Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on fractions of a circle.

For today's lesson, you'll need something to write on and some thing to write with, and a calculator.

If you can take a moment now to clear away any distractions, including turning off any notifications.

And if you can, try and find a quiet space to work where you won't be disturbed.

Okay, when you're ready, let's begin.

Okay, so time for the Try This task.

So you are told that the circumference of this circle is 60 centimetres.

Now you need to estimate these lines here.

How long do you think they are? So pause the video and have a go.

Pause in three, two, one.

Okay, so I think this one is about 1/4 of the full circle.

So I think this one is about 1/4 of 60, 60 divided by 4, 15.

This is about 1/2 of 60, so 1/2 of 60, 30.

I'd probably say this is about 1/3, and you might've thought it was slightly different, and it looks maybe a tiny bit less than 1/3, but I'd say it's close enough to 1/3, so I'd say 1/3 is 20.

So if you have something around 20, or maybe a bit less, I'd say that's pretty good.

This looks about 3/4 to me.

So if that's 3/4, 1/4 was 15, so 3/4 is 60 minus 15, 45.

Hmm, what about this one? It's pretty tiny, isn't it, this one.

Well, if you think something like that is about, that there is 1/4.

So that's 15.

I'd say maybe that's a bit less than halfway, or that's about halfway there, so I'd say this is around six or seven centimetres.

So maybe 1/10 of a circle.

Now, if you thought it was slightly different, don't worry.

Well it's just, it's getting similar answers to my answers.

So really well done.

Okay.

What fractions of the circle do you see here? And use the degrees to help you.

So you may wish to pause the video and work this out.

Okay, welcome back.

Hopefully, you paused this and you saw that this was a whole.

This was 1/2, 1/4, 3/4.

Hmm, what about this one? How many 60 degrees go into 360? Six, so it is 1/6 of the circle.

How many 120s go into 360? Three, so it is 1/3 of the circle.

Now, you could have got 2/6 for this, because that's double that, and 2/6 is correct.

because 2/6 is just the same as 1/3.

Okay, what about this one? Well, this one is 60 less than a full circle.

So 60 less than a full circle is 5/6 because 60 is a 1/6.

So 6/6, a whole, take away 1/6, 5/6.

Hmm, how many 30s go into 360? How many threes go into 36? 12, so it's 1/12.

And you might have used your answer for this one.

This one being 1/6.

So half of 1/6 is 1/12.

You might've used that one to help you there.

Okay, so really well done if you've got that correct.

Now we're going to use these fractions.

Imagine the full circle has a circumference of 60.

Can you work out these lengths here? So the arc lengths.

So you may wish to pause the video and have a go.

Pausing three, two, one.

Okay.

Welcome back.

So this one is 60.

This one is half of 60.

1/4 of 60.

3/4 of 60.

Okay, if 1/4, 60 divided by four, is 15.

Well, this one is 15 times three, 45.

Well, this one, this is 1/6 of the circle.

So that's 10 centimetres.

This one is 1/3 of the circle.

So that is 20 centimetres.

This one is 5/6 of the circle.

So 60 take away 10.

And this one, well, that's 1/12, so that's half of that one, or 60 divided by 12, which gives me five centimetres.

Okay, really well done if you've got those correct.

Now we're going to be using this idea, but we're going to be looking at more complicated fractions.

So what fraction is the shaded sector? Well, how many degrees are in a circle? Well, it's 72 out of 360.

Now we do not need to simplify that, okay.

We can just say it's 72 out of 360.

And now, we know the fraction, so because we know the fraction, we can find out the arc length of the shaded sector.

So we can find out this length here.

So how do I work out the circumference of the full circle? Well, we need the diameter times pi.

Hmm.

We don't have the diameter.

We have the radius.

What would the diameter be? 10, so we do 10 times pi, which equals 10pi.

And now we do the fraction of the circle times by the circumference of the full circle.

And that gives you your answer.

Now we wouldn't get something particularly nice here in terms of pi.

So what I'm going to do is I'm going to work this out on a calculator, and you can use the fraction button to help you.

72 over 360, times by 10pi, and that gives you the answer of 6.

28.

Okay, and that's my answer to two decimal places.

So your turn.

Work out the arc length of that sector.

Pause the video and have a go.

Pausing three, two, one.

Okay.

Welcome back.

So hopefully, you did 113 divided by 360, times by 10 pi, which should give you 9.

86.

Really well done if you've got that correct.

If you didn't, maybe just check your calculation and check you didn't make a mistake.

So it was 113 divided by 360, times 10pi.

Okay, so sometimes for areas of sectors, you might not get the full circle but we always still need to work out the circumference of the full circle.

So the circumference of full circle equals, so the radius is six, so the diameter is 12.

So it's 12pi.

Okay, now what fraction do we have? Well, we have 102.

That should be a two, sorry.

Out of 360.

So we do, what fraction of the circle we have, times by the circumference of the full circle, and that gives us our answer.

So 102 divided by 360 times 12pi, and we get 10.

68 to two decimal places.

Okay.

So your turn.

Work out the arc length of this sector.

Pause the video and have a go, in three, two, one.

Okay.

Welcome back.

Hopefully, you did this calculation.

Hopefully, you found that the circumference of the full circle equals 16pi.

So then you did 220 over 360 times 16pi, which gives you the answer of 30.

72 to two decimal places.

Okay, so really well done if you got that correct.

Okay, so now it is time for the independent task.

So pause the video and have a go, and resume once you've finished.

Okay.

Welcome back.

Here are my answers.

You may need to pause the video to mark your work.

Okay, so now it's just the Explore task.

So we have this grey line is 30 centimetres and you need to work out the length of each of the colours.

So I'm just going to flick through the colours.

So we have the green, like that.

The blue, like that.

The pink, like that.

And the turquoise? Yeah, the turquoise, like that.

Okay, so pause the video to complete your task and resume once you've finished.

Okay, so hopefully you got that they were all 15pi.

Wow.

That's interesting, isn't it? That's kind of like last time when they were all the same.

So what can you say about arc lengths then? What kind of generalised statement can you make, because of this, and because of what we did last time? What if it wasn't a semicircle? What if it was 1/3 of a circle or 1/4 or a circle? How might that look? Would that still have the same pattern? Hmm.

So that's something for you to think about.

So that is all for this lesson.

Thank you very much for all your hard work, and I look forward to seeing you next time.

Thank you.