Lesson video

Hello everyone.

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

In today's lesson on transformations, we'll be looking in depth about rotation by using more mathematical language to describe a rotation size and the direction in which a rotation turns.

Pause here to familiarise self, with some of the keywords we'll be using today.

Let's split this lesson into three cycles.

With the first describing the direction of a rotation, a shape can be rotated in two directions.

Both these words, you might be familiar with, clockwise and anti-clockwise.

Here's a shape and we want to turn it clockwise.

Clockwise can be described, as the top of an object being rotated right.

But there's a second half to this explanation, we also need to be aware of.

It is also the bottom of an object being rotated left.

Have a look here.

On the other hand, we have anti-clockwise, which can be described as the top of an object being rotated left, but also the bottom of an object being rotated right.

Have a look.

Here's a quick check for understanding about the directions of rotation.

In which direction will this trapezium rotate? I will do the rotation and then have a pause to think about which answer it is.

Well done if you got anti-clockwise as your answer.

Okay if this arrow currently pointing south was to rotate clockwise, which compass direction would it pass first? Would it be N for north, E for east, or W for west? Pause and have a think.

Remember, clockwise is in this direction, therefore the first direction it would pass is west.

Very well done if you spotted that.

Now onto some independent practise.

For each of these, I just want a one word answer, clockwise or anti-clockwise.

Pause the video to see which direction each of those shapes is turning in.

Okay, next question.

An arrow is currently pointing in the eastern compass direction.

If the arrow rotates anti-clockwise, which compass direction will it pass first? Think about which direction anti-clockwise is.

And for question five, the arrow is currently pointing north.

The arrow will rotate clockwise until it points north again.

In which order will it pass through all of the other compass directions? There should be three, or four if you include north answers in a specific order, pause the video, have a look at those two questions.

Okay, time for some answers.

For question one anti-clockwise.

That is because the arrow is pointing from the bottom of the shape going right.

For question two, it is also anti-clockwise because that arrow is pointing from the top of the shape going left.

However, question three, the arrow is pointing in the opposite direction from the top going right and therefore the answer is clockwise.

For question four, the answer is north anti-clockwise means it's gonna go from east up to north, then to west, then to south, and so north is the first compass direction it passes through.

And for question five, remember the definition of clockwise.

If you have the top of an image, you then turn that top of the image going right.

With the top of the image being north, going right means east, then down to south.

Then from the bottom you go left.

So from south to west, so north, then east, then south, then west, is the correct order.

And in our second cycle of the lesson, we'll be looking at the size of the rotations that we create.

The size of a rotation can be described using angles, written in degrees.

A quarter turn is a 90 degree rotation, whilst a half turn is a 180 degree rotation.

We say that a three-quarter turn is 270 degrees and a full turn is a 360 degree rotation.

Here's an example of how we use these angles in a full description.

So here's an object and an image.

They have been rotated by less than 360 degrees.

There are two ways of describing this transformation.

Let's have a look at the first one.

The object has been rotated by some size clockwise.

Notice how the arrow starts pointing up and then points right.

That means we're going clockwise by a quarter turn which is 90 degrees.

For statement number two, well we've described it clockwise, so the only other option is to describe it, anti-clockwise.

Okay, a check for understanding.

On the left we have a set of different angles, and on the right we have their descriptions in words.

Match the size, measure degrees with its description.

Note that some of the numbers don't match with the description.

Pause the video and see if you can match them all.

And the answers are as follows.

Remember, a full turn is a full 360 degrees, a half turn is 180 degrees.

Notice that half of 360 is 180.

The numbers and the descriptions do match mathematically.

Three-quarter turns is 270 degrees and one quarter turn is 90 degrees.

Again, notice that three-quarters is going to be three times one quarter.

Three times 90 is 270, and again, the numbers match.

Okay, next question.

Here's an animation of this arrow rotating.

By how many degrees does this arrow rotate? I'll play the animation and pause the video afterwards to find your answer.

Notice how there is a little pause after each 90 degrees of the rotation.

There were three lots of 90 degree rotations.

I'll play it again, once, twice, three times.

As we said before, three-quarters of a turn is 270 degrees.

Super well done if you've got that right.

Heres another animation, but this time I want more detailed answers.

By how many degrees does the shape rotate and in which direction? Again I'll play the animation and then pause the video to figure out which two answers describe the rotation.

Well done if you got the two parts of your answer, 180 degrees, half a turn, and it was rotating clockwise.

Okay, next check.

If this arrow were rotated clockwise by 90 degrees, in which direction would it end up pointing left, down, or right? Pause the video to try and mentally visualise that rotation.

So remember, clockwise is this direction, and if I were to rotate it 90 degrees, it would end up pointing right.

Super well done if you got that correct.

Okay here is more practise for you.

For each of these questions, I want you to write down either up, down, left or right, depending on the description for each arrow.

Pause the video and have a think about which direction each arrow will end up pointing in.

Okay question number two, and here is where we will start to use these full sentence descriptions of a rotation.

This object has been rotated by less than 360 degrees to get the image.

By putting down either an angle in degrees to describe the size of the rotation or clockwise or anti-clockwise to describe the direction of the rotation, complete both of these possible statements.

And question three is exactly the same again.

An object and an image that'd be rotated by less than one full turn, by putting in an angle and a description of the direction clockwise, anti-clockwise, complete both of those sentences.

Question four is very similar to the previous questions, however, I'm starting to remove more and different words from the descriptions.

In a similar way to before, fill in all the missing words, but be careful about what one of those missing words is.

Pause the video and be careful with how you complete each description.

And the answers.

For question one, if I were to rotate that arrow currently pointing up anti-clockwise, it would end up pointing left.

Again remember the definition, if you're rotating anti-clockwise, you are taking the top of the image and rotating it left, so from pointing upwards, it would rotate left.

For B, a 180 degree rotation, either clockwise or anti-clockwise would point in the complete opposite direction, which is left.

For CA 270 degree rotation anti-clockwise is the same as going through three one quarter rotations.

So currently pointing left after one quarter, it would point down, two quarters, it would point right and so three-quarters of a full turn two 70 degrees, it would end up pointing up.

And for D, a 90 degree rotation clockwise would make the shape end up pointing left.

Well done if you got all four and that you weren't caught out by the fact that, three of the four answers were left.

Okay, question number two, for statement one, we would say that the object was rotated clockwise by a quarter turn, so 90 degrees.

For statement number two, you didn't even really need to use the shape because statement one described clockwise, and so the opposite direction to clockwise is anti-clockwise.

The missing word for statement two.

For question number three, the object has been rotated 90 degrees, but neither the statement talks about a direction, so you had to be careful about which direction the object was rotating in and by which angle.

The object rotates 90 degrees anti-clockwise.

Therefore, statement two has to be a clockwise description and that will be three-quarters of a full turn, 270 degrees clockwise.

Question number four, clockwise direction rotation meant it was rotating in a three-quarters of a full turn rotation, which is 270 degrees.

In a similar vein for statement two, the anti-clockwise rotation would've been 90 degrees.

The missing word you should have been careful about was always mentioning that the transformation is a rotation.

Saying the word rotation is just as important about giving the direction and the size as well.

And onto our third and final cycle of the lesson.

This cycle is about describing more complicated rotations, compared to the more straightforward quarter half and three-quarter turns that we were looking at before.

So we are aware that one full rotation is 360 degrees, but what if we were to go through more than one full rotation? Well the degrees, the angle that we would describe it with would be a number greater than 360 degrees.

For example, here is one full turn and a quarter of an extra turn.

This would be 360 degrees plus 90 degrees, which is 450 degrees.

So one and a quarter full turns would be 450 degrees in total.

So here's a quick check, one and a half full turns, would be equal to how many degrees? There are two answers, one which is the un-added answer and one which is the answer added together.

I recommend looking at the words one and a half full turns and figure out if that corresponds to A, B, or C, and then do the addition for D, E, F and G afterwards, pause the video and have a think.

Okay, one full turn is 360 degrees, so B is instantly ruled out.

And a half turn is 180 degrees extra.

So C is the correct answer for that left hand set of answers.

360 plus 180 is 540, and so G is the correct answer added together.

Now we've talked about having one full turn.

Well actually we can have more than one full turn.

So in this question we have two and a quarter full turns, which of the answers unadded, is correct on the left and then adding more together what is the correct answer on the right? Pause the video, have a think.

So two full turns is two lots of 360 degrees.

So A is ruled out because it only mentions one full turn of 360 degrees.

A quarter full turn is 90 degrees, so we've got 360, 360 and 90 that is the answer B.

Similarly, if I were to add up 360, 360 and 90, I would get the answer of 810 degrees, which is the answer F.

So far we've been looking at all different types of rotation, all different sizes of rotation, except all of our answers have been multiples of 90.

90, 180, 270, 360, 450, 540, they are all multiples of 90.

Rotations can be any size, not just multiples of 90 degrees.

So what we're going to do now is estimate the size of the rotations when they are not multiples of 90 degrees.

You can estimate the size of these unfamiliar rotations by comparing them to the more familiar quarter turn 90 degrees, half a turn, 180 degrees.

Three-quarter turns 270 degrees, and the full turn 360 degrees.

You can also compare it to zero degrees, no rotation.

So here's an example of an object that has been rotated to get its image, but the image hasn't been rotated by a nice 90 degree, or multiple of 90 degrees.

So how do we estimate how much this object has been rotated by? Well, let's compare it to two rotations we are more familiar with.

Here's a quarter turn, 90 degrees.

We can see that our rotation is a little bit more than that 90 degree rotation, but a little bit less than the 180 degree rotation.

So any answer sensibly in the middle of 90 and 180 is a good estimate with the best estimate being around 135 degrees, 135 degrees being perfectly in the middle of 90 and 180.

Okay, quick check, which of these values is a sensible estimate for the size of this anti-clockwise rotation? I'll put up some helpful animations partway through the pause in this video to give you some hints.

Pause at a suitable moment to figure out your answer.

And the answer is 45 degrees.

We haven't rotated 90 degrees to get a quarter turn, and so it's going to be an answer less than 90 degrees, which is 45 degrees.

Second check, which of these values is a sensible estimate for the size of a rotation between half a turn and three-quarters of a turn? Think about which angles correspond to half a turn and three-quarters of a turn and figure out which of these answers is roughly halfway between.

Pause the video to figure it out.

So a half a turn is 180 degrees and a three-quarter turn is 270 degrees.

Which of these answers is halfway between 180 and 270? The answer is B.

Very well done for all the effort you've put in today, we are onto the last set of practise questions.

For each of these descriptions, I want you to write down a calculation of at least two angles and then the total angle that describes each of these rotations.

For some, you'll have to add at least two angles together, and for others you'll have to find a number roughly halfway between two angles.

Pause the video and write down all of your calculations and then your answers to each of these four descriptions.

Okay for question number two, I've given you one fully correct statement.

The object has been rotated 100 degrees clockwise.

By thinking about clockwise and anti-clockwise directions and the quarter, half and three-quarter turns, what angle would be best suited to go in statement number two? Pause the video and have a think.

Okay for question number three, I've given you less information than I have for question number two, you have to fill in more yourself.

For statement number one, I have said the object has been rotated 140 degrees, but I've not given you a direction.

I want you to think about which direction would be roughly a 140 degree turn.

It might help to think which two nice rotations, 140 degrees lies between.

Similarly, for statement number two, I've not given you either a direction or a size figure out both yourself.

Pause the video now and good luck.

And question four, the final question.

If this arrow is rotated anti-clockwise through two full turns and three-quarters of another, so that is a lot of rotation, which direction will the image of the arrow be pointing up, down, left or right? In total, how many degrees will the arrow have rotated? It might help for you to write down individual rotations, individual angles, and then add them all up to get the full description of two full turns and three-quarters of another.

I would like you to give a description of a different anti-clockwise rotation that would result in the arrow pointing in the same direction.

Furthermore, for the last question, part D, I want you to give a different description, but this time for the clockwise rotation, take your time there's a lot to digest with question number four, pause the video and good luck.

Okay, onto the answers for question one, between a quarter and a half turn, a quarter is 90 degrees, a half turn is 180 degrees, and so a number anywhere between 90 and 180 is correct.

With the midpoint being 135 degrees.

Well done if you got any number between 90 and 180.

Two full turns is just going to be one full turn plus one full turn.

360 plus 360, which is 720.

No rotation is zero degrees and a quarter turn is 90 degrees.

So any number between zero and 90 would be perfect with 45 degrees being the exact midpoint answer.

One and three-quarter turns would be 360 degrees plus 270 degrees, 360 plus 270 is 630 degrees.

Well done if you've got those correct.

Okay question number two.

If the object has been rotated through 100 degrees clockwise, then it would've been rotated anti-clockwise by the rest of one full turn.

If one full turn is 360 degrees, I take away the a 100, to get 260 degrees, which is the rotation anti-clockwise.

Same logic for question number three.

If it's been rotated in one direction by 140 degrees, 360, take away 140 is 220.

So I know that the degrees for the second statement is 220, but what about the direction? Notice how you've got the top of the object now pointing towards the bottom left, imagine a 90 degree rotation, the top of the object would be pointing straight left.

If it's going to be pointing bottom left, it'll be a little bit more than 90 degrees, which is 140, and the direction that that would be turning is anti-clockwise.

So 140 degrees would be anti-clockwise with the opposite direction, clockwise being statement number two.

Number four, which direction would the image of the arrow be pointing in? It would be pointing left.

Well done if you figured that out.

Now the description is two full turns and three-quarters of another.

Two full turns will be 360 plus 360, and three-quarters of another would be 270.

So 360 plus 360 plus 270, is 990 degrees.

For part C, you didn't need to do the two full turns, you could just have no full turns and just rotate it by three-quarters of a full turn.

Similarly, you could have rotated it by any number of full turns, 10 full turns, doesn't matter.

As long as you've got the three-quarters of a turn, as part of your description, the number of full turns does not matter.

And it's the same logic for Part D, you can have any number of full turns, zero, seven, 10, your choice, as long as it mentions one quarter or a quarter of a full turn, that is the part of the rotation that's important.

Well done if you spotted those very specific details.

And that is it for today's lesson where we have learned to describe rotations by their direction, clockwise and anti-clockwise, and we've used those directions to solve problems about compass directions North, southeastern, and west.

We've also learned how to describe rotations of different sizes, both in words and using numerical angles.

We've also covered the fact that rotations can take any size, not limited to angles that are multiples of 90 degrees or less than 360 degrees.

So thank you very much for joining me today, and I hope to see you soon for some more maths.

Have a good day.