Lesson video

Hello and welcome to today's lesson on ratio and proportion.

That by the end of the lesson today, you will be able to use representations to calculate unknown values in ratio contexts.

So you might be using a drawing of the situation to represent the problem, or you might be thinking about using something like a bar model perhaps.

So we're going to look at how we can use representations to calculate unknown values and ratio problems. So let's make a start.

The key word we're going to be looking at using today is the word variable.

So you might want to pause the video here and have a think about what variable might mean.

You might think the variable sounds a little bit like something you've used before, maybe in a context that's not math.

So let's have a look at what we're going to, how we're going to think about variables in this lesson.

So a variable is a value which can be represented by more than one number in a situation or in an equation.

So if you think about the work you've done on ratio before, we've often been given a ratio, but then the situation has changed.

We've got to keep that ratio true, but the values within our situation may change.

So as we go through today's lesson, particularly in the second part of the lesson, really think about what is it that is staying the same when we're thinking about these ratio contexts and what is it that is changing, what can we vary and how does the way that we vary it matter for other values within our ratio context? So we're gonna be thinking about how things change, how things stay the same, and how we can calculate missing values within calculations, thinking about the variables.

So two parts to our lesson today.

In this first part, we're really gonna be concentrating on using a bar model to represent a ratio problem.

So we've used them perhaps before just to represent a situation.

We're really gonna look at how a bar model can help us to represent a problem and support us in finding a way to solve that problem and come up with an answer.

Today's scenario is around a school quiz.

So a school is holding a quiz for teams of adults and children.

Each team in a school quiz is made up of two adults and three children.

So I wonder how we could talk about that perhaps using some language we've used before and how we might be able then to represent it.

Let's just think about some language that we might have used before to think about ratio.

Is that looking familiar to you? So we've got this scenario where each team in a school quiz is made up of two adults and three children.

So can we fill in this stem sentence to help us to make sense of what's going on? And using that language of for every, which is so important when we're thinking about ratio and describing the two factors, the two variables perhaps in our scenario.

So we can say that for every two adults, there are three children in our team.

We could also say, for every three children, there are two adults.

Okay, so that's using language to think about the problem, perhaps some language that's familiar to you, but we think we're gonna use bar models.

So let's have a think about how we could draw a bar model to represent this problem.

Okay, so there's our information again.

In each team, each team in a school quiz is made up of two adults and three children.

How many are in each quiz team? How many people have we got in each quiz team? So let's see if we can draw some representations to help us to think about that.

So we're gonna start with a bar model, something that might look quite familiar, and our bar model showing us that two parts of the bar are adults, and three parts of the bar are children.

Is it accurately done? I think it might be, but what we can see is that, that there's a smaller, what we'd say proportion.

A smaller part of the team is adults and a larger part of the team is children.

And we know that our ratio is for every two adults, there are three children.

So can we draw that a different way? So let's think, sticking with a bar model, but just maybe arranging it slightly differently.

So here I can clearly see that my parts of my bars are equal sizes.

This time I've separated my bars a bit.

So I've got one bar showing me the two adults and one bar showing me the three children within a team.

So you might wanna have a look at those, think about which one makes most sense to you, which one you think helps you to understand the situation better.

And then I'm going to ask you a question.

So where is the whole team in each of the bar models? Where is the whole team in that top bar showing the two adults part of the bar and the three children part of the bar? And where is the whole team in that bottom bar where we've got the adults and the children sort of separated out? What do you think? Well, let's have a look.

So in the top image, our whole team is the length of the whole bar, isn't it? So we can see that we've got two adults and three children.

So we've got five people in one quiz team and we've added an extra bar along the top there to show that whole team representation at the top with two adults and three children.

And at the bottom, if you can see, I've now moved my two bars so that they are next to each other.

So we can see those five parts in the bar.

Before, we'd have had to look at the two bars together.

Now we've just lined them up so that we can see the two adult part and the three children part, and we can see that idea of five people in a team all together.

And that's gonna be quite important because we might be thinking about how many adults, how many children, but we might be thinking how many teams. So let's see if we can use these bar models to help us to solve a problem.

Right, so our challenge here is to change the bars to show that there are three teams taking part.

If you'd like to have a go yourself, then pause the video.

So our bar model at the moment shows us there are five people in one quiz team, but I need to show that there are three teams taking part.

So if I've got three teams, how many people have I got taking part? Well, if I've got three teams, I've got three times as many people taking part.

So I've got three whole teams. So three times five, I've got 15 people in the three quiz teams. But if I've got three times as many teams, I've got three times as many children.

Three times three is equal to nine and I've got three times as many adults.

Three times two is equal to six.

So my numbers have changed.

I've kept my bar the same because I've still got that idea that two parts of my bar are representing the adults and three parts are representing the children.

But now, rather than having one adult and one child in each part, I've got three adults and three children in each part of my bar.

So I've got three times as many people taking part.

Let's have a look using the other representation.

So here we've got this idea that I've got those two adults and three children taking part.

So each part is representing one adult or one child, but I want three times as many.

I want three whole teams. So there's my one whole team.

How can I change this to show that it's representing three whole teams? So I've got two lots of three adults, six adults in total, and three lots of three children.

I've got nine children in total.

And let's just think what our problem was around how many, we had three quiz teams. So if there are three quiz teams, there are three times as many people taking part, three times as many adults and three times as many children, and three times as many people all together.

So I've multiplied everything by three.

You might want to have a think about which representation you find helps you most to make sense of the problem and to help you to solve the problem.

So a chance for you to have a think about which representation you like, maybe have a go at using both of them.

And there's a question here to look at.

So we've got the same school quiz teams, the teams made up of two adults and three children.

How can you change the bars to show that there are six children taking part? So this time, we are changing, we are varying the number of children.

So what will the rest of the bar model look like if there are six children taking part? So again, you might want to pause the video and have a go before we look at the answer together.

So let's look at that first representation where we've got those bars split up to represent the number of adults, the number of children, and then the team as a whole.

So we were told that we were changing the bars because there are now six children taking part.

So we had three children there, we now want six children.

So we've got two times as many children.

We've doubled the number of children.

So that means if we double the number of children, in order to keep our ratio the same, to keep the balance in the team the same, I need to double the number of adults as well.

So we had two adults originally.

I'm going to double that, two times two is equal to four.

And what about the total number of people taking part? Well, in my one quiz team, there were five people.

I've now got two times as many.

So I've got two quiz teams, which is 10 people.

So we've changed that bar model to show that there are now six children taking part, and that means there are four adults and 10 people all together.

What about the other bars where we had the sectioned up with two, showing two groups of one adult and three groups of one child? Let's have a look at that bar model.

So can we see the six children taking part in this bar, in this representation using a bar? And we can, we've got three groups of two.

So each of those individual one children has been doubled.

We've got twice as many children, so six children.

So therefore, we've got to double the number of adults and we can see that that will mean there are four adults.

And if we look there, we've got five groups of two.

So we've doubled the number of people altogether.

We can see the 10 people all together in our quiz team as well.

So again, pause and have a think about which bar model works best for you in representing the problem and helping you to come to a solution.

Time for you to have a practise and to draw some bar models of your own to justify your decisions.

So we've got some decisions to make here.

So you've got three statements, A, B, and C, and you've got to decide which of the following statements is true.

So we're still thinking about that same school quiz team made up of two adults and three children.

But A says, if there are five teams, there will be 15 children.

Can you decide whether that's true or false and draw a bar model to justify your decision? B states, if there are 10 adults, there will be 13 children.

Again, can you draw a bar model to justify whether you think that's true or false? And C states, if there are 14 children, there will be seven teams. Is that true or false? And can you think, draw a bar model to help you to justify your decision? So how did you get on? Let's have a look at the solutions and the way you may have justified your decisions.

So A said if there are five teams, there will be 15 children.

And yeah, that's true.

How can we draw a bar model? I've chosen to draw the bar model with the quiz team along the top and then the adults and the children shown us those different size, those bars underneath to show that for every two adults, there are three children.

So it said if there are five teams, there'll be 15 children.

So I need five times as many teams, I need five times as many adults, I need five times as many children.

So if I multiply all my original numbers by five, I can see that there were originally three children.

Five times three is 15.

So yes, there will be 15 children in five teams. Okay, B says that if there are 10 adults, there will be 13 children.

So 10 adults.

Let's have a think.

How many adults are there in one team? There's two adults in one team.

So if I need 10 adults, I've got five times as many adults.

Oh, well, that's the same bar model as we had for A, isn't it? And we know from our answer to A that if there were 10 adults, that meant five teams, there would be 15 children.

So no, if there are 10 adults, there'll be 15 children, not 13.

Because if there are 10 adults, we've got five times as many people, five times as many teams, five times as many adults, and we would need five times as many children.

So B is false.

And C states that if there are 14 children, there will be seven teams. Hmm.

Well, seven and 14 look quite, quite friendly numbers together, don't they? But does that really work? So let's just have a think.

There'll be seven teams. So that means seven times as many people involved in the quiz.

So there would be seven times as many children and three times seven is 21.

So there wouldn't be 14 children, there would be 21 children.

So C is false as well.

We can also think about multiples.

We may have talked about, you may have talked about multiples with ratios, that if you've got the ratio for every two adults, there are three children in a quiz team, however many quiz teams you have, there will be a multiple of three children if we've got complete teams. And 14 is not a multiple of three.

So therefore if we had 14 children, we'd have to play around and maybe have an extra person or a person missing depending on our numbers.

We wouldn't have a complete number of quiz teams because 14 is not a multiple of three.

Okay.

So we've used bar models to represent our ratio problems and used them to begin to help us to solve them, to think about those multiplication that we need to use in order to solve the problems. So we're going to move on now in the second part of the lesson and think about that idea of variables.

Think about the values that are changing, and we're going to look at representing ratio problems with three or more variables.

So let's have a look, see what that will look like.

So let's start by reminding ourselves what a variable is.

So a variable is a value that can be represented by more than one number within a ratio problem.

So let's think about the variables we've been thinking about with our quiz teams. So we've been thinking about a quiz team made up of two adults and three children.

Thinking about that ratio, for every two adults, there are three children.

So what have our variables been in this problem? And how many have we had? So how many variables are there? What could we change? Well, we could change three things because we could vary the number of teams, the number of adults, and the number of children.

So we've been looking at a problem with three values that we can vary.

And if you think about it, we can vary one of them.

And then how we vary one value tells us how we vary the other values.

Okay, let's look at a new problem and think about the number of variables we've got.

We're gonna make sandwiches, so we're going to make cheese and tomato sandwiches.

And so our ingredients, our recipe tells us that for every cheese and tomato sandwich, you need two slices of bread, three slices of cheese, and four slices of tomato.

So how many variables have we got in this problem? You might want to pause and have a think.

Okay, what did you come up with? In this problem, we've got four variables because you can vary the number of sandwiches you make, the number of slices of bread you have, the number of slices of cheese you use, and the number of slices of tomato you use.

So there are four things that can change, that can vary in this ratio context, but remembering that once we've varied one thing, that tells us how we vary the other aspects or the other ingredients, in this case, in our sandwich.

Okay, you're getting pretty good at drawing these bar models to represent ratios.

So can you have a go at drawing a bar model to represent this sandwich ratio that for every cheese and tomato sandwich, you need two slices of bread, three slices of cheese, and four slices of tomato.

Pause the video so that you can have a go at drawing the bar model.

Okay, how did you get on? Here's the bar model that I drew.

Okay, so I've represented one sandwich, so I've bracketed it all together to show that I've got one sandwich here.

And the parts of my sandwich, because ratio is about parts within a whole is two slices of bread.

And for every two slices of bread, I have three slices of cheese and four slices of tomato.

So you can see that I've made the bar represent the fact that I've got a smallest number of slices of bread, the second largest number of slices of cheese, and the biggest number of slices of tomato.

So I've divided my bar up so that I can see, the, sort of, relative sizes of those numbers as I work with my bar model.

Okay, so where are the variables represented in the bar model? Well, my whole bar represents a sandwich, and then each part of the bar represents an ingredient within the sandwich.

So my variables were the number of sandwiches.

So that's my bar, my total, the slices of bread.

So I've got a part of the bar representing the bread.

I've got the cheese was another variable.

So I've got part of the bar representing the cheese, and I've got a, my final part of the bar representing our final variable, the number of slices of tomato.

And it's your turn to do some practise.

So we're sticking with that same way of making a cheese and tomato sandwich.

And we're gonna use that bar model to help us answer these four questions.

So part A asks us how many sandwiches can you make from a loaf of bread with 26 slices? And what else do you need? Remembering that we're following this recipe for a cheese and tomato sandwich.

So how many sandwiches can you make from a loaf of bread with 26 slices? And what else do you need to make those sandwiches? Part B says, how many slices of cheese do you need to make a 100 sandwiches? Part C, if you use 28 slices of tomato, how many slices of cheese will you need? And D says that a pack of cheese has 25 slices.

How many sandwiches can you make? So your second task is to think about Aisha's pancake recipe.

So Aisha is making pancakes, and for every two eggs, she uses three spoons of sugar and six spoons of flour.

So think about the variables we've got in there, how many variables have we got for Aisha's pancakes? And draw a bar model to represent the recipe.

And when you've drawn that bar model, there's a second part to this task.

So part B of the task relating to Aisha's pancakes asks you to use the bar model to complete these sentences in different ways.

So we're thinking about each of those different elements of the recipe that Aisha can vary.

So if Aisha uses a certain number of eggs, she will need how many spoons of sugar and how many spoons of flour? So that's varying the number of eggs.

The second sentence, Aisha varies the number of spoons of flour she uses.

So if she varies the number of spoons of flour, how many eggs and spoons of sugar will she need? And the final sentence, she's varying the number of spoons of sugar she's using.

So when you've drawn your bar model to represent Aisha's pancake recipe, use that bar model to help you to complete these sentences in different ways.

Okay, let's have a look at how you got on with our sandwich problems. So we've drawn our bar model, and A, question A asked us, how many sandwiches can you make from a loaf of bread with 26 slices? And what else do you need? So 26 slices of bread.

What do we know about these sandwiches? Well, we know that for every cheese and tomato sandwich, you need two slices of bread.

So if I've got 26 slices, how many sandwiches can I make? Well, I reckon it can work out how many twos, yes, 26 is a multiple of two.

13 times two is 26.

So therefore, I can make 13 sandwiches with my 26 slices of bread.

So I was using two slices of bread in the original bar model, I've now got 26.

I've got 13 times as many slices of bread because I'm making 13 sandwiches.

So I need 13 times as much cheese and 13 times as much tomato.

And 13 times three for the cheese is 39 slices of cheese.

And 13 times four for our slices of tomato means I will need 52 slices of tomato in order to make the 13 sandwiches that I can make from a loaf of bread with 26 slices.

Part B asked, how many slices of cheese do you need to make a 100 sandwiches? Oh, gosh.

Well, it's interesting.

If you look at my bar model, I've not changed all the sections of the bar model here because the information I have is that I'm making a 100 sandwiches.

So I'm making a 100 times as many sandwiches, so I'll need a 100 times as much cheese.

So I'll need three times a 100, which is 300 slices of cheese, a 100 times as much.

The question doesn't ask me about the slices of bread and the slices of tomato.

So I've left those blank on my bar model because this time, I'm just interested in how much cheese I will need.

And C asked if we use 28 slices of tomato, how many slices of cheese will you need? So I've got 28 slices of tomato.

I need four slices of tomato for each sandwich.

So four slices, I can make seven sandwiches because four times seven is equal to 28.

So 28 slices of tomato is enough for seven sandwiches, seven times as many sandwiches, I'll need seven times as many slices of cheese.

I needed three.

So three times seven is 21, so I will need 21 slices of cheese.

It doesn't ask me about the bread.

So I've left that part of my bar blank.

It's still there because it's part of my sandwich, but it's not part of the answer this time.

So I don't need to calculate the number of slices of bread I need.

And the final part asked us about a pack of cheese.

So this pack of cheese has 25 slices in it.

How many sandwiches can you make? Do you remember we talked a bit about multiples when we've talked about ratio? and I'm looking at this and thinking, well, I need three slices of cheese for each sandwich.

25 isn't a multiple of three, is it? So I'm possibly going to have some, oh, I am, I'm gonna have some cheese left over.

So what can, well, how many sandwiches can I make? How many complete sandwiches can I make? Well, 25 isn't a multiple of three, but 24 is, and that will make eight sandwiches because three times eight is 24.

So I can make eight sandwiches, but I will have one slice of cheese leftover.

And that sometimes happens, doesn't it? That's real life.

We don't always have exactly the right number to make our sandwiches.

Okay, I hope the bar model helped you to solve those problems. So our second task was Aisha and her pancakes.

And A asked you to draw a bar model to represent the recipe.

So our recipe says that for every two eggs Aisha uses, she uses three spoons of sugar and six spoons of flour.

So my bar model very simply has two parts for the eggs, three parts for the sugar, and six parts for the flour.

We don't know in this case how many pancakes she makes with this, all we know is those three parts of the recipe.

So we've got three things that we can change, three variables.

So the second part of our question, you are asked to use the bar model to complete the sentences in different ways.

So those sentences were saying, if Aisha uses this number of eggs, she uses this number of spoons of sugar and this number of spoons of flour.

And then we changed the sentences around to start by deciding how many spoons of flour she needed.

So what else would she need? And then how many spoons of sugar she needed and how much else she would need.

Now obviously, there are lots and lots of different answers you could have come with and lots and lots of different sentences.

So what I want you to think about is how you completed those sentences and what you noticed about the values that you put into the sentences.

So I've given an example here.

If the number of eggs that you decided on was four times the original value of her two eggs, so four times two, that would be eight eggs.

Then the number of spoons of sugar and flour will also need to be four times the original values.

So if Aisha uses eight eggs, she will need four times as many spoons of sugar.

Four times three is 12.

So 12 spoons of sugar.

And four times six spoons of flour, which would be 24 spoons of flour.

So hopefully, what you can see in your sentences is that whatever number you decided to multiply that, in that first amount by, you then multiplied all the other amounts by the same factor in order to work out how they needed all or how many they needed altogether.

If you decided maybe she used 15 spoons of sugar, five times as much sugar, she would've needed five times as many eggs and five times as many spoons of flour.

So look back at your sentences and see if you can spot that common factor, that number that you've multiplied each value by in order to keep the recipe correct.

Thank you for all your work in this lesson.

I hope you've been able to move your use of bar models on to help you to solve some slightly more complicated ratio problems involving more variables.

And hopefully, you've now got a better understanding of what a variable is.

A variable is a value that can be represented by more than one number within a ratio problem.

As I say, you've used bar models to represent ratios and that we've begun to use some multiplication and division to calculate unknown values.

And that the bar model helps you to see the relationship between the variables in a ratio problem and is a useful way of helping you to make sense of a problem and to find the calculations that you need in order to solve that problem.

Thank you for all your work today, and I look forward to seeing you again.

Bye!.