Lesson video

Hi there, and welcome to this lesson with me Dr Saada.

In today's lesson, we will be looking at proportional relationships.

All you need for today's lesson is a pen and a paper.

So pause the video, go grab these, and when you're ready, let's make a start.

To start today's lesson, I would like you to try this.

This 12 step staircase is 324 centimetres high.

How high is the sixth step? How high is the ninth step? Which steps are the easiest and the hardest to work out the height of? So if you're feeling super confident about this, and you think you can tackle this question on your own, please pause the video now and have a go at this.

If not, don't worry, I'll give you some support in three, two, one, there we go.

So, the question says 12 step staircase, and the 12 steps are 324 centimetres high.

We want to find out the sixth step.

Well, if the whole staircase is 12 steps, and we want to find the sixth step.

That's saying I want to find that six out of 12, which is halfway over.

What is half of 324? Think about it carefully.

You can do this without a calculator.

You can say, what's half of 300? What's half of 20? And what's half of four? Add them up.

Okay, I think with this hands, you should be able to make a start on your own.

So pause the video now and have a go at this.

All right.

Now, we should be able to mark and correct the work.

We said half of 324, I want you to work that out.

Half of 300 is 150, half of the 20 is ten, and half of the four is two.

So if we add them up, that gives us 162 centimetres.

Did you get that? Good job.

How high is the ninth step? So the ninth step is nine out of 12 steps.

And I can simplify this fraction to three quarters.

So really what I want is three quarters of 324.

Now there are so many different methods for finding three quarters of 324.

I love bar models, so I'm going to use the bar model for this.

I'm going to start by drawing a bar, and I'm going to label it 324 and I divide it into four equal parts because I want to find three quarters.

Now to find one part, I need to do 324 divide by four, and that gives me 81.

This tells me that I have 81 in here, 81 in here, and 81 in here.

Obviously, I still have an 81 in the last bit.

I only want the three quarters, that's 81 times three.

And therefore, the answer is 243 centimetres.

Now which steps are the easiest and the hardest to work out the height of? Obviously, finding half is one of the easiest.

Finding three quarters, we just did that it's not that difficult.

So finding the ninth step, okay? Now, finding a quarter? What will the quarter be? Good job.

So finding the third step is not going to be difficult.

We already actually have it, we know it's 81.

Finding maybe one out of 12.

So the first step, we need to divide by 12 to find two or the second.

We double it, really good.

So we can do this, which one would be the hardest maybe? Very good.

So the hardest would be, for example, the fifth step, because I don't have something that I can half or double, or just divide by three to find.

So the fifth step could be a bit tricky.

The eighth step, the seventh step.

So some of them will be a bit trickier than others to find out.

Okay, some really good job, let's move on to the Connect task.

So for our Connect task, we're going to continue thinking about the Try this question.

So we're going to continue to think about the 12 step staircase question.

I've given you a table here.

It has steps and heights, and it's got zero, one, two, three, five, six and 12 for the number of steps, and it has a height of 324 for the 12 steps, none of the other heights are being given for you.

So we're going to work them out ourselves.

So we're going to look at how can we represent problems like this using bar models for today's lesson.

So I can say what I want to work out the height of six steps.

So I'm going to start with the bar model.

I'm going to have 12 equal parts to represent the 12 stair steps.

And these are 324 centimetres.

I don't want the 12, I'm interested in six of them.

So I draw a bar that has been divided into six equal parts.

And I want to know how much this is.

Now obviously, the easiest way is to say half of 324 equals 162 centimetres and we know what the height of six steps are.

Now the other way is finding one equal part.

So in that bar model, the first one where it says 324, thinking about, okay, I have 12 equal parts, if I divide those parts, so if I divide 324 by 12 parts, I get 27.

So I know for example, that each one of these is 27.

I know I have 27 here, I know I have 27 here, and so on.

Each of them, this is 27, this is 27, this is 27.

Therefore, I must have 27 in each of these, okay? And if I do 27 multiply by six, that will give me 162.

So I can go to the table now and write down 162.

Okay, next one.

Let's look at four at the fourth step.

So, again, I'm going to start our model.

I'm going to start a new one, because I've written on the first one.

If you are working this in your book, you can always try and use the same one if you have enough space on it.

So 324 is the 12 steps, I want four.

So what's the answer to this one? Now I can say 324 divide by 12 and this tells me that each equal part is 27 centimetres.

So I can write 27, 27, 27, 27, either at the map or 27 multiplied by four, it's entirely up to you, it's whatever you feel confident with, okay? And that is four out of 12 of 324.

If I add these up or multiply 27 by four, that gives me 108.

And I can go to the table and write down that it is 108.

Now also, it's important to note that four out of 12 is the same as a third, so it's a third of the 324.

So I could have also done a bar model and divide it into three equal parts and found only one part.

So I'm going to write the 108 here.

Okay, so I could have also done because I know that this fraction in this fraction are equivalent, I could have drawn a bar model and divide it into three equal parts and said the whole thing is 324.

Divide that by three to find one part, which will be 108.

Okay, now let's move on to the next one.

So we have done, what half of 324 is.

Now can we work out what the third step would be, the height of the third step? So three out of 12.

Three out of 12 simplifies to a quarter.

Do I have any information that I've already calculated that can help me finding a quarter of the whole thing? Good thinking.

I have a half of 324, makes 162.

Well if I know what a half is and I want to find a quarter, I just have to half the half, okay? Cause a half of half is a quarter.

So now I can see the quarter of 324 is equal to 81 centimetres.

So I had 162, half of that 160, half of it is 80.

Half of the two is one.

So I end up with 81 centimetres, okay? So I have 81 centimetres or the other way it could have been that I go back to the bar model.

And I say, well if each of these is a quarter, each of these is 27, I want a quarter.

So that's one, two, three and let me draw a line or something here and say what I want 27 plus 27 plus 27, or 27 multiplied by three and that would have given us 81.

So now I can go to the table and write this down.

Okay, the next one, can we find step two? The height of step two? What do I already know that I can use to help me with step two? Okay, good job, good thinking.

I already know what four out of 12 is.

So if I know what four out of 12 is, I should be able to find two out of 12 because it should be half of it.

So I can go and say, well, two out of 12 of 324 is 54.

Because all I have to do is half 108 by hand already.

Now again, the other way is to the bar model and say, well, I only want up to two here, I only want two sets, 27 plus 27, which is 54.

So I can write this down.

Okay.

Now, if I know this, can I work out this step one, the height of step one? Good.

I can say one out of 12 of 324.

And that should really be half of what I already have, half of the 54.

Half or 50 is 25, half of four is two is 27.

Or I can just look at the bar model and I've got my own set in the bar model.

Okay, so that's why with the bar model is hard for you don't have to even think about fractions if you want that.

So we can do this.

And now for step zero, what's happening with step zero? It's going to be zero because if I have one part is 27, no parts is zero and that means we're still at the ground level, there's no height there, okay? Now let's have a look at five, step five.

What would step five be? You can really get to there.

There are a couple of ways of doing it.

I can think about it as five out of 12 of 324.

Or I can go to the bar model and say, well, I want five parts.

So I want one, two, three, four, five.

Each part is 27, so 27 multiplied by five, and that would give us? 135.

And we are done.

Okay, there's something really, really important I want you to notice from this task, and from the table in particular.

Now if we look at the table, to get from one to 27, what do we do? We multiply by 27.

From two to 54, multiply by 27.

Three to 81, we multiply by 27.

Four 108, exactly the same thing.

12 to get to the 324? Excellent.

So in every case, we are multiplying by 27 or multiplying by a constant number, that number is not changing.

It works every time.

I mean even in the case of the zero, we have zero if you multiply by 27 that gives you a zero.

It works also for the five for the six, we had seven, eight, anything.

That is a constant change here.

The two quantities are either step, number of steps and the height are set to be directly proportional, because one quantity is always a constant multiple of the other.

There are all of these numbers are multiples of these numbers up there, okay? And it's always that constant relationship of multiplying by 27.

So the number of steps and the height of the steps are directly proportional to one another.

And this is what allows us to use those bar models really nicely, or to refer to fractions when we're trying to do the working out.

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

So both questions using at least two methods.

Question one, seven apples cost five pounds 95.

Each apple costs the same amount.

What is the cost of nine of these apples? Question two, seven apples cost five pound 95.

Each apple costs the same amount.

If I spend 11 pounds five pence on these apples, how many have I bought? What is the same and what is different about these two questions? Now this is what I would like you to do.

Pause the video and have a go at the two questions using at least two methods.

Try and use a bar model as one of your methods.

Once you're done, you can press play and we can mark and correct the work together.

So pause the video in three, two, one.

Welcome back.

How did you find that question in the independent task? Okay, I'm going to go through the solutions and I would like you to mark and correct your work as we go along.

Are you ready? Let's make a start.

Seven apples cost five pounds 95, each apple cost the same amount.

What is the cost of nine of these apples? So I'm going to use method one as bar models cause I absolutely love bar models.

I think they make problem solving a lot easier and really visual.

So method one, I'm going to start with a bar model.

I have a bar here that I divided into seven equal parts to represent the cost of that seven apples, 5.

95.

I know that each part is equal because the question said each apple costs the same amount.

I want to find out the cost for nine apples so I'm going to draw a bigger bar and divide it into nine equal parts.

And I want to find how much the total cost for the nine apples is.

So to start with, I can see that the 5.

95 is equal to five seven equal parts so I can say 5.

95, divide that by seven and that gives me 85 pence.

So I know now that in each of these spots, I can put an 85 pence.

Which also tells me I can put 85 pence here.

And now to find the cost of the nine apples, all I have to do is eighty five pence multiplied by nine and that is seven pounds 65.

So I know that total cost is seven pounds 65.

I could have also done 85 pence plus 85 pence plus 85 pence.

If you're not really confident with multiplication, you can do that.

Okay, let's look at method number two.

So I started by writing this question or using and thinking about ratio.

So the way I represent ratio questions.

So number of apples to the cost, seven apples, five pound 95.

Can I find out the cost of one apple, which is very similar to what we've done in the bar model that when we found one part, okay? From seven to one, I divide it by seven.

So I need to do the same to the cost and divide that by seven, which gives us 85 pence.

Therefore, to find nine I'd multiply by nine and multiply here by nine and that gives me seven pounds 65.

And this is a really nice way of checking that your answer is correct, because if you end up with a different answer in each of them, you know that you went wrong somewhere.

I wonder how many of you used the bar model and used it correctly? Remember with the bar model there's so many different representations.

Yours could be slightly different to mine, doesn't have to be exactly as mine but you should end up with the same answer.

And I wonder how many of you use ratio to solve this? Okay, I have actually done a third method because I thought it was a really interesting one and it will relate to things that we are going to learn in this unit.

So I started with a table.

So I created this table and I said number of apples, cost and then I wanted to work out the cost divided by the number of apples.

So I have been given that there are seven apples, I know that the cost of these is five pound 95.

So if I divide the cost by that number of apples that will give me basically the cost of one apple, which is 85 pence.

I want to find nine, obviously one apple is 85 pence so I need to do exactly the same thing so that I have 85 pence as the cost of the apple.

And now the total cost is seven pound 65 pence.

Okay, so now the three methods have given me exactly the same answer, really good.

Because they've given me the same answer, I can now say the cost of the nine apples is seven pounds 65.

And the beauty of using three different methods is that if one doesn't work, or you think that oh, you know what I've reached the point where I cannot work this out, you can use another one, or you can use more than one method to check if your answer is correct.

Okay, well done.

Again, now we're going to look at question number two, seven apples cost five pound 95.

Each apple costs is the same amount.

If I spend 11 pounds and five pence on these apples, how many have I bought? We're going to use the same methods that we have used for the previous question.

So for method one, we used the bar models, we're going to use that.

For method tool, we used on refer to ratio.

And we found the cost of one unit, and it's called the unitary method.

So we're going to find one, the cost of one and use that to help us.

Now we're going to use a table and look at the proportion between one value or one thing and the other.

So as I start with method one, so we have bar models.

We're going to start it in a similar way to the previous one, I have a bar, I divide it into seven equal parts, and that is representing five pounds 95.

Now this time, if I spend 11 pounds and five pence, I don't know how many apples, I want to work at the number of apples.

So I know I'm spending more than 5.

95.

So my second bar is going to be a bit bigger.

I don't know how many equal parts I'm going to divide it into.

Okay, so with the first one, we know that each equal part is 85 pence, because we've just worked that out earlier.

Now my second bar a bit bigger.

Now how many equal parts I'm going to divide it into? I know that it is equivalent or it's worth 11 pounds and five pence.

So now I want to know how many equal parts I have to put inside it to know how many apples I need.

Now I know that each equal part is 85 pence.

So really, I want to know how many lots of 85 pence there are in 11 pound and five pence.

To do that, I can divide 11.

05 divide by 0.

85.

So I'm finding out how many lots of 85 pence will make 11 pounds and five pounds, and the answer is 13.

So I can start by dividing it into 13 equal parts.

And if I want to really double check my answer, I can go now and 85 pence in each, multiply 85 pence by 13 or add it 13 times and it should give me 11 pounds five pence, which means that I bought 13 apples.

Let's look at methods two.

So I'm going to start by writing it as a ratio, number of apples to cost.

Seven apples cost five pound 95.

Now remember the first step from the previous question, we found the cost of one apple, and that is called finding the cost of the value of one unit, okay? So to do that, I divide by seven.

So that gives me 85 pence per or for each apple.

Now, what do I know? I don't know how many apples I'm going to buy.

I know that I'm going to spend, the cost is going to be £11.

05, okay? The point is missing , so let me just correct this.

Okay, let's talk through it.

So what do I have here? I had 85 pounds.

Now I knew that the total cost from the question is 11 pound and five pence.

Sorry, I don't know what happened to the 05, it disappeared.

So 11 pounds five.

Now I need to think what have I done from to get from 85 pence to get to 11 pounds and five pence? I multiplied by 13.

I figured this out by doing the inverse, which is I say what is 11.

05 divide by 0.

85? And you can use a calculator for that.

It will give you 13.

So you know that from here to there to here, you multiply by 13, and the other way around you divide it by 13.

So you can use it either way.

Now if I multiplied here by 13, I need to do exactly the same thing to this part of the ratio.

So, I have one, divide that by 13, that gives me 13.

So I know that I have 13 apples or I bought 13 apples.

Now, let's look at method number three.

And that was by creating the table.

The table shows the number of apples, the cost and the cost divide by the number of apples.

So basically the cost per one apple.

So seven apples cost 5.

95.

And this tells me that one apple cost 85 pence.

Now what do I know? The cost of the apples is not going to change so that is still 85 pence.

I know that the total cost that I've spent is 11 pounds five pence, and I need to work out this number here.

Now how do I work this out? I can say £11.

05 divide by something gives me 85.

What is that something? How can I rearrange this so I can work out the value that goes in this box? I could say 11.

05 divide by 85 pence and need to find that number, and that number is 13.

So again, this here I showed us that we can use a variety of methods to answer questions like this.

Some methods probably easier.

My favourite method is the bar model, but I can use all the others just to help me check my answers, that I'm not making any silly mistakes anywhere.

What's your favourite method? Okay, a huge well done if you had any of these methods written down and you were having a go at the Independent task.

It's really important that if you have not, if you don't have one of these methods, or your answer is incorrect, that you mark and make some notes as we go along.

cause these can help you with your future learning.

And now it is time for your Explore task.

Let's read it together.

In this skyscraper that's 300th step is 75 metres high.

How many ways can you complete these number frames using a different digit in each box? The height of step something is something.

If you're feeling super confident about this, please pause the video and have a go at this now.

If not, I will do support in three, two, one.

Okay, and for support.

75 metres is equal to 75 multiplied by 100 to 7500 centimetres.

we're converting this into centimetres because when we're writing the sentence here, we need centimetres, we need it to be in centimetres.

Now I can think about this like this and say, number of steps to the height, one.

The 300th step is 7500 centimetres.

So what two digits can I maybe work out? I can work out the 30th.

That's one of the easiest to work out.

What did I do to get from 300 to 30? Good job.

I divided by 10.

So I can do the same thing here, divide by 10 and that gives me 750, which is a three digit just like we have to have here.

So two digit by a three digit.

So now I can put that in the box.

Now, with this hand, can you come up with slightly different statements? Can you try and challenge yourself so you can get different numbers in each box? Pause the video now and have a go at this.

I wonder what kind of numbers and statements you came up with? How many did you work out? Okay, these are some of the possible solutions, the height of step 10 is 250.

The height of step 20 is 500.

And the height of step 25 is 625.

And you could work so many more.

You could work out so many more and have so many statements.

However, did any of you manage to find ones where the digit in each box is different? I would love to see some of your work to see what you came up with for your statements.

This brings us to the end of today's lesson, you have done some fantastic learning and you should be super proud of yourself.

I want you to take two minutes to think about everything that we've learned in today's lesson, reflect back on it, and write down the three most important things that you've learned from today's lesson.

And then I would like you to complete the Exit quiz.

That's all from me today.

Enjoy the rest of your learning for the day, and I will see you in the next lesson.

Bye.