Lesson video

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

Saada.

In today's lesson, we will be looking at graphs of proportional relationships.

For this lesson, you will need a pen and paper, a pencil, and a ruler.

So please pause the video, go grab these and when you're ready, come back here and click presume so we can make a start.

A shop sells ribbon for three pounds per metre.

The table shows the cost y pounds, for different lengths in x metres.

Complete the table and then plot the graph of the quantity and the cost.

Now the quantity will need to go on the x-axis and the cost will be on the y-axis, I've given you the table.

I highlighted one cell in that table with grey, I wonder if you know why? If you're feeling super confident, please pause the video now and have a go at this question.

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

Okay, so we know that the shop sells the ribbon for three pounds per metre.

So for one metre, you're going to pay three pounds.

For y you're going to write down three.

Now what about for two metres, how much would we pay? If one metre we pay three, for two metres surely we'll pay double that and that would be? Excellent.

What about for three metres? And compare that to one metre.

Try and think about the tables that we completed for our last lesson.

Where we drew arrows on top of the table and underneath it, when we were looking at proportional relationships.

With this hint, you should be able to make a start now, off you go.

This task should take you about five minutes to complete.

In particular, if you have it printed, if you do not have it printed, you will need to construct your table and you will need time to draw the x-axis on the y-axis.

It may take a little longer, please pause the video and complete the task to the best of your ability.

Resume the video once you're finished.

Hi there, how did you get on with this task? Really good.

So this is my table here.

I looked at the information that we have been given, that one metre costs three pounds.

I use that information to help me complete the table.

You can see that I have greyed the first cell for y/x because I'm dividing by zero.

You cannot do that.

If you're really interested in finding out more about it, please go and research and you'll find lots of interesting information out there.

Okay.

What did you notice from your table? Come on say it.

Really good.

The y divided by x is always a constant value so this tells me that there is a direct proportion.

So this tells me that that is a direct proportion between x and y, really good.

Did you plot the graph? Excellent.

So your graph should look like a straight line.

It should look like a linear graph, it should cross the x.

Did you draw your graph, okay.

Was it similar to my graph? Really good.

So your graph has to be a linear graph, it must cross or pass through the origin at zero zero.

It must do that in order for it to be a direct proportion graph.

Look at that gradient on my graph, it's three and y divided by x is three.

So having the table can tell me how the graph is going to look like or having the graph can tell me what the table is going to look like.

Now I can see that the gradient of the graph here.

Now, I can see that this line here has a gradient of three because difference in x is one, the difference in y is three and remember we learned about gradients before.

If you're not sure or if you cannot remember it, please go back to the previous lessons.

So, again, we know that the equation of a straight line is y equals mx plus c.

We have been given that c, that y ordinate is going to be zero.

Again, we can take any two points to try and find the gradient of the graph.

But we can see here by looking, okay the gradient is three.

So I can say that y is equal to 3x plus zero.

So we don't really need to write it down.

And from that three, I know the cost per one metre from the gradient.

So I can either have the graph, find the gradient, have the equation.

Use that gradient to help me find the cost of one thing, whatever it is, or I can use the table to help me with that.

Now let's have a look at some examples in context.

x represents the number of units of gas used in a house in a month.

y represents the cost in pence.

There is a charge of 12 pounds each month, even if no gas is used.

Each unit of gas costs 25 pence.

Cause even if I don't use gas at all in my house I do not cook, I do not need to choose any heating, I will still have to pay some sort of money for the gas company because they will have a connection charge.

So now if I spend zero in the table, I'm going to still end up paying 12 pounds per month according to this scenario.

Now, what happens if I use five units of gas? Well, five units, each unit is at 25 pence.

Five times 25 is 125 pence or one pound 25.

One pound 25 plus the 12 pound is 13 pounds 25.

So I'm going to pay 30 pounds 25.

Now, if I try and calculate y divided by x, I divide 12.

35 divided by five.

That gives me 2.

65.

What does this really mean? In context, it means how much I'm paying for one unit, okay? Because at the moment when I use five units only, I'm not paying 25 pence on the per unit.

If you take into account that 12 pounds that I'm paying originally.

Now, what about if I use 10 units, what happens? Well 10 units will cost 25 times 10 is two pound 50.

Two pound 50 plus the 12 that will give me, will cost 14 pounds 50.

So I will need to pay 14.

50.

Okay, what does that mean? How much am I paying per unit of gas in that month, I'm paying one pound 45 pence for each unit.

If we look at these numbers here, they are not constant, this tells me that this scenario is not discussing a direct proportion.

When the number of units doubled the cost did not double.

So it's definitely not directly proportional.

Now we can use a graph to help us also see this.

If I look at this graph, I have my x-axis and y-axis.

It's a quick sketch.

I have 12 pounds there I have five and I have 10.

And I floated at five, I'll put the point at roughly 12.

35 and at 10 I've put it at about a 14.

50.

The I got a ruler and joined them and you can see that they make a lovely nice straight line.

However, that line does not cross the y-axis at zero zero.

If it does not cross the y-axis at zero zero, then this is not direct proportion or not showing direct proportion.

Our graph for direct proportion will look like this.

I will have x and the y and it will have a straight line and it will start at the origin or it will pass through that origin.

Now let's have a look at another example.

x represents the number of text messages, y represents the cost.

Each text message costs the same.

10 text messages costs 1 pound 80, complete the table.

So, zero text messages, am I going to get charged? Not really.

It didn't say anything here.

They said that each text message will cost exactly the same.

So no text message will cost no money.

Now 10 text messages is one pound 80, so can we use that to calculate five? Yeah.

If we half it, that would give us five, which is 90 pence and I'm putting one pound 80 for 10.

Now what's the cost per text message.

90 pence divide by five is 18 pence and one pound 80 divide by 10 is also 18 pence.

Now from those numbers, that's a constant value.

So we know that we have a direct proportionship between the cost of messages and the number of text messages sent.

So they are directly proportionate.

Now, if I show you this graph, I have the x-axis, I have the y-axis.

I will just put zero and the five and the 10.

And I did a quick sketch to show where that one pound 80 and the 90 pence are.

Plotted the points and you can see that they make a straight line that passes through the origin this time.

The line passes through zero, zero.

and therefore I can tell that the two variables here are then directly proportional to one another.

Again, in our last example for today's lesson is about a topic that we always discuss just before we go on holiday and that is exchange rates.

So the graph here shows the exchange rate between British pounds and US dollars.

The question is asking us to use the graph, to find certain amounts when we convert them.

Before we do that, I would like us to look at the graph here.

What do we notice? What kind of graph is this? Have a little think and say it too the screen.

Really good.

So we can see here that we have linear graph and the graph intersects the y-axis at zero, zero at the origin.

So this graph here shows us direct proportion between the two relationships.

It shows a direct proportion between the British pounds and the US dollars.

Now, the question says, use the graph to find two pounds in dollars.

So I go to two pounds, which is the x-axis, I draw a vertical line until I reach the graph.

And then I draw a horizontal line and read the y-axis.

And the y-axis here shows it's about $2.

75.

Four pounds 50 to US dollars.

We do the same thing, go to four pound 50, draw a line, reach the graph, read the value and it's about six point, I would say 6.

25.

Sorry.

I don't know why it says seven here.

It should say 6.

25.

Just bear with me.

So 6.

25 US dollars.

Okay.

Now next one, $6 to British pounds.

So this time we want to convert from dollars, dollars are on the y-axis.

So we'll go to the y-axis and we find six, we draw a line and when we hit the graph, we go down to the y-axis and read it off.

So if I do that, that gives me this line here, which is about four pounds 30 or four pounds 25.

How can you use the graph to convert $10, sorry 10 pounds to US dollars.

Because if we look here at the graph, the 10 pounds is somewhere here and we cannot see the rest of the graph, so we cannot use it.

What can we do instead? Really good thinking.

Okay, so we can use any values that have been given to us because this is direct proportion.

It really isn't going to make a difference.

So I can, for example, go here and say, well, I know that this relationship here, I know that $eight is equal to 5.

75 pounds, seven point 75 pounds and then I can use that to help me, or I can use one of the previous ones that we have already calculated.

How can you use your graph to convert $80 to pounds? So similarly, if I want $80, I don't have $80.

What can I do? I can use the eight again, so I can use the eight and read it off from here and just multiply it by 10.

So now, because I've got that direct proportion, I have that direct proportion relationship.

I can do all sorts of things with the graph.

The most important thing to remember is that the graph must passes through the origin and the gradient of the graph tells you a lot of information about the relationship.

Now, it is your turn to have a go at the independent task, please make sure that you're reading the questions carefully and attempting every single part of the question.

You should have this printed off, so you can see the graph and use it.

If you do not have it printed off, just make sure that you're reading the scales properly.

It is time now for you to have a go at the independent task, please pause the video and complete it to the best of your ability.

It should take you about 10 minutes to complete.

Resume the video once you are finished.

Welcome back.

How did you get on with this task? Did you manage to complete it? Really good.

So that answers the question one, a is 69 to $70, b 57 to 58, for c and d it's 42 pounds and $73 and four e it's $960.

How did you do on this one? All correct, excellent.

Really, really good.

Now let's look at question two.

I guess for question two, if you used the graph correctly, the first part should've been 10 kilogrammes equals 22 pounds.

The second part 10 pounds is equal to 4.

5 kilogrammes.

For part c, the difference and you needed to give it in both units.

7.

75 kilogrammes and 18 pounds.

And for the last one you needed to find out that 75 kilogrammes is equal to 165 pounds and then decide and make a statement.

That yes, Sofia is correct.

So you've done really well.

Well done.

And your last task with today's lesson, let's do it together.

Antoni wants to convert 321 Australian dollars to Euros.

How many different ways can he use the conversion graph to help him? So there's so many things that you can do with this conversion graph.

I want you to have a little thinking about all of this.

I want you to try and find as many methods as possible before you come back for the solutions, off you go.

Hi there, how did he get on with this task? How many methods did you manage to find.

Okay really good.

So there's so many things that you could have done with this task and I'm just going to share with you my thoughts on this task, okay.

So I looked at the graph and I thought, let me find a point that tells me, or helps me with the conversion, right.

And I found this point at this point.

Now this point tells me that two Euros are equal to three Australian dollars.

So then I thought, okay, let me use this information from the graph and write it down, so I wrote this as my first method.

Two Euros is equal to three pounds.

I want to find what 321, so what did I do from three to get to 321, I multiplied by 107.

So I did the same to the Euros and the answer is 214.

That was one of my methods.

Now the second method, started off with the same, again originating from the same point, Euros to Australian dollars is two to three.

I want to know for one Australian dollars, what am I going to get in terms of Euros? So divided by three, I found that it will be 2/3 and I left it as a fraction because I don't need to do anything with it now I don't need to round it.

And then from one, so that's the unitary method.

I found one Australian dollar.

So I found one Australian dollar, how much Euros I would get for that? And then I converted all of that into Euros.

Now, the last one and it's the most interesting one I think.

I worked out the gradient, the gradient of the graph is 1.

5.

So they gradient of the line is 1.

5, which tells me that the equation is equal to y equals 1.

5x.

So when y equals 321, I can write this equation 321 equals 1.

5x and now I can be arranged to solve for x.

x is equal to 321 divided by 1.

5 and therefore x will still give me the same number 214.

And it's a lovely method to use when you have the graph without having to refer back to all of that, what did I do here? What did I do there? So that last method is definitely my favourite, which one is yours? This brings to the end of today's lesson and each well done on all the effort that you have showed throughout this lesson.

Please do not forget to complete the exit quiz.

This is it from me for today.

Enjoy the rest of your learning and I'll see you next lesson.

Bye.