Lesson video

Hello, my name is Mr. Tilstone.

It's really great to see you today.

I hope you're having a good day.

Today, we're going to be looking at coordinates.

And I bet you are getting pretty good at coordinates.

I bet you know quite a bit already.

Well, let's see if we can take that knowledge even further.

Today, we're going to be focusing on something called translation with coordinates.

So if you are ready to begin, let's do that.

The outcome of today's lesson is this, I can draw and translate simple shapes on the full coordinate grid.

You might have had some previous experience with translation.

And that's our keyword.

So my turn, translate, your turn.

My turn, translation, your turn.

So they're different forms of the same word.

Let's find out what they mean.

Here's a reminder.

Translation is a type of transformation.

So you may have come across different kinds of transformation such as reflection, rotation.

This is one type of transformation.

Every point of a shape moves the same distance in the same direction.

And we've got an example of that there.

So have a look at what you notice there.

This is translation.

Our lesson today is split into two cycles.

The first will be translation across the y-axis and the second, translation across the x and y-axis.

So for now, let's begin by looking at translation across the y-axis.

And in this lesson you're going to meet Alex.

Have you met him before? He's here today to give us a helping hand with the maths.

What can be said about shape B? Have a look at that.

What do you notice? That's what good mathematicians do, they notice things.

What can you say about it? Anything at all.

Well, you might have said it's congruent.

Remember that word? Congruent to A as they are the same shape and size.

Did you notice that? Its position on the y-axis is the same.

B is not further up or down than A.

So the y-coordinates are the same.

Its position on the x-axis is different.

A is further to the left and B is further to the right.

Well done if you said any of those things.

Translating a shape means moving it without flipping or rotating it.

In this case, shape A has been translated to the right to shape B.

There are other directions and we're going to explore that.

But this one is to the right.

So here's shape A.

And we can take that exact same shape and sort of slide it along and overlay it onto shape B.

And you can see they are exactly the same.

They are congruent.

And they haven't been flipped or rotated or anything like that, just moved to the right.

That's translating.

Now this time, a four-quadrant grid has been used and I'm sure that by now you're getting very familiar with and very confident with four-quadrant grids.

Now, this is not a translation.

Can you see why it's not a translation? Well, the two shapes are not congruent.

Can you see why? The shape is the same.

They're both right angled triangles, but the size is different.

Well done if you spotted that, the side lengths are different.

And if we have a look at this shape as it moves to the right, you'll see it doesn't overlay that shape perfectly.

So it is not a translation.

No.

What about this one? Have a look.

Do you think this is a translation? See what you can notice.

No, it's not a translation because the two shapes are not congruent.

Can you see what's stopping them being congruent? They look very similar, don't they? But they're not, the angles are different.

So here we go.

If we slide the shape along to the right, you can see it doesn't quite overlay it because of those slightly different angles.

So that's not a translation.

It almost was, but it's not.

No.

What about this one? Do you think this one is a translation? What do you think? Hmm.

Try and visualise that shape on the left being slid to the right.

No, it's not a translation.

Even though the two shapes are congruent.

That is a reflection, which is a different type of transformation.

It's a reflection but it's not a translation.

And let's see that.

Here we go.

So we start here and we slide along and you can see no, it doesn't overlay it.

So that's not a translation.

That doesn't fit the criteria of a translation.

No.

What about this one? What do you think? Would you say this is a translation? Hmm.

No, it's not.

Even though the two shapes are congruent, that's not a translation.

Let's have a look.

It's a rotation.

So it's had some kind of transformation but it's not a translation.

And you can see, as we slide that shape over, that's not a translation.

No.

There are various ways to describe the amount of space involved in the translation.

Could say the shape has been translated two places to the right.

You could say that, it's correct, but it's a bit informal.

That word places is a little bit informal.

It's not wrong.

What about this? The shape has been translated two units to the right.

Again, that's correct.

It's a little more mathematically precise to say units.

Units is a useful general term, meaning one of something.

And the shape has been translated two to the right.

So we're not saying places or units, just two to the right.

Think that's correct? Yeah, it is.

It's concise but it's still mathematically acceptable.

So you can say it without the word units, that's fine.

This time the shape is on a four-quadrant coordinate grid.

The shape is translated four units left this time.

So we've looked at translations to the right.

This time it's being translated to the left.

Now one strategy is to translate each vertex.

So that shape's got three vertices and we're going to translate all three of them.

So here's one of the vertices and we're going to start translating it.

One place.

Two places.

Now let's pause there, see what you've noticed.

Notice that the translation has crossed the y-axis so it doesn't skip over it, it crosses it.

That's included as part of the translation.

Let's carry on.

So do remember, this is really important.

The axis counts as a point where X is zero.

So needs to be included in the steps of the translation.

Don't skip over it.

So let's carry on.

Three and four.

So now it's been translated four places, four units to the left.

Here we go.

Let's put a little mark there.

A little dot so that we know where it is.

And then we can move on to a different vertex.

We're going to do the same thing.

One, two, three, four.

And we didn't really need to count because we could use the other vertex and we know it's in a vertical line.

But we've counted anyway, it's fine.

Let's put a dot there.

One more vertex to go.

And it's this one.

One, two, three and four.

Put a dot there.

So we've translated each of the vertices.

All we've got to do now is join the vertices together and we've got a perfect translation.

Four units to the left.

Alex translates a different shape across the y-axis.

Do you agree with Alex? He says, "I have translated the shape five units left." Hmm, have a look.

Is that five units to the left? What do you think? What do you notice? Look where his number one is.

Hmm, I don't think he's right there, do you? Alex counted the vertex the first step on the translation, but it's not, that's incorrect.

You don't count that as the first one.

That's where it should be.

Now it's translated five units to the left.

So he's had another go at translating across the y-axis.

And this time he says, "I have translated the shape four units left." Do you agree? What do you think? There's no numbers there.

Have a look.

Has he translated four units left? Hmm.

I can see an issue.

Look at the vertex where he started.

Look at the vertex where he's finished.

Are they the same? Mm-mm.

Alex has translated the top left vertex to the top right vertex.

So it's not the same vertex at the moment.

The shape is translated five units to the left.

Alex can check by looking at one of the vertices from each shape and looking at the numerical difference between the x-coordinates.

So originally this vertex was at two, four and now it's at negative three, four.

Hmm, so think about the difference between two and negative three.

"I translated the shape five units left." Because the difference between two and negative three is five.

So the coordinates themselves are a good way to check.

The y-coordinate is still the same, but the x-coordinate is five less.

When a point is translated horizontally, the x-coordinate changes.

When a point is translated left, the x-coordinate decreases.

When a point is translated right, the x-coordinate increases.

Alex is translating this shape five units right.

He uses a different strategy.

If you've got a different strategy, what else would you do or could you do to translate that? We've looked at the strategy of going from vertex to vertex and plotting all of the vertices.

What else could you do? He says, "I'm going to translate just one vertex and then form the rest of the shape around it." Okay, so he is going to make a congruent shape based on one vertex.

So he is picked one of the vertices, could have been any of those vertices.

He's picked that top right one and he's going to translate it five units to the right.

So at the minute it's on negative one.

Think about what five more than that would be.

It'll be four.

So that's been translated five units right.

Just that vertex.

And then Alex must ensure congruence.

Each side length in the original shape must be exactly the same in the translated shape.

So here we go.

So that side length is the same and that's the same as it's corresponding side length in the other shape.

And so is that, and then we join back to the start.

And would you agree, we've got a congruent shape that's been translated? So I quite like that strategy, that's good.

I think that's saved some time.

What do you think? Well done, Alex.

Okay, we've got rectangle.

This shape is translated five units left.

What will the coordinates of the new shape be? Hmm, what do we think? How could we find out? What strategies have you got here? How will the y-coordinates change here? How will the x-coordinates change? Well, let's use a table and let's investigate that.

What do you notice about the coordinates? So we've got the original shape in the left-hand column and because the rectangle is translated, five units left the x-coordinate and the translated shape will be five less.

So the x-coordinate has been coloured purple here on your screen.

We're going to make each one five units less.

Subtracting five from each x-coordinate will give the translated x-coordinate.

So let's see how good you are at subtracting from positive numbers to give negative numbers.

So two, subtract five.

The y-coordinates will not change as the shape is not being translated up or down.

So they will remain the same.

So here, we are gonna start with negative three, four.

What about the next one? Five less than four.

Negative one, four.

How about the next one? Five less than two.

We've already done that one, haven't we? That's negative three, one.

And five less than four, we've already done that one too.

That's negative one, one is the translated coordinate.

So here we go.

That's our original shape, that's our translated shape and that's where it would appear on our four quadrant grid.

Here's negative three, four.

Here's negative one, four.

Here's negative three, one.

And here's negative one, one.

Let's do a little check for understanding, shall we? Let's see how much you've taken in there.

Describe the translation from A to B.

Pause the video and have a go at that.

How did you describe it? There's a few ways that we could describe it.

I wonder which one you came up with.

Well, we could pick any of the vertices, any of the coordinates and see how it's been translated.

So let's start with that one.

That's been translated negative one to three.

So shape A has been translated four units to the right.

And Alex says, "It doesn't matter which vertex you start from." We could have said the same thing that that vertex has been translated four units to the right.

So therefore the whole shape has been translated for units to the right.

It's time for some practise.

Number one, describe each translation.

And you've got a stem sentence to help there.

So A to B is mm units to the left or to the right.

So you can circle the correct word there, left or right, and fill in the gaps with a number.

And then C to D mm units to the mm.

So this time you've got to write the number and the word.

And then E two F, you've not got any help at all.

But you can use the stem sentences above to help you.

And number two, translate each shape according to the command.

So A is going to be three units to the right, B, five units to the right, and C five units to the left.

Number three, a square has the following coordinates and the square is translated five units to the left.

So think about what's going to change about those coordinates and how it's going to change.

What are the new coordinates? I don't think you need a coordinate grid for that.

Just think about the coordinates.

And then number four, a right-angled triangle has the following coordinates.

So this is a different shape this time.

A right-angled triangle.

The triangle is translated seven units to the right.

What are its new coordinates? Good luck with all of that.

Work with somebody else if you can.

And share ideas, and strategies, and all that kind of thing.

Pause the video and I'll see you soon for some feedback.

Welcome back.

Let's give you some answers.

So from A to B, that's two units to the left.

C to D is eight units to the left.

And E two F is three units to the right.

So well done if you've got those.

Number two, translate each shape.

A was three units to the right, that's how you needed to translate it and that's what that looks like.

B, five units to the right, looks like that.

And then C, five units to the left, looks like this.

And well done if you drew your shapes in their translated positions.

Number three, a square's got those coordinates.

The square is translated five units to the left.

What are its new coordinates? So each of those x-coordinates is going to decrease by five.

The y-coordinate isn't going to change.

So that gives you negative three, negative two, negative one, negative two, negative three, negative four, and negative one, negative four.

And number four, a right-angled triangle has the following coordinates.

The triangle is translated seven units to the right.

What are its new coordinates? So once again the y-coordinate is not changing but the X one is.

So those negative x-coordinates are going to become positive, they're going to be increasing by seven.

So that leaves us with two, negative one, two negative, five, and five, negative five.

Very well done if you got those.

And if you did, you are definitely ready for the next cycle.

And that is translation across the x and y-axis.

How could you describe the movement of the triangle from A to B? What do you think? Have a look at that, from A to B.

As well as left and right, translations can also involve up and down movements.

And this one does.

A has been translated five units down.

And there's lots of ways we can check that.

You can take any of those vertices and count to the corresponding vertex.

The triangle has been translated across the x-axis, so it's gone across that x-axis.

How can that translation from A to B be described? Have a look at that.

Hmm.

How would you describe it? This is what Alex has to say.

He says, "I can just imagine it's sliding down like a diagonal line." But translations can have more than one instruction and translation can only happen horizontally or vertically.

So Alex needs a rethink there.

What else could he say? He says, "Ah, I need a two step strategy then, I think." I think you do, Alex, yes.

So now that we know that you can't translate diagonally, how can the translation from A to B be described? Hmm.

Alex says, "I'm going to use this vertex here to think about my translation." So he's going to start with this one.

The top of that triangle A.

"First I'm going to visualise the horizontal movement needed." So that is where the translated horizontal vertex would be.

The vertex has been translated five units right.

"Now I can think about the vertical movement." And what could we say about that? So we started by translating it five units to the right.

Now what? So that's where that same vertex is on the new shape.

The vertex has been translated six units down.

So there's been more than one part to that translation.

Five units right, six units down.

"I can say the translation is five right, six down." How have the coordinates of the vertex changed? Let's have a look.

So the original coordinates were negative four, four.

And the new coordinates one, negative two.

So this time you can see X and Y have both changed.

The x-coordinate has increased by five because the translation was five units to the right.

The y-coordinate has decreased by six because the translation was six units down.

So both have changed.

Shape A is translated five units left and six down.

What is new position? What strategy have you got here? Alex says, "I'm going to translate this vertex first." Let's have a look, this one.

So first five units left.

That's the first part of the instruction and that's where that would be.

That's the first part of the translation.

Now I move it six units down.

Is Alex correct? Have a look, have a count.

What do you think? "Oops, I miscounted and included the point as a first step." Remember he was doing that before.

That's an easy mistake to make.

At least he knows he's done it.

There, that's where it should be.

"Now it has been translated six units down.

I'll do the other vertices." So that one would be there.

This vertex, which is three down from the one vertically above it, would go there because it's three down from the one vertically above it.

And this vertex here would appear just here to make the same rectangle.

Alex joins his vertices and checks his new shape is congruent.

There we go.

Well done, Alex.

Alex makes a prediction about the coordinates of shape A and B.

He says, "I translated five left.

So the x-coordinates will be five less in B than in A." What do you think? Do you agree with that? Let's have a think again.

He says, "I translated five left.

So the x-coordinates will be five less in B than in A." Do you agree? Well, here we go, so that's the first shape.

It's got a vertex of one, four.

And the second shape, negative four, negative two.

Five less than one is equal to negative 4.

Three, four and negative two, negative two.

Five less than three is equal to negative two.

The rectangle is now in a different quadrant.

So it has negative coordinates.

And you might have noticed it's in the negative negative quadrant.

So negative X value, negative Y value.

Let's have a check.

Let's see if you've taken that on board.

A square has a following coordinates.

A square, remember, think about the properties of a square.

Three, negative one, five, negative one, three, negative three, five, negative three.

The square is translated six units left and five units up.

What are its new coordinates? So think about what you need to do to each of those coordinates.

Pause the video and have a go.

Welcome back.

Let's have a look, shall we? The translation is left, which means the x-coordinate will reduce.

The second part is up, which means the y-coordinate will increase.

So subtract six from the x-coordinate and add five to the y-coordinate.

Is that what you did? Let's see what that gives you.

That gives you negative three, four, negative one, four, negative three, two, and negative one, two.

Very, very well done if you've got those right, you're ready for the next step.

And these are the final practise tasks.

Number one, describe each translation.

Now just like before, you've got some stem sentences there and then some reduced stem sentences, and then finally, no stem sentence at all.

See if you can figure it out.

Number two, one vertex has been translated for each shape.

Describe each translation and complete the shapes.

Have a good think about that.

Number three, translate each shape according to the command.

So A is four left, six down and B is six left, six up.

That's got quite a few vertices, hasn't it, B? So be careful.

Number four, translate each shape according to the command.

C is nine right, two up and D, seven right, six down.

And both of those are quite tricky because they've got diagonal lines.

So again, be careful.

Think about the vertices.

Number five, a rectangle has the following coordinates.

Two, three, I've, three, two, four, five, four.

The shape is translated seven units to the left and seven units down.

What are its new coordinates? Pause the video.

And good luck with that and I'll see you soon for some feedback.

Welcome back.

How did you get on? Number one, describe each translation.

So A to B is four units to the right, then two units down.

C to D is six units to the left, then two units up.

Then E two F, there was no scaffold there for you at all.

So well done if you said something like 10 units to the right and then one unit up.

And number two, one vertex has been translated for each shape.

Describe each translation and complete the shapes.

So for A, that was six units to the left and four units down.

And the translated shape looks like this.

And for B, that's two units to the right and six units up.

And the translated shape looks like this.

Well done if you've got those.

Number three, translate each shape according to the command.

So four left, six down looks like this.

And six left, six up for B looks like this.

And that's actually on the x-axis.

C, nine right, two up looks like this.

And that one crosses the x-axis.

And D, seven right six down, that's where the translated shape is.

And number five, that rectangle has been translated seven units to the left and seven units down.

So we're going to subtract seven from each x-coordinate and seven from each y-coordinate.

And that gives us negative five, negative four, negative two, negative four, negative five, negative three, and negative two, negative three.

We've come to the end of the lesson.

There's been a lot of new knowledge to take on board today and you've done really well.

You've made great progress.

Today, we've been drawing and translating simple shapes on the full coordinate grid.

When a shape is translated on a four-quadrant coordinate grid, the shape is congruent.

A translation can be one step or two step, and are usually given as mm units up, down, left or right.

Remember, sometimes we don't say the word units, it's still correct.

The translations might also be thought of as horizontal and or vertical.

And that's the end of the lesson.

A pat on the back, please and a big sigh of relief.

And that lesson is finished.

So very well done.

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

But until then, enjoy the rest of your day.

Whatever you've got in store, I hope it's a successful one.

Take care and goodbye.