Lesson video

Hello there.

You made a great choice with today's lesson.

It's gonna be a good one.

My name is Dr.

Ronson and I'm gonna be supporting you through it.

Let's get started.

Welcome to today's lesson from the unit of geometrical properties and Pythagoras's theorem.

This lesson is called rotational symmetry, and by the end of today's lesson we'll be able to recognise rotational symmetry in shapes.

This lesson will introduce a new key word and that is rotational symmetry.

Rotational symmetry describes when a shape appears the same after some rotation, and the order of rotational symmetry is the number of times the image appears the same as the object in one full term.

Now that's not clear to you straight away, don't worry about it.

We'll see plenty of examples of this during today's lesson.

We'll also be using this previous keyword as well during today's lesson, and that is the centre of rotation is the fixed point about which an object is rotated.

This lesson contains two learn cycles.

In the first learn cycle, we'll be unpacking what rotational symmetry is and trying to understand more of its properties and how to go about identifying rotational symmetry in shapes.

And the second learn cycle will be finding shapes that have particular rotational symmetries or even designing some shapes that have particular rotational symmetries.

Let's start off with though, identifying rotational symmetry in shapes.

Here we have two rectangles that are congruent.

Laura is going to rotate the left rectangle a full turn.

How many times will it look like it is in the same orientation as the rectangle on the right? What do you think about this? Well, let's see what Laura thinks.

Laura says, "I've marked the top of the rectangle so I can see when I've done a full turn" and then she begins to turn it.

It looks like it's in the same orientation as a rectangle on the right at this point here.

So that's once she turns it some more and then it looks like it again here.

Now I know it says the word top on there, but Laura wrote that in order just to keep track of where she is, originally that word wasn't there.

So if you imagine that word top wasn't there, the rectangle on the left would look very much like the rectangle on the right.

And then we get to one full turn and it looks like Laura says, "There were two times when the rectangle on the left looked like the rectangle on the right while I was rotating it.

Why was that?" Why do you think it is? Pause the video while you think about that and press play when you're ready to continue.

Well, this all has something to do with rotational symmetry.

Rotational symmetry describes when a shape appears the same after some rotation.

And the order of rotational symmetry is a number of positions in which a shape appears the same as it did originally when rotating through a full turn.

So in the previous example, we were comparing two congruent rectangles that were not initially in the same orientation and the order of rotational symmetry told us how many times one shape appeared the same as the other shape during that full turn.

But here we can also think about rotational symmetry when looking at a single shape.

By considering how many times during a full turn it appears the same as it did in its start and position.

And when we do this, the centre of rotation is in the middle of the shape for this process.

So here we have a rectangle like we saw earlier.

This has rotational symmetry of order two.

If we rotated this rectangle around, it would look the same as it does now on two occasions, once at the start and then again after turning 180 degrees.

So that's why it has rotational symmetry of order two.

We can find rotational symmetry of a shape by drawing a congruent shape on a piece of tracing paper.

So for example, here we have a triangle.

If we wanted to investigate its rotational symmetry, we could place some tracing paper over it and trace over that triangle.

And now we have two congruent triangles.

The one on the left is the original and the one on the right is the one on the piece of tracing paper.

We can then place a tracing paper back over the triangle and rotate it around, counting how many times it fits perfectly over the original during that full turn.

So let's do that.

It fits perfectly once when it's in this position.

Then as we start to turn it around, well it doesn't quite fit here, but it fits at this point.

So that's twice.

And then again here, so that's three times.

And now we're back to the start.

So this shape has rotational symmetry of order three.

We could also surmise that this triangle is equilateral because each vertex fits perfectly on top of each of the other ones, meaning that all the angles must be the same.

Now, you might be thinking that you already recognise that this triangle had rotational symmetry or order three before we started using the tracing paper.

So you might be thinking, "Well, what's the point of the tracing paper? Well, yes, tracing paper may not always be necessary for shapes where a rotational symmetry is clear to see, but rotational symmetry is not always quite so clear.

So it can be helpful when rotational symmetry is less clear.

For example, let's take a look at this hexagon.

Jun says it looks like it has rotational symmetry of order six, but let's check it with some tracing paper.

We trace over it and then we start to count the rotational symmetry.

We can count it once here in its starting position.

Then as we turn it around and see how many times it fits over itself again, ah, it doesn't quite fit here.

It must not be a regular hexagon, meaning that all the lengths are not the same and all the angles are not the same.

That's why it doesn't quite fit over itself here.

It doesn't fit here either.

It does fit here.

So that's order of two so far, Doesn't fit here and doesn't fit here.

So while this shape initially looked like it had rotational symmetry of order six, it actually has rotational symmetry of order two.

If you think about that previous triangle we saw, it had rotational symmetry of order three because it was equilateral.

All the angles were the same, all the lens were the same.

But if it hadn't quite been equilateral, if it was just a little bit off, then it's order of rotational symmetry would've been different.

So the tracing paper can be particularly helpful on occasions where we're not entirely sure whether a shape is how it appears to be to the naked eye.

Here we have Alex.

Alex is investigating rotational symmetry of the shape below, and here's what he does.

He traces over with tracing paper and starts to count its rotational symmetry.

He says "One, two, three." He surmises that the order of rotational symmetry is three, but that's not right.

What mistake did Alex make? Pause the video while you think about that and press play when you're ready to continue.

It looks like the mistake that Alex made was that he counted the starting point twice.

He counted it once at the beginning, before he started rotating the tracing paper, and then he counted it again at the end, after he did a full turn.

So his answer was more than what the rotational symmetry is.

And this is a really common mistake that people make.

They count the starting point twice in that same way.

So it is really important that you are consistent with when you choose to count the original position that the shape is in, either always count it at the start before you begin rotating the shape, like we've done so far.

Or always count it at the end once you've done a full rotation.

Whichever system you use, stick with that system.

If you are always doing it in the same way, you'll probably be less likely to get mixed up and count it twice.

Let's see what Lucas and Sofia are up to.

They are finding the order of rotational symmetry for the shape below.

Lucas says, "I think it will have rotational symmetry of order zero because it won't look like it's in the same position again in any other position." And Sofia says, "If we draw a congruent shape on tracing paper, it would fit perfectly on top of it once.

So it has rotational symmetry of order one." What do you think? Who do you agree with, Lucas or Sofia? Pause the video while you think about this and press play when you're ready to continue.

Well, all shapes have rotational symmetry of order of at least one.

That is because when you make a congruent shape on a piece of tracing paper, you can always fit that shape over its original one at least once every time.

So Sofia was correct in this case.

So let's check what we've learned.

Which shape has rotational symmetry of order four? Is it A, B or C? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is C.

That shape is a square, meaning that all four of its sides are the same length.

All four of its angles are the same as well.

They're all 90 degrees.

And then when you rotate that shape 90 degrees, it will look the same as it does right now.

So it'll look the same four times as we do that full turn.

Whereas with A and B, those two have rotational symmetry of order two.

Here's another question.

The shape below has rotational symmetry of order, what goes in the blank there? Pause the video while you write that down and press play when you're ready for an answer.

The answer is two.

The shape has rotational symmetry of order two, it will look like it does right now here and then once you've turned it 180 degrees.

True or false, a shape can have rotational symmetry of order zero? Is that true or is it false? And choose one of the justifications below.

The answer is false, because all shapes have rotational symmetry of at least one.

Over to you now for task A.

This task has two questions and here is question one.

Here we have eight shapes and you need to write down the order of rotational symmetry for each shape and tracing paper may help you with this task or you can use any other equipment that you think might help you instead.

Pause the video while you have a go at this and press play when you're ready to continue.

And here is question two.

You've got the letters from A to Z here, all capitalised.

You need to sort the letters into the table according to the order of rotational symmetry.

If you think a letter has order one, place it in the far left column.

If you think it has order two, the next one and so on.

Pause the video while you have a go at this and press play when you're ready for some answers.

Let's see how we got on with that then.

Here's question one.

We had to write down the order of rotational symmetry for each shape.

We could do that by using tracing paper, placing the tracing paper over the original shape, tracing it to create a congruent shape, and then turn the paper 360 degrees to see how many times it fits in.

You could also measure the length and the angles using the rule and protractor and work things out that way, if you wanted to.

Or if you have an editable version of this slide deck, you could take this slide, duplicate these shapes and rotate them yourselves on the computer and see how many times it fits over too.

However you do it, let's take a look at the answers.

In question A, all those lengths are the same and all the angles are the same.

It's a regular pentagon and it has order rotational symmetry five.

In B, well, all the lengths are the same, but not all the angles.

So this only has rotational symmetry of two.

It's a rhombus.

And in C, well, it's an scalene triangle.

None of the lengths are the same.

None of the angles are the same.

It only fits into itself in this position.

So it has order one.

For D, it's an isosceles triangle, and that means two of the angles are same and two of the lengths are same, but it only has order rotational symmetry one, because if you look at that top vertex, which is the different angle to the other two, that vertex will only fit into itself in this position.

So that's why its order is only one.

For E, we have a regular octagon, That means it has order rotational symmetry eight because all its angles are the same and all its length are the same.

So it'll fit into itself eight times.

And then for F, it is an octagon, but it's not a regular octagon.

Four of its length are equal to each other and four of its other lens are equal to each other as well.

This shape has rotational symmetry of order four.

And then for G, well, this one also has rotational symmetry of order four.

Every time we turn that ninety degrees or a quarter turn, it'll fit into itself.

And then for H, that shape looks quite a bit like the shape in G, but it's not quite the same.

It's not quite consistent all the way round.

This one only has order two.

And in question two, we have to sort the letters into the table according to the order of rotational symmetry.

And for ones which have order one, we have A, B, C, D, E, F, G, J, K, L, M, P, Q, R, T, U, V, W, and Y.

Most of the letters have order one.

You might be thinking, "Which ones are missing there?" Well, H, I, N, S, X and Z, they have order two.

And O has order greater than four if you draw it as a circle.

However, we need to bear in mind that sometimes different fonts or different people write letters in different ways.

So in particular, X and O, the order rotational symmetry may differ depending on how they're written or the font that they are in.

Well done so far, it's now time to get creative as we move on to the second learn cycle, which is finding shapes with particular rotational symmetries.

Here we have Andeep and Izzy.

Andeep and Izzy are designing a puzzle board for preschool children.

Let's take a look at it.

It's a wooden board where there are some spaces for shapes to go, like little indents in the board, we have some puzzle pieces which are different shapes.

And you may be thinking, what are those circles in the middle of each of those puzzle pieces? Well, if we turn one of those puzzle pieces on its side and get a side view, we can see that each of those circles is a handle, so that the child can pick up the shape and place it into its space and then also take that shape out of the space again.

And children solve the puzzle by placing the pieces in the correct spaces.

Sometimes you see these puzzle boards with some kind of picture or pattern printed on top of them.

Here it looks like Andeep and Izzy are still in the design phase.

That's probably why it's still playing.

So Andeep and Izzy tested a puzzle with some children and discuss what they saw.

Andeep says, "The children matched the pieces to the shapes easily, but they found some of the pieces easier to insert into their spaces than others." I wonder why that is? Izzy says, "Which puzzle piece did they find the easiest and which puzzle piece did they find the most difficult?" What do you think? Maybe pause the video while you consider what you expect the answer to be, and then press play when you're ready to continue.

Andeep says, "They found the regular octagon the easiest to insert into its space.

They struggled the most with the trapezium." Now that might be what you expected or it might have surprised you.

Izzy says, "Why do you think they found the regular octagon easier to insert than the trapezium?" Maybe pause the video and think about this and press play when you're ready to continue.

Andeep says, "The regular octagon has rotational symmetry of order eight.

So there are eight different ways that the piece fits into its space." And Izzy says, "Ah yes, the trapezium has rotational symmetry of order one.

So there is only one way that it can fit into its space." They decide to rearrange the spaces on the board so that the pieces are in order from the easiest to the most difficult to insert in their spaces.

So we can see here at the moment, they've got A, B, C, D for where they're gonna put the spaces they want.

They want the highest order rotational symmetry to go in A, because that will be the easiest.

You can place that shape in lots of different ways, and they want the one with the lowest order rotational symmetry to be in space D, because that one will be the most difficult because it only fits into its space in one way.

So which order should those shapes go in? Pause the video while you think about this and press play when you're ready to continue.

So we'll start off with the regular octagon, which has order eight, and then the square which has order four.

And that triangle, so long as it's equilateral has order three and then the trapezium, which has order one.

Let's look at another situation now.

Here we have Jacob who is designing a logo.

He wants the outline of the logo to have rotational symmetry of order six.

Now, Jacob not just going to go and draw a random shape hoping that a symmetry of order six, he's gonna be methodical about it.

So here's what he does.

He says, "There are 360 degrees in a full turn.

So I'll start by dividing 360 by six to get 60 degrees." That means I want the shape to look the same every time I rotate it 60 degrees.

He then says, "I'll draw a shape that has an angle of 60 degrees at a vertex." And that's what we can see here.

He then says, "I'll create five more shapes that are congruent to the first one." Now, he might do this using tracing paper or by cutting out the original shape and drawing around it multiple times, like the original shape is the mould.

Or he might go about using a computer.

He could do this by selecting the original shape he drew and then duplicating it multiple times.

However he does it, he then says, "I can then rotate the shape and arrange them around the centre point." And now he says, "I have my logo where the outline has rotational symmetry of order six." So let's check what we've learned.

Which shape would have rotational symmetry of order six from the shapes below? You got a irregular heptagon.

B, irregular hexagon, C, a regular octagon or D, a regular pentagon.

Pause the video while you make a choice and press play when you're ready for an answer.

The answer is B.

A regular hexagon would have rotational symmetry of order six.

Sam has designed a logo.

She's going to do this by making shapes that are congruent to the one below and then fitting them together around a point.

When she's done this, what will its order rotational symmetry be? Pause the video while you write down what you think and then press play when you're ready for an answer.

Well, we have that 45 degree angle there, and we know that Sam is gonna create congruent shapes and rotate it around that point at the bottom.

So what we wanna know is how many 45 degree angles fit into 360 degrees? Well, 360 divided by 45 is eight.

So its rotational symmetry will be of order eight.

Alex is designing a logo.

He wants to have rotational symmetry of order 12.

What angle should he use at the vertex of each congruent shape? Pause the video while you write down your answer and press play when you're ready to hear what the answer is.

We'll do 360 divided by 12 to get 30 degrees.

And now it's over to you for task B.

And I hope you're feeling creative.

This task contains two questions and here is question one.

Aisha is designing a puzzle board for preschool children where they place shapes into spaces on the board.

She wants to arrange the spaces so that the shapes are in descending order of rotational symmetry.

That way it goes from easiest to most difficult to place inside the spaces.

What you need to do is draw six shapes on the board so that each shape has a lower order of rotational symmetry than the previous one.

Pause the video while you have a go at this and press play when you're ready for question two.

And here is question two.

Design and draw a custom logo with rotational symmetry of order five.

Now you may use tracing paper or a protractor to help you during this task.

Alternatively, you may wanna use some computer software.

However you go about doing it, pause the video while you have a go and press play when you're ready to look at some answers.

I hope you enjoyed that.

Let's now look at some answers.

Now, there are lots of different ways we can answer these questions and lots of different shapes we can draw.

We can't go through them all, but what we can do is look at some examples.

So here we are to draw six shapes where each shape had a lower order rotational symmetry than the previous one.

Here's an example.

We could start off with a circle and then a regular octagon, then a regular hexagon, and a square, an equal triangle, then a scaling triangle.

But there are other options we could have.

Maybe compare your answer to somebody else if you can, and see if you agree with each other, or check your answer by using tracing paper over each of the shapes and counting its rotational symmetry.

And then for question two, once again, there are lots of different ways we can do this.

And I hope you came up with some really creative ideas yourselves.

But hopefully what a lot of us did was we created an initial shape that had a vertex which would go at the centre of the overall shape, and that would have an angle of 72 degrees, for example, something like this.

And then by using that plus four more congruent shapes piece, them all together around a point to create an overall shape that has rotational symmetry of order five, like the one we can see on the screen here.

Answers can be checked using tracing paper.

Great job today.

Let's now summarise what we've learned in this lesson.

Rotational symmetry describes when a shape appears the same after some rotation.

And congruent shapes can be used to investigate rotational symmetry like by using the tracing paper.

When exploring rotational symmetry, the centre rotation is always at the centre of the shape.

And when identifying the order rotational symmetry, you must be careful not to count the starting point twice, either count it at the start before you begin turning it or count it at the end after you've done a full turn.

Whichever way you do it, try and be consistent so you always know what you're doing with that.

And all shapes have rotational symmetry of order, at least one because it always fits into itself at least once.

Thank you very much, all the best.