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

In today's lesson we'll be looking at direct proportion.

For today's lesson, you will need a pen, a paper, and a ruler.

So, if you do not have these handy, please pause the video, go grab them, come back, make sure you're sitting somewhere quietly so you can concentrate on today's lesson.

And when you're ready, click resume, and we can make a start.

Your first task for today's lesson is to find how many different arrows can you place between these boxes? Move the boxes around.

Do you need different arrows? Please pause the video and have a go at the try this task.

This should take you five minutes to complete.

Please resume once you're finished.

How did you get on with this task? There are so many possible solutions for this one.

This is what I have done.

I looked at the 30 and the six and I thought from 30 to get to six, what do I need to do? Well, I can divide by five, but that was not given to me on one of the arrows.

Instead I can multiply by a fifth because dividing by five I'm multiplying by a fifth or equivalent, and the same thing applies for the three and 15.

So, to get from the 15 to the three, I will need to multiply by a fifth as well.

Now, looking at the six to get to the three.

And the 30 to get to the 15, in both cases, either I divide by two or multiply by half.

One of the other possible solutions would be this, what I've done with this one is I moved around the numbers.

So, I started with the 15.

So, I put the 15 and the 30 at the top, and then underneath them the three and the six to see whether that would then, to see whether that would make a difference or not.

I wonder how you did that.

Let's have a look at this question together.

The cost of apples, x, is directly proportional to the mass, y.

The cost of three kilogrammes is six pounds.

Complete the tables shown below using two different methods.

We have been given two tables.

The first table has one, two, three, four, five, for the mass of apples in kilogrammes.

And we need to find the cost.

Now, the only piece of information we have been given is the six pounds, which is the cost of three kilogrammes, but we have been told that they are directly proportional.

So, this is one way of looking at it.

I can think about it like this.

What did I do from three to get to six? I multiplied by two.

This tells me that I just need to multiply here by two as well.

One times two, and that gives me the cost for one kilogramme of apples.

It also tells me the cost now for two kilogrammes of apples, and it does the same for four kilogrammes and for five kilogrammes.

Just by looking at one relationship, going down from one to the other, the vertical relationship or the vertical multiplier.

What happened? And I managed to answer it.

Now, how can I answer using a slightly different method? Let's have a look.

From one to get to three what did I do? I multiplied by three.

So, whatever the cost was for one, I must have multiplied it by three to get the six pounds.

So, that multiplicative relationship is not just going vertically, but also horizontally.

It's going both directions.

One to get to three are multiplied by three.

So, something times three gives me six, that something must be two.

So, I can write down two pounds.

Now that agrees and works well with the first table, doesn't it, it's the same as the answer we had from table one.

Now, let's have another look.

From one to two we multiply it by two.

So for the cost, we must also multiply by two and that gives us four pounds.

So again, the numbers are agreeing with the first table.

Let's have another look at another one.

From one kilogramme to four kilogrammes, we multiplied by four.

So, I need to multiply the cost of one by four.

And that gives me eight pounds.

And for five, I just don't have space to show you the arrows, but from one to five, I multiply by five.

So, multiply the two by five.

And that will give me 10 pounds and we have the same numbers.

And the point of this example was just to show you that I can multiply in both directions.

I can multiply this way.

So, going vertically and I can multiply going horizontally If I have something in direct proportion.

And now let's have a look at this question.

The cost of text messages, x, is directly proportional to the cost of text messages, y.

The cost of three text messages is one pound 80, complete the tables shown below.

And again, we have two tables and we're going to complete them using two different methods.

So, if I look at the first table, I have three text messages cost one pound 80.

So, if I look at the relationship this time, I'm multiplying by 0.

6.

I'm multiplying by 60 pence, because what I'm really saying here, if three text messages cost me one pound 80, then each text message, one text message is costing 60 pence.

So, the multiplier this time is not an integer and that is absolutely fine.

It does not have to be a whole number.

So, I can do the same thing here and say that 60 pence is the cost for one text message.

For five, multiplied by 60 pence, that gives me three pounds.

And if I have seven text messages, again, multiply by 60 pence, that gives me four pounds 20.

Now let's check.

Does it work across as well? So, to get from one to three, multiply by three.

So, something multiplied by three is one point 80.

What is that something? Yeah, that is 60 pence.

So, it works here.

From one to get to five, we multiply by five, so I need to do the same thing for the cost, multiply by five.

And that gives me three pounds.

Again, my numbers are matching with the first table, and for seven, we multiply by seven, so 60 pence multiplied by seven.

That gives me four pounds 20.

So again, I've had the same two numbers using the two different methods.

And this is just to show us that again, we can do the multiplication vertically or horizontally.

It really doesn't matter.

Sometimes you would find that it's a lot easier to do it horizontally.

Sometimes you will find that it's a lot easier to do it vertically.

In all cases, doing it both ways means that you are checking your working out and that you're not making any silly errors, 'cause if you do it both ways and end up with a different number, you know that you went wrong somewhere.

And then you can pause and double check your answers and work it out.

It's time for you, so have a go at the independent task.

Please read the questions carefully and try and answer them to the best of your ability.

If you're are finding the questions a little tricky, go back to the examples that we have done.

Copy them down into your book and use them to help you and guide you with the independent task.

The independent task should take you about 10 minutes to complete.

Please pause the video and complete it, resume once you're finished.

Welcome back.

How did you find this task? Really good, okay.

So, these are some of my answers.

Can you please start marking and correcting your work, I've completed the table here for you.

For part B, one metre cost three pounds so 60, I multiplied that by three and that gave me 180.

And you can see that I did my multiplication horizontally.

For part C, for 560 metres.

Again, I did it horizontally because it's easier to multiply this way by three, rather than to multiply vertically by 560.

Okay, and my answer was 1,680.

For the last one, again, I found that horizontally and actually doing the division was easier.

So, it's three divided by three.

It got me to one, so 980, I need to divide that by three.

And that gave me a number.

My answer was not a whole number and that's fine because the question didn't say to the nearest metre, it just said how much you can buy.

So, I did 980 divided by three to find the number of all the amount of metres that I can buy with 980 pounds.

I didn't round it up because the question didn't say rounded to the nearest metre or centimetre or anything, it just asked for how much we can get back.

We it's, it just asked how much can we get? How many metres can we get for 980 pounds.

Did you get this correct? Well done.

Okay, now let's have a look at question two together.

One litre of fruit punch contains 200 millilitres of apple juice and is enough for 12 people.

How much apple juice is that in 1.

5 litres of juice? So, let's make a start on the first part.

One litre means 1000 millilitres.

We need to make sure that the units are the same before we start doing any calculation.

And now I said, well, fruit punch to apple juice, 1000 millilitres, 200 millilitres.

So, what did I do? I divided by five.

So, I did my calculation going horizontally, not vertically, from 1000 to 200, I divide by five.

So, I divided 1,500 by five, and that gave me 300 millilitres.

And I knew that that's the answer.

Now for part B, Harry used two litres of apple juice for the same recipe.

How many people was this enough for? So, we're looking at people, we're are not looking at the fruit punch and the apple juice, we're looking at Apple juice to people.

Now, what information did we have before? We knew before that the apple juice to people, the ratio was 200 to 12.

So, now I want two litres.

Two litres is 2000 millilitres.

So, what did I do to get from 200 millilitres to 2000? I multiplied by 10.

So, I multiplied the number of people by 10.

And that gave me 120 people.

And you can see here that with the first one, it was a lot easier to do my calculation horizontally because I could divide by five.

Whereas with this one from 200 to 12, I don't know what I need to multiply or divide by.

I will have to work it out, but I can visibly see that from 200 to 2000, I multiply by 10.

It's so much easier to spot so that it's easier to go to the 12 and say, let me multiply that by 10.

And that's my answer, 120 people.

Question three, Karen bought 12 individual packs of crisps priced at 70 pence each.

How much money would she have saved if she, if she had bought multi pack bags at six each, at cost of two pounds 79? So, as she bought 12, 12 multiplied by the 70 pence, it means that they cost her eight pound 40.

Now, if she buys the packs and she wants to 12, she will have to buy two of them.

So that's two pounds 79 months multiplied by two, that's five 58, and you can already see that she would have saved quite a bit of money.

If you do the subtraction here, you can work out how much she would have saved.

And you can see that she would have saved enough to buy another pack.

This brings us to our explore task.

Let's read it together.

Pete is painting the walls in his bedroom.

So far, he has used five litres of paint, and covered 24 metres squared of wall.

What area will 10 litres cover? What area will 30 litres cover? How much paint will cover 240 metres squared? In each case, explain how you worked out the answer.

What area will seven litres cover? How many methods can you use to solve this? So, I want you to really think about all the different methods that you can use to work it out, whether it's tables, whether it's bar models, whether it's writing it down as ratio, anything that, anything in any method that you can use, I want you to think about it.

But most importantly, I want you to be able to explain your method.

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

If not, don't worry, I'll give you some support.

Support in three, in two, and in one.

Okay, so.

As a hint, look at this, we have been told five litres covered 24 metre squared.

What about 10 litres? What's the relationship from five to 10? Did you spot that? There you go, that's your first clue.

Then after this from five to 30, what have we done? That's your second clue.

Seven litres, well, we only have five litres, but five to seven, it's not, from five to seven is not an easy multiplier.

It can be done, but it's not an easy multiplier.

Can we use the unitary method? Can we find one litre? What would be the one litre cover? from one litre can you work out seven litres? How much would seven litres cover? With this hint, you should be good to go.

Please pause the video and complete the explore task.

Try and use as many methods as you can.

This should take you 10 to 15 minutes.

Once you're done, please resume the video.

Welcome back.

How did you get on with this task? How many methods did you use? I would love to know.

Okay, so this is some of my working out for this question.

So, I decided to use a table.

I said, paint and area on the wall.

Five is 24 metres squared.

So, for 10 I'm multiplying by two.

So, that was easy, I could have doubled it.

For 30, I didn't even bother working it out from the five, I used the 10.

I knew that 10 litres would cover 48, multiplied that by three.

And that gave me the 30 litres.

And for 240, I looked and I was like, oh, I can see a connection there between the 24 and the 240, I'm multiplying by 10.

So, I need to multiply the amount of paint by 10.

And that gave me 50 litres.

I wonder if you did the table too.

Now for the seven litres, I thought, let me start by drawing a bar model, divided into five equal parts.

That's five litres, one of, one of the parts.

Well, if the five litres cover the 24, then 24 divided by five, that's 4.

8 metres squared.

So, one litre covers 4.

8 metres squared.

And now this makes it a lot easier for me to find seven litres.

I also tried with the ratio methods.

So, I did five litres covers 24 metres squared.

One litre covers 4.

8 metres squared.

So, seven multiplied by seven and found out the answer.

Now there was one more interesting method.

Just thought it looked like a ratio.

So, I wrote down five litres will cover 24 metres squared.

Now, seven litres.

What did I do from five to 24? What is my multiplier? And I know that some of us find it really difficult if it's not integers.

Well, actually it's not that difficult.

If I have five and I want to get to 24, I need to multiply by 24 out of five as a fraction.

And now seven multiplied by that 24 out of five as a fraction will give me the answer.

And that answer matches with the answer I got from that other methods.

Those are some of my thoughts.

You could have done so many things with this question.

And as I said, I would love to see you work on this.

This brings us to the end of today's lesson, a huge well done on all of your efforts and all the mathematical thinking that's been happening, especially in that last explore task.

Please remember to complete the exit quiz, to show what you know.

Enjoy the rest of your day.

And I'll see you in the next lesson, bye.