Lesson video

Welcome to today's lesson.

My name's Ms. Davies.

I just want to start by saying well done for choosing to learn using this video.

I hope you find lots of interesting bits as we explore this lesson.

Let's get started.

In this lesson we're gonna be looking at the negative rate of change of a graph.

If you haven't already had a look at positive rates of change on a graph, that might be something you want to do prior to looking at this lesson.

By the end of the lesson, you'll be able to calculate the negative rate of change, and we're gonna use this word gradient from a graph.

So some keywords that you might be familiar with, but we need to have a really clear idea of for today's lesson.

The most important one is this idea of a gradient.

So the gradient is a measure of how steep a line is.

It is calculated by finding the rate of change in the y-direction with respect to the positive x-direction.

Couple of other things that we're going to use today.

We're gonna use the idea of an absolute value.

If you haven't come across it before an absolute value of a number is its distance from zero.

So for example, the number five and the number negative five are both five away from zero so they both have an absolute value of five.

Negative three would have an absolute value of three.

And lastly, this idea of parallel.

So two lines are parallel if they are straight lines that are always the same distance apart.

So this lesson's gonna be broken down into two parts.

Firstly, we're gonna look at calculating the negative gradient from graph, and then we're gonna have a look at calculating the gradient from any graph.

So we'll look at some positive and some negative gradients coming in at the end.

So to recap, a gradient describes the change in y for an increase of one in the positive x-direction.

Izzy's had to go at finding the gradient of this line.

You can see her working out to the left.

And she says the gradient of this line is two.

Is she correct? What do you reckon? Right, it's quite possible that Izzy has only done positive gradients so far.

So let's see how this one differs.

Where she's gone wrong is she's measured one in the negative x-direction, not the positive x-direction.

Whenever we measure gradient, we go one right and then see how to get back to our line.

Whereas Izzy has moved one left and that's where she's got in a muddle.

You might be able to see why that has happened.

So this is what Izzy's thinking.

If I move one in the positive x-direction, my vertical line never reaches my graph.

What do you think Izzy needs to do? Well done if you've spotted that she needs to move down this time.

Once she's moved one in the positive x-direction, her vertical line needs to be drawn downwards.

And see that now we've annotated our graph, we've moved one right, so positive one in the x-direction, but then we've moved down two, so negative two in the y-direction.

And that's how these negative gradients are going to differ from positive gradients.

So the gradient at this line is negative two.

Y decreases as the graph moves to the right so that means the gradient is negative.

Let's have a look at the gradient of this line.

So if we move one step in the positive x-direction, we're always moving one step right and then we draw our line downwards this time, we get a gradient of negative three.

Let's see what this is gonna look like in a table of values.

So this is a table of values where x increases by one each time.

So negative one, zero, one, two, three.

And then our y-values increase by four.

Because our y-values increase this is gonna be a positive gradient.

For linear relationships which form graphs with negative gradients, the y-values will decrease as the x-values increase.

So again, we need to make sure that we have an x step of one, but this time we can see our y-values are going down by three each time.

As the value of x increases by one, the value of y decreases by three.

So the gradient is negative three.

Right, let's see if we're confident with this idea of negative gradients then.

So you've got A, B, C, and D.

Which of them have negative gradients? Off you go.

Well done if you spotted that it is B, the blue graph and C, the purple graph that have the negative gradients.

A and D, so that pink and green graph have positive gradients, they're increasing as x increases.

Let's try the same with a table of values.

So which of these tables of values will form graphs with a negative gradient when plotted? Pause the video, give it a go.

Well done if you spotted that it's B and C.

The values of y are decreasing as x is increasing.

Make sure that you're confident with that one before you move on.

So let's see where some of the difficulties can come with negative gradient.

So Andeep has tried to measure the gradient of this line.

Well watch his working out and then we'll see what we think.

Have a look at what he's done.

What has he done well, given some successes and where has he made a mistake? Can you suggest what he can do to get the correct answer? Off you go.

Right, he's actually done loads of things really well.

So he's definitely chosen to move in the positive x-direction.

He knows that y is decreasing.

You can see that he's written negative one and he's got a negative gradient.

He's also been really clever and spotted the scale on the x-axis.

Can you see how he's written plus four even though it's only gone along two big squares, you spotted that the x-axis is going up in twos.

So that is an overall movement of four in the x-direction.

He's even filled his table in correctly.

So we've got a change in x, four, change in y, negative one.

That is correct.

Did you spot where he went wrong? It's that last step.

What he's supposed to be doing is making the change in x one 'cause that's how gradient is defined.

So what he needs to do is he needs to divide both sides of his ratio table by four to work out what the gradient is.

He should then get an answer of change in x one, change in y negative a quarter.

Well done if you spotted that mistake and corrected it for Andeep.

Perfect, so Andeep's taken your advice and he now knows the gradient is negative a quarter.

Well done.

So what we're gonna do now is I'm gonna show you an example then I would like you to have a go at one yourself.

Watch what I do carefully and then use the same concept when you try one.

So we're gonna calculate the gradient of this line.

I've noticed that going across one doesn't put me on a nice integer coordinate.

So I've drawn a line that goes right two.

I've drawn a line that goes down negative three and you can see that I've met my line again.

I'm gonna put that in a ratio table.

So change in x is two, change in y is negative three.

And I know to get the gradient I need to work that to one.

So if I divide both sides by two, then change in x is one and change in y is negative three over two.

There's no need to change it to a decimal.

Fractions are a lot easier to work with most of the time.

So my gradient is negative three over two.

What I'd like you to have a go at doing now is calculating the gradient of this line.

See how you get on and then we'll work through it together.

Lovely.

Let's look at what we've done.

I'm gonna show you the answer first and then I'll talk through it.

So just watch.

Did you get the gradient, negative two thirds? Hopefully you did.

So we've seen that we've moved right three, down two to get to our line again, placing it into our ratio table.

We've got change in x is three, change in y is negative two.

And then converting that to a gradient.

We know that x needs to be one, so I need to divide by three.

Negative two divided by three is negative two thirds.

Fantastic.

Let's have a look at this one.

So this time I'm paying particular attention to my scales.

So that's a movement of plus two.

Looking at my y-axis, that's a movement of negative eight.

Putting it into my ratio table, I need to divide by two to get negative four.

So my gradient is negative four.

Have a go at this one yourself.

Well done.

So looking at the scale, I've used these two points.

So I've used 0, 50 and I've gone right two squares, that's a movement of 10.

And then I've got down three squares, that's a movement of negative 30 looking at the y-axis.

Putting it into my ratio table and then dividing by 10 I get a gradient of negative three.

Well done if you got that one.

Time to have a practise then.

So for each one, find the gradient of the line, think about drawing on your arrows to help you and then there's space to use a ratio table if you wish.

Off you go.

Okay, second set, same idea.

Please do feel free to draw yourself out a ratio table to help particularly with some of the trickier ones.

Off you go.

Lovely.

Let's have a look at our answers.

So the first one, you should have negative three.

The second one, negative one.

The third one, negative four.

And the fourth one, negative two.

I didn't feel I needed a ratio table for those ones because they all had an x step of one and a y step of one, so I could just use the squares to count.

I did feel like I needed ratio tables for this one.

So the first one I've moved right four and down one to get back to my line.

So that is a gradient of negative a quarter.

Well done if you remembered the negative.

Well done if you've got your fraction correct.

F, I've moved right five to go down three.

Do take your time with these.

Check what happens when you move right one.

If you don't get back to an integer, try moving right two, then moving right three, then moving right four and eventually you'll find a movement, which gets you back to an integer point.

So this time I had to move right five to go down three to get back to an integer point.

Converting that to a gradient then I've got negative three divided by five or negative three fifths.

Right G, paying attention to my scales.

So if I look at my y-axis, I've got 0, 10.

And then if I go across three big squares, I can go down two big squares.

So that's a movement of 10, 20, 30 in the x-direction, negative five, negative 10 in the y-direction.

Divided by 30 then, so I've got one and then negative a third.

So my gradient is negative a third.

Well done if you've got that one, there's lots of bits that could have caught you out.

And H, again, you're gonna want to look at your scales.

So from negative 20 to negative 10, that's plus 10 in the x-direction.

And then from one to zero, that's minus one in the y-direction.

So I've got 10 on negative one.

So if I divide those both by 10, I get one and negative a 10th.

Well done if you've got all or most of those right.

You are now an expert at finding negative gradients.

So now that we're an expert in finding negative gradients, we're gonna have a look at the gradient from any graph.

So knowing the gradient of a graph can give us information about what the graph looks like.

So four pupils have had a go at drawing a graph on the axis below.

You've got A, B, C, and D.

Jun says, "The gradient of my graph is negative one." Which graph must be Jun's? Well done if you did that without having to do any working out.

There's only one graph with a negative gradient and it is that purple one labelled B.

So Jun is just gonna sum up the difference between a positive and a negative gradient.

So this is what he says.

"Lines with positive gradients increase as the graph moves to the right.

Lines with negative gradients decrease as the graph moves to the right." I hope we agree so far.

Aisha says, "So I don't need to worry about the negative numbers or the positive numbers.

I can calculate the gradient using the absolute values and decide if it's positive or negative from the shape of the graph." You might want to read that again.

What do you think Aisha means by that? And do you agree with what she's trying to do? Pause the video and have a think.

So what Aisha means, this idea of an absolute value is its distance from zero.

So three and negative three have the same absolute value.

They have an absolute value of three.

Five and negative five have an absolute value of five.

Okay, so it's essentially the value not considering its direction from zero, just using the positive direction.

So what Aisha is saying is she could find the change in X and the change in Y and not worry about whether she's going left or right, whether it's positive or negative.

She can just look at the values.

She can find the gradient from the positive values and then she can use the graph to decide if it's increasing or decreasing.

And then she knows if it's supposed to be positive or negative.

Let's have a look at where that works.

So we've got one and three.

And remember what I said? Aisha was saying she's not gonna worry about which direction she's going in.

She's gonna sort that out at the end.

So she's got a change in x of one and a change in y is three.

But she knows that the gradient's negative 'cause it's going down as we're moving right.

So she knows that's gonna have to be negative three rather than three.

Aisha's method does work.

And knowing whether your gradient is supposed to be positive or negative can be really, really useful and it helps you check your answers.

I wonder if you can see any difficulties with Aisha's method.

What do you think? So Aisha's method does produce the correct answer, but remember when we're working mathematically, it's not just about getting the right answer, it's about our working out and somebody being able to follow our arguments and also making sure we don't make any mistakes as our maths develops and as our questions get more complex.

So a couple of things that could happen if she uses this method.

She might get her x and her y-directions confused.

Okay, so you know we've got a one and a three written on our graph.

She might write her gradient as a third rather than three 'cause she might get them the wrong way round.

What may also happen is that once she's done her hard work, she might forget that the gradient's supposed to be negative and then the problem with that is none of her working out supports her argument that it's gonna be a negative gradient.

Lastly, this might not work as well if she's not given the graph.

So you might look at some questions later on where you're finding the gradient just from coordinates.

And then without having the graph there, it won't be easy for her to see whether it is increasing or decreasing and get the positive and the negative the right way round.

Okay, let's look at some more things we can tell from the gradients.

So as well as knowing whether it's increasing or decreasing, knowing the gradient can tell us other features of the graph.

Have a look at A, B, C, or D.

Which one do you think is the steepest? So pink is the steepest, it has a gradient of five, then B, the blue one has a gradient of three, then D, the green one has a gradient of two and then C, the purple one has a gradient of one.

What do you notice about the gradients of the steeper lines? Good spot if you noticed that the steeper lines had larger gradients.

We're gonna investigate that now with negative gradients.

Which is the steepest this time? Pause and have a look.

Lovely, so there's our gradient in order from steepest at the top is C, the pink one with a gradient of negative five down to the least steep, which is B, the blue one, which has a gradient of negative a half.

With negative gradients I wonder if you noticed that the steeper lines have smaller values for the gradient.

Can you see that C, negative five is actually a smaller number than negative two, isn't it? But it's a steeper line.

So just bear that in mind when you're thinking about steepness.

When you've got negative gradients, smaller numbers, so negative five for example, smaller than negative two, but the gradient is steeper.

What do you think is steeper? A graph with gradient one or a graph with gradient negative one? Pause the video and come up with your answer.

Well done if you noticed that they have the same steepness.

They are equally steep and they are reflections of each other.

If you calculate the absolute value of the gradient, then the larger the absolute value, the steeper the gradient.

So essentially what we're saying is if you ignore whether it's positive or negative and just look at the absolute value, the bigger the absolute value, the steeper the gradient.

So graph of gradient negative two would look steeper than a graph with gradient of one.

Let's put that to the test.

So here are some gradients.

Can you put them in order of steepness? Start with the steepest.

Off you go.

So the steepest, gradient six, then two, then one, then 0.

5.

Bit more of a challenge now.

Which ones do you think is steepest this time? Again, put them in order.

Well done if you thought about looking at the absolute values.

Even though it's the smallest number there, the steepest gradient is negative five, then four, then negative three, then two.

Last one, have a go at putting these in order of steepness starting with the steepest.

Right, some fraction work there.

If you're not sure about the size of the fractions, you might have wanted to convert them over a common denominator to help you.

Well done if you spotted that negative one is going to be the steepest.

All the other values are between negative one and one.

Then negative two thirds is the next steepest.

Three fifths is the next steepest, negative a half is the last or so the least steep.

Well done.

That was a tricky one.

Lots of skills you're bringing together there.

Lastly, we're gonna have a look at this idea of parallel.

So lines are parallel if they are always the same distance apart.

So looking at this graph A and C, so the purple and the green are parallel.

They are always the same distance apart.

The gradient of both lines is two.

So both A and C goes right one, up two, right one, up two.

And because of that they remain parallel for the entirety of the line.

B has a gradient of one, that one's not parallel to A and C.

So Izzy says, "A line with gradient three and a line with gradient negative three have equal steepness.

Will they be parallel?" Well, let's have a look.

So there's a gradient of three and a gradient of negative three.

She's absolutely correct.

They are equally steep but they are not parallel.

One has a positive gradient and one has a negative gradient.

In fact, you can see that they cross quite clearly in the middle.

Parallel lines will have exactly the same gradient.

So what we can do is we can calculate a gradient to see if lines would be parallel if plotted on the same axes.

So let's have a look at these three.

So that one has a gradient of negative two.

This one also has a gradient of negative two, even though it's been zoomed out so the squares are smaller.

And this right one is gonna be a little bit trickier to work out the gradient.

So I've gone right five squares, which is 10 in the x-direction and then down four squares, which is negative 20 in the y-direction.

Pay particular attention to your scales.

That also has a gradient of negative two.

All three lines then are parallel, but they're just drawn on differently scaled axes.

If I put them all on the same set of axes, they would look parallel.

What about these lines? Let's see if these are parallel.

Okay, I'd like you to pause the video.

Have a look at the working out.

Do you think those lines are parallel? Well done if you spotted they are not parallel.

They have different gradients.

One goes right four, up three, one goes right three, up four.

Even though it's not clear because the axes have been scaled, by using our gradient method to work out the gradient, we can see they are not the same.

True or false then, any two straight lines will intersect, that just means cross, somewhere on a coordinate grid? What do you think? Well done if you've spotted that is false.

I'd like you to read the justifications and decide why.

Okay, so that first one, the intersection point could be a really large value and that doesn't fit on the axis.

That's not the point we're getting at here.

The point we're getting at is that lines with the same gradient will not intersect.

So most lines will intersect somewhere and it's right that that might be off the graph, that might be at a really large value or really small value.

However, not all lines will intersect.

The lines are parallel and they have the same gradient, they will not intersect.

What about this one? A line with a positive gradient cannot be parallel to a line with a negative gradient.

What do you think? Well done, that one is true.

What is the reasoning this time? Is it that positive gradients are steeper than negative gradients? Or that lines have to have the exact same gradient to be parallel? It is that second one.

Positive gradients are not necessarily steeper than negative gradients.

We know it's the absolute value which determines the steepness.

Lines do have to have the exact same gradient to be parallel.

Negative three and three that we looked at earlier are not parallel.

They're going in opposite directions.

Time to have a practise then.

So you've got four lines, A, B, C, and D and four gradients.

Can you match each line to its gradient? You'll notice that there's no numbers on the axes.

Okay, that's because you're using the features of the graph to work out which one is which.

So each of those four, A, B, C, and D on the graph match with one of the gradients, A, B, C, and D to the right.

Give that one a go.

Well done.

So this time I'd like you to work out which lines are parallel.

They're all drawn on slightly different axes.

So you're gonna need to calculate the gradients and match any parallel lines.

Okay, give that one a go and then we'll look at the last question.

Finally then, Andeep is trying to prove that this shape is a parallelogram.

He started it and what we need to do is finish off his working.

So he says opposite sides in a parallelogram are parallel.

Therefore they must have the same gradient when drawn on a graph.

What he's done so far is he's shown that the left and the right hand sides are parallel because they both go right one, down two.

So they both have a gradient of negative two.

See if you can finish off to show that the entire shape is a parallelogram.

Well done.

Let's have a look at our answers.

The gradient of B, the green line is one.

The gradient of A, that blue line is three.

The gradient of D, that purple line is negative two.

And the gradient of C, that pink line is negative a half.

You need to be thinking about your steepness in order to help you with those.

So parallel lines then.

A had a gradient of negative a half.

B had a gradient of a half.

C, negative 2.

D, negative a half.

E two and F two.

So lines that would be parallel if plotted on the same axis are A and D and then E and F.

And finally helping Andeep out.

So he's shown us that the left and the right hand sides are parallel.

So we need to show that the other two sides are parallel.

So for the top we have to go right three and up two, and for the bottom we have to go right three and up two.

So I've shown it on a diagram, but I've also added it into a ratio table.

So the change in x is three and the change in y is two.

So it has a gradient of two thirds and that's the same for the top and the bottom.

If they're parallel and the other two sides are parallel, we have a parallelogram.

Well done then guys, we are now experts in calculating positive and negative gradients and all those things that tells us about the graphs, loads of skills used in that lesson.

Just to recap then, if y decreases as x increases, the gradient is negative.

Negative gradients can be calculated by finding the amount y decreases when x increases by one.

We've seen how graphs with the same gradient are parallel and also how the absolute value of the gradient lets us compare steepness.

Well done and I really look forward to seeing you again.