Helicopters

My background is as an engineer, not a mathematician, so this is going to be an unusual maths lesson in that I'm going to be using maths as engineers use maths, really as a tool to make things work better and make them in the first place. So the thing we're going to be making work is this helicopter. To get it working you just need to fold the wings out and put a paper clip on the bottom:

What we're going to be doing first of all is watching and explaining, which we need to do if we're going to do some maths. We watch and explain, and look for patterns and then put numbers to the patterns.

The first thing to do is to get the helicopters working and then look at them. The first questions will be:

• which direction do they turn in?
• And do all the helicopters turn in the same direction, or do some turn in different directions?

I want you to watch and tell each other what you see and what you think is happening. You're going to be guessing and testing - a lot of maths starts with guesswork and then you do the sums to check if your maths is right. And you're going to be measuring. So we're not going to be doing any difficult maths, but we're going to be using it in a slightly different way.

So watching and telling someone else is going to be a key activity and you're going to be watching and telling each other. One of the first questions is what direction does your helicopter turn in? So as it's falling to the ground, which direction is it turning in when it's dropped and does it turn in the same direction each time? And do all of the helicopters turn in the same direction?

• I've discovered it rotates clockwise and anticlockwise depending on which way you fold it.
• I've discovered if you don't fold the wings so far back it rotates an awful lot faster.

I'm very impressed with your abilities to watch and tell. First of all you've found out that you can control the direction the helicopter turns in by changing the wings over - you can actually change the direction it's turning in. Also if you make it more of a Y shape it spins faster. If you have a flatter Y, it goes slower, and if you have a T shape sometimes it doesn't work at all, particularly if you bend it down. So you can actually make it spin faster or slower by making it more of a Y or T shape.

When I drop this one (the diagram above shows the wings turned to you, with the rest of the helicopter away from you) will it turn clockwise or anticlockwise? Don't try it straight away - just think about it for a minute or two, guess and then tell each other. After you've done that, try it out and see how good your guesswork is.

• If the right-hand side is facing forward it will go clockwise, if it's facing backwards, it will go anticlockwise.

That's very good, thank you. But I want to put a question to you all because I started off by showing you this and asking if it will go anticlockwise or clockwise. If you said it's going to go clockwise you'd be absolutely right. And if you said it's going to go anticlockwise, you'd be absolutely right. Oh! I'm going to say that again. If your prediction was that this helicopter is going to go anticlockwise you'd be absolutely right, and if you said it's gonig to go clockwise, you'd be absolutely right. Now that's a bit of a puzzle, and I want you to think about that for a minute or two.

• We haven't come up with anything - we're just a little confused!

Often engineers get very confused about things too, and even mathematicians can get confused about things. But I want you to do a little experiment, which involves working in pairs. I want one person to lie on the floor, and the other to drop the helicopter down towards them. The person on top is going to say which way it's turning, and the person lying on the floor looking up at it is going to say which way it's turning. That might give you a clue as to why I claimed that when it's dropped it turns both clockwise and anticlockwise.

I hope that you've found out that clockwise and anticlockwise are what we call relative terms. So if you were watching the helicopter fall away from you, if you were standing above it, you would see it turning one way, and if you were lying on the floor you would see it turning the other way. So clockwise and anticlockwise are accurate descriptions as long as you know what you're looking at.

I want you to use your imaginations now. Look at a clock or a watch for a minute then look away. Now you're going to use your imaginations. You know that clocks and watches go clockwise - that's where the term comes from - but I want you to imagine that you're behind the clock-face - go on an imaginary journey into the inside of the clock and look at the clock face from the inside. Now which way are the hands turning? Are they turning clockwise or anticlockwise? Ah! You've got the answer. If you're looking from inside a clock to the outside, it's going anticlockwise.

What I wanted you to do in that activity was to understand the idea of relative terms. In other words, clockwise is a relative term. It's not an accurate term unless you explain where you're standing, where you're looking. So next time someone says "that's turning clockwise" or "that's turning anticlockwise", you can say, "actually you're wrong, it's not, it's turning both clockwise and anticlockwise"!.

I want to explain just a little more about what we're going to be doing. You must have seen the News and other TV programs where you see aeroplanes dropping relief supplies to people on the ground, people who've been stranded because of floods, or people who are in areas where there's no food, and they're desperately hungry. Aeroplanes fly over the area, and if it's possible to land they do. Often, however, the country is very rough and so they drop relief supplies by parachute.

Today's question is - Could we use something like this paper helicopter (but much bigger) to drop relief supplies? Why should we want to do that? Well, each time a parachute is dropped, it costs about £50 for the parachute and sometimes the bag of rice or grain or medical supplies is hardly worth that value. So if we could use something cheaper we could actually afford more money for food and medical supplies. So the idea we are going to investigate is - Could we flat-pack these helicopters in an aeroplane, load them up with a 50kg bag, say, of food or medical supplies which could be dropped in a controlled way to the people on the ground in areas where they are needed?

I want to make two points about this. One is that you're using a very small- scale model to investigate this problem, and this is something engineers do a lot. Secondly, relative to your helicopter one which would carry a real 50kg load would need to be much bigger - one is relatively large, the other relatively small. (It's that word "relatively" again).

What would happen if you were to put a 100kg load on a helicopter. Would it still work? If we think of one paperclip as representing a 50kg load, imagine what would happen if you put double that load on the helicopter. In other words, test it by putting two paperclips on the helicopter. Will it still work? Will it work in a different way?

Back to guessing and testing. The question is about whether when you add a second paperclip, does the helicopter go faster, slower or the same? It's important to guess before you test. If you want to be a good engineer or scientist or mathematician, you need to be good at guessing, and the more guessing you do, the better you become at it.

This is a picture of the Leaning Tower of Pisa in Italy, which you might recognise, and the picture on the left is of a man called Galileo. Galileo was very interested in dropping cannonballs from the Leaning Tower of Pisa because he wanted to see if a large cannonball would fall at the same speed as a small cannonball. If you want to do a bit of background study, you might find out a little bit more about Galileo's experiments because what you're doing is quite similar.

You have a helicopter with two paperclips on it, and I asked the question what will happen? Some of you might guess that it goes faster, some of you might guess that it goes slower, some of you might guess that it goes at the same speed. When you test it out we have a problem that the results are often very messy. It's often difficult to tell what is happening.

I found out that it went faster.

That's good. Could I just ask, when you say it went faster, in which way did it go faster? Did it go faster downwards, or did it spin faster?

Faster spinning.

Faster spinning? That you, that's interesting.

It got to the point where it couldn't handle any more weight.

That's interesting as well, OK.

I want to talk about how we could be sure about whether it was going faster or slower with two paperclips, because sometimes it goes so fast it's difficult to measure. If it was going a little slower it would be easier to measure but it's still difficult to time it, and to know whether it's going faster or slower. So I wonder how many of you thought of trying a race - racing the helicopter with two paperclips against the helicopter with one paperclip. You could then see if one went faster and you could actually measure the result.

Now although I've shown one reaching the ground first in this slide, I think that whether you put two paperclips on or one paperclip, they fall at exactly the same speed. Does anyone agree with me that they fall at the same speed? I'm going to say that's a "yes". The people with their hands up are saying "yes". But some of you disagree. The problem is that the only way to be sure is to do lots of tests and lots of measurements. Then if we take all of those tests, see which results actually give the better indicator. If you've done averages, you could take the average of all the tests in order to get one important result. So just testing once often isn't enough. We have to do lots of tests, lots of measurements and then take the average results.

I want to ask another question now because if I'm right that a helicopter with two paperclips will fall at the same speed as one with one paperclip, does it mean that one with four paperclips will still fall at that same speed? I want you to guess - will it fall at the same speed, twice as fast, half as fast ... ? Then test your guess. Just think, if it still works with four paperclips, that would be equivalent to putting a 200kg load on one helicopter, and this would start to look quite a good idea. But it might fall so fast it would burst the bags of flour or rice or destroy the medical supplies because they hit the ground so fast. Have a guess first, will it still work? Would you risk it? Then test your guess.

It would go faster.

It will go faster. Can I just ask, does that mean faster to the ground or faster spinning, but the same speed going to the ground?

Faster spinning.

Faster spinning again. That's interesting.

I found that the one with four paperclips fell to the ground faster. It spun a lot faster so it didn't get much wind resistance which would make it fall faster.

That's an interesting finding as well. Thank you.

If it was a real-life one, it probably would have exploded.

Right, OK.

I think they hit the ground at the same time, if you chuck them both in the air, they hit the ground at the same time.

I discovered that the one with four paperclips went faster spinning and dropping down.

Thanks for those reports. What I sense is that you're getting different results. The problem in doing investigations of this kind is that you will often get different results and you have to try and make measurements in order to be sure. I'd like you to continue with this in your project work. Continue with adding paperclips and see what effect it has on the way the helicopters spin and fall to the ground. What I would agree with is that the more paperclips you add, the faster the helicopter spins. But does it go faster down to the ground? In my experience - your experience might be different - if you have a helicopter with one paperclip and one with two, the one with two paperclips starts falling faster. As soon as it starts spinning faster, however, it falls at the same speed. So what I think happens is the one with two paperclips starts falling faster initially, then starts spinning faster, and once that happens, they then fall at the same speed. This is something you can investigate.

A good question to investigate would be whether you could keep adding more and more paperclips and have the helicopter still working? A parachute is limited in the number of bags of supplies it can carry. But if this is something we could develop, say a helicopter would take a very heavy load and still be cheaper, that could be quite exciting.

I want to go back to the idea of how big a helicopter could be if it was going to be used for real. So I'm going to go back to the idea of size and shape. Instead of measuring something working we're going to look at the helicopter and see what we mean by bigger or smaller.

What happens if we change the shape of the helicopter, and if we change its shape how do we describe that? So we could say - will the helicopter work better with larger wings? But what do we mean by larger wings?

Here we have one helicopter (design 1) and if we're to make the wings larger (design 2), how much larger would they be? Think about this and discuss it with each other.

Let's just look at one wing. How much bigger is the wing of helicopter 2 relative to helicopter 1?

It's one quarter bigger.

That's a very good way of describing it, thank you. Any other ways of describing it?

It's 2 by 4 - the smaller one's 2 by 4 and the bigger one is 2 by 5, the larger one is one quarter bigger.

Thank you, that's a good, accurate description. Any others?

The wing span of the 2 by 5 is bigger and would be more useful for carrying heavier weights because it would cause more wind resistance and stop the rice or medical supplies being destroyed when they hit the ground.

That's a great idea. Thanks for that.

If we're going to make something bigger, bear in mind that if we were going to do this project for real and design something which would be capable of dropping medical supplies, we'd have to describe it very acccurately. So, to say it's a quarter bigger, we'd have to say in which direction. Given design 1 which is 4 by 2, we could make these square cms, so it would be 4cm by 2cm. Then we could make design 2 5cm by 2cm. Then we could talk not just about the length but the area as well. So we have to get the shape right, and we have to get the area right.

Now I want you to see if you can make a helicopter half the size of the one you've got. Just take a pair of scissors and don'e measure anything, just guess and cut.

What I'd like you to think about and work on later, is: if you made one half the size, how can you be sure it's half the size? How would you measure it? How could you use maths to be absolutely sure?

I've been describing the helicopter by showing it to you and I've also been describing it in the form of a drawing. Now I'm going to show you a design for a helicopter in the form of a code. Can you guess what shape this helicopter is? Can you break the code? We don't have time to talk about this, but you could discuss it later.

There's the complete wing. We've been using coordinates, which you can connect to show the whole wing. The coordinates give a code which can be easier to transmit over the phone or electronically than drawing.

One of the interesting things about the helicopter shape is that they fit together - they tessellate. If you were going to make a lot of them you could save material by fitting them together, so there's no waste.

So what have we been doing? Remember watching and explaining - looking at something and trying to explain what's happening. Looking for patterns - going faster, spinning faster, falling faster, trying to measure what the pattern is going to be. That might involve estimating or guessing and then finding averages of results that are different each time. Then finding the right words to describe things because clockwise and anticlockwise can be a little bit misleading because they are relative terms. And we've been thinking of ways of measuring speed, changes in size and scale.

I hope you've got lots of ideas to work on and that you'll be able to use your maths to get accurate measurements and descriptions, and accurate ways of describing shape and size.