two fraction proofs

the first is from Fawn Nguyen
the second is from Tony Gardiner

a ppt is here

the second proof involves multiplying out brackets

two geometrical proofs

an intention for this work is to put up statements, in a deductive order and for students to think about and then hopefuly give reasons for these statements

a ppt for the regular pentagon is here
a ppt for a triangle (and congruence) is here
both of these are entirely rigorous

it's just that one of them is false....

the first is from Tony Gardiner, involving a regular pentagon
leading to (a well known) result derived from similar triangles
I very much like the way that Tony has structured this, so that statments build upon each other

in this case, students establish that EB is parallel to DC by considering angles















the second is fairly well known
it helpfully uses congruency conditions

a recap on congruency conditions for triangles before using these

summing the cubes

Nicomachus appreciated this rule (~60 to 120 CE)

thanks to David Wells for the numerical approach

a ppt is here

this task can begin with an exploration of numbers
with a result being easy to find

it can be justifed using numbers

there is a novel diagrammatic representation

algebra can be tackled (if you can expand a cubic bracket) to justify the uniqueness of the first few results

a result that is needed later

see youtube

or nrich 325

how can you be sure that this process continues?

an interesting alternative diagram
possibly using Cuisenaire rods

algebra used to justify that you need to start at 1
(for the first two, three and four consecutive cube cases anyway)

the three consecutive cube case is easiest:

some square number patterns

all due to David Wells

proofs could involve expanding brackets and simplifying expressions
or use the difference of two squares equivalence
or could be considered by using diagrams

power proofs

a ppt is here

working out rules for powers

using Pascal's triangle, establish that because if 'n' is a digit, n^5 ends in the same digit 'n'
then so does (n + 10)^5
and so does (n + 10k)^5

red and yellow spheres

picking twice, with replacement

a ppt is here

exploring whether or not you are more or less likely to obtain two spheres of the same colour (RR or YY) than two speres of different colours (RY or YR)

 or use a tree diagram

options with a total of 9 spheres still

various totals

you can just consider half of the diagram

is the fawn area larger or smaller than the red plus the yellow?

fold once

fold twice

approached algebraically

an alternative algebraic justification

utilising pythagoras to justify the result

equable triangles

the proof that there are only five equable triangles (integer sided) was done in 1904

this version of a proof is largely due to David Wells with small bits of help from me

it is complex but involves only GCSE tools