Lesson video

Hello and welcome to this lesson on ratio and proportion.

In today's lesson, we're going to be thinking about how we can represent ratio in different ways.

So there's a key phrase we're going to be using today in our lesson, and that's many to many.

When we've looked at ratio, we've realised that ratio is about comparing the parts within a whole.

And we've thought about how the ratio helps us to understand how those different factors in the whole relate to each other.

And today we're going to be thinking about many to many relationships.

So let's have a think about what that means.

So we're going to be using many to many to describe ratios where each value is more than one.

Some of the ratios you might have come across say for every one thing, there are a different number of other things.

What we're going to be thinking about today is this idea of many to many where each value is more than one.

And the example we've got there on the screen is a group of children.

And for every three boys, there are four girls.

So each value within the whole is greater than one.

And those are the sorts of ratios we're going to be focusing on in our lesson today on representing ratio in different ways.

So there are two sections to our lesson today.

And in the first section, we are just going to be looking at how we represent ratio relationships.

And as I've said, we're going to represent those with objects, with drawings.

And we're going to move on and think about bar models.

So in this scenario, we've got Aisha and Alex, and they're sharing marbles.

And the information we have is that for every two marbles Aisha has, Alex has three marbles.

So we've got that many to many idea.

And some of you might look at that and think, well, that's not very fair 'cause Aisha is not going to have as many marbles as Alex.

And sometimes we think about sharing as being equal, don't we? And so sometimes when we're thinking about sharing in a certain ratio, it's not always what we might consider to be a fair share.

So we're going to put that aside and we're gonna think about the way that these marbles are being shared between Aisha and Alex, how we can represent them and how that can help us to solve some problems. So here we've just got a picture and we can clearly see above the picture of Aisha are her two marbles and above the picture of Alex are three marbles.

But how else could we represent this relationship? Oh, and remembering, of course, this is a many to many relationship.

So what we've done here is put the marbles into a bar model.

And so you can clearly see that we've just drawn a bar model around the marbles that we already have.

And we can clearly see, still see Aisha's two marbles and Alex's three marbles.

From that relationship, for every two marbles Aisha has, Alex has three marbles.

So each section of our bar at the moment represents one marble.

So two sections representing Aisha's two marbles and three sections representing Alex's three marbles.

I can simplify it still further and take the pictures away and put a one in to represent each of those one marbles.

So we can see that Aisha has two sections of the bar for her two marbles, and Alex has three sections of the bar to represent his three marbles.

Put the marbles back in just to show that relationship and how the two representations can work together.

So all together, in that first time they've shared their marbles, we've shared out a total of five marbles.

Always important to know not only the values of the parts but to know the value of the whole in a scenario when we are working with ratio as well.

Okay, so let's have a look and see what's changed here.

So this time, we can see that Aisha has four marbles and Alex has six marbles.

So how does that relate to the bar model? So we can see that each of those sections of the bar model represented one marble that first time.

But now each of those sections represents two marbles.

So we can see that each box now has an extra marble in it for each of them.

We've shared them out again.

Aisha now has four marbles and Alex has six marbles.

So we've taken that basic pattern of for every two marbles Aisha has, there are three marbles of five marbles in total.

And we've now sort of doubled that, shared them out again and we've doubled the values.

So each section of our bar is now worth two marbles.

Okay, so a little something for you to have a go at just to check that we're making that move from drawing marbles to representing the marbles that we're sharing out in that ratio on a bar model.

So how many different ways can you complete the sentence at the bottom there? There are hmm marbles in total.

So how many different total marbles can we make? And what will the bar model look like each time? Remember, we've looked at a bar model where each section represents one marble and then we looked at it again where each section represented two marbles and we could see that total number of marbles increasing.

So pause the video and see how many different ways you can complete the bar model and the sentence with a different total number of marbles.

How did you get on? I had a go and came up with a couple of bars to have a look at.

So remembering that for every two marbles Aisha has, Alex has three marbles.

Let's have a look.

You may have come up with these bar models as well.

I decided that, oh my goodness, I was going to make each of those boxes 10 marbles.

So originally< it was worth one marble.

So Aisha had two and Alex had three.

This time, I've gone 10 times bigger.

So Aisha has 20 marbles, Alex has 30 marbles and there are 50 marbles in total.

And then I thought, well, what about another number? What other number could I put in there? I could put any number in there as long as I'm putting the same number in each box because then I'm saying that for every two boxes of marbles that Aisha has, every two sections Aisha has, Alex has three.

So for the other one, I decided on six.

So instead of each section representing one marble, each section now represents six marbles.

Aisha still gets two parts, two of those sections and Alex still gets three.

And there are still five sections in total.

So this time, I've got five lots of six.

So I've got 30 marbles in total.

Okay, bit of practise for you.

Now we're thinking about those bar models and a chance for you to draw your own bar model.

There's a task for you to do first though.

You've got four different ratios there.

You can choose whatever context.

You might decide to carry on using marbles, or you might decide to use something else.

But you've got four different ratio sort of scenarios there.

And your first job is to choose a many to many ratio.

Remember we looked at those words at the beginning of the lesson and talked about them with Aisha and Alex's marble.

So choose a many to many ratio represented as a bar model.

You could write a story using marbles or another object.

And part B of your task is to list at least three different total quantities of marbles if that's what you've chosen that would work with your ratio.

Pause the video now and have a go.

And then we'll have a look at what we've got together.

I hope you enjoyed thinking about your ratios.

Let's just look at that first step, which was to choose a many to many ratio first that you were going to represent as a bar model.

So which of these are many to many ratios? If you remember that definition, the many to many ratio was when each part that we were considering was greater than one.

So we can see in those bottom two, we've got that idea of for every one, there are three, for every two, there is one.

So we can remove those two.

And these are the many to many ratios as each value is more than one.

So I had, for every three orange marbles, there are four blue marbles.

That was the three to four ratio, which was one of the choices.

And then for the other one, I stuck with marbles again.

And this was for every seven somethings, there are three somethings.

So I said for every seven orange marbles, there are three blue marbles.

And then I drew the bar around my marbles to show how my original ratio would translate into a bar model.

So then I'm going to think about listing these quantities that would work with those ratios.

And I'm going to try and think is there anything I notice about the totals that I make in each case? Let's have a look.

So as you see, I've removed my marbles this time and I've got my bar model there saying for every three somethings, there are four somethings.

So you might be able to put in what whatever you decided on as your context for your ratio.

And that's the top bar.

And then the bottom bar is set up to show for every seven somethings, there are three somethings.

And we think of those somethings as those sections of the bar model.

So for every three, there are four.

And for every seven sections, there are three sections.

Whatever number I put into those boxes in that first bar model, I'm always going to have seven times that number.

So if I went back to my marbles, I might put a five in each of those boxes and therefore, I would have seven lots of five.

So I'd have seven times the number that I put in the box.

So I think I can say that my total number of objects, if I'm sharing marbles in the ratio of three to four, for every three orange marbles, there are four blue marbles, my total number of marbles will always be a multiple of seven 'cause I've always got those seven sections of my bar to represent my ratio.

So what about the for every seven, there are three? What would we have then? Well, I've got 10 sections in that bar, haven't I? So whatever number I put into the different parts of the sections of my bar model, I'm going to have 10 times that number, 10 lots of that number.

So my total number of objects, marbles or whatever we chose will always be a multiple of 10.

Okay, so we've represented ratio relationships using bar models and seen how we can move from the object to the pictures to use a bar model.

And now we're going to move on and think about representing ratio using multiplication equations.

So let's go back to Aisha and Alex and that ratio that for every two marbles Aisha has, Alex has three marbles.

So last time, we had our bar divided into five different parts, each representing that sort of one marble share, one marble's worth or one share.

And for every two Aisha had, Alex had three.

This time, you can see we've modified our bar slightly and we've just looked at Aisha's share being, in this case, two marbles and Alex's being three marbles.

So we can see from this image that Aisha has two marbles, Alex has three marbles and the total number of marbles is five.

Okay, so for this time, they've shared the marbles out again.

And so for every two marbles Aisha has, Alex has three marbles, we've done that again.

So what can we see now in our representation and how can we make that relate to the bar model? So we can see here that Aisha now has four marbles.

So her share of the marbles, her two parts of the marbles is worth four.

And Alex has six marbles.

His three part share of the marbles is now worth six and the total number of marbles is 10.

If we have a look at that, for every two marbles Aisha had, Alex has three marbles and we had five in total.

Now Aisha's got four, Alex has got six and the total number of marbles is 10.

So we've doubled everything up, haven't we? We've multiplied by two.

For every two marbles Aisha has, Alex has three marbles.

But what if this time Aisha has eight marbles? What will the picture look like? You might want to pause, have a think or maybe even have a go at drawing it for yourself.

So let's have a think.

What would the picture look like if Aisha has eight marbles? Wow, added a lot of marbles in there.

But can you see that Aisha now has 2, 4, 6, 8 marbles.

We've now got four groups of the marbles.

What about the bar model this time? Aisha's still got that two parts of the whole number of marbles.

So this time, she's got eight marbles.

Alex has got that three part share of the marbles.

But remember, we've now got four sets of marbles.

So he's got 12 marbles and in total is? Well, all together, we've got eight marbles and 12 marbles.

We could look at it that way.

Or we've got four lots of the five marbles.

So whichever way we're gonna look at it, we've got 20 marbles.

But what calculations did we use to work out that Alex has 12 marbles and that the total number of marbles is 20? Again, you might want to pause the recording just to have a think about that.

So pause if you'd like to and then we'll have a chat about it together.

Okay, so what calculation did we use to work out that Alex has 12 marbles? Well, let's start with Aisha.

She's got eight marbles, we knew that, and in the original share, she got two marbles for every three marbles that Alex got.

But in eight marbles, she's got four times as many marbles.

So she's got two multiplied by four marbles.

Two times four marbles, which is eight marbles.

So what about Alex and his 12 marbles? What calculation can we do to work that out? Well, Alex gets three marbles each time and they've shared them four times.

So three multiplied by four is equal to 12, three times four is 12.

What about that total number of marbles? And we can see the total as an addition.

We could say that we can see eight add 12 is equal to 20, but we're meant to be representing using multiplication here.

So what multiplication can we think of that will get us to that total of 20 marbles? Well, we looked at the fact that in the original ratio, for every two marbles Aisha has, Alex has three marbles, we're thinking about five marbles in total.

We know now that to get Aisha's eight marbles, she's got four times as many marbles.

So there are four times as many marbles in total.

So we can think in a multiplicative way that this is five multiplied by four equals 25 times four.

What does the four represent in those equations and what does the five represent in the bottom equation? Again, you might want to pause and have a think about the answer to those questions.

Let's have a think.

What does the four represent in the equations and what does the five represent in that bottom equation? Well, the four represents the number that the original amount has been multiplied by.

So in that top equation for Aisha's marbles, in the original story, Aisha, for every two marbles Aisha got, so two was her basic share but we know that there are four times as many shares.

They've shared it out four times.

So two times four is equal to eight.

For Alex, in the original sharing out, he got three marbles but he's now got four times as many as well.

So three times four is equal to 12.

The total number of marbles, originally it was five but we know we've got four times as many.

So five times four, the five being the number of marbles in the basic ratio multiplied by that four because we've got four times as many is equaling 24, represents the number the original amount has been multiplied by.

And the five in our final calculation represents the total number of marbles in that basic ratio.

So two plus three, for every two marbles Aisha has, Alex has three marbles, giving us those five marbles.

Now, you'll notice here that our ratio has changed.

So now for every three marbles Aisha has, Alex has four marbles, and we've got that represented with pictures of the marbles and with that bar model with Aisha's three marble share and Alex's four marble share.

But what we know this time is that Alex has 20 marbles.

So what multiplication equations would you use to work out the missing numbers? So to work out how many marbles Aisha has and how many marbles there are in total now.

Again, you might want to pause the video at this point so that you can have a think and then we'll look at the answers together.

So this time, we know that Alex has 20 marbles.

Alex in the original ratio had four marbles.

So how many times more marbles as Alex got? Well, we know that four multiplied by five is equal to 20.

So four multiplied by five gives us Alex's 20 marbles.

So now we know that that sort of missing value that we're looking for is that times five.

There are five times as many marbles.

So to work out Aisha's marbles, we know she had three in the original sharing but we know that we've got five times as many.

So three times five is 15, and the total number of marbles, what did we have in that total number? For every three marbles Aisha has, Alex has four.

So we've got a total of seven this time but we now know that there are five times as many marbles.

So seven times five is equal to 35.

And we could also check that by looking at the total number of marbles represented in that bar.

And we've got a 15 and a 20.

And we also know that 15 plus 20 is equal to 35.

So you've got a ratio here of triangles and stars and some questions there asking you.

So if there are 12 stars, how many triangles will there be? If there are 12 triangles, how many stars will there be? If there are 15 stars, how many triangles will there be? And if there are 20 triangles, how many stars will there be? And you've got some multiplication equations down the right-hand side and your job is to match the equation to the question.

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

If there are 12 stars, how many triangles will there be? Well, our ratio says that for every three stars, there are four triangles.

So if there are 12 stars, we've got four times as many stars, so there'll be four times as many triangles.

And so four multiplied by four is equal to 16.

So if there are 12 stars, there'll be 16 triangles.

Let's look at the second question.

If there are 12 triangles, how many stars will there be? So this time we want 12 triangles.

Well, in our original ratio there are four triangles, so that's three times as many triangles.

So we'll need three times as many stars.

So three multiplied by three is equal to nine.

If there are 15 stars, how many triangles will there be? So three stars in our original ratio, three times five is equal to 15.

So there are five times as many stars.

So there'll be five times as many triangles.

Four times five is equal to 20.

And then for our last question, if there are 20 triangles, how many stars will there be? Let's look back again.

There were four triangles originally, there are now 20, so that's five times as many.

So there'll be five times as many stars.

Three times five is equal to 15.

So I hope you managed to match the equations to the questions.

Okay, so in this task, we've got a recipe for pancakes and Aisha is making pancakes.

And for every two eggs she uses, she uses three spoons of sugar.

So we've got some stem sentences there, sentences with gaps in for you to complete in different ways.

And then to think in part B about what multiplication equations could represent your sentences.

Have a go at filling those gaps in, writing those sentences in different ways and thinking about the multiplication equations that could represent your sentences.

There were lots of different possibilities for this.

I hope you had fun thinking about pancake recipes.

We've presented the information here.

I've presented it in a table, and it's quite likely that you will see the values that you put into your sentences in this table.

So to begin with, we've got that bit that for every two eggs, she uses three spoons of sugar.

And then I've just gone on and thought, well, maybe she used twice as many eggs.

So four eggs, she'd need two times, twice as many spoons of sugar.

Maybe you decided that she used 10 eggs.

So if she used 10 eggs, that's five times as many as the original recipe.

So she'd need five times as many spoons of sugar.

So if she used 10 eggs, she'd use three times five, which is 15 spoons of sugar.

Which times tables have helped you in working this out I wonder? I wonder if you've used the two and the three times table to help you.

If you look at the values in the table, you can see those tables represented quite clearly.

We've got multiples of two for eggs, multiples of three for sugar.

Now, we focused on many to many ratios today.

So that two eggs and three spoons of sugar.

But I've posed a little challenge for you at the bottom there.

How many spoons of sugar does Aisha need if she uses one egg? So we've gone back to a not a many to many ratio this time, but she's just using one egg.

Well, hang on a minute.

Two eggs, we're halving it now.

We were only using one egg, so she only needs half the number of spoons of sugar.

So she'd need 1.

5, one and a half spoons of sugar.

Fantastic work today.

Well done.

I hope you have enjoyed exploring ratio through different representations and through using multiplication.

And I hope you can see that you can now use multiplication to calculate missing values in ratio problems. You can work out the value of a part using multiplication and that you can also work out the total number using multiplication.

And I hope that the bar models have helped you to see those problems come to life on the paper and give you a way of representing values when we perhaps don't want to draw eggs and sugar and marbles.

Thank you for your hard work and I hope I'll see you again soon.

Bye.