1. Section 2.4 Numeration Mathematics for Elementary School Teachers - 4th Edition ODAFFER, CHARLES, COONEY, DOSSEY, SCHIELACK

2. The word symbol for cat is different than the actual cat A symbol is different from what it represents

3. Here is another familiar numeral (or name) for the number two Numeration Systems Just as the written symbol 2 is not itself a number. The written symbol, 2, that represents a number is called a numeral.

4. Definition of Numeration System An accepted collection of properties and symbols that enables people to systematically write numerals to represent numbers. (p. 106, text) Hindu-Arabic Numeration System Egyptian Numeration System Babylonian Numeration System Roman Numeration System Mayan Numeration System

5. Hindu-Arabic Numeration System Developed by Indian and Arabic cultures It is our most familiar example of a numeration system Group by tens: base ten system 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Place value - Yes! The value of the digit is determined by its position in a numeral Uses a zero in its numeration system

6. Definition of Place Value In a numeration system with place value, the position of a symbol in a numeral determines that symbols value in that particular numeral. For example, in the Hindu-Arabic numeral 220, the first 2 represents two hundred and the second 2 represents twenty.

7. Models of Base-Ten Place Value Base-Ten Blocks - proportional model for place value Thousands cube, Hundreds square, Tens stick, Ones cube or block, flat, long, unit text, p. 110 2,345

8. Models of Base-Ten Place Value Colored-chip model: nonproportional model for place value chips from text, p. 110 One Ten One Hundred One Thousand 3,462

9. Expressing Numerals with Different Bases: Show why the quantity of tiles shown can be expressed as (a) 27 in base ten and (b)102 in base five, written 102 five (a) form groups of 10 we can group these tiles into two groups of ten with 7 tiles left over (b) form groups of 5 we can group these tiles into groups of 5 and have enough of these groups of 5 to make one larger group of 5 fives, with 2 tiles left over. 27 No group of 5 is left over, so we need to use a 0 in that position in the numeral: 102 five 102 five

10. Find the base-ten representation for 1324 five Find the base-ten representation for 344 six Find the base-ten representation for 110011 two = 1(125) + 3(25) + 2(5) + 4(1) 1324 five = (1×5 3 ) + (3×5 2 ) + (2×5 1 ) + (4×5 0 ) = 125 + 75 + 10 + 4 = 214 ten Expressing Numerals with Different Bases:

11. Find the representation of the number 256 in base six 6 4 = 1296 6 3 = 216 6 2 = 36 6 0 = 1 6 1 = 6 256 - 216 40 -36 4 1(216) + 1(36) + 0(6) + 4(1) = 1104 six 1(6 3 ) + 1(6 2 ) + 0(6 1 ) + 4(6 0 ) Expressing Numerals with Different Bases:

12. Change 42 seven to base five First change to base 10 42 seven = 4(7 1 ) + 2(7 0 ) = 30 ten Then change to base five 5 3 = 125 5 2 = 25 5 0 = 1 5 1 = 5 30 - 25 5 - 5 0 30 ten = 1( 5 2 ) + 1 ( 5 1 ) + 0 ( 5 0 ) = 110 five Expressing Numerals with Different Bases:

13. Expanded Notation: 1324 = (1×1000) + (3×100) + (2×10) + (4×1) 1324 = (1×10 3 ) + (3×10 2 ) + (2×10 1 ) + (4×10 0 ) Example (using base 10): or This is a way of writing numbers to show place value, by multiplying each digit in the numeral by its matching place value.

14. Egyptian Numeration System Developed: 3400 B.C.E One Ten One Hundred One Thousand Ten Thousand One Hundred Thousand One Million reed heel bone coiled rope lotus flower bent finger burbot fish kneeling figure or astonished man Group by tens New symbols would be needed as system grows No place value No use of zero

15. Babylonian Numeration System Developed between 3000 and 2000 B.C.E There are two symbols in the Babylonian Numeration System Base 60 Place value one ten 42(60 1 ) + 34(60 0 ) = 2520 + 34 = 2,554 Zero came later Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Babylonian system:

16. Roman Numeration System Developed between 500 B.C.E and 100 C.E. (one) (five) (ten) (fifty) (one hundred) (five hundred) (one thousand) Group partially by fives Would need to add new symbols Position indicates when to add or subtract No use of zero 900 + 90 + 9 = 999 Write the Hindu-Arabic numerals for the numbers represented by the Roman Numerals:

17. Mayan Numeration System Developed between 300 C.E and 900 C.E Base - mostly by 20 Number of symbols: 3 Place value - vertical Use of Zero Symbols = 1 = 5 = 0 Write the Hindu-Arabic numerals for the numbers represented by the following numerals from the Mayan system: 0(20 0 ) = 0 6(20 1 ) = 120 8(20 ×18) = 2880 2880 + 120 + 0 = 3000

18. Summary of Numeration System Characteristics SystemGroupingSymbols Place Value Use of Zero Egyptian By tens Infinitely many possibly needed No Babylonia n By sixtiesTwoYesNot at first Roman Partially by fives Infinitely many possibly needed Position indicates when to add or subtract No Mayan Mostly by twenties Three Yes, Vertically Yes Hindu- Arabic By tensTenYes

19. The End Section 2.4 Linda Roper