Post on 31-Dec-2015
Chapter 3
Numeration
And
Computation
5
• 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!
• Fe• Fi• Fo• Fum• Fiddle• Fruit• Folks• Fist
Fe fiddle-fiddle-feFi fiddle-fiddle-fiFo fiddle-fiddle-foFum fiddle-fiddle-fumFiddle fiddle-fiddle-fiddleFiddle-fe fiddle-fiddle-fiddle-feFiddle-fi fiddle-fiddle-fiddle-fiFiddle-fo fiddle-fiddle-fiddle-foFiddle-fum fiddle-fiddle-fiddle-fumFiddle-fiddle fiddle-fiddle-fiddle-
fiddle
Fe fi-fiddle-feFi fi-fiddle-fiFo fi-fiddle-foFum fi-fiddle-fumFiddle fo-fiddleFiddle-fe fo-fiddle-feFiddle-fi fo-fiddle-fiFiddle-fo fo-fiddle-foFiddle-fum fo-fiddle-fumFi-fiddle 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 SystemPage 144
• Additive System
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
• 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 <
1
60x60x60
216,000
60x60
3600 60
Babylonian Number System
Number Symbol
1 l
10 <
0
1
60x60x60
216,000
60x60
3600 60
Babylonian Number System
< < < l l l l < < < < < < l l l l l l l
< < l l l l < < < < l l l l l
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.
Mayan Number System
Number Symbol
0
1
5
20x20x20x18
144,000
20x20x18
7200
20 x 18
360
20
1
Mayan Number System
Roman Numerals
Roman
Symbol
I V X L C D M
Hindu
Arabic
1 5 10 50 100 500 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
IIIIIIIV (the one that comes before 5)VVI (the one that comes after 5)VIIVIIIIX (the one that comes before 10)X
I XII XXIII XXXIV XLV LVI LXVII LXXVIII LXXXIX XCX C
I X CII XX CCIII XXX CCCIV XL CDV L DVI LX DCVII LXX DCCVIII LXXX DCCCIX XC CMX C 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 NumbersPage 149
Homework QuestionsChapter 2
C
BA
BA
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
millions thousands thousands thousands hundreds tens ones
____ , ____ ____ ____ , ____ ____ ____
0123456 10101010101010
Base 10 Number System
hundred ten hundred ten
millions millions millions 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
10010
Googol-plex
googol10
The I Hate Mathematics Book by Marilyn BurnsMath for Smarty Pants by Marilyn BurnsSpaghetti and Meatballs for All! by Marilyn BurnsThe m&m’s Counting Book by Barbara Barbieri McGrathCounting Kisses by Karen KatzMath Potatoes by Greg TangMillions 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 thirty-eight”
Reading Numbers
• 25, 638, 304
• “Twenty five million, six hundred thirty-eight thousand, three hundred four”
Models for Numeration Lab
FLAT
LONG
BEAN
LONG-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
12341011121314
20
21
22
23
24
30
31
32
33
34
40
Count in Base 5
12341011121314
20
21
22
23
24
30
31
32
33
34
40
41
42
43
44
100
Base Six
Six Digits: {0, 1, 2, 3, 4, 5}
216’s 36’s six ones
. . . _____ _____ _____ _____ _____
64 63 62 61 60
Count in Base 612345101112
13
14
15
20
21
22
23
24
25
30
31
32
33
34
35
40
41
42
43
44
45
50
51
52
53
54
55
100
101
.
.
.
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 Twelve1
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
29
2T
2E
30
31
.
.
.
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
1001
1010
1011
1100
1101
1110
1111
10000
10001
10010
10011
10100
10101
10110
10111
11000
11001
11010
11011
11100
11101
11110
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 Division382 = _____five
5 )382 (beans)
(longs)
Short Division382 = _____five
5 )382 (beans) remainder 2
76 (longs)
Short Division382 = _____five
5 )382 (beans) remainder 2
5 )76 (longs) remainder 1
15 (flats)
Short Division382 = _____five
5 )382 (beans) remainder 2
5 )76 (longs) remainder 1
5 )15 (flats) remainder 0
3 (long-flats)
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
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
Day 3
Homework QuestionsPage 154
Homework QuestionsPage 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 modelTake 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
LF
F F
F
Note your Final Answer
1221three + 122three
2120three
LF
F F
FF
LF
Subtract
432six – 144six =
F F F F
432six – 144six =
= 244six
F F F F
LAB
201three 2312four 255six
+102three +203four +134six
111two 2333four 11011two
+101two +333four +10101two
1221three 2312four 1001four
-122three -203four -112four
1010two 101ten 1001three
-101two -11ten -112three
Day 4
Homework QuestionsPage 177
Worksheet Questions
Scratch Addition
2395
789
5463
1284
985
+677
4567
2396
569
392
1974
+568
Napier’s Bones
82
2026
21
1
4
8
4
2
3
421
3
8
21
4
0
4
000
0000
6
6
6
64 x 36
82
2026
21
1
4
8
4
2
3
421
3
8
21
4
0
4
000
0000
6
6
6
64 x 36
82
2026
21
1
4
8
4
2
3
421
3
8
21
4
0
4
000
0000
6
6
6
22
1
43
18
6
64 x 36 =2304
82
2026
21
1
4
8
4
2
3
421
3
8
21
4
0
4
000
0000
6
6
6
2
30 4
22
1
43
18
6
Lattice Multiplication98 x 47
7
4
89
Lattice Multiplication98 x 47 = 4606
4
6
0 6
53 6
6
23
63
7
4
89
Lattice Multiplication576 x 49
9
4
675
Lattice Multiplication576 x 49 = 28,224
2
2 2
54
02
82
4
4
9
4
675
53
6
2
4
8
Egyptian Multiplication22 x 28
22 x 28 = 616
1 282 564 1128 22416 448
448112+56616
22 x 28
1 282 564 1128 22416 448
22 x 28 = (16 + 4 + 2) x 28
22 x 28
1 282 564 1128 22416 448
22 x 28 = (16 + 4 + 2) x 28= (16 x 28) + (4 x 28) + (2 x 28)
22 x 28 = 616
1 282 564 1128 22416 448
22 x 28 = (16 + 4 + 2) x 28= (16 x 28) + (4 x 28) + (2 x 28)= 448 + 112 + 56= 616
Egyptian Multiplication48 x 65
Egyptian Multiplication48 x 65 = 3120
1 652 1304 2608 52016 104032 2080
2080+1040 3120
Russian Peasant Multiplication32 x 45
Russian Peasant Multiplication32 x 45 = 1440
32 45
16 90
8 180
4 360
2 720
1 1440
32 45 32 x 45
16 90
8 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 45
16 90
8 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90
8 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90
8 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180(4 x 2) x 1804 x (2 x 180)
4 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180(4 x 2) x 1804 x (2 x 180)
4 360 4 x 360
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180(4 x 2) x 1804 x (2 x 180)
4 360 4 x 360(2 x 2) x 3602 x (2 x 360)
2 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180(4 x 2) x 1804 x (2 x 180)
4 360 4 x 360(2 x 2) x 3602 x (2 x 360)
2 720 2 x 720
1 1440
32 45 32 x 45(16 x 2) x 4516 x (2 x 45)
16 90 16 x 90(8 x 2) x 908 x (2 x 90)
8 180 8 x 180(4 x 2) x 1804 x (2 x 180)
4 360 4 x 360(2 x 2) x 3602 x (2 x 360)
2 720 2 x 720(1 x 2) x 7201 x (2 x 720)
1 1440 1 x 1440
Russian Peasant Multiplication48 x 65
Russian Peasant Multiplication48 x 65 = 3120
48 6524 13012 2606 5203 10401 2080
2080 +1040
3120
48 65 48 x 65
24 130
12 260
6 520
3 1040
1 2080
48 65 48 x 65
24 130 24 x 130
12 260
6 520
3 1040
1 2080
48 65 48 x 65
24 130 24 x 130
12 260 12 x 260
6 520
3 1040
1 2080
48 65 48 x 65
24 130 24 x 130
12 260 12 x 260
6 520 6 x 520
3 1040
1 2080
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
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
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
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)
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≈500Round 587 to the nearest hundred:587≈600Round 549 to the nearest hundred:549≈500Round 550 to the nearest hundred:550≈600Round 551 to the nearest hundred:551≈600
5-Up RulePage 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 MusicThe 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 21Now, at last we’ve all begun.4 times 7 is 28Let’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)