Lesson video

Hi there and welcome to another lesson with me Dr.

Rim Saada.

In today's lesson, we're going to look at equations of lines.

All you need for this lesson is a pen and paper.

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

The try this for today's lesson is the following.

You've been given a grid, you have x and y-axes given.

The first student says, "My point has y-coordinate three." And another student says, "My point has x-coordinate negative two." Show all the points each student could be describing on this grid.

If you're feeling super confident about this, please pause the video and have a go at it.

If not, don't worry, hold on and I'll give you some support.

Okay, if you want some support, let's have a look at the first student.

The first student is saying, "My point has y-coordinate three." So what point could he be thinking about? He could be thinking about zero, three.

Now x-coordinate is zero, he didn't say anything about the x-coordinate so it really doesn't matter what x-coordinate we go for, but the y-coordinate is three.

And I can go here and plot it.

The second student said, "My point has x-coordinate negative two." So I need to think about a point that has an x-coordinate of negative two.

I'm thinking negative two, zero.

And I can go to the grid and plot it here.

Now, with this hint I want you to think about what other points could the students be thinking of? Write their, coordinates of the points and then put them on the grids.

Pause the video now and have a go.

Okay, and now let's mark and correct the work.

So the first student could have been thinking about a point to the coordinate one, three, two, three, three, three, four, three.

Do you have these? Do you have five, three? What about negative one, three or negative one, sorry, negative two, three, negative three, three, negative four, three, negative five, three.

There are so many points that we can come up with.

And so far, I have only listed points that have coordinates of integers.

Did you think about fronting one with a decimal or a fraction I wonder.

If you did really good.

I could have written something as well like 0.

5, three and so many others.

Okay, so these are all the points that he was thinking about.

Now, looking at the points, how can you describe them? What do they form? Looking at all these points stick to each other, what do they form? Really good, they form a straight line, what kind of straight line? A horizontal line, really good.

Now let's look at the second set of points.

So the second student said, "My point has x-coordinate of negative two." We said it could be negative two, zero.

What else would you have? Really good negative two, one.

What else, negative two, two.

How many points do you have, I wonder.

Do you have as many as me or do you have more? Negative two, three.

Ooh this is a really interesting point, isn't it? This point is a common point that both of them can be thinking of.

Negative two, three, it satisfies this description of students number one, that it has a y-coordinate three, and it satisfies the description of student number two that it has an x-coordinate of negative two.

So this point works for both students.

What other points did you write? Negative two, four really good, negative two, five, negative two, negative one, negative two, negative two, negative two, negative three, and negative two, negative four.

And you could have written negative two, negative five.

Again, you could have done one with a fraction over the decimal like, for example, negative two, negative 4.

5 or 4.

7, and so on.

Really good, now looking at the second set of points, what do they form? Really good, they form a straight line.

What kind of straight line? This time, it's a vertical line, really good description, well done for getting this done.

And let's move on to the connect task.

For our connect task, you have been given a grid.

How could you describe the lines to someone who cannot see them? So you have a grid, it has four lines on it.

What equations describe each line? So, if we look at them, we can see that, okay, we've got straight lines, we can describe them as horizontal or vertical.

But we want to get a bit more mathematical here.

So in order for us to do this, we're going to look at one line at a time and see what can we make of that line.

Let's make a start with the green line.

I want to describe it a bit more I need to know the coordinates of some of the points that are on that line.

So let's take a point.

This point here has the coordinate negative two, two.

Let's take another one.

I have this point.

It has the coordinate of one, two.

Can you, have you started making any of observations? What's common between the two points here so far? Let's choose another one and see if we can make a proper observation about it.

So this point here 3.

5, two.

What's the same or what's different that three points that I chose that are on the green line? Have a little think, really good.

The three points that I chose here, they all have different x-coordinates, but they all have the same y-coordinate.

What's the y-coordinate? It's two, excellent.

So they all have the y-coordinates, two.

If I want to describe this, I can say every point on the green line has a y-coordinate of two.

Now, what equation describes that line? But I'm saying, y-coordinate is two, y-coordinate is two.

How can I write this down algebraically? I can say, the line has an equation y equal two.

Okay, you can see it is a straight line, it's horizontal, it passes through the two, all the points in that line have a y-coordinate of two, therefore that line has an equation of y equal two.

Let's look at the second line.

Let's look at the red one.

And let's start by taking a point on it.

So let's say this point.

Negative three, zero.

Let's take another point and look at the coordinates.

Negative three, one.

Let's take one more point, shall we? This one, negative three, two.

What's the same, and what's different about these three points? The coordinates of these three points, really good.

The coordinates of all of these three points, the x-coordinate is always negative three, the y-coordinate is different.

So if I want to describe them, I can say that every point on the red line has an x-coordinate of negative three, it doesn't matter where I take the point on that line.

Even if I go further if I extend that line above the grid, so it's not just enclosed in the grid, any point on that red line is going to always have the coordinate the x-coordinate of negative three, okay? So now if I want to write down the equation that describes that line, I can say that the line has an equation, x equal negative three.

Really good job.

Now, I want you to have a little think about the blue line for me please.

Choose a point, and write the coordinates down.

Choose another one on that blue line and write it down on your piece of paper.

Are you starting to see something? Can you write the sentence to describe? Describe that line using the sentence, really good.

Now, do you think you can tell me what the equation is? Okay, let's do it together and check if you're right.

So I'm going to choose the point.

My point might be different to yours.

So I went and chose this point and it has a coordinate of four, one.

Another point, coordinate of four, zero.

I'm going to go for a third one, four negative 1/2, what's the same and what's different? Every time I have an x-coordinate of four.

So I can write down every point on the blue line has x-coordinate of four.

Now the equation of the line therefore is? Neither line has an equation x equal four.

So well done if you are correct.

Let's do the same for the orange one.

Perhaps now you can start writing down the description without having to choose the points.

Just imagine if you were going to choose a point on that line, what kind of coordinate would you get? Really good.

Okay, how would you write it down as a sentence? What's the description every point? Really good, excellent.

So every point in the orange line has an x-coordinate of negative one.

Sorry, every point on the orange line has a y-coordinate of negative one.

So every single point there will have y-coordinate of negative one.

So we have one, negative one, two, negative one, and so on.

See I've made mistake, you need to make sure that you avoid those mistakes or you always check by choosing some points.

Okay, so now I can write down that the equation, the line has an equation y equal negative one because every point on that orange line is going to have something negative one as a coordinate.

Really good, well done.

It is time now for you to have a go at this independent task.

You have three questions, please pause the video and answer the questions.

And when you're ready, press play and we will mark and correct the work together off you go.

Let's mark and correct the work.

Question one, give the equations of the four coloured lines in the diagram.

Let's just start with the orange line.

If you look at the orange line, if you pick any point on that line you will always have the coordinate of negative two something.

Negative two, zero, negative two, one negative two, three and so on.

Therefore, every line on that point has an x-coordinate of negative two, and therefore the equation of the line is really good.

X is equal to negative two.

Let's look at the green line.

So if I look at the green line and pick any point on it it will always have the coordinates of one, two, two, two three, two four, two it will always have a y-coordinate of two.

Therefore the equation of the line is, excellent.

It is y equal two.

Now let's have a look at the red line.

Any point in the red line will have what kind of coordinate? One, zero one, negative one, one, negative two, therefore the equation of the line is great job, x is equal to one.

And the last line, what do you have? Really good, if you look at that line at any point, at any given point, if you choose any point, you will always have the coordinate of one, negative three negative one, negative three, negative three, negative three.

So the y is always, the y-coordinate is always negative three, therefore the equation of the line is y equal negative three.

Really good, I wrote down threes, let's correct it.

It should be negative, sorry, I missed that negative sign.

So y is equal to negative three.

Okay, and question number two.

The diagram shows two sides of a rectangle, which enclose 14 squares and can be drawn on the grid on this grid, what equations would form the other two sides? So we have two sides already of the rectangle that have been given to us.

The first side is this one here.

What is the equation of this line? Excellent, it's x equal negative three.

X equal negative three.

And equation of this line here, really good, y equal negative one.

So we already have, two sides of the rectangle.

So we need the other two to finish off that rectangle.

Now, we know that rectangle will enclose 14 squares in order for us to have 14 of these squares inside, what kind of rectangle will I need? Really good, either I need one by 14, or I will need a two by seven.

Now, the question clearly says, it can be drawn on this grid so we can fit it on this grid.

Can I fit the one by 14 on this grid? The answer is no, because this here from I only have from negative three to five, the five is the highest.

So, I'm not going to make 14 squares.

So, I cannot do a one by 14, can I make a two by seven? Really good I can, so, if I want to draw a two by seven, then I need one, two well that's easy.

That's a two by seven.

So I need seven across as one, two, three, four, five, six, seven.

So my next line will be somewhere here.

So let's just start by drawing the line.

So I want to do something to add one at the top here.

So I know I have a rectangle inside that is two.

What's the equation of this line here that I just drew? Very good, the equation of that line is y equal one, 'cause every single point on that line is going to be to have the y-coordinate of one.

Now, we said we wanted two by seven.

So we have our two.

If I got seven across my next line would be here.

And what's the equation of this line? Good job, equation of this line is x equal four, okay? And now inside these four lines, four straight lines, I have inside the rectangle that is a two by seven.

I know it's two here.

I know it's seven there.

I know that by counting the squares, but not only by counting the squares, I know x three, x equal negative three is here, sorry.

And x equal four and from negative three to get to four, I have seven spaces from negative three to zero, zero is my reference point that get three that's the three units and from zero to four that's another three, another four.

So altogether I have seven.

And similarly, up along the y-axis, I went from negative one to one, which is two.

Negative one to zero is one unit from zero to one, that's another unit.

So I made a two by seven.

Really good, now for question three, I'm just going to go to the next slide, because I need some space to show you how to work it out.

Let's read question three together.

A square encloses 16 small squares in the grid.

The left hand side is x equal 11.

And the base is y equal negative five.

What equations would form the other two sides? So let us just start looking at what information have we been given? We know that we need a grid.

So I'm going to start with drawing my x and y-axes.

The question says we have 16 small squares in the grid.

So I'm going to draw a square that has 16 smaller squares inside it so what are the dimensions are going to be of that square? The main one, is going to be four by four to get the 16 small squares, right? The question says one of the sides is x equal 11.

So I'm not doing this accurately, it's a sketch just to help you work things out.

So I'm going to go to my x-axes, find where 11 is, mark it, get a ruler, draw a line and label it with the equation of that line, which is x equal 11.

Now, the base is y equal negative five, so I'm going to go to the y-axis.

I'm going to find where roughly y equal negative five is, use a ruler and draw the line and I'm going to label it with an equation of that line.

So I'm going to write y equal negative five.

So now I know that I have to these other two sides of the square and I want to know where the other two sides are, okay? I want a four by four.

So I'm going to go and say to myself, well, let's look first at the x equal 11.

If I'm here, and I want to go this way I'm going to go across by four because I want to create a four by four, okay? Where would I get two? If I'm at x equal 11.

If I go forward, really good.

I will need a 15, I'm going to take the ruler and draw the line and I'm going to write down the equation of the line x equal 15.

Now, I know that x equal 11, x equal 15 are two sides of the square.

I also know that this side here, y equal negative five is also another side.

So it means I need to cut it off somewhere here because at the moment, I have a four across, so here I have four but here from zero to five, to negative five.

I have five at the moment.

I don't want five I want only four.

So if I want four, where would I draw my line and I want this one to be here to be included.

I want this line to be included.

So I need to go from negative five up four places and that gets me two.

Really good, it will get me two negative one.

And I can now take a ruler and draw a line and the equation of this, and now I have the equation of this line y equal negative one, okay really good.

So what am I going to do next? Excellent, so I have y equal negative one, and here I have a four by four and I've managed to answer the question I have all the four all of the four equations I have, this is one of them.

This is my second equation.

This is my third equation.

And this one here is my fourth equation.

Excellent job well done if you had this correct.

And our explore task if two questions have a go at.

The first one, find the equation four lines that form a square which encloses nine squares.

And I have given you here as part of the grid that will help you answering the question.

For question two, find the equations of four lines that form a rectangle which encloses 12 squares.

And I gave you a little bit of a hint, we'll start by starting that grid and showing you a rectangle there.

Now if you're feeling super confident about this, please pause the video and have a go.

If not, I'll give you a bit of support.

Okay, so for support, I will I want you to think about the first one.

So find the equations of four lines that form a square, so you want to form a square that encloses nine squares, so inside it is nine squares.

Therefore, this square must be a three by three.

Without the x-axes lines are not fixed, we need a set such that the x lines are three units apart and the y lines are also three units apart.

So you choose what your starting point is.

X equals something and you want the other one to be three apart and you choose, y is going to be the other y is three apart.

With this hint, you should be able to have a go.

So pause the video and have a go.

Okay, now let's answer it.

Let's correct it together.

So as I said, the first one, it's entirely up to you how you start it, I'm going to start with this.

I chose x equals one.

I wanted to see apart so the next one will be x equal four.

And for the y, again it's the same thing.

I chose y equal one and y equal four and I made them three apart.

So it's entirely up to you.

Your numbers could be different.

There are so many answers for this one, provided you have the numbers are three apart and three apart.

Now for the second one, we want to make 12 squares.

It's a rectangle, so it's not as easy as the square where the square it's fixed number.

Okay, so 12 squares, I can have a 12 by one, I can have a six by two.

I can have a four by three.

I can have a three by four, a two by six, or a one by 12.

So it's entirely up to you how choose or what you choose to go for, okay? There are so many different ones that you can make, I'm going to look at the six by two, okay? So I'm going to choose x equal one and I'm going leave six places.

So the next line is that x equal seven.

And because I'm going to make a six by two.

If I choose y equal three, I'm going to also choose y equal one.

And they are two apart.

So I've made the 12 squares.

Really, really interesting to see you what numbers you came up with, and how many you managed to draw because you can come up with so many with this question.

This brings us to the end of today's lesson where we learned about equations of straight lines.

We learned how to use them to help us construct or draw squares and rectangles.

So we've done some fantastic learning today.

You should be so proud of yourself.

Please remember to complete the exit quiz.

And that is it from me for today.

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