Lesson video

Hello, my name is Dr.

Rolenson, and I am happy to be helping you with your learning during today's lesson.

Let's get started.

Welcome to today's lesson from the Unit of Constructions.

This lesson is called, "Securing the Skill of Using a Pair of Compasses." And by the end of today's lesson, we'll be able to use a pair of compasses to draw different circles.

Here are some previous keywords that you may be familiar with and we'll be using again in today's lesson.

So you may wanna pause the video at this point, while you remind yourselves the meanings of these words before pressing Play to continue.

This lesson contains two learning cycles.

In the first learning cycle, we'll be drawing diagrams with multiple circles and seeing how they interact with each other.

And the second learning cycle, we're looking specifically at drawing circles that intersect exactly once.

But let's start off with drawing diagrams containing multiple circles.

Let's remind ourself how to use a pair of compasses to draw a circle with a specific radius.

The needle of the compass goes in the centre of the circle and the distance between the needle and the pencil tip is equal to the radius of the circle, because the pencil will be at the edge of the circle, and then the pencil is rotated around the needle to inscribe the circle.

So we can do that together now, by taking a plain piece of paper and drawing a circle with a radius of five centimetres.

on our plain piece of paper, mark a cross in it somewhere to be the centre of the circle.

Open up the pair of compasses so that the distance between the needle and the pencil tip is five centimetres.

Or you might find it easier to draw a five centimetre line first, and open up your pair of compasses so that the needle is at one end of the line, and the pencil tip is at the other end of the line.

Whichever way you do it, place the needle on the cross and then rotate the pencil around it to draw a circle like so.

You may wanna pause the video at this point while you have a go at that and press Play when you're ready to continue.

Let's check how we're getting on with that.

So Sofia has got her pair of compasses ready to draw a circle.

Which of these statements will be true? Is it A, the radius will be nine centimetres, B, the diameter will be nine centimetres, C, the radius will be eight centimetres, or D, the diameter will be eight centimetres? Pause the video while you have a go at this and press Play when you're ready to continue.

The answer is C.

The radius will be eight centimetres.

We're looking at radius here rather than diameter, because the distance between the needle and the pencil tip is a distance between the centre of the circle and its edge.

So we're talking about radius.

And we can see in this case, the needle's not quite at zero on the ruler, the needle's at one, so the difference between one and nine is eight.

So that pair of compasses must be open to eight centimetres.

Let's now check how well we can draw a circle with a specific radius before we continue.

Take a plain piece of paper, mark a cross in it somewhere near the middle of the page, it doesn't have to be exactly in the middle, but make sure you leave yourself plenty of space around it.

And using a pair of compasses, draw a circle with a radius of eight centimetres with a cross at the centre.

Pause the video while you have a go at this and press Play when you're ready to check it.

Okay, let's check how we've done with that.

So on our circles, draw two radii, preferably in different directions, and use a ruler to accurately measure the length of each radius and check are both of the radii in your circle eight centimetres? Pause the video while you check that and press Play when you're ready to continue.

If both radii are not perfectly eight centimetres, don't worry, with more practise, you will get better at using a pair of compasses.

So, Andeep, Izzy, Jun and Sam each draw two circles.

One circle has a radius of five centimetres and the other circle has a radius of eight centimetres, and your challenge is gonna be to recreate the drawings for each pupil.

So let's hear about what their drawings look like.

Andeep says, "My circles do not touch anywhere on the page." Izzy says, "My circles intersect at two points." Jun says, "My circles intersect at two points.

The centre of one circle can be found somewhere inside the other circle." And Sam says, "My circles do not touch anywhere on the page.

The centre of each circle can be found somewhere inside the other circle." So here's what they all say.

Pause the video, take some paper and try and recreate the images for each pupil.

Pause the video and press Play when you're ready to continue.

Okay, let's now take a look at what they've done.

Here's Andeep's drawings.

He's got two circles like this and he says, "My circles do not touch anywhere on the page." Here's Izzy's drawings.

She's got two circles that look like this and she says, "My circles intersect at two points." And we can see those two points are the points where those two circles cross with each other.

That's where they intersect.

And here's Jun's drawing.

His is a bit like Izzy's, in that they intersect at two points, but Jun also says, "The centre of one circle can be found somewhere inside the other circle." Can you see how the centre of the small circle is inside the bigger circle? And here's Sam's drawings.

Her circles do not intersect anywhere on the page.

And she also notices that the centre of each circle can be found somewhere inside the other circle.

The small circle has its centre inside the big circle, but also the big circle has its centre inside the small one as well.

Now here we have Alex.

And Alex has drawn a circle with a radius of eight centimetres.

He prepares to draw a circle with radius five centimetres by marking a point for its centre.

The distance between the two centre points is two centimetres, and before Alex draws his circle, him and Lucas are trying to decide whether or not the circles will touch.

Alex says, "I think they will, because the radius will be five centimetres, but the point is only two centimetres from the centre of the first circle." But Lucas says, "I think they won't, because the radius will be five centimetres, but the point is six centimetres away from the nearest point on the first circle." Who do you agree with? Pause the video while you think about this and press Play when you're ready to continue.

Let's take a look at this now in a bit more detail together.

Here's Alex's original circle, and we've made it a bit bigger on the screen so we can see it a bit more clearly.

The radius of this large circle is eight centimetres and the centre point of the small circle is two centimetres along this radius.

Which means there are six centimetres left to go, until we get to that circle's edge.

Now the radius of the small circle is five centimetres, so that means it will not be far enough to meet the edge of the large circle, so the circles will not touch.

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

Which statement about the two circles is true? Is it A, the circles do not touch, B, the circles touch once, or C, the circles intersect at two points? Pause the video while you have a go at this and press Play when you're ready to continue.

The answer is C.

The two circles intersect at two points.

Which statement is true about the two circles? A, neither circle has its centre point inside the other circle.

B, one circle has its centre point inside the other circle, or C, both circles have their centre points inside the other circle? Pause the video while you have a go at this and press Play when you're ready to continue.

The answer is B.

One circle has its centre point inside the other circle.

The small circle in particular has its centre inside the bigger circle.

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

This task consists of one question and here it is.

For each part of this question, need to draw two circles, one circle with a radius of five centimetres and the other circle with a radius of seven centimetres.

And you'll draw 'em both on the same page.

And each time, before you draw the circles, mark the positions of the centre points with a cross or a point and draw each pair of circles according to the conditions in each of the five parts of this question.

So A, B, C, D, and E, each of those parts give you some conditions for how to draw your pair of circles.

And that's what I'd like you to do.

Pause the video while you're having a go at these and press Play when you're ready to look at some answers.

Okay, well done with that.

Let's check how well we've got on with it.

We're gonna check our answers by not only looking at what they look like, but measuring the distances between the two centre points in each pair of circles.

So in part A, the circles do not touch and neither circle has its centre point inside the other.

So it should look like the image we can see on the screen now, where you've got two circles on the page somewhere that don't intersect at any point, yours might be in a different orientation, that's fine, but when you measure the distance between the two centre points, that distance should be greater than 12 centimetres to make that happen.

And then in part B, the circles intersect at two points.

Neither circle has its centred point inside the other circle.

So it should look something like this.

We've got two circles intersecting at two points.

The center's not inside each other.

It could be in a different orientation, that's fine.

But when we measure distances between those two centre points, it should be greater than seven centimetres and less than 12 centimetres.

And in part C, the circles intersect at two points and one circle has its centre point inside the other circle.

The diagram should look something like this.

Again, it might be in a different orientation, but the check it's done accurately.

Measure the distance between those two centre points.

And you should find it is greater than five centimetres but less than seven centimetres.

And part D, the circles intersect at two points and both circles have their centre point inside the other.

That should look something a bit like this.

And if it's done accurately, the distance between the two centre points should be greater than two centimetres, but less than five centimetres.

And finally, E, the circles do not touch.

Both circles have their centre point inside the other circle.

That should look something like this.

And if you measure the distance between the two centre points, it should be less than two centimetres if done accurately.

You're doing great so far.

Let's now move on to the second learning cycle, which is drawing circles that intersect exactly once.

Here we have Laura.

Laura draws two circles.

One circle has a radius of four centimetres and the other circle has a radius of eight centimetres.

Here's what her drawing looks like.

She says, "My circles touch each other exactly once." How far apart are the two centre points? Pause the video while you think about this and press Play when you're ready to continue.

Well, let's take a look at the two radii.

One radius is eight centimetres and then the other radius is four centimetres.

So to get from one centre point to the other, we're gonna travel eight centimetres first to get to the point where they intersect, and then we're gonna travel another four centimetres to get to the other centre point.

So altogether, they must be 12 centimetres apart.

And here we have Jacob.

Jacob wants to redraw the circles below, so that they touch exactly once.

How far apart should he draw the two centre points? Pause the video while you think about this and press Play when you're ready to continue.

Well, it'll be easier to see this if the radii are horizontal because that way they're going towards each other.

And then we can see then if we put these two circles so that they intersect exactly once, then the distance between the centre points will be six and four or four plus six to make 10 centimetres.

However, this is not the only way for these two circles to intersect exactly once.

We could have it, so that one circle is inside the other like this.

And in this case, we've got the distance from the centre to the edge of the large circle is six centimetres.

And the distance from the centre to the edge for the small circle is four centimetres.

So the two centre points must be six, subtract four is two centimetres apart.

Here we have Aisha.

Aisha draws two points that are nine centimetres apart.

She wants to draw a circle around each point with them touch each other exactly once.

What radius could she use for each circle? Pause the video while you think about this and press Play when you're ready to continue.

Aisha could use any pair of positive numbers that sum to nine.

For example, a circle that has a radius of six centimetres and a circle has a radius of three centimetres or maybe five centimetres and four centimetres or even decimals would work as well.

If they sum to nine, she'll create an image that looks a bit like this, with the two circles intersecting exactly once and the centre of those circles will be outside each other.

But she could also have a situation like this, where one circle is inside the other.

And to create something like this, she needs to use any pair of positive numbers with a difference of nine.

For example, if the large circle had a radius of 12 centimetres and a small circle had a radius of three centimetres, that would work too.

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

The two centre points in each diagram are 10 centimetres apart.

All the lengths are given in centimetres on these diagrams. In which diagram could the measurement be correct? Is it A, B, or C? Pause the video while you make your choice and press Play when you're ready for an answer.

Okay, the answer is B, 'cause we can see we've got four and six makes 10.

So here's another one.

Once again, the two centre points are 10 centimetres apart and the lengths are given in centimetres.

So which diagram could the measurements be correct? Pause the video while you have a go at this and press Play when you're ready for an answer.

The answer this time is C.

If you thought A, you might be thinking about adding the four and six together, however, because we've got one circle inside the other, the distance between those centre points is the difference between the radii.

And then in B, you might just be thinking about the fact we've got the 10 there perhaps.

But in C, one radius is 14 and the other one is four.

And because they're inside each other, the difference between 14 and four will be the distance between those two centre points.

It might be easier to see that, if for part C you imagine the radius of 14 going upwards from the centre point rather than downwards.

And one more question.

How far apart would the centre points of these two circles need to be for the circles to touch exactly once? Pause the video while you work this out and press Play when you're ready for an answer.

Well, there are two answers you could give for this.

You could do eight plus five to make 13.

So it can be 13 centimetres apart.

And it will look like the image on the left.

Or you can do eight, subtract five to get three centimetres.

So it'd be three centimetres apart, like the one on the right.

Okay, over to you now for Task B.

This task contains three questions and here is question one and two.

In question one, for each question, draw two circles, one with a radius of five centimetres, and the other one with a radius of seven centimetres.

Mark the positions of the centres each time.

Perhaps do that before you draw the circles, and draw each pair of circles according to the following conditions.

In part A, it says he wants 'em to touch exactly once with neither circle having its centre point inside the other.

And part B, the circles need to touch exactly once, with one circle having its centre point inside the other.

And then in question two, draw two points that are 12 centimetres apart and then draw a circle around each of those points so that they touch each other exactly once.

And then I want to write down what is the radius of each circle.

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

And here is question three.

We've got a complex diagram here.

We have points A, B, C, and D, which all lie on the same straight line.

And you've got a circle around each of those four points.

What you need to do is draw an accurate version of this diagram.

Pause the video while you do this and press Play when you're ready to check your answers.

Okay, how did we get on with that? Let's check now.

In question one, to check our answers, we'll have measured distance between the two centre points in our pairs of circles.

In part A, the circles touch exactly once and neither has its centre point inside the other.

So the image should look something like this.

Could be in a different orientation, that's fine, but the distance between those centre points should be exactly 12 centimetres.

And in part B, the circles touch exactly once and one circle has its centre point inside the other, that should look something like this.

And the distance should be exactly two centimetres between those two centre points.

And then in question two, you had to draw two points that are 12 centimetres apart, and then draw a circle around each point so that they touch each other exactly once, and then write down what is the radius of each circle.

Well, once you've done that, you could've had any pair of positive numbers that sum to 12.

For example, six and six as your radius.

Or you could have any pair of positive numbers with a difference of 12.

For example, 14 and two.

So check your answers by either adding them together if your two circles are kinda side-by-side or find a difference between them, if your circles have one inside the other.

And then in question three, you had to make a accurate version of this diagram.

To check your answer, measure the distance between point A and point D.

If you've done it accurately, those should be 17.

5 centimetres apart.

Fantastic work today.

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

A pair of compasses could be used to draw a circle with a specific radius, but also we could use a pair of compasses to draw lots of circles on the same page.

In some cases, they'll intersect twice, some cases they won't intersect at all, and in some cases, they'll intersect each other exactly once.

And if you're finding using a pair of compasses tricky at this point, that's absolutely fine.

It's perfectly normal when you first start using a pair of compasses to find it a little bit tricky.

But the more we practise with them, the easier we find it.

Thank you very much.