Lesson video

Hi there.

My name's Ms. Lambell.

Made such a good choice deciding to join me today to do some maths.

Let's get cracking with it.

Welcome to today's lesson.

The title of today's lesson is "Direct Proportion in Context".

And that's in the unit, understanding multiplicative relationships, in particular, with percentages and proportion.

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

A quick reminder about what direct proportion is.

If two variables are in direct proportion, they have a constant multiplicative relationship.

Today's lesson, I've split into two learning cycles.

In the first one, we will concentrate on conversions, and in the second one, we will look at other contexts involving direct proportion.

Let's get going on that first one.

So we're concentrating here on conversions.

Conversions between units of measure is an example of variables which are directly proportional.

For example, converting between metric units, converting between millilitres and litres, or kilometres and metres, converting between units of time, so converting between hours and minutes, or seconds and days, converting between different currencies.

So if I've got an exchange rate, I can convert between pounds and euros, for example.

Converting between metric and imperial units, so maybe converting between miles and kilometres.

Since variables which are directly proportional to each other sharing multiplicative relationship, we are able to use a ratio table to find other information.

Aisha says, "My dad has a really good way of remembering the miles and kilometres conversion fact." And Jacob says, "Oh, please tell me.

When I run, I measure it in kilometres, but my mum always wants to know what it is in miles." His mum wants to know in miles, she's not really sure about kilometres.

But Jacob, I expect, he's got a smart watch, or a pedometer, or something that gives his result of his run in kilometres.

Aisha's dad's way of remembering then the conversion is this.

"If you had a 5 mile wait, in kilometres, it would be 8." Oh, that is pretty cool actually Aisha, 'cause it rhymes.

So it should be nice and easy to remember.

Let's just say it again.

"If you had a 5 mile wait, in kilometres it would be 8." And Jacob agrees with me.

"Yeah, that's brilliant.

Really good way of remembering." Aisha and I wants to know a bit more about Jacob's running.

So she says, "How far did you run this weekend, Jacob?" Jacob says, "I ran 3 kilometres on Saturday and 4 kilometres on Sunday." How far did Jacob run this weekend, but in miles? So we want to convert it so he can tell his mum, so she's got an idea of how far he's run.

We can represent this on a ratio table, because miles and kilometres are directly proportional.

"If I had a 5 mile wait, in kilometres it would be 8." There's my conversion.

That's what I need to start with.

I need to start with that equivalent that I know.

We can see that in total, at the weekend, Jacob ran 7 kilometres.

We are looking now then for the multiplicative relationship between 8 and 7, which is 7/8, multiply by 7/8.

So therefore, I need to multiply by the miles, sorry, by 7/8.

So 5 multiplied by 7/8 is 4.

375.

This means that Jacob ran 4.

375 miles at the weekend.

Oh we got Pedro Panda now.

Wonder if you've seen Pedro Panda before.

I expect you have.

Another rhyme now.

This time to help us remember the metric/imperial conversion for mass.

And it's this.

"A kilo of Pedro Panda's bamboo, when measured in pounds, is 2.

2." So again, it rhymes to help us.

"A kilo of Pedro Panda's bamboo, when measured in pounds, is 2.

2." Nice.

"Jacob, do you know how heavy you were when you were born?" So, Aisha knows that she was 8 pounds and 3 ounces.

Wondering if you know how heavy you were when you were born.

And if you do, do you know it in pounds and ounces, so in imperial measurements? What do you know it in kilogrammes metric? Jacob knows his in the metric unit of kilogrammes.

He says "Yes", that he was 3.

63 kilogrammes.

We want to work out who is heavier at birth.

Now because we've got one in imperial and one in metric, we can't spot quickly which person was heavier at birth.

We need to have them in the same units.

We know that kilogrammes and pounds are directly proportional, and I'm going to use my rhyme.

"A kilo of Pedro Panda's bamboos," that's 1 kilo, "when measured in pounds, is 2.

2." And I know Jacob's weight in kilogrammes, so I wanna make sure I put out the correct place in my ratio table, so on the kilogramme side.

Now, I'm looking for my multiplicative relationship.

This time, I've decided to move horizontally.

So I'm going to try and keep reminding you, you can go whichever way suits you, horizontally or vertically.

One multiplied by 2.

2 is 2.

2.

So, therefore, I need to multiply my number of kilogrammes by 2.

2, giving me 8 pounds.

We can now see that Jacob was 8 pounds at birth, so Aisha was heavier.

She was 3 ounces heavier.

Not very much heavier, but just a little bit.

Your turn now using our rhyme, "A kilo of Pedro Panda's bamboo, when measured in pounds, is 2.

2." I'd like you to decide which of the following is correct.

If Pedro Panda eats 110 pounds of bamboo in a day, how many kilogrammes does he eat in a day? Pause the video, come back when you've got that answer.

And what did you decide, A, B, C, or D? It's B.

Let's take a look at why.

We do 110 divided by 2.

2, because this time we're going back from pounds to kilogrammes.

So that's 50 kilogrammes.

Did you get it right? Super great.

Aisha and Jacob now are looking at this question.

"Calculate the area of this rectangle." Aisha says, "The area of a rectangle is the length multiplied by the width." Is Aisha right? Yeah, she is, isn't she? Multiply the length by the width.

Jacob says, "I'll work that out." Oh, I bet Jacob's got his calculator to hand.

It is 1.

24 multiplied by 5.

62, which is 6.

969 metres squared.

So he's gonna put the answer onto the answer line, 6.

969.

Is Jacob's answer correct? No.

The units on the answer line are centimetre squared and not metre squared.

Jacob's worked out the answer in metres squared, but that's not what we were asked to do, looking at the question.

Aisha says, "We can change our answer.

Centimetres and metres are directly proportional." Which is true, isn't it? "Therefore, we can use a ratio table." Also true.

Brilliant, Aisha.

Jacob says, "Okay, we will do that.

And there are 100 centimetres in a metre." He's remembered the conversion 100 centimetres in 1 metre.

Here's Jacob's original answer, and here is the ratio table, 1 metre is 100 centimetres.

Yep, agreed.

And then we've got 6.

969.

Moving horizontally, my multiplicative relationship is multiplied by 100, so I multiply by 100.

So our answer actually should have been 696.

9.

Aisha says though, "That still does not seem right." Hmm, wonder why Aisha thinks that.

Why might Aisha say this? What do you think? Well, luckily, Aisha's gonna share her ideas with us.

She says she's estimated the area by rounding 1.

24 metres to 100 centimetres.

Right, she knows the answer needs to be in centimetres, so she's converting at the beginning, and she's rounded 5.

62 metres to 600 centimetres.

This gives an area of 100 multiplied by 600, which is 60,000 centimetres squared.

Let's just check back through Aisha's thinking and Aisha's conversions.

1.

24 metres is definitely closer to 1 metre than 2 metres, which is 100 centimetres.

5.

62 metres is closer to 6 metres than 5 metres, which is why she's chosen 6 metres, converted it to 600 centimetres.

And this gives an area, yeah, because the area of a rectangle is length times width, and 100 multiplied by 600 is definitely 60,000.

"Ah," Jacob said, "you're right Aisha, my answer of 696.

9 centimetres squared is not even close to 60,000 centimetres squared." Ah, I wonder why.

What's going on? We're gonna take a look at what is wrong with this method.

Let's start by looking at metre square.

So on the left, I've labelled it with metres, and on the right I've labelled it with centimetres.

What's the area of the square on the left? This area is 1 multiplied by 1, which is 1 metre squared.

The area of the one on the right, is 100 multiplied by 100, which is 10,000 centimetres squared.

Are these exactly the same squares? Yes, they are.

So therefore, they must have the same area.

This must mean that 1 metre squared is equal to 10,000 centimetres squared.

You know 1 metre squared is equal to 10,000 centimetres squared, which is what Aisha just said.

Jacob says, "Which makes sense, as both the length and the width have been multiplied by 100." And Aisha's response is, "Yes, that means that the overall effect on the area is 10,000." Yeah, 100 multiplied by 100 is 10,000, isn't it? Let's go back and look at our original ratio table.

This was the ratio table that we used when we realised that our answer was not in centimetre squared, it was in metre squared, and we multiplied by 100.

Jacob says, "Looking at this now, it was obviously not going to work.

This ratio table is converting metres to centimetres." Aisha says, "But we were converting metres squared to centimetres squared, so we do need a whole new ratio table." Let's take a look.

Here was our value.

We worked out 1.

24, or Jacob worked out 1.

24 multiplied by 5.

62 was 6.

969.

That was in metre squared.

We've now just shown that 1 metre squared is the same as 10,000 centimetres squared.

So this is what our ratio table should have looked like, our multiplicative relationship, and we're gonna do the same to our area giving us 69,690.

Aisha's estimate was 60,000, wasn't it? So it's correct now.

It's in the right sort of area.

It's in the 10,000s.

You are now gonna have a go at this question.

I've given you there the conversion so that you don't need to look back at a previous slide or previous things in your book.

So I'd like you, please, to convert 542,800 centimetre squared into metre squared.

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

You shouldn't need a calculator for this because you're all multiplying by powers of 10.

Okay, and the answer was C.

And as a calculation, that would be 542,800 divided by 10,000, giving us the 54.

28 metre squared.

And now this one, please.

This time we're going from metre squared, so you hopefully you've drawn out a ratio table, and this time we're going from metre squared to centimetre squared.

Our multiplicative relationship is 10,000, so it was D, 0.

852 multiplied by 10,000 is 8,520 centimetre squared.

"Calculate the volume of this cuboid, give your answers in metres cubed." Aisha says, "Let's work out the proportional relationship between centimetres cubed and metres cubed." All three dimensions will be multiplied by 100, meaning, the overall effect will be multiplied by 1 million.

What do you think? Yeah, I think Jacob's right, isn't he? If I multiply the length by 100, the width by 100, and the height by 100, the overall effect is going to be multiplying by a million.

The volume of this cuboid.

Remember, to find the volume of a cuboid, we do length multiplied by width, multiplied by height.

Does it matter if I have those in a different order? No.

Because if I flip that, where I stand it, so it's taller, its volume doesn't change.

It just looks slightly different on the table.

That is 108,000 centimetres squared.

Here is my ratio table showing my conversion between metres cubed and centimetres cubed, making sure that I put my 108,000 in the centimetre cubed side of the ratio table, and then I'm looking for that multiplicative relationship.

Here, I've decided to show that as a division, I could show it as a multiplication, but I'm showing it as a division.

We know that we can flick between multiplication and division when we're looking at multiplicative relationships.

The answer is 0.

108, and that would be metres cubed.

If you are not confident with that, you could always convert all of your values before you do your calculation.

Now, your turn.

Draw yourself a ratio table and convert 13.

24 metres cubed into centimetres cubed.

Great work.

And the correct answer here was C.

We multiply 13.

24 by 1 million, which gives us 13,240,000 centimetres cubed.

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

Now the images are not to scale, that's really important.

I've given you a conversion of 1 metre is equal to 3.

3 feet, so that is going to be your conversion that you are going to use in the top row of your ratio table.

I would like you to write the following in order from the shortest to the tallest.

You need to make sure that you convert them all into the same unit.

That's entirely up to you whether you want to convert them into feet, or if you want to convert them all into metres.

Once you've got them all in one unit, you'll be able to order them from the shortest to the tallest.

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

Question number 2.

You're going to work out the maximum number of boxes below that will fit into the box above.

You've got a cube with dimensions of 40 millimetres.

How many of those cubes can you fit in the big box at the top? And for part B, you've got some boxes there with the dimensions.

How many of those could you fit inside the larger box at the top? Good luck and come back when you're ready.

You can pause the video now.

And question number 3.

I'd like to fill in the missing values, so, using those ratio tables.

Good luck and come back when you're ready.

You can pause the video now.

Question number 5.

Can you please calculate the volume of this cuboid give your answer in centimetres cubed? Pause the video and then when you come back, we'll check those answers.

The correct order were the garage, then the elephant, then the teepee, and then the bus, and then the giraffe.

Question number 2.

a.

you could put fit in 24,000 boxes, and b.

16,000 boxes.

3a.

was 32,000, b.

12.

5, c.

23.

987, and d.

740,000.

Question number 4, is 1.

09 metres squared.

Remember, there, you needed your answer in metre squared, and also, to two decimal places.

Well done with those.

And question number 5, the answer is 504,000 centimetres cubed.

Well done if you've got that one right.

Now, we're going to move on to looking at other contexts that involve direct proportion.

As well as conversions, there are other contexts which involve direct proportion.

So for example, changing between different currencies, scaling recipes, scaling the cost of particular items. Here, we have an example about cost of pencils.

The cost of pencils is directly proportional to the number of boxes.

One box costs 3.

24 pounds, what is the cost of 18 boxes of pencils? I think you'll be able to work this out yourself.

So I'm gonna pause a moment and give you a chance to work that out.

Right, we want the cost of 18 boxes, so 18 needs to make sure it goes on the boxes side of my ratio table.

I'm looking for my multiplicative relationship.

I've decided to go vertically.

If I went horizontally, it would be multiplied by 3.

24.

So I'm multiplying the top row by 18, giving me an answer of 58.

32 pounds.

18 boxes of pencils cost 18 pounds, sorry, 58.

32 pounds.

How did you get on? See, I knew you'd get it right.

The number of dollars received when exchanging currency is directly proportional to the number of pounds.

Here is our ratio table.

What is this ratio table telling us? It's telling us that if I purchase 15 pounds worth of dollars, I'm going to end up with $18.

90, and we want to know how many dollars we would get for 100 pounds.

How much will we get for 100 pounds? Again, I'm gonna pause because I think you're so good at ratio tables now.

You probably don't need me to go through these examples, so I'm gonna give you an opportunity to have a go yourself.

Multiplicative relationship.

I've multiplied by 100/15.

I could convert that to a decimal, I could convert it into a simplified fraction, but I don't need to.

My multiplicative relationship on the right is going to be the same, meaning, I get $126 for every 100 pounds.

And our recipe problem.

The amount of milk needed to bake scones is directly proportional to the number of scones being made.

We can see here that 15 scones requires 180 millilitres of milk.

A cafe has 4.

5 litres of milk.

How many scones can they make? 4.

5.

My multiplicative relationship, I've decided to move horizontally, multiply by 12.

This time I'm going in the opposite direction, so I need to apply the inverse, which is divided by 12, and I get 0.

375.

Does my answer look right? No.

What mistake have I made? I've forgotten to change the 4.

5 litres into millilitres, which was the unit that was used in the recipe.

It's really important to check that you are using consistent units when working in questions.

It should have been looked like this, 400, sorry, 4,500 millilitres, that's equivalent to 4.

5 litres of milk.

We know our multiplicative relationship hasn't changed because we haven't changed the 15 or the 180.

We know we're doing the inverse, and so therefore, we can make 375 scones if they have 4.

5 litres of milk.

So really important to make sure you check that you are using consistent units throughout the question.

And because the recipe was in millilitres, it makes it more sense to change our amount of milk into millilitres too.

I could have done it the other way round.

I could change the recipe into litres and leave it as 4.

5.

Now, we'll do this one on the left together, and then I know you are super ready to give the one on the right a go independently.

The cost of batteries is directly proportional to the number bought.

12 packs cost 4.

56 pounds.

How many packs can you buy for 95 pounds? Ratio table with our equivalent at the top, 12 packs is 4.

56 pounds.

We've got 95 pounds to spend, so we need to make sure that that goes in the pounds column, the cost column, then I'm looking for that multiplicative relationship.

Now, here, I type that into my calculator and it automatically gave me my answer in its simplified fractional form.

So I could do the same to the left-hand side.

The packs giving me 250.

You can buy 250 packs for 95 pounds.

Your turn.

The cost of batteries is directly proportional to the number bought.

15 packs cost 5.

25 pounds.

How many packs can you buy for 210 pounds? Pause the video, draw your ratio table, find that multiplicative relationship, and find out for me how many batteries we can buy for 210 pounds.

You can pause the video now.

Okay, how did you get on? Here, we've got 15 packs cost 5.

25 pounds.

We want to know how many we can buy for 210.

Let's look for that multiplicative relationship.

It's multiplied by 40, so I multiply by 40.

I can buy 600 packs.

Now, you can have a go at Task B.

Question number 1, I'd just like you, please, to work out the following.

Pause the video, make sure you show me all your workings, and then come back when you've got all of those correct answers.

Okay, question number 2.

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

I'd like you to rank them from the cheapest to the most expensive.

And I've given you the conversion rate, the exchange rate, for pounds to dollars and pounds to euros.

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

Remember, I don't just want to see the order.

I want to see those ratio tables with the converted values so that I can be confident you understand how to put those in order rather than just guessing.

But of course, I know you wouldn't just guess.

Question number 3, I'd like you to calculate the ingredients needed for the following number of cookies.

So our recipe is for 15 cookies.

You're going to work out the recipe for 18, 72, and 6.

And then part D, you have plenty of sugar, flour, and chocolate chips, but unfortunately, you only have 1 kilogramme of butter.

What's the largest number of cookies that you can make? You've got enough of all of the other ingredients.

Pause the video and come back when you've got all of those recipes.

Okay, let's check those answers then.

Question 1a.

22 pounds and 10 pence, b.

7.

40 pounds, c.

5.

55 pounds, d.

15.

99 pounds.

Question 2.

They go in order from the cheapest to the most expensive, USA, UK, France, and we can see there that if we convert the US dollar price of $60.

80 into pounds, that's 47.

50.

And if we convert 57.

23 euros into pounds, that's 48.

50 pounds.

And the recipe, the cookies, making me a bit hungry now.

I keep talking about cookies.

So a.

butter 90 grammes, flour and sugar both 150 grammes, chocolate chips 120 grammes.

b.

for 72 cookies, we need 360 grammes of butter, 600 grammes of both sugar and flour, and 480 grammes of chocolate chips.

For six cookies, we need 30 grammes of butter, 50 both of sugar and flour, and 40 grammes of chocolate chips.

And part d, I've got plenty of sugar, flour, and chocolate chips, but only 1 kilogramme of butter.

1 kilogramme is equal to 1,000 grammes.

I've divided that by 75 because that tells me how many lots of the recipe I can make.

And then I've multiplied that by the number of cookies that one recipe makes, which is 200.

So the 13.

3, recurring, represents the number of times you could make the recipe.

And since you can scale the recipe 13 and one third times, you could make 200 cookies.

Alternatively, you may have worked it out this way.

You may have done 75 divided by 15, to work out how much butter you need for one cookie, and then take in your 1,000 and divided it by the number of grammes of butter for one cookie, giving you 200.

Either way, you get the same answer of 200 cookies.

Now, we can recap our learning and we can summarise it.

Conversion between units of measure, are an example of variables which are directly proportional.

Make sure that we remember that 1 metre squared is equal to 10,000 centimetre squared, and 1 metre cubed, is equal to 1 million centimetres cubed.

As well as conversions, there are other contexts which involve direct proportion.

And the ones we've looked at today, are changing between different currencies, scaling recipes, and scaling the cost of particular items. And ratio tables can be used to solve direct proportion problems. Fantastic work today.

Well done.

And hopefully, I'll see you again really soon.