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Hi there and welcome to another next lesson with me, Dr Saada.

In today's lesson, we will be looking at the constant of proportionality.

For this lesson you will need a pen and a paper, so go grab these and when you're ready, we can make a start.

The cost of this chain is directly proportional to its length.

Complete the table for the cost of the multiples of five.

Create and complete table for the cost of multiples of two.

You have been given here a table and it tells you the relationship between the length and the cost.

Part of that table is missing and you need to complete that.

Just try this task, it should take you about seven minutes to complete.

Please pause the video and complete it, resume once you're finished.

How did you go on with this task? Really good.

So these are my answers for the first part.

Can you please mark and correct your work.

For the second part, create and complete the table for the cost of multiples of two.

Well first I need to know what the cost of two is going to be.

So I know that five centimetres cost 15 pence, so I can use the unitary method to find the cost of one centimetre.

One centimetre is going to cost the three pence.

Now I can use that information to help me construct and complete table of multiples of two.

And this is my table for multiples of two.

So two centimetres cost six pence, four centimetres cost 12 pence, six centimetres cost 18 pence and so on.

Please pause the video if you need to mark and correct your work.

Let's have a look at this question here.

The price of posting a parcel is proportional to the mass of the parcel.

What does this mean? When proportional means that one quantity is a constant multiple of the other.

So if one is zero, the other is zero, if one doubles the other quantity doubles.

If one triples, the other triples.

If one halves, the other halves and so on.

Calculate the cost of sending the parcels below.

And we'd been given some parcels.

We know the mass of these parcels.

We have only been given that that parcel that is three kilogrammes, cost 18 pounds.

So we are going to do this by constructing a table.

It has the mass m in kilogrammes and price p in pounds.

I have converted here everything into kilogrammes, because we don't want to be mixing between the units.

You want to be consistent with one of them.

So I decided to go for kilogrammes.

It would have worked if I went for grammes as well.

And I wrote down underneath three, 18, because that's how much it costs to post that parcel.

Now, what have I done from three to get to 18? I multiplied by six.

So really I did m, the three, multiplied by six and that gave me p.

So m times six equals p.

I can even write it as p equals six m.

Now that six there, that I'm multiplying by, is always going to be the same for any given parcel.

And that is the constant of proportionality.

That number there, number six.

So therefore I multiply this by six, I know that they'll cost me 60 pence to post 100 grammes parcel.

For 500 grammes it would cost me three pounds.

For 8.

5, if I multiply that by six, it'll cost me 51 pounds.

And for 15 kilogrammes, it cost me 45 pounds.

So what's really really important here to note, I can write the relationship, between the two quantities here.

So between the two variables, between the mass and the price as an equation.

I can say that the price p is equal to six lots of m and that six lots of, that's the constant of proportionality that we're looking at today.

Now that number doesn't always have to be a whole number.

Let's have a look at this example.

The cost of ice cream is proportional to the number of ice creams you buy.

Without calculating the cost of one ice cream, calculate the cost of two, eight, and 12.

Again, now it said clearly without calculating the cost of one ice scream.

So we're not going to go and use the unitary method in this case.

To start with, we're going to assume that there are no offers for buying more ice cream.

It's directly proportional.

We're not getting these from a pack.

This is a question that I just made up, just for this lesson, okay.

Now I can start by creating a table and say number of ice cream and the cost.

Two ice creams, I don't know how much they cost.

Five, I know that they cost seven pounds, eight and 12.

Now I need to think from five to get to seven, what have I done here? I multiply by seven out of five, 7/5.

Or if you prefer to use decimals, multiply by 1.

I prefer to use fractions.

Now, what this is telling me that the constant of proportionality is 7/5, okay or 1.

So I need to multiply the number of ice cream by that number, by that constant in order to get the cost.

So I can go and say, while two multiplied by seven out of five or by 1.

And that gives me 2 pounds 80.

8 multiplied by 7/5 is 11.

20 and 12 multiplied by 7/5 is 16 pounds 80.

I've now calculated the cost of ice cream without having to calculate the cost of one, okay? I use the constant of proportionality.

Now, if I tell you that the number of ice cream can be written as X and the cost can be written as Y, can we write maybe any equation to show that relationship between X and Y? Or how do we get from X to Y? To can get to Y we need to have X, and then we need to multiply X by 7/5.

So I can say when X multiplied by 7/5 is equal to Y, therefore you can write it as Y equals 7/5 of X or 7X out of five.

So you can write it as Y equals 7/5 of X or Y equals 7X out of five.

And that 7/5 is the constant of proportionality.

And as you saw here, it does not have to be a whole number.

In our first example, it was an integer and here it's not, okay? And now if I draw a graph to show this, the gradient of my graph is going to be 7/5.

And remember in previous lesson, we looked at graphs of direct proportion, and we said that the gradient was really really important.

Well, the gradient tells us the constant of proportionality.

It is time for you now to have an independent task.

You've three questions to complete.

Please read the questions carefully and do this to the best of your ability.

The independent task should take you about 10 minutes to complete.

Please pause the video, resume once you finished.

Welcome back, how did you get on with this task? Really good.

Come on let's mark and correct the work together.

Question one you're told that three chocolate bars cost two pounds 55, seven chocolate bars cost five pounds 95, and you needed to fill in the gaps in the arrows.

Now, the question has already converted everything into pence for you.

So 255 divided by three, that gives you 85.

So you know that the cost of one chocolate bar is 85 pence.

So to get to the 255, you needed to multiply three times 85, to get to that five pounds 95.

You needed to multiply seven by 85.

Now, what if I tell you I want to buy 12 chocolate bars.

It has to be 12 times 85 equal, and the answer to that bit is 1,020 pence.

It's not showing up.

There we go.

And if I have n chocolate bars, they cost m pence what's the relationship? Well, I'm going still to multiply the n by 85, just like we did in the previous two examples in the tables.

So I can say that m is equal to 85 lots of n, 85n.

For question number two, you were given a table that had the number of mugs and the costs, and you had some of the numbers there.

Now, how do you get from four to one pound 60? We multiply by 0.

Now, if I think about the number of mugs as m and their cost as c, I can write the equation 0.

4m equals c.

to get to c I have to multiply m by 0.

4, and we'll have to use this because the question made it really clear that we need to do this without finding the cost of one.

So we need to use the constant of proportionality.

Now, if I put two pounds 80 into that equation and rearrange to find m, I would find that m is equal to seven, nine, and then 10 in that table.

You could have also done that by just looking at the table and saying, well, to get from one pound 60 to four, I need to divide by 0.

So you divide everything by 0.

Last one, the number of hours Alex works is directly proportional to the pay he receives.

Can we complete this table? Well, if you look at this relationship here, to get from 17, from the number of hours to the pay 212 pounds 50, you multiply by 12 pounds 50, which tells you that the constant of proportionality is 12 pounds 50.

If I have letters for the variables here, so for hours worked if I have h, and for pay if I have p, and I want to write this as an equation, I need to say that total hours worked at p, p is equal to 12 pounds 50 times the number of hours.

So I can write it as p is equal to, and that should have said 12.

So let me come back to this.

Well, 50 multiplied by h, and now I can use that formula, rearranged to find h.

And these are the answers.

Did you get all of this correct? Really good job, well done.

This brings us to the explore task, the relationship between the height of these shapes and their perimeter is directly proportional.

Fill in the tables.

The first table is given to you about a regular triangle.

Regular means all the three sides are equal.

So the regular triangle, the height and the perimeter, the perimeter being rounded to two decimal small places.

The second shape is a regular pentagon.

And the last one is a circle.

Can you estimate the perimeter of a heptagon with a height of one metre? What about a nonagon? if you're feeling super confident, please pause the video and make a start.

If not, I'll be giving you a hint in three, in two, and in one.

So the question is saying that the perimeter is directly proportional to the height of the shapes.

So think about the table.

Think about, to get from the height to the perimeter, what am I doing? I'm not supplying by 3.

461, what do I need to do to the rest? What is that constant of proportionality? Now, if you don't want to do it this way, and you find it easier to do it this way, you also can do that from one to two I doubled that.

So if I double the height what's going to happen to the perimeter? By multiply by five, what's going to happen? Explore task should take you about 10 to 15 minutes to complete.

I would suggest that you do some research about regular heptagon and regular nonagon, and about their height and their permitter.

Please pause the video and complete the task.

Resume once you finished.

Welcome back.

How did you get on with the explore task? Do we have the same answers? Okay.

There are so many ways that you could have answered this question.

I use the constant of proportionality to answer the questions here.

So I looked at the relationship between the height and the parameters, and I wrote down the formula that connected them.

And then I used that to help me find the perimeter, in every case.

Did you research the perimeter of a heptagon with a height of one metre? What did you get? Really good.

So I did some research too and I found that the perimeter of a heptagon with one metre is equal to 3.

20 to two decimal places.

I meant to estimate it before doing the research, because I looked at the tables that we had.

And I found out that for a triangle, I saw a shape with three sides.

The perimeter was 3.

46, for the pentagon five sides, the perimeter reduced and became 3.

25.

So I knew that mine is going to be for a heptagon, less than 3.

25, and I was correct.

Did you find that for a nonagon? I would love to see your answers for this task.

This brings us to the end of today's lesson.

Well done on all the fantastic learning that you have done.

Please remember to complete the exit quiz, to show what you know.

Enjoy the rest of your learning for the day.

I'll see you next lesson.

Bye.

I've finished the video