Lesson video

Hello, and welcome to this lesson on ratio and proportion.

In the lesson, we're going to be thinking about using knowledge of multiplication and division to solve a range of problems in different contexts involving scaling.

Let's have a look at what we're going to be doing today.

To begin with, we're going to be using multiplication and division to solve scaling problems. You may have looked at some maps and plans with scales.

Why is it useful to have maps and plans with a scale? What do they allow us to do? Well, having scale on maps and plans means that you can draw an accurate representation of something much larger.

We can't draw it in real-life size, so we need something that has a scale so that we get an accurate representation of a much larger object or a larger space on a manageable-size piece of paper that we can use.

That's what the whole point of having scales and scale drawings and maps is all about.

Here are the measurements on the plan for a house.

This plan is 20 centimetres long, 12 centimetres wide, and then, we've got those other measurements showing us the width of a bedroom, the width of a bathroom, and so on.

So, these are the measurements on the plan.

How big do you think the house is and what would the scale look like? You might want to pause the video here so that you can have a think about how big this house is.

I wonder what you came up with.

I wonder how many different ways you can complete this sentence.

"For every centimetres on the plan, there are metres or centimetres in real life." We've got a STEM sentence there that we could fill in differently depending on the scale we have decided to use or depending on the size of the house.

I wonder how many different ways you can complete that sentence, perhaps using ideas from your discussion about how big the house is.

Again, you might want to pause the video here before we go on and look at some examples together.

I wonder what you came up with.

Here are three examples.

A says, "For every one centimetre on the plan, there's one metre in real life." So, quite a simple scale for us.

B says, "For every two centimetres on the plan, there's one metre in real life." And C says, "For every five centimetres on the plan, there is one metre in real life." I wonder if you came up with any of those as a scale.

My question is, which scale would represent the biggest house? A, B, or C, which one will be the biggest house? Again, you might want to pause the video to have a think and then we'll discuss the answer together.

What did you think, then? It's an interesting one, because in A, B, and C, the number of centimetres gets bigger.

But what does that mean in terms of the scale and the size of the house that this is a plan for? Well, let's look at A.

For every one centimetre on the plan, there's one metre in real life.

That would mean that the house would be, it's 20 centimetres wide, so it would be 20 metres wide in real life.

In B, for every two centimetres on the plan, there's one metre in real life.

20 centimetres is 10 lots of two centimetres, so it would be 10 metres in real life.

The house in B would actually be half the size of the house in A, or half the width of the house in A, certainly.

What about C, then? C says, "For every five centimetres on the plan, there's one metre in real life." So, our width of 20 centimetres, how many five centimetres are there? That's four, so our house in C would actually only be four metres wide, so a lot smaller.

It's interesting to make sure that we really think about how when we change the scale, what does that mean for the size of something in real life and what does that mean in terms of interpreting this plan and allowing us to visualise what the real thing looks like? I hope you came to the same conclusion, that A would represent the biggest house in real life.

Okay, so let's have a think about some different scales.

We've taken just the kitchen and lounge from our plan and the fact that on the plan, it is 12 centimetres long.

I've got some different scales for you here to look at.

In a moment, we're gonna pause the video and you are going to see if you can match the scale to the real-life length of the kitchen and lounge.

Okay, so pause, have a think, and then we'll look through the answers together.

Okay, so in that first scale, for every one centimetre on the plan, there are 50 centimetres in real life.

We've got 12 centimetres on the plan, so we want 12 lots of 50 centimetres.

Well, 50 centimetres is half a metre, isn't it, so 12 lots of half a metre would be six metres in real life.

There we go, for every one centimetre on the plan, there are 50 centimetres in real life, so 12 centimetres will be six metres.

For every two centimetres on the plan, there's one metre in real life.

So, if I've got 12 centimetres, every two of those is worth one metre.

That's six lots of two, so that's six metres as well.

That's interesting.

Same distance, six centimetres.

I wonder why that is.

Hang on a minute, let's think about that.

If one centimetre in in the first scale represented 50 centimetres, if I double my one centimetre, that will be worth two centimetres, would be worth one metre, which is the same as the scale in the second one.

We've got two scales there that would actually represent the same distance in real life.

Okay, let's look at the third one.

For every three centimetres on the plan, there are two metres in real life.

I've got 12 centimetres on my plan.

How many threes are there in 12? That's four times three, so I want four times two metres, so that's eight metres.

And then, my final scale.

For every six centimetres on the plan, there are two metres in real life.

How many six centimetres have I got in 12? That's two.

Two times six centimetres is 12 centimetres and two times two metres is four metres.

Well, that's an interesting one, because those bottom two, the last one is half, so how does that relate? Three centimetres is two metres, and then six centimetres is two metres.

I've got a doubling there representing a halving of my size.

Oh, that's interesting, wasn't it? I've doubled the number of centimetres that represent two metres, so therefore, I've got half as many metres in real life.

Lots of really interesting connections to make when we look at those scales.

Okay, so let's have a think about the calculations that that we did.

I've drawn some mini-tables here, the sorts of tables that you might have seen when you were comparing recipes and things like that and collecting tables of values relating to ratios.

Let's have a think about what we've got.

We know on our left-hand column, here, one centimetre represents 50 centimetres in real life.

And we've now got 12 centimetres, and we worked out that that was 600 centimetres, or six metres.

So, what did we do? We had to multiply by 12, and then we multiplied by 12.

One centimetre times 12 is 12 centimetres.

50 centimetres times 12 is 600 centimetres.

Okay, let's look at the second one.

This time, for every two centimetres on the plan, there's one centimetre in real life.

But we know that there are 12 centimetres on our plan.

Two centimetres multiplied by six is 12, so one metre multiplied by six is six metres.

And let's think about what the table's gonna look like for our third one.

Three centimetres on the plan is equal to two metres in real life.

I've got four times as many centimetres on my plan.

Three times four is equal to 12, so two times four metres is equal to eight metres.

And then, finally, for every six centimetres on the plan, there are two metres in real life.

I've got 12 centimetres, so that's two times my six centimetres.

I've doubled my number of centimetres on the plan, so I need double the number of metres in real life.

Two metres multiplied by two is eight metres.

We can really start to think about the calculations that we've done and how we're using our multiplication, in this case, to solve these scaling problems. We did do some division as well.

If we think about it, we worked out how many lots of two centimetres there were in 12, how many lots of three centimetres there were in 12.

So, we were doing a bit of division, but possibly maybe thinking about it in terms of multiplication.

Time for you to do some practise.

I'm going to ask you to draw a scale plan of the room that you are in, and you're going to draw it on a piece of A4 paper.

You need to decide what a suitable scale would be so that your plan fills most of the paper, allowing you some space to write the measurements on, of course.

Have a think about the scale that you're going to use, explain why you chose that particular scale, and maybe explain why other scales didn't work and you decided to use the scale that you went with.

You might want to pause the video now so you can do that before we carry on with the lesson.

How did you get on? You might have used a scale of one centimetre on the plan is equal to one metre in real life, or you might have decided that, for the room you're in, a better scale might have been one centimetre on the plan represents 50 centimetres in real life, or maybe a different one.

Again, which of these rooms do you think is bigger and why? That's an interesting one, isn't it? If we're using one centimetre to represent one whole metre, that might be because we've got to fit more metres on our page.

I think that if we've used one centimetre for one metre, we might be in a bigger space than if we'd used one centimetre to represent half a metre in real life.

Anyway, I hope you've enjoyed drawing a scale drawing of the room that you're in, and I hope that you picked a good scale so that you filled your A4 paper.

Okay, so we've been using multiplication and division to solve scaling problems so far.

But for this next part of the lesson, we're going to think about where scaling maybe involves working with fractions as well.

So, let's have a look at some other scales.

Sometimes, a scale will lead to working with fractions.

How could we work out the missing values on this scale? Let's have a look.

What can we see and what do we know? Well, we know that the only bit of complete information we've got is right at the end of our double number line.

We know that 15 centimetres on the drawing is equivalent to five metres in real life.

So, for every 15 centimetres on the drawing, there are five metres in real life.

But the values we've got to work out are three and four, and then we've got that gap at the beginning that we need to have a think about.

What can we work out quite easily from this one? Well, I'm seeing a link, I think, between 15 centimetres on the drawing and three centimetres on the drawing, so let's have a think about these two, here.

15 divided by five is equal to three, so if I divide by five, I can work out what the value is for three centimetres on the drawing.

Five divided by five is equal to one.

So, I know that three centimetres on the drawing is equal to one metre in real life.

That gives me quite a good starting point, maybe, for working out those other missing values.

Let's have a look at those ones.

Okay, so if I look at that scale on the number line on the top, there, for the centimetres on the drawing, I think I'm looking here above that green box at a one centimetre, so let's fill that bit in.

Three divided by three is equal to one centimetre, so if I can divide my number of metres in real life by three, I can work out what one centimetre on the drawing is worth.

And that's sometimes useful to know.

One divided by three, well, one divided by three, one divided into three equal parts, one of those parts would be 1/3, so one divided by three is equal to 1/3.

I now know that one centimetre on the drawing is equal to 1/3 of a metre in real life.

Let's look at how I can use that information to fill in another missing value.

Okay, so let's have a look at this.

I now need to know what four centimetres is worth.

Well, I know I can use multiplication here, because four centimetres is four times one centimetre, so I can multiply my value for one centimetre by four.

My one centimetre on the plan was was equal to 1/3 of a metre in real life, so I want four times 1/3.

Let's think about that with those STEM sentences we've used before.

There are four lots of one centimetre on the plan, so there will be four lots of 1/3 of a metre in real life.

And this is 4/3, or 1 1/3 of a metre.

Okay, so time for you to have a bit of a check.

Here are some missing boxes.

Can you use some multiplication and division and think about handling fractions in order to work out the values in those missing boxes? You might want to pause the video here before we go through them together.

Okay, so let's have a look.

How did you work out those missing values? Well, I started here because I can see that three multiplied by three is nine.

I've got three times as many centimetres on my drawing, so I'm gonna have three times as many metres, so I've got three metres.

That's a nice, straightforward one.

No fractions involved, there.

I'm using whole numbers at this point.

But what about that equivalent to five centimetres on the drawing? How am I going to work that out? Well, I can see that I can multiply my one centimetre on the drawing by five to get me five centimetres on the drawing.

Then, I've got to multiply 1/3 of a metre by five, and 1/3 multiplied by five is 5/3, or 1 2/3 of a metre.

We've used multiplication here to help us and we used division to help us to get to that 1/3 of a metre being equivalent to one centimetre on the drawing.

We talked a lot about multiplication and division, but we can't fail to notice that there are those relationships along each number line.

For each extra centimetre on the drawing, we get an extra 1/3 of a metre.

So, if we knew that four centimetres on the drawing was represented by 1 1/3 of a metre, then we would know that 5/3 could be represented by one more 1/3.

So, adding on 1/3 to go from 1 1/3 to 1 2/3 or 4/3 to 5/3.

The multiplication and division are really important so that we can understand those relationships.

But we can sometimes use an addition or subtraction relationship if we're just working along one step on our number line, or our double number line in this case.

Okay, time for you to do some practise.

We've gone back to our plan of our house, but this time, we've changed the measurements on the plan.

There are three new scales here for A, B, and C, and your task is to draw the double number lines to represent each scale and then to work out what the real-life measurements are using those different scales.

Okay, how did you get on? Let's have a look.

This is A.

A said, "For every two centimetres on the plan, there's one metre in real life." Our double number line, we can start off by highlighting that two centimetres on the plan is one metre in real life.

And then, we can use multiplication and division and maybe a bit of addition and subtraction to fill in the gaps to work out the values that we need.

You can see here that we filled in those different values.

We've used our number line carefully to put in the values that we need.

We've got the values for four and five, six, nine, and 15 centimetres, and we've used our understanding of the relationship between the two centimetres on the plan representing one metre in real life and the fact that we've taken that down to one centimetre on the plan representing half a metre in real life in order to help us to come up with those real-life measurements.

And we've done the same here for B.

For every three centimetres on the plan, there are two metres in real life.

This time, we've got thirds involved instead of halves because we've got that three centimetres on the plan.

So, one centimetre on the plan is worth 2/3 metre in real life.

A slightly more complicated one, there.

Let's have a look at C.

Nothing's changed for C.

I've put C up there, but the double number line has stayed the same and all the measurements have stayed the same.

How can that be? What's going on, here? For B, it said, "For every three centimetres on the plan, there are two metres in real life." And for C, it says, "For every six centimetres on the plan, there are four centimetres in real life." I wonder if you've seen what I've seen.

Can you see that both of those scale relationships are represented by the same double number line? And we had that once before, didn't we, when we were looking at the dimensions that we could use what looked like a different scale, but it actually represented the same relationship.

And we can see that here very clearly.

If every three centimetres on the plan represents two metres in real life, if we have twice as many centimetres, six centimetres, we've got twice as many metres, four metres.

That is exactly the scale that we've been given, here.

So, B and C are, in fact, the same scale.

And so, therefore, the double number line will look the same and the measurements on the real, the measurements of the real room or the real house will be the same as well.

Thank you for your hard work in this lesson.

I hope you're feeling more confident in using multiplication and division to interpret scales, to work out real-life dimensions from those on a plan, and to work out how to create a plan from real-life dimensions.

A scale sometimes means that you need to work with fractions, so I hope that you've felt confident in working with those fraction scales today.

And then, also, realising that scales can look different, but actually represent the same thing, so that we can write scales that mean the same in translating from a plan to real life, but might look slightly different in the way that they're written down so we can find equivalent scale.

Again, thank you for your work, and I look forward to seeing you again.

Bye!.