Lesson video

Welcome to today's lesson on checking the understanding of perimeter and area.

By the end of today's lesson, you can explain the properties of various polygons.

Now, polygons and area and perimeter are not new concepts, as you have encountered these before in your studies at key stage two.

However, it's important to check our understanding of these things because if we want to learn further and increase our knowledge, it's important to make sure we have those fundamentals.

In this case, how solid is our understanding of perimeter and area? Well, in order to check that, we're going to start by looking at polygons.

Now you might think to yourself, I know what those are.

But just in case, let's check.

These are the words that we're going to be using today in our lesson.

We'll be using the words polygon, quadrilateral, regular, irregular, and parallel.

In a moment, I'm going to ask you to pause the video and write down what you think each of these words mean.

It's okay if you don't know one because we will be going over them.

But sometimes, it's a good idea to think about what we already know or what we think we understand so that we can check this and help to further our understanding.

Pause the video now and write down what you think each word means.

A polygon is a flat, two-dimensional, closed figure made up of straight line segments.

You're going to see lots of examples of polygons today.

Lots of polygons have other, more mathematically-specific names so that you know exactly which polygon we're referring to.

For example, a rectangle is a polygon.

So is a hexagon, and we'll see more of these today.

Quadrilaterals are polygons that have exactly four sides.

You might know the names of lots of quadrilaterals, such as a square, rectangle, and there are many more.

A regular polygon has sides that are all equal and the interior, or inside angles, are also all equal.

So an example of a regular polygon could be a square.

Now, an irregular polygon has sides that are not equal, or the interior, that is the inside angles, are also not equal.

Now, two lines are said to be parallel if they are straight lines that are always the same non-zero distance apart.

It's important, then, to realise that parallel lines must be straight.

This is something that can be easily forgotten because we focus on the fact that the lines cannot cross, i.

e.

, they stay the same distance apart, and we sometimes forget that the word straight is actually important here to define what kinds of lines they are.

So what's actually going to happen in our lesson today? Well, you can see the different sections we'll be moving through.

And we're going to start by defining what a shape is.

Now, there are an infinite number of shapes in the world.

Some shapes can be grouped into families because they share similar characteristics.

So an example of this would be the family of quadrilaterals.

Any polygon with exactly four sides is a quadrilateral.

However, just like families, we don't all look the same, and so our quadrilaterals don't either.

Here's just some examples of shapes that belong to the family of quadrilaterals.

As you can see, they've all got exactly four straight sides but they do not look the same.

Can you recognise some of the shapes here? Now, these shapes also fit into a broader category.

So although these all belong to the family of quadrilaterals, they're still all polygons.

They're just very specific polygons.

So not only do they belong to a really broad family, known as the family of polygons, they also belong to a much closer-knit family, which is the family of quadrilaterals.

Now we're going to play a little game.

Sara here is thinking of a shape, and she's going to give us some clues.

What we'll ask you to do is have a look at the clue and then either write down or draw what shape you think Sara might be thinking of.

So let's look at her first clue.

My shape has four sides.

Pause the video now and either write or draw what shape Sara might be thinking of.

Let's see what you've come up with.

So the shape had to have four sides.

You might be thinking, well, how do I know exactly which shape it is? And the answer is we don't.

Not from the clues Sara's given us.

There are actually loads of correct answers here.

As long as you've drawn a quadrilateral, that means a shape with four sides, you're gonna be right.

You might have written down the name of a specific quadrilateral, and that's absolutely fine too.

Both approaches are correct.

Here's just some examples.

We've got a parallelogram, a square, a rectangle, and a trapezium.

You might be thinking, that trapezium looks a little funny, it's upside down.

But actually, there's no right way up to draw a trapezium.

Let's play again.

What shape might Sara be thinking of now? My shape has four interior angles which are all right angles.

Hmm, slightly tricky at this time.

She's given us a bit more information.

And that means I can't just pick any old shape.

I've got to be a little bit more specific.

Pause the video now and either write or draw the shape that Sara might be thinking of.

Let's see what you've put.

She could either be thinking of a square or a rectangle here because she needs a quadrilateral with four interior angles that are all right angles.

So as long as you've drawn or written down the name for one of these, well done.

Finally, let's think about Sara's last shape.

My shape has four sides which are the same length.

Again, pause the video and either write down or draw what Sara might be thinking of.

In this case, there are only two options.

It's either a square or it's a rhombus.

Well done if you got those right.

If you didn't get those right, do they make sense now? Can you see that in both these shapes, all of the sides have exactly the same length? Now it's time to practise.

In this task, you are to describe one of these shapes to either a partner or write down a description of this shape and check that it only describes exactly one of these.

How many clues does it take to make sure that when you read those clues, you know exactly which shape you're talking about? Which of these shapes do you think requires the most clues? Pause the video now while you have a go.

Welcome back.

Now, there are many possibilities, depending on which shape you chose.

So just here are some examples.

If you'd given the clue, my shape is not a polygon, then it had to be the circle.

So just one clue needed.

If you'd picked the hexagon, you might've said, "My shape has six sides." And again, just one clue needed for that shape.

However, if you'd picked the parallelogram, as you can see, there are a lot more clues needed.

You could have said, "My shape has four sides," but we had more than one quadrilateral there, so that doesn't narrow it down.

I've still got a few to choose from.

There are two pairs of parallel equal sides.

Well that helps, but at the moment, it could be a parallelogram or it could have been the rectangle.

I still need more information.

Ha, here we go.

The interior angles are not right angles.

At this point, it's definitely the parallelogram.

But you can see, that shape required a lot more clues in order to get down to exactly one shape that was being described.

Now we're gonna move on and look at how we can group our shapes by these properties that we've been talking about.

Now, we as mathematicians can group our shapes based on properties such as how many sides they have, whether they're polygons or not, and whether they're regular.

Remember, regular shapes have all their sides, all their interior angles exactly the same.

Now, all triangles are polygons.

Well, that makes sense, they have straight sides and they're a closed shape.

Definitely a polygon.

By closed, I mean their sides match up.

There are no gaps.

Now, an equilateral triangle is very special because it belongs to the family of polygons, like all triangles do, but it is also a regular polygon.

So it belongs to the family of regular shapes.

Because in an equilateral triangle, all of its sides are the same length.

Now, rectangles also belong to the family of polygons.

And a square belongs to the family of regular polygons.

In fact, a square can be thought of as a regular rectangle, which makes sense, doesn't it? Because if we think about how we describe a rectangle, a square can be thought of as a rectangle that happens to be regular, i.

e.

, its sides are the same length.

That can be a little tricky to get your head round because we used to think of squares and rectangles as different things, whereas actually it's just a special case of the other.

Now, quick check, a pentagon has to be a regular shape.

Do you think that's true or false? Now, we want you to justify your answer.

So if you think it's true, then that's because you're saying pentagon's have five straight sides.

And if it's false, perhaps you think the shape there justifies that it can be a pentagon, but be irregular.

What do you think? Pause the video while you make your selection.

Well done.

The shape you can see in B is a pentagon, it's an irregular pentagon.

Being a pentagon just means you need five straight sides.

It doesn't mean they all have to be the same length.

What about this one? A kite is a regular quadrilateral.

Is that true because a kite has four straight sides, or is it false because a kite has two pairs of sides that are the same length, but all four sides are not? Pause the video while you make your selection.

Well done, it's definitely false.

We know from the shape of a kite that all four sides do not need to be the same.

They have two pairs of sides at the same length.

It's time now for our second task.

What I'd like you to do is to put each polygon in the correct place on the diagram.

So if it's a regular shape, it must go inside the regular circle.

If it's a quadrilateral, it needs to go inside the circle that represents the quadrilaterals.

And if it's an irregular shape, it goes in the bottom circle.

If it's an irregular quadrilateral, it needs to go in the space where those two circles, the irregular circle and the quadrilateral circle, where they overlap.

Pause the video while you have a go at this task.

Welcome back.

This is where those shapes should have been placed.

So the equilateral triangle is a regular shape but it is not a quadrilateral, and it's not irregular.

So it has to go into the section where it's by itself.

Where the regular and quadrilateral circles overlap, we can put our squares and our thrombuses, because they they're both quadrilaterals and they're both regular shapes.

Where the quadrilateral and the irregular circle overlap, we can put in our kites and trapeziums, because in both a kite and trapezium, not all the sides are the same length.

And then at the bottom, we have our irregular octagon.

Now we needed to say that the octagon was irregular because otherwise we wouldn't have known where to put it.

So because it's an irregular shape, it must go within that circle.

But it's not a quadrilateral, hence why it's in the part that isn't overlapping with any other circle.

Now let's try part B of the task.

Why are some of the spaces empty? Pause the video while you write down your thoughts.

What did you put? Now, you don't have to write the exact same thing as me.

You might have the same meaning, but different words.

I've said that some of the spaces are empty because you can't have a shape that is both regular and irregular at the same time.

Either all its sides are equal or they're not.

It can't be both at the same time.

So that space has got to be empty.

I've also said that quadrilaterals have to be either regular or irregular.

They can't be neither.

So this is why it's not possible to have a quadrilateral that isn't regular or irregular.

So that one's got to be empty too.

Now we're gonna move on and look at our third section, and this is how we find missing lengths with our shape properties.

So we're gonna use what we've learned so far and apply it to finding a missing length in a polygon.

So for a shape to be regular, we know that all of its sides must be the same length and all the interior angles must be the same size.

Now, showing that for every side of a shape can produce a very messy diagram.

As you can see down here, that is a lot of two centimetres all the way around my shape.

Not only is it messy to look at, it took a long time to draw, too.

And I don't really wanna spend my time doing all of that.

Surely there's a quicker way.

And here it is.

Same shape, but look at what's changed.

The notation I've got here shows us which lines are the same length.

Lines that are marked are the same.

And what we mean by that is they have the same length.

So here, we can see that there is a mark, just one, that crosses through one of our sides, and all other sides that have the same mark are the same length.

I hope you can agree that the shape on the right looks a lot less cluttered than the shape on the left.

It was also a lot quicker to draw.

Each subsequent set of lines of the same length are marked in a similar way.

You can see the rectangle on the left.

The two shorter sides are known as X.

Remember, we can use a letter to stand for an unknown number.

So in this case, we're showing that this could be any side length.

It doesn't have to be a particular number for this to work.

Those two sides are opposite each other.

In fact, they're parallel and they're also the same length.

So in the diagram on the right, you can see that we've marked those two sides with just one line intersecting each.

And we've done this at right angle, so it's very clear.

Now, the side that's marked B, we can see it's longer.

It's a different length to X.

We don't know what the number is, but we know it's not the same.

You can see the diagram in the right tells us that those two opposite or parallel sides are the same length, but we need to show they're not the same length as the sides marked X.

So this time, we're using two lines to represent the fact that these are paired together, they are the same length, but they're not the same length as the sides marked X.

So it's a less cluttered diagram.

But it's also showing us which sides are the same and which sides are different.

Now what I'd like you to do is to fill in the missing lengths here.

So you have two statements.

Look at the diagram and write down which length is the same as the one marked with the letter.

Use that notation to help you.

Pause the video and fill in the gaps.

Welcome back.

Let's see what you put.

So the length marked W is three centimetres long.

We can see that the length had two lines that were crossing it, and the only other side that has the same two lines crossing it is the three centimetre line.

So these two must be the same length.

Now, the length marked K has one line crossing it and the only other side that also has this notation is the 4.

5 centimetre line.

So these two must be the same length.

Now, sometimes the lines used to indicate the same length are not needed because you can use the properties of a shape.

So for example, in a rectangle, we know that the pairs of sides are the same length.

Now, see if you can work out if this statement is true or false.

If one side of a square is four centimetres long, then all sides of a square are four centimetres long.

Now, is that true because a square is a regular shape, or false because a square is an irregular shape? Pause the video and make your choice.

Of course, it's true.

A square is a regular shape, and so I know all sides are the same length.

I didn't need a drawing or notation marked on it to know that this is true because I know my properties of a square.

In this one, I'd like you to write down or state the missing length for each shape.

What we mean is what is the length that each letter is representing? Pause the video and have a go now.

Welcome back.

Let's see how you did.

In our first shape, we know that the length E is the same as the side that has a length of 10 because they're both marked with the same notation.

And likewise, E must be 17.

In B though, there's no notation marked on.

Instead, we're given the statement that this shape is a parallelogram.

In a parallelogram, we know that the parallel sides have equal length and this means that the side marked F must be the same as the side marked 11.

So F must stand for 11.

In C, again, no notation here.

But what we're told is that this is a three-unit square, which means each side is three units long.

Well, that means G must be three, and so must be H.

So no notation, but again, using the knowledge of the shape properties to work out what the missing length must be.

This brings us now to our last section, when we're looking at missing lengths in composite rectal linear shapes.

Don't panic if those phrases don't sound familiar.

I think you'll recognise this once you see a picture.

Composite rectilinear shapes are shapes made from two or more rectangles.

So you can use your knowledge of the property of rectangles to reason about these missing side lengths.

Let's have a look.

Here's an example of a composite rectilinear shape.

Does this sort of shape look familiar to you? We can see it's made up of putting two rectangles together.

We're going to use our knowledge of rectangles to work out what the length marked X must be.

If we create a parallel arrow, so parallel to the 10 centimetre line, going from the base of the shape all the way up to the top, we know that's the same length, it's got to be 10.

Well, hang on a second.

I've already got a two-centimeter length at the bottom and then I've got this length marked X.

Well then, the missing length must be eight centimetres because 10 is the total height of my shape.

I already have a distance of two, what else do I need? I need eight more.

Now it's your turn to have a go.

What is the length of the side marked Y in this composite rectilinear shape? There are four options here.

Pause the video and select the one you think is correct.

That's right, it's five centimetres.

We know that the entire length across is seven.

At the top, we can see we've got two.

The Y is our remaining distance.

So if we've got two centimetres already and we need to get to seven, how much more do we need? We need another five.

Time for your final task.

For each one, write down the length of each marked side.

In other words, what does the W stand for, what does the N stand for, what does the Zed stand for, and what does the N stand for? Pause the video while you have a go at the task.

Welcome back.

Let's see how you got on.

The entire distance across our shape is five.

At the top, we can see the one length, which is parallel and we've also got that length of M.

So we can do five takeaway one to find our missing length of four.

Now we can consider the height of the shape and look at W.

The entire distance is seven.

And we've already got a distance marked there of 1.

5, with W being the remaining amount.

To find the remaining amount, we're gonna subtract.

So seven take away 1.

5 gives us 5.

5.

Let's look at B.

In this shape, we can see that we have Zed as the entire distance across the bottom.

Now at the top, we are using our previous knowledge, so what we looked at in the previous section, to see that the top three lines are marked as the same length.

So if the length of one of them is three, we know all of them have length three.

So that's three, add three, add three, giving us a total distance of nine, which is what Zed must be.

Now let's look at N.

Again, we need to use our knowledge of our previous section to see that the length of 2.

4 at the bottom is the same as the other marked side.

So the two bottom sides must be 2.

4, and the middle section is 3.

2.

N is our total distance across.

So let's sum this.

3.

2 at two lots of 2.

4, gives us a total distance of eight.

Let's summarise what we've done in today's lesson.

We've looked at the fact that quadrilaterals are a family of polygons.

They're a particular family in that they all have exactly four sides.

A shape can belong to more than one family.

So for example, it can be regular, and/or a triangle.

So for example, an equilateral triangle is both a triangle and a regular shape and a polygon.

So it belongs to three families.

The properties of a polygon can be used to calculate the length of unknown sides, and that's what we did in our final two sections today in our lesson.

Well done.

You've worked really hard today and I appreciate how much effort you've put into your lesson.

I look forward to seeing you in our next lesson, where we're going to go on a little bit further and secure some deeper understanding of perimeter and area.

Goodbye for now.