awareness 2 some examples in rod pre-number…

some algebra before arithmetic pre-number awarenesses

examples of free play awarenesses:

rods are good things to play with

we can make pictures

we can build models and buildings

we can play games

we can invent games

we can share our ideas

we can learn from each other

rod colours are in families

rods can be packed away in family colour order if we like

rods can be packed away in other ways

rods have a definite regular order of size

I can recognise rods by how they feel

rods can be named by colour and in other ways

directed informal awarenesses:

rods of same colour are same length

rods of same length are same colour

groups can be made of the same colour

groups can be made of the same lengths

language (word concept) awarenesses:

trains can be made in a variety of ways

rods can be swapped for others

staircases can be made in a variety of ways

mats can be made with rows of rods of the same colour

we can answer questions by making patterns of rods

awarenesses relating to adding and its inverse:

I can find one rod to fit two others

I can find two or more rods to fit one

I can find rods to fit long trains

I can find the difference between two rods in different ways

I can find the difference between two trains of any length

awarenesses relating to equivalence, multiplication and factors:

I can find one rod which fits two or more rods of the same colour

I can find rods of the same colour to fit some other longer rods

I can find rods of the same colour to fit some, but not all trains

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:

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:

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…

dave eats a 1 rod and britney spears

That’s a bit annoying Dave… we can’t see all the complements to 10 now, we’ll have to imagine that one above the 9, as well as the missing one above the 5…

what can you see?

complements to 10 and the sum of all the numbers to 10

a nice rectangle 5 x10 + 5 = 55

the old devil al-jebr would see:

a half of any number multiplied by the same number plus a half of that particular number is;

the sum of all numbers up to that number.

so, add all the numbers up to 100:  50 x 100  + 50  makes 5050   nice…

the power of al-jebr

by the way, here’s the same infamous Dave eating Britney:

video

( thanks to MMc and J )

48

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…

TRY IT

dust lies on top of tables…

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…

dust – your first view of the number 6

Here is  a six rod, with a mat pattern beneath.

Each line of the pattern can only have rods of the same colour. Only ‘same-coloured’ rods that fit the 6 are allowed this time and for this reason:

These same- coloured rods are FACTORS of 6… (2 and 3 are factors of 6)

6 is called the PRODUCT

The towers to the right say ‘three twos’ and ‘two threes’. They are ‘crossed rods’ and we read them as multiplications. They are equivalent to the line of red rods and the line of green rods to the left.

They are DUST

They are FACTOR TOWERS in their simplest form

They show the ATOMS of 6, if you like…

The INNER STRUCTURE of 6

6 can be re-constructed from this DUST in EVERY (multiplicative) WAY

The TOWER OF DUST shows the INNER STRUCTURE of the NUMBER 6

(in the multiplicative domain)