Lesson video
In progress...Loading...
Hello, Mr. Robson here.
Welcome to Maths.
Today we're looking at graphs of direct proportion.
Direct proportion is everywhere in this world, so it's good to know a bit about it.
Shall we take a closer look at what the graphs look like? Absolutely.
Our learning outcome is that we'll be able to recognise direct proportion graphically, and interpret graphs that illustrate direct proportion.
Keywords.
Two variables are in direct proportion if they have a constant multiplicative relationship.
Two parts to today's learning.
We're gonna begin by graphing direct proportion.
If two variables have a constant multiplicative relationship, they are in direct proportion.
Let's consider if T and C are directly proportional in these two cases.
In the left hand table to turn 2 into 80, we multiply by 40.
To turn 4 into 160, we multiply by 40.
Looking at the vertical relationship, to turn 2 into 4, we multiply by 2.
To turn 8 into 160 we multiply by 2.
We've got constant multiplicative relationships.
Therefore this is direct proportion.
Let's contrast that with the table on the right hand side.
To turn 2 into 110, we multiply by 55.
To turn 4 into 170 we multiply by 42.
5.
Vertically on the left hand side, 2 multiplied by 2 makes 4.
Whereas on the right hand side we have to multiply by 17 over 11 to turn 110 into 170.
There's not a constant multiplicative relationship going on between these variables.
Therefore, this is not direct proportion.
Let's check you've got that.
Sort the following into examples and non-examples of tables showing direct proportion.
Is each one an example of direct proportion, or is it a non-example? Pause and decide now.
Welcome back.
I hope you said A is an example of direct proportion.
3 multiplied by 12 makes 36.
12 multiplied by 12 makes 144.
T and C are directly proportional.
B is a non-example.
It's not a constant multiplicative relationship going on between T and C there.
C was an example.
There's a constant multiplicative relationship between T, and C in that example.
The two examples we saw earlier are costs charged by Karaoke venues.
T stands for time.
C is the cost.
Venue A charge 40 pounds per hour.
Venue B charge 50 pound hire fee, and then 30 pounds per hour.
We can graph each of these models to compare how a direct proportion graph differs to a not direct one.
Venue A at 40 pound per hour gives us that graph.
Venue B with a 50 pound hire fee, and then 30 pound per hour, gives us this graph.
Venue A is an example of direct proportion just as the table is an example of direct proportion.
B is an example of not direct proportion.
What I'd like from you now is a little input.
What is the same between the two graphs? What is different between the two graphs? Pause and make a few observations.
Welcome back.
I wonder what you spotted.
Did you notice the same between the two graphs? They're straight line graphs.
What does that matter? Well, that matters a lot.
It means there's a constant rate of change.
Both graphs have a constant rate of change.
So what was different? A key difference was that our graph of direct proportion began at the origin.
The graph of not direct proportion did not.
Beginning at the origin, and maintaining a constant rate of change is a key feature of a direct proportion graph.
Other features come out when we look at certain coordinates.
If I look at the coordinate 3, 120, and 6, 240, we get this relationship.
Twice as many hours costs twice as much money.
That is an example of direct proportion.
I could look at those two coordinates in a different way.
I could say we take the T coordinate, and multiply it by 40 to get the C coordinate, and that happens all over the line.
There's a constant multiplicative relationship between time and cost.
That is a feature of direct proportion.
Contrast that with the graph of venue B, which is not direct proportion.
We have this 50 pound hire fee, and then a 30 pound per hour charge.
By contrast, this graph doesn't begin at the origin.
That changes this.
If we look at 3 hours costing 140 pounds, 6 hours costing 230 pounds, it's no longer the case that twice as many hours is twice as much money.
When we look at the relationship between the T coordinate, and the C coordinate, there's no longer a constant multiplicative relationship.
It's different between these two coordinates.
It's different between every coordinate on this line.
When there's no constant multiplicative relationship between time and cost we don't have direct proportion.
We see a lot of straight line graphs in maths, but they only represent direct proportion if they begin at the origin.
Our graph on the left hand side is an example of direct proportion.
The graph on the right is not.
Talking of graphs.
Look at these five.
I'd like to check that you understand what is, and what is not a direct proportion graph.
I'd like you to sort these five into examples, and non-examples of graphs showing direct proportion.
Which are examples? Which are not? Pause and decide which fit into which category.
Welcome back, I hope you said the first one is absolutely an example of direct proportion.
Begins at their origin, has a constant rate of change.
B was a non-example.
But why? It begins at the origin.
It's because there's not a constant rate of change.
That's why it's not direct proportion.
C is an example of direct proportion.
Begins at that origin.
Constant rate of change.
D is not, it does not begin at the origin.
E is also a non example.
Whilst it begins at the origin, there's not a constant rate of change along the whole length of the line.
Graphs of direct proportion begin at the origin, and have a constant rate of change.
This graph represents prices charged by a bike hire company.
The cost is in direct proportion to the length of time the bike is hired for.
How might we find their hourly rate? The good news is like many problems in maths, there's multiple ways to solve this.
One simple way is we could read from one hour on the horizontal axis.
Oh look, one hour is equal to 8 pounds.
Their hourly rate must be 8 pounds.
We could look for the multiplicative relationship in any coordinate pair.
Let's pick this pair, 3 and 24.
How do we turn the T coordinate into the C coordinate? Funnily enough, we multiply by 8.
Is that gonna be true for every coordinate? Of course it is.
This is an example of direct proportion.
The multiplicative relationship between the T coordinate.
and the C coordinate will always be 8.
Yet another way to find the hourly rate would be to calculate the gradient.
We can pick any two moments on the graph to help us calculate the gradient.
All we need to know is the vertical change, and the horizontal change.
In this case, we've got a vertical change of 24 divided by the horizontal change of 3, and funnily enough we find a gradient of 8.
Direct proportion graphs have the form Y equals KX, whereby the gradient is the multiplier.
In this case, the graph is C equals 8T with C and T the variables, and 8 the multiplier.
Quick check you've got that.
Y is in direct proportion to X.
What is the gradient of the line? Three options to choose from.
Pause and make the right one now.
Welcome back, and we went for option B.
The gradient is 15.
We can read that from the graph.
The gradient's 15.
This is the same as the multiplier.
Pick any coordinate pair, and you find the multiplier between the X coordinate, and the Y coordinate is also 15.
Another little check.
Y is in direct proportion to X.
What is the equation of the line? Three options.
Pause, take your pick.
Welcome back.
Lots of ways we can figure this out.
We need to know the gradient.
There's the gradient.
Alternatively, we could have considered the multiplier between coordinates.
We know it's option A, Y equals 15X.
Graphs of direct proportion come in the form Y equals KX where the gradient is the multiplier, K.
Practise time now.
Question 1, which graphs show X and Y in direct proportion? In each case, I'd like you to write a sentence to justify your answer.
Pause and do this now.
Question 2, this graph shows a directly proportional relationship between X and Y.
For Part A, calculate the gradient.
For Part B, use any coordinate pair to demonstrate that the multiplier is the same as the gradient.
Then finish off with Part C, writing an equation for the line.
Pause and do those things now.
Feedback time now.
Question 1.
I hope you said A is not direct proportion, and you might have written, "Because the Y intercept is not the origin." B is an example of direct proportion.
How do we justify that? We say that it's a constant gradient, or a constant rate of change and the Y intercept is at the origin.
C is not direct proportion.
It's not a constant gradient.
That's how we justify that answer.
C is not direct proportion, also not a constant gradient.
Question 2, Part A, I asked you to calculate the gradient.
We can do this by picking any two coordinates on the line, and dividing the vertical change by the horizontal change.
In my case, I did 160 divided by 20 and got 8.
Wherever you read that gradient from, you should have got 8.
Part B, use any coordinate pair to demonstrate that the multiplier is the same as the gradient.
I'm gonna do this with a coordinate pair 10, 80.
To turn 10 into 80, I need to multiply by 8.
Oh look, it's the exact same result as my gradient.
Any coordinate you picked should have given this result of 8.
With all that in mind, how do we write the equation for the line? Well, it must be Y equals 8X If the multiplier is 8, we take X, multiply it by 8, and we get the Y coordinate.
It must be the line Y equals 8X.
Onto the second half of the lesson now.
In context.
The multiplier, the gradient, will not always be an integer.
This conversion graph converts British pounds into Euros.
We've multiple methods to find the exchange rate.
We can find the gradient.
I need the vertical change over the horizontal change.
I get 1.
2.
We could do it by coordinates.
The coordinate, 100, 120.
120 divided by 100 is 1.
2.
That's the multiplier, and we know that the gradient, and the multiplier will be the same, so no surprise, and that's 1.
2.
Therefore, the exchange rate must be one pound is equal to Euros 1.
20.
Quick check you can do that for a different currency exchange this time.
Use two different methods to find the exchange rate between British pounds and Chinese Yuan.
Pause and do this now.
Welcome back.
Let's see how we did.
Hopefully one of your methods was finding the gradient.
If I look at the vertical change of 183 here, the horizontal change of 20, I found a gradient of 9.
15.
The other method would've been to look at the coordinates.
We look at the coordinate pair 40, 366.
That's the multiplier.
The exchange rate, therefore must be that one pound is 9.
15 Yuan.
We frequently see direct proportion in the context of purchasing.
Example.
For a manufacturing firm, the cost is in direct proportion to the quantity of bolts they purchase.
50 bolts cost 35 pounds.
How much would 200 cost? We can do this numerically.
If 50 bolts costs 35 pounds, we don't need 50 bolts.
We need 200 bolts.
We need to multiply everything by 4.
50 by 4 makes our 200 bolts.
35 pounds, 4 makes 140 pounds.
We know it'll cost 140 pounds.
You can also see this graphically.
This same mathematical relationship, but on a direct proportion graph.
There's our 50 bolts costing 35 pounds, and that's a graph of direct proportion, constant rate of change, intercept at the origin.
We need to repeat this step of 50 bolts, and 35 pounds, 4 times.
Four equal steps like that gets us to this point, 200, 140.
Oh look, 200 bolts costing 140 pounds.
Quick check you can repeat that skill now.
For a manufacturing firm, the cost is in direct proportion to the quantity of nuts they purchase.
20 nuts costs 8 pounds.
How much would 100 be? For Part A, I'd like to calculate this numerically, and for Part B, I'd like you to describe how you could show the exact same result graphically.
Pause and do that now.
Welcome back.
Part A numerically.
Let's start with 20 nuts costing 8 pounds.
We don't want 20 nuts, we want 100, so we need to multiply by 5 on both sides of our ratio.
8 pounds multiplied by 5 pounds becomes 40 pounds.
100 nuts are gonna cost us 40 pounds.
Part B, I asked you to describe how you could share the same result graphically.
Your description should have said something along the lines of, "Five equal steps of 20 along, and 8 up." Let's see what that looks like.
1, 2, 3, 4, 5 equal steps, of 20 long, and 8 up reaching that point.
100 bolts costing 40 pounds.
A graph can also help us immediately see when two variables are not in direct proportion.
A taxi firm charge 8 pound for a 2 mile journey, and 14 pounds for an 8 mile one.
Those two coordinates reflect those two charges.
How can we immediately see that this relationship is not in direct proportion? There's not a relationship of direct proportion between the miles travelled, and the pounds charged.
Well done.
You're shouting at the screen, "That line does not intercept the axes at the origin.
That's how we know." The straight line would not pass through the origin, so this is not direct proportion.
We didn't have to do any number work.
We could immediately see from the visual image that this was not going to be direct proportion.
Quick check you've got that.
How do we know that X and Y are not in direct proportion for this pair of coordinates? Pause, tell the person next to you, or say it loud to me at the screen.
Welcome back.
I hope you said something along the lines of, "A straight line would not intersect the Y axis at the origin so X and Y are not in direct proportion." Again, no number work necessary there.
We could see it from the visual imagery on the graph, that is not a direct proportion graph.
In some cases we'll not be given the graph.
The first coordinate I have is 8, 71.
The second quarter I have is 11, 92.
For this pair of coordinates, are X and Y in direct proportion? It'll be inefficient to draw the graph, so we need other methods, and it's nice to know we've got a few.
Method one, we could use the multiplicative relationship.
To turn 8 into 71, I'd need to multiply by 8.
875.
To turn 11 into 92, I'd need to multiply by 8.
36.
I've truncated that number.
But importantly we can see that's not a constant multiplicative relationship.
This is not direct proportion.
If they were in direct proportion, the multiplier turning our X coordinate into our Y coordinate would be the same.
It would be constant.
In this case it's not.
That's method one.
Lovely to know we're always armed with multiple methods to solve the same problem in mathematics.
We could look at the equation of the line.
To find the equation of the line, the intercept of these two coordinates, firstly, we need the gradient.
The gradient is the change in Y over the change in X.
The change in Y is 21.
Change in X is 3.
21 over 3 is 7.
We've got a gradient of 7.
We now need to find the Y intercept, and we'll have the equation of the line that goes through these two points.
So Y equals 7X plus C with C being the Y coordinate at the intercept.
Substitute in a known coordinate.
We know that when X equals 8, Y equals 71, so we can substitute that in.
And we'll find that C equals 15.
Aha! A Y intercept at 0, 15.
Is that direct proportion? Well done.
It absolutely is not.
We know that a direct proportion graph must have the form Y equals KX with a Y intercept at the origin.
In this case, we've got Y intercept at 0, 15, so we know it's not direct proportion.
Quick check now.
Let's look at this problem that Izzy's been wrestling with.
For this pair of coordinates are X and Y in direct proportion? That's the question that was posed to Izzy.
Izzy started with, "I need the gradient." Good start Izzy.
That's the change in Y over the change in X giving a gradient of 9, okay? Yes, they are in direct proportion with a multiplier of 9, okay? Is Izzy correct? What do you think? Pause, have a conversation with a person next to you or a good think to yourself.
Is Izzy right? Is this direct proportion, a constant multiplier of 9? You tell me.
Pause, see you in a moment.
Welcome back.
I hope you said, "No, Izzy's not correct." There's multiple things you could do to check she's correct.
One is to check the Y intercept.
If we know that the equation of the line is Y equals 9X plus C, i.
e.
that's a gradient of 9, the X coefficient, let's find that C, and it'll reveal the Y intercept to us.
When we substitute in a known coordinate, so when X equals 17, we know that Y equals 150, we find that C equals negative three.
That is not an intercept at the origin.
That therefore will not be direct proportion.
Practise time now, Question 1.
Using this conversion graph, show two methods for finding the exchange rate from British pounds to Turkish Lira.
Use the information on the graph, and I'd like to see two methods.
Pause and do that now.
For Question 2, a mountain bike hire firm charge 100 pound for a 7 hour hire, and 60 pounds for a three hour hire.
Use this grid to make a sketch which demonstrates why the hire charge and time are not in direct proportion, and write a sentence to support your answer.
Pause and do that now.
Question 3.
Which of these pairs of coordinates have a directly proportional relationship between the X and Y variables? Three pairs to consider.
Which are in direct proportion? Which are not? You'll need some good mathematics to support your answer in each case.
Pause, give this a go now.
Feedback time now.
Question 1 asked you to use a conversion graph, and show two methods for finding the exchange rate from British pounds to Turkish Lira.
It's clearly a graph of direct proportion so we can consider the gradient.
Reading from any two coordinates, you'll find that your change in Y over your change in X, or your change in Lira over your change in pounds is 41.
2.
Alternatively, you could have picked coordinates, and looked for the multiplier that links the pound coordinate to the Lira coordinate.
In this case, 7,210 divided by 175 was 41.
2.
No surprise to see that multiplier is the exact same value as our gradient.
Therefore, our exchange rate must be that one pound is equal to 41.
2 Turkish Lira.
Question 2, a mountain bike hire firm charging 100 pound for 7 hours, 60 pound for 3 hours, and we're going to make a sketch to demonstrate why this is not an example of direct proportion.
There's not a directly proportional link between the amount of time we hire this bike for, and what we are charged.
Firstly, we're gonna need some numbers on those axes, and we're gonna plot the two known values, 7 hours as 100 pounds, 3 hours as 60 pounds and our straight line.
That's how we show it's not direct proportion.
You'd need to back that up with a sentence to explain what's going on here to any reader.
You might have written, "Constant rate of change, yes, but not passing through the origin.
Therefore, this is not directly proportional." Question 3, three pairs of coordinates, and I asked you to consider which ones have a directly proportional relationship.
You had two methods to do this.
I'm gonna go for the equation of the line method.
For the first pair of coordinates, the change in Y over the change in X, so we also get a gradient of 12.
When we put that into our Y equals MX plus C, to find the Y intercept, we find that C is indeed 0.
That'll give that line a Y intercept, 0,0.
That's the origin.
That pair are in direct proportion.
You might have looked at the constant multiplier between the coordinates, and that method would've been fine, as long as you said, "This is an example of direct proportion." For B, the gradient was 9.
The Y intercept when you substitute that back into the general form of a straight line, was 40, or should I say it would intercept the Y axis at 0, 40, therefore not direct proportion.
For C, the gradient was 6.
5.
When we substitute that back into the equation of a line, we find our C value to be 0.
That means we've got a Y intercept, 0, 0.
That's the origin.
This is a case of direct proportion.
That's the end of the lesson now, sadly.
What we've learned is that we can recognise direct proportion graphically, by the key features of a constant rate of change, and a Y intercept at the origin.
We can interpret graphs in context that illustrate direct proportion such as currency conversion graphs.
Hope you enjoyed this lesson.
Hope you found it interesting.
Hope you enjoyed the links to the wider world of mathematics beyond algebra and graphs.
I'll look forward to seeing you again soon for more lovely mathematics.
Goodbye for now.
