Lesson video

All right.

Well done and thank you for loading the video for this lesson.

My name is Ms. Davies, and I'm going to help you as you work through this topic.

But feel free to pause things, rewind things, so that you're really comfortable with the ideas that we're exploring, and I really hope there's bits and pieces that you find really enjoyable and really interesting as we work through.

Right.

Let's get started then.

Welcome to a very exciting lesson where we're problem solving with plotting coordinates.

You are going to be using your knowledge of plotting coordinates to solve lots of different problems today, hopefully, some nice challenging ones, some nice fun and interesting ones as well.

We're going to break this down into two different types of problems. We're going to start by looking at calculating the horizontal and vertical distance between coordinates, and then we're going to have some fun finding the coordinates where two lines are going to intersect.

So calculating horizontal and vertical distance.

So we can use our knowledge of properties of shape, so we're returning to an idea of shape, to solve shape problems with coordinates.

So what shape have I drawn below? And how do you know? Perfect.

we have a rectangle.

I know it's a rectangle because opposite sides are the same length, and it has four right angles.

If it was on a tilt, it would still be a rectangle if it had four right angles.

So we're going to look at calculating the horizontal and vertical distance between coordinates to help us solve some shape problems. How can I express the distance between these coordinates? What do you reckon? Right.

You'll notice they both have the same y values.

They both have a y value of 3.

So to define the distance, we want to concentrate on the x values.

So the x values have a difference of 4.

We can calculate that by doing 6 minus 2 to give us 4.

We can say then that the horizontal distance is 4.

We're not using units at the moment, because we haven't got any units in our question.

So 4 is fine as a distance in this case.

We're going to bring our rectangle back in then.

So what are the side length of this rectangle? I'll talk you through the first bit.

So let's just look at this vertical distance, this right hand side.

So the vertical distance is the difference between 1 and 4.

We can calculate that by doing 4 subtract 1 or 3.

That side length is 3.

Equally, the parallel side on the other side will also be 3.

Let's look at the horizontal distance.

So this time the y values are the same.

So if we look at the x values, we want the difference between -2 and 3.

Be careful when you're subtracting your negatives here.

Because I've drawn it to scale, you might want to start by just counting the squares, but you're going to want to be able to do it using a more accurate method for when the scales aren't drawn on for you, okay.

So in order to calculate the difference between -2 and 3, we can do 3, subtract -2, and that gives us 5.

Double check by counting the squares in this case.

Yep, we've got a distance of 5.

Right.

This time I've plotted a square.

What properties does a square have? So all angles are right angles still just like our rectangle and all sides are the same length this time.

And that's where we're going to use this idea of distance between coordinates to work out the length of the sides, and if it's a square, all sides are the same length.

So let's use the coordinates that I've marked on for you.

The y coordinates are the same.

So we're looking for the difference between the x coordinates.

So we want the difference between negative 5 and negative 2.

So you can do that by starting on negative 5 and counting up until you get to negative 2 and seeing how far you've moved.

Using a method, you can do that as negative 2 subtract negative 5.

Just be really careful with your negative values there.

So if you're doing negative 2 subtract negative 5, that's like saying negative 2 add 5, which gives us 3.

Just check that that makes sense looking at our diagram.

Has that got a difference of 3? Yes it does.

If I know that side length is 3 and it's a square, all the other sides must also be 3.

So I've marked those on my diagram and when you are looking at problem solving, guys, you want to draw as much as you can on your diagram, write as much working out as you can down to help you 'cause sometimes it's easy to look at a problem and not know how to get to the next step.

But by putting some of the things that you know on your diagram, it's easier to see how to work towards that next step.

So now we know all the sidelines are 3.

We can use that to find the other two coordinates.

Let's start with that top left coordinate.

The x value is going to be negative 5 because it is vertically above the -5, -2.

So it's still going to have an x value of negative 5.

The y value you can calculate 'cause you know the side length is 3.

So we're going to do negative 2 and we're going to add on 3 to work out what the y value is.

So that y value is going to be 1.

We can do the same for the top right corner.

So the x value has to be negative 2 because it's vertically above the -2, -2.

The y value, well it's in line with, it's perfectly in line with the one we just worked out, which was -5, 1.

So that means the y value of that coordinate is also 1.

So that's going to be -2, 1.

Right.

A bit of a check for you then.

What are the sidelines of this rectangle? Give it a go.

Well done.

It doesn't matter which order you try to work them out in.

I'm going to start with the the top line of our rectangle, which is the horizontal distance.

So the y values are the same.

So to find the horizontal distance, I'm looking at the difference between the x values.

So I'm going to do 10, subtract 4, so that distance is 6.

The horizontal size have a length of 6.

Right.

Then let's look at that vertical line.

So the x values are the same.

So I'm looking at the difference between the y values now.

So I'm looking at the difference between 3 and negative 1.

So that's 3 subtract negative 1 which is 4.

Then I'm just going to check that by thinking about a number line.

So I'm going to think about negative 1 on a number line, 3 on a number line and what the distance between them is.

And I do have to add on 4, don't I, to get from negative 1 up to 3? That means the vertical sides have a length of 4.

Perfect.

We're going to put that into practise now then and have a go at some problems. So what you're going to try and do is work out the missing coordinates in each shape.

These two shapes to start with are a square.

So remember what we said about properties of a square? you've been given two coordinates, you need to use them to find the other two.

I recommend that you, once you've worked out the length of your side, you draw it on your diagram to help you.

You could also have a go at sketching this roughly to see if the coordinate you've gone with makes sense.

Right.

Off you go.

I hope you enjoy playing about with those and then we'll look at the next couple together.

Well done, guys.

So another couple of problems this time we're looking at rectangles.

Now, the two rectangles in the first diagram looking at 2a, those two rectangles are the same size, okay.

So you need to use the fact that they're the same size to work out the missing coordinates of D.

If you're not sure where to get started, again, I suggest you work out the length of the sides of the rectangle and then remembering that both rectangles are the same size.

Once you've got your answer, have a think about whether it's a sensible answer.

Does it work on a coordinate grid? 2b is very similar, okay? It got a little bit trickier because you've got to work out the bottom right-hand corner, which might just require you to do an extra couple of steps.

However, I still recommend you write on when you've worked out the length of the sides, label them on your diagram to help you work out that missing coordinate.

All right.

Give it a go Looking at 2c then.

So very similar to the questions that you've just had, we've got two rectangles that are the same size, but this time you've got an extra piece of information, so we know that the length is double the width.

So you've got two coordinates on your picture already.

So you're going to want to start with those two coordinates and then remember that that length is going to be double the width.

The other thing to be careful with this one is to just be aware of your negative values.

Think about which quadrants you are plotting your coordinates in and therefore, what sort of x and y values they should have.

Just to check whether you're on the right lines with your negative coordinates.

2d is really similar but we're working with triangles now.

So again, the two triangles are identical.

You'll notice that one of them sort of been rotated, okay, to get the other one but they are identical triangles.

Give that one a go and then we'll look at our answers.

Right.

Well done if got some or all of these right? They were really nice problems that hopefully you enjoyed playing around with.

So the first one, you should have got 7, 9 for your top right corner and 7, 4 for your bottom right corner.

For B, if you'd worked out that your side length was 7, you get 4,2 for your top right corner and 4, -5 for your bottom right corner.

Let's look at the next lot.

So if we work out that top length is 18, that's actually the only piece of information we need.

We don't need to work out that the other length is 8.

If you have, that's absolutely fine.

Because it's on the same horizontal as the line 4, 7, we know the y coordinates got to be 7, but that x coordinate has got to be 18 bigger than the 22.

So you do 22 plus 18, then we get our 40, 7, So, it's 40, 7.

Okay.

Looking at 2b.

You do need to work out the length of both sides this time.

So the length of the top was 11 and the length of the left hand side or the right hand side 'cause at the same it should be 7, right? You could have done this a number of different ways.

I've done it by working out the top right coordinate first and then working out e.

So that top right coordinate has a y value of 2 'cause it's on the horizontal of -17, 2, but it has an x value of 5 'cause it's 11 bigger the negative 6.

So negative 6 plus 11 is 5.

I've written that down on my diagram so I can then use that.

So I know that top one's 5, 2.

Then I've got a vertical distance of 7 to get to e.

So I know that my x coordinates going to be 5 because it's going to be the same as the coordinate directly above it.

My y coordinate is going to be 2 minus 7.

So that gives me negative 5.

Amazing if you got that one, 5, -5.

Well done, guys.

Let's look at 2c then.

So the length is double the width was the key piece of information here.

So if you worked out the length is 8, that means the width is 4.

What that means we can do is, we can work out that central coordinate by taking away 4 from negative 8.

So that won't be -2, -12.

And then if we take away another 4, that'll get us down to B.

So -2, -16.

It's going to have an x coordinate of negative 2 because it's on the same line as the -2, -8 that's above it.

And then let's look at 2d.

So we've got two triangles this time and we'll need to work out the side length of that first triangle.

So that vertical side between -9, -4 and -9, 0 has a length of 4.

And then the other side length between -9, 0 and 3, 0 has a length of 12.

Now this triangle has then been been rotated, so the length next to the 12 on the top is going to be 4.

So if we add 4 onto 3, 0, we get the coordinate 7, 0.

Notice that they're all sitting along the x axis.

If that top corner is 7, 0, we want the one directly below it.

But we know that that side length is 12.

So we've got 7, 9 and then we need to take away 12 and that gives us 7, -12 for a.

Well done, guys.

Some really nice problems there.

I hope you found some of them at enjoyable.

We're going to have go now at finding the coordinates where two lines intersect.

We can find where two lines intersect by finding coordinate that follows the rules for both lines.

So here are some coordinates that follow the rule x equals 2.

We know that that means they'll sit on the line x equals 2.

Do any of these coordinates also follow the rule y equals 3? What do you think? Yeah, well done.

The coordinate 2, 3.

Because it follows the rule x equals 2, it'll be on the line x equals 2.

And because it follows the rule y equals 3, it'll be on the line y equals 3.

That means the two lines will cross over.

They will intersect at the coordinate 2, 3 'cause it's the coordinate that's on both lines.

Right.

So if you can suggest some coordinates that are on the line, y equals x plus 2, write a few down.

you may have come up with some similar to me.

Right.

Do any of yours or do any of mine also follow the rule y equals 2x? Well then, if you spotted that 2, 4 follows both rules, that means both lines will intersect at 2, 4.

'Cause that coordinate fits both rules, that coordinate will sit on both lines, which means that's where the two lines will cross over when we draw them.

Right.

Here's a coordinate.

See if you can find two rules which apply to that coordinate.

Off you go.

Yeah.

Well done.

X equals 4 and y equals 6.

you might have gone even further and thought of other rules that work for that coordinate.

For example, W was x plus 2, okay.

There's not just two rules that follow a certain coordinate.

There might be many rules that work for a certain coordinate.

Right.

Which two rules apply for that coordinate? Lovely.

We've got y equals x plus 2 and we've got y equals 5.

So this time can you find two rules which apply to the coordinate -3, 3.

Off you go.

A trickier one, this one.

So well done if you spotted.

If it's y equals x plus 6 and y equals negative x.

two rules which apply for the coordinate -2, 4.

What do you think? We have y equals x plus 6 again and then y equals negative 2x.

Good spot if you've got both of those.

So you're going to have a go at practising this yourself then.

So for each question, I'd like you to find a coordinate that follows both rules.

you can use the axes to help you issue like.

So the first one you want to coordinate that follows the rule x equals negative 3 and the rule y equals 4.

Give that one a go.

Perfect.

Well done.

Same idea but I've got some slightly trickier rules for you to use now, okay.

Use the axis to help you if you like, play around with some different values.

See if you can find the coordinate that works for both rules.

Right.

I wonder if you managed to find the coordinate that worked for both rules in all of those.

It's time to really put your brain to the test now then.

So for each pair, just like before, I want to know where the lines will intersect.

So you're looking for a single coordinate.

If you're finding one of 'em a bit tricky, you might want to list some coordinates that follow that rule to help you and then see if you can find one that matches with the other rule.

Off you go.

I'd be really, really interested to see how many of these you managed to find.

Fine or change of the lesson then.

So just like before, we've got different rules for you to try and find where the lines intersect.

Again, I really recommend writing out some coordinates that follow the rule if you're finding one particularly tricky.

Off you go.

Well done, guys.

I wonder how many of those you manage to find.

It's quite a nice feeling when you find the one coordinate that fits both the rules.

If you do have graphing software available, you might have wanted to type these into Desmos and see where those lines meet.

It's a nice challenge to see if you can do it without that graphing software as well though.

So a, you should find-3, 4, b, 2, -1, and c, -3, 0.

For d, you should have got 2, 4, 4, 4 and 1, 2.

I wonder if you did that just looking from the lines or whether you've started to think about how you can draw a line in and then see where they cross.

A mixture of both is going to make sure you don't make a mistake.

And then we're getting onto the trickier ones, weren't we? So we've got 4, 9, 1, 2, 3, 6, 5, 10, <v ->1, -2,</v> and 0, 0.

And there'll only be one answer for each, 'cause if you think of two straight lines, they're only going to cross at one coordinate, okay.

And if you found that special one coordinate, then well done.

Our last set then.

So the x and the y need to add to give a 6, but the x and the y have to be the same.

Our only option is 3 and 3.

So well done if you got 3, 3.

For the next two questions we've got y equals negative x.

So you might have wanted to just write out some coordinates where y equals negative x and then we also want the y to be the x subtract 4.

So we need that difference of 4 between our coordinates.

This is going to help us for h and i.

Make sure you've got them the right way around.

So h, y equals x minus 4.

So you need 2, -2.

For i, y equals x plus 4.

So you need -2, 2.

For j, y equals 4x and y equals x plus 6.

So we need a number that when we times it by 4, it's the same as adding 6.

Well done if you spotted that it's 2, 8.

And the very last one that was put in there is a little bit of a tricky one.

So absolutely outstanding today if you've got that one first time.

Y has got to equal half of x and you need to be able to add 3 to x to get y.

If you think about that, if you have positive values, if you are half them, they actually always get smaller, don't they? So there's no way that y could be 3 bigger than x.

That's where our negative values come in.

When you half a negative value, it actually gets bigger.

So half of negative 6 is negative 3 and and that's the same as adding 3.

If you didn't get some of those, that's absolutely not a problem today.

We've just been developing our algebra skills, playing around with this idea of graphs and coordinates.

And I hope there's been some bits that you've found quite fun.

Let's see what we've learned then.

So we can work out the horizontal or vertical distance between coordinates, and we did that to solve some great problems with rectangles and squares and triangles.

We can find where two lines intersect by finding a coordinate that follows both rules.

And what we've developed on top of both those things is this idea of just giving things a go and solving some problems. We know that the more that you write down on your page, the more it gets your brain thinking and it might show you the route to get to your solution.

Thank you for participating in that lesson today.

It'd be lovely to see you again.