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 SYMBOL NUMBER SYSTEMS ARE ADDITIVE! 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. Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach Scan the QR Code Facebook: A Learning Place 1 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 65728. 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...! Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach Scan the QR Code Facebook: A Learning Place 2 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! Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach Scan the QR Code Facebook: A Learning Place 3

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