Lesson video

Hi there and welcome to another lesson with me Dr.

Saada.

In today's lesson we will be looking at direct proportion.

All you need for today's lesson is a pen and paper.

Pause the video, go grab these and when you're ready, you press play and we can make a start.

To start today's lesson, I would like you to try this.

Yasmin is buying some chain for a project.

Three hardware shops each have different deals.

First one shop A chain costs eight pence per two centimetres.

In shop B chain costs 40 pence to start and then two pence per two centimetres.

Remember per means for every.

Shop C chain costs 10 per two centimetres for the first 10 centimetres, then four pence two centimetres.

I've started three tables for you, one for each shop, copy and complete these tables, which shops should Yasmin buy her chain from? I want you to pause the video and have a go at these questions.

When you're finished press play so we can make a start.

So pause the video in three, two, one.

Now it is time for us to look at the solutions to the try this so let's mark and correct our work.

For the first table from shop one, the chain costs eight pence per two centimetres.

So if we look at the length, where it says two centimetres, I can put eight because that's how much it costs.

I don't need to write pence in there because it already said cost in pence.

So I can write the numbers down without the units.

Well if two centimetres cost eight pence, how much would four cost? Excellent double it 16 and therefore six centimetres cost to 24.

I had already filled that part for you.

What about eight? Really good, well done 32, what about 10? Excellent 40 and 12 is 48.

For 14, good job.

Trying to find the answers to these, you may have done it in so many different ways.

The first one was given to us.

So we would have done this from the information given in the question here.

To find four, you could have doubled the two so you could have thought about it like this, but I'm doubling this, doubling the length, so I need to double the cost.

You also may have thought about it as well, two to get to eight what have I done? I've multiplied by four.

So I'm going to do the same thing here four to find the cost, I'm going to multiply by four.

Not sure how many of you saw that relationship but it's something important to notice, okay? Some of you to work up eight would have ended up doubling the four.

And some of you to find that the six would have added the two plus the four, so added the eight plus 16.

So there are so many ways of completing this first table, and that's fine.

Because the chain cost is not changed, it's every two centimetre you pay this much.

But let's look at the second one.

The chain costs 40 P, to start with, so you're going to pay 40 pence plus two pence per two centimetres.

So to buy the two centimetres length, you need the 40 plus the two that gives us 42 pence and that's why we have 42 pence here.

Now to buy four centimetres, we still need to have the 40, we need to pay 40 pence, and then we need to pay two, two pence for two, and then another two, okay? So in total, we pay 44.

Number six, okay, for six centimetres, we still have the 40 plus two plus two plus two because we need six centimetres, which is 46.

Really good to get this correct.

And it was already done for you, for 10 it's 50, for 12 what did you get? Good job 52 , for 14, excellent 54 and for 16 what did you get? Really good job 56.

Now some of you may also have noticed that every time I'm buying something, I'm actually increasing by two, by two pence.

So that's another pattern that some of you may have looked at, and that's fine if you did it this way, okay? Now the last one shop C, chain costs 10 pence per two centimetres.

So every two centimetres I buy, I have to pay 10 pence for it.

For the first 10 centimetres, so only for the limited amount, so for 10 centimetres, then I start paying four pence for each two centimetres, it becomes more expensive.

So let's work this out.

If I buy two centimetres, guess if I buy two centimetres, how much would I pay? Really good, I would pay 10 pence for it 'cause it's at 10 for two centimetres.

If I buy four really good, I'm going to end up by paying 20 double it.

For six it's 30, for eight I've given it 40 and 50 for 10 good job.

So we will pay 50 for 10, okay for 10 centimetres.

Now for 12, excellent.

We it's been given.

So let's explain where did this come from? So to buy that 12, I need 10 centimetres and the 10 cost 50.

And then I need to pay for two centimetres.

But remember for two centimetres above 10 centimetres anything above 10, we need to pay four pence for the two.

So that's the 50 plus the four that this is equals to 54.

What about 14? What did you write down? Good job, 54, 58 and for 16,62 enter.

Now let's have a little chat about which shop should Yasmin buy the chain from? From A, from B or from C, what do you think? Okay, really good.

It really depends on how much she wants to buy in this case, okay? 'cause if she wants to buy anything up to eight centimetres, she's probably better of with shop A 'cause after eight centimetres, it's 32 here, up to eight, it's 48, eight is 46.

So it's cheapest to buy it from A.

If she wants to buy 16, where is it cheapest from? This is 64, this is 56 and here only got 62 pence.

So it's probably cheaper to buy the 16 from shop B.

So it really depends on what, on how much she wants for the project, so it depends on the amount, okay? That she needs for the project that she's doing, okay.

Each will going to get these tables correct, let's move on to our next task.

Which of these statements do you agree with? Shop A costs the same amount for every length.

So the cost and the length are directly proportional.

Shop B is all also charges the same amount per length.

So cost and length are directly proportional.

Or shop in shop C, the costs per length changes after 10 centimetres, so cost and length are not directly proportional.

Now in order for us to decide which statement we agree with or not, we need to do a bit more digging.

We need to really look at each statement and see what it really means, okay? So I'm going to take each statement on different slide and we're going to look at each of these independently and decide if we agree or disagree with each of these statements.

So let's make a start on the first one, Okay.

So what do we know about shop A? We know that shop A sells the chain costs eight pence per two centimetres in that shop again.

And we have this table that we worked out during the try this.

So it tells us the costs of two centimetres, four, six, eight, 10, 12, 14 and 16 centimetres length of chain, okay? So it tells us the cost of each of them, we've worked them out earlier.

Now, in order for us to have a deeper analysis of what this actually means, and whether we agree with this statement or not that shop A costs the same amount for every length, we need to do some sort of calculation to see whether these are directly proportional or not.

So I'm going to calculate the cost divided by the length.

The cost divided by the length is actually telling me what is pence per centimetres, so it's telling me how much money I spent for one centimetre.

So if we look at the first one, eight divided by four, by two is four.

So I pay four pence per centimetre, that's what it means.

16 divide by four is also four.

So I pay four pence per centimetre.

What so far this is telling me, is that I pay four pence four cents for one centimetre, what that I buy a two centimetre long piece or a four centimetre long piece? Is that always going to be the case, plus the length matter? Let's see, next one.

24 divided by six is also four, 32 divided by eight is also four.

40 divided by 10 is also four.

So so far, I know that I'm going to pay four pence per centimetre if I buy a two or four or six, eight or a 10 centimetre long chain.

What about if I wait a bit more? Okay, bit longer piece.

48 divided by 12 is also four.

56 divided by 14 is also four, 64 divided by 16 is also four and even if I carry on, it's always going to be four.

The cost divided by the length for this particular shop, is always going to be four.

I'm going to pay four pence per centimetre.

And the reason for that is that the chain cost eight pence per two centimetre That's it, it's a standard that isn't, or if you buy more, if you buy less, this is what happens that isn't the starting cost plus an additional cost.

So if I look at these numbers here, again, these numbers the cost divided by the length, or either the cost per centimetre is always constant.

It's always the same.

So now I can say the cost per length is a constant value.

What does that tell me? Well, it tells me then the cost and the length are directly proportional, okay? So the cost and the length are directly proportional in shop A.

Let's have a look at the second statement in shop B.

Okay, so we agree with statement number one Let's check statement number two.

Do we agree with it or not? Statement number two is about shot B, and it's also saying that the length and the costs are directly proportional.

So we're going to test if that is the case or not.

Shop B gives us the following offer.

Chain costs 40 pence to start, and then two pence per two centimetre.

So we have a starting cost of 40 pence and then it depends on how long the chain we want how long we want the chain to be, we pay two pence for every two centimetres.

Now this here the table is that we completed in the try this for the cost and at different lengths.

So we're going to do the same thing, we're going to find the cost divided by length, so we're going to find the cost per one centimetre.

So find the cost for one centimetre , 42 pence for two centimetres.

So for one we divide 42 divided by two and that gives us 21 pence.

So if I buy a two centimetre piece, I'm actually paying 21 pence for each centimetre.

But what if I buy a four centimetre piece, I pay 44 and divide 44 by four, I'm paying 11 pence per centimetre.

Next one 46 divided by six.

I'm paying 7.

6 recurring or 7.

67 if you want to do decimal places, that's how much I'm paying per one centimetre.

What do we notice so far? Really good that the cost per centimetre is not constant.

It varies with that length of the chain.

The smaller the chain, the more I'm paying per centimetre.

The bigger the chain so far the smaller or the less I am paying per centimetre.

Let's carry on and see if that pattern continues.

48 divided by eight, that is six, so I'm paying six pence per centimetre, for the next one on paying five, okay.

52 divided by 12, 4.

3 recurring or 4.

33, so I'm paying even less per centimetre.

Next one 3.

86 per centimetre and what is 56 divided by 16? Okay, really good, that's a 3.

5, so I'm paying 3.

5 pence per centimetre.

So again, let's look at these numbers.

Are they the same? No.

Is it a constant change? No, it's not.

I'm paying more per centimetre if I buy a smaller piece, I am paying less per centimetre if I buy a bigger piece, okay? So, can we say that the shop is charging us the same amount per length? Do we agree with that first part? Is the shop charging us the same amount per length? The answer is no.

It's clearly charging us different amount for different lengths, okay? And is the cost and the length, are they directly proportional? No.

We discussed the the previous one, that for shop A that were directly proportional 'cause that constant number we had value the value of four it was constant, in this case these numbers are different.

So, the cost per length is not a constant value, the cost per length is not a constant value, and therefore, I can safely say that the cost and the length are not directly proportional in shop B.

Okay, so we agree with the first statement.

We disagree with that second statement.

Let's have a look at that third one.

So the third statement in shop C, the cost span name changes after 10 centimetre.

So the cost and length are not directly proportional.

Shall we check? Are they directly proportional or not? We already have a bit of a hint that you know, it does change after a certain length so it may not.

And let's look at our table.

So again, this is the table that we created in a try this, tells us the costs of two, four, six, eight, 10,12, 14 and 16 centimetres.

So we're going to do exactly the same thing and find out the cost of a per, and the cost divided by the length, so the cost per one centimetre.

So two centimetres costs 10.

So we're going to find the cost of one, so we're going to divide by two.

So 10 divided by two is five, so we're paying five pence per one centimetre.

Okay, the next one, 20 divided by four is five.

So we are again paying five pence per centimetre.

This is really interesting, isn't it? So so far, we have two numbers of other centres struggle.

The third one, 30 divided by six is five.

So again, we're paying five pence per centimetre.

So so far, whether we pay we buy a two, a four or a six centimetre long chain, we're paying five pence per each for each centimetre.

All right, next one, 40 divided by eight.

Really good, that's also five and 50 divided by 10, that's also five.

So so far, what we can see looks like direct proportion.

Do we need to continue with the table? Do we need to check for every single other value? Can we just make the assumption now that it's directly proportional? Excellent, we cannot just assume it is.

We need to check the rest.

Because the shop already told us really clearly that at first it costs this much after that, equals this much, so let's have a look.

54 divided by 12, there we go is 4.

5 now pence per centimetre, so we buy a bit longer, we're paying less.

Next one, 58 divided by 14 is 4.

1, and 62 divided by 16 is 3.

9.

So by looking at this, the first part, it looked like we've got the same constant value, but if we look at these numbers here, actually, the cost per centimetre is changing.

Therefore the cost per length is not a constant value.

It has to be a constant value throughout, so it's not and therefore we can say that the cost and the length are not directly proportional in shop C.

Now it is your turn to have a go at this independent task, you have two questions to answer.

Please read the questions carefully and attempt them.

Once you finished, you can come back here where I will go through the solutions so you can mark and correct your work.

While answering the questions, try and use a table or a bar model to present the questions and that will help you answer the questions.

So, pause the video now and have a go at this and for support, let's read the question together.

A meal in a restaurant costs a fixed price for each person.

For seven people the total cost is 161 pounds.

What is the total cost for eight people? So to start this question, I drew one bar, divided it into seven equal parts and labelled it with 161 at the top.

Then for the second part, I do another part another bar model divided into eight equal parts, and I said, that's the bit I need to work out.

So I need to go to the first bar, 161 divided into seven equal parts, I need to find that one equal part, I need to find what one part is, how would I do this? Excellent, I will need to do 161 divide seven, and that will tell me what will go in one part.

I can use that in the second bar and put it all the way through to find out the cost for eight people.

Now with this hint, you should be able to make a start.

Pause the video now and have a go.

Welcome back, let's start answering the questions together.

So we have our bar model, for the first one, 161 divided by seven gives me 23.

So I know that one part in that bar is 23.

So I know it's 23 though out here as well to work out the answer and I need to do 23 multiplied by eight.

Excellent or you can even add it up eight times if that's what you want to, but a more efficient way is to multiply it.

So I can say or when I multiply, I can find find out the answer is 184 pounds.

So I can say now that the total cost for eight people is 184 pounds.

Question two; Seven chocolate bars cost 3.

15 all together.

How much will three chocolate bars cost? How much will 10 chocolate bars cost? Okay, so let's do one part at a time.

With this one, I'd like to do it slightly differently.

Remember all these methods that we have discussed, you can still use a bar model and get the correct answer.

I'm going to use a table.

So I'm going to start by sketching the table, and the table would have chocolate, the cost and the cost per chocolate.

So I'm going to have how many chocolate bars do I have? I've been told in the question that seven chocolate bars cost 3.

15.

So the cost of one chocolate bar is 3.

15 divided by seven, 0.

45 pence.

So I have 0.

45 pence.

Now, what other information have I been given in the question? I've been given that I need to work out the cost of three and the cost of 10.

Now the cost per one chocolate is still the same nothing here and the question told me that they will give me a discount or it will change up it's constant, it's the same thing ,the same price.

So I have three chocolate bars that I need to work out and I have 10 chocolate bars that I need to work out.

The cost of in both cases 0.

45 pounds per chocolate bar.

So to work this out, I need to do three multiplied by 0.

45 Okay, or 0.

45 pence multiplied by three or 45 pence plus 45 pence plus 45 pence that gives me 1.

35 pence for three chocolate bars.

And for 10 I need to do the same work.

If you've done this correctly, so that is 4.

50.

So now I can say that three chocolate bars will cost 1.

35 and 10 chocolate bars will cost 4.

50.

This brings us to our explore task for today's lesson.

Xapier makes two pitchers of juice.

How do you know his statement is incorrect? So that's true the statement what is he saying? He's saying the amount of cordial is directly proportional to the amount of water and we have been given a table here.

Now the table shows the amount of cordial in millilitres and the amount of water that has been mixed together.

Change one of the numbers to make the relationship directly proportional.

So you have two things to do with this task.

First, you need to be able to tell me why is his statement incorrect and the second thing because it's incorrect, how can you change one of these numbers to make that values have to make the relation directly proportional? Pause the video and have a go at this now.

Okay, welcome back.

Let's have a look at how do we know that his statement is incorrect? If I look at the table and look at the relationship that has been given to me back, what do I have? To get from 300 to 1000, 1200 millilitres, I multiply by four.

And remember how we said in order for two things to be directly proportional it has to be a constant relation multiplicative relationships, you have to be multiplying by the same number at every time.

300 times four is 1200, is 600 multiplied by four 2000? No, 600 multiplied by four is 2400.

In fact, what they've done here is multiplying by 10 after three or if you want it as a decimal multiply by 3.

3 recurring.

And we cannot do this, if we make something four times bigger we have to make the other thing must also become four times bigger and this is how we know that the relationship is not directly proportional.

In fact, we don't know that for the 600 cordial, we need 2400 millilitres of water to make it directly proportional.

So that would be one of the answers for the second part.

Now, I also want to point out something else.

Not sure if any of you noticed this, but to get from 300 to 600, I multiply by two.

So if I double the amount of cordial that I have in the juice, then I need to double the amount of water that I'm using.

So again, 1200, if I double it, that should give me 2400.

And which is the same answer we had from multiplying by four , if we go from the cordial into the water.

So if you look here, the person has multiplied by five after three.

So didn't multiply by two, didn't double the amount of water, like he has doubled the amount of cordial and this tells me again that this is not directly proportional.

And now let's look at some of the numbers that you could have have changed, you could have changed some of these numbers.

So the first one that we said you could have done it as with other 400.

But these are some of the other possible solutions that you may have.

I wonder which ones you wrote down.

This brings us to the end of today's lesson, and you did well than all the learning that you have done during this lesson.

Please remember to do the exit quiz.

And I'll see you next lesson, bye.