Lesson video

Hello and welcome to this lesson on ratio and proportion.

In this lesson, we're going to be solving problems involving scaling and ratio.

We're going to look at how these two things are very closely related, how we can use similar representations to make sense of the problems that were set and how multiplication and division link to the solving of problems in both the context of scaling and ratio.

So let's look at what's in this lesson.

So we're gonna start off by representing problems involving scaling and ratio.

So some of these representations will be very familiar to you.

So there might be bar models, double number lines, but looking at how we can represent both scaling and ratio problems in similar ways.

So in this scenario, Sam is making orange squash and they use one part orange for every four parts water.

And hopefully you can see that language of ratio in there, the for every.

So you are asked how you could represent this.

So what could you draw? What have you used in the past to represent ratio that would help you to make sense of what's going on as Sam makes their orange squash? So you might want to pause the video at this point and have a go at drawing your own representation and then we'll look at some together.

So I started by drawing a bar model and in my bar, I've split it into five parts to represent the one part orange with an O and the four parts of water with a W.

And I could now put some numbers into this to help me to sort out what happened if Sam scaled up their recipe to make more orange squash or even to make less orange squash.

So I could use a bar model to help me.

I could also draw a double number line.

Now, double number lines are clever, because they show us both values all in one.

So the top parts of my number line shows me how many parts of orange I've got and the bottom part of my number line shows me how many parts water I've got.

So you can see there that where I've got a one for my orange, on my lower number line, I'm showing a four for my water.

So I can see that they use one part orange for every four parts of water.

And the double number line then allows me to record other values that would work if we kept this ratio of one part orange for every four parts water.

So I wonder what representations you drew.

Did you draw something different? So let's have a look, see how we can use these then to go on and solve some problems. So here's our first problem.

Sam is making orange squash.

It's the same ratio, the same recipe.

They use one part orange for every four parts of water.

So the question asks us, "How much orange will they need to make one litre of squash?" And how can the representations help you to work this out? We'd have to think about this, because in the recipe it just says one part orange for every four parts water.

It doesn't give us any number of millilitres or litres.

So how are we going to use our representations to help us to solve this problem? Let's have a look.

So in my bar model, I have my one part orange and my four parts water.

I didn't have that bar showing the whole quantity, but now I can put that bar in to say that I want to make one litre in total.

So I can now see with my bar that I could maybe start to put some values in to help me to solve this problem, because I need to know how much orange I'm going to need.

So what can I see there? Well, I can see that that one litre is going to be split into five parts.

Four of them will be water and one of them will be orange.

It might help me to think of my one litre as 1000 millilitres at this point and then I can think about splitting that 1000 millilitres up to tell me how much I'm going to have in each of those sections of the bar.

So I can divide my 1000 millilitres into five equal parts.

And I know that if I divide 1000 into five equal parts, each of those parts is worth 200.

So each of those parts is worth 200 millilitres.

So Sam will need 200 millilitres of orange to make one litre of squash, because one part of that one litre is orange.

So the bar model has helped me to see that I need to divide my 1000 millilitres by five, because the orange squash is made up of five parts.

One of them is orange and four of them are water.

What about using the double number line to help me? Now, in the bar model it was quite easy to see where the whole amount was, but where is the whole amount that we make in our double number line? Well let's have a look at that basic ratio.

One part orange for every four parts of water.

So how many parts have I got in total? Well, I've got five units of squash.

I don't know what those one and four parts are yet.

I haven't been told that they're a certain number of millilitres or litres, but what I do know is that in my orange squash, there are five parts altogether.

One of them is orange and four of them are water.

So I've got that five total units of squash.

So what if I look at doubling all of that? So if I've used one unit of orange, I'm now using two, so I've got twice as much orange.

So my four units of water are now eight units of water, twice as much water.

And you might have used a STEM sentence in other lessons to help you to think about that idea of if I've got twice as much of one part, I need twice as much of the other part and I will also have twice as much of my total.

So if I've used two parts orange and eight parts water, the total amount of squash is now 10 of whatever unit I've used.

Now, we know that one litre of squash is 1000 millilitres.

Ooh, thousands and tens, I can do some multiplication there quite easily, can't I? So I've put a dotted line in my double number line, dotted part, because I'm going to move to another part.

But let's see how those parts will relate together.

If I take my 10 units and multiply them by 100, I know that 10 multiplied by 100 is equal to 1000.

So 10 multiplied by a hundred will give me my 1000, and in this case it's my 1000 millilitres.

So if I then multiply my amount of orange that I've used by 100 as well, if I've got 100 times as much, I need 100 times as much orange.

So my total amount of squash will be 1000 units and my orange will be 200.

But I know that this time, my units have been defined as millilitres.

So if I make 1000 millilitres of squash, I will need 200 millilitres of orange, exactly the same result we got from the bar model but thinking about it using our double number line.

We'd also work out that we'd need 100 times as many units of water, so 800 millilitres of water, but the question doesn't ask us that, the question's really only interested in the orange, but we can use it as a check.

So we can see that that 200 plus 800 would equal 1000.

So we've definitely made those 1000, in this case, millilitre units of squash.

So we're going to use those representations, 'cause that's what we're looking at, representing problems involving scaling and ratio.

So can you use a representation to decide which of the recipes on the right-hand side of the screen will give orange squash that is the same strength as Sam's, that is stronger than Sam's squash or that is weaker than Sam's squash? So sometimes if you put a bit more orange in, you can make a stronger orange drink.

If you put a bit less orange in, you make a weaker orange drink.

So you've got four recipes there and you need to decide which of these recipes will be the same as Sam's squash, so equal to for one part orange if there are four parts water, which will be stronger and which will be weaker.

So you might want to pause the video here and have a go and then we'll look at the answers together.

Okay, so I've matched those statements to the recipes.

I wonder if you've got the same.

So for one that's the same strength as Sam's squash, we've got two parts orange for every eight parts water.

Well let's look at the original.

That was one part orange for every four parts water.

So if we've used twice as much orange, we'll use twice as much water, so two parts orange for eight parts of water.

So that would be the same.

Those two ratios are equal and we can say that this would be the same strength as Sam's squash.

So what about the the top one, one of the ones that's stronger? So that top one says one part orange for every three parts water.

Well, we know in the original recipe it was one part orange for every four parts water.

So there's less water for our part of orange in this top recipe, so therefore it will be stronger squash than Sam made.

The third one down says two parts orange for every five parts of water.

Well, we know that if we double the amount of orange in Sam's recipe, we'll need eight parts of water.

This only has five parts of water for two parts of orange, so it must be stronger squash.

And for the weaker squash at the bottom there, two parts orange for every 10 parts of water.

Well again, if we doubled our orange, we'd need to double our water and that would've given us two parts orange for eight parts of water in Sam's original recipe.

This has two parts of orange for every 10 parts of water, so it must be weaker than Sam's squash.

So I wonder how could you represent the recipes to justify your answer? So this part of the lesson is all about representing problems. So I wonder what representation we could use to justify those answers even further.

Let's have a look.

So this is over to you for a bit.

So you're going to think about representing these ratios to show whether they are the same as, stronger than or weaker than Sam's squash recipe, which remember, was one part orange for every four parts of water.

So pause the video now, have a go at your representations and then we'll look at some together.

Okay, so how did you get on? I chose bar models for this.

I wonder what you chose, but we're gonna have a look at some bar models to justify these responses.

So remembering Sam's original recipe was one part orange for every four parts water.

So there's Sam's recipe at the beginning.

So let's think about that first statement which says one part orange for every three parts of water.

So let's draw a bar model to represent that.

And we can see that there is less water for one part of orange.

So this is going to be stronger than Sam's recipe.

So one part orange for every three parts water is stronger than Sam's recipe.

What about two parts orange for every eight parts water? So we could look at our bar model this way, two parts and then eight parts of water, but is there a way we can organise this information better so we can compare it more easily with Sam's? So I'm just gonna redraw the bar, see if you can see what I've done.

So I've just shifted the letters around so that I can see that one part orange for four parts water, one part orange for four parts water.

I've still got two parts of orange for every eight parts of water, but I've sort of rearranged it to make two, a repeat of Sam's recipe.

So we can really see that there's twice as much orange and there are twice as many parts of water.

So this means that Sam's recipe, if we use two parts orange for eight parts water, this will be the same strength as Sam's recipe.

We might have made a bit more of it, but it will taste the same, it will be the same strength.

So two parts orange for every eight parts water is the same as Sam's recipe.

What about two parts orange for every five parts of water? Let's have a look.

So again, I've put my two parts of orange and then my five parts of water, but again, that makes it quite difficult to compare to Sam's original recipe.

So let's redraw the bar.

So we can see at the beginning there, we've got one part orange and four parts water.

And then we've got another whole orange but only one more water.

So this is going to be stronger, because I've got two parts of orange but only five parts of water.

So this will be a stronger orange-tasting orange squash than Sam's original recipe.

And finally, I've got two parts of orange for every 10 parts of water.

So let's have a look again.

Two parts orange and 10 parts of water.

So if I redraw that bar and sort of share my water out with my orange.

This time, I can see that for every part of orange I've got five parts water.

So that's going to make a weaker squash, a weaker tasting squash than Sam's original recipe.

So two parts of orange for every 10 parts of water is weaker.

So we've looked at this with both a bar model and also thinking about the idea of, well if I double the amount of orange, I need to double the amount of water, so I can compare the recipes that way as well.

Okay, so we've represented problems using scaling and ratio, we've used bar models, we've used double number lines.

So in this second part of the lesson, we're going to go on and think about how we can use a ratio table in order to help us to organise the information to solve problems, and we may take the information to put into the ratio table from another representation, but the ratio table will help us to organise the values that we know about in order to calculate the values that we want to find out.

So let's have a look at how we can use ratio tables.

So we've got a new scenario here, a new context.

We're making a friendship bracelet and we've got plain beads and patterned beads.

And to make a friendship bracelet, you need three plain beads for every four patterned beads.

So the question asks us how many plain beads do you need to make a bracelet with 28 beads? So again, we've got a part, part and a whole here.

We've got one part of our bracelet is plain beads, one part is patterned beads and then we've got a whole number of beads in a bracelet.

Where does the scaling come in? Well, this time we are being asked to make a bracelet with more beads than that basic ratio that we've looked at.

So we are going to scale up the number of beads, multiply up the number of beads that we've used to get to a bracelet with 28 beads, but we're going to keep that same relationship between the beads that for every three plain beads, there will be four patterned beads.

So let's have a look, we've got a double number line here and I've got my number of plain beads on the top of my number line, the number of patterned beads on the bottom, but I'm also keeping a check of the total number of beads, because that's important to us in this problem.

So for every three plain beads, I use four patterned beads and therefore I've used seven beads in total.

But we want to look at a bracelet with 28 beads and work out how many plain beads we need.

So let's put a 28 on our double number line.

It doesn't matter exactly where it is, we're thinking about that relationship between the numbers that we've got.

So I've put a dotted line in there.

So if I have seven beads in total in my basic ratio, what have I done to my seven to get to 28 beads? So we could think of this in multiplication, we could think of it as a division, but there's a multiplicative relationship between seven and 28.

And if I think about my seven times table, I can see that actually it's seven multiplied by four.

So I've got four times as many beads in this bracelet.

So if I've got four times as many beads, I'm going to need four times as many plain beads.

So let's have a look.

We're gonna think though now about how to turn this into a ratio table.

So I've kept my double number line there, because that's a representation that we're a bit more familiar with.

So I know that I need four times as many beads in total, so I'm going to need four times as many plain beads.

So rather than draw a double number line, I could think about organising this information into a table.

Let's have a look.

So I've drawn a very simple table.

So in my left-hand column, I've got my number of plain beads.

So I know in my basic ratio this is three and on the right-hand side, I've got the total number of beads, because that's what I'm interested in this time.

That's where the information is and those are the values I'm working with.

So the total number of beads when there are three plain beads, is seven.

And I know that I'm going to use 28 beads.

So I know that seven multiplied by four is equal to 28.

If I've got four times as many beads in total, I will need four times as many plain beads and three times four is equal to 12.

So if I make a bracelet with 28 beads, 12 of the beads will be plain, because I've got four times as many beads in total, so I need four times as many plain beads.

So we've got the same instructions to make a friendship bracelet here that you need three plain beads for every four patterned beads.

So the question this time asks, "If you use 20 patterned beads, how many beads will there be in total?" So this time, we're given information about a part of our bracelet, just the patterned beads and we've got to work from that to work out the total number of beads if we use 20 patterned beads.

So we've got a double number line here.

So we've got a question mark there, 'cause we don't know what the total number of beads is, but this time we know that the number of patterned beads is 20.

So what we're trying to do, is to find that relationship from the bit we know about, the four patterned beads in the original instruction and the 20 patterned beads that we're actually using.

So we've got to think about what's happening along that arrow, and we might think about it in terms of multiplication or division.

But remember, we're looking for that multiplication.

How many times more patterned beads have I used? And therefore, I can work out how many times more beads there will be in total.

So let's have a look at how this will work in a ratio table.

So I've taken the information from my double number line and I've put it into a ratio table.

So this time, I've got the number of patterned beads in my left-hand column in my table and in my right-hand column, I've got the total number of beads.

Those are the values I'm interested in.

I'm told how many pattern beads there are and I'm asked how many there will be in total.

So those are the bits of information I'm really interested in.

This time, I'm not interested in the plain beads.

So how can I use this table to help me? So what have I done to that four to turn it into 20? Remember, we're really thinking about multiplication and division.

Well, I can see that four multiplied by five is equal to 20.

I've got five times as many patterned beads.

If I've got five times as many patterned beads, there will be five times as many beads in total.

So seven multiplied by five is equal to 35.

So if I use 20 patterned beads, I will have 35 beads in my bracelet in total.

Time for you to have a go.

This time, we're told we've got 42 beads in total on our bracelet.

This time, how many more patterned beads will there be than plain beads? So you might want to pause the video here and have a go yourself and then we'll look at the answer together.

Okay, so we've got a bit more information to think about here, because we know that our basic instruction said for every three plain beads, there are four patterned beads, and that gives us seven beads in total.

This time, I know that the total number of beads is 42, but I'm asked for how many more patterned beads there will be, so I'm asked for that difference between the number of plain and patterned beads.

So let's think back to that top row of my ratio table that I've drawn.

So I know that in my original instruction, for every four patterned beads there are three plain beads.

So the difference between the number of beads is one.

There's one more patterned bead in the original instruction and the total number of beads there is four plus three, which is seven.

So the information in the question is that there are 42 beads in total.

And I've worked out here that seven multiplied by six is equal to 42, 7 times six is 42.

So if there are six times as many beads in total, there will be six times as many patterned than plain beads, but there will be six times the difference as well.

So what I can actually do is make my ratio table a lot smaller and I can forget about the plane and patterned bead numbers, because that's not what the question's asking me about.

It's asking me how many more patterned beads there are than plain.

So it's asking me for the difference.

So if I know that seven multiplied by six is 42, there are six times as many beads in total, the difference will be six times as well.

So one multiplied by six is six.

So in this bracelet of 42 beads, there will be six more patterned beads than plain beads.

Okay, time for you to do some practise and to have a go at using a ratio table to solve scaling and ratio problems. So part A is around a map.

Part B is about Sam and Aisha sharing some marbles.

And part C is about beads on a bracelet.

So have a go at solving those problems using a ratio table, pause the video and then we'll have a look at the solutions and the ratio tables together.

Okay, how did you get on? I wonder if your ratio tables look the same as mine.

So this is A.

So this is about a map and it said that four centimetres on the map represents 10 kilometres in real life.

So part one says if the route is 12 centimetres on the map, how far is it in real life? So my ratio table has distance on the map on one side and distance in real life on the other side.

We know that our scale is for every four centimetres on the map, there are 10 kilometres in real life.

So my top row shows that scale that I've been given at the beginning of the problem.

So distance on the map is four centimetres, the distance in real life is 10 kilometres.

The information I'm given is that the route is 12 centimetres on the map.

So the distance on the map is 12 centimetres.

Well, I can see that four multiplied by three is equal to 12.

So I've got three times the distance on the map, so I will have three times the distance in real life and 10 kilometres multiplied by three is 30 kilometres.

So the route in real life is 30 kilometres.

It's a long walk to go on.

Part two says that the same route is represented on a map where the scale is five centimetres for every 10 kilometres.

So how far is the route? This time, we know that the route is 30 kilometres long.

So our 10 kilometres in real life has been multiplied by three and is 30 kilometres.

So this time, how far is it on the map? Well, if 10 kilometres is represented by five centimetres and I've got three times as many kilometres, I'm going to have three times as many centimetres on the map.

So on this map, our route would be five times three centimetres long.

Our route would be 15 centimetres long.

So it will be the same distance in real life, but the route actually is longer on the map.

B is about Sam and Aisha sharing marbles.

So for every three marbles Sam gets, Aisha gets five.

Okay, so part one says if Sam has 15 marbles, how many does Aisha have? Our basic ratio is for every three marbles Sam has, Aisha has five.

So Sam has 15 marbles, which is five times as many marbles.

So if Sam has five times as many marbles, Aisha will have five times as many marbles.

Five multiplied by five is 25.

Let's look at part two.

It says if Aisha has 15 marbles, how many do they have altogether? So we know how many they have altogether in our basic ratio, because we know that for every three marbles Sam has, Aisha gets five, and so that gives a total of eight marbles.

This time we know that Aisha has 15 marbles.

So Aisha has three times as many marbles.

So the total will be three times as many marbles and three times eight is equal to 24.

Does it matter how many Sam has this time? It doesn't this time.

We don't need to know how many Sam has.

We could have worked out that Sam has nine marbles and then added together Sam's and Aisha's marbles, but we do know that if we scale up, multiply up the number of marbles that one person has, then we will scale up the total number of marbles by the same factor.

So let's have a look at part three.

So part three said if they have 80 marbles altogether, how many more marbles does Aisha have than Sam? So we know in that first share that for every three marbles Sam has, Aisha has five marbles, the total number is eight.

The difference between the number of marbles they get is two.

Aisha gets two more than Sam each time they share the marbles.

And the question is asking me how many more marbles? So it's asking me about the difference and the information I have is the total number of marbles.

So the number that Sam and Aisha has doesn't really matter.

I could work out how many Sam and Aisha have and subtract to find the difference, but I know that if I scale up the total number of marbles, I will scale up the difference by the same factor.

So I can actually get rid of the columns for Sam and Aisha and just look at the total number of marbles and the difference between the number of marbles they have.

So I started with eight, I've now got 80 marbles.

So what have I done? I have multiplied by 10, I've got 10 times as many marbles.

So if I've got 10 times as many marbles in total, I know that the difference between the number of marbles they have will be 10 times bigger as well.

So my difference will be 20 marbles.

So part C is about bracelet making.

So for every two plain beads, there are three patterned beads on a bracelet.

So part one says there are 30 beads altogether.

How many beads are plain? So I'm looking at the total number of beads and the plain beads.

I don't need to know about the patterned beads at this point, I can just focus on the how many beads altogether and how many plain beads.

So I might draw my full ratio table to start with, but the information I have is about how many beads altogether and the question is how many plain beads? So I can ignore the patterned beads in my table and just focus on the information that I've got here.

So I know that I had five beads altogether in my basic instruction, but I've used 30.

So what have I done? Well, I've multiplied by six, because five multiplied by six is 30.

I've got six times as many beads altogether, so I will have six times as many plain beads, so I'll have 12 plain beads.

Part two says there are 45 beads altogether, how many beads are patterned? So this time I'm not interested in the plain beads.

So I can get rid of the plain beads from my ratio table and I can look at the information I've got there, my basic share that when there were five beads altogether, three of them were patterned.

But I haven't got five, I've got 45 beads.

So what have I done? Well, I've multiplied by nine, because five multiplied by nine is 45.

So if I've got nine times as many beads altogether, I will have nine times as many patterned beads and three multiplied by nine is 27.

Let's think about part three.

So now I'm looking at that difference in the number of beads and I'm looking at how many beads there are all together.

So that's the important bit.

But I think I'm gonna draw the full table just to make sure I've got the right information.

So I know that for every two plain beads there are three patterned beads.

That means there are five beads altogether and the difference between the number of beads is one.

So that's all my information from that basic instruction.

What I know from the question is that that difference is now four.

So I've got four more patterned beads than plain beads and I want to know how many beads there are altogether.

So I can ignore just the basic number of plain and patterned and just look at the total number and the difference, because I know that if I make the difference a number of times bigger, then there will be that number of times bigger beads altogether.

So let's have a look at that.

What we've done.

So one multiplied by four is equal to four.

So I've got to multiply the total number of beads by four.

I've got four times as many in the difference, so I've got four times as many beads altogether and four multiplied by five is equal to 20.

Thank you for your hard work in this lesson again.

Okay, I hope that you can now see how you can identify values on a scale or a double number line or maybe even in a bar model and arrange them into a ratio table.

Now, the ratio table allows us to organise the information so that we can see the relationships between values and spot the multiplication or division equation that we need to solve the problem.

So I hope that you are feeling more confident about taking values from representations, putting them into ratio tables and using multiplication and division to solve problems involving scaling and ratio.

Thank you and I hope to see you again soon, bye.