Lesson video

Hi, I'm Mrs. Wheelhouse, and welcome to today's lesson that's on the unit, Maths in the Workplace.

In this series of lessons, we're exploring how maths is used in different careers.

So let's get started.

In today's lesson, we're going to look at some of the ways that maths is used by engineers.

On the screen you can see the phrase fractional form, which we're going to be using today in our lesson.

Now, if that's familiar to you, brilliant, but if not, feel free to pause the video now while you have a read through the definition.

Our lesson today is broken into two parts, and we're going to begin by looking at precision in engineering.

So what do you think engineers do? Well, engineers take the principles of science and mathematics and apply them to develop solutions to problems. They build bridges over rivers and gorges.

They construct tunnels under rivers, seas and cities.

They construct buildings that are sky high in compact spaces.

Fundamentally, they solve problems for humanity.

Problem solving is the very essence of what maths is.

Now, a key aspect of any work carried out by engineers is precision.

If they are not precise, the bridge falls down, the building collapses, or the plane falls out of the sky.

Look at your ruler, what are these tiny precise measurements in between each centimetre? That's right, they are millimetres, and there are 10 millimetres in one centimetre.

Now millimetres and centimetres are metric measurements.

Turn your ruler around, what does the 'in' stand for, and to what family of measures does it belong? Pause the video and chat with the person next to you or have a think for yourself.

Welcome back, what did you say? Well, 'in' stands for inches, an imperial measurement.

Historically in the UK, we use the imperial measurement system.

What do you think we call these small precise measurements in between the inches? No, we don't call them milli-inches, But if you did say that, I can see why.

They actually don't have a name, they're described as fractions of an inch.

So this is half an inch, a quarter inch.

What do you think you'd call these measures? Pause the video while you work this out.

Welcome back, let's see what you put.

Well, the first one would be one eighth of an inch.

The next one is three quarters of an inch, and then the last one is seven eighths of an inch.

You might find a ruler even more precise than this one.

So look, we even have even more tiny measurements in between.

So what do you think this new fraction is? Well done if you said a 16th.

One 16th of an inch is about as accurate as can be cut by human hand.

So old rulers were rarely more accurate than this.

What do you notice about the denominators of these fractions? Well, two for eight and 16 are all powers of two, and this is deliberate because half of a half is a quarter, half of a quarter is an eighth, and half of an eighth is one 16th.

These powers of two combined with one other key fact allowed early engineers to accurately communicate with one another.

At its widest point, an adult thumb is approximately one inch in diameter.

Let's just do a quick check that we got all of that.

So, could you please fill in the blanks? Pause and do this now.

Welcome back.

Let's see what you put.

Half of a half is a quarter, half of one eighth is one 16th, and half of one 16th is one over 32.

Now see if you can remember.

The adult thumb is approximately what in diameter? If you can remember that, you should be able to fill in these blanks.

Pause and do this now.

Welcome back.

The length that you can see at the top of the screen is approximately four inches long.

Remember, the adult thumb is approximately one inch in diameter.

So at the bottom of the screen, this length is approximately two and a half inches long.

Well done if you got those right.

Now, this rule of thumb enabled engineers in the UK to measure, cut and communicate to a good degree of precision.

We can measure precise length like one and five eighths of an inch.

Here we go, so our two out thumbs are two inches in total, so therefore in between them is one inch and there's the two inch.

Well then halfway between one and two is one and a half, and now I can see one and three quarters and therefore in between the one and a half and one and three quarters is one and five eighths.

Now I'd like you to have a go.

Where do you think two and one eighth of an inch will be? Pause and try this now.

Welcome back.

Let's see what you did.

Well, there's our two inches, there's the three inches.

I can therefore work out where two and a half is, where two and a quarter is, and therefore where two and one eighth is.

Well done if you got that right.

Now for longer measurements, carpenters used to carry another interesting tool, a five foot folding rule.

This was known as a carpenter's rule or a Mason's Rule, and that's it stretched out all the way.

This length is four feet long.

This length is two and a half feet long or two foot six inches.

There are 12 inches in one foot.

So let's see if you can remember this.

This is a something folding rule and there are something in a foot.

Pause the video and see if you can work out what goes in the blanks.

Do this now.

Welcome back, let's see what you put.

Well, this is a five foot folding rule.

Now you might have gone for the name feet and it's okay if you didn't, but all I needed was how much can you see there? Now there are something in a foot.

There are 12 inches in a foot.

Now measuring in feet and inches and being accurate within one 16th of an inch was fine for the challenges of past.

Buildings were only a few stories high and they still stand if they're built to this level of precision.

However, today we build skyscrapers, almost a kilometre tall, bridges that suspend a million tonnes of load, and jets and rockets, which travel at almost a thousand miles per hour.

These modern demands mean that a 16th of an inch will not suffice.

Engineers levels of precision today have to be far more accurate than that.

You can see on your ruler that one millimetre is smaller than one 16th of an inch.

That makes it a more accurate measure to use.

However, that's actually still not accurate enough for modern day engineering challenges.

Do you know of any metric measure of length that is smaller than a millimetre? Hmm, pause the video while you think about this now.

welcome back.

Let's see what we can come up with while we have a metre, a centimetre, a millimetre.

And remember, a centimetre is 100th of a metre.

A millimetre is 1000th of a metre.

Well, we could actually introduce a micrometre and that's 1000000th of a metre.

We can even go smaller to nanometers or even picometers.

Gosh, they're tiny.

Now, laser cutting technology means modern engineers can cut within 25 microns, micrometres in accuracy.

Wow, that's precise.

Now engineers will describe these measures using standard form.

So a centimetre could be written as one times 10 to the negative two metres, and then we can apply standard form to the others.

Now, what I'd like you to do is write the standard form equivalent please for one nanometer and one picometer.

Pause and do this now.

Welcome back, what did you put? We should have had one times 10 to the power of negative nine and one times 10 to the power of negative 12.

Well done if you got those.

Now remember, engineers could use laser cutting devices to be precise within 25 microns.

What is this in standard form? Now I've given you a quick reminder that one micron or micrometre is one times 10 to the power of negative six metres.

So does this mean that 25 microns can be written in standard forms A, B, or C? Pause the video and make your choice now.

Welcome back, you should have gone for B.

By using the standardised international unit of metric measure, the metre and the global language of mathematics, standard form engineers all over the world can now accurately communicate with one another.

It's time now for your first task.

For question one, could you read the following measures and write down what they are? Pause and do this now.

Question two, use a rule of thumb to measure out the below.

So for part A, one and three eighths of an inch and part B, two and three 16 of an inch, pause and do this now.

Question three part A, how many millimetres on a centimetre? Part B, how many micrometres in a millimetre, Part C, how many micrometres in a centimetre, and part D, engineers can laser cut to precision of 2.

5 times 10 to the negative five metres.

What fraction of a centimetre is this? Pause the video while you work these out now.

Time to go through our answers.

For question one part A, the arrow was indicating two and a half inches on the ruler.

Part B, the arrow is indicating one and three quarter inches on the ruler.

And for part C, the arrow is indicating one and three eighths of an inch on the ruler.

Well done if you've got those right.

Question two, using the rule of thumb, let's see where you should have put these.

So there's our one and three eighths of an inch, and for part B, there is our two and three sixteenths.

Now remember if got very close to this, that's brilliant, well done.

So for question three part A, how many millimetres in a centimetre? Well, there are 10.

Part B, how many micrometres in a millimetre? Well, then we needed to take the millimetre, and divide it by the micrometre measurement to show that there are a thousand micrometres in a millimetre.

And then how many micrometres in a centimetre? There are 10,000.

In Part D, we asked what fraction of a centimetre is this? So when we work this out, we discovered that the fraction is one 400th.

Well done if you got that right.

Can you imagine one 400 of a centimetre? That is the kind of precision engineers operate to.

It's now time for the second part of our lesson, and that's looking at other applications of maths and engineering.

Now a Mason's Rule is often a five foot folding rule, and it was five for a reason.

I wonder why that might be.

Well, we're building the wall early in the process, a mason would measure three foot up the wall and four foot from the base.

They would then see if the five foot rule fitted.

Why do you think they did this? Pause the video and have a quick discussion.

Welcome back.

Did you spot why? This is Pythagoras' theorem in action? Three squared plus four squared is equal to five squared, and we refer to this as a Pythagorean triple or triplet.

So this shows a right angle triangle.

More importantly, it also tells you that your wall is being built perfectly upright.

If the wall were not right, the five foot rule would not fit.

The rule needs to meet with both the point at four foot from the base, and the point 3 foot up the wall in order to make the 3, 4, 5 right angle triangle.

Let's check you've got that.

These bricks are being laid at the corner of a new building.

How do we know that the two walls are not perfectly perpendicular? Pause the video and work this out now.

Welcome back.

What did you say? Well, you might've said the rule does not fit to make the 3, 4, 5 right triangle.

So we know the corner is not a right angle, and you can see that because the five foot rule reaches above the three foot distance.

Now, ancient Egyptian engineers used to make a 3, 4, 5 triangle out of rope marked out by knots.

When they pulled the triangle tight, they knew they had the perfect right angle.

Do you think they used feet? <v ->Well, the ancient Egyptians predate feet</v> by thousands of years, but so long as you use the same unit of measure between the knots, the ropes will work.

Now, ancient Egyptian engineers were the first great builders in human history.

They used 3, 4, 5 triangles to check that everything was right as they built.

Interestingly, they were doing this 2000 years before Pythagoras was even born.

Let's do a quick check, which of these ropes could the Egyptians have used to check that things are right? Think about how you'll know.

Pause the video and do this now.

Welcome back.

Let's see if you've got this right.

A, absolutely can be used because three squared plus four squared is equal to five squared, and we can see that with the knots on our ropes.

For B, however, I've got four squared plus five squared equals six squared, that's not true, so that's not a right triangle.

And what about C? I have six squared plus eight squared equals 10 squared.

And that's true, so I could have used that one.

Well done if you got these right.

Now, ancient Egyptian engineers and many since have used 3, 4, 5 triangles because this is the smallest triplet and therefore efficient to make.

Why don't you get some string and try to make your own 3, 4, 5 triangle to check that things are right in your classroom or in your house.

You don't have to use feet.

You can make a three inch, four inch, five inch triangle.

If you do this, you'll be behaving like the earliest great engineers in human history.

It's now time for our final task.

For question one, show whether each of these walls is upright.

Pause and do this now.

Question two, explain how we know this corner is not right.

Pause and do this now.

Question three, engineers today use Pythagoras to calculate lengths.

Find all the missing lengths of the girders in this bridge.

The design is symmetrical.

Pause and do this now.

Welcome back, let's go through these answers.

So you had to show whether each of these walls was upright.

In other words, Pythagoras theorem is needed to justify here.

Or three square plus four squared is five squared.

We know that's a right.

For part B, 30 square plus 40 squared is equal to 50 squared, so that's also right.

And then in part C, five square plus 12 squared equals 13 squared, so that's also right.

Note that it didn't matter what we were measuring in here as long as the units were consistent within our diagram, we had absolutely no problem using Pythagoras.

If they hadn't been consistent, we would need to have done some conversion first.

Question two, how do we know this corner is not right? Well, you might have said in order to be right, the Mason's Rule should fit to the points measured at three foot and four foot to make a right 3, 4, 5 triangle.

Because the rule does not fit here, we know the third length is not five foot and therefore this is not a right angle triangle, so therefore the corner is not a right angle.

In question three, I asked you to fill in all the missing lengths.

So I started by filling in the ones I'd already been given.

Remember the design is symmetrical after all.

The next thing I did was work out the height, that right angle triangle we can see, and that was 12 metres.

So then I filled that in.

I've actually now filled in all of the horizontal and vertical girders.

So now let's fill in the ones that aren't.

Because the design symmetrical, I know that halfway across is 24 metres, so now I can find the length of that diagonal, which is 26.

I've now only got two more girders to go.

While I know that the distance between the two uprights is 30 metres and I know that the height is 12.

Now using that I get a length of 32.

31 metres.

So I'm going to round here and say it's 32.

3 metres to one decimal place.

Well done if you've got these all right.

Let's sum up what we've learned today.

We have seen some of the ways in which maths is used by engineers.

We understand the need for precision and that engineers use standard form to communicate really precise numbers.

We know that for thousands of years, engineers have used Pythagoras theorem to check for right and non-right angles in their work.

Well done, you've worked really well today and I hope you've enjoyed learning just some of the ways in which maths is used by engineers.

I look forward to seeing you again for some more maths.

Goodbye for now.