transform up or down to decade boundaries – do many

**do not be afraid to use the open space on paper**

Reply

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’

remember:

**a numeral is the symbol for the idea called number**

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:

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?**

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**

**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…