Lesson video

Hi there.

My name is Mr. Tilstone.

Something that I'd like you to know about me is that I really love maths, so you can imagine how excited and how delighted I am to be here with you today teaching you this lesson.

It's going to be a number three lesson today.

We're going to focus on shapes, particularly parallelograms, and something that's special about a parallelogram that we're going to investigate.

If you're ready, I'm ready.

Let's begin! The outcome of today's lesson is I know that any parallelogram can be decomposed and the parts rearranged to form a rectangular parallelogram.

You might've had some very recent experience of decomposing and then recomposing shapes to form a new shape with the same area.

Well, we're going to be extending that thinking a bit today with parallelograms. It might be a while since you've last thought about parallelograms, so we're going to have a little reminder about what they are.

And that's our key word, just the one.

My turn, parallelogram.

Your turn.

Can be a bit tricky to say that word, can't it? What is a parallelogram? Well, it's a quadrilateral, so it's a four-sided shape with two pairs of parallel and equal sides.

So this is an example of a parallelogram.

Not all parallelograms look exactly like this.

This is one example.

Our lesson today is split into two cycles.

For the first part, we're going to be thinking about what is a parallelogram.

Looking at some examples, looking at what makes 'em a parallelogram.

And in the second we're going to be decomposing parallelograms, taking them apart to explore something special about them.

So let's start by focusing our thoughts on that question, what is a parallelogram? In today's lesson, you're going to meet Alex.

Have you met Alex before? He's here today to give us a helping hand with our maths, and very good, he is too.

The Class 6 teacher at Oak Academy has challenged the pupils to create parallelograms using number rods.

Have they succeeded? Now, before we look at their examples, have you got any number rods in your classroom? If you've got any in front of you, because you could do the same, why don't you do this challenge too? Let's see if you know what a parallelogram looks like.

Can you create one? Can you create more than one using those coloured rods? But let's have a look at the Oak Academy pupils' responses.

This is one.

Is this a parallelogram? If we take away the number rods from around it, this is a shape that's been created.

What do you think? Are we looking at a parallelogram or not? No, we're not, for lots of different reasons.

It is a quadrilateral but it's not a parallelogram because none of the sides are parallel.

None at all.

So here's another one of the pupil's responses to the challenge.

What do you think to that shape? Is that a parallelogram? Again, I can see it's a quadrilateral.

Four of the number rods have been used to create four sides, but is it a parallelogram? That's the number rods taken away.

That's a shape that we are looking at.

Parallelogram or not? No.

Do you know why? Only one of the pairs of sides are parallel, and that's these.

It's got to be two pairs of parallel sides to be a parallelogram.

It's a quadrilateral, and specifically it's a trapezium, but it is not a parallelogram.

It doesn't meet all of the criteria.

What about this shape here? Is this a parallelogram? What do you think? Thumbs up if you think it is.

Thumb down if you don't think it is.

Well, let's have a look.

Let's have a look at the checklist.

What makes a parallelogram? This is a shape that it's created.

In a parallelogram, opposite sides are equal.

Would you say that's true of this shape? Yes, I would.

We can see that here.

Those two sides are opposite and parallel.

And so are those.

They are different pair of sides.

They're parallel to each other.

Are the opposite sides equal in length? Yes, they are! You can see that.

Those opposite sides are the same length as each other.

And are the opposite angles equal? Let's have a look.

Yes, those two opposite angles are equal.

And so are those.

So yes, it gets a big tick.

It fits all three of those criteria.

It is a parallelogram.

What do you notice here about these different responses? Are these parallelograms? In all of them, are the opposite sides parallel? Yes.

In all of them, are the opposite sides equal in length? Yes.

In all of them, are all the opposite angles equal? Yes.

So it ticks all three of the criteria.

They're all parallelograms, but what do you notice? These parallelograms are the same, but they're in different orientations.

So parallelograms can be in all sorts of different positions and orientations.

And what do you notice this time about these parallelograms? Let's use that checklist again.

Opposite sides parallel, are they? Yes.

Opposite sides equal in length, are they? Yes.

Opposite angles, are they equal? Yes! So these are all parallelograms. What do you notice this time? What about these parallelograms? They've got the same pairs of side lengths, and we can see that from the number rods that have been used, but different equal pairs of angles.

So the opposite angles are equal, but they're not the same in each parallelogram.

What do you notice this time? Hmm.

These look different, don't they? They've used the same number rods, the same colours.

Let's use that checklist.

Let's see if they're parallelograms. Opposite sides are parallel.

Is that true? Yes, they are.

Opposite sides equal in length.

Is that true? Yes, it is.

Opposite angles, are they equal? Yes, they are! So they are parallelograms. What was different this time about these parallelograms? What shapes are these? They're rectangles, but they fit the criteria of a parallelogram.

So all rectangles are also parallelograms. One of the opposite pairs of angles are right angles and the other opposite pair of angles are also right angles.

But they still fit all three of those criteria.

Here you go.

Look.

All rectangles are parallelograms. All of them.

Every single one.

If it's a rectangle, it's also a parallelogram.

What do you notice this time about these parallelograms? Hmm? Shall we use our checklist again? Are the opposite sides parallel? Yes.

Are the opposite sides equal in length? Yes.

Are the opposite angles equal? Well, there's already a little bit of a clue there, isn't there? Because the right angle symbol's been drawn in.

Yes.

Squares also fit the criteria of parallelograms. Squares are a type of rectangle, and we already know that all rectangles are parallelograms. So all rectangles are parallelograms and therefore all squares are parallelograms. You might have noticed one of those squares was in a slightly different orientation to the other, but it's still a square.

What do you notice this time? Hmm.

Opposite sides, are they parallel in both examples? Yes.

Opposite sides are equal in length? Yes.

Opposite angles equal? Yes! So they're both parallelograms. These are rhombuses, a special kind of quadrilateral.

So they fit the criteria of parallelograms and the squares are kind of rhombus for that same reason.

They're a parallelogram with sides of the same length.

Those squares are a type of rhombus.

Let's have a check.

So hopefully you've got some number rods in front of you.

Use number rods to create three different parallelograms. Pause the video and have fun exploring that.

What did you manage to come up with? Did you manage to share with each other and compare what you got? Let's have a look.

This is just some examples.

There are many.

You might have some different ones.

Did you, for example, create any rectangles? Did you create any rhombuses? Maybe you made a square that's a kind of rhombus and a kind of parallelogram and a kind of rectangle? You might have made one that's like this.

That's not a rhombus, but it is a parallelogram.

It's time for some independent practise.

So Task A, number one, write three properties of a parallelogram.

You might remember them.

If you can't remember them, can you think about what they are? What are those three things that need to be true for it to be a parallelogram? What are those three things that we tested all those shapes against? Number two, tick the parallelograms. So have a good look at all of those different shapes.

I can see some parallelograms, I can see some that are clearly not parallelograms, and I can see some that are almost parallelograms but not quite.

So you've got to tick the parallelograms, please.

Number three, complete these partially-drawn parallelograms. Some of them have more than one possibility.

a doesn't, there's only one way to complete that.

b, also just one way to complete it.

But c, there's lots of ways to complete that to make a parallelogram.

So if you've done one, maybe do a different one and even a different one again.

And we've got some more partially drawn-parallelograms. Here for d.

This is on a different kind of paper, different isometric paper.

Have fun exploring all of that.

And I'll see you soon for some possible feedback.

Welcome back.

How did you get on? Let's have some answers.

So the three properties of a parallelogram, you might have worded this slightly differently, but we're looking for something like this.

Opposite sides are parallel, opposite sides are equal in length, and opposite angles are equal.

You might've also mentioned about them being quadrilaterals as well.

And here are the parallelograms that you can see.

I can see one in the middle near the bottom that's almost a parallelogram, but it's not joined up.

So that stops it being a parallelogram.

That sort of plus shape on the bottom right.

It's got lots of parallel sides, but it's not quadrilateral, is it? So that can't be a parallelogram.

So lots of reasons why the ones that weren't parallelograms aren't parallelograms. Number three, complete these partially-drawn parallelograms. Some have more than one possibility.

Well, this is how to complete a.

b goes like this.

You need two sides for b.

And c, there's more than one way to do this.

This is one of them.

That's one example.

And for d, on this different kind of paper, this is how to complete d.

Just one way.

Just one way to complete e as well and this is it.

For f, there's more than one way because there's only one line there.

Lots of ways we could go with that.

That's one possible example.

There were lots and lots of others.

And the same for g.

That's just one of many examples of how you could complete that.

But all of those shapes have got two pairs of parallel sides.

Are you ready to move on to decomposing parallelograms? You're doing really well so far.

Let's go.

Alex begins with a parallelogram.

Here we go.

He's made a parallelogram.

He decomposes a parallelogram into smaller polygons by cutting.

And you're going to be doing this soon yourself.

What do you notice? So he's going to decompose.

He's going to turn it to different shapes and rearrange.

Look at this.

So he's cut off the triangle on the end and he's cut off the triangle on that end as well.

So he's decomposed it to three different shapes.

And the shapes are two identical triangles and a rectangle.

Let's have a quick check.

Trace round one of your non-rectangular parallelograms that you drew in that last cycle.

Or if you like, you can just create a new one.

But you're creating a parallelogram that's not a rectangle.

Can you decompose it just like we just did into two identical triangles and a rectangle? So get those scissors out and off you go.

Did you succeed? Have you ended up with a rectangle and two triangles? Here's an example.

This is a parallelogram.

It's not rectangular.

It's a parallelogram though.

Cut that triangle off.

Cut that triangle off.

And we've got a rectangle and two triangles, so it can be decomposed.

Alex composes a polygon from the decomposed parts.

What has he done? Hmm.

What have you noticed? It's got the same area, hasn't it? The area hasn't changed.

What about now? Has the area changed? No.

What about now? Has the area changed? No, but what do you notice about this particular shape? Something interesting about this.

Alex has made a rectangle from the three parts of the parallelogram.

So he cuts off that triangle, he cuts off another triangle, and then he rearrange those shapes to form a new rectangle.

So he's got a rectangular parallelogram now.

So over to you for a check.

Can you compose a rectangle from the three parts of your parallelogram? So you've got in front of you that rectangle and two triangles.

Can you turn it into one rectangle? Has the area of your original parallelogram changed? Pause the video and explore.

Let us have a look.

Did you manage to do that? Well, here's an example.

Here's what you could've done.

Here's how you could have arranged your triangles and your rectangle to form a new rectangle.

Has the area changed? No, it hasn't.

It's exactly the same.

We haven't added anything new and we haven't taken anything away.

Can any parallelogram be composed into three parts and recomposed as a rectangle? So here's a parallelogram.

It's in a different orientation, isn't it? Alex says, "I'm going to rotate it so it's easier to see where I might split it to make two identical triangles." He tries it with this parallelogram.

Here we go.

It's rotated.

If I cut the parallelogram here, I have a rectangle and two identical triangles.

Just like before.

Yes, I can even recompose this parallelogram into a rectangle.

I can visualise that as well, but let's prove it.

There we go! A new rectangle has been formed from the rectangle and two triangles.

Alex thinks he can show this relationship even more efficiently.

Let's see what he does here.

He decomposes the parallelogram into one right-angled triangle and a trapezium, and then he moves the triangle to the opposite side of the trapezium.

Let's see that.

So he's cut off that triangle from one side.

It's a right-angled triangle.

So now look, yeah, he's got a triangle and a trapezium.

And then he's moved the triangle to this side.

Ah, yes! And that's also made a rectangle.

So he's able to do that with just one triangle cut off.

Has the area changed? No.

He's again ended up with a new rectangle.

The area of the parallelogram and the rectangular parallelogram are exactly the same.

Nothing has been added, nothing has been taken away.

It's just been moved.

And over to you for a check.

Cut out a new parallelogram and make just one cut to decompose it into a triangle and a trapezium.

So you should end up with a right-angled triangle and a trapezium.

And then move the triangle to create a new rectangle.

Let's see if it works for you.

Okay? Ready to explore? Off you go.

Let's see.

This is one example.

So here's a parallelogram and we've cut off the right-angled triangle from the right-hand side and we've moved it into that space.

And lo and behold, we've got a rectangle.

Any parallelogram can be made into a rectangle that has the same area.

That's what I want you to learn at this point in the lesson, and it's an important point so we're going to say it again together.

Are you ready? Let's go.

Any parallelogram can be made into a rectangle that has the same area.

Right, just you're going say it now.

All right, ready? Go.

It's time for some practise.

Number one, redraw the parallelograms to form rectangles with the same area.

So, in the case of a, look, moving that triangle over into the gap, what would the new rectangle look like? And do the same with b.

Number two, cut out some parallelograms and turn them into rectangular parallelograms. Explore the triangle part that you remove.

Can you find an example where the part is scalene? So a scalene triangle that's been removed, an isosceles triangle that's been removed and repositioned, and an equilateral triangle.

Can you do all of those? Let's see.

Let's investigate.

Let's find out.

Pause the video.

Have fun with that.

And I'll see you soon for some feedback.

Welcome back! How did you get on? So for number one, you Were doing some drawing.

You're redrawing these parallelograms. And that's what the first one would look like.

Those two have got exactly the same area.

The triangle's been moved from the left into the gap.

And the same with b.

And number two, you were cutting out some parallelograms and turning them into rectangular parallelograms by cutting out the triangles.

So the triangular part, did you manage to do one that was scalene? Well, you could.

Here's an example.

Move it into the gap and we've got a rectangle.

Isosceles, so two of the sides are same.

Yes, that's possible.

Here's an example.

And that's made a rectangle.

So parallelogram has been turned into a rectangular parallelogram.

Equilateral.

No, that's not possible.

The triangles created by parallelograms are always right-angled triangles.

The angles in an equilateral triangle are always each 60 degrees.

So it is not possible to do that with an equilateral triangle.

We've come to the end of the lesson.

I've had so much fun today, and I hope you have too.

So today we've been looking at the idea that any parallelogram can be decomposed, taken apart, and the parts rearranged to form a rectangular parallelogram.

It's possible to decompose and recompose any parallelograms to turn them into rectangular parallelograms without changing the area.

And here's one example just here.

A huge well done on your achievements and your accomplishments and your learning successes today.

You've been absolutely fantastic and I really do hope that I get the chance to spend another maths lesson with you in the near future.

But until then, enjoy the rest of your day, whatever you've got in store.

Take care and goodbye.