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.

So pause the video and go and grab these if you do not have them handy.

And when you're ready we can make a start.

Antoni goes to the fair.

He can choose to pay two pounds for each ride or he can pay five pounds for a wristband and each ride costs one pound.

Antoni says, "I want to go on seven rides," which way will cost the least amount of money? My hint for you for the try this task is to try and draw bar model to represent the problem.

Now pause the video and complete the task.

This should take you five minutes, resume the video once you're finished.

So the question told us very clearly that Antoni wants to go on seven rides and told us two options that he has.

For the first one, he needs to pay two pounds for each ride.

So I started this by drawing a bar model dividing it into seven equal parts.

I put a two pound in each part and therefore I knew that the total cost if he goes for option 1 will 14 pounds.

For the second one, he can pay five pounds for a wristband and then an additional cost for the rides.

But each ride will only cost now one pound if he has the wristband.

It's almost like having a discounted price because you're a member of something.

So I started by drawing a bar model and I put five pounds in it 'cause I know he has to pay that for the wristband.

And after that, he is going on seven rides.

Each ride, he will pay one pound.

So I've got seven parts, seven equal parts and I put one pound in each of them.

I added this up and that turned out to be 12 pounds.

And this showed me that it is cheaper for Antoni to go for option 2, pay five pound for the wristband and then pay one pound for each ride.

Again, now let's look at this in a bit more depth.

Option 1, Antoni can choose to pay two pounds for each ride.

Let's complete the table.

If he goes on one ride, he's going to pay only two pounds.

If he goes on two, he's going to pay? Really good, four pounds.

Three rides he will pay? Four, five, six rides and seven rides he will pay 14 pounds which is what we calculated in the try this task.

Now I want us to calculate the cost per ride, so how much it costs for one single ride.

I know that this was given to us.

But this will be really, really important.

We will move onto option 2.

So the cost per ride, if he goes on one ride he will only pay two pound for one ride.

If he goes on two rides, he will pay four pounds.

So for per one ride he is still paying two pounds only for each one.

If he goes on three he will six as six divided by three is two.

He's still paying two pounds only per ride.

He's not getting a discount for example because he is going on more rides.

He's not getting any offers.

He's still paying the same amount for every one ride, okay? For when he goes on four rides he still pays two pounds, five rides, he's still pays two pounds per ride.

That is not changing.

12 divided by six that's two.

It's not changing.

14 divided by seven is two.

So with option 1 he always paid two pounds per ride.

It doesn't matter whether he went on one or seven.

Let's plot those numbers on the graph.

So I have a grid here.

I'm going to plot that number of rides on the x-axis and the cost on the y-axis.

Okay, I'm not going to be able to plot all of them but I want to plot the first few.

So for one ride, he pays two pounds.

So my first point is going to have a coordinate of 1,2.

And second point is going to have a coordinate of 2,4 the next one, 3,6.

And there we go, these are my first four points just because I don't have enough space on the screen to put the rest of the points.

I'm going to grab a ruler and join them and then make a nicer straight line, yes.

And it passes through the origin.

So this is really, really important.

It passes through that point 0,0.

Now with option 2 what happens? If he goes on one ride how much is he going to end up paying? He's going to pay for the wristband, that's five pounds plus one pound for the ride, that is six pounds in total.

So if he goes only on one ride he will still pay six pounds.

So how much is he paying per one ride in that case? The full six pounds.

Whereas with option 1 he was only paying two pounds per ride.

Okay, what about if he goes on two rides? He will pay the five pound for the wristband and two pounds.

So that's five plus the two that's seven pounds.

Three, how much will he pay for three rides? Good job.

And four? Really good, he would pay nine pounds.

For five rides, how much will he pay? Excellent.

10, and for six rides? 11, and for seven? Really good, and we can see the difference between the first table and the second table already.

We can see that for five rides it doesn't matter whether he goes with option 1 or option two, right? But more than five he's better off with option 2.

Less than five he's better off with option 1.

So we can already start seeing some patterns in here.

Let's calculate the cost per ride in that second option.

So the second option, if he goes only on ride it's six divided by one, which is six.

For two rides it's seven divided by two, which is three pound 50.

So he's paying three pounds 50 for one ride.

Next one, eight divide by three, 2.

6.

Next one, nine divided by four is 2.

25.

10 divided by five is two.

And we can see that that number per ride the cost per ride is decreasing as he goes on more rides.

For six, 11 divided by six is 1.

83 and for the last one is 1.

7, okay? Now let's plot the points for the set from the second table on the graph to see if the graphs are similar or not.

So we're going to again plot the number of rides on the x-axis and the cost on the y-axis.

So the first point we have the coordinate 1,6, the second one we have 2,7, and then 3,8 and then 4,9.

Now I'm going to join them using a ruler.

Again, I can see that I have a straight line.

The difference is this time that this line does not cross through or does not intersect through our axes at the origin, it doesn't intersect at 0,0.

It crosses the y at five.

Okay, let's go back to the tables and just look at one thing.

The cost per ride in option 1 was always what? Two, did not change, it was a constant value.

The reason for this is that the number of rides and the cost in option 1 are directly proportionate as one increases the other increases by the same multiplier.

So if the number of rides doubles, the cost doubles.

If the number of rides triples then the cost triples and so on.

Let's look at the cost per ride in option 2.

Was it the same? No, it changed it.

It wasn't a direct multiplier.

The cost per ride was changing all the time and it will continue to change even if we carry on calculating up to 100 rides, okay? And the reason for that is that this one is not directly proportional.

The option 2 does not show us a direct proportion between the number of rides and the cost.

We can see from the graphs as well, which graph shows direct proportion and which one does not.

The pink line there, the pink straight line shows direct proportion.

It starts at zero.

It crosses the y-axis at zero.

So it shows me that these two relationships or these two variables that I have are directly proportional.

The blue graph does not cross the y-axis at zero and therefore, it does not show direct proportion even though it looks like it's a really nice and neat straight line.

It does not cross the y-axis at 0,0, and therefore, it does not show direct proportion.

Now, it is your turn to have I go at the independent task.

If you're feeling super-confident about this please pause the video now and make a start.

If not, don't worry.

I'll be giving you a hint in three, in two and in one.

Okay, so you have been given three tables in question 1.

You need to fill in the missing values in the tables and you need to talk about what kind of connection the two variables in each table have.

So if we look at the first table it's showing us the number of mugs, the cost and the cost divided by the number of mugs.

The first column has already been completed for you.

So you can see that four mugs cost one pound 60 and the cost of one mug is 0.

40, which means 40 pence.

Now, I want you to think about these two numbers here.

What did I do from one number to the other? What do I do from one pound 60 to two pound 80? So whatever happened to the cost needs to happen to the number of mugs, because the cost per mug is constant, it's 40 pence as well, that table is not changing.

If you want to do this one I would think about this.

And for this one, think about the cost divided by something is equal to 40 pence.

The cost of three pounds 60 divided by something is equal to 40 pence.

How do I find what that something is? I need to rearrange to work out that something.

Now, I'm going to give you a hint that you should use all the time, not just in this lesson.

If you're struggling with a question like this and you don't know how to rearrange it, am I going to divide them, am I going to multiply? Write a similar question with numbers that you're really confident with.

For example, write down I know that six divided by two equals to three.

I wrote this one, because I have a division.

So I tried to write the division question that I'm really, really confident with.

Now I ask myself if I want to write two equal something what am I going to write? I'm going to write down that it's six divided by three equal to two.

And therefore, now I can be arrange the previous equation.

And I can say that in order for me to find out that question mark, what do I need to do? Really good.

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

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

Please pause the video and complete it to the best of your ability.

Resume the video once you're finished.

Welcome back.

How did you get on with this task? Okay, the solutions to question number 1 are here for you.

So please mark and correct your work.

If you need to pause the video while you're marking and correcting, you can do that.

We can see from all the tables that that third row is always constant.

The cost divided by the number mugs was always 40 pence, which tells me that the number of mugs and the cost of the mugs are directly proportional.

The number of pens and the cost are also directly proportional.

because the cost per pen was always the same.

And the total number of hours worked and the cost of how much money you earn is also directly proportional, because that was 12 pound 50 per hour every single time.

Did you get these correct? Really good, well done.

Let's go through question 2 together.

The charge for Anna's phone bill is directly proportional to the number of megabytes of data used.

When Anna uses eight megabytes, she pays 24 pence.

How much will Anna pay if she uses 39 megabytes? Now what does directly proportional means? Means if one quantity increases, the other increases in a similar by a similar multiplier.

So if something doubles, the other thing doubles if something halves, the other thing halves, okay? So now let's write this down.

I'm going to do it using this method where I write it in a similar way to ratio.

Data to cost.

Data, eight megabytes, so cost 24 pence.

And I'm going to find one unit.

i.

e.

I'm going to find the cost of one megabyte.

So if I know the cost of eight, I can divide by eight to find the cost of one.

So I'm going to divide here by eight.

And that tells me that for one megabyte she has to pay three pence.

Now I'm not interested in one megabyte.

I want to find 39.

But having the one will really help me find the 39.

So now I can say, well, what is the cost of 39 megabytes? I'm multiplying by 39.

So I need to multiply that three pence by 39.

And that gives me 117.

Now I can say that Anna pays 117 pence or one pound 17 if I convert it into pounds.

Let's have a look at the second part.

John has the same phone contract.

The same phone contract means they're going to pay the same amount for each megabyte.

He gets a bill for 21 pounds.

How much data did he use? So again, one megabyte, three pence.

Now, what we know here is that he pays 21 pounds.

And I'm going to convert it into pence just to make sure that I am using everything in the same unit.

It's either I convert that into pence or the three pence into pounds.

It's easier for me to convert everything into pence in this case.

Now, what do I do to get from three to 2,100? I'm multiplying by 700.

Three times the seven is 21.

We've got 100 times bigger.

So we made it 2,100.

So I multiply by 700.

So I need to multiply here by 700.

This tells me that John uses 700 megabytes in order for him to get a bill of 21 pounds.

Did you get that correct? Good job.

Again, now we're going to continue with the same theme of number of rides and the cost of number of rides in a fair.

I want you to look at the two tables here.

The first table here shows the cost of number of certain number of rides in fair number 1.

And the second one for fair number 2.

I want you to work out which rule connects the number of rides to the cost for each fair.

I want you to describe what is the same and what's different in fair 1 and fair 2.

And I want you to think about if you draw these in, on a graph what would be the same and what would be different? Think about all the things that we've discussed in today's lesson.

Now, if you're feeling confident please pause the video and make a start.

If not, I'm giving you a hint in three, a hint in two and a hint in one, okay? So look at the tables here and look at the relationship.

What is happening every time from the number of rides to the cost? What are we doing? Is there a certain multiplier that we are using? If yes, what is it? And then write down when I ask you to write down the rule it means I want you to write down for me something that starts with C equal and tell me what the rule is.

How do you get to C from n, if we knew n? So now if I tell you 10 rides, how can you send me quickly without having to list all of them? You have to do the same thing for the second one.

Looking at these numbers here tells us what? Really use those numbers to help you with writing the rules.

Now with the first rule it's quite easy.

The second one might be a bit tricky.

So I want you to think about nine, 13, and 17.

Think about them as a sequence.

We've done sequences before.

Think about it.

If you're writing the nth term, what would you write? How would you write it down? Now with this hint you should be able to make a start.

Please pause the video and complete the explore task which should take you about 15 minutes to complete.

Resume the video, once you've finished.

There are so many ways that you could have gone about the explore task.

I'm just going to share here, my thoughts on this.

So I looked at the first table then I thought, well what am I doing to get from the number of rides to the cost? Every time I'm multiplying by four, one times four is four, two times four is eight, three times four is 12, which is really just the cost per one ride.

So if you did that third row that's also correct to find the cost of one ride then use that to help you.

So I said, therefore, the cost C is equal to four rows of n and it's four multiplied by the number of rides.

Did you get that too? Really good.

Now with the second one it wasn't as easy.

It's C equals five plus 4n or C equal 4n plus five.

If you use my hint about sequences and you wrote down nine, 13, 17 and worked out the nth term, you will have worked this out to be correct.

Now, the other method is to think about the numbers that we already had.

So if you think about those numbers, four, eight, and 12 and how do we get to nine, 13 and 17? From four to nine, we're adding five, from eight to 13, we're adding five, and from 12 to 17, we're adding five.

So I can say that the first part of the rule is going to be the same.

C is equal to 4n, and I'm doing something extra to it and I'm adding five.

So you may have written C equal 4n plus five, that's also correct, because addition is commutative, so whether we say C equals five plus 4n or 4n plus five, they're both correct.

Now, describe what is the same and what is different? What did you write down I wonder? Okay, these are some of the things that I thought.

Well both fairs, the rides cost four pounds.

We're still paying four pounds for each of them, because we have that multiplier of four.

What is it different is that in the second fair, for fair number 2, you have to pay five pound extra for entrance.

So you're either paying it at the entry or you're paying it for a wristband, but there's some sort of five pounds extra being charged there.

And now, what about the graphs? Did you draw this on a graph? What did you get? Okay, so both of the graphs are linear graphs.

They both had the gradient of four.

And again, that four, because of the multiplier because of we're multiplying by four.

Now, what was different, is that the first one And y intercept at 0,0, so the origin.

And we discussed that if the graph is linear and it crosses at the origin at 0,0 it means that it's the two relationships, so the two variables are directly proportional.

The second one intercepted at 0,5, which tells me that the second one is not directly proportional.

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

I hope you enjoyed the lesson as much as I did.

Now I want to remind you to complete the exit quiz to show what you know.

Take care and I'll see you next lesson.

Bye.