ALGEBRAIC GENERALISATION Unlock stories by generalising number properties

ANDREW WILES Why is this man so famous?

FERMAT’S LAST THEOREM

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.

PROOF  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

 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.

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?

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!