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Easy and engaging proofs – Dr. Paul Brown
Proof is most interesting when you do it yourself. For our students to create proofs, they need
motivation – and confidence that the task can be accomplished. Successful experience at interesting,
easy problems can nurture both of these traits. This is the reason proof should be introduced as early
as possible, perhaps using some of the ideas outlined below.
Proof that only a few calendars are necessary
My calendar collection goes back to the 1980s. Most years, I can pull out an antique calendar that
matches the current year. In order to be able to do this every year, how many calendars do I need in
my collection?
When I produce a relevant but out-of-date calendar and ask this question, my students always
marvel at my miserly behaviour! Then they enthusiastically attack the task, arguing about leap years
and whether it matters that my system cannot cope with public holidays.
Eventually we reach the conclusion that ordinary years can start on any of the seven days of the
week, and leap years the same. Fourteen calendars suffice. The majority of the proof consists of
definition and reformulation of the problem.
Proof that polygons have the same number of vertices as sides
“Triangle” means “three-angle” and refers to a shape that has three corners (“vertices”). By
inspection, such shapes also have three sides, so they could just as well be called “trilaterals”.
What happens when you cut off one of the vertices of a triangle? When I urge my students to do this
with scissors, I always counsel that it does not hurt the triangles.
Students will quickly realise that cutting off one vertex creates two new vertices, and the cut itself is
a new side. A polygon of n vertices, and presumably n sides, becomes a polygon of n–1+2 = n+1
vertices and n+1 sides. What the students may not realise is that they have performed a proof by
induction.
A dice trick
A tower of ordinary dice is built, and capped with a coin. To the astonishment of all, I can see right
through the coin and predict the number on the upward face of the top die. By walking around the
tower of dice, I predict the number on the upward face of the next die. Exhausted by the mental
stress, I sit down – but continue to predict the upward face of each die.
When you can see two adjacent faces of a die you can determine the other four. The proof is easy:
opposite faces have a sum of seven, and the 1, 2 and 3 faces are always manufactured anticlockwise
around their common vertex (Gardner, 1964). With five minutes of practice, anybody can perform
the trick. The proof is a good example of the “pigeonhole principle”, a useful technique in many
contexts.
Derivation of the formula for the area of a circle
One of my favourite lessons is the “scary teeth”! Once the students have a clear idea that 2πr is the
length of the circumference, they can establish that the area of a circle is πr2 by rearrangement of the
sectors:
A circle
Semicircles
Sectors
Scary teeth
Cut sectors starting from the circle’s centre, carefully stopping just short of the edge, then pull to
form the “scary teeth”. The “scary teeth” fit together to form a parallelogram which has base of half
the circumference, and is one radius high. The more sectors, the more convincing is the
parallelogram. Area = base × height = 0.5 × 2r × r = r2
Interior Angles in a Triangle
Students are sometimes asked to rip the corners off a triangle and place the pieces together to
establish that the sum of the interior angles of a triangle is 180. This activity often provides a
disproof, as crooked cuts and poorly arranged scraps of paper can produce a result that does not
look like a straight angle!
Once students understand the geometry of parallel lines, a genuine proof that the sum of the interior
angles of a triangle is 180 is available to them:
A
A
A
A
A
C
B
C
B
C
B
C
B
C
Here, A stands for “Alternate” and C for “Corresponding” angles on parallel lines.
This and other very accessible proofs are discussed in Brown (2008) and follow-up assessment ideas
are suggested.
It’s a draw
If the game concluded in a draw, how many possibilities are there for the halftime score? Regardless
of the sporting code involved, for a game where scores increment in single points a draw of n-all can
occur after, remarkably, the square of n+1 different halftime scores.
A diagrammatic proof can be formed by extension of the idea below:
For a ‘three all’ draw, each team must have started on 0, gone to 1 and then 2, and finally 3.
This can be shown as a point with four lines coming from it, like this:
Team B
0
1
2
3
Team A
0
1
2
3
Another nice proof by diagram shows that the sum of the first n odd numbers is the square of n:
An excellent collection of such visual proofs has been made by Nelsen (1993). By the way, did you
notice that the two examples above are really the same thing?
Difference Squares
Four numbers are written to form a square. The differences between the numbers at adjacent
vertices are calculated and written halfway along the sides, forming a square. The process is
repeated, finding absolute differences on each iteration. What results from this?
Obviously, the first step is to experiment. Here is the start with the numbers 1, 4, 6 and 9:
I ask students to prove that any starting values eventually give four zeros. It is a task that requires
generalisation, perhaps along these lines: “The difference between any two non-zero values is less
than the greatest of them, therefore each iteration causes all values to be smaller, and to eventually
become zero”.
It is also important that students refine and explore their own conjectures. For example, what if one
of the numbers is vast while the others are tiny: will the required number of iterations be enormous?
Squares
Squares are both shapes and numbers. The “Squares” CD-ROM (Brown, 2007) introduces many
situations where geometric and algebraic proofs arise. It includes assignments on “It’s A Draw” and
“Difference Squares”, as mentioned above, in nine chapters of material suitable for secondary
students of all ages. “Squares” includes sixteen reproducible posters about squares, showing such
things as a square-wheeled tricycle, a drill bit that produces square holes, and how to draw lines of
exact irrational lengths. These posters can be printed or projected on to interactive whiteboards. The
intention is to stimulate mathematical thinking, and to provide many contexts in which opportunity
for proof arises.
Getting through the curriculum
As you have seen, proof is not the same as memorisation of the classics, with abstruse symbols and
no context. Instead we should consistently offer proofs of all types and of increasing formality as we
cover ordinary curriculum content, interspersed with a few engaging non-routine tasks.
Much of the content we teach can be related to proof, and much of the pattern-recognition we
encourage can be generalised and proved. Proof need not be an extra topic in the curriculum, just
the unique approach that is essential to mathematics.
Recommended Resources
A tiny but invaluable book, perfect for the teacher and for the school library:
Polster, B. (2004). Q.E.D.: Beauty in mathematical proof. New York: Walker.
The classic dialogue on the nature of proof:
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge:
Cambridge University Press.
A free online mathematics magazine for school students (Search for “Proof”):
http://plus.maths.org
Alexander Bogomolny’s brilliant interactive collection of all things mathematical, including many
proofs:
http://www.cut-the-knot.org
Calendar recycling website:
http://www.timeanddate.com/calendar/repeating.html
References
Brown, P. (2007). Squares: Interesting algebra and geometry activities for secondary students.
Adelaide: Australian Association of Mathematics Teachers.
Brown, P. (2008). Opportunity to create proof. In J.Vincent, J. Dowsey & R. Pierce (Eds.), Connected
maths: Proceedings of the Mathematical Association of Victoria Annual Conference 2008 (pp.
54 – 60.) Brunswick, Victoria: Mathematical Association of Victoria.
Gardner, M. (1964) The ambidextrous universe: Left, right, and the fall of parity. Harmondsworth,
Middlesex: Penguin Books.
Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. Washington DC: Mathematical
Association of America.