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Hi there, welcome to another maths lesson with me Dr.


In today's lesson, we will be looking at surds from tilted squares.

For this lesson, you need a pen, a paper and a ruler.

So, please go and grab these, and when you're ready, let's begin.

I'd like you to start today's lesson by looking at these squares here.

For each square, I would like you to find the length of one side from the given area.

I want you to think about what do you notice about the area and side lengths.

Which one is the odd one out and why? This task should take you about five minutes to complete.

If you are struggling and would like a little hint, I would suggest that you start by looking at the smallest square.

That's the square that has an area of four centimetres squared.

Now think about the properties of a square.

All the sides are equal.

And to calculate the area of a square, we need to multiply one side by the other.

So what number multiplies by itself to give you four? With this hint, you should be able to have a go at the task on your own.

Please pause the video and complete the task.

Resume once you're finished.

Welcome back.

How did you get on with this task? Did you manage to find the length of one of the sides of the squares for each square that was given to you? Okay, let's go through some answers.

So we started with the smallest one, it had an area of four centimetres squared, that side length must have been two centimetres.

Two centimetres multiplied by two centimetre gives us four centimetres squared.

Next one.

You may have done them in a slightly different order, I did the 16 centimetres squared, and the side length there is four centimetre.

Again for the same reason, four centimetre multiplied by four centimetre, gives us 16 centimetres squared.

Next one I went to the, 25 centimetres squared.

And that gives us a side length of five centimetre.

36 centimetre squared, we have six centimetre multiplied by six centimetre, will give us a 36 centimetre squared.

And the smaller one, nine centimetres squared.

I had three centimetres multiply by three centimetre.

Did you get all of these correct? Good job.

Now, I left the 10 centimetres squared for now.

I left it for a reason, but I don't know what that side length is.

Because if I try and think about my numbers, I cannot think of a whole number multiplied by whole number, an integer multiplied by an integer to give me 10.

So, when I wanted to answer the question: "What do you notice?" I wonder what you wrote down.

Because what I noticed is, that all of the areas here for every single square, was a square number apart from the 10.

16 is a square number, the four is a square number, 36 is square number, 25 and nine are all square numbers.

So, I wrote down that they are all square numbers apart from 10.

Which one is the odd one out? Therefore, I decided that 10 centimetre squared, was the odd one out.

And the reason for that, not just because the area is a square number, but also all the other squares had integer length for the side lengths apart from the 10 centimetre.

I cannot find a whole number or an integer as a side length.

So, I said it does not have an integer side length.

Now, I'm not going to tell you now what the side length for that one is, because that's what we're going to look at in today's lesson.

I wonder if you wrote something down, and if you did, you can come back to it, and correct it or work it when we go through this.

So let's look at this in a bit more depth.

You have a tilted square here.

What is the area of the square? You should be able to answer this from our last lesson.

So if you want, you can pause the video now, and have a go at this.

Or just have a little think about it and write down your answer.

Okay, what did you write down? Okay.

So, there are so many ways to find out the area of this tilted square as we discussed in our previous lesson.

We discussed the three, at least the three different methods.

This is how I have done it.

I thought, let me find the area of the biggest square, which is a four by four, so that's 16.

And now let me find the area of one of those triangles here.

So, that's 1/2 times the base times height.

The base of this triangle is one and the height is three.

1/2 multiplied by three is 1.


But I have four triangles.

So, I said 16 minus 1.

5 times four, which is 16 minus the six, and that gave me 10 units squared.

Now if you've done it using different method, you may have split that and tilted the square into four, into five shapes.

So into a smaller square and four triangles.

Well, you may have done it by counting the squares.

You should have ended up with 10 units squared as well.

How can I use the side length to find the area of a square? Well, let's look back at maybe simpler squares to start with.

If I have a three by three square, that gives me an area of nine.

If I know that the area is nine, I can find out the side length by square rooting it.

So square root of nine is three.

So, I know that each side of the square must be three.

If I have another square, so let's assume I have a five by five, that gives me an area of 25.

If I knew that the area was 25 and I wanted to work out the side length, I can square root the 25 and that would give me five as the side length.

So, is it going to be any different if I have any different number? Is it going to be different if I have a number which is not a square number, as the area? Really good.

So, in this case, I need to think about what number multiplied by what number gives me 10.

And I can write it in decimal, but it's not going to be precise, because it will involve some rounding up.

So instead what I can write is, that it's a square root of 10.

So the length of one side of the square is the square root of 10.

And I leave it as the square root of 10 in surd form.

So this is called in surd form.

What this means is, I'm leaving it with the square root, because it's an irrational number.

And this is something you're going to learn about.

Rational and irrational numbers.

So, it's an irrational number, it means it's a decimal that does not terminate.

I don't want to write it and round it, so I can leave it in surd form.

It's more precise.

And now it is your turn to have a go and practise, what we have just done.

I'd like you to look at this diagram here.

You have been given four tilted squares.

For each of the squares, I would like you to find the area and the side length.

And I want you to leave your answer in surd form.

So if your answer is not an integer for the length, I'd like you to leave it in surd form.

So, square root of something.

The Independent task should take you about eight minutes to complete.

Please pause the video, and complete it to the best of your ability.

Resume the video once you're finished.

Welcome back.

How did you get on with this? Did you manage to find the area for each square? What about the side length? You left it in surd form? Okay in that case, let's mark the work, and see how you did.

So for the first one, I had an area of five square units.

You could have done it in any of the methods that we discussed earlier.

If the area is five, this tells me that the side length must be the square root of five.

Okay, the second one.

What did you get? An area of 10 square units and it's really the same or very similar tries.

It's really the same square that we looked at earlier, isn't it? Okay, and therefore the side length must be square root of 10.

The third one, what did you get? Excellent job.

Eight square units, and therefore, the side length is the square root of eight.

And the last one gave me two square units, and therefore the side length must be square root of two.

From this, we also need to learn and understand that what this means, for the first one, it means that the square root of five multiplied by the square root of five, gives me five.

For the second one, square root of 10, multiplied by square root of 10 gives me 10.

So that's how we end up with that area.

We multiply one side by the other side.

Square root of eight, multiplied by the square root of eight is eight, and square root of two multiplied by the square root of two is equal to two.

Okay, really good.

Now let's look at the Explore task and look at the area of tilted squares and surds in a bit more depth.

So, here I've given you three tilted squares.

I would like you to find the area of each of the squares.

I would like you to find the side length, of each of the squares.

I also want you to think about what might the next square in this pattern look like? How could you describe it? What would be the area and the side length of the next one? Maybe even the one after, if you really want to challenge yourself.

This task should take you 10 to 15 minutes to complete.

As I said earlier, you need to find at least the next square in the pattern.

But if you want to challenge yourself, find the next two or even the next three.

Look at patterns and what is happening, and list anything that you notice down.

Pause the video to complete the task, and resume once you're finished.

Welcome back, how did you get on with the Explore task? Okay, so there are so many ways you could have started this task.

And I'm going to just go through my thoughts about this task and how I went about it.

So first, I wanted to find the areas and the side length of each of the squares that have been given.

So I looked at the first one, and I found out that the area is five.

I did that by counting the squares, and therefore the side length must be square root of five.

The next one, I had an area of 10 and therefore, the side length must have been square root of 10.

The one after that, the area was 17.

I did this one, by splitting the shape into a square, and four triangles.

You may have done it slightly differently.

This taught me that, the side length must be square root of 17.

What might the next square in this pattern look like? And in order to do this, I started by thinking what shape is it going to be? It's obviously going to be a tilted square.

The squares are getting bigger, so it's going to be bigger than the pink one.

But I thought, how would it look like? Well, for the first one, to get from one vertex to the other, I went up one, and two to the right.

In the second square, I went one up, and three to the right.

Next, I went one up, and four to the right.

So the next square, will start at the same location.

I'll go one up, and how many to the right? Really good.

I'll go one up and five to the right.

And then I said, "How would I describe it?" If I would describe it in terms of the area, I would say it has an area of what? Well, let's look at the numbers.

The area of the first square is five.

The area of the second one is 10, the area of the third one is 17.

So, let me write them down.

5, 10, 17.

What would happen next? What comes next? Well, one of the things that I noticed here, is that all of these numbers, are one more, than the square numbers.

If think about square numbers, what do we have? We have four, 9, 16, extra one will be 25.

To get from four to five, I added one.

From nine to 10, I added one.

From 16 to 17, I added one.

Therefore, the next one from 25, If I add one, it must be? Really good job.

Must be an area of 26 square units for the next square.

Even without drawing it, I was able to say it.

What would be the one after that? What's the next squared number after 25? Very good, 36.

Add one, must be 37.

You can actually go now and draw it, and double check.

You can, list the area of the next to zero, four, and then draw them, to check if this works.

And it will.

Okay, what will be the side length of the next one? Well if it has an area of 26, then the side length must be, the square root of 26.

This is how I went about the Explore task, I wonder how you did it.

This brings us to the end of today's lesson, so huge well done for your effort.

I want to remind you to complete the exit quit, so you can show what you know.

Enjoy the rest of your learning for the day, and I'll see you next lesson.