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

expanding and simplifying

starfish

one incorrect simplification

the powerpoint is here

multiple proofs

intended to lend some purpose to expanding and factorising...

maybe more interesting for students to create their own versions after a while

they could substitute a few numbers into the expressions to check that the statements seem to hold

expanding and factorising steps

algebraic product puzzles

find the gaps

equivalent things

the intention of this first resource is that students substitute the given input values
and then, maybe with some nudging, appreciate that there are shorter rules that give the output numbers from the input numbers

and that these simpler expressions can be obtained by cancelling the algebraic fraction

the following resources involve justifying then creating equivalent ('same only look different') expressions to the one in the middle

students can be asked to create their own expressions or versions

Eudoxus' ladder

Eudoxus of Cnidus (now the tip of SW Turkey) was a renowned mathematician (amongst other things) who lived around 400 BC

one of the methods he (reportedly) developed or adopted when studying irrationals was to use a number series (called his 'ladder') in order to approximate to the square root of 2:

how is the 'ladder' formed?
how can it be used to approximate to the square root of 2?









'Reaching the Core of AS Mathematics', available from the ATM, interestingly links this blended recursion 'ladder' to the expansion of:

work out and simplify this expression for n = 2, 3, 4 etc

what has it got to do with Edoxus' ladder and why?

in the limit,

how does this provide an approximation to the square root of 2?

equivalence route

find a route across the grid
so that the two expressions in a box are the same

shrink these