Lesson video

Good to see everyone.

I'm Mr. Gratton and it's good to see here for another lesson of maths.

And in today's lesson on transformations, we'll be understanding the nature of reflections, analysing its properties to identify what changes and what doesn't change when the object is reflected.

There will be two keywords used throughout the lesson: reflection and sense.

Pause here to familiarise yourself with these definitions, but we will go into much greater detail about what these words mean during the lesson.

Speaking of sense, our first lesson cycle will be about understanding the meaning of the word sense by looking at reflections in some more visual contexts.

On this slide, there are three types of cats all subjected to one transformation.

Which of these transformations is the reflection and how do you know? So the left hand cat has been rotated and the right cat has been enlarged.

The centre cat is the reflection.

You might see similar scenes to this if a cat were to stand by some water, a puddle or a river, for example, you can see the reflection of the cat in the water and it would look something a little like this.

What other words would you associate with the word reflection? Here are some that I could think of, flip, mirror and equidistant.

Sam's description here is accurate.

The image of the cat has flipped in the reflection, possibly made in some water.

The reflection is a mirror image of the cat.

Those are some contexts you could use these words.

All reflections can be seen, can be created in a mirror.

That is where the mirror word is so useful in everyday language.

Let's look at some further examples that we can properly analyse in some detail.

Which of these rectangles with the word frog on it have been reflected and which ones have been rotated? And more importantly, how can you tell? What can you do to make some more of the words readable? And for which ones is it impossible to make readable without trying to modify in your head what the letter F, for example, looks like.

You can tilt your head left or right to identify the word frog.

Maybe have to tilt your head quite a bit to see the word frog in its normal orientation.

For those where rotating your head does not help and you have to modify in your brain what the letter F looks like, those are most likely to be the reflections.

If I tilt my head left, I can tell that the first example is a rotation.

If I tilt my head right by a little bit more, I can tell the fourth, the right most example is a rotation.

However, no matter how much I tilt my head left or right, the second and the third examples will never, ever look exactly like the word frog, F R O G.

Therefore, these are most likely to be the reflections.

This ability to easily identify a word without being clever and figuring out mentally by adjusting what the letters should look like that is after it has been transformed depends on the sense of the object and whether this sense has changed or not.

An object maintains sense after transformation if the object is still identifiable as exactly the same object without any further transformations being applied.

Again, for some words you might be able to figure out what the word says, so I might be able to figure out that word on the screen says frog, but the letters of the word will never ever directly read F R O G without it looking backwards.

The F written backwards and the R written backwards, for example, in the orientation and direction that the word frog should do no matter how much you tilt your head left or right to help.

Which transformations maintain this idea of sense, the readability of the word and which ones don't? Well, actually all of them maintain sense except for one.

Reflection.

Right.

Let's have a look at translation.

First of all, the moving transformation as you move the word frog, it still very clearly says the word frog.

No head tilting, no anything needed.

It still says the word frog.

With rotation, you can still identify the letters as F, R, O and G, even if it requires you to start tilting your head a bit more, the closer the rotation gets to a 180 degree half turn.

Because the word is still as readable with a bit of head tilting help, perhaps the sense of the object has stayed the same.

Have a look at this and see if you can still identify the word frog as it rotates.

There you go.

I have snapshotted for you the orientation of the word frog at different stages of the rotation.

At each stage, you can still easily identify and read the letter F as the letter F, even if it has been rotated.

This is the same for all the letters, R, O and G as well.

Therefore, the sense of every single one of those letters has remained the same.

Now on to reflection.

Why doesn't it maintain sense? When a word has been reflected, the letters of the word will have been transformed so that it is impossible to identify the letters as exactly the same letter.

No matter how much you tilt your head, the sense of those letters has changed because it will never look like the original again without a second reflection.

For these two reflections, can you tilt your head in any direction to make the letter F look exactly like this upright F? Tilt your head left, tilt your head right.

Can you make those letter F's look like the big one on the bottom of the screen? No, it is impossible to do so and therefore, the sense of the letter F has changed.

You'll also notice that the letter F here is the letter F written backwards.

If you see a letter written backwards, that is an easy way of telling that the sense of that letter has changed.

You might have spotted that some letters like the letter O in frog did look the same after the reflection.

In these cases, the letter O did change sense, but you just can't see it because of the symmetries of the letter O.

This is because reflection always changes the sense of any object even if the image after the reflection looks exactly the same.

This becomes easier to visualise if I change the letter O to be in a different font.

As you can see that fancy curly O has changed sense compared to its original form.

Also, notice that the letter O seems to be written backwards.

Remember what I said earlier? If a letter is written backwards, then it has definitely changed its sense.

The letters R and G as well as the F or clearly have been written backwards as well.

So to summarise, I can tell that these two reflections have changed their sense compared to the original for two reasons.

One, no matter how much I tilt my head left and right, those two reflections will never look exactly like the original.

And two, the letters F, R, O and G all look like they have been written backwards compared to the original word frog.

Right.

Let's check your understanding of sense, in which of these pairs of words has the sense changed? Pause the video to give yourself some time to tilt your head to see if you can figure those out.

Plan, sign and cosh have all had their senses changed.

This means they definitely have been through a reflection.

The others, they have been rotated.

And time for some independent practise.

The word zone at the top is the subject.

Below are five images, some have the same sense because they've been translated or rotated whilst some others have had their sense changed because of a reflection.

Pause the video to identify the sense changes.

And for those whose senses stayed the same, tell me whether they've been translated or rotated.

Pause the video now.

Next set of questions on this L-shaped polygon.

Very similar tasks to before.

Which of these have had their sense changed because you can no longer tilt your head to make it look like the letter L and identify which have been translated or rotated.

Pause the video to give these two questions a go.

Onto the answers.

The bottom two image are the reflections.

No matter how much you tilt your heads, the letters Z and N will never look like the proper letters Z and N.

Therefore, they've had their sense changed and they are reflections.

You can also tell by the rightmost correct image that the letters Z and N have clearly been written backwards because they have been written backwards.

I know their sense has changed and therefore, they have been through a reflection.

The top left and the middle are rotations because you can tilt your head quite a bit for the left hand one and you can then read the word zone exactly like the object.

The right hand one has just been translated 'cause it just says the word zone exactly like the object just in a different location.

For the L shapes, the top left and the bottom middle are the reflections.

Again, you can't tilt your head in any way to make them look like the letter L.

With the other three, you can.

Well done for all the work you've done on sense so far, which is one of the properties of reflection.

In this next part of the lesson, we'll be formalising all the other properties of reflection and checking to see whether shapes are reflections or not.

Based on how these properties have changed or haven't changed.

When an object is reflected, the length of its sides remain invariant, meaning that the side lengths do not change in length.

However, the sense of these sides will change.

Which of these two diagrams shows a reflection and for which one does it not and why not? Pause the video to have a quick look.

The left hand one is the reflection whilst the right hand one is an example of a non reflection.

For the left, all the sides remain the same lengths but have changed location due to that reflection whilst the right one has shrunk with a height going from five to four units.

In a similar vein, the angles in the image of a reflection are always invariant.

That means the angles remain the same but will be in a different location.

Which of these is the reflection and which one isn't? Pause the video to have a quick look.

Again, the left hand one is the reflection.

This is because the angles of 55, 107, 63 and 135 are all the same in the object and the image.

Notice though how the orientation of these angles has reversed.

55, 107, 63 and 135 is the order of the angles going clockwise in the object and those same angles go anti-clockwise in the image 55, 107, 63 and 135 going in that opposite direction to the object above it.

This is always true for reflections.

The angles stay the same, but the orientation of the angles reverses.

This property of reversing orientation is consistent with all parts of the shape.

Even if it's hard to see.

Here, you can see that an object, a square looks identical to its image, but its orientation has actually changed even though you can't see that change.

It's easy to see this change if we label the vertices of the object and the image.

Here, the objects have vertices labelled anti-clockwise, A, B, C and D starting at the bottom left hand corner.

Whilst in the reflection the vertices are labelled clockwise, A prime, B prime, C prime, and D prime.

Also note, A on the object is the bottom left while A prime on the image is on the bottom right.

Okay, check time.

Complete the labelling of the vertices of this image, remembering to put a prime dash symbol next to each letter in your image.

It might help for you to sketch the image to help in your labelling.

Pause the video to give the sketching and labelling of that right hand image a go.

And here is the correctly labelled image.

Well done if you labelled all four vertices correctly.

Next check.

Which of the images A, B, C or D is a reflection of the object that trapezium at the top? Pull the video and there may be more than one right answer.

A and C are the correct answers.

B and D have had their side lengths altered in some way.

Sophia is incorrect when she says, if you reflect the arrow, it will have the same orientation as it will still face in the same direction.

Come up with an explanation as to why her statement is incorrect.

Pause the video to construct your answer.

Sophia is incorrect because all reflections change orientation even if the image looks the same.

You can also reflect that arrow horizontally so that the image of the reflection points left not right.

Okay.

Final set of practise questions.

Question one is all about labelling the image with the correct vertices.

Remember, you need to include all the prime dash indicators for all the letters of any image.

Pause the video to complete the labelling of all reflected images.

And question number two, match the objects to the reflected images.

There are two images that do not have a match.

Pause the video to begin pairing.

Question three, put a tick in the correct boxes to define the properties of reflection.

You might note that some small clarifications need to be made.

Pause the video to complete the table summarising everything that we've done on reflections today.

Here are the answers for question one.

Notice how the orders of the letters for each pair of shapes is the same.

It's just the orientation has reversed from clockwise to anti-clockwise or vice versa.

Anti-clockwise to clockwise.

So for example, A, B, C, D on that first object has its letters labelled anti-clockwise for the object.

Whilst the reflection A prime, B prime, C prime, D prime is orientated clockwise for the image.

For question two, C pairs with F.

D pairs with G, E pairs with H, A pairs with J, and B pairs with K.

They are all the reflection pairs.

I and L are the odd ones out 'cause they do not have any pair and this is the completed table.

Side lengths and angles are always invariant.

However, position, orientation and sense always change, not usually change.

No matter what the reflection is, where it's done or what the object is.

Position, orientation and sense always change even if visually it does not look like it has.

That is brilliant work throughout today's lesson on introducing and understanding the concept of sense, how it applies to reflection, and how other properties of reflection help you to identify what are reflections and what are non reflections.

Thank you so much for joining me today and hope to see you soon for another lesson on maths.

Have a good day.