Lesson video

Hi, how are you today? I hope you're having a good day.

My name is Ms. Coe and I'm really excited to be joining you for this maths lesson.

In this maths lesson, we're going to be looking at geometry.

Now you might wonder what geometry means.

Geometry is all about shape, so I'm really excited to say that this lesson is all about exploring different shapes.

I really hope you'll enjoy learning about this as much as I will enjoy teaching you.

By the end of this lesson, you'll be able to say that you can make and draw triangles on circular geoboards.

We have a few key words for this lesson.

I'm going to say them and I would like you to say them back to me.

Are you ready? My turn.

Vertex, your turn.

My turn.

Vertices, your turn.

My turn.

Perpendicular your turn.

Excellent work.

Let's see what those words mean.

A vertex is the point where two lines meet and the plural of that is vertices.

So in the triangle you can see that we have highlighted one vertex and we know that a triangle has three vertices.

You may have called these corners, but try really hard to use the proper mathematical term.

Two lines which meets at a right angle are perpendicular.

In this lesson today, we're going to be making and drawing triangles on circular geoboards, and we have two cycles in our lesson today.

In the first cycle, we're going to look at constructing triangles, so thinking about any sort of triangle.

And then in the second cycle we're going to focus on constructing triangles which have perpendicular sides.

If you're ready, let's get started with the first cycle of our learning.

In this lesson today, you're going to meet Jacob and Sofia and they are going to be sharing their examples of their work and posing some questions for us to think about.

Let's start here by thinking about triangles.

Which of the shapes here are triangles? How do you know? Take a moment to have a think about these.

What do you know about the properties of a triangle that will help you? Let's see what Sofia has to say.

"A triangle has three straight sides and three vertices." Does this help us eliminate some of the examples? "These two then are definitely triangles." Which one do you think she means? That's right.

Those two examples are definitely triangles.

They have three straight sides and three vertices.

"This shape here though has four sides, so it is not a triangle." I wonder if you can name that shape.

It's not a triangle.

It has four sides, but I noticed something else about the properties.

That leaves two to think about.

This shape, well, the bottom of it, says Sofia, looks like a triangle, but it actually has five vertices, so it's not a triangle.

If it has five vertices, it is a pentagon because that is a five sided shape.

What about the last example? Well, this one could be a triangle, but the lines need to join to make sides.

It is not closed, therefore it's not a polygon, which means it's not a triangle.

Time to check your understanding.

Select the shapes which are triangles.

You have A, B, and C to think about.

<v ->Pause the video here.

</v> (no audio) <v ->Welcome back.

</v> I hope you looked really closely and thought about the properties of a triangle.

How did you get on? A and C are both triangles.

I wonder why and I wonder why B is not a triangle.

Well, B is not a triangle because it doesn't have straight sides.

It looks sort of triangular shaped, but it has curved sides.

And look at those vertices.

They are not sharp points, therefore it is not a triangle.

A and C are triangles because they have three straight sides and three vertices.

Well done if you said that.

Jacob has used a circular geoboard to draw some shapes.

He says that they're both triangles.

Do you agree? Now a circular geoboard is simply a circle with fixed points as you can see here.

You may have seen these before and you may have used elastic bands or string to make shapes, but you can also draw them as well.

So Jacob has drawn what he thinks are two triangles.

What do you think? Do you agree with him? Look closely at them.

Well actually this first one is not a triangle.

Why not? Well, a triangle has three straight sides and three vertices, and I can see there that the bottom side is made of the curve of the circle.

Therefore it is not a straight side, so it's not a triangle.

Jacob now draws two different triangles on the same circular geoboard.

What is the same about these triangles? What is different? Have a really close look at the two triangles.

They're definitely triangles.

They have three straight sides and three vertices, but what's the same and what's different? I wonder what you notice.

Well, two of the vertices are in the same position on the circular geoboard, which means that one of the sides on both triangles is the same length.

We can see that this vertex and this vertex are in the same position on both geoboards.

So that means the line or side connecting them is the same length in both triangles.

One vertex has moved one position around the circular geoboard.

So the last vertex that we haven't talked about started off in one position and you can see if you look on the second triangle, it has moved one position around the geoboard.

So that means that Jacob has drawn two triangles, but the properties of those triangles are different because of the position of that last vertex.

The vertices and some of the size are different sizes.

I wonder if you can notice anything about the vertices.

For example, if you look at the top vertex, you can see that on the first triangle is quite small, whereas the second one is larger.

Neither of them are right angles, they're both smaller than right angles, but I can see just by looking that the second triangle has a larger vertex on the top than the first one.

Time to check your understanding.

Jacob draws a third triangle, so we now have A, B, and C.

A and B you've seen before and C is the new one.

I would like you to discuss the similarities and differences between them.

If you have a friend nearby, perhaps you can have a conversation with them.

Pause the video here.

(no audio) Welcome back.

What kinds of things did you notice? What kinds of things did you say? You might have said that each triangle has two vertices in the same position on the circular geoboard.

One of the vertices has changed position.

It has moved one space around the circle each time.

If you look really closely at triangles B and C, you can see that they have the same side lengths and the same sized angles in them, so they're the same triangle.

They just look a little bit different.

Time for your first practise task.

For question one, I would like you to construct different triangles using each geoboard.

So the first job is to make some different triangles using the circular geoboards.

Then I would like you to think about what is the smallest triangle that you can make.

So you might need a few different circular geoboards there.

What is the smallest one that you can make? How do you know it's the smallest? And then for question three, I'd like you to think about can you make the same triangle, so the same properties of the triangle from different starting points.

I hope you really enjoy drawing lots of different triangles.

Remember to use a ruler and really carefully join the points of the circular geoboard.

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

(no audio) Welcome back.

How did you get on? I hope you enjoyed drawing different triangles.

Remember there are lots and lots of different triangles that you could have drawn on this circular geoboard.

Here are some of them.

Did you make these triangles? What's the same about them and what's different? Well, I can see that all of them have one vertex in the same position on the circular geoboard, but the others are different.

I can also see that they have slightly different properties.

Some of the side lengths are different across all three triangles.

You may have talked about the differences in your triangles.

If we think about question two, I asked you to create the smallest triangle.

So we're looking for the one that took up the least amount of space on the geoboard.

Remember, we still needed to use the points on the geoboard as the vertices of the triangle.

So this one is the smallest triangle that you could make.

Well done if you drew that triangle.

Remember though that yours may have been in a different position around the circular geoboard.

And for question three we asked you to think about drawing the same triangle on one geoboard.

So thinking about different positions.

For example, if we took the smallest triangle, there are lots of different positions that we could put that triangle in.

It's the same triangle, but it's used different points for the vertices.

You can see by looking that they take up the same amount of space and they have the same properties.

So those triangles are identical.

You may also have drawn some that crossed over each other.

So for example, this triangle can be shown at different points on the circular geoboard.

We could have shown it there or we could have crossed it over and shown another example here.

Well done if you managed to find answers for all three of those questions.

And in the process drew lots and lots of different triangles on your circular geoboards.

Let's move on to the second cycle of our learning where we're thinking about constructing triangles, but this time we're thinking about perpendicular sides.

So now Sofia and Jacob are exploring drawing on a different circular geoboard.

So the first one that they used is on the left hand side and the new one is on the right hand side.

What do you notice? What's the same and what's different? That's right, the new circular geoboard has more points around the circle.

They're still equally spaced, but there are more points, which means there's more spaces to draw vertices around this circle than there was in the original geoboard.

Geoboards can have a different number of equally spaced points on them.

So the first cycle we used a nine point geoboard and the second one has 12 points around there.

Jacob and Sofia practise drawing perpendicular lines on the Geoboards.

Take a really close look at Jacob's example and Sofia's example.

Do you think that they have been successful in drawing perpendicular lines? I wonder.

Sofia says that she can use a right angle checker to see if this is a right angle.

If there is a right angle, remember, that means that the lines are perpendicular.

They have to meet at a right angle.

So Sofia uses her right angle checker on Jacob's lines.

What do you notice? Oh dear, I don't think Jacob has got a right angle.

"Sorry, Jacob," says Sofia, "The angle is greater than a right angle, so that means the lines are not perpendicular." Jacob is going to use a right angle checker on Sofia's example to see if she has a pair of perpendicular sides.

What do you think? Do you think they meet as the right angle? So we can use the right angle checker to see that Sofia is absolutely correct.

She has a right angle, which means her para sides are perpendicular.

Well done, Sofia.

This is a right angle.

So you have drawn a pair of perpendicular lines.

Time for a check of your understanding.

Look at example A and B, I would like you to explain whether the lines are perpendicular or not.

Pause the video and take a moment to have a think.

(no audio) Welcome back.

What did you think? Let's start with A.

A is not a pair of perpendicular lines.

If we use a right angle checker on these lines, we can see that the angle is smaller than a right angle.

If the angle is smaller than a right angle, then it is not a right angle, which means the sides are not perpendicular.

What about B then? B is an example of perpendicular lines.

Again, we can use our right angle checker to see that they do meet at a right angle, therefore these sides are perpendicular.

Now you may have noticed that one of the sides was very short compared to the other.

Does that matter? Absolutely not.

What is important is that they meet at a right angle and that's what makes the lines perpendicular.

Well done if you said that and you reasoned about these two sets of lines.

So Sofia knows she's got a pair of perpendicular lines and she uses them to make a triangle.

Let's see what she does.

There we go, she draws one final line to make a triangle.

Remember, a triangle has three straight sides.

She says, I know the sides are perpendicular because they form a right angle.

We can double check if we need to with our right angle checker.

And then remember, we can show that it is a right angle by drawing a small square.

A right angle can be thought of as a square corner.

And so the symbol we use to represent that right angle is a small square like the one you can see there.

She has made a right angled triangle, a triangle with a right angle, and she wonders whether she can make a different triangle using that same pair of perpendicular lines.

What do you think? Time for your second practise task.

I would like you to construct triangles with perpendicular sides using the geoboards.

Can you draw more than one triangle on each board? So look really carefully.

Remember, you need to make a right angle in order to have a pair of perpendicular sides.

If you've got a right angle checker, you might want to use that to double check that you've got a right angle.

For question two, I would like you to think about this question.

Can a triangle have more than one pair of perpendicular sides? Now, you might decide immediately yes or no.

I would like you to use the geoboards to show your ideas.

So if you think it can, then can you draw me some examples? And if you think it can't, can you draw some examples to show why not? Good luck with those two tasks and I will see you shortly for some feedback.

Pause the video here.

(no audio) Welcome back.

How did you get on? I hope you enjoyed drawing lots of different triangles with perpendicular sides and had a really good think about whether a triangle can have more than one pair.

So remember, for question one, there are lots of different triangles that you could have drawn.

You may have drawn a triangle with the same properties as Sofia's.

It just looks different because we've moved it around the circular geoboard.

So here we can see that we have a triangle and we've clearly marked the pair of perpendicular sides by showing where the right angle is.

Two of these triangles can be drawn on one geoboard.

So maybe you explored drawing the same triangle on one geoboard.

Can you visualise what this one would look like? It could look like this.

Here we have drawn the same triangle with the same properties and we've drawn it twice on one geoboard.

You might have drawn a different triangle with a pair of perpendicular sides.

Remember, you might have used different points on the geoboard.

But what is really important is that you've checked really carefully that that pair of lines are perpendicular, that they form a right angle.

You can see we've marked the right angle here on another example.

Maybe you drew this triangle too.

For question two, did you realise that you cannot make a triangle with more than one pair of perpendicular sides? Let's explore why.

Here is one pair of perpendicular sides.

You can see from the highlights that this pair of perpendicular sides forms three vertices, three points on the circular geoboard.

So making one pair of perpendicular sides gives us three vertices.

And how many vertices does the triangle have? Oh yeah, that's right.

They have three.

So if we tried to make another pair of perpendicular sides, as you can see here, this would actually give us a shape with four vertices.

That isn't a triangle.

That particular example is a rectangle.

So if you had another pair of perpendicular sides, you would need another, a fourth vertex.

That means it is no longer a triangle, so it is not possible to make a triangle with more than one pair of perpendicular sides.

Well done if you realised that and you drew some examples to show why not.

We've come to the end of our lesson, and I hope you've enjoyed making and drawing triangles on different circular geoboards.

Let's summarise what we've learned today.

A triangle is a 2D shape with three straight sides and three vertices.

A triangle can have a pair of perpendicular sides.

One of the vertices therefore will be a right angle.

So that's a great way to check, but a triangle cannot have more than one pair of perpendicular sides.

Thank you so much for all of your hard work today.

I really hope you've enjoyed drawing those triangles and I hope to see you in another maths lesson soon.

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