofia used 36 tiles.

And the square array of 36 counters or tiles here shows us that 36 is a square number.

Jacob has also made a square design.

Oh, he hasn't arranged his tiles yet though, has he? He says, "I have used 16 tiles to make my square.

How many rows of tiles, and how many tiles in each row?" What do you think, you might want to pause and have a think about this before Jacob shares his thinking with us.

Well, 16 is a square number, isn't it? Jacob says, "You could just try different rows, but that's not very efficient." He says, "I know that 4 times 4 is equal to 16 and I can write this as 4 squared is equal to 16." Sofia says, "This means that there are four groups of four tiles or four rows of four tiles.

The square array of 16 counters or tiles shows us how 16 is a square number.

And Jacob says, "My design is four rows of four tiles." 4 squared is equal to 16.

Time to check your understanding.

Which array best represents the question below? So here's the question.

I have used 25 tiles to make my square.

How many rows of tiles, and how many tiles in each row? A, B, or C? Which one represents the problem best? Pause the video, have a think.

And when you're ready for some feedback, press Play.

How did you get on? It's C, isn't it? 25 is a square number, and it's formed by multiplying the side length five by itself, 5 times 5.

So we will make a square array with five rows of five tiles.

B was a bit sneaky there.

One of the things that we can get wrong is when we see that little 2 above the 5 to show 5 squared, we can think about multiplying by 2, but we are not.

We're remembering that it's 5 multiplied by itself two times, 5 times 5.

And A, we just have the wrong factor in there.

We were looking at a square, but only four tiles.

We'd have had tiles left over if we'd only used 16 squares, wouldn't we? So C was correct, well done if you got that right.

Jacob makes a different design.

This time, he says, "I have used 18 squares to make my design.

Have I made a square design?" Is 18 a square number? Oh no, 18 is not a square number.

The closest square numbers are, as Jacob says, he says, "I know 4 times 4 and this is equal to 16.

So I can write this as 4 squared is equal to 16.

And I also know 5 times 5 is equal to 25.

And I can write this as 5 squared equals 25." So 4 squared is equal to 16, and 5 squared is equal to 25.

So this means your design will not be a square, 18 is between 16 and 25.

But there are no whole numbers left to square between four and five, are there? So 18 is not a square number.

Or in this arrangement of 18 counters or tiles shows how 18 is not a square number.

He says, "My design looks like a compound shape," a shape from two shapes stuck together.

So we have got a square in there of 16, but then we've got two extra tiles for Jacob's 18 tiles.

And time for you to do some practise.

So in task B, you're going to solve the following problems using your knowledge of square numbers.

So in A, Sofia's design is a square, one side is eight tiles long.

How many tiles has she used? In B, Jacob's design is a square, one side is nine tiles long.

How many tiles has he used? And then in C, Sofia decides to remove a square from the middle of her original design from part A.

One side of the square she removes is three tiles long.

How many tiles are left over? Oh, that's interesting.

You might want to do some drawing there.

And for D, Jacob adds to his square design in B by adding another square onto the side.

One side of the extra square is six tiles long.

How many tiles has he used altogether? And in E, Jun's joining our tile-making party today.

Jun's tile design has 81 tiles.

Sofia says, "You can represent this as 9 tiles times 2 tiles." Is she correct? So have a go at those five parts of your question, and when you're ready for the answers and some feedback, press Play.

How did you get on? So in A, Sofia's design is a square, one side is eight tiles long.

How many tiles has she used? Well, that's 8 squared, 8 multiplied by 8.

So 8 times 8 is equal to 64.

And that's the total number of tiles she used.

In B, Jacob's design is a square, one side is nine tiles long.

How many tiles has he used? Well, that's 9 squared, which is 9 times 9.

9 times 9 is 81.

So the total number of tiles used was 81 in Jacob's design.

Then Sofia decides to remove a square from the middle of her original design in part A.

In part A, she had an 8 square, she used 64 tiles, but she's removed a square.

And one of the sides of the square she's removed is three tiles long.

So how many tiles are left over? So she had 64 tiles and she's removing three squared number of tiles.

Well, 3 squared is equal to 9, isn't it? And 64 subtract 9 is equal to 55.

So she uses 55 tiles.

Well done if you work that one out.

Did you draw a picture to help you? I had to picture something in my head to help me.

Now Jacob, in D, is adding onto his design.

So his design was 81 tiles because it was 9 squared, and he adds on an extra square.

And one side of the extra square is six tiles long.

So that must be 6 squared for the full square.

So he's now got 9 squared plus 6 squared.

So that's 81 plus 36, and that's equal to 117 tiles.

So that's how many tiles he used altogether.

That's a lot of tiles, Jacob, well done.

And then Jun came along to play with the tiles as well.

And his design has 81 tiles.

Sofia says you can represent this as 9 tiles times 2 tiles.

Is she correct? Oh, she's not, is she? She hasn't thought carefully about how the little two works.

9 squared represents 9 times nine, not 9 times 2.

9 times 9 is equal to 81, so Jun also used 81 tiles.

Well done if you've got all of those right, and well done for thinking hard about your squared numbers.

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

We've been identifying and using square numbers to solve problems. So what have we learned about? We've learned 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 when both factors have the same value, the product is called a square number.

Thank you for all your hard work and your mathematical thinking in this lesson.

I hope I get to work with you again soon, bye-bye.