ALGEBRAIC GENERALISATION Unlock stories by generalising number properties

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.

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?

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!