### place value - A Learning Place A Teaching Place

```PLACE VALUE
Before Place Value number systems were created, people used Symbol Value number systems.
This means a symbol retained its value regardless of its place in the number. In Roman Numerals
the value of the symbol X is always 10, for example XI, IX, CCCDXVII - the value of the X is always
10. Symbol Value number systems are additive - they use addition and subtraction to calculate the
value of the number. For example, CCCDXVII is 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1 = 367
In Place Value number systems, the value of a symbol changes dependent on its place in the
number. For example in the current day place value number system (Hindu Arabic), the symbol
(numeral) 3 has a value of 3 ones in 23 and 3 tens in 32. In Place Value number systems, we also
add the values of the symbols to calculate the value of the number. For example, 23 is 20 + 3 and
32 is 30 + 2
PLACE VALUE NUMBER SYSTEMS ARE BOTH ADDITIVE AND MULTIPLICATIVE!
PLACE VALUE NUMBER SYSTEMS - A BRIEF HISTORY
The Place Value number system the world uses today is one of many created and used by different
civilisations. Some of the different Place Value number systems in use include:
Origin
Base
Hindu Arabic
Chinese (Abacus)
10
Sumerian
Babylonian
Mesopotamian
60
The base refers to the number the civilisation decided to
multiply by to calculate the value of subsequent places, for
example in the Mayan Place Value number system, the
first place had a value of 1, and the next place had a value
of 20 (1 x 20). Their symbols included
Numbers were written from bottom to top. Below you can
see how the number 32 was written:
Mayan
Mesoamerican
North American
Danish
Irish
Gaulish
20
North America
Central America
4
Yuki Tribe(California)
8
Papua New Guinea
27
20s
1s

(1 x 20 = 20)
(2 x 1 + 2 x 5 = 20)
Place Value number systems were created by civilisations
who were involved in scientific study, particularly
Astronomy. Famously, the Mayans created calendars using
their number systems which traced the path of planets
through the skies. Interestingly, some have interpreted the
end of the Mayan calendar to be an ominous prediction!
Our own calendar ends every year - on December 31 - and
then a new calendar begins the next day!
The Hindu Arabic Place Value number system is the one that has survived and is used
internationally today. This is not because it was the best one (Base 12 would make multiplication
easier as 12 has more factors than 10!), but because of the Hindu's proximity to other civilisations
who were also involved in scientific study. The Hindu's Place Value number system, created
between 300 and 500CE, soon spread west through Muslim civilisations, where the numerals were
changed - thus the number system became known as the Hindu Arabic number system.
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The number system allowed the Hindu and Muslim civilisations to make huge advances in
Mathematics and Scientific study during the European 'dark ages' - so called partly due to the lack
of progress of European civilisations (some would say regression) during the same period.
Following time spent in Muslim Algiers, Italian Mathematician Leonardo Fibonacci recognised the
superiority of the Hindu Arabic Place Value number system and introduced it to Europe through
the publication in 1202CE of his Liber Abaci (Book of Calculation) - and the dark ages ended and
the renaissance began!
From then it was widely used in European mathematics, and with the invention of the printing
press in Europe around 1440CE, it replaced Roman Numerals in general use. As the Chinese were
also using a Base 10 number system, transition to the Hindu Arabic Place Value number system
was natural
Initially, the Hindu Arabic Place Value number system was used solely for whole numbers. The
value of the lowest place was ones. In this way, all other values could be calculated from the ones
place (see Place Value is Multiplicative below). Fractions were used using different notations including numerator and denominator
Extending the Place Value system to include fractions, began with Jewish and Persian
Mathematicians who recognised that using different systems for recoding whole numbers and
fractions made calculation difficult. When the Place Value number system was extended to include
fractions, it soon became less obvious which place was the ones place and so some kind of
notation was needed to identify the ones place (and to separate the whole numbers from the
fractions). For example 13 as 1 ten and 3 ones, and 13 as 1 one and 3 tenths, look the same
Initially different people used different notations to identify the ones place. In his booklet De
Thiende ('the art of tenths'), first published in Dutch in 1585, Flemish Mathematician Simon Stevin
used numbered circles, for example 65∙728, he recorded as 65728. Others used a bar over
the ones place to identify it. European countries finally settled on a comma(,) however English
speaking countries used the comma to separate groups of numbers (for example 1765293 was
recorded 1,765,293) so Britain used a mid dot (∙) and the United States used period (dot) on the
base line. Australia adopted the British mid dot. All notations are 'decimal' points because of the
decimal nature of our numbers system (see Place Value is Multiplicative below). Modern
computers generally use a dot on the base line
TEACHING PLACE VALUE
Place Value is Additive, which means that the values of the digits are added together to
determine the value of the number. Simply naming the place of a digit is not demonstration of an
understanding of additive place value. For example, identifying that the in 23 the 2 is in the tens
place and the 3 is in the ones place does not demonstrate understanding that we have 2 tens and
3 ones - and that we are adding the 2 tens to the 3 ones to calculate the value of the number
Additive Place Value begins in Kindergarten as students develop their understanding that teen
means 10 and… It continues into Year 1 and Year 2 as students develop their understanding that
10 can be seen and described in 2 ways - as 1 ten and 10 ones. From Year 1, students develop their
capacity to understand and describe place value flexibly as they partition two- and three-digit
whole numbers into standard and non-standard place value parts. For example, 54 can be seen
and described as 5 tens and 4 ones, 4 tens and 14 ones, 3 tens and 24 ones, 2 tens and 34 ones, 1
ten and 44 ones, and 54 ones! 327 can be seen and described as 3 hundreds and 2 tens and 7
ones, 32 tens and 7 ones, 327 ones, 2 hundreds and 12 tens and 7 ones, 2 hundreds and 2 tens
and 107 ones...!
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Additive Place Value continues into Year 3 and beyond, as students extend their capacity to
understand and describe place value flexibly as they partition two- and three-digit numbers
including decimal fractions into standard and non-standard place value parts. For example, 5∙4 can
4
be seen and described as 5 ones and 4 tenths (5 ones and 10), 4 ones and 14 tenths (4 ones and
14
), 3 ones and 24 tenths (3 ones and
10
44
24
), 2 ones and 34 tenths (2 ones and
10
54
34
), 1 one and 44
10
tenths (1 one and 10), and 54 tenths (10)! 3∙27 can be seen and described as 3 ones and 2 tenths
and 7 hundredths (3 ones and
327
2
10
and
7
), 32 tenths and 7 hundredths (32 tenths and
100
hundredths (100), 2 ones and 2 tenths and 107 hundredths (2 ones and
Value is Multiplicative below for more on decimal fractions)
2
10
and
107
100
7
), 327
100
)...! (see Place
Understanding of additive place value is essential for students to add, subtract, multiply, divide
and create fractions intelligently using partitioning (in both standard and non-standard place value
parts), number relationships and properties
Place Value is also Multiplicative which means that the values of places are 10 times
greater as we move to the left and 10 times lower as we move to the right. The decision to
increase 10 times as we move places to the left and to decrease 10 times as we move places to the
right is not a mathematical one - the other way would have worked just as well! As long as we all
agree that this is how we will assign values to numbers, we will all read numbers in the same way
This understanding that we are multiply and dividing by 10 as we move between places is vital to
understanding decimal fractions. The value of the place to the right of the ones place is calculated
1
1
by dividing 1 by 10 to get 10 (tenths). Further dividing 10 by 10 gives the value of the next place to
1
the right - 100 (hundredth). Symbols were needed and created to identify the ones place, (see
Place Value Number Systems - A Brief History above) resulting in a decimal point
The decimal point has no greater importance than to identify the ones place, thus separating the
whole numbers from the fractions. It is a decimal point simply because our Place Value number
system is decimal. The prefix (first letters) in decimal are 'dec' - meaning 10 (think decade - 10
years, and decagon - a shape with 10 angles and sides). Values of places to the right of the decimal
point are calculated in exactly the same way as values of places to the left - we simply divide by 10
as we move places to the right and multiply by 10 as we move places to the left
The decimal point cannot move - as then it would no longer identify the ones place nor separate
whole numbers from fractions! Understanding that the values of the places increase 10 times as
we move to the left, and decrease 10 times as we move places to the right develops
understanding that when multiplying and dividing by 10, the numbers move between places regardless of whether we are multiplying or dividing a whole number or a decimal fraction!
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