Lesson video

Hello there, my name is Mr. Tilstone.

It's great to be with you today.

I hope you're having a lovely day.

Let's see if we can make it even better by having a successful maths lesson.

Today's lesson is all about coordinates and I imagine you are getting to be quite the coordinates expert, you've got lots of skills and lots of knowledge, and hopefully feeling very confident about the whole thing.

Let's see if we can put that to the test with some problem solving situations.

If you are ready, I'm ready, let's begin.

The outcome of today's lesson is I can solve problems involving missing coordinates.

Our keywords are as follows: My turn, congruent, your turn.

And then my turn, vertex, your turn.

That's the singular and the plural, my turn, vertices, your turn.

Let's have a little reminder about what those words mean.

If two shapes are the same shape and size as each other, they are congruent and you can see an example there.

And the vertex is where two or more line segments meet.

The plural is vertices.

Our lesson today is split into two parts, two cycles.

The first will be used known coordinates to find unknown coordinates, and the second find missing coordinates of congruent shapes.

Let's begin by focusing on using known coordinates to find unknown coordinates.

And in this lesson, you're going to meet Aisha and Andeep.

Have you met them before? They're here today to give us a helping hand with the maths.

What is the same and what is different about these coordinate grids? Have a good look.

Do what good mathematicians do which is to notice things.

What can you notice? Anything the same? Anything different? Well, Aisha says, "Both have axes labelled from -5 to 5." Did you notice that the x and y axes are both labelled that way? And then Andeep says, "One has grid lines," the one on the left, "and one does not." Did you spot that, the one on the right? What about these two coordinate grids? What's the same and what's the difference? So same again, please, see what you can notice.

How do they differ? How are they similar? Well, Aisha says, "Both have -5 and 5 labelled on each axis." Did you spot that? Yep, those numbers appear on both.

Andeep says, "One shows the values in between and one does not." So the one on the right doesn't show the values in between 0 and 5.

They're there but it doesn't show them.

We can say that this grid is partially labelled and lots of our grids today are going to look just like this partially labelled.

So this is an example of a partially labelled grid.

The coordinates of point a have been given.

So point a is at 4, 2.

It's in the positive-positive quadrant.

Is it possible to work out the coordinates of point b? Hmm, so have a look at that, what do you know? What can you work out from that information that's been given, hmm? Andeep says, "The x-coordinate is already known." Did you spot this? "The point is on the y-axis, so it is 0." It's 0 across if you like.

And Aisha says, "The line through both points is horizontal, so the y-coordinates of both points are in line with each other.

They are the same!" Did you notice that? Let's have a look at that.

Here we go.

So that forms a horizontal line, so they must be the same as each other.

So what must it be? We've got that information already.

That must also be 2 up.

So it's 0, 2.

Here is one vertex of a square.

We need more information, don't we? We need to know something about some other vertices.

What about this? Here's another vertex of the square.

Aisha says, "Part of this coordinate is known and part is unknown.

What do we know?" Hmm, have a look, see what you think.

What coordinate do we know or can we figure out based on its position, based on what we know about the first coordinate that was given.

Well, let's draw that horizontal line through them again.

Does that help? Is that more of a clue? The y-coordinates must be the same.

Well, we've got the y-coordinate, haven't we? Of that first vertex.

So we should know the y-coordinate of this.

So therefore, it's something, 2.

We are told the side length of the square is 2 units.

How does this help us find the missing x-coordinate? So if we know this square's got a side length of 2 units, I think we can then figure out that other coordinate there, the missing one.

Aisha says, "The missing x-coordinate must be 2 further than the known coordinate." And then a nice simple sum here, 1 + 2 equals 3.

So it must be 3, 2 'cause it's 2 along from that 1, there we go.

So that's got coordinates of 3, 2.

Here's a different vertex of the square.

Hmm, I wonder if we can figure this one out.

Part of this coordinate is known just like before and part is unknown.

What is known? What can you see that's known? What about this? The vertical line runs through these coordinates, so the missing x-coordinate must also be 1.

They're the same distance across.

So that's 1, something.

Now can we figure out that missing y-coordinate? Have you got any ideas? What do we know about this square? I'll give you a hint.

What do we know about the side length of the square, hmm? Is there enough information to work out the missing y-coordinate, what do you think? Aisha says, "The side lengths of the square are each 2 units." Yes, we were told that, weren't we? "So the y-coordinates must be 2 units apart." Hmm, so here we go.

These are 2 units apart.

2 + 2 equals 4.

So the missing y-coordinate must be 4 because it's 2 away from the 2.

So there we go.

So that has got coordinates of 1, 4, that vertex.

Is there enough information to work out that final coordinate? What do you think, hmm? I think there is.

And Aisha agrees.

"Yes!" She says.

"It is on the same vertical line as 3, 2, so the x-coordinate must also be 3." So we knew that already.

We didn't have to do any working out.

See if you can apply that same logic to work out that missing y-coordinate.

Is that on the same line as anything? It is, it's on the same horizontal line as 1, 4.

So the y-coordinate must therefore be 4.

That makes that 3, 4.

So when we broke that up into little steps, we could figure out what the missing coordinates were.

The final coordinate of the square is 3, 4 and that is the square.

It took some thinking about what we got there.

Well, let's have a check.

If you know the x and y-coordinates of vertex a, so have a look at that, vertex a, what other coordinates do you know? So you know the x-coordinate of vertex, mm, which one? And you know the y-coordinate of vertex, mm, which one? See if you can figure that out.

If you've got a partner with you, discuss that.

See if you can come to a consensus on that one.

Pause the video.

Did you manage to agree on an answer here? Let's have a look.

So you know the x-coordinate of vertex d, they are the same.

And you know the y-coordinate of vertex b, they are the same.

With partially labelled axes, coordinates can be used to transform a shape.

What are the new coordinates of the shape if it is translated 5 units to the left and 3 units down? So hopefully you've got a bit of experience and a bit of confidence with translation.

See if you can figure this one out.

Well, Andeep says, "5 units to the left means each x value will be 5 units less." Hmm, so look at the x values.

Can you subtract 5 from each? This is where it will be after that first translation.

And then, 3 units down means each y value will be 3 units less.

So see if you can subtract 3 from each of the y values.

So that's where it will be.

But what are the coordinates? Andeep says, "I need to subtract 5 from each x value and 3 from each y value in every-coordinate." And when you do that, here are the coordinates that you get.

So that 1, 4 becomes -4, 1.

The 3, 4 becomes -2, 1.

The 1, 2 becomes -4, -1.

And the 3, 2 becomes -2, -1.

It's time for some practise.

Number 1a, find and write the missing coordinates of the rectangle.

Complete the sentences to support your thinking.

A, mm, line runs through points a and c and points b and d.

So what kind of line? And a, mm, line runs through points a and b and point c and d.

What kind of line? Think about the kinds of lines we've looked at so far when two coordinates have been in line with each other.

And b, translate the rectangle 3 units to the right and 1 unit up.

What are the new coordinates of the shape? Number 2, two coordinates of a square are 2, -3 and 7, -3.

What could the other two coordinates be? Remember, it is a square.

Think about the properties of a square.

Think, how long is the length of each side? Sketch the squares you could make on the grid and write the coordinates of the vertices.

And finally question 3: An isosceles triangle has two coordinates of -2, 3 and 2, 3.

And you can see that on the image.

This gives the one side with a different length.

What could the third coordinate be to show the two sides of the same length? And is there more than one answer? Sketch the triangles that you could make on the grid and write the coordinates of the vertices.

Good luck with that.

If you can work with somebody else, please do and share ideas with each other.

Pause the video and I'll see you soon.

Welcome back, how did you get on with that? Would you like some answers? Let's go.

So number 1a, a vertical line run through points a and c and points b and d and a horizontal line run through points a and b and points c and d.

And that's going to help us to find those missing coordinates.

So we've got -4, 6 in the top left.

2, 6 was already given in the top right.

2, -4 in the bottom right and -4, -4 in the bottom left.

And then when we translate that rectangle 3 units to the right and 1 unit up, the new coordinates are as follows! That's what it would look like, and they are the coordinates.

So 3 units to the right means we add 3 to the value of each x-coordinate.

And 1 unit up means we add 1 to the value of each y-coordinate.

And two coordinates of a square are 2, -3 and 7, -3.

What could the other two coordinates be? This is one possibility.

So in this case, the side length is 5, so we're going to use that to help us.

Each side is 5 units long.

That was one possible square.

You might have drawn one with a given line as a base of the square rather than the top of it.

So there are different positions that that could be in.

And last but not least, question 3, an isosceles triangle, so two sides of the same length and one different, has two coordinates of -2, 3 and 2, 3.

So the third coordinate could be anywhere on the y-axis as that is halfway between the given points.

So the coordinate will therefore be 0, and then any number.

So well done if you've got one possibility and well done if you've got more than one possibility.

You're doing really well! Lots of thinking involved in this lesson, isn't there? The key, the secret is to take your time and think about what you're doing and notice what you can.

Let's move on to finding missing coordinates of congruent shapes.

These triangles are congruent.

They are the same as each other.

Is there enough information to work out the missing coordinate? Have a look, see what you can notice.

See what coordinates we already know, what coordinates we already have, hmm.

Well, we could draw this vertical line.

That would help.

So what coordinate do we know there? These x-coordinates are the same and we know the x-coordinate, so that's also -4.

Now have a look at this side length.

This side of the triangle goes from 0 to 5 on the y-axis.

The same side length on the other triangle will give a y-coordinate of -5.

So -4, -5 'cause it's got a side length of 5 and it goes from 0.

So there we go, we can use that to help us with this.

That's gotta be -5, it's 5 down just as other one was 5 up.

These y-coordinates are the same.

We can draw a horizontal line to show that.

So they must be the same.

And we know the y-coordinate of that vertex.

So therefore, the y-coordinate of that vertex, it's the same, it's -5.

Just one to go.

The shortest side of the triangles is 2 units longer.

We know that by looking at the original triangle, it went from 0 to 2.

So it's 2 units long.

So therefore, this one's got to be 2 units long as well.

And it starts at -4, so it's -4 + 2.

The x-coordinates must have a difference of 2, so it must be -2, -5.

So that seemed complex, but when we broke it down into little steps and noticed things about those congruent triangles, it became a lot easier.

And just as Aisha says, "We did have enough information!" Two congruent rectangles this time have been placed onto the coordinate grid and the coordinates of some of the vertices are given.

Can you work out the missing coordinates? So have a good look.

See what you can notice.

Think about the side lengths.

Think about what's in line horizontally or vertically with known vertices and coordinates.

"The x and y-coordinates of this vertex are already shown elsewhere on the grid!" They are, so we can fill them in.

That's probably where I would start.

So they are the same.

So if that's 8, that's 8, and these are the same.

So if it's 7 on the known one, it must be 7 on this one too.

So that's at 8, 7.

Okay, we're getting there.

That's one hurdle completed.

Can we work out the side lengths of the shape to find the other coordinates of the rectangles? Hmm, is there enough information for that? Well, the long side starts at 0 and ends at 7.

So it's 7 units long.

Let's have a look at the short side.

The short side starts at 3 and ends at 8.

So it's 5 units long.

Aisha says.

"Let's add in the side lengths to help us work out the missing coordinates." So there we go, so we write all of those in.

That's going to help us work out what those coordinates are.

So there's 7 and 5.

Okay, what about this one? This point must be 3, 0.

3 + 5 equals 8, so the side length is correct as well.

So that's at 3, 0.

That means a coordinates of this vertex must be -4, 0 because 3 subtract 7 is -4.

It's on the x-axis and makes alongside 7 units.

So that's -4, 0.

We're getting there.

And what about this one? So we know how long the short sides are, how many units.

This vertex is above the last one.

So it has the same x-coordinate of -4.

So they're in line, they're in a vertical line.

That's gotta be -4.

The y-coordinate must be 5 to make the short side 5 units long.

The coordinates are -4, 5.

So that's what I mean.

When we broke that down into little tiny steps, it worked.

It was doable.

Let's have a check.

Two congruent rectangles have been placed on a coordinate grid.

Is there enough information to work out the missing coordinates? Pause the video.

Exchange ideas with a partner if you possibly can and give that a go.

What did you think? Was there enough information there or not? No, there wasn't.

The short side of the rectangle has been found.

It's 3 units long.

We can work that one out, but the long side has not yet been found.

There isn't enough information for that.

One more set of coordinates is needed from one of these two vertices.

Either of those green vertices there, and then we can work it out.

It's time for some final practise.

Number 1, the triangles are congruent, so they're the same.

Find and write the missing coordinates.

So remember to look for coordinates that are on the same vertical line as each other, coordinates that are on the same horizontal line as each other.

Remember to think about the side lengths, what's the short side of the triangle, et cetera, et cetera.

Number 2, the rectangles are congruent, they're the same.

Find and write the missing coordinates.

Number 3, the squares this time are congruent.

Find and write the missing coordinates.

Okay, this is challenging in the nicest possible way.

This is going to make you think.

So do take your time with it, persevere with it, persist with it, and I promise you'll get there.

Pause the video add off you go.

Welcome back, how did you get on with that? Let's have a look.

So the triangles are congruent, they're the same.

Find and write the missing coordinates.

Here they are! A vertical line runs through these x-coordinates, so they're the same.

The triangle in the first quadrant has a vertical side that goes from 0 to 7 on the y-axis.

So it's 7 units longer and we can apply that information with the other triangle as well.

The vertical side of the other triangle will therefore go from 0 to -7 on the y-axis.

And the triangle in the first quadrant has a short side that goes from 0 to 3 on the x-axis, so the short side is 3 units long.

<v ->8 + 3 equals -5.

</v> Number 2, the rectangles are congruent.

Find and write the missing coordinates.

Some coordinates are in line with other known coordinates, so are the same.

So we know that that one's 7, it's got to be.

That one's 10, it's got to be.

This one's -3 because it's in line with the other -3.

The long side of the rectangle goes from 0 to 10, so it's 10 units long.

And we can use that information.

7 subtract 10 equals -3.

So that must be -3.

So that must be -3 as well 'cause it's in a vertical line.

The final coordinate is known as it's on the x-axis already, so that's 0.

And finally number 3, the squares are congruent.

Find and write the missing coordinates.

So you can look for x and y-coordinates that are on the same lines as others.

And I think you're probably getting really good at that by now.

You can also work out that the side length of the square is 5 units because the difference between 1 and 6 is 5.

And check that that is true with the coordinates that you have found.

So let's have a look.

This has got to be 3 'cause it's in a vertical line with the other 3s.

That's got to be 6 'cause it's in a horizontal line with the other 6.

That's got to be -4 because we know the side length is 5 and that's 5 down from 1.

1 subtract 5 is -4.

So that's got to be -4 as well because it's in a horizontal line with the other y-coordinate.

That's got to be -2 because 3 subtract 5 is -2.

And therefore, that must also be -2 because it's in a vertical line with the other -2.

We've come to the end of the lesson.

Today's lesson has been solving problems involving missing coordinates.

I've really enjoyed the high level of challenge in today's maths lesson and I hope you have to.

Coordinate grids are not always presented with grid lines and the axes are not always fully labelled, and that's been the case with your coordinate grids today.

When this is the case, it's essential to consider the known coordinates to work out missing coordinates.

Missing coordinates sometimes appear on the same horizontal or vertical line as known coordinates, so they are the same.

And personally, that's the part that I find the easiest when there's missing coordinates.

Sometimes, we need to use known vertices to calculate unknown side lengths when working out missing coordinates.

You've been amazing in today's lesson.

You've really pushed yourself and you've worked so hard, fantastic! Give yourself a very, very well deserved pat on the back and breathe a huge sigh of relief.

Your math lesson is over.

I hope to get the chance to spend another math lesson with you in the near future.

But until then, enjoy the rest of your day.

Whatever you've got in store, be successful at whatever that is.

Take care and goodbye!.