Lesson video

Hello there, how are you today? My name is Ms. Coe.

I am really, really excited to be working with you on this lesson as part of your geometry unit.

Now, you may not know the word geometry, but geometry is all about shapes and lines, and so we're going to be doing lots of exploring with shapes and lines and the properties of shapes.

I'm really excited about this lesson and I hope that you are too.

By the end of this lesson today, you'll be able to say that you can decompose a polygon in different ways and identify some properties of polygons.

In this lesson today, we have some keywords that we're going to think about.

I'm going to say them and I would like you to say them back to me.

Are you ready? My turn, polygon, your turn.

My turn, quadrilateral, your turn.

My turn, parallel, your turn.

My turn, decompose, your turn.

Now, you may be familiar with some of these words, but let's take a look at what they mean.

A polygon is a 2D shape made up of three or more straight lines.

And a quadrilateral is a polygon, but it's a special polygon, it has four straight sides and four vertices.

Parallel lines are straight lines that are always the same distance apart, they never get closer together or further apart.

And we'll be looking at some examples of those in this lesson.

And to decompose means to break something into parts that are the same as the original when you put them back together.

And don't worry if you're not sure about that word, because that's what we're going to be doing in this lesson.

So in this lesson today, we're going to be investigating different ways of decomposing a polygon.

And we have two cycles for this lesson, the first cycle, we're going to explore the fact that polygons can be split or decomposed in different ways.

And then, in the second cycle, we're going to talk about right angles and polygon sides, and you may have a little bit of knowledge about right angles already.

So if you are ready, let's get going with the first cycle.

In this session today, you're going to meet Laura and she's going to be asking us lots of questions and helping us with our mathematical thinking.

So let's start here.

This is a pentagon.

A pentagon is a polygon with five straight sides and five vertices.

How could you split it using one straight line and what new polygons will you create? Hmm, I wonder if you can visualise, if you can imagine splitting that pentagon using one straight line.

Can you imagine where it might start or where it might end? And then, can you think about the parts that you've made and what new polygons you're creating from it? Let's look at one example.

We could draw a line connecting two vertices, so you can see there we have a line that started at one vertex and ended at another.

We have split the shape into two parts using a straight line, so we fit the description.

What new polygons have been made? What can you now see inside the pentagon? Let's label them A and B to think about them more closely.

Ah, thanks, Laura.

Polygon A is a triangle, it has three sides and three vertices, and if we look really closely at that part of the pentagon, we can see that because of the line that we drew, it now has three sides, three vertices, so that's a triangle.

What about B then? Thanks, Laura, again.

Polygon B is a quadrilateral because it has four sides and four vertices.

Laura thinks a little bit more closely about Polygon B, and she says, "It doesn't have any right angles," remember right angles are a square corner, "but it does have a pair of parallel sides.

These are sides that stay the same distance apart." I wonder if you can look carefully and spot the pair of parallel sides in Polygon B.

So we can say that the pentagon can be decomposed into a triangle and a quadrilateral.

Remember, decomposing is just splitting things up into smaller parts.

So we can say that, in this case, we have decomposed the pentagon into a triangle and a quadrilateral.

We know that some quadrilaterals have special names, you know that a square and a rectangle are special types of quadrilateral.

Well, this quadrilateral has a special name too, and this is called a trapezium.

I'm going to say that word again and I'd like you to say it back to me.

My turn, trapezium, your turn.

Great job.

So a trapezium is a quadrilateral, a four-sided shape, with exactly one pair of parallel sides, and you can see that we've highlighted them there.

A trapezium has only one or exactly one pair of parallel sides, we can see that the other two sides are not parallel, they get closer together.

Good thinking, Laura, let's see if we can keep a lookout for trapeziums while we decompose our shapes.

And it'd be great if you could say and use that word while we're decomposing our shapes.

So let's go back to our pentagon.

We started off by drawing a line from one vertex to another, but we could also split it by drawing a line from a vertex to a side.

As you can see here, we've started at the vertex, and this time we've drawn the line to just one of the sides.

What new polygons have been created this time? Take a close look.

Let's label them A and B to help describe them.

Laura has noticed that Polygon A is a triangle, it has three sides and three vertices.

Hmm, we made a triangle last time.

So does that mean that B is a trapezium? Well, no, Polygon B this time has five sides so it is a pentagon.

It is a smaller pentagon than the original pentagon, but because it has five sides and five vertices, it is still a pentagon.

So the pentagon can be decomposed into a triangle and a smaller pentagon.

It does seem that there is more than one way to decompose this pentagon.

We could also draw a line connecting two sides, as you can see here, what new polygons have been made this time? Hmm, let's again label them A and B to help us describe them.

Laura notices that Polygon A is a quadrilateral, it has four sides and four vertices, but it doesn't have any pairs of parallel sides so it is not a trapezium.

And I can see that it's not a square or a rectangle either because it doesn't have any right angles in it.

Polygon B is another smaller pentagon, let's check, one, two, three, four, five, yep, it has five sides and five vertices.

So we can say that the pentagon can be decomposed into a quadrilateral and a smaller pentagon, that's another way of decomposing the pentagon.

Time to check your understanding.

We have used a different line to decompose the pentagon.

I would like you to complete the sentence, "The pentagon can be decomposed into a mm and a mm." Now we know we've thought about this in different ways, but I would like you to concentrate on the shape that you can see on the screen.

What shapes has the pentagon been decomposed into? Pause the video here, discuss with a friend, if you can Welcome back, how did you get on? We can say that the pentagon can be decomposed into a triangle and a hexagon.

Let's look closely at that.

We can see that this polygon inside the pentagon has six sides and six vertices.

That means it's a hexagon, remember, that's the special name for polygons with six sides and six vertices, and the smaller parts has three sides and three vertices so it's a triangle.

Let's take our learning a little bit further.

We've been looking at decomposing the pentagon using one straight line.

But what if we used two straight lines? And there's a little bit of a rule there, the lines can touch one another, but they cannot cross one another.

Hmm, so I wonder if you can visualise or imagine drawing two lines on this pentagon? Remember, they can touch one another, but they can't cross one another.

What different shapes might we create? Well, here is one example.

So in this example, we have drawn two straight lines from the same vertex.

One of the lines has gone across to another vertex and one of them has gone to the side.

Let's label the polygons that we've made A, B, and C so that we can think about them.

What polygons can you see in the pentagon? Well, Laura has noticed that this time, the pentagon has been decomposed into three polygons so we've decomposed it into three different parts.

Polygons A and B are different types of triangles, they have three sides and three vertices so they're both triangles, even though they look a little bit different.

What's about Polygon C? Polygon C is definitely a quadrilateral, it has four sides and four vertices.

Hmm, I wonder, is it one of our special quadrilaterals? Is it a trapezium? Does it have exactly one pair of parallel sides? What do you think? Absolutely, Laura, Laura can see a pair of parallel sides and they're the horizontal sides that you can see in shape C.

She can only see one pair so that means that Polygon C is a trapezium.

Well done if you spotted that.

So this time we can say that the pentagon can be decomposed into two triangles and a trapezium.

I wonder, do we get other shapes if we draw other lines? Time to check your understanding.

We have decomposed the shape in a different way, I would like you to describe the shapes that the pentagon has been decomposed into.

Remember we've been using words like sides, vertices, and we've been thinking about parallel sides, sides that stay the same distance apart.

Can you spot any of those in your discussions? Pause the video and have a think.

Welcome back.

Laura said, that she can see three polygons, all of them have four sides, so they're all quadrilaterals.

I wonder, are there any special quadrilaterals in there? Well, no, none of the quadrilaterals have any pairs of parallel sides so none of them are trapeziums and none of them are squares or rectangles because I can't see any with four right angles.

So they're all quadrilaterals, so we can say that the pentagon has been decomposed into three different quadrilaterals.

Well done if you spotted that.

Time for your first practise task, I would like you to decompose each of the polygons in different ways using straight lines.

You can use one, two, or even three straight lines to decompose your shapes.

And I'd like you to label the new polygons that you make.

So you've got shape A there, which is a triangle, and shape B, which is a rectangle, and you also have shape C, which is a hexagon.

So really carefully think about the different ways that you can decompose these shapes.

Pause the video here and have a go at that task and I'll see you shortly for some feedback.

Welcome back, how did you get on? Now remember, that there are lots and lots of different ways to decompose the shapes so we are just gonna show you some examples.

Here are some ways you may have decomposed the triangle using one line.

And I wonder if you notice that if you split or decompose the triangle using only one line, that they usually make two triangles.

You can see here all of our examples, we've decomposed the larger triangle into two smaller ones.

That doesn't always happen.

Let's look at this example.

Here, we have used one line to decompose the triangle and we do have one smaller triangle.

But look, we have a quadrilateral and I can see that exactly one pair of sides is parallel, so we have a trapezium as well.

And also, if you use more than one line, like in this example, we can see that the triangle here has been decomposed into two smaller triangles and a quadrilateral.

Well done if you experimented with different lines and the different shapes that you can make with a triangle.

Here are a couple of examples using one line for decomposing the rectangle.

So when you are using one line, you may have made triangles or smaller rectangles like these examples.

What about these examples here? Maybe you made some like this using more than one line.

I wonder if you spotted any of our special shapes.

Did you make any trapeziums? The polygons that we've shown here are trapeziums because they have exactly one pair of parallel sides.

So well done if you spotted any trapeziums. And, finally, here are some examples with our hexagon.

You may have found that the hexagon can be decomposed into trapeziums and other polygons, and we had lots of fun spotting the different trapeziums that we could see here.

We also noticed that if we used three lines from one vertex, we could decompose the hexagon into four triangles.

I wonder if you found that example too.

And this example here we have decomposed the hexagon into a pentagon and a quadrilateral.

So there are lots and lots of different ways of decomposing this hexagon.

I hope you found lots of different shapes.

Let's move on to the second cycle of our learning where we're thinking about right angles and polygon sides.

Laura has started with an octagon.

Remember, an octagon is an eight-sided shape and she has drawn two straight lines to decompose the octagon.

This time the lines cross.

What do you notice? What can you see? Laura has noticed that where the lines cross in this example, she can see four right angles.

They look like square corners and we can use a right angle symbol like this to mark them.

So you can see in the middle of the octagon, we have four right angles where the lines have crossed.

And what do you notice then about the polygons that the octagon has been decomposed into? Can you name the shape? Well, Laura has noticed that there are four identical polygons, that means they're the same.

Each of the polygons has five vertices and five sides so each of these polygons are pentagons.

So we can say that the octagon has been decomposed into four identical pentagons.

Each of the pentagons has at least one right angle, we've marked one of them on the shape.

Let's take a closer look at that pentagon.

When two lines meet at a right angle, they have a special name and that name is perpendicular.

I'm going to say that again and I'd like to say it back to me.

My turn, perpendicular, your turn.

Great job.

So if two lines meet at a right angle, they are perpendicular.

These two sides here are a pair of perpendicular sides because sides connect, they meet to make a right angle.

So Laura says, "Well wait, actually, I can see another right angle in the pentagon." Can you spot another right angle in the pentagon? So that means that this pair of sides here are also perpendicular.

Laura spotted the right angle and realised that the two sides that make the right angle must be a pair of perpendicular sides.

Can you spot any other perpendicular sides in this pentagon? Well, actually there's another right angle over here, so that pair of sides must be perpendicular as well because they meet and form a right angle.

Good spot, Laura.

So there are three right angles in this pentagon, that means there are three pairs of perpendicular sides.

Laura draws two straight lines to decompose this octagon, the lines cross.

What do you notice this time? Look closely, can you see any right angles? Let's label the polygons A, B, C, and D to help us think about them.

This time, the octagon has been decomposed into a triangle, two quadrilaterals and a pentagon.

I wonder if you can see which shape is which.

Polygon A is a triangle, it has a right angle and therefore, it has a pair of perpendicular sides.

So we can say that Polygon A is a triangle with a pair of perpendicular sides.

Polygon D is a pentagon, it has more than one right angle, so it has more than one pair of perpendicular sides.

We've just shown one of the right angles there so you can see the sides that connect together to make that right angle form a pair of perpendicular sides.

But there are others in that pentagon.

Polygons B and C have one pair of parallel lines, so they are trapeziums. And did you notice that they are identical? They're the same.

Did you spot those parallel lines? Can you spot anything else about Polygons B and C? "But wait," says Laura, "the trapezium also has a right angle, so it has a pair of perpendicular lines too.

' In fact, it has two right angles, I wonder if you can spot the second one.

So we can say that this is a trapezium and it has two pairs of perpendicular sides.

So different shapes can have a pair of perpendicular sides, it doesn't have to just be quadrilaterals, triangles and pentagons and lots of other shapes can have pairs of perpendicular sides too.

Time for a quick check of your understanding.

How has the octagon been decomposed this time? Can you use our special words parallel and perpendicular to describe the shapes? Pause the video and have a go.

Welcome back, how did you get on? Did you use our special words parallel and perpendicular? Well, Laura decided to label them A, B and C to make it easier to talk about.

She noticed that the octagon had been decomposed into three polygons, two quadrilaterals, and one pentagon.

Now let's see what she had to say about parallel and perpendicular sides.

She noticed that the two quadrilaterals are trapeziums, they have exactly one pair of parallel sides.

She also noticed that Polygon C has three pairs of perpendicular sides because she can see three right angles.

So remember, if you can see a right angle, it means the shape has at least one pair of perpendicular sides.

She also noticed that Polygon B, which we know as a trapezium, also has two pairs of perpendicular sides because that has two right angles.

So that's a trapezium with parallel sides and perpendicular sides.

Well done if you spotted all of that.

Time for your second practise task, I would like you to decompose each of the polygons that you can see here, but your challenge is to make at least one shape with a pair of perpendicular sides.

Remember, you're looking out for right angles, mark the pairs of perpendicular sides that you find.

For question two, you can see that we have identical hexagons here.

I would like you to see how many different ways you can decompose this shape to make at least one shape with a pair of perpendicular sides.

Good luck with those two tasks and I'll see you shortly for some feedback.

Welcome back, how did you get on? Now remember that, for these shapes here, there are lots of different ways that you could have decomposed them and they might have been different to ours, but we've got some examples here.

You can see that we have shown the perpendicular sides using the right angle symbol, but also using arrows to show which sides are perpendicular.

If we look at the pentagon at the bottom, we can see here that we used a couple of lines to decompose the shape, and then we decompose that shape into a triangle and two quadrilaterals.

Well done if you noticed that those two quadrilaterals are also trapeziums because they have exactly one pair of parallel sides, and we can also see that we have marked one of the pairs of perpendicular sides.

You may have spotted other perpendicular sides.

Remember, you might have decomposed your shapes in different ways.

Again, for question two, there are lots of ways of decomposing the hexagon, but here are some that we came up with.

Again, you may have noticed these pairs of parallel lines or you might have spotted different pairs of parallel sides.

Let's take a look at the first example.

In the first example, we have decomposed the shape into two smaller shapes.

I can see a triangle and a smaller hexagon.

We've marked one of the pairs of perpendicular sides that you can see, but you might have spotted another right angle and another pair of perpendicular sides.

Well done if you've marked lots of different perpendicular sides on your shapes and found lots of different ways to make right angles and therefore a pair of perpendicular sides.

We've come to the end of our lesson, and I hope you've really enjoyed investigating different ways of decomposing different polygons.

Let's summarise what we've learned.

Shapes can be decomposed into other shapes, a polygon can be decomposed in different ways and different polygons.

Lines are parallel when they are always the same distance apart.

And a trapezium is a special type of quadrilateral, which has exactly one pair of parallel sides.

Pairs of perpendicular sides meet at a right angle, different polygons can have one or more pairs of perpendicular sides.

Thank you so much for all your hard work today, and I look forward to seeing you again in another maths lesson soon.