Lesson video

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

Saada.

In today's lesson, we would be looking at finding the length of a line from tilted squares.

All you need for today's lesson is a pen, a paper, and a ruler.

So go grab these, and when you're ready, let's make a start.

I would like you to start today's lesson by looking at the line segments that I have given to you here on the grid.

List the line segments from the shortest to the longest.

I want you to think about what is it that made you make that decision, and what was your reasoning behind it? If you're feeling confident about making a start, please pause the video and have it go at this.

If not, I'm going to give you a hint in three, two, one.

Okay, so if you want the hint, think about each line segment here.

And think about one end of the line segment and what was the journey that we took to get to that other end of the line segment.

So for example, with this blue one here, how did I get from one end to the other? I went down one and across three.

Can you do the same for the others and use that to help you decide which one is the shortest and which one is the longest? The try in this task should take you roughly five minutes to complete.

So please post a video and complete it to the best of your ability.

Resume once who finished.

Welcome back, how did you find this task? Did you manage to list the line segments from the shortest to the longest? Good job.

Which one was the shortest? And which one was that longest segment? Okay, really good.

Let's go through the solutions for the drivers.

So you can mark and correct your work.

So for green one, I went one across and one down.

For the next one I went two across and one down.

Next one, I went two across and two up.

For the purple one, it was three across and two up.

The blue one we had already done.

It's one down and three across.

And the last one is one across and four up, okay.

So I want you to imagine that we have a robot on one end of the line segment, and we want to send the robot from one end to the other.

Think about the journey, what are we going to tell the robot? So we will have one by one for the green.

Whereas for the second one, for the tilted one we will have two by one.

So then the robot is covering more distance in the second one.

Therefore, this is the shortest, and this is, the second one.

After that, it's this one, number three.

And what was your fourth one? Really good, so the blue one is the fourth, the fifth is the purple and the last one is this one here, okay? Really good.

How did you decide, what was your reasoning for listing them in that particular order? Okay, really good.

So this is mine.

By counting how to get from one end of the line segment to the other.

That's what helped me decide which one was the shortest and which one was the longest? I guess I would like you not to have a look at the grid here, you have been given a line segment.

And the question is saying, find that length of the line shown on the grid using the area of tilted squares.

So if I want to know what the length of the side here without actually measuring it, I will need to draw a square around it, find the area of that square, like we did in the last lesson, and then I can square root the area of the square to find the length of one of the sides of the square.

So this incline here can be one of the sides of the square.

So now if I want to draw a square around this, first, I need to know what did I do to get from one end of the line segment to the other? So I went down one and three across, now I need to think, okay, what do I need to do to get to the next vertex of the square? What I need to do exactly the same thing.

I need to do a one and a three movements, so I can go three up and one across.

And that gives me my second line, and I know where my Vertex for this square is.

Now I always like to draw the two lines adjacent to the line that's already been given to me.

So I'm going to go back to the line segment and look at the other end and draw the next line from there, So I'm going to go here, and say one across and three up.

And that's my next line.

I find this is the easiest way of doing it.

Now I can easily connect the two here.

What's the area of this tilted square? Well, if I split this shape up here, I have the area of the square is going to be the area of the square that is in the middle plus the area of the four triangles.

So I held the square in the middle is two by two that's four, the area of the triangle is 1/2 multiplied by base multiplied by height.

The base is one and the height is three.

So that gives me 1.

5.

I have four triangles, So that's 1.

5 multiplied by four, which is six.

Now I can add them up.

The area the square plus the area of the four triangles, gives me 10 units squared.

So now I know that the area of the pink square is 10 units squared.

Therefore, the length of one side is going to be really good.

It's going to be the square root of 10.

So the square root of 10 units, whatever those units are, and I'm to leave my answer in surd form, it's more precise.

And we've discussed that also in our previous lesson, let's have a look at another example.

So if I have this side here, this line here, and I've wanted to find the length of the line again using filtered squares, I can start by saying how to.

What did I do to get from one end to the other? I went up two and across two.

So I need to do the same thing now from one of the endpoints.

I'm going to do it from here, that's two and then two.

And that tells me whether vertex or the next vertex will be squared.

As at now I can draw a line.

Now, again, I'm going to go to the other end because I prefer to draw two lines connected to the line that has already been given to me.

So I'm going to go two down and two across, and that tells me where the next line is going to be.

Now I can simply join them.

Now I need to find the area of this square.

There are so many ways you can find the area.

You can do it by counting the squares, you can do it by splitting the shape into smaller shapes.

I'm going to split it up like this.

And I have here four equal triangles.

So I'm going to find the area of one triangle, I have four of them.

So the area of one triangle is 1/2 base times height, And the base in here is two and the height is two.

So 1/2 multiplied by two multiplied by two is going to be two units squared.

So I know that I have one triangle, I have four of them.

So I need to multiply by four.

That's two multiplied by four and that's eight units squared.

And this gives me the area of that pink square of the tilted square.

Now I will define the length of the line.

So the line is actually one of those sides of the square, therefore, the length of the line is going to be the square root of eight units.

And again, I would leave it in surd form, it's more precise.

Now it's time for you to have a go at the independent task.

You have key questions which require you to find the length of various lines.

You will need you to use tilted squares and the area of tilted squares in order for you to answer the questions.

Now, if you need a bit of support, I would suggest that you go back to the examples and maybe copy one of them down and see how we've done it, and try and do exactly the same thing.

The independent tasks should take you roughly 10 minutes to complete.

So please pause the video and complete it to the best of your ability.

Resume the video once you finish.

How did you go on with question? Was very similar to the examples we have done together, right? Okay, really good.

So I'll make you go through the answers, you can mark and correct your work.

So the first line here, I started by counting.

What did I do to get from one end to the other end on the line? And I went four across and two down.

Then I drew a square using this information and then calculated the area of this squared.

And the area of that square was 20 units squared, and therefore I knew that the side length is the square root of 20.

Did you get that? Really good.

Next one, I did exactly the same thing and I found that I needed to go one down and four across.

And then I went and I calculated the area, which was 17 units squared and therefore I knew that the side length is the square root of 17 units.

Did you get that too? Excellent, well done.

Let's have a look at question number two.

So with question two, I gave you a set of axes.

I gave you an X and Y axes.

And I gave you a line segment then I asked you to use the line, to draw a square on the grid.

I started by counting, what did I do to get from one end to the other? And that was four across and two up.

So I did the same thing and found the next vertex connected the lines.

So I went up four the style and two to the left, instead of two to the right, because imagine if you go two to the right, you're not going to create a square.

And then I went back to that line segment that was given to me and use that to help me draw the next line.

I went one, two, three, four, one, two, and I drew there.

Next point or where the vertex is connected them.

And now I was able to close that square.

Now I needed to write down the coordinates of the vertices.

The first vertex is already given to me at coordinate of , the second one at , the next one that i did was , and the last one was at.

Did you get that? Did you get a slightly different answer? Because there is another possible solution.

So what I could have done is I could have done this then four down and two to the right to make a square, but in the other direction.

And I could have done the same here, four and then two, and that would give me the second line of the square, and then join them up.

So some of you may have done that first pink squares some of you may have done the one in blue, they're both correct.

If you've done both then brilliant.

You've managed to spot that there's more than one possible answer for this.

Now, what are the coordinates if you constructed the blue one? The first two coordinates are the same, but the next two are different.

So we had five and.

Good job if you did both of them.

Now, regardless of whether you've have done the pink one or the blue, they are both equal.

And if you calculated the area of the square, you would have ended up with 20 units squared.

And therefore this tells us that the line is, or has a length of the square root of 20 units, okay.

Excellent job if you got this correct.

Now let's use what we have learned about finding the side length of tilted squares in order to try and find lengths of different triangles.

So in this explore task, I've given you here a grid and it has four triangles on it.

And I want you to list the side lengths of each triangle, so every single side for every single triangle.

What do you notice? If you're feeling super confident, please pause the video now and have a go at this.

If not, I will be giving a hint in three, in two, and in one.

Okay, so if you will start with a blue one, for example, you need to think about this side first, what's the length of this side? Well, this one is easy.

You can do it by actually counting the squares, then do the same for this side.

Again, you can count one, two.

Now for this one here, you cannot just count.

You need to find the area of the tilted square.

Once you have the area of the tilted square that I already drew around the triangle for you, you can square root the area, just like we did in the previous questions.

Once you've squared root then you know what the length of that slanted side of the triangle is? So with this hint now, you should be good to go.

The explore task should take you about 15 minutes to complete.

So please pause the video and have a go at this.

Resume once you're finished.

How did you get on with the explore task? Okay, really good.

Let's go through some of the answers together and that way you can mark and correct your work with me.

Okay, so the first triangle, if you look at the blue one, these two sides are easy to find.

We can just count the squares, so each of them is two.

To find the slanted side, we will need to find the area of the slanted square here, of the tilted square, sorry.

And the area of the tilted square is eight.

Therefore, this side length here is this square root of eight.

Now square root of eight is in surd form, and I'm going to leave it in surd form because it's more precise.

But roughly, what is square root of eight if we don't have a calculator to calculate it.

Well, we know that the squad or mine is three, so the square root of eight is just below or less than three.

It's just less than three.

It's definitely more than two, right? Because square root of four is two.

So it's closer to three.

So we know that this side is just less than three.

Next one.

If I look at this triangle here, this side is one and this side is two.

To find the slanted side I need to find the area.

The area was five, and therefore this here is the square root of five.

Again, the square from the form is two.

So we know that this side is a bit more than two, right? Because it's the square root of five, so it's a little bit longer than two.

Next one.

This side here is four this side is two, the area is 20.

Did you get that? Excellent job? Now therefore, this side here is equal to square root of 20.

Again, what is the square root of 20 roughly? Well, I know that the square root of 16 is four, so the square root of 20 is going to be bigger, right? The square root of 25 is five, so it's somewhere between four and five.

Let's look at the next one, this idea is three, this is one.

What did you get for the area of that tilted square? Really good.

So the area here is 10 and therefore the side length is square root of 10.

Again, what is roughly the square root of 10? Well, we know that this web nine is three, so it's a bit more than three.

What did you notice? What did you write down? Okay, really good.

So one of the things I noticed is that the slanted square, sorry, the slanted side of the triangle was the longest in each case.

So in each of these cases, when I looked at approximately what does that square root equate to, it was always bigger than the two shorter sides.

So the two other sides of that triangle, and this is why I'm calling them the shorter sides.

So the slanted side in these triangles are always the longest.

This brings us to the end of today's lesson.

Now there are two things that I would like you to do for me now.

The first thing I would like you to have a think about the most important thing that you've learned from today's lesson.

Say it to this screen.

Really good.

And if you want, you can write it down as an inflexion in your book or on the paper that you are doing the working out on.

So you can write down, in today's lesson I learned.

And tell me what you've learned.

And the second thing is I would like to remind you to complete the exit quiz to show what you know.

This is it for me for today, enjoy the rest of your learning for the day, and I'll see you next lesson.

Bye.