Lesson video

Hello, I'm Mrs. Lashly.

I'm looking forward to guiding you through your learning today.

Okay, so today's lesson, we're gonna look at the nature of a translation and appreciate what changes after a translation and what is invariant.

Some words that you've used before that we'll be using again throughout the lesson is the idea of congruent.

So one shape can fit exactly on top of another using rotation, reflection, or translation, then we would say the shapes are congruent.

And orientation.

So it's hard to describe orientation, but a picture says a thousand words.

So example of orientation changing on the left, and an example of orientation not changing on the right.

So today we are gonna be working with the word translation a lot.

So translation is a transformation in which every point of the shape moves the same distance in the same direction.

And our first part of the lesson is all about translation, that key word.

So all of the L shapes in that figure are congruent.

And that means that if you could cut them individually out, they would all stack on top of each other.

You may need to turn them over, you may need to rotate them, twist them around, but they would stack up neatly on top of each other.

So which of these congruent L shapes can be slid, so without doing any turning or flipping, onto another one? So if you could imagine sliding them around, but not rotating, not flipping, which ones would land up on top of each other? So you hopefully you came up with two pairs that could do some sliding onto another, and they would be ones that have been translated.

So translation is where there is no flipping, no twisting, just movement.

So C could slide onto E without having to be flipped or twisted.

So just think about that.

Could you imagine moving that onto E? It could be that E could slide onto C.

You may have done it that way round, and that's exactly the same and that's fine.

Or F could slide onto D.

And again, you may have said D onto F.

So they are two pairs that have been translated.

So let's see it in action.

C to E.

Or E to C.

No twisting, no flipping, just sliding.

F to D or D to F.

They are translations.

There are quite a lot of transformations mathematically.

We are gonna focus on translation.

And when a translation takes place, the object is moved to create its image.

So the object, the original shape is then moved, that's what the transformation is, and it becomes its image.

If the image and the object are not congruent, then it cannot be a translation.

So an object and an image are always congruent if a translation has taken place.

So here are some examples of translated objects.

A reminder about the notation there that the A prime is that's the image of vertex A.

So A, the vertex A on the rectangle, that's on the object.

And then A prime is the vertex on the image.

It's the corresponding vertex.

So that rectangle is just moved to the right.

And then we've got some triangles, N and the image of N, and that's just moved down.

And then we've got the image of P and P, and that has moved to the left and down.

So they are congruent to each other.

They're exactly the same shape and size, but they're just in a different position.

They've moved.

So here's some non examples of translated objects.

Okay, so let's have a look at why they are not translated objects.

So with the rectangles, if I can slide that blue one on top of the purple, we can clearly see that they are not the same size.

Even if I bring it down, then the overlap comes at the bottom instead of the top.

So they're the same width, but they're not the same length.

The triangles at the top.

Again, by just moving vertically and horizontally by sliding can one fit and lay exactly on top of the other.

And in this case, it's another no, so they're not translated.

And the last two triangles at the bottom.

Can the blue lay on top of the purple by just moving horizontally and vertically? Again, that angle and that top edge is too large.

They do not lay exactly on top of each other.

So a check.

To be a translation, the image must be a different size, congruent, or a different orientation to the object? Pause the video and have a go at that.

Hopefully you've gone for congruent.

So to be a translation, the image must be congruent to the object.

So you're gonna do a bit of practise.

I want you to draw four more copies of the shape so it could be a translation.

Pause the video and have a go at that.

And task two, you're gonna write down the pairs that are a translation of each other.

So again, think about sliding.

Which ones could slide, without twisting, without flipping, on top of another one? Write down the pairs.

Pause the video and come back when you're ready.

So here we've got an example of four that you may have drawn.

You could have drawn four anywhere you liked.

They could have overlapped, but they needed to be congruent to that original trapezium.

And question two, A and F are pairs that are translated.

B and H, C and G, D and E.

Again, the order of your letters do not matter.

So if you did F and A, that's the same as A and F.

Okay, so we're gonna move on to the second part of today's lesson, and that's now thinking about the orientation and position of the image.

So the object and image are congruent if a translation has taken place.

We focused on that in the first part of this lesson that if it's not congruent, then it cannot be a translation.

If they are congruent, then a translation may have taken place.

Here we've got an example of a translation that's taken place.

We've got a purple triangle labelled A, B, C, and then the green triangle, A prime B, prime, C prime, all meaning the image of A, the image of B, and the image of C.

So the colour itself doesn't matter in maths.

It's just a way of indicating that they are slightly different.

The purple and green have just been a colour of choice here.

It's not significant.

Whereas the labelling, the A and the image of A, or A prime, is what we would use more often in maths to identify which one is the image and which one is the object.

So Alex has said to Lucas that his shape is congruent to Lucas's.

Their two shapes are congruent to each other, so therefore, it is a translation.

What do you think about that? Lucas said it can't be.

You cannot slide one on top of the other completely.

So going back to that concept of sliding, moving, it cannot move onto the other one.

Aisha said, my shape's congruent to yours, so it's a translation.

What do you think Lucas is gonna say to her? Again, it cannot be.

You cannot slide one on top of the other completely.

The objects and image are congruent if a translation has taken place.

The image will not be mirrored if a translation has taken place.

And that's because you couldn't slide it on top of the other without having to then do a flip.

So if it is mirrored, it cannot be a translation and the image will have the same orientation as the object.

Because if the orientation has changed, then that means there's potentially it has been twisted or it has been flipped.

We've already mentioned if it is flipped, then it cannot be a translation.

And if it's been twisted, then it cannot be a translation.

So if the orientation changes, a translation has not taken place.

When an object is translated, so if we think about moving, sliding one shape, a congruent shape onto another congruent shape, what changes and what is invariant? So just a reminder, invariant means doesn't change.

So do the lengths change when you translate a shape? No.

Do the angles change within the shape? No.

Does it become mirrored? No, because that would mean a flip has taken place.

Does the orientation change? No.

Does its position change? Yes.

So if a translation has taken place, it will be congruent.

Therefore, it's lengths and its angles haven't changed.

It will not have been mirrored, and the orientation will not have changed.

What will have changed is its position.

So all the others are invariant.

So the only property that isn't invariant when a translation is completed is.

Pause the video and when you're ready, come back for the answer.

The position.

So all the others are invariant.

They do not change after the transformation, but its position does change.

If a translation has taken place, then we want to think about where the image of each of the vertices is located.

Remember that a translation is a transformation where all vertices will have moved in the same direction by the same distance.

So here we've got an example where we've got quite a complicated polygon, and one of the edges on the image has been given.

And we want to figure out where the rest of the image would be located and labelling the vertices as we go.

So we have a given edge on the image, and it's important that we think about the labels of the vertices because there are two edges, which are vertical by any length of two.

There's the edge AB and the edge GF.

So in order to find the correct position, we need to think about which image edge we have.

And that's G prime F prime.

So when we're locating the other vertices, we need to consider the edge GF on the object and relative to that.

So if we start by thinking about A, that is two to the left of G.

So the image of A needs to be two to the left of G prime.

If we now think about the next nearest vertex, which is E, that is one to the right and one up from F.

So the image of E will be one to the right and one up from F prime.

D is one below E.

So the image of D will be one below the image of E.

And now we've placed three more vertices.

We can continue that until we can find the full image.

We can form the image by building up around the vertices.

And if you did that, this is where your image would be located.

Labelling the vertices, A prime, B prime, C prime, D prime, E prime.

So it's important that you think about the edge that's given in order to get the position correctly, especially when two of the edges are the same in terms of length and direction.

Here's a check, a true or false.

So firstly, the vertex label doesn't matter in a translation, true or false? And then justify your answer.

Pause the video whilst you have a think about that, and then come back for the answer.

Hopefully you went for false.

The vertex label is very important in a translation.

And that's because if they're not in the same position relative to their vertex, then a translation has not taken place.

You're gonna do a bit of practise.

And so the first question, you need to complete the shape so that a translation has taken place.

So you've got the object labelled with its vertices, and you've been given part of the image that you need to then complete the shape, including the labels for the vertices.

Pause the video, and then when you're finished with that, come back.

Okay, so part two of your practise is you need to identify which property of the image is incorrect for it to be a translation.

So why can these not be translated? Pause the video, think about that, and then come back when you're ready.

And task three, complete the table.

So what's changed and what hasn't? And then decide whether a translation has taken place.

Pause the video, and when you're ready, come back.

Okay, so question one.

You needed to complete the shape so that a translation has taken place.

You needed to draw the image.

I also asked you to label the vertices.

So on part A, the rectangle had translated.

It's moved right and down.

And you were given the bottom length and the edge on the left hand side.

So you needed to add in the top and the right hand side to make sure it's congruent, make sure it's exactly the same size, and then the image of A and the image of D.

So A prime and D prime.

On B, you were just given the, you were given the bottom edge between R and P, but R prime and P prime 'cause it was the image.

And hopefully you've successfully drawn the rest of that image in and written Q prime for the image of Q as that vertex.

And then part C, there was enough space for you to potentially draw your rectangle in the wrong position.

So that's where the vertices come into play.

You were given Z prime and W prime, which means that Y and X need to go above them.

And this is where the image should be.

If you've drawn the rectangle below that line that was given, then you've actually not done a translation because the orientation would've then changed.

The position of those vertices have changed.

And therefore, its orientation has changed as well.

So this is where it should have been.

So part C, make sure you have drawn your rectangle in the correct position for that given edge.

Section two, you need to identify which property made it not a translation.

So remember, with a translation, the only property that changes is its position.

So if the lengths, the angles, if it becomes mirrored or its orientation changes, it cannot be a translation.

So for part A, it has been mirrored.

So that one cannot be a translation.

It wouldn't just slide on top of it.

You'd need to flip it as well.

For part B, it's quite subtle, but hopefully you noticed that the lengths have changed.

So it's still an L shape, but if you were to measure all of those edges, they wouldn't be identical.

And so therefore, the lengths have changed, it's not congruent anymore, and then a translation cannot have taken place.

And C, it's orientation.

So once again, thinking about that idea of can they slide on top of each other? Well, no.

It cannot slide on top of each other.

You'd need to twist as well.

So when we're talking about the moving and the sliding, the only options you've got are moving horizontally and moving vertically.

So no twisting involved.

Question three, you need to decide whether translation has taken place by completing the table, and then thinking about what it should have said if a translation had taken place.

So the lengths have changed.

Again, if you use the grid behind it, you could have looked to see if the lengths have changed.

You could measure it with a ruler.

The angles have also changed.

So that again is much more subtle.

But with a protractor, you'd be able to measure that more accurately.

Had it been mirrored? No.

Has the orientation changed? No.

Has its position changed? Yes.

So what stops this being a translation is that actually they're not congruent.

So if the object and the image are not congruent to each other, then they cannot, a translation cannot have taken place.

So it can only be a translation if the object and image are congruent.

So during this introduction to translation lesson, we have looked at the fact that translation is a transformation.

And that is where every point of the shape, every vertex of the shape has moved by the same amount in the same direction.

The image is congruent to the object, and that means that its lengths and its angles do not change.

Its orientation will be the same between the image and the object.

So if a translation has taken place, its orientation does not change, and the only property that does change is the position.

And so translation is all about moving the shape horizontally and vertically.

Well done today.

I look forward to working with you again.