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Hello and welcome to another video.

In this lesson we'll be talking about the gradient.

I am Mr. Maseko.

For today's lesson make sure you have a pen, a pencil, and something to write on, a ruler will also be really useful.

So if you don't have those things, please get them now.

Okay, now that you have your things, let's begin today's lesson.

So we first start with our try this activity, so can you fill in the gaps using the instructions that the students are giving you? Pause the video here and give this a go.

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

Well, for the first student who says, "In my graph, the y-ordinate is double the X." So what's double zero? Well that is zero.

Double one, two, that's four, six, eight, that's 10.

Well the next student says, "In my graph, the y-ordinate is triple." Well triple zero is still zero, three, six, nine, 12, 15.

Okay and the third student says, "The y-ordinate is five times the X." Zero, five, 10, 15, 20, 25.

So, what is the same and what's different in these graphs? Well the first thing that's the same is all of them go through the point zero, zero.

What's different? Well the first graph goes up in twos, the y-ordinates go up in twos, the second graph, the y-ordinate goes up in threes, and the third graph, the y-ordinates go up in fives, okay.

What else can we say? Well you could also say the gaps between the y-ordinates for all the graphs is consistent, the gaps between the y-ordinates for all the graphs are consistent and now that should be ringing some bells in your lessons if you think back to what we talked about in the previous lesson.

Name each of the students' graphs.

Naming graphs, that was two lessons ago.

The y-ordinate is double the X so how can we write that? So we could say the y-ordinate is equal to two times the x-ordinate 'cause it's double the x-ordinate.

And the second one is the y-ordinate is equal to three X.

And then this one is the y-ordinate is equal to five lots of X.

Good, well done.

So let's connect this together.

So here we have three lines and these lines represent the students' graphs.

Pause the video here and see if you can spot which line belongs to which student.

Okay, so what have you come up with? Well, if you look at the coordinates that are plotted, on the purple line there's a coordinate one, two, two, four, three, six, four, eight, five, 10, so what's that? Good, that is Y is equal to two X.

And then on the green line, you have one, three, two, six, three, nine, and that is Y is equal to three X.

And the orange line, one, five, two, 10 and that's Y is equal to five X.

See what we had before? All these lines go through the point zero, zero, okay.

What's happening with these lines? What can you see? When you go from two X to three X to five X, what's changing on these lines? What is changing on these lines from the line two X to the line three X to the line five X, what's changing? Yes, it's the steepness, the steepness of the line is changing, really well done.

So, in linear graphs, the coefficient, that is the number that's multiplying the X, so the coefficient of X determines how steep the line is, this is referred to as the gradient.

The coefficient determines how steep and that steepness we call the gradient.

So if you look at this, spot something, on the purple line when that x-ordinate increases by one, the y-ordinate increases by two.

Hmm, okay.

When the x-ordinate increases by one, the y-ordinate increases by two and that's consistent throughout that entire line Y equals two X.

When the x-ordinate increases by one, the y-ordinate increases by two, the line is Y is equal to two X.

Are you making the connections? What did we say? The coefficient of X determines the steepness and that steepness we call the gradient.

So we know that the gradient of this line is two 'cause we know from the coefficient of X the gradient is two.

Can you make that same connection with the other lines? Let's look at that orange graph and fill in the gaps on this statement.

On the orange graph, every time we move up one in the x-direction, we move up blank in the y-direction, so this graph has a gradient of we fill that in.

So every time we move up one in the x-direction, we move up, we can check, we move up one in the x-direction and we move up one, two, three, four, five in the y-direction, so this graph has a gradient of, got it, five.

How else could we have shown that this graph has a gradient of five? By looking at the coefficient of X 'cause it's Y is equal to five X, so that coefficient of X tells us the steepness which is the gradient.

The green line, we should be able to do this without looking at the graph now.

On the green line, every time we move up one in the x-direction, we move up three in the y-direction.

Now how do we know this? Because the coefficient of X is three so the gradient is three and we can check this on our graph, every time we move up one in the x-direction, we move up three in the y-direction for the line Y equals three X.

Okay, you're making some really important connections here, so let's see if we can apply this to a different-looking problem.

So this student asks, "What would happen if every time I move up one "in the x-direction "we move down two in the y-direction?" Well, we can try this, let's plot some points.

Let's start with the point zero, zero, we move up one in the x-direction and we move down two in the Y so we're at a point here.

So do it again, one in the X, then down two, we have a point there.

One in the X, down two, we'll point here, then if we draw that straight line, what do you notice when you draw that straight line? What's different between that straight line and the other straight lines? You see, all the other straight lines tended upwards, they all looked like this, but this straight line is tending downwards.

For the other straight line, we moved up in the X and we moved up in the Y, but for this one we moved up in the X, we moved down in the Y.

So what would the gradient of this line be? We moved up one and we moved down two, the gradient of this line would be, so the gradient would be equal to negative two because every time we move one up in the x-direction, we move down two in the y-direction, so the gradient is negative two.

Pause the video here and give this a go.

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

So we have plot the following lines on the coordinate grid and state the gradients.

Well line one has the coordinates negative three, three, so negative three, negative three and one, one, well that's one, one and the point four, four.

Once you've got your points, you're going to join them with a straight line using a ruler, so a straight line using a ruler.

Then let's determine the gradient of this line.

Well we can pick any point on this line, so let's do this point, one, one.

When we move up one point in the X, we move up one in the Y.

We move up one in the X, we move up one in the Y, so this line has a gradient and has a gradient of one 'cause we move up one in the X and we move up one in the Y, so the gradient is one.

Really well done if you got that.

Now let's do the second line, let's do that second line.

Well that second line has points at negative one, negative four, so negative one, negative four and then we have zero, negative one and then we have the point two, five.

And then again what we do is we join these with straight lines.

Now normally when you plot straight lines, remember always use Xs, for the first line I used points and that shouldn't be the case, any time you're plotting straight lines, always use small Xs where your points are.

Okay, and then we now want to determine the gradient.

Well, let's pick this point here, when we move up one in the X, we move one, two, three up in the Y, we move up one in the X, we move three up one in the Y so the gradient is, good, the gradient of this line, the gradient is three.

I'm going to leave this line on there.

Let's do the gradient of line three, first we plot it, we have negative two, five, we have zero, one, and then we have two, negative three and then we join this up with a straight line.

Okay, so what happens now? Well if we start from this point which is negative two, five, this time when we move up one in the X, we move down two in the Y, up one in the X, we move down two in the Y.

Up one and down two in the Y, that means the gradient is, good, the gradient is negative two, the gradient is negative two because we move up one in the X, we move down two in the Y.

So we have a linear graph, has a gradient of three and goes through three, seven and four, Y.

The gradient is three, if you look from three to four, the x-ordinate has increased by one.

So if the gradient is three, what does the y-ordinate go up by? Good, the y-ordinate has to go up by three, so seven add three, Y would be, so Y would be 10.

Really well done if you got this.

You have made some excellent progress.

So here's an explore task, find the gradient of each of these lines.

So use the same principle we've just been going through, find the gradient of each of these lines and list some linear graphs that would have the same gradient as one of these graphs.

So pause the video here and give that a go.

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

Let's start with the orange line.

Well let's pick a point on the orange line, we'll pick this point which is two, zero.

When we move up one in the X, we move one, two, three, four up in the Y, so the gradient of the orange line, the gradient of that orange line is four, good.

Okay let's do the green line.

Well for that green line let's pick this point, we move up one in the X and then we move, one, two, three up in the Y, so the gradient for this one would be three.

And then let's do the red line.

Well for the red line I'm going to pick this point, we move up one in the X and then in the Y we move up by 1/2, that's halfway between six and seven, we move up one in the X and then in the Y we move up by 1/2 so the gradient of that line is equal to 1/2.

It is possible to have fractional gradients.

And now let's do the purple line, I'm just going to pick this point here.

Well we move up one in the X, we move down one in the Y, up one in the X, we move down one in the Y, we've moved down so the gradient is going to be negative, so this gradient is negative, but negative what? How many spaces have we moved down? We moved down one which means the gradient is equal to negative one.

List some linear graphs that would have the same gradient as one of these graphs.

Well, if we have a gradient of three, we want any graph where the coefficient of X is three, so you could have a graph that is Y is equal to three X plus one.

Remember the coefficient of X determines the gradient, so any graph where the coefficient of X is three would have the same gradient as the green line and you can do the same thing for the orange line, the purple line, and the red line.

If you want to share your work, ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Thank you very much for participating in today's lesson, I hope to see you again next time, bye for now.