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.

Welcome to our lesson on rate of change from a coordinate pair.

By the end of this lesson, you'll be able to calculate the rate of change, or the gradient from two coordinate pairs.

A couple of keywords we're gonna use today.

The first one is parallel.

Two lines are parallel if they are straight lines that are always the same, non-zero, distance apart.

Crucially though, we're gonna have a look at this idea of gradient.

If you haven't looked at gradient already, then I do suggest that you pause the video and take some time just to check you know how to calculate the gradient of a line before carrying on with today's lesson.

So what the gradient is, is a measure of how steep a line is and it's calculated by finding the change in the y-direction with respect to the positive x-direction.

And that's gonna be really crucial for what we're looking at today.

Our lesson is gonna be split into three parts and we're gonna start with reading the axes accurately to calculate the gradient.

We can calculate the gradient of a line if we know how much y changes when x increases by one.

So here's our graph.

Notice that there are no labels on our axes.

And Jacob says the gradient could be one.

Lucas says the gradient could be two, and Sofia says the gradient could be a half.

In fact, all three of them could be correct.

I wonder if you could see how.

I'd like you to pause the video.

Have a think.

What could the scales be on each axis to make each pupil correct? Okay, so to make Jacob correct, the scales on the x and y-axis have to be the same.

So they could both be increasing by one each time.

They could both be going up in steps of five.

As long as the x and the y-axis had the same scale, then that would show a gradient of one.

Lucas would be correct if the scale on the y-axis was double the scale on the x-axis.

For example, if the x-axis had a step of 0.

5 and the y-axis had a step of one, then the gradient would be two.

The example I'm gonna show you, the x-axis has a step of one, but the y-axis has a step of two.

You might wanna pause and just check you agree that that has a gradient of two.

And finally, Sofia, she would be correct if the scale on the y-axis was half the scale on the x-axis.

So, for example, if the x-axis had a step of one and the y-axis had a step of 0.

5, or this example here where the x-axis has a step of two and the y-axis has a step of one.

That now has a gradient of a half.

The important bit being that we need to know the scales on our graphs if we are using them to find the gradient.

There are other pieces of information that we can be given, which would help us find the gradient without the scales on the axis, and that's what we're gonna look at now.

If we know a coordinate on the line, we can work out the scale and therefore work out the gradient.

So we've got the coordinate here, one, six.

We know then that the x-axis must be increasing by one and the y-axis by two, so that that coordinate is now one, six.

Once you've got that information, we can calculate the gradient.

So you've got an increase of one in the positive x direction, an increase of six in the positive y direction, so our gradient must be six.

Here's another set of axes.

This time we've got the coordinate two, negative 10.

Let's see if we can work out the scales.

So this time, the x-axis needs to have a step of 0.

5 because it has to get to two in four squares.

The y-axis must have a step of negative two.

We can use this now to calculate the gradient.

If we go across two in the positive x direction and down 10 in the negative y direction, then we can divide those both by two to get our gradient.

So when x increases by one, y decreases by five and we have a gradient of negative five.

What we've shown here then is if we have a line that goes through the origin and we know one of the other coordinates, we can work out the gradient from there.

We don't actually need the axes if we know the origin and one other coordinate.

This time, we've got two coordinates on our line, but we also still don't know the scale.

So let's have a go at calculating the gradient.

Looking at the coordinates, I've got one, two, and two, four.

That means that horizontally they have a distance of one, so that's an increase of one in the positive x direction.

And then if we look at the y coordinates, that's an increase of two in the y direction.

That means that without knowing the scale, we can see that they have a gradient of two, or that line has a gradient of two.

Let's see if you can try this yourself.

So A, B, and C have been plotted on this set of axes.

A is the origin, so zero, zero, B is one, three, and C is three, six.

Can you calculate the gradient of the purple line? That's the one that goes through A and B.

And then the green line, that's the one that goes through A and C.

So you are using the origin and one other point to work out the gradient.

Give it a go.

Let's look at it together then.

So to get from A to B, if we just look at the coordinates, don't worry about the squares.

From zero zero to one, three, x has increased by one.

From zero, zero to one, three, y has increased by three.

That means our gradient must be three.

Let's do the same with our green one.

So don't worry about can to the squares.

Just look at the coordinates.

We've gone from zero, zero to three, six.

That means x has increased by three and y has increased by six.

Putting that into a ratio table, that gives us a gradient of two.

Well done if you've got both of those.

If you're not sure, pause the video, have a look at those, and maybe rewatch the explanation so that you're super confident.

Let's put this into practise.

So the points A, B, C, and D have been plotted on the same axes.

You can see that we don't have any numbers.

They're just on the same set of axes.

What I want to know is, which is steeper, the line that goes from A to B or C to D? And then you've got two other questions about the steepness of the lines.

And then d, can you move that coordinate D to make the shape a parallelogram? Thinking about the fact that parallelograms have parallel sides.

Give that one a go and then we'll look at the next set.

So this time I'd like you to calculate the gradient of each line.

Each line goes through the origin, so you're given that it goes to the origin and then you've got one other coordinate.

Can you use that to work out the gradient? Off you go.

Well done.

Last set to look through.

So for this section, you've been given two coordinates on the line.

And again, I'd like you to find the gradient.

Feel free to use a ratio table or draw the lines on the diagram to help you get that change in x and change in y.

Off you go.

Let's look at our answers then.

So A to B is steeper than C to D because it goes right two and up four, whereas C to D goes right three and up three, so essentially a grading of one, just looking at the squares.

A to C is steeper than B to D.

Again, it goes right two, down three, whereas B to D goes right three, down four.

And lastly, A to B is steeper than A to C.

They both go right two.

There's a bigger jump to get to B than there is to get down to C.

Good spot if you noticed that the corner of D needs to move one square left and one square up to make a parallelogram.

It'd marked on the grid for you to see.

So our gradients then, so A, we've gone right one, up five.

So a gradient of five.

B, we've gone right two, up four.

So that means for every one we've gone right, we've gone up two, gradient of two for B.

C, we've gone right six but down 30.

So that means for each one that we've gone right, we've gone down five, which gives us a gradient of negative five.

And finally, we've gone right 30 from the origin and down 10, that means we get a gradient of negative 1/3.

Well done if you got that one.

For this set, it might have been useful to notice that the x values are only increasing by one.

So from two to three, we've got a change of one in the positive x direction.

So we just need to look at the change in y.

So from eight to 12 is four, so a gradient of four.

In F, from 18 to 20, that's two.

So a gradient of two.

In G, from negative 10 to negative 15, that's a decrease of five.

So our gradient is negative five.

And H, we've gone from negative four to negative 24.

So our gradient is negative 20.

And we could use that method because the x values had only increased by one each time.

So reading the change in the y values gave us our gradient.

We're gonna use those skills now then to calculate gradients from coordinate pairs.

So Sofia says, "If I draw a line through the coordinates three, four and six, one, the gradient of the line will be negative." How could we see if Sophia is correct? Pause the video and have a think.

Do you think she's correct? How could we check? What we can do is we can have a go at plotting them roughly.

It doesn't have to be exactly and seeing where they are in relation to each other.

So there's three, four roughly, it doesn't need to be exact.

Six one must be further to the right 'cause it's got a higher x coordinate, but it must be lower 'cause it's got a smaller y coordinate.

So roughly there.

Sketching a line through them shows us that this line would have a negative gradient.

If we draw a line through the coordinates, negative five, four and negative eight, negative two, what do you think the gradients gonna be in terms of positive or negative? Pause the video, have a think, and then see if you're right.

Let's have a look.

So negative five, four, it needs to be left of the y-axis within an x coordinate negative five and four, and then negative eight, negative two.

So it needs to be further to the left and it needs to be further down.

So roughly there, negative eight, negative two.

This time the gradient is going to be positive.

So it's really useful to be able to have a rough idea of where coordinates are going to go.

So we can have an idea if our gradients are gonna be positive or negative.

So Sofia has plotted these coordinates.

I would like you to work out which coordinate matches with each point on the graph.

Once you've done that, she draws a line through two of the coordinates to make a negative gradient.

Which two coordinates do you think it could have been? Pause the video, have a good think, and then we'll look at it together.

Well done.

There's lots of coordinates to try and match up there.

So A must be negative five, one, B must be negative seven, three, C must be negative two, four, D, negative three, negative five, and E, negative six, negative one.

They don't have to be exactly in the right place, they just have to be in the right place in relation to each other.

Lots of answers for the second part.

So you could have had A and D, that would've had a negative gradient.

B and A, B and D, and B and E would all have had negative gradients.

And then D and E would also have a negative gradient if you drew a line between it.

The reason we've reviewed our sketching skills is that we can use this to help us calculate the gradient of a line if we know two points on the line.

So we're gonna have a look at working out the gradient of the line which passes through seven, 10 and four, one.

We don't need a graph, but we're gonna sketch them roughly on a graph just to give us a bit of an idea of what it's gonna look.

So we are expecting the gradient to be positive.

Now we can look at the actual coordinates to see if we can work out the exact gradient.

So from four to seven we have an increase of three in the x direction.

From one to 10 we have an increase of nine in the y direction.

Really important now to be using these ratio tables so we don't get in a muddle.

If we divide both values by three, that'll tell us the change in y for one increase in x.

So the gradient must be three.

And then just look at your graph and see if that makes sense.

Lucas is gonna give it go.

The gradient of the line which goes through negative four, five and negative seven, nine is 1.

3.

We're gonna have a look at his working, then I want you to think about, or think about as we go through, is he correct? Right, he has made at least one error there.

Can you spot where he is gone wrong? Pause the video and have a look again.

So firstly, he's plot is coordinates wrong to start with.

They should be that way around.

Negative seven, nine should be higher than negative four, five, and that means straight away, we can see that we should end up with a negative gradient.

Now, if we look at it, we've got plus three in the positive x direction, minus four in the y direction.

So let's change that in our table.

And this last bit was a little bit more subtle.

I wonder if you spotted it.

Negative four divided by three isn't exactly negative 1.

3.

It might have been better if Lucas had left his answer as a fraction.

So I would write this gradient as negative four over three, and then that's an exact answer.

We haven't had to round it off because it's a recurring decimal.

We can actually calculate the gradient of the line without sketching the points as long as we know two coordinates.

So we're gonna work out the gradient of line joining the coordinates three, negative two and negative seven, three.

Now, remember, gradient is looking at when x increases by one.

So I generally prefer to rewrite the coordinates so it has the smallest x value first.

So I'm gonna rewrite them.

So negative seven, three is first and then three, negative two is second.

Then what I want to know is how I got from negative seven to three, what's that change in x? So that's three subtract negative seven, which gives me 10.

Just have a look at your values.

It is an increase of 10 from negative seven to three.

Let's do the same with our y values.

So negative two subtract three is negative five.

So from three to negative two, we've subtracted a five.

Now I can use my ratio table to give me a gradient.

So divide both values by 10.

So when x increases by one, y decreases by a half.

So our gradient is negative a half.

What we're gonna do now is I'm gonna give one a go.

I'd like you to watch carefully what I've done and then you're gonna try similar one on your own.

So calculate the gradient of the line through negative 10, negative nine and negative six, seven.

Now, I do advise sketching the points.

So I've done that to start with, making sure they're in the correct place in relation to each other.

I know that my gradients gonna be positive, so that's gonna help me check my answer.

If I look at the change in the x coordinates, from negative 10 to negative six, that's an increase of four.

From negative nine to seven, that's an increase of 16.

Putting that in my ratio table, I can see that when x increases by one, y increases by four.

So my gradient is four and I'm expecting it to be positive, which it is.

Time for you to have a go.

Feel free to sketch the points to help you.

Use my scaffolding on the left hand side.

Fantastic.

So negative eight, five is probably in your top left hand corner, or near enough.

Two, negative five is gonna be towards that bottom right hand corner.

We can see then that the gradient is gonna be negative before doing any calculations.

To get from negative eight to two, that's an increase of 10.

To get from five to negative five, that's a decrease of 10.

So the change of x is 10, the change in y is negative 10.

That gives me a gradient then of negative one.

We said our gradient was gonna be negative, so that's looking good.

Time for you to have a practise.

So I'd like you to find the gradient of the line joining the two coordinates given.

I recommend using the axes to roughly plot your points and there's space underneath to use a ratio table if you wish.

Give it a go.

Well done.

So in this second set, I'd like you to decide which line would be steeper.

So, for example, for A, you've got the line joining zero, zero and five, seven, or the line joining zero, zero and five, negative four.

Try to think of a justification or a reason for your answer.

Fantastic.

Let's have a look at our working then.

For the first one, I would've written the coordinates the other way around.

If you've plotted them up, that makes it a little bit easier.

So the x values have increased by three to get from four to seven, and the y values have increased by six to get from two to eight.

Putting that into our ratio table, we get a gradient of two.

For the second one, the x values have increased by two.

The y values have decreased by six.

Again, putting it into our ratio table, we can see the gradient is negative three.

For C, you should have a gradient of six, x increases by two, y increases by 12, and D, you should have a gradient of negative three over two.

You might wanna rewrite them so that the negative five tens on the left, then we've got an increase of eight in the x coordinates and a decrease of 12 in the y coordinates.

So we've got negative 12 divided by eight, and that does simplify down to negative three over two using equivalent fractions.

So, firstly, the line joining zero, zero and five, seven will be steeper.

The change in x is the same for both pairs, so from zero to five in both pairs.

But the change in y is larger for that first pair.

So from zero to seven rather than from zero to negative four.

For question two, the line joining zero, zero and three, four will be steeper.

If they have the same gradient, the second coordinate would be six, eight in that second pair.

So it'd be zero, zero and six, eight would get you the same gradient.

Being it's only six, seven, that one won't be as steep.

For question c, notice that there's an increase of one in the x coordinates, but three in the y coordinates.

But for the second set, there's a change of one in the x coordinates, but four in the y coordinates.

So that must be steeper.

And finally, the line joining negative four, four and five, negative four is steeper because there is a change of nine in the x coordinates, which is the same as the second pair, but there's a decrease of eight for the first pair, whereas there's only a decrease of seven for the second pair.

There's a lot of thinking going on in that activity.

So well done if you wrapped your head around that idea of steepness.

We're gonna have a look at this final part of the lesson.

So we're gonna look at coordinates that lie on a straight line and using some ideas of gradient to see if we can tell whether coordinates are on a straight line.

The gradient between any two points on a straight line is the gradient of that line.

So here's a line and Jacob says, "I'm gonna calculate the gradient between the points zero, negative four and one, negative two." Just have a look at the graph and see where zero, negative four is and one negative, two is.

He does that by finding the change in x.

So that's an increase of one.

And then look at the change in y.

That is an increase of two.

So the gradient between those two coordinates is two.

Lucas says, "I'm gonna calculate the gradient between the point two, zero and four, four." Let's find 'em on our graph.

There's two, zero, there's four, four.

This time his change in x is two, whereas his change in y is four.

Using a ratio table, we know that our gradient is going to be two.

The gradient was the same for both Jacob and Lucas 'cause they both picked points which were on the line.

True or false, you can pick any two points on a line to calculate the gradient of the line? What do you think? Well done.

That is true.

Jacob and Lucas just showed us that.

Which one is the correct justification? Perfect, it's the fact that the gradient is the same everywhere on the line.

We can use this fact to see whether three coordinates are on the same line.

Let's try this one.

Are one, one, two, four, and five, 13 on the same line? Start by finding the gradient between the first two points.

So our gradient is three.

Then let's check the gradients between the second two points.

Our change in x is three and our change in y is nine.

So our gradient, if we divide those both by three, our gradient is three.

The gradient is the same between the first two coordinates and the second two coordinates, so they must be on the same straight line.

Remember that points that lie on a straight line will follow the same linear relationship.

You probably had a go at plotting some linear relationships before where you've come up with coordinates that fit a rule.

Well, if those three coordinates are on the same line, they will fit a rule.

Most of the time they're quite tricky to spot, which is why we have other methods for finding out the rule.

However, it's a good way of checking whether we've got a correct rule.

So these coordinates, for example, fit the rule y equals three x minus two.

Right, Sofia says, "I want to see if zero, negative five, six, negative two and 14, two are on the same straight line." She's gonna find the gradient between zero, negative five and six, negative two and zero, negative five and 14, two.

Will that work? Pause the video and have a think.

You might wanna sketch them out.

Yeah, that will work, as long as one of the coordinates is the same in both calculations.

So both calculations, we've used the coordinate zero, negative five.

Let's have a go at using Sophia's method then to see if they do lie on the same straight line.

So you have a gradient of a half between the first two and we have a gradient of a half between the first and the last.

All three points lie then on a straight line.

The rule, if you're interested, is y equals a half x minus five, but you don't need to be able to work that out yet.

Lucas then says the gradient between one, two and two, four is two.

He says the gradient between six, six and seven, eight is also two.

These four points must lie on a straight line.

What do you think about Lucas' statement? Okay, Lucas is incorrect.

He's correct about the gradients.

They both have a gradient of two.

However, lots of points have a gradient of two between them.

It doesn't mean they're on a straight line.

It might mean they're on a parallel line, but not the same straight line.

Let's see if we can show this is incorrect.

So we could pick a different pair and test the gradient between them.

If they're on the same straight line, the gradient should be the same between all of the coordinates.

So let's try one, two and seven, eight.

So we have a change of six in the x direction and six in the y direction.

The gradient linking those then would be one, whereas the gradient linking the other two pairs is gonna be two.

You might have also noticed the rule, one, two and two, four both follow the rule y equals two x.

That's not true for the other coordinates.

So the other coordinates could lie on a straight line with a gradient of two, but it's not the same one as one, two and two, four.

For which options can be sure the coordinates lie on the same line? Off you go, well done if you spotted it's the top and the bottom.

So the line between A and B has the same gradient as B and C.

All three of those would then form a straight line.

If A and B has the same gradient as C and D, we don't know that they form a straight line.

If you draw lines between A and B and C and D, they'll be parallel, but they're not the same straight line.

The line between B and C has the same gradient as the line between A and C, then yes, they'll all be on the same straight line because C is in both of those calculations.

Final practise then.

By testing the gradient, see whether each set of three points lie on a straight line.

Feel free to sketch your points on an axis and use ratio tables to help you with your gradients.

So, the first one, no, the gradient between one, two and two, six is four, but between two, six and four, 12 is three.

B, yes, the gradient between any two points is one.

C, yes, the gradient between any two points is negative two.

and d, no, the gradient between negative five, zero and negative three, negative four was negative two, but the gradient between negative five, zero and three, negative eight is negative one.

You might have chosen different sets of coordinates to try, that's absolutely fine, but you should have the same answers.

There's a lot of thinking in that lesson today.

We've really stretched our understanding of gradient.

So well done.

We have looked at how the gradient of the line could be calculated given two coordinates on that line.

We know that sketching can help us get an idea of whether it's supposed to be positive or negative.

Technically, we don't have to sketch them.

We can work out the gradient just from two coordinates.

But the sketching did help us avoid mistakes.

And that last bit that we investigated, three points A, B, and C will be on the same straight line if the gradient between A and B is the same as B and C.

So a little bit of problem solving we're bringing in there at the end.

Fantastic work today, guys.

I look forward to seeing you again.