Numeration

1
Chapter 3

• Numeration
• And
• Computation

2

• 5

3

• Americas Funniest Home Videos
• Tallys on a staff
• Pebbles in a pouch
• Abstract idea of three-ness evolved
• New Guinea
• iya one
• rarido two
• Additive Number System

4
Count!

• Fe
• Fi
• Fo
• Fum
• Fiddle
• Fruit
• Folks
• Fist

5

• Fe fiddle-fiddle-fe
• Fi fiddle-fiddle-fi
• Fo fiddle-fiddle-fo
• Fum fiddle-fiddle-fum
• Fiddle fiddle-fiddle-fiddle
• Fiddle-fe fiddle-fiddle-fiddle-fe
• Fiddle-fi fiddle-fiddle-fiddle-fi
• Fiddle-fo fiddle-fiddle-fiddle-fo
• Fiddle-fum fiddle-fiddle-fiddle-fum
• Fiddle-fiddle fiddle-fiddle-fiddle-fiddle

6

• Fe fi-fiddle-fe
• Fi fi-fiddle-fi
• Fo fi-fiddle-fo
• Fum fi-fiddle-fum
• Fiddle fo-fiddle
• Fiddle-fe fo-fiddle-fe
• Fiddle-fi fo-fiddle-fi
• Fiddle-fo fo-fiddle-fo
• Fiddle-fum fo-fiddle-fum
• Fi-fiddle fum-fiddle

7
Written Number System
8
Additive Number System

• M M M N N N N l l l
• M M N N N l l l l

9
Use a Multiplier

• M M M N N N N l l l
• M M N N N l l l l

10
Positional Number System

• M M M N N N N l l l
• M M N N N l l l l

11
Egyptian Number SystemPage 144

• Additive System

12
Million Man!

• How Much is Million by David M. Schwartz
• www.davidschwartz.com
• If You Made a Million
• The Magic of a Million Activity Book
• Millions to Measure
• If You Hopped Like a Frog
• G is for Googol

13

• Millions Poster
• Collecting a Million Pennies
• Sharing Millions with the Elementary School
• Collecting a Million Pennies in High School
• Spending a Million Dollars

14
Babylonian Number System
15
Babylonian Number System
16
Babylonian Number System

17
Mayan Number SystemPage 146

• As early as 200 BC, these resourceful people had
  developed a remarkably advanced society.
• They were the first Native Americans to develop a
  system of writing and to manufacture paper and
  books.
• Their calendar was very accurate, with a 365 day
  year and a leap year every fourth year.

18
Mayan Number System
19
Mayan Number System
20
Roman Numerals
21
Roman Numerals

• Addition Principle
• Subtraction Principle
• The only things that can be subtracted are 1, 10,
  and 100 (I, X, and C).
• You show subtraction by placing a smaller symbol
  to the left of a larger symbol. You may only
  subtract one symbol at a time.
• You will write one place value at a time.

22
Roman Numerals

• Subtraction Principle
• I can only be subtracted from V and X
• X can only be subtracted from L and C
• C can only be subtracted from D and M

23

• I
• II
• III
• IV (the one that comes before 5)
• V
• VI (the one that comes after 5)
• VII
• VIII
• IX (the one that comes before 10)
• X

24

• I X
• II XX
• III XXX
• IV XL
• V L
• VI LX
• VII LXX
• VIII LXXX
• IX XC
• X C

25

• I X C
• II XX CC
• III XXX CCC
• IV XL CD
• V L D
• VI LX DC
• VII LXX DCC
• VIII LXXX DCCC
• IX XC CM
• X C M

26

• Write 1469 using Roman Numerals

27

• Write MMMCMXCIX as a Hindu Arabic Number.

28
Multiplication Principal

• 649

29
Multiplication Principal

• 649 DCXLIX
• 649,000 DCXLIX

30
Multiplication Principal

• 649 DCXLIX
• 649,000 DCXLIX
• 649,428

31
Multiplication Principal

• 649 DCXLIX
• 649,000 DCXLIX
• 649,428 DCXLIXCDXXVIII
• 649,000,000

32
Multiplication Principal

• 649 DCXLIX
• 649,000 DCXLIX
• 649,428 DCXLIXCDXXVIII
• 649,000,000 DCXLIX

33
Hindu-Arabic NumbersPage 149
34
Homework QuestionsChapter 2
35
(No Transcript)
36
(No Transcript)
37
Venn Diagram Lab Answers
38
Test - Chapter 2

• http//mcis.jsu.edu/faculty/mjohnson/ms133r2.html

39
Day 2
40
Set Theory Test
41
Base 10 Number System

• 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

42
Base 10 Number System

• hundred ten
• millions thousands thousands thousands
  hundreds tens ones
• ____ , ____ ____ ____ , ____ ____
  ____

43
Base 10 Number System

• hundred ten hundred
  ten
• millions millions millions thousands
  thousands thousands hundreds tens ones
• ___ ___ ___ , ___ ___ ___ , ___ ___ ___

44

• Ones
• Tens
• Hundreds
• Thousands
• Ten Thousands
• Hundred Thousands

45

• Millions
• Ten Millions
• Hundred Millions
• Billions
• Ten Billions
• Hundred Billions

46

• Trillions
• Ten Trillions
• Hundred Trillions
• Quadrillions
• Ten Quadrillions
• Hundred Quadrillions

47

• Quintillions
• Ten Quintillions
• Hundred Quintillions
• Sextillions
• Ten Sextillions
• Hundred Sextillions

48

• Septillions
• Ten Septillions
• Hundred Septillions
• Octillions
• Ten Octillions
• Hundred Octillions

49

• Nonillions
• Ten Nonillions
• Hundred Nonillions
• Decillions
• Ten Decillions
• Hundred Decillions

50
Googol

• 10,000,000,000,000,000,000,000,000,
• 000,000,000,000,000,000,000,000,000,
• 000,000,000,000,000,000,000,000,000,
• 000,000,000,000,000,000,000

51
Googol-plex
52

• The I Hate Mathematics Book
• by Marilyn Burns
• Math for Smarty Pants by Marilyn Burns
• Spaghetti and Meatballs for All!
• by Marilyn Burns
• The mms Counting Book
• by Barbara Barbieri McGrath
• Counting Kisses by Karen Katz
• Math Potatoes by Greg Tang
• Millions of Cats by Wanda Gag

53
Expanded Notation

• Expanded Notation tells what the number means.
• 25,683

54
Expanded Notation

• 25,683 20,000 5,000 600 80 3
• 25,683 (2 x 10,000) (5 x 1000)
• (6 x 100) (8 x 10) (3 x 1)
• 25,683 (2 x 104) (5 x 103) (6 x 102)
• (8 x 101) (3 x 100)

55
Reading Numbers

• 25, 638
• Twenty five thousand, six hundred thirty-eight

56
Reading Numbers

• 25, 638, 304
• Twenty five million, six hundred thirty-eight
  thousand, three hundred four

57
Models for Numeration Lab
58
8 beans, 6 longs, 5 flats
Exchange pieces for an equivalent collection (one
that has the same number of beans) using the
least number of pieces. 1 Long-flat, 1 Flat, 2
Longs, 3 beans
59
2 Long-flats, 3 Longs, and 4 beans
How many beans total? 269 beans
60
Make a collection of 42 beans using the least
number of pieces possible.
61
Make a collection of 42 beans using the least
number of pieces possible.
62

• Begin with 1 Long-flat. Trade in as needed to
  give away 12 beans.

Whats left?
63
4 Flats, 2 Longs, 3 Beans
64
Base Five

• Five Digits 0, 1, 2, 3, 4
• 125s 25s fives ones
• . . . _____ _____ _____ _____ _____
• 54 53 52 51 50

65
Count in Base 5

• 1
• 2
• 3
• 4
• 10
• 11
• 12
• 13
• 14

20 21 22 23 24 30
31 32 33 34 40
66
Count in Base 5
20 21 22 23 24 30
31 32 33 34 40
41 42 43 44 100

• 1
• 2
• 3
• 4
• 10
• 11
• 12
• 13
• 14

67
Base Six

• Six Digits 0, 1, 2, 3, 4, 5
• 216s 36s six ones
• . . . _____ _____ _____ _____ _____
• 64 63 62 61 60

68
Count in Base 6
25 30 31 32 33 34 35 40
41 42 43 44 45 50 51 52
53 54 55 100 101 . . .
13 14 15 20 21 22 23 24

• 1
• 2
• 3
• 4
• 5
• 10
• 11
• 12

69
Base Twelve

• Twelve Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T,
  E
• 1728s 144s twelve ones
• . . . _____ _____ _____ _____ _____
• 124 123 122 121 120

70
Count in Base Twelve
29 2T 2E 30 31 . . .

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8

9 T E 10 11 12 13 14
15 16 17 18 19 1T 1E 20
21 22 23 24 25 26 27 28
71
Base Two

• Two Digits 0, 1
• 8s 4s twos ones
• . . . _____ _____ _____ _____ _____
• 24 23 22 21 20

72
Count in Base 2

• 1
• 10
• 11
• 100
• 101
• 110
• 111

1000 1001 1010 1011 1100 1101 1110 1111
10000 10001 10010 10011 10100 10101 10110 10111
11000 11001 11010 11011 11100 11101 11110 11111
73

• 1324five is read one, three, two, four, base
  five
• Expanded notation will tell us what it means.
  (This is the same thing as converting to base 10,
  because base 10 is what we understand.)
• 1324five (1x 53) (3x 52) (2 x 51) (4 x 50)

74

• 1324five (1x 53) (3x 52) (2 x 51) (4 x 50)
• (1 x 125) (3 x 25) (2 x 5) (4 x 1)
• 125 75 10 4
• 214
• 1324five 214ten

75

• 1324five (1x 53) (3x 52) (2 x 51) (4 x 50)
• (1 x 125) (3 x 25) (2 x 5) (4 x 1)
• 125 75 10 4
• 214
• 1324five 214ten
• 1324seven

76

• 1324five (1x 53) (3x 52) (2 x 51) (4 x 50)
• (1 x 125) (3 x 25) (2 x 5) (4 x 1)
• 125 75 10 4
• 214
• 1324five 214ten
• 1324seven(1x 73) (3x 72) (2x 71) (4x70)
• (1 x 343) (3 x 49) (2 x 7) (4 x 1)
• 343 147 14 4
• 508
• 1324seven 508ten

77
Start with base 10

• 382 _______five

78
Put 382 beans in groups of 5

• 382 ______ five

79

• 382 ____2 five
• 76 longs and 2 beans left over.

80
Put 76 longs in groups of 5

• 382 ____2 five
• 76 longs and 2 beans left over.

81

• 382 __ 12 five
• 76 longs and 2 beans left over.
• 15 flats, 1 long left over, 2 beans left over.

82
Put 15 flats in groups of 5

• 382 __ 12 five
• 76 longs and 2 beans left over.
• 15 flats, 1 long left over, 2 beans left over.

83
Put 15 flats in groups of 5

• 382 _ 012 five
• 76 longs and 2 beans left over.
• 15 flats, 1 long left over, 2 beans left over.
• 3 long-flats, 0 flats left over, 1 long left
  over, 2 beans left over.

84
Put 3 long-flats in groups of 5.Not enough you
are finished.

• 382 3012 five
• 76 longs and 2 beans left over.
• 15 flats, 1 long left over, 2 beans left over.
• 3 long-flats, 0 flats left over, 1 long left
  over, 2 beans left over.

85
Short Division382 _____five

• 5 )382 (beans)
• (longs)

86
Short Division382 _____five

• 5 )382 (beans) remainder 2
• 76 (longs)

87
Short Division382 _____five

• 5 )382 (beans) remainder 2
• 5 )76 (longs) remainder 1
• 15 (flats)

88
Short Division382 _____five

• 5 )382 (beans) remainder 2
• 5 )76 (longs) remainder 1
• 5 )15 (flats) remainder 0
• 3 (long-flats)

89
Short Division382 _____five

• 5 )382 (beans) remainder 2
• 5 )76 (longs) remainder 1
• 5 )15 (flats) remainder 0
• 5 )3 (long-flats) remainder 3
• 0

90
Short Division382 _____five

• 5 )382 (beans) remainder 2
• 5 )76 (longs) remainder 1
• 5 )15 (flats) remainder 0
• 5 )3 (long-flats) remainder 3
• 0
• 382 3012five

91
Day 3
92
Homework QuestionsPage 154
93
Homework QuestionsPage 161
94
Go over Labs
95
Adding Bean Sticks

• 324five 243five

96

• 324five 243five

97
Make Exchanges

• 324five 243five

98
Make Exchanges

• 324five 243five

99

• 324five 243five
• 1122five

100
Use your base five pieces to find each of the
following

• 43five 24five

101
Use your base five pieces to find each of the
following

• 43five 24five 122five
• 313five 233five

102
Use your base five pieces to find each of the
following

• 43five 24five 122five
• 313five 233five 1101five
• 304five 20five 120five 22five

103
Use your base five pieces to find each of the
following

• 43five 24five 122five
• 313five 233five 1101five
• 304five 20five 120five 22five 1021five
• 1000five 100five 10five

104
Use your base five pieces to find each of the
following

• 43five 24five 122five
• 313five 233five 1101five
• 304five 20five 120five 22five 1021five
• 1000five 100five 10five 1110five

105
Take away modelTake away 3 beans

• 232five 143five

106

• 232five 143five

107
Take away 4 longs

• 232five 143five

108

• 232five 143five

109
Take away 1 flat

• 232five 143five

110

• 232five 143five

111

• 232five 143five
• 34five

112
Use your bean sticks to complete the following

• 1142five 213five

113
Use your bean sticks to complete the following

• 1142five 213five 424five
• 2331five -124five

114
Use your bean sticks to complete the following

• 1142five 213five 424five
• 2331five -124five 2202five
• 4112five 143five

115
Use your bean sticks to complete the following

• 1142five 213five 424five
• 2331five -124five 2202five
• 4112five 143five 3414five

116
LAB

• 1221three 122three

117
Note your Final Answer

• 1221three 122three
• 2120three

118
Subtract

• 432six 144six

119

• 432six 144six
• 244six

120
LAB
121

• 201three 2312four 255six
• 102three 203four 134six
• 111two 2333four 11011two
• 101two 333four 10101two

122

• 1221three 2312four 1001four
• -122three -203four -112four
• 1010two 101ten 1001three
• -101two -11ten -112three

123
Day 4
124
Homework QuestionsPage 177
125
Worksheet Questions
126
Scratch Addition

• 2395
• 789
• 5463
• 1284
• 985
• 677

127

• 4567
• 2396
• 569
• 392
• 1974
• 568

128
Napiers Bones
129
64 x 36
130
64 x 36
131
64 x 36 2304
132
Lattice Multiplication98 x 47
133
Lattice Multiplication98 x 47 4606
134
Lattice Multiplication576 x 49
135
Lattice Multiplication576 x 49 28,224
136
Egyptian Multiplication22 x 28
137
22 x 28 616

• 1 28
• 2 56
• 4 112
• 8 224
• 16 448
• 448
• 112
• 56
• 616

138
22 x 28

• 1 28
• 2 56
• 4 112
• 8 224
• 16 448
• 22 x 28 (16 4 2) x 28

139
22 x 28

• 1 28
• 2 56
• 4 112
• 8 224
• 16 448
• 22 x 28 (16 4 2) x 28
• (16 x 28) (4 x 28) (2 x 28)

140
22 x 28 616

• 1 28
• 2 56
• 4 112
• 8 224
• 16 448
• 22 x 28 (16 4 2) x 28
• (16 x 28) (4 x 28) (2 x 28)
• 448 112 56
• 616

141
Egyptian Multiplication48 x 65
142
Egyptian Multiplication48 x 65 3120

• 1 65
• 2 130
• 4 260
• 8 520
• 16 1040
• 32 2080
• 2080
• 1040
• 3120

143
Russian Peasant Multiplication32 x 45
144
Russian Peasant Multiplication32 x 45 1440

• 32 45
• 16 90
• 8 180
• 4 360
• 2 720
• 1 1440

145

• 32 45 32 x 45
• 16 90
• 8 180
• 4 360
• 2 720
• 1 1440

146

• 32 45 32 x 45
• (16 x 2) x 45
• 16 90
• 8 180
• 4 360
• 2 720
• 1 1440

147

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90
• 8 180
• 4 360
• 2 720
• 1 1440

148

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• 8 180
• 4 360
• 2 720
• 1 1440

149

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180
• 4 360
• 2 720
• 1 1440

150

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• 4 360
• 2 720
• 1 1440

151

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• (4 x 2) x 180
• 4 x (2 x 180)
• 4 360
• 2 720
• 1 1440

152

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• (4 x 2) x 180
• 4 x (2 x 180)
• 4 360 4 x 360
• 2 720
• 1 1440

153

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• (4 x 2) x 180
• 4 x (2 x 180)
• 4 360 4 x 360
• (2 x 2) x 360
• 2 x (2 x 360)
• 2 720
• 1 1440

154

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• (4 x 2) x 180
• 4 x (2 x 180)
• 4 360 4 x 360
• (2 x 2) x 360
• 2 x (2 x 360)
• 2 720 2 x 720
• 1 1440

155

• 32 45 32 x 45
• (16 x 2) x 45
• 16 x (2 x 45)
• 16 90 16 x 90
• (8 x 2) x 90
• 8 x (2 x 90)
• 8 180 8 x 180
• (4 x 2) x 180
• 4 x (2 x 180)
• 4 360 4 x 360
• (2 x 2) x 360
• 2 x (2 x 360)
• 2 720 2 x 720
• (1 x 2) x 720
• 1 x (2 x 720)
• 1 1440 1 x 1440

156
Russian Peasant Multiplication48 x 65
157
Russian Peasant Multiplication48 x 65 3120

• 48 65
• 24 130
• 12 260
• 6 520
• 3 1040
• 1 2080
• 2080
• 1040
• 3120

158

• 48 65 48 x 65
• 24 130
• 12 260
• 6 520
• 3 1040
• 1 2080

159

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260
• 6 520
• 3 1040
• 1 2080

160

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520
• 3 1040
• 1 2080

161

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520 6 x 520
• 3 1040
• 1 2080

162

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520 6 x 520
• 3 1040 3 x 1040
• 1 2080

163

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520 6 x 520
• 3 1040 3 x 1040
• (2 1) x 1040
• 1 2080

164

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520 6 x 520
• 3 1040 3 x 1040
• (2 1) x 1040
• (2x1040)(1x1040)
• 1 2080

165

• 48 65 48 x 65
• 24 130 24 x 130
• 12 260 12 x 260
• 6 520 6 x 520
• 3 1040 3 x 1040
• (2 1) x 1040
• (2x1040)(1x1040)
• 1 2080 (1x2080)(1x1040)

166
Mental Math

• The ability to make accurate estimates and do
  mental arithmetic is increasingly important in
  todays society.
• It is essential that the basic addition and
  multiplication facts be memorized since all other
  numerical calculations and estimations depend of
  this foundation.

167
Mental Math

• This should NOT be rote memorization of symbols.
  Students should experience the facts by frequent
  use of manipulatives, games, puzzles, and problem
  solving activities.
• In the same way, students learn basic properties
  of whole numbers and use them to figure out any
  fact they may have forgotten.

168
Mental Math

• One digit facts and the properties of whole
  numbers are the basis for mental calculations.

169
Mental Math

• Using Easy Combinations
• 35 7 15

170
Mental Math

• Using Easy Combinations
• Using Adjustments in Mental Calculations
• 57 84

171
Mental Math

• Using Easy Combinations
• Using Adjustments in Mental Calculations
• 57 84
• 83 - 48

172
Mental Math

• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
• 352 647

173
Mental Math

• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
• 352 647
• 739 - 224

174
Mental Math

• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
• 352 647
• 739 224
• 4 x 235

175

• 8 3 4 6 7 12 4 3 6 3

176

• 25 x 8

177

• 4 x 99

178

• 57 - 25

179

• 47 x 5

180

• 286 347

181

• 493 x 7

182
Rounding

• When we are asked to round 5,842 to the nearest
  thousand, it is because we want something close
  to 5,842 without any small pieces. We dont want
  anything any smaller than a group of a thousand.
• 5,842 is between 5,000 and 6,000. Which one is
  it closest to?

183

• 5,842 to the nearest thousand
• 5,842 6,000

184

• 67,498,499 to the nearest thousand
• 67,498,499 is between
• 67,498,000 and 67,499,000
• Which one is it closer to?
• To the nearest thousand
• 67,498,499 67,498,000

185

• Round 524 to the nearest hundred
• 524500
• Round 587 to the nearest hundred
• 587600
• Round 549 to the nearest hundred
• 549500
• Round 550 to the nearest hundred
• 550600
• Round 551 to the nearest hundred
• 551600

186
5-Up RulePage 201
187

• Round 549 to the nearest hundred
• 549500
• Round 550 to the nearest hundred
• 550500
• Round 551 to the nearest hundred
• 551600

188
Round 29,853 to the position indicated.

• Ten thousand
• 30,000
• Thousand
• 30,000
• Hundred
• 29,900
• Ten
• 29,850

189
Approximate By Rounding 2,954 482 82

• Round to the nearest thousand
• 3,000 0 0
• Round to the nearest hundred
• 3,000 500 100
• Round to the left-most digit
• 3,000 500 80

190
Round to the left-most digit to find approximate
answer.

• 2681 241
• 3000 200 3200
• 2681 241
• 3000 200 2800
• 2681 x 241
• 3000 x 200 600,000
• 57801 336
• 60,000 300 200

191
I have . . . Who has . . . ?
192
Math and MusicThe Magical Connection!

• Scholastic Parent and Child Magazine
• Spelling
• Phone Numbers
• School House Rock

193
Skip to My Lou

• Chorus Times facts, theyre a breeze
• Learn a few, then work on speed.
• Times facts, youll be surprised
• By just how fast you can memorize.

194

• 3 time 7 is 21
• Now, at last weve all begun.
• 4 times 7 is 28
• Lets sing what we appreciate.
• (Chorus)
• 5 times 7 is 35.
• Yes, by gosh, were still alive.
• 6 times 7 is 42.
• I forgot what were supposed to do.
• (Chorus)