Lesson video

Hello and welcome to this lesson on ratio and proportion.

By the end of this lesson, you should be able to explain how and why scaling is used to make and interpret maps and plans.

Now, scaling may be a new idea, a new concept to you.

So let's have a look at the language we're gonna be using in this lesson and how it relates to ratio and proportion.

So the keywords we're going to be using in this lesson are scale, scaling, and double number line.

So let's look at what those words really mean.

So a scale is used to show how the distances on a map or plan relate to distances in real life.

Representing one distance using another is called scaling.

And scaling distances involves multiplying or dividing values by the same factor or divisor.

I mean if you think about a map, they're much smaller than real life, and a plan is much smaller than the thing that it's representing.

So somehow we've got to work out how the distances on the map or the plan relate to distances in real life.

And this is where a scale comes in.

So a scale is often shown on a double number line.

Now I'm sure you are familiar with single number lines, but a double number line is where we put two together.

So that one number line shows the distance on the ground and the other number line shows the distance on the map.

So in this scale the double number line shows that for every one centimetre on the map there are two kilometres on the ground.

So the top number line represents how many kilometres there are on the ground, and the bottom number line represents how far that is on the map.

So we're going to be investigating double number lines, interpreting them, and making use of them to think about how we can make sense of scales in terms of maps and plans.

So, let's get in and have a look at a map and see how scale works.

So there are two parts to our lesson.

We're going to be thinking in the first part about using that language of for every, for distance and scale.

If you've done some work on ratio, you might be familiar with that phrase, for every, so you'll be able to use it this time to think about maps and scale in relation to maps and plans.

We've got a map here of Sandy Island.

So here's the map.

How big do you think Sandy Island is? What do you think? Have you got enough information? What else do you need to know? Well, Sandy Island could be any size, couldn't it? We don't know what the scale is on our map.

So we need to know about the scale of the map, what the distances on the map mean in real life in order to work out how big Sandy Island is.

So, we're going to assume that the grid is a one centimetre grid.

It might not look like one centimetre the way you're looking at it on your screen or on your paper, but we're going to say that this grid is a one centimetre grid.

And we've been given the information that for every one centimetre on the grid there are two kilometres on Sandy Island.

So just take a moment to have a think about that, look at the map, the map of the island, and think about how many centimetres there are and what that means in terms of how many kilometres there are on Sandy Island.

So if we know that for every one centimetre on the grid, there are two kilometres on Sandy island, can you work out, estimate how wide the island is? So you might want to pause the video here and have a think about estimating how wide Sandy Island is.

So I've marked the width of Sandy Island with a big purple arrow.

So, what do we know about this scale that will help us to work out how wide the island is? Well we are told that for every one centimetre on the grid, there are two kilometres on Sandy Island.

So how long is that purple line on our map? Well, on the map, the island is about eight centimetres across.

Well, we'll say roughly, because the sides of the island aren't absolutely straight.

So the island is about eight centimetres across, and we know that for every one centimetre on the grid, there are two kilometres on Sandy Island.

So we've got eight lots of two kilometres, two kilometres multiplied by eight.

And so that equals 16 kilometres.

So we can say that Sandy Island is about 16 kilometres wide.

So, let's have a think about using what we've learned about the map scale for Sandy Island to estimate some other distances on the island.

So we're still thinking that for every one centimetre on the grid, on the map, there are two kilometres on Sandy island, and there's a sort of stem sentence there to help you to think if there are so many centimetres on the map, so it will be so many kilometres on the island.

So can you use that information and that thinking to estimate these distances? So number one asks you to estimate the distance from the tree to the campsite.

And you can see a tree and a tent on the island, that distance.

Number two asks you to estimate the distance from the campsite to the volcano.

So if you go south from the campsite, you'll come to something that looks like a sort of mountain with things flowing down it.

So we've got lava flowing down the outside of our volcano here.

So that's our campsite to the volcano.

And then question three says the mountains to the volcano.

So the mountains are the ones with the white snow peaks on them, and that will be a definite estimate because as you can see that journey isn't along the grid lines.

So, think about how you are going to make that journey and how you will estimate that distance on your map.

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

Question one asked you to estimate the distance from the tree to the campsite.

So it says it's about three centimetres on the map, so it'll be about six kilometres on the island because remember every one centimetre on the map represented two kilometres on the island.

So we've got three multiplied by two kilometres, three lots of two kilometres, so six kilometres on the island.

You might have gone for a little bit more than that if you included the extra half square, but I made it three centimetres.

So, we are estimating.

Question two asked you to estimate the distance from the campsite to the volcano.

This time I've gone for 2 1/2 centimetres on the map.

So what's 2 1/2 multiplied by two kilometres? So two kilometres multiplied by 2 1/2.

So I reckon that's four plus one kilometre, five.

So roughly five kilometres on the island.

And question three asked you to estimate the distance from the mountains to the volcano.

Well, I've estimated about five centimetres on the map.

Tricky one this because it is going on the sort of diagonal across the grid.

So I've estimated about five centimetres on the map.

So five lots of two kilometres, 'cause remember for every one centimetre on the map, there are two kilometres in real life on the island.

So five lots of two kilometres is 10 kilometres.

Time for you to do some practise now.

We've called this one changing scales.

Now so far we've worked with a scale that for every one centimetre on the map there are two kilometres in real life, but this time we're changing the scales and in fact the scales might be different for quite a few of these questions.

So we're gonna give you some information and from that you're going to work out either what the scale is or what the distance is in real life.

So A says if the distance between the treasure and the tree is eight kilometres, what is the scale of the map? So this time you're given a distance in real life and you're asked to work out what the scale of the map is.

In B, if the island is 24 kilometres wide, what is one centimetre worth on the map? Part C then helps you to do D and E as well.

So part C says, if the distance between the campsite and the boat is five kilometres, how far is it from the boat to the tree? So using one distance to work out another.

In D, you are asked to say what is the scale of the map in part C? So if we take part C, what is the scale of the map? And for E, if the scale is the same as in part C, how far is it from the campsite to the volcano? So you might want to pause the video now, have a go at those questions, and then we'll have a look through the answers together.

How did you get on? Did you manage those different scales? So A said if the distance between the treasure and the tree is eight kilometres, what is the scale of the map? Well the treasure and the tree are a nice neat two squares apart, aren't they, two centimetres on the grid, on the map? So, if that's two centimetres, two centimetres equals eight kilometres.

So for every two centimetres there are eight kilometres.

So for every one centimetre there are four kilometres.

So this time the scale of the map is for every one centimetre there are four kilometres.

So let's look at B.

If the island is 24 kilometres wide, what is one centimetre worth on the map? Well, from our first exploration of this map, we reckon that the island is about eight centimetres wide on the map.

So eight centimetres on the map is equal to 24 kilometres in real life.

So for every one centimetre there are three kilometres.

Can you see how we're using our multiplication and division here? So if eight centimetres is 24 kilometres, if I divide by eight, I will work out what one centimetre is worth.

So 24 divided by eight is three.

So there are three kilometres for every one centimetre on the map for B.

So part C said if the distance between the campsite and the boat is five kilometres, how far is it from the boat to the tree? Well the campsite to the boat is quite a neat one centimetre I think.

So that one centimetre on the map is equal to five kilometres.

So how far is it from the boat to the tree? Well I estimate that the boat to the tree is about 4 1/2 centimetres.

So we need 4 1/2 multiplied by five.

Well I know that four times five is 20, 1/2 of five is 2 1/2 so 22 1/2 kilometres roughly.

So D asks, what is the scale of the map in question three? Well we kind of quite neatly worked that out, hasn't it? Because we worked out that for every one centimetre on the map there are five kilometres on the island.

So our scale is one centimetre for every five kilometres.

And if we take that scale on into E, how far is it from the campsite to the volcano? Well there are about 2 1/2 centimetres on the map from the campsite to the volcano.

So 2 1/2 multiplied by five, two multiplied by five is 10, and 1/2 of five is 2 1/2.

So 10 and another 2 1/2, 12 1/2 kilometres, 12.

5 kilometres on the island if we've got 2 1/2 centimetres on the map.

Okay, so we've thought about that language of every in terms of interpreting distance and scale.

And now we're going to go on and use a double number line to represent the scales on maps and plans.

So let's have a look.

So here we are back on Sandy island and we're going to have a think about this as a, this scale idea, as a double number line.

So here we've got a scale representing kilometres on the island.

So from zero up to 20.

We're now going to add a second number line which talks about centimetres on the map.

So here we've got our centimetres on the map.

So we've got two straightforward number lines representing distances, but what we want to do is bring them together so that we can create a double number line so that we can relate the distances on the island to the distances on the map.

So let's bring these two number lines together.

There we go.

So we've got our double number line here.

The top showing us the kilometres on the island and the bottom scale showing us centimetres on the map.

So we can see here that one centimetre on the map is equivalent to two kilometres on the island.

If we look at that lower scale for centimetres on the map and the upper scale for kilometres on the island.

So here we've got a map scale on a double number line with some missing values and we're asked to work out what the missing values are and how we know.

So you might want to pause the video here to have a look and then we're gonna go through and work out the values of A, B, C, and D together.

Okay, so let's have a look at these missing values together.

Now I'm sure you've seen those patterns in the top number line and the bottom number line.

And just by counting along you can work out what those missing values are.

But as we talk through them, I want you to really think about the relationship between the A missing in the kilometres and the two centimetres below it, the B missing in the kilometres and the four centimetres below it.

And then for the C centimetres missing in the 10 kilometres and the D centimetres missing and the 16 kilometres.

We can see those patterns along the number lines, but let's look at those patterns between the two number lines and using those to help us.

So A is the number of kilometres represented by two centimetres.

Well we know that one centimetre represents two kilometres, so twice as many centimetres must represent twice as many kilometres.

So A must be equal to four kilometres because two times two kilometres is equal to four kilometres.

What about B then? That's four centimetres.

So that's four times as much as one centimetre.

So four multiplied by two is equal to eight.

So B is eight kilometres because four lots of two kilometres is equal to eight kilometres.

What about the other way round though? How can we work out how many centimetres on the map from how many kilometres on the island? So C is the number of centimetres that represents 10 kilometres.

Well we know that one centimetre represents two kilometres or two kilometres is represented by one centimetre.

So how many lots of two kilometres have we got to make 10 kilometres? Well it's five times, isn't it? So we've got five times as many kilometres, so we must have five times as many centimetres, so that must be five centimetres.

So C is five centimetres because 10 kilometres is five lots of two kilometres and we know that for every two kilometres on the island, there's one centimetre on the map.

So let's apply that same thinking to work out D.

We've got 16 kilometres on the island, we know that every two kilometres is one centimetre.

So how many lots of two kilometres have we got? Well two times eight is equal to 16.

So we've got eight times as many kilometres, so we must have eight times as many centimetres on the map.

So D is eight centimetres because 16 kilometres is eight multiplied by two kilometres, eight lots of two kilometres.

That thinking between the double number lines is going to be really important when you are working out missing values in scales.

Okay, so some for you to have a go at.

So to check about that thinking, thinking between the two scales, not just along, we can see the patterns along.

Let's look at those relationships between.

So again, to have a look at these and see if we can describe the scale and fill in the missing values.

So you've gotta work out what the scale is from the double number line and then fill in the missing values.

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

Okay, so that first scale, our one centimetre is representing five kilometres.

So our basic scale is for every one centimetre there are five kilometres.

And if you've been working with ratio, you'll recognise that phrase for every from the work on ratio.

So for every one centimetre there are five kilometres.

So our first missing value was that 15.

Well we can see the pattern 5, 10, 15, 20, 25, 30 and so on along the kilometres.

But how does the 15 relate to the three centimetres? Well if I know that for every one centimetre there are five kilometres, if I've got three centimetres, I've got three times as many centimetres.

So that'll be, the distance will be three times five kilometres.

So 15.

If we carry on on that missing values on the top we can see the same thinking for that 30 kilometres that was missing, it's six centimetres worth of kilometres.

We know that one centimetre is worth five.

So six times as many centimetres is worth six times as many kilometres, so 30 kilometres.

So what about those missing values in the centimetres? So we were looking for how many centimetres represents 20 kilometres.

So again, we know that every one centimetre represents five kilometres.

So how many fives in 20? 20 divided by five is equal to four.

So it must be four centimetres.

We've got four times as many kilometres, so four times as many centimetres.

And then for the final missing value was the number of centimetres representing 45 kilometres.

Again, we know that relationship of for every one centimetre there are five kilometres.

So if I've got 45 kilometres, that's nine times as many kilometres.

So I must have nine times as many centimetres.

So nine centimetres.

Let's have a look at the bottom double number line there.

This time we didn't have that 500, did we? We knew that for every two centimetres there were 1,000 metres.

So therefore for every one centimetre we've got half the value.

For every one centimetre there must be 500 metres.

So how many centimetres represent 1,500 metres? Well 1,500 is three lots of 500, three times 500.

So that will be represented by three centimetres.

But how many metres are represented by four centimetres? Well we know for every one centimetre there are 500.

So if I've got four centimetres then I must have four times 500 metres, which is 2,000 metres.

And then our final missing value was how many centimetres represent 4,000 metres? Wow.

Well I know that one centimetre represents 500 metres, but I also know that two centimetres represents 1,000 metres.

So I've got four times 1,000, so I must have four times the two centimetres.

So 4,000 metres will be represented by eight centimetres.

Okay, I hope you are starting to see that relationship between the two parts of the double number line.

Time for you to do some practise.

So we've not given you a double number line here, but we are asking you to think about the scale on this drawing of a car.

So we're told that one centimetre on the drawing represents 30 centimetres in real life.

So can you fill in the real life measurements? You've been given the measurements on the plan and the gaps are the measurements in real life for you to fill in.

Can you use that information about scale to work out the real life measurements? Second part of this task gives you the double number line.

So can you complete the double number line to show the scale being used in the drawing? And remember, think about the relationship between a centimetre measurements in real life and the centimetres on the plan as well as just thinking about using the patterns along those number lines.

And we've got another task for you to do.

This is a plan of the downstairs of a house perhaps and you've been given some measurements for the plan and you've been given a double number line.

So question one asks you what is the scale for the plan? And question two asks you to fill in the gaps on the scale in the double number line.

So how did you get on with this first part of task one? I'm going to bring you back to that stem sentence we were using about if something is 10 times as much then the other value will have to be 10 times as much as well.

So let's see if we can use that thinking to help us with this.

So let's look at the overall width of the car, that top arrow.

So the 12 centimetres on the plan, we know that for every one centimetre on the drawing there are 30 centimetres in real life.

So 12 lots of 30, 12 times 30, 12 times three is 36, 12 times 30, 360 centimetres.

So we can use that thinking with all of those other measurements.

Two centimetres for the centre of the wheel to the outside, to the back of the bumper.

So we've got twice as many centimetres, so therefore we'll have twice as many centimetres in real life.

So 60 centimetres.

And then for the gap between the distance between the centres of the wheels, eight centimetres, so eight times as many, eight times 30 is 240.

The gap between the, the distance between the centre of the wheel and the front bumper is the same as the centre of the wheel and the rear bumper.

The height of the car, we've got a half in there, haven't we? So we've got 4 1/2 times,.

so four times 30 plus 1/2 of 30.

Four times 30 is 120, 1/2 of 30 is 15.

So 135 centimetres.

I hope you were able to use that thinking to help you and that thinking will help you with the double number line.

And here are those equations that we used.

12 times 30, two times 30, two times 30 again, eight times 30 and then 4.

5 times 30.

So then you were asked to complete the double number line and, yes, you can go along and just continue that pattern of adding 30 each time, but it's useful to look between, looking at those gaps.

So that 60 is represented by two centimetres, we know that one centimetre represents 30.

So twice that, double that value, will be double the number of centimetres and we can see that going through.

What's the relationship between the three centimetres and the 90 centimetres? Well three multiplied by 30 is equal to 90.

We've got three times as many centimetres, we've got three times the number of centimetres in the actual car.

So just have a look through, check that you got the the right values and really think about that relationship between the number of centimetres in real life and the number of centimetres on the plan.

So now we've got a plan of part of a house.

So we're asked what is the scale for this plan? Gosh, where do we start here? Well I dunno where you started, but I started at the end of the number line.

So I looked at this here where I've got some whole numbers and I've got some numbers that look quite friendly to me.

So I can see here that 15 centimetres on the drawing is equal to five metres in real life.

I wonder what one metre in real life is worth then? Can I think where that's going to be? I'd need to divide by five.

So let's think about dividing by five.

Well 15 divided by five is three.

So if I can connect these two values on my double number line, I might be able to work out what one metre in real life is worth on the drawing.

So let's see.

So 15 divided by five is equal to three.

So 15 centimetres divided by five is three centimetres.

And if I divide my five metres by five as well, I will find out my value for one.

So three centimetres on the drawing is equal to one metre in real life.

So for every one metre in real life there are three centimetres on the drawing.

So I can say that that is my scale.

For every 15 centimetres on the plan there are five metres in real life.

For every three centimetres on the plan there is one metre in real life.

So I've been able to use that idea of dividing by five, division, to be able to scale my values to work out what one metre in real life is going to be represented by on the plan.

So can I use this information to fill in some gaps? At least I know what one metre is in real life and I can do some other calculations now to fill in those gaps.

So what other relationships can I see? Well, I can see here that three times two is equal to six.

So if three centimetres on the drawing was worth one metre in real life, if I double that then six centimetres on the drawing is going to be worth two metres in real life.

Okay, what else can I see? Well I can see I multiplied by three here.

Three multiplied by three is equal to nine.

So if three centimetres on the drawing was worth one metre, then multiply that by three, nine centimetres on the drawing is going to be worth three times as many metres.

So three metres in real life.

What else can I see? I've got a times four relationship there, haven't I? Three times four is 12.

So if I've got four times as many centimetres, I've got four times as many metres.

So four metres in real life.

What about five? That's quite an interesting one.

I wonder how I'm going to think about going to five.

Well, I wonder if I could work out what one centimetre on the drawing is worth.

So if I divide by three, three divided by three is equal to one, then I've then got to divide one metre by three.

Well one metre divided by three is going to give me 1/3 of a metre.

So to have a think about that.

I can go into fractions here.

I know that three centimetres on my drawing is worth one metre.

So if I make that three times smaller, divide by three, then one centimetre will be worth 1/3 of a metre in real life.

And now I can work my way up from my one centimetre to my five centimetres by multiplying by five.

And if I've got five times as many centimetres, I need five times as many metres.

So, oh gosh, five times 1/3, well five lots of 1/3 is 5/3, and 5/3 is the same as 1 2/3.

So if I have five centimetres on my drawing, I will have 5/3 or 1 2/3 of a metre in real life.

Thank you for your hard work today.

I hope you've begun to get an understanding of how scale allows large things to be represented on smaller maps and plans.

We've used that language of for every, sort of linking our work on scaling to our work on ratio as well and seeing how multiplication and division can help us to convert measurements in real life to measurements on a map and the other way round, measurements on a map to measurements in real life by scaling the values up and down.

Thank you again and I look forward to seeing you soon.

Bye.