Lesson video

Hiya, my name's Ms. Lambell.

I'm really pleased that you've decided to join me today to do some maths.

I'm really looking forward to working alongside you.

Welcome to today's lesson.

The title of today's lesson is describing exchange rates as a ratio, and this is within our unit understanding multiplicative relationships, fractions and ratios.

By the end of this lesson, you'll be able to use a ratio to describe exchange rates.

An exchange rate is the rate at which one currency is exchanged for another.

The two currencies are in proportion because they have a constant multiplicative relationship.

Today's lesson, I've decided to divide into two separate learning cycles.

In the first, we will be looking at representations of exchange rates, so different ways of representing an exchange rate.

And in the second learning cycle, we will concentrate on writing those exchange rates as a ratio.

We're ready to get started on that first learning cycle.

Let's go.

Representations of exchange rates.

The relationship between the number of boxes of pens bought and the total cost is shown on the graph below.

Does this graph show a multiplicative relationship? It does.

As the number of boxes doubles, so does the cost.

That's a really, really good way of checking whether something has a multiplicative relationship.

As one thing doubles, does the other? This is a graph that shows the relationship between two currencies used in a computer game.

There's zigs and zags.

Does this show a multiplicative relationship? Well, let's check.

I can see here that two zigs is equal to three zags, two zigs equals three zags.

Now I'm gonna double the number of zigs.

Four zigs is six zags, so yes, it does.

I doubled the number of zigs and that doubled the number of zags.

We could also represent this on a double number line.

You're really, really familiar with double number lines now and we could represent this with a double number line.

Here is my double number line and we can clearly see from this that two zigs, if we find two on the zig line, is equal to three zags.

And again, if we find four on the zig line, we can see that that is equal to six zags.

So these are two alternative ways of representing the same information.

Izzy and Jun are going on a school trip.

Gosh, they're lucky.

I wonder where they're going.

They need to exchange their spending money from pounds into euros.

Oh, they're going somewhere in Europe.

Izzy says, "I changed 100 pounds and I got 117 euros." Jun says." I've got 75 pounds.

How many euros will I get?" And we can represent this graphically or on a double number line.

So just like with the zigs and the zags, but here we've got real-life currencies, not computer game ones.

Let's take a look at that.

Often, exchange rates are shown graphically.

So we can see here a graph that shows pounds against euros.

This is the point that we know and use to draw the line.

We knew that Izzy had 100 pounds and that was exchanged for 117 euros and we know that zero pounds is equal to zero euros.

So therefore, we can draw a line that joins zero, zero up to this point that we know, which is Izzy's known exchange.

We could estimate the value of 75 pounds.

So Jun says he has 75 pounds and wants to know how many euros he's going to get.

So we could estimate that from this graph.

We're going to find 75 pounds and that will be on the horizontal axis because that's our pounds axis.

So let's find 75.

Well, it's halfway between 70 and 80, so it's here.

So notice, I've drawn a nice perpendicular line from the axis up to my purple line, which is my exchange rate line.

And then I'm going to horizontally go across and I'm going to read off that value.

And we can see that it's approximately 87 euros, and that's only an approximation.

It's not very easy with the graph I've given you to read off an exact value, but we got an approximate idea of how many euros Jun is going to get.

Alternatively, we could represent this on a double number line and we'll now take a look at that.

So we've still got the same thing.

We've got Izzy changing 100 pounds and got 117 euros and Jun wants to know how many euros he's going to get.

Here's my double number line.

I've got pounds and I've got euros.

I'm going to represent my pounds on the top line and euros on the bottom.

Now, we know that 100 pounds is equal to 117 euros and Jun has 75 pounds.

So that's why I've placed it to the left 'cause 75 is less than 100, and we want to work out the equivalent in euros.

And like I said, you are super, super good at double number lines now, so we know how we're going to solve this problem.

We are looking for the multiplicative relationship between 175.

And remember, if you're not sure, do the inverse.

We do 75 divided by 100 and I've just written that as 75 over 100.

It does simplify and I could write it as a decimal, but actually here I've just decided because I didn't have my calculator at hand that I'm going to just rewrite my division as a fraction 'cause we know we can do that, they mean the same thing.

Moving from the right to the left, I am multiplying by 75 over 100, so I need to do the same to the euros.

That means 87.

75 euros.

Jun will get 87.

75 euros.

So our graph gave us a very good idea but we didn't have the exact answer of this 75 cents at the end.

So like I said, notice in here that we are able to find the exact value, whereas with the graph, we were only able to find an approximate value.

Exchange rates are always changing.

They change all of the time.

What would happen to the graph if the exchange rate changed and Izzy got 185 euros? Would it get steeper? Just think about that a moment.

What have you decided on? Do you think it's going to get steeper? Let's take a look.

The green line here shows our original exchange rate.

We can see Izzy was getting 117 euros.

The purple line shows the new exchange rate.

Now she's got 185 euros.

The higher the exchange rate, the steeper the graph is going to be.

The purple line is steeper than the green one.

So the higher the exchange rate, the steeper the line.

Given that these are plotted on axes that have exactly the same scale, I'd like you please to match each exchange rate to the correct graph.

Pause the video.

No guessing, I want you to reason this.

Remember, the higher the exchange rate, the steeper the graph.

Good luck with these and I'll be here waiting when you come back and we'll check those answers.

Good luck.

Let's check those answers.

A was with F.

This graph.

B matched with D and C matched with E.

We can see the steepest graph was E and that was the highest exchange rate, 175.

Then the next steepest was D and that was the next highest exchange rate.

And then the shallowest was F with the lowest exchange rate.

If we know a conversion of currencies, we can draw our own conversion graph.

I changed 100 pounds and I got $125.

So we are looking at dollars now and I can see here that we've marked on 100 pounds and $125.

And remember that zero pounds is equal to $0.

So that's how I'm able to plot that purple line.

We're gonna estimate now how many dollars we would get for 60 pounds.

So we need to make sure we're on the right axis first.

And that's the horizontal axis, that's where the pounds are.

We find 60, we draw a line.

Remember, it needs to be perpendicular to that horizontal axis up to the purple line, which is our exchange rate line.

And then we can read across.

You would get approximately $75.

Remember, it's only an approximation at this point on this graph because of the scale that I've decided to use because I need you to be able to see it from where you are.

Now we're gonna estimate how many pounds you would get for $100.

So this time I know the dollars, so I need to find 100 on the dollar axis and that's the vertical axis.

So I'm going to find my line.

Remember, it needs to be perpendicular to the axis and I'm gonna follow across until my purple line and then I'm gonna read off, and I can see here that we would get approximately 80 pounds.

Now you're ready to have a go at this check for understanding.

In this check for understanding, I'd like you to spot the mistake.

The question is use the graph to estimate the number of pounds you would get for $50 and the answer that this person has given is 40 pounds.

You can now pause the video, work out the mistake, and even better if you could tell me what the correct answer is.

I'll be here waiting for you when you get back.

Good luck.

Well done.

What did you come up with? And this is a really, really common mistake, which is why I wanted to highlight it.

Here the person has worked out what $40 is.

They've used the wrong scale to start with.

If we look, the dollars is on the vertical axis, so this person should have found 50 on the dollars axis and then red down to find out what the value is and we can see that the correct answer would've been approximately 63 pounds.

Well done if you got that right.

Like I said, it's a really common mistake.

You must make sure you look carefully at which axis you are starting from.

You're now ready to have a go at some questions independently.

I think I've got four of these for you to do.

So you can pause the video when you've done this question and then come back.

Remember, you'll need to plot the line yourself.

So remembering that zero pounds is $0 and that 10 pounds is $15.

So mark across where 10 and $15 is and then draw a nice straight ruled line joining those points together, remembering to go across the entirety of your graph.

Good luck and then I'll see you in a moment when you come back.

Well done.

Now let's take a look at question number two.

Exactly the same thing.

The exchange rate has changed.

It's now that eight pounds is equal to 10 euros.

So mark that point on, draw your conversion line and then have a go at the two questions.

Good luck and pause the video.

And part C.

You're good to go on this one now.

I'll see you in a bit.

Well done.

And D.

Great work.

Let's check those answers.

So this should be what your line looks like.

And then eight pounds is equal to $12 and $6 is equal to four pounds.

Part B, that's what your line should look like.

And six pounds is anywhere between 7.

2 euros and 7.

8 euros.

And four euros is between three pounds and £.

3.

40.

C, five pounds is anywhere between 6.

2 euros and 6.

8 euros.

And 16 euros is between £12.

20 and £12.

70.

And part D, £11 is $12.

20 to $12.

70 and $8 is £6.

80 to £7.

20.

So as long as any of your answers are within those range, they are absolutely fine.

Now we can move on to looking at writing exchange rates as a ratio.

Andeep, Jun and Sofia are going on holiday.

Lucky them.

They all exchange some pounds for euros, but who got the best deal? Let's have a look and see what they got.

Andeep said, "I changed 120 euros and I got 100 pounds," "I received 30 euros for 20 pounds." "I received 90 euros for 60 pounds." So there are the different amounts of money.

Now, it is very difficult at the moment to compare them probably because they didn't exchange the same amount of money, but I want you to just think about how might we go about comparing these? Well, we can write them as ratios.

Let's take a look at what that looks like.

Same information there.

So Andeep's as a ratio is eurps to pounds is 120 to 100.

Jun is 30 to 20 and Sofia's is 90 to 60.

Has that helped? No.

We need to create equivalent ratios where one of the currencies is the same in all three.

Wonder what we could do.

I've noticed that they're all multiples of 10, so we could find how many euros each of them would have received for 10 pounds.

Here's Andeep's.

How many euros for 10 pounds? Jun and Sofia using the exchange rate that they got.

How do we get from 100 to 10? We divide by 10.

So if I'm dividing the number of pounds by 10, I must divide the number of euros by 10, giving us a ratio of 12 to 10.

What do I need to do with Jun's? I need to divide by two.

So divide the euros by two.

This gives us a ratio of 15 to 10.

And Sofia, 60 divided by what is 10? That is six.

Divide 90 by six and we get the ratio of 15 to 10.

Who got the best exchange rate.

Andeep received 12 euros for every 10 pounds.

Jun and Sofia both get 15 euros for every 10 pounds.

Therefore Jun and Sofia both got the best deal.

If they'd not all been multiples of 10, we would need to create equivalent ratios where a currency is set to one Same processes we went through when we were trying to find how much 10 pounds would get us.

I'm dividing by 100, so I divide by 100.

This is my new ratio.

I divide by 20.

So I divide 30 by 20.

There's my new ratio.

And then the final one, we already know it's the same as Jun's but let's just look.

60 divided by 60 is one.

So 90 divided by 60 is 1.

5.

When we have the ratio in the form one to n, this is the exchange rate, it's giving us the exchange rate.

For every one pound, I'm getting 1.

2 euros.

Or for every one pound, I'm getting 1.

5 euros and that's the exchange rate.

We could also have worked out the exchange rate of euros to pounds.

Let's take a look at that.

Here we have the same information, but what we're looking at now is working out the exchange rate for one euro and how many pounds.

This time, we need to divide 120 by 120 to get one on the euros.

And so I'm gonna do the same division there and that gives me one to 5/6.

Jun's I'm gonna divide by 30 and I get one to 2/3.

And we know that Sofia's is the same.

And remembering that when we have the ratio in the form one to n, that this is the exchange rate.

This is exchange rate for euros to pounds.

Here are the exchange rate we've just calculated.

So we've got Andeep's, Jun's and Sofia's.

We've got them as one pound to euros and one euro to pounds.

Izzy says, "Earlier you said Jun and Sofia had the best deal, but now Andeep's is better." What do you think about Izzy's comment? This is why it's important we know what the ratio is telling us.

This means for every one pound, you get 1.

2 euros.

This one means for every one pound, you get 1.

5 euros.

You're getting more euros for each pound.

If we look at this exchange rate, this is saying for every 83 pence, you get one euro.

And this one is saying for every 67 pence, you get one euro.

Each euro costs you more here.

So each euro costs you more here, which is why that exchange rate is not as good because you are paying more for each euro.

True or false? The exchange rate shown on the left is a better deal than the exchange rate shown on the right.

Pause the video.

Remember, you need to be able to justify your answer.

So choose true or false and also your justification.

When you're ready, you can come back.

You can pause the video now.

Great work.

What did you decide? It's quite difficult.

I find it quite difficult to get my head around this.

Let's take a look.

It was true.

And the reason was is each euro costs less.

If we look at the first one, each euro costs you 78 pence.

And in the second one, each euro costs you 87 pence.

The exchange rate for pounds to Australian dollars is one pound equals $1.

92.

Andeep says, "I have 120 pounds spending money for my holiday.

How many dollars will I get?" And we can set this out using a ratio table because we have expressed it as a ratio and we know that in a ratio table, we need to find the multiplicative relationship and my multiplicative relationship is multiplied by 120.

So I need to multiply the number of dollars by 120, giving me $230.

40.

The exchange rate is still the same.

One pound is equal to $1.

92.

Andeep says, "I didn't spend all of my money and I have brought home $48.

How much will I have after I've converted it back into pounds?" Let's have a look.

Here's my ratio table.

We've got the ratio of the exchange rate at the top and this time, we've got $48.

So making sure that 48 goes in the dollar column.

I'm looking for my multiplicative relationship.

And actually, it's easier here to look at the multiplicative relationship horizontally rather than vertically because we know that this is multiplied by 1.

92.

So our ratio table, we can either work with a multiplicative relationship horizontally or vertically and sometimes it won't really matter which you do, but other times, it may be easier to do one rather than the other.

Now, this time though, I'm going backwards.

I'm going in the opposite direction.

What do I need to do if I'm going in the opposite direction? Yes, that's right.

We need to do the inverse.

We need to divide by 1.

92 and that gives us 25 pounds.

Now, it's worth saying here that actually, Andeep probably wouldn't get 25 pounds back exactly because the companies who exchange the money take a fee themselves.

But throughout this lesson, we are just going to assume that it's free, they've got a special offer on and they're not charging you for exchanging your money.

We've now got a different exchange rate and we are looking at Japanese yen.

One pound is equal to 182 Japanese yen.

We're gonna convert 400 pounds into yen.

So start with your exchange rate on the top.

Make sure that you put your value in the correct column.

Then look for that multiplicative relationship and make sure you do the same to both, 72,800.

I could also have worked horizontally.

Now we're going to take a look at exchanging back from yen into pounds.

So imagine we've been on holiday and we've not spent all of our money and we're going to convert 50,960 yen back into pounds.

We start with our ratio table with our exchange rate in.

We've got our 50,960.

And here again, this way horizontally is going to be much easier.

My multiplier is 182, but remember, we're going back in the opposite direction.

So we must do the inverse, which is divide by 182 and that gives us 280 pounds.

We'll get 280 pounds back.

I'd like you now to do this check for understanding.

So you're going to pause the video and then you're going to work out what the mistake is and then when you are done, you can come back.

Good luck.

What mistake did you come up with? Hopefully it's the same as mine.

We needed to use the exchange rate to convert 46 pounds, sorry, 46 euros into pounds.

And I can see here that the 46 has been put in the pounds column instead of the euros column.

So that's the mistake.

Really important we get things in the right columns.

The 46 was 46 euros, so should have been in the right column underneath the 1.

15.

Really important to make sure that we get everything in the correct columns.

Now you're ready to have a go at task B.

I'd like you please to find the missing values in each of the tables, given the exchange rate in the top row.

You can pause the video now.

Obviously, you're gonna need to use your calculators for this.

That's absolutely fine.

But make sure you keep recording of the workings that you did and when you're ready, you can come back.

You can pause the video now.

I'll be here waiting when you get back.

Well done.

And then now let's have a look at question number two.

Andeep is comparing prices of sunglasses in the UK, the US and France.

We want to rank the following from the cheapest to most expensive.

So again, pause the video, come back when you're ready.

How did you get on with that one? Let's check our answers.

1a is 322 euros.

B, £39.

50.

C, 2,775 yen.

D, 25 euros.

E, £12.

50.

F, 87 pounds.

And then in order, so we converted them all into pounds.

We've got the middle one, which was in the USA and that was £45.

60.

And then the French, when we converted it was £45.

20.

So in order from cheapest to most expensive was the UK, then France and then the USA.

Well done.

We can now summarise our learning from today's lesson.

So we've been concentrating on looking at exchange rates and we looked at exchange rates shown graphically and we can estimate conversions using a graph.

But we also looked at then finding exchange rates on a double number line and then we finished up by using the ratio tables.

Remember, an exchange rate can be written as a ratio.

Thank you for sticking with me for today's lesson.

You've done fantastically well and I really look forward to seeing you again shortly.

Goodbye.