Lesson video

Hi there, my name is Ms. Lambell.

You've made a really good choice deciding to join me today to do some math.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Inverse proportion in a context." And that is within our unit understanding multiplicative relationships with percentages and proportion.

By the end of this lesson, you will be able to recognise inverse proportion graphically and use inverse proportionality in a range of contexts.

Here's a definition of inverse proportionality, but I'm not going to stop and read it through with you now because we'll look at this as we progress through this lesson and I don't wanna give away any spoilers.

Today's lesson is split into two learning cycles.

In the first one, we're going to be concentrating on looking at graphs of inverse proportion and then using what we've learned by inverse proportion.

In the second learning cycle, we will look to find missing values.

Let's get started with that first one.

Graphs of inverse proportion.

It takes 3 people 2 days to paint a fence.

How long will it take 6 people? Sam says, "If you double the number of people then you will double the time." Do you agree with Sam? Izzy doesn't think that Sam is right.

Let's see what Izzy has got to say.

She says, "How can the job take longer if more people are doing it?" Hmm.

Let's take a look.

Well, simplify the problem first, so that we can decide what's going to happen.

It takes one person six days to paint this fence.

How long will it take two people? We'll start by thinking about how many panels one person can paint in a day.

Here, we can see that there are six panels and it's going to take 1 person six days.

This means that they would be able to paint one panel per day.

One person can paint one panel in one day.

We're now going to look at how long it will take two people? That was the question.

And to see this difference between the two people, I'm going to use different shading.

Person one is shaded and person two is unshaded.

Person one.

On day one, paints this panel.

And person two on day one paints.

And person two on day one paints this panel.

That's day one done.

Person one comes back on day two and paints this panel.

And person two comes back on day two and paints this panel.

And then, on day three they both return and person one paints this panel, and person two paints this panel.

We can now see that the job is going to take three days in total.

It would take two people, three days, to paint the same fence.

The same problem.

It takes one person six days to paint this fence.

How long will it take three people to paint the same fence.

Person one is going to be dark shading.

Person two will be unshaded and person three will be light shading.

Person one on day one, paints one panel.

Person two on day one and person three on day one.

Now, they return on day two.

Person one paints this panel on day two.

Person two paints this panel on day two.

And person three paints this panel on day two.

It would take three people, two days, to paint the same fence.

Let's put the information we have into tables.

We've got the number of people is one, and that takes six days, and then two people would take three days.

And then one person takes six days.

So three people would take two days.

Sam says, "As the number of people increases, the number of days decreases." Izzy says, "That does make sense.

It must mean that the number of people and days are not directly proportional." Because we know if variables are directly proportional and as one thing increases, the other increases.

Sam says, "I remember we could draw graphs of direct proportion and they look like this." This is a graphical representation of direct proportion.

It has a constant positive gradient and it has the intercept of zero.

The graph starts at the origin.

Izzy says, "Direct proportion means as one variable increases, the other increases at the same rate.

This relationship in the fence problem shows that as one variable increases the other decreases." Maybe the graph will look like this.

So Sam is suggesting maybe, this is what a graph showing this may look like.

What do you think? I wonder if you agree with Sam.

Sam and Izzy are working on this problem that their teacher has set them.

They've got to find as many rectangles as they can with an area of 24 square centimetres.

So we've got our width and our length, and we know the area needs to be 24 centimetres squared.

And we know the rectangle area, sorry.

We know the formula for finding the area of a rectangle is length multiplied by width.

Sam says, "1 centimetre by 24 centimetres gives an area of 24 centimetres squared." Yep, that's right.

Izzy says, "3 centimetres by 8 centimetres gives an area of 24 centimetres squared." Yep, I agree with that.

3 x 8 = 24.

Sam's made another suggestion.

"2 centimetres by 12 centimetres gives an area of 24 centimetres squared." And Izzy says, "4 centimetres by 6 centimetres gives an area of 24 centimetres squared." Sam says, "That's all of them, isn't it?" Do you agree with Sam? Do you think that Sam and Izzy have found all of the rectangles we could draw with an area of 24 centimetre squared? Actually, Sam says, "Thinking about it, I think there could be some with decimal dimensions." So Sam's thought about what he said and he's now revising that.

He thinks that maybe there could be some decimal dimensions.

Izzy says, "Yes.

2.

4 multiplied by 10 gives an area of 24 centimetres squared." Now Sam says, "Surely there are an infinite number of rectangles dependent on the degree of accuracy used to measure the dimensions." Do you agree with Sam? "Yes, Sam, that must be true." Because we can always measure to a greater degree of accuracy.

Let's take a look at the graph representing this problem.

I've got my length and my width.

And we'll plot some points that we know.

So we'll use those dimensions that Sam and Izzy we're using.

Let's start with one and 24.

A length of one, a width of 24 gives an area of 24 centimetres squared.

We can also reverse the coordinates because we don't care whether the rectangle is longer than it is wider, or it is wider than it is long, as long as its area is the same.

So therefore, we would plot the point 24, one.

And now two and 12.

Let's just double check two multiply by 12 is 24.

So yes, it does have an area.

Let's plot two, 12 on our graph.

And again, we can reverse those coordinates and plot 12, two.

Moving on now then to three and eight.

Three multiplied by eight is 24, good.

So I'm gonna do three and eight.

Three, eight, and then eight, three.

And now, four, six.

Let's plot four, six And reverse the coordinates and plot six, four.

Now, we're going to plot 4.

8, five.

Just check for me is 4.

8 multiplied by five = 24.

4.

8 multiplied by 10 is 48, and then half it is 24.

So 4.

8 and five.

And then reverse those coordinates and plot in five and 4.

8.

Sam says, "It looks like this is forming a curve." Do you agree with Sam? It does form a curve.

Sam is right.

This is what our graph looks like if we plot every dimension that we can, we end up with a curve.

This was what Sam thought originally.

So Sam's previous idea was pretty close.

However, it forms a curve and not a straight line.

But I really admire Sam for having a thought about what might happen.

This is an example of inverse proportion.

As one variable increases, the other decreases in proportion.

In this example, as the width of the rectangle increases, the length decreases to keep the area at 24 centimetre squared.

Can you think of any other examples of inverse proportion? As a speed of a car increases, the time taken to cover a certain distance decreases.

The number of people doing something and the time it takes to do it.

As the number of people increases, the time it takes to finish decreases.

We can use this graph to read off possible dimensions of our rectangle.

If the length is 10 centimetres, we can draw a line to the curve and across and we can read off the value.

The width is 2.

4 centimetres.

So it was possible to read values from the graph representing our rectangle.

This is the graph representing the fence problem that we started with.

How many people will I need if I want to finish the job in five days? Five days, we find five and our days axis, which is the D, and then I follow down.

Why is that not appropriate? You can't have one and a bit of a person.

So in the rectangle problem, it was okay, because we could measure our dimensions to a greater degree of accuracy.

But here, it's not possible to have one and a bit of a person.

So the context is really important.

We need to look at the context to see whether we can read off any value from our graph.

You are now gonna have a go at this one for me, please.

I want you to match each graph to the correct description.

One of the graphs shows direct proportion, one shows neither and one shows inverse proportion.

Pause the video and when you've made your decision, just come back.

Okay.

Direct proportion was the middle one.

Constant positive gradient, intercept zero, so it starts at the origin.

Neither was the last one.

Has a constant positive gradient, but its intercept is not zero.

And then inverse proportion is the one we've just looked at, so it forms a curve shape in that quadrant of the graph.

Task A now, you're going to label each of the graphs as direct proportion, inverse proportion, or neither.

Pause the video and then when you've got your answers, come back.

Okay, question number two.

I'd like you to decide if the following situations are directly proportional as one variable increases, the other increases.

Or inversely proportional, as one variable increases, the other decreases.

You've got the number of taps and the time taken to fill a bath.

The number of Euros exchanged and the number of pounds.

The number of boxes of tiles bought and the cost.

The number of days to dig a hole in the number of people.

And the time taken to get to Edinburgh and the speed of the car.

Pause the video, decide which ones are direct, which ones are inverse, and then when you're ready, come back.

Okay, let's check those answers.

So A was neither.

I just look, we can see that it doesn't start at the origin.

B is inverse proportion.

C is neither, so that was the one.

Remember that Sam thought it might be, and then we found out it isn't.

D does start at the origin, but it doesn't have a constant positive gradient.

E is direct because it does start at zero and has a constant positive gradient.

And F is another example of inverse proportion.

And then onto question two.

A is inverse.

B direct.

C direct.

D inverse.

And E inverse.

How did you get on with those? Great work.

Well done.

Now, let's move on then to our final learning cycle for today, and that is using inverse proportion to find missing values.

So we've just looked at how we can do that with the graph, but we might not always want to draw a graph out.

So let's take a look at how we can do it without the graph.

Sam says, "When we worked on direct proportion, we used ratio tables." Yeah, we did, didn't we? So Izzy's agreeing with Sam.

"Yes, we did." So she assumes that we can use them for inverse proportion as well.

We'll take a look at using the rectangle example we have already looked at, so in the first learning cycle.

How many rectangles can you find with an area of 24 centimetres square? That was the question that we were looking at.

Let's start with the dimensions of six by four, which we know has an area of 24 centimetres squared.

Here's my ratio table.

So we've got six and we've got four.

We know that the width is three.

Let's look for that relationship between the six and the three.

And my multiplicative relationship is to multiply by 0.

5.

I multiply by 0.

5 and I get two.

Sam says, "Hold on, hold on.

That doesn't give a rectangle with an area of 24 centimetres squared." "Oh, yes, Sam, you're right.

That is an area of six centimetres squared." So Izzy's agreeing with Sam, but actually that doesn't satisfy the question that we were looking at.

We were trying to find rectangles with an area of 24 centimetres squared.

Sam says, "I can see why it doesn't work like a direct proportion problem." And Izzy says, "Why is that, Sam?" "Direct proportion is when one variable increases the other also increases." "You are right, Sam.

An inverse proportion is when one variable increases the other decreases." "That means that we cannot do the same to both sides of the ratio table." Here, we've gone back to our ratio table, we've half the width.

"If the width of the rectangle is three centimetres, what would the length be?" "It would be eight centimetres to give an area of 24 centimetres squared." "The length has doubled." "We multiplied by 0.

5, which is the same as dividing by two." "So when we divided the width by two, we multiplied the length by two." And Izzy says, "Of course, that makes sense.

As they are inverses of each other and as one increases the other decreases." "And the problem is an inverse proportion problem." So recognising the use of inverse in both of those.

Let's go back to our original problem.

So that was the fences.

If it takes three people two days to paint a fence, how long will it take six people? Three people, take two days.

And we want to work out for six people.

What have I done? What's my multiplicative relationship between three and six, that's multiplied by two.

It's an inverse proportion problem as I increase the number of people, the number of days is going to decrease, so I need to do the inverse.

I'm going to divide by two.

So it's going to take one day.

In a factory, 10 machines take five days to produce a lorry load of items. How long will it take two machines? Here are Sam's workings.

So we've got machines, there's 10.

Days is five.

And we want to know for two machines.

So two machines, we're looking for that relationship.

10 divided by 5 is 2.

So therefore, we need to do the inverse to the number of days.

5 x 5 = 25.

And here are Izzy's workings.

So Izzy has done 10, and five, and two.

And 10 are a multiplicative relationship between 10 and two is multiplied by 0.

2.

So I divide by 0.

2.

Remember I'm doing the inverse operation.

That's 25.

Whose method is correct? They're both correct.

Both methods are correct.

Sam has just decided to start with divide, whereas Izzy decided to start with multiply.

How long will it take three machines? We're sticking with the same problem.

The factory 10 machines take five days.

We want three machines.

And Sam is saying, "Hmmm, 10 is not a multiple of three.

That means I cannot divide." And Izzy says, "Sam, remember, any pair of numbers have a multiplicative relationship." We can multiply by three over 10.

So I divide by three over 10 because this is an inverse problem, so I'm doing the inverse operation giving us 16.

7 to one decimal place.

It will take three machines, 16.

7 days to produce a lorry load.

A and B are inversely proportional to each other.

Find the missing value in the table.

We know they're inversely proportional to each other because we are told that.

We are looking for the multiplicative relationship between 15 and 24.

And I've done 24 divided by 15 to give me 1.

6.

If they're inversely proportional, I must use the inverse operation.

So I'm gonna divide by 1.

6.

Eight divided by 1.

6 is five.

Sam says, "I've just had a thought when we looked at the rectangle, the product of each row had to be 24 as this was the area of the rectangle." Izzy says, "That's interesting, Sam.

Is it the same here?" What do you think? Well, 15 x 8 = 120.

And 24 x 5 = 120, so it is true.

This is a really good way to check your answers.

We'll have a go at the one on the left together and then I'd like you to have a go at the one on the right independently.

A and B are inversely proportional to each other.

Find the missing value in the table.

What is my multiplicative relationship? I multiply by 5 over 16.

I'm gonna do the inverse operations, so I'm gonna divide by 5 over 16, giving me 64.

Your turn.

A and B are inversely proportional to each other.

Find the missing value in the table.

Pause the video and come back when you've got your answer, please.

Now, let's check your answer.

The multiplicative relationship between 25 and 32.

32 divided by 25 is multiplied by 1.

28.

Inverse because they're inversely proportional.

So I'm gonna do 128 divided by 1.

28 giving me an answer of 100.

Now you are ready to have a real good go at Task B.

Just as we did in that check for understanding, you are going to please find the missing value in each table, and I have told you that each of the relationships between A and B show an inverse proportional relationship.

Pause the video and when you've got your six values, come back.

Great work.

Question two, use a ratio table to answer the following.

A, it takes four people two days to paint a house.

How long will it take eight people? B, it takes 10 factory workers four hours to pack a lorry full of boxes.

Two people are on holiday.

How long will it take to fill the lorry now? C, it takes three days to fill a pond if five taps are used, how long will it take if four taps are used? Give your answer in hours and minutes.

And D, four machines work all day every day.

They make one million items in seven days.

The company buys one new machine.

How long will it take to make one million items now? Give your answer in hours, sorry, days, hours and minutes.

You'll need a calculator for these.

Remember to draw out those ratio tables and remember that all of these are inverse proportion problems. Good luck with these.

They are quite challenging, especially the last two but I know that you are up for a challenge.

So pause the video and I look forward to seeing you when you join me to check those answers.

Let's check our answers then.

Number one A, the missing number was 10.

B, the missing number was 13.

C was 192.

D was 20.

E was 8.

75.

And F was eight.

I hope I didn't catch anybody out by asking you to find numbers that weren't necessarily in the bottom right-hand box of the table.

I'm sure I didn't.

And then onto question number two.

A is one day.

B, five hours.

C is 3 hours, 45 minutes.

Now, if you didn't manage to change it from a decimal, your decimal answer there would've been 3.

75.

And then D, the correct answer was five days, 14 hours and 24 minutes.

And if I remember rightly, if you weren't quite able to convert that into today's hours and minutes, I'm pretty certain that the decimal answer was 5.

6.

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

It's been quite a challenging one, I think you'll agree, but hopefully now you've got your head around what we mean by inverse proportion.

It's fairly easy to calculate missing values.

Inverse proportion is when one variable increases the other decreases.

In this example, as with the width of the rectangle increases, the length decreases to keep the area at 24 centimetres squared.

The graphs of inverse proportion take on this shape.

A ratio table can be used to solve inverse proportion problems, remembering to use the inverse operations.

Like I said, I think today's lesson was quite challenging, so well done.

You've done fantastically well and I look forward to seeing you again really soon to do some more math.

Goodbye.