Lesson video

Hello there.

My name is Mr. Tilston.

It's lovely to see you today, and I hope you're ready and rearing to go for today's maths lesson, which is all about shape.

Let's do this.

The outcome of today's lesson is, "I know that two congruent triangles can be arranged to compose a parallelogram." And I'm going to bet that that word, "Congruent" Is new to you.

Well, don't worry by the end of the lesson, you'll be really familiar with it.

So our key words are, my turn, congruent, your turn.

My turn, parallelogram, your turn.

What do those words mean? I bet you know parallelogram, but what about congruent? If one shape can fit exactly on top of another using rotation, reflection, or translation, then the shapes are congruent.

And we're going to look at lots of examples of that.

And a parallelogram is a quadrilateral with two pairs of parallel and equal sides.

You might have had some recent experience of exploring parallelograms. Rectangles are an example of parallelograms as well, and so are squares.

And our lesson today is split into two parts, two cycles.

The first will be, "What is congruence?" We're really going to unpick that word.

And the second will be, "Composing a parallelogram from triangles." If you're ready, let's start by asking the question, "What is congruence?" In today's lesson, you're going to meet Alex and Sam.

Have you met them before? They're here today to give us a helping hand with our maths.

The class six teacher asks the pupils to create triangles using number rods and classify them.

You might have some number rods in front of you now.

Fantastic.

If you have, why not have a go yourself before I show you some examples> Here are some of the responses from the Oak pupils.

Here's one.

It's equilateral.

All of the sides are the same length, and so are the interior angles.

What about this one? It's a triangle, isn't it? What kind of triangle? Two of the rods are the same colour, and one's different.

So two of the side lengths are the same and one's different.

What kind of triangle is that? That's an isosceles triangle.

What about this one, what kind of triangle is this? It's got three different coloured rods, so three different side lengths.

What could it be? What's the name of it? Scalene.

And what about this one? There's two things you could call this.

It's an isosceles triangle again because, just like before, two of the sides are the same length and one's different.

But you could also say it's a right-angled triangle 'cause you've got the right angle in it.

Alex and Sam made these triangles.

What do you notice about them? What can you say about those triangles? Are they the same? Are they different? What do you think? Well, their triangles are exactly the same shape and size, the same number rods have been used to create them, so we know that.

But they've been rotated.

The different orientations have not changed the size or shape.

So they are the same.

They're still equilateral triangles, both of them.

And let's introduce that special term then.

The mathematical term for this is congruence.

Can you say that word with me? Congruence.

Just you.

Those shapes are congruent to each other.

They have congruence.

Are these triangles congruent to each other? What do you think? Are they the same? Both triangles have the same side lengths and the same interior angles.

One's a reflection or maybe a rotation of the other.

So yes, they are congruent.

You think you're starting to get the idea about what congruence is now? Are these two triangles congruent? What do you think? What do you notice? Have a look.

Look at the number rods that have been used to make them.

Are they congruent? No.

Both triangles are isosceles.

They've both got two long sides and a shorter side, and their long sides of the same as each other, but one of the side lengths is different.

So they are incongruent.

That's the opposite of congruent.

They're incongruent, they're not congruent.

Are these triangles congruent? What do you think? They're the same kind of triangle, aren't they? They're both equilateral.

Are they congruent? Both triangles are equilateral.

Hmm.

But their side lengths are different, so they're incongruent.

That's not the same size, and that's one of the things that congruent shapes have to be.

They have to be the same size.

It's the same type but not the same size.

So it's incongruent or not congruent.

Let's have a check for understanding.

Let's see if you're getting the idea of congruence.

So true or false, it is possible for an equilateral triangle and an isosceles triangle to be congruent.

Is that true or false? Now, you might want to pause a video there or I'm going to give you a couple of possible justifications if you want a little bit more of a clue.

So here's the justifications.

A, as long as they have the same perimeter, they are congruent.

Or B, two congruent shapes have to be identical, so two different types of shapes cannot be congruent.

Okay? Pause the video, what do you think? Let's have a look.

Is that true or false? Did you manage to come up with an agreement with your partner if you have got one? That's false.

And the justification for that is that two congruent shapes have to be identical.

They've got to be the same shape.

So two different types of shapes, and in this case, equilateral and isosceles triangles are different types.

Two different types can't be congruent.

It's got to be the same type of shape to be congruent.

Time for some independent practise.

So number one, what does it mean if two shapes are congruent? Can you explain that in your own words? Can you explain that in as much detail as you can? And number two, use number rods if you've got them to create some pairs of congruent triangles.

So you might want to think about things like rotating and reflecting one of the triangles, that kind of thing.

How many pairs of congruent triangles can you create? Have fun exploring that.

And number three, in each row, can you tick the congruent triangles? And number four, draw some pairs of congruent triangles on the grid.

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

Welcome back.

How did you get on with that? How did you find that? Do you think you're starting to understand congruence? Well, let's find out, let's give you some feedback.

Number one, what does it mean if two shapes are congruent? You might have worded this slightly differently, but they're exactly the same shape and size, although they may have been rotated or reflected.

That's two congruent shapes.

And then using number rods to create some pairs of congruent triangles.

Well, basically any two triangles that are identical, they might be repositioned, different orientations, that kind of thing.

But here is an example here.

You might notice that the same number rods have been made for both, and that shows their congruence.

And the congruent triangles in each row.

For A, it was these two.

They're exactly the same, one's slightly rotated.

And for B, it was these two.

One's been reflected.

And for C, it's these two.

They're exactly the same shape, just rotated.

Number four, some pairs of congruent triangles.

Basically any two triangles are identical in type and size.

For example, these isosceles triangles, they're congruent.

Okay, I think you're getting good with congruence and we're going to use and apply that in the next cycle, which is going to be about composing a parallelogram from triangles.

So can you remember some recent work, hopefully that you've done on parallelograms? Let's get that back into your mind, and here we go.

Alex composed different polygons from two congruent, and you know what that means now, two congruent, right-angled triangles.

What can we say about the area of each polygon? So let's have a look.

So he's put them together, he's composed a new shape, a new polygon.

What could you say about that polygon? What about that one? The area is always the same.

It's been decomposed and composed differently, but the area hasn't changed.

Nothing's been taken away, nothing's been added, nothing's been made bigger or smaller.

It's just been recomposed.

Alex composes this polygon from the triangles.

What properties does it have? Can you classify that polygon? Hmm.

What kind of polygon is that? It's a very common one, isn't it? Two congruent right-angled triangles can be used to compose a rectangle.

A rectangle is a parallelogram with four equal angles of 90 degrees.

So if you've got two congruent right-angled triangles, you can form a parallelogram, a rectangular parallelogram.

And then Alex says, "I wonder if the opposite is true.

Can any rectangle be decomposed into two congruent right-angled triangles?" Hmm, that sounds like it's worth investigating.

Let's find out.

It's an interesting question.

Well, that's exactly what Alex is doing.

He's drawn a line along the diagonal from vertex to opposite vertex, and he's cut along the line.

Here we go.

Just like that.

And that's formed two triangles.

What could you say about those triangles? The rectangular parallelogram was decomposed into two congruent right-angled triangles.

So, so far what Alex wondered is true, a rectangle can be decomposed into two right-angled triangles.

Let's do a check.

Investigate whether this works with your own rectangles.

Try rectangles with different side lengths.

What happens with a square, for example? That's a kind of rectangle.

Okay, pause the video and see what you can find out.

What did you find out? Did you manage to investigate some different ones, like a square, for example, and maybe some very narrow thin ones, anything like that? Well, here we go, here's a square.

And that can be decomposed into two right-angled triangles.

So yes, it's true.

Any rectangular parallelogram, including squares, can be decomposed into two congruent right-angled triangles.

But that's only true of rectangular parallelograms. Did you notice you always cut one diagonal to decompose the parallelogram? Hmm.

Alex now uses two congruent scalene triangles.

He thinks he will still compose parallelograms. Can you visualise any? So in your mind, can you put those two triangles, those scalene triangles that are congruent together to make a parallelogram? Hmm.

Well that's a parallelogram, so that's one way to compose it.

And that's a different parallelogram.

Does you picture that one? That's a different way to compose it.

Alex draws and cuts out a parallelogram that is not a rectangle, so to see if he can decompose it into two congruent triangles as well.

So he's going to cut out a parallelogram.

Just like this.

Can he decompose it into two congruent triangles? What would you do? How would you cut it>? Like that.

From vertex to vertex across that diagonal.

And then that's created two triangles.

And what can we say about them? They're scalene, and they're congruent.

So any parallelogram can be decomposed into two congruent triangles.

They are formed by cutting along a diagonal just like we just did.

So that's true of rectangular parallelograms and non-rectangular parallelograms. Let's do a check.

Which of these statements are true? It might be more than one.

A triangle can be composed from two parallelograms. A parallelogram can be composed from two congruent triangles.

A parallelogram can be decomposed into two congruent triangles.

Have a think about that, picture that, visualise that, discuss that and see what answers you can come up with.

Pause the video.

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

Well, B's true.

A parallelogram can be composed from two congruent triangles.

So if you've got two congruent triangles, two triangles that are the same, you can put them together in certain ways to form a parallelogram.

And C is also true.

A parallelogram can be decomposed into two congruent triangles by cutting along the diagonal.

Time for some practise.

Number one, which of these parallelograms has been correctly decomposed into two congruent triangles? And explain why or why not for each one.

Number two, draw and cut out some parallelograms. Can you decompose them into two congruent equilateral triangles? Can you do that with two congruent isosceles triangles? And can you decompose 'em into two congruent scalene triangles? Investigate that.

Pause the video, good luck with your explorations, and I'll see you soon for some feedback.

Welcome back.

How did you get them? What did you find out? Well, let's start with number one.

These are two congruent quadrilateral.

So they are congruent, but they're not triangles.

Each of them's got four sides.

The parallelogram has not been cut along a diagonal.

These shapes are incongruent.

One is a triangle and one is a trapezium.

They're not congruent shapes.

What about this one? Yes, these are two congruent scalene triangles.

The parallelogram has been cut along the diagonal, so everything's been done properly and that has left us with two congruent scalene triangles.

And draw and cut out some parallelograms. Can you decompose 'em into two congruent equilateral triangles? Yes.

The parallelogram would need to be a rhombus though for that to be true.

Can you decompose 'em into two congruent isosceles triangles? Is that possible? Yes, it is.

That's an example just there.

And can you decompose 'em into two congruent scalene triangles? Yes, you can.

And that's one of many examples, and maybe you found some different ones.

We've come to the end of the lesson, and I've had so much fun today exploring those parallelograms and the idea of congruence and the idea that parallelogram can be decomposed into two congruent triangles.

So any parallelogram can be composed from two congruent triangles or decomposed into two congruent triangles.

So that includes rectangles and it includes squares, which are a kind of rectangle.

All of those you can cut along the diagonal from vertex to vertex and turn it into two congruent triangles.

And this applies to rectangular parallelograms, including squares.

I've really enjoyed spending this time with you, and I hope you've enjoyed today's lesson.

Hope I get the chance to work with you again in the near future on a different maths lesson.

In the meantime, I think a little pat on the back is in order to celebrate your accomplishments today.

And enjoy the rest of your day, whatever you've got in store.

If you've got some more lessons coming up, hope they're good lessons, and I'll see you again hopefully for some more maths.

Take care and goodbye.