Download Chapter 3

Chapter 3
Numeration
And
Computation
5
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America’s Funniest Home Videos
Tally’s on a staff
Pebbles in a pouch
Abstract idea of “three-ness” evolved
New Guinea
• “iya” – one
• “rarido” – two
• Additive Number System
Count!
•
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•
•
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Fe
Fi
Fo
Fum
Fiddle
Fruit
Folks
Fist
Fe
Fi
Fo
Fum
Fiddle
Fiddle-fe
Fiddle-fi
Fiddle-fo
Fiddle-fum
Fiddle-fiddle
fiddle-fiddle-fe
fiddle-fiddle-fi
fiddle-fiddle-fo
fiddle-fiddle-fum
fiddle-fiddle-fiddle
fiddle-fiddle-fiddle-fe
fiddle-fiddle-fiddle-fi
fiddle-fiddle-fiddle-fo
fiddle-fiddle-fiddle-fum
fiddle-fiddle-fiddle-fiddle
Fe
Fi
Fo
Fum
Fiddle
Fiddle-fe
Fiddle-fi
Fiddle-fo
Fiddle-fum
Fi-fiddle
fi-fiddle-fe
fi-fiddle-fi
fi-fiddle-fo
fi-fiddle-fum
fo-fiddle
fo-fiddle-fe
fo-fiddle-fi
fo-fiddle-fo
fo-fiddle-fum
fum-fiddle
Written Number System
Number Symbol
Additive Number System
M M M N N N N ^ ^ ^ ^ l l l
M M N N N l l l l
Use a Multiplier
M M M N N N N ^ ^ ^ ^ l l l
M M N N N l l l l
Positional Number System
M M M N N N N ^ ^ ^ ^ l l l
M M N N N l l l l
Egyptian Number System
Page 144
• Additive System
Million Man!
• How Much is Million by David M. Schwartz
• www.davidschwartz.com
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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
• Millions Poster
• Collecting a Million Pennies
• Sharing “Millions” with the Elementary School
• Collecting a Million Pennies in High School
• Spending a Million Dollars
Babylonian Number System
Number Symbol
1
l
10
<
60x60x60
60x60
216,000
3600
60
1
Babylonian Number System
Number Symbol
1
l
10
<
0
60x60x60
60x60
216,000
3600
60
1
Babylonian Number System
<<< llll<<<<<<lllllll
<<llll
<<<<lllll
Mayan Number System
Page 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.
Mayan Number System
20x20x20x18
Number Symbol
144,000
0
20x20x18
1
7200
5
20 x 18
360
20
1
Mayan Number System
Roman Numerals
Roman
Symbol
I
V
X
L
Hindu
Arabic
1
5
10
50
C
D
M
100 500 1000
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.
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
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
I
II
III
IV
V
VI
VII
VIII
IX
X
X
XX
XXX
XL
L
LX
LXX
LXXX
XC
C
I
II
III
IV
V
VI
VII
VIII
IX
X
X
XX
XXX
XL
L
LX
LXX
LXXX
XC
C
C
CC
CCC
CD
D
DC
DCC
DCCC
CM
M
• Write 1469 using Roman Numerals
• Write MMMCMXCIX as a Hindu Arabic
Number.
Multiplication Principal
• 649 =
Multiplication Principal
• 649 = DCXLIX
• 649,000 = DCXLIX
Multiplication Principal
• 649 = DCXLIX
• 649,000 = DCXLIX
• 649,428 =
Multiplication Principal
• 649 = DCXLIX
• 649,000 = DCXLIX
• 649,428 = DCXLIXCDXXVIII
• 649,000,000 =
Multiplication Principal
• 649 = DCXLIX
• 649,000 = DCXLIX
• 649,428 = DCXLIXCDXXVIII
• 649,000,000 = DCXLIX
Hindu-Arabic Numbers
Page 149
Homework Questions
Chapter 2
A
B
C
A
B
Venn Diagram Lab Answers
Test - Chapter 2
• http://mcis.jsu.edu/faculty/mjohnson/ms133r2.html
Day 2
Set Theory Test
Base 10 Number System
10 digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Base 10 Number System
hundred
ten
thousands thousands thousands
millions
hundreds
tens
ones
____ , ____ ____ ____ , ____ ____ ____
10
6
10
5
10
4
10
3
10
2
1
10
10
0
Base 10 Number System
hundred ten
millions millions millions
hundred
ten
thousands thousands thousands
hundreds tens
ones
___ ___ ___ , ___ ___ ___ , ___ ___ ___
• Ones
• Tens
• Hundreds
• Thousands
• Ten Thousands
• Hundred Thousands
• Millions
• Ten Millions
• Hundred Millions
• Billions
• Ten Billions
• Hundred Billions
• Trillions
• Ten Trillions
• Hundred Trillions
• Quadrillions
• Ten Quadrillions
• Hundred Quadrillions
• Quintillions
• Ten Quintillions
• Hundred Quintillions
• Sextillions
• Ten Sextillions
• Hundred Sextillions
• Septillions
• Ten Septillions
• Hundred Septillions
• Octillions
• Ten Octillions
• Hundred Octillions
• Nonillions
• Ten Nonillions
• Hundred Nonillions
• Decillions
• Ten Decillions
• Hundred Decillions
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
100
10
Googol-plex
10
googol
The I Hate Mathematics Book
by Marilyn Burns
Math for Smarty Pants by Marilyn Burns
Spaghetti and Meatballs for All!
by Marilyn Burns
The m&m’s Counting Book
by Barbara Barbieri McGrath
Counting Kisses by Karen Katz
Math Potatoes by Greg Tang
Millions of Cats by Wanda Ga’g
Expanded Notation
Expanded Notation tells what the number
means.
25,683
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)
Reading Numbers
• 25, 638
• “Twenty five thousand, six hundred thirtyeight”
Reading Numbers
• 25, 638, 304
• “Twenty five million, six hundred thirtyeight thousand, three hundred four”
Models for Numeration Lab
BEAN
LONG
LONG-FLAT
FLAT
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
2 Long-flats, 3 Longs, and 4 beans
How many beans total?
269 beans
Make a collection of 42 beans using
the least number of pieces possible.
Make a collection of 42 beans using
the least number of pieces possible.
Begin with 1 Long-flat. Trade in as needed to
give away 12 beans.
What’s left?
4 Flats, 2 Longs, 3 Beans
Base Five
Five Digits: {0, 1, 2, 3, 4}
125’s 25’s
fives ones
. . . _____ _____ _____ _____ _____
54
53
52
51
50
Count in Base 5
1
2
3
4
10
11
12
13
14
20
31
21
32
22
33
23
34
24
40
30
Count in Base 5
1
2
3
4
10
11
12
13
14
20
31
41
21
32
42
22
33
43
23
34
44
24
40
100
30
Base Six
Six Digits: {0, 1, 2, 3, 4, 5}
216’s 36’s
six ones
. . . _____ _____ _____ _____ _____
64
63
62
61
60
Count in Base 6
1
2
3
4
5
10
11
12
13
25
41
53
14
30
42
54
15
31
43
55
20
32
44
100
21
33
45
101
22
34
50
.
23
35
51
.
24
40
52
.
Base Twelve
Twelve Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E}
1728’s 144’s twelve ones
. . . _____ _____ _____ _____ _____
124
123
122
121
120
Count in Base Twelve
1
2
3
4
5
6
7
8
9
15
21
29
T
16
22
2T
E
17
23
2E
10
18
24
30
11
19
25
31
12
1T
26
.
13
1E
27
.
14
20
28
.
Base Two
Two Digits; {0, 1}
8’s
4’s
twos ones
. . . _____ _____ _____ _____ _____
24
23
22
21
20
Count in Base 2
1
10
11
100
101
110
111
1000
10000
11000
1001
10001
11001
1010
10010
11010
1011
10011
11011
1100
10100
11100
1101
10101
11101
1110
10110
11110
1111
10111
11111
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)
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
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 =
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
Start with base 10
382 = _______five
Put 382 beans in groups of 5
382 = ______ five
382 = ____2 five
76 longs and 2 beans left over.
Put 76 longs in groups of 5
382 = ____2 five
76 longs and 2 beans left over.
382 = __ 12 five
76 longs and 2 beans left over.
15 flats, 1 long left over, 2 beans left over.
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.
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.
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.
Short Division
382 = _____five
5 )382 (beans)
(longs)
Short Division
382 = _____five
5 )382 (beans)
76 (longs)
remainder 2
Short Division
382 = _____five
5 )382 (beans)
5 )76 (longs)
15 (flats)
remainder 2
remainder 1
Short Division
382 = _____five
5 )382
5 )76
5 )15
3
(beans)
remainder 2
(longs)
remainder 1
(flats)
remainder 0
(long-flats)
Short Division
382 = _____five
5 )382
5 )76
5 )15
5 )3
0
(beans)
(longs)
(flats)
(long-flats)
remainder 2
remainder 1
remainder 0
remainder 3
Short Division
382 = _____five
5 )382
5 )76
5 )15
5 )3
0
(beans)
(longs)
(flats)
(long-flats)
382 = 3012five
remainder 2
remainder 1
remainder 0
remainder 3
Day 3
Homework Questions
Page 154
Homework Questions
Page 161
Go over Labs
Adding Bean Sticks
324five + 243five
324five + 243five
Make Exchanges
324five + 243five
Make Exchanges
324five + 243five
324five + 243five
1122five
Use your base five pieces to find
each of the following:
1. 43five + 24five
Use your base five pieces to find
each of the following:
1. 43five + 24five = 122five
2. 313five + 233five =
Use your base five pieces to find
each of the following:
1. 43five + 24five = 122five
2. 313five + 233five = 1101five
3. 304five + 20five +120five + 22five =
Use your base five pieces to find
each of the following:
1. 43five + 24five = 122five
2. 313five + 233five = 1101five
3. 304five + 20five +120five + 22five = 1021five
4. 1000five + 100five + 10five =
Use your base five pieces to find
each of the following:
1. 43five + 24five = 122five
2. 313five + 233five = 1101five
3. 304five + 20five +120five + 22five = 1021five
4. 1000five + 100five + 10five = 1110five
Take away model
Take away 3 beans
232five – 143five
232five – 143five
Take away 4 longs
232five – 143five
232five – 143five
Take away 1 flat
232five – 143five
232five – 143five
232five – 143five
34five
Use your bean sticks to complete
the following:
1. 1142five – 213five =
Use your bean sticks to complete
the following:
1. 1142five – 213five = 424five
2. 2331five -124five =
Use your bean sticks to complete
the following:
1. 1142five – 213five = 424five
2. 2331five -124five = 2202five
3. 4112five – 143five =
Use your bean sticks to complete
the following:
1. 1142five – 213five = 424five
2. 2331five -124five = 2202five
3. 4112five – 143five = 3414five
LAB
1221three + 122three
F
LF
F
F
Note your Final Answer
1221three + 122three
F
LF
F
LF
F
F
2120three
Subtract
432six – 144six =
F
F
F
F
432six – 144six =
F
F
F
F
= 244six
LAB
201three
+102three
2312four
+203four
255six
+134six
111two
+101two
2333four
+333four
11011two
+10101two
1221three
-122three
2312four
-203four
1001four
-112four
1010two
-101two
101ten
-11ten
1001three
-112three
Day 4
Homework Questions
Page 177
Worksheet Questions
Scratch Addition
2395
789
5463
1284
985
+677
4567
2396
569
392
1974
+568
Napier’s Bones
6
0
4
0
1
1
2
0
6
2
8
4
0
0
0
1
1
3
3
4
0
6
2
2
2
2
0
4
8
2
6
0
4
8
64 x 36
4
6
0
0
1
1
2
3
3
4
0
0
0
1
1
0 2
6 2
2 2
0
6
2
8
4
0
4
8
2
6
0
4
8
64 x 36
4
6
0
0
1
1
2
3
3
4
0
0
0
1
1
0 2
6 2
2 2
0
6
2
8
4
0
4
8
2
6
0
4
8
1
8
1
2
3 6 2 4
64 x 36 =2304
4
6
0
0
1
1
2
3
3
4
0
0
0
1
1
0 2
6 2
2 2
0
6
2
8
4
0
4
8
2
6
0
4
8
2
1
8
1
2
3 3 6 2 4
0
4
Lattice Multiplication
98 x 47
9
8
4
7
Lattice Multiplication
98 x 47 = 4606
9
3
4
6
8
3
6
2
5
6
3
0
6
6
4
7
Lattice Multiplication
576 x 49
5
7
6
4
9
Lattice Multiplication
576 x 49 = 28,224
5
2 2
0
7
2
6
2
8
8 4 5 6
3
2
2
5
4
4
9
4
4
Egyptian Multiplication
22 x 28
22 x 28 = 616
1
2
4
8
16
28
56
112
224
448
448
112
+56
616
22 x 28
1
2
4
8
16
22 x 28
28
56
112
224
448
= (16 + 4 + 2) x 28
22 x 28
1
2
4
8
16
22 x 28
28
56
112
224
448
= (16 + 4 + 2) x 28
= (16 x 28) + (4 x 28) + (2 x 28)
22 x 28 = 616
1
2
4
8
16
22 x 28
28
56
112
224
448
= (16 + 4 + 2) x 28
= (16 x 28) + (4 x 28) + (2 x 28)
= 448 + 112 + 56
= 616
Egyptian Multiplication
48 x 65
Egyptian Multiplication
48 x 65 = 3120
1
2
4
8
16
32
65
130
260
520
1040
2080
2080
+1040
3120
Russian Peasant Multiplication
32 x 45
Russian Peasant Multiplication
32 x 45 = 1440
32
16
8
4
2
1
45
90
180
360
720
1440
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
(4 x 2) x 180
4 x (2 x 180)
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
(4 x 2) x 180
4 x (2 x 180)
4 x 360
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
(4 x 2) x 180
4 x (2 x 180)
4 x 360
(2 x 2) x 360
2 x (2 x 360)
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
(4 x 2) x 180
4 x (2 x 180)
4 x 360
(2 x 2) x 360
2 x (2 x 360)
2 x 720
32
45
16
90
8
180
4
360
2
720
1
1440
32 x 45
(16 x 2) x 45
16 x (2 x 45)
16 x 90
(8 x 2) x 90
8 x (2 x 90)
8 x 180
(4 x 2) x 180
4 x (2 x 180)
4 x 360
(2 x 2) x 360
2 x (2 x 360)
2 x 720
(1 x 2) x 720
1 x (2 x 720)
1 x 1440
Russian Peasant Multiplication
48 x 65
Russian Peasant Multiplication
48 x 65 = 3120
48
24
12
6
3
1
65
130
260
520
1040
2080
2080
+1040
3120
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
6 x 520
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
6 x 520
3 x 1040
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
6 x 520
3 x 1040
(2 + 1) x 1040
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
6 x 520
3 x 1040
(2 + 1) x 1040
(2x1040)+(1x1040)
48
24
12
6
3
65
130
260
520
1040
1
2080
48 x 65
24 x 130
12 x 260
6 x 520
3 x 1040
(2 + 1) x 1040
(2x1040)+(1x1040)
(1x2080)+(1x1040)
Mental Math
• The ability to make accurate estimates
and do mental arithmetic is increasingly
important in today’s society.
• It is essential that the basic addition and
multiplication facts be memorized since all
other numerical calculations and
estimations depend of this foundation.
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.
Mental Math
One digit facts and the properties of whole
numbers are the basis for mental
calculations.
Mental Math
• Using Easy Combinations
35 + 7 + 15
Mental Math
• Using Easy Combinations
• Using Adjustments in Mental Calculations
57 + 84
Mental Math
• Using Easy Combinations
• Using Adjustments in Mental Calculations
57 + 84
83 - 48
Mental Math
• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
352 + 647
Mental Math
• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
352 + 647
739 - 224
Mental Math
• Using Easy Combinations
• Using Adjustments in Mental Calculations
• Working From Left to Right
352 + 647
739 – 224
4 x 235
8 + 3 + 4 + 6 + 7 + 12 + 4 + 3 + 6 + 3
25 x 8
4 x 99
57 - 25
47 x 5
286 + 347
493 x 7
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 don’t 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?
5,842 to the nearest thousand:
5,842  6,000
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
Round 524 to the nearest hundred:
524≈500
Round 587 to the nearest hundred:
587≈600
Round 549 to the nearest hundred:
549≈500
Round 550 to the nearest hundred:
550≈600
Round 551 to the nearest hundred:
551≈600
5-Up Rule
Page 201
Round 549 to the nearest hundred:
549≈500
Round 550 to the nearest hundred:
550≈500
Round 551 to the nearest hundred:
551≈600
Round 29,853 to the position
indicated.
• Ten thousand:
• 30,000
• Thousand:
• 30,000
• Hundred:
• 29,900
• Ten:
• 29,850
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 =
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
“I have . . . Who has . . . ?”
Math and Music
The Magical Connection!
• Scholastic Parent and Child Magazine
• Spelling
• Phone Numbers
• School House Rock
“Skip to My Lou”
Chorus: Times facts, they’re a breeze;
Learn a few, then work on speed.
Times facts, you’ll be surprised
By just how fast you can
memorize.
3 time 7 is 21
Now, at last we’ve all begun.
4 times 7 is 28
Let’s sing what we appreciate.
(Chorus)
5 times 7 is 35.
Yes, by gosh, we’re still alive.
6 times 7 is 42.
I forgot what we’re supposed to do.
(Chorus)