What Can We Expect from GCSE Extended Pieces of Work?Author(s): Colin DixonSource: Mathematics in School, Vol. 16, No. 4 (Sep., 1987), pp. 26-27Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214368 .

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What can we expect from GCSE

extended pieces ofworlk. ofworlk.

by Colin Dixon, Whitley Bay High School

Having launched directly into a GCSE scheme offering 38% coursework, I was interested in getting my fourth year started with extended pieces of work both from their point of view and mine.

We had a preparatory lesson when I discussed about 20 possible starters but was careful to emphasise that there was nothing to prevent them coming up with their own ideas: in fact I suggested that a piece of work entirely of their own stood a good chance of being awarded a better grade than a development of something I had suggested. Also, as I had a set of talented youngsters (the sort of group I would have previously entered for GCE a year early), I made it clear that I would expect a high standard of work.

As circumstances had it, I was away from school for a full week and I gave the class the opportunity to have something written up for me when I returned. I realised, of course, that under normal circumstances I would have been able to get involved with each project and prompt where necessary. So it was with some trepidation that I returned to the classroom to assess what had happened.

Apart from three pupils who had found the task too hard

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on their own, I was pleasantly surprised at some of their efforts. About a third of the class (30 pupils) had opted for their own idea and about another third would have benefited from some help from me. But I can hardly express my sheer delight at the high standard of quite a lot of the work. Such was the quality of the work that most of it was quite acceptable as an extended piece of work to be submitted for the final grade. One in particular I would like to share with you.

Investigation by Thomas Randell A man with a lorry lives at the end of a motorway. Along the motorway are sited five depots. Travelling between home and a depot, or depot and depot, is considered to be a journey. He visits the depots (not necessarily all of them) in order to pick up jobs but by the tenth journey he must be back home. He is not allowed to pass a depot without calling in, whereupon he receives payment for the particular job he is given. For example, in the diagram below we see the amounts he receives at each of five depots.

Mathematics in School, September 1987

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Home A B C D E ....... ... ......-- - ---- ... ...,. .

If he went

H-A-B-C-D-C-B-A-B-A-H

he would receive

11 + 4 + 2+ 10 + 2+ 4+ 11 + 4+ 11 = 59

But he could have made 71. Can you see how? The purpose of the investigation is to find the maximum

amount possible to be earned, whatever the prices offered. The algorithm which Thomas came up with was as

follows. Beneath the depots A, B, C, D, E compute the following

amounts: 5A, 4B, 3(C-A), 2(D-B), (E-C) respectively. So in the example given we have:-

H A B C D E

-- o--( @---- o @ @----1 5A 4B 3(C - A) 2 (D - B) (E - C) 55 16 -27 12 12

Starting from the left, add into the total any positive amount that is reached, e.g. 55 + 16. When a negative amount is reached, add the numbers to its right until a positive amount is formed, then add this positive amount into the total. In the case above, - 27 + 12 + 12 = - 3, and as this is still negative it is neglected. Thus the maximum amount obtainable is 71. Note that nothing was suggested as to which depots should be visited.

Another example

H A B C D E

5A 4B 3(C - A) 2(D - B) (E - C) 25 48 12 -20 21

Maximum possible is 25 + 48 + 12 + 1 = 86. By trial we find the route to be H-A-B-C--D-E-D-C-B- A-H.

Extension

The investigation was then extended by suggesting that some jobs might lose money.

H A B C D E ,@I

. ........

5A 4B 3(C - A) 2(D - B) (E - C) 35 -16 3 2 2

-13

-11

-9

Mathematics in School, September 1987

So 35 must be the maximum. By trial we see the route to be H-A- H-A-H-A-H-A-H-A-H.

Suggestions for further investigations were (i) six depots, 12 journeys?

(ii) n depots, 2n journeys? (iii) branches off the motorway?

H A B C, E

.F

D

What intrigued me was how Thomas hit on the algorithm in the first place, and why did it work? The solution came to me whilst fishing - you will have gathered that the cerebral activity was more likely than matters piscatorial!

Solution Depending on which depots are visited, a total of aA + bB + bC + dD + eE will be accumulated.

Suppose A is the furthest depot visited i.e. the only depot visited, then the total will be 5A.

If B is the furthest visited, then this is because it is worth visiting, and the journeys will be H--A-B-A-B-A- B-A-B-A-H and the total will be 5A + 4B.

If C is the furthest visited, then because C is worth visi- ting, it should be visited as much as possible and the jour- neys would be H-oA-~B--C-B-C-BC-B-B-A-H, and the total will be 2A + 4B + 3C = 5A + 4B + 3(C-A).

If D is the furthest visited then it is because it is worth visiting so the journeys will be H-A-B-C-D-~C-D- C-B-A-H, and the total will be 2A + 2B + 3C + 2D

= 5A + 4B + 3(C-A) + 2(D-B). If E is the furthest visited then the journeys must be

H-A-B-C-D-E-D-C-B-A-H and the total will be 5A + 4B + 3(C-A) + 2(D-B) + (E-C).

It should now be clear that all optimum journeys will be of the form:-

H

with the oscillation only between the last two depots visited. It should also be clear that the algorithm can be extended

quite easily to deal with n depots and 2n journeys. Three cheers for GCSE!

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