Lesson video

Hello, one and all, and welcome to this maths lesson on transformations.

I'm Mr. Gratton and today we will be identifying what transformations have taken place when given both an object and its image.

Pause here for a quick check on some of the keywords that you'll need for today's lesson.

And the key word of invariant point is a point on a shape that has not changed location after the object has been transformed.

First up, let's have a look at how we can describe in detail a transformation that has taken place.

Here we have an object and an image after a transformation.

Sam claims the object was rotated whilst Jun claims the object has been reflected instead.

Pause here to think about or discuss who you agree with.

Actually, I agree with Lucas, who claims it could have been either a rotation or a reflection and he can describe each transformation to justify his claim.

Sometimes it is possible to describe a transformation in multiple different ways.

You'll see this happen for different transformations when an object has different types of symmetry, rotational or lines of symmetry.

For this object, the transformation could have been a reflection in the line y = 8, or a rotation of 180 degrees with a centre of rotation at the coordinates 5, 8.

When describing a transformation, it is important to give as much detail as possible.

This is especially important when there are multiple different possible transformations.

So, how do we identify if a transformation is, for example, a translation? Well, a transformation may be a translation if it looks like it has moved without its orientation, sense and size changing at all when you compare it to its original objects.

Now that we know the transformation is a translation, we can describe the translation by first of all, saying it is a translation.

I know it might seem a little bit obvious, but stating clearly what transformation is happening, well, it isn't just important, it is essential when fully describing a transformation.

Secondly, we need to give detail of the translation where the easiest way is through the use of a translation vector.

For this particular translation, a good description is a translation by the vector -2, 4.

Two clear points that fully describe what is going on.

Right, pause here to identify all of the statements that correctly describe this particular transformation.

This is a translation by the vector -3, -2.

Next up, how do we identify if a transformation is a rotation? Well, a transformation may be a rotation if the image looks like its orientation has changed without its sense changing at all when you compare it to its original object.

Now that we know the transformation is a rotation, we can describe the rotation by saying the transformation is a rotation, again stating what transformation is occurring.

Secondly, we need to give the angle of rotation in degrees.

Thirdly, we need to state whether the rotation is going clockwise or anti-clockwise.

That's the direction.

And lastly, we need to state the centre of rotation as a pair of coordinates.

For this particular rotation, a good description is a rotation by 180 degrees, either clockwise or anti-clockwise, with a centre of rotation at the coordinates 4, 9.

Three clear points that fully describe what is going on.

It will be 4 points rather than 3 when the rotation is not a 180 degree rotation, and then the direction matters.

Right, for this object and image, pause here to fill in the blanks to fully describe the transformation.

This is a rotation of 90 degrees anti-clockwise with a centre of 5, 12.

Alternatively, you could have had a 270 degree clockwise rotation instead.

Our third transformation, how do we identify if this is a reflection? Well, a transformation may be a reflection if the image looks like its orientation has changed and its sense has changed when compared to its original object.

Now that we know the transformation is a reflection, we can describe the reflection by saying it is a reflection and giving a description of that line of reflection such as with the equation of a line.

For this particular reflection, a good description is a reflection in the line y = x.

For a reflection, there are only 2 clear points needed to fully describe what is going on.

Onto the next check.

Pause here to match the description to the transformation.

Image X is a reflection in the line y = 6.

Image y is a reflection in the line x = 5, and Z is a reflection in the line y = 11.

And lastly, how about identifying an enlargement? Well, a transformation may be an enlargement if the image looks like its size has changed without its sense changing at all when compared to its original, unenlarged object.

Now that we know the transformation is an enlargement, we can describe the enlargement by saying it is an enlargement and then giving a scale factor of enlargement and then describing the centre of enlargement as a pair of coordinates.

For this particular enlargement, we have an enlargement with a scale factor of 2, and then by joining corresponding points on both the object and image, we can find the centre of enlargement as the coordinates 8, 1.

Three clear points that fully describe what is going on with an enlargement.

Next up, pause here to write down the missing part of the description for this transformation.

We've given you all of the clever details, but the obvious part is missing, saying it is an enlargement.

Usually you state the type of transformation first before all of the other details.

Furthermore, pause here to fill in the blanks to fully describe this transformation.

This is an enlargement with a scale factor of 3 and a centre of enlargement at the coordinates 1, 12.

For this object and image, Jun asks whether it is possible to the mapping from object to image if more than one transformation has taken place.

Laura thinks that this is possible, but only if the details of one of the two transformations is already given to us.

Laura uses her understanding of the properties of transformations to identify that one of the transformations is a rotation since the orientation of the image is different to the orientation of the original object.

Exactly what the details of the rotation are, however, are unknown because the other transformation isn't clear.

It is not known yet.

But now we are given that the first transformation was a 90 degree clockwise rotation around the centre of 1, 7.

So what could the second transformation have been? Well, let's first of all perform this rotation and then figure out what the second transformation could have been by comparing the first stage image after the first transformation to the final image after this currently unknown second transformation.

So we have a rotation with centre 1, 7, going clockwise to get this middle image.

Pause here to think about or discuss suggestions for a transformation that takes us from the middle image to the final image.

The second transformation is a translation by vector of -2, -3.

For this two transformation check, the first transformation is a translation by vector of 0, 4.

Pause here to describe in as much detail as you can the second transformation.

The image after the first transformation is this.

And so the second transformation is an enlargement and the final image has lengths that are 4 times bigger, so the scale factor is 4.

The shared vertex between the middle and final image is the centre of 9, 5.

Amazing effort so far.

Time for some independent practise.

For question one, pause here to complete the description for each transformation using the information at the bottom.

And same again, pause here for parts 1C and D.

For question 2, pause here to write down full descriptions between these shapes.

And finally, question 3, pause here to write down transformations between these shapes, knowing that for each object image pair, two transformations have taken place.

Right, onto the answers.

For question 1A, we have a reflection in the line of reflection y = 17.

For 1B, we have an enlargement with a scale factor of 2.

5 and a centre of enlargement at 5, 12.

For part C, we have a rotation of 180 degrees and a centre of 5, 8.

And here are some possible answers to 1D.

Pause here to check if yours matches one of the ones on screen.

For question 2A, we have either a reflection in the line y = x, or a rotation by 90 degrees anti-clockwise with a centre of 9, 9.

For 2 B, we have an enlargement by a scale factor of 3 and a centre of 12, 15.

For 2C, we have an enlargement by a scale factor of one third with a centre at 21, 9.

And for 2D, the two possible single transformations are a rotation by 180 degrees with a centre of 13, 14 or a translation with a vector of - 6, 4.

For 3A, the second transformation was an enlargement by a scale factor of 3 with a centre of enlargement of 10, 1.

and for 3B, we have a rotation by 90 degrees anticlockwise with a centre of 8, 13.

Now that we are familiar with how to describe all sorts of different transformations, let's dig a little bit deeper and describe certain transformations with certain properties, when the object and its image have an invariant point.

Here we have an object and an image after an unknown transformation.

Jun claims that every point on the object changed location after it was transformed into its image.

Pause here to think about or discuss whether you agree with Jun's observation.

I think Laura's statement is accurate.

Whether or not every point on the object has moved really depends on what transformation took place.

If the shape was rotated, then one point actually did not change location.

Can you spot which point Laura is talking about? So if a translation occurred, then no point stayed in the same location.

Let's have a look.

As you can see, every single point on that object moved to become this image.

However, if a rotation occurred instead, then one vertex on the object is in the same location as its image.

Let's have a look.

As that object is rotated, there is one single point that stayed in the same location.

If a point on the object stays in the same location before and after a transformation has occurred, then that point is called an invariant point.

If that transformation was a rotation, then that vertex at 5, 9 did not change location and so is called an invariant point.

Sam believes that a translation can never have an invariant point.

Whilst Jun holds hope that there may be one vector for even one shape that results in an invariant point.

Pause here to think about or discuss who you agree with.

There is one specific translation, a translation by the vector 0, 0.

That means that every single point on the object is in the same location as its image because that vector describes no movement at all.

Therefore, every single point on the object is an invariant point after a translation by the vector 0, 0.

However, a translation by the vector of anything other than 0, 0 means that exactly zero points on the object are invariant points.

With translation in variant points are clearly either all or nothing.

Right, for this check, we have an object that will be translated by the vector, a, zero for sum number a.

Pause here to identify which statement is correct.

Translation only results in an invariant point if the vector is 0, 0, so a must equal 0 for this to be the case.

Next up, pause here to think about or discuss.

Is there a rotation where every point on the object results in an invariant point.

If you rotate the object by 360 degrees, the image will fully overlap the object, with each and every point rotated back into the same location.

Therefore, every single point is an invariant point.

This is also true for rotations of multiples of 360 degrees.

Sometimes the vertices of an object may look like invariant points after a transformation has taken place when actually they are not.

For example, we have an invariant point at 4, 4 for this rotation.

On the other hand here we have a non-example of an invariant point at the vertices of the object at least.

Whilst the image perfectly overlaps the object, no vertex on the image is in the same location as its object.

Vertices on the image overlap with different non-corresponding vertices on the object.

To make this a little clearer to see, let's perform the rotation again, but this time with a vertex labelled A, like so.

And then we perform the rotation.

Vertex A was at 4, 5 on the object, but is now at 0, 3 on the image.

This means vertex A is not an invariant point on the perimeter of the object.

Whilst it is true that no vertex on this object is invariant, this is not true for the whole shape.

If the centre of rotation is inside or on the perimeter of an object, that centre of rotation will always be an invariant point.

The centre of rotation for this shape is 2, 4, which looks to be the centre inside of the rectangle.

Since the centre of rotation is inside the rectangle, that centre is also an invariant point.

Invariant points can be inside an object, not just the vertices of an object.

Okay, for this check, pause here to identify the transformation that has an invariant point and write down the coordinates of that invariant point.

Transformation B has an invariant point at 4, 5.

This is because 4, 5 is a centre of rotation where that centre lies on the perimeter of that triangular object.

Some transformations can have a collection of invariant points.

We mostly see this after a reflection.

For example, for this reflection, there is one invariant point at 5, 3 because one vertex lies on the line of reflection at x = 5.

However, for this reflection and entire side of the object lies on the line of reflection, x = 4, resulting in a collection of invariant points.

If an edge of an object lies on the line of reflection, then every single point on that edge is an invariant point.

So, on this edge there is a collection of infinitely many invariant points on that object found on the line x = 4.

Okay, here's a different possibility.

It is also possible to have a collection of invariant points inside of the object rather than just on the edge of an object, such as with this triangle.

We have this triangle that is reflected in this line of reflection, x = 3, resulting in this image.

We can see here that there is a collection of invariant points on a segment of the line x = 3 that is inside of and on part of the perimeter of that triangular object.

Right, for this next check, pause here to match the statement to the incomplete transformation.

Here is each transformation taking place, and here is the completed transformation.

We can see that C has a collection of invariant points whilst D only has one invariant point.

Last up, enlargements.

Similar to rotations, enlargements may have an invariant point at its centre of enlargement, but only if the centre of enlargement is on the perimeter of or inside of the object.

For example, if this object is enlarged by a scale factor of 3 with a centre at 2, 2, then the point 2, 2 is an invariant point on the object.

However, if the object is enlarged with the centre at 2, 1 instead of 2 2, then there are no invariant points on the object.

This is because in this example, the centre of enlargement isn't inside or on the perimeter of the object, rather it is outside of the object.

But what if we have the object and image and we do not know the location of the centre of enlargement to identify if it is an invariant point or not? Well, we can find out where the centre enlargement is by joining corresponding vertices between the object and image with a ray or a line like this to find the centre.

The centre of enlargement for this example is inside the object.

Therefore, there is one single invariant point at the centre of enlargement at 6, 7.

Can an enlargement ever have more than one invariant point? The answer to that is no, because every single point on an object that isn't the centre will be enlarged away from its original location.

That is of course, unless the scale factor is one and then every single point on the object is invariant because the object and image overlay each other completely.

Okay, for this final check, pause here to choose the correct statement for this enlarged object.

We can draw the rays that connect corresponding vertices to find out that we have a centre of enlargement that is outside of the object, therefore A is correct.

There are no invariant points on this object because the centre of enlargement is outside of the object.

Great stuff, onto the practise.

For question one, translate object A and identify any invariant points with image B.

For question 2, reflect object A and identify any invariant points with image C.

Pause now to try these two questions.

For question 3, rotate object D and identify any invariant points with image E.

And for question 4, object F has already been enlarged.

By considering the centre of enlargement, identify any invariant points.

Pause now for these two questions.

And finally, question 5, pause here to analyse object H and image I.

Amazing effort on all of these transformations.

The answers are, for question one, there are no invariant points.

For question 2, there is a collection of invariant points along y = 17.

For question 3, there is one invariant point at 19, 15.

And for question 4, there are no invariant points because the centre of enlargement is outside the object.

And finally, question 5, for part A, the translation is by vector 2, 2, and for part B, if there was a rotation, it will have been by a 90 degree clockwise rotation with centre 5, 4.

And for part C, for a collection of invariant points, we could have had a reflection along that shared edge where the line of reflection has equation x + y = 9.

Well done everyone, and thank you all so much for your effort today in a lesson where we have observed that it is possible for an object and its image to have been transformed in more than one way, and to fully describe each type of transformation, we require a different set of descriptions.

Furthermore, we have been introduced to an invariant point, a point on an object that stays in the same location before and after a transformation has occurred, and that after a transformation, some objects have no invariant points, one invariant point, a collection of in variant points or simply that the entire object is invariant.

Once again, I appreciate so much all of the effort that you've put in into today's lesson with me, Mr. Gratton.

Until our next maths lesson together, take care and have an amazing rest of your day.