Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

We're looking at graphs of inverse proportion today.

This is a really cool bit of maths.

So what are we waiting for? Let's get stuck in.

A learning outcome is that we'll be able to recognise inverse proportion graphically and we'll be able to interpret graphs that illustrate inverse proportion.

Key words for today, inversely proportional.

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

Two parts to our learning today.

Firstly, we're going to look at plotting inverse proportion.

If we're going to understand what it means to plot inverse proportion, it's useful to look back at something that came before this, something with which you'll be very familiar.

Let's look at this example.

Car is driven at 50 miles per hour.

How far will it travel in one hour, two hours, three hours, et cetera? Well, in one hour, one hour at 50 miles per hour, that's 50 miles.

Two hours at 50 miles per hour, that's a hundred miles.

150 miles, 200 miles, 250 miles.

This is a classic example of direct proportion.

For every increase of one hour in time, there's an increase of 50 miles in distance.

Both variables increase at a constant rate.

Direct proportion can be written as an equation of the form y equals kx, where k is the multiplier.

In this case, d equals 50t.

You've got a multiplier of 50.

You take time, multiply it by 50, you get your distance.

Direct proportion can be graphed.

This graph will have those coordinates.

One hour of time, 50 miles in distance.

Two hours of time, a hundred miles in distance.

And we get that straight line.

This is the graph d equals 50t.

This direct proportion graph demonstrates that for every hour driven, the car travels a further 50 miles.

Key features of a direct proportion graph: A straight line representing a constant rate of change.

And a y-intercept at the origin zero, zero.

That y-intercept at the origin zero, zero is a very logical feature.

Zero hours would see the car travel zero miles.

That's why we get that intercept there.

Other key features of a direct proportion graph: A constant multiplicative relationship.

What's meant by that? Let's look at a couple of coordinates.

2, 100 and 4, 200.

We'll put those into a table.

We see a constant multiplicative relationship.

This is the graph d equals 50t.

T multiplied by 50 makes d.

Jun and Aisha are discussing this problem.

A car is driven 60 miles.

How long will it take at 10 miles per hour, 20 miles per hour, 30 miles per hour? Sounds familiar, doesn't it? Sounds familiar to Jun.

Jun says, "I've seen problems like this before.

This is direct proportion." Aisha says, "I'm not so sure.

In direct proportion, as one variable increases, the other increases.

I don't think that happens in this case." Do you agree with either pupil? Pause and have a think about their two statements.

Welcome back.

I wonder what you thought.

You agreeing with Jun? You agreeing with Aisha? Jun says, "Let's work it out and explore what's going on." Well done, Jun.

That's the spirit.

Exploring what's going on in maths is a lovely thing to do.

We start with 60 divided by 10 equals six hours.

Well, of course, 60 miles, 10 miles per hour, is going to take you six hours.

Then 60 divided by 20 equals three hours.

If we're travelling at 20 miles per hour, it'll take us three hours to travel 60 miles.

Finishing off, 60 divided by 30 equals two hours.

And then when you look at those results, Aisha spotted something.

"I was right.

Can you see how time decreases as speed increases? This is not direct proportion.

This is something else." Well spotted, you two.

Let's compare this new relationship to the one we saw earlier.

The one we saw earlier about the car being driven at 50 miles per hour and how far it will travel in one hour, two hours, three hours.

That was an example of direct proportion.

In direct proportion we get a relationship in the form y equals kx, where k is the multiplier.

This case was d equals 50t.

Take time, multiply it by 50, and you get your distance.

It was all about multiplication in the first example of direct proportion.

That's because two variables are in direct proportion if they have a constant multiplicative relationship.

Let's contrast that with the one that we've just had Jun and Aisha looking at.

A car is driven 60 miles, how long will it take it? 10 miles per hour, 20 miles per hour, 30 miles per hour.

Can you see the difference in this case? This is a relationship in the form y equals k over x, y equals k divided by x.

Specifically in this case, we had the example t equals 60 divided by s.

We had to take the 60-mile journey, divide it by our varying speed to find our time.

This was all about division.

Can you see the difference between our direct proportion example and our inverse proportion example.

This is an example of inverse proportion.

A relationship in the form y equals k over X or by a constant is divided by one variable to find the other variable.

As a result, as one variable increases, the other variable decreases.

Let's check if you've got this.

Into which category does this example fall? The number of sweets you have and the number of packets you buy.

Is that direct proportion, inverse proportion or no proportional relationship whatsoever? Pause and take your pick.

Welcome back.

I hope you said that's A, direct proportion.

If we have more packets, will we have more sweets? Yes.

Will the change be in direct proportion? Yes, assuming there is a constant number of sweets in each packet.

This is a relationship in the form, y equals kx.

In this case we'd have s equals k multiplied by p, where s is the number of sweets you have, p is the number of packets you buy, and k is the constant, the number of sweets in each bag.

Let's check this one.

Into which category does this example fall? The number of sweets you get if you share one packet with multiple friends.

Is that direct proportion, inverse proportion, or no proportion or relationship whatsoever? Pause and take your pick.

Welcome back.

I hope you said B, inverse proportion.

Why is that? The question you should be asking yourself is, if there are more friends, will we each get more or less sweets? The answer is less.

As the number of friends increases, the number of sweets we each get decreases.

This will be a relationship in the form y equals k over x.

Specifically in this case, it'll be s equals k over f.

S will be the number of sweets you get, f is the number of friends you share with, and K, the constant with a number of sweets in the packet.

We can graph inverse proportion.

Aisha and Jun consider a cycle ride of 24 miles and length of time it takes at varying speeds.

And Aisha proposes, "If a direct proportion graph looks like this," yes, that's a direct proportion graph, "then I think a graph of inverse proportion must look like this.

Hmm, well, it seems logical.

We're doing a cycle ride of 24 miles.

The length of time it takes is going to decrease as our speed increases.

That seems pretty logical, Aisha.

Jun says, "Whilst it's good to hypothesise, Aisha, you're not right on this occasion.

Let's do a table of values and I'll demonstrate." Good.

It's nice to check out what I should hypothesised.

This is a relationship of inverse proportion.

Y equals k over x.

If we're gonna do a table of values, we need to know how this relationship looks in this context.

This example will be t equals 24 over s.

Why? Time is going to vary.

Speed is going to vary.

But the 24 miles will remain constant.

If we travel at a greater speed, we'll do it in less time, but we have to do that 24 mile distance.

The 24 miles will remain constant.

So if t equals 24 over s, we can use that relationship to plot a table of values.

Let's start with 24 divided by 24 equals one and pop one in there in our table.

What's that mean? It means 24 miles at 24 miles per hour takes one hour.

Speed of 24 miles per hour, we'll have a time of one hour.

24 divided by 12 is two.

Pop that in the table.

What does that mean? It means 24 miles at 12 miles per hour takes two hours.

We'd keep going doing that and populate the whole table.

We're ready to put this onto a graph now.

Let's plot the coordinate 24, 1 at a speed of 24 miles per hour and time of one hour.

12,2, the speed of 12 miles an hour, time of two.

Let's plot 8,3, 6,4, 4,6, 3,8, 2,12, 1,24.

Graphs of inverse proportion form a smooth curve in the first quadrant.

Notice this is not a straight line.

The points are forming a curve.

Jun says, "You were right to say the graph decreases, but it's not a straight line." Let's compare the graphs of direct and inverse proportion.

These are the two examples we've seen so far.

On the left is our direct proportion travelling at 50 miles per hour.

On the right is example that Jun and Aisha just plotted.

24 miles at varying speeds.

When we compare them, we've got a straight line for direct proportion versus a curve for inverse proportion.

Direct proportion sees our graph increasing.

Inverses proportion sees our graph decreasing.

And direct graphs touch the axis at the origin.

Inverse graphs do not touch the axis.

I wanna look at this last point a little more closely.

Inverse graphs do not touch the axis.

Or if you've seen a graph before, that touches neither axis.

Have you? In this case we applied the relationship t equals 24 divided by s.

Take 24, divided by our speed, we find the time it takes.

If you think about that s value, dividing by zero is undefined so our s value cannot be zero.

You can't substitute zero in there.

It'll be undefined.

That means our curve cannot touch s equals zero.

S equals zero is that line there or the t axis in this case.

At the other side, t can't ever equal zero.

Why is that? No value for s will give us a t value of zero, so our curve cannot touch t equals zero.

That's the line, t equals zero.

No matter how larger value you substitute in for s, you could substitute a billion in there, and we'll still have a t value of 24 over a billion.

That's small but it's still greater than zero.

We'll never touch that axis.

I wonder what it feels like to travel at a billion miles per hour.

Hmm.

Let's check you've got this.

I'd like you to sort the following into examples and non-examples of graphs showing inverse proportion, which ones are inverse proportion and which ones are not.

Pause and consider each one now.

Welcome back.

I hope for A, you said that's a non-example.

It's an example of direct proportion but it's not an example of inverse proportion.

I hope you said for B, that's a non-example.

It's decreasing but it's not a curve, and that line touches both axis.

Inverses proportion would not.

C absolutely was an example of inverse proportion.

D, a non-example.

It was not a curve.

E was also a non-example.

It looks so close to our actual example of inverse proportion.

What's its crucial failure? It can't touch the axis.

That curve on E touches the axis, whereas we know inverse proportion would not.

Practise time now.

I'd like you to match each scenario to a graph.

There's four graphs.

There's four scenarios.

Which one models which? Pause and match those up.

Question two, I'd like you to complete the table of values and plot this inverse proportion graph.

This is a model of the time taken to walk or run 12 miles versus the speed it's done at.

Complete the table of values, plot the coordinates, draw that graph Feedback time now.

We should have matched graph A to the fourth scenario, the conversion of kilogramme into pounds.

Graph B was our second scenario.

The taxi firm charging 5 pound fixed fee and then two pounds per mile.

C was the top scenario, the bath emptying at two litres per second, then slowing to one litre per second.

Leaving D to be the time taken to build a wall versus the number of builders.

Let's take a closer look at each example.

For A, the conversion of kilogrammes into pounds.

Why is this? 'Cause it's direct proportion as a constant rate of positive change.

For every extra kilogramme you get, you'll get 2.

2 more pounds, and that will happen constantly.

Also, the line intercepts the axis at the origin.

Why is that? Because zero kilogrammes is zero pounds.

That's why that matches that scenario.

For B, this one was not direct proportion.

It's a taxi firm charging five pound fixed fee, then two pound per mile.

It's a constant rate of positive change but not direct proportion because it does not intercept the axis at the origin.

Zero miles won't cost you zero pounds.

You still gotta pay that five pound fixed fee.

That's why that scenario matched that graph.

For C, a bath emptying at two litres per second, and then slowing to one litre per second is a constant rate of negative change, which changes to a lesser rate of change at a given moment.

It's a straight line graph.

That's how we know this is one is not inverse proportion.

D was the graph of time taken to build a wall versus the number of builders.

As a number of builders increases, the time taken decreases, but neither time nor builders can be zero.

That's why that line doesn't touch either axis.

Question two, I asked you to populate the table of values.

We would need to use a relationship t equals 12 over s.

That's our 12 miles being divided by our speed to give us the time it takes.

That'll give you those values in your table.

You'd p plot these coordinates.

1,12, 2,6, 3,4, 4,3, 6,2, 12,1, and then you'd plot that nice smooth curve.

Onto the second part of the lesson now, reading and interpreting.

Interpreting information is a key skill in mathematics.

This graph shows the number of days taken for people to dig a hole.

How long would it take five people? Aisha says, "The answer is three." What's missing from Aisha's answer? Pause, tell the person next to you.

Welcome back.

I hope you said two things.

I hope you said units of measure and also context.

Aisha corrects herself and says, "It would take five people three days to dig the hole." This answer contains units of measure and context.

This is a more accurate interpretation.

Quick check you've got that.

When reading and interpreting an inverse proportion graph, it's important that we include which of the below? Units, context, the coordinate, what should we include? Pause, tell the person next to you.

Welcome back.

I hope you said absolutely option A.

We definitely want units.

That adds accuracy.

Option B, let's add the context.

This aids understanding for the person we're communicating to.

For option C, you don't necessarily need it.

If you correctly communicate the units and context, you wouldn't need to tell 'em the coordinate.

Aisha and Jun are reading this graph about the number of sweets you receive when sharing a packet with friends.

How many sweets would you each get if two people shared? I can find that on the graph.

Jun has a go.

He says, "I shall read from here.

That's 2,10.

We would get 10 sweets each." Makes sense.

Aisha says, "I shall read from here, 2,10.

We would get 10 sweets each." What do you notice? Pause, tell a person next to you.

Welcome back.

I wonder what you noticed.

Aisha noticed something.

Aisha said, "That's amazing, Jun.

You read from the wrong axis but still got the right answer." Jun says, "Yes, Aisha, this is really interesting.

The coordinates on this graph seem to have some sort of symmetry.

We have a coordinate of 1,20, a coordinate of 20,1.

A coordinate of 4,5, a coordinate of 5,4.

Graphs of inverse proportion have a line of symmetry, y equals x, so the coordinates have symmetry.

That is a line of symmetry.

On this graph, of course, it's the line s equals f.

But for any graph of inverse proportion, the line y equals x will be a line of symmetry.

That means our coordinates also have symmetry.

"That's cool," say Jun and Aisha.

And I agree.

Quick check you've got that.

This graph shows time, t, in hours to an 18 kilometre obstacle course event at various speeds, s, in kilometres per hour.

Lovely graph of inverse proportion.

What does this coordinate tell us? The coordinate, 3,6.

Does that mean it takes three hours at six kilometres an hour, or it takes six days at three kilometres an hour, or it would take six hours at three kilometres per hour? Which is it? Pause and take your pick.

Welcome back.

Hope you went for option C, it would take six hours at three kilometres an hour.

That vertical line reads from our speed axis, and that's a speed of three kilometres per hour.

Reading across horizontally to the t axis, that's six what? Six days, six hours.

We were told in the question that time is in hours.

if you paid attention to that, you saw it was option C and not option B.

Well done.

Practise time now.

For question one, this inverse proportion graph plots the number of builders, b, versus the time taken, t, in days for the digging of some foundations.

Interpret these coordinates in context, the coordinates at a, b, c, and d.

I'd like you to be sure to write full sentences and include units of measure in your answers.

Pause and do this now.

Welcome back.

Let's see how we got on.

For A, 2,30, we should have said it will take two builders 30 days to dig the foundations.

It's not enough to say two builders, 30 days.

You must write a full sentence.

It's far clearer communication.

For B, you should have said it will take six builders 10 days to dig the foundations.

For C, 10,6, the other way around, it'll take 10 builders six days to dig the foundations.

Lovely symmetry between coordinates B and C.

And for D, it would take 30 builders two days to dig the foundations.

That's the end of the lesson now, sadly.

What we've learned is that we can recognise inverse proportion graphically.

A graph of inverse proportion will not be a straight line.

It'll be a smooth downward curve which does not intersect either axis.

We can interpret graphs that illustrates inverse proportion and we know to include units and context in our answers.

I hope you've enjoyed this lesson as much as I have.

I shall look forward to seeing you again soon for more wonderful mathematics.

Goodbye for now.