Lesson video

Hello, I'm Mrs. Lashley.

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

Okay, so in today's lesson, our aim is to look at whether we can identify if a shape has been transformed or not.

So words that we've met previously and that you should be familiar with, transformation, so is a process that could change the size, orientation, or position once it's been performed.

Congruency, where one shape and another shape are said to be congruent.

If they could stack on top of each other, fit exactly on top of each other after a reflection, a rotation, or a translation.

And invariance is after a transformation, if that property hasn't changed.

So our lesson has got two parts to it.

The first part is identifying which transformation has taken place.

And then the second part is to recognise and describe the transformations fully.

So we're gonna get started with the first one, which is which transformations.

So a transformation can change one or more properties of the object.

And those properties are the lengths, the angles, its position, orientation, and sense.

So with a translation, those five properties, let's think about whether they do change after a transformation of translation.

So firstly, does the orientation change? No.

What about the lengths? Is the object and the image the same size? Yes.

The position, does the position change? Yes.

The angles, the angles within the shape, within the object, within the image; after a translation, do the angles change? No.

And what about the sense of the shape? Well, the sense is the same.

So that one doesn't change either.

So moving on to the second transformation of rotation.

So here we've got a diagram that shows an object and an image and after a rotation has taken place.

So once again, the orientation.

Has the orientation of the shape changed? Yes.

The lengths.

If you were to measure the edges, have they changed? No.

What about the position? Has the position changed? Yes, it's been rotated, so it's moved to a different place.

The angles within the shape, have they changed? No.

And what about its sense? No, that hasn't changed either.

So a rotation, the orientation changes, its position changes, but its lengths, its angles and its sense stay the same.

They're invariant.

Moving onto the third transformation of reflection.

So again, the diagram to indicate a reflection.

Has the orientation changed? Yes.

Have the lengths changed? No.

The position? Yes.

The angles? No.

The sense? Yes, the sense of the shape has changed after a reflection.

Onto the fourth one.

The fourth is enlargement.

Remember, the enlargement doesn't always indicate that it's got larger, but it can get smaller too.

So has the orientation of the shape changed? No.

Have the lengths changed? Yes.

Has its position changed? Yes.

Have the angles changed? No.

And what about the sense? The sense hasn't changed either.

So every time you are met with a diagram that shows a transformation has taken place, if you can identify the properties that have changed or haven't changed, that will help determine which transformation has taken place.

So one of the figures below has been selected by Aisha.

There's four to look at.

And by asking a series of questions, her friends are going to find which one she selected.

So Andeep is gonna pose the first question to Aisha.

Is the image congruent to the object? So remember, that congruent means that if we could cut them out individually, they would fit on top of each other after a rotation, a reflection, or a translation.

She says it is.

So the object and the image are congruent.

So which figure can it not be? Cannot be three, because they were not congruent.

The lengths have changed.

Has the orientation changed? Yes, it has.

So which figure can it not be, if the orientation has to change? It means it can't be figure one.

So we're left with figure two and figure four.

Has its sense changed? No.

So just its orientation has changed, but its sense hasn't.

So that means that she originally selected figure four.

We're correct, that's good.

So check for you.

We've got Lucas who says, "Is the image congruent to the object?" And Alex says, "No, it is not." So which one is it? Hopefully you went for D, because D has got a change of lengths.

The size has changed and that's why they're not congruent.

Whereas A, B, and C, all of those are congruent pairs.

Another check.

So as the something has changed, a something may have taken place.

So which property of the five has changed that informs us what might have taken place? So the sense has changed and reflection is the only transformation of those four that changes the sense.

So you might have put position, but that doesn't help you decide which transformation has taken place.

So because the sense here has changed, a reflection may have taken place.

So here we've got a square, the object square, which is the purple one, labelled A, B, C, D.

And it could have been transformed in various ways to become the green one, the image.

If we now have the labels for the vertices on the image, we can now decide which transformation has taken place.

So the corresponding vertices, the image of A and A, the image of B and and B, are in the same sort of space.

They go around the shape in the same order, starting at the same point.

So A is the top-left, image of A is the top-left.

Going anti-clockwise, we then get the image of B, the image of C, the image of D.

And so all that's happened is it has moved.

So this one could have been a translation because a translation doesn't change its orientation, but it does change its position.

But it could have been transformed in other ways too.

So once again, now we've got the labels for the images vertices, we can think about what transformation may have occurred.

So now if you look, the vertices are not in exactly the same position.

So if you go clock anti-clockwise around the object, it goes A, B, C, D.

If you go anti-clockwise around the image, it goes D prime, C prime, B prime, A prime.

So D, C, B, A.

So it's sort of reversed.

And then if you look at where they are, A and D, and the image of A and D are still at the top, and B and C, and the image of B and C are still at the bottom.

So it hasn't rotated.

This one could have been a reflection and that reflection line would be right in the centre to be equidistant from both.

But what about if the vertices, the image of each vertex was like this.

So now, D, that originally on the object was at the top of the shape, is now at the bottom, and B was at the bottom of the object is now at the top of the image.

So there's been some sort of movement.

And so this one is a rotation, or it could have been a rotation.

So a check for you.

The vertices of the image are always in the same position regardless of the transformation, true or false? And then justify your answer.

Hopefully, you said it was false.

You just saw in those examples that depending on which transformation has an effect on those vertices as well.

And so the justification, each transformation causes a different effect on the image and therefore its vertices may change position.

So the trapezium along with its image can create these three shapes.

And so the purple is the object, the green is the image.

And there has been a transformation that was caused by combining them.

They're making three shapes.

So we want to know which type of transformation has taken place in each case to make the new shape.

So if we look at this one, so it's actually created, it starts, the object is a right trapezium.

And with its image, it's now an isosceles trapezium.

So it's still a trapezium, it's an isosceles trapezium.

And so which type of transformation has taken place? Well, a reflection has happened here.

So along that edge, a reflection, and it would've created this isosceles trapezium.

If we focus on this one, so here we've got the object and the image again.

They are combined to create a new shape.

And this shape's a parallelogram.

How, how did the object get transformed to become its image? Well, this would've been 180 degree rotation, again, twisting again along that edge so that the edges match up to create our new shape.

And the third shape was a rectangle.

So once again, you've got an object and the image combined and they've made this rectangle.

What transformation would've allowed that to happen? And this would've been another rotation of 180 degrees, except from it's on a different edge, on that slanted edge, that would match up to make a rectangle.

So here's a check for your understanding.

So which transformation could not have taken place to create this shape using the object and image: translation, rotation, or reflection? So it cannot have been a translation.

And that's because a translation does not change the orientation or the sense of the of the object.

And here, they do not, both the object and the image do not have the same orientation.

So now a bit of practise for you.

So question one, you need to consider the changes and decide which transformation could have taken place in each case.

And there might be more than one for some of them.

Pause the video, give it a try, and then when you're ready, come on back.

Question two, slight change to question one.

So by considering the invariant properties, so remember, that means ones that do not change after a transformation, decide which transformation could not have taken place in each case.

There might be more than one.

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

So question three, you need to identify the type of transformation that's taken place to create the new shape.

So we've got a right-angle triangle that has been transformed six times to create a new shape.

So the new shape is there on the right-hand side, and the shape with its object and image, to show you its creation is on the left-hand side.

So six questions.

You need to decide which transformation has taken place to the right-angled purple triangle in order to create a new shape.

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

And then question four, you need to add the labels to the images vertices.

So you've been told a reflection has taken place.

So where would the image of A, B, where would the image of C and etc, for each one.

Pause the video, give it a try, and then when you're ready, come back and we'll go through the answers.

So question one.

This one, you need to decide which transformation could have taken place by considering the changes to the properties.

So if the lengths have changed, then that means that the image will not be a congruent shape to the object and therefore, only an enlargement could have happened.

Because a reflection, a rotation, and translation all have a congruent image.

If the senses changed after a transformation, then it's only a reflection that could have happened.

If the orientation changes, then it could be a rotation or a reflection.

And if the position's changed, it could be any of the four.

All of them will move them to a different position in some way.

Question two, this time you need to consider the invariant properties, so the properties that didn't change after the transformation.

And the question is, which transformation could not have taken place? So if the lengths are invariant, if the lengths do not change, then it cannot be an enlargement because an enlargement changes the lengths.

If the sense is invariant, if the sense doesn't change, then it cannot be a reflection because a reflection changes the sense.

If the orientation is invariant, then it cannot be reflection nor a rotation because they change the orientation.

And if the angles are invariant, then all four transformations keep the angles the same.

So there isn't a transformation for that.

There is no answer.

Question three, you need to identify what had happened.

So for the triangle, it would've been a reflection along that sort of vertical length.

For part B to make a rectangle, it'd been a rotation about the slanted edge.

For C, to get a parallelogram, it would've been a rotation again.

For D, an isosceles triangle there, a reflection across the bottom edge.

E, we get a kite, and that's a reflection along the longest edge of the original object.

And F, we get a different, another parallelogram, and that's from a rotation.

So reflection, rotation, rotation, reflection, reflection, and rotation.

And then question four, here are where the image vertices should have been after the given transformation.

Okay, so we are moving on to the second part of this lesson where we're gonna recognise and describe the transformations that have taken place.

So on the screen you can see a diagram where we've got two congruent shapes and we're gonna look at whether we can recognise a transformation and also describe it.

So Sophia has started a description.

She said that the sense has changed, so she's thinking about the properties of that shape.

So the sense has changed, the position has changed, the orientation has changed, but the lengths and the angles are invariant.

So Lucas has said that that means it must be a reflection because a reflection changes the sense.

It's the only one of the four transformations that will change the sense.

So the sense of the shape has changed.

So it's understandable that Lucas has decided this must be a reflection.

However, if we put a mirror line on there, which you probably would've assumed it was diagonal, it doesn't end up where the pink one is.

It's got the same sense, it's got the same orientation, but its position is not quite right.

And so this is where we use the properties to help us gauge which, if any, transformation may have taken place.

The position of its image is also important.

So like we just saw, the sense had changed, the position had changed, its orientation had changed, and they have to have happened for it to be a reflection.

But if its position is not correct, then it isn't a reflection.

So, an enlargement from a set point.

The invariant properties will be that its angles haven't changed, the sense will not have changed, the orientation will not have changed.

So if you enlarged a shape by a positive scale factor, then those three properties are unchanged.

The changed properties will be its position and the lengths because it's an enlargement, so the lengths will either get larger or get smaller depending on the scale factor.

And the position is also relative to its set point.

So we've got the left-hand one and the right-hand diagram here.

So the left-hand one shows a correct set point and the object and the image.

The ray lines are there to show us that the corresponding vertices both live upon those ray lines that meet at the set point.

Whereas the second diagram on the right, the position of the image is incorrect.

It's the correct size, but its position is incorrect.

So if you went through the five properties, have the angles changed? No.

Has the sense changed? No.

Has the orientation changed? No.

Has its position changed? Yes.

And has its lengths changed? Yes.

So the right-hand diagram meets the five properties that it should for enlargement.

It is an enlargement, it's just not an enlargement from a set point.

And that can be seen by the rays.

The rays do not go through the corresponding points on the object and the image.

So we need to make sure the position is correct, before we can ultimately say an enlargement, or a translation, or a reflection, for example, have taken place.

So here we've got a diagram that shows us an object and an image.

And the only property that's changed is its position, because the lengths are the same, the angles are the same, the orientation is the same, and its sense hasn't changed.

So they are all invariant.

The position has changed, so this is a translation.

Hopefully you remember, that the triangle can be described have being translated by a vector.

And in this case, the object has become the image by moving three to the right and down by one.

So here we've got another diagram and we want to think about what has changed to be able to start to recognise which transformation may have occurred.

So the orientation and position have changed.

The lengths, the angles, and the sense are invariant.

So because its position and its orientation has changed, then it is a rotation.

And in the description, we need to know where it rotated from, which direction, and how much the turn was.

So it's full description would be the object has rotated 90 degrees anti-clockwise about the point A.

Okay, so once again, if you look at the object and the image, think about the properties that have changed and the properties that have not.

So in this case, the orientation, the position, and its sense has changed.

And so it is a reflection.

And the description that we would give, the object has been reflected in the X-axis.

And remembering that the X-axis is the horizontal one, and it's reflected about that.

That it's equidistant if you choose corresponding points, that they are equidistant from the reflection line.

And again, going through the properties, have the lengths changed? Yes.

Have the angles changed? No.

Has its position changed? Yes.

Has the orientation changed? No.

Has its sense changed? No.

So the main thing here, what has changed is the lengths.

And so the position and lengths have changed.

It has been enlarged.

A description for this, the object has been enlarged by a scale factor two.

So all of the lengths have been doubled and the distance from the centre of enlargement has also doubled, and that centre of enlargement is the origin or the coordinate (0,0).

So a quick check for you.

So what information do you require in the description of a rotation? Hopefully, you went for B.

So for a rotation, there are three important things you need to say, and that's the centre of rotation, the direction, so either anti-clockwise or clockwise, and finally, the amount of turn.

So now it's time for you to practise.

So the first task, you need to fill in the gaps using the diagram to help you.

Pause the video, and when you're ready, come back for some further questions on that.

So there's three more questions to do with that diagram.

Same idea, fill in the gaps, and when you're ready, come back for another question.

So question two, three different transformations have taken place to create an image, which combined with the object forms a square.

So what are they? And you need to describe each fully.

So that rectangle, the purple rectangle, has been transformed and combined with its image, a square has been created.

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

So question one, you should have filled in D on the first gap.

So shape A has been reflected in a diagonal line passing through (-3, 3) and (1, -1) onto shape D.

For part B, you should have filled in J and K.

So shape J has been reflected in the y-axis onto shape K.

Don't be put off by the numbers alongside the y-axis.

There is still a gap of two and a gap of two either side.

So we know that they need to be equidistant to be a reflection and there are two squares for either side of the y-axis.

The numbers are a little bit off-putting.

Part C, shape B has been translated by (-2, -6) onto shape E.

So it may have taken you a little bit of time there, but firstly, you needed to find any that could have been translated.

So think about the orientation and the sense and then finding one that was two to the left and six below the other.

Then the further three questions.

So shape H has been enlarged by scale factor 2 from (-2, -1) onto shape J.

Okay, so scale factor 2.

And then E, shape H has translated by (0, 8).

You needed to figure out what that missing number was in the vector onto shape A.

So again, you needed to find something that was directly above or below, because the zero was the horizontal displacement.

So that was your clue for that one.

And F, shape C has been rotated 180 degrees about origin onto J.

So again, a rotation of 180 degrees from the origin.

So thinking about some that could have done that.

And then question two.

So question two, the order that you put the three doesn't matter, but these are the three that you should have had.

So you should have had a translation.

So just moving it and moving it to the right by two, so (2, 0).

You should have had a reflection along the right edge of the shape.

And you should have had a rotation 180 degrees about the midpoint of the right edge, with the 180 degrees, because it doesn't matter of direction.

If you have mentioned clockwise or anti-clockwise, that would still be correct.

So rotating 180 degrees clockwise is the same result as rotating 180 degrees anti-clockwise.

So if you've added the direction there, you're still correct.

So in today's lesson about transformed objects, we wanted to be able to identify that transformation had taken place and describe it.

So when you are identifying them, you need to consider the properties and compare it to the object.

If the lengths have changed, then an enlargement may have taken place.

If the sense has changed, then a reflection may have taken place.

If the orientation has changed, then it may have been a rotation.

And if its position has changed, then a translation may have taken place.

Thank you for working so hard today.

Hope to work with you again soon.