Category Archives: loci and constructions

Tethered Goat

No 4 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is a classic task for working systematically:


This is a classic question that I usually use when teaching loci and constructions.  I usually scaffold it with a diagram, but I like the approach here that promotes students to introduce their own diagram as a way of getting closer to the problem. Next time I use this I will make sure I give it plenty of time, get students discussing the problem in pairs and really resist providing a diagram and thus removing some of the joy of discovering it!

A goat is tethered by a 6 metre rope to the outside corner of a shed measuring 4 metres by 5 metres in a grassy field. What area of grass can the goat graze?

If I felt the need to make this question easier,  I might start with the rope being 4 metres long, then try 5, then try 6.  In fact, this could be taken a lot further, what happens as the length of rope is increased further?  Try 10m, 50m, 100m.  What happens when the rope gets really long?

Or, as a way in to this problem I might ask:

A goat is tethered by a 6 metre rope to a simple post in an open field. What is the area of grass it can eat?

I will have this question ready written, but hidden on the same slide on the board so that I can show it to the whole class if I feel they need it. But alternatively I might just suggest this to individual students if they are really struggling and making no progress.  Either way, the point is that it is not another question that I want them to work on, it is a problem solving strategy that I am exposing them to: If the problem seems too difficult, try to find a simpler scenario as a way in to the problem. Solve that, and then think about what knowledge and methods you used and how they could be applied to the main problem.