ALGEBRAIC GENERALISATION Unlock stories by generalising number properties.
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Transcript of ALGEBRAIC GENERALISATION Unlock stories by generalising number properties.
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ALGEBRAIC GENERALISATION
Unlock stories by generalising number
properties
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ANDREW WILES
Why is this man so famous?
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FERMAT’S LAST THEOREM
No positive integers satisfy the equation:
n > 2
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ON DOING MATHEMATICS…
Perhaps I can best
describe my
experience of doing
mathematics in terms
of a journey through a
dark unexplored
mansion.
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FINDING THE FURNITURE…
You enter the first room
of the mansion and it's
completely dark. You
stumble around bumping
into the furniture, but
gradually you learn where
each piece of furniture is.
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THE LIGHT GOES ON
Finally, after six months or so,
you find the light switch, you
turn it on, and suddenly it's all
illuminated. You can see exactly
where you were. Then you move
into the next room and spend
another six months in the dark.
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So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of -- and couldn't exist without -- the many months of stumbling around in the dark that proceed them.
AFTER 7 YEARS WILES PROVED FERMAT’S LAST
THEOREM
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ALGEBRAIC GENERALISATION
Aim:
• To explore algebraic generalisations of number strategies
Success Criteria:
• I can generalise from a number strategy
• I can explain why an algebraic identity is always true
• I can use identities to manipulate algebraic expressions
• I know key algebra vocabulary and recording conventions
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EGG TECHNIQUE
E – Explain the strategy or method used to
solve the problem.
G – Give other examples that use the same
strategy or method.
G – Generalise – use algebra to show the
underlying structure.
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PROOF
Show that the sum of consecutive
numbers is always odd
Show that the sum of three
consecutive numbers is always
divisible by three
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SOPHIE GERMAIN
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I used to come up to my study, and start trying to find patterns. I tried doing calculations
which explain some little piece of mathematics. I tried to fit it in with some previous broad
conceptual understanding of some part of mathematics that would clarify the particular
problem I was thinking about. Sometimes that would involve going and looking it up in a
book to see how it's done there. Sometimes it was a question of modifying things a bit,
doing a little extra calculation. And sometimes I realized that nothing that had ever been
done before was any use at all. Then I just had to find something completely new; it's a
mystery where that comes from. I carried this problem around in my head basically the
whole time. I would wake up with it first thing in the morning, I would be thinking about it
all day, and I would be thinking about it when I went to sleep. Without distraction, I would
have the same thing going round and round in my mind. The only way I could relax was
when I was with my children. Young children simply aren't interested in Fermat. They just
want to hear a story and they're not going to let you do anything else.
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FACTS
Took 358 years before it was proved
It took 7 years for Andrew Wiles to prove it
The proof is 150 pages long
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WHO IS NEW ZEALAND’S MOST FAMOUS
MATHEMATICIAN?
Vaughan Jones
Only winner of
Fields medal (the
mathematics
equivalent of the
Nobel Prize)
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HOW DID HE WIN IT?
Vaughan Jones
was attending
a conference
in Mexico…
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His car
broke
down…
WHAT DO MATHEMATICIANS DO?
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He started
looking at a dot
pattern on the
cover of a maths
textbook…
WHAT DO MATHEMATICIANS DO?
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WHAT DO MATHEMATICIANS DO?
He began
experimenting with
the mathematics
that he saw in the
dot pattern…
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WHAT DO MATHEMATICIANS DO?
And noticed a
link between
the dots and
knots…
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WHAT DO MATHEMATICIANS DO?
This lead to him developing a formula for describing knots:
V(T) = (1/t) (t – 1 – t – 3 – t – 1 + t – 2 + 1) = t – 4 + t – 3 + t – 1 Which is now called the Jones’ polynomial
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WHAT DO MATHEMATICIANS DO?
And he won the Fields Medal.
WOW!