Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on finding a formula.

For today's lesson, you'll need a pen and paper or something to write on and with and a calculator.

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

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

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

Okay time for the try this task.

So remember last time when I showed you the demonstration of how we could get an area of a circle, well, I want you to use what we looked at then.

Okay can you use this idea to think about what is the same and what is different about each student responses? Why did they all give the area? So pause the video and have a go, pause in three, two, one.

Okay, welcome back.

So, in statement here, half the circumference, what can you say how that is half the circumference? So half the circumference, that's write that down though.

Half of the circumference times by half of the diameter.

So you can think of that as half of pi times diameter, which is just the conference times a half times the diameter.

Okay.

This pi r times r , what's pi r, which is half of the circumference.

So that there, we could think of that as pi r Okay.

And times the radius and the radius is just half the diameter.

So I guess they give us the same thing.

And what about this half pi d so half pi times diameter, times the radius.

Well, that's kind of like what happened here.

So we can say that half pi d is equal to the same as pi r 'cause half the diameter is the radius.

So we end up all getting the same responses.

So even though these are different ways that you say it, and you might remember it, they all get the same thing.

Okay, they all give the area of the circle.

So whichever one helps you remember it, remember that? Okay.

Okay now another one that people like as people like to think of pi r times r as pi r squared.

So some people like that.

And if you think that's easier to remember than that, well then remember that, but if you think that's easy to remember, then remember that.

So whatever makes sense to you from this diagram, try and use that one.

Okay.

So find the missing length to one decimal place.

Ah, so this time where our are is not in terms of pi, so we're not going to get, a nice value of x.

So we've got area, which is pi r times r which is 18.

So now we're going to divide both sides by pi.

So that gives us r times r on this side, they simplify to one and here, well I have to use a calculator, So I'll do 18 divided by pi, I get a really long number 5.

729, and that number is infinitely long.

So when I have a big long number like this, we'd like to keep that on our calculator, we don't like to cut it short to two or three decimal places.

So now here I've got our times up, which is the same as r squared.

So to get to just r, I need to square root it.

So I only do this because that is equal to r squared, so to undo the squared, I square root it.

So that just gives me r and now, because I've kept that number on my calculator, I can just do the square root of ans.

And that gives me 2.

392, ops sorry, one decimal place that gives me 2.

4 and the unit is centimetres.

Okay.

So that's really important that we keep it as ans for as long as we can.

And if you can't do the answer on your calculator, try and have like four or five decimal places.

Okay, so your turn, calculate that the missing length when the area is 36.

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

Okay welcome back.

Now, here we should have had area equals pi r times r and that is equal to 36.

So we divide both sides by pi and we get, and you know what, I'm going to write this as r squared here, r squared equals, well what's 36 divided by pi 11.

459155, duh duh duh duh Okay.

Now what I'm going to do is I'm going to square root both sides, and that gives me r so square root of ans.

And I get to one decimal place 3.

4 centimetres.

So really I don't know if you've got that.

Now, somebody who's might have thought that this radius would have doubled because the area doubled, well, it doesn't, it doesn't quite work like that with area.

We're not going to explore this too much, this lesson, but I'm just going to tell you, this is all to do with scale factor.

Okay.

What would actually have to do to get the area doubled? Sorry, what would actually have to do to get the radius doubled is our area four times bigger.

So you might want to check that now, you might want to check what the radius would be of a circle with area 72, and you will learn why that works in the future.

Okay, this time find the missing length to one decimal place.

And we've got the circumference, not the area.

Okay, so we've got this circumference, not the area.

So as I follow through circumference, well, we can think our circumference is pi times diameter.

So to find out what the diameter is you to divide both sides by pi, we get diameter equals.

so as I dial on my calculator 18 divided by pi, 5.

729 et cetera.

Now I am not going to remove that number from my calculator because I need to find the radius because this is the radius and not the diameter.

So I divide that by two to find the radius, and because that equals two.

So I divide by two, so I divide this number by two, and that gives me, ans divided by two is 2.

to decimal place 2.

9 and that's centimetres.

Okay so what did we do? we wrote our formula for circumference, we divided both sides by pi, then we've got the diameter, so we divide by two to find the radius.

So your go pause the video, and have a go in three, two, one.

Okay, welcome back.

Hopefully you did this equals 36 36 divided by pi is 11.

459 duh duh duh and then you divide that by two to find the radius, radius equals 5.

7.

Now you can't actually see it here because I've rounded this and I've rounded this, but the radius now is actually double because that wasn't exactly 2.

9, and that wasn't exactly 5.

7, what's happened to my radius is my radius is doubled.

So if my circumference doubles my radius doubles, that's really strange.

Why does it happen for circumference but not happen for area? And this is all to do with scale factors because with circumference we just times in one length whereas areas we times in two lengths.

So area, the scale factor is different, and I don't want to go into too much know now, but just know it's different for area than it is for perimeter.

Okay so given the circumference, find the area hoof quite a tricky one this, isn't it? So we have the circumference or pi times diameter equals 28 pi.

So what is the diameter? Divide both sides by pi, so we get our diameter is 28.

So now that means, and this just means implies, so that implies our radius is 14.

So if our diameter is 28, that implies that our radius is 14.

So now we can find the area 14 pi times 14, and I do not know 14 squared I'm sorry.

So that is 196 pi 196 pi centimetres squared.

Okay so well done, if you know your faulting times table better than me.

Okay so your turn.

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

Okay welcome back.

Hopefully you found this.

So divide by pi to get the diameter is 14, which means or implies that the radius is seven.

So the area of that circle is seven pi times seven, or seven squared, if you want to think of it like that, 49 pi.

Okay so really well done if you got that correct.

Okay so what is different this time? Well, this time we've got the area and we want the circumference.

So we've got the area, which is pi r times r and that is equal to 1681 pi.

So we divide both sides by pi, which gives me r times r which I'm actually going to write as r squared here.

So r times r is r squared.

So r squared is equal 1681.

So I square root both sides to find r, that gives me 41.

So that means our radius is 41, but we have not finished halfway.

We need to find this a circumference.

So the circumference is pi times diameter, so two times the radius gives us 82 which is our diameter.

So our diameter, and I'll just draw an arrow to let you know that I've gone over here.

Our diameter times by pi gives us a circumference of 82 pi.

So your turn, pause the video and have a go, pause in three, two, one.

Okay welcome back.

Hopefully you got this.

Divide by pi r times r is r squared.

So I'm going to write it like that, So square root of 289, I think it's 17, but let me check 289, 17 yeah.

17 equals r.

So now we've got our radius.

So to find our circumference, we need to do pi times diameter.

Diameter is double the radius, so it's 34.

So we get, and let's write that like that 34 pi.

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

Okay so now it's time for independent tasks, so I would like you to pause the video and have a go resume once you're finished.

Okay, here are my answers.

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

And I just want to quickly draw your attention to four and five.

So on this question, it says a bike wheel with diameter 58 centimetres spins as it travels one kilometre, how many full turns does it do? So it made 48.

8 turns, but we rounded down because we wanted to know how many full turns, so that.

8 of a turn is not a full turn.

So that's why it's only 448 full turns.

And on the last one, he needs 9.

503 bottles to cover the lawn, but you can't go out and buy.

503 of a bottle.

So he has to buy 10, so in that situation we round up, so well done if you notice that, I imagine quite a few people would have made mistakes or errors at that point.

Okay so now it is time for the explore task.

So the area of the first grey circle, that little circle there is pi, the area of this circle, the teal dotted circle and the grey circle is four pi.

So that means that the area of the teal circle, is three pi.

Why is that? Because four pi takeaway pi is three pi.

Okay so how many areas can you find? So I want you to pause the video to complete your task and resume once you've finished.

Okay so here is what I found going from that inner one to the outer one, we get pi three pi five pi seven pi nine, 11, 13, 15, 17 pi, 19 pi.

What's going on there? They're all increasing by two pi, they're all odd numbers.

Why, okay what would happen? What sequence or what pattern would we get if we did two rings at the time.

So if we did the grey and the teal, and then we did the teal and the blue, and then did that one and that one and that one and that one, what would we get there? What is the radius of each one? What if the radius was two? How would that change? Would our pattern change? Would it stay the same? What if it was three? Would our pattern change? or would it stay the same? Well, two rings or what, how many different ways could we combine rings to get an area of 20? So there's loads of questions that you could think of here.

And I guess it is a case of just going to explore a bit more, maybe you've worked out some of these already, maybe you found out some really interesting stuff.

There are just a few ideas that I had that you may be interested in.

Okay so that is all for this lesson, thank you very much for your hard work, and I look forward to seeing you next time.

Thank you.