algebraic generalisation

ALGEBRAIC GENERALISATION

Unlock stories by generalising number properties

ANDREW WILES

Why is this man so famous?

FERMAT’S LAST THEOREM

No positive integers satisfy the equation:

n > 2

ON DOING MATHEMATICS…

Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion.

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.

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.

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

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

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.

PROOFS

Show that the sum of consecutive

numbers is always odd

Show that the sum of three consecutive

numbers is always divisible by three

SOPHIE GERMAIN

OTHER FACTS

Took 358 years before it was proved

It took 7 years for Andrew Wiles to prove it

The proof is 150 pages long

WHO IS NEW ZEALAND’S MOST FAMOUS

MATHEMATICIAN?

Vaughan Jones

Only winner of Fields medal (the mathematics equivalent of the Nobel Prize)

HOW DID HE WIN IT?

Vaughan Jones was attending a conference in Mexico…

His car broke down…

WHAT DO MATHEMATICIANS DO?

He started looking at a dot pattern on the cover of a maths textbook…

WHAT DO MATHEMATICIANS DO?

WHAT DO MATHEMATICIANS DO?

He began experimenting with the mathematics that he saw in the dot pattern…

WHAT DO MATHEMATICIANS DO?

And noticed a link between the dots and knots…

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

WHAT DO MATHEMATICIANS DO?

And he won the Fields Medal.

WOW!