Lesson video

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

Saada.

In today's lesson we will be looking at proportion in context.

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

If you don't have these handy, please pause the video, go grab them, and when you're ready, we can make a start.

Here you are given the ingredients for a bacon and egg pie that serves four people.

Zaki wants to make enough pie for two people.

What are the quantities of each ingredient that Zaki will need to use? Yasmin is cooking for six people.

What are the quantities of each ingredient that she will need to use? To try this task should take you about five minutes to complete, so please pause the video and complete the task.

Resume once you're finished.

How did you find this task? Okay, so Zaki wanted to make enough pie for two people.

The ingredients were given to serve four people.

So, all you had to do is really half everything.

I've halved everything and these are my answers.

Did you get the same answers? Really good.

Now, Yasmin wanted to make enough pie to serve six people, and there are so many different ways that you could've gone about this.

You could have calculated what is required for one person first, and then multiply it by six to find for six people.

You could have used the two, the information for two people and multiply it by three, so tripled it because you're tripling the number of people.

I have 300 grammes for puff pastry, three tablespoons oil, 300 grammes of bacon, three onions, three eggs, 240 mil of cream, and 60 grammes of cheese.

Did you get all of this correct? Really good.

What a great start to the lesson.

Okay, and keeping on the same theme, Antoni wants to make a pie for mom, dad, and herself.

Work out the quantities of each ingredient that she will need, and what made this challenging? Again, we have the same ingredients list that serves four people, and we want to know how much she will need.

Now, she is making this for herself, mom and dad, so for three people.

So, what I can do, we can use a unitary method to find for one person, and from that, then find for three people.

Or we can multiply by 3/4.

If we know that we have a recipe that serves four people and we want to serve three people, that's a three out of four.

That's 3/4, so we can multiply by 3/4 if we want to.

So, let's have a look first at the unitary method to start with.

So, I can say, "Well, okay, I'm going to work out what?" How much do I need to serve one person, and then how many do I need to serve three? 'Cause once I know one, it's easy to work out three.

For the first item, 200 grammes of puff pastry, you need 200 for four people, but we want for one, so we're going to divide by four.

If we divide it by four, that gives us 50 grammes.

Now for three, we need to multiply it by three, and that gives us 150.

And notice here, what have I done? First step, divide by four.

Second step, multiply by three.

That is the same as multiplying by 3/4.

I'm multiplying by three and I'm dividing by four.

Next one, two tablespoons of oil.

That's for four people, so to find for one person, divide by four, that gives us half.

Then multiply it by three, that gives us 1 1/2 tablespoons, which can be a bit tricky to measure.

Now, 200 grammes of bacon, divide by four.

That gives us 50 for one person, 150 grammes.

And for two onions, so for one person, we only need half an onion, and therefore for three people, we need 1 1/2.

Eggs, we need two eggs for four people.

So, if I divide that by four, I get half an egg for one person.

And again, for three, we need 1 1/2.

We do the same thing with 160.

Can you think about what would be the first step? What's the number? Really good.

So 160, we divide it by four.

We get 40 mil for one person, but we went for three people, so how much would we need? Excellent job.

120 mil.

Okay, and the last thing is the cheese.

You've got 40 grammes of cheese for four people.

How much would you need for one person? Excellent.

And for three people, you need to multiply that by three, and that gives us 30 grammes of cheese.

Really good.

Now, let's have a look at the different method.

So, this one, we've done it almost like by creating two columns or something that looks like a table, so let's look at a different method.

I can say I need 200 grammes of puff pastry for four people.

How much do I need for three? Well, from four to get to three, I multiply by three, so I need to do the same to the 200, and that will give me the 150 grammes that is needed.

This method is very similar to what we have done there with the two columns when we just listed them down.

Now, you could have also looked at this and said, "Well, you know what? "Let me find an easier way," because multiplying by 3/4 can be a bit challenging.

What have I done to get from here, from the 200 to the four? I divided.

What did I divide by? Excellent.

So, once you know how much you divided by here, you know what you need to do for the next set of numbers, so you can use that.

So, you don't have to only work vertically, you can also work horizontally, okay? So, we've divided here by, let me write it down.

By 50, so I know that I need to have something here that divides by 50 to give me three, and that something must be 150.

Okay, now let's look at one more example where we can do the written method.

So, we'll use the cheese one, okay? So, for cheese, we knew that we needed 40 grammes for four people.

What have I done? Divided by 10, so I know that I have something divided by 10 will give me three, and that's something must be 30 grammes.

This way, you can see that it's actually easier to do it across rather than vertically, rather than having to deal with fractions.

Okay, let's have a look at this example.

Max is going on holiday.

He has budgeted to take 180 pounds.

The exchange rate is one pound equal 1.

12 euros.

How many euros will Max have to spend? So, the first part of the question is telling me this key information.

It's telling me that he's taken 180 pounds.

It's giving me the exchange rate, and it's asking me for euros, so I know I need to give an answer in euros.

So, I can go and start answering the question.

Pounds to euros, one to 1.

12.

I want 180, so I'm multiplying by 180, and I get my answer, which is 201.

60.

Now, it's really important that we structure our work neatly.

I'm writing the pounds under the pound sign, The euros under the euro sign, to make sure I don't get things mixed up.

It's always a good idea to write the units or the names of the variables before you start writing any numbers so you know that you're writing the number under the right place.

Okay, now I could have done this slightly differently.

I could have said, "Well, okay.

"Pounds to euros is one to 1.

12, "and going across I'm multiplying by 1.

12, "so I know that I need to multiply 180 by 1.

12." I end up with the same answer.

It's a really good way of checking your answer is correct, and also, sometimes it's just the calculation is easier if we go across.

In this case, we ended up doing the same calculation anyway.

Sometimes it's just easier, so double check which way do you prefer.

Now, the second part.

In July, 2015, Max could have exchanged his pounds for euros at a rate of one pound equal to 1.

44 euros.

How many more years would Max have got if he had gone on holiday then? So, the exchange rate was slightly different.

It was one to 1.

44.

He wants 180, so if we multiply that, it is 259.

20.

So, if he had exchanged his pounds in 2015, that's how much euros he would have got.

Have we answered the question? Not yet because the question said, how many more euros would he have got if he had gone on holiday then? Well, if he went on holiday then he would've had 259.

20.

If he's going on holiday now he's getting 201.

So, the difference between them, I can do my subtraction here, and the answer is 57.

60.

So, Max could have got a lot more euros if he exchanged his money back in 2015.

How much is one euro worth in pounds? So again, there a couple of methods to do this, but we can start like this.

One pound is equal to 1.

12 euros.

I want to find what one euro is in pounds.

So what have I done from here to there? I divided by 1.

12.

From euros to pounds I divided by 1.

12, so I need to do the same thing now.

Divide by 1.

12, and that gives me 0.

89, so it's 89 pence.

So, one euro is equal to 89 pence.

So, here we go.

We can see that we can use direct proportion in real life, when we are planning for holiday exchanging money or helping our friends and family plan for the holidays.

Now it is time for you to have a go at the independent task.

You only have two questions to answer.

If you are feeling confident, you can pause the video now and have a go at this.

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

Okay, so if you need a bit of support, let's read the first question together.

This is a recipe for fruit punch for six people.

Jack made fruit punch for 10 people.

How many millilitres of orange juice did he use? You've been given the recipe there.

That recipe is for six people, you want it for 10 people.

So, what would be a good idea here is to find how much we need for one person first.

So you can start by writing this down.

Orange juice, the number of people.

We need 1,500 for six people.

How much do we need for one? Once you know for one, you can easily work out how much we need for 10 people.

For the second question here, Philippa went on holiday to Sweden.

The exchange rate is one pound to 1.

2 euros.

She exchanges 550 into euros.

So, you're looking at something really similar to the example that we have just done.

You need to start by writing pounds to euros.

Write down the exchange rate as it is given, and use that to help you work out how much 550 pounds is in euros.

Now, with this hint you should be able to make a start.

The independent task should not take you longer than 10 minutes to complete.

So, please pause the video and complete it to the best of your ability.

Resume the video once you're finished.

Welcome back.

How did you get on with this? Okay, let's Mark and correct the work together then.

The first question here you had orange juice for six people.

1,500 was used.

You wanted to find for 10 people, and my answer was 2,500 millilitres.

Now, there are so many ways you could have done this.

I did it in a single step by multiplying by 10 out of six.

You could have first used the unitary method to find for one person how much orange juice is needed, and then multiply that by 10.

You may have decided to start by dividing by three to find for two people, and then multiply by five to find 10.

So, there's so many different ways that you could have answered this question.

For the second one, you were looking at exchange rates between pounds and euros.

We had one pound equal to 1.

2 euros.

We needed to find out how many euros we will get for 550 pounds.

So, what we have done here, we multiplied by 1.

2.

So, we multiply by 1.

2 and that gives us 660 euros.

For the second part, Philippa returned home with 78 euros that were left with her from the holiday, but now the exchange rate is different.

So, one pound is equal to 1.

25 euros.

We want to know how much is 78 euros in British pounds.

So, what have we done from here to there? We divided by 1.

25, so we need to do exactly the same thing here.

And if you do this correctly, the answer is 62 pounds 40 pence.

So how much money, how much pounds would she get back? That's how much money she would get back.

interesting to know that when you go on holidays the exchange rate is very different when you sell or buy different currencies.

This brings us to our explore task.

Let's read it together.

The price of posting a parcel is proportional to the mass of the parcel.

What does this mean? Calculate the cost of sending the parcels below.

You have been given four parcels.

The first one is 100 grammes, the second one, 500 grammes, the third one, one kilogramme, and you know that this one will cost you six pounds to post.

And the last one is 2.

5 kilogrammes.

Draw a graph to represent this information.

What shape will the graph be, and why? Write that equation, linking the mass of the parcel m to the price p.

Now, if you're feeling super confident about this, you can pause the video now and have a go at the question.

If not, I will be giving you a hint in three, in two, and in one.

Okay, so my first hint is in this question here, the mass is given in a mixture of grammes and kilogrammes.

We need to do something about it.

So you either convert everything into kilogrammes, or you convert everything into grammes to help you make a start.

Now, thinking about this, you can start by converting kilogrammes to grammes.

One kilogramme has how many grammes? Really good.

It has 1,000, therefore the 2.

5 kilogrammes will have 2,500.

Now, once you've written it like this, you can start writing down the information and the proportion that you know.

What do you know? What relationship do you know about the mass and the price? We know that if I have 1,000 grammes at one kilogramme, it will cost me six pounds to post it.

So, if I have 100, how much would it cost me? Remember the question said that the price of posting is proportional, so there's a constant multiplier between the mass and the cost.

Now, with this hint you should be able to make a start.

Now I have another hint for you.

When you are going to draw the graph, you need to have the mass on the x-axis and the price or the cost on the y-axis.

The explore task should take you 15 to 20 minutes to complete because it may take you a bit longer to draw the graph and the axes.

So, please pause the video and complete the task.

Resume once you're finished.

Welcome back.

How did you get on with this? Okay, really good.

Let's mark and correct the work together.

So, the cost of posting the parcels are shown here.

So, for the 100 grammes it will cost 60 pence.

For the 500 grammes it'll cost three pounds, and for the 2.

5 kilogrammes, it'll cost 15 pounds to post.

Did you get this correct? Very good.

The second part, draw a graph to represent this information, and what shape will it be? So, let's talk first about the shape.

What shape will it be? It will be a linear graph.

Really good.

It will intersect the y-axis at the origin at zero, zero.

We know this because there's a direct proportion relationship between the mass and the price, okay? And if there's a direct proportion relationship there, then we end up with a linear graph passing through the origin.

So, if you plotted your graph you should have something that looks like this.

If you calculated the gradient for the graph, it should be six.

Did you do that? Really good.

You could have done that by choosing two points, worked out the difference in y and the difference in the x coordinates, divided the difference of y by the difference of x, or by drawing triangles around two points.

So, whichever method you feel more confident with.

Now, if I was going to write down an equation of that straight line, remember y equal mx plus c.

What would be the equation of this line? Y equal m is the gradient, so six x plus c.

Well, c, it crosses at the origin, so it's just y equal six x.

Now, next part said write an equation linking the mass of the parcel m to the price p.

So, we can write down that m multiplied by six is equal to p.

So, now I can write an equation.

P is equal to six m, and the six here is the gradient, but it also represents the constant of proportionality.

Remember, we've learned about this in our previous lesson.

This brings us to the end of today's lesson.

You've done some fantastic learning, so a huge well done.

Please don't forget to complete the exit quiz to show what you know.

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

Bye.