Lesson video

Hello and welcome to this lesson on comparing gradients.

I am Mr. Maseko.

Before you begin this lesson, make sure you have a pen, a pencil, a ruler, and something to write on.

Okay.

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

First, let's try this activity.

Move a vertex on this triangle to make a right-angled triangle.

Now if you look, the first student says you can move point B to the point two, three.

So if we take point B, and move it to the point two, three, that would make a right-angled triangle here.

So that's where point B now is.

So you moved it to two, three, and that makes a right-angled triangle there.

So, that same way move one of the points on that triangle to make right-angled triangles.

See how many points you can come up with.

See how many different right-angled triangles you can make.

Pause the video here and give this a go.

Okay, now that you've tried this, let's explore some of the answers that you may have come up with.

We'll focus on point A first.

Well, the easiest right-angled triangles to make are the ones where we have a horizontal and a vertical line making a right angle.

So we can take point A, and move it to either one of those orange Xs.

And we'll have a horizontal and a vertical line making a right angle.

So here, if point A moves there, we have a horizontal and vertical line making a right angle.

And there's a right-angled triangle.

It's the same thing if point A moves here.

We have a horizontal, and a vertical line making a right angle.

The same thing can be said for point B.

But point B would move to one of the purple Xs to make a right-angled triangle with a horizontal and vertical line.

So instead of moving point A to one of the orange Xs, you can move it to anywhere on the orange line.

But let's see this.

Let me take, so any one of those orange lines.

Let me take this point on the orange line.

If we move point A there, we would have this right-angled triangle.

See that line BC stays the same.

And then that's where our new point A is.

So, we have this right-angled triangle, with a right angle at point B.

Well, let's explore this.

Why would that be a right angle? Well, let's see if I get rid of that drawing.

Let's look at this line, BC.

And this orange line.

Let's explore the gradients of those lines.

We'll pick a point a on BC, so I'll pick this point.

When we move up one in the X, the Y ordinate increases by two.

So the gradient of BC, we'll write it here.

The gradient of BC, of the line BC, is equal to two.

Well now let's look at the gradient of that orange line.

Let's pick that point.

When we go up one on the X ordinate, the Y ordinate goes down by a half.

You've got a half a box down.

So the gradient of the orange line, well that is equal to, we've gone down by a half.

So, it is equal to negative a half.

The gradient of that orange line.

The gradient of BC is two.

The gradient of the orange line, that it makes a right angle with, is negative a half.

That's interesting.

Now this are the orange line is parallel to the other orange line.

And two lines that are parallel always have the same gradient.

Good.

Well, let's look at point B.

Well, point B can move to anywhere on the purple line, and you'd make a right-angled triangle.

So if we take point B, and put it on the purple line, let's say here, you would have this, with a right angle at point A.

Or if you moved it onto the second purple line, so if you moved it here, you would have.

The line AC is still there.

We have a right angle at point C.

We know the gradients of those purple lines are the same, 'cause they're parallel.

But what is the gradient of the purple line? Well, the gradient is the purple line, pick a point on the purple line.

When we increase the X ordinate by one, the Y ordinate increases by one.

So the gradient of the purple line is equal to one, because when the X ordinate increases by one, the Y ordinate increases by one.

So, if you think back to the previous thing we did for the orange line, and the line it made a right angle with.

When the gradient of the orange line was.

So, if you think back to the orange line and the line it made a right angle with, the gradient of the line BC was two.

The gradient of the orange line was negative a half.

And then here, the gradient of the purple line is one.

What would the gradient of the line it makes a right angle with be? So that line AC.

So the gradients of AC.

Well, if you look, when we go pick a point on AC, and I will pick this point.

We go up one on the X, and then the Y ordinate decreases by one.

The gradient of AC is equal to negative one.

So you got two lines that make a right angle.

One gradient is positive and the other is negative.

And that's the same thing that happened previously.

Okay.

It is a coincidence.

Well, let's explore further.

Let's see.

Let's look at point C where we can move point C to see if this little conjecture we've made, is correct? Well, let's look at point C.

Well point C, well, that can move to anywhere on the blue line.

So, anywhere on that blue line, we can move point C.

And our right angle will between the line AB, and that blue line.

There's our right angle.

And there's point C.

That's where we've moved it.

Well, the gradient of the line AB, the gradient of AB.

What is the gradient of AB? 'Cause we're just exploring with gradients.

What's the gradient of AB? Well, let's work it out.

Pick a point on AB.

So any points, let's pick point A.

When we go up one on the X, we go up by a half on the Y.

So the gradient is equal to a half.

Okay.

What would the gradient of the blue line be? That AB makes a right angle with.

What conjecture have we already made? Well, we can assume that it's going to be negative.

So the gradient of the blue line will be negative, but negative what? Well, will it be negative? Well, let's see.

Well on the blue line, if we start from point B, we go up one on the X, and we go down two on the Y.

So that gradient is negative two.

So just, there is a pattern happening here and I want you to see whether you can spot the pattern.

And if I go back, we'll just look at them.

These two perpendicular lines, BC and the orange line, the gradient of BC was two.

The gradient of the orange was negative a half.

The gradient of the purple line was one.

The gradient of AC was negative one.

The gradient of AB was a half.

The gradient of the blue line was negative two.

So, it looks like that pattern we spotted so far where the gradients of perpendicular lines, one is positive and the other one is negative.

But there's something more going on, and let's explore further and see if we can figure it out.

Here's an independent task for you guys to try.

Pause the video if you don't want a clue.

But if you want to clue, keep watching.

Okay.

So, we're looking at the gradients of the line segments.

So, we want the line LK.

So that's just going from point L to point K.

And you just draw the straight lines.

So using a ruler draw a straight line from L to K.

So what is the gradient of that line? Well, when you go up one on the X, you go up one on the Y.

Remember, that horizontal axis is always X, and the vertical direction is always Y.

It doesn't matter that this is not a normal coordinate grid that you're used to.

Vertically, we always call that Y.

Horizontally, we always call that X.

So now pause the video here and give that a go.

Now, what you should have come up with are these gradients.

The gradient of the segment LK is one.

The gradient of the segment LAV is negative one.

A14 is negative a half.

LF is two.

Five K is a half.

N17 is a quarter.

I17 is negative four.

Now, give a line segment that will be parallel to LK.

So parallel to LK.

We want one that moves.

So, what's the gradient of LK? It's one.

So we want a line that goes, when it goes one up in X, it goes one up in the Y.

So we could go something like this.

So from P, so P13.

That would be parallel to LK.

Because the gradient of P13, you go up one in the X, you go up one in the Y.

That's also one.

Now, question three is a bit trickier.

Which was perpendicular? And you could have just done this by inspecting your lines, you would have seen that if you look at I17 and N17, will they make a right angle? And you can see that 'cause you've drawn the lines.

If you look at A14, and LF.

Look at A14 and look at LF.

Well A14 and LF, at the point they meet they also make a right angle.

So these lines, these are perpendicular.

The same can also be said for LK and AV.

When they meet, they would also make it right angle.

Those would also be perpendicular.

Now look at their gradients and see if you can spot something.

Spotting patterns is what we're trying to do.

Just exploring with gradients.

And that's the same thing you're going to be doing in that explore task.

For each of those segments, work out their gradients.

Pause the video here and give this a go.

So, these are the gradients you should have come up with.

Look at the perpendicular lines and look at their gradients.

What do you notice? What we've already made the conjecture that perpendicular lines one is positive.

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

But there's more going on here.

What can you see? Well, look at the ones that become fractions.

So if you look at one is gradient of AB.

The gradient of AB is three.

The gradient of BC, the perpendicular line, is negative one over three, or negative a third.

The gradient of HG is negative two.

The gradient of IH is a half.

What are you noticing? It kind of looks like one of them is always a fraction that kind of uses or that uses the other gradient as its denominator.

And one is positive, one is negative.

It's a pattern that we're spotting, and the more we explore with gradients, the more of these patterns we're able to spot.

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

Thank you for participating in today's lesson.

Bye for now.