short cuts

this is a task very similar in purpose to 'both ways'

the powerpoint is here
(download for the animations to work)

it's a good way to introduce expressions, with some conventions for writing a sequence of operations as an expression and then manipulating such algebraic expressions

initially students input a range of numbers and work out the outputs
















"try some simple numbers then some harder ones - maybe big numbers, fractions, decimals, negative numbers"

for integer inputs, by looking at the output numbers, students can find a 'short cut' rule
this is helped by the output numbers all being even
"what happens?"

rather than needing 3 operations you can use a 'shortcut', of two operations

in this case the shortcut is x 2, + 2
or (deliberately designed to have an alternative)
+ 1, x 2













students can then explore a shortcut rule using 'm' for a million, 'b' for a billion etc. as an input number

here's the script:

"let's check out the rule (shortcut of x 2, + 2) for a really big number"
"what's a really big number?"
"no, 3425 is too small"
"OK, a million is good - how many noughts in a million?"
"oh, that's too big, I think I'll just write 'm' for a million"

"a million (m) add 3 is?"
[m + 3] sometimes they say m3 to which you appeal to worldly conventions
"then multiply by 2 is?"
[2m + 6]
"subtract 4 is?"
[2m + 2]

"does this fit with the rule?"
"why is this the same as + 1 then x 2?"

this encourages writing expressions in an appropriate manner 

later work can involve using an expression as an input e.g. 3m + 2 

other sets of three (or more) operations can be introduced:

and shortcuts established:
x 5 then + 10
or
+ 2 then x 5

all of the expressions can be written in two ways

a possible introduction to using brackets to stage operations

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

equivalence route

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

factorising expressions

when factorising e.g. 8n + 12 it can be helpful to decide the highest common factor (4 in this case) and then cross off both numbers, writing how many times 4 goes into each number instead



this then is the factor inside the bracket: 4(2n + 3)

when factorising expressions with powers of symbols, it can be helpful to write out the expression in full, e.g.