Lesson video

Hello there.

My name is Mr. Goldie and welcome to today's maths lesson.

And here is the lesson outcome.

I can describe the shapes made when rectangles are cut on the diagonal.

And here is the key word for today's lesson.

Just one key word today.

Can you repeat it back after me? So the word is "diagonal".

Let's take a look at what that word means.

So a diagonal is a straight line connecting one vertex to another but is not the side or edge.

We'll take a good look at what diagonal means in today's lesson.

And here's our lesson outline.

So the first part of the lesson is making one diagonal cut and the second part of the lesson is making two diagonal cuts.

Let's get started.

In this lesson you will meet Aisha and Laura, who'll be helping you with your maths today and asking you some questions as well.

Aisha is thinking about different ways to cut this rectangle into two equal parts.

" I can use a vertical line to cut the rectangle into two halves," says Aisha.

So here's a vertical line.

" This gives two smaller identical rectangles," says Laura.

Here they are.

" I can use a horizontal line to cut the rectangle into two halves," says Aisha.

This time Aisha uses a horizontal line.

" This also gives two smaller identical rectangles," says Laura.

See the two rectangles made when the larger rectangle is cut on that horizontal.

" Is there another way to cut the rectangle into two equal parts?" Says Aisha.

" The rectangle can also be cut diagonally from one vertex to the opposite vertex." So remember when it's cut diagonally, it doesn't mean it's along the side or an edge.

So the rectangle can be cut like this or like this.

Those are both diagonal lines.

So we've got a cut going from one vertex to another, but it's not along the side or the edge of the shape.

Which rectangles have been cut diagonally? What do you think? Remember, diagonally means from one vertex to the opposite vertex.

So have a good look.

There are three shapes there.

Which of them have been cut diagonally? Pause the video, see if you can mark out the answer.

And welcome back.

Let's take a look.

So our first shape here, shape "a", is it cut diagonally? Yes, it is.

Shape "b", is that cut diagonally? No it isn't.

Now this line does not cut the rectangle from a corner to the opposite corner.

In fact, it's going from one side of the shape, isn't it, to the other side? If you've rotated that rectangle, you could quite easily see it could be a horizontal cut or a vertical cut.

Let's take a look at shape "c".

Has shape "c" been cut diagonally? Yes it has.

We've got a cut going from one vertex to the opposite vertex.

So very well done, if you were to tell the "a" and "c" one diagonal cuts and the "b" was not.

Aisha thinks about the shapes that have been made, " Each half of the rectangle is a triangle." " Each half is a right-angled triangle," says Laura.

So each triangle that has been made there has a right angle, this shape here, this triangle here has a right angle and the other triangle also has a right angle.

" Each triangle is identical, they are the same shape and size, says Aisha.

Aisha cuts another rectangle into two equal parts.

" I'm going to make one diagonal cut," says Aisha.

" What do you notice about the shapes that are made?" It may be really helpful to talk to a friend if you've got one available nearby.

Pause the video, see what you can come up with.

And welcome back.

How did you get on? What did you come up with? Let's take a look, see what Laura and Aisha have to say about these two shapes.

So first of all, Aisha says, " The starting shape was a square.

The shapes made up both right-angled triangles." Okay, so hopefully you worked out they were both triangles and hopefully you spotted as well that both triangles have a right angle.

This triangle here has a right angle, this triangle here has a right angle.

" Each triangle also has two sides the same length." Well I dunno if you spotted this.

So these two sides on the triangle are the same length as each other.

So very well done if you spotted some or all of those things about those triangles.

And let's move on to task A.

So in task A, you're going to be investigating whether Aisha and Laura are correct.

Asia says," When you cut a rectangle diagonally, you always get two right-angled triangles." Laura says," When you cut a square diagonally, you get right-angled triangles with two equal sides." Are they both correct? Does that always happen? So remember a diagonal is a straight line connecting one vertex to another but is not the side or edge.

Now once you've worked out whether Laura and Aisha are correct, can you explain your findings? Can you say why they are correct or why they are incorrect? So here are the shapes for investigation.

These are the shapes you're going to be trying or you can try some different rectangles of your own instead.

So don't forget, some shapes have gotta be rectangles and some shapes have gotta be squares.

And on this shape here, be careful to cut from one corner to the opposite corner of that shape that has been rotated around, so be careful with that one.

So pause the video, see if you can work out whether Aisha and Laura are always correct, whether they're sometimes correct or whether they are never correct.

And welcome back.

How did you get on? What did you manage to find out? Let's take a look and see whether you got the right answers.

So first of all, let's look at what Aisha said.

So when you cut a rectangle diagonally, you always get two right-angled triangles.

Aisha is correct.

When you cut any rectangle diagonally, you always get two right-angled triangles.

And this is why it's a bit tricky to explain, but listen really carefully and see if you can understand why.

One vertex of the rectangle remains the same in each triangle.

So here we have one of the right angles in the rectangle.

When we take the rectangle and we cut it into two separate triangles, that right angle has stayed the same.

All the vertices of a rectangle are right angles.

So each triangle has one right angle.

So very well done if you manage to have a go trying to explain that it's quite complicated.

And let's move on to what Laura said.

So Laura said," When you cut a square diagonally, you get right-angled triangles with two equal sides." We've looked at one example earlier on and that did happen.

Does it always happen? Well there's a square and there's a diagonal line running across the square.

Laura is also correct.

Two sides of the square make two sides of each triangle.

A square has all sides the same length.

So the triangle has two sides of the same length as well.

So you can see there that these two sides on the square are the same length.

The square has all sides the same length.

And when we cut it into two triangles, this triangle here has two sides of the same length as well.

Those two sides there haven't changed and the other triangle will also have two sides the same length as well.

So very well done if you spotted that Aisha and Laura were both correct and well done if you had a go trying to explain why they are correct.

That is some difficult mathematics there, trying to explain what you've noticed.

But that is what maths is about.

Maths is all about trying to explain what you have noticed, explain why something is true or not true.

And let's move on in our lesson two part two.

The part two is making two diagonal cuts.

Aisha uses two diagonal cuts to divide this rectangle into four parts.

" This rectangle is a square.

It has four equal sides," says Aisha.

So she cuts it diagonally across this way and she also cuts it diagonally that way, as well.

Aisha has divided the square into four triangles.

Aisha and Laura discussed the triangles that Aisha has created.

Each triangle is exactly the same shape and size.

They have two sides of equal length.

So let's have a look at those four triangles.

So they're all exactly the same and each of them has two sides the same length.

" Each shape is a right-angled triangle," says Laura.

Because they're identical, that first triangle here is a right angle, so all of them must have a right angle.

Start with a square.

" Make two diagonal cuts to divide the square into four triangles." So make sure when you start this, it is a rectangle with four equal sides.

" What do you notice about the four triangles?" Let's start with the square.

Cut it diagonally one way, cut it diagonally the other.

What do you notice about the four triangles you end up with? Pause the video, see what you can work out.

And welcome back.

How did you get on? Did you start with the square? Did you cut it diagonally two ways? Did you end up with four triangles? Let's take a look, see what you should have discovered.

So first of all, here is one of the triangles that that square would've been divided up into.

All four triangles are identical to each other.

Now hopefully when you cut them out, if you cut them out really, really carefully and you lay them over the top of each other, you should have noticed they're all exactly the same size.

Each triangle has one right angle.

Very well done, if you cut your square really carefully into four triangles and you managed to work out that the four triangles were identical, they each have a right angle and maybe you spotted this as well, they each have two sides the same length.

Aisha wonders whether this always happens.

" Would I always get four identical right-angled triangles if I divide any rectangle using two diagonal cuts?" We've had to go already investigating squares.

Does it work with any rectangle? " Let's change this square into a different rectangle," says Laura.

Let's take the square, let's stretch it out and make it into a different rectangle.

Hmm, Laura's having a good look at that rectangle and she says," The diagonal lines have moved.

They make different angles where they cross each other." Aisha uses two diagonal cuts to divide the rectangle into four parts.

" I cut from one vertex to the opposite vertex." So she cuts it one way like this and she cuts it one way like this.

" Aisha has divided the rectangle into four triangles." Aisha and Laura look at two of the triangles.

They're gonna start off by looking at these two first of all.

" These two triangles are identical.

They are exactly the same shape and size." " We can show this by rotating one of the triangles." Let's take this triangle here, rotate it around, and we should be able to see these two here are exactly the same shape and size.

One of the angles on these two triangles is larger than a right angle.

So if you use the right angle measurer here, we can actually see this angle here is larger than a right angle.

Angles can be larger than right angles.

" Each triangle also has two sides that are equal length." These two sides here are also equal length.

Aisha and Laura look at the other two triangles.

" These two triangles are identical.

They are exactly the same shape and size." " Every angle on these two triangles is smaller than a right angle." So Laura goes through and she measures each of the angles in the triangles.

So every angle is smaller than a right angle.

Angles can be smaller than right angles.

" Each triangle also has two sides that are equal length," says Laura.

These two sides here are the same length as each other.

And because the other triangle is identical, that also has two sides that are equal length.

Aisha thinks about cutting rectangles diagonally.

" When I divided the square using two diagonal cuts, I got four identical triangles." All four of the triangles looked like this.

" When I divided a rectangle using two diagonal cuts, I got two pairs of identical triangles." So two of the triangles looked like this and two look like this.

" I wonder if this always happens?" Says Aisha.

So when you cut a square into four triangles using two diagonal cuts, do you get four identical triangles? When you cut a different rectangle, a rectangle that isn't a square using two diagonal cuts, do you get two pairs of identical triangles? That's what Aisha's wondering.

Does that always happen? Well that's what you're going to be investigating for task B.

What happens if you divide a rectangle using two diagonal cuts? Can you complete the two statements? So Laura says," When you cut a square using two diagonal lines, what happens?" So try out some different squares, have a go cutting 'em using two diagonal lines.

What do you notice? Does it happen with all squares? Aisha says," When you cut a rectangle that is not a square using two diagonal lines, what happens?" What happens, Aisha? Could you complete that statement? So again, this time have a go using rectangles that are not squares.

Cut them using two diagonal lines.

What do you notice? And does this happen every single time you try it? So that's part one of task B.

Have a go trying to complete those two statements.

And then here's part two of task B.

So which statements are sometimes true, always true or never true? And you're going to be sorting the statements into the correct place on the table so you can see there it says sometimes true, always true, never true.

And investigate by using different rectangles.

Don't forget some of those rectangles have got to be squares.

And here are that statements you're gonna be sorting.

" When you make two diagonal cuts through any rectangle, you end up with four triangles." " The triangles will all have an angle greater than a right angle." " The triangles will have right angles." " Each triangle will have two sides the same length." " The four triangles are identical." " The triangles will have angles smaller than a right angle." So which of those are always true, sometimes true or never true? Pause the video and have a go at task B.

Welcome back.

How did you get on? Did you manage to do both parts of task B? Let's take a look to see whether you've got the right answers.

So here are the answers for part one of task B.

So what happens if you divide a rectangle using two diagonal cuts? So you may have completed Laura's statement like this, " When you cut a square using two diagonal lines, you get four identical right-angled triangles." You may have completed Aisha's statement like this, " When you cut a rectangle that is not a square using two diagonal lines, you get two pairs of identical triangles." So very well done if you wrote something similar to that.

Let's move on to part two of task B.

When you make two diagonal cuts through any rectangle, these things are sometimes true.

The triangles will have right angles, so sometimes they'll have right angles, sometimes they won't.

" The four triangles are identical." Sometimes they're all identical to each other, sometimes they are not.

Depends whether you've investigated a square or a different type of rectangle.

These things are always true.

You end up with four triangles.

Each triangle will have two sides of the same length.

It doesn't matter if you start with a rectangle or a square, each triangle will have two sides the same length and the triangles will have angles smaller than a right angle.

Every triangle will always have an angle smaller than a right angle, so that's always true.

And finally, never true.

This is never true.

" When you make two diagonal cuts through any rectangle, the triangles will all have an angle greater than a right angle." Some triangles might, but not all of them.

That can never be true.

They won't all have an angle greater than a right angle.

So very well done if you sorted those statements into the correct places on the table, that's excellent work.

So a square will give you four identical right-angled triangles.

When you make two diagonal cuts through a square, you'll end up with four identical right-angled triangles.

When you cut other rectangles using two diagonal lines, you get two pairs of identical triangles that do not have right angles.

Very well done today.

I'm sure you've done some really good mathematical thinking and you've had to reason about different shapes and about whether things always happen, whether they sometimes happen or whether they never happen.

Very well done if you try to explain some of your ideas as well, that's excellent.

Let's move on to our lesson summary.

So an angle can be exactly a right angle.

An angle can be less than or greater than a right angle.

When you cut a square using two diagonal lines, you get four identical triangles.

When you cut a rectangle that is not a square, using two diagonal lines, you get two pairs of identical triangles.