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So welcome to your 10th and final lesson in the coordinates and shapes topic.

Today, we'll be learning to solve problems involving circles.

The equipment that you will need is a pencil and piece of paper, a piece of string, and a ruler.

Pause the video and get your equipment together if you haven't done so already.

So for today's lesson, we're going to solve problems involving circles, relationships between the circle parts, then solve problems involving circles before moving on to an independent task and then a final, final knowledge quiz.

We'll start by looking at the relationship between the parts of a circle.

So think back to yesterday's lesson and also using the clues on this circle here, what is the relationship between the diameter, which is the measurement going all the way across from boundary to boundary through the centre, and the radius, which is going from the centre to the boundary of the circle? That's right.

The relationship is that the diameter is double the radius.

So the diameter is the radius times two, and the radius is half of the diameter, so the radius is the diameter divided by two.

So now I'd like you to pause the video and construct a circle with a radius of five centimetres.

So remember, measure your string so that the distance from the centre to the pencil is five centimetres.

And then carefully draw your circle.

Great work.

Now, we're going to use that circle in our next part of the lesson.

So we're going to look at the relationship between the circumference and the other measurements.

So remember the circumference is the distance all the way around the boundary of the circle.

So first of all, just check your measurements of your circle.

It doesn't matter if it's not exact.

We just want roughly a circle with a diameter of 10 centimetres and a radius of five centimetres.

So just roughly check that, and then write your measurements on the circle like I've done.

So now you're going to measure the circumference of the circle and you can do this, as we practised yesterday, carefully by placing a piece of string around the boundary of the circle.

And then once you have put the string all the way around, you measure the string from the start of it to where it met the beginning again around the circle, and then measure that using your ruler.

And it might be helpful to have another person there to help you hold the string.

So you should have found that your circumference was roughly 30 centimetres.

So this squiggly equals means approximately.

So the circumference of this circle is approximately 30 centimetres.

So now we can start to think about the approximate relationship between the other parts of the circle.

So think about that the diameter is 10 centimetres and the circumference is 30 centimetres approximately, so what has happened to 10 to get to 30? Well, we all know 10 times three equals 30, which means that the circumference is roughly or approximately equal to three times the diameter of the circle.

And if it's roughly three times the diameter, then it's about six times the radius of the circle.

So let's use that to help us answer some questions.

So if the circumference is approximately, and that is the key word here, because this is not absolutely accurate, it's an approximation.

If the circumference is approximately three times greater than the diameter, let's find the approximate circumference of the circles below.

So circumference is approximately equal to the diameter times three.

So use this equation to find the approximate circumferences of these three circles.

Pause the video while you make some notes.

So for the first one, we already had the diameter.

So to find our approximate circumference, you were to do seven times three, which gives you an approximate circumference of 21 centimetres.

The next one was slightly more complicated.

So either, you could have done the radius times six, because we know that circumference is approximately equal to the radius times six, or you could have found the diameter by doing the radius times two, and then multiplying that by three.

Either way, you should have had an approximate circumference of 36 centimetres.

Again, for your third one, you have the radius.

So you could have either multiplied your radius by six or multiplied it by two to get the diameter and then multiply that by three to find the circumference.

And you would have come to the approximates circumference of 78 centimetres.

Now we're going to move on to solving problems with circles.

So this is going to be up in the corner to help us to remember.

To find the diameter, we multiply the radius by two, and to find the radius, we divide the diameter by two.

So Chloe invents a board game and makes a spinner.

The diameter of the outer circle is 12 centimetres.

The inner circle has the diameter of 1.

2 smaller than the outer circle.

She makes an arrow spinner, which is the same length as the radius of the inner circle.

How long is the arrow? So we're being asked to find what is the radius of the inner circle.

So we'll start with what we know.

We need that the diameter of the outer circle is 12 centimetres.

So we annotate that onto the diagram.

Then we also mean given the information that the inner circle has a diameter, which is 1.

2 centimetres smaller.

So I know that 12 centimetres subtract 1.

2 centimetres is equal to 10.

8 centimetres.

So the diameter of the inner circle is 10.

8 centimetres.

And I've been asked, what is the radius of the inner circle? So radius equals diameter divided by two.

So I'm going to do 10.

8 divided by two, which is equal to 5.

4 centimetres.

So the radius of the inner circle is 5.

4 centimetres.

So that is the length of the arrow that she needs to make her spinner.

Let's do another circle problem together before you go on to do some independently.

This time you're using coins.

Now, remember, that a coin is actually a cylinder because you can measure the depth of a coin, you can measure its thickness.

We're looking at the circular face of the coin.

And this is a two pence coin and a two pence coin has a radius of approximately 13 millimetres.

So Metila makes a line of nine two pence pieces with no gaps between the coins.

And we want to know how long is the line of coins.

So let's start with what we know.

We know that the radius of the 2p is 13 millimetres.

But we need use the diameter as the measurement from boundary to boundary for this question.

So we know that the diameter is two times the radius.

So the diameter is 26 millimetres.

Now with a problem like this, it's helpful to do a sketch.

So we know we're looking at a two pence piece, which is 26 millimetres across, and we have a line of them, where there's nine coins with no gaps in between.

So if we put that together as a diagram, we've got all nine two pence coins, which are 26 millimetres in diameter.

So to find the length, either we need to repeat, do repeated addition, adding 26 together nine times, or we do 26 multiplied by nine, and you can use column method for this.

And your answer is 234 millimetres.

I hope it's now fine for you to complete some independent learning.

Pause the video to complete the task and click restart once you're finished.

In question one, this is another question about rows of coins, so we know what we're looking for.

Luke creates a line of five pence pieces and it is 10.

8 centimetres long.

So we want to know how many five pence pieces has he used.

So if we think about which measurement we're interested in, we're looking at the diameter.

If the radius was nine millimetres, the diameter is 18 millimetres.

So we want to know how many lots of 18 millimetres are there in 10.

8 centimetres.

Now, it's always helpful to be in the same units.

So I'm going to convert these to 108 millimetres.

So I want to now find out how many lots of 18 millimetres in 108 millimetres by doing 108 divided by 18, which gives me six.

So he has used six five pence pieces.

In question two, you were asked to use your knowledge of circles to solve these problems. So, first of all, what is the length of the column of coins and the row of coins? In your table, you were only given the radius.

So you needed to make sure that you calculated the diameter in order to find the length of the column and the row.

The length of the column all the way down the column was 122 millimetres.

And the length of the row all the way across was 117 millimetres.

So you were finding the diameter of each coin and then adding them together.

In the second question, three coins of equal value were added to the left of the two pence piece.

So here, to increase the row, the length increased by 5.

4 centimetres.

So what coin has been added? Again, I like to work in the same unit.

So we're looking at 54 millimetres.

So we're thinking which three coins of the same value would add together to get 54 millimetres.

So I know it can't be 1p because three times 20 is 60.

I know that it can't be 2p because three times 26 is 78.

And I know that it is 5p because three times 18 is 54.

So this is three lots of 5p that have been added on.

In the last part of the question, four coins were added in the positions marked with xs on the diagram, where the new row is now 9.

9 centimetres, and you needed to think about what coins could have been added.

So I'm going to convert this into millimetres, 99 millimetres.

So the row is 99 millimetres long, and we know that this 5p coin has a diameter of 18 millimetres.

So I'm going to subtract that, which means that these three coins, sorry, these four coins, their diameters needed to add together to make 81 millimetres.

And you may have found different combinations.

It was basically trial and error.

But one way that I found of doing it was to use a 5p, 5p, 1p, and 10p.

And you may have found a different combination using trial and error to approach that problem.

So this question, Youcef makes a line of five five pence pieces, okay? So he has got no gaps again between the coins and you need to work out how long is the line of coins.

Pause the video and work out the length of the line of coins.

So you know that the diameter is the measurement that you're interested in.

So you will have done the radius multiplied by two to give you the diameter of 18 millimetres.

And I hope that you drew a sketch to support your working out.

So you know that there was a line of five pence pieces with diameter 18 millimetres.

So you were looking at multiplying 18 by five to give you 90 millimetres.

For question three, you were asked to calculate the approximates circumference of each coin, remembering that the circumference is approximately equal to the diameter multiplied by three.

So we've been given the radius.

So we needed to use that to calculate the diameter.

say for the 10p, the radius was 12.

5 millimetres, so I can put that on here, and then doubling it to find the diameter, that's 25 millimetres, and multiplying it by three and it gives us the approximate circumference, which is 75 millimetres.

For a 2p, the radius is 13 millimetres.

So the circumference is 26 and multiply that by three gives us an approximate circumference of 78 millimetres.

For the 5p, the radius was nine millimetres.

So the diameter, sorry, 18 and multiplying that by three gives us 54 millimetres for the approximate circumference.

And for the 1p, the radius was 10 millimetres.

Therefore, the diameter was 20.

And multiplying it by three gives us an approximate circumference of 60 millimetres.

This question you were asked to find the approximate circumference of the entire target board.

So we need to annotate it with what we know.

The smallest circle has a radius of one centimetre.

So we'll add on that we also know that the diameter is two centimetres.

And the largest has a radius of 11 centimetres.

So then we know that the largest has a diameter of 22 centimetres.

So if we find the difference between the diameter of the largest circle and the smallest circle we do 22 subtract two gives us 20 centimetres, then we're looking at the difference here.

We're going to divide that 20 by one, two, three, four circles to find what the jumps in the diameter should be each time.

So if we divide 20 by four, then the diameter should increase by five centimetres each time.

And let's check if it works.

So if that was two centimetres, then the next greatest one, if we're going five centimetres greater should be seven centimetres.

Then the next greatest one, five centimetres greater, would be 12 centimetres.

Then the next one, through the centre would be 12 add five is 17 centimetres.

And then finally, 17 add five is 22 centimetres.

So they increase by five centimetres each time.

And then the, so that the diameter of the, all the way across, sorry, the greatest circle is 22 centimetres.

Now we were asked for the approximate circumference of the target board.

So if the greatest circle has a diameter of 22 centimetres, then the approximate circumference is 22 multiplied by three, which gives us 66 centimetres.

So for your final question you were asked whether it's possible to create a line of coins, exactly 12 centimetres long.

So again, I converted this to millimetres so that we're dealing with the same unit.

And I also added a column onto this table because I know that I'm using the diameters.

So the diameter is double the radius.

So for a 1p is 20 millimetres.

For a 2p, it's 26 millimetres.

For 5p, its 18 millimetres.

And a 10 is 25 millimetres.

So you were looking for combinations of coins to add together to get 120 millimetres.

So an example that I found was using two 10p, sorry, two 2p, and then two 10ps, and a 5p, and I'll show you why that works now.

So a 2p has a diameter of 26 millimetres, so 26 and 26.

A 10p has a diameter of 25 millimetres, so 25 and 25.

And then the 5p has a diameter of 18 millimetres.

So I know that 26 add 26 gives me 52.

25 add 25 gives me 50.

So already I'm on 102.

And then I add that 18, and that gives me 120.

Another way of doing it would be to use four lots of 10p, so that's 25 times four.

And then also adding on a 1p, which is 20, and 25 times four is a hundred, and add the 20 is 120.

So you may have found other ways of doing it, but they are two possible combinations that I've found.

Now it's time to pause your video and complete the final quiz, and then click restart once you're finished.