Lesson video

Hello there.

My name is Mr. Tilstone.

I'm a teacher and although I teach all of the subjects, the one that I enjoy the most is definitely maths.

So it's a real pleasure to be here with you today teaching you this maths lesson.

Let me ask you a question.

How much do you know about area? Let me ask you another question.

How much do you know about parallelograms? Let me ask you one last question.

How much do you know about the area of parallelograms? And I bet the answer to the last one is not much at all.

So let's investigate that.

If you are ready, I'm ready.

Let's begin.

The outcome of today's lesson is I can explain how to calculate the area of a parallelogram.

We've got two key words, my turn, parallelogram.

Your turn.

My turn, perpendicular.

Your turn.

Now I think there's a good chance you've heard both of those words before.

You may remember what they mean or you may not.

So let's have a little reminder.

A parallelogram.

Here's an example on the screen, is a quadrilateral with two pairs of parallel and equal sides.

So not all parallelograms look exactly like that, but that's one example.

And then two lines are perpendicular if they meet at a right angle.

So right angles are key to perpendicular lines.

Our lesson today is split into two cycles.

The first will be the perpendicular height of a parallelogram, and the second, calculate the area of a parallelogram.

So if you are ready, let's begin by thinking about the perpendicular height of a parallelogram.

And that's probably new terminology to you.

In this lesson, you're going to meet Alex.

Have you met Alex before? He's here today to give us a helping hand with the maths.

In order to calculate the area of a parallelogram, its perpendicular height needs to be known.

And like I say, you might not have heard that before.

So let's investigate that.

To find this measurement, select one side.

This is called the base.

I bet you've heard the word base before.

What do you think the base is on this shape? You might be surprised.

That's one example of the base, the bottom.

Any perpendicular line drawn from the base to the height of the opposite vertex shows the perpendicular height.

So let's see that in action.

Here's a line, it's perpendicular.

It forms a right angle.

That's a perpendicular height.

So it's coming from the base and it's forming a right angle.

That's a perpendicular height.

That's a different example of a perpendicular height.

Again though it's coming from the base and it's forming a right angle that's a perpendicular height.

Here's another example.

I think you're starting to get the idea a little bit.

That's perpendicular height.

Even outside the parallelogram, that's still the perpendicular height.

If we were to extend that base, we could form those perpendicular lines.

The base is the side from which the perpendicular height is found.

Now, you might have thought that the bottom was a base and you would've been right, but there are other bases on this shape.

Let me show you what I mean.

In this case, look, this side could be the base and a perpendicular line could be drawn from there.

The other sides are not based in this case because the line showing the height is not perpendicular to each side.

So in this case, two different lines are bases.

For a different side to be the base, a new perpendicular height needs to be found.

So we've got that same shape, but it's been rotated.

Now have a look.

You think you can see where you might be able to form a perpendicular height.

Where could a base be? Hmm.

Well, the base could be the bottom or the top of that shape as we're seeing it now, they're both the bases.

There we go.

We've made a perpendicular line.

So it's got a perpendicular height.

There we go.

And again, that could be the base, but it's still got the same perpendicular height.

Rotating the parallelogram so that the base is vertical or horizontal makes it easier to determine the perpendicular height.

So if your shape isn't already rotated that way, you could do it yourself.

You could turn your page around.

Even when the base is not horizontal, the perpendicular height can still be determined.

So, on the left we've got an example where the base is horizontal.

On the right, it's not horizontal, but it's still got a base.

And like I said before, you could turn your head a little bit and see it as a base or you could turn the page until it's horizontal.

But even if you don't do that, it's still the base.

Okay, that's probably a lot of new information.

Let's do a little check.

Is Alex correct? Alex says, "I have drawn a line "to show the perpendicular height of this parallelogram." What do you think is that perpendicular height? Give him some feedback.

Pause the video, have a discussion if you can.

I'll see you shortly.

What do you think? I think he's got some bits of that right, but not everything.

No, what he has done is established a base.

So well done, Alex, but the line must form a right angle from the base.

That is not a perpendicular line, is that I can't see a right angle there.

So good try.

Could he fix it, do you think? Could he change something? He's got the base.

Could he make a perpendicular height? Yes.

Now we've got a rectangle.

Now we've got a perpendicular height.

Well done, Alex, for persevering.

It is time for some practise.

Tick the parallelograms which correctly show the perpendicular height of the parallelogram.

So I'm looking at that now and I'm seeing that some of them have got a perpendicular height and some of them haven't.

See if you could spot them too.

And number two, determine a base for each parallelogram and then draw a line to show the perpendicular height.

Remember, it may be easier to rotate so that your base is horizontal or vertical.

So you might want to just turn your page around a little bit so that you can see that, that's my top tip.

And there are different possible answers for this one.

So you might have a different answer to the person next to you and that's fine.

Okay, good luck with that.

Pause the video and I'll see you soon for some feedback.

Welcome back.

How did you get on? Do you think you are starting to get the hang of these perpendicular heights? Let's have a look.

So the parallelograms, which correctly show the perpendicular height of the parallelogram are as follows.

Well, they're all parallelograms, but this one has got a perpendicular height.

A base has been established and a perpendicular line has been drawn from it.

So that's got the perpendicular height.

And the one next to it has also got a perpendicular height.

And you could show that outside the parallelograms. This one's done has made right angle.

It's got a perpendicular line.

And this one too.

You needed to rotate it possibly to to see it, but once you've established a base, you can then establish that perpendicular height.

Not quite on the next one, but the final one, yes.

Here's a base and here is a perpendicular height.

Forms a right angle.

So well done if you've got those.

And this is just one way of using one side as a base.

Did you notice that the interior angles of D are just smaller than a right angle? So it was almost there.

It was a little bit of a tricky on that one.

But let's have a look at the others.

So that's one possible way to establish a perpendicular height.

For A, that's got a right angle.

That's one way to do it from B.

That's a very long base in this case, isn't it? And you could maybe choose that as a base for C.

You might have chosen a different one, but as long as you then make a perpendicular line from it that's got a right angle, it's a perpendicular height.

And D, that's slightly tricky one.

Here we go, we can draw a perpendicular height coming from the base just like so.

And for A and D, you could use the adjacent size as a base and at a different perpendicular height.

So they would also have been bases that you can form perpendicular height from.

And some examples have been shown there.

For B and C, you could use a different side, but the height will be drawn outside of the shapes perimeter so it's less efficient.

Right, let's move on.

Now we're going to look at calculating the area of a parallelogram.

You might have had some recent experience of cutting up parallelograms and rearranging the parts of it.

If you have, try and reactivate all of that knowledge 'cause it's going to be helpful in this next cycle.

And this is what I'm talking about.

Alex remembers a relationship between parallelograms and rectangular parallelograms because remember, all rectangles are parallelograms. So let's have a look.

Is this jogging your memory using the scissors? Could we cut part of this parallelogram off and turn it into a rectangular parallelogram? What could we do? Well he decomposes, that's a good word, the parallelogram into one right angle triangle and a trapezium.

And then he moves the triangle to the opposite side.

Let me show you what we mean.

So he's cut that part of the parallelogram off.

He's got a right angle triangle there.

He's going to manipulate it, going to move it over to the other side.

There we go.

And what do you now, we've got a rectangle.

So it's now a rectangular parallelogram.

His new shape is a rectangle.

The area of the parallelogram and the area of the rectangle are equal, wouldn't you agree? Nothing's been added, nothing's been taken away, it's just been recomposed.

Alex knows that a rectangles area can be calculated by multiplying the length and width.

And I bet you knew that too.

So let's have a look.

So let's put some dimensions onto this rectangle.

It's not to scale but it's 20 centimetres by 10 centimetres.

So Alex knows that we can multiply those numbers together.

That's fairly gentle arithmetic.

Hopefully, you can do that too.

10 times 20 or 20 times 10 equals 200 centimetre squared.

Now the rectangle can also be described as having a base of 20 centimetres and a perpendicular height, that you've just explored of 10 centimetres because you can form a right angle there.

Now Alex recomposes a parallelogram, he's going to move that triangle again.

He knows that the area is still 200 centimetres squared 'cause nothing's been added, nothing's been taken away.

So that's 200 centimetres squared, and that's also 200 centimetres squared.

Wouldn't you agree? Nothing's changed, just repositioned.

So can you see it's still got a base of 20 centimetres and it's still got a height of 10 centimetres? Base of the parallelogram is 20 centimetres, perpendicular height is 10 centimetres.

Let's have a check for understanding.

See if you've managed to assimilate that new knowledge.

The parallelogram is decomposed and the parts recomposed as a rectangle.

What's the area of that parallelogram? Explain why.

Pause the video and give that a go.

What do you think? Did you manage to come up with an agreement with the people you're working with? Hopefully you did.

It's got the exact same area as a rectangle.

Now hopefully, this was a known times tables fact.

Hopefully this came to you automatically 'cause you're good at your tables.

12 times 6 is 72.

So that's got an area of 72 centimetres squared.

Now remember, the parallelogram next to it is no different in terms of area, it's just been recomposed.

So then that must also have an area of 72 centimetre squared.

So multiplying that base by that perpendicular height gave the area of that parallelogram.

To calculate the area of a parallelogram, multiply the base by the perpendicular height.

That's the main learning that I'd like you to take away from this lesson.

So that's really important, and I'd like to say it again, please.

This time will you say it with me? Are you ready? To calculate the area of a parallelogram, multiply the base by the perpendicular height.

Brilliant.

Now just you say it.

Are you ready? 3, 2, 1, go.

Let's give an example.

So this is the base.

The base is five centimetres.

Let's establish a perpendicular height.

There's lots of places we could do that.

Here is one of them.

So it's got a perpendicular height of three centimetres.

What's the area of that parallelogram? Remember, multiply the base by the perpendicular height.

We've got those pieces of information with us.

3 multiplied by 5 is 15.

So that's got an area of 15 centimetres squared.

Even when the base is not horizontal, just like here then, the area of the parallelogram can still be calculated.

And remember, little top tip, you can rotate your page if you like, but you don't have to.

If you recognise that as a base and you can see a perpendicular height, you don't really need to do that.

So to calculate the area of a parallelogram, multiply the base by the perpendicular height.

I can see a base, I can see a perpendicular height.

What can you do with those numbers? Multiply them together.

Again, that arithmetic, not too difficult I hope.

What do you think? The base is 10 centimetres, the perpendicular height is 6 centimetres, we multiply those together and we've got 60 centimetres squared is the area of that parallelogram.

Let's do a check for understanding.

Let's see if you're getting this.

I think you probably are.

Is Alex correct? He says the area of my parallelogram is 20 centimetres squared.

Is he right? And if so, why? Or is he not right? And if so, can you give him a little bit of feedback, please? What might he have done? Pause the video.

What do you think? Was he right or not? No, he wasn't.

What's he done, do you think? Let's have a look.

What should he have done? No, although he has correctly identified the base, which he did before.

So well done, Alex.

He seems to know about bases and the perpendicular height.

Well done, Alex, for that.

He's added them together, but that's not what you do, is it? You multiply them together just like you would to find the area of a rectangle.

So what should it be? 5 times 15.

5 times 15 or 15 times 5 is 75.

So that's 75 centimetres squared.

Well done if you got that.

Let's do another check.

If calculating the area of this parallelogram, which two values are needed? So there's no numbers on here, but would you need value A? Would you need value B? Or would you need value C? Which two of them are needed? Pause the video, have a think.

Well, there's a little bit of what you might call a red herring there.

Some information that we don't need to work out the area of the parallelogram.

A is needed.

That's the perpendicular height.

B is needed, that's the base.

And they must be multiplied together.

But C, which is the side length, that's not the base, is not needed and will not help.

So you don't need that value.

You just need to multiply A by B.

Time for some more practise.

Work out the areas of these parallelograms. They're not drawn to scale, but we've got some information there.

And on some of them there is that red herring information.

So there might be one of the measurements you don't need.

So do be careful.

Remember, you're looking for the base and the perpendicular height each time.

Got some more examples here.

And number two, a parallelogram has a base of 12 centimetres and a perpendicular height of 4 centimetres.

First one, what's the area of that parallelogram? What can we do with those numbers? And then the second question is, can you draw that parallelogram using accurate base and height lengths? Now there are many, many possible answers to that question.

So if you've got one, why don't you find another and another and another and another.

Keep going.

Pause the video and I'll see you soon for some feedback.

Welcome back.

How did you get on? Let's have a look.

So number one, the areas of these parallelograms are as follows.

And with A, there was a red herring there, we didn't need that side length of five centimetres.

We just needed the base and the perpendicular height.

That was 4 and 10 or 10 and 4.

So the 10 centimetres multiplied by the 4 centimetres gives us 40 centimetres squared.

And for B, the two bits of information were there, for the base was 15 centimetres, the height was 6 centimetres, multiply them together and that gave you 90 centimetres squared.

That's not a known times tables fact, but hopefully that came to you fairly quickly.

Maybe you used partitioning, 6 times 10, 6 times 5, added them together.

And for C, there's a red herring here again, something you don't need.

We don't need the five centimetres, we do need the eight centimetres, that's the base, and the three centimetres, that's the perpendicular height.

Multiply them together.

That is a times tables fact.

And that's 24 centimetres squared.

Maybe you needed to tilt your page for that and maybe you didn't.

And for D, we've got another times tables fact here.

The base this time is in a different orientation, it's vertical.

But we've got our base and we've got our perpendicular height.

That's 12 and 4, multiply them together, that's 48 centimetres squared.

And again, if you know your times tables, that was hopefully instantaneous.

And for A, did you notice that the units are changed? They're now metres, not centimetres, but we still use the same method that is multiplying the base, which is seven metres by the perpendicular height, which is six metres.

And that's another times tables fact, 7 times 6 is 42.

So that's 42 metres squared.

And then the parallelograms got to base of 12 centimetres and a perpendicular height of 4 centimetres.

Hopefully, you could visualise that in your head, and hopefully you remember what to do.

Even though you couldn't see the parallelogram, you multiply those two lengths together and that gives you 48 centimetres squared.

It was a times tables fact again.

And then the parallelogram that you drew might have looked very different to this, but this is one example.

Here's our base, 12 centimetres.

Here's our perpendicular height.

You didn't need to go there, it could have been anywhere along that line or even just outside it.

And here is a different 12 centimetre line that's also the base, remember.

And again, that could have gone a little more to the left or a little more to the right, but that's where I put it.

And then all we've gotta do is join them together.

So we've got those two new side lengths, and now we've got a parallelogram that's got a base of 12 centimetres and a perpendicular height of 4 centimetres.

So therefore it's got an area of 48 centimetres squared.

We've come to the end of the lesson.

I've really enjoyed today.

I hope you have too and I hope you've learned lots of new things.

Today we've been explaining how to calculate the area of a parallelogram.

The area of any parallelogram can be found by multiplying its base by its perpendicular height.

So we've got an example here.

In this case the base is 10 centimetres, the perpendicular height is 4 centimetres.

Multiply those numbers together, we've got the area of 40 centimetres squared.

Perpendicular height can be found by selecting a base and drawing a perpendicular line from the base to the opposite vertex.

Moving the base so that it is horizontal or vertical makes it even more efficient, and that's my top tip.

Well done on your achievements and your accomplishments today.

I've got no doubt that you've learned some new things there.

So well done.

Give yourself a little pat on the back.

It's very well deserved.

Hope you have a great day, whatever you've got in store for the rest of your day.

Until the next time, take care, and goodbye.