Lesson video

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Hi there and welcome to another maths lesson with me, Dr.

In today's lesson, we will be looking at graphs of inverse proportion.

For this lesson you will need a pen a paper, a pencil and a ruler.

So if you do not have these handy, please pause the video, go grab them and when you're ready, we can make a start.

To start today's lesson I would like you to have a go at the following question.

The speed of a journey at constant speed is inversely proportional to the time taken.

The journey takes four hours at 30 km/h.

I've given you here at a table.

I want you to find out the time, if the speed is two, three, five, six, eight, 15, and six km/h.

Once you've finished, I would like you to draw a graph of speed against time on a graph paper, if you have one.

If not, you can do it on square paper.

The Try this task should take you about 10 minutes to complete.

Please remember to use a pencil for your graph.

Pause the video now and complete the task, resume once you're finished.

Welcome back.

How did you get on with this task? Really good so, the question already told us that the speed is inversely proportional to the time taken.

So I know that the product of speed multiplied by the time will always be the same number, the same answer, that same constant of proportionality.

So looking at the table, I know if the speed is 30, the time taken is four hours.

What's the product? 30 multiplied by four is 120.

So I know that the product of any given speed multiplied by the time here is going to be 120.

So I can go to the first one and say, okay, at a speed of two, two multiplied by what number gives me 120? And the answer is, good job.

Three multiplied by what number gives me 120? And the answer is 40.

Three multiplied by 24 is 120, so I'm keeping that constant the same.

Six multiplied by 20 is 120.

Eight multiplied by 15 is 120.

15 multiplied by eight is 120 and six multiplied by 20 is 120.

Now you may have used a slightly different method.

One of the other methods that we have discussed in last lesson.

And that should have given you exactly the same answers.

The second part, I asked you to draw a graph of speed against time I did it on graph paper but if you don't have graph paper, you could have done this on a square paper.

If you've done that, you should have a graph that looks like this one.

Did you get that? Okay.

I hope that you connected your graphs using a nice smooth curve, because you can see here that you do not have a straight line.

Now I want you to have a little think about this graph and compare this to the other graphs that we used or looked at when we were looking at direct proportion.

What's the same and what's different? So have a little think.

Okay, really good.

So, what's different here is that this graph looks like a curve rather than a linear graph.

It's not a straight line graph.

It is not in the format of Y equal MX plus C.

Really good.

Does not cross the Y axis or the X axis.

It gets really, really close to both Y and X axes, but it has not touched them.

Whereas with direct proportion, it was a linear graph.

It crossed the Y axis at the origin.

Really good.

What's else? Okay.

So what's the same is that both of them show us some sort of relationship between the X and the Y.

With direct proportion graphs, it showed, the graph showed us that as the X increases the Y increases.

And here it's showing us as X increases, as we go along the X axis here.

So as X is getting bigger and bigger and bigger, what's happening to the Y value? It's dropping and dropping and dropping it's going down.

Okay? So in this, both the graphs showed us a relationship or proportional relationship, however one of them was direct and one was inverse.

Okay.

So out of today's lesson, I really want you to be able to recognise the graphs of inverse proportion and see how they are different from the graphs for direct proportion.

So I have a graph here that I'm showing to you.

It's got, I've got the X and the Y axes.

I have a straight line, It's a linear graph passing through the origin, okay? So we know for direct proportion that if we divide Y divide by X, we always get a number.

I called it K, which is a constant number.

And that is the constant of proportionality.

That number is always the same.

Now if I take this, Y out of X equal a constant number equals K.

If I rearrange this to write it down as Y equal something I would get Y equal KX.

Now Y equal KX is a linear graph.

You remember equations of straight lines that are in the format of Y equal MX plus C.

Well C in this case is zero because the graph is crossing that Y axis at zero.

So this is a linear graph.

It passes through zero, zero the origin.

And it has a gradient of K whatever that number is That's the gradient and we looked at this in previous lessons.

So this graph is for direct proportion.

Now for inverse proportion, what do we know? We know that in inverse proportion, YX is equal to a constant, it's the product that is equal a constant value.

The product of the two quantities that we're looking at.

Now, let's rearrange this and write it down in the format of Y equal something That would be Y equal K out of X.

Now what happens as X gets bigger, when X is increasing, K is the same.

Yes, because that's the constant.

If X is bigger and bigger and bigger then Y it's going to get smaller and smaller and smaller.

Remember if you are dividing by a bigger number, then our answer gets smaller.

So it's like saying, K out of 10 or K out of 100.

Now it's out of 100, that number is going to get smaller and smaller.

So if I was going to draw a graph like this, I know that as the X gets bigger, as I move to bigger X on the graph, Y value will get smaller and smaller and smaller.

And it will look like this.

And I'm talking about if K, sorry, why did they write A? So I'm talking about when K is greater than zero, okay? So I'm talking about K being equal, greater than zero.

What's going to happen? Is this here.

The graph will look like this.

What's happening here as X is increasing, the Y is decreasing, decreasing, decreasing, but it's never touching the X axis, it doesn't get there.

Now, so this is how the graph is going to look like.

This is our inverse graph or a graph that shows the inverse proportion.

If K, the constant of proportionality.

Let me just correct this.

So if K is smaller than zero, so we looked at K being bigger than zero so if K is negative is smaller than zero, the graph will look like this this.

Okay.

So I really need you to be able to recognise the difference between the two graphs.

It is time now for you to have a go at the independent task.

You have two questions to complete.

If you're feeling confident, please pause the video now and have a go at this.

If not, I'll give you a hint in three, in two, and in one.

So the first question is asking you to identify which graphs are showing direct proportion, which ones are showing inverse proportion, and which one are showing none.

You will find it helpful to look back at your notes from the previous slide and use that to help you identify the graphs.

For question number two, you need to use your knowledge about speed, distance, time to complete the table and then on the graph.

Now you should know that speed is equal to distance divided by time.

The table is asking you to find out the the time taken.

So you need to rearrange that equation.

Time is equal to distance divided by speed.

So please write that down.

Time equal distance divided by speed.

Now this equation should help you fill in that table, and then you should be able to draw your graph.

With this hint, you should be able to have a go at the questions.

The independent task should take you about 10 minutes to complete.

Please pause the video and resume once you're finished.

Welcome back.

How did you get on with this? Okay.

So if we look at the first graph here, it's showing us inverse proportion, very similar to the sketch I did with you earlier on during the lesson.

Graph B is not showing inverse, and it's definitely not showing direct proportion.

We know that because it does not pass through the origin so it's neither.

Graph D is showing direct proportion.

It's a linear graph, it passes through the origin.

And the last one again is neither because you can see here, you have one line and then another line connected.

So it's not showing direct or inverse.

Did you get these correct? Well done.

Let's have a look at question two.

Bill and Ben are planning 240 miles journey from London to Paris.

They want to know how long it will take if they travel at different speeds.

And I wanted you to complete the table.

I gave you various speeds.

Now you've given the distance.

That distance for the journey is always going to be 240, it's not going to change.

To find the time we need to divide distance by that speed.

So 240 divided by five, that gives us 48.

So, if they're travel at five mph, they will take 48 hours to complete the journey.

240 divide by 10 is 24.

240 divide by 20 is 12.

240 divide by 30 is eight.

240 divide by 40 is six.

240 divide by 48 is five.

So if they travel at 48 mph, they will take five hours to complete the journey.

Again you see that this is the opposite of if they travel at five.

So if they travel at five mph, they take 48 hours.

If they travel at 48 mph, they take five hours.

Okay.

For 60, 240 divide by 60 is four, and 240 divide by 80 is three.

Did you get the table correct? Really good.

Now I gave you a hint about how the graph should look like.

The speed needed to go on the X axis and the time needed to go on the Y axis.

Now, this is how my graph looks like.

Yours should look very similar.

Okay.

For our Explore task, I want you to have a look at this and tell me whether you agree with Antoni.

Whether you think he's right or not.

Antoni says if this shows direct proportion, then this shows inverse proportion.

Now you've been given the two graphs.

I want you to have a little think about the two graphs.

Do you agree with Antoni, and why? Do you disagree with him, and why? what do you think the reason for Antoni's thinking that this second graph is an inverse proportion graph.

What you can do as well with this task is you can take these two graphs and sketch them on your piece of paper.

Make some numbers on the X and Y axis, and look at the constant of proportionality.

Now we know that for direct proportion, Y divided by X should always be a constant number.

And we know that for inverse proportion, X multiplied by Y should be a constant number.

Can you explore this a little bit further for me.

Please pause the video and have a go at the Explore task.

This should take you between five to 10 minutes to complete.

Resume the video once you're finished.

Welcome back.

So tell me, do you agree with Antoni? Is he right? And how did you know that? Really good.

So the second graph, I disagree with Antoni here because the second graph does not show inverse proportion.

In fact, we spent the whole lesson looking at inverse proportion graphs.

The second graph shows a linear graph.

It has, it's a graph of Y equal MX plus C.

And I think Antoni made this mistake because he looked at that direct proportion graph, which had a positive gradient in this case.

So, it was going up like this.

And he thought, well, this has a negative gradient, it's going down.

And therefore it must be the inverse but we know that that's not the case.

Now, also one of the other things that I did was this, I thought okay you know what let me sketch a graph that looks like the graph that he thought was inverse proportion.

So I sketched a graph similar to that, and I've put some numbers on the graph.

And then I took the coordinates and I multiplied X by Y to see if I was going to get a constant.

Because remember if it's inverse, X multiplied by Y should be constant.

So one of the coordinates was one, four so I multiplied one by four, that gave me four.

Two, three.

That gave me a six.

Four multiplied by one, that gave me four.

And five multiplied by zero.

That gave me zero.

So, I know that the product of X and Y is not the same for every single point on this graph.

And therefore, I definitely know that it's not a graph of inverse proportion.

Did you get that? Really good.

Well done.

Some fantastic learning there so well done.

This brings us to the end of the lesson.

I would like to remind you to complete the exit quiz to show what you know.

Enjoy the rest of your day and I'll see you next lesson.

Bye.