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Hi there, my name is Miss Lambell.
You've made a really good choice deciding to join me today to do some maths.
Come on, let's get started.
Welcome to today's lesson.
The title of today's lesson is "Graphing Direct Proportion," and that's in the unit "Understanding Multiplicative Relationships" and concentrating on percentages and proportion.
By the end of this lesson, you'll be able to recognise direct proportion graphically and be able to identify features of the graph.
So if I show you a graph, you will be able to tell me if it does show variables that are directly proportional to each other.
The keyword for today's lesson, obviously then, is going to be direct proportion.
Two variables are in direct proportion if they have a constant multiplicative relationship.
In today's lesson, there are two learning cycles.
In the first one, we're going to look at features of graphs, and in the second one, graphs representing direct proportion.
Come on then.
Let's get started.
Features of graphs.
We know that we can show any variables which are in direct proportion on a graph.
We have seen this previously when looking at things like conversions, so converting between units of measure.
And those could be metric units of measure, so converting between centimetres and metres, or it could be converting between metric and imperial, so be converting between the number of miles and the number of kilometres.
Also with conversions, we could think of that as exchange rates, so converting between the number of pounds and the number of dollars.
And costs.
If we know the cost of one item or more than one item, then we can scale this to find the cost of any items. Saying that word scale has just reminded me of another example, and that's with recipes, so we can scale recipes to ensure that we've got enough of whatever it is we are making.
In this lesson, we're going to concentrate on what the key features of a graph showing direct proportion are.
Before we do that, we will use a ratio table to check whether variables are proportional to each other.
If two variables have a constant multiplicative relationship, they are in direct proportion, a constant multiplicative relationship.
We are going to consider if A and B are directly proportional to each other.
Here's a table showing A and B.
I'm looking for my multiplicative relationship.
What's the multiplicative relationship between 3 and 15? Yes, that's right.
It's multiplied by 5.
I now need to check what the multiplicative relationship between 12 and 60 is.
What's my multiplicative relationship here? Yeah, you're right.
Multiply by 5.
That's a constant multiplicative relationship.
They both multiply by 5, so these are directly proportional to each other because they have a constant multiplicative relationship, and we're referring to these variables now as being directly proportional to each other.
And this table, what is the multiplicative relationship between 6 and 15? Now, I've purposely chosen here to go horizontally rather than vertically because 6 and 7, the multiplicative relationship between those, gives me a horrible decimal, but remember I could leave it as a fraction.
But I thought it would be also useful to remind you that if we're looking at checking whether something is directly proportional, we can move either vertically or horizontally.
6 multiplied by 2.
5 is 15.
7 multiplied by 3 is 21.
Does this show that A and B are directly proportional? No.
You're right.
There isn't a constant multiplicative relationship, so therefore they are not directly proportional to each other.
That multiplicative relationship must be constant.
Just a quick check for understanding then.
I'd like to sort the following into examples and non-examples of tables showing direct proportion.
So is it an example of one showing direct proportion or not an example, a non-example.
Pause the video, and then when you've decided, come back and we'll check.
How did you get on? Let's look at the first one.
That is an example.
If I look, I've multiplied 12 by 8 to get to 96, I've multiplied 6 by 8 to get to 48, or I could move vertically, 6 is half of 12 and 48 is half of 96.
b is a non-example.
If I look at my multiplicative relationship, it's not the same.
And then c is an example, okay? So c does show an example.
Hopefully you've got all of those right.
A box of pens costs 3 pounds.
The number of boxes and cost are directly proportional.
If I increase the number of boxes of pens I buy, then the cost is going to increase at the same rate.
Here's our graph showing the cost and the boxes, and this shows a directly proportional relationship.
Plumber A charges 40 pounds per hour.
Plumber B charges a 50 pound call out charge plus 30 pounds per hour.
Are the number of hours and charge directly proportional? So A and B.
Just like you to think.
Do you think they are directly proportional? Both of them, neither of them, or one of them? We'll take a look at their graphs.
Here is a graph that shows plumber A.
We know that every hour is 40 pounds, so I use that to mark on 1 and 40 as a coordinate point.
So 2 is gonna cost me 80 pounds, and 3, 120, and so on.
And this is the graph for plumber B.
Now, plumber B, if I work out how much I'm actually going to pay if plumber B comes for one hour.
The charge he charges or she charges.
A 50 pound call out charge plus 30 pounds per hour, so for one hour it's going to cost me 80 pounds.
2 hours is going to cost me the 50 pound out charge plus 2 lots of 30, so 2 hours is going to cost me 110.
So hopefully you can see now how I've plotted those graphs.
Do we think this graph shows direct proportion? Let's check using our method.
So our method, remember, is to read off one value, and I've chosen 1 just 'cause that's nice and easy.
So 1 hour is 40 pounds, and then double the number of hours and read off the price.
My multiplicative relationship moving vertically is multiply by 2 multiply by 2.
Yes, plumber A's charge is directly proportional to the number of hours.
Let's take a look now then at plumber B.
What do you think? Does this one show direct proportion? We'll check using our method.
1 hour costs 80 pounds.
2 hours costs 110 pounds.
Our multiplicative relationship between 1 and 2 is multiply by 2, and the multiplicative between 80 and 110 is multiply by 1.
375.
Therefore, does that show that those values, or variables I should say, are directly proportional? And the answer is no.
So our check was to double the number of hours and see if the price doubled, and here the price does not double.
Now I'd like you to consider this question.
What is the same and what is different about those graphs? We know that the one on the left shows a directly proportional relationship and the one on the right doesn't.
Pause the video.
Have a think about this.
If you've got somebody there with you, have a chat with them about it, and then come back when you're ready, and we'll see what you've come up with.
Okay, they both have a constant positive gradient.
Now, when I'm saying those words there, they should be very familiar to you.
Positive gradient sloping upwards.
Constant, it is just one straight line.
The one on the left starts at the origin.
It starts at the 0.
00, and the one on the right doesn't.
The left one we already established does show direct proportion, and the one on the right does not show direct proportion.
Let's take a look at these two graphs now.
Graphs showing direct proportion have a constant positive gradient with an intercept of zero.
Let's look firstly at the graph on the left hand side.
Does it have a constant positive gradient? Does it have an intercept of zero? This is a graph showing direct proportion because it has an intercept of zero and we can see that it has a constant positive gradient.
Now let's consider the graph on the right hand side.
Does it have a constant positive gradient? Does it have an interceptive zero? This is a non-example because it doesn't show direct proportion.
Here we can see that the intercept is not zero.
It doesn't matter that I've not given you a scale there.
We know that it is not going to go through the origin because we can clearly see that it's above there.
It has a constant positive gradient, but here the intercept is 0.
5 rather than zero.
Key things then.
If it shows direct proportion, it has a constant positive gradient and the intercept will be zero, or the origin.
Here's a check for you.
I'd like you to sort the following into examples and non-examples of graphs showing direct proportion.
Pause the video, and when you've decided which is which, come back and we'll check.
Great work.
a is an example.
It's got a constant positive gradient and the intercept is zero.
b, now you have to be careful here.
Hopefully you weren't sat too far away from the screen.
We can see here it's got a constant positive gradient, but it doesn't start at the origin.
The intercept is not zero.
c is an example.
It has a constant positive gradient and the intercept is zero.
d is a non-example.
We can see it has a constant positive gradient, but its intercept is not zero.
And finally e is a non-example.
It has an intercept of zero, but it doesn't have a constant positive gradient because we can see that the gradient of the line changes as we move across the graph.
Your turn to have a go at this task.
Firstly you are going to decide which of the following tables show that A and B are in direct proportion, and remember to give a reason for your answers.
Pause the video, and when you're done, come back, and we'll have a look at question number two.
Well done.
Question number two.
Which of the following graphs show two variables that are in direct proportion? Give reasons for your answers.
Pause the video, and I'll be waiting when you get back.
Well done.
Here are our answers.
So a is correct.
If we look, the multiplicative relationship between A and A is multiply by 4, and B and B is multiply by 4.
b, we can clearly see there that it is not the same multiplicative relationship, so therefore they are not in direct proportion to each other.
c, again, they're not.
Here, I've decided to get move horizontally rather than vertically.
It doesn't matter.
You may have a different value there.
We can see it's not constant.
And then d is showing A and B are directly proportional because it has a constant multiplicative relationship.
And then onto question two.
a was not because it had a constant positive gradient, but it did not have an intercept of zero.
b was showing direct proportion because both of those things were satisfied.
Constant positive gradient, intercept of zero.
c, we can see it's got a constant positive gradient, but we can clearly see the intercept is not zero.
And d, yes, it has an intercept of zero, but remember here the gradient changes.
The gradient is not constant throughout the entire graph, so it does not show direct proportion.
Now we'll have a look at graphs representing direct proportion.
So we've already looked that, but we're going to take that a little bit further now.
The table below shows two variables, x and y.
Are they directly proportional? What did you decide? Let's look for the multiplicative relationship between x and y.
5 multiplied by something is 15.
Well, that's 3.
Remember we need to check though, so let's check using some other values in our table.
1 multiplied by 3 is 3.
3 multiplied by 3 is 9.
Yes, there is a constant multiplicative relationship of multiply by 3.
What is the multiplicative relationship between x and y? Well, y is equal to 3 multiplied by x.
I started with x, I multiplied it by 3, and then that gave me my value for y.
So y equals 3 multiply by x 'cause generally when we're talking about graphing things, we write them in the form y equals.
Let's plot the graph that represents this relationship.
Here's my graph.
I've got x and y.
Zero, zero is a point on my graph.
I know that 1, 3 is also on my graph.
I know that 2, 6 is on my graph.
And I could continue going all the way across my graph and then remember to join those points together with a ruler and a straight line.
What is the gradient of the line? So gradient, you might have to dig back in your memory to when you last did gradient.
What's the gradient of the line? Let's take a look.
So if you haven't quite remembered, we're gonna go through it now anyway, so don't worry.
The gradient is my change.
As I move one square to the right in the x direction, I'm moving 3 squares up in the y direction.
The gradient of the line is 3.
What do you notice? The multiplicative relationship is the same as the gradient.
y equals 3 multiplied by x, and the gradient of the line was 3.
Here we have Jun.
She says, "I wonder if the multiplicative relationship between x and y is always the same as the gradient?" What do you think? Let's take a look at another example.
Here we have a table of values.
Our multiplicative relationship between x and y is multiplied by 1.
5.
Just check that works with some other values.
2 multiplied by 1.
5 is 3.
6 multiplied by 1.
5 is 9.
Yes, that is the correct multiplicative relationship between x and y.
Here is my graph showing multiplicative relationship or showing those values plotted onto a graph.
Let's look at the gradient.
Here, I've decided to go across 2 in the x direction, and that means that I've moved three in the y direction.
So my change in x is 2 and my change in y is 3, but our gradient is always the change in x is 1.
So what have I done to get from 2 to 1? I've divided by 2, so therefore I'm going to do 3 divided by 2, and that's 1.
5.
The gradient is the same as the multiplicative relationship, and you may already be thinking to yourself, "Well, of course it is," because of you know how you find your values of x and your values of y.
Below is a graph showing the relationship between the number of cookies and the amount of chocolate chips.
Quite fancy a chocolate chip cookie.
How about you? What is the multiplicative relationship between them? So I'm choosing a point on my line, and that point is 2, 26.
Notice I've chosen one where I have integer coordinates.
That's important.
You can do it without, but actually it just makes life easier if you choose coordinates that have points that have integer coordinates.
And then this is the other one I've chosen, which is the coordinate point 4, 52.
I'm looking for my change in the x direction, that's plus 2, and my change in the y direction, which is plus 26.
By drawing out my ratio table, change in cookies is 2, which means the change in chocolate chips is 26, but for the gradient we always need to know the change in 1 moving horizontally.
So 1 I've divided by 2, so I divide by 2 to give me 30.
The multiplicative relationship between the number of cookies and the amount of chocolate chips is number of cookies multiplied by 13 equals the grammes of chocolate chips.
We know that graphs showing direct proportion have a constant positive gradient and an intercept of zero.
This means that they can all be written as an equation in the form y equals kx, where k is referred to as the constant of proportionality.
The constant of proportionality is the multiplicative relationship between the two variables.
We've already established that.
It's the rate of change, and remember that's what a gradient is showing us, and the gradient, which we've also established.
Let's take a look at the graphs we looked at earlier.
The multiplicative relationship between x and y is y equals 3x.
So we looked at that earlier.
We came up with that.
All I've done here is missed out that multiplication symbol, but we know that that's okay when we're talking about algebra.
The rate of change in y as x increases by 1 is 3.
The gradient of the line is 3.
The constant of proportionality is 3.
The equation of this line is y equals 3x.
Let's take a look at this one.
Here, we didn't know the change in x of 1.
We knew it was change of x is 2 and then a change in y was 3.
For the gradient, we need to know the change in x of 1, so we've filled in our ratio table.
The multiplicative relationship between x and y is y equals 1.
5x.
The rate of change in y as x increases by 1 is 1.
5.
Every time I go one square to the right, I go up 1.
5 squares.
That's what we mean by the rate of change.
The gradient of the line is 1.
5.
The constant of proportionality is 1.
5.
The equation of this line is y equals 1.
5x.
So we can see that actually it's the same in all of them.
It's just different ways that we can refer to that value of 1.
5.
And here is our cookie and chocolate chip example.
Same thing.
We already looked at that ratio table.
The multiplicative relationship between x and y is y equals 13x.
The rate of change in y as x increases by 1 is 13.
Every time I add another cookie in, I need another 13 grammes of chocolate chips.
The gradient of the line is 13, the constant of proportionality is 13, and the equation of this line is y equals 13x.
Now a check for you.
I'd like you to pause the video and decide what you think needs to go in each of those spaces.
Good luck with this.
Remember, if you need to just go back and have a look at the previous slides again, of course you can.
How did you get on? Let's take a look.
I've drawn on my lines so I know my change in a and my change in b.
My change in a was 4, my change in b was 2, but remember, we need to know the change for the gradient.
We need to know the change in a of 1, so I have divided 4 by 4 to get to 1.
2 divided by 4 is 9.
5.
We're looking here at this type of graph at the multiplicative relationship because it has a constant positive gradient and an intercept of zero.
We're looking at the multiplicative relationship between the a and b, and that is that b equals 0.
5a.
The rate of change of b, sorry, the rate of change in b as a increases by 1 is 0.
5.
The gradient of the line is 0.
5.
The constant proportionality is 0.
5.
The equation of this line is b equals 0.
5a.
Hopefully now you can see the link between all of those different ways of saying the same thing essentially.
Now what I'd like you to do is to calculate the value of k.
Remember, that's your constant of proportionality, so for each of these graphs.
Pause the video, and come back when you're ready.
Okay, question number two.
Here, I'd like you to match the table, to the graph, to the equation.
So matching one table, one graph, one equation.
Good luck.
Pause the video, and then when you get back, we can check those answers.
Great work.
Question number one, a, k was 2.
b, and I'm going down, k was 2.
5.
You may have that as a fraction.
I didn't say whether I wanted you to write it as a fraction or a decimal.
c is 0.
25 or 1/4, and d is 0.
2.
Again, you may have that as a fraction of 1/5.
Question number two.
Table a matched with graph C and equation iii.
Table b matched with graph D and equation i.
c, table c, sorry, matched with graph B and equation iv, and table d matched with graph A and equation ii.
How did you get on? Amazing.
Let's now summarise our learning.
The key features of a graph showing direct proportion are really important to remember.
It has a constant positive gradient and the origin is the y intercept, so it has an intercept of zero, so it must start at zero, zero.
As soon as we see a graph that doesn't start at zero, zero, then we know it cannot possibly be showing direct proportion.
Because of this, we know then that all graphs showing direct proportion can be written in the form y equals kx, where k is the constant of proportionality.
The constant of proportionality is the multiplicative relationship between the two variables, the rate of change, so as I increase 1 in the x direction, what is my increase in the y direction, which is another way of working out the gradient.
Well done on today's learning.
You've done fantastically well.
Quite a challenging one there I think, so well done for sticking with me.
I look forward to seeing you really soon.
Goodbye.
