Lesson video

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

Sada.

In today's lesson, we would be learning about tilted squares.

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

So please go and grab these.

And when you're ready, let's make a start.

We're learning about tilted squares, this lesson, so we'll need to start by actually making some squares.

So your first task is to look at this grid here.

You have been given a three by three grid, and I want you to think about all the different ways that you can join up four vertices on this grid to make a square.

So how many ways can you do it? I want you to draw all of them.

Once you're done, I want you to calculate the area of each square.

This task should take you about five to seven minutes to complete.

Please pause the video and complete the, try this task to the best of your ability.

Resume once you're finished.

Welcome back.

How did you get on with this task? How many squares did you manage to draw? There are lots of possible answers.

You may have had a square like this.

I've joined here, the vertices of some of the squares with that already on the grid to make a squared where you may have something like this, or even a bigger square like this, or even a bigger one, but on top.

Okay.

So there are so many different possible answers for this one.

And all three, you finish to complete the area for the squares that you created.

Now we have a three by three grid.

Inside the grid we have a tilted square.

I want you to think about if I ask you, can you calculate the area of the tilted square? Can you think about a method for calculating the area there.

Have a little think and say it to the screen, if you already thought of one method.

Can you think of another? Okay.

Really good.

Well, there are some clues on the screen here for you.

So let's look at the methods that these three students are coming up with and see if you thought of similar methods to them.

The first student here says I counted the squares.

So if you go to that tilted square, you can see one square right in the middle and around it you can see four triangles.

What each triangle, if you take one piece from one place and replace it, that connected to the other piece, it forms a square.

Let me show you what I mean.

So there's a square here and you have those four triangles.

If you take this piece here and move it there, that will form a square.

If you move this here and put it here, it makes a square.

If you don't move this from here.

So we do that.

It forms a square.

And this piece to here, it forms a square, and you ended up with five squares.

Did you think of this method? It's a really good one, isn't it Let's look at the second student and what he said.

He said, I found the area of the triangles and the small square and small squares.

So I asked that is very similar to the first student, but you used a bit more of a method for the area for it.

So had the template squared divided it into smaller shapes.

So he ended up with one square right in the middle and calculated the area of that.

It's it's one because it's a one by one, then calculated the area of each triangle to calculate the area of the triangle.

It's half base times height.

The base is one and the height is two.

So the area of a triangle is half times, one times two, which is just one.

We have four triangles.

So that gives us an area of four plus the area of the square, which is one that gives us a total of five units a square, which is exactly the same answer that the first student got by counting the squares.

Now let's look at the third method.

I calculated the area of the four triangles between the big and the small square.

I subtracted that from the area of the bigger square.

What does this mean? Let's have a look at the grids.

The third student started by calculating, the area of the big grid, which is a three by three.

So that area for that is nine then said, well, you know what? If I take away, if I subtract these four triangles, imagine I remove those four triangles that I've just numbered.

I'll be left with just the tilted square.

So I can say that it's the area of the tilted squared is the area of the grid, subtract those four triangles.

So the area of the square, the bigger square is nine.

The area of each triangle is half times base times height.

It's the same half times, two times one.

And that gives us one.

There are four of those triangles.

So that's an area of four units squared.

Now nine subtract that four we'll get five squared units, a square Now this confirms that we had the same answer with three different methods.

So it doesn't really matter which method you use, as long as you are being really careful and doing your counting properly or the calculation properly, Which method do you prefer? Okay.

And which method do you think is the most efficient? Any idea? Okay.

So it really depends because if I have a really small tilted square, I could probably just do it by counting.

But if I have a really big one, it will take me very long time to count.

So it's about thinking about which method you feel most comfortable with and which method is more appropriate for the question that you are trying to answer.

And now it's time for you to do the independent task.

What are you going to practise finding the area of tilted squares, just like we have discussed in the previous question, there are so many different methods that you can use.

So you choose the method that you feel most comfortable with.

Okay.

And the method that you prefer.

If you really want to challenge yourself, use more than one method.

That way you're challenging yourself.

And you're checking your answer, because if you use more than one method and end up with the same answer and your answer is more likely to be correct, The independent task should take you about 10 minutes.

Please pause the video and complete the two questions.

Resume once you've finished, How did you get on with the independent task? Hopefully you've managed to use more than one method just to double check that your answers are correct.

Really good.

Let's have a look at the solutions to question one.

So for question one, you needed to find the area of the tilted squares in each of those grids.

The first one was really easy.

All I had to do is I could have done it by counting.

I split it.

So each two small triangles make one square.

So I have two units a square.

The second one, how did you do it? Okay, good job.

I split it this way and combined to find the squares again.

I added and that gave me five units squared.

Next one.

Okay.

Very good.

So I here calculated the area of the smaller shapes that were created.

I looked at the square that I have inside and it's a two by two.

So it has an area of four units squared.

And then I looked at the triangles half times base times height.

The base was one and the height was three.

So that gave 1.

5.

I had four of them.

So 1.

5 plus another 1.

5 plus another 1.

5 plus another 1.

5 or multiply by four that gave six and did that to the area of the square.

And that gave 10 units squared.

Did you get that? Good job.

Next one, I did something similar.

So I split the shape up.

And then in here I had a three by three.

So I knew I had an area of nine.

I needed to calculate the area of the triangles.

So the area of each triangle was half base times height.

Now the base in here was one and the height was four.

So half of four is two.

So I knew that the area of one triangle is four, but I had the four of them.

So four times two is eight.

And now if I add them up the eight plus the nine gives me 17 units squared.

Did you get that? Good job And you may have done yours slightly differently or used a different method.

And that's fine.

You should have ended up with exactly the same answers as me.

Now for the second one said, the question said, continue the pattern and draw the next tilted square inside the grid.

Find the area of the tilted square that you drew.

Okay.

So let's look at the pattern.

What what's happening in, in the square, in the tilted squares.

In question one, I started from one vertex to another and I wanted to see what was the journey.

I went one up and one across.

Then I went forward the second one.

I went one up and two across.

And that's just me getting from one vertex to the other.

Then I went to the second one to see if there's a pattern.

I went up one and three across.

So what do you think is going to happen for the next one to get from one vertex to the other.

What am I doing? Really good, one up and four across.

So one and four across.

And that gets me to that second vertex.

So now if I go to this grid here and I wanted to make a start in exactly the same location.

So let's say, I'm going to start here.

What do I need to do to get to this second vertex on that square? So if this is the first vertex, what do I need to do? Up one and across five.

Across one across two, across three, across four, and now across five.

So that's the pattern and the pattern for that as well.

If you see, if you look at the number of the grids, you look at the grid.

We had a two by two, three by three, four by four, five by five.

And this one is a six by six, So the pattern is going to continue.

So I can start here and say what, there we go.

This is where my second one's going to be.

Okay.

And you should, you should have really used the border to, to connect those two vertices.

Now, how do I get to the next vertex? It's the same gap.

Isn't it? The same distance between them and that distance is going to be measured by going up down, across or to the right or to the left.

So I need to do the same journey.

So I need to go one and four.

Sorry.

One and then five.

So one across and then five down.

And that gets me to the next vertex.

Now I can use the ruler and connect them.

Obviously, mine here is not with a ruler because I'm trying to do it to show you, but you should be doing this, using a ruler on your piece of paper.

Now to get from that vertex to the next one, again, we need to go one, one down and four, this time four to the left bottom.

All right.

So it will one, two, three, four, and then we go, now we can just connect these here.

And we have our tilted square.

Now, did you calculate the area for this one? When did you get as your answer? Okay, really good.

I also looked here and I really wanted to look for patterns here for the area.

So I thought I'll give the first one.

I had a two by two and I didn't, it didn't form a square, like in the middle with triangles around it.

They were not square, but we had the triangles around, the small triangles.

In a two by two.

I had a one by one square formed.

In a three by three, I had the two by two square formed in the middle.

In a five by five, I had three by three square being formed.

So this one here, next one up, what square should I be forming inside it? Even without having to actually cut it down.

It should form a four by four square, right? Which gives us 16 units.

And if you drew that, you would know that this is correct.

And then we can look at the area of the triangles around it.

Now in the previous ones, what did we get? The area of the triangle was half base.

The base was always one.

And the height changed in the three by three grid.

The base was two.

In the four by four, The base was three.

In the five by five, the base was four.

So now in this one, six by six, the base is going to be? Really good.

So now the area of the triangles need to be half base times five, half one times five, which is 2.

5, but I have four triangles.

So 2.

5 multiplied by four is 10.

And now I can add them up 10 plus the 16 is 20, 26 units as square.

And I remember I did it without actually this time, because I could see the pattern.

I didn't need to cut the shape.

But you can do this.

And you would find that we have one, two, three, four.

So it's a four by four, as I said, and it's correct.

Well done if you had this correct, really good job, Now it's time for us to look at the area of tilted squares at a greater depth.

Let's read the question together.

Zaki uses a five by five grid and makes some squares by moving the vertex around the edge of the grid.

It's actually a four by four.

Sorry, one mistake.

You can see what he has done here, started with the following square and then started making the square different sizes and just moving it around.

I want you to calculate the area of each of those squares here.

So the green square and how many different areas have been made? So using this method of just starting initially with the full on square, and then moving, rotating around how many different areas that he make.

Can you repeat this with some larger grids? If you have some spare paper at home, draw grids, maybe do a six by six, seven by seven, draw five by five.

And tell me, what do you notice? What kind of squares are you making? Are you making more or less squares? And what kind of areas are you coming up with with bigger grids? You need to spend at least 10 minutes on the explore task.

In particular, on the second part, where you're going to draw your own tilted squares on a five by five grid, six by six grid, seven by seven grid, and start looking for patterns in the number of squares that you're making and the areas that you're creating.

So please pause the video and complete the task.

Resume once you're finished.

Welcome back.

How did you get on with this task? Really good.

Let's mark and correct the work.

So for the first part, you had different squares that you needed to calculate the area for For the first one.

It's the easiest as four by four.

So the area was 16 units squared.

The second one was a tilted square.

You could have used various methods to find that the area of that tilted square, as we discussed earlier in the lesson.

We have done that correctly.

Your answer should be ? Good job if you had 10 units squared.

What about the next one? When did you get? Really good? So eight units squared and next one, 10 units squared.

So even though it was tilted or rotated differently inside the bigger grid, you can see that the triangles, the four triangles that we have outside are identical to that second shape that we have.

So it had the same area of 10 units squared And the last one we just went back to 16 units squared.

So how many different areas are made? So we have here a four by four grid and in the four by four grid, we managed to make three different areas.

We made the 16, we made 10 and we made eight units.

Did you repeat this for larger grids? Did you try it for four by five, six by six? What about seven by seven? There are so many things when you do explore.

If you actually draw the grids and start drawing the squares.

And it's quite interesting to start making observations about the things that are happening and what patterns can you spot in there.

Now, when I did it I noticed that the larger the square grid, the more squares which can be made.

And one of the other things that I noticed that every time the grid increased by two units, the number of the squares created increased by one.

So for a four by four here, we had, how many squares did we create? Okay.

We created three different areas.

So when, when not five by five, but six by six, we managed to create one more.

So it was four.

Okay.

So there are so many things that you could have noticed.

I would love to know what you wrote down and what observations you've made.

You have done something amazing learning today, looking at tilted squares and the areas and using different methods to calculate the area of tilted squares.

If you would like to show your work with Oak National, please ask your parents or carer to share your work on Twitter, tagging @OakNational and hashtag #LearnwithOak.

I would love to see some of your work, in particular your answers to the explore task.

This is it from me, for today's lesson.

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

Bye.