How Small Is Dutiful?

How Small Is Dutiful?Author(s): Donald CrossSource: Mathematics in School, Vol. 16, No. 3 (May, 1987), pp. 10-11, 29Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214211 .

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Hotu Sm all is Dntlff by Donald Cross, Exeter

One summer when Pythagoras was on holiday in Crete with his three sons, he sat on the beach admiring a large right- angled triangle he had traced on the sand. Suddenly his three sons came running towards him from the chalet higher up the beach.

"Mind where you tread," he called to them. "And don't be so boisterous."

"Not more right-angled triangles," groaned his oldest son. "We're on holiday, dad."

"I've got a problem for you," his father said. "Just the sort for the three sons of a genius."

"Let's have it," said the second son, resignedly. "If we don't solve it now, we'll have to do it later."

"Quite right," said his father. "'You have a duty to me and to mathematics to keep thefamily name alive."

10

"Dutya" inquired the youngest son. "What do you want us to doa"

"Look at my triangle," said his father. "I want you to draw a scalene triangle in the sand having two resemblances to it: the same perimeter, and the same area arising from the addition of the squares drawn on the three sides of the triangle, as you get from the three squares drawn on mine. There is more than one solution, and I shall award a prize to the triangle having the smallest internal area."

Reluctantly, the two older sons set to work. From time to time, they looked with some astonishment at their younger brother who had drawn a single straight line, and sat looking intently at it. When they themselves had produced one solution each, they summoned their father.

"'Very good," he said to the oldest son when he had checked his triangle. "'Very good."

Mathematics in School, May 1987



To the second son, he said, "Excellent. The area of your triangle is even smaller than that of your brother's."

"Do I get the prize thena" he asked.

"'We must first hear from your younger brother."

"But he has drawn only one line," protested the two brothers.

"Hush!" said their father. "Let him speak."

The youngest brother had a twinkle in his eye. "I'm afraid I have failed in my duty to you, father. My 'triangle' has no area whatsoever. Can I claim that it fulfils your two other requirementsa"

"Why do you aska" his father queried.

"If it does, there is an even smaller right-angled triangle leading to a similar solution. However, the smaller right- angled triangle has this drawback that it doesn't yield the kind of solutions my brothers have just offered."

"He's cheating," said the second son. "How can you possibly say that a straight line is a trianglea"

"Of course you can't," said the oldest son. "Everybody knows that any two sides of a triangle must be together greater than the third side. That obviously means they cannot together be equal to it."

All this time their father had been watching his youngest son who stood quietly with a look of particular satisfaction on his face.

"Is there anything else you want to say, my sona" he asked.

"Yes," said the youngest son, laughing. "I will not press the matter. As it happens, I can draw two 'proper' solutions, both with smaller internal areas than either of my brothers', so I am able still to claim the prize."

"Fm proud of you, my sons," their father said. "You have all done well. But don't rest on your laurels. You are old enough now to draw your own right-angled triangles. Produce one now (the smallest possiblea) that will yield only one acceptable solution to my original problem."

Several hours later, Pythagoras judged the results of his three weary sons' labour.

"Pride goeth before a fall," he said to his youngest son. "You have not won on this occasion."

"But both my right-angled triangle and my scalene triangle are smaller than theirs."

" True," said his father, ""But they have stuck to the rules, and you have not. They have found separate solutions in which only one answer is possible. Flushed with success, you failed to notice that there were two solutions to the right- angled triangle you chose."

His youngest son looked glum.

"Don't be too upset," his father said. "'m really very pleased with you all."

"Oh, good," said the oldest son. "Can we go for a swim nowa"

" Yes, of course," said his father. " You can begin to

relax now. You'll find your problem for tomor- row, compared with today's, little more than a mere doddle:

Draw a right-angled triangle and a scalene triangle having unequal perimeters such that the sum of the squares on the three sides of the one triangle is equal to the sum of the squares on the three sides of the other triangle. Make all six sides a different length. Now off you go, and enjoy yourselves."

Hints and Solutions on page 29.

Mathematics in School, May 1987 11



scheme has proved to be popular with all years and abilities and it is noticeable that many pupils respond much better to a piece of work which is identified as a Task than to an equivalent piece of routine work.

Coursework Our Award Scheme was already in place and running when we went for a Mode 3 examination several years ago, and it seemed logical to use the scheme to provide the Coursework element that we wanted. So the Task Lists for the fourth and fifth year (Green and Gold) levels were given the additional designation "CSE Coursework" and were al- located 30 per cent of the marks. In general it has served us very well and done what we wanted. Not that we were ever completely satisfied with what we were using. There was a little uneveness about the sizes of the various tasks that bothered us a bit, but we never got up enough steam to do more than just tinker with it when really some major re- writing was called for in places. Overall it was a good system, and we felt that the inequalities were not doing the individual candidates any major disservice.

Then came GCSE and coursework became mandatory. Well it could be put off for a few years, but since we were already doing it there seemed little point in taking that option. That provided the stimulus to sharpen up our act and so, those lovely two days last summer were devoted almost entirely to looking at the fourth and fifth year tasks.

However, part of our work entailed thinking about what coursework ought to be. To me it looks as though the model we have is different in kind and in spirit from that allowed in the various syllabuses. They call for two or three specific units of work to be done at some time in the course, and I am concerned that these do not become just a couple of lumps attached as an afterthought, and that they are not woven into the fabric of the course at all. Is that what coursework is intended to bea Surely that is not in keeping with the spirit of the National Criteriaa Those same syllabuses do say that the "units" can be made up of several smaller pieces of work (which is what we are doing) but, the

few indicators I do have, point to very few taking up that offer.

I think this could be a pity for many reasons. First of all, several small tasks being done throughout the course are more easily perceived to be a part of that course - even though the tasks vary widely in their content and purpose; the binding nature of the structure gives the whole thing a coherence and credibility that might otherwise be missing. Secondly, small tasks seem to fit much more easily into the classroom scene, allowing greater opportunities for various approaches, making it easier to deal with individuals who have fallen behind, and needing no continuity so that everyone is "in with a chance" for each task. Thirdly, small tasks lend themselves much more readily to being set as a series of separate homeworks, and a loud "hurrah" for that; at long last credit can be given for all those joyless hours of toil to which we condemn children; credit which is accrued and used in the final assessment. Fourthly, changes are much more easily accomplished; a few of the tasks can be altered every year to take account of changed thinking or interests.

But that is enough of moralising! I only set out to give a brief account of what we do, not to run a crusade. Can we offer some practical helpa Yes, for a small charge. For anyone who wants to get started, it is quite a chore to think up 100 (5 x 20) small tasks. So we can let you have copies of our five task lists. Of course, not all of the tasks would suit your purpose, but even those that do not can still serve to act as "idea-generators". There are also blank certificates, pupil record cards, teacher record sheets and an explanatory sheet (a handout for parents), all of which would serve as masters - possibly after some adaptation - so that you can reproduce your own. You can also have some of the actual worksheets used; which ones will depend upon availability at the time, but there will be sufficient to serve as examples of what might be done.

Interesteda Then send a3.00, cash with order, cheques payable to "Dawlish School" to Mathematics Department, Dawlish School, Elm Grove Road, Dawlish EX7 OBY.

IYow Smam is Dt~iPuIa

I - SOLUTIONS AND HINTS

I I suggest the following answers to the competitions Pythagoras proposed:

(i) I believe that the three sides of his first right-angled triangle measured 91, 84 and 35 units.

(ii) The three sides of the scalene triangle proposed by his oldest son probably measured 95, 79 and 36 units.

(iii)The three sides of the scalene triangle proposed by the second son almost certainly measured 100, 71 and 39 units.

II (i) The three sides of the "joke" triangle proposed by the youngest son measured 49, 56 and 105 units.

(ii) The three sides of the small right-angled triangle he humorously had in mind measured 5, 12 and 13 units, with a "scalene triangle" of sides 7, 8 and 15!

III (i) The first serious solution the youngest son offered was a triangle of sides 101, 69 and 40 units.

His second solution had sides of 104, 61 and 45 units.

It is likely that the oldest son's solution to the second conundrum was the following: Right-angled triangle: sides of 65, 56 and 33 units Scalene triangle: sides of 68, 51 and 35 units.

(ii) The second son's solutiona Probably right-angled triangle with sides of 53, 45 and 28 units and scalene triangle with sides of 56, 39 and 31 units.

The youngest son's mistaken solution may well have been: Right-angled triangle: sides of 35, 28 and 21 units Scalene triangle: sides of 36, 25 and 23 units.

The second solution (which he missed) gives a scalene triangle with sides measuring 33, 31 and 20.

IV I'm not sure whether Pythagoras was teasing with his final problem, but personally I found it the most taxing of all.

The best I can do with it is to offer the following equation: (27x2 + 366x + 1120)2 + (36x2 + 678x + 3102)2 + (45x2 + 762x + 3298)2 = (27x2 + 358x + 1070)2 + (36x2 + 614x + 2702)2 + (45x2 + 818x + 3648)2 Where x;> 0

(It turns out that, in a sense, Pythagoras was teasing. There was no need for me to make such heavy weather of Pythagoras' last problem. The smallest (a) solution lies squarely (sic) in the territory of the youngest son's mistake: 352 + 282 + 212 = 202 + 232 + 392)

Mathematics in School, May 1987 29



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How Small Is Dutiful?