Lesson video

Hi there, my name is Ms. Lambell.

You've made a superb choice deciding to join me today to do some maths.

Let's get cracking.

Welcome to today's lesson.

The title of today's lesson is, "Checking and Further Securing Understanding of Direct Proportion in a Context," and this is within the unit compound measures.

By the end of this lesson, you'll be able to recognise direct proportion in a range of contexts, including compound measures.

Some keywords that we'll be used in today's lesson are rate, exchange rate, and direct proportion.

A rate is a relationship between two units, for example, miles per hour, which is abbreviated to MPH, and this is a rate called speed, which relates to distance and time.

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.

Two variables are in direct proportion if they have a constant multiplicative relationship.

Today's lesson is split into two separate learning cycles.

In the first one, we will just concentrate on currency conversion, so using those exchange rates, and then we'll move on to looking at proportional relationships as rates in the second learning cycle.

Let's get going with that first one, currency conversion.

Here we have Andeep and Izzy.

Let's see what they're chatting about.

Andeep says, "Hi Izzy, I noticed something when changing my foreign currency back at the airport after my holiday." "What was that Andeep?" "When I changed my money before I went, the exchange rate was one pound equals 1.

17 euros.

When I changed at the airport whilst I was waiting for my flight, the exchange rate was one euro equals 0.

86 pounds.

Izzy says, "Exchange rates are always changing" and Izzy's right, exchange rates are always changing.

Andeep says, "I know that, but why was one of them written as one pound equal to euros and the other as one euro equal to pounds?" Izzy's response to that is, "It will depend on the currency of the country you are currently in." "Ah, yes, that makes sense," Andeep says.

Let's take a look at the exchange rate in the UK.

One pound equals 1.

17 euros.

This could also be written as a unit ratio.

It could be written as one pound to 1.

17 euros.

What does that mean? It means that for every one pound you'll get 1.

17 euros.

Let's take a look at the exchange rate in the European country.

One euro equals 0.

86 pounds, and this could be written as a unit ratio, one euro to 0.

86 pounds.

What does that mean? This means that for every one euro you'll get 0.

86 pounds.

Or might be easier to think of it as, for every one euro, you'll get 86 pence.

I'd like you now to have a go at this check for understanding.

You're gonna use the ratios to complete the sentences, pause the video, and then when you've got your answers, pop back.

Great work, let's check those answers for you then.

So we should have for A, this means for every one pound you will get $1.

26.

B, this means for every $1, you will get 0.

92 euros.

C, this means for every one euro you will get $1.

08.

And finally D, this means for every $1 you will get 0.

53 pounds.

Izzy and Jun are going on a school trip.

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

Izzy says, "I changed a hundred pounds and I got 117 euros." Jun says, "I have 75 pounds, how many euros will I get?" We can represent this exchange graphically on a double number line or in a ratio table.

Let's take a look at that.

Often, exchange rates are shown graphically.

We can see here the graph of the exchange rate that Izzy got when she exchanged her money.

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

So we can see here that 100 pounds is equal to 117 euros, and we can use this to draw the line because we know that zero pounds equals zero euros, and therefore we can join a line from zero, zero going through the point we know.

Does this graph show a directly proportional relationship between pounds and euros? Yes, it does.

How do we know that it shows a directly proportional relationship? And the reason we know that is because the Y-intercept is zero and the relationship shows a constant positive gradient.

And those are the two things that we need a graph to show that the variables are in direct proportion to each other.

Now, we could use this line to estimate the value of 75 pounds.

We find 75 on our pounds axis, we draw up to our exchange rate line, and then we can read across.

75 pounds would be approximately 87 euros.

An alternative representation would be a double number line, and we will take a look at that now.

Here's our double number line.

We've got our pounds and we've got our euros, and we know that Izzy changed 100 pounds and got 117 euros, and Jun wants to change 75 pounds, and we want to know how many euros he's going to get.

This is how we're gonna set up our double number line.

Now you are super good at double number lines, so we now know we are looking for the multiplicative relationship between 175 and that's multiplied by 75 over 100.

So I'm gonna multiply my euros by 75 over 100, giving me 87.

75.

Jun will get 87.

75 euros.

Notice that by using the double number line, we are able to find an exact value.

We're only able to find an approximation on the graph because of the scale of our graph, but here, we can find the exact value.

We can also represent this in a ratio table.

A ratio table really, is a double number line, but we just represent it in a table form.

We've got a hundred pounds is equal to 117 euros and the 75 pounds that Jun is going to change.

The same multiplicative relationship.

Of course it is, because we are using 100 pounds and 117 euros, so we know that we're going to get 87.

75 euros.

That's how many euros Jun will get for his 75 pounds.

As we've mentioned already, exchange rates are always changing.

What would happen to the graph if the exchange rate changed and Izzy got 185 euros? Would it get steeper? We can see here the graphs, the green line shows the original exchange rate and the purple line shows the new exchange rate, and we can clearly see that the purple line is much steeper than the green line, the original exchange rate.

So the higher the exchange rate, the steeper the graph will be.

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

So you're going to pause the video and then when you've got your answers you can come back and we'll check those for you.

Pause the video now.

Great work, let's check those answers.

So A matched with graph F, B matched with graph D, and C matched with graph E.

How did you get on? Of course you got them all right, well done.

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

Izzy's now off somewhere where they have dollars and she's got 100 pounds and she's exchanged it and got $125.

we can represent this on a graph.

We know that zero pounds is equal to $0 and we know that 100 pounds equals $125.

Those are my two points and then I can draw my straight line Joining those points.

Remember, that line extends beyond that point because we could change any number of pounds or dollars that we wanted to.

Jun wants to change 80 pounds and Andeep wants to change 150 pounds.

Can this graph be used to find how many dollars Jun and Andeep get? Alex says, "We can use it to find 80 pounds but not 150 pounds because that is off the graph." Laura's response to that is, "If we found the exchange rate, we could find any number of pounds." We will put this information into a ratio table to help us find the exchange rate, which is also the unit ratio of pounds to dollars.

We know that Izzy exchanged 100 pounds and got $125.

We want a unit ratio in pounds so that we can convert 80 pounds and 150 pounds into dollars.

What's my relationship between 101? I divide by 100, I need to do the same on the right hand side of my ratio table, giving me 1.

25.

As a unit ratio in pounds, the ratio is one pound to $1.

25 and this means that you will get $1.

25 for every pound that you exchange.

Now we can work out how much money or how many dollars Jun is going to get.

One multiplied by 80, so we are gonna multiply 1.

25 by 80.

June is going to get $100.

Now let's take a look and see how many dollars Andeep is going to get.

Andeep is exchanging 150 pounds.

I've multiplied one by 150, so I'm gonna do the same to the dollars, giving me 187.

50.

Jun is going to get $100 and Andeep will get $187.

50 or $187.

50.

Now, looks like they've got some money left so they've been off somewhere and they've got some money left, and they're going to exchange the money back.

Jun wants to change $20 and Andeep wants to change $18.

We could use this ratio table, but as we need to find more than one value, it would be more efficient to find the exchange rate, and that's the unit ratio in dollars because we are converting from dollars into pounds.

We know that 100 pounds is equal to $125.

We want to find the exchange rate in terms of dollars.

I'm going to divide 125 by 125 and then I need to do that to my pounds, giving me 0.

8.

We now know that 0.

8 pounds is equivalent to $1.

We can now work out our equivalent values in pounds for our $20 and our $18.

One multiplied by 20, so I'm gonna multiply 0.

8 by 20, giving me 16.

And let's repeat that now for Andeep.

Andeep wants to exchange $18.

One multiplied by 18 is 18, so I multiply 0.

8 by 18, giving me 14.

40 pounds.

Jun will get 16 pounds and Andeep will get 14.

40 pounds.

Now it's time for you to do a check.

I'd like you to write a unit ratio in euros for the first two, and for the final one, I'd like you to write a unit ratio for pounds.

Pause the video, give these a go.

You may use a calculator and then when you've got your answers, come back and we'll check those for you.

Good luck.

How did you get on? Super, well done.

We take a look here, there's my ratio table.

So I'm working out, I want to know the exchange rate or the unit ratio for euros, so I'm gonna divide by 100, giving me one euro to $1.

63.

Now let's take a look at Andeep's statement.

There's my ratio table giving us that one euro equals $1.

09.

Remember, it's okay to have that ratio written the other way around, but you must make sure you include the units.

Sofia, if we look at Sofia's here, we can see there's our ratio table giving us one pound to 1.

2 euros.

Now, I'm sure you've got all of those right, so you are ready now to move on to the first independent task of today's lesson, task A.

So you're gonna do exactly the same thing as you've just done.

You are going to work out the unit ratios, here though, I'd like you to decide what the appropriate unit ratio is and then you're going to use that to answer the questions.

Pause the video now and then when you've got those answers come back, and I'll reveal the next two questions.

Super, and parts C and D.

And question number two.

"At the airport you can pay in pounds or dollars.

Using the same exchange rate, calculate the cost of these items purchased in a cafe in pounds." So two sandwiches per $8 each, one coffee for $4.

25, 1 cold drink for $2.

50, and one slice of cake for $6.

50.

And you are going to use the same exchange rate that you will get on the T-shirt.

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

Super, and then question three.

"The exchange rate in London is one pound to 1.

16 euros.

The exchange rate in Madrid is one euro to 86 pence or 0.

86 pounds.

Sam wants to change some pounds into euros.

In which city would Sam get the most euros?" Lucas's statement is incorrect.

I'd like you to show me why Lucas's statement is incorrect.

And Lucas says, "The exchange rate is the same in both cities." Please show me why Lucas is incorrect.

Pause the video and then come back when you're ready.

Superb work on those, well done.

Let's check those answers.

One A part one, 928 euros, part two, 317.

84 euros.

B part one, $139.

52, and part two, $1,907.

50.

Now C part one, is 28,704 baht, and part two is 552 baht.

And then onto D, part one is $198, and part two is $957.

How did you get on? Great work, and now question two.

The most efficient way to do this was to work out the total cost of all of the items purchased in the cafe in dollars first, which was $29.

25.

Use the exchange rate given on the T-shirt giving us that $1 is equal to 0.

8 pounds so the cost in pounds was 23.

40.

And then finally question three, we can see here you should have this ratio table and you are working out one pound is equal to how many euros and we can see that Lucas is right if we round or truncate the number of euros.

That would give 1.

16, showing that it's the same exchange rate.

However, if we consider the most accurate exchange rate, he's incorrect, and Sam should choose to exchange their money in Madrid.

Great work on those questions.

Let's move on now to that second learning cycle.

And that is, proportional relationships as rates.

A tap fills at a rate of 12 litres per minute.

How long does it take to A, fully fill a bath with a capacity of 180 litres and B, fully fill a bucket with a capacity of 13.

5 litres? Why would it be useful here to find the time per litre? This will make it easier to workout how long it will take to fill various items. We know the rate is 12 litres per minute, 12 litres per one minute.

We've said that it's going to be easier because we're trying to work out how long it's going to take to fill a capacity of 180 litres and 13.

5 litres.

If we know how long it takes per litre, then that is going to make these calculations much easier.

So we're going to work out how long it takes to fill one litre.

We're going to look for our relationship and that's divided by 12, so I'm going to divide by 12.

And I've decided here to leave my answer there, as a fraction.

We now know that we are looking to the bath first.

We're trying to fill up the bath and that's 180 litres.

So I'm now looking for that multiplicative relationship which is multiplied by 180, 12th multiplied by 180.

Remember, you could think of that as a 12th of 180 if you didn't have a calculator.

But for today's lesson, you may use a calculator all the way through, that gives us 15 minutes.

So to fully fill the bath would take 15 minutes.

And then we can look at part B, we are now filling the bucket.

The bucket had a capacity of 13.

5 litres, one multiplied by what is 13.

5? That's 13.

5.

We do the same to the time, therefore we're gonna do one 12th multiplied by 13.

5, which gives me 1.

125 minutes.

"Lucas's dad wants to know how many gallons of fuel it takes for the distances he needs to travel if he buys this car." So on a Monday, travels 16 miles, Tuesday, 37 miles.

On a Wednesday, where he goes a long way, 154 miles.

And on Friday, 25 miles.

Looks to me like you must work from home on a Thursday.

And we can see that this car is advertised as 40 MPG.

What do you think MPG stands for? It stands for miles per gallon and this represents the rate of miles covered per gallon.

What is the most efficient way of answering this question? As the MPG, the miles per gallon doesn't change, we find the total number of miles and then we can use the MPG.

I'd like you please to calculate the total distance travelled by Lucas's dad during that week.

And you should get 232 miles by find in the sum of those four values.

We can represent this rate as a ratio table.

We know that we are doing 40 miles per gallon, so 40 in the miles and one in the gallons column.

Let's work out how many gallons we need to drive one mile.

So we're going to divide by 40.

Remember to do the same on the right, that's 0.

025.

Remember there, you may have written one 40th, one over 40.

We know that Lucas's dad needs to travel 232 miles.

So now we can use our unit ratio to multiply to find how many gallons multiply by 232.

So let's multiply the gallons by 232, giving us 5.

8.

He will need 5.

8 gallons of fuel.

This graph represents the fuel economy of a different car.

We can see our horizontal axis is litres and we can see that our vertical axis is kilometres.

And we can see from the graph that 20 litres will mean we can drive 400 kilometres.

We're going to use the graph to write a unit ratio for fuel and for distance.

Actually, I think you could probably have a go at this yourself before I go through it.

So pause the video, and see if you can find me please, a unit ratio for fuel and for distance, and then when you're ready, come back.

How did you get on? Did you get two unit ratios? Let's take a look and see if those are right, which I'm sure they are.

Start with the ratio table.

We know that 20 litres means that we can go a distance of 400 kilometres, if we want the unit ratio for fuel we want to find out how far we can travel on one litre.

So we're gonna divide by 20, so we end up with 20.

On one litre of fuel you'll be able to drive 20 kilometres.

Use the same starting table because we know that the fuel economy of the car has not changed.

So 20 litres takes us a distance of 400 kilometres, but this time we want our unit ratio in distance.

So we need to know how many litres we will use to travel one kilometre.

So I divide by 400, giving us 0.

05.

Now, could you please have a go at this check for understanding, I'd like you to complete the sentences, pause the video, and come back when you've completed both sentences.

Super, we should have for every one litre of fuel you can travel a distance of 20 kilometres.

And table B, for every one kilometre travelled, you need 0.

05 litres of fuel.

Which table is most useful for answering the following question? "How many litres of fuel do you need to travel 50 kilometres?" And that would be table B.

Because our unit ratio is in distance, which table is most useful for answering this question? "How far can you travel on 35 litres of fuel?" Now you probably said table A because I said the last one was table B, but let's have a look.

We know how many litres of fuel we have, so therefore it's easier to use the one where we know how many kilometres we go for one litre, so table A.

Let's have a go at this question together, and then you can have a go at the one on the right hand side yourself.

Here, we're going to choose the most appropriate table to calculate how far we can travel on five litres of fuel.

Table A is the most useful as the unit ratio is in litres and we know how many litres of fuel we have.

So we're going to use table A.

We know that one litre of fuel means we can travel 20 kilometres and we have five litres of fuel.

So we are looking for that multiplicative relationship.

And we can see that you can travel 100 kilometres on five litres of fuel.

Your turn now, same two tables.

I'd like you please, to choose the most appropriate table to calculate how much fuel you need to travel 36 kilometres.

Pause the video, and then when you've got your answer, come back and we'll check.

Which table did you decide? You should have gone for table B.

You can use table A, but we are looking at the most efficient method and that would be table B.

And that is because it's a unit ratio is in kilometres and we know how far we are travelling.

We're travelling 36 kilometres.

So here's my ratio table using table B.

We are nowhere travelling 36 kilometres, so my multiplicative relationship multiply by 36.

We end up with us needing 1.

8 litres of fuel to travel 36 kilometres.

Here, we have another situation where we can think about this as rates.

Here is part of an electricity bill.

What I'm gonna ask you to do in a moment is to pause the video and decide where you think electricity charges come from.

So that's the 2052 multiplied by 0.

245 equals 502.

74 pounds.

Pause the video, and look at the bill, and decide where each of those values has come from.

I'll be waiting when you get back.

Let's take a look, where did you decide that 2052 had come from? It was the units of electricity used.

And where does that come from? How was that calculated? Because there is nowhere on that bill that has that value apart from using it in that calculation.

So where did it come from? It was the new reading, subtract the previous reading and this will tell us how many units were used in that time period.

8,904, subtract 6,852 is 2052.

That's the number of units used during this time period.

What about the 0.

245, what did you decide that was? And that's the price per unit.

We can see the cost is 24.

5 pence per KWH and that stands for kilowatt hour.

Why is it different to the cost? The cost is given in pence and the calculation used the cost in pounds.

We probably wouldn't want to calculate a bill in pence, we would want to give it in pounds.

And then finally, what did the 502.

74 pounds represent? And this represented the total cost.

Your turn now, I'd like you please, to calculate the cost of electricity using this bill.

Pause the video, you may use your calculator, but make sure you write down all your steps of working to show that method clearly, and then when you get back, I'll be here waiting to go through the answer with you.

Great work, units used would be the new reading, subtract the old reading or the previous reading, giving us a total of 10,078 units used.

Cost per kilowatt hour is 28.

5 pence, which is 0.

285 pounds.

We multiply the number of units by the cost per unit, giving us a total cost of electricity of 2,872.

23 pounds.

Is that what you got? Of course you did.

Now we're ready to do task B.

The final task for today's lesson, so well done.

Let's keep going just for a little bit longer.

Question number one.

"A tap fills at a rate of eight litres per minute.

How long does it take to A, fully fill a bath with a capacity of 150 litres? B, fully fill a bucket with a capacity of 12 litres?" Pause the video and come back when you've got those answers.

Question two, "Lucas's dad wants to know how many gallons of fuel it takes for the distances he needs to travel if he buys this car." So we can see the distances travelled and we can see that the fuel economy of this car is 25 MPG.

And remember, that stands for miles per gallon.

Pause the video, come back when you've got that answer.

Question three, "This graph represents a fuel economy of a vehicle." I'd like you to use the graph to answer the following.

A, how much fuel do you need for 156 kilometre journey? B, how far can you travel on 20 litres? And C, the vehicle has a full tank of fuel and can travel 750 kilometres, what is the capacity of the tank? Pause the video and come back when you're ready.

Question four, here is part of an electricity bill.

I'd like you to calculate please, the cost of electricity using the bill and I'd like you to give your answer to the nearest penny.

Again, pause the video and then when you come back, I'm pretty sure this is the last question, so you'll be nearly done.

So just stick with me a little bit longer.

Pause the video now.

Great work, let's check through our answers.

Question one A, 18.

75 minutes and B 1.

5 minutes, or you may have put 18 and three quarter minutes, and one and a half minutes.

Question two, he will need 11.

44 gallons of fuel.

Question three part A, 8.

32 litres are needed for a journey of 156 kilometres.

B, 375 kilometres is how far you can travel on 20 litres.

C, the fuel capacity of the tank is 40 litres.

And finally, question number four.

The number of units used was 1,114.

To work out the total cost, we multiply that by the cost per unit of 0.

274 pounds, meaning the cost of electricity for this time period was 305.

24 pounds.

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

Any two variables that are directly proportional to each other share a multiplicative relationship and can be written as a unit ratio in either variable.

For example, we could write one pound to $1.

25 or 0.

8 pounds to $1.

It is useful to think of rates as per or for every.

For example, one pound to $1.

25 means $1.

25 per pound or $1.

25 for every one pound.

When working with metre readings such as electricity, gas, or water, the number of units used is found by subtracting the previous reading from the new reading.

Well done with everything that you've done during today's lesson.

I've been super impressed and I really look forward to you joining me again really soon to do some more maths.

Take care of yourself and goodbye.