the ferrer diagram of 12 as monochrome trains…

‘Ferrer’ means ‘complete breakdown’ that is all.

This below is a breakdown of 12 showing its prime factors:

12

Each row contains ‘trains’ of the same colour. You would ask children to ‘make as many trains of the same colour as you can find that will fit underneath twelve’

(or something else of the same meaning…)

you can call 12 a composite number if you wish…

if a number only has trains of white and itself it is called a prime number…here’s a few:

ferrer 2 comp primes

however the main thing here is to notice that

the number twelve has a very rich internal structure

If you are completely and utterly aware of this, many things become easier to see, conversely, or should I say, inversely,

if you are not totally and instantly aware of the inner structure of 12

THERE WILL BE PAIN…

(particularly with operations involving fractional parts)

let’s hear it for the rods…

quite a long one…

twosevenths

twofifthslong

‘two thirds of three quarters of four fifths of five halves of two sevenths of seven one-ths of one sixth of six fourths of four thirds of 3 fifths is two fifths’

huatou: learn to read the rods

two fifths of five sevenths

twofifths

if the yellow rod is 1, the red is two fifths

if the black rod is 1 the yellow is five sevenths

read straight from the rods: two fifths of five sevenths is two sevenths

the 5 is common to both and cancels out

(the 5 is the ‘denominator’ of the first and ‘numerator’ of the second fractions)

the ‘x’ is an operator, in this case called ‘of’

the flip law 2

it works for multiplication and addition but not subtraction and division.

in multiplication it works for fractions as operators too

play yourself…you must own flip it as second nature

ps the normal name for flip it is the commutative rule

pps dressing is not commutative – you don’t put your socks on over your shoes

48

 

 

48 6x8 2x3x2x2x2

 

The LHS shows 6×8

The RHS shows 2x3X2X2X2  dust ( the 8 is 2x2x2 and the 6 is 2×3 )

As far as the number 48  is concerned the order of rods in the tower is irrelevant, but this needs ‘proving’. Take my word for it at the moment.

48 6x8 2x2x2x2x3

 

So long as the tower is constructed using the rods on the right, the order is irrelevant.

So, as 2x2x2x2x3 is the dust, this means we combine these a pair at a time in any order:

try it yourself..that’s best…but

here’s my mind at work for example:

start with 2, double it double it double it, that’s 16, times 3 is 48 (2 4 8 16 48)

2 threes are six, double it, double it, double it, that’s 48  (6 12 24 48)

2 twos are 4, two fours are 8, three of them is 24, double it, 48

and so on…..

IF YOU HAVE THE TIME AND THE SPACE IN SCHOOL TO DO THIS TILL THE COWS COME HOME AND YOU ARE LITTLE, AND YOU START SLOWLY WITH THE NUMBERS UP TO 10 AT FIRST, STUDYING THE NUMBERS ONE BY ONE FOR A DAY OR TWO EACH FOR EXAMPLE WITHOUT STRESS, YOU WILL ‘GET A FEEL’ FOR THE NUMBER YOU ARE STUDYING WHICH WILL BE VERY POWERFUL IN YOUR FUTURE STUDIES OF THE NUMBER SYSTEM AND OPERATIONS YOU WILL NO DOUBT BE REQUESTED TO CARRY OUT…

(In general, the present school arrangements almost totally inhibit this…)

ps 6×8 is one piece of your ‘tables’, using the dust you see and get the ‘feel’ for 6×8, 8×6, 3×16, 16×3, 2×24, 24×2, 4×12, 12×4, never mind ‘half of 48 is 24’, ‘half of 12 is 6’, ‘half of 48 multiplied by 2 is 48’, ‘a quarter or fourth of 48 is 12’, ‘an eighth of 48 is a half of 12’…and so on till the cows come home…

yap yap yap…

TRY IT

 

 

 

 

little blueboy piggy gives a tip about a certain ‘opposite’…

boypigblue‘the opposite of multiplication is fractions as operators…’

the big bad wolf says, ‘nah, its division…’rods159

the three little pigs

illustrations by Georgien Overwater

story by Susanna Davidson