Chapter 3

Chapter 3

NumerationAnd

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-feFifiddle-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

http://www.davidschwartz.com/

• 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 Symbol1 l

10 <

1

60x60x60

216,000

60x60

3600 60


Number Symbol1 l

10 <0

1

60x60x60

216,000

60x60

3600 60


< < < 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 SystemNumber Symbol

0

1

5

20x20x20x18

144,00020x20x18

720020 x 18

360

20

1

Mayan Number System

Roman Numerals

Roman Symbol

I V X L C D M

HinduArabic

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 =


• 649 = DCXLIX

• 649,000 = DCXLIX


• 649 = DCXLIX

• 649,000 = DCXLIX

• 649,428 =


• 649 = DCXLIX

• 649,000 = DCXLIX

• 649,428 = DCXLIXCDXXVIII

• 649,000,000 =


• 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

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 tenmillions thousands thousands thousands hundreds tens ones

____ , ____ ____ ____ , ____ ____ ____

0123456 10101010101010

Base 10 Number System hundred ten hundred tenmillions 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 LabFLAT

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 Twelve12345678

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 11100101110111

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= 2141324five= 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= 2141324five= 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= 2141324five= 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= 5081324seven= 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


382 = __ 12 five


15 flats, 1 long left over, 2 beans left over.

Put 15 flats in groups of 5

382 = __ 12 five



Put 15 flats in groups of 5

382 = _ 012 five



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



3 long-flats, 0 flats left over, 1 long left over, 2 beans left over.

Short Division382 = _____five

5 )382 (beans) (longs)


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


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


5 )382 (beans) remainder 2 5 )76 (longs) remainder 1 5 )15 (flats) remainder 0

3 (long-flats)


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

0


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


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


1. 43five + 24five = 122five

2. 313five + 233five =



2. 313five + 233five = 1101five

3. 304five + 20five +120five + 22five =



2. 313five + 233five = 1101five

3. 304five + 20five +120five + 22five = 1021five

4. 1000five + 100five + 10five =



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 =


1. 1142five – 213five = 424five

2. 2331five -124five =


1. 1142five – 213five = 424five

2. 2331five -124five = 2202five

3. 4112five – 143five =


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


Worksheet Questions

Scratch Addition

2395 78954631284 985+677

45672396 569 3921974+568

Napier’s Bones

822

02621

1

4

84

2

3

421

3

821

4

0

4

000

0000

6

6

6

64 x 36

822

02621

1

4

84

2

3

421

3

821

4

0

4

000

0000

6

6

6

64 x 36

822

02621

1

4

84

2

3

421

3

821

4

0

4

000

0000

6

6

6

22

1

43

18

6

64 x 36 =2304

822

02621

1

4

84

2

3

421

3

821

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

62

36

3

7

4

89

Lattice Multiplication576 x 49

9

4

675

Lattice Multiplication576 x 49 = 28,224

2

2 25

40

28

2

4

4

9

4

675

53

6

2

4

8

Egyptian Multiplication22 x 28

22 x 28 = 6161 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 281 282 564 1128 22416 448

22 x 28 = (16 + 4 + 2) x 28= (16 x 28) + (4 x 28) + (2 x 28)

22 x 28 = 6161 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 4516 908 1804 3602 7201 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 6524 13012 2606 5203 1040

1 2080

48 65 48 x 6524 130 24 x 13012 2606 5203 1040

1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 5203 1040

1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 520 6 x 5203 1040

1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 520 6 x 5203 1040 3 x 1040

1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 520 6 x 5203 1040 3 x 1040

(2 + 1) x 1040

1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 520 6 x 5203 1040 3 x 1040

(2 + 1) x 1040

(2x1040)+(1x1040)1 2080

48 65 48 x 6524 130 24 x 13012 260 12 x 2606 520 6 x 5203 10403 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 Combinations35 + 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 + 8483 - 48

Mental Math

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

352 + 647

Mental Math


352 + 647

739 - 224

Mental Math


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 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 thousand3,000 + 0 + 0 =

• Round to the nearest hundred3,000 + 500 + 100 =

• Round to the left-most digit3,000 + 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 surprisedBy 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)

Chapter 3

Documents

n n n n

mayan number systempage

lpositional number systemm

hindu arabic number

legyptian number systempage

number system10 digits

leap year

day year