Lesson video

Hello, Mr. Robson here.

Great choice to join me for Maths today.

I love coordinates and I'm pretty sure you will too.

So our outcome for today is that we'll be able to plot coordinates in the first quadrant.

Keywords are gonna play a key role in this lesson.

Quadrant is that first keyword.

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

The Cartesian coordinate system I'm sure you've seen before, and I hope you can see the horizontal x axis and the vertical y axis.

They divide the coordinate grid into our four quadrants, our first quadrant, our second quadrant, our third quadrant, and our fourth quadrant.

They're quadrants.

Secondly, origin, where the x and y axes meet right in the middle.

The coordinate pair (0,0), we call that point the origin.

The next word is plotting.

When we draw on a graph or a map, that's known as plotting.

Today, we'll be plotting coordinates.

Finally, we'll be doing translations.

To translate a shape is to perform a translation, a transformation in which every point of a shape moves the same distance in the same direction.

Three stages to this lesson.

We're plotting coordinates in the first quadrant, we're using coordinates to plot shapes, and translating shapes.

Let's get going with plotting coordinates in the first quadrant.

This is a Cartesian coordinate grid.

We need to be able to label it with these key features, the x axis, the y axis, the origin, and the first quadrant.

Pause this video and tell the person next to you or tell me back on the screen where those fourth labels are going to go.

Firstly, we can label the x axis.

Then we can label the y axis.

It's really important whenever you draw a coordinate grid, you label your x and y axes.

Thirdly, let's label the origin.

It's useful to remember the coordinates of the origin (0,0).

And then finally, we can label the first quadrant.

It's this first quadrant that we're gonna plot coordinates in today.

Coordinates are a system used to define the position of a point on a Cartesian coordinate grid.

A Cartesian coordinate grid.

Cartesian is quite an unusual word.

In fact, it's a name.

It's a name from a mathematician called Rene Descartes, who was around some 400 years ago in France.

Descartes did something incredible.

He fused geometry and algebra.

Geometry is a Greek word.

Algebra comes from "al jebr." Al jebr being an Arabic word.

We've been using these elements of maths for hundreds of years, but independent of each other.

Rene Descartes took the two, fused them, and created the Cartesian coordinate grid, which has enabled some incredible advancements in mathematics, and particularly in science.

We wouldn't have planes in the sky or be talking about missions to Mars if this part of our mathematical learning journey had not happened.

So on a Cartesian coordinate grid, what language would you use to describe this position? Two along, three up? It is, isn't it? It's two a long and three up.

Well, no, 'cause that's too long, and it's three up, but it doesn't reach that position.

So do we need to be more specific? Two right and three up.

Right is more specific than along.

If you say along, along to the left, along to the right, two right is far more specific.

Ah, I see a problem.

That is two right and three up, yet it does not reach that coordinate position.

Do we need to be more specific? Again, the answer's yes.

Two right, three up from the origin.

When we talk about coordinates, we're describing their position in relation to the origin.

"Two right, three up from the origin" is a correct way to describe that position.

Could we describe it with even slightly more technical language? Could you describe it in a sentence using x and y, making reference to that x axis and that y axis? Pause this video.

Tell the person next to you or tell me back on the screen.

"Two in the x direction and three in the y direction" is better language than "two right and three up." In fact, to clarify even further, we'd be saying, "Two in the positive x direction, and three in the positive y direction from the origin." We've plot positive coordinates in the y direction and the x direction, they will appear in this first quadrant.

So quickly to check that you've understood what we've covered so far, which sentence best describes this coordinate? One along, four up.

One in the x direction, four in the y direction.

Four in the x direction, one in the y direction.

Or one in the positive x direction, four in the positive y direction.

Pause this video.

Tell the person next to you or tell me back on screen.

The sentence which best describes the coordinate is "One in the positive x direction and four in the positive y direction." One along and four up, not specific enough.

Is it right, is it left, that along? We need to be specific.

"One in the x direction, four in the y direction" does describe the coordinate, but I'd like to see maybe, "One in the x direction, four in the y direction from the origin." And I'd quite like the addition of positive x direction and positive y direction.

When you come to plot in all four quadrants in the future, the difference between the positive and the negative direction is going to be huge.

So let's start using the accurate language now.

"One in the positive x direction, four in the positive y direction" is good language to use to describe this point.

Let's go back to this coordinate, two to the right and three up from the origin.

Could you describe this position in a sentence using x and y? Well, I thought we'd already covered that.

Two in the positive x direction, three in the positive y direction.

Is there another way we can describe it using x and y? Have a think.

Tell the person next to you, tell your teacher, or tell me back on screen.

A way we could describe this is to say, "The x coordinate is two and the y coordinate is three." Notice now we've started at the point and we've read down to the axes.

Reading down to the x axis, we can see the x coordinate to be two.

Reading across to the y axis, we can see the y axis to be three.

When you come to use this part of mathematics in map reading, it's this way in which you'd read your coordinate position on a map.

It's a useful way to be able to read coordinates on a Cartesian coordinate grid, and in fact, it defines the way that we write coordinates.

We write coordinates with the x coordinate first and the y coordinate second.

So we'd call this coordinate (2,3).

The two represents the x coordinate.

That's the coordinate we're reading from the x axis And the three represents the y coordinate.

That's the coordinate we're reading from the y axis.

Let's just check you've understood that.

Which sentence best describes this coordinate? The x coordinate is five, the y coordinate is two.

The x coordinate is two, the y coordinate is five.

Or five right, two up.

Pause this video.

Give it a go.

We'd say the x coordinate is five, the y coordinate is two.

People do get mixed up with coordinates.

Some people would've said option B, but it's really important that we read the x coordinate from the x axis.

Go from the coordinate down to the x axis, it's a five.

The x coordinate is five.

And when we read across to the y axis, the y coordinate is two.

So that's the way that we write coordinates, the x coordinate followed by the y coordinate, and we use brackets, and we separate the x and the y coordinates using a comma.

It's really important whenever you plot a coordinate and label it, you use this format, brackets, x coordinate, comma, y coordinate, close those bracket.

All right, let's just check that you were paying attention there.

How do you correctly write a coordinate pair? x coordinate first, y coordinate second.

x coordinate second, y coordinate first.

Or either way.

It doesn't matter.

Pause this video.

Tell the person next to you or say it back to me on screen.

It's the x coordinate first and the y coordinate second.

It might help you to remember that the alphabet ends x, y, z.

The x comes before y in the alphabet.

The x coordinate comes before the y coordinate when we're communicating, writing, reading coordinates.

So plotting.

How do we plot coordinates? How do we plot the coordinate (3,1)? Well the x coordinate's three, so we go three in the x direction.

The y coordinate's one, so we go one in the y direction from the origin.

And we reach that point (3,1).

Once we plotted it, we can label it to correctly communicate the point that we've plotted.

If that's the coordinate (3,1), what's going to be different if we plot the coordinate (3,4)? That's a tiny change, but what will be the impact? Pause this video.

Tell the person next to you, tell your teacher, or say it back to me on screen.

The coordinate (3,4) still has an x coordinate of three, so we still travel three in the positive x direction from the origin.

The change was in the y coordinate.

It's no longer a one, it's a four.

So that journey in the y direction is longer, so we reach that point there.

We travel three in the positive x direction, four in the positive y direction.

We can plot that point, and importantly, label it correctly, brackets, x coordinate, comma, y coordinate, close brackets.

The x coordinate there remained the same.

We moved further in the y direction.

I'd just like to check you've understood that now.

Could you sketch a grid, zero to five on the x axis, zero to five on the y axis, and plot the coordinate (4,2)? You'll want to pause this video while you do that.

The coordinate (4,2), four in the x direction, two in the positive y direction, plotting there and labelling it (4,2).

If I said plot the coordinate (4,0), (4,0), where would that be? Pause this video.

Plot that coordinate.

So we still had four in the positive x direction, but then crucially, zero in the y direction.

Pay careful attention to this because it's one that students frequently get wrong when there's a zero for the x coordinate or the y coordinate.

In this case, four is in the positive x direction and zero means no movement in the y direction.

So the coordinate remains there on the x axis.

Make sure you've got that in the right position and make sure you've labelled it (4,0).

Plot the coordinates (2,5), (5,2), and label them.

Pause this video.

Give that a go.

They should be here, two corners plotted and labelled (2,5) and (5,2).

If you can spot the difference between this, it'll enable you to be really accurate in your plotting and reading of coordinates.

What's the same? What's different about the two? Not the same.

There's a two under five.

The difference is crucial.

The order in which we see those two and five within our brackets determine the direction of the two and the five.

For the coordinate (5,2), we have a positive five in the x direction, positive two in the y direction.

For the coordinate (2,5), it's a positive 2 in the x direction and a positive five in the y direction.

Let's check you've got that.

Andeep plots coordinate (1,2).

What has he done wrong? Pause this video.

Explain to the person next to you, explain to your teacher, or say it back to me on screen.

Well spotted.

He moved two in the x direction and one in the y direction.

He's accidentally plotted the coordinate (2,1) instead of the coordinate (1,2).

I hope you can see the difference between those two coordinates.

Okay, time for some practise now.

Task A, question one.

The coordinate (0,0) where the x and y axes cross is called the.

One of our keywords for today is required there.

Insert that keyword into that blank space.

Next, you see the first quadrant for values up to x is five and y is five.

I'd like you to plot these coordinates and label them, the coordinates (3,5), (5,3), (5,5), and (0,5).

Plot those and label them.

Pause this video.

Give that a go.

Question three.

Izzy has plotted the coordinate (3,0).

Write a sentence to explain her error to her.

Pause this video.

Give that a go.

Okay, time for some feedback.

The coordinate (0,0) where the x and y axes cross is called the origin.

The origin, really important word in coordinate geometry that one.

Plotting these coordinates and labelling them.

(5,5), five in the positive x direction, five in the positive y direction.

(5,3) is five in the positive x direction, three in the positive y direction.

Whereas (3,5) was three in the positive x direction, five in the positive y direction.

(0,5), zero followed by five.

If zero is the x coordinate, we have no movement in the x direction, and then a movement of five in the positive y direction.

Then you plot that point there, you will notice that point sits on our y axis.

Points that are set in our y axis are gonna become really, really important in your future learning on coordinate geometry, so pay particular attention to that coordinate (0,5).

Back to Izzy.

She had plotted the coordinate (3,0) and I asked you to write a sentence to explain her error to her.

Your sentence might have read, "(3,0) is three in the positive x direction and then zero in the y direction." Izzy did it the other way around, zero in the x direction, three in the y direction.

You might have taken her book and corrected her and shown her that her coordinate is (0,3) and where the coordinate (3,0) would rightly have been plotted.

Okay, that was plotting coordinates in the first quadrant.

I hope you enjoyed it.

Next, using coordinates to plot shapes.

Plot these coordinates.

What do you notice? You might want to draw a grid and plot them on paper or on our whiteboard.

You might want to just point at the screen and say, "That coordinates there.

That coordinates there." Pause this video.

Give it a go.

There's our four coordinates.

Did you notice something? They formed a square.

They formed the vertices, the corners of a square.

What about these coordinates? Once you've plotted them, you notice the same thing happens.

A square, different position, different size of square, different area, but a square nonetheless.

So which additional coordinate would you need to plot here to form square? Pause this video.

Tell the person next to you or tell me back on screen.

I hope you plotted that coordinate and said we'd require the coordinate (4,1) and communicated it correctly, brackets, x coordinate first, y coordinate second.

We need the coordinate (4,1) to plot a square.

Plot these coordinates and see what you notice, (2,1), (4,3), (2,5), (0,3).

Plot those.

Pause this video.

Give it a go.

So something's changed.

They form a square, but it's a tilted square.

So where should we plot point D to make sure A, B, C, D forms a square? Pause this video.

Tell the person next to you or say it back to me on screen.

We need to plot D at (2,0) in order to form a square, and you would've noticed that square 2 was tilted.

Okay, let's practise that.

I've given you two coordinates, the coordinates (3,2) and (5,2).

They're already plotted for you.

What other pair of coordinates would we need to plot in order to form a square? Pause this video.

Give it a go.

Okay, feedback.

We could have plotted those two coordinates at (3,0) and (5,0) and formed that square.

But that wasn't the only answer, was it? We could also have plotted those two coordinates, (3,4) and (5,4) and also made a square.

So there were two answers, yeah? I hope you are shouting at the screen, "I have a third option." Well done if you do.

We could apply those two coordinates and found a tilted square.

Three answers to that one.

We could also have had (4,1), (4,3), and made a square which was tilted.

Okay, that was an introduction to how we can use coordinates to plot shapes.

Now we look at how we can translate shapes.

Okay, the coordinate (1,2).

What will happen if we add three to only the x coordinate? We currently have an x coordinate of one, y coordinate of two.

If we add three to the x coordinate, what's going to happen? Pause this video.

Say to the person next to you, tell your teacher, maybe say it back to me on screen.

When we add three to the x coordinate, the one which was the x coordinate becomes four, we get the point (4,2).

Crucially, what's happened by adding three to the x coordinate, the point has translated three to the right, or to say it a bit more accurately, three in the positive x direction.

We've performed a translation of that coordinate.

So what would happen if we added three to the y coordinate instead.

I like to test mathematical intuition in my students, so pause this video and make a suggestion to the person next to you, to your teacher, back to me on screen.

It doesn't matter if your suggestion is wrong.

What do you think will happen if we add three to the y coordinate instead? The x coordinate of one doesn't change.

The y coordinate of two becomes a y coordinate of five and we've translated that point, three up, or three in the positive y direction.

Okay, let's check that sunk in.

If we add positively to the value of the x coordinate, a point will translate in which direction? The positive y direction, the positive x direction, or diagonally.

Pause this video.

Tell the person next to you.

It's a positive x direction.

It was like that first example we saw.

When we add to the x value of the coordinate, we move in that positive x direction.

Oh, shapes.

This time plotting three coordinates to form a triangle.

Triangle ABC has the below coordinates, (1,5), (1,2), (2,2).

What happens if we add three to each x coordinate? Well, those coordinates will change to (4,5), (4,2), (5,2).

But what if we plot those coordinates? What impact will that have on triangle ABC? Pause this video.

Make a suggestion to the person next to you or a suggestion back to me on screen, and I don't care if your suggestion's wrong.

Let's try and suggest what might happen.

We get these coordinates, and when we join them to make our triangle, we can see that triangle ABC has been translated by three in the positive x direction.

Point A has moved three in the positive x direction, point B has moved by three, point C has moved by three.

By adding three to each of the x coordinates, we've translated our shape three in the positive x direction.

Let's start again and look at this slightly differently.

Triangle ABC has the below coordinates, (1,5), (1,2), (2,2).

That's the same triangle.

But what if we don't add three to all the x coordinates? What if I said, "Let's leave that first coordinate the same and add three to the x coordinate for coordinate B and coordinate C?" What impact is this going to have when we plot these coordinates? Pause this video.

Make a suggestion to the person next to you, to your teacher, or maybe to me on screen.

Point A didn't move.

Point B and C translated three to the right.

So when we now join a triangle together, wow, that's changed.

We had a right triangle.

We now have a very scalene triangle indeed.

Point B and C translated three to the right.

Point A didn't move.

So we no longer have a translation of our shape.

We have to change all the coordinates by the same rule in order to successfully translate our shape.

Let's just check your understanding of that.

True or false? Changing all the coordinates of a shape will translate it.

Changing all the coordinates of a shape will translate it.

Is that true? Is that false? I would like you to justify your answer whether you think it's true, whether you think it's false.

Would you justify by saying, "Yes, because all the vertices will move?" Or would you justify it by saying, "Yes, but only if you follow the same rule for each coordinate?" So, true or false? And please justify your answer.

Pause this video.

Give it a go.

The justification should be, "Yes, but only if you follow the same rule for each coordinate." If you try to justify with, "Yes, because all the vertices will move," we have to specify that movement.

Remember, a lot of this lesson has been about being specific.

If I changed all your vertices, I could move them in the x direction, the y direction, the positive x direction, the negative x direction.

by adding or subtracting from its coordinates.

They have to follow the same rule.

So it's justification B, need the same rule for each coordinate.

Time for a practise task.

Why will the below not create a translation of rectangle ABCD? Rectangle ABCD, you can see plotted on the coordinate grid.

If we change those coordinates to (4,5), (3,1), (5,1), (5,5), why will that not create a translation? Pause this video.

Give that problem a go.

You could have this by plotting them.

If we plot the point at (4,5), (3,1), (5,1), and (5,5), we see that the four points did not follow the same rule.

Most of them moved by three in the positive x direction, but one of them only moved by two in the positive x direction, which changes everything about our shape.

We've gone from having a rectangle to having a trapezium.

It's really important that you remember that the points, if we're going to translate a shape, have to follow the same rule.

In summary, today, coordinates can be plotted in the first quadrant by following the positive x direction and positive y direction from the origin.

And a set of points can be used to create shapes.

Those shapes can be translated by applying the same rule to each coordinate.

I hope you've enjoyed today's lesson.

I thoroughly did.

See you again soon.