Why do this problem?
This problem offers the opportunity to practise calculating
areas of circles and fractions of a circle in the context of an
optimisation task.
Possible approach
Start by showing the diagram from the problem, and ask
learners to think on their own for a few moments about where
the goat might have been tethered to yield the area shown.
Next, allow them to discuss their ideas with their partner,
and finally share convincing arguments with the whole class.
Having determined where the hook is, the challenge is to work
out the area available to the goat and then to consider
different positions of the hook in order to find the maximum
possible area.
This is a fantastic opportunity to talk about the benefits of
factorising, working in terms of $\pi$, and only using the
calculator at the very end of the computation.
Once learners have established where to fix the hook for
maximum goat nutrition (!), move on to other lengths of rope.
Suggest that pairs of learners work with different lengths of
rope, and finish by sharing their findings with the rest of the
class.
Key questions
What can you say about the radius of each part of a circle?
How does this help you to pinpoint where the hook must be?
How does the space available to the goat change if the
hook is moved?
Possible extension
What happens if the rope is longer than the sum of the sides
of the shed?
Investigate what happens to the area available for sheds of
different dimensions, or sheds which are not rectangular.
Possible support
The activity can be modelled by building the frame of the shed
from multilink cubes, and learners could use string to work out
the shapes of the regions that could be made available to the
goat, when fixing the hook at different points.