##### 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.

Good stuff here. I also like these tethered goat-type problem. Another issue, both in my teaching and when doing long-term SoW planning for mixed-attainment groups was to consider a) what underlying knowledge students would need to be able to make a start on working on and developing such problems; b) where I might use such problems in a SoW and c) progression of the development of students’ skills and knowledge. Clearly students would need to know about how to calculate areas of circles etc so I would use such problems having explored and caused students to derive A=pi r^2. A lovely aspect of these problems is having different conditions the number of posts and the placing of the posts. Of course with other conditions the loci of ellipses can be traced out so plenty of extensions beyond GCSE level

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Thanks Mike. If students haven’t yet covered area of a circle, there is still merit in posing the question, what would the grass look like. No need to actually calculate a numerical answer for area.

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