Lesson video

Hello, I'm Mr. Gratton, and thank you so much for joining me for today's lesson.

In today's lesson on transformations, we'll be introducing rotations, learning about the nature of rotation and in a real-world context, and analyse which properties of a shape remain invariant after rotation has occurred.

Some of this might already be familiar to you, but let's work together to frame it in a new and more mathematical way.

Right.

Several key words will be used during the lesson.

These are object, image, congruent and invariant.

If you're unfamiliar with these words and their definitions, pause here.

Here is the definition of rotation, our main focus for the lesson.

Okay, let's look at the lesson in two parts, with the first focusing on identifying rotations in the real world.

The word rotation might be familiar to you, but what other words describe the same thing? Turn, spin, and revolve.

So for a key, you might say that you turn a key to lock a door.

Well, this is the same action as rotating the key to lock the door.

You will have rotated in your life and not even realised it.

Jigsaw puzzles, they will require you to rotate pieces around in many different ways to see if they fit in different orientations.

It's rarely the case that all pieces will fit into the puzzle without needing to rotate them first.

Here's an example of when it doesn't work when you don't rotate.

See, the piece overlaps the puzzle, but it doesn't fit it.

However, rotation allows it to fit perfectly.

See? Time for you to practise some rotations.

Grab a pencil, lie it flat on a table with the writing tip pointing away from you.

And the rule is the pencil cannot be lifted off the table in any way.

Instruction one, rotate the pencil so the tip is pointing towards the left.

In how many ways can you make that happen? Pause and see if you can figure that out.

Actually, there're infinitely many ways of doing this rotation.

The easiest way is to do a quarter turn in the direction on screen.

Instruction number two, reset your pencil so the writing end is again pointing away from you.

Rotate your pencil so the tip is now pointing to the right.

Can you do this in two different ways, and can you describe what you have done? The two easiest ways are to rotate it a quarter turn in the direction on screen, also known as clockwise, or three quarter turns in the opposite direction, also known as anti-clockwise.

Okay, instruction number three, rotate your pencil so it is now pointing towards the door.

What is the most complicated rotation you can perform to do this? And can you describe what you've done? Pause now to give it a go.

There are several possible rotations that you could do to achieve this.

Most of these involve taking your pencil and rotating it through one full turn or two full turns or three full turns first, before finally doing the turn that points it towards the door.

Let's solve some basic problems with rotation.

At the moment, none of these jigsaw pieces fit into the puzzle as they are.

We will need to rotate them a quarter turn first in order to make it fit.

Here's what they look like after the rotation.

Which piece fits into the puzzle now? Yes, c fits.

Well done if you spotted that.

We can always use rotation to try and solve puzzles like this.

What about a different type of problem? I want to rotate this car sticker so its wheels are on the road like a normal car.

Which direction will the car face, left or right? Can you rotate it in your head to figure that out? Let's have a look.

The car will face left.

Right.

Let's check your understanding.

Which of these shapes, a or b or c or d, fit onto the shape at the top after a rotation? Pause and have a look.

And the answer is d.

Let's have a look at how it fits.

There's the shape.

Rotate it half a turn and it fits into that space.

Okay, time for some independent practise.

Laura has a lot of car stickers.

She wants to rotate all of them so that the wheels are touching the road.

Which cars will face left and which cars will face right after Laura rotates the cars? Pause and have a go.

Time for you to create your own drawing.

Sketch your own car or any vehicle that you want in a weird orientation so that it's facing in some random direction.

Can you identify whether when you rotate your drawing it will be facing left or right when the wheels of your vehicle are on the road? Extra challenge, can you draw two different vehicles that will point in the same direction? Pause and good luck with your drawings.

So for Laura's stickers, these are the directions that all cars face.

A, B, C, E, and G will point left.

D, F and H will then point right.

You could have drawn any amazing car.

Here're some examples of ones that I've made.

Really good work on all that you've done on rotations in the real world.

Let's look at cycle two where we look at properties of rotations and all the words linked to those properties.

For this part of the lesson, we will need to use the word invariant, meaning a property of the shape that has remained the same after you have transformed it in some way.

When an object is rotated, its lengths are invariant, meaning the lengths do not change in its size.

Which of these is an example of rotation, and which one isn't an example? Have a think.

The left one is the rotation and the right one is not.

This is because all the side lengths of the left are invariant after the half a turn's rotation has occurred.

Whilst the height of the right image is one square greater than its object, the shape has been stretched a little and therefore it is not a rotation.

The side lengths are not invariant.

Similarly, angles also remain invariant after rotation.

Which of these is an example and which of these is not an example of rotation whilst looking at the angles? The left one is again the example.

Notice how the angles are both the same and also follow the same order.

When reading clockwise, the angles are 82, 64, 124 and 90 in that order.

Looking at the object, that starts in the top left going clockwise.

82, 64, 124 and 90 is again the order clockwise, but now starting on the bottom right in the image.

The angles and the order of the angles always stays the same with rotation.

The right pair is not an example because very clearly the angles do change between the two shapes.

There are practical ways of checking the invariants of the sides and the angles of a shape.

And that is to use mathematical equipment, rulers to measure the sides and protractors to measure the angles.

Using a ruler, which of these is not a rotation of this left rectangle? Let's have a look with a ruler, 22, 22, 25.

5 and 22 centimetres.

The third shape is not a rotation because its side length is different from the side length of the object.

The other two images are rotations because they share the side length of 22 centimetres.

Okay, check time.

Which of these is a rotation of the top trapezium? Pause and have a think.

And the answer is b, as at least one side length has changed in the other two shapes.

Can you spot which of the side lengths have changed for a and which for c? For a, the bottom length should have been two squares across, not three.

And for c, the top is now six squares across, not five.

Well done if you spotted those.

Onto the next question with Sam.

Sam has taken this object, a rectangle with side lengths of three centimetres and five centimetres, and claims the image is a rotation because the side lengths are invariant.

They stay three and five centimetres.

Is their statement true or false? Have a think and also see if you can justify your answer.

The answer is false because the angles have clearly changed.

They were all once right angles because it was a rectangle.

Now they are clearly not.

This is a parallelogram instead.

We want to make sure rotation is not mixed up with any other transformation.

And a property that is less commonly focused on is the mirror image, usually associated with reflection.

Rotations never ever become mirrors of their objects, and here is a way to tell.

Take this rectangle.

I have written the word rotate on it.

And as I rotate the rectangle, you can still read the word written on it.

If it does get tricky to read, you can tilt your head to make it a bit easier.

Try tilting your head right.

You might be able to read the word rotate still.

However, if you mirror a word, whilst you might still be able to figure out what the word says, it will never actually clearly and properly say the word rotate.

It will always be a distorted, mirrored version of that word.

Rotations never are mirror images.

You will always be able to figure out what the word says, even if it means tilting your head a little bit to do so.

Okay, quick check.

Which of these has been rotated, and which has been mirrored to try and trick you? Pause and have a look at them all.

Well done if you spotted that a and c are the rotations.

B is a mirror image.

Okay, one last property.

I've alluded to this word once or twice before during the lesson, and that word is orientation.

When an object is rotated, its orientation changes even if the image looks the same.

Here's an example.

After my rotation, the orientation has changed, right? But the square still looks the same, so I'm kind of confused.

Let's now do this same rotation again, but with each and every one of the vertices labelled.

Will the vertices be in the same location after my rotation or will they end up in a different place? Let's have a look.

Note, whilst the shape may look the same, it is still a square.

The vertices have changed their orientation.

Notice that the A on the object is on the bottom left, but the A prime on the image is on the top left.

The vertices have now rotated to a new location, even if sometimes that location is where a different vertex used to be.

There is however one exception to this rule.

That exception is with a full turn of rotation.

When a full turn occurs, let's look at it in parts, quarter turn, half turn, three quarter turns, one full turn.

Only after that one full turn will there be an orientation that is invariant to its object.

After one full turn, all the vertices will be in exactly the same location.

This also occurs with multiples of a full turn, so two full turns, three full turns.

As long as it's exactly a full turn and not a full turn and a bit, the orientation will be invariant.

Okay, last few checks.

Whose statement is correct, Sophia who says, "As you rotate the arrow, it will start to point in a different direction and never point in the same direction" or Lucas who says, "After a full turn of rotation, it will return to the same orientation and point in the same direction." Pause and have a think about what they're saying.

The correct person is Lucas.

Sophia is correct in the sense that the arrow begins to point in different directions and the orientation will start to change, but she's wrong when she says it will never point in the same direction again.

Because Lucas is correct that the arrow will point in the right direction, right, after a full turn of rotation has occurred.

Okay, next statement, and Alex's statement is incorrect.

Which of these might be the correct explanation as to why? And there may be more than one correct answer.

Pause and read what Andeep and Alex have said and look at the possible explanations as to why Alex is incorrect.

Both a and c are correct.

An image always looks the same after a full turn of rotation, but it may have rotated less than or more than one full turn such that the image looks the same as the object but may have a different orientation.

Final sets of practise tasks.

Which of these trapezia are rotations of the top trapezium? There may be more than one right answer.

And for question two, there will be other rotations not shown here.

Can you draw them in this space on the right? Pause to give these two questions a go.

Question three, you might need some equipment to help you answer.

Which of these rectangles below are rotations of the rectangle at the top? Pause and take your time to measure each and every shape carefully.

Question number four, the object has been rotated to become its image, and Andeep says, "The object and image both look the same, therefore they have the same orientation." Explain why Andeep might be incorrect.

Pause and have a think about how you can explain this.

And the last question, put tics in the correct places in the table to describe all the properties of rotation we have discussed today.

Pause and fill the table.

And the answers a and c are both correct.

Well done if you spotted both of those.

A is a half a turn's rotation of the object at the top, and c is a quarter turn.

For question two, the one where you had to draw it yourself, you could have rotated it by a quarter turn in the opposite direction to see.

Very well done if you managed to draw that accurately.

Onto question number three, using a ruler, you'd have seen that the first, third and fourth of the images had the same length as the object.

However, image two was too short and image five was too long.

For question number four, orientation is only preserved if the object has been rotated through one full turn or two full turns or three full turns or multiples of exactly a full turn.

If it has been rotated by less than that, for example, a half a turn for this image, the shape may look invariant, but it will still have a different orientation.

Super good spot if you managed to understand and write down that difference.

As a summary of everything that we've learned today, the lengths and angles are always invariant, whilst the position and orientation usually change.

I say usually because they are invariant, but only after a full turn.

Every other degree of rotation means that the position and orientation do change.

And that summary concludes today's lesson on introducing rotations.

Super well done for working hard throughout the entire lesson.

In today's lesson, we have covered placing rotations into real-world contexts through puzzles and problems. We've also identified what is a rotation and what is not a rotation.

We have also learnt to understand properties of rotation such as the invariant lengths, the invariant angles, and that rotations are never a mirror of the object you started with.

And finally we have visualised orientation, when it does change and when orientation is preserved.

Thank you so much for once again joining me in a lesson.

Have a great rest of your day, and I hope to see you soon for some more maths.