How Numbers Work
Numbers were invented by Humans. In the early days there were
simple bead counts. Counting is almost as fundamental to
language as words. Some cultures had little use for numbers
and their mathematics is quite primitive.
Australian Aborigines used "1,2,3,many". I explain this as...
if a hunter sees 1 kangaroo it could be the last, so he leaves
it. If he sees 2 kangaroo he can kill one and leave the other.
If he sees 3 kangaroo he can have a feast. More than 3 is not
an issue and there is simply more. He can report all this
using language. Note there is no 0, zero. It makes no sense to
return from a hunt and confess to see"NO KANGAROO". His gin
(wife) would think he had been in the bush too long and
probably knock him on the nut. Notice also there is no
negative numbers. Like zero it makes no sense to come home and
report "NOT SEEING 2 KANGAROO". His gin would definitely knock
him on the nut becasue not only is he not seeing kangaroo he
is now counting the things he can not see! Fractions make no
sense either because things happen in KANGAROO units and are
never seen as HALF a KANGAROO. There is no need to add or
divide so maths for the Australian Aborigine was pretty
primitive. They did draw a lot of pictures but it is
archeologists who scrapped away thousands of years of
habitation and tagged a time line. They did count hiigher than
3 by "subitising". [Subitising is a term that was coined by
the theorist Piaget and defined the ability to instantaneously
recognise the number of objects in a small group without the
need to count them.]. Looking at a flock of birds or a herd of
kanagroio teh aborigine can associate 5, 10 etc to the group.
We all do this.
Well, it is a good story anyway. See "First Footprints" a
book,and a documentary, about the 80,000year old journey
of the Australian Aborigine. https://www.imdb.com/title/tt3286844/
The Romans used a peculiar system that prevented operations
like adding and multiplying. They used symbols I, V, X, L, C,
D and M for 1,5,10, 50, 100, 500 and 1000. The by combining
these they made 4 = IV or 1 less than 5. Centuries are written
on most big screen films using Roman Numerals so 1978 =
MCMLXXVIII. It is more clear if we expand as M CM LXX VIII.
The Romans had a few other things on their minds than doing
maths but a lot of our language comes from them. A Centurion
was one of 100 soldiers who made up a fighting unit as a 10x10
Who invented the Place-Value system? A simple Google Search
reveals most things.
"Only three cultures, as far as we know, invented a place
value numeration system: the Mayans, the Babylonians, and the
Hindu people of India. The current, almost planetary wide,
place value numeration system is derived from the Hindu
system. It was transmitted to western Europe by the Moslem
"The most commonly used system of numerals is the Hindu–Arabic
numeral system. Two Indian mathematicians are credited with
developing it. Aryabhata of Kusumapura developed the
place-value notation in the 5th century and a century later
Brahmagupta introduced the symbol for zero."
Place value means just that. The value depends on the place of
the number. In the illustration of 369 we see 369 is actually
300 + 60 + 9 and we say three humndred and sixty-nine but we
could say 3 hundreds, 6 tens and 9 ones.
Bilbo in the book The HOBBIT has an interesting and better
interpretation of placevalue numbers. He counts one, two,
three, ...eight, nine, ten, tenty-one, tenty-two,
...tenty-eight, tenty nine, two-ty, twoty - one,...ninety -
nine, tenty-ten and so on. The Maori counting is similar with
11 being te ko ma tahi or 10 + 1. The Chinese system is
similar. The words we use for teen-numbers has its roots in
French, Italian and the Romans. Where on earth did eleven and
twelve thirteen come from. There was an ancient Anglo Saxon
word called fife or 5. Hence fifteen = fife and ten.
Problem Based Learning with Place-Value.
Problem-based learning is fun. There are no long boring
exercises that you have to complete before you are allowed to
go and play! in PBL you get to collaborate and share and talk.
It is fun. As a teacher I am never convinced that what I am
teaching is being learned and what is being learned has
anything to do with what I am teaching! So you explore and ask
questions and learn whatever it is you need to learn. That is
how it works. I can still test you for knowledge, skills and
problem solving. So, have a play with these problems and do
not forget you can share pictures and ideas and questions with
me at email@example.com
1. Take a look at The Original Mind-Reader.
Link = http://www.flashlightcreative.net/swf/mindreader/
Try and figure out how all this works! It caused havoc on
the web when first released with people thinking it was
infecting computers. It is a simple, well fairly simple,
explanation based around place-value and designed in a
brilliant way by Andy with some random effects added. Very
entertaining and I use this a lot. By struggling to make
sense of this problem you grow your brain mathematically.
2. Think up
a three digit number, for example, 234.
Now write twice to make the 6 figure number 234234. Divide
this 6 figure number by 7, and then answer by 13 and then that
answer by 11. The number you get should be 234 or the same as
you started with? How does this work? This problem is a very
cool problem and has connections to many other place-value
problems. (Use a calculator!)
3. Here is a simple one. In the number 123,456,789, 000, or
123 billion, 456 million, 789 thousand, is there a "3". I
could ask is there a 1, a 2, a 3, a 4 or any digit? What is
down a 4 digit number and best at this stage
if all the digits are different. (For example 3761.) Now
scramble the same digits to make another number. (For example
7136.) Now find the difference between the two numbers. (Hint
large - small). (7136 - 3761 = 3375).
There are a couple of curious things about this
- It is divisible by 9 even though neither 7136
nor 3761 were divisible by 9. I was fascinated by this as
a young lad and spent a long time trying to figure out
- If you add the individual digits of 3375 we get 3+3+7+5
= 18 which is always a multiple fo 9 and if we keep adding
we get 1+8=9 and 9 is what we will always get. Adding the
digits gives a term called the "digital root" . Here is
alink on n-Rich for Digital Roots - https://nrich.maths.org/5524and
and another https://banking.currentaffairsonly.in/digital-root-or-seed-number-trick-for-bank-exams/
All just for fun.
5. Here is a heap of place value problems. https://nrich.maths.org/public/leg.php?group_id=1&code=6
6. Rules of Division.
Best here I think! https://www.mathsisfun.com/divisibility-rules.html
7. Here is test
of what you know about place-value.
Write down the numeral (use digits) for "eleven thousand,
eleven hundred and eleven".
All of the above works because we use a Base 10 Place-Value
Counting system. We can use any whole number greater than 1
for a base. Computers use Base 2 so their number look like
110101001 and only use 2 digits 0 and 1. Base 10 uses 10
digits 0,1,2,3,4,5,6,7,8,9. Base 16 uses
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F and is also used in computer
Reflection is always good. So what did you learn? I liked the
Mind Reader the best.
always that anyone can learn anything if you put
your mind to it.