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’

no wonder adding up is tricky…

If you are asked to do an addition you are really being asked to

add the coefficients of a polynomial equation

Look , here’s what 326 really means:

polynomial1

3 is the coefficient of the second power, 2 is the coefficient of the first power and 6 is the coefficient of the zeroth power.

THAT’S WHY ITS HARD FOR LITTLE KIDS TO UNDERSTAND AT FIRST

huatou: why polynomial?

the flip law 1

This rule is profound and will change many things, illustrated here with a few rods:

SONY DSC

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

2x  3

with little children, the easiest and most meaningful form is by using this choice in the attaching of the operator

because:

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

2  x3

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…

disambiguation 2 the ‘rod’ domain and the ‘marks on 2D paper’ domain

If you are keen to work with Cuisenaire rods and arithmetic, you should be aware that you will be working in several different areas, or domains. If you are using real rods, this is the ROD DOMAIN. It exist in the familiar world of 3D objects all around us. What we are going to be looking at are “Parallel worlds”, or “Similar worlds”. We are going to gradually (I can’t stress that word enough) move over into other worlds, which are similar, but not the same. Same means, “another one or an identical thingy”.

Similar means “somewhat like….” So we start with the rod domain.

We might start making marks on paper. This is not the rod domain, we have switched over to the flat 2D world of “marks on paper”. This is a flat, more abstract world where we make marks that represent real things. You can make marks of different natures. You could make ‘drawing’ marks, ‘letter’ marks or even, ‘numeral’ marks. These each have different characteristics. Imagine you are a child. They are not real things in the sense of the real 3D things you are used to. Your teachers might think they are a bit removed. You might think it seems a lot removed from what you are used to and are familiar with. In fact if you are not really ready to operate in this world, it will become a very confusing world, a world where you can feel inadequate, stupid and incapable of seeing precisely what you are being asked. You will have entered a FEARFUL SPACE OF HORROR. You will not want to come here again. You will dread it and just like a snail who’s touched something it doesn’t like, you will RETRACT YOUR FEELERS. Remember this. It’s not where I meant to go, but as it is so important I went there to emphasise it anyway. I was becoming paragraphically ambiguous. However, if the above does happen, IT IS A CRIME AGAINST HUMANITY, really, a criminal offence that should be punished. I’m not joking. It happens all time because teachers in particular either don’t know what they are doing and/or don’t have enough time to do this domain switching properly and gradually. Lets start again with this domain disambiguation……already we have seen, the pure rod domain, the rod plus speech domain, the drawings on paper domain, the written language domain, the numeral domain………..You see, it is a bit complicated for a little child, as it is for you maybe…..