Pens are sold in boxes. Each box contains 6 pens. How many pens are there in 5 boxes.

If 4 ice creams cost £4.80, what is the cost of 2 ice creams?

4 pens cost £4.40. What is the cost of 1 pen?

4 bottles of sports drink cost £2.40. How much will 16 bottles cost?

4 bottles of sports drink cost £2.40. How much would 11 bottles cost?

Looking at proportional relationships

Key Stage 3

Maths

Direct and indirect proportion

﻿Hi there, and welcome to this maths lesson. My name is Dr. Saada, and in today's lesson, we will be looking at proportional relationships, in particular, the unitary method. Don't worry if you don't know what that means yet, you will by the end of today's lesson. You will need a pen and a paper for this lesson. So pause the video, go grab these, make sure you're sitting somewhere away from distractions and when you're ready, we can make a start. I'd like you to have a go at this question for me. Let's read it together. Four oranges cost 92 pence. What's the cost of one orange? Draw a bar model to help you work out the cost of eight oranges. Describe how the bar model can help you work out the cost of three oranges. Please pause the video to complete this task. It should take you about five minutes. Resume once you're done. Welcome back, how did you get on with this task? Really good. Come on, let's mark and correct the work. So what's the cost of one orange? 92 divided by four is 23 pence. Did you get that? Good job. Draw a bar model to help you work out the cost of eight oranges. So I drew a bar model, divided it into four equal parts to represent the cost of each orange. And I said that to the four parts are equal to 92. I want eight, so I draw another bar with another four equal parts, labelled it with 92, and therefore, now the total cost of the eight oranges is going to be 184 pence, or if I convert it to pounds, it's 1.84 pound. Next one, describe how the bar model can help you work out the cost of three oranges. So you've got four equal part to represent the cost of each one orange. Four oranges cost 92 pence. So now, I can work out the cost of one part as 23. If I want to find the cost of three oranges, I can say, well, this is three oranges here, that's 23 times three, which is 69 pence. or I can even say that the whole thing is 92, let me take away that one 23 that we have to leave us with the cost of the three oranges, and that gives us again, 69 pence. So I end up with the same answer. Did you get this? Really good job. Let's move on to the next task. Let's read this question together. Four bottles of sports drink costs 2.40 pounds. How much will 12 bottles cost? So there's so many ways that I can answer this question. I can start, I love bar models, so I can start by drawing a bar model, divide it into four equal parts and say that the four equal parts are equal to 2.40 pounds. Each equal part here represents the cost of one bottle. How much will 12 of these cost? So I can work out now the cost of one of these. And if I want to do that, I need to do 2.40 pounds divided by four, and that gives me 60 pence. Remember, I either write it as 60 pence or 0.60. Never write it as 0.6. When you're talking about money, you need two numbers after that decimal point. So now I know that this bit here is 60 pence. I want to find that's the cost of one bottle, I want the cost of 12 bottles. So I can say 12 multiplied by 60 pence, and that is equal to 7.20 pounds. So I've answered the question. I could have also done this by having a bar that is 2.40 pounds, another bar that is 2.40 pounds, and a third bar of 2.40 pounds. That would have given us 12 bottles and I could have just added the 2.40 pounds plus 2.40 pounds plus 2.40 pounds. We would have ended up with exactly the same answer. There's more than one way of doing this. Now let's look at another method of doing it. I can write this in a similar way to how we write ratios. So I can say four bottles cost 2.40 pounds. 12 bottles will cost what? What have I done from 4 to 12? I multiplied by three, so I need to multiply here by three. And that gives me 7.20 pounds. Okay, next one. How much would nine bottles cost? Well, nine bottles would cost me, nine multiplied by the 60 pence, which is 5.40 pounds. I know this, because I know the cost of one bottle. Jo spends 11.40 pounds on bottles of sports drink. How many bottles did she buy? So now what do we know? We know that she spent 11.40 pounds. We know that one bottle cost 60 pence but we don't know how many parts there are. So what are we trying to find out according to our bar model? We're trying to find out how many 60 p's go into 11.40 pounds. So really, I want 11.40 divided by 60 pence to know how many parts I should have and it tells me here if I did the division that I need, 19 bottles, I need 19 equal parts that would give me 11.40 pounds. Remember, I can always check my answer by doing the inverse by doing the multiplication. Now there's another way of doing this. I can say that four bottles will cost me 2.40 pounds. What is the cost of one bottle? We already know that, but we divide here by four. So we need to divide there by four. And the cost of one bottle is 60 pence. This is something that we already know. Now, I don't know how many bottles I have, but I know that I, or Jo, sorry, spent 11.40 pounds. Now to you get from 60 pence to 11.40 pounds I multiply by 19. In order to find out what you multiplied by you do the inverse. You do 11.40 pounds divided by 0.60. And you find out what the number is. You will multiply it by 19, which tells me I need to multiply here by 19. And that tells me that we have, or Jo has 19 bottles. This goes to show that we can solve the same question using a variety of methods. And it's a method that you feel most comfortable with. I love bar models. I find them really, really helpful and quite visual. So I'm going to recommend that you use that for today's lesson. Now it's your turn to have a go at the independent task. You have three questions to complete. I would like you to use a bar model to help you answer the questions. If you want to challenge yourself, you can use more than one method, then compare your answers. If you answer the same question, using two different methods and end up with the same answer, the chances is that your working out is correct. Please pause the video and complete the independent task. This should take you about 10 minutes to complete. Resume the video once you're finished. Welcome back. How did you go on with the independent task? Did you manage to draw bar models for all three questions? Really impressed. Let's do this together. Pens are sold in boxes. Each box contains six pens. How many pens are there in two boxes, 12 boxes, 30 boxes and 120 boxes? So start with, for the first part, I drew a bar and I said, well, there are six inside one box. So in two boxes, there must be 12 pens. For 12 boxes, I drew this time a bigger bar and I said, well, in one box there are six. So I need 12 boxes, so 12 equal parts. 12 multiplied by six is equal to 72 pens. Did you get that too? Really good. For the next one, for 30 boxes and 120 boxes, I thought, "Well, you know what? It's going to take me too long to draw a bar that has 30 equal parts, 120 equal parts." So let me use a different method along with the bar models. So I started by writing it like I would write a ratio, pens to boxes. And I know that in one box, there are six pens, so I wrote six and one. I want to know how many pens that are in 30 boxes. So 30, what did I do? I multiplied by 30. So I need to do the same on the other side. So six times 30 is equal to 180. And this is why it's really helpful to know one, because when I know one unit, one box, how much it has, it makes it easier to calculate the rest. Now I want to calculate and work out how many pens there are in 120 boxes? I could go back to one box and say one time 120 gives me 120, so six times 120. But then I thought to myself, "Do you know what? I know the relationship between 30 and 120." I know to get from 30 to 120, I need to multiply by four, so I can do the same to the 180 multiply that by four, which gives me 720. And that tells me that there are 720 pens, 120 boxes. Did you get that at all? Really good. Let's move on to question number 2. Three packs of crisps cost 1.20 pound, how much will six packs of crisps cost? So I started by drawing a bar model and I said, well, we have three packs, so three equal parts that cost 1.20 pound. So one part will cost me what? And it'll cost me 40 pence. I know this, because 1.20 pound divided by three is 40 pence. Then I said, okay, well I need now the cost for six of them. So I drew a bar that had six parts. Each one part is 40 pence, so multiply 40 pence by six. And that is 2.40 pounds. After that also, just left that first bar and doubled it. So I could have said 1.20 pound multiplied by two. How much will eight packs of crisps cost? Okay, and here I thought, I'm not going to start drawing again. I can use what I already have. I have a bar here that has six equal parts. I want eight, so I can add one, two. And I know that each of them is 40 pence, because one pack of crisp costs 40 pence. So now all I need to do is add that 2.40 pounds plus the 40 pence, plus the 40 pence, or I could say 40 pence multiplied by eight. And that would give me the answer to the question. 3.20 pounds is the cost of eight packs of crisps. Question number 3. Eight small cheese pizzas cost 24 pounds, how much would six small cheese pizzas cost? So again, with this one, I started with a bar model, divided it into eight equal parts. I said, the whole thing is 24 pounds. I want to know the cost of one pizza. I want to find one unit. So 24 divided by eight equals three. So I know now that the cost of one cheese pizza is three pounds. I want to know the cost of six of them. So I can put three and six of the parts and say, well, it must be three multiplied by six, which is 18 pounds. And now I found the cost of six small cheese pizzas. Did you get this? Really good. So what was really, really important about all of those questions here is always finding the cost or the value of one unit. And this is why this method is called the unitary method. You find one unit first and use that to help you answer the question. Let's look at this in more depth now. If four smoothies cost 6.80 pounds, what is the cost of seven smoothies? And I showed you here three different methods for working out this question. The first one shows a bar model. It doesn't have all the numbers in there, but it's the start of a bar model. The second one is a table. And the third one is writing it as we would write ratios. Now, I want you to choose any two methods and explain how these methods presented the same maths. When you're doing this, I'll also like you to start thinking about which method is your favourite and why? Which one is easier? Which one is more efficient? Off you go. Please pause the video and complete the explore task. This should take you about eight to 10 minutes to complete. Resume once you're finished.

Hello everyone. It's Ms. Jones here. I'm going to be going through the solution to your let's explore task for you. Okay, so we had three different ways of representing this problem and you needed to look at them and think about how they were representing the maths and what's going on here. So the first way we have is the bar model. So we see here, we've got two bars. So this one representing the cost of four smoothies. So it's got four equal parts. This one representing the cost of seven smoothies. So if we're using this representation, we know that the cost of four is 6.80 pounds. So we've got four equal parts. We can divide by four to work out the cost or the value of one of these bars. So if I wanted to work out then the value of seven smoothies, I could multiply 1.70 pound by seven, and that would've got me 11.90 pounds, okay? Should put my pound sign in there. Now, the other way we've got here is a table. So again, we've got here a column that's representing four smoothies. We've got here, 6.80 pounds. We know if we divided that by four, we'd get 1.70 pound. And then a column for seven smoothies. So what we need to do here is think about, okay, we've got the cost of one smoothie. We can multiply by seven, so again, get 11.90 pounds, okay? So we've got the same answer. We've represented it two different ways. Now the third way of representing the maths here was using this method, which relates to ratio. So we've got here again, four smoothies is equivalent to 6.80 pounds. One smoothie would be 1.70 pound. What would seven be? So we can see here what's happened again. It's been divided by four and the same has happened on this side divided by four. And if I want to work out seven, we can see one's been multiplied by seven. So I need to do the same here. Multiply 1.70 pound by seven. We already know what that is from our work earlier, and we'd get 11.90 pounds. So you can see that actually we're getting the same answer each time. But what else did you think was similar and what was different? You might've noticed that each time we were starting with 6.80 pounds dividing by four to find out the cost of one smoothie and then multiplying by seven to then find out the cost of seven smoothies. And you could use each of these representations when you're working out problems similar to this. But have a think about which one you prefer to use, which one provides the best visual representation and you think you could use perhaps in a more complex problem to help you understand and visualise it, okay? So take a moment to think about that before we finish, okay? I'm going to pass you back over to your teacher to end the lesson. Thank you.

This brings us to the end of today's lesson. You've done some fantastic learning today. I want you to spend two minutes thinking about the one most important thing that you've learned from today's lesson and maybe write it down as a reflection. And I want to remind you to complete the exit quiz, to show what you know. This is it from me for today. Enjoy the rest of your day. I want to see you next lesson. Bye.

Looking at proportional relationships

Looking at proportional relationships

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UnitMaths / Direct and indirect proportion

Maths