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Hi, there.
My name's Ms. Lambell.
You've made such a fantastic choice deciding to join me today to do some Math.
Come on, let's get going.
Welcome to today's lesson.
The title of today's lesson is "Checking and securing understanding of direct proportion graphs." And that's within the Unit: Direct and inverse proportion.
By the end of this lesson, you'll be able to identify direct proportion graphs from their features and you'll be able to use the graph to create an equation to algebraically model the relationship.
Some keywords that we'll be using in today's lesson are direct proportion, gradient and intercept.
Here's just a quick reminder of those just in case you need it.
Two variables are in direct proportion if they have a constant multiplicative relationship.
The gradient is a measure of how steep a line is.
It is calculated by finding the rate of change in the y-direction with respect to the positive x-direction.
And an intercept is a point where a line or curve meets a given axis.
Today's lesson is split into two learning cycles.
In the first one, we will review the features of direct proportion graphs.
And in the second one, we will look at finding the equations of direct proportion graphs.
Let's get going with that first one.
So we're gonna be reviewing features of direct proportion graphs.
Let's go.
We know that we can show any variables which are in direct proportion on a graph.
We have seen this previously when we looked at conversions, costs, speed, et cetera.
There are lots of variables if they're directly proportional that can be represented by graph.
In this lesson, we are going to concentrate on what the key features of a graph showing direct proportion are.
And you may already have an idea about this 'cause like I said, this is a check-in and securing understanding lesson.
Before we do that though, 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.
We'll consider if A and B are directly proportional to each other.
Here's a ratio table with A and B.
Let's look for the multiplicative relationship between the two A values.
Six multiplied by something is 30 and we know that that's five.
Now let's find the multiplicative relationship between the two B values.
Nine multiplied by something is 45.
And again, we know that that's five.
We can see it has a constant multiplicative relationship.
Let's check what happens as we move from A to B.
What is the multiplicative relationship between A and B for the top row.
Six multiplied by something is nine.
And that's not as easy to do, is it? So remember, you can always reverse and do nine divided by six to find that multiplier.
And that's 1.
5.
We're gonna do the same thing now with the bottom row.
30 multiplied by something is 45, and that is multiply by 1.
5.
Again, we can see that the relationship between A and B as a constant multiplier.
The multiplicative relationship is constant, therefore A and B are directly proportional to each other.
Let's take a look at this ratio table representing A and B.
I'm going to look at the horizontal movement, so I'm looking at the multiplicative relationship between A and B on the top row.
20 divided by eight is 2.
5.
Eight multiplied by 2.
5 is 20.
10 multiplied by something is 30.
Well, we know that that's three.
We can clearly see here that that multiplicative relationship is not constant.
Therefore, A and B are not directly proportional to each other.
Your turn now.
Which of the following tables show values of A and B in direct proportion? So you're going to pause the video.
You're going to look for those multiplicative relationships.
Remember, if it's constant, they are in direct proportion.
If it's not constant, then they're not in direct proportion.
Pause the video and when you come back, we'll check those answers for you.
How did you get on? Let's check.
A was correct.
A and B were directly proportional to each other.
B, they were not directly proportional to each other.
And in C, they were directly proportional to each other.
How did you get on with those? You got them right? Brilliant, of course you did.
A box of pens cost 2.
50.
The number of boxes and the cost are directly proportional.
Here is a graph representing the number of boxes against the cost.
You can see that one box cost 2.
50, so two boxes would cost five pounds, three boxes cost 7.
50 and four boxes would cost 10 pounds.
We can see that we can represent this graphically.
This graph shows a proportional relationship or I should say directly proportional relationship.
Now let's take a look at this scenario.
Electrician A charges 45 pounds per hour.
Electrician B charges a 40 pound call out charge plus 35 pounds per hour.
Are the number of hours and charge directly proportional? What do you think? Well, let's take a look at the graphs.
Here is a graph representing Electrician A.
We've got our hours and then the charge.
And here is the graph for Electrician B.
Again, we've got the hours and the charge.
Let's firstly look at Electrician A.
Does this graph show direct proportion? Well, let's check using our method and I'm not sure I've mentioned it already, but remember our method is, is if I read off one hour, then two hours should be double if they are directly proportional to each other.
Let's use the graph to read off the cost of one hour using Electrician A.
So we find our value.
Remember we draw a line up from the one hour to our line and then across and we can see that one hour costs us 45 pounds.
Now let's use the graph to work at how much it's going to cost us for two hours.
Two hours is 90 pounds.
One multiplied by two is two.
And 45 multiplied by two is 90.
Electrician A's charge is directly proportional to the number of hours.
Now let's compare that to Electrician B.
What do you think about this one? Do you think this one shows direct proportion? Again, let's check using our method.
Let's read off what one hour costs.
One hour costs 75 pounds.
Now let's read off what two hours cost.
And we can see there that it's halfway between 100 and 120, so two hours cost is 110 pounds.
My multiplier between one and two is two, but my multiplier between 75 and 110 is 22/15.
How did I get that 22/15? Yeah, that's right.
I did 110 divided by 75 and I decided to leave my answer there as an exact fraction.
We can see here that Electrician B's charge is not directly proportional to the number of hours.
If I double the time, it does not double the cost.
That's a really good way remember of checking whether two variables are directly proportional to each other.
What is the same and what is different about these two graphs? Pause the video, have a think about that, and then when you've got some ideas, come back and we'll talk about them.
(birds chirping) What did you decide? Well, they both have a constant positive gradient.
We can see that it's one straight line and it's positive sloping upwards.
The one on the left starts at the origin.
It starts at zero, zero, but the one on the right does not.
The one on the right starts at 0, 40.
The first graph, Electrician A, we know shows direct proportion.
We've worked that out using.
Reading off our values for one and two hours.
And this graph for Electrician B does not show direct proportion and we've shown that again with our one and our two hours.
Graphs showing direct proportion have a constant positive gradient with an intercept of 0, 0.
The one on the left, is an example of a graph showing direct proportion.
We can see that it starts at zero, zero.
It intercepts the y-axis at zero, zero.
This graph has a constant positive gradient and the y-intercept at the origin.
If we compare that to the graph on the right hand side, we can see that this is a non-example of a graph showing direct proportion.
We can see clearly that the graph does not start at the origin.
It has a constant positive gradient, but the y-intercept is not at the origin and therefore does not show direct proportion.
Looking again at Electrician B's graph, although the electrician's total charge is not directly proportional to the number of hours, their hourly cost is.
Translating the line, we can show this.
So if we translate the line, eventually it will end up at zero, zero.
Although, this graph now shows the proportional relationship between time and cost, reading from this graph would not give you the total cost.
You would have to remember to add on the fixed charge of 40 pounds.
Which of the following show graphs of variables in direct proportion? Pause the video, make your decisions and then come back.
And what did you decide? A, yes it does.
It's a constant positive gradient and it starts at zero, zero.
B, looks like it possibly might be, but we can see that actually it doesn't start at zero, zero, it doesn't start at the origin.
C is correct.
Constant positive gradient.
And it starts at zero, zero.
D, it doesn't start at zero, zero.
And E, although it does start at zero, zero, it doesn't have a constant positive gradient.
The gradient changes partway through the graph.
A water tank with a capacity of 15,000 litres is leaking at a rate of one litre per hour.
Here is graph representing the litres in the tank in thousands measured against the number of hours.
Jacob says, "I have drawn a graph." We can see Jacob's graph.
Well done, Jacob, "to represent this." Jacob says, "I have drawn a graph to represent this." And we can see here Jacob's graph, well done, Jacob.
"I thought this would be a direct proportion problem as the rate at which the water is lost is constant and over time the amount of water lost is increasing." Sofia says, "Me too, Jacob.
This graph has a constant gradient but it is negative so it does not show direct proportion." Do you think that this problem is a direct proportion problem? A water tank with a capacity of 15,000 litres is leaking at a rate of one litre per hour.
Do you think that's a direct proportion problem? What did you think? Let's move on and have a look and see what Jacob and Sofia say.
Sofia says, "This is really puzzling me.
After three hours it will have lost three litres and after 12 hours it will have lost 12 litres.
This is direct proportion." Jacob says, "That's it, Sofia.
If we plot the graph of water lost rather than water left in the tank, we will have a graph with a constant positive gradient." Let's look at the graph of the litres lost over time.
Now we can see that because we've changed what we are plotting against the hours and here we are looking at litres lost, we can see that it shows a graph of direct proportion.
Sofia says, "That is better.
I'm glad we sorted that out." And Jacob says, "So we need to think carefully about what variables we plot on the graph." Your turn now to have a go at Task A.
Which of the following tables show that A and B are direct? Which of the following tables show that A and B are in direct proportion? You must give a reason for your answer, and if they are in direct proportion, I would like you, please, to find the value of B when A is five and if necessary.
Now it's your turn in Task A.
Question number one, which of the following tables show that A and B are in direct proportion? I'd like you to give a reason for your answer and for those that show direct proportion, I'd also like you to find the value of B when A is five.
And if appropriate, we'll give your answer to one decimal place.
Pause the video, come back when you've got your answers.
You may use this calculator here.
Good luck.
Well done on those.
And question number two, which of the following graphs show variables that are in direct proportion? And again, I'd like you please to give a reason for your answers.
Pause the video and then when you're ready, come back and we'll check the answers to questions one and two.
Well done.
Now let's check our answers.
Question One.
A.
We can see the multiplicative relationship is multiplied by four and multiplied four.
So yes, it is a directly proportional relationship.
A is five B would be 69.
2.
B, Here we are multiplying by 0.
25 or divided by four, and this side we're multiplying by four, so therefore it does not show a directly proportional relationship.
C, to go from A to B, I've multiplied by 0.
4 and I've multiplied by 0.
3.
Remember here, you may have decided to work vertically rather than horizontally.
It is not a constant multiplier, therefore, it does not show a directly proportional relationship.
And finally, D, I've decided to go horizontally.
My multiplier is 1.
8 and my multiplier is 1.
8.
So yes, it does show a directly proportional relationship between A and B.
When A is five, B is nine.
Question number Two.
A, did not show a directly proportional relationship.
It has a constant positive gradient, but the y-intercept is not the origin.
B, did show a directly proportional relationship.
It had a constant positive gradient and the y-intercept was the origin.
C, did not.
It had a constant positive gradient, but the y-intercept was not the origin.
It was very close to the origin, but it wasn't the origin.
And then D was no.
The y-intercept is the origin, but the gradient is not constant.
We can see that the gradient changes along the graph.
Now we can move on then to that second learning cycle.
And in this set learning cycle, we are going to be concentrating on equations of direct proportion graphs.
The table below shows two variables x and y.
Are they directly proportional? What do you think? If we look, we can see to go from x to y, I'm multiplying by four.
Let's just check that.
Two multiply by four is eight, four multiplied by four is 16, five multiplied by four is 20.
Yeah, to go from X to Y, I have multiplied by four.
What then is the multiplicative relationship between x and y? And you might be thinking to yourself, "Hang on a minute, haven't you just answered that question?" But think about how you might write this.
Y is equal to four multiplied by X.
Y is always four times bigger than X.
We can plot this on a graph to represent the relationship.
And we'll plot in the coordinate points, Zero, zero, one, four, two, eight, three, twelve and four, 16.
I haven't been able to fit the five, 20 on my graph, but that doesn't matter.
We can see then that we end up with this linear graph.
What is the gradient of the line? Can you remember how to calculate the gradient? Let's calculate the gradient.
We need to know the change in y for every one increase in x.
I'm going to start at the 0.
28 and I'm going to move one place to the right and then I'm going to look to see what the change in Y is.
And we can see here that the change in Y is four.
It's gone from eight to 12.
So the gradient of the line is four.
Did you decide the gradient was four? Well done.
What do you notice? And you should notice that the multiplicative relationship is the same as the gradient.
The gradient was four and we were multiplying X by four to give us Y.
What's Jun saying? "I wonder if the multiplicative relationship between X and Y is always the same as the gradient?" What do you think? Here we've got a table of values of X and Y.
What's the multiplicative relationship here? We're multiplying by 2.
5.
Here is the graph.
What about the gradient? We're checking to see if the gradient is 2.
5.
Here, my change in X is two and my change in Y is five.
Putting this into our ratio table, the change in X is two and the change in Y is five.
But remember the gradient is the change in X is one.
So we're going to divide by two, divide by two, and we can see that yes, the gradient is the same as the multiplicative relationship.
Below is a graph showing the relationship between the number of necklaces and the number of red beads needed.
What is the multiplicative relationship between them? So we take a point that we know the exact coordinates of, that's important to choose a point.
We know the exact coordinates of that point are 2, 26.
And we choose a second point.
Again, one that we know the exact coordinates of.
The exact coordinates of this point are 4, 52.
We are looking at the change in X.
It's gone from two to four, so that's plus two.
And we are looking for the change in Y and it's gone from 26 to 52, so that's a change of 26 putting this into our ratio table.
If the change in necklaces is two, the change in red beads is 26.
But for the gradient, we need the change in necklaces to be one.
So I'm dividing by two, which gives me 13.
The multiplicative relationship between the number of necklaces and the number of red beads is, number of necklaces multiplied by 13 equals the number of red beads.
We know that graphs show in direct proportion have a constant positive gradient and a y-intercept of zero.
This means that they can all be written as an equation in the form y = kx, where K is the constant of proportionality.
The constant of proportionality is, the multiplicative relationship between the two variables, the rate of change, and the gradient.
Let's take a look at the graphs we looked at earlier.
Here's the first graph we looked at.
The multiplicative relationship between X and Y is y = 4x.
The rate of change of Y as X increases by one is four.
The gradient of the line is four.
The constant proportionality is four.
The equation of this line is y = 4x.
Hopefully, you can see the link between all of these.
Here's the second graph we looked at.
The multiplicative relationship between X and Y is y = 2.
5x.
The rate of change of Y as X increases by one is 2.
5.
Every time we go across one, we go up by 2.
5.
The gradient of the line is 2.
5.
The constant proportionality is 2.
5.
The equation of this line is y = 2.
5x And then we've got our beads and necklaces problem.
There is our ratio table.
The multiplicative relationship between X and Y is y = 13x.
So the rate of change of Y as X increases by one is 13.
The gradient of the line is 13.
The constant proportionality is 13.
The equation of the line is y = 13x.
I'd like you now to pause the video and fill in the gaps, please.
I'll be waiting when you get back.
How did you go? Filled them all in? Superb.
Let's take a look then.
So we had our change in X was three.
And if the change in X was three, the change in Y was two.
But remember, for the gradient, we need to know the change in A as one.
And so that is going to be.
I've divided by three, two divided by three is 2/3.
The multiplicative relationship between A and B, is B equals to 2/3a.
The rate of change of B as a increases by one is 2/3.
The gradient of the line is 2/3.
The constant of proportionality is 2/3.
The equation of this line is b = 2/3a.
How did you get on? Yes, of course, you got it all right.
Now then we can have a go at Task B.
For this question, I'd like you please to calculate the value of K, the constant of proportionality.
Using these graphs, please find the value of K.
Pause the video and then when you've got your four answers, come back and I'll reveal question number two.
Well done.
And question number two.
Here, you need to match each equation to its correct graph.
Again, pause the video and then when you're ready, come back.
And now for the final question for this task and this lesson is question number three.
Explain how you know that Y is not directly proportional to X.
Pause the video, write your explanation, and then when you are ready, come back and we'll check those answers.
Great work.
Now let's check our answers.
Question one.
Part A, K was three.
B, K was 1.
5, or you may have decided to write that as 3/2.
C is 0.
5 or you might have written as a fraction as of a 1/2.
And D is 0.
4.
Question two.
A match with graph C.
B, match with graph D.
C, match with graph B.
And D match with graph A.
And then question three, this graph does not show that Y is directly proportional to X as the y-intercept is not zero, zero.
So you should have something that referred to the y-intercept not being zero, zero.
Or you may have chosen to write the word origin, instead of zero, zero.
Now we can summarise our learning from today's lesson.
Graphs showing direct proportion have a constant positive gradient and a y-interceptive of zero, zero.
Which remember is the origin.
This means that they can all be written as an equation in the form y = kx.
Where K is the constant of proportionality.
The constant of proportionality is the multiplicative relationship between the two variables, the rate of change and the gradient.
Great work today.
You've done fantastically well.
As always, look forward to seeing you again really soon.
Take care of yourself.
Goodbye.
