Lesson video

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Hello, Mr. Robson here.

Great choice to join me for maths today, coordinates, yay.

Let's get going.

Our learning outcome for this lesson is that we're able to plot coordinates in any quadrant.

Quadrants an important word, which I'll be referring to several times through this lesson.

A definition of it would be a quadrant is any one of the 4 areas into which a space is divided by the X and Y axes in the Cartesian coordinate system.

So this is a lesson on securing understanding of plotting coordinates.

We'll start by plotting coordinates in all four quadrants.

Then we'll draw some axes and then we'll plot shapes in all four quadrants.

Let's get going with coordinates in all four quadrants.

But what is a quadrant? Quadrant, you don't hear that word very often, but it's a really important one in the realm of coordinates.

So I'd like to set you this challenge.

What other quad words do you know? I know three.

Do you know more than three? Pause this video, see if you can think of some.

Quad words that I know; quadrilateral, one of my favourite mathematical words, quad-bike and quadruped.

So what does quad tell us? The Cartesian coordinate grid system we see here, we think of in terms of its four quadrants.

The first quadrant, the second quadrant, the third quadrant, the fourth quadrant.

I need you to know these.

So I'm gonna just run through them one more time and ask you to pay careful attention to how I revealed them.

They went anti-clockwise, the first quadrant, the second quadrant, the third quadrant, the fourth quadrant.

You definitely want to pause this video and sketch that down or take a really secure mental image of that.

I need you to know where those quadrants are.

So what's the difference between the quadrants? Let's consider the X axis for a moment.

The X axis is the horizontal one.

Let me isolate that X axis and say to you what's the difference between these two points? Which two points? These two points.

Pause this video, tell the person next to you what's the difference between those two points? I hope you said one is plotted in the positive X direction and one is plotted in the negative X direction.

So what about the Y axis? Well, the same thing.

When we go up from the origin, we go in a positive Y direction.

When we go down from the origin, we go in a negative Y direction.

So the difference between these quadrants is a positive or negative direction in the X direction or the Y direction.

So we need to be able to label our four quadrants with these 4 labels.

Positive X direction, positive Y direction, positive X direction, negative Y direction, positive Y direction, negative X direction, negative Y direction, negative Y direction.

Can you put those 4 labels on our coordinate grid? Pause this video and do so.

Let's start with the easiest one, positive in the X direction, positive in the Y direction, we're in that first quadrant.

If we're positive in the X direction but negative in the Y direction, we end up in that fourth quadrant.

If we're positive in the Y direction and negative in the X direction, we're in the second quadrant.

And then when both the X and Y directions are negative, we're in our third quadrant.

Let's check you've got that.

I'd like you to match the quadrant to the correct description of the coordinates.

Pause this video, take a moment, match the quadrant to the description.

Our first quadrant positive X coordinate, positive Y coordinate.

Our second quadrant would see us with a negative X coordinate and a positive Y coordinate.

Our third quadrant would see us with a negative X coordinate and a negative Y coordinate.

While the fourth quadrant sees us with a positive X coordinate and a negative Y coordinate.

So how about plotting points in those quadrants? We should be familiar with coordinates and the idea that we see the X coordinate first, the Y coordinate second in that order, just like the alphabet X, then Y.

The two coordinates separated by a comma.

So brackets 3, 2, close bracket.

The X coordinate is the 3.

The Y coordinate is the 2.

So to plot that, coordinate 3, 2.

From the origin we're travelling 3 in the positive X direction, 2 in the positive Y direction, there's 3, 2.

What's different? What about plotting -3,2? The coordinate -3, 2.

Something's changed, what's changed? What impact will it have on where we plot that coordinate? I'd like you to make a suggestion to the person next to you, to your teacher, or suggest it back to me on screen.

Pause this video and make your suggestion.

The difference crucially is the X coordinates now -3, which means we're travelling in a negative X direction from the origin.

That's that -3 X direction from the origin.

Still a positive 2 in the Y direction, leading us to that point there -3, 2.

Let's check that you've got that.

Which of these coordinates will be in a negative X direction from the origin? Is it 5, -1, 5, 1 or -5, 1? Pause this video, give that question a go.

<v ->5, 1 was the correct answer.

</v> Some people might be fool into taking that top one, 5, -1, but crucially we were talking about the X direction.

And the coordinate 5 -1, the 5 is the X coordinate and it's positive.

That would be in a positive X direction.

The -1 would take you in a negative Y direction.

We were focused on the X direction and the only coordinate that moved us negatively from the origin was -5, 1.

So we should have picked C, -5, 1.

So we plugged the coordinates.

3, 2 and -3, 2.

What about these two coordinates? 3, -2 and -3, -2.

Where would they be? Pause this video and make a suggestion as to where those two coordinates would be.

The coordinate 3, -2 with a positive 3.

So moving positively in the next direction and a -2, so we're moving negatively in the Y direction from the origin.

Positive 3 in the X direction, -2 in the Y direction.

Take us to that point there.

That's the coordinate 3, -2.

<v ->3, -2 negative X direction,</v> negative Y direction.

Taking us to that point, -3, -2.

As long as you pay particular attention to is that X coordinate positive or negative? Is that Y coordinate positive or negative? You'll be plotting your coordinates in the correct quadrant.

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

In which quadrant will you find the coordinate 1, -4? The second quadrant, the third quadrant or the fourth quadrant? It's the fourth quadrant.

We've a movement of 1 in the positive X direction followed by a movement in the negative Y direction.

Taking us into that fourth quadrant.

Let's check another one.

What about this coordinate, - 11, -14? In which quadrant would you find that coordinate? The second, the third, the fourth? Pause this video, make a suggestion.

It's the third quadrant.

Whilst a coordinate grid on the screen is not big enough to plot this coordinate, I hope you appreciate that.

coordinate -11, -14 is negative in the X direction, negative in the Y direction from the origin.

So it must be in that third quadrant.

Time to practise now.

For this first question, I'd like you to plop these coordinates.

1, 4, -1, 4 <v ->4, 1, -4, -1</v> 4, -1 and 0, -4.

Pause this video and give those coordinates a go.

Second question, in Laura's book you see that she's plotted the coordinates 2, -1 and -3, 4 like this.

She's made some errors.

What feedback will you give her to correct her work? Write a sentence or two that would help Laura.

Pause this video and have a go at this question.

Plotting the coordinates 1, 4 is in the first quadrant, we find -1, 4 and -4, 1 in our second quadrant.

Negative in the X direction, positive in the Y direction.

Pay particular attention to the difference between -1,4 and -4, 1.

<v ->1 in the X direction, -4 in the X direction.

</v> X coordinate first, Y coordinate second and you won't go wrong.

The next coordinate -4, -1.

We find that down in the third quadrant where both the X coordinate and the Y coordinate are negative.

Then on our fourth quadrant we find the coordinate 4, -1.

4 positively in the X direction, -1 in the Y direction.

Taking us to that point there.

The last coordinate I want to give a special mention to 'cause it's one of the most frequently errored ones 0, -4.

We take the X coordinate first, we have no movement in the X direction, neither positive nor negative, just no movement.

And then we go -4 in the Y direction.

Taking us to that point there, 0, -4.

Where your learning on coordinate geometry is going is gonna be really important.

Now you can identify coordinates that are on this Y axis.

So if you solidify that now 0, -4 is in that position, that's gonna benefit you hugely going forward in your learning.

Onto Laura now, she plotted 2, -1 and -3, 4 with some errors.

What did we suggest to her? There were lots of ways you could have explained her error to her.

You might have included the following, follow the axes in the correct direction depending on the value of the coordinates.

If you'd pointed out specifically where the coordinates 2, -1 and -3, 4 were, you could have shown her the difference between the positive direction for that positive two X coordinate and the positive 4 direction for that positive 4 Y coordinate.

That would've helped Laura out.

Drawing axes, it's hugely important that we can do this accurately.

We'll use it in many subjects in mathematics, not just in coordinate geometry and statistics.

You'll use it in geography, you'll use it in science.

So let's accurately draw some axes.

Let's start with a question, what's this coordinate? Pause this video and make a suggestion to the person next to you or perhaps to me on screen.

We don't know is the answer.

We don't know that coordinate.

Without grid lines and accurately labelled axes, it's really difficult to read or plot coordinates accurately.

If I give you the grid lines and I label those axes, you could tell me that coordinate, but without them we couldn't.

What's wrong with this coordinate? What coordinate? This coordinate.

There's something unusual here.

Pause this video and see if you can spot it.

So it looks right, but it's not right.

It looks like it's 2 in the positive X direction and 3 in the positive Y direction from the origin.

But it doesn't read like it.

It reads like it's the coordinate 3, 3, but it looks like it's the coordinate 2, 3.

What on earth is going on here? Have you spotted it? It's the axis.

It's the way they're labelled.

That one shouldn't be there.

The point where the X and Y axes meet is the origin and we label it with 0 or with the coordinate 0, 0.

In this case, the X axis is accidentally marked with a 1 in the position of the origin.

That's why this coordinate looks all sorts of wrong.

So when we draw axis, it's really useful to start with that point, the origin, and then we can work outwards and count in equal steps.

Then we'll be accurately plotting axes.

So start with the origin.

A little check for you here.

What's wrong with these axes? Similar error, different direction.

The origin was marked as 1 on the Y axis this time.

So all our labels in the positive Y direction as well as the label on the origin were incorrect.

We'd need to take our rubber, get rid of those, we plot the origin and then count up in the positive Y direction.

Being able to spot errors in axes will enable you to draw them accurately in many subjects.

So what's wrong here? Pause this video, see if you can spot the error.

It's the scale on the X axis.

The origins marked fine, the Y axis is fine.

It's the X axis.

They're not equal steps.

In the positive direction, we step away from the origin one at a time.

In the negative X direction, we're stepping -2, -2, -2.

It's inconsistent and it can't be.

It has to be the same all the way along that axis.

So if it were the same scale all the way along the axis, what would we put in these blank spaces to keep the scale on the X axis consistent? Pause this video, write down what you would label those blank spaces with.

So the scale is 2, we're going in steps of two, so we'd be 2, 4, 6, 8, 10 in the positive X direction.

It's fine that the scale on the Y axis is different, steps of one versus our X axis steps of two.

The most important thing is the scale is consistent the length of the axis.

Okay, what mistakes been made here? There's something wrong.

Can you spot it? Pause this video, have a good look.

It's the X axis in the negative direction, isn't it? It's in reverse.

It's the right labels in the wrong order there.

So what should they read? Pause this video, sketch down the way they should be correctly labelled.

I hope you wrote -5, -4, -3, -2, -1 and labelled them like so on your axis.

Okay, some practise now, I'd like to fill in the blanks to label this coordinate grid.

I'd like to keep the scale in ones, steps of one in both the X axis and the Y axis.

Remember my top tip from earlier, don't forget to start with the origin and work outwards.

Pause this video, give that task a go.

Question two, the scales changed, but I think we can still fill in these blanks and label our coordinate grid.

Remember, the scale on the X axis has to be consistent all the way along the X axis and the scale on the Y axis has to be consistent all the way along the Y axis.

Pause this video, give it a go.

Question 3, we've got Jacob's book this time.

He's drawn this coordinate grid.

What advice would you give him to improve his work? See if you can spot more than one thing you would like to improve about his work and write a few sentences to help Jacob out.

Pause this video, give it a go.

Feedback, filling in the labels to complete the coordinate grid.

Start with the origin and work outwards.

Let's label our origin then.

Let's work positively along the x axis, negatively along the X axis, positively along the Y axis, negatively along the Y axis.

Pause this video and just check that your labels are the same as mine.

For the second question, the scale had changed.

Again, it's useful to start of the origin.

When I label the origin 0, you can see immediately the 2 in the Y direction tells us the scale of the Y axis.

It's gonna step up in twos.

So in the positive Y direction we can go 2, 4, 6, 8, 10, and then reflect that in the negative Y direction.

<v ->2, -4, -6, -8, -10.

</v> steps of two on the Y axis.

But those steps of two don't work on the X axis, we have some clues on the X axis.

The easiest one I think I've spotted is there's two steps to get from the origin to 10.

So that must mean each step is 5, so 5, 10, 15, 20, 25 in the positive X direction, that works.

So it must be steps of 5 in the negative direction, <v ->5, -10, -15, -20, -25.

</v> Again, pause the video, check that your labels look the same as mine.

Question 3, Jacob's book.

Several bits of advice we could have given him.

Your advice might have included label the X and Y axis.

You haven't seen a coordinate grid yet in this lesson that hasn't had the X and Y axis labelled.

Be sure to do that every time.

In the case of Jacob, he's aligned his X axis with a line in his book, but he hasn't done the same with the Y axis.

He's left it floating between two lines.

That could lead to all sorts of inaccuracies in plotting and reading coordinates.

So make sure your Y axis lines up with the lines in your book.

Lastly, when you're labelling, start with the origin and work outwards.

I think when Jacob labels his origin, he's gonna find a scale problem on his X axis.

Starting out the origin and counting outwards in equal steps will help his accuracy.

That was drawing axes.

We'll finish with plotting shapes in all four quadrants.

Let's plot these coordinates and let's see what you notice.

2, 2, -3, 2, <v ->3, -2, 2, -2.

</v> Plot those coordinates.

Pause this video, give it a go.

We get those coordinates and you notice rather joyously they form a rectangle and then you notice even more joyously that rectangle is parallel to the X axis and parallel to the Y axis.

Or should I say the sides of the rectangle parallel to the X axis, parallel to the Y axis.

That's gonna enable us to solve all sorts of interesting problems. But first we need to spot patterns in these coordinates.

Do you see any relationships, any patterns? Pause this video and have a really good look.

Did you spot the matching X coordinates? The X coordinate for those two vertices was 2.

The X coordinate for those two vertices was -3.

They were the same direction, whether in the X positive direction or the negative X direction, they were the same direction from the origin.

Same for the Y coordinates.

We see a 2 and a 2.

Both of those vertices are positive 2 in the Y direction.

The vertices at the bottom of our rectangle were -2 in the Y direction.

Now what that means is I could have admitted to tell you that coordinate and asked you to find that coordinate.

I could even have removed the labels and the grid lines and you could still have told me what this fourth coordinate of this rectangle is because we know it must be 2 in the positive X direction and -2 in the Y direction.

So all of a sudden we didn't need the labels and the grid lines provided we had certain information about this rectangle.

But surely if I only give you two coordinates, that's not enough information.

Or have you got enough information? Can you tell me the missing two coordinates of this rectangle? Pause this video, make a suggestion to the person next to you, to your teacher, or maybe suggested to me on screen.

So the top left vertices, well it must be negative to in the X direction.

For the bottom right vertices, the X direction must be positive one.

The Y directions, well we've got a positive 4 and a -3.

And by just knowing two coordinates of that rectangle, you can tell me the other two coordinates.

Let's check that you've got that.

If I give you these coordinates on a rectangle, can you tell me the fourth coordinate that we must add in order to form a rectangle? Pause this video, give it a go.

The coordinate was -1, -4, <v ->1 in the X direction,</v> just like the vertices in the top left and then -4 in the Y direction, just like the vertices in the bottom right, thus forming that rectangle.

Some practise now, plot these coordinates.

What shape do they fall? That's the coordinates, 3, 1, -2, 1, <v ->2, -3, 3, -3.

</v> I'll give you a moment to do that.

Pause this video.

Question two, fill in the blank coordinates on these rectangles.

Pause this video, give it a go.