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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 maths.

Come on, let's get going.

Welcome to today's lesson.

The title of today's lesson is checking and securing understanding of inverse proportion graphs.

And that's with the new unit direct and inverse proportion.

By the end of this lesson, you'll be able to identify inverse proportion graphs from their features.

You will also be able to use a graph to create an equation to algebraically model a relationship.

A quick recap then of what we mean by inversely proportional.

Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.

So direct proportion is there when there is a constant multiplicative relationship between the two variables.

The difference with inversely proportional is that there is a constant multiplicative relationship between one variable and the reciprocal of the other, so that's the slight difference between the two.

Today's lesson, I've split into two separate learning cycles for you.

In the first one of those, we will look at reviewing the features of inverse proportion graphs.

You will have seen inverse proportion graphs before, we just want to make sure that you are happy with what they look like and what their main features are.

And in the second learning cycle, we will concentrate on finding the equations of those inverse proportion graphs.

Let's get going with that first learning cycle then.

Reviewing features of inverse proportion graphs.

Andeep has raised 105 pounds.

Well done, Andeep.

I wonder how he raised it.

Did he do a sponsored silence? Did he do a run? We're not sure, but he's raised 105 pounds, which is superb.

He decides to share the money equally between two charities.

So he's got two favourite charities and he decides to share the money equally between those two.

How much money will each charity receive? I'd like you to work that out for me please.

And of course you will have done 105 divided by two, meaning that each of Andeep's two favourite charities will get 52.

50 each.

But what will happen to the amount of money each charity receives if he shares the money between more charities? So he decides to share it between three charities or five charities or even 10 charities.

What will happen to the amount of money that each charity receives? As the number of charities increases, the amount each receives will decrease.

That makes sense, doesn't it? If you are sharing an amount between more things, then each is going to get less.

This is an example of inverse proportion.

Can you think of any other examples of inverse proportion? Pause the video and write down any other examples of inverse proportion you can think of.

I'm wondering what you came up with.

There's a couple that I thought of.

The time taken to run a race and the speed at which the race is being run.

The quicker you ran the race, the faster you must have been travelling.

The cost to hire a Banksy castle and the number of people who are hiring the castle.

Obviously, if one person is hiring the castle it's going to cost a lot, whereas if actually there's 10 people, they're going to share that cost between 10 of them.

So as the number of people increases, the cost per person decreases.

And the old classic one, the number of people to dig a hole and the time taken to dig that hole.

We've all seen those sorts of problems haven't we? We know that we can draw a graph of variables that are directly proportional to each other.

Therefore we must be able to draw graphs of variables that are inversely proportional to each other.

You'll be really familiar with drawing graphs of variables that are directly proportional to each other and the features of those.

They have a constant positive gradient and an intercept of 0, 0 or the origin.

Let's take a look at what the graph of Andeep's charity problem would look like.

Not a problem is it though.

Let's take a look.

I've chosen some different numbers of charities.

Obviously, we could've chosen other ones as well, but I've just chosen these as a few.

If Andeep shares his money between three charities, each charity would get 35 pounds.

And we can mark this with a cross.

Now, what about if he shared it between five charities? 105 divided by five means each charity would get 21 pounds.

And again, we can mark that.

Between six charities, each charity is going to get 17.

50.

Let's mark that one.

What about if it was seven charities? Each charity now would only get 15 pounds.

And again, let's mark that point.

I've now decided to skip to 15.

Okay, I could've done ones in between, but just for ease of plotting this on the graph, I've chosen 15, which would mean that each charity gets seven pounds.

And we can plot that point.

20 charities.

Each of them now is only getting 5.

25.

And there's that point.

And if he was going to share it between 30 charities, each one would get 3.

50.

Now representing this with 3.

We can see that this is forming what? What's the graph going to look like if I join all of those points up? That's right, it's going to be a curve.

This is what it looks like.

So we've got the number of charities and the amount of money that each charity receives.

In the last example, we were using the graph to model a real life situation.

We know that graphs exist in an abstract sense and then have an infinite nature.

What do you think the equation representing Andeep's charity money problem is? We were taking the amount of money raised, and remember this was 105 pounds.

And we were dividing it by the number of charities that we were going to share the money between.

Let's let x represent the number of charities and y represent the money received by each charity.

Can you write me now an equation in terms of x and y to represent Andeep's charity money problem.

So therefore the equation is? That's right, it's y equals 105 over x.

To find the amount of money that each charity gets, we take the total amount of money, which in this case was 105 pounds, and we divide it by the number of charities, which we have said was going to be represented by x.

We get the equation y equals 105 over x.

Now, what type of graph is y equals 105 over x? It's a particular type of graph.

Can you remember what that type of graph is? It's reciprocal graph.

Any equation of the form y equals k over x produces a reciprocal graph.

Can you remember the key features of a reciprocal graph? Don't worry, if you need a refresher we're going to take a look now.

Let's use our equation for Andeep's situation, so y equals 105 over x.

However, we are no longer going to consider the context of the fact that it was money raised and it was sharing it between different charities as we cannot give the money to, for example, half of a charity or 0.

09 of a charity.

That is not possible in real life.

So we are going to consider this equation now in its abstract sense.

We'll take a look at what happens when x is less than one.

What do you think is gonna happen when x is less than one? Well, let's take a look.

Here is my table of values.

Remember, we're using the equation y equals 105 over x.

If x is 0.

5, we're going to find y by doing 105 divided by 0.

And that gives me a value of 210.

If X is 0.

1, we do 105 divided by 0.

1, which is 1050.

Now, x is 0.

01.

Going to do 105 divided by 0.

01.

Remember, that's the same as multiplying by 100.

So we get when x is 0.

01 y is 10,500.

What about when x is 0.

001? 0.

0001 or 0.

00001? Pause the video and I'd like you to fill in those ones for me.

And hopefully, you spotted that there was a pattern each time I was multiplying by 10.

You should have then when x is 0.

001 y is 105,000.

When x is 0.

0001 y is 1 million and 50,000.

And when x is 0.

00001 y is 10,500,000.

Is that what you got? Yes, of course, you did.

Will the graph ever reach an end point? No, it will carry on infinitely.

If we zoom in to look at part of the graph, the line appears to be straight and vertical.

However, this is not the case.

It is still a curve.

But because it increases so steeply, it begins to look like a straight line.

Sticking with our same equation, y equals 105 over x.

What happens when x is less than zero? X is less than zero.

So what happens if x is negative? Have you got any thoughts on what it might look like? What might happen? Let's take a look.

Here's our table of values.

When x is -1, y is 105 divided by -1, which we know is -105.

What about if x is -10? 105 divided by -10 is -10.

Now let's say that x is -100.

105 divided by -100 is -1.

05.

And again, I'm gonna ask you please to work out for me what y is when x is -1,000, -10,000 and -100,000? Pause the video, work out those answers for me and then come back when you're ready.

And what did you get? Should have when x is -1,000, y is -0.

105.

When x is -10,000, y is negative -0.

0105.

And when x is -100,000, y is -0.

00105.

What would that look like if we were to plot that on a graph? This is the graph of y equals 105 over x.

The key features then of a graph representing inverse proportion are.

It will only appear in two quadrants.

You can see here that it has appeared in two quadrants.

It will not intercept either the x or the y-axis.

We've just looked at y.

It's because it keeps getting closer and closer and closer but never actually gets there.

Your turn now.

I'd like you please to write a sentence to support your answer to does this graph represent inverse proportion? Pause the video, write your sentence, and then come back when you're ready to check.

No, it doesn't represent inverse proportion.

And that's because one of the lines intersects with the x-axis.

We can see here that the line intersects the x-axis.

Why might someone say that it does represent inverse proportion? I can see a few people making that mistake.

Why do you think they might make that mistake? And I think that's because it is a reciprocal graph, but graphs representing inverse proportion remember, are reciprocal graphs that do not cross either axis.

So if somebody quickly just glanced at that, they would recognise it's a reciprocal graph, but we need to check carefully that it doesn't intersect either the x or the y-axis.

Task A now.

Question number one.

You're going to match each graph to the correct statement.

Y is directly proportional to x.

Y is inversely proportional to x.

And C.

The graph is neither directly or inversely proportional.

So pause the video, decide which graph is which, and then come back when you're ready.

And now let's check those answers.

A, was neither directly or inversely proportional.

We can see the graph has a constant positive gradient, but the intersect is not zero, zero or the origin.

B is directly proportional.

It's got a constant positive gradient and the intercept is zero zero.

And then C, we can see here this is a reciprocal graph that does not intersect either the x or the y-axis.

So this is a graph that shows y is inversely proportional to x.

Now let's move on then to looking at equations of graphs.

Now we can move on then to looking at that second learning cycle and that's equations of inverse proportion graphs.

Match the graph to its equation.

Y equals two over x.

Y equals five over x.

Laura says, "Hi, Lucas.

Do you have any idea how I know which graph is which?" Wonder if you've got an idea.

Well, let's see what Lucas is saying.

Lucas says, "I think that the first graph is equation A, as it is closer to the origin." Do you agree with Lucas? Lucas is correct.

The smaller the dividend, the closer to the origin the graph will be.

Two is smaller than five, therefore it is going to be closer to the origin.

So graph A is the left hand one and B is the right hand one.

Match these equations to their correct graphs.

Pause the video, make your decision, and then come back and we'll check those for you.

What have you decided? Well, we know the smaller the dividend, the closer to the origin the graph is.

Y equals 0.

5 over x is B.

We can clearly see that that's the one that has the closest.

The lines are closest to the origin.

Y equals eight over x is the one that that's furthest away.

So that was a C.

And then Y equals three over x matches with A.

Laura says, "We could find the equation of a graph representing direct proportion." And Lucas says, "Yes, we could.

So surely we can find the equation of a graph representing inverse proportion." Yeah, that makes sense, doesn't it? If we can find the graph of something representing direct proportion or sorry, the equation of the graph, then we should be able to do that for inverse proportion as well.

Laura says, "To find k, we found the gradient of the graph.

What will we do here? It's a curve." So Laura's right, wasn't it? When we were looking at graphs of direct proportion, k, the constant of proportionality, was the gradient of the graph.

But she spotted a problem here because it's a curve.

Lucas says, "Let's think about this.

The equation will be y equals k over x." Yep, we know that because we know that it's a reciprocal graph.

Then Laura says, "We can use this.

When x equals one, y will be equal to k." When x is one, y equals k over one.

Oh yeah, y equals k.

And Lucas has said the same as me, of course.

Let's draw a graph we know the equation of and test this out.

This is the graph of y equals four over x.

Yeah, that's the graph of y equals four over x.

Now Laura was saying that when x equals one, y will be equal to k.

Let's read off what y is when x is one.

When x is one, we can see that y is four.

When x is one, y was four, so Laura was right.

Well done, Laura.

So because we are doing k over x, and if x is 1k divided by one is just k we end up with y equals k.

Now, let's have a go then at task B.

The first thing I'd like you to do is to sort the graphs into ones that show direct proportion, ones that show inverse proportion, and ones that show no proportional relationship at all.

And then when you've done that, I'd like you please to match each of the equations to the correct graph.

So I've given you here the equations y equals 3.

5 over x.

Y equals two over x.

Y equals x over two.

Y equals 0.

5 over x.

And y = 2x + 0.

So you're going to match each of those equations to its correct graph, and remember also to decide whether they're showing inverse, direct, or no proportional relationship.

Good luck with those.

Pause the video and I'll be waiting for you when you get back.

And question number two.

This time, I would like you please to find the equation of the graphs.

So here, we've got some direct proportion graphs and some inverse proportion graphs.

You know how to find the equation of both of those types of graph.

So you're gonna pause the video, you're gonna find the equations of each of those graphs, and then when you're ready come back and we'll check those answers for you.

Good luck with this.

I know you can do it.

Okay then, how did you get on? Well done.

Question number one.

A showed inverse proportion.

We can see that it's a reciprocal graph.

And that was equation y equals two over x.

Read off, when x is one, y is two, so k is still y equals two over x.

Graph B showed direct proportion.

We could see there was a constant positive gradient and the intercept was zero zero.

If we find the gradient of this graph, the change in y as we increase x by one and that is 0.

And I've written that as y equals x over two.

Remember, that's just saying y equals x divided by two, which is the same as half of x, which is the same as 0.

5x.

C is inverse proportion.

Again, we can see that's a reciprocal graph.

Now, we need to find the equation.

When X is one, what is the value of y? And that's 0.

So k is 0.

5, y equals 0.

5 over x.

D was neither.

Which of those equations shows neither inverse or direct proportion? And that is y = 2x + 0.

And finally, graph E was inverse proportion.

Again, a reciprocal graph.

We read off the value of y when x is one to find our value of k and that's 3.

So y equals 3.

5 over x.

And onto question two.

Find the equations of these graphs.

A is y equals 0.

4x.

If we read off the y value and x is one, we can see it's 0.

K is 0.

4, giving us the equation y equals 0.

4 over x.

B is y equals 4/3 x, if we define the gradient of that graph, I would probably look at the 0.

34 and the 0.

00.

So the change in x is three, the change in y is four, but we want the change in x to be one.

So we divide by three, which is how we end up with y equals four over three x.

C is y equals 3/4 x.

So here, I would probably use the 0.

3, sorry, four three and zero zero.

The change in x is four.

The change in y is three.

So I divide three by four to get the change in x one, which gives me y equals 3/4 x.

And finally, the last one, is y equals four over x.

We find the value of y when x is one and we can clearly see that that's four.

So k is four.

Giving us the equation y equals four over x.

Now, we can summarise the learning from today's lesson.

So the first thing we looked up was that the key features of a graph representing inverse proportion are that it's a reciprocal graph.

That it will appear in two quadrants.

And it will not intersect either the x or the y-axis.

And there's an example there of a graph representing inverse proportion.

The equation of a graph representing inverse proportion has the form y equals k over x.

And remember, k can be found by reading off the value of y when x was equal to one.

The smaller the dividend, the closer to the origin the graph will be.

Those are the key things that we've looked at during today's lesson.

You've done fantastically well.

I'm really pleased that you decided to join me today.

What a fantastic choice that was.

I'd like you really to take care of yourself and hopefully, I'll see you again really soon.

Goodbye.

I've finished the video