Lesson video

Hi there.

My name's Ms. Lambell.

You've made a superb choice deciding to join me today to do some maths.

Let's get going.

Welcome to today's lesson.

The title of today's lesson is "Checking and Securing Understanding of Scaled Drawings," and that's within the unit Bearings.

By the end of this lesson, you'll be able to interpret scaled drawings in a variety of contexts.

Some keywords that we'll be using in today's lesson are proportion and ratio.

You should be familiar with these, but it's always worth a recap.

Variables are in proportion if they have a constant multiplicative relationship.

A ratio shows the relative sizes of two or more values and allows you to compare a part with another part in a whole.

Today's lesson, I've split into two separate learning cycles for you.

In the first one, we will just concentrate on interpreting scales.

And in the second one, we will look at actually then using those scales.

Let's get going with that first one, interpreting scales.

Here are the maps of two islands.

The arrows show the widths of the widest part of the two islands.

Which island is wider? Jun says, "Well, that is obvious.

The arrow on the right is longer, so the island on the right is wider." Do you agree with Jun? And I'd also like you to explain your answer.

What did you decide? Well, hopefully you said that you didn't agree with Jun.

We cannot be sure as we do not know the scales of the maps are the same.

We've not been given a scale on either of these maps and so the scales could be different and therefore we can't make comparisons.

Now I've given you the scale of each map.

The scale of the left-hand map is one centimetre equals 25 kilometres, and the scale of the right hand map is one centimetre equals 20 kilometres.

Maps must have a scale so that we can interpret them.

If there is no scale, then we are not able to make comparisons between them.

We need a scale to make sure that we can make comparisons.

I've measured the width of the map for island A, and it's four centimetres, and I've also measured the width of the map for island B, and that's five centimetres.

Which is the widest island? I'm gonna give you a moment to think about this because remember this is a check-in and securing lesson, so it is something you should be familiar with.

Let's go through it then.

Here's my ratio table from island A.

I've got my map on the left-hand side and my real on the right hand side.

Notice I've also included the units so it's obvious what units I'm working with.

So the scale of island A is one centimetre equals 25 kilometres.

I've measured the map as four centimetres, so the map is four centimetres.

I'm looking for that multiplicative relationship between one and four, and that's multiplied by four.

Therefore, I need to multiply the real length in kilometres by four.

25 multiplied by four is 100.

I wonder if that's what you got.

Of course you did.

Let's now look at the island B.

Here's my ratio table for island B.

So I start with my scale.

One centimetre equals 20 kilometres.

And on the map, I measured five centimetres.

Let's look for that multiplicative relationship between one and five.

That's multiply by five, so I need to multiply the real length by five also and I get 100.

So we can see that actually both of the islands are the same width.

So until I had that scale I was not able to compare the widths of each of those islands.

We are more likely to see scales written as a ratio, and you'll probably be familiar with this.

Let's see what Laura's got to say.

"One centimetre equals 25 kilometres would be 1:25 when written as a ratio." Do you agree with Laura? As always, I like you, please, to explain your answer.

Did you agree with Laura? Laura, unfortunately, is incorrect.

If there are no units in the ratio, then they must be the same unit.

1:25 means one centimetre equals 25 centimetres, not 25 kilometres.

Have a go at this check for understanding.

Which of the following is the correct ratio of one centimetre equals 25 kilometres.

Pause the video, work out your answer, and then come back and join me when you're done.

What did you come up with, A, B or C? Well, the correct answer was C, and let's take a look why.

Here I've got my place value table.

I know that I'm working with kilometres.

My original ratio was kilometres and I need to convert this into centimetres 'cause my one is in centimetres.

And remember, if there's no units, they need to be the same units.

Effectively what I need to do is to convert 25 kilometres into centimetres.

I've highlighted my kilometres.

I'm gonna add my decimal point, and I'm gonna put in my 25 kilometres.

We want to convert that to centimetres.

So let's highlight the centimetre column and put my decimal point.

Now, remember the two and the five must stay in the same columns.

We can't change their columns 'cause otherwise we're changing the value of the 25 kilometres.

So I've got my two and my five, and then I need to make sure I've got my placeholders, and we can now see that 1:2,500,000 is the equivalent ratio of one centimetre equals 25 kilometres.

We can now see that this is the correct unitary ratio equivalent to one centimetre equals 25 kilometres.

Did you get C? Of course you did.

The scale on the map is 1:50,000.

The distance between two places on the map is nine centimetres.

What is the distance in real life in kilometres? Let's take a look at this one Again, here's my ratio table.

I've got my map to real.

Notice this time I don't need to put the units because the units are the same in both.

My ratio or my scale is 1:50,000 and I know that the distance between the two places on the map is nine centimetres.

So we're looking for that multiplicative relationship.

Multiply by nine.

Multiply by nine, we get 450,000.

It is 450,000 centimetres.

Ah, have you got any idea how long that is? 'Cause, currently, I haven't really got an idea of how long it would take me to get there.

Let's have a look at something more useful.

It would be more useful if we had this as kilometres because then we could make a judgement about how long it might take to walk there or to drive there.

Again my place value table, I'm starting this time with centimetres, so we put our decimal point in, and then we are putting in our digits into the correct columns.

I want to convert this to kilometres.

So let's highlight the kilometres and add in our decimal point, and then we can see that the answer is 4.

5 kilometres.

Right, I've got a sense now of how far it is between those two places, 4.

5 kilometres, and I could definitely walk that far.

The scale on a map is 1:300,000.

The distance between two places on the map is five centimetres What is the distance in real life in kilometres? So we're going to do this one together, and then I know you're going to be ready to do one independently, the one on the right hand side.

Start off with your ratio table, with your scale written in, We know it is five centimetres on the map.

The map is on the left-hand side of my ratio table.

That multiplicative relationship here, I can hear you all shouting, multiply by five, 300,000 multiplied by five is 1,500,000, and that is 15 kilometres.

And if you need to there, remember you could use your place value table to make that conversion.

Now it's your turn.

The scale on the map is 1:500,000.

The distance between two places on the map is nine centimetres.

What is the distance in real life in kilometres? Pause the video and have a go.

Come back when you're done.

You can use a calculator for this question if you want to.

I'll be here waiting when you get back.

Like I said, pause the video now.

Starting with our ratio table with the scale 1:500,000 and we know that the map distance is nine centimetres.

Looking for that multiplicative relationship, multiplied by nine.

That's 4,500,000 which is 45 kilometres.

How did you get on? Well done.

You're ready now then for task A.

You are going to complete the missing information in each row of the table.

So pause the video and then come back when you're done.

Well done.

Let's check your answers.

I'm going to go down the column and read out the map diagram ones first.

So the missing values, well, 1.

5 centimetres, five centimetres, 12 centimetres, 3.

5 centimetres, and six centimetres.

And now going down the column for the real only giving you the missing ones.

We've got 15 kilometres, 120 metres, 21 kilometres, and 0.

315 kilometres.

How did you get on? Super, well done.

Now we can move on to that second learning cycle.

We're going to be looking at using scales.

Let's go.

A road on a map measures 10 centimetres and represents 6,000 metres.

What is the scale of the map? This time, we know that 10 centimetres is 6,000 metres, and we want the scale of the map.

When we have a scale, we want a unitary ratio.

So that's why I've put one in the map column.

We're looking for that multiplicative relationship.

We're multiplying by 1/10.

we're multiplying by 1/10, which is 600.

Don't forget also you could divide by 10 here, divide by 10 rather than multiplying by 1/10.

whichever you feel most confident with.

So we've got 600 metres.

The scale is one centimetre to 600 metres.

Why can't we just write 1:600? And the reason is 'cause the units are different.

If we wrote 1:600, that would mean one centimetre is 600 centimetres, not 600 metres.

Often scales do not contain any units as we've already mentioned.

What would the scale be in the form one to something without units? Here's the scale we ended up with 1:600.

I need to convert 600 metres into centimetres.

I know that one metre is a hundred centimetres, so 600 metres, my multiplicative relationship is multiplied by 600, so I multiply by 600, giving me 60,000.

I can now write my scale without the unit.

It's 1:60,000.

A road on a map measures five centimetres and represents 2,500 metres.

What is the scale of the map? The first thing we need to do is to work out 2,500 metres in centimetres.

2,500 metres in centimetres, well, we know that one metre is equivalent to 100 centimetres.

Looking for that multiplicative relationship.

Notice I'm going from the bottom to the top this time.

It doesn't matter which way round I decide to use that multiplicative relationship as long as I go in the same direction both times.

So I'm multiplying by 2,500, giving me 250,000.

I now know that five centimetres in on the map is 250,000 centimetres in real life.

So I'm converting now.

Now I'm looking for that multiplicative relationship.

So multiplied by a fifth or divide by five, whichever you prefer, and I end up with 50,000.

My scale is 1:50,000.

So there were two different ways of doing it.

You can either convert first so that you've got the same units or you can do that conversion at the end.

Again, it's whatever you feel most comfortable with.

Have a go at this check for understanding.

You need to match each of the scales on the left-hand side to the correct ratio.

Pause the video and come back when you are done.

Super work, well done.

So three centimetres equals 15 metres is 1:500.

Six centimetres equals three kilometres is 1:50,000.

Five centimetres equals 250 metres is 1:5,000.

And then, obviously, the final one, four centimetres to 20 kilometres, is 1:500,000.

How'd you get on? Well done.

The scale on a map is 1:50,000.

The distance between two places is 5.

3 kilometres.

What is the distance on the map? So this time we're going back the other way from the real distance to the distance on the map.

Firstly we need to convert 5.

3 kilometres into centimetres.

And again, we can do this with our place value chart if we need to.

So I'm starting with kilometres, and I'm put my decimal point, and then I've got 5.

3.

I'm converting it into centimetres.

So we place our decimal point.

Remember the digits need to stay in the same columns, and then my placeholders.

It is 530,000 centimetres.

Now I can work out the distance on the map.

I've got my scale 1:50,000 and I know in centimetres that the real distance is 530,000.

So I'm looking for that multiplicative relationship.

That's not obvious, is it? So remember what we do here is we divide 530,000 by 50,000, and that gives us 10.

6.

one multiplied by 10.

6 is 10.

6.

It will be 10.

6 centimetres on the map.

Remember you can check your answers using the multiplicative relationship between the map and the real distance.

What is my multiplicative relationship between one and 50,000 that's multiplied by 50,000? This time my arrow is going in the opposite direction.

What do I have to do when my arrow's going in the opposite direction? Yeah, I have to do the inverse operation.

So I'm going to divide by 50,000, and that means I get 10.

6.

Remember you multiplicative relationships can either be vertically or horizontally, and I would always encourage you to do it one way and then check using the other.

Let's have a go at this one together, and then you can have a go at the one on the right hand side independently.

The scale on a map is 1:300,000.

The distance between two places in real life is 7.

5 kilometres.

What is the distance between them on the map? I've got 1:300,000.

Now I've converted my 7.

5 kilometres into 750,000 centimetres.

You may need to use your place value chart to do this.

Remember that's absolutely fine, but you may now be confident with just converting between the two.

I'm looking at that multiplicative relationship, again, not obvious.

So I'm going to do 750,000 divided by 300,000, which is 2.

5.

One multiplied by 2.

5 is 2.

5.

On the map, the distance between the two places is going to be 2.

5 centimetres.

Your turn now.

The scale on a map is 1:400,000.

The distance between two places in real life is 7.

2 kilometres.

What is the distance between them on the map? Pause the video, have a go, and then when you're done, come back.

Super work, well done.

Let's check that answer.

Hopefully you converted your 7.

2 kilometres into 720,000 centimetres.

Looking for that multiplicative relationship, it was multiplied by 1.

8.

One multiplied by 1.

8 is 1.

8.

So on the map, the distance is going to be 1.

8 centimetres.

And you got that, didn't you? Yes, of course you did.

This is a scale model of a car.

The scale is 1:43.

What is the real length of the car.

Right, in order to answer this question, we need to know the length of the car in centimetres.

You tell me.

What is the length of the car in centimetres? It's seven centimetres.

Use the scale to find the real length of the car.

We've got 1:43 is the scale, and the length of the model car is seven centimetres.

We're looking for that multiplicative relationship and I've decided to move vertically.

I could do one multiplied by 43.

So multiplying by 7, 43 multiplied by seven is 301.

The real length of the car is 301 centimetres or you would probably more likely refer to the length of the car in metres, so it's 3.

01 metres.

What about the real height of the car? What is the height of the car in centimetres? Now you tell me.

The height is 3.

2 centimetres.

This time I'm gonna ask you to do this yourself before I go through it.

I'd like you to use the scale to find the real height of the car.

Pause the video and come back when you can tell me how tall a real Mini is.

Let's take a look.

So scale hasn't changed.

It's still a scale of 1:43, and we know that the height is 3.

2 centimetres.

My multiplicative relationship is multiplied by 3.

2.

43 multiplied by 3.

2 is 137.

6.

The real height of the car is 138 centimetres, and I've rounded that to the nearest centimetre.

Or, like I said before, probably, the dimensions of car would be given in metres, so it would be 1.

38 metres.

And again, that's to the nearest centimetre.

We cannot measure the width of the model from this picture, but I know that the width of the car in real life is 1.

42 metres.

What is the width of the model? Use the scale to find the model's width.

Again, I'm gonna ask you to pause the video and give this a go yourself first.

Here's my ratio table, and hopefully you spotted this time that the 1.

42 was the real distance, so it needed to go on the right hand side of the ratio table.

My multiplicative relationship is multiplied by 71 over 2,150.

How did I get that? I did 1.

42 divided by 43.

That's my multiplicative relationship.

Is that the correct multiplier? Was that the multiplier that you got? This is the correct multiplier for this ratio table, but, unfortunately, it's not the correct multiplier to solve the problem.

What mistake have I made? Scale means one centimetre equals 43 centimetres, therefore we need to convert 1.

42 metres to centimetres to be able to solve the problem.

What I should have written was 43, sorry, one to 43 and 142.

Really important that those units are the same.

Now I can look for that multiplicative relationship.

Multiply by 142 over 43, and that is 3.

3.

The width of the model is 3.

3 centimetres and that's given to one decimal place.

Laura and Jun are working on a project where they need to make a scale drawing of a bedroom, cut out a rectangle of card to represent the bed and place it into the room.

The dimensions of the bedroom are 2.

5 metres by 3.

5 metres.

The dimensions of the bed are 190 centimetres by 90 centimetres.

Laura draws the plan of the bedroom and Jun cuts the template of the bed.

Here is Laura's plan of the bedroom, and here is the template of the bed.

Laura says, "This cannot be right." Why might Laura say it is not right? The width of the bed is less than half of the width of the room.

If we look at the width of the room, it's 2.

5 metres, which is 250 centimetres, and 90 is less than half of that, but we can see on the drawing that actually it's more than half of the width.

What might have happened? They did not agree on a scale they were going to use before drawing the plan and the template.

They've used two different scales.

Really important here, they should have decided on the scale they were going to use first.

What scale did Laura use to draw the plan of the room? The plan is drawn on centimetre squared paper.

Let's take a look.

The plan is five centimetres equals 250 centimetres in real life.

The width of the room was 2.

5 metres and that was represented by five squares on the grid.

So now know that five centimetres is equal to 250 centimetres.

Here, I've divided by five.

So I decided here to do divide rather than multiply.

I could multiply by 1/5 remember.

250 divided by five is 50.

A scale was 1:50.

What should the dimensions be for the template of the bed? Well, let's start with the width.

We now know that our scale is 1:50 and we know that the bed was 90 centimetres wide.

We're looking for that multiplicative relationship, which is multiplied by 1.

8.

One multiplied by 1.

8 is 1.

8.

The width of the bed should have been 1.

8 centimetres.

Let's make the width of the bed 1.

8 centimetres.

And now let's look at the length.

Here's my scale, 1:50, and I know that the length of the bed is 190 centimetres, so I'm looking for that multiplicative relationship, which here is 3.

8.

One multiplied by 3.

8 is 3.

8.

The length of the bed should be 3.

8 centimetres.

Now let's make the length of the bed 3.

8 centimetres.

And Laura says, "That looks better." So it's really important if we're gonna be combining things that we use the same scale, which of the following have used a different scale? I'd like you to pause the video, make your decision, and then join me when you're done.

What did you get? A, we were multiplying by five.

3.

2 multiplied by five is 16.

So the scale was 1:5.

The second one, 10.

5 multiplied by five is 52.

5.

Again, the scale was 1:5.

The next one, nine multiplied by 4.

8 equals 43.

2, so the scale was 1:4.

8.

And then the final one, 0.

7 multiplied by five is 3.

5.

Given a scale of 1:5, we can now clearly see that C was the one that had used a different scale.

And of course I know you've got that.

Finally, we can move on to task B.

Question number one.

I'd like you to write the map scale of the following as a simplified ratio in the form of map to real life.

Pause the video and then when you're done, you can come back.

Well done.

And question number two.

A map has a scale of three centimetres to 15 kilometres.

The distance between two towns in real life is 42 kilometres.

What is the distance on the map between these two times? And I'd like you please to give your answer in centimetres.

Again, pause the video and come back when you're done.

Great work.

And question number three.

Jun and Laura placed their rulers on a map.

The map scale is 1:625,000.

What is the approximate real life distance in kilometres between Cheltenham and Oxford? Pause the video, and then we've got your answer.

You can come back.

And then, finally, for today's lesson, your final question is, what is the approximate real distance in kilometres between Swindon and Reading.

super work.

Let's check those answers.

Question number one.

A, the simplified ratio is 1:500,000.

B, The simplified ratio is 1:200.

Question two, one option is to correctly write the ratios and then convert from centimetres and kilometres.

Three centimetres is equal to 1,500,000 centimetres, and then 42 kilometres is 4,200,000 centimetres.

And then you can see my table there, so the distance would've been 8.

4 centimetres.

Alternatively, you could have labelled the units in the ratio table.

And personally, I think that that one's a lot easier as we were given the scale three centimetres to 15 kilometres.

Question three, part A, the approximate distance between Cheltenham and Oxford is 60 kilometres.

And you can see my working out there if you need to pause the video and take a look at that.

And then for part B, Swindon and Reading is 58.

125 kilometres approximately.

Let's summarise our learning from today's lesson.

Map scales can be represented with a scale including units.

So for example, one centimetre equals 18 kilometres.

More commonly, they are represented as a ratio with no units.

So for example, 1:60,000.

A ratio table is a useful way to convert between scales.

And remember, a really good thing to do is to check your answers.

So use the multiplicative relationship maybe vertically, and then check with the horizontal or the other way round.

Well done with your learning today.

You've done fantastically well.

I hope to see you again really soon.

Take care of yourself.

goodbye.