Lesson video

Hi, everyone, I'm Miss miles and welcome to your maths lesson.

Before we start, a did you know fact.

Today's fact is, did you know that the current number system that we use is the Hindu Arabic number system that was actually invented in India in the sixth or seventh century.

Okay, back to today's math.

For this lesson, you need a pencil or a piece of paper and a clear workspace.

In today's lesson, we will be identifying diagonals.

And when you're ready, let's get started.

Okay, so what is a diagonal? The diagonal of a shape is a straight line segment that joins one vertex to another.

So if you look at the rectangle on screen now, you will see that there are two diagonals drawn onto it.

The blue one and the green one and they join from one vertex to another.

Okay, let's have a look at this a little bit closer now to see what we can find out about the diagonals in this rectangle.

The diagonals of a rectangle are congruent.

That means they are the same length.

So if you look at the two diagonal lines on screen and the two rulers that have been placed on, we can see that both of the diagonals are roughly 14 centimetres, meaning they are congruent.

What do you notice about the diagonals in a rectangle? Is there anything else so you can spot? Looking at the rectangle on screen with the rulers on and also the red oval that has been marked on.

Pause your video and see what you can spot.

Off you go.

Okay, now, hopefully you will have noticed by looking really carefully at the red oval that's been marked on screen.

Our two diagonals cross at roughly seven centimetres.

Now we identified earlier that both of the diagonals were 14 centimetres each.

And they cross at exactly seven centimetres each, meaning that they cross that halfway.

And that means they bisect one another.

So what diagonals are these shapes from? On screen now there are four sets of lines coming from four different shapes.

And we need to work out what shapes they are from.

And I'm going to show you how to do that.

I've chosen one of the sets of diagonals, chosen the third one and I have found out what shape they are from.

So I know that the beginning of one of the lines would be a vertex because diagonals join one vertex to another.

So I know that at the top of that line there will be a vertex.

So in order to work out what shape it is, I have to draw lines to connect the vertices.

And I've done that all the way around.

I now know that it has given me a kite.

So what you need to do now is you need to look at the other three sets of diagonals on screen.

You need to draw them and identity what shape they come from.

Pause your video now and have a go at that for me.

Okay, let's have a look at our answers now.

So our first shape as you can see is a square.

So we took our first vertex here and we connected it to the vertex over here and we carried on and hopefully you will have found that you had a square.

The next one is a trapezium.

And again, we joined our vertexes together to find that we had a trapezium.

We'd already done the kite together and the final shape is a parallelogram.

Okay, now what we're going to do, we're going to have a look at these shapes in a little bit more detail and we are going to work out what we can deduce about them.

So what can you deduce about the diagonals of these quadrilaterals? So we're going to take our closed kite now that we looked at together earlier.

And we're going to see what we can deduce about it.

So, here is my kite and it has got some extra markings drawn on for us to look at.

So I know that first things first, the diagonal lines are of different lengths.

So here's my one diagonal and here's my other.

And they are of different lengths to one another.

I also know that the diagonals intersect at right angles.

So here they are intersecting at right angles.

And those yellow lines have shown that because of this the diagonal lines are perpendicular to one another.

And the diagonals do not intersect at halfway.

And that's because this line is shorter than this line, okay? Now, the two blue lines that have been drawn on show me that from the vertex to the point where the two lines meet is the same length on either side and that's what the blue lines mean.

Now let's have a look at our other quadrilaterals now.

So we come back to our square, our trapezium, our kite and our parallelogram, all quadrilaterals, all have four sides, all have four vertices.

You now need to see what you can deduce about the diagonals in these quadrilaterals.

So we've already looked at the kite, but you need to look at the other three shapes.

What can you deduce? At the bottom of the screen there is some key vocabulary that you can use to help you.

Look at the lines that have been drawn on in the markings.

Pause your video there and have a go at that for me.

Okay, now, let's go through them and have a look what we can deduce about the diagonals in these shapes.

So starting with our square.

So I can see by looking at my yellow markings here that they intersect at a right angle and they are perpendicular.

I can also see by the blue lines that they are all an equal distance from the vertex to the point where they cross, which means that bisect one another because they intersect at half way.

Okay, let's have a look at our trapezium now.

So our trapezium has no additional marking added on and maybe you'll have noticed that.

And that's because a trapezium doesn't always have to look like this.

A trapezium must have one pair of parallel sides.

But it can look a variety of different ways.

So we can't say it will always the diagonals will always intersect a right angle.

And we cannot say that they will always be the same length.

Okay, finally, let's have a look at our parallelogram.

So, the lines marked on my parallelogram show me that this line here from the vertex to the point where the two diagonals intersect and from these vertex to the point where they're parallel where the diagonals intersect are the same length.

And it also shows me that this section and this section are the same length.

They do not meet at a right angle and they are not perpendicular.

Okay, did you get the same answers as me? Let's move on.

Okay, so let's have a look at this now.

What do you notice? On screen, there is a quadrilateral and it has one of its diagonals marked on with the green line.

And you need to consider some questions about it.

So what do you notice about the two triangles that are formed? And what do you notice about the angles? Also, what do you notice about the shape as a whole? Pause your video and see what you can find.

Okay, let's go through some of the possible answers together.

So we can see that the angles here, so this angle here and this angle here, opposite ones, they are of equal size and they have been marked by the red curve.

And now we've marked on some extra angles here.

The blue ones are also of equal size, showing us that the two triangles that have been created by the diagonal line are congruent; they are the same.

Now let's have a look at the shape as a whole.

Now, I can see that it has opposite sides that are the same length.

They're not all the same length.

Opposite sides of the same name and they have been marked on by the lines on screen, okay? Also, if I look at the purple lines that have been added on here, I can see that it has two sets of parallel sides.

So I wonder what shape this is.

Now I know that it's got two sets of parallel sides and I know that it has opposite sides of equal length.

And that tells me that this is a parallelogram.

Because a parallelogram has opposite sides of equal length and has two sets of parallel lines.

It is not a rhombus because a rhombus would have all sides of equal length.

Okay, let's take our understanding of diagonals and shape properties and apply it to an independent task.

Regular pentagons have more than one diagonal from each vertex.

Two examples have been done for you.

So if we have a look here.

Here is a vertex that has been on identified and two diagonals have been drawn on to two other vertices.

If you connected all of the diagonals how many parts will be inside be broken up into? How many different shapes can you find within? And what do you notice about the innermost shape? So what you need to do are you need to get a piece of paper and a pencil and if you find it literally tricky to draw a pentagon, you could trace the one on screen.

You need to put the two diagonals that are on there already and then find another vertex.

So perhaps this one and draw on the two possible diagonals from that vertex, then find the next vertex and draw on the two possible diagonals from there.

And continue until all of the diagonals are being drawn on and then answer the questions on screen and see what you can find.

Pause your video there and have a go that for me.

Okay, let's have a look at our answers now.

So here is the pentagon with all of the diagonals drawn up.

So first question was, how many parts would the inside be broken up into? And the answer is 11 parts, okay? Let's count them together.

One, two, three, four, five, six, seven, eight, nine, 10, and 11.

How many different shapes can you find within? Two different types of isosceles triangle and one pentagon.

So here's one type of isosceles triangle and here is another type of isosceles triangle.

An isosceles triangle has two sides that are of the same length.

And here is the pentagon in the middle.

And what do you notice about the inner most shape? So we know it's a pentagon, don't we? It is another regular pentagon that has been flipped around or rotated.

Did you get the same answers? Have a look at your answers and see how you did and if you need to make any changes, please do that now.

Well done for completing today's lesson.

Don't forget to complete the end of lesson quiz and I'll see you again very soon.