In this problem students estimate in a situation that is familiar to them, but one that they may not have considered before. Some students may be challenged by the measurements involved, and by a result that may not be geometrically intuitive.
The power of algebra is also demonstrated in a situation that is not about solving equations.
Puck is a character in Shakespeare’s play Mid-Summer Night’s Dream. In Act II, Scene I of the play he flies off and puts a girdle round the Earth.
How big is his girdle?
Puck magically sews an extra 6m to his girdle and then makes it into a circle around the earth. How far above the Earth is it?
(Assume that the Earth is a sphere with no mountains etc.)
- Discuss what a girdle is and how to find the length of Puck’s original girdle?
- Ask students to vote as to the size of the solution.
How far out from the earth would the girdle be after 6 metres has been added?
- Allow groups time to invent and carry out a method for solving the problem.
- Summarise and share the various methods.
If Puck had repeats his two-girdle experiment on the Moon, how high above the Moon would the second girdle be?
The circumference of a circle is 2πr. In this problem π is 3.14 is used. The Earth's radius 6378km. So the circumference is 2π(6378). This is approximately 40053.84km. So this is the length of Puck’s original girdle.
Now look at this the other way round. Start with a girdle of 40053.84+ 0.006 = 40053.846km. What is the radius of a circle with this as its circumference? Surely it will hardly lift off the Earth’s surface?
Let’s see. Now 2πr = 40053.846, so r = 40053.846/2π = 6378.00096. So the radius of the new girdle is 0.00096km more than the first one. That is 0.96m. That’s almost a metre. Does this mean that adding 6m to the Earth’s circumference gives a circle with radius nearly a metre more than the Earth? That the girdle would stick up a metre all the way round the Earth?
The fact is that it doesn’t matter how big the first circle is, the second girdle will always stick 0.96m above the surface.
Suppose that the original sphere had radius r. Then it’s first girdle would be of length 2πr. Add 6m to that to give 2πr + 6, where we are now working in metres. What is the radius of a circle with circumference 2πr + 6? It’s (2πr + 6)/2π = r + 6/2π. So the increase is just 6/2π which is 0.96m - just under a metre. That calculation is good for any value of r, whether it’s large like the radius of the Earth or small like the radius of a tennis ball.