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Pig and Other Tales Sample Pages
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collection of sample pages (117KB Acrobat file).
(The file is taken from
the MA CD-ROM for ICME 2000 produced by Bill Richardson.)
The pages shown here are © The Mathematical Association; their
formats have been adapted for display on the Internet.
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'PIG' AND OTHER TALES
A Book of Mathematical Readings
with
Questions and Specimen Answers
for
Students of A Level and Scottish Higher
Edited by Doug French and Charlie Stripp
The Mathematical Association
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Introduction
The readings in this book have been chosen from past
issues of the Mathematical Gazette, since it was felt that they
could profitably be studied by A level and Scottish Higher Mathematics
students. They cover a wide variety of topics, and whilst some are more
difficult to understand than others, most should be accessible to the
majority of such students.
The readings are accompanied by questions on their
content. The questions are structured to help the student get the most out
of each reading. The skill of reading and understanding mathematical or
scientific articles is an important one, and is not one that is generally
addressed. However, some A level boards are now introducing comprehension
exercises into their examinations, for which these readings should be
especially relevant. Some advice to students on reading mathematical
articles is given on page 1.
As well as questions, specimen answers are also included,
which should help to stimulate discussion as well as enabling students to
check their solutions.
It is intended that the materials should be photocopied
to allow their use to be as flexible as possible. Permission is given by
the Mathematical Association to purchasers to make copies for use in their
institutions.
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THE MATHEMATICAL ASSOCIATION
READINGS WITH QUESTIONS AND SPECIMEN ANSWERS
FOR A LEVEL OR
SCOTTISH HIGHER STUDENTS
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Advice on Reading Mathematical
Articles |
iv |
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| 1. |
Pig |
1 |
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S Humphrey |
Mathematical Gazette, 63, December
1979 |
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An analysis of the probabilities
associated with a simple game which uses two dice. |
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| 2. |
The Trouble with
Asymptotes |
7 |
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J Deft |
Mathematical Gazette, 72, October
1988 |
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A comparison of three proposed
methods for finding the equation of the oblique asymptote to the graph of
a rational function. |
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| 3. |
The Glass Rod Problem |
12 |
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J Haigh |
Mathematical Gazette, 65, March 1981 |
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A neat pictorial solution to a
probability problem. |
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| 4. |
Bride's Chair Revisited |
17 |
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R Webster |
Mathematical Gazette, 78, November
1994 |
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An interesting property arising from
the diagram used in Euclid's proof of Pythagoras' Theorem. |
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| 5. |
Morley and More |
22 |
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N Reed |
Mathematical Gazette, 65, March 1981 |
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A proof of Morley's Theorem, which
states that an equilateral triangle is formed by joining intersections of
lines trisecting the angles of any triangle. |
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| 6. |
a. |
Imagine the Roots of a
Quadratic |
26 |
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C R Holmes |
Mathematical Gazette, 74, October
1990 |
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| b. |
The Complex Root of a Quadratic
from its Graph |
26 |
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S F G Wessels |
Mathematical Gazette, 65, March 1981 |
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Two related ways of displaying the
complex roots of a quadratic on a graph. |
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| 7. |
Newton-Raphson and the
Cubic |
32 |
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R Dunnett |
Mathematical Gazette, 78, November
1994 |
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An interesting result which arises
when the Newton-Raphson formula is used to solve a cubic. |
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| 8. |
Bending the Sheet |
37 |
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H M Cundy |
Mathematical Gazette, 72, October
1988 |
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Graphical and numerical methods
applied to solving a practical problem. |
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| 9. |
The 14-15 Puzzle |
42 |
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A K Austin |
Mathematical Gazette, 63, March 1979 |
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A proof by contradiction used to show
the impossibility of solving a simple puzzle. |
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| 10. |
How to win the National
Lottery |
46 |
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N Mackinnon |
Mathematical Gazette, 78, November
1994 |
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An analysis of some aspects of the
National Lottery using simple probability. |
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| 11. |
Mathematics and the Motion of the
Human Body |
55 |
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T Roper |
Mathematical Gazette, 74, October
1990 |
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Two mathematical models of walking
which involve an application of motion in a circle. |
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| 12. |
Kurschak's Tile |
62 |
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J L Anderson and K Sydel |
Mathematical Gazette, 62, October
1978 |
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A fascinating dissection which
provides an elegant way of finding the area of a regular dodecagon. |
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Article and Comprehension Questions |
The book includes specimen
answers as well. |
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Chapter 2: THE TROUBLE WITH
ASYMPTOTES
J Deft
Mathematical Gazette, 72, October 1988
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Considerable staffroom discussion was generated
recently by a question which asked for a sketch of
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Clearly there is an asymptote x = 2, and
another oblique asymptote whose equation has to be found. |
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"Easy," said Aloysius. "When x is very
large, the most significant term on the top line is x2,
and on the bottom is x. The fraction then becomes (near enough)
x2 / x, and so the asymptote is y =
x." |
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"I don't do it that way," said Balthazar.
"Obviously |
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and if we divide the top and bottom lines by
x we get |
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When x is large, 3 / x and 2 /
x tend to zero and we are left with the asymptote y =
x + 4." |
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"I don't get either of those answers," said
Cordelia. "I agree with |
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but then by algebraic long division I get |
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and when x tends to infinity this gives
an asymptote y = x + 6." |
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As far as the sketch is concerned, of course,
it really doesn't matter - the graph disappears from the top right-hand
corner of the paper at an angle of approximately 45° - but clearly only
one of these answers can be right. Some simple numerical work shows that
the correct solution is in fact Cordelia's, and it is then fairly easy to
see that each of the other methods depends on the erroneous assumption
that the limit or tendency of a quotient is equal to the quotient of the
limits. Nevertheless, the discussion drew quite forcefully to our
attention the unreliability of the "quick methods" that we had been using
happily for many years. |
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JOHN DEFT |
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Questions on Chapter 2: THE
TROUBLE WITH ASYMPTOTES |
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| 1. |
Where does the curve cut the x and
y axes? |
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| 2. |
Why is there an asymptote at x = 2? |
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| 3. |
What happens as x gets closer to 2 from
above and below? |
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| 4. |
Illustrate Aloysius's argument numerically and
explain how it does show that y is very large and positive when x
is very large and positive. |
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| 5. |
Explain in the same way what happens to
y when x is very large and negative. |
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| 6. |
Sketch the curve on the basis of the
information established so far and compare with a curve plotted on a
graphical calculator or a graph plotter. |
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| 7. |
What is meant by an oblique
asymptote? |
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| 8. |
Show that the line y = x cuts the
curve when x = -0.5 and, assuming the two branches of the curve are
on opposite sides of the asymptote, explain with reference to your sketch
why y = x cannot be the oblique asymptote. |
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| 9. |
Explain Balthazar's argument. |
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| 10. |
Show that the line y = x + 4 cuts
the curve when x = -5.5 and explain why it cannot be the oblique
asymptote. |
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| 11. |
Explain how Cordelia showed that |
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| 12. |
Why does this show that y = x + 6
is the oblique asymptote? |
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| 13. |
What happens if you try to find where y
= x + 6 cuts the curve? |
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| 14. |
What does it mean by the erroneous
assumption that the limit of a quotient is equal to the quotient of
the limits?
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