Lesson video

Hello, my name is Dr.

Rowlandson and I'm thrilled that you're joining me in today's lesson.

Let's get started.

Welcome to today's lesson from the unit of constructions.

This lesson is called, Understanding Construction of a Circle, and by the end of today's lesson, we'll recognise a circle as a collection of points that are equidistant from a fixed point.

Here are some previous keywords that you may be familiar with already and will be reusing again in today's lesson.

So you may want to pause the video at this point, while you re-familiarize yourselves with these words, before pressing play to continue.

This lesson contains two learning cycles.

In the first learning cycle, we're gonna be focused on properties of a circle, and in the second learning cycle, we'll be focused on properties of an arc of a circle.

But let's start off with properties of a circle.

Let's start off with an investigation.

Take a plain piece of paper and mark a cross somewhere near the middle of the page.

Doesn't have to be exactly in the middle, but give yourself plenty of room around it.

Then take a ruler and mark a point that is exactly 5 cm away from the cross.

And then using your ruler, mark some more points that are exactly 5 cm away from the cross.

What do you notice about your points so far? Pause the video while you think about this, and press play when you're ready to continue.

Aisha says, "The points follow the path of a circle." I wonder what it would look like if we plotted 1000 points.

Imagine what that would be like.

Aisha says, "The points would look like a circle", something like this.

That's because when we plot multiple points that are all equidistant from a single point, they follow the path of a circle or an arc, because a circle is a shape such that every point is equidistant to its centre.

For example, here in this image we've got a circle and all the points along it are 5 cm away from its centre.

For example, a side view of a wheel is a circle.

That's because each radius is the same length all the way around the circle.

For example, it might be 13 inches.

Why do you think it's important for every point on the edge of a wheel to be equidistant to its centre? Pause the video while you think about this, and press play when you're ready to continue.

Well, Andeep says, "Imagine what it'd be like if every point wasn't equidistant to the centre." It probably wouldn't feel very comfortable to ride on, would it? So side views of bicycle wheels is one example of a circle in real life, but can you think of anything else that forms or moves in a circle? Pause the video while you think about this.

Maybe talk to someone else about it if you can, and press play when you're ready to continue.

Some examples could be found in these images on the screen here.

On the left, we have the London Eye, which every pod is equidistant to a fixed point on the centre of the wheel.

In the middle, we have a clock, where the tip of each hand remains equidistant from a fixed point.

And on the right we have not so much the moon, because that's a sphere, but the craters on the moon.

The edges of an impact crater are approximately equidistant from the centre of the crater, not exactly, but approximately.

Let's think about this in a bit more detail then.

In which circle is the centre plotted in the correct position? Let's investigate this.

One method could be just to simply look at each diagram and see which one looks more correct.

The point in the left diagram looks more like it's in the centre of the circle than in the right diagram, so we might conclude that the one on the left is correct and the one on the right isn't.

But sometimes our eyes may deceive us when we're looking at things.

It may not always be quite so clear as it may be in this situation, and you never know when you're looking at an optical illusion, so it might be worth being a bit more sure about our answers.

So how could we be more certain that the one on the left has its centre point exactly in the middle rather than the one on the right? Pause the video while you think about this and press play when you're ready to continue.

A more reliable method might be to start measuring things with a ruler.

We could measure the radii in different directions.

So for example, the radius from the left of the circle to the centre is 5, from the centre to the right is 5.

Let's just check it in a few other ways.

From the top left to the centre is 5, and then the other way is 5 as well.

So we've measured four radii and each time they're 5 centimetres.

So we could feel quite confident that this circle has its point in the centre.

Whereas on the right hand diagram, if you measure the radius here, we get 5.

3 centimetres and then we get 4.

6 and then we get 5.

5 and then 4.

5.

Because all those radii are different, we know for certain that that centre point is not in the centre of that circle.

However, no matter how many times I measure the radius in different direction of the circle, I can't do it in every single possible direction because there are just so many.

There are infinitely many.

Then there might be one point on that circle which is not quite at the same distance away from the others.

So how could we be even more certain than that? Pause the video and see if you can think of a different method before pressing play to continue.

A third method could be using a pair of compasses.

We could take our pair of compasses, put the needle on the centre, put the pencil on the edge of the circle somewhere and then draw a circle with it, and see if it traces over the rest of the circle perfectly all the way round.

On the left hand diagram, it's done that.

If you do it on the right hand diagram, well, I put my needle in the centre and the pencil somewhere on the edge of the circle and then draw a circle with my pair of compasses and I'll get something like this.

We get a circle that doesn't overlap the original one perfectly, therefore we can deduce that that centre point is not actually in the centre of that circle, so it's not a centre point.

Here's another situation.

We've got a pair of axes here on a coordinate grid and we define the coordinates of 12 points, which are 5 units away from the origin.

Now the origin is a coordinate (0, 0), and we can find one of these points quite easily.

For example, the coordinate (5, 0), is exactly 5 squares across from the coordinate (0, 0).

And we could do that in a few other directions.

We could go upwards to (0, 5), or we can go left 5 units to get to minus (-5, 0), or we can go down 5 units to get to (0, -5).

So we've got those four coordinates pretty easily and we're convinced that all of them are 5 units away from the origin.

But the questions asked for 12 coordinates.

So how could we find some more? Pause the video while you think about this, and press play when you're ready to continue.

What we could do is take our pair of compasses, put the needle in the origin, because that is gonna be the centre, put the pencil somewhere 5 units away, for example on the coordinate (5, 0), and draw a circle.

Everywhere along that circle will be exactly 5 units away from the origin, because all those points in that circle are the same distance away from the centre.

So then we can look along that circle and see if we can find some more coordinates.

For example, here we have two coordinates, (4, 3) and (3, 4).

We probably wouldn't have thought of those just by looking at the numbers, because it's not obvious that those are 5 units away, just based on the numbers.

But looking at the circle, we can see that they must be the same distance away from the origin as is the coordinate (5, 0), which is 5 units away.

Let's look for some more coordinates.

In the left we have these ones, (-3, 4), and (-4, 3).

We've got these two coordinates here, (-4, -3) and (-3, -4), and two more coordinates here, (3, -4) and (4, -3).

There's also all the other coordinates along the circle which have decimal ordinances as well, but here are 12 coordinates with integer values that are all 5 units away from the origin.

So let's check what we've learned so far.

Every point on a circle is mm from its centre.

What is the word that goes in that blank? Pause the video while you write the word down, and press play when you're ready to continue.

The word is equidistant.

Every point on a circle is equidistant from its centre.

True or false? AB is equal to AC.

So what that means is, the distance from A to B is equal to the distance from A to C.

Is that true or false? And choose one of your justifications below.

Pause the video while you have a go, and press play when you're ready for an answer.

The answer is true and we know that because every point on a circle is equidistant from its centre.

Okay, it's over to you now for task A.

This task contains three questions and here is question one.

First, mark 8 points that are exactly 7 centimetres away from this cross.

And then draw a circle with radius 7 centimetres with a centre at that cross.

And then write down what do you notice about the points from part (a) and the circle from part (b).

Write a sentence or so for part (c) there.

Pause the video while you have a go at this, and press play when you're ready for question two.

And here is question two.

Aisha marks a cross on a piece of paper and then marks a point that's exactly 5 units away from it.

She asks everyone else to mark a point 5 units away from the cross.

Now, think about how you could use a pair of compasses to check whether everyone's point is in the correct position or not.

And then once you've done that, decide whose point's too close and whose point is too far.

Pause the video while you have a go at these questions and then press play when you're ready for question three.

Okay, and here is question three.

You've got a pair of axes on a coordinate grid and you need to find the coordinates of 12 points, which are exactly 10 units away from the origin.

Pause the video while you have a go at this, and then press play when you're ready to go for the answers to all three questions.

Well done with that.

Let's now work through this together.

In question one, you mark 8 points that are exactly 7 centimetres away from the cross.

It won't look exactly like this, but it could look something like this.

Eight points, all the same distance away, all 7 centimetres away.

And then part (b), you draw a circle, that might look something like this.

Then part (c), what do you notice? Well, all the points from part (a) lie on the circle that you drew in part (b).

And question two, how could you use a pair of compasses to check whether everyone's point is the correct distance away from the cross? Well, we could put the needle of our pair of compasses on the cross at the centre, put the pencil at the point that Aisha made, and then draw a circle and see if it goes through everyone else's points.

If it doesn't, then they've probably put it in the wrong place.

Or in other words, draw a circle with the cross at the centre and Aisha's point on the circle.

If we do that, it looks something like this, and then we can see whose point is too close to the cross.

Well that's Jacob, because his point is inside the circle.

And whose point is too far from the cross? Well that's Sofia, because her point is outside the circle.

And finally here is question three.

We need to find the coordinates of 12 points, which are exactly 10 units from the origin.

You can do it by drawing a circle that is 10 units in radius, with the centre at the origin.

And once we do that, we can see we've got these 12 points here, which all have integer values for the ordinance.

Now there are other coordinates that have decimal values, but these will be the easiest ones to find.

Great work so far.

Let's now move on to the second part of today's lesson, which is looking at the properties of an arc of a circle.

An arc of a circle is part of the circle's circumference, which means that every point on the arc of a circle is equidistant to its centre, or at least what the centre would be for the whole circle.

So for example, on the image on the right, every point along that arc is 5 centimetres away from the cross, which would be the centre of the whole circle.

For example, a wrecking ball swings in the arc of a circle.

So Laura says, "While the chain is taut, the ball is always the same distance away from the fixed point at the tip of the crane." Can you think of any other examples of things that form or move in an arc? Pause the video while you think about this, and then press play when you're ready to continue.

Well, some things could be anything to do with a pendulum.

For example, sometimes clocks have a big pendulum, which swing from left to right, they swing in an arc.

Or if you take an object like a pencil, for example, hold it at one end and twizzle it like this, the tip of the pencil will swing in an arc as well.

The curve below is an arc of a circle, and we've got those three points there, A, B, and C.

One of those three points is the centre point of the circle which that arc belongs to.

So which point would be the centre of the circle and how could we find it out? Pause the video while you think about what method or strategy we could use to find which of those three points would be the centre of that circle for the arc, and then press play when you're ready to continue.

One thing we could do is use a pair of compasses.

We could place the pencil tip of the compass on the arc, somewhere along it maybe at one of the ends, and then place the needle of the pair of compasses on one of the points, such as point A.

And then from there, draw a circle and see if that circle overlays over the arc.

In the case of point A, no, it doesn't it.

So we can deduce that point A is not the centre of the circle for that arc.

We can do it again for point B.

That hasn't worked either, has it? So we can deduce that point B is not the centre, and point C, yes, that's worked.

So we can deduce then, that point C is the centre.

Okay, let's check what we've learned so far.

Every point on an arc of a circle is mm from the centre of its complete circle.

Pause the video while you write down the word that goes in that blank, and press play when you're ready for the answer.

The answer is equidistant.

Every point on the arc of a circle is equidistant from the centre of its complete circle.

And here's another question to think about.

A seesaw tilts around a fixed point to raise and lower the seats at each end.

Which image below shows the path that the seats move? Is it A, B, or C? Pause the video while you have a go at this and press play when you're ready for an answer.

The answer is C.

The seats will move in an arc and the centre of the circle for the arc will be the middle of that seesaw there, because that's the fixed point.

And finally, we have three arcs here, A, B, C, and D, a cross, which is the centre point where the needle of the compass is and a pencil of the compass, poised and ready to draw an arc.

Your question is, which arc will the pair of compasses draw? Is it A, B, or C? Pause the video why you make a choice, and press play when you're ready for an answer.

The answer is B.

The reason why we can see that, is because each end of that arc is the same distance away from the cross, which will be be the centre.

Whereas with arc A, one end of that arc is much closer to the compass needle than the other end.

And the same with arc C as well.

And with arc D, we can see that one end of that arc is at the point where the needle would be, so the pencil would never get to that.

So we can see it's definitely not arc D, so it must be arc B.

Okay, it's over to you now for task B.

This task contains just one question, and here it is.

The curves that are labelled W, X, Y, and Z, are arcs from four circles, and the points that are labelled A, B, C, and D, are four possible centres for these circles.

What you need to do is match the arcs with their centres.

So pause the video while you have a go at this, and then press play when you're ready to go through some answers.

Okay, great job with that.

Here are our answers.

Point A is the centre of the circle that has arc Y on it.

Point B is the centre with X on it.

Point C is the centre with W on it, and point D is the centre with Z on it.

We can check each of these using a pair of compasses by putting the needle on the point, putting the pencil tip on the arc that we've matched it with, and then drawing a circle around that point.

We should see that the arc sits perfectly on part of that circle.

Excellent work today.

Spot on.

Let's now summarise what we've learned in this lesson.

The collection of points an equal distance from a given point form a circle, and a pair of compasses can be used to draw a circle that has a particular radius.

The centre of a circle is a point that is the same distance from every other point on that circle, and that's where the needle would go on your compasses.

But remember that drawing with a pair of compasses can be really tricky to begin with.

It can be a bit fiddly when we first start using them, but with more practise over time, it starts to feel easier and we get more accurate and smooth with our drawings.

So the more we practise, the better we'll get.

Thank you very much.