Lesson video

Hiya, my name's Ms. Lambell.

I'm really pleased that you've decided to join me today to do some maths.

I'm really looking forward to working alongside you.

Welcome to today's lesson.

The title of today's lesson is "Describing More Conversions Using Ratio." This is within the unit "Understanding Multiplicative Relationships to Fractions and Ratio." By the end of this lesson, you'll be able to use ratio to describe conversions.

Two key words that we'll be using in today's lesson are proportion and ratio.

Here's a quick reminder, the definitions of those two things.

Variables are in proportion if they have a constant multiplicative relationship, and a ratio shows the relative sizes of two or more values, and allows you to compare a part with another part, or a whole.

I've divided today's lesson into two separate learning cycles.

In the first one, we're going to concentrate on looking and using map scales, and then in the second one, we will look at other units of measure.

Let's get started on that first one.

So like I said, we're gonna be concentrating on map scales.

Here are maps of two islands, with the main road shown.

Which road is the longest? So here are the two maps, and we want to know which road is the longest.

Aisha says, "Well that's obvious." Alex says, "I'm not sure it is." So Alex is not sure that it's obvious.

What do you think? Aisha wants to know why Alex says that he's not sure.

Let's see what Alex has got to say.

He says, "We don't know the scale of the maps." Now we've got a scale, we can see that the first one is every one centimetre is equal to 30 kilometres, and the scale of the second one is that one centimetre equals 18 kilometres.

Maps must have a scale.

If there is no scale, comparisons between them cannot be made.

The road on the map for island A is 2.

5 centimetres long, and the road on island B is four centimetres long.

Which has the longest road? Let's take a look at that.

Here we have our ratio table, and I've got my map on the left hand side, and I've got my real distance on the right hand side.

We use the scale given to us, which is one centimetre equals 30 kilometres, and then we know that on island A, the length of the line is 2.

5 centimetres.

So on the map it's 2.

5 centimetres.

We need to look for our multiplicative relationship.

So our multiplicative relationship between one and 2.

5 is multiplied by 2.

5.

So on the map we're multiplying by 2.

5.

This means in real life we are going to be multiplying by 2.

5.

30 multiplied by 2.

5 is 75.

It's going to be 75 kilometres.

Now let's look at map B, or island B, I should say.

This time, the scale is one centimetre equals 18 kilometres, and we know that the road is four centimetres long on the map.

Again, that multiplicative relationship between one and four, nice and easy, multiplied by four.

So we need to multiply the 18 by four, giving us 72.

Now we can see that actually, island A has the longer road.

So even though it looked shorter, because of the scale of the map, it actually was longer.

We can now answer the question, "The road on island A is the longest." A road on a map measures 10 centimetres long, and it represents 6,000 metres.

What is the scale of the map? Let's have a look.

So we've got our map and our real.

10 centimetres equals 6,000 metres.

We want to know the scale.

Now on a map scale, we will want the scale showing us one to something, and in this case, we're gonna be doing one centimetre.

What is my relationship between 10 and one? My multiplicative relationship between 10 and one is I've multiplied by one 10th.

I need to do the same on this side then, with the real distances.

6,000 multiplied by one 10th is 600.

The scale is one centimetre to 600 metres.

Why can't we just write one to 600? Yeah, the reason is, is that the units are different, so I must have those units there.

Otherwise it's going to look like one centimetre equals 600 centimetres.

Often though, scales do not contain any units.

So if you look at a map, often you'll see no units.

What would be the scale in the form one to something, so without the units? Let's take a look.

We've got our map and we've got our real.

We know that it's one to 600.

We've just worked that out.

We need to convert so that the units are the same.

We know that one metre is equivalent to a hundred centimetres, so let's work out what 600 metres is in centimetres.

So my multiplicative relationship here is multiplied by 600.

I'm going to do the same to the centimetres.

So it's 60,000.

The scale would be one to 60,000.

There is no need now for units, because we've got both of them in centimetres.

A road on a map is five centimetres long, and that represents 2,500 metres.

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

2,500 metres in centimetres.

We know that one metre is equivalent to a hundred centimetres.

My multiplicative relationship, notice here, I'm going from the bottom to the top, that's absolutely fine, is multiplied by 2,500.

I'm going to do the same to the centimetres and I get 250,000.

Now we know that on the map it's five centimetres, and we know what it is in real life in centimetres.

Now we can work out the scale.

My multiplier that takes me from five to one is one fifth.

I'm going to multiply 250,000 by one fifth, to give me 50,000.

The scale is one to 50,000.

Let's just recap what we did.

We needed to change, firstly, 2,500 metres into centimetres, and then we could work out what that scale was.

I'd like you here please, to match each of the following to the correct scale.

Pause the video, and when you are ready, come back and we'll check those answers.

Good luck with these.

Well done.

Let's check those answers.

First one, three centimetres equals 15 metres was one to 500.

Six centimetres equals three kilometres was one to 50,000.

Five centimetres equals 250 metres was one to 5,000.

And then the final one, four centimetres equals 20 kilometres was one to 500,000.

How did you get on with those? You got them all right? Brilliant, well done.

The scale on a map is one to 50,000.

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

What is the distance in real life, in kilometres? So we're now going to use the scale.

We've looked at finding out what scale is.

Now we're going to use it.

Here's my ratio table.

I've got one to 50,000.

Seven centimetres is on the map, so that's gonna go on the left hand side.

My multiplier is seven, so I get 350,000.

What are the units of the 350,000? They have to be the same as the units that we started with.

So this is centimetres, it's 350,000 centimetres.

The question, however, asked us to give the distance in kilometres, so we now need to convert this into kilometres.

Let's take a look at how we're gonna do that.

You've done this previously.

So a quick recap on how we do this.

We can use our place value table.

350,000 centimetres.

So centimetres is where I'm starting.

Let's pop in our decimal point, and then we're going to put in our 350,000, remembering to start from the decimal point.

So I've got four zeros, five, and then three.

We want to convert that into kilometres.

So we're looking at kilometres now.

So I need to put my decimal point in, and then put my digits in.

I don't need to put those trailing zeros in, because 3.

50000 is the same as 3.

5.

So the answer was 3.

5 kilometres.

Let's have another go at one together, and then I know that you'll be ready to try one of these independently.

The scale on a map is one to 300,000.

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

What is the distance in real life, in kilometres? We start with our scale, and we start with our distance on the map, which is four.

Multiplier here is four.

So we multiply by four.

We get 1,200,000.

but we need to convert that.

Remember that's centimetres.

So we need to convert that into kilometres, and that is 12 kilometres.

Now, I'd like you please to have a go at this one.

The scale on the map is one to 500,000.

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

What is the distance in real life, in kilometres? Pause the video now, have a go, and when you're ready, come back and we'll check that answer for you.

Great work.

Let's check your answer.

One to 500,000.

We've got three.

So my multiplier was three.

I'll get 1,500,000, which is 15 kilometres.

We've now used the scale to find a distance in real life.

Now we're going to look at what happens if we know the distance in real life, what that's going to be on the map.

So our scale is one to 50,000.

The distance between two places is 5.

3 kilometres.

What is the distance on the map? Firstly, we need to convert 5.

3 kilometres into centimetres.

5.

3 kilometres, so we have kilometres, put our decimal point, 5.

3.

Then we want centimetres, so pop in your decimal point.

53, and this time we do need those placeholders.

Okay, 'cause they're not after the decimal point.

We now know that it is 530,000 centimetres.

Now we can work out how far it's going to be on the map.

So my scale was one to 50,000.

This time, I knew the real distance, which was 530,000 centimetres.

We need to look for that multiplicative relationship.

Remember, if it's not obvious, you can divide 530,000 by 50,000.

We end up with 10.

6.

So here I'm gonna multiply it by 10.

6.

So the distance on the map is going to be 10.

6 centimetres.

The scale on the map is one to 300,000.

The distance between two places in real life is 4.

5 kilometres.

What is the distance between them on the map? I've already converted 4.

5 kilometres into centimetres.

I need to make sure that that is on the real life side of my ratio table.

I'm looking for that multiplicative relationship.

Remember, if it's not obvious, you can do that division, and that's multiplied by 1.

5.

So the answer is 1.

5 centimetres.

I'd like you now to have a go at this one.

So the same scale, but this time the distance is 7.

2 kilometres.

Pause the video, good luck, come back when you're ready.

How did you get on with that one? Let's check.

7.

2 kilometres is 720,000 centimetres.

We look for that multiplicative relationship, which is multiplied by 2.

4.

So we now know that it's 2.

4 centimetres.

You are now gonna have a go at this task.

You're going to complete the missing information in each row.

So here you're gonna do all of the three things that we've just been through.

You're gonna find the scale sometimes, sometimes you're gonna find the distance on the map, and sometimes you're gonna find the distance in real life.

And if I've asked you to find the distance in real life, I've given you the units that I would like your answer in.

Of course, here you can use a calculator.

Good luck with this, and I look forward to seeing you when you come back.

Let's check those answers.

The missing value in the first row was 15.

15 kilometres.

The second row was one to 2000.

The third row was 50 centimetres.

Fourth row was 180 metres.

The fifth row was one to 300,000.

The sixth row was 15 centimetres.

The seventh row, 2.

5 centimetres.

The eighth row was one to 3,500,000.

And the final row was 0.

39 kilometres.

How did you get on with those? We are now ready to move on to the next learning cycle.

So in this learning cycle, we're going to be looking at different units of measure.

The approximate equivalent of inches and centimetres is that one inch equals 2.

54 centimetres.

Aisha is talking to her dad about his favourite teddy bear from when he was little.

He remembers that it was 17 inches tall.

Aisha measures her teddy, and it's 45 centimetres tall.

Whose teddy is the tallest? Let's take a look at how we're going to answer this question.

The first thing we need to consider, is will it be easier to convert Aisha's teddy's height or her dad's? It's going to be easier to convert her dad's from inches into centimetres.

And that's because we know what one inch is.

We could convert it the other way, but actually, it's easier if we convert from inches into centimetres.

Let's take a look.

The ratio table, with the equivalence in the top row.

One inch is 2.

54 centimetres.

We knew that Aisha's dad's teddy was 17 inches, so that's gonna go in the inches column, and then we'll again look in that multiplicative relationship, which is multiplied by 17.

So I multiply it by 17.

I get 43.

18.

Aisha's dad's teddy was 43.

18 centimetres.

Aisha's teddy was 45 centimetres.

Now we can answer the question.

Aisha's teddy was taller.

Not much taller, but Aisha's teddy was slightly taller.

We've got Alex now.

Let's see what Alex is up to.

I know that the distance between two villages is five miles, and this is equivalent to eight kilometres.

The distance between Lands End and John 'o Groats, as the crow flies, so that means if I could just fly directly there, so not going on roads, is 601 miles.

How far is this in kilometres? And Alex says, "I have to work out how far one mile is, so that I can convert this." Do you agree with Alex? You could, but you don't have to.

So we're gonna have a look at how we could do that without knowing what one mile is, in terms of kilometres.

The distance between them was 601.

How far is it in kilometres? This is the question we are answering.

I know that five miles is equivalent to eight kilometres, and I know that as the crow flies, the distance is 601 miles.

I'm looking then for that multiplicative relationship.

Remember, if you're not sure you can divide, and here I've not bothered to work out what that is.

I've just literally written it as a fraction.

So I'm going to do the same on this side, and I get 961.

6.

So the distance between Lands End and John 'o Groats as the crow flies is 961.

6 kilometres.

Aisha and Alex completed a bike ride at the weekend.

Aisha says she rode four miles.

Alex said he rode six kilometres.

We want to know who rode the furthest.

So at the moment, it's difficult to tell, because they are in different units.

So we're going to convert them so they're in the same unit.

And I've decided to convert my miles into kilometres, or Aisha's four miles into kilometres.

Using my equivalent of five miles is eight kilometres, this time I want to find out what four miles is.

My multiplicative relationship between five and four is 0.

8.

Therefore I need to multiply the kilometres by 0.

8, also, which is 6.

4.

Now we can see who rode furthest.

Aisha rode furthest, she rode 6.

4 kilometres, and Alex rode six kilometres.

So Aisha rode the furthest.

If two things have a multiplicative relationship, you can find any pair of equivalent values.

"Zings" and "zangs", I've made them up, have a multiplicative relationship.

What is the missing number of zangs? We know that the relationship is this.

Three zings is equivalent to 18 zangs.

We want to know what 20 zings is, in terms of zangs.

"The multiplier for 30 to 20 is a fraction," Aisha says.

And Alex says, "But the multiplier from three to 18 is an integer.

Can we use the multiplicative relationship between the zings and the zangs?" Yes, of course we can, because we know there's a multiplicative relationship we can work between just the zings, or we can work between the zings and the zangs.

Here it may be easier to find the multiplicative relationship between three and eight, which is multiplied by six.

So to go from zings to zangs, I multiply by six, so I get 120.

It's always worth looking to see whether the multiplicative relationship is easier moving horizontally rather than vertically, and vice versa.

What I'd like you to do now is to decide, in each of these following ratio tables, if it is easier to use the multiplicative relationship horizontally, so moving across, or vertically, moving down.

Pause the video, have a go at these.

When you're ready, I'll be here waiting to check all your wonderful answers.

Good luck.

Super.

Let's have a look and see how you got on.

So the first one, definitely be easier to work horizontally.

The multiplier from five to 15 is a nice integer.

It's three.

The second one would be better if we moved vertically, because the multiplier between six and 36 is an integer.

The third one, again, vertically.

Fourth one would be horizontally.

Horizontally for the fifth one.

And then finally, actually the last one, we could go horizontally.

My multiplier there is nine.

Or vertically, my multiplier there is 11.

Always worth, like I said, having a quick check as to see which is going to make your life just a little bit easier.

We're gonna end up with the same answer either way, so it doesn't matter.

It's just sometimes making our life easier sounds like a good thing.

We're gonna find the missing value.

So here I can see that my multiplicative relationship between five and 600 is going to be an integer, because 600 is a multiple of five.

Five multiplied by 120 is 600.

So moving from the left to the right, in the ratio table, I'm gonna be multiplying by 120, giving me 3840.

You are now gonna have a go at this one, so you can pause the video, and then come back when you're ready.

Remember to look and decide which is the most efficient way of finding that missing value.

Good luck.

You can pause the video now.

Great work.

Let's check your answer.

Hopefully you decided that it was easier to go from 12 to 48, because my multiplier is four, and so my answer was 28.

You could have gone from 12 to seven, but it would've just meant that you ended up with a fractional multiplier.

Now you're ready to have a go at these questions.

So what I'd like you to do is to find the missing value in each of the following ratio tables.

You can pause the video, and then when you're done, come back and we'll check those answers for you.

Super work.

Now for question two and three, question two, I'd like you to write the following order from the lightest to the heaviest.

The images are not to scale, okay? So that piece of cheese is not the same size as the brick.

Using the conversion that one kilogramme is equal to 2.

2 pounds.

So you're going to convert them all into the same unit so that you can write them in order from lightest to heaviest.

And in question three, I would like you to write them in order from shortest to tallest, using the conversion one inch equals 2.

54 centimetres.

You can pause the video now.

Good luck with these, and I'll be here waiting when you get back.

Great work on those.

Now for question four.

Write the following in order from the closest to the furthest from this sign, using the conversion that five miles is equal to eight kilometres.

Pause the video, and as always, I'll be here ready to check your answers when you get back.

Let's check those answers then.

The first one, the missing numbers, I'll go across.

We're at 84.

30, 40, and then on the bottom row, 6270, 40, and 297.

Question number two, in the correct order, was cheese, sack, the brick, and the books.

And the correct order for three was feather, then the teddy bear, and the flour, and then the robot.

Obviously there, I'd like to see all of your working out, not just a list that you may have guessed.

But you wouldn't do that anyway.

And question number four, the correct order was town B, town A, town D, and then town C was the furthest away from the sign.

Now we can summarise the learning that we've done during today's lesson.

Firstly, we looked at map scales, and we looked at the fact that sometimes they may include units.

So for example, one centimetre equals 18 kilometres.

But more commonly, they are expressed as a ratio with no units.

So for example, one to 60,000.

For every one centimetre, that is 60,000 centimetres in real life.

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

And there's an example there.

A ratio table is also a really good way of converting between other units.

Well done on today's lesson.

You've worked really, really hard.

I've really enjoyed working alongside you.

I look forward to seeing you again really soon.

Goodbye.