Dissecting and ConservingAuthor(s): John BradshawSource: Mathematics in School, Vol. 31, No. 4, Dissection Puzzles Special Issue (Sep., 2002), pp.3-6Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30212196 .

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Disseti and Conservin by John Bradshaw

Conservation of Area and Volume

While relatively basic concepts, such as conservation of area and volume, are discussed in the context of early child development, people tend to assume that they have already been internalized by the time pupils reach KS2. Consequently, they are not to be found as such anywhere in the National Curriculum, although surely employed repeatedly in both mathematics and science. For example, in KS3 under 'Mensuration', "Pupils should be taught to ... find ... areas enclosed by circles, recalling relevant formulae". Hopefully few, if any, pupils are today simply told the formula for the area of a circle, merely so that they can recall it on demand? Presumably, they will encounter a demonstration or investigation based on a series of dissections and rearrangements of a circle, such as this:

C/2 r

r

Circle (disc) Rectangle

It does, of course, rely upon the pupils' acceptance of the principle of conservation of area. This is usually implicit in the teacher's demonstration and associated explanation, rarely if ever being made explicit. Occurring as it usually does in KS3, it is probably also one of the first occasions a pupil encounters the principle of summing to infinity. Again, probably implicitly. Nevertheless, if done well, and hence memorably, it does also serve as a superb early pre-calculus introduction to integration.

An even simpler, area-conserving dissection demonstrates the formula for the surface area of an open cylinder:

, ---r"Cp

h - h

Open Cylinder Rectangle

Not much further along in the school syllabus will be found the necessity to calculate volumes of cylinders, and again an appropriate dissection can demonstrate the sense of the resultant formula.

r C/2

-

f V I h

h

/

Tangrams

However, before all that, it is quite likely that children will have enjoyed 'playing' with other, more recreational dissections, such as those already described by Peter Ransom in this issue. The traditional puzzle of this type is the Tangram.

The Tangram is made from cutting a square into seven rectilinear shapes, as shown on the left. Starting with a 4 x 4 or 8 x 8 square grid is easiest, as it is constructed by bisection and quadsection. Traditionally, all seven pieces must be used, with no two pieces overlapping. An infinite number of geometrical, representational andfanciful shapes can be made.

Certainly, I have always enjoyed such lessons myself, justified as much for the intrinsic pleasure as for other, more worthy ends. Little need be said about the fun - either one enjoys such things or not, but there is a high probability that the majority of the class will do so, as have generations before. It seemed to me to increase the fun, however, if one 'hyped it up' a little beforehand. However, attributing its creation to 'the great god Tan', who caused it to be written down in seven golden books, as Sam Loyd would have had us believe, is surely stretching it a little too far. In America from the end of the nineteenth century, Loyd's book, The Eighth Book of Tan (Loyd, 1903), played its part in popularizing the puzzle and obscuring its origins, wherein he further suggested that thousands of years ago, the Chinaman Li Hung Chang had proved Pythagoras' Theorem by means of the Tangram - although I gather he failed to include that proof in his book ...

A more likely derivation is that the name by which we now know it was adapted from the O.E. trangame or trangram: 'a toy; something intricately contrived; a puzzle' - words likely to have been themselves derived from the even earlier trangrain: 'a strange thing' (c.f. Websters Dictionary, 1996).

Irrespective of its real origins, the geometrical aspects have appealed to mathematicians and teachers throughout the ages, and in 1817, a teacher, W. Williams, published New Mathematical Demonstrations of Euclid. This was subtitled: Euclid rendered clear andfamiliar to the minds ofyouth, with no

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other mathematical instruments than the triangular (sic) pieces, commonly called the Chinese Puzzle. Subsequently, Charles Dodgson is known to have owned a 'play-book' containing 323 Chinese tangram designs, bearing the title: The Fashionable Chinese Puzzle. Whether he compiled or copied it himself is unclear - it seems not to have been published, although a version of it with the same title was definitely published in New York in 1818, with 39 new puzzles added. Subsequently, however, the English puzzler and writer, Henry E. Dudeney, bought the puzzle-book from Dodgson's estate and, like Loyd, then brought the puzzle to a much wider audience. He it was who invented the two shapes shown here, inspired by the stories of Dodgson's alter ego. (We shall hear more of Lewis Carroll's fascination with dissection puzzles in John Sharp's article.)

However, the more plausible origins hold enough fascination in themselves. It is just possible that it is over 2000 years old. Its Chinese name - Ch'i ch'ae pan, translating to Seven-board of cunning, or Seven ingenious plan - certainly dates from the Chu era (740-330 BC). Although the earliest known publication clearly relating to the puzzle only dates from the early nineteenth century, there are much earlier references to it, including an early Chinese encyclopaedia which referred to it as "... a game for women and children".(!) A more widely known publication, Ch'I Ch'iao t'u ho-pi, in which the puzzle is called The Board of Wisdom, appeared in 1813 and may have provided the original from which Dodgson's 'play-book' was copied.

It was recorded in the published version of The Fashionable Chinese Puzzle that: "This ingenious contrivance has for some time past been the favourite amusement of the Ex-Emperor Napoleon, who, being now in a debilitated state, and living very retired, passes many hours a day in thus exercising his patience and ingenuity".

So, the puzzle has a good pedigree and warrants at least some time being spent on it. Certainly it is still well known and has even been incorporated into trade marks. In fact the University of Teesside's logo is a tangram, cleverly incorporating both the U for University and the T for Teesside, and represented their

TEESSIDE

original seven academic Schools. This shape appears on p.108 of Joost Elffers' encyclopaedic work Tangrams (Elffers, 1976), along with the rest of the alphabet and more than 1600 other configurations.

Perhaps unsurprisingly, Tangrams have also featured in fictional literature. One of Robert Hans van Gulik's Judge Dee murder mysteries, The Chinese Nail Murders, set in China in the 7th Century AD, has a mute child witness who communicates with gestures and Tangram pictures. A last enigmatic arrangement of the Tangram finally provides a solution for Judge Dee.

Fine, but what other justifications are there for incorporating Tangrams into maths lessons? One could, as have Michel Dekking and Jaap Goudsmit (in Elffers, 1976), go along the route of classifying the shapes, or polygons, into 'convex' and 'concave', and attempt to prove that there are in fact just thirteen possible convex polygons that can be made from the seven Tangram pieces. Dekking and Goudsmit then go on to define a measure of how 'spread out' a Tangram shape can be. They call the two small isosceles triangles in the set, from which the other five pieces can be constructed, 'basic

triangles'. They then define the 'convexity number' of a Tangram shape as being the minimum number of 'basic triangles' needing to be added in order to make the shape into a convex polygon. Hence the original square, or a large isosceles right-angled triangle, would have a convexity number of 0, whereas the one shown here would have a convexity number of 19. They state that, theoretically, a connected Tangram can be at most 56-convex, but while that upper limit is in reality undoubtedly unachievable, it

nevertheless encompasses far more shapes than could reasonably be listed - and that the highest known at the time of writing was a 41-convex Tangram, being discovered by two Dutch students, and illustrated in Elffers' book.

However, as with the simpler dissections described in the previous article, rather more modest ends can be achieved in the classroom and for many pupils an engaging, and for some a challenging, task is to recreate the dissection for themselves. I have found that an 8 cm by 8 cm square grid, drawn by the pupils on thin card and subsequently cut out, is convenient. (Admittedly, one can buy ready-made Tangrams, in card, plastic - and even in coloured high-density foam, for those who wish to geometricize in the bath ...)

Having made seven reasonably accurate pieces, merely shuffling them and reconstructing the square without looking at the original configuration is a challenge in itself. There is only one way to do this, discounting rotations and reflections. And then there are the challenges of constructing a triangle; a rectangle; a parallelogram; a trapezium and other quadrilaterals; semi-regular pentagons and hexagons; both convex and re-entrant; et al., as well as the traditional panoply of anthropomorphic figures.

Looking more closely at the components we have: two small isosceles right-angled triangles, a middle-sized one and two large ones, together with a square and a parallelogram - five triangles and two quadrilaterals. Surely, the use of correct mathematical language is always worth practising?

Concave Tangram, with 19 basic triangles added to form a convex polygon

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But then:

* What is the area of the original square? * What are the areas of each of the seven pieces? Three

different formulae needed here; or possibly pupils may use, and be able to justify, their own methods. An interesting way into the topic, or a fun way to revise?

* What is the sum of the areas of the seven pieces? Does it match the original area? (... and if not, why not?)

To my mind, the logical sequence expounded above warrants being set out neatly in detail in the pupils' exercise books - it can then be seen as a neat, satisfying, closed and internally consistent piece of work.

If Pythagoras' theorem is already in place, a useful rider here could be to ask questions about the perimeters of the different shapes. The area is constant, and the perimeter has a minimum value, certainly, but does it have a maximum value? Perhaps perimeter could provide an alternative measure of how 'spread out' a Tangram is?

Mathematics, Not Magic!

The whole emphasis here is that the mathematics is explicit - OK, there are puzzles involved in trying to figure out just how to make the shapes out of the seven pieces, but it's not magic, it's all explainable. (Hopefully) intuitively, the area is conserved, and the mathematics does indeed confirm that. There are things to be investigated which may initially be puzzling, but they can soon be clarified.

A colleague is fond of asking what the difference is between a magician and a mathematician. Despite the alluring alliteration, exemplified by Martin Gardner's Mathematics, Magic and Mystery (Gardner, 1956), the roles of the magician and, in particular, the mathematics educator are poles apart. The magician essentially tries to surprise, mystify and entertain, but never to explain. The maths teacher may well try to surprise and entertain, but primarily to enlighten and, if necessary, to explain - but never deliberately to mystify!

Nevertheless, there are slightly more puzzling puzzles to be unravelled, even with Tangrams. For example, these two figures are each made according to the traditional rules - all seven pieces are used in each case, with no overlapping. However, one of them has a foot, while the other does not. Where does it come from?

Cognitive Dissonance

There is no paradox really, they are merely different but valid Tangram configurations, and a still-consolidating grasp of the concept of the conservation of area need not be shaken so long as a satisfying solution is finally found. However, once the concept is well and truly in place, irrevocably internalized and secure, then more fun can begin ...

Retaining the principle that a good teacher seeks to interest, intrigue and engage their pupils in mathematical activity, with the ultimate aim of further enlightenment, then apparent paradoxes often do the trick(!) Whole books have been written on this theme of course, the classic being Eugene Northrop's Riddles in Mathematics (Northrop, 1944). Of these, a common device is one that seems to contradict the 'law' of conservation of area.

One such was possibly devised, but certainly presented by Sam Loyd in his own inimical way, calling it the Gold Brick Puzzle (Gardner, 1959, quoting Loyd, 1914).

Loyd's contrived context was of a (presumably naive) farmer buying a 'gold brick' from a (presumably crooked) 'top-hatted stranger'.

Although naturally originally presented in inches, the square 'brick' here has a side of 24 cm (devaluation?) and so an area of 576 cm2.

1 cm 23 cm

B

24 cm

24 cm A

The square is cut along the diagonal, and the top triangle slid up by one unit. The small triangle left projecting at A is cut off and inserted in the space now made at B.

However, we now have a 23 cm by 25 cm rectangle, which only has an area of 575 cm2.

Where has the farmer's 1 cm2 of gold gone?

Whole-class lessons are not necessarily being dealt with here now. I usually reserve such challenging puzzles for differentiated activities, and they are particularly suitable for individual homework tasks - although I have in the past had to pay for it at subsequent parents' evenings!

Finally, another of this sort of counter-intuitive dissection puzzles, and one that has considerable educational potential, is shown here.

It has probably been employed by Sam Loyd and undoubtedly many others, but I first came across it in the context of yet another opportunity for duplicity - strange how often that theme recurs.

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The story went that a silversmith beat out a thin sheet of silver, in the shape of an 8 cm square, so having 64 cm2 of British Standard Thickness, or whatever, sheet silver. He then dissected it according to the left hand diagram, and rearranged it as below, into a 5 cm by 13 cm rectangle. Clearly, he now had 65 cm2 of silver and so had gained 1 cm2 of silver. Repeating the process 64 times, he profited by a whole new sheet of silver - money for nothing!

5 cm 3 cm 5 cm

5 cm

8 cm

5 cm

This one never seems to fail to intrigue people in general, leave alone just our pupils. The range of solutions received is always a delight to see, and sharing the explanations is without doubt the best part of the exercise. Solutions generally, however, tend to have had one of three methods: large scale, accurate drawings or actual cut-outs; consideration of the ratio of the triangles involved; or appeals to trigonometry to compare how the angles match - or otherwise! All three approaches are of course valuable in the maths classroom, not least for revision purposes, but also to once again attempt to refute some people's 'black and white', 'one right way to do it' stereotype of mathematics.

Gilding the Lily?

Personally, having got 'the answer', I can never leave well alone, and ask if the dimensions involved (3, 5, 8, 13) are significant? Are there alternatives? - apart from mere multiples, of course. It usually does not take long for

someone to make the link to the Fibonacci sequence, and so we can then explore larger diagrams, for which the original square has a side which is an even Fibonacci number. If it is already known that the ratio of adjacent terms of the sequence tends to a limit - and it's possible that some KS4 pupils will have met the Golden Section - then it becomes clear that the larger the original square is, the better will be the 'fit' of the pieces, or in other words, the mismatch will decrease - and the puzzle will become even less amenable to solution by practical methods. For example, a dissection with dimensions 13, 21, 34, 55 - so having an original square of side 34 units - can still be rearranged to show a discrepancy of 1 square unit. However, the proportional difference in area will be that much less, as will the angular 'error'. Great stuff for the Maths Club!

However, this too will be met again as part of John Sharp's tour of dissection puzzles, in the following article.

There is, of course a wealth of information about the Tangram and associated puzzles - a recent gem being William Gibbs' article about his Tangrami Square (Gibbs, 1999) which entangles Tangrams with origami! A few other sources of inspiration are listed below. M

References

Elffers, J. 1976 Tangrams - the Ancient Chinese Shapes Game, Penguin. Gardner, M. 1956 Mathematics, Magic and Mystery, Dover reprint. Gardner, M. 1959 Mathematical Puzzles of Sam Loyd, Dover reprint. Gibbs, W 1999 'Tangrami Square', Mathematics in School, 28, 4. Loyd, S. 1903 The Eighth Book of Tan, Dover reprint 1968. Loyd, S. Jnr. 1914 Cyclopedia of Puzzles, privately published. Northrop, E. E 1944 Riddles in Mathematics, Pelican. van Gulik, R. H. 1961 The Chinese Nail Murders, Penguin. 1818 The Fashionable Chinese Puzzle, A. T. Goodridge & Co., NY

The Puzzling Place www.puzzlingplace.co.uk University of Teesside www.tees.ac.uk

Keywords: Tangram; Dissection; Area.

Author John R. Bradshaw, Mathematics Department, St. Martin's College, Lancaster

LA1 3JD. e-mail: j.bradshaw@ucsm.ac.uk

Prime Problems by Keith Parsons

(1) Find four numbers A, B, C and D which satisfy the following conditions:

(i) Each number is a prime, less than 1000.

(ii) Between them, the four numbers contain each of the digits 0-9 inclusive once only.

(iii) (A +B) = (C+D).

Note The zero must be in one of the numbers and is not to be obtained by rearranging the equation to (A +B) - (C+D) = 0.

(2) Find five numbers A, B, C, D and E which satisfy the following conditions:

(i) Each number is a prime, less than 1000.

(ii) Between them, the five numbers contain each of the digits 0-9 inclusive once only.

(iii)A+B+C-D=E.

(3) Find five numbers A, B, C, D and E which satisfy the following conditions:

(i) Each number is a prime, less than 1000.

(ii) Between them, the five numbers contain each of the digits 0-9 inclusive once only.

(iii)A +B - (C x D)=E.

Solutions: page 45

Keywords: Prime numbers.

Author A. Keith Parsons, 2 Warwick Grange, Solihull, West Midlands B91 1DD.

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