1

GeneralisingGeneralising

Gosford HillGosford Hill

May 2010May 2010

2

Remainders of the Day (1)Remainders of the Day (1) Write down a number which when Write down a number which when

you subtract 1 is divisible by 5you subtract 1 is divisible by 5 and anotherand another and anotherand another Write down one which you think Write down one which you think

no-one else here will write down.no-one else here will write down. How can you check quickly How can you check quickly

whether someone’s example whether someone’s example number is correct?number is correct?

3

Remainders of the Day (2)Remainders of the Day (2)

Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2

and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33

and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4

Why must any such number be Why must any such number be divisible by 3? divisible by 3?

4

Four ConsecutivesFour Consecutives

Write down four Write down four consecutive numbers consecutive numbers and add them upand add them up

and anotherand another and anotherand another Now be more extreme!Now be more extreme! What is the same, and What is the same, and

what is different about what is different about your answers?your answers?

+ 1

+ 2

+ 3

+ 64

Can 44 be the sum of four consecutive numbers?

5

Up & Down SumsUp & Down Sums

= =

1 + 3 + 1= 2x2 + 1 = 22 + 12

= =

1 + 3 + 5 + 3 + 1= 3x4 + 1 = 32 + 22

6

One Sum DiagramsOne Sum Diagrams

1

1

(1- )2

Anticipating,not waiting

1-2

7

Reading a DiagramReading a Diagram

a

a

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

x2z + x(1-x) + (1-x)2(1-z)

xz + (1-x)(1-z)xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-

z)

8

Which looks bigger …Which looks bigger …

Yellow Annulus Yellow Annulus or or Light Green Light Green CircleCircle??