Lesson video

Hello, Mr. Robson here.

Checking and understanding the plotting of coordinates today.

You've made a great choice joining me for this lesson.

Let's get started.

Our learning outcome for today is that we'll be able to plot coordinates in any of the four quadrants.

What do we mean by quadrant? It's an important word, which I'll use a few times during this lesson.

A quadrant is any one of the four areas into which a space is divided by the x and y axes in the Cartesian coordinate system.

The Cartesian coordinate system being that grid there.

The x and y axes divide it into our four quadrants.

That's the first quadrant, the second quadrant, the third quadrant, and the fourth quadrant.

That's the word quadrant.

There'll be two parts to our lesson today.

Firstly, we're going to look at plotting coordinates in all four quadrants.

Alex plots the coordinate (4, 3).

"That's four right, three up," Alex says.

"There it is." What has Alex done wrong? Pause the video and tell the person next to you.

It has to be four right and three up from the origin.

I hope you said something along those lines.

Alex has just gone four right, three up from any starting point.

It has to be four right and three up from the origin, the origin being that coordinate (0, 0) where the x and y axes cross.

So that would be the coordinate 4, 3.

Four right and three up from the origin.

We could make that language even more precise by saying four in the positive x-direction and three in the positive y-direction because we talk about x-direction and y-direction when plotting coordinates.

I'd just like to check that you've got that.

Which sentence best describes this coordinate? Is it two right, five up? Two in the x-direction, five in the y-direction? Five in the x-direction, two in the y-direction? Or two in the positive x-direction, five in the positive y-direction? Which sentence best describes the coordinate? Pause this video.

Tell the person next to you.

It was D, two in the positive x-direction and five in the positive y-direction.

It's really important we use that language, positive direction and negative direction, when we're plotting coordinates.

Alex says, "That's the coordinate bracket 5 1 closed bracket." Hmm.

Alex has made more than one mistake.

What are they? Pause this video and tell the person next to you.

Coordinates are written in the form x, y with a comma in between.

Alex didn't have a comma in the coordinate that he wrote.

It's important that we recognise that they come in the form x coordinate first and y coordinate second.

In this case, the x coordinate is one and the y coordinate is five.

We can read those coordinates by looking from the point when I go down to the x-axis, it's an x coordinate of one.

When I go across to the y-axis, it's a y coordinate of five.

Alex had the two numbers the wrong way round.

He should have written bracket one comma five closed brackets.

Make sure you write your coordinates like that, which means something's missing from these coordinates.

But Aisha says, "I know the missing coordinates." What do you think she has noticed? Pause this video.

Tell the person next to you.

They all share the same y coordinate of eight.

That line there pointing at the y-axis, we can read a y coordinate of eight for all three of those points.

So if that coordinate is (7, 8), then that coordinate must be (2, 8) and that coordinate must be (11, 8).

They had a common 7 coordinate.

Let's check if you can repeat that skill.

Here's four coordinates with some missing information.

What are those missing coordinates? Pause this video and give this problem a go.

We should have noticed that there's a common x coordinate for three of these coordinates.

The upmost coordinate is (7, 10).

That x coordinate of seven must be repeated there and there.

We have coordinates (7, 6.

5) and (7, 3).

From there, I hope you notice there's a common y coordinate for those two coordinates.

That would make that coordinate (3.

5, 6.

5), the 6.

5 being their common y coordinate.

Moving on to something slightly different now.

What's different between these two coordinates? Pause this video and tell the person next to you.

I hope you said something along the lines of one of them is three in the positive x-direction and three in the positive y-direction from the origin.

That's the coordinate (3, 3).

The other one is three in the negative x-direction and three in the negative y-direction from the origin.

That's the coordinate (-3, -3) appearing in the third quadrant.

We won't always have positive values in our coordinates.

We'll see them with negative values, which we'll see them plotted in the second, third, and fourth quadrants.

So what is true of this coordinate? Four options there.

Choose one.

I hope you went for option B.

It's got a positive x-direction and a negative y-direction.

You see that reflected when I tell you that is the coordinate (+3, -2).

How about this coordinate? Four options again.

Pause and take your pick.

Option C, negative x-direction and positive y-direction.

The coordinate was (-4, +5).

Jun says, "This is the coordinate (-3, +2)." Is Jun correct? It looks correct.

Is it correct? Pause this video and have a conversation with the person next to you.

The truth of this problem is it might be, but we don't know.

Until we know the scale on the axes, we cannot be certain.

It looks like a scale of one, but that doesn't necessarily mean it is.

If we put a scale of one on both the x and y axes, then indeed that is the coordinate (-3, 2), but that scale won't always be the case.

If we change the scale, like so, still got a scale of one on the y-axis, but pay attention to that x-axis.

The scale is now moving in fives, steps of five.

This is no longer the coordinate (-3, 2).

It's the coordinate (-15, 2) because we changed the scale on the axes.

Scale is really important, so let's check that you've got that.

If coordinate A is (10, 5), what would that make coordinate B on this grid? Is it (9, 4)? (5, 4)? Or (1, 4)? Pause this video and have a little think about that problem.

It's (5, 4).

It's helpful if I show you the scale on those axes now.

If coordinate A is (10, 5), then that means we've got steps of one on the y-axis, but steps of five on the x-axis.

From there, you can see that coordinate B is (5, 4).

If I changed the problem a little bit, it looks the same, but something has significantly changed.

If I said to you coordinate A is now (5, 10), what would that make coordinate B? Three options for you.

Do you think it's (5, 4), (1, 4) or (2.

5, 8)? Pause this video and have a little think It's (2.

5, 8).

Why would that be? Well, let's apply that scale.

It's not quite steps of five on the x-axis.

It's steps of 2.

5.

A non-integer scale, that would make the x coordinate of B 2.

5.

If the y coordinate of A is 10, then that's steps of two on the y-axis, so B would have a y coordinate of eight, hence B is the coordinate (2.

5, 8).

Another useful skill is to be able to estimate coordinates.

If coordinate A is (-2, 4), what do we know about coordinate B? Pause this video and make a few suggestions to the person next to you.

There's a few things we know.

It's in the first quadrant, so that tells us that both the x and y coordinates are positive.

We also know that the y coordinate must be less than four.

How do we know that? Because coordinate A has a y coordinate of four and coordinate B is below that closer to the x-axis, so we know the y coordinate must be less than four.

We also know something about the x coordinate just through estimation.

We can estimate that the x coordinate is going to be greater than two because if you look at the two coordinates, B is further away from the y-axis than A, so that distance in the positive x-direction must be greater than two if coordinate A is two in the negative direction.

Let's check that you've got that.

Which of these is a good estimate for coordinate B? If we know coordinate A to be (-2, 4), which of these would be a good estimate for B? (1, -7), (1, -3), or (3, -5)? Pause this video and make your choice.

(1, -3) would be the best estimate.

Why? Well, (1, -7) that's too far in the negative y-direction.

Option C is too far in the positive x-direction.

Coordinate B is closer to the y-axis than coordinate A, so that x coordinate must be less than two, hence (1, -3) is the best estimate of those three.

How about this problem? Which of these is a good estimate for coordinate A? Is it (-9, -12), (-9, -7), or (-3, -12)? Pause this video and have a think about that one.

(-9, -12) is the best estimate.

(-9, -7) well that's not far enough in the negative y-direction.

We can see that A is below coordinate B, so the y coordinate must be lower than negative seven.

As for option C, -3 is not far enough in the negative x-direction.

Coordinate A is further from the y-axis than B, so it must be less than -3.

Hence, (-9, -12) was the best of those three estimates.

Practise time now.

Question one, there's a coordinate grid.

I'd like you to plot those four coordinates.

Pause this video and give that a go.

Question Two.

Well, we have no axes.

Does that mean we can't find these missing coordinates? We can, you're given some information, but not all the information.

My hint for you would be think about the scale.

Pause this video and fill in those missing coordinates.

Question three, and this is a scale problem.

If coordinate A is (-6, 4), what are coordinates B and C? That's question A.

Question B and C are similar.

The thing I'll warn you about is they all require different scales on the axes.

Pause this video and give those three problems a go.

Feedback time.

The plotting of those four coordinates, you might want to just pause this video and check that your coordinates are in the same position as mine.

For question two, I hope you noticed a shared x coordinate in that position there.

The x coordinate that was missing there must be 83.

And then we can look at the scale in the x-direction.

Three steps to get from an x coordinate of 80 to an x coordinate of 83.

Well, that tells us it's a scale of one in the x-direction so we can fill in all our x coordinates now.

That missing one must be 79.

That missing one must be 85 because of that scale of one in the x-direction.

What about the scale in the y-direction? Well, the y coordinates we were given were 10 apart from -25 to -35, they're 10 apart, but in five steps, so each step must be two.

We've got a scale of two in the y-direction.

That enables us to spot that that y coordinate must be -31.

There's a shared y coordinate there.

And then one, two, three steps of two down from that position, that y coordinate must be -37.

Question three.

Quadrant A is (-6, 4).

Well, B must be (-12, 0).

If A is -6 in the x-direction, that's steps of three on the x-axis, so B must have an x coordinate of -12.

B is on the x-axis, so it's y coordinate must be zero.

C is (-9, -8).

Again, steps of three on the x-axis, -3, -6, -9, if it's x coordinate.

And if A has a y coordinate of four, that's a scale of four on the y-axis.

The two steps down from the x-axis would take us to -8, so a y coordinate for C.

Part B.

If coordinate C is (-6, -40), what are A and B? The x coordinate of -6 would see steps of two on the x-axis and a y coordinate of -40 would see steps of 20 on the y-axis, so coordinate A would be (-4, 20) and coordinate B would be (-8, 0).

Part C might have caused you a few problems because if we know coordinate B is (-40, 0) , the only thing we're given information about is the scale on the x-axis.

The x coordinate of C would be -30, the x coordinate of A would be -20, but we aren't given a scale on the y-axis so we can't find those exact coordinates.

Okay, second part of the lesson now.

We're gonna take our knowledge of coordinates and we're gonna do some problem solving with shapes.

I told you it was beautiful.

Two identical squares and we know coordinate B is (12, 12).

How do we know coordinate A? Pause this video and tell the person next to you.

I hope you said something along the lines of it must be coordinate (6, 6) because it's half the distance in the x-direction and half the distance in the y-direction.

What if I change that image to three identical rectangles and I tell you that coordinate A is (8, 10)? Can we use that information to find coordinates B and C? What does your mathematical intuition tell you? Could we find coordinates B and C with the limited information we have? Pause this video, make a few suggestions to the person next to you.

We can.

We're gonna repeat that same eight in the x-direction, 10 in the y-direction journey to get from coordinate A to coordinate B, and then from coordinate B to coordinate C.

If we make the journey again from A to B, that will take us from an x coordinate of eight to an x coordinate of 16, a y coordinate of 10 to a y coordinate of 20.

And again we'll do another eight in the x-direction, 10 in the y-direction to get from coordinate B to coordinate C, that would take us to (24, 30) as a coordinate.

Same problem.

Different problem.

Three identical rectangles and I've given you the coordinate of one vertex on the first rectangle.

How is this problem different? Can we find coordinates B and C? Pause this video, make a few suggestions to the person next to you.

You would've noticed that one of the rectangles has a different orientation.

You might have said it's pointed in a different direction.

Mathematically, we'd say it's got a different orientation, that middle rectangle.

The A to B is now +5 in the x-direction and +2 in the y-direction.

So to get from coordinate A to coordinate B, we move five in the x-direction, two in the positive y-direction, taking us to coordinate (7, 7).

Then to get from coordinate B to coordinate C, it's a two in the positive x-direction and a five in the positive y-direction.

That would take us to coordinate (9, 12).

Let's just check you've got that now.

What are the coordinates of point A? Three identical squares.

Coordinate C is (12, 12).

What are the coordinates of point A? (1, 1)? (3, 3)? Or (4, 4)? Pause this video and have a little think.

It was option C (4, 4).

Why? Because it'll be one-third of the journey to (12, 12).

One-third of 12 is four, so it must be four in the positive x-direction, four in the positive y-direction.

Another beautiful problem.

What are the coordinates of point B? Three identical rectangles.

We know coordinate A is (5, 0.

4).

So what would coordinate B be? Would it be (15, 1.

2)? (15, 12)? Or (9, 6)? Pause this video and have a little think.

It was option A (15, 1.

2).

How do we know that? Because it's three times that journey.

Three identical rectangles.

We're told coordinate A is (9, 8).

What are the coordinates of point B? Would it be (18, 16), (18, 24) or (27, 24)? Pause this video and have a little think.

It's B (18, 24).

If I put the scale on the x-axis, coordinate A is nine in the positive x-direction, so that would tell us 9, 18, 27 on the x-axis.

Coordinate A is eight in the positive y-direction, so we could put 8, 16, 24 on our axis and then you can see coordinate B is (18, 24).

Different problem now.

If we know, and this is a square with coordinate A (3, 0) and coordinate B (0, 0), do we know all four coordinates? Yes, we do.

C must be (0, 3) because it's a square.

Those lengths are three.

D must be (3, 3) because there's a matching x coordinate with A and a matching y coordinate with C.

So point D must be (3, 3).

A fairly straightforward problem.

Not a straightforward problem.

I've got coordinate B (0, 0), the origin.

I've got coordinate A (3, 1).

The question tells us it's a square.

Do we have enough information to know all the coordinates of this square? What do you think? What is your mathematical intuition telling you? Can we find coordinate C? Can we find coordinate D? Pause this video and tell the person next to you.

We do.

It's a three one movement between the vertices.

What do I mean by that? Well, to get from B to A, we have to move three in the positive x-direction and one in the positive y-direction.

With all four lengths of this square being the same, that movement must be repeated to get us between the vertices.

We have a tricky problem first though because that three one isn't a positive three, positive one.

We have a three movement in the y-direction or a positive three movement in the y-direction, but the one movement isn't positive anymore.

That one movement must be a negative one in the x-direction to get us from point A to point D.

So what does that tell us point D is? It must be the coordinate (2, 4).

It's positive three in the y-direction from one, one plus three being that y coordinate of four, and it's negative one in the x-direction from three.

Three minus one is two, hence coordinate D is (2, 4).

Once we know that, can we find coordinate C? Indeed we can because that movement three one from B to A must be the same as that movement three one from C to D, so we subtract three and we subtract one and we can read that C coordinate of (-1, 3).

A trickier problem that, but achievable.

A similar problem, this one, but a slight tweak in that we're now talking about a rectangle.

We've got three coordinates for this rectangle.

Can we find coordinate D? Is there anything we did in that square problem that might help us to find this missing coordinate? Pause this video, make a few suggestions to the person next to you about how we might solve this.

We can solve it because the journey from point B to point C must be the same as the journey from point A to point D.

What do I mean by that? I mean, this movement here, to get from coordinate B to coordinate C, we added two to the x coordinate, we added six to the y coordinate, or we moved positive two in the x-direction, positive six in the y-direction.

That same movement will be reflected here, so we'd add two to the x coordinate add six to the y coordinate.

That tells us that point D is (44, 50).

Alternatively, you might have said, well, the journey from B to A must be the same as the journey from C to D, so there's a second method to doing that.

On this rectangle, which journeys will match on both distance and direction? So option B is, B to A and C to D.

There's B to A and there's C to D.

They're the same distance, they're the same direction.

Option A wasn't right because A to B is a different direction,.

Whilst it's the same distance, to move from A to B is to travel in a different direction as moving from C to D.

The last one was correct as well.

C to B and D to A were the same journey in distance and direction.

Another check.

If this is a rectangle, what are the coordinates of point C? Pause this video.

Have a go at this problem.

It was B (20, 58).

Why? Because the journey from coordinate A to coordinate B is positive five in the x direction, negative 14 the y-direction, that journey must be repeated from coordinate D to coordinate C.

That would tell us coordinate C is (20, 58).

Practise time now.

Question one.

Find the missing coordinates.

We have three identical squares.

We're told coordinate A is (9, 9).

I'd like you to find coordinate B, coordinate C, and coordinate D.

Pause this video and give this problem a go.

Question one part B.

We've got four identical rectangles and we're given one coordinate.

Coordinate D is (2, 7).

I'd like you to find the missing coordinates A, B, C, and E.

Pause this video and give this problem a go.

Question two part A.

Find the missing coordinates of this square.

Pause this video.

Give this problem a go.

Question two part B.

Find the missing coordinates of these identical rectangles.

Pause.

Give it a go.

Feedback time now.

Question one part A, we were told coordinate A was (9, 9).

That helps us to identify coordinate B.

That would be half the distance in both directions, so coordinate B must be (4.

5, 4.

5) because we know they're identical squares.

If coordinate B is (4.

5, 4.

5), coordinate C must be (-4.

5, -4.

5).

You can almost think of that bottom left square as a rotation of the first square.

That would leave us with coordinate D zero because it has no movement in the x-direction and -4.

5 in the y-direction.

Question one part B.

If D is (2, 7), we've got the length of this rectangle.

They must be two and seven.

If we apply that seven in the positive x-direction and that two in the positive y-direction, we find that missing coordinate E to be (9, 9).

We can find the remaining coordinates because if D is (+2, +7), B must be (-2, -7) and C must be (0, -7).

Coordinate A was the trickiest one because we need that journey of -2, that journey of -7, and that journey in the 7 direction of -2 to tell us that coordinate is (-9, -2).

Question two part A.

Find the missing coordinates of this square.

We have a journey to get from B to A of positive five in the x-direction and negative two in the y-direction.

If we consider that journey again to get to point C, well, that would become positive two in the x-direction and positive five in the y-direction.

That tells us coordinate C is (2, 5).

Coordinate D, the positive two in the x-direction and positive five in the y-direction is the same to get from B to C as it is to get from A to D.

So when we put that same movement on coordinates A and D, well, that tells us that coordinate D is (7, 3).

Question two part B.

Start with the simplest thing we've got.

We've got two coordinates F and G, and we can see the difference between them or the journey between them.

We've got a movement of positive 14 in the x-direction, positive three in the y-direction to get from coordinate F to coordinate G.

That tells us all that we need to know about the shorter lengths of these rectangles.

The short lengths are related by these two movements, positive 14, positive three, or positive three, negative 14.

That's the key to unlocking this problem because we can see that positive three in the x-direction, negative 14 in the y-direction is that journey from coordinate C to coordinate D, so coordinate D must be 27 plus 3, 30, 19 minus 14, five.

We've got coordinate D.

We can do the same thing to move from coordinate B to coordinate A.

Move positive three in the x-direction, negative 14 in the y-direction.

Coordinate A must be (-2, -3).

And then coordinate E is the coordinate, the movement from E to D is the same as the movement from F to G, so it must be 14 less on the x coordinate and three less on the Y coordinate.

Coordinate E must be (16, 2).

That's the end of the lesson now.

We can plot coordinates in all four quadrants and identify the scale of unlabeled axes if given a coordinate.

It's possible to estimate a coordinate even when we don't have axis labels or grid lines, and we can use this knowledge of coordinates to carefully consider the journey between vertices of shapes like squares and rectangles to find missing vertices.

I hope you enjoyed this lesson and those problems as much as I did.

I'll see you again soon for more mathematics.