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.

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Geometry Problems

Having now taught this lesson, I’ve edited a few things. 

This problem was posted on Twitter last week by @solvemymaths.  I must confess I was pretty slow to solve this! It’s a lovely problem – so easy when you know how.  I created this Geogebra file.  It didn’t help me solve it directly but it did answer the question of whether this is a fixed shape or not, which I was struggling to visualise from the diagram. It isn’t fixed and that gave me an idea for a lesson. Rather than just give my students the problem, we will draw it first to provide that hook, the hunger to answer the question, “Why does that happen?”. I also provides a bit of practice using a compass, a skill that is needed at GCSE and one which we don’t spend much time practising.

Here’s how I would now run the lesson (key maths words italicised):

1, On plain paper everyone draw a circle using a pair of compasses.  You chose the radius, anything between, say 1cm and 6cm. Make sure you write down your radius. Calculate the area of this circle.

2, Then, using a ruler, carefully draw a tangent, doesn’t matter where it is on the circle.  Make the line nice and long, using a 30 cm ruler if you have one.

3, Then, using compasses again, mark 5cm either side of where you tangent touches the circle. Label the points A and B.

4, Next draw a second concentric circle, going through A.  It should also cut B if you have drawn accurately. You should have a shape looking something like this:

or this 

5, Now, calculate the area of the big circle.  You’ll obviously need to measure the radius of the big circle first.

6, Now find the area of the orange ring. Compare your area with that of others around you.

Where you go next will depend on the class. I’m not going to give you a full solution here, that would spoil the fun! But the answer is 25π…