Lesson video

Hi there.

My name is Ms. Lambell.

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

Let's get cracking.

Welcome to today's lesson.

The title of today's lesson is Compound Measures for Density, and this is within the unit, Compound Measures.

By the end of this lesson, you'll be able to use compound measures for density.

Keywords that we'll be using in today's lesson are density and mass.

The density of a substance is measured as the substances mass per unit of volume.

The mass of a substance is a measure of the quantity of matter in the substance.

Today's lesson is in two learning cycles.

In the first one, we will look at calculating density, and in the second one we will broaden that out looking at solving problems involving density.

We'll get going on that first one, and we'll look at calculating density.

The density of gold is 19.

3 grams/centimeters cubed.

The dimensions of a gold bar in the shape of a cuboid are 25 centimetres by seven centimetres by 3.

5 centimetres.

What is the mass of the gold bar? Sam says, "Hi Izzy.

I came across this question when I was doing some revision.

I've not been taught anything about density yet.

Have you any idea how to answer this question?" Izzy says, "I have not been taught about density either.

I am sure we can work this out if we think about what the unit means." I like that Izzy, thinking about what that unit means.

Sam says, "Well, km/h means kilometres per hour, so this must mean grammes per cubic centimetre.

Do you agree Izzy?" Izzy says, "Yes, it must mean that." So if we work out how many cubic centimetres the gold bar is, we can find its mass." Calculate the volume of a cuboid.

I know you know this.

That's right, volume of a cuboid is at length multiplied by its width multiplied by its height.

I'd like you now and you may use a calculator to calculate please, the volume of the gold bar.

Don't forget to give the units of your answer And you should have 25 multiplied by seven multiplied by 3.

5, and that is 612.

5 centimetres cubed or cubic centimetres.

Sam says the volume of the gold bar is 612.

5 centimetres cubed, Izzy says, "And each cubic centimetre has a mass of 19.

3 grammes.

We can draw then a ratio table.

We know that each centimetre cubed has a mass of 19.

3 grammes.

We also know the volume of the gold bar, because we've just calculated that, we're missing one piece of information from our ratio table.

We can find it.

The multiplicative relationship multiplied by 612.

5.

I need to do that to the mass, giving me an answer of 11821.

25.

The mass of the gold bar is 11821.

25, don't forget grammes, we're talking about its mass, and because the units of measurement for density were grammes per cubic centimetre, my mass must be in grammes.

Sam's now saying this must mean that we can work out the density of an object if we know it's mass and volume.

Izzy says, "Yes, you are right, Sam." The volume of a cannonball made of iron is 700 centimetres cubed.

The mass of the cannonball is 5,500 grammes.

What is the density of iron? Sam says, "I think we need to find the mass per cubic centimetre," and Izzy agrees.

We know that the mass of the cannonball is 5,500 grammes, and we know its volume is 700 centimetres cubed.

We need to work out the mass per cubic centimetre, so one cubic centimetre I've divided by 700.

I need to do that to the mass, giving me 7.

86.

The density of iron is 7.

86 grammes per cubic centimetre.

Sam says, "I saw a video clip at the weekend.

It showed an iron cannibal floating in mercury.

Izzy's questioning that.

Surely not.

How can something that heavy float in a liquid, hmm? How can an iron cannibal float in mercury? Hmm.

Well, let's take a look.

25 cubic centimetres of mercury has a mass of 340 grammes.

We're gonna calculate the density of mercury.

We know that 340 grammes has a volume of 25 cubic centimetres, and the density we will be measuring in grammes per cubic centimetre with dividing by 25.

340 divided by 25 is 13.

6.

The density of mercury is 13.

6 grammes per cubic centimetre.

We already worked out that the density of iron is 7.

86 grammes per cubic centimetre as the density of mercury is greater than the density of iron, the iron cannonball floats.

Wow, still messes with my head a bit.

I think I'll have to go and watch that video that Sam's talking about.

The density of gold is 19.

3 grammes per cubic centimetre.

A gold ring is taken to a jeweller to see if it is real gold.

The mass of the ring is 4.

8 grammes, and its volume is 0.

3 centimetres cubed.

Is the ring real gold? Let's set ourself up with a ratio table.

We know that the mass of the ring is 4.

8 grammes, and the volume is 0.

3 cubic centimetres.

Density, we're going to be measuring in grammes per cubic centimetre, so we need to find how many grammes one cubic centimetre is, so I'm going to divide by 0.

3.

Divide by 0.

3 gives me 16.

The density of the ring is 16 grammes per cubic centimetre, which is lower than the density of gold.

Therefore, that ring is a fake.

Your turn now.

I'd like you to match each with its correct density please, pause the video, and then come back when you're ready.

Let's check in with those answers.

The first one is 25 grammes per cubic centimetre.

The second one is 24 grammes per cubic centimetre.

The third one is 270 kilogrammes per cubic metre, and the final one is 240 kilogrammes per cubic metre.

Notice here we've introduced another unit of density, but I'm sure that didn't put you off.

Now, we're ready to do task A, or you are ready to do task A should I say, calculate the density of each of the following metals, and then I'd like you to rank them in order from the least to the most dense.

Pause the video, you may use a calculator, but remember to show me all steps of your working.

Good luck with these, and when you're ready, come back and we'll move on to the next question.

Question number two, the density of platinum is 21.

45 grammes per cubic centimetre.

A necklace is taken to a jeweller to check if it is a genuine platinum necklace.

The mass of the necklace is 12.

87 grammes and its volume is 0.

6 cubic centimetres.

Is the necklace genuine? Draw a ratio table to support your answer, pause the video and come back when you're ready.

And question number three, Izzy finds three different planks of wood in her shed.

She weighs and measures them, which plank of wood has the lowest density? Again, draw your ratio tables to support your answer.

You can pause the video, and I'll be waiting when you get back.

Let's check those answers then.

Copper is 8.

96 grammes per cubic centimetre, titanium 4.

51, sodium 0.

97, zinc 7.

14, and lead 11.

3.

Therefore, in order from the least to the most dense, it's sodium, titanium, zinc, copper, and finally, lead.

Question two.

Yes, the necklace was genuine.

Using those values, we do get a density of 21.

45 grammes per cubic centimetre, so it was a genuine platinum necklace.

Finally, question number three.

You can see the three ratio tables there.

The third plank was the one with the lowest density.

Now, let's move on then to looking at problems involving density.

This is a stainless steel screw, and it has a mass of five grammes.

If the density of stainless steel is 7.

9 grammes per cubic centimetre, what is the volume of the screw? We know the mass of the screw is five grammes.

The unit of density is grammes per cubic centimetre.

This shows the number of grammes per centimetre cubed.

One centimetre cubed has a mass of 7.

9 grammes.

We now have one missing value in our ratio table, so we can find that missing value.

Let's look for the multiplicative relationship between 7.

9 and five.

Now that's not obvious, so I'm going to do five divided by 7.

9 and I'm going to leave that as a fraction because otherwise it gives me a horrible decimal.

I'm gonna multiply by five over 7.

9 giving me 0.

63.

The volume of the screw is 0.

63 centimetres cubed.

If the density of this brick is 1,700 kilogrammes per metre cubed, so notice here we're using kilogrammes and metres cubed.

The process is going to be the same.

What is the mass of this brick? Firstly, we need to find the volume of the brick, and I know you know how to do that, because earlier we found the volume of that gold bar.

I'd like you please to calculate the volume of the brick in cubic metres, because our density is kilogrammes per cubic metre.

I'd like you please to find the volume of the brick in cubic metres.

Pause the video, decide what your answer is, and then come back and check in with me in a moment.

Volume of the brick, and I've converted each of my measurements from centimetres into metres by dividing by 100.

0.

065 multiplied by 0.

215 multiplied by 0.

103.

That gives us a volume and notice it's in standard form, but we're not going to worry about that, because we know how to deal with numbers in standard form, but I'm going to leave it in that form for the moment.

The volume of that brick is 1.

44 multiplied by 10 to the negative three metres cubed.

We're told in the question that the density of the brick is 1,700 kilogrammes per cubic metre.

That gives me the first row of my ratio table.

We also know the volume of the brick.

We've just calculated that as 1.

44 multiplied by 10 to the negative three.

I'm looking for that multiplicative relationship and to get from one to 1.

44 multiplied by 10 to the negative three, I need to multiply by 1.

44 multiplied by 10 to the negative three.

I now need to do that to the mass.

If I do that, I end up with 2.

447.

The mass of the brick is 2.

447 kilogrammes and I know it's kilogrammes, because my unit of density was kilogrammes per metre cubed.

Spot and correct the mistakes.

The density of lead is 11.

2 grammes per cubic centimetre.

What is the mass of this cube of lead in kilogrammes? You're gonna pause the video, you're going to find the mistakes, and also, of course, I want you to correct those mistakes.

Pause the video, and when you come back, we'll check that you've managed to spot and correct all of those mistakes.

Good luck.

And what did you decide? 0.

6 metres, and here we had our volume was in cubic centimetres, so the first thing I needed to do was to make sure I was working in centimetres when finding the volume of that cube of lead, 0.

6 metres is 60 centimetres, so I should have done 60 cubed, which is 216,000.

That means now my multiplier is incorrect, it should have been 216,000, which means my mass is incorrect.

It should have been 2,419,200 grammes.

And checking the question, the question wanted to know the mass of the cube of lead in kilogrammes, I now need to convert that into kilogrammes, which is 2419.

2 kilogrammes.

Now task B, complete the missing values in the table.

Pause the video and come back when you're ready.

Question number two, a lorry transports sheets of wood.

Each sheet measures 3.

6 metres by 2.

5 metres by 0.

12 metres.

The density of wood is 450 kilogrammes per cubic metre.

The lorry can carry 12 tonnes of the sheets, calculate the maximum number of wooden sheets the lorry can carry.

Pause the video, now take your time over this question.

It seems like there's lots and lots of information there,, but actually you've done all of the things you need to to be successful at answering this question.

I have every faith in you.

Pause the video and then come back when you're ready.

Question number three.

These steel rods are packed into boxes.

We can see the dimensions of the steel rod.

We can see it has a diameter of 12 millimetres and a length of 5.

4 metres.

The mass of each box cannot exceed 150 kilogrammes.

The density of the steel is 7.

85 grammes per cubic centimetre.

I need you to calculate please, the maximum number of steel rods the box can safely hold.

Here, it's really important that you look carefully at your units, what units are being used, and if you need to do any conversion of units, make sure you convert them before doing any calculations.

This is a challenging question, so have a real good go at it.

I know that even if you find this tricky that you are gonna persevere with it, and that perseverance will see you through to the correct answer.

Pause a video and then come back when you're ready.

Well done, almost there.

We can now check those answers.

Some of those were quite challenging, so well done if you've stuck with me right the way through to the end, which of course I know you have.

Question number one, the missing mass for object A was 135 grammes.

The missing volume for B was 600 cubic centimetres or 600 centimetres cubed.

The missing density for object C was 980 kilogrammes per metre cubed.

The missing volume for D was 6.

2 centimetres cubed and the missing mass for E was 4733.

52 kilogrammes.

Really important there to make sure that you've got your units correct, although actually I think I did give them to you in the table, didn't I? Question two.

The first calculation you needed to do was to work out the volume of one sheet of wood, and that was 1.

08 cubic metres.

The mass of one sheet of wood is going to be 1.

08 multiplied by 450, which gives us 486 kilogrammes.

You probably there drew a ratio table to get that.

Now you did need to know this conversion and that's that one metric tonne is equal to a thousand kilogrammes.

The lorry can carry 12 tonnes.

That means the lorry can carry 12,000 kilogrammes.

We can now divide 12,000 by the mass of one sheet of wood, which is 486 kilogrammes, giving us an answer of 24.

691, and that number carries on.

This means the maximum number of sheets the lorry can carry is 24.

Remember here, we know that in general 24.

6 would round up to 25, but if we went to the 25th one, then unfortunately the lorry would be overloaded, and we wouldn't want that, because that might cause an accident.

Question three, first thing you needed to do was to calculate the volume of one rod.

Here you needed to be careful with your units.

The density of the steel was given as grammes per cubic centimetre.

Therefore, we need to find the volume of one rod in cubic centimetres.

12 millimetres is 1.

2 centimetres, so I've done pi multiplied by 0.

6 squared, because I needed the radius and the radius is 0.

6, and I've multiplied that by 540, because 5.

4 metres in centimetres is 540.

This gives me the volume of one rod as 610.

7256119 cubic centimetres.

Now I can work out the mass of one rod, and as I said previously, you've probably drawn a ratio table there, so we take the volume of one rod and multiply it by the density, 7.

85, given us 4794.

196 grammes.

The mass that each box cannot exceed is given in kilogrammes, so I've decided to change that mass into kilogrammes by dividing by 1000 to give me 4.

794 kilogrammes.

This means the maximum number of rods is going to be the mass that the box cannot exceed, which is 150 kilogrammes divided by the mass of one rod, which is our 4.

794, and that equals 31.

28, and that carries on, again, the maximum number of rods the box can safely hold is 31, we would need to round down in that situation.

Summarising the learning from today's lesson, the density of a substance is measured as a substances mass per unit of volume.

The main units of measurement of density are /cm cubed, which is grammes per cubic centimetre and kg/m cubed, which is grammes per cubic metre.

The mass of a substance is a measure of the quantity of matter in that substance.

Ratio tables are an excellent way to solve problems involving density, and we can see here the example that we went through of checking whether the ring was a genuine gold ring or whether it was a fake.

I remember it was a fake.

Well, we got there.

Some of those questions in task B were challenging, so I'm really pleased that you've managed to stick with me right to the end, and I'm sure that you will see the benefits of that in the future.

Super proud of you, take care of yourself, goodbye.