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Chapter 3

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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

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

Fefiddle-fiddle-fe

Fifiddle-fiddle-fi

Fofiddle-fiddle-fo

Fumfiddle-fiddle-fum

Fiddlefiddle-fiddle-fiddle

Fiddle-fefiddle-fiddle-fiddle-fe

Fiddle-fifiddle-fiddle-fiddle-fi

Fiddle-fofiddle-fiddle-fiddle-fo

Fiddle-fumfiddle-fiddle-fiddle-fum

Fiddle-fiddlefiddle-fiddle-fiddle-fiddle

Fefi-fiddle-fe

Fifi-fiddle-fi

Fofi-fiddle-fo

Fumfi-fiddle-fum

Fiddlefo-fiddle

Fiddle-fefo-fiddle-fe

Fiddle-fifo-fiddle-fi

Fiddle-fofo-fiddle-fo

Fiddle-fumfo-fiddle-fum

Fi-fiddlefum-fiddle

M M M N N N N ^ ^ ^ ^ l l l

M M N N N l l l l

M M M N N N N ^ ^ ^ ^ l l l

M M N N N l l l l

M M M N N N N ^ ^ ^ ^ l l l

M M N N N l l l l

- Additive System

- 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

< < < l l l l < < < < < < l l l l l l l

< < l l l l < < < < l l l l l

- 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.

- 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.

- 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

IX

IIXX

IIIXXX

IVXL

VL

VILX

VIILXX

VIIILXXX

IXXC

XC

IXC

IIXXCC

IIIXXXCCC

IVXLCD

VLD

VILXDC

VIILXXDCC

VIIILXXXDCCC

IXXCCM

XCM

- Write 1469 using Roman Numerals

- Write MMMCMXCIX as a Hindu Arabic Number.

- 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

- http://mcis.jsu.edu/faculty/mjohnson/ms133r2.html

10 digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

hundredten

millions thousands thousands thousands hundreds tens ones

____ , ____ ____ ____ , ____ ____ ____

hundred ten hundredten

millions millions millions thousands thousands thousands hundredstensones

_________ , ______ ___ , _________

- 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

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

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 tells what the number means.

25,683

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)

- 25, 638
- “Twenty five thousand, six hundred thirty-eight”

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

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

How many beans total?

269 beans

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

What’s left?

Five Digits: {0, 1, 2, 3, 4}

125’s 25’s fives ones

. . . _____ _____ _____ _____ _____

5453525150

1

2

3

4

10

11

12

13

14

20

21

22

23

24

30

31

32

33

34

40

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

Six Digits: {0, 1, 2, 3, 4, 5}

216’s 36’s six ones

. . . _____ _____ _____ _____ _____

6463626160

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

Twelve Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E}

1728’s 144’s twelve ones

. . . _____ _____ _____ _____ _____

124123122121120

29

2T

2E

30

31

.

.

.

9

T

E

10

11

12

13

14

15

16

17

18

19

1T

1E

20

21

22

23

24

25

26

27

28

1

2

3

4

5

6

7

8

Two Digits; {0, 1}

8’s 4’s twos ones

. . . _____ _____ _____ _____ _____

2423222120

1000

1001

1010

1011

1100

1101

1110

1111

10000

10001

10010

10011

10100

10101

10110

10111

11000

11001

11010

11011

11100

11101

11110

11111

1

10

11

100

101

110

111

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

382 = _______five

382 = ______ five

382 = ____2five

76 longs and 2 beans left over.

382 = ____2five

76 longs and 2 beans left over.

382 = __ 12five

76 longs and 2 beans left over.

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

382 = __ 12five

76 longs and 2 beans left over.

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

382 = _ 012five

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.

382 = 3012five

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.

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

324five + 243five

324five + 243five

324five + 243five

324five + 243five

324five + 243five

1122five

- 43five + 24five

- 43five + 24five = 122five
- 313five + 233five =

- 43five + 24five = 122five
- 313five + 233five = 1101five
- 304five + 20five +120five + 22five =

- 43five + 24five = 122five
- 313five + 233five = 1101five
- 304five + 20five +120five + 22five = 1021five
- 1000five + 100five + 10five =

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

232five – 143five

232five – 143five

232five – 143five

232five – 143five

232five – 143five

232five – 143five

232five – 143five

34five

- 1142five – 213five =

- 1142five – 213five = 424five
- 2331five -124five =

- 1142five – 213five = 424five
- 2331five -124five = 2202five
- 4112five – 143five =

- 1142five – 213five = 424five
- 2331five -124five = 2202five
- 4112five – 143five = 3414five

1221three + 122three

1221three + 122three

2120three

432six – 144six =

432six – 144six =

= 244six

201three2312four 255six

+102three+203four+134six

111two2333four 11011two

+101two+333four+10101two

1221three2312four 1001four

-122three-203four -112four

1010two101ten 1001three

-101two -11ten -112three

2395

789

5463

1284

985

+677

4567

2396

569

392

1974

+568

128

256

4112

8224

16448

448

112

+56

616

128

256

4112

8224

16448

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

128

256

4112

8224

16448

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

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

128

256

4112

8224

16448

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

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

= 448 + 112 + 56

= 616

165

2130

4260

8520

161040

322080

2080

+1040

3120

3245

1690

8180

4360

2720

11440

324532 x 45

1690

8180

4360

2720

11440

324532 x 45

(16 x 2) x 45

1690

8180

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

1690

8180

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

8180

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

8180

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

(4 x 2) x 180

4 x (2 x 180)

4360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

(4 x 2) x 180

4 x (2 x 180)

43604 x 360

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

(4 x 2) x 180

4 x (2 x 180)

43604 x 360

(2 x 2) x 360

2 x (2 x 360)

2720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

(4 x 2) x 180

4 x (2 x 180)

43604 x 360

(2 x 2) x 360

2 x (2 x 360)

27202 x 720

11440

324532 x 45

(16 x 2) x 45

16 x (2 x 45)

169016 x 90

(8 x 2) x 90

8 x (2 x 90)

81808 x 180

(4 x 2) x 180

4 x (2 x 180)

43604 x 360

(2 x 2) x 360

2 x (2 x 360)

27202 x 720

(1 x 2) x 720

1 x (2 x 720)

114401 x 1440

4865

24130

12260

6520

31040

12080

2080

+1040

3120

486548 x 65

24130

12260

6520

31040

12080

486548 x 65

2413024 x 130

12260

6520

31040

12080

486548 x 65

2413024 x 130

1226012 x 260

6520

31040

12080

486548 x 65

2413024 x 130

1226012 x 260

65206 x 520

31040

12080

486548 x 65

2413024 x 130

1226012 x 260

65206 x 520

310403 x 1040

12080

486548 x 65

2413024 x 130

1226012 x 260

65206 x 520

310403 x 1040

(2 + 1) x 1040

12080

486548 x 65

2413024 x 130

1226012 x 260

65206 x 520

310403 x 1040

(2 + 1) x 1040

(2x1040)+(1x1040)

12080

486548 x 65

2413024 x 130

1226012 x 260

65206 x 520

310403 x 1040

(2 + 1) x 1040

(2x1040)+(1x1040)

12080(1x2080)+(1x1040)

- 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.

- 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.

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

- Using Easy Combinations
35 + 7 + 15

- Using Easy Combinations
- Using Adjustments in Mental Calculations
57 + 84

- Using Easy Combinations
- Using Adjustments in Mental Calculations
57 + 84

83 - 48

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

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

739 - 224

- 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

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

Round 549 to the nearest hundred:

549≈500

Round 550 to the nearest hundred:

550≈500

Round 551 to the nearest hundred:

551≈600

- Ten thousand:
- 30,000

- 30,000

- 29,900

- 29,850

- 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 =

- 2681 + 241 =
3000 + 200 = 3200

•2681 – 241 =

3000 – 200 = 2800

•2681 x 241 =

3000 x 200 = 600,000

- 57801 ÷ 336 =
60,000 ÷ 300 = 200

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

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)