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

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

Written Number System NumberSymbol

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 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 NumberSymbol 1 l 10 <

Babylonian Number System NumberSymbol 1 l 10 < 0

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 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 NumberSymbol 0 1 5

Mayan Number System

Roman Numerals Roman Symbol IVXLCDM Hindu Arabic

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

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.

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

Venn Diagram Lab Answers

Test - Chapter 2

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 ____, ____ ____ ____, ____ ____ ____

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

Googol-plex

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, , ,683 = (2 x 10,000) + (5 x 1000) + (6 x 100) + (8 x 10) + (3 x 1) 25,683 = (2 x 10 4 ) + (5 x 10 3 ) + (6 x 10 2 ) + (8 x 10 1 ) + (3 x 10 0 )

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

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.

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... _____ _____ _____ _____ _____

Count in Base

Count in Base

Base Six Six Digits: {0, 1, 2, 3, 4, 5} 216’s 36’s six ones... _____ _____ _____ _____ _____

Count in Base

Base Twelve Twelve Digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E} 1728’s 144’s twelve ones... _____ _____ _____ _____ _____

Count in Base Twelve T E T 1E T 2E

Base Two Two Digits; {0, 1} 8’s 4’s twos ones... _____ _____ _____ _____ _____

Count in Base

1324 five 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.) 1324 five = (1x 5 3 ) + (3x 5 2 ) + (2 x 5 1 ) + (4 x 5 0 )

= (1 x 125) + (3 x 25) + (2 x 5) + (4 x 1) = = five = 214 ten

1324 five = (1x 5 3 ) + (3x 5 2 ) + (2 x 5 1 ) + (4 x 5 0 ) = (1 x 125) + (3 x 25) + (2 x 5) + (4 x 1) = = five = 214 ten 1324 seven =

1324 five = (1x 5 3 ) + (3x 5 2 ) + (2 x 5 1 ) + (4 x 5 0 ) = (1 x 125) + (3 x 25) + (2 x 5) + (4 x 1) = = five = 214 ten 1324 seven =(1x 7 3 ) + (3x 7 2 ) + (2x 7 1 ) + (4x7 0 ) = (1 x 343) + (3 x 49) + (2 x 7) + (4 x 1) = = seven = 508 ten

Start with base = _______ five

Put 382 beans in groups of = ______ five

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

Put 76 longs in groups of = ____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 = __ 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 = _ 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)remainder 2 76 (longs)

Short Division 382 = _____ five 5 )382 (beans)remainder 2 5 )76 (longs)remainder 1 15 (flats)

Short Division 382 = _____ five 5 )382 (beans)remainder 2 5 )76 (longs)remainder 1 5 )15 (flats)remainder 0 3 (long-flats)

Short Division 382 = _____ five 5 )382 (beans)remainder 2 5 )76 (longs)remainder 1 5 )15 (flats)remainder 0 5 )3 (long-flats)remainder 3 0

Short Division 382 = _____ five 5 )382 (beans)remainder 2 5 )76 (longs)remainder 1 5 )15 (flats)remainder 0 5 )3 (long-flats)remainder = 3012 five

Day 3

Homework Questions Page 154

Homework Questions Page 161

Go over Labs

Adding Bean Sticks 324 five five

Make Exchanges 324 five five

Make Exchanges 324 five five

1122 five

Use your base five pieces to find each of the following: 1.43 five + 24 five

Use your base five pieces to find each of the following: 1.43 five + 24 five = 122 five five five =

Use your base five pieces to find each of the following: 1.43 five + 24 five = 122 five five five = 1101 five five + 20 five +120 five + 22 five =

Use your base five pieces to find each of the following: 1.43 five + 24 five = 122 five five five = 1101 five five + 20 five +120 five + 22 five = 1021 five five five + 10 five =

Use your base five pieces to find each of the following: 1.43 five + 24 five = 122 five five five = 1101 five five + 20 five +120 five + 22 five = 1021 five five five + 10 five = 1110 five

Take away model Take away 3 beans 232 five – 143 five

Take away 4 longs 232 five – 143 five

Take away 1 flat 232 five – 143 five

34 five

Use your bean sticks to complete the following: five – 213 five =

Use your bean sticks to complete the following: five – 213 five = 424 five five -124 five =

Use your bean sticks to complete the following: five – 213 five = 424 five five -124 five = 2202 five five – 143 five =

Use your bean sticks to complete the following: five – 213 five = 424 five five -124 five = 2202 five five – 143 five = 3414 five

LAB 1221 three three

Note your Final Answer 1221 three three 2120 three

Subtract 432 six – 144 six =

= 244 six

LAB

201 three 2312 four 255 six +102 three +203 four +134 six 111 two 2333 four two +101 two +333 four two

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

Day 4

Homework Questions Page 177

Worksheet Questions

Scratch Addition

Napier’s Bones

64 x 36

64 x 36 =2304

Lattice Multiplication 98 x 47

Lattice Multiplication 98 x 47 = 4606

Lattice Multiplication 576 x 49

Lattice Multiplication 576 x 49 = 28,224

Egyptian Multiplication 22 x 28

22 x 28 =

22 x x 28 = ( ) x 28

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

22 x 28 = x 28 = ( ) x 28 = (16 x 28) + (4 x 28) + (2 x 28) = = 616

Egyptian Multiplication 48 x 65

Egyptian Multiplication 48 x 65 =

Russian Peasant Multiplication 32 x 45

Russian Peasant Multiplication 32 x 45 =

x

x 45 (16 x 2) x

x 45 (16 x 2) x x (2 x 45)

x 45 (16 x 2) x x (2 x 45) x

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90)

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90) x

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90) x 180 (4 x 2) x x (2 x 180)

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90) x 180 (4 x 2) x x (2 x 180) x

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90) x 180 (4 x 2) x x (2 x 180) x 360 (2 x 2) x x (2 x 360)

x 45 (16 x 2) x x (2 x 45) x 90 (8 x 2) x 90 8 x (2 x 90) x 180 (4 x 2) x x (2 x 180) x 360 (2 x 2) x x (2 x 360) x

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

Russian Peasant Multiplication 48 x 65

Russian Peasant Multiplication 48 x 65 =

x

x x

x x x

x x x x

x x x x x

x x x x x 1040 (2 + 1) x

x x x x x 1040 (2 + 1) x 1040 (2x1040)+(1x1040) 12080

x x x x x 1040 (2 + 1) x 1040 (2x1040)+(1x1040) 12080(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

Mental Math Using Easy Combinations Using Adjustments in Mental Calculations

Mental Math Using Easy Combinations Using Adjustments in Mental Calculations

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

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

Mental Math Using Easy Combinations Using Adjustments in Mental Calculations Working From Left to Right – x 235

25 x 8

4 x 99

47 x 5

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, = Round to the nearest thousand 3, = Round to the nearest hundred 3, = Round to the left-most digit 3, =

Round to the left-most digit to find approximate answer = = – 241 = 3000 – 200 = x 241 = 3000 x 200 = 600, ÷ 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)