DOCUMENT RESUME

ED 127'192' SE 021 229.. .

AUTHOR Adams, Patricia A., 51. Nyberg, Luanne, Ed.TITLE Change and Calculations: MINNEMAST Coordinated ,

Mathematics - Science Series, Unit 24INSTITUTION Minnesota Univ., Minneapolis. Minnesota School

. Mathematics and Science Center.SPONS AGENCY, ,National Science Foundation, Washington, D.C.PUB'DAft . 71 -

NOTE . 158p.; For related documents, see SE021201-234;r Photographs may not reproduce well

AVAILABLE/FROM MINNEMAST, Minnemath Center, 720 Washington Ave.,S. E.., Minneapolis, MN 55414

EDRS PRICE MF-$0.83.HC-48.69 Plus Postage.DESCRIPTORS *Algorithms; Computers; *Curriculum Guides;

Elementary Education; *Elementary School Mathematics;*Elementary School Science; Experimental Curriculum;*Interdisciplinary Approach; Learning Activities;,Mathema'tics Education; Measurement; primary Grades;Process Education; Science Education; Units of Study(Subject Fields) ..

. 'IDENTIFIERS * MINNEMAST; *Minnesota.ylatj!!matics and ScienceTeaching Project

ABSTRACTThis volume is the twenty-fourth in a-series of 29

coordinated.MINNEMAST units in mathematics and science forkindergarten. and the primary grades. Intended for use by third-gradeteachers, this unit guide proiides a summary and overview of theunit, a. list of materials needed, and descriptions of four groups of .

lesSons,and activities. The purposes and procedures for each activityare dis-cugsed. Examples -of-que-stton-s--ancd--d-i-sciision-topicsand in, several cases ditto masters, stories for reading aloud, andother instructional-materials are included in the book: This unitcovers both comnytaticnal and measurement ideas. In .the two sets oflessons on computation, tlifeclass simulates a computer in activitiesdesigned to promote understanding of addition and subtraction in a

.place value system. Measurement activities are related to' liquidvolume, length and time. Two job booklets, one on pouring and theother on balancing as methods of measurement, provide activities for

'independent work by Students. (SD)

**********************************************************************i.* Documents acquired by ERIC include many informal unpublished* materials not available frOm other sources. ERIC makes every effort ** to obtain the best copy available. Nevertheless, items of marginal ** reproducibility are often encountered and this affects the. quality ** of the microfiche and hardcopy reproductions.ERIC makes available * .

* via- the ERIC Document Reproduction Service (EDRS) . IDES is not* responsible for the quality of the original document. Reproductions ** supplied by EDRS are the best that can be made frci the original. ************************************************************************

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CAMPOILA'T.IO NS..

MINNEMAST COORDINATED MATHEMATICS-SCIENCE SERIES

c.E

1.

2:

WATCHING AND WONDERING

CURVES AND SHAPES

3. DESCRIBING 4tND'CLASSIFYING

c.E4. USING OUR SENSES

A'z 5. INTRODUCING MEASUREMENT

6, NUMERATION

7 . INTRCDUCING SYMMETRY

g.

B. OBSERVING,PROPERTIES

9. NUMBERS AND COUNTING

O. 1 DESCRIBING LOCATIONS

1 1. INTRODUCING ADDITION AND SUBTRACTION

12, MEASUREMENT WITH REFERENCE UNITS

1.3, INTERPRETATIONS OF ADDITION AND SUBTRACTION

14, EXPLORING SYMMETRICAL PATTERNS

15, INVESTIGATING SYSTEMS

c.E16, NUMBERS AND MEASURING

amt

17, INTRODUCING MULTIPLICATION AND DIVISION

O 18. SCALING AND REPRESENTATION

-20 19. COMPARING CHANGES

20. USING, LARGER NUMBERS

21. ANGLES AND SPACE

22, PARTS AND PIECES

23, CONDITIONS AFFECTING LIFE

24. CHANGE AND CALCULATIONS

25 MULTIPLICATION AND MOTION

26, WHAT ARE THINGS MADE OF?5 27, NUMBERS. AND THEIR PROPERTIES

28. MAPPING THE GLOBE

29/ NATURAL SYSTEMS

OTHER MINNEMAST PUBLICATIONS

The 29 coordinated units and several other publications are available from MINNEIyIAST on order.Other publications include: ..

STUDENT MANUALS for Grades 1, Vand 3, andprinted TEACHING AIDS for Kindergarten an Grade I.

, 4. -

LIVING THINGS INFIELD AND CLASSROOM(MINNEMAST Handbook (or all 'grades)

ADVENTURES INSCIENCE AND MATH{Historical stories for teacher or student)

QUESTIONS AND ANSWERS ABOUT MINNEMAST .

Sent free with price fist on request

OVERVIEW 3

(Description of content of each publication)

MINNEMAST RECOMMENDATIONS fOR SCIENCE AND MATH IN THE INTERMEDIATE GRADES(Suggestibns for programs to succeed the MINNEMAST Curriculum in qrades 4, 5 and 6)

o

9

Cww....NTGCALCULATIONS

UNIT

NIMAST

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MINNESOTA MATHEMATICS AND SCIENCE TEACHING PROJECT720 Washington Avenue S. E Minneapolis, Minnesota 55455

r rteplYne2.,

MINNIRlyEAST

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4.;/':.'..,,, ....

.. .6....e..,,, t., JArte`,, (Y...t. ,...,.

,--...%.,. ,i". Q .DIRECTOR jAMESH. 'WERNTZ, JR.

Professor of Physics . . i.-

University of Minnesota l', '1, '-. ';, .

ASSOCIATE DIRECTORFOR SCIENCE

i f,t; c-'-'------7ASt5triiTE-DIF.E$TOR WELLS HIVELY II

FOR RiSEARCHIXND EVAI,Ti.4,TI.ON Associate Professor of Pgychology1

i ,-, University of Minnesotaj1 -

ers--/,-e,--4..4vcs .

E < pt forithe'rights.to materials reserved by others, the publisheragtt'dp7iight owner hereby grants permission to domestiC persons of..

,

,,,ztr:rtle United States and Canada for use of this work without charge in'''-' Pi,glish language publications in the United States-and Canada after

-cjuly 1; 1973, provided the publications' incorporating materials covered)

)4: _-c \1-) ,- byk these copyrights contain acknowledgement of them and a statement--, , \

il-) - .1'7,1 tnat the publiqation Is endorsed neither by the' copyright owner nor the4/., I' NatiOnal Science FOundation.- For conditions of use and permission to...use \inaterials contained herein for foreign publications or publications,in of er than the English language, application must be made to: Officeof the University Attorney, University of Minnesota, Minneapolis,Minnesota 55455.

ke Minnesota Mat ematics and Science Teaching Projectdeyeloped. these ma erials under a grant from theNational Science Fo ndation.

© 1 69, University o Minnesota. All rights reserved.N

. ROGER S. JONESAssociate, Professor of PhysidsUniversity of Minnesota

Third Printing1971

5

C GE ANDCALCULATIONS

si sie,11,i

-;. This unlit was developed by MINNEMAST on the.,..,, basis d experiences of the many teachers who- ." scspo 0;',1

taught an earlier version in their classrooms. .

-,I"14 --.._., ::-/

%I-

PAUL REDINAssociate Professor of MathematicsBethel College

ROY W. MUCKEY,InstructorHemline University

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EDMUND C. BRAY Assistant to DirectorCutriculuM Materials.Development

PATRICIA A. ADAMS Editbr,LUANNE NYBERG EditorMELISSA MILLER Assistant

SONIA FORSETH Art DirectorCHARLES BIORKLUND IllustratorJUDITH L. NORMAN Illustrator

STEVE SORMAN Illustratort

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CONTENTS

Materils List

IntreL 17.tion

vi

viii

Section I. Addition 1

Lesson I The TPieces Computer 3

Les son 2 Adding and Regrouping with the Computer 17

Lesson 3 Place Value 22Lesson 4 Addition with a Numeral Computer 29Lesson 5 Developing the Vertical Form for Addition 36Lesson 6 Addition Practice Games, 40

Section 2. SubtraCtion 44

Lesson 7 Computer Subtraction 46.

Lesson '8 Computer Subtraction with Regrouping 53

Lesson 9 Developing the Vertical Form for Subtraction 57

Lesson 10 Subfraction with Regrouping in Vertical Form 63

Lesson.11 Addition and Subtraction Games 68

Section 3. Measurement . 71

Lesson 12 Liquid Volume Measure 73

13 Lencith "90__LessonLesson 14 Time 96

Section 4. The Pouring System and the Bala'rice. Beam System 100

Pouring System Job Booklet Reductions 107

Balance Beau! Job Booklet Reductions 125

Appendix (Sheets A and B) 144

total numberrequired toteach unit

Complete Listof Materials for Unit 24

(Numbers based on class size of 30.)

item

30 **Student Manuals.

* marking pen

lessons inwhich item

is used

1, *masking tape 1,3,7,9,12,14;Section 4_-

30

1

30

30

15 pr.

I

19'

9

4

2

1

\

* *Sheets A and B I

abacus (optional) ,1 ,'scissors 2

* small plastid bags 2-14

*dice labeled 0-5 and 4-9 6

*ruler or straightedge 7"

*half-pint containers 12

* pint containers 12

*quart containers 12

*half-gallon containers 12

gallon container 12

1 *water dipper of cup (measuring) '1 2,Section4

1 *yarn or string 12

I large pail (optional) I 2,Section 4,

-I sponge (optional) 12

15 * yard and meter sticks 13

1 clock with second hand 14

2 * plastic tubes Section 4

30 lengths * centimeter tape Section 4

I. * butcher tray Section 4/

30 ea. color, *red, black and silVer paper clip's Section 4

I * beam support block Section 4

*kit items as well as`,** printed materials available from Minnemath. Center,

A720 Washington Avenue S .E. , Minneapolis, Minn.. 55455t.

vi

o-INTROD1.10TION4 .

4

In the fipt two sections of the unit, the children set up acomputer to work addition and subtraction pioblems. Theseproblems scan be written in several ways:

I. horizontatly 25 + 71

2. vertically+71

3. in T notation 2T1 + 5TO+'7T1 + ITO

Regardless of the form of the problem, the same rules arefollowed. We add or subtract the numbers in each place(5 + 1 and 2+ 7) to get the sum or differenCe. Though mostpeople prefer to write addition and subtraction problemsvertically and to work from right to left, this need not bedone. No matter how the problem is set up, we still followthe rule that the numbers in each place mustbe added orsuttcacted to find the answer.

In Section 1 algorithms (rules) for addition are developed withheavy emphasis on the place value system, The childrenreview T notation for expressing place value. In T notationthe number 385 would be rewritten as 3T2 + 8T1 + 5TO. SinceT Stands for 10, the ten's place is called T1.because theelements have been grouped by ten just once. The one'splace is called TO because no groups of ten have been madein this place, and the hundred's place is called T2 becausewe have grouped by tens twice. Each place is ten times asgreat as the one to its right (TO = 1, T1 = 1.OTO or 10,T2 = 10T1 or 100). First the children do addition using a .

T-piece compliter. Thea they add,Using numerals in theircomputer. Finally the vertical form tbr addition is developedfrom an addition. chart which represents the computer.

10viii

4a

Section 2 develops the subtraction algorithm, again with......emphasis on plaCe ,value.: The children set up a computerand work subtraction problems with T notation. The progres-sion of subtracting with a computer, then a chart and finallyin the standard vertical fo m is similar to the format and'procedures for addition in ection 1.

Section 3 involves measure ent systems of length, volumeand time duration. These s stems will contrast the placevalue of the base-ten numeration system the children workedwith in Sections 1 and 2. The children will pee that measure-ment units for length and time duration are not regular. That'is, there are 12 inches in one toot but only three feet 'in oneyard and there are 60' Minutes in one hour and 24 hours in oneday. For volume measures the children set up a 'base-twocomputer and find that those units are regular: 'two pintsequal one quart, two quarts equal one half-gallon, etc.Besides learning different measurement units, the childrenshould. compare the different place values of measurementsystems to the base-ten numeration system.

Section 4 is made 'up of two job booklets which thy childrenwill\complete .independently. 'One of the booklet is apouring,system and the, other presents a balance eam system,By pouring water from dne measured tube into an ther orbalancing a meter stick, the children generate, o dered pairs.They graph these pairs and, use them to predict ew states oftheir systems. The booklets are designed to be self-instruc-tional With a minimal amount of help from you.

Each booklet Should be .completed in one week.

ix

1

a

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NOTES ON TEACHING-THIS UNIT

This LuiliTstIrbe-taugh4concurrently with Unit 23, Conditions,Affecting Life, 'Starting one week 'after Unit 23. Sections ,1',through 3 should be taught for apprOximately°30 minutes eachday, for fodr weeks. The Job Booklets of Section 4 will takefwo weeks to complete. For the first week, half of the class

-completes the Pouring System 'Booklet while the other halfcompletes the Balance Beam System Booklet. During thesecond week each group of children completes the otherbooklet.

For Lesson 12 of Section 3, you or the children must provide,one half-gallon container and one gallon container. Milkcartons or plaStic pitchers work well.

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SECTION I *ADDITION

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PURPOSE

- To develop an understanding of and .facility -in tis-ing.place value (T- notation).

.To introduce and provide practice with the additionalgorithm.

To prdvide game situations for enjoyable practice withbasic addition facts and the addition algorithrri.

COMMENTARY.

In Lessons Land 2 a classroom computer is set up using chili:dren as components. Each desk represents one place in'theplace value system. The children develop rules which tellthe computet how to add using T. pieces. It does so by

57, physidally combining pieces which represent the number of I.

elements in each place. The rule of combining.the elements!in each place is always followed, though the form for writingaddition problems changes as the section progresses.

An..algorithm for regrouping is developed in Lesson 2. The 1

children learn that they must trade 10 pieces for I of the nextlarger piece6e, For example,. 10 TO pieces, are traded for I Tpiece; When th computer 'adds a problem such as 9 + 8, thT0 place will receive first 9 T° pieces and then 8 TO pieces.The child at thal place trades 10 of, his 17 T° pfeoces, for I T1piece. He then gives -the 1-T1 piece tothe child at the next

t1 place, and keeps the 7 T° pieces at .the-TO place.

rn Leison 3 the children explore the place Value system furtherby makktig representations of the T3 ,and T4 places .. They tapetogekhei -1 0 T2 pieces/to make a T3, piece and then 10. T3 pieces; .to make a T4 *-piece. The idea that the place value system goeslir: on ah0',on is developed

"=-.

In Lesson 4 the rules for the T:piebes computer'are :modifiedS Q that nuinerak, written on slips of paper are accepted by thecomputer and added mentally by the children who are computercomponents. Whet the problem, 9 + 8 is added with the numeral

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computer, the child at the T° place will receive a Slip ofpaper marked 9T° and another marked 8T°. He adds thesementally and fills out one slip fo'r each place value repre-sented in the sum. That is, the sum 1.7 is rewritten asITI 7T0. He passes the IT I slip to the II place. ThoUghthe form of regrouping has changed, first using .T pieces andthen using numeral- slips, the rule that 10 elements aregrouped and passed to the next larger place remains.

When 'the children are familiar with the addition process using.the numeral compute, the vertical form for adding is developed.In Lesson 4.a Chart is) draWn on the ch,alkboard which representseach place in the computer. The children add a problem usingthe computer while you add it using the chart. You stress thefact that the place value addition is the same using both methods.In Lesson 5, .lines are removed from the chart until only the.vertical form for writing addition problems remains.

Ganies in Les Sons 3 and 6 provide an enjoyable way for the .

kahildren to practice the basic addition. facts. These gamescan be modified to include "" more diffidult problems as the chil-dren become proficient. Since most of the games require onlya few 'minutes to play, .one or more of them should be usedalmost every day throughout the year.

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2.

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Lebson I: THE T-PIECES COMPUTER

This lesson b gins a review of addition and place value usingthe. T notat on approach that the children worked with ins Unit20. You may find that they need to regain,tamiliarity.with thissystem: For thiS reason Material from second grade has beenincluded in the activities.. Those children who do not recallT notation will have considerable experience with 'it in thefirst few lessons. Be sure to use T notation language your-self but dO riot derhand thaethe children use it exclusively.

,....-.:... .The children make tileir own classroom computer in which theyserve as components. A set Of rules is devised that tells the,computer how tq ad numbers and respond with the sum. Theoperation of the computet stresses place value and .provides

. .practice using thl addition facts. ,,,,

The- time_aklowed for thiS lesson is-two'. days. If the children t,

e------remember T notation from th second grade; Activity A shouldgo very quickly and)eiti- loud d start Activity B on the same day.If 'many children in your class do not understand T notation;spend the first day on Activity A only.. ,The rate of progressthrough these first thtee Lessons will depend on the abilities ofthe children. Feel free to speed up the 'Introduction to the,com-puter in Activity B if you think the children understand the pro- .cedure. (:.

MATERIALS

3 latge sheets of paper, one each labeled T I ,

marking .pen

masking tape

T2

set of T pieces made from 5 copies 'of Sheet A and I copyOf Sheet B .(a,vailable from MINNEMAST)

abacus (optional)

-- for each.childsheet. of paper

Worksheets I -.3

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3

..

'PREPARATION.

Make a set of T piedes. by cutting up five copies of Sheet Aand one copy of Sheet B. should now have one set of Tpieces, containing 19 T2 'pieces,, 18 T1 pieceS' and 20 TO

pieces. ,(Full-size ,copies''of,Sheets A and B can be found inthe Appendix.) Lab41 three large sheets of paper TO, T1,. and,.T2

4

Unit 24Sheet A (5 per child)

,

Cut-on dark lines. Cui"2-T2 pieces from each Sheet,AA,..,

T2 Piece' 'I T2 \piece...: -

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PROCEMIRE

Activity A

Read. the story, "The Amazing Machine" to the class. The. .

story is on page 7. .

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THE AMAZING MACHINE

Freddy was shopping at the supermarket. He found all the

things on his mother's list and wheeled his cart up to the check-out counter, It was fun to watch the clerk add up his purchaseson the cash register. The machine showed a total of $3.86.

Freddy gave the clerk the five-dollar bill his mother had given

him. With a ,clatter and a Clang the machine registered the

mount of change Freddy was to receive -- $ 14. Fourteen centscame sliding down a little chute forhim to pick up, and the clerk.gave him a dollar bill. He put all t money in his pocket" and

,took his, beg of groceries out throug the door that opened by. it-self when.he came to it.

All the way home Freddy thou ht about the. machine. It

gave the right answer every time. How did it do it? How did itknw whento add anCl.r\hen to subtract?. Freddy wondered whyfit never muitiplied'the numbers when the clerk pushed the buttons.

And why did it always send`jUst the right coinedown the chutei

Freddy couldn't wait to ask someone how machines like the cash

regi-ster v7brk7"--

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.41

d

Ask the children if they have ever watched a machine thatcomputed for us, like the cash register, and wondered howit worked.

WHAT OTHER MACHINES CAN YOU THINK OF THATCOMPUTE FOR VS AS CASH REGISTERS DO? (Addingmachines, odometers, pedometers, abaci, computers.Do not expect the children to use thesewords.)

You might :nention that the addition slide rule used in Unit 13is a kind o computer.

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Have the children turnto Worlsiheet I. Discuss ,the machines with themand explain that numbersare put in each machineand answers come out.,Clarify the fact thatmachines do only whatthey are instructed to do.They can only follow therules someone has madefor them. Tell the classthat you have a plan tomake a computer right inthe classroom but you,will need everyone's helpto do it.

dR,

With masking tape, tape the T signs to the chalkboard inthis order:

T2 T1

dTO

(Save these labels 'for Activity B.)

22

,Ask the children what To, T1, and T2 represent. If yourclass does not remember that these are names for the ones,tens and hundreds places, you may want to use the followingreview. If the review is not necessary, go onto Activity B.

Hold up a TO piece 0 and a T1 piece u-HOW MANY OF THESE PIECES (TO) DOES IT TAKE TO MAKETHIS PIECE (TI)? (10.)

Holding the T1 piece, count with the class the 10 TO, sectionsmarked on it;' Then hold up the T1 and Tapieces and ask:

1111111111111011111111N11111111111111111111111111111110111111111111111111111118111111111111111111.111111M

HOW MANY OF THESE PIECES (T1) DOES IT TAKE TO MAKE:.. THIS PIECE (T2)? (10. )

Count the 10,TIstrips with the class. Explain that becauseit takes 10 small pieces to make along piece and it takes10 of those strips to make the next sized piece, we say weare "grouping by tens:"

Hold up the appropriate pieCes again during the following. _dis_cussion. \--

WHEN WE GROUP 10 SMALL PIECES WE GET A LONGPIECE. WE C#LL THIS PIECE 111. THE T TELLS US THATWE ARE GROUPING BY TENS AND THE 1 TELLS U5 THAT WEHAVE GROUF,:E0' BY TEN JUST ONCE.

NOW SUPPOSE WE GROUP 10 T1 PIECES -- WHAT CAN WECALL THIS NEW PIECE? (T2.)

THE T TELLS US THAT WE AE GROUPING BY TENS -ANpTHE 2 TELLS US THAT THE PROCESS OF GROUPING BYTENS HAS BEEN poNg TWICE. FIRST WE GROUPED 10 OFTHESE PIECES (show a TO piece) AND GOT ONE. OF THESEPIECES (show a -T1 piece).. THEN WE GROUPED 10 OF THESE

a

2 3

4

PIECES (show a T1 piece) AND GOT ONE OF' THESE PIECESIshow a T2 piece).

Now hold up a TO piece and ask:

WHEN WE ARE GROUPING THIS WAY, WHAT CAN WE CALLTHIS PIECE? FIRST LET'S, WRITE DOWN T. TO TELL USTHAT WE ARE GROUPING BY TENS. NOW HOW MANY TIMESHAVE WE GROUPED BY TENS? (None; zero times.) WHATCAN. WE WRITE. TO TELL US THAT? (TO.)

Explain to the children how to read T notation:

T° is "T to the zero power."

T 1 is "T to the first power."

T2 is "T to the sedond power," etc.

Tell them that we can also 'say it a shorter way:

.11TV is "T to the zero."

T1 is "T to the one," or "T to the first ;"

T2 is "T to the two," or "T to the second:" .

Label the place's, on an al:)ctis TO, T1 , T2, and higher if youdesire. Also label the places I , 10e 100, etc.

3

Show a nurnber on the abacuS and have each.child write thenumeral In the following form on a piece of paper.

34S = 300 + 40 + 5

345 = 3T2 + 4.T1 + ST°

2.4

10°

The relationship between T notation a\i.d the 'e paraded' base-ten notation should be shown and discu\Ssed e en.if ariabacusis not available.

Have each child-write a 3-digit Inumber on small piece'ofpaper and pass it to the child bdhind him. Each student writesthis number in expanded base-ten notation (549 = 500 + 40 + 9)and in T notation (549 = 5T2 + 4T1 + 9T0). The student thenhands the paper back to 'the child. who wrote it. This childchecks to decide if the notations. have been written correct-ly,. Go around the-room and quickly-check those the childrenthink are wrong:

Activity B

Tell ihee children they are now going to ,build the computer youdiscussed ire Activity A. Remove the T signs from the chalk-board. Place three desks at the front of the room' and ldbelthem as shown in the photograph below. Explain to the 'Oil-dren that the desks are part of the' computer. (Save theseIldbeilsfor subsequent lessons.) You may want to'draw an imaginal-yrstart button on the chalkboard next to the computer desks.

N

0

'Ask one student to sit at each of t1 three desks. These chil-dren are part of the computer. Explain that computers will ,do1NOnly what they are instructed to do. Tell the childr'en that youhave made up some rules for the computer to follow.. Thenwrite these rules on the board in your own words:

Rule I. The computer is to add the numbers fed intoit.

Rule 2. Each desk can take only one kind of piece.The TC3 place can take only TB, pieces, the T'place can take only T I pieoes and the T2'place, can, take only T2 Pieces.-

Rule 3. Each place must- tell its contents when theProgrammer calls for it.

Choose three more children to help operate the computer byacting as Programmer; Banker, and Reccirder. The Recordershould use the chalkboard and the Banker should be, given allthe TO, T-1, and T2 pieces. The jobS of the Programmer,Banker, and Recorder are explained inthe example below.file rest of the children should act as Repairmen. Tell theRepairmen to .watch carefully to see if the Programmer, Banker,or Recorder make any mistakes. Repairmen should be allowedto correct "anyerrors they find. (The first problem will prob-ably go quite slowly., until the Children learn' the procedure.)

LET'S SEE HOW OUR COMPUTER WORKS. LET'S HAVE ITADD 21 AND 32.''

Recorder:

12

The Recorder writes the. problem on the chalk-board. (The box shows what is written at

21 each step and need:not be drawn+32 on the chalkboard.)

Ask: HOW DO WE WRITE 21 IN 'T NOTATION?,,. (2i = 2T1 + 1T°)

The Recorder writes this on the chalkboard. He should havewritten:

21 =+ 32

2T1.+ 1TO

%

".

. HOW CAN WE, SHOW THIS NUMBER WITH THE TU, T I ANDT2 PIECES)v.P,

0

Banker: The .Banker holds up the proper pieces for the Tplace" (in this case for the T° notation of 21

.

Programmer: The .Programmer takes the pieces, checks to besure they are .the correct number, and valtig., and

`feeds them to the computer (places them On 'theT° desk).

Computer T0: The child sitting at this place checks the pieceshe it given and nods if he thinks they are the

Banker:

correct ones.

While the Programmer and the T° place child arechecking the TO- pieces; the Banker picks up theT1 pieCes and passes them to the Programmer.

Programmer: He checks the T pieces and feeds them into thecomputer.

Computer T I : This child checks the pieces and nods if hethinks they are .correct.

'Note: In this problem there are no T2 pieces, but ifthere were the Bankerwouldhand them to theProgrammer, who would check them and handthemi.to the T2 child who would check them andnod if they were correct.

27

. r.

Now ask: HPW.DO WE WRITE 32 NOTATION?

(32 = 3T 1 + .210).

Recorder: The fecctrder writes this on the chalkboard.

21 ! IT0+32=

2

Thg children go "through the sameprocedUre to feed the Tpieces for 32 into the'.compiuter.

Programmer: When the T pieces representing 21 and 32 havebeen fed iritothe computer the Programmerptishes.an imaginary, start button. .

Computei: . The students at each place again' count theirRieceb and then, 'one at a time starting with theT° place, they tell the total number of piecesthey have. (3,pieces for the TO place,' 5 for the

. T 1 and 0 for tlie T2.) -

.

Recorder: When the T° place child calls out the total num-,

ber-of pieceshe has, the Recorder writes' it inthe proper place on the chalkboard. He does the ,

same-for the T1 place. T2 hag--nopieces so theRecorder- doessot write anything.

'14

If

, 4 1 = 2T1 + 1r-.4. 32 = '3T1 + 2TO

1.5T) +

Now the Recorder converts the T notation answerto the standard notation. (53) and .writes it underthe original problem. The completed problem shouldlook like this:

21 =2T1 + ITO

+32 = 3T1 + PTO

53 = 5-T1 + TO

0

*4.: ,, %

Review What the O'ormitifeffiii-done. i has added the piecesrepreiAintirig the -first number..(21) to' pieces of the secondnumber'(32) to give the _Sinn lof the two numbers (53).

Choose three.diffeient.bhildren to `sit at the. desks and three. more to a'ct'ai'Progranimer, hanker and Recordert Itet them'ahooSe another' problem that does not, involve regrouping andhelp them. go through the addition procedure with the computer.

' The. proceduze should go- faster this time. 'Work-enough prob-lems, such as3f +.462, 291 + 4 and 128 i 40,, se that each .child has a chenbe to be a part of the computer set tip. Youmay want to work a few more problems with those children whohave difficulty the firit 'time. Alf the children should becomeproficient at adding; with the computer. Remind the Repairmento report, any mibtakethey notice.

O

4

4

I

1 5';--

I I

Save the T place labels and T pieces for future us.

gave the children complete Worksheets 2 and 3. These*worksheets provide review of the .concept :T notationand give practice in converting numerals to-notation;Have all the children do the bonus problems on Worksheet 3,with youit,help if necessary. You maiWant,to:Provide a fewmore problems of this type so that the children understandthat the T2, place; for example, stands for the hundred' splace even when' TI and TO are .not: written sout. If the chil-,dren do not read the problems carefully they may answer8t2.= 8 instead of scro. Ocdasionally you should refer to the

//1 places as the "hundred's place," "ten's place end "one'splace." .

llorkstairet 2um 14

"flit In"the blanks. In the lSal box, draw the T piecestoe the rauisse.

4

12 ploce TI piece 1° piece

t4 =_/_-_/i _q__70 25 . ..1 T1 .510

"`ijicr'.11142:..- C',T1 . 0 70. .. .3 '

324 s 3 :12 .- 2 TI 9 70

.1 .

.313334-.312.i.%'3"ik '2 ',7°

,t..3...

16

worksheet 3Unit 34

Nem

m412 pi... can be cut Into /0 TT pieces.

One TI piece Lim be cut Intel:Ill° pieces.

Fill In the blanks.- . 1?

36 .3 rt - 6'70.

v . o /2. a. /I ..,,7 i°e '

16 = f ti i°

54' Il 512 414 * 216

792 51.12 71 al° / . 072 Of 910

540 Z 3 la 4 71 01° is° . 112 014 OT°

632 e 42 4j .1 7° a" 212 6TI OT°.

156 si :12 . 3" T1 ._10 3-03 e ma 0T1 37°-

.

'BONUSs1 jp,,,. at f = via 410 ; 412 fITI

liee 314 610 ae sr? 47° 3I° . 312 ITI

.."910)

"

e et

9 ,

4,1

Lesson 2: ADDING AND REGROUPING WITH THE COMPUTER

Ip this lesson. the children work in, computerteams to solveaddition problems, some Of which involve regrouping, Thislesson will take two class periods, .one for each activity.

MATERIALS

-- for each child 2.--=

set of T. pieces cut from 5 copi s of Sheet.A and I dopy ofSheet B

scissorssmall plastic bag

Worksheets 4 7

-- for each

.several sheets ofdaPer/.. PROCEDURE

Activity A/ .

Give. each child five copies of She t A and one copy ofSheet B. The T pieces should be gut apart along the heayylines. -Eaoh.child/Should have la/T2 pieces', 18 TI piecesand 20 T° pieces in.all.' His-T pieces should be stored in

; a.plastic bag.t

Divide theprass into teams of three. Each team will set,up its own cOmputation machine consisting of a Computer,a Programmer and a Recorder. Ask the PrOgrammer to turnto Worksheet 4 in hie' Student Manual and tear out the desk-cOmPuter, / .1

Explain th&re s of each assignment:

R: The Recottr writes the problem in both standard, andT hotatio on sheet of paper.

$

31

1

13: The Prograihmer pl.atS The pieces representing, the addends of the .problem into the desk computer and preSses

.;.-

the start button. ,...7------:,.C: The Computer computes (tounts,the pieces'if necessary)

and- tells the Recorder the number of 1' pieceshe nowhaS in each place. ..

R: The ReCorder writes the answer, first in T notationand then-in standard natation.

worksh ot 4 ttol 24

O

Deik :Computer

To denloristrate the-proper procedure and recording tech-niq,ues t the.-cla-

's-s7

write this problem on the chalkboard-i : .,---------'iTrIkerti al form: 143

I." . +245

Choos e,. one team to help with the demonstration while theoher tams go:thtough the problerfi at their desks. The chil-

gen, ai the desks should, check their work againSt the worko(iheldemonhtration team at each step. Allow time for di's- ,

4 .

auspig and correcting errors. ,

18.

. f

I

I

I

Step The Recorder writes the problem-and converts it to...T,_notation. The Recorder_on_the demonitrationteam should write it on the chalkboat\d-wcrile-the

--'other Recorders write. it on their paper.

143 = `LT2 + 4T1 + .3T°+245-=2T2-+-471--ir-5T0--

Step 2:

Step 3: .

The Programther gets T pieces from his plastic bag.(the bank) to represent the first number, 14'3.

He feeds the pieces into-their proper placeS in thedesk computer. Then he gets the pieces that rep-resent 245 and feeds them. into the computer.

The child who hag the job of Computet counts the'total number `of pieces in.each_plaCe.and tells theRecorder how many he 'has of each kind of piece:f!.or thisf problem it should,be 8.52p_ieces_a_DPieces and 3 T2' pieces.

The Recotder writes 3T 2 + 8T1+ 8T° under the 'Tnotation -problem and puts 388 undet the standardnotation problem.

0

Give trie teams two more problems that do not require regrouping.. The team members should change jobs so that each chil,has a chance:to do each job. Kaye about the room while thchildren are working and check their methods, work and an-swers. At tle conclusion of this activity, select saRecordeco read his 1164' s results to the class so that all can veritheir answeqs. If 'some groups finish early, you may Want Logive them additional

_problems.

1'9

Have each grouphwork the probl m, 246 + 325, and.give yourtheir answers. The,answer the are ltkelYAoget Is:

/ ..Activity B . .,

5T2 + 6T1 + VI.T°,*.; ,, ,. ..

If any team regrouped t- get,pT4 + 7T1 + IT:0, ask the rest ofthe children to try to figure 31.1t tidily they got such an answer.If `no-team got the correct answercask someone to tome upand write 5T2 + 6T1 + I ITO/in Standard notation. :Ige will'probably write. 5 611. Explain that'this.answer is ratich too.large and'suggest that' the children cheqk>their work.

..5

Go over the pieces iii.each 'place of one team's computerwith the class while th!e other teams check their computers.Someone should discover that 10 of the TQ pieces cah'beexchanged for one T1 piece with one T° piece, left over: Put..10 .of the TO pieced together to .show this. b Rewrite5'1'2 + 6T1+_1 IT° a's 5T2 + 6T1. + .IT1°+ IT°. insert paredhe,.,ses around the 6T1 + 1121. Then add to get 5T2 + 7T1 + IT°..or 571,

t".

WOULDN'T IT BE EASIER IF THE MACHINE WOULD DOTHIS REGnOTIFING FOR US?,

Remind the children that the machine will do what we tell itto do. Ask,them to develop a rule for regrouping.. It shouldbe similar to the following rule:"

.There may be no more thari nine pieces in each place.If there are 10 or-more pieces'in a place, 10 'piecesmust be exchanged for one of the next larger pieces.The new piece must be entered into 'the phoper place.

Have-each-team-try-adding 246 + 325 again and help them toregrbup,

Let each team apply the regrouping rule to several problemssuch, as 46 + 38, 132 + 49, 269, + 150. The Children shouldchange jobs for each proble-n. When the children are readY:let them' try aoproblem requiring regrouping in both the TO

and the T1 places for example, 49 + 76 or 98 +

20,

r,

Review the'ryles-that-- our corn-_-...Puters must folloCy and, have de

teams 'complete Worksheets 5heir-desk computers.

H-Eachtbhild-should. serve as Re-:COrder. to 'fill in his own work-sheets. Worksheet 7 should be'completed individualy, It re=-quires the-children to build on.their familiarity with the placevalue system. Some of the chil-dren:may dot be able to completeit.' Do not press thetn. Thismaterial willbe covered inLess On 3.

worksheet

tout 24,

tee )04 computer to help you add.8

write the problem

in T notation if )ou ant.t

516tel

Loj

Workiheet 5Unit 24

Name

Use your computer to work these problems. 1"

1,

.257 1,1 7 To

341 = 3 la ;I "I To

598 312 . 9 TI P. To

343 ' 3 72 4 71 -5' 76_12..T?37.7 '3 12 7 T1 .2, -113

Add. Use your computers. Use T nOtaiion if you want.

368227

593

182

309

Y91

265

lit

a

Morkehjet 7Unit 24

Fill In the blanks.

Nast

74

It ti.kes 72i_of the 76 pieces to make--.1.--11 piece.

-t

It taker /0" of the T1 pieces to meke.L.T2 piece.

It takesu, of the 12 pieces to sake / 73.plece.

It takes' /0 of the 73 pieces to ismke_t_T4 piece.

Ittake) /0 of the 76 pieces to make 1 Piete.

It take.: 42 of the 75 pieces to make 1 le6 piece.

it.takoe 10 of the 76 pieces to make 1 -77 piece.

2I

Les son 3: FLA.CE VALUE

This le'sson develops the .idea of place vafue as a repeat!npattern generated by the rule that each place is, ten times,larger than the place to its right. In the firSt activity thechildren regroup the TO, Ti and T.?, plades and see a need for"pieces to the left of T2. In Activity g the children tape to-gether T pieces to produce values to the left of T2; Thenthese pieces are taped to the wall, providing a visual repre-sentation of the relationships among T places. The lastthree activities include worksheets and games that providepractice with simple addition facts°. Spend two days on thislesson. Activities A and B should 'be done on the same day.'

MATERIALS

labels marked TO, T1, T2 (from Lesson 1) and .0,. T4, andT5

masking tape

wall or chalkboard sp*ace about 2 yards wide

-- for each child- 7-, .

plastic bag with set of T :pieces

Worksheets 8 - 12

PROCEDURE

Activity A

Setup a three-desk' computer. Assign a child to each deskplus a ProgramMer and a 'Recorder to operate it. Give themthe problem, 562-4- 659. The4tecorder writes the peblem inT notation and the Programmer feeds the pieces into the Com-=puter. The children playing the role of computer must add upthe 'number of pieces in each_place and exchange ten of their'pieces-for the next larger piece from. the Programmer.

22

In this problem thchild in the TO place will have 11 pieces.He exchanges 10 of these for a T1 ,piece and then gives theT1 piece to the child at the T1 place. The child.at, the Tiplace exchanges ten of his pieces and passes the r2 piece to

3 u"

the T2 place. When the. student in the T2 place finds thathe must exchang§ tedofhis pieces for the next largerpiece, he will be faced i.vith the question of what to usefor the next larger. piece.

(If the children at the TO and T1 places had trouble re-grouping, go through a feW more problems before explainingwhat, the child at the T2 place ShoUld do. The childrenshould not try to regroup the T2 pieces until they can do .itfor TO and Ti..)

Ask the class to help the child in the T2 place with his'regrouping. The children should realize that they have noT3 piece. }false the question of what this Piece should looklike and ask the children if they would like to make: one.(They will do this in the next acti,vity.. SaVe the ,desk labelsfor Activity B.)

Activity B4.,. A A

L: YOU will need masking tape, place value labels for TO throughTs, and wall or chalkboard space about two-yards wide. Eachchild should have his. bag With a set. of T pieces.

Tape a TO piece to the wall orcha4board,and ask how thisplace should be labeled. Tapethe TO sign above the piece.Ask the children what piece belongs in the next placja,.andtape a T1 piece to the' left ofthe To pied,e'. Label the T1place. Have the children sug-gest which piece-goes.in the ,

next, lace and tape a T2 Piecethere. Tape the T2 sign aboveit.

0

WHAT WOULD A PIECE FOR THE NEXT PLACE LOOKLIKE?HOW MANY TO PIECES DID IT TAKE TO MAKE A T1 PIECE?(10.) HOW MANY T1 PIECES DID IT TAKE TO MAKE AT2 PIECE? (10.) SO HOW'MANY T2 PIECES WILL ITTAKE TO MAKE A PIECE FOR THE NEXT PLACE? (10.) ,

3`7

23

"No

.4

))

Tape together 10 T2 pieces to make a T3 piece and put it upto the left-O-Rhe T4 piece. Ask the class how this shouldbe labeled, and tape the label above it.

WHAT WOULD WE CALL THE NEXT PLACE? (T4.) WHATWOULD'A T4 PIECE LOOK LIKE? LET'S TRY TO MAKE ONE.

Have the-children work on the floor in groups of four or five.Each child should bring_ his T pieces. Pass arqund a rollof masking tape and scissors so that each group can cut offnine two-inch pieces. Have the children take their T2 piecesout of the bag, lay 10 of them upside down on the floor andtape them together to make a T3 piece. As soon as a groupfinishes a T3 piece, have them bring it to you so that youcan tape it to the wall.o, Several groups may have to make twoT3 pieces, so that there will be 10 T3 pieces in all, enoughto make a T4 pike.

When the T4 piece is completed,.tape the T4 label above it.Your T4 piece should be 50 inches high and 50 inches wide,

3

3 8

.HOW Would? THE NEXT PIECE BE MADE?) (By taping10 T4 pieces in a row.) WHAT SHOULD THIS NEXT PLACEBE LABELED? (Tape the T5 label to the left of the T4label.).

Ask the. children if they could continue this pattern and wiletherthe place pattern could go on in the other direction (to theright of TO). Let them speculate about what one of these piecesmight look like. If some of the children are interested,_ letthem explore this question, with your during free, time.They should find that there is no limit to the place values ineither direction. If one TO piece were cut into 10 equalstrips, each of those strips would represent a 11-1 piece. Inlater units or grades the children may learn that T-1 = 10=1 =

11and 73 =10-3 _

1000

Any children who did not complete Worksheet 7 in the pre-vious lesson should do so.now-.

Activity

Tell the children that when areal computer works it doesmore than our desk models. It stores information in itsmemory bank. This memory bank is, in soma ways, like ourbrains. We can store facts in our brains and remember themwhen we need to. We are going to test our own computers,our brains, to see how any addition facts they, have stored.

Have the children comi5leteWorksheet 8 without the computerif they can. Although the worksheet includes most of theaddition facts, you may want to make supplementary 'work-sheets which reverse the order of the addends and coverproblems like 4 + 4, that are not inclUded.

VT6rksheets, 9, '10, 1 1 and 12 involve word 'problems illus-trating regrouping. Assign these according to the readingand mathematics skills of your class. You may want to setaside time to help children Who have reading difficulty.

WI

25

;IL

.1

-te

torksbeetthit 24

t 21

3

34

62

57

2. 7 5

ti

4 8 7 5 2 42 5 8 9 .5 6

4 '7C 5. /0

C 6 5 1 4 35 2 3 I .6 9

9 /0

'6 8 ,4 4 5 78 9 8 7 . 3

1,2 11 lo

E 6 3 1 2 23 5 7 9 3 I

// 7

9 8 94 6 I ' 6 3 7

3 3 16,

Iforkalreet 10Unit 24

Name

*lark had lot or baseball cards. When he bundled

In groups-or ten be had 3 bundles of ten

ind 5 cattail E tirf res36"Ha. many did he hays altogether?

Sad also hal some baseball conk. His had 4 bunilleeof tinoona and extras

E zy,122 its1!0. many did Mee allosehtar7

Sam and Irma decided to keep 'halt. cards In the sass

box. Hoe many bundles,or tan did they Mrs altogether?

Hoe many extra cards ltd they hayss7.22._.

Could they get anothir bases of ten from the

4oe hoe many bundles of ten do they have?

Hoe many sutra? 3

BONUS93Ito* many baseball cards do they have IlogeSither7...

261 man

sorksheet 9 %elseEnt t 24

...1:1* T2 piece =7:1=0 a 71 Piece 0. 10 piece

tau=

The 141 piece represent the number I .

-14) Pieces can be put together to make one 71 piece,

One 71 piece is thi 'same so 9 71:1 pieces.

The 71 piece represents the number i° .

...L.71,pleces can be put together to make one T2 pitck,,,

The T2 ptece the sass as i° 71 ;lees's..0u

The T2 piece represents the numbs r /a°

1Q T2 pieces can be put together to make one 73 piece.

One 73 piece is the sue se_g_../ 72 pieces.

BONUSThe 13 Piece represents the number i"a

Vorkeheet II HasaUnit 24

OW and At, slitter, Jan, also had collected baseballl

cards. mil had 27. Hoy many bundles of ten could he

Itowaany extra would he hays? 7 .Hint: 27 271 '771:1.

Jan had 35. Ho. 5311y b4.01d ills Or ten could *Moltke?

3 Ito. many extra .ould she have? .

0

8111 and Jan decided to put their cards together. Ile*

inany,bili tai or ten ecsald they hers ItottethettAL

Ito many extra? Could they make another bundle

or ten from Nov hoe many bundles'ot ten

do they hare?_4_... Iloy many extra? Z_Z

)t, BONUSHoe many baseball cards do they hove altogether? 6.4

1.

O

.1aerkshee t 12 NamMit 34

km and Perk and 3111 and Jar: Nettled I. put all etthelrlhweball rants together.'

Sae and Mirk had 1'3 or 47 beadles of ten and.1....sore. 3114 and Jan had ....4.,..er_f_)_bendlee it tenard-7.1.eore.

then they put per together hoe raw bundles it 'tendld they bie?2_!. Hoe many extraTA:.

is can represent ,bundles-by le. 12 0 1°.

1Wcan cheiye Id, TI to .1 .12 'V Tt 0 P.71w 5 extras 5 1°.

is could add these to get 12 y ti S 10.

So and Mark and 3111 andJan have ig5 baseballcard. altogether.

Li

%.")0.()'

Activity D: Relay Game

This game-is a favorite with the children and should beplayed all year. It can be varied t6 accomodate more dif-ficult addition. problems as well as subtraction, multiplica-tion and division.facts;

'Have the children sit at their desks. The child in the firstdesk of a vertical column of desks stands beside the childIn,the second desk of that column. You present a simpleproblem (such as 5 + 5) to these two ohildren. The child ofthis.pairwho calls out the answer first is the winner andmoves to stand beside the child in the third desk. You pre-sent a simple problem to this pair of children. The one whoanswers first gets to compete with the child in the fourthdesk and soon. The object of the game is to see if some-

t, one can move all the way around the room and back to hisown desk. To do so, he would always have to answerevery problem .faster than the child, he is competing with.

27

..

Activity E: Mental Arithmetic

Many teachers 'have found this review technique to be highlysuccessful. Do the activity for one br two minute. each day.

Tell the class thatyou are going to give them problems tocompute with their own computers their brains. Start outwith simple problems such as 4 +.7, and work up to problems°such as.-12 + 8 or 3 + 5 + 7. Ask the children to raise theirhands when they have the answer, and call on one child togive the' answer.. Look for ,speed and accuracy.

28

ti

13,

Lesson 4: ADDITION WITH A NUMERAL COMPUTER

< ?In this lesSon a classrooinr computer is designed which asnumerals rather than T pieces. The children:devise rules bywhich the new computer can perform- regrouping operations.Then the steps in computer addition are recorded in chart form.This activity provides a transition between adding WitliTpieCes andadding in vertical form.

MATERIALS

- 2 sets of T2, T1, TO desk labels- demonstration T pieces

several hundred pieces of paper (about 2" k 2")

Worksheets 13 - 17

PREPARATION

Before the lesson beginS, prepare several hundred slips , ofpaper by cutting'up larger :.beets or providing, scratch pads.

PROCEDURE

`Activity A

Set up and label three desks for a computer as you did inLesson 3. Assign a Recorder, a Programmer and three chil-dren to sit at the desks: Have the coMputer,,work an' addi-tion problem such as 432 + 516 using T pieces. After theproblem is comi>leted, discuss with the class th,e value of,amore efficieht computer that doesn't need to use T pieces.Suggest trying to make one that uses just numerals.'

Arrange another three-desk computer next to the first one.Label the desks TO, Ti , and T2, put several pieces of paperon each desk and assign a child ,to sit at each desk. Writethe following problem in verticallform 'on the chalkboard:325 + 462. Have the PrOgrammer,i,put 325 into the first com-puter with T pieces. Ask the children how 325 might be ,

Plo,

YJ

1`.

.;

it

4 .,

2.9

",

A

t..

4

Y

':

30-

4

put into the numeral computer. If no child suggests it,propose that a Programmer write the numera's 3, 2, and 5on slips of paper. Choose another Programmer to come tothe numeral computer and fill out the slips. Each slip shouldbe labeled with the proper place notation. The Programmerthen feeds these slips intothe numeral computer at the properplaces.

3

Have the T. computer Programmer enter 462 into his computer..with T pieces while the numeral computer Programmer feedsin numeral slips for 462.

Without going through the procedure, review what the personin each place of the T computer does with his pieces:

.1.. He combine's his pieces,2. Checks to see if he has more than nine pieces,

Exchanges 10 for a T piece of the next larger size ifnecessary, aria

Tells how many pieces he has when the Recorder callsfor the, answer.

Ask the children to suggest how the new computer might com-bine the. numbers in each place. (Each child can combine hisnumbers mentally, using the addition facts',.and write the .an-swer "On a slip of paper.) For example, the child at the Toplace would add 5 and 2, and mark a slip of paper with thetotal4(7) and the place value (TO). He would hand it to,theProgrammer yheri it is called for. The Programmer take's eachslip tkthe-Recorder who then- writes,each numeral on thechalkboard.. (Choose a Recorder for the numeral computer.)Similarly; the...chi.ldrenit the T1 and T2 positions completeSlips for the Progranimer 'and the Recorder writes these numer-als on the board. Let the children at both computers.com-plete the problem.

El

f

NOwtell the children that the computers are going to have arace. Hand each Recorder a slip of paper with this problemwritten- on it 487 + 512. Both Recorders start at the. sametime. The Recorder for the T computer writes the problem inT notation on the chalkboard while the other Recorder writes itin standard vertical form.

The Programmer for each computer starts his job as soon ashe sees the problem. The T computer Programmer counts outthe pieces and feeds thein into the correct places. The num-eral computer Programmer fills out the slips and hands them tothe students at the computer desks. Each child at the T com-puter deiks counts his total number of pieces -and tells theProgramnier this. number. The/Programmer tells the Recorderwho Must write it in T notation and then convert it to Stan-dard notation. Ai,the same time the children at the numeralcomputer add the niimbers on their ,slipS, fill out new slipsshowing the sum and,the place value. They hand these slipsto the. Programmer, who then carries them to the Warder.The Recorder writes the 'total in Standard notation. The otherchildren in the classroOm act as Repairmen to catch mistakesand also watch to see which computer finishes first: Thework of each Recorder should look like this:

487 4T2 + 8T1 + -7TO+512 +5T2+ 1T1 + 2TO

999 9T2 +.9T1 + 9TO = 999

The class should compare the efficiency of the two machinesand d9 another problem, if necessary, to confirm theof the numeral computer. Ask why the numeral computer isfaster (because it is making use of the addition facts thateachChM' has stored in his brain

Have the children summarize the rules that ea-c-h-filasce-of-thenumeral d om pilfer is following when ii-Ladirg:::-These:shouldinclude: each place accepts the proper numerals, adds them,..writes the sum on a slip of paper, and feeds it out on 'demand.

3

Write on the chalkboard alprOblem that involves r_groupillgi274 + 119. Ask the children td develop a rule for the nutneral,computer to deal with this situation._ This rule.might be: 4

No place may feedout a numeral larger than 9. If the ,

sum of its numerals is more than 9, two;slip4 Must befilled out, one for each of the plade values, in the sum.The larger place value slip must be given to the nextplace tothe left.

Tell the children at the T- notation computer that they willnow become parts of a numeral computer. Have the childrenin each numeral comPuter work the problem you wrote on theboard (274 + 119). They will find that the sum, 13T0, mustbe written as "IT) ± '3110. For each computer the IT' slip -ispassed to the T1 place. The T1 place mustadd this' slip to.its other two VT1 and 1T1) and fill out a slip for the proper;sum (9T1).

Choose new children to sit at each cotnputer. Give both com---:,.puters the same problem, one whi.Ch requires regro,uptng.in the 4

.TO and T1' places. Remind the children to Write and label a *`

slip for each of the places in the,sum):and to hand the properslipsto the place tb.their left.

If you feelthe children areready -to their own, they,

- cari now form grpups of three: Give each group 40 or 50slips of paper. The groups are to solve addition prjblems

`using their desk computer and numeral slips. VIritea fewsa p.le problems such as .2013, 29, 560 on the dhalk-

- +463 +92 +377

32

; 4 *5

board and circulate to check that the numeral slipsarebeing marked and regrouped properly.

Activity B

'Draw a chart on the chalkboard to introdude. a system forrecording -the--Aeps that the computer goes through when it ,.`adds. Write a problem beside it. .--

,.

,.. 0. V

O

4.

n

Organize the children into groups of three and have one childin each grdup take out his desk cOmputer. You will act asRecorder for all groups. One child in each group fills outslips for the lirat.addend and another child fills out slipdfor the second addend. These slips are then put in the cor-rect places of the desk computer. Eavch child acts as oneplace of the computer, adding the numerals in his place.Starting at the TO place, call on a different group for eachplace to tell you its .answer. As the children tell you thesum for each place, write these on the chart. When a num-ber is regrouped and a slip is passed to-the next place., ,enter

.a I in the next place on the chart. Your chart should finallyhook like this.

ios+237

12- Ti T°. .

r

06.1, 3

3 zi 2

43.3

Workehert 13

13.11 24

Add. Use your numeral computer.

44t

518

321 a

_.11211

17 .

434 a

I27

71 10r3

Itecord Mat goesInto the computerhere.

!hat iONCADUI Of thecomputer here.

.14

1111iL

0

Worksheet t4Untt 24

Work their, problems. Use your desk computer if youneed It. 4

Johnnie could Ilft 42 Pounds.Pete could lift 54 pound*.Hoe such could they lift altogether? -----

4-

M.17 *etched 4$ poulds ten year. ago.

She has soloed IS Pound*.Hoy much, does she sreIgh?..---.1.ound*.

Lk.

* .

When Oath VAR mix years old she *am 44 Inches tall.In ten years she has gross. 7 Indies.

liw tell la she? f""

17$

34'

ti

Q

Dg as many problems with the numeralcomputer and the chart as you fear arenecessary fbr the children to understandthe chart. Choose new children to runthe computer each time,

Have the children turn to Worksheet 13.If they need help let them work a fewprobkems with friends. They shouldcomplete 'most of the worksheet bythemselves. The children should doWorksheets 14 - 171,individually ontheir own time. Sotne children mayneed help in reading ;some of the prObl:-lems.

Note: Save the desk computers forLesson 7.

Workrw1,1 14 iconttnuedlMt! 24

Harli and sally save pennies.Mark ha. l63 penntes. Sally has 219 penntesIto. many pennies do they have togetheellLe_pennies

01.I/ ItS

f

),: BONUSMerest's! has 37 cents. Dilly has 29 dente and lasshas 42 Cents. If they put all their money togethermould they have enough to lay their teacher ahandkerehlef that costs 98 cents? Vet

Hoe much money do they have?-22.--cent*.

Can they also buy her a card that coats 10 cents?

Ns

Ilertahnet 15Unit 34

Nose

The kids at s'etweei bellied to see how sassy Lopata*

sticks they could collect. Bills and klary thirdgrade clits collected 475 atlas. That would be

4

Vont buntlee.IT2 bundles% 11 bundles and

.IA1 coolst* our sticks with the pieces se used

to our cowoutr.'

one t a r.) = 1 TO

one bundle of ten stick)

one slant bundlelien bundles of tent

a1 T.

Ac could represent 4 gland bundles by T2.

We could represent 7 bundles of ten by 7 71.

ee could represent 5 extra it tIcki by 5 T°.

12

brashest 17-thlt 24

For axlairt. only:

Mr. Walters, the popctele man said that If the thirdgrads closets coutd collect enough popsicle sticks,to sake a surer bundle, he bold give each thirdKreider a postdate.

Everyone wanted to,anow how big super bundle le.

A mull bundle ITII Is 0 sitClu.

A giant bundle (121 to 0 *wall bundles or /°°sticks.

A super bundle 1131 Is 0 ;tent bundles or '00small bundles or sticlu.

A pull bundle 410 ettekal

A giant bundle 1100 stick.)

A super bundle (1000 sti;ikei

t.

Tu

rZ

r3

.7

Cork%heel 16(bit 24

'Sago ;

s7

The alas. across the hall collected 319 *ticks. This

would /sake -/ plant bundles or 3 12. i Fndtelal

of ten or and 9 extras or 9 To.

One Class had 4T2 771 570.

The other had 372 Ill 9143.

That equals. / T2 bundles, 8 Ti bundles and

14 TO extras.

WM the 1 extra slicks wake another bundle of ten)

rt5 3'Then we will have altogether 7 slant bundle*

I121. 9 bundles of ten 4141 and ...±_extras (Pl. in

T plexes this would be.-=_7-1:2 9 T1 4 10 or

79V altogether.

Worksheet 17 Icontinuedl24

Two ofithe classes toicellar bad utmost 000 sticksor_(_ slant bundles (T bundled.

The "thee third crad2 clues h.n1317.stleks.This sould (giant bundles) I Ti(bundles or tent and 7 ,P lextrast.Did the three %losses together lase at least 10 slantbundle./ Y43

Did they hose eunuch sticks to Like I super bundle,Yes

The I test two a kisses had exact ly 794 .1 Irks.The thud alas. hod 317 at Is ks.Vow piny did they Noe .110.1ether?-1111

49

e,

36

Lesson 5: DEVELOPING THE VERTICAL FORM FOR ADDITION

.-In this lesson the numeral computer technique for addition isconyerted to the vertical form for addition.

MATERIALS

Worksheets 18-20 .

PROCEDURE

Discuss with the clast the fact that we often need to doaddition,problems but we do not always have access to acomputer. Develop the idea that each child can be his owncpmputer by using his own memory device (his brain) and byfollowing certain rules. Put the' problem 263 + 324 in verticalform on the board and t.:1 aw a computer chart next to it.

Enter the numerals in the chart as you review their placevalues in the standard algorithm. Then add the numeralsin each place of both problems, stressing their place valueagain. The completed record should look like the one in

photOgraph_below.,,

50

t

Work as many problems using .the standard alqorit4-1 and thechart as you feel the children need to understand that thenumerals of each addend on the chart-ere in the dame placeand same relation as they are when they are written in theusual Vertical form.

CAN WE SHOW WHAT WE DOcVEN WE ADD WITHOUTUSING THE CHAIM LET'S T11.

Put thiS problem on the chalkboard:

265+722

WHAT NUMERAL WOULD GO INTO THE TO PLACE IN OURFINAL ANSWER? (7.) WHY? (5 + 2 = 7.)

WHAT NUMERAL WOULD GO INTO THE T1 PLACE IN OURFINAL ANSWER? (8.) WHY? - (6 + 2 = 8.)

WHAT DOES THIS ADDITION SENTENCE MEAN IN TERMS OFPLACE VALUE? (6T1 2T1 8T1 or 60 + 20 = 80.)

WHAT NUMERAL WOULD GO INTO THE T2 PLACE IN QURFINAL ANSWER? (9.) WHY? (2 + 7 = 9 )

WHAT DOES THIS MEAN IN TERMS OF PLACE VALUE?(2T2 ± 7T2 = 9T2 or 200,+ 700 = 900.)

WHAT IS OUR ANSWER? (987.)

Notice that the order in which we add the columns does notmatter when we do not'have to regroup. But, when regroup-ing is necessary, we should add the TO column first, thenthe column, etc.

Now put this problem, which involves regrouping, on thechalkboard in vertical form: 438 + 249. Use the standardalgorithm which begins by adding the first place on theright (the TO place).

r I1.

37

"%.

38

WHAT_NUMERAL WOULD GO INTO THE TO PLACE IN.OURFINAL ANSWER? (8 + 9 = 17. We regroup 17 to 1 T1 + 7110.Be sure the children understand this transition of 17 intoT notation. The 7 is recorded in the TO place and the 1 isentered at the top of the T1 column. The children shouldsee that the 1 in this case means 1T1 or 10.)

WHAT NUMERAL WOULD do.INTO THE T1 PLACE IN OURFINAL ANSWER? (8. ) WHY? (1 111. + 3T1 + 4T1 = 8T1 .The 8 is recorded in the T1 place. Be sure the childrenunderstand that the 8 means 810 or 80.)

WHAT NUMERAL WOULD GO INTO THE T2 PLACE IN OURFiNAt'ANSWER? (6.) WHY ?' (411-+ 2T2..7-= 6T2. The 6 isrecorded the T2 place because in thisefexample the 6stands for 600.)

your final work should look like this:

438'+ 249

687

Worksheets 18 and 19 should be done individually duringfree time. Each problem on Worksheet 18 is to be added.using both the addition chart and the vertical form.

All the problems on Worksheet ,19 involve regrouping. Youmay want to go through the first problem with the class tosee that they understanid hOw to regroup using the standardalgorithm. If the children are still having difficulty regroup-ing after they have completed Worksheet 19, you may wantto'use the problems on Worksheet 20 for class demonstration.Otherwise, the children can complete Worksheet 20 on theirown.

Worksheet.4' that 24

Name

Add. Do not use your desk computers.

26t314

575

2 6 1 \41

5 7 sr

265722

oy4

:,

i.5.

.2

, $ 7

43A412

21-1-.

ri4.2

1343.. 31.4

, .21t '

1

7i 4

s/

9 f.

IlorkShete%\k NameUnit 24

Follow the steps to do this problem:2;5

792

Step t. Start with the 10 plate. Add. 5*rite your answer In the box 7above.

12

Step 2. Did you pill the 2 In th;'70 plate?

Did you write the I above the 6 to remind'yourself, that you regrouped? YeS

Step 3. to to the T1 place and Add.

H*rite 9 In the Ti place to thebon above.

Step 4. Co to the,12 plate and add. 2S

*rite 7 In the 72 plate. 7

Step 5. In your answer 792111--

\ /Do these Mohacs.., 36 76 1:47

: 35 Litt 'LSD' '07I WV 370

/\e % I A9Ai2$7 . 51S

ig,2 4.3- 7/7

Worksheet 20Untt 24

Add.

Wm,

473

t

6349

/3.2

..%

46 4336 iiPz 57

. ,'64'647

3444 57

t:C5 /2.4

se4967 667

r50 53()

c.

39

47,

40

Lesson 6: ADDITIO& PRACTIC GAMES

This lesson provides.practicein addition that should bepleasant for the children. Spend.this class-etiodducing tte games, and encourage the children to play themin free time throughout the year.

MATERIALS,

15 pairs of dice (one of each pair labeled 0-5, the otherlabelee4-9) .

pencils and paper

Worksheet 21

PROCEDURE

Activity A

Begin by reviewing*the addition algorithm. Have a child goto the chalkboard and work a problem that requires regroupingwhile the other children do it at,their seats. Do as many imore problems as you think the children need. Then play thefollowing game.

Let each column of chil-dren be .a team. Write a.number between lend 20on the chalkboard in' frontof each team. The firstchild on each team goesto the chalkboard,, writesanother number under thefirst and adds the two.Then the seconcrchildgoes to the board, writesa number under the firstanswer and adds thesetwo. This continues un-til each person in theteam has had a chance toadd a pair *of 'numbers.,,,tThe team whose finaltotal is closest -F.) 100 isthe winner.

54 0

4+7

) ,-

For example:

c.

13 (the number you put on the chalkboard)

+20 (the number the first pUpii pUtS on the hoard)

33 (the answer the first pupil gets)

+ 5 (the number the second pupil puts on the, board)

38 (the answer the second pupil computes)+1` .(the number the third pupil adds)

55 (the answer.for the third Pupil, etc;)

. ;

One minute should be long eriough for each pupil, but thisis not a race against time, ,so allow more time if your classneed's it: Have each team of children check its work at thechalkboard. Allbw them to telithe person at the board if his,Work is wrong', but not what the mistake is.

This game provides an opportunity for you to determine whichchildren in the class are having trouble regrouping. You canprovide more practice and individual attention for those chil-dren whd need it. When you play the game a,Second time,start from the back of each column.eo that every child gets achance to regroup with larger numbers.

Activity B

Let the children choose partners. -Give each pair of childrena die with the numerals 0, 1, 2, 3, 4, 5 on the face. Thefirst player throws the 0 °- S die twice. The first throwthe ten's place digit, the. second throw is the one's placedigit. The player records this number on his paper. Thenthe second player generates a number the same way. Bothplayers repeat this procedure and record their numbers'in thecolumn form.

Sample: ist Player 2nd Player

'4}.+ 24 + 04

5 5 41

sit

0

42

"

4,*"

I

Each player adds his numbers. The player whose sum islarger scores one point. If a player makes an error and hisopponent notices the error,. the opponent wins a point. Thefirst player with 5 points wins the game. The next gameshould be played the same way, but with the die'inarked4 - 9. You can vary the gaine by using the die marked 0 - 5for the first set of numbers and the die marked 4 - 9 for thesecond set of numbers and vice versa. Three and four-digitnumbers should also be used. Afterthey have played thegame once or twice, let the children decide how many digitsto use for each number. These variations will give the .stu-dents practice with all of the addition-facts and many oppor-tunities to regroup.

Activity C 0

Computational skilld are developed by frequent practiCe .

over an extended period of time. This practice need not betedious, but should be provided often and with emphasis onaccuracy.

r "0 V

se

4.7

4

o

Worksheet 21 provides practice in computational skills. Itwill be used each day. for orkpweek. Explain to the childrenthat.the class is going to have a contest in computation for,one week. Have them turn to Worksheet 21 and look at the'left-hand column, whiCh contains the practice problems.These problems are similar to each day's test problems. Setup a certain time to give the test each day and tell the chil-dren to complete each day's practice problems on their own\ before the test. Put the answers to the practice problems onthe chalkboard or bulletin board each day.%(under a coyer pieceof paper) so that the children can check their own answers,

When all the children havedone the practice problemsfor the day, write the testproblems on the chalkboard.The children copy these ontheir worksheets and' solvethem. Correct the problemstogether and have the chil-dren record their points asindicated on the worksheet.You may wish to devise areward system for thosescoring the most points dur-ing the week, or for thosescoring.abov.e a Certainnumber-of points. You orthe children can make upsimilar worksheets through-out the year, varying thedifficulty as is appropriatefor different students.

.

Torkshost 21Unit 24

Practice Problems

N...

.Pointfor mach My

Test Problem Problem Point.

bay i

46

2?72

524375

S 22

432669

. ,,

153523

1676

Day 2

720

' tt/?IS

5042.

93

.

603372

464086 19'75

Day 3o

37

_,,11

rta

6435

t-iv

5318

71

7274----146

2

Day 4

27

20123

o

784111,2

4876

124

83mg

172 2

Day 5o

325427 9,3

`

253

6

383

36

'106627733

416674

31092

kb Score

43

r 4...

44

SECTION 2 SUBTRACTION'

PURPOSE

To introduce and provide, practice with the subtractionalgorithm.

To provide 'game situations for enjoyable practice withbasic subtraction facts.

COMMENTARY

In this section the subtraction algorithm is,.developed and thenapplied to problems written in the vertical form. In Lesson 7,the children program the T-pieces computer to do subtraction.They divide each place of the computer in. half horizontally.The minuend is in the top or "start" level and the subtrahendin the bottom or "subtract" level. Subtraction is done in thecomputer by a comPatisor process., Each piece in the bottomlevel is matched with one in the top-leVel:-To-find-t1;e_dif7ference, the computer Counts the number of unmatched piecesin the start level.

In Lesson 8, the rule for regrouping is developed.. One piece.from the next larger place than the place lacking pieces isexchanged for ten of the n'eeded pieces. The new pieces areentered in the start level of their place. Then the pieces arepaired and the difference is. noted.

The physical process of subtraction represented by the T-piecescomputer is replaced by a mental piocess when the computer is'converted to accept numerals in Lesson 9. The child at eachplace does the subtraction in his head and writes the differenceon a slip of paper.

When the children are familiar with the, subtracting and regroup-ing algorithms,, the vertical form for 'subtraction is developed.In Lesson 9 a representation of the numeral computer is drawnon the chalkboard. The minus sign replaces the "start" and"subtract" labels. Place value is stressed while problems areworked using this chart.

In Lesson 10 lines are removed from the chart until only thevertical roiiiiremaja_.s. Then the class develops.a notation,for regrouping and works-p ractice problems using, the verticalform.

Games in Lesson 11 provide enjoyable practice-o additionand subtraction facts.

0 5045

46

Lesson 7: COMPUTER SUBTRACTION'S.

This lesson extends the computer rules so that problemsinvolving subtraction can be solved. It is the first of fourlessons that 1.vill.develop the procedure for' ubtraction firstwith the T pieces computer and finally in vertical form.

MATERIALS

computer desk labels T2, Ti and TO

6 pieces of 3" x 5" paper (3 labeled "start" 'and 3 labeled"subtract")

demonstration T pieces

masking tape

-- for each child

desk computer (Worksheet 4)

ruler or other straightedge

2 pieceS of 3" x 5" paper

bag with set of T pieces

Worksheets 22 and 23

G

71,

PROCEDURE

Activity A

Discuss with the class how the T pieces computerworked foraddition. (It would take only certain pieces in each place,combine the pieces, regroup if necessary and produce theanswer.) Then ask the class:

WILL THIS COMPUTER LET US SOLVE A PROBLEM SUCH AS,5 --3? (No.)

WHAT IS DIFFERENT ABOUT THIS PROBLEM FROM THEOTHERS WE 'DID WITH OUR COMPUTER? (It involvessubtraction.)

Place three desks at the front of the room; labeling them T2,T1 and -T°', as in previous lessons . With masking tape,divide each desk top into two sections and label as shown.(Save these labels for later lessons.) Also draw this three-desk computer on the chalkboard.-

START

IsuCTRAcij

T°

STA RTI

IsuentActi

Use the following procedure to develop the subtraction pro-cess.

THE COMPUTER MUST SOLVE THE PROBLEM, 5 - 3. HOW-136-WE-WRITE THIS IN T NOTATION? Write both equa-.

tions on the chalkboard:

5 = 5T0-3 = -3T°

- 61. 47

Choose'five children to work the computer (a Recorder, aProgrammer and three Computers). Tell the children theywill work at the desk computer doing the same thing thatyou do at your computer drawn on the chalkboard. The'Recorder copies the problem in vertical and T notationforms on the chalkboard.

WHAT NUMBER WOULD THE COMPUTER LOOK AT FIRST INOUR PROBLEM, 5 - 3? THAT IS, WHAT NUMBER WOULDIT START WITH? (5 or 5T°.)

Enter the pieces representing 5 (the first number or minuend).in the start level of the To place in your computer on thechalkboard. You_can draw_in_the_piexesbutitidill-be-better-to tape the paper T pieces in place. (Remind the other chil-

--dren in-the classroom to *etch closely so that they will be,..able to work in groups later.) The Programmer counts out, theTo pieces and feeds them into the To place of the desk com-puter.

WE-,HAVE NOW TOLD THE COMPUTER TO START'IATITH 5.WHAT IS THE NEXT THING WE-MUST TELL IT TO DO?(Subtract 3.)

Enter the pieces representing 3 (the number to be subtracted--or the subtrahend) in the subtract level of your computer.The Programmer does the same at'the To desk.

WE MUST 'NOW PROVIDE THE COMPUTER WITH RULES ITCAN USE ALL THE TIME TO SOLVE SUBTRACTION PROB-LEMS.

Suggest a rule similar to this and write it on the chalkboard:

One by one, take the pieces in the start level and pairtherm with pieces in the subtract level.

48

riTA RT

urnrsUBTRAcr 1

EEO

Draw lines pairing the pieces in your computer. , Or, if youtaped T pieces to the board, rearrange them.so that they arepaired up. Instruct the child at the TO place of the desk com-puter to pair his pieces. Tell him not to remove the piecesfrom the start level, but only slide them together so there isan obvious physical pairing of the pieces. Each piece in thesubtract level must be paired with a piece ip/the start level.

When you have .performed these operations ask the class:

HOW MANY UNPAIRED PIECES DO 1,'E HAVE LEFT IN THESTART LEVEL OF THE T° PLACE OF THE COMPUTER?(Z T° pieces.)

WHAT IS THE SOLUTION TO THE PROBLEM , 5 - 3? (2.)

tli.th child at the TO place of the desk computer how manyunpaired eces he has left. He tells the Recorder this

'number anethe,Recorder writes the answer on the phalkboardin T notation arid't en converts it to standard decimalnotation.

Mention that when working 1.-1.1 groups the Computer tells theRecorder the number of unpaiied pieces remaining in the startlevel at each place value. Select five new children tooperate the computer. You may want to do a problem on the

6 t.)

ISuBTRAcTI

50

chalkboard while the .children do it at the computer so thatall the children in the classroom can see what is being done.Feel free to go through this procedure faster if the childrencatch on quickly. Write this problem on the chalkboard nearyour computer: r

348 = 3T2 + 4T1 + 8T0-235 = -2T2 + 3T1 + 5T0

The Recorder alsO writes the problem on the board. As youfeed the pieces for 348 (the minuend) into the start level ofyour computer, the programmer feeds the pieces into the cor-ract places of the desk computer. Then you feed the piecesfor 235 (the.subtrahand) into the subtract level of your com-puter while the Programmer does the same with the deskcomputer. Ask the children in the classroom what to do next..They should tell you to pair the pieces in the subtract leVelat each place with pieces in the start level of the same place.The child at each desk pairs his pieces and tells the Recorderthe number of unpaired pieces remaining in the start levet ateach place.. The Recorder writes each number on the chalk-board under the problem. While he is doing this, you; pairttl? pieces in your chalkboard computer. The children in theclassroom should act as Repairmen and check your workTosee if it is correct. The children at the desk computer checktheir work against yours to see'if they are correct. Afterpairing the pieces in each pla.ce these pieces remain:I T2 + I T1 + 3T0. The children should have little difficultytranswsingithis into base ten notation, 113.

Have each child take out his desk computer and draw adouble line through each place compartment of the computerto separate the start level from the subtract level. Use3" x 5" pieces of paper to label the divisions "start" and"subtract." These labelS can be placed next to the computer.The new desk computer should look like this:

START

O

rtTf" T1 TO

6

V

Divide your class into groups of three. Put at least one child-who participated in the,demonstration into each group to helpthe others. Let the children choose)obs and work severalproblems such as 74 32 and 629 - 512. Walk around theroom checking to see that each group is using its subtractioncomputer correctly.

Point out to the children that in the TO place of their deskcomputers, they will have to match groups Of pieces insteadof pairing individual pieces since only three pieces can beput flush with the dividing line.

5

Activity B

F.ac_h_group of three children should now complete Worksheet22 together. The members of the group take turns being Pro-grammer and Computer. Each child acts as his own Recorderand writes the computations on his worksheet. The childrenshould complete Worksheet 23 independently.

Vetteheet 20Ualt 44

Ube year Seek eseputor to subtract. Use T notation It

yes mot to.

,l ..

140136

13

37 .

- 241

12.

yrs- 364

5g,

so- 47

-iL

1411 ,.

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Iles your desk computer to subtract.

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Lesson 8: COMPUTER SUBTRACTION WITH REGROUPING

In this, lesSon a rule is devised which tells the,computerthowto subtract when regrouping is necessary. In Activity A .thechildren cothplete a worksheet of subtraction problems usingthe facts they have store heir own memories. In ActivityB the children use their comp rs to regroup in,subtractionproblems. ,I,

MATERIALS

2-forach child =-

..desk computer

set of T piecesWorksheets' 24 and 25

%Outlast 24Unit 24 Nair

Subtract. Do not use your (copular.

0 9 a 7411/

5 6 a 9 7-3 -5 -5 -7 -6.

C.', , 9 5 9 7-5 -9 -5 '

7

9.4

a4!, 9 7 9 6.7 -3 -7 -2 -4

E 7 8 7 b 7-3 -2

F 14 15 16 13 12.6 -9 -5 .5 -7a

PROCEDURE

Activity A

To check the children's'knowledge of subtractionfacts, ask them to doWorksheet 24. Introducethis test as you did withthe addition test by men-tioning that the .children'sbrains are like computers,and can be tested to findout what information they-have stored.

tl

.

53

LSU$TRACTI

54

S.

Activity B

fl

Q..

Ask the children'to form groups of three and see that eachchild has his desk computer and T pieCes. Have each childin the group work this problem with his" desk computer:43 - 26. The children in each group can consult each otherabout this problem.

As they proceed, the children "will find that they do not haveenough pieces in the start level .of the TO-place to pair witheach piece in the subtract level of that place.

T1 0

START-

nm

Remind the children of the.rule for subtracting with thecomputer: each piece ih the subtract level must have amate in the start level. Ask the children for suggestionson how to pair the pieces in the two levels of the TO place.If no one suggests exchanging a Ti piece for TO pieces, ask.them What they did with the extra TO pieces when. they wereadding. Remind them that they exchanged 10 TO pieces for

1 Ti piece. Ask the children how they could reverse this ...

process to get some more- TO pieces to Solve the problem.When they have discovered what' to do, formulate a rulethat tells the computer how to regroup for subtraction prob-lems: 4

To regroup, the computer must know which T place needsmore pieces. Then the computer removes one piece fromthe start level of the place to the left of the one that needsmore pieces and exchanges it faten of the next smallerpieces. These smaller pieces are entered into thestart level of the T place that needs the extra pieces.

63

0

a

Now have the children ag64n try the problem, 43 26, usingthe new rule for subtracting with the computer. Each computershould look like this:

1 0T.

All the pieces in the subtract level of both places cari nowbe paired with pieces in the start level of both place4; Thechildren should,count the unpaired piecs in each place todetermine the ailswer." They will see tkiat, _V T1 °and 71 TO

pieces remain unpaired and that IT' -1-f7TO = 17.

0

55

tifricaroz_frt.r4^4.40.1

O

0

.Oo0

Two children in each group can put away their desk corn-. putersoancfT `pieces," so that only one computer is Out foreach group. Remind the children to rotate jobs of Pro-grammer," Recorder and Computer. Then have the groupswork as many prbblems ou feel necessary for the classto master the regrouping procedure. Go aroun e rchecking the work and helping those groups that do notunderstand the prOc81.Trer,-Suggest problems such as:

64 482 328 . 452 '''...",reto-Wawstla-mot.re39 /7 257 ''-236 -376

,After you are assured that the children know how to subtractwin regrouping, assignXbrk-slieet 25 to be done individuallyduring free time_with-lhe desk computers: If there is time,you may-want to let the children start the worksheet duringclass time to determine whether any children are havingtrouble regrouping;

co

PoriaMot >r NomUnit 114

etrbtroct. L. your ceoputor.'

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iS\7 01

Lesson 9: DEVELOPING THE VERTICAL FORM FOR SUBTRACTION

In this lesson the rules for the subtraction computer arechanged so that it will handle numerals rather than T pieces.The arrangement. of levels and columns in the desk computerused in Lesson 8 provides a smooth transition. to the develop-ment and use of the algorithm for subtraction in verticalform.

MATERIALS

labels marked T2, TI,,,T°

masking tape

3 "start" and 3 "subtract" labels (from Lesson 7)30 to 40 slips of paper

Worksheet 26

PREPARATION

.44

Before the lesson, prepare 30 or 40 small slips of paper fromlarger sheets or provide a scratch pad for your demonstration.

PROCEDURE

ActiVity A

Discuss the inadequacies of the T pieces computer for sub-traction (it is slow., we will not always have 'one available).Ask the children to think of a way we might change the com-puter so that it accepts numerals instead of T pieces.Remind them that they used an addition computer that ac-cepted numerals in Lesson 4.

Set up the three-desk computer used in Lesson 7 by dividingeach desk into sections with masking tape, and labelingeach section "start" or "subtract." Select a Programmer,a Recorder and three Computers.

Present this problem for the compUter to solve: 469 - 327.Establish the procedure for putting numerals into the

57

computer. The Recorder writes the problem on the chalkboardand rewrites it in T notation. His record should look likethis:

469 = 4T2 + 6T171- 9T°-327 = 3T2 + 2T1 + 7T0

Then the Recorder transfers the numeral and place valuenotation for each place onto separate Slips of paper.

. ,

0

58

O

9;

The Programmer enters the first number (the minuend) byputting the appropriate slip of paper in the start level ofeach place. The number to be subtradted (the subtrahend) isentered.in the subtract level of the computer.

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9116701ct

'ismer

IUCPAC-11T°

After the Programmer has pushed the start button, the Com-..,puter at each desk mentally subtracts and then writes the .

answer on a slip of'paper, labeling it with the proper placevalue. Each Computer hands his slip to the Programmerswhenit is called for. The output for the sample problem is:

The Recorder writes the answer in T notation under the Tnotation problem on the chalkboard, and records the differencein vertical notation (142) under the original problem.

Give the children as many examples of subtradting withthe numeral computer as you feel are necessary for them tounderstand the process. Suggest various types of 'problems,but do not include any that involve regrouping. Give asmany children as possible a chance to be part of the computer.

ti

4.

Activity B

'Discuss with the children the fact that we do not alwayshave a computer available to help us work subtraction

rt

59

problems. Tell them that you have an idea for making achart that can be used to help subtract.

Put a sample problem on the chalkboard. Next to it make asketch of each. place and level of the subtraction computer.As the children tell you where to enter each numeral, writeit in.

, -......4 4"

T0

6..,-------- ---g' STA RT

a-- / 0 a 5pagiRACT

/7

Point to the TP Place in the computer and ask what we do. with the numerals there. (We subtract 2 from 8.)

Write in the +result below the bottom line of the computerI

in the T0 column. Enter the differences in the T I and T2places in the same way. Ask the children to compare thecomputer with the original problem to see if they noticeanything interesting about the placement of the numerals.(The numerals in the computer are in.the same places as

0 the numerals in the problem.)

Erase the lines that separate the start and subtract levelsof the computer and ask: .

HAVE I CHANGED THE PROBLEM BY REMOVING THESELINES? (No.)

Erase the! start label and ask:

IF I TAKE AWAY THIS LABEL, CAN I STILL TELL WHICHNUMBER ISTARTED WITH? THAT IS, CAN I TELL WHICHNUMBER I WANT TO SUBTRACT FROM? (Yes, it isalways on top.)

r17

PA" A

1 °IX

o

Discuss the original problem:

1------==.MTh' EXAMPLE, THAT ittt'S'""ti-S-T-Hil a ARE GOING TO

.e WE KNOW THATTRACT THE SECCX

SHOULDN'T ADD THESE NUMBERS?NUMBER FROM THE FIR I+Ifl

b(The minus sign tells us\o subrt.) CAN WE REPLACE .toti.,4, ,

THE WORD SUBTRACT IN R20nRUI ,3641E FarT111*- .t7P'" ',$4,1

, ,4AgrErase the subtrabt label and r;i7lZe''"it .4rinu014n.

mr.A.,,,,Draw-a1,14e beneath the lower numerals of he prOSIFITL-,--,,,v_.\ .,_.,,,separ ting them frOm tireg ace for the. answ Make thnecess ry changes in your c board compute o makedtN

look .lik this: ',.t-.0 '\

40.04.-A1"

6 I

'44

I 0 oZ-

3 6

Explain that nothing has been changed by altering the com-puter. Verify this by again subtracting the numerals ineach T place. Have several children do problems at theboard using the simplified chart.

When you are sure the children understand the use of thechart, ask:

Write a problem on the chalkboatd-in-vertical formdAtforexample, 633 - 50Z, and si.fblraet---,dectly.

OUT USING THE CHART? LET'S TRY.CAN WE SHOW WHAT WE DO WITH -

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children of the place value subtraction: that is, 6 5 = I

is reall 6T2 - 2 or 600 - 500 = 100. 6 - 5= I is

en do Wo lisee 1,26 using either the sub-or the vertica

ave;k,/:ti:liz,....4.,,..tisa4 tra ).0T

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trach Vie the charts If y need then-

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--lesson 10: SUBTRACTION WITH REGROUPING IN VERTICAL FORM

In this lesson the students use the subtraction chart to help, them with problems involving regrouping. Then they learn to

use the vertical form to solve subtraction regrouping problems.

MATERIALS

Worksheets 27, 28 and 29

x 11" paper for..w.,;11..<<--p'4`

PROCEDURE

Activity A

Draw a subtraction chart on the chalkboard and pose a prob-lein which involves regrouping, for example, 482 - 265. Askthe'ciiildren to copy the problem and solve it using a similarchart they can draw on a sheet of paper.

42a.

T2 T1 TO

5

The question of how to subtract 5 from 2 will arise.Review subtraction regrouping with T pieces. One ofthe next larger pieces was removed and exchanged for10 of the heeded pieces. Explain that since we cannotsubtract S from 2, we must take a T1 piece away and ex-change it for 10 T° pieces. Cross out the 8 in the T I place

63

a

.11 c

and put in a 7 so that everyone will know we have taken 1 T1

away. Emphasize that you take the T1 piece from the topnuniFOT,fffeo-n-e-751 are subtracting tam arid-not-from-ttre--bottom number. Then, on the chart, add the 10 TO to the 2 TOalready there, making 12 TO.' Now we can take STO away,leaving 7. Subtract in the other places, reminding the chil-dren to subtract 6T1 from 7T1 , not from the 8T1 that iscrossed out. Your completed chart should loot like this:

-.265

T2 T1 al To/04-.2=1.1

7ry

6 3--

Activity B

Explain to the children that we do not need the chart to dusubtraction that involves regrouping. Do a sample problemwith the class using vertical form, but stressing the placevalue of each operation.

As you work the problem,point out the place thatneeds more numerals (theTO place in this example).

Cross out the numeral youwill take 10 from (the 2 inthe T1 place). Write inthe remaining numeral,(1),to remind yourself thatyou have taken away I T1.

64

3 2 6-2 1 7

3z 6-2I 7

Remind the children that theI T1 or 10 must be added tothe T° place. Show them howyou wrote this on the chart:cross out the 6 and writeabove it, 10 + 6 = 16. Nowthe children know that theycan subtract 7 from 16.

Tell the children there is away to indicate re- 1/b---grouping in t ce. 34 0

-2 I 7

lo.i.6=163

-2 I 7

We-can do the addition inour heads (10 + 6 = 16) andonly put down the sum (16).

1),

Now ask:

1

CAN ANYONE THINK OF AN EVEN SHORTER WAY TO SHOWTHAT WE HAVE REGROUPED TO THE T° PLACE? D

If no one thinks of it, showthe children that they canwrite a I next to the 6 toindicate -16 rather than cross

-2 1 7out the 6 and rewriting 16.The children shoul,c1 learn todo the problems this way.

Se sure the children understand that when they regroup a TIto the T° place, they are adding 10 to the number alreadyin the T° place.

Complete the problem. Now have the children work severalproblems, preferably without the chart. If they need thechart let them use it for one or two problems. Then eachchild should work a few problems without the chart .beforegoing on to the next part of this activity. Write problemslike these on the chalkboard in vertical form: 47 -319,627 - 243, 250 - 135. Go around the room checking workor let a few children come to the board to demonstrate thesubtraction regrouping procedure.

65

p

bb

--

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1111.11

44,

When the children feel confident about regrouping, pose aproblem on the chalkboard that involves regrouping in morethan one place, for example, 322 - 158. Work the problemwith the class, letting them tell you how to proceed.

If some children are confused about subtracting in the T1

place remind them that they must regroup from the next largerplace. They must take I T2 from the T2 place. A, T2 equals10 T1, so they add I OT1 to the number already in the T1 place.

Activity C

Have the children turn to Worksheet 27 and work the firstProblem or two with them. They should be able to completethe rest of the worksheet themselves. Worksheet 28 providesmore practice with subtraction. Encourage the children tobecome efficient in using the subtraction algorithm with prob-lems written in vertical form rather than dependingon thesubtraction chart.

0

The practice and test problems on Worksheet 29 should beused for one week. Each day the children do the practiceproblems and then you give them the test problems.

lierkareest 27Unit 14

Mae

fisittract Use-the charts only If you need thee.

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946 i 134

- lid - b

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Porkshoot 211Unit 24

Subtract.

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. 646- 726/ /,"

703 452- 402 - 348

3o1 i ay

4639 406- 3217 - 392

900 1342- 6P- si

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IN tt INI 10

Practice 'Probleses

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Pointsfor each IV

Test Problees Problem Points

Ilicy 1

41- 34

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610

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Day 2

72-36

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Day 5

622- 3116

234

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'b Score

67

.

68

Lesson 1 1: ADDITION AND SUBTRACTION GAMES

Games and a worksheet give the children practice withaddition and subtraction problems. Th9, games "Fast Math"and "Mental Arithmetic" can be varied ..nd played as oftenas you wish.

MATERIALS

Worksheet 30

PROCEDURE

Activity A: Fast Math.

Divide the class into two teams. Each team may select aname, for example, "Control" and "Chaos." If you have anodd number of children, ask one to serve as Recorder. Seateach team so that the first personon Control is matched withthe first person on Chaos, etc. Since the pairs will be corn-peting , match children of equal abilit.

The first pair go t6 the chalkboard, sta don either sideof a scoreboard drawn there. Give the pair a problem.Both are to write the problem, add it, and record the answer.The firstrto write the correot answer wins a team point. Thepoint is recorded on the scoreboard under the team name.When they have finished, the first pair return to theirdesks and the second pair go through.the same prodedure.This continues until everyone has done a problem. Playthe game again, this time using subtraction' problems .

You may want to stop during thejgarne to discuss any proll-lems that arise: Such discus'sionsaar. be a good leargingexperience.. Children who are slow-Writers will be at adisadvantage, but may find they Can still win by doing thecomputation rapidly. Both their writing speed and computa-tional skills may benefit from the competition.

c°Ad.

4

/.,..r \.zThis game can be used in several ways. hildren who ftn

.

their work early can get together in smqyvroups t8 play 4.4t round or two. You can divide the childr44 into,teams ac-cording to abilities and needs, and assign the problems ac-cordingly. Children:Who need remedial Work should be es-pecially encouraged to play the game on their own.. Foranother variation, alternate addition and subtraction problemsduring, the same round.

Activity B:, Mental Arithmetic

The Mental ArithmetiC'gdme from Lesson 3 can now be expand-ed to include subtraction.' Pose problems such as,, 7 -, 2,8 1 .f r - 9. :`When the children ; acome proficient at add-ing' and'subtracting nientalfr,1 you can vary the game y ask-inging problems that involve c:iloperations,, such as10 - 6 + 3 and 8 + 10 8. YC.: can use this activityd ringunstructured time, for exb ple, when tithe children are 1. pingup to leave the room or when they are getting out new ),k

materials: Spend no more than two or three minutes each4, ,,time you play.

1.,

1

. . t., 169

SiI4

c

N,

ActivityC

\Workshe t 30 gives the children practice in recognizingaddition lnd subtraction combinations which yield 10's or100's. As the children complete the workiheet ask themif they con find a pattern-that helps them finish the problems

terthe pattern in the first problem4s:

3 7 8 -L..\i): 2`4 + 6;

10 10 10

---'7i-previile-more practice with the types of Combine ions,use similar problems during the Fast Math game.

.

O

30

/'iw

/3

4

30

100

or subtract.

Num

6 3 7 2 .E

7 0 4 0 6 0 . 7 0 3 0 . e 0 . 2 0 a

200 300 o

10 . 30 ao 90

90

82

- 90 60 - 30 30 60 o o 00

30 43 57 21 79 o

-.-, No.

SECTION 3 MEASUREMENT

PURPOSE

To review units of measure and notation for these units.

- To observe the relationships between units in each system.of measure and compare them with other systems ofrelationships (place value systems, etc.).

7 To establish a feeling for relative size among somecommon units of measure.

COMMENTARY

In Section 3 the students investigate several systems ofmeasurement. Physical representations of these measuresare used to establish a feeling for the size of each measureand for the relationships between the measures. The chil-dren should be encouraged to play and experiment with thematerials during free time.

The study of measures of volume, length and time 'durationis included here to emphasize and contrast the relationshipsamong the units in each of these systems with those of ourbase-ten numeration system. The relationships betweenmeasures of volume are regular, as are the relatiOnshibsbetween place values in the base-ten numeration system.The relationships between measures of length and time arenot regular. These similarities and differences should beemphasized.

Lesson 12 deals with the measure of liquid volume. Theunits that are studied include the half-pint, pint, quart,half-gallon and gallon. The children set up a base-twocomputer to help them in the addition of the units of liquidvolume measure. The students draw concept tree diagramsto illustrate.the relationships among these units. Lesson12 will take two class periods. You must supply one half-gallon container and one gallon container.

In Lesson 13 the children study the British syStem for mea-suring length. They compare the British system of inches,

71

72

feet and yards with the metri-cTsystem and find that therelationships among the metric units are regular: each unitis ten times as large as the next smaller unit. The childrenmake a concept tree illustrating the irregular relationshipsamong the British units. The children do not set up computersin this lesson, but feel free to encourage them to do so ontheir own. As a class the children should discuss how thesecomputers would work even if they do not actually set themup. This on will take two clan 0 periods.

Lesson 14 is an investigation of the measurement of timeduration. The children make physical representations ofseconds, minutes and hours to illustrate the relationships:among theSe units. Endourage the children to discuss howa computer for time duration would add amounts of time.They should be encouraged to set up such a computer, or atleast make Up the rules for it, even though they do not act-ually set up the computer in this lesson. The lesson willtake one class period.

The abacus may be used to provide a representation of therelationships that exist among the units of measure in these- ,measurement systems. For example:

C'

Section 3 attempts to emphasize the relationships amongunits of measure within any given measurement syst6m.In all of these investigations, the students should beencouraged to finish the following statement. unitsOf this size are equal in value to one unit of the next largersize.

80

LessoR12:'LIQUID VOLUME MEASURE,Though met:, of the world uses metric units, to measure liq-uid volume, the United States uses the half-pint, pirit,quart, half-gallon and gallon. In this lesson the childreninvestigate the relationships among these common units ofliquid measure. The lesson should take two clasp periods.

In Activity A,the children study the relationships among thehalf-pint, pint, quart, half-gallon and gallon containers.In Activity B they build a base-two computer that addsvolume measures. These two activities should be done thesame day.

Activity.0 is optional, depending on whether or not yourclass .needs more practice in adding with a base-two com-,puter asethey did in Activity B. You may wish to us_e_theactivity for remedial work with some children or as a ten-

_minute review at the beginning of the class period on thesecond, day.

In Activity D the children make a diagram called a concepttree which provides a visual representation of the relation-ships among the different sized containers.

A time schedule for this lesson could be:

Day 1: Activity AActivity B

Day 2: Activity C

activity D

MATERIALS

10 minutes20 -30 minutes

-(Optional or 10 minutes ifused for review)30 minutes

desk labels marked "Galion," "Fgallon," "Quart,""Pint," "4-pint" and "Bank"'

- containers (with sizes labeled): 19 half-pint, 9 pint,4 quart, 2 half-gallon and 1 gallon

73

74

masking tape

water dipper or cup

yarn or string

- water10 20 -slips of paper

L. large bucket or pail and a sponge (optional).

Worksheets 31.- 35

PREPARATION

For this lesson you or the children should bring in onehalf-gallon container and one gallon container. Beforethe lesson, make one each of the desk labels mentioned ----in the Materials List. Using masking tape, also labelthe size of each container. Place the containers on thetable you will be using for the bank. The children willneed a supply of water for this lesson. If you do not havea sink, have a bucket of water in a place that is conven-ient for the children. Since there is bound to be-sorne spil-ling, have a sponge ready.

PROCEDURE

Activity A

This activity illustrates relationships among the various liq-uid measures. The children will have an opportunity tohandle the containers and talk about the relationships they

<,discover. The activity serves as an introduction to units ofvolume measurement and will provide an opportunity to com-pare a base-two system with the base-ten numeration system.

Place one labeled container of each size or a table. Thewater supply should be nearby. Have two students come tothe table and assist you. One student will be the Experi-menter and`bne will serve as Recorder. Write the title,"Liquid Measure," on the chalkboard. In ardolumn to theleft on the board write " I pint," " I quart/ "i-gallon,'' and" I gallon."

8 3

a

Have the Children turn toN- Worksheet 3I'which is a

copy of-the liquid- Inea-Stitechart; you ha-e dfawn on thechalkboard. The childrenat their desks will fill in-fhb worksheet as the Experi-menter and Recorder demon-strateat the front oftheroom. Explain that the Re-corder will be responsiblefor entering information onthe chalkboard chart.

*orkblitet

Unit 24

Liquid Ma.ure

/int. Pint. Quart, Galion

I pint

I qunrt m

Kelton m2

I cotton

Give the Experimenter the half-pint container. Then askthe class:

HOW MANY TIMES WILL THE EXPERIMENTER HAVE TOEMPTY THE HALF -PINT CONTAINER INTO THE PINT CON-TAINER-IN ORDER TO FILL THE PINT CONTAINER?

Guesses may vary. Have the Experimenter fill the half-pintcontainer with water and pour if into the pint container. Theother children in the class keep track by counting each half-pint that is poured into the pint. Each child draws a repre-sentation of each half-pint on his worksheet. The Recorderdraws and labels a representation of the half-pint container

8D

1.

75

on the chalkboard to the.right of the one-pint measure.Each time the Experimenter fills and empties the half-Tintcontainer, the Recorder and children at their desks should.sketch and label a representation of the container. Whenthe pint container is full ask:

HOW MANY HALF-PINTS DID IT TAKE TO FILL THE PINTCONTAINER? (2.)

The Recorder should repeat this as he checks his diagram.(Reading from left to right, "one pint equals two half-pints.")

Now use the pint container and ask the class to guess howmany times the pint will have to be filled and emptied intothe quart container. Follow the same procedure using the.quart to fill the half-gallon and the hall-gallon to fill thggallon. The Recorder and the children at their desks drawand label representations of the containers. The chart shouldlook like this:

Ask the children to imagine a computer set up to add volumemeasures.

WHAT WOULD SUCH A COMPUTER LOOK LIKE? (Therewould be five desks labeled "i-pint," "pint," "quart,"" -i-gallon" and "gallon.")

(You should erase the liquid measure chart at this time.)

76

(is) 0

Draw a chart representing this computer on the chalkboard.Tell the children that last week you bought a half-gallon ofmilk, a quart of orange juice and a hall-pint of whippingcream. Write these amounts in the top line of the chart.Have the children- tell you where each numeral goes. Thensay that yesterday you bought a quart of milk and a half-pint of coffee cream. Put these figures in the chart. Thenask the children the total quantity of liquid you bought, butdo not write down their answers.

G 0 P 2P

/ / 0

0

/

O

OUR COMPUTER NEEDS SOME RULES TO TELL IT HOWTO ADD LIQUID VOLUME MEASURES. LET'S SET UP'ACOMPUTER AND MAKE UP THE RULES SO THAT IT CANADD THE QUANTITIES IN THE CHART.

Leave the chart on the board and go on to Activity B.

Activity B

Briefly review how the T piece(Desks were labeled to represenwere selected from the, class tomers or Computers.) Recall thait was told to do. Emphasize tcertain rules for the computer t

Rule 1. Each place coul

.;

omputers were set up.place Values and childrenct as Recorders:, Program-the computer did only.whatt it was necessary,to makefollow. The rules were:

accept only one d of piece.44,

77

78

0

Rule 2. Each place could not have more than nine-piebes. (If there were more than nine, tenpieces had to be traded in for the next largerpiece.)

Rule 3. Each place must tell its contents ta the Pro-grammer when he calls for it.

Have five children push their desks together in front of theroom. They will act as Computers. Bring oat the desklabels and masking tape. By studying the column headingsthe children should be able to tell you where each label isto be placed. Tape the labels to the desk as the childrenlocate the proper positions. Give each child a few smallblank slips of paper.

Select one student to serve as the Banker. Have him 'someto the bank and group the containers according to size Withthe smallest containers on the right side of the table. (Thecontainers should already be labeled.) Ask the children ifthey agree with the grouping and ordering the Banker has done.

ARE THE CONTAINERS GROUPED IN ORDER ACCORDINGTO SIZE? (Yes.)

or 6 4

VIM

Ao

Ask the danker to read to the class the label on the smallestcontainer. Point to the title in the right-hand column of the

'chart on the chalkboard. The Banker should continue toread the 'labels on the containers in each size category asyou point to its place on the chart. Then ask:

SHOULD;WE FEED T PIECES INM OUR COMPUTER?(No, we 'should use half-pints, .pints-,'quarts, -half-gallons and gallon's.)

Choose a Recorder and two Programmers,. Have the Recorderread the quantities in the first row of the chart (one half-pint,one quart and cane half-gallon) while one Programmer gets thecontainer corresponding to each measure from the bank.These containers should be filled with water and put in the

zproper places in the computer. /

IPTI :01047.;'

.1 0/^44ii4;:r a ;

14 1

. A.

*44- I

11jtJ

a

79

1.

O

80

to

Ihe other Programmer should then go to the bank and get thecontainers that illustrate the amounts fOr, the second:row asthe Recorder reads these quantities. The Programmer fillsthese with water and puts. them. in the computer. Now thecomputet must find the-total. Before it can do so, it musthave a.set2o_f_retres to f011ow: Keeping in mind the rules thatthe addition and subtraction computers followed, ask.thechildien to sug-test-rales for this computer. These shouldinclude:

Rule I,. Each place in the computer earl. accept only.one kind of unit.

_Rule 2. The computer counts the number of containersin each place.

0

Rule 3.. The *computer feeds out the number of .units ineach place.

When the computer follows these rules it will feed out thesetotals. (Write them on the chart.)

O

G IG Q P 2P`/

0

Q 09 /

0 i 0 (9--

Suggest a rule for the computer to regroup:

Rule 4. If any place in the computer.contains enoughof one kind of unit to form the next`largerunit, it must exchange the correct number ofsmaller units for one unit of the next sizeand put that unit in the proper place.

kt,

o . 1

O

7

4

Remind the children that in the T pieces computer, 10 piecesof one kind were exchanged for 1 of the .next larger pieces.But this computer is different. Lead the children to suggestfilling the smeller containers with water and then pouringthe water into the larger cont&iner, In this way they candeterthine.fOrsiire how many smaller-containers equal'slarger one.

There are two half-pints in the computer, so the child at thatplace asks the bank for the next larger unit, one pint: HeOurs one Of his half-pints into the pint Container and dis-covers that the pint container is only about half full. If hethinks another half-pint will fit into the pint, let him pour itin. He will see that two half,Tint:s equal one pint. Then heputs the lilted pint container into the proper place in the com-puter and returns the half-pints to the bank. Emphasize thattwo half-pint containers hold exactly the same amount ofwater as one pint container, that is, these two measures areequivalent. (If the two half-pints did not. completely fill thepint container, have the children spedulate as to why theydid not. Perhaps the two half-pint containers were not com-pletely full or some water may have spilled while the childwas pouring-.)

The last rule for this computer-should now be revised to say:4 ett

RUle 4. If two containers app'eat in any one place atthe computer these two must be exchangedfor one of tilt. next larger containers. An eScrception is gr-ilon's place which can hdld morethan two containers.

The child at the quart place of the computer exchanges the .two quart containers for one half-gallon container. Againemphasize that these are equivalent in" volume.. When thenew half-gallon containefis pit in the proper place, twohaif-gallon, containers can be exchanged for a gallon con-,Miner. i - ,

. .

The children at the computer starting with the half -pint'place, feed out the numerals 0, I , 0, 0, Irby writing them

81

t.

O

on slips of paper. Each slip should also be labeled with0

the place. value. The Recorder fitlI1 in the chart. on thechalkboard.

Choose other children to operate the computer and worK afew more prOblem.. The children should learn to disCriminate.visually among the sizes cd the containers, so remove thelabels a&- soon as- possible.. . , .

While the children. are ttsifig the computer to do problems,discuss with them its similarity to the addition and subtrac-tion computers they'used that were base-ten computers.The rules by which tho.3e computers and this one operate%are the same except that this computer must regroup whenit has two or more containers in a place. The base-ten com-puter did not regroup until it had ten or more piedes in a.place. Let the children discuss the idea that,this Computercould be used to add and subtract numerals using any base,with only slight changes in the rules to allow forregroupingin the proper base.

Activity C

In this activity the children do more 'addition problems,withvolume containers and the cOmputor. If the children in yourclass did well' on Activity B, you can omit thiS activity, orperhaps go through the procedure once as review' befdre start-

; ing Activity D. - ..4,

Place all the containers li ted in the beginning of this les--;son on 'a table. -pes:ignate five desks to be the'compUteand label them with the place! Value names, The-childrenat these desks can serve as Computers. Select a child tobe Recorder and have him draW a diagram Of.the computeron the chalkbbard: The children who -Cloilizt have a roleshould act as Repairmen, watching carefully to see that thecontainers are fed into the proper places. Randomlyldistr'il;-'1.11<4y container to each Repairman. Call on Owed studentsto "fe Id" the container's they have into the computdt one ata time at the proper place. It is important that no two chil-dren you call on have the same size container. c

82

1

:,

The Recorder should write on his chart in the proper placethe number of containers that have been entered or fedinto the computer. For example:

&

Q p ..

I .a.

Call on two to four more children to feed their containersone at a time into the computer. The Recorder should writein the second part of the problem on his diagram.

G:

iG2 Q - P1P-2

a i

Now that the computer has been programmed, have eachplace add the number of containers it has. Retnind the Com-puters that each place must trade in its containers if it hastwo or more. If a place needs to exchange two of its con-tainers for the next larger, that child Should find the propercontainer on someone's desk and exchange it for his two.He shoulcC'take the larger container b7 to the computer andput it in the next larger place. Whi the computer is re-grouping, the Repairmen should watch for errors.' You maywant to allow the person finding an error to replace the personwhO made the mistake.

L.

83

a

',527o When the computer has finished its o.omputation, the Record-,er calls on each place to tell its answer. The solution tothe sample problem is:

84

S

In some problems the gallon's place may end up with morethan two containers. Remind the children that the number ofgallons can go over two without regrouping. (A barrel isusually a 31i-gallein cpntainer and a hogshead is a 63-galloncontainer, but the children do not need to, know this.)

Have the class do several problems following this procedure,Let different children operate the computer each time so that,evetVone in the class gets a chance.

. I "

sr*

Activity D

In Units..2, 8 and 21 thechildren used concept trees/ to classify curves and sortsets of objects infotubsets.

c° The relations among thevarious liquid measures canbe described by building aconcept tree.

`

Place I9 tjalf-pint con-tainers in row on atable or onIthe floor. Havethe children turn to Work-sheet 32 which they willcomplete a you demon-strate*. Th n ask:

HOW MANY HALF-PINTCONTAINERS DOES ITTAKE TO MAKE A PINTCONTAINER?

Worksheet 32Unit 24

Same

.

a

(It may be necessary to recall the regrouping the childrendid in Activity B.) Then place one empti pint container infront of two half-pint containers in your display. Ilse yarnor string to connect the two half-pints to the pint. The chil-dren should draw_ a pint container on their worksheet that

. depicts the same relationship; They should draw lines tothe pint 'container.

HO3Ar MANY'MORE TIMES CAN WE SHOW THAT TWO HALF-PINTS MAKE A PINT? (8.)

-,Have a child come up and put 8 pint containers in the propetrelationship to 16 half-pint containers in the display. Againhe can use yarn to connect the containers. Similarly have .

the children -draw eight more pint containers each in relation

85

' 86

to two half-pint container on the worksheet and connectthern.with lines.

Note that one half -pint container is left over and cannot bepaired. Ask the. class:

HOW MANY PINTS DOES IT TAKE TO MAKE ONE QUART?(2.)

Select another Child to come to the display and depict thisrelationship. He should place one quart container in frontof each pair of pint containers and connect them with yath.The other children represent the relationship on their work-sheets by drawing four quart containers and connecting eachquart to the two pint containers it equals. Again, one pint

IGO2.;

.7

cannot be paired. Follow this procedure and combine quarts. to make half-gallons and half-gallons to make a gallon. Have

the children complete Worksheet 32 as'you complete the dis-play. The completed display should look like the diagrambelow.

1 R.

I QT

a<7al.

Review the tree diagram and What it shows.

WE BEGAli,WITI.1 19 HALF-PINTS. WHAT DO THEYEQUAL?

Answers may be given in terms of pints, quarts, half-gallonsor gallons. 'Accept all answers but be.sure that the childrendo not forget the pint and half-pint that could not be paired.

. /

87

IF THERE WERE MORE HALF-PINTS COULD WE BUILDANOTHER BRANCH FOR OUR TREE? (Yes.)

Continue asking the children_ what each level of the treeequals in terms of the next larger level. Summarize the treediagram:

WE HAVE FOUND THAT 19 HALF-PINTS EQUAL ONEGALLON, ONE PINT AND ONE HALF-PINT.

.

Help the children start Work-sheets 33 and 34. Remind them(if they forget)-that they mustregroup when there are two ormore containers in one place.Be sure the children understandhow to use the. concept tree tofind the relations among thevarious units of liquid ,measure.Have the class cOmplete these

tt" two worksfieets and Worksheet35 by themselves.

Vorkeheet 33Unit 24

Change hilt -Dints to larger unite. Reeelaber the rulefor regrouping.

3 half-Wes

half-We

7 halr,pinte

10 hilt -pinta

12 hilt -pinta

1

14 half-pinta

10 hilt -pinta

gallon i gallon quart pint i pint

1

. ,

10 2

88e

Sorkaheet 341.-111t 24

hare

Sai the concept tree on the hoird to help answer thequest ions.

1Hoe 441Y quarts are there mink 41i1011. ..

I tattoo It the aaseIloe eolne pints are there In urn ¢al lunt_8.I gallon It, the ease eel...pints.floe autos ways .an you sake the same 41.. I bolt -.intim,'One ,edy Is to use one quart ait1 two plot..

roil'esany other ways,san you ...Ike one 1,111-e.,1 ion) I t. Ithem he

!pt ,,41'.P "3 t i,.t Pio( i7,,to

li t Ikatfpiwt5 3 no

4t,t$ a .0.4 tspst.f tp`..1401 1.".

!half s.b

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0

Worksheet 35Intl 2.1

same

tont ept t rev

Draw to tontoIner on the

tontclit tree to represent

S halt-pints. I ow witty

twit-pints till I till one i Pint*. *"!

to eels

\ow draw in tontoiners

ND/N51ot the pints that 5

1111. lowm.ins pints "mid son 1,11 with tin

. wire then .111v k

now

1Pow 0.111% l/Int .111 till 11110 to t there

pints . II Mire .111.C. thaw,

I mt.coms. II, ft et1,1;10( tb onnri th,1 .oit3t fw f Med.

q holt.isints Iry lM s..w0 .0 J... .m.14. Y 111015 Id

103

0

.0

'89

tot

90

Lesson" 13: LENGTH

This lesson begins with an act1vity.that presents a briefhistOry of measurement. It demonstrates the necessity ofstandardized systems of measurement and motivates thechildren to explore the two SysteRs that are in use today.The students compate the British and metric sy tems, andare introduced to the units of measurement, used in each.The first two activities should take one class period.

In Activities C and E the children investigate the relation-ships among the units of measurement in the British 'system.Activity D is an optional activity that encourages further'investigation of the metric system and the relationshipsamong its units.

MATERIALS

- foot ruler for each'child- yardstick for every pair of children

- meterstick- paper- Worksheet 36

PROCEDURE

Activity A

Before standardized measures were introduced, the lengthsof various parts of the body were used as approximate mea-sures. In this activity, the children will use their-feet,thumbj, hands and arms.to measure things in the classroom.

Explain to the .class that a long time ago these measureswere used;

yard = the distance from thetip of the nose to theend of the middle fin-ger on the hand of anoutstretched arm.

10

S

0

foot =. the length of anyavailable foot.

palm = the width of fourfirigers when thefingers are closed.

inch = the width of thethumb.

Have the children choose partners. For each measurementthe children make; one child ineach pair should lay downthe appropriate appendage while the other marks where themeasure falls. Have the.pairs.keep 'records- of their pea-

-sures on paper.

Ask each pair' to measure the width of a student's, desk inpalms and inches and record their results.. Then have sev-eral different pairs measure the width of the door in palms,the length'of the chalkboard in yards, the length of theradiator or. other object on the floor in feet, and the length-of-a--textbook iminches. Choose children who vary in sizeto measure the same object.

103a.

'91

' 92

I

t,

Have all the pairs that measured the sFme object read theirresults. There_should be some variety among the answers.Ask why these discrepancies obcurred (people-are differentsizes); Lead the children to the ides that a more accuratesystem.ameasurement is needed for precise measuring.

Activity B .

. .Tell the children that a lOng time ago people discoveredwhat the class ha& just discovered: that a more precisesystem-of measurement 4as'needed. Write the-words"British" and 'Metric" on the_chalkboard and explain thatthese are names of two systems. Of measurement. Show-thechildren a yardstick and write "yard" -under the proper head-

.

ing. Expkain that this is one of the measures in the Britishsystem: Then show the children a meterstick, compare it-withithe. yardstick and write, "meter" under 1!Metric." Ex-plain that this is one of the units in the metric system.

Write the other units of measure used iri each system be-neath the proper heading.. Ask the children to examine,these closely and tell you'how many of each' kind of unit ittakes to make the.next larger unit. As you enter the mea-sure, draw lilies on the board to show the lengthi of thosemeasurements that will fit on the chalkb'eard. The completedchart is shown below.

British' Metric

12 inches3 feet

5280 feet

.

===

.

43I footI yardI mile

.

.

.., .

10 millimeters10 centimeters10 decimeters

...._10 meters10 dekameters1;0 hectometers

.

.

- == .

=

I centimeterI decimeterI meter, .

1 dekameter.I hectometet1 kilometer

10

4

Point out to the children that the relationships among themetric units are regular. Each unit is 10 times .larger thanthe next smaller unit. Ask thechii0reri what a computerusing the,metric system would look Ake. Ask how Ranymetric units could be in each place of the computer beforethey would have to regroup. (9. If there. were 10Sinits, thecomputer could exchange them for i of the next larger units.)

In contrast, point out that' the British units are not regular.For example, three feet make one yaid, but.12 inches makeone foot. Ask the children to imagine a computer-using theBritish system; Lead them to see that they would have tohave a different rule for regrotiping at each place.

Activity

Choose an object in the room, for instance ydur desk top,and ha've a child measure its length in inches with a yard-stick.stick. Have him write the length of the object .in iriches onthe cthalkboard:

1'

!Then have each student take a blank sheet of paper and'make the same number 'of dots as inches in the object alongone edge.of the paper. You should represent the inches with

!dots on the chalkboard. Then ask how we could tell thenumber of, feet .in the object. (Connect every 12 inches.)Draw lines to make a tree diagram which illustrates thatevery twelve inches are equivalent to one, foot, and havethe children do the same on their p'apers. When they havefinished,' ask them to count the number of feet and remain-inginches on their. charts.

In the tree diagram below, there are 38 inches or 3 feet, "2inches; or one yard, 2 inches.

111111IIIIIIIIII 1 1 1 1 1 1 I I 1 I 1 1 I. 1 .1

10 7

Have the children choose two other objects to measure.Ask them to make diagrams to represent the lengths theyfind in yards , feet and inches. The children may chooseto measure and compare their own heights.

Activity D

This.activity is optional. Repeat Activity C using the meter-stick for measuring lengths. Have,the children draw a con-cept tree to illustrate the length of objeCt they measured.The cOmpleted.tree will,show the rerKtionships among theunits of measurement in that system. Compare these withthe relationships among the British system of measure.

1 a ,e%

-94

//'

C

Activity E

In this activity the children have the opportunity to.practicetheir measuring skills. Worl,thheet 36 provides a list ofthings tc be measured. The children should measure eachitem to the nearest units indicated at the right of the work-sheet. For example, the width of a window is to be mea-sured to the nearest foot while the length of the chalkboardis to be measured to the ne'are.st.yard. Have the childrenwork .individually or in pairs.

The last fourdines on the Worksheet are blank so that youor the children can choose some items to Measure. Askeach child or pair of children to decide which unit of lengthto use to measure each object. After all the measurementshave been taken, children can check with each other to seeif theYfound the same measurement for each object. Havethem justify the appropriateness Of the units they choosefor the last objects.:.

Worksheet 36Un It 2d

Same

Measure. Use ruler or a yardstick.

Length of the teacher's desk (ruler! - - foot

width of the rooe,IyardalIckl

Width of one wIndor 't ruler)

Length of this worksheet Irulerl

width of 'you'e Msk Irulerl

Length of table (ruled

yards

;Litt

Inch*,

Inches

Length of the chalkboard lyardatIckl _tuskWrite In the object. its oeseureeent, andthe unit yoomeasured with.

a

4

Sr,

95

r.

O

96

Lesson 14: TIME

Clocks measure time in seconds, minutes and hours. We'also use, day, week, month, year and century to describeunits of time. In-this lesson, the children investigaterelationships- among some of the units of time measurement.

- MATERIALS

- 2 or 3 sheets of paper per child

- clOok with second hand

-- masking tapeWorksheets 37 and 38'

PROCEDURE

Activity A

Discuss with the Children the relations among some of theunits of time measure. As various units are mentioned,list them on the chalkboard. Your list might include:

60 seconds -= 1 minute

,60 minutes = 1 hour

24 hours =, 1 day

365 days = _ tyear3,66 days = 1 leap year

100 years = -1 century

We are a-cCtistomect3to watching a clock to determine thepassage' of a certain amount of time. It is interesting totry to judge. Some amount of time with our eyes closed.Tell the children that they will try to judge when 60 secondshave paSsed.. ..\ .

Ask. the children to. close iheir eyes and keep them closedfor 60 seconds starting when,you say "go:" Tell them'to-open their eyes whenthey.think.the time has passed. Whenthey have their eyes closed you will be writing numerals ontheChalkboard to indicate the number of seconds that.havelapsed. Start writing 5-secOnd intervals after 1-5 or 20 sec-

.onds have gone. by. As each child opens his eyes he notes

d.

Zr.

,the numeral you have just written to discover how long heactually had his eyes closed. Caution'the children not to

- talk after they open their eyes so they-do not give away thetime. There will probably be much variation in the amountof time the children keep their eyes closed.

Ask the children if they think they 7eould,come closet tojudgng"60 seconds if given another chance. Repeat theprocedure three or four times to allow each child to modifyhis techniques for estimating the length of a minute's du-'ration. After a few trials you may want to ask those chil-dren who came close to 6.0 seconds what methods theyused to make judgments,' but do not spend too much timeon this.

Activity B

Give each student two or three sheets of paper. Appoint onechild to be the clock watcher. He is to call out each sec=and in one minute. When you say'"gof " the clock watcheris to start hiS coun , and each child is to make one tally per.second for 60 se-con s.,on the edge of one sheet of paper.Some of the children will need to use a second sheet ofpaper while others --will crowd all their marks into a smallspace along the edge. Point out that these diffetences donot matter; each child's marks represent, one minute. - b

,

-The children ca illustrate that ittakes 60 seOorids to make'one minute by,dawing-dines to show that all the'secondtallies produce one minute-tally.

1'1 I I 1111111111111 11 f1 11111 1111111 1. 1 1111

1

4

11:i

SQ

97

17

c.

98

Ask:

0

HOW MANY MINUTES DOES IT TAKE TO MAKE ONE HOUR?(60.)

HOW CAN WE USE THE MINUTE REPRESENTATION WEHAVE MADE TO ILLUSTRATE ONE HOUR?

Allow the children to give their ideas, and then suggestthat an hour could be illustrated by taping 60 minute repre-sentations to the chalkboard. Have the children bring their.minutes to you. Since there will not be enough to representan hour, have the class repeat the procedure to generatemore minute representations.

4

When 60 minutes haven been made, the representation of anhour can be constructed by.taping them to the chalkboardand connecting each one to a central point with chalk lines,This procedure may seem to be busy work, but do 'not hastento drop the activity. Although students have occasion todiscuss the relationships that exist between units of timemeasure, they seldom see these relationships other than ona crock face.

Ask the children how this system of time measurement wouldwork in a computer. Point out, that there are 60 seconds inone minute and 60 minutes in one hour, but there ate 24 hours

one day. The children will see that a computer would needone rule when regrouping the ,seconds and minutes, but itwould need different rules for regrouping hours into days,,days into weeks, weeks into months, and months into years.Ask the class to state these rules. If you have time, set Upa time measurement computer and have the children do someproblems that involve regrouping.

1!2

)

o

To review the relationships among time units, complete thefirst part of Worksheet 37 with the children. Worksheet 38reviews material covered in Section 2.

Worksheet-,37tht t

F,111 In the blanks,

second,' -Lelnute LLeinutes ..Lhour'LI hours = day 3" dive': -1-year

.J00 ye, j_. century

75 seconds inut and eeconds.

100 isinutes andil-inutes.14 days _:..7_1uika

26 hour,8 _t_deys and hour..

Johnnie ran for 2 minutes and 27'aecceWs and then he

meted. Her than ran for 1 minute and 15 **condo. Hoe

Iona did he run altogether', J ""'I" and 41?"4-

Miry picked raspberries for 1 hour and 20 minutes In the

morning and t hour and 501eloutss to the atterneee.

long did she pick berries that lay? -3'40." low9000,011 .

1.4

Worksheet 36Wit 34

Can you do these?

3 hours2 hours

57 minutes6 minutes

feet 7 inches2 feet 5 Inches

b hours . -,_minutes i_teet o Incite'

4yards 1 toot 7 Inches

l_yard 2 feet Inches11 hours

- 1, hour-811,puitvatss50 Mutes

ys rd 40 hours I nut's

Fill In the blanks.

II halt-pints =.1..halt-gallons,..t_pinte andhalt- pinta.

h atch is longer. a meter or a yard? Ate,.

which Is more ties, a minute or 'smelt -.46*ot

Which holds more, a pint or a wart? urt

W hich Is longer, 3 feet or 1 yard? Sore

hatch is gore time, 1 hour or 70 minutes? 7" 1"t4.

r,

99

100

1

SECTION 4

PURPOSE

THE POURING SYSTEM AND THE BALANCE BEAM SYST;11,

To introduce the childien to self-instructional materials.

- To provide the children opportunity to analyze and comparetwo different systems. .. . .. .

To prOvide practice in using graphing to analyze systemsand Co store data.

COMMENTARY

This section consists of two bookletS in the Student Manual,both of which are designed to be read and Completed by thestudents. witha minimum of help from\you.

1. You Will want to teed through\ the copy of the book- ,`+ lets with answers, but you should also complete both

--;4 booklets on your own. to familiarize yourself with the

, content before introducing them to the class. Thiswill help you anticipate problems the children might

''c', encounter. , , i

\ ; i\'

2.. The:necessary materials for the bOoktets should be_pla4cr,:ip a convenient spot in the classroom. The ,,,,

L., , .

,,,i,students.work in pairs. Each ..child fills'in his ownanswers in the booklet in his manual.

\

3. Half the class begins with the BalanCe' Beath Systemwhile the Other half begins on the PoUring Systembecause there isnot-enough equipment for all chilt.dretriO do the same booklet t'the.;,same time. As

. .each pair finishes on they go on to theother One. Ealch boo : should. take no more than oneweek: Even if some ibhildren; do not finish in thattime, they must stall oh-the other booklet. They- canComplete the bOokletS in their own time- if they wish.Those children who finish both booklets quickly canspend the remaining time playing the mathematicalgames frorn Lessons 3, 6 and 11.

vb

'C

4. A letter to the students appears just before thebooklets in,the Student Manual. The whole clissshould read the letter together. Each pair ofstudents work together with the pouring and balanc-ing, but each child should decide his own answers..Then the two compaie answers and go over the work,if necessary, until they agree.

5. The students should work swiftly but carefullythrough their booklets. You should check eachpair's work daily to be sure that they are notmaking errors such as working with an unbalancedbeam, spilling water from a measured tube, orwriting ordered pairs incorrectly; You may want towalk- around as the students experiment, checkingtheir procedures and written work.

Though the children may not have worked with self-instruc-tional materials before, they will be working this way oftenin third grade units. Encourage them to work out difficultiesthey may have by themselves.

Because the children will have to become accustomed toworking on their own, the firSt job booklet might go more

;slowly than the second.

$tudents.with reading problems might be grouped togetherso that you can help them get started'on each job. At notime should one student in any pair be alloWed to dominate'.the work to the extent that the other student doesn't get achance^to generate his own answers. If this occurs, havethe children change partners.

As the students pour water and balance the beam, they areasked to find and describe many different states of thesystems in which:

a. The dePth,in either tube is a whole unit.

b._ The depth in either tube is not a whole unit.

c. The beam is balanced.

d. The.beam is not balanced.

115

N\

10I

r.

.,102

Once they have generated these states, the children describethem by one of these two techniques:

1. Ordered pairs.. The ordered pair that describes thestate of this system (the depth of the Waterjn-eachtube) is (7,3). For this example the childrenwould be asked to find and describe all the statesof this system of 10 units of water. Other stateswould be (8,2), (9;1), (10,0), etc.

( 7,3 )

re.

Points on a grid. The children are asked to,findmany different,combinations of places to hang two,paper clips so that the beam balances. They, then-graph these, combinations on a grid. One point,thatdescribes a-balanceestate is at (20, 80k Later'an extra clip is added and the Children are asked tofind and graph balanced states, Of this new system.

S

20 80

I iii( 20,80.)

1111

IIIIII . ' 111

.ill

111 Ilcr

to

IIo' 1. ac 30 yo 50 (Do To go

In the Pouring System boOklet the children should discoverthat when they change the state of a system, the total amountof water does not change. If the children are asked to writefour ordered pairs for a system containing nine units of water,they should discover that the Sum of each ordered pair willequal nine.

With the Balance Beam System booklet the children learnthat to balance the, beam with two paper clips., the orderedpair must total 100.. For example, if they are told that thered clip is at 70, they can predict that the black clip mustbe at .30.

Another-way the children can predict.the position of theclip when given* the position of the red clip is to determinehow far from the center" the red clip is. If it is at 70, theysee that it is 20 units from the center. Then they will dx4cov-er thatto balance the beam, the other clip must also be 20units from the Center, at the `30'30 mark..

During later jobs in the Balance Beam Booklet the childrenuse three paper blips: one red, one black, and one station-ary silver'clip. When the silver clip is in position, say at60. the children must determiriewhere to put the other twoclips. They lAiilless'ee that the distance of one clip from thecenter of the beam must equal the distance.of the other clipplus the distance of the silver clip. In other words, thedistance of the cfip on one side must equal the combineddistances of the two clips on the other side. The orderedpair (10, 80) balances the beam when the silver clip is at60. The silver clip is 10 units from .the center and the clipat the 80 mark is 30 units from the center. These two clipsarea total diStance of 40 units from the center. The thirdclip at 10 is also 40 units from the center, so both sideSbalance.

40'

distance of redclip from centerof beam

.

30

distance ofblack clipfrom center.

., 1 0'

distance ofsilver clipfrom center

I O3

3%

t

"27

104

In somesome jobs the children are asked to write several orderedpairs that wilt balance the beam When the silver clip is at acertain point, say 60. After Wilting several* such pairs, thechildren are asked to add the numbers in each pair. Theydiscover the sum of 6ach`pair is the same when the beamis bal ced and the silver clip is stationary.

MATERIALS

-- for each pair of children --

Pouring System:

2 plastic tubes,

measuring -cup,':

butCher ,tray/7

maskingApe

Balance' Beam System:

meter stick

inche'S in diameter and 4 inches long

wooden block

black)*6 paper. clips, (2 red, 2 silver ;2 black)

straightened paper clip

masking tape.

-- for the class --

pail (optional)

PREPARATION

,Straighten one paper clip for each pair of ,children who will

, be working on the Balance Beam System. The clip willsupport the meter stick which acts as a ,balance arm.

A completed beam balance appears on page 126.118

.4

Each child-working on the Pouring System will need two.plastic tubes, 1:i inches in diameter and 4 inches long.For every tube, cut off a ten:-unit length of centimetertape. Tape it`to the outside of the tube, making surethat the zero mark rests where the tube cavity beginsrather than at the base of the tube. For a completed tube,see page 108.

A

No,

base

0-

I OS

.

UNIVERSITY 0:y4inmsota,

Office of the Director

Dear Student,

MINNESOTA SC1100L4MATILEMATI.CS-ANO SCIENCE CENTER720 WASHINGTON OE isiuE S.E. ivNIINNEAPOLIS,MINiIESOTA 55414

Your teacher has told you to turn to either the Pouring System-bookletor the Balance Beam System booklet in your -Student. Manual. You andyour partner should work together yoUr booklets;.. .Read each pagevery carefully and try Willi in e-very blank and to answer:all thequeitions. Be sure to discuss allyodi ans;Wers with ybui partner.If you and your partner disagree, the twbr of you 'should,-work the prob-lem again tc find the correct answer.

Sometimes it may beard to.find..the-'answer to a question. Whenthis happens, read-the question again.~ Tty working 'the problem a-different war,' Dqn't go .to-yourteacher unless you and yoi.r part=ner just can!St find the answer.- For' some 'questions, there.may be

,More than one correct answer.'-F,

. From time t, me, the teacher will look at the work yOu've done..She will beThecking to see that you have done the jabs correctlyand that you are working as fast as you can. Each booklet shouldtake one week. If you don't get donewith one of the booklets.:intime, you can finish it in your spare time.

Have fun.

Sincerely,

The Authors

12

..)

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p

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stem

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(5,4

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firs

t mem

ber

of (

5,4)

isJr

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..

seco

nd m

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(5,

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kr

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e fir

st m

embe

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es tI

stat

e

of tu

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5.

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sec

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mem

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(,4)

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my

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pai

r (4

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n (B

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pair.

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f1;s

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be r

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4,5)

is

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sec

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mem

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4,5)

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desc

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2:7

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4.8

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wro

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units

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3.6)

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the

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) is

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pair.

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3 m

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e ar

e 3

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he 6

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6.3)

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he

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,

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3.6)

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49

5

Job

7

7

a

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wro

te(

7,

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re a

re7

units

of w

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in tu

be

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rote

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rs a

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units

of w

ater

in tu

be5

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re,

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its o

f wat

er in

tube

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111.

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I wro

te (

7, 2

).

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ir.

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first

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ber

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e

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red

pair

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ribes

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e of

tube

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r de

scrib

es

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tube

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rote

( Z

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) is

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pair

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t mem

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of th

e

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red

patr

des

crib

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e

stat

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tube

B

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sec

ond

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ber

desc

ribes

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stat

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Mak

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look

like

'thi

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ous

syst

em c

onta

ins

u be

and

tube

The

re a

rev

units

of w

ater

in th

is s

yste

mN

o w

ater

can

be

addc

i to

or.r

emov

ed fr

om y

our

syst

em.

Use

onl

y th

e w

ater

Iry

tube

s A

and

B.

Pou

r th

e w

ater

bac

k an

d fo

rth

betw

een

and

in.

Dra

w a

pic

ture

of e

ach

new

sta

te o

f you

r sy

stem

.

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inni

ng s

tate

ew s

tate

New

sta

teN

ew s

tate

,New

sta

te

Job

10

Iis

a -

who

le n

umbe

r.

2 is

a w

hple

num

ber.

iS n

ot o

who

le n

umbe

r.

is n

ot a

who

le n

umbe

r.

5 is

a y

ew-0

num

ber.

31 is

hit

a ifr

imia

. num

ber.

AB

Put

you

r sy

stem

in th

e st

ate

(2, 0

).

Pou

r w

'attC

r ba

ck a

nd fo

rth

betw

een

A a

nd B

.

Can

you

get

the

sam

e st

ates

as

show

n be

low

')

Circ

le a

ll th

e w

hole

inia

tbef

b.

A s

Writ

e th

e or

dere

dpai

rs th

at d

escr

ibe

each

sta

te. F

ill in

the

blan

ks.

ST

AT

E R

ST

AT

E X

2 is

a w

hole

num

ber.

0 is

a w

hole

num

ber

I is

au.

140

1-is

a W

A 0

1-0

.-nu

mbe

r

num

ber.

ST

AT

E(0

0 is

a,

(Vhf

e,

num

ber .

A..i

s,

A4

Inu

mbe

r.I

53,

ti

er

Ijo

b!!

Put

you

r sy

stem

inA

-8IA

. sta

te 1

4,3)

:

4,00

4

You

r sy

stem

sho

uld

look

like

this

:

4

B

3

The

wat

er le

vel I

n ei

ther

tube

atio

uld

be a

t a w

hole

num

ber

mar

k.

Dra

w a

'pic

ture

of-

each

new

sta

te o

f you

r sy

stem

.

Writ

* an

ord

ered

pal

l' to

des

crib

e ea

ch s

tate

.

I

1111

1010

3 )

1-)

,)

B ,)

54

AP

ut y

our

syst

em in

the

stat

e (3

M.*

You

r sy

stem

sho

uld

have

..Lun

its o

f wat

er in

tube

A a

nd_Z

_unt

ts,o

f,w

ater

In tu

be B

.

No

wat

er c

an b

e ad

ded

or r

emov

ed fr

om y

our

syst

em.'

your

wat

er, b

ack

and,

folth

bet

wee

n

Rec

ord

each

of t

he s

tate

s in

whi

ch th

e w

ater

leve

l in

one

tube

is e

iwho

le u

nit.

Beg

inni

ng s

tate

sa

Job

13I.

c

adde

nd +

add

end

= s

um

5 =

4I

svr

= &

Wen

d +

add

end

7 =

+ 5

sum

= a

dder

cli(

cleh

d3

+ 2

= 5

add

end'

+ a

dden

di=

son

3 +

.6 =

9ad

dend

+ a

cicl

ervA

= s

um

Use

the

num

bers

0,1,

2,3,

4,5,

es a

dden

ds.

Add

:any

two

of th

e ad

dend

s to

get

the

sum

of 5

.W

rit' a

dditi

on s

ente

nces

to s

how

the

diffe

rent

way

s.

+ l=

s4-0

=.5

"

'3 4

-.2

r

I fou

nd s

ix w

ays

to a

dd th

e ad

dend

s to

get

the

sum

of S

.F

alse

56

Job

19P

our

five

units

of }

wat

er in

toan

d rio

units

into

The

sys

tem

-cci

niai

ns a

tota

l of

units

of w

ater

.

No

wat

er c

an b

e ad

ded

or r

emov

ed fr

om th

e sy

stem

.

The

re m

ust a

lway

s be

,a to

tal o

fth

-un

its o

f wat

er

Pou

r th

e w

ater

bac

k an

d fo

rth

betw

een

t-.,1

and

Afte

r ea

ch p

ourin

g, th

e w

ater

leve

l In

at le

ast o

ne tu

be

.sh

ould

be

at th

e w

hole

uni

t mat

k.

Beg

in

Rec

orda

ll th

e po

ssib

le s

tate

s of

ryou

r,`"

sylte

m.

Cou

nt o

nly

thos

e st

ates

whe

rilW

ater

leve

l is

at a

uni

t mar

k.

Sh.

A3

I°M

eted

Pai

rA

dditi

on S

ente

nce

ling

stat

e

1,.5

"O

5:4-

0 =

s-..

14

-.,..

I.;

11)1

144

1=5"

32.

. .

13,

0...

,_,.-

.34

1.=

5

2._

....,

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......

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..::

2.i3

2+3=

5

.0x

.0)

5-

0,4:

5"-

.--

3-

,, 7 lT

here

sho

uld

be 6

pos

sibl

e st

ates

, Did

you

find

'ther

n al

l'If

y.)u

sito

not

find

6 s

tate

s, g

o ba

ck a

nd fi

nd th

e O

ther

s.

Pr

es

No

r.

ILto

4

-

T .

4.14

:713

27A

.B(D

I:sa

t you

r sy

sten

tlfns

m s

tate

(3,

3)

.10

poi p

our

wat

er b

alk

ann

fort

h 1.

.etw

eer.

r.j

lust

thtn

k ab

out d

otn9

It.

OT

hInk

onl

y or

.tats

in w

bzch

the

wai

t r l'

.' et

I.. I

nwho

lt, to

ots.

Ros

t-oa

t tho

se

stat

es e

sti:r

t.:cl

ip-I

ds a

nd a

s po

int,

..

st:

." t tik

ul

44

Er.

4

S

.

cf.

(3, 3

)

4.

(4,a

)

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:

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ates

ture

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rth

b.tw

eer

I fou

nd th

ese-

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10;

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l:

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rded

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eral

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rr,

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firs

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sec

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is a

(nu

mer

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r) p

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you

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ame

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on th

e gi

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ates

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the

char

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ved

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they

lie

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stem

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nly

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er to

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tes.

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the

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t sta

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on th

is

The

poi

nt (

6)

desc

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ld b

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ate

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ent s

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e

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stat

esw

e stat

esM

ary

take

turn

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e

of o

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m. W

hen

thin

k w

e've

foun

d al

l the

we

quit,

A.

0 74.(

4 .4 eA eM

The

n w

e co

unt t

he In

itial

s to

see

how

man

y di

ffere

nt s

tate

sre

we

foun

d. W

e're

not

tryi

ngto

see

who

won

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ount

Mar

y'7

initi

als.

A4

eM *A *AM

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74

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dob3

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our

syst

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the

stat

e (7

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0 O

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er s

houl

d re

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intu

it be

side

the

poin

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per

son

shou

ld c

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e th

e st

ate

of th

e sy

stem

.

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wat

er c

an b

e ad

ded

to o

r re

mov

ed fr

om th

e sy

stem

.

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ind

the

poin

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t to

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t04

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e tu

rns

chan

ging

sta

tes

and

two

tng

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h th

is s

ente

nce;

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sta

tes

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my

part

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tet t

al n

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m

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tate

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Job,

°Writ

e yo

ur p

artn

er's

nam

e he

re,r

.Fl

eter

,

0 G

et th

ese

mat

eria

ls fr

om y

our

teac

her.

One

At o

f mat

eria

ls ls

eno

ugh

for

both

of y

ou.

hole

met

er s

tick

1

1z

1

.(c

alle

d be

am)

woo

den

bloc

k

ket4ii

kii

groove

6 pa

perclips

,.4,

,41

5f"

2're

d2

silv

er2

blac

k

3

I str

aigh

tene

d pa

'Per

clip

mas

king

tape

11

®P

ut y

our

bala

nce

beam

toge

ther

tike

this

:

Pap

er c

lip g

oes

thro

ugh

hole

in b

eam

.P

aper

dill

, res

ts in

gro

ove

in b

lock

.

78

lob

I con

tinue

d

0 P

ictu

rede

scrib

es o

ur b

alan

ce b

eam

..

not b

alan

ced

not b

alan

ced

,ba

lanc

ed

OT

he b

alan

ceb.

tam

we

mad

e is

bal

ance

d.

0 If

your

bea

m is

bala

nced

go o

n to

Job

2.

If yo

ur b

eam

Is p

lea

bala

nced

do

wha

t tie

pic

ture

s be

low

sho

w y

ou.

Li Li

ghtly

stic

k so

me

mas

king

tape

on

the

arm

that

is u

p.

The

n le

t go

to s

ee if

It b

alan

ces.

If th

e be

am b

alan

ces,

go

on to

job

2,If

it do

esn'

t bal

ance

, mov

e th

e ta

peba

ck a

nd fo

rth

unlit

bal

ance

s.

Be'sure to' stick the masking tape on

tightly once the !vain is balanced.

JobZ

®B

e su

re y

our

bala

nce

beam

is b

alan

ced.

Hav

e th

e 0-

Hly

sid

e fa

cing

you

.

0,0

010

as

()T

he p

aper

clip

will

slid

e ov

er th

een

ds o

f the

bal

ance

bea

m.

to 7

0 10

10

I

Han

g a

red

pape

r cl

ip a

t the

16

mar

k.

Han

g a

bloc

!, pa

per

clip

at t

he 8

0 m

ark.

I

Is th

e be

am b

alan

ced/

Yes

In th

e ch

art b

elow

, rec

ord

the

posi

tions

of t

he c

lips.

-

@C

hang

e th

e po

sitio

ns o

f the

bla

ck a

nd r

ed c

lips.

Rec

ord

the

resu

lts.

^

Mal

ty P

os$4

12 O

lsw

elis

Pos

ition

of

red

clip

Pos

ition

of

bloc

k cl

ipDescribe the Beam

B .

bala

nced

NB

i. n

ot b

alan

ced

..

/6,

goN

g

-.

,

..,-

---

_.

,/ Y

.

.

p_

se

1.11

=11

.

cV 4

.Job

2 c

ontin

ued

0H

ang

one

red

cup

and

one

blac

k cl

ip.fr

om th

e be

am.

Cha

nge

the

posi

tions

of t

he c

lips.

Rec

ord

the

diffe

rent

sta

tes

in w

hich

ntpc

hea

rib b

alan

ced.

Any

Ord

ered

pai

r 74

a,t-

fora

L5

/AI

00, e

C

Pos

ition

of

Pos

ition

of

Pos

iti,n

of

Pos

ition

of

Red

Clip

Bla

ck C

lipR

ed C

apB

lack

Clip

Gam

e1.

One

of y

ou b

egin

s by

cho

osin

g a

num

eral

whe

re

the

red

clip

will

be

hung

.

2.T

hen

the

othe

r pe

rson

writ

es th

e nu

mer

al w

here

th41

aslc

clip

mus

t be

hung

to b

alan

ce th

e be

am.

3.H

and

the

red

anda

he b

lack

clip

at t

he n

umer

als

you

Wro

te. H

ow d

id y

ou d

o/ D

id th

e be

am b

alan

ce/

4.T

aket

urns

writ

ing

num

eral

s an

d ha

ngin

g th

e re

d

clip

and

'the

blac

k cl

ip.

N.)

co Jof,

3.

We

call

the

bala

nce

beam

a s

yite

m.'

Mat

ch th

e na

mes

with

the

part

s of

the

syst

em.

The

sta

te o

f a s

yste

m is

the

cond

ition

of t

he s

yste

m.

Des

crib

e th

e st

ates

of t

he s

yste

m b

elow

:

red

blac

k

red

clip

at3

-bla

ck c

lip a

t65

/

blac

k

,re

d,

red

clip

at

blac

k cl

ip a

t25

--

red

blac

k

red

clip

at

blac

k cl

ip a

t9C

2

82

Job

3 co

ntin

ued

Fro

m n

ow o

n, tr

y to

find

onl

y st

ates

In w

hich

the

beam

is b

alan

ced.

O ©

),

Pla

ce th

e re

d "c

lip a

t 39:

Pla

ce th

e bl

ack

clip

so

that

the

beam

is b

alan

ced.

, red

clip

at

3 1

blac

k at

6'f

Pla

ce th

e bl

ack

clip

at 7

2.re

d at

ip,a

tc2

52bl

ack

atP

lace

the

red

clip

to b

alan

ceth

e be

am.

72

Fin

d on

ly b

alan

ced

stat

es c

kf,y

our

syst

em.

Des

crib

e ea

ch b

alan

ced

stat

e./f

ogy

prz,

i-rit

et' 7

107.

.15

/90

i3ev

/vec

724.

red

clip

at

4bl

ack

atre

d at

blac

k at

O)0

)."*. ' re

d at

blac

k at

red

atbl

ack

at_

_-

0re

al, a

tbl

ack

atre

d at

Wan

t: at

......

0-

83

L11

Mak

e yo

ur s

yste

m lo

ok li

ke th

isI

vesc

rthe

the

stat

e of

you

ryl

tem

.I

edbl

ack

red

clip

at

(9bl

ack

clip

at

/

0 D

id y

ou w

rite

(19

.81

tode

scri

be th

est

ate

of y

our

syst

em?

Yes

No

'Tw

ee n

umbe

rs li

ke 1

9 an

d 81

are

cal

led

a pa

ir (

*num

bers

.

The

19

in (

19,

8')

desc

ribe

sth

e pd

sitlo

n of

the

red

clip

.r

The

81

hi (

19. 8

1) d

escr

ibes

the

posi

tion

of th

eild

ek c

lip.

iO

Supp

ose

we

agre

e to

wri

te o

ur p

airs

of

num

bers

.ina

spec

ial o

rder

.

The

fir

st n

umbe

r de

scri

bes

the

posi

tion

of th

e re

d cl

ip.

The

sec

ond

num

ber

desc

ribe

s th

e.po

sitio

no?t

he b

lack

clip

.

(1-1

o, 6

0)T

he f

irst

mem

ber

of th

is 'p

air

The

sec

ond

mem

ber

of th

is p

air

Is 6

0T

he 4

0 de

scri

bes

the

pgsi

tion

of th

e re

d cl

ip.

The

60

desc

ribe

s th

e po

sitio

n of

the

clip

(40,

60)

is c

alle

d an

ord

ered

pai

r of

num

bers

.

RB

RB

RB

0 Pu

t you

r ba

lanc

e be

am in

the

stat

es (

40,6

0),

(75,

25)

,(1

8, 8

2).

84

5O

Rem

embe

r ou

r or

der

ford

escr

ibin

g.th

e st

ate

of o

ur s

yste

m.

4T

he f

irst

mem

ber

8f th

e or

dere

d pa

ir d

escr

ibes

the

posi

tion

of th

e re

d cl

ip.

The

sec

ond

mem

ber

of th

e or

dere

d pa

ir d

escr

ibes

the

posi

tion

of th

e bl

ack

clip

.

0 Pi

ck th

e or

dere

d pa

ir o

f nu

mbe

rs th

at d

escr

ibes

eac

h st

ate.

6'

94.

Bla

ck

(5,9

0) o

0

90 Red

(90,

5),

Red

(95,

8) o

.

95

Bla

ck

(8,9

5)

(43

_._.ffl

Red

rr., B

lack

..

4:11

Pr

(99

. 6)

0 Y

our

answ

ers

shou

ld b

e (6

.94)

(B

0,5)

(8,

95),

0 W

rite

ord

ered

pai

rs to

des

crib

e th

e st

ates

bel

ow.

496

I, 3

3I

,20

80.1

288

laR

edB

lack

419'

4

4:1

Bla

ck S20

Red

o?c9

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ar

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1=1

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71J2

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edI

(0/0

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lac)

k

f

as

a

Job

6 (thi

nk (

6., 7

5)'d

escr

ibes

a b

alan

ced

stat

e

rlert

used

my

riala

nce

beam

to c

heck

you

r or

dere

d

pair.

(6,

71)

does

not

desc

rib^

a ba

lanc

ed s

tate

Hel

en

,0

Tak

e tu

rns

writ

ing'

orde

red

palis

.that

des

crib

eba

lanc

ed s

tate

s.

Che

ck e

ach

orde

red

pair

with

you

r ba

lanc

e be

am.

8492

04i.

Thg

t tO

reS

/op

is 'C

orry

?-t

-

Sta

te o

f a S

ySte

m

Red

, B

lack

BB

alan

ced'

NB

AN

ot B

alan

ced

6,75

NB

Sta

te o

f a S

yste

m

Red

, Bla

ckB

!. B

alan

ced

NB

= N

ot B

alan

ced

0, P

ut y

our

beam

bal

ance

in fo

ur d

iffer

ent s

tate

s th

at a

re b

alan

ced.

Writ

e th

e or

dere

d st

air

for

thes

e st

ates

.O

ily7%

;7"

7.6

4S/a

pes

0r,e

ct

0(-

1)(-

7/)4

(7

772)

Look

car

eful

ly a

t the

ord

ered

pai

rs y

ou w

rote

.

Can

you

dec

ide

ahea

d of

tim

e w

here

to p

ut th

e ci

lks

r-03

h,t

the

beam

is b

alan

ced,

If yo

u ha

vp a

way

, try

it a

few

tim

es to

be

sure

it w

orks

. ,

0 T

ry to

thin

k of

ano

ther

, way

to d

ecid

e ah

ead

of ti

me.

Tal

k it

over

with

yeu

t,.P

attn

et.

Flit

in th

e ch

art.

Bal

ance

d S

tate

s

S°11

_N

ot B

alan

ced

Sta

tes

Bed

Bla

ckA

dditi

on S

ente

nce

r/o

. 90.

/0 4

. 79=

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qty

poi..

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roe

1

at,5

,00

is c

b.re

cer

Red

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dditi

on S

ente

nce

20' ,

; 90

2.16

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0 =

//0

atyl

itv.,

ma,

- d

sso

t*A

s /

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6 ee

olec

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.

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1mm

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Use

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rayo

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the

line

labs

.

10.

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-U

se a

bla

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ayon

for

the

blac

k lin

e.

.

IAA

wj d

RE

D.

r4_.

R.1

.0I

01

75-

'

CD

with

line

with

1@

ti

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ered

pai

r Is

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tem

is __.-

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(zt-

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25 o

n th

e R

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line

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cray

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n th

e B

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cray

on.

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if jo

b 8

cont

inue

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5 on

the

RE

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w a

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egm

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to th

e to

p of

the

grid

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ind

75 o

n th

e B

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ne.

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w a

BLA

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line

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men

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the

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the

grid

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ind

the

Inte

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grid

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the

prev

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page

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our

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to s

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Wan

d D

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or

NO

T B

ALA

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ED

BN

B

Alio

,bo

8B

415;

63-

C''

7o, 3

0

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0,/0

89

0 0 A

/I

a.

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mmEMMEM

IIMMUNEMMEM

MINNIMMENUMERMUNERWM

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MEMSMWOMMOMMIMMMME

Ausimummlimmommorno

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miumpmummmummammuma

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ARIMMEMOMMOINI mama

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MEMMUNNMMUMMOINMMEMAN

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MUSEMEMEMMOIMMEMOMMENME

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mommo oomi

N6

/tie

A1.

6

z. .

0

Job

10B

e su

re-y

our

Lea

rn b

alan

ces

whe

nth

ere

are

no c

lips

on it

.

0W

hen

a,pa

tr o

fnu

mbe

rsha

ve a

sve

cial

ord

er. w

e co

il th

eman

ort

ee4

dpi

leT

he f

irst

mem

ber

of th

e pa

ir d

escr

ibes

the

pbst

tion

oft

-&-

&.c

lip.

11

The

sec

ond

mem

ber

of th

e pa

ir d

escr

ibes

the

posi

tion

"of

the.

..I la

ticlip

.sy

Use

one

red

clip

and

one

bla

ckcl

ip.

Bal

ance

the

beam

for

El

diff

eren

tst

ates

.R

ecor

d th

ese

stat

es o

nth

e ch

art.

Red

Bla

ckO

rder

ed P

air.

Add

ition

Sen

tenc

eLe

tter

Na.

:

.(

,)

t'-

.A

ny p

aar

Ph&

,-.

4-o+

0.15

IOd

8<

Isco

rrec

t.

E

.'

G H

92

ft

Job

10 c

ontin

ued

LOea

teth

e8

poin

ts th

atde

scri

be th

e st

ates

Of

your

sys

tem

on

the

grid

bel

ow.

Wri

te o

rder

ed p

air

nam

es a

nd le

tter

nam

es.

®-

Dra

w a

line

con

nect

ing

the

poin

ts y

ou m

arke

d on

the

grid

.

Wha

t do

you

notiz

e? D

o th

e po

ints

lie

on a

ra

curv

ed li

ne?

(3,4

) is

-cal

led,

an t.

vier

ed p

air

beca

trse

we

agre

e th

at:

The

fir

st m

embe

r of

the

pair

des

crib

es th

epo

sitio

n of

the_

C-V

t-' .

clip

i

The

sec

ond

mem

ber

of th

e pa

ir de

scrib

es th

epo

sitio

n of

the

tee

14-

clip

,

ft

Job

1.1

4ha

_ts

. ti

a...U

se O

ne r

ed c

lip a

nd o

ne b

lack

dip

,F

ind

stat

esw

here

the

beam

Is b

alan

ced:

00 r

d C

.Pe"

(;"

'LLe

tter

Red

flack

d re

a=

.

A.

()

)i'

74

8 '

.

3)

'D

.

....

....

..

.....

...

94.4......

.

4

$.5

5 4

1.,*

PI

ist

Job

I con

tinue

d

Use

414.

011

otay

on.

Mai

k po

int*

tf..

on th

e 9f

hlrik

iSb

poin

ts a

le o

n a

stra

ight

line

.Y

eN

o

Use

one

red

clip

and

one

bla

.-i.

clip

.F

ind

4 st

ates

an

whi

ch y

our

syst

em is

not

bal

ance

d.

R,y

it.1

-Pai

r Lk

a,t

S o

es0

5 C

.OrP

eCit'

Lette

r;ta

me

Pea

Bla

ckO

rder

ed P

air

Add

ition

Sen

tenc

e

It(

5)

s(

5-F

==

.

TC

)i-

- -=

'-'

U

,

.,)

.-,

Q

Use

an

oran

ge c

olor

td.m

ark

poin

ts (

R. S

. T, U

) on

the

grid

.T

hese

poi

nts

are

on a

str

aigh

t lin

e.Y

es

line

0 P

oint

s w

hich

stra

ight

repr

esen

t a )

3ala

nced

sys

tem

fall

on-c

urve

d lin

e.

do

Poi

nts

whi

ch r

epre

sent

anunbalanced

syst

em0

fail

on S

titra

ight

line

.'

Try

som

e m

ore

stat

es w

here

you

rsy

stem

\i?ba

lanc

edor

unb

alan

ced.

Loca

te m

ore

poin

ti on

the

grid

.if

t. 9

5

1

I

V-

@ U

se o

ne r

ed c

lip a

nd o

-e b

lack

clip

,W

itte

orde

red

patr

s to

rep

rese

nt th

e st

ates

of t

hii s

yste

mT

his

char

t sho

ws

orde

red

pairs

for

bala

ncea

'and

unb

alan

ced

stat

ee.w

e fo

und.

Bal

ance

d

Red

Bla

ckin

itiltI

on S

ente

nCe

60)

yo60

+%g0

= 1

00ct

:ol p

.a.ir

thi:t

' r. w

eal*

100

isi t

at?

SO

C

.

'

O

Unb

alan

ced

Red

,B

lack

-A

dditi

on S

ente

nce

.

30 S

O; +

30 8

0=in

g)

O. n

i 1..

rt..

.ts d

..4.

no 4

- to

te-(

foe

:3.

CiO

rre,

Ct.

. J

Whe

n th

e be

am is

bal

ance

d, th

e su

m-o

f the

two

mem

bers

of th

e or

dere

d pa

ir is

.

0W

hen

the

beam

is u

nbal

ance

d, th

e su

m o

f the

two

mem

bers

of th

e or

dere

d pa

ir ca

nnot

be

Fin

d so

me

mor

e st

.itas

Bal

ance

d

96

41C

V

fil,

Unb

alan

ced

r

Job

12 c

ontin

ued

;

Map

of P

airv

ille

P

--.. 15

th A

ve.,.

_

14th

Ave

nue

..

.IP

.

13th

Ave

nue

a.

4...

.,

in14

.1

.

f2th

Ave

nue

,.

...

r) iii

'

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<p)

15 b <p)

,tr 1- <p)

,

% t7P

.

V

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4C".

) f.'

a

0 T

orn

is 'r

otan

ding

on

the

com

er o

f10

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eet a

nd/4

1-A

venu

e:

OM

aly,

is s

tand

ing

on th

e co

rner

of,

8S

tree

t and

/3 A

venu

e.

One

dog

isre

gale

the

corn

er o

fif

Str

eet a

nd 1

3 A

venu

e.'

-I c

®T

he c

ar is

sto

pped

at t

he c

orne

r of

2S

tree

t and

l'_ A

venu

e....

..r

,iA

'

0 P

iece

an

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h S

tree

t, 12

th L

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l

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id ,s

0 T

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ss o

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nt M

is (

116.

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venu

e).

. ill

,I

-

i.

974

.

,.4

;.,

.,

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,'

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a.

-a

01

Job

15ra

t m y

our

own

cn.m

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se-p

ur b

alun

ce.b

earn

to c

heck

you

Bita

nced

(qc)

,1.

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,

, 35

)+

21 A

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., 18

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aWer

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ot

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.=..-

----

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-- 0

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=

(,

IL )

4^=

.

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=.

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98

The

sta

te is

(30

, 70)

.Is

the

beam

tela

nced

2

9. 4

r" a

....

AM

RA

ND

The

sta

te is

(43

. 63)

.N

oIs

the

beam

bal

ance

d?Y

es

'The

red

clip

As2

aZur

tits

from

aSO

.

The

bla

ck c

lip is

...t.f

ikun

its fr

om S

O.

4,3

41..1

411

The

red

clip

is_2

__un

its fr

om O

.

The

bla

ck c

lipfr

om S

O.

a

4.

Joli

13 c

ontin

ued

4.

lb*

it0

0'th

ere

are3

appl

es in

this

pic

ture

.T

here

Nie

4",

ora

nges

In th

is p

ictu

re.

;W

e w

ill w

rite

an,(

appl

e. o

rang

e) o

rder

ed p

air

to d

escr

ibe

the

pict

liro.

Tho

ord

ered

pai

r w

e w

rite

is(

3_4

11

_W

,e c

an w

rite

an (

oran

ge: a

pple

) or

dere

d,,p

air

to d

escr

)be

the

pict

ure,

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ord

ered

pai

r w

e w

rite

is(

1-1

Writ

e or

dere

d pa

irs to

(A

scrib

e ea

ch p

ictu

re:

:111

0111

P-,

.##1

#0i

\1/4

4ruc

k., a

irpla

ne)

orde

red

pair

23

.

..(b

oy. (

p) o

rdet

ed. p

air

.

.4.

,c2.

--.

-,

.)

,,(a

irplin

e. tr

uck)

ord

ered

pai

r.

t.?-

'.g

.--

,

(girl

. boy

) °a

lert

) lri

r

..tg

-it-

.r

-

/

,

-I

-- 1

. aF

yr-

44

-4 1.

c t

Pla

ce:*

silv

er c

flp a

t60

on y

our

beam

.

DO

NO

Tis

lavE

TH

IS C

LIP

FO

R T

HE

RE

ST

.or

TH

IS m

ar..

010

le1

i se

pip

a_..

0-

.....,

,,N.a

ga

loo

Fin

d si

x or

dere

d pa

irs th

at r

ave

bala

nced

sta

tes

of th

e be

am'

witl

) th

e si

lver

clip

at 6

0,

CY

.I1A

s.t t

otal

s-t

ie.i

soc

g,re

ot44

.sm

*,bl

vtat

Red

Bla

ckA

dditi

on S

ente

nce

V4,

-..

n-

(V

4 -

,.

)a

.

Red

Bla

ckA

dditi

on S

ente

nce

2)

;)

' ._.

"t.-

- =

--

1 1

l)

Ns

The

silv

er c

lip is

at

O.

'Put

the

red

clip

4t 7

0.

I pre

dict

that

the

blac

kcl

ip m

ust b

e at

"AO

if I w

ant t

he b

eam

toba

lanc

e.

Che

ck y

our

pred

ictio

n.

4....

Thr

e be

am is

bal

ance

d If

- th

e re

d cl

ip is

ata

Land

the

blac

k cl

ip is

at 1

0. "

( 7o

,re

d"

blac

k

The

silv

er c

lip is

at 6

0.

Put

the

blac

k cl

ipit

90.

I pre

dict

that

the

red

Clip

mus

t be

at-

..0If

the

beam

is to

-

bala

nce.

Che

ck y

our

pred

ictio

n.

The

bei

m is

ifth

e rid

clip

is a

t_e_

and

the

blac

k cl

ip is

at

70.

( 0

*q0

)-

red

blac

k .

100

1,

7

job-

14

cont

inue

d

- G

AM

E[P

RE

DIC

T /O

N I

,

0 T

ette

the

silie

r cl

ipI

off t

hebe

am.

.

Che

ck th

e be

am a

l see

,s,

that

.,1t,i

s,,b

alnc

ed_.

)yith

-.

out c

lips.

-4

- :-

-

.P

lace

',si

lver

clip

at

."po

int 7

0; K

eep

the

'iil

ver

clip

on

the

bean

ifo

r..1

Fi r

est o

f thi

s ga

me.

'C

.

silv

er

1I.

..

I che

cked

' our

pre

dict

ion,

_.-%

-...

Joe

(20,

60)

desc

ribes

a b

alan

ced

stat

e.t.

Will

ie-

Who

se

wou

ld 1

pla

ce th

e

red

clip

and

the

blac

k cl

ip"

.to

bal

ance

the

beam

?-

:-

....s

era.

:4.

1 pr

edic

o -6

0 )

-_--

.....,

io.

I., f

Ij re

oE

.,,,

S.,

,.

-r

Writ

e yo

ur p

redi

ctio

n fo

r st

ates

in w

hich

the

syst

em is

bal

ance

d.T

est e

ach

lited

lctic

in. a

mt p

;(1.

t I.a

tts

ta.1

4 to

zo

co,r

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TE

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red

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red

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* * ,)

.

4.

I*

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..

/

I.

k.

..

.4t)

,50

10-li

Job1

5)

Put

the

silv

er c

hi::

at 6

0.P

ut y

our

syst

em in

the

stat

e (t

o, 8

0)re

d, b

lack

?23

,,Red

40to

+ la

3040

50ie

70

$1,

90

iou

LJ

,

Silv

erB

lack

- 40

10.

.30

(dis

tanc

e of

red

) (

dist

ance

of s

ilver

) /d

ista

nce

of b

lock

)cl

ip fr

om c

ente

rC

lip fr

om c

ente

r\ c

lip fr

om c

ente

r

2,T

he d

ista

nCe

of th

e.

the

,dis

tanc

e of

the

two

clip

on

one

side

'm

ust e

qual

clip

s on

the

othe

r si

de.

3.T

he r

ed c

lipis

Y 0

units

)ror

nth

e ce

nter

of t

he s

tick.

4.T

he s

ilver

clip

is-

10"n

its fr

om th

e ce

nter

of t

he s

tick.

5.. T

he'b

lack

clip

is-3

0un

its fr

om th

e ce

nter

of t

he s

tick.

6.W

hen

the

snee

r cl

ip is

in 6

nd th

e sy

stem

'is In

the

stat

e (1

'0.8

0)is

the

beam

bal

ance

d?Y

eN

o

7.Pl

ace

the

silv

er C

lip a

t 60.

Pla

ce th

e re

d cl

ip a

t 25.

Whe

re s

houl

d th

e bl

ack

clip

go?

I-05

8.H

INT

:th

e di

stan

ce o

f the

'the

dist

ance

s of

the

two

clip

Wor

n th

e ce

nter

)=

clip

s (f

rom

thi c

ente

r) o

non

one

Sid

e.'

the

othe

r si

de.

=1

.1c

,LS

73;7

"--

,S

ilver

7131

=9.

Sho

uld

the

blac

kcl

ip b

e pu

t at 6

5 to

bala

nce

the

beam

?Y

eN

o

102

4*,

job

35 (

cont

inue

d)

a

1

'0 2

030

40

50 6

0 70

80

90 1

00I

-LI

It

The

sta

te o

f thi

s sy

stem

is (

20. 7

0. T

he s

ilver

clip

fs a

t 60.

The

lred

clip

isa2

2.un

itsfr

om th

e ce

nter

.

Is th

e be

am b

alan

ced?

The

bla

ck c

lipis

4Q.u

nits

from

the

cent

er.

No

tT

he s

ilver

clip

is O

uni

tsfr

om th

e ce

nter

.

010

1520

30 4

050,60

707

5 80 9

0 10

0.

6.

81"

,00

4,4.

a

The

sta

te o

f the

sys

tem

is (

75'

15).

The

silv

er c

lip is

at 6

0.th

e,be

am b

alan

ced?

esN

o

The

bla

ck c

lip

from

the

cent

er

The

red

clip

from

the

cent

er

The

silv

er c

lipis

laun

itsfr

omth

e ce

nter

103

Job

16Pl

ace

the

ilium

clip

at 7

0. U

se o

ne r

ed c

lip a

nd o

ne b

lack

clip

.

0,F

ind

4

IDis

tanc

e fr

oin

cent

er

A (

cRo

60)

30i_

20

red

blac

k'sl

iver

B (

4)

_re

dbl

ack

silv

er

C (

,)

4-re

dbl

ack

silv

er

P

,..

÷re

dbl

ack

silv

er

,4

T.

4:-

44

1;

1-1

---i

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'--,

t,

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---#

,

1

f

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IIII

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iI

711

H1

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trr

,I

it.

t.

i, .

:..

4

1,.

11

1'1

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

1111

111

IIII

I11

1111

1111

1111

11

1111

1111

1111

1101

1111

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L

i1

1i ,

4112

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1111

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1111

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a

f,

1I

#

c ,

..r.

_ A

t Cu

le. c

r.C

e L

a L

e I.

,e o

n A

ror

iu o

r il

A

Job

16 c

ontin

ued

.

Use

a g

reen

Cra

yon

to lo

cate

poi

nts

(A, B

, C, D

) on

the

grif

d.

Do

thes

e po

ints

fal

l on

a st

raig

ht li

ne?

No.

Lin

d 4

stat

es in

"whi

ch th

e be

am is

unb

alan

ced.

01, ,

h,..

4 p

0 j :

, b

/ C.

Io.

rNsu

itAS.

Dis

tanc

e fr

om C

ente

rre

dbl

erak

silv

er

,+

F.,

)=

.+

G.

.*

ri, \

_S=

1-

0 U

se a

n or

ange

cra

yon

to lo

cate

poi

nts

(E, F

. G. I

I) o

n th

e gr

id.

The

se p

oint

s fa

ll on

a s

trai

ght l

ine.

Tru

e

0 fl

it in

the

blan

ks.

The

bea

m is

bal

ance

d.

The

silv

er c

lip is

at 7

0.

The

red

clip

is a

t 25.

The

bla

ck c

lip m

ust b

eat

-41.

_:,_

.

The

bea

m is

bal

ance

d.

The

silv

er c

lip is

at 7

0.

The

bla

ck c

lip is

at 7

5.

The

red

clip

mus

t be

at--

6.7

....

The

bea

m is

fl

bala

nced

.T

he b

eam

is u

nbal

ance

d.

The

silv

er c

lip is

at 7

0.T

he s

ilver

clip

is a

t 70.

The

red

clip

is a

t 15.

The

bla

ck c

lip is

at 6

9.

The

bla

ck c

lip m

ust =

I,

The

r9s

1 cl

ip :r

ust n

ot b

ebe

at _

1.1.

5.--

-..

at

rs

105

C:0

O

0 M

ove

the

511v

er c

lip to

a n

ew p

ositi

on o

n th

e be

am.

The

pos

ition

of t

he s

ilver

clip

g

The

sliv

er c

lip s

tays

at_

for

the

rest

of t

717:

17).

40

Fin

d so

me

stat

es w

here

the

beam

is b

alan

ced

and

som

e w

here

it is

not

bal

ance

d.\

.

..'

Bal

ance

d, m

o..,,

,j po

sscb

it.

Unb

alan

oced

4..n

f.,.4

/G11

.

A(

,.

D E,

)

// .

:__+---==/.

.Fc

5)

==

--.4

-

M(

JQ

.

=:-

-:

N(,

,J

___1____=___

1 =,

'z P

C)

).1_

.7.=

___ ___

0C

,)

____-.1-___=

.7

.

R(

.,

___+____=

0 U

se a

gre

en c

rayo

n to

loca

te p

oint

sA

t. B

, C, 9

, E, F

) on

the

grid

on

the

next

pag

e./

If Ahis system is balanced, the

Points lie onastraight line.

Whe

n th

e be

am is

bal

'anc

ed. t

he s

um o

f the

, tw

o m

embe

rs o

f the

ord

ered

pai

r is

'06

False

\

Slo

b 17

con

tinue

d

Use

.an

oran

ge c

olor

to lo

cate

poi

nts

(M, N

, 0, P

, Q, R

).

If this syStemAS unbalanced, the

points lie on a Straight lirie.

Whe

n th

e be

am is

unb

alan

ced,

the

sum

of t

he tw

om

embe

rs o

f the

ord

ered

pai

r is

not

True

Yy4t`

it1i

001

.

um.

......

....m

III

SMdMAKIXU,sTAKMOIWPOr%41mMw.r

107

job

18

IPut

the,

silv

er c

lip a

t 40.

Do

not

mov

eth

e cl

ip.

The

tabl

es s

how

the

posi

tion

of e

ither

the

red

clip

or

the

blac

k cl

ip.

Pred

ict w

here

the

othe

r cl

ip m

ust b

e pl

aced

to b

alan

ce ti

le b

eam

.

Che

ck y

our

pred

ictio

n. I

f yo

ur p

redi

ctio

n is

not

cor

rect

,%

vrite

in th

e co

rrec

t pos

ition

.

BalancedSystem with Silver Clip at 40

my

pred

ictio

n°

test

Red

Bra

ckC

orre

ct if

ositt

oil .

30R

Y(

36,

gO )

4og

od(-

7o ,I

cr)

35(

36, 7

5)

o5(

5'5

'0?-

5 )

6941

(.0

141

).2

-gt8

1.2.

( as.

, 7.

2-)

7,2-

3 8

( 7.

2, 3

S)

my

pred

ictio

nte

st

Red

Rla

ck.

Cor

rect

Pos

ition

o

3o3P

,.

( go

,_3

)

/./.0

.7t,

(40,

70

)

96c'

',_(

q o

1 ao

)

377/

(31

),7!

)

35('

5.1

25 )

-73

37(

73)0

7 )

92/2

(qr 1

P.--

)

rom

mill

a

tol

-Job

19

0, R

emov

e al

l clip

s fr

om th

e ba

lanc

e be

am..

0T

he c

ente

r of

the

bala

nce

beam

is a

t poi

ntj.l

.

OU

se o

nly

the

red

clip

and

abl

ack

clip

.

Find

sev

eral

sta

tes

in w

hich

the

beam

is b

alan

ced.

rill

in th

e ta

ble.

al.,

pa,jr

t hi

t 1i 0

eS

Cor

re.e

..-t

100

(R

ed

1

Ble

ckT

he r

ed c

lip is

_uni

ts f

rom

SO

.

The

bla

ck c

lip is

-Uni

ts f

rom

50.

(I

)T

he r

ed c

lip is

_uni

ts f

rom

SO

.

The

bla

ck c

lip is

_uni

ts f

rom

j50.

1)

,

.T

he r

ed c

lipis

_uni

tsfr

om S

O.

The

bla

ck c

lip is

_uni

ts f

rom

50.

i

"'T

he r

ed C

lip is

.---

.uni

ts f

rom

.50.

The

bla

ck c

lip is

_uni

ts f

rom

50.

()

).T

he r

ed c

lip is

-uni

ts f

rom

SO

.

The

bla

ck c

lip is

_uni

ts G

orn,

50.

(1

)T

h re

d cl

ip is

units

fro

m 5

0.

The

bla

ck c

lip is

-uni

ts f

ront

,S0.

log

A

Job2

0

0 T

he o

rder

ed p

airt

iL.,2

14ca

n be

use

d to

des

crib

eth

e st

ate

of th

is s

yste

m,

0 T

he fi

rst m

embe

r of

the

orde

red

pair

30 is

the

posi

tion

of th

er1.

A_c

lip.

0 T

he s

econ

d m

embe

r of

thU

ord

ered

pai

r is

10

7 a

is th

e po

sitio

n of

the

hia-

t4-

clip

0 W

hy is

(30

.70)

an

orde

red

pale

)B

ecau

se w

e ag

ree

to w

rite

the

pair

in th

e or

der

of(r

ed p

ositi

on, l

elet

- k-

pos

ition

).

Loca

te th

ese

poin

ts w

hich

desc

ribe

stat

es o

f a b

alan

ced

beam

,>y,

..tem

,

A=

(31,

76)

8=

4.0)

C =

b.=

est,

FLA

CK

,00 ab 70 wo

5.0

'O so li

A

B

0J

Jo J

e to

heto

itst

Ye

too

> R

p

"Pis

,

dob2

1

asss

Thi

s is

the

last

Job

of t

his

book

let.

Wha

t did

you

lear

nfro

m th

is b

ookl

et?

Do

this

job

co h

elp

you

find

out.

0 D

o no

t use

the

bean

i to

chec

k yo

ur a

nsw

ers

in th

is lo

b.Ju

st ta

lk th

ings

ove

r w

ith y

our

part

ner,

'

0 W

e w

rite

Pai

rsof

num

bers

to d

escr

ibe

stat

es o

f our

sys

tem

.T

hese

pai

rs m

ust f

ollo

w a

spe

cial

brd

er.

The

ord

er is

2(1L

1clip

pos

ition

,610

.t k

clip

pos

ition

0 T

hink

abo

ut u

sing

one

bla

ck c

lip a

nd o

ne r

ed c

lip.

Fill

in th

e fo

llow

ing

char

t.LL

t S

ees

tpa

. tha

t so{

Isis

cor

rt.c

.t.C

71/0

(3 c

orre

ctB

eam

Bal

ance

d

atny

rct

,,rrt

i.

tai 1

080e

amN

ot B

alan

ced

C)

A)t

B(

).

C(

)

EI

)/

E(

)

/II2

M(

/

N( %

)o

0 P(

/

0(

lob

2I c

ontin

ued

OU

se a

blu

e cr

ayon

to m

ark

poin

ts (

A.,1

3;C

-, U

. C)

on,th

o,gr

i,1 b

elow

..

The

se p

oint

sdo

not

fall)

on

a st

raig

ht li

ne.

0 Ils

e d

gree

n cr

ayon

to m

ark

poin

ts (

M, N

, 0, p

, 0)0

1)'..

te g

rid b

elow

.

The

se p

oint

s (f

all,

LAcK

'

on a

str

aigh

t lin

e.

:...

.-

So

.....

--

V=

.4I

t t

tt

4-

- -

.

oit

;

t

Tt

-4

--.

t

.....

..

:.

-

.....

-r...

......

... =

4-

-tt

TP.

It

t.

..--

,..... 4

tt-

5oo

3.X

MO

61se

SS.0

OS

707f

7.JS

loS

0 S

how

this

Job

to y

our

teac

her.

She

wan

ts to

kno

who

w y

ou d

id.

Cut on dark lines.

Cut 2 te

pieces-from

each Sheet A.

1 T2,, piece

1 T

2, p

iece

d i

4

....)

a..

0.

I

e,

cn CD

CD CJI

CyV

CD

'I

I1

i

I

..

do

'1