‘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
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
This rule is profound and will change many things, illustrated here with a few rods:
language and rod domain (with a bit of number):
two threes is just as long as three twos, and also they have the same volume
number domain with some signs:
2 x 3 = 3 x 2 = 6
al-jebr domain with signs:
a x b = b x a = c
more and more abstract, more and more general
disambiguation blurb: in the ‘real’ world, two green rods is not the same as three reds. This is why some people object to agreeing that 2 x 3 is the same as 3 x 2. They are correct. However, in the number domain, the answer, which is a pure number, is not affected by the order. The product as a number is invariant to the transformation. If you want another example, it is as invariant as taking a homotopy group functor on the category of topological spaces. You probably don’t need this information. As most people working on calculations are looking for ‘the answer’, one can say that for all intents and purposes, calculationally speaking,
the order of operations in multiplication is irrelevant
further more, if you wish to mention it, and which also makes no difference to the product, the number sentences, transformations or equations, whatever you want to call them, contain the sign ‘x’. This sign is called an operator and it has to be attached to something. It has to be attached either to the first numeral or the second in this case. If it is attached to the first numeral, like this ‘2x’, this ‘whole’ is again called an operator, in this case a ‘doubling’ operator. In language it says ‘two lots’ or ‘two groups’, so this, 2x 3 says
two lots of three or two threes
with little children, the easiest and most meaningful form is by using this choice in the attaching of the operator
a) just like in reading, one reads the first numeral first and
b) one doesn’t have to hold the first number in the mind to the same degree as the form below whilst reading the second number. (Saying ‘two threes’ seems somehow less complicated for little learning minds than saying ‘two multiplied by three’)
c) one doesn’t even have to mention the ‘x’ in language, merely recognise it ….two threes
nevertheless the product is still invariant to order
if the operator is attached to the 3, we get 2 x3 which says:
two multiplied by three
most teachers call this the ‘correct’ way, but it is just one way
the x3 becomes a ‘trebling’ operator
so, in summary, and for the benefit of little children:
THE ORDER IS IRRELEVANT
and this 2 x 3 with the operator ‘x’ in the middle, at first means ‘two threes’ to little people
later, with much practice, it looks like ‘two threes and three twos’ at the same time
(ps you can introduce all the other ways of saying it whenever you feel it’s appropriate)
JUST UNDERSTAND WHAT’S GOING ON…
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.
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…
Here’s 8 with its factors: ‘two fours’ and ‘four twos’ which you see to the right.
Remember if you can find rods of the same colour which are the same length as another rod, as in the picture to the left, they are called factors of that number.
At the extreme right is the DUST of 8, ‘two times two times two’, 2x2x2
This is the ATOMIC STRUCTURE OF 8 in terms of multiplication.
Why is it useful and very very good indeed?
Because from the dust, 2x2x2, you can, if you feel like it:
Build ALL combinations of factors of a product
THIS BEATS ‘tables’
DUST EATS ‘tables’
DUST IS ABOVE ‘tables’
DUST BEATS ‘tables’
DUST LIES ON TOP OF ‘tables’ AS WE KNOW ONLY TOO WELL!
ps if you keep saying ‘tables’ it sounds weird too…
illustrations by Georgien Overwater
story by Susanna Davidson
Sometimes you are in great danger of either running out of rods or wanting to show how to understand a better way of writing out or showing large numbers. See this:
Put the 5 across the 10. Get used to this, its so useful. Its just another way of showing the same thing. If we agree on it, it works, that’s all.
Here’s another that comes from the ‘real’ world, though you still see it on your flat screen:
This is not two sixes, its six times six, six multiplied by six, its six sixes, its 36.
We have gone UP rather than staying on the flat. We have moved into another dimension. COOL…
The height of our little 6 rods is 2. There are two layers. To use an even cooler way of saying the same thing, so long as we agree, which we do, you can do this:
This means 6 times 6, 6 multiplied by 6, six sixes or:
SIX RAISED TO THE POWER 2
Another way of saying this is six squared.
Take a look at googol and googolplex:http://wp.me/p3kPBg-8Q