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teachwire.net/secondary DISCUSSION Q What did you find out? What happens as the rope gets longer? What happens if you change the tethering point from O to A or B? How did you work it out? Students might begin by making rough sketches of the locus for different lengths of rope. Alternatively, they could produce accurate scale- drawings – with, say, 1 GOING DEEPER Confident students could explore what happens with ropes longer than 10 metres or sheds that have different dimensions. They could also invent their own tethered goat problems, perhaps with more complicated shed shapes, or even obstacles in the field! ABOUTOUR EXPERT Colin Foster is a Reader in Mathematics Education at the Mathematics Education Centre at Loughborough University. He has written many books and articles for mathematics teachers. His website is www.foster77.co.uk , and on Twitter he is @colinfoster77. ADDITIONAL RESOURCE There is a classic, difficult goat-tethering problem that is well-known in recreational mathematics – further details can be found via bit.ly/goat- problem 91 Q Suppose that the rope is 3 metres long. Make a sketch that clearly highlights the grass that the goat can reach. Then calculate the area of the grass that the goat can eat. Students need to see that the goat can reach three quarters of a circle with a radius of 3m centred on O , and can therefore eat 3 4 TT 3 2 = 21.2 m 2 (correct to 1 decimal place) of grass. Q What happens for other lengths of rope? See Fig 1 – what happens if the goat is tethered at A or B, rather than O? From which tethering point can it eat the most grass? 4 m Fig 1 B A O 6 m cm representing 1 m – perhaps using centimetre- squared paper, to make drawing the shed easier. They will need to think carefully about what happens when a rope snags on a corner of the shed. For the goat tethered at O , and a rope of length r metres ( r for r ope or r adius), there are three initial cases, as shown in Fig 2. This makes a continuous, piecewise function . Confident students might want to check, via substituting in, that ‘adjacent’ formulae give the same value where they ‘meet’ (at r = 4 and r = 6). They could also attempt to sketch a graph as shown in Fig 3, in which the three cases are represented by different colours. Similar thinking will give similar equations and curves for tethering the goat at A or at B . We can see from the graph that for any given length of rope, the goat tethered at O gets the most grass. Length of rope (m) Area of grass (m²) Fig 2 Fig 3 0 ≤ r ≤ 4 Area = 3 4 TT r 2 Area = 3 4 TT r 2 + 1 4 TT ( r – 4) 2 4 ≤ r ≤ 6 Area = 3 4 TT r 2 + 1 4 TT ( r – 4) 2 + 1 4 TT ( r – 6) 2 6 ≤ r ≤ 10 tethered at O tethered at A tethered at B