Lesson video

Hello, I'm Mrs. Lashley.

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

So today we're going look at how we can describe when an object has been moved or changed.

A word that you should be familiar with is orientation.

And here you can see an example of where the orientation has changed.

An example where the orientation hasn't changed.

We're gonna work with that word during the lesson, so it hopefully should become more clear if it's not at the moment.

So during today's lesson we've got some keywords.

The first of the keywords is the object.

And the object is the starting figure before a transformation has been applied.

Once the transformation's been applied, then we call the image.

The image is the resulting figure.

So the object is what we start with.

And the image is after the transformation.

A transformation is a process that may change the size, that orientation, or the position of a shape.

We're also gonna be using the word congruent, but we're gonna explore this more during the lesson.

And a property of a shape is said to be invariant if it doesn't change after the shape has been transformed.

So if there is a property that hasn't changed, even though a transformation has taken place, then that property is said to be invariant.

So we've got two parts to the lesson today and we're gonna start with congruency.

So on the screen you've got pairs of non congruent shapes and pairs of congruent shapes.

And I want you to look at both of those and think about what does it mean to be congruent.

So if you come up with something, does it hold on your non congruent shapes and your congruent shapes? What does congruent mean? Well, on the pairs of non congruent shapes, we can see on the top one that you've got two shapes, two quadrilaterals, four-sided shapes, all of which have got four right angles.

So their angles are the same, however, one is a square and one is a rectangle.

So it's shape, the dimensions are not the same, although the angles are.

Below it, we have got two isosceles triangles.

So this one, the shape is the same, but their angles would be different and their lengths.

So although the shape's the same, they're not congruent.

And lastly, the two without the grid background, they have all got edges of four and two centimetres, but the shapes are different.

So the order of the edges, the green one, the parallelogram goes four, two, four, two, whereas the kite goes two, four, four, two.

So although there's four sides all of the same length, the shape is not the same.

Whereas the congruent shapes, we've got two right angled isosceles triangles.

If we were to measure the angles, they'd be the same angles in both triangles.

And if we was to measure the lengths, the lengths would be the same.

And then we've got two rectangles.

Again, measure the lengths.

The lengths are the same.

Measure the angles.

The angles are the same.

All that's different is there orientation.

So, but they are still congruent.

So two or more shapes are said to be congruent.

If one shape can fit exactly on top of another, using rotation, which is twisting, reflection, which is to flip or translation, which is to move.

So which of these triangles are congruent? So just have a think.

Reread that definition of congruency.

And on that diagram, which of those would be congruent triangles? So which of these triangles are congruent? Well, let's cut them out and have a go.

So B fits under A, with a little bit of twisting, C fits there too.

D, twisting doesn't work.

Ah, but if we flip it, then that also fits underneath A.

E doesn't, too small F twisting and it fits nicely.

G, some twisting, fits.

H, no, there's some overlap.

So here we can see that they are nicely stacked, they're congruent, and the other two are not.

A, B, C, D, F and G are identical.

The E and H aren't.

So just a quick check, these rectangles, which of those are congruent? Pause the video and give it a try.

Okay, so I'm hoping you've gone for A, B, and E.

So you can use the grid to support you with counting to check the lengths are the same.

You're told that their rectangles, which means all their angles are 90 degrees.

So it's just a case of checking their lengths.

C is clearly too wide.

D is too narrow and F is too long.

So if we were to cut them out, there would be an overhang.

You wouldn't be able to stack them on neatly.

Are these two squares congruent? Hopefully you've said no because although the shape's the same, they've got four right angles, they're not the same size.

The lengths, if we use the grid, there's four squares for the left hand square.

So its edge length is four, whereas the other one, the edge length is six.

They're different size.

If we were to cut them out, they wouldn't stack on top of each other.

How about these two triangles, are they congruent? The answer is yes, they are congruent.

So they are right angled triangles.

The other two angles within the triangle are labelled 59 degrees and 31 degrees.

And then the edges are three centimetres, five centimetres and 5.

8 centimetres.

But if you remember back to the example of non congruent, the parallelogram and the kite, even if the edges are the same, we need to check they're in the correct position.

So the five centimetres is the length between the 90 degree angle and the 31 degree angle.

And that's the same on both triangles.

If we check the 5.

8 centimetres, the longest edge, it's between the 59 degrees and the 31 degrees.

Therefore, these triangles are exactly the same.

If we were to cut them out, they would neatly stack on top of each other.

So another check for you, Laura said shapes can be congruent to each other even if flipped or rotated.

Do you agree? And if you do justify your answer, so she's correct, she's right.

Flipping and turning a shape doesn't change the angles or lengths, it just changes its orientation and whether it's mirrored or not.

But again, keep thinking about if you could physically cut them out, would they stack on top of each other so they look like one shape? So the angles and lengths are still the same, regardless if you flip or rotate.

So if we want to think about this idea of a question where we want to create a congruent rectangle, we've been given the tr the rectangle that we're trying to match and that rectangle is two squares wide and three squares long or high.

The other edge that's been given is the two.

So what we need to do is find a way of creating the two by three rectangle.

Where could you put the other two vertices? So that the rectangle that's created when you join up the four vertices is identical to the given two by three.

Hopefully you've got an idea of where you would put them.

So you may have gone on the left hand side and then we've got our two by three.

It's a rotated rectangle, but we've already established that just because it's rotated doesn't mean it's not congruent.

But there is an is an alternative.

You may have gone on the right hand side and so another two by three rectangle.

So you're now gonna do a bit of practise for yourself.

So which of these triangles are congruent? So pause the video here, have a go and then come back when you're finished.

Question two, you need to decide if they're congruent or not.

And if not, why? Again, pause the video and give it a try.

And then question three, similar to that rectangle example, you need to decide where you could put the third vertex to make the two triangles congruent.

So one edge has been given, you need to decide where the third vertex, which would join with them other two to create a congruent triangle to the given one, pause the video and give it a try.

Okay, so some answers now.

So question one, which of these triangles are congruent? Remember if they're rotated or flipped, it doesn't matter.

They just have to be identical in size and angles.

So A, B, E and F are congruent to each other.

C is too short, D is too long and G and is also not got the correct dimensions.

Question two, so 2A, they are congruent.

If you've measured the length with your ruler measured the angles of your protractor, you would see that they all match up so they are congruent.

Parallelograms B, these are not congruent squares because their sizes are different.

The lengths, if you measured it, the lengths would be different.

Part C, these are not congruent as well.

So this one, hopefully you were checking the positions of the lengths, not just the four numbers.

So the three centimetres is next to that 152 degree obtuse angle.

And on the other side it was 5.

5 centimetres on the left hand quadrilateral.

Whereas on the right hand quadrilateral it was the 5.

4 centimetres and that's why they are not congruent.

Their edges are not in the same places.

Question three, there's four places you may have chose so that there's four dots that indicate the vertices that you could have plotted.

If you joined them up with lines, you would've got the same triangle.

Okay, so we're gonna move on to the second part of the lesson now, which is to do with describing the changes that have taken place.

So here you can see triangle A and it's been transformed in four different ways.

I just want you to look at that and think about what it might mean to transform.

Hopefully you've come up with something that suggests that the original shape, the object is changed.

So it might have got smaller, it might have twisted, it might have flipped, it may have just moved.

So transformations change a property or many properties of the shape.

So when they're applied we need to consider which is the object, the original shape before the transformation, and which is the image, the result after a transformation.

So if you think to yourself, when you look in the mirror and you see your, your reflection, your mirror image, you are the object, you are the thing that is looking into the mirror.

What you see is your image of yourself.

So you are the object and the reflection is the image.

So here is a similar idea of reflection.

Imagine that grey dotty line is the mirror, then the object, you look into the mirror and what you see is the image.

So for throughout the lesson we're gonna be using purple to indicate object.

It'll be labelled in some way as well and green to be the image.

So here are some examples of objects and images.

So this one, again, the purple is our object and it has been twisted or rotated to become the green image.

Its size, its angles are the same, but it's just now being moved by being rotated.

Here we've got the purple triangle to be the object and the green to be the image.

So this one, the orientation hasn't changed, it has just moved to another position.

So I mentioned that we're either gonna label them object or label them image, but actually mathematically, we have a more precise way of doing this.

And that's to label the vertices.

So if the vertex on the object was known as A, then the image of that point A would get this, this looks like an apostrophe, we can say A prime and that means the image of point A.

It's the corresponding point on the image after a transformation.

So look out for that during the lesson as well.

So that's gonna indicate to you which one is the object and which one is the image.

If it's got the prime notation, then that is the image after a transformation.

So on your check, I just need you to think about which word is missing and which label of the vertex is missing.

So hopefully you said this was the object, the purple, the ABC triangle is the original shape, no transformation has happened yet.

And then the image needs to have the vertex B prime to indicate that that is the corresponding point, the image of vertex B.

So when the object has been transformed, the result is the image.

So after a transformation, we call it the image and we can describe the transformation based on how the object has changed.

So the object, the original has had a transformation and then we get the image out and our description will help us with the transformation.

So here we've got this image of A and A prime.

So we know that the purple, the vertex A is on the object and A prime is on the image.

And Andeep has said that the object has moved to the left and down.

And that is a description of what has happened, what the transformation has done.

It has moved the object to the left and down.

And if we consider the individual properties of the image compared to its object, it can help you with your descriptions.

So Lucas has said, has the orientation changed? And Jacob has says yes, so the orientation has changed.

Lucas has then asked Jacob, have the lengths changed? So have the lengths of the triangle changed? And the answer there is no, has its position changed? Yes, it's not exactly where it started.

Have the angles changed? No, is it mirrored? Is it like you're looking in the mirror? Yes, so a description for this transformation that's turned the object into its image could be the lengths and angles are the same.

The position and the orientation have changed and the image is mirrored.

It has been flipped.

And properties that stay the same after a transformation are known as invariant.

So another transformation has taken place.

We've got the object, we've got the image.

Jun has said, has the orientation changed? Sophia said no.

Have the lengths changed? No.

Have the angles changed? No.

Has the position changed? Yes, is it mirrored? No, so a description for this transformation, Jun has gone or, the only change is the position.

All the other properties are invariant.

The object has moved to become the image.

So that word, invariant, means when they do not change after a transformation.

So can you look at this transformation? The object has become the image after a transformation and complete the table for each property.

So hopefully you went for lengths is no, so the shape has just twisted.

Now, Sam has described the image compared to its object.

So Sam has said, the lengths and position have changed, but all other properties are invariant.

So I want you to think about where that hot air balloon could be on the screen and what it might look like.

Did you think because of the description that it needed to be somewhere else on the screen and also, it needed to have changed size.

It said the lengths and positions have changed.

So everything else is the same but its size, the lengths of it have changed and where it is has changed.

Okay, so is is described the image compared to its object.

So the image is congruent to the object, its position and orientation has changed, its isn't mirrored.

So again, I want you to read that description again and think about where the rocket ship could be and how it will look.

Okay, so it needed to have sort of twisted because its orientation has changed, the fire and the burn at the back of the rocket needed to stay sort of shorter on the left and longer part on the right.

And that's because it wasn't mirrored.

If it had been mirrored, then the the long part, which is to the right of the flame would need to now be on the left.

But the description told us it hadn't mirrored the word congruent, going back to the first cycle in the lesson, that word congruent means that if I were to cut them out, they would lay on top of each other with no overlap or excess is identical shape.

It's just moved and twisted.

So a check, true or false, the description states how the image has changed and then I want you to justify your answer.

It's false.

Hopefully your justification was that it states how the object has changed.

So our description states how the object has changed to become the image.

Now you're gonna go bit of practise last part of the lesson.

So I just want you to match the descriptions to the correct figure, the figure being the sort of images, pause the video, have a try and then we'll move on.

Question two, you need to complete the table regarding each of the properties.

So is it yes or no? And then use that to support you with writing a description.

Pause the video and have a go.

That might take you a little bit longer.

And then finally write a description for the two transformations for part A and for part B.

It might be that you want to go through that table from question two again for each of them to support you with your description.

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

So question one, question one.

You needed to match up the descriptions to the correct figure.

So the left figure would match with the image is congruent and has the same orientation as the object.

It's not being rotated or flipped.

So its orientation hasn't changed.

The middle one is the con image is congruent but mirrored.

So it's being mirrored to look in the other direction, but it's the same size and same shape.

And the last one, the image is congruent but the orientation has changed from the object.

Question two, you need to complete the table and then write yourself a description.

Your description's not gonna be exactly the same as the one on the screen, but as long as you're hitting the sort of similar idea.

So the orientation had changed, the lengths hadn't the position changed, the angles didn't change, and it has become mirrored.

So the description I went for is the lengths and angles are in variant, its orientation and position has changed.

It is a mirror image of the object.

And then question three, your descriptions might have been for part A, the object and image are congruent, but in a different position.

And for Part B you may have gone for the image has a different orientation to the object.

They are also congruent.

So you may have mentioned that as well.

So during the lesson we have looked what congruency really means and that's it's not changed size but may have been translated, rotated, or reflected.

The description states how the object has changed to become its image.

And when a transformation has taken place, some of the properties change and some are invariant, where invariant means does not change after a transformation.

Well done today, I look forward to working with you again.