Lesson video

Hello, and welcome to another video.

In this lesson, we'll be going through our second lesson on gradient.

Again, I am Mr. Maseko and it's really nice to see you today.

Make sure you have a pen, a pencil, something to write on and a ruler.

So pause the video here and go get those things if you don't have them.

Okay, now that you have those things, let's get on with today's lesson.

Try this activity.

A student says the point E is at the point eight, four.

Now, this is a unit grid.

If the point E is eight, four, what are the other coordinates on the script? Pause the video here and try this now.

Okay, now that you've tried this, let's see what you've come up with.

Well, we know that E is eight, four, that's what our student has said.

So this is the unit grid.

So that's plus one on the x, plus another one on the x, and we've got one up on the y.

So to get from E to F, we're going two left and one up, two in the x direction, and one up in the y direction.

So we've added two to the x, so this has to be 10.

And we've added one to the y, so that's 10, five.

And then point D, well, on point D, that is one less on the X but still one up on the Y, so that is seven, five.

So on that red line, the y ordinate is always five if the point E is eight, four.

And then let's go to the point B, that is two left from seven, five, so we take two away from the x ordinate, so our point is five, five.

Now the point A, we take one from the x ordinate, and then we take three from the y ordinates.

So that is four, five, take away three is? Two plus four, two.

And our last point is that point C.

Well, if we look at it, we go from point B, we go for one and 1/2.

That's about halfway in between that boxes, one and 1/2 in the Y.

So that is and then we go half a box in the x.

So 5.

5 in the x, could be added 1/2 and then 6.

5 in the y could be added 1.

5.

So that point is 5.

5, 6.

5.

So what if we ask, what is the gradient of each of these lines? Think back to what we did when the last lesson.

How did we find the gradient? What did we do? Well, we looked at what happens if we increase the x ordinate by one.

So what happens to the y ordinate? So let's do this for the green line.

So let's pick a coordinate on the green line.

I'm going to pick this coordinates here, which is two, one.

If I increase my x ordinate by one, what is my y ordinate increased by? Well, that's one, two, three.

The next point on the y is three up, if I increase my x ordinate by one, the y ordinate increases by three.

So the gradient of the green line is three.

Let's do the orange line, pick a point on the orange line.

I'm going to pick this point here.

What if we go increase the x ordinate by one, the y ordinate increases by 1/2.

So the gradient of this line is 1/2, the gradient that line is 1/2.

Let's do the purple line.

For the purple line, If we increase the x ordinates by one, the y ordinate doesn't increase by one, it decreases by one.

So the gradient is, for the purple line, the gradient is negative one.

Good, the gradient is negative one.

Now what about the red line? What is the gradient of that red line? What do you think? Well, the y ordinate doesn't increase.

Anytime you increase the x ordinates by one, the y ordinate stays the same.

So the gradient of that horizontal line, the gradient is zero, good.

So get this pen writing, the gradient is zero.

So let's look at naming these graphs.

Well, in order to name these graphs, what we know about them already? We know their gradients, and we think back to last lesson, the gradient of the graph is always the coefficient of x.

So if we take that green line with a gradient of three, we know it will be y is equal to three x.

But is it just y equals three x? Well, let's pick a point on this line, or that point there is two, one.

So three lots of two, what's three lots of two? Or three times two gives us six.

If the graph was y equals three x, that point would not be a two, one, it would be a two six.

So what do we do to six to get one? Well, we do it, take away five.

So that graph is three x take away five.

Now you can check the name of this graph using another point.

So let's say the points four, seven.

Well, let's check it.

What's three times four, three times four, that gives us, what gives us 12? And 12 take away five gives us seven, so that works.

And that would work for all the points.

That green line is y is equal to three x takeaway five.

What about the red line? It's just a horizontal line, we've done this before as well.

The names of horizontal lines, well, horizontal lines, what stays the same on horizontal lines? The y ordinate, and this one is just y is equal to four.

And then if we look at the orange line, well, what's the gradient of the orange line? We know the gradient of the orange line was 1/2.

So like the green line, we can say y is equal to 1/2 of x, or x divided by two.

Y is equal to half of x, or x divided by two.

But it's that it? Well, lets pick a point on this line.

Well, we can look at the point two, one.

X is two, two divided by two gives us one.

Well, that seems to work but let's check it with another point.

Let's take this points six, three.

X is three, six divided by three, what? Six divided by two, that is three.

Well, that was for that point.

Well, let's check another one just to be sure.

Let's take the point 10, five.

We'll take point 10, five.

10 divided by two gives us five.

And then this is the equation or the name of the orange line.

The green line, y is equal to three x take away five, remember the gradient was three.

So the coefficient of x has to be three.

For the red line, it's just y is equal to four because is a horizontal line and the y ordinates stays the same, and it has a gradient of zero.

So we can't write zero x because that's just nothing, so it's just y equals four.

And then the orange line is y is equal to x divided by two.

Or 1/2 of x 'cause the gradient is 1/2.

And the purple line, we know the gradient is negative one.

So we know that its y is equal to negative one x.

Well, let's see what we do.

Let's pick a point, the point is one, eight.

Well, if we do negative one x, or negative one, how do we go from negative one to eight? Well, we add nine.

So that'll be negative one x add nine.

Now it's not usual you see this written like this, this would usually be written as y is equal to nine takeaway x.

And you can see that the gradient is negative one.

Remember, when there's a one in front of a letter in algebra, we don't write it on.

So just 'cause you can't see it there, there would be a one there.

It'll be written as y is equal to nine, take away one x.

But because we are good mathematicians, we know that we don't need to write this one.

So those are the names of these graphs.

And you can see that gradient is included in the names of these graphs.

And these names, the names of this graphs we call them the equations of the lines.

So the names of graphs are called the equations of the lines.

And that is how we're going to refer to them from now on.

So what did we notice about the gradient and the equation? Well, let's take that line y equals three x minus five, the gradient, we knew we worked this out, the gradient was equal to three.

What do you notice about the gradient of the equation? The gradient is always the coefficient of x.

Can we check this? Well, we can look at the orange line, which was y is equal to 1/2 of x.

But what's the coefficient of x there? Well, the coefficient of x here is 1/2 and the gradient was equal to 1/2.

The gradient was equal to 1/2.

So what I want you to do now is to try this independent task on your own.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you come up with.

A line of the equation y equals four x minus one, what's the gradient.

It's four, how do you know? Because the coefficient of x is four.

A line of the equation y equals nine take away five x, what's the gradient? Is it five? No, it's not five, it's negative five.

'Cause the coefficient of x is negative five.

And we want to draw two lines with a gradient of two.

Well, how do we know a line has a gradient of two? When we move up one in the x ordinates, we go up two in the y ordinates.

So if we pick any points, we pick any starting points, we move up one in the x, and we move up two in the y, and that gives us a line with a gradient of two.

See up one in the x, up two in the y.

So that gives us a line with a gradient of two.

You can do this from anywhere on the graph.

So you just, let me pick this points, move up one in the x, up two, up one in the x, up two, up one in the x, up two in the y, and that line also has a gradient of two.

So there are two lines with gradients of two.

Really well done if you've got something that looks like this.

Now what's special about those two lines? They have the same gradient and if you look at them, they're moving in the same direction.

They're not getting closer to each other, sort of like train tracks.

So what can we say about those lines? The special word we can use in maths for them, these lines are parallel.

Good, those lines are parallel.

And parallel lines always have the same gradient, which is why they never meet.

'Cause they have the same gradient.

Now let's look at these lines.

Now, I want you to try this explore task, organise these linear equations into groups that have the same gradients.

And can you create another linear equation to go with each group? Pause the video here and try this now.

Okay, let's see what you come up with.

Well, I'm going to start with equation A, equation A has a gradient of three.

How do you know, because the coefficient of x is three.

Any the other line with a gradient of three? Well, that would be E, which has the coefficient of x being three.

So we have A and E would've come together.

Let's look at equation C.

Well, equation C you've got x/2.

So what's the coefficient of x when is x divided by two.

That is 1/2, because this is the 1/2 of x.

So this can also be written as 1/2 of x plus 14, so we know the gradient is 1/2.

So what other lines have a gradient of 1/2? Well, that would be D, D also has a gradient of 1/2.

And that's really important.

You can either see this as written as x divided by two or 1/2 multiplied by x, that is the same thing, that means the same thing.

So we have C and D, those have the same gradients.

Let's look at equation B.

We are looking for cross with a coefficient of x that is one, there's no number in front of the x.

So we know the coefficient is one.

So let's look around.

Well, the one that springs to me is J.

It looks like the coefficient of x is one.

But the gradient of that line isn't one.

And we'll find out why in a later lesson, the gradient of this line will be negative one, because this has to be written as y equals five take away x.

And we'll find out more about why we write as y is equal to five takeaway x in order to find the gradient at a later lesson.

But this gradient would actually be negative one.

And that's the same gradient as G.

So if you look at this, same gradient, so G and J can get grouped together.

Here we have y equals two x 'cause x plus x is two x any lines with the gradient of two x? Well, that is H, F and H go together.

And if you look at this, B is on its own and then it doesn't have a pair.

'Cause that's the only one the gradient of one.

So let's pick another line that has a gradient of one.

we could have said y is equal to x plus four.

Now B also has a pair.

So when you're trying to figure out lines to go in other groups, just make sure the line that you've picked has the same coefficient as that group, the same coefficient for x as that group.

Now, if you want to share your answers, ask your parent or carer to share your work with Oak tagging @OakNational and #Learnwithoak.

Thank you very much for participating in today's lesson.

I will see you again next time.

Bye for now.