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Documents acquired by ERIC include many informal unpublished* materials not available frOm other sources.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. ************************************************************************
tfr
BEum Imo adir G
. ,. "
.1(*IN
air am .1114E0
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
//: N
l; I
MINNESOTA MATHEMATICS AND SCIENCE TEACHING PROJECT720 Washington Avenue S. E Minneapolis, Minnesota 55455
r rteplYne2.,
MINNIRlyEAST
37,,,,4
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
tv
0
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
June
May
Apr
ilM
arch
Febr
ury
Janu
ary
Dec
.N
ovem
ber'
.,
,,
Obt
ober
Sept
embe
rIn .
.
.
7'.-
.
..
-.
"
I1
j1
1.I
I,
II
June
/viy
Apr
ilM
arch
Febr
Try
Janu
ary
Dec
.N
ovem
ber
Oct
ober
Sept
embe
r
C,
co
1.A
CD I6 yO
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
o Q
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.
O
P
V.)
rr
._.
..-
..
"
,. .
..-.
.-
UN
IT 2
'3-
,'
Thi
s un
it ge
nera
lly s
houl
d be
taug
ht th
ree
days
a w
eek
for
.
peri
ode
of 4
0 to
50
-min
utes
:.
'.
.U
NIT
24
-°°
,'v
,
Thi
s ur
ilt g
ener
ally
sho
uld
be.
taug
ht e
very
day
for
peri
ods
of a
ppro
xim
atel
y 30
min
utes
.-
..M
on, -
Tue
s.c7
Wed
.-T
hurs
.Fr
i:'
Mon
.T
ues.
Wed
.T
hurs
..
Fri.
1st -
wee
ki
Les
son
1
*L
essb
n'''
2
*j..
.ess
on.
3q
' ' , '.
.
,.
t2n
d w
eek
.
--
-
--
Les
son
,4
t 'Les
son
-5
,
-L
esso
n.
'IL
esso
n,2
.
Les
son
'3-
,,3r
d w
eek
Les
son
.6,
Les
son
7L
esso
n-8
'
.,-L
esso
n3
,L
esso
n4
Les
son
5-
, less
on 6.' ..-
,4th
Wee
k.
.'
Les
son
'J9
Les
son
10.
Les
son
7e
Les
son
8i
.L
esso
n9
Les
son
'I0
Les
son
11
*
5th
mee
k-
Les
son
.
11L
esso
n12
Les
son
.13
'
46.,
,'
.Lesson
1 4
Les
son
-
12L
esso
n13
less
on.
14.
6th
weo
ek-
Les
sOn
"
15L
esso
n..1
6L
esso
nlt-
Les
son
098"
-
**Po
urin
gSy
stem
'Po
urin
gSy
stem
Pour
ing
-SyS
teri
v:.
,
Pour
ing
;.SY
si.e
inpo
urin
gSy
stem
'
'':7t
h w
eek
'-L
esso
n1'
9.
'
.-
'
**B
alan
ceB
eam
Syst
em
Bal
ance
---B
eam
Syst
em
Bal
ance
, 'Bea
mSy
stem
.,
Bal
abbe
Bea
m-
Sys
tem
Bal
ance
Bea
mSy
stem
*It
is im
port
ant t
ote
achL
esso
ns1,
2 a
nd 3
on'
the
days
'Spe
cifi
ed:
**H
alf
of th
e cl
ass
does
one
-s,
ystb
mon
e w
eek,
then
the
othe
r ..S
yste
mth
e fo
llow
ing;
wee
k.-
C/3 rnC.)
1-3 t, Cl)
C)
ET
" t t3 0 1-3
tr3
,
C.) Z :73
.
Cf)
ca
°,
C) 0 to0 1-3
r4 k
, Argus N
isi
tite
4-
t+ it
tfp
"13\
A
.
SECTION I *ADDITION
0
.
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
. , .
114.
r.
.
..
-- .
0
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.
' ;
2.
T
O
1'
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
I t
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...: -
i:
i
.
1
r---.....
___t_...
I.,
!i
,
I
r- -t-
0. .
,..L
--a-
r
III
18.
4-
PROCEMIRE
Activity A
Read. the story, "The Amazing Machine" to the class. The. .
story is on page 7. .
4
4ry
19
A.
$ 1.14
A
it,2.0
4 ;
t ( f 4.,
F
4ceor_te ," ". ,
-.MEOW 4. kr: IT
, 4 4
. P('N"
4 \-
f' '7'
4. ..,,.,..:,2.,,..,. 4(`-:', .4. '',;r:(-1:..?:=:; A: 1 sgl s.''' -1...I
,t'''',0;1':,r745,4:;1:.::t4'1'-'4-''esev7, !O, 1-.1.4:".
..2 *,-LN,!..4...,,,,',..:st r.f.).-ttl,:::: ,,r,-, - 11)':4 1 .--t79 "t ...1i- I to. ,L, l ..,,,y,:, :V ''' 1, 4 ,....
'*"." .z'it'' .0' ist;;I:,\;_. 'i--""I i174:fi
,,,,4,-,f::1,4--
14,''''f41N..`)- 7.,,,,,k...
iihal 1/2c,
\-,
I
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"--
4-x
.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.
rka tUn I t 24
-rIrlii+ Asa at
Same
I
cf..,1
NOtli10,11 Pd.
ptImprirririttiu7111117111t1
tait14.0
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 ,.
t0Ailto
6 u"
J2
'brashest 23Uott 24
Naas
Iles your desk computer to subtract.
640
T212Iv ...
.
496- 321
115
736436
30,
475 c
- 156
3Z0
. ...
297 °
- 164
33
111'6'...Y
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.'
f3 a.2, -If .f 35'
3I22!
O
4160at1412
621 411 1411-
723A
56
PC4:14544.'''''CAZ*SeatArAtia"!60.4'fl'''''''C'°'
-j;t:1*klte.261.'"*:°44741.411\
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.
r/ler "era 79RT 6,
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'" ',$4,1
, ,4AgrErase the subtrabt label and r;i7lZe''"it .4rinu014n.
mr.A.,,,,Draw-a1,14e beneath the lower numerals of he prOSIFITL-,--,,,v_.\ .,_.,,,separ ting them frOm tireg ace for the. answ Make thnecess ry changes in your c board compute o makedtN
look .lik this: ',.t-.0 '\
40.04.-A1"
6 I
'44
I 0 oZ-
3 6
Explain that nothing has been changed by altering the com-puter. Verify this by again subtracting the numerals ineach T place. Have several children do problems at theboard using the simplified chart.
When you are sure the children understand the use of thechart, ask:
Write a problem on the chalkboatd-in-vertical formdAtforexample, 633 - 50Z, and si.fblraet---,dectly.
OUT USING THE CHART? LET'S TRY.CAN WE SHOW WHAT WE DO WITH -
Ns. _
A4Nevtl
"44,
vt'4.
"."
..Ste.Stra.i&
//
children of the place value subtraction: that is, 6 5 = I
is reall 6T2 - 2 or 600 - 500 = 100. 6 - 5= I is
en do Wo lisee 1,26 using either the sub-or the vertica
ave;k,/:ti:liz,....4.,,..tisa4 tra ).0T
"c-''s---r; 43?,nAr. 4'i,%\e\N4*.liIt karaa,,,4,47
iA'2,;Z1:"*"-.t, ,,,,S,"" 1,......n....,-,...........,............._,?_!__......7t:,...... .
''"46.,...''',.....,-,......
''';'444t*:*ii.6!4A,."014." 4.44:6S.. %.,.6 \----...............
Ve: '1,`rN,..
t466 kio',4 **V Air i4.4.41,14Xi,SP,614,1,,AkV2,- 4. ovio,00,,,,114,4.w.,..mor-1,7.1
:11A. '416 6 6............4................6%*, op....u. 4/ V ; t"it0 00 000/00.
trach Vie the charts If y need then-
(C42.'? '.;?*It.`fp,t,,,,t ,`...,26,,,yrcpitiLl7V1131'
'N
C.+.1i0j
IA- 145
T2 7C--Al It ?a ri 10
5.12
MED
17,2C
"I
44
I
2$
1.4ii/
Af
:', IIMIllili
39)
,
r s , ,A, 465
V6- Le
12 11)
-928
fia3;
T2 ri 10
.0011.0.0 V 0.0 0 0 0.0.0ir
':7),/
onws*.g...r........,..........................&.................***..................
,
vigtavapotau4,6014,a, ,,,,,,m6v011.0.4,4,0,16),,,,,ht.oz,,ros.tufe,04.1,41",atomluvi ,4 6c11,446,14,1 e. -6661.444 4.1 4.'442.466tt 43,:4t6444.,:wr -tmta
62
*z
":1
'4.rAPSel
--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
--
.44.1o., 61/4
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.
4- 217
36
1;12
72 11 /41
J'4 j 5'
_1._,I=1 --7 ------
.7.2 r
9112 72
rt . 10
-,221 - 6144' / i
r2 ii 113 12 11 70
4111 GOO 73!1 - Ne t_._ _1-17 r-- id d
72 11 70 12 rt 101
946 i 134
- lid - b
_so
Porkshoot 211Unit 24
Subtract.
ease
-4i- 23
.24/
____..i
. 646- 726/ /,"
703 452- 402 - 348
3o1 i ay
4639 406- 3217 - 392
900 1342- 6P- si
519' , ks:7
IN tt INI 10
Practice 'Probleses
Naas
Pointsfor each IV
Test Problees Problem Points
Ilicy 1
41- 34
Pi
3,2- 272
610
5447
--V...45
44$
1 .... .._=363Nu
Day 2
72-36
f6'
A..̀....Iv '.. ,
Sli- 316
aoo
977- 3011
4$- 29'if I619
Day 3
444- 2113
?.9 I
X716- 474
31.1.
.
463. 1711
694., Jim
31023C6
taw 4
512- 417
ITS
., 436- 172
466
492. 149
634-292
2 .303 342
Day 5
622- 3116
234
413-144
/ 49
402. 1997155
90e- 32317-7
'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
r
it
srN
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
..)
ir7--v 4 7..". 4. . 7.
p
145, .1
1:'!S47C
,",1:--;/;
3fc, ;:c;
tx:. - 41.:*;;.::%, 19 .4, v*.1;;7-.: :^ ir-I%-
'
..
Job
I'D
escr
ibe
each
Pic
ture
, .
1 1e
/iver
,",;
(Z
'' 'b
oys
,..>
giri
s
".."
. ...
,-"
.1'; .
4.."
-.." 7.
-N
V
(2_
,..qo
f 1,
-..
. -
,_:
_5_.
,,,p1
-)
..%
,1
.1se
,
..1
(
-tlit
,4,
..1et
:'`4
-.,,,
- X
XX
X'
XX
XX
.t
(E
_ i"
,
0 00
0,00
000
00.
a,,,
j,)
)41
.)
.
C_I
L_-
rock
ets
,__
_+._
il4la
-\
,-
r' k
-
(-
,_4
. Id r4 C
Gun
its I
n A
,_../
._.u
n4,5
in ,i
ri a., ,
7re
I"
in
0, ti
A A
.
,,
..7 -
tut "
tr. P
)
.,
Get
tbss
ie m
ater
ials
fro
m y
our
teac
her.
k
Use
the
two
piec
es °
nape
.L
atte
r th
e tu
bes,
A e
nd B
.Pu
t som
e w
ater
in th
e po
urin
gcu
p.
Fill0`
'to th
e 5
mar
k.
44
L
O
te.
1Y1
qy
and
I are
Din
ners
.
0 W
e bo
th tr
y to
do
all t
he J
obs.
.ru
efa
lse
We-
com
pare
all
our
answ
ers.
(3)
If.w
e ha
ve d
iffer
ent a
nsw
ers,
we
disc
uss
then
i to
see
who
is C
orre
ct6
fals
e
rais
e
Pou
r w
ater
into
%lik
e
1
Mak
e
rF
them
VIM
:.lo
ok ,B
17 'LI
Z r
.,
. 0
i 6
AIM
. 9", e
4.. I
,.
-e
; ..
.'
Des
crtb
ean
d 17
31
'Dep
t) o
f wat
erin
rA--
--A
1.6
/9un
its.
.uni
ts.
-
.1-
'..1
.,. D
epth
of w
ater
inT
oIS
rD
epth
of w
ettr
irat A
l I1
units
.
units
..
.....
..
Dep
th o
f wat
er in
ffg
I.t..
...i
FT
hD
epth
of w
ater
in....
A1
units
.'
units
;
-....
.-c-
.D
epth
of w
ater
inA
rt
A Dep
th o
f waf
er In
Ifs
0.
units
.`un
its.
.--4
4A
Dep
th o
f wat
er in
pt14
u.
..
....-
; 464
_.,
A"
f=:,
e.
I.il
-.C
O.
.3.
.9
0
''
1
Job
4D
escn
bo th
e ,ta
te o
f e tr
ry.
.s-
,.
1 ct
.
E.
ini
111
A13
A
if 3
'
(6,1
3)is
an
orde
red
r.dr
.
6 m
eans
that
Wer
e ar
e.
1.0
units
of w
ater
in tu
be
3 m
eans
that
ther
ear
r:N
3in
flub
C
units
of w
ater
-e
'
Jab5
A
The
ord
ered
pai
r,(5
.4-')
des
crib
es th
est
ate
Of m
y sy
stem
:
.T
he o
rder
ed p
air
(5,4
) is
an.
(A,B
)or
dere
d pa
ir:-
The
firs
t mem
ber
of (
5,4)
isJr
Tte
..
seco
nd m
embe
r of
(5,
4) is
kr
Ttr
e fir
st m
embe
i'des
crib
es tI
stat
e
of tu
be/4
5.
The
sec
ond
mem
ber
desc
ribes
the
stat
e,of
tube
/3
So
(,4)
is c
alle
d an
(A
l-) ty
pe o
for
dere
d pa
ir.
The
ord
ered
pai
r (4
.5)
desc
ribes
the-
otat
e of
my
syst
em.
The
ord
ered
pai
r (4
,5)
is a
n (B
,A)
orde
red
pair.
The
f1;s
t mem
be r
ot (
4,5)
is
The
sec
ond
mem
ber
of (
4,5)
is.5
-
The
firs
t mem
ber
desc
ribes
the
stat
e
of tu
beIf
The
sec
ond
mem
ber
desc
ribes
the
stat
eof
tube
_R
L
So
14.5
) is
cal
led
a (B
,B..)
tV,p
e of
orde
red
pair.
WF
itu (
A.B
) an
d (B
at)
pairs
to d
escr
ibe
the
syst
ems.
BA
)A
9a-
3B
A
( 3)
2:7
)
4.8
,Job
e)
r.
(j)11
111.
wro
te (
6.3)
to d
escr
ibe
the
stat
e of
the
syst
em.
0 (6
,3)
is c
alle
d an
ord
ered
pai
r.
0 T
he 6
gle
ans
ther
ear
e
'un
its o
f wat
er in
tube
A
0 T
he 3
mea
ns th
ere
are
3
units
of w
ater
to tu
be B
Tom
wro
te (
3.6)
to d
escr
ibe
the
stat
e of
the
sYst
eid.
(1)
(3.6
) is
an
orde
red
pair.
The
3 m
eans
ther
e ar
e 3
units
of w
ater
In tu
be
0 T
he 6
mea
ns th
ere
ate
(C.
units
of w
ater
in tu
beA
,
8111
wro
te (
6.3)
. Was
he
corr
ect',
,
Tom
wro
te (
3.6)
. Was
he
corr
ect"
,
49
5
Job
7
7
a
Bill
wro
te(
7,
The
re a
re7
units
of w
ater
in tu
be
The
re a
re2
units
of w
ater
in tu
be
TO
Mw
rote
(2
.7
Tite
rs a
re2
units
of w
ater
in tu
be5
The
re a
re,
7un
its o
f wat
er in
tube
A-
2.1
111.
NN
W
I wro
te (
7, 2
).
(7,2
)1s
anor
dere
d pa
ir.
The
first
mem
ber
of th
e
orde
red
pair
desc
ribes
the
stat
e of
tube
4T
he s
ectA
id m
embe
r de
scrib
es
the
stat
e of
tube
E3
'I w
rote
( Z
(2; 7
) is
an
orcl
ere4
pair
The
firs
t mem
ber
of th
e
orde
red
patr
des
crib
es th
e
stat
e of
tube
B
The
sec
ond
mem
ber
desc
ribes
the
stat
e of
tube
A
50
Job
Fill
in th
e ch
arts
,
/111
.4..
g.I:
Elt
tl. 4.,
.Z
.; 2E
DO
.1
kr)I
..3I
,
4....
(11 1'
- °I
C4
.5-
.00
4.0
trI
. t1
al'C
O'
='.
OM
NI
it.1
j.1,
,,.,..
.:r ..:
::7.5
,
1
7,
,J
ev, 1
x
:".7
.7.7
IMIN
IN""
""""
""...
*at
azT
;1: A
II
......
..
<0
). .
..i...
#)...
......
..
i:... ea ia
. rr,
"Ci
.,
ta 0
0.
cral
%.4
.1
-
1
2. ED
0 0. . .
7
-6gy
,k
(7)
1'
C7,
CD
liVIO
-sI IM
OIf
teIM
MII
IMIT
h1i
diii4
4,4,
:..1`
70.
IS V ..
.../
.
a 51
Mak
e yo
ur s
ysie
ms
look
like
'thi
s.Y
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
.
Beg
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._
....,
!...
......
c.,5
,7,,,
.,..
..::
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
)
(Lei
:
.1'
1*.
.CD
.NO
S p
ouLw
ates
ture
k .s
nd fo
rth
b.tw
eer
I fou
nd th
ese-
stlte
s..
-
C" t
151O
1
10;
-4,*
.3,
3ge
l:
'5 (
''*I
f.a
:.:'''
. t,.
....,
'..
(p)
The
se s
tate
s ar
e th
k sa
ne a
s th
e on
es I
reco
rded
on
Ow
um
..
*. #
The
grid
poi
nts-
fan
in a
str
aigh
t lin
eN
o
,t1
!At 1
-
= J
ob 2
8F
I
I can
ye
grid
to r
ecor
d th
e st
ate
(2i,2
)
tz},z)
clot
h in
P u
.-,
Loca
te,th
e gr
id-p
oint
s th
at c
an b
e us
ed to
dei
crin
e th
et.
h..
Writ
e or
dere
d pa
irs to
oes
cktb
e th
e v.
:wet
.
.
(.3
di.
Ilic.
.
/.
tl) ..1
'.
(I .
gal..
..-
i0
.0.
. 6,
. ( 4
,
4,
.=.
......
.W'
RN
.
..
is
0 ,4
11'0
8
8
I ,.r
.
...,
ft *.
-=
1 2 3
0
s.
.
rn
.rob
)7
RT
he fi
rst m
embe
r is
a la
tter.
The
sec
ond
mem
ber
14 a
num
eral
.
A,3
.is(le
tter,
num
eral
) pa
rr,
The
firs
t mem
ber
is a
num
eral
.
The
sec
ond
mem
ber
is d
7.R
is a
(nu
mer
al. l
ette
r) p
arr.
Writ
e a
(lette
r, n
umer
al)
orde
red
pa)r
nex
t to
each
on th
e 9r
id.
It
A-1
QL-
2)
(A3)
(t.'
(.13
.4)
V ..
4.
...,
I
->
2
BC
DE
Dld
you
writ
e;(B
. 2)
(D,
(r 2
)(D
.3 )
(G.3
)
Use
som
e of
the
orde
red
pairs
bel
ow
to n
ame
the
on th
e gi
ro.
(Z(
A.9)
( r
4) 0
49
y(N
I
(K 7
) (
1.4
3 )
No
t,
Job
i8 I dra
w a
lino
seg
.nen
t and
.C4t
i Is
y st
ony.
; lin
e,
Now
i'll
flow
oth
er li
ne s
egr,
,,ne
,
,-,0
,,.to
the
<ar
tinor
, ,o,
e
.
.,ifi
rC
r it\<
*1-
,,
41
I cad
the
tart
Ing
hny
iqr
,---
.--"
'
ltt.
1...
ha+
,..4
1ca
,,...
'N,
The
oth
er !m
e";
4)e
lam
ed,1
1'4.
,,,C
iv. t
hen'
pat
t)f.
.
-.-
/-1,
2.3
.1...
..-..,
'..-_
_....
,___
_,
1...-
,
.
Joe
:-
.B
ill,
.
21
45
I,
4 5
b
In th
era
cc 1
..,:tt
ift: 4
.in
!ht r
. ,I.,
t,ho
wfa
r t,
Bill
has
con
3;ii-
iVs.
ear:
. rt.1
1,,
-.),
%%
ate
a nu
n-.e
ral
foe
has
ton
_5_
,-4.
,,,..
.at t
hq ti
t,,,,
,,t.,,
t,....
Tom
has
rrn
,.11
in.',
.
.#5
I4
6
6
01,
Jobi
9
.
:*
,,,/ ,
,, # , "
did
each
run
ner
-num
eral
at
arro
w,
the .1
1.1
go"
tip
7
lliIl
lr'
.I
1I',
'fill
show c, ;i=
inho
w
0
the -
empt
yfa
r
- 2OO
,
How Writ
e
of
...
each
- --
.... 0
.
0
far a -
. I
each
boy
ran.
-4 ......
.
2
-4 3
ti ... 456
I ou
t the
ane
w
.
.
saili
ngpl
ace,
line
in
.
.
How
far
-did
rill i
n th
e--
----
---i-
-
each
boy
get
"0
s..
I.
s4
04 3
30
2I
iI
if i
i0
Suir
tino
line
0St
art
s .
How
far
Fill
in,th
efrom - 0
the
0 34' s
.
h.d0
IIIg
-.
star
t
ti 5
are
the
arro
ws?
.
,-.
-.
..dó
0I
2
0111
r011
111D
3
I.i
I,,
it
.0St
art
,
4
X
Job2
0 rind)
:Iry
J1
wo-
-.1
,pa
irrit
IV',
(.,.;
14".
qr.-
.,
1,
, 1t.:l
,Wn
a.I.
, r.
' Poo
r I.,
,'A
.,..
Af,
4'
; It,"
-_..
,r.:.
%.
'--,
Ad -
".--
)
1,, ,
X a
lo.s
.,,:
14,0;
11-.
0_
I,,,
Pt t
',
--4-
v
IRE
-,--
-/1
11
ER
VIN
IIIIII
IIII
INN
1
..II
II.
3k.
..
I se
e th
ataw
ay tr
orn
,
B she'I'
AV
'i Ufi
ttssr
arrr
oglin
er
r , "
i4,1
,,,
,(x
41.j.
!',r
ig
IIII
I_
10E
111
1102
1113
131
,I
lid,..
.
fr-
..
..,.
lkill
l.
UR
Nt
23
Wri
te in
the
firs
t rne
rd...
-r o
f ea
ch ,o
n-tr
ed p
'ito
ir-
'''''
ill.
dr.T
7___
(4,1
1){X
;V)
I,
,..
..i
.
4,
3
1
Job2
1 Inte
rsec
fron
B is
2 u
nits
from
the
othe
r st
artin
g lin
e.
3 2 031
A4T
2
So
I writ
e 2
es th
e se
cond
rrie
rnbe
r of
my
orde
red
pair.
8
SrA
RT
Writ
e th
e se
cond
mem
ber
of e
ar/o
rder
ed p
air
nam
e.
3 2
olic
y)-
.oh
o/'
da,_
-.:
llo..,
.
4S
tAR
TIn
icLI
NE
Writ
e bo
th r
nellb
ers
of th
e or
dere
d pa
irs.
4 64
?Ise
esi
dres
s of
say
hou
se a
s (4
-sa
.D0
1 ite
e-et
Poi
nt A
or
Poi
ntW
rite
year
eni
evei
her
e.
My
hous
e as
et P
oint
. 5.
The
add
ress
et P
oint
B,is
The
add
ress
_at
Poi
nt A
15,
ca.
..0
( bi
t).
21 0
c 3
5)
4 3 2 I
AbE
lmj
0i7' 3 2 01
00I
23
1.40
..,4
0i
23
45
6
i (#
s 3
)3 f 0.
( 6
,3 1
1,
!4
, I.
)
1-6-
o,0
23
5ii
.
t.
:rH
Job2
3,
Put
l.et
ir ..---
,
. the
re a
re
syst
emA
Z , a A
.....1
)Li
in
1'
. 4 . units
this
ft a i .1-
. fi.
of w
aterst
ate.
^
inbe
A.
The
ord
ered
pai
r 64
Adu
scrib
es th
eLt
ate
of m
y sy
stem
.
The
lust
mem
ber
of th
e or
dere
d
pair
is4
s.,
The
sec
ond
mem
ber
of th
e Z
irder
edP
au is
2.M
ere
are
.a.
.un
it oW
ater
in tu
e B
.
In e
n (A
. B)
orde
red
pair,
the
first
me
ber
alw
ays
desc
ribes
tube
,.,
the
seco
ndm
mbe
r al
way
s de
scrib
es tu
beO
A
In e
n (B
. A)
orde
red
pair,
. the
firs
t mem
.Ith
e se
cond
mem
beral
way
s de
scrib
es tu
beal
way
s de
scrib
es tu
be%
a
4
I say
that
the
poin
t (4.
;4.
desc
ribes
the
stat
eofm
ksy
stem
.A
.1
Inut
s of
wat
erin
tube
A.
1111
11M s BIM I
t5.
66
to
J062
/4
Dra
w a
rrow
s to
find
the
poin
t whi
ch r
epre
sent
s ea
ch s
yste
m.
S -
-:
. 4
3
A.
AB
1
Ei 1
.
.
4
' .
#/
43.,
1
Y;,.
171
n>
01
1>
'j
12
34
S
.11
.
1
.
..
r, .I. na
2 i(
11 ir
it )
( "1
9 0
)
1
'
LI).
1
0
0el
i2
i
fx
PO CD Jo
b25
Pou
r w
ater
bac
k-an
d fo
rth
betw
een
El a
nd C
IU
se o
nly
who
le n
umbe
r le
vels
.
Rec
ord
thes
e st
ates
on
the
char
t and
grid
bel
ow.
.
Who
le N
umbe
r Le
vels
5, 3
-4
11,
.
3, '5
4 7o, 1,
0
6,Z
r
410
.
.9 7
.5
0.
( ch
s) c
"r
4 3 2
IP)
C5
3)
0
I1.
34
Look
at t
he p
oint
styo
u m
arke
d on
the
grad
:
Do
they
'lle
on a
cur
ved
line/
Do
they
lie
en a
str
aws
tine,
ttrit
S6
78
910
7,
No
68
120
Put
you
r sy
stem
in th
e st
ate
(.4,
4 )
:
0U
se o
nly
this
wat
er to
form
new
sta
tes.
.
Fin
d on
ly s
tate
s th
at h
ave
the
wat
er it
:el
of a
...h
ole
unit.
,Rec
ord
tnec
t sta
tes
on th
is
The
poi
nt (
6)
desc
ribes
one
stat
e of
you
r sy
stem
.'
V.O
n th
e to
ld b
elow
. loc
ate
and
nam
e th
e ot
her
stat
e. y
vo
10 9 7 6 S
GR
ID
nom
imam
ININ
IIII
1111
11M
IM
II
MN
III
M'
MIR
I WM
MII
IMR
NM
IIIIB
M11
1111
1111
1
9E
llEll
IliH
IA
o )
12
3 4
'S6
76
9 to
0.
ip,r
;.1,..
.....
mite
red
Pai
r
.ii
k/.4
,
3.7
.3;7
.
AiA
,/
/1,
1
010
d,10
100
/0/0
I/
Yt
.2-
2,-3
,-
73
1,3
The
se a
re th
e st
ates
you
shou
ld h
ave
liste
d.
0.
(6.)
41)
(a, 1
0
(Cr)
(54)
1,4
..(44
,4)
((ye
,(7
,3)
(3.7
)04
0)
. 69
iI
Ikg
s. ,a
r.
ck,1
3.27
T.
A 6
'.
.t--
ftdb
..0
1:ut
you
r sy
stte
rt'!n
the
stifl
e(3
,3)
A'
0.1,
0so
t- p
our
wat
er b
edk
ant,
fort
hbo
twae
r.iF
tj,,
6'Ju
st ti
ni.k
abo
ut d
oing
it.
...r-
..
#.
'..-
-1:
...
CyT
hnuc
onl
y of
" -,
tam
s !n
wl,a
ch th
e w
et. r
1t.v
e 1
lin
who
tee
stin
ts.
Pe-
..rd
thos
e' s
tate
s ac
dre
ere;
J.dt
a an
d ar
pO
lt.t,
Itti
.
..
. h.
..,...
,.t.
..:.k
.:
4.(1
1,11
1
. .
.'V
I
.
iii
t .
6
(Ls)
(3,3
)
Ippo
w p
stg.
wat
eceu
rck
and
fort
h be
twae
r1"
,,;',
5v.
.
I fou
nd th
eses
tate
s..
0'C
,§.
4,4
3,3
4.4
5,1
14,0
..
0 T
hese
sta
tes
are
th2
sate
as th
e:on
es I
rec
orde
don
UK
- gr
in.
..,
C:b
.th,1
The
grid
poi
nts-
fall
in a
str
aigh
t lin
e.Ig
o"
t
t
Job
2.8
/.ca
n va
a a
rid to
rec
ord
the
stat
e
ii
3cat
e.th
e gr
id, p
oint
s th
at c
an b
e-us
ed to
des
ctlu
e th
e 1,
1te
t tie
writ
s or
dere
d pa
irs to
cte
swbe
tha
st.u
.s.
(3 ,1
-k.
...
)-
Cl. i
.i i
,3Y --
t/"
1' 2
-0.
10s
.
. AA
l
( 4
,i ?
.
..=
.It
,'..
.
li
.r o
'.s'
qc, )
$
,,r
.s
,e
-ow
oom
enna
;0
Se
.
,C
e.
tN
.4
Job2
9 ..
1111
tune
A tv
the
ten
;gut
m40
..
Kit
nu n
ut,,t
tat,,
b.,
a
a'
J
/
4.10
1?)3
9
1
A8
Till
you
r sy
stem
in s
tars
(,0
).
Thi
nede
e_tu
nits
of K
aty
sn tu
be A
.
The
re o
te_p
_ un
tts P
I ant
i r a
n tu
be p
.
Fltv
f tha
tpoI
nt o
n th
e 'g
rid b
elow
.
Poi
sr a
ll
Nov
. the
re
tube
A.
Noo
; the
,'
tt,t,e
B.
FIn
d th
e:.h
art
belo
w.
the
Wat
erfr
omA
into
8.
ere
_a_
vita
!, of
xat
et In
.;
are
units
of %
Net
er In
-Pou
r th
ese
Fin
d ,u
s ^
We'
The
y ne
ed.
Rec
ord
all
(7-2
ten
one: no
t
your
At
unit%
diffe
rent
be w
hole
stew
sof.v
ete
...to
ter
ntr,
on-
on th
e
be, z
grid
;se
t
too,
tt t).4.
1 t,
,ttL
ot./
A B
orde
red
pair
(a, b
) on
the.
6A
f4 4
41
1
5
441
Iqto
4
ts-c
iottJ 1,
5/
2(Z
AN
".
5-.1
)
(4.0
).4
4,41
0
14
S6
Lt')
11..4 .C
f.th.
tjr.
3).
Dra
w a
str
aigh
t ils
r th
roug
h th
e tw
o po
ints
(O
;4.)
and
(Gip
),.(
441.
) ,c9,
q)it (k
V.
,
Pou
r th
e w
aler
,bac
k en
d fo
rth
betw
een
A a
nd B
.
.Rec
ord
seve
ral d
iffer
ent s
ites
on th
e gr
id.
Do
all t
he 1
was
you
net
tled
fell
on th
e lin
e se
voan
tNo
.'S
S
7273
.Job
31a
,r11
logi
n lir
/put
ting
lirm
in V
IM*.
it is
().
,
Arle
yA
I
my
5101
1001
0k
'
.
,.
Mar
yS I
Now
VII
reco
rd th
atth
e gr
id. P
ll w
rits
em)
near
the
poin
t.
CI
stat
e.on
igy
initi
al
0
0I
23
4 50
Now
the
mus
t
has
Mn
it's
stat
e
use
inits
syst
emA
1111
4say
turn
.pi
t cha
nge
Of t
he s
yste
rh.
Ial
l the
wat
er M
ary
.
8 6N
eW s
tatV
s...
7A
nn 6
I fin
d (4
My
initi
al Is
r/1
5-)
on th
e gr
d.A
for
Ann
.'w
'-
-0 2 I
23
0.5
67
LI
We
stat
esw
e stat
esM
ary
take
turn
. cha
ngin
g th
e
of o
ur s
yste
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
. I c
ount
Mar
y'7
initi
als.
A4
eM *A *AM
)°Y
74
o
dob3
.2A
rut y
our
syst
em to
the
stat
e (7
, 3)
0 O
ne p
artn
er s
houl
d re
cord
this
sta
teon
the
Quo
.W
rite
Irar
intu
it be
side
the
poin
t.
l0T
he o
ther
per
son
shou
ld c
hang
e th
e st
ate
of th
e sy
stem
.
No
wat
er c
an b
e ad
ded
to o
r re
mov
ed fr
om th
e sy
stem
.
® F
ind
the
poin
t tha
t des
crio
es th
tsne
w s
tate
. Wttt
eyc
ur In
atal
n x
t to
It.
t04
Tak
e tu
rns
chan
ging
sta
tes
and
two
tng
poin
ts.
0 F
inis
h th
is s
ente
nce;
12 I t 10 9 8 7
0
6- 4 3 2
My.
sta
tes
4
my
part
ner's
sta
tet t
al n
umbe
r of
's m
ites
sie
foun
d.
m
A
m
A
m
4'
rn,
0
DIO
-You
find
at l
east
II s
tate
s?
23
4S
67
8, v
in
Nrk
,75
1;!;..! --74.4""rr: 11,V..-V4',4*4 44,..rp- V$1-rt,' j,4 -4
r
0. . ,.. ..+-* -,.:-...,$,t . :.:,.;;;;1,-.iir.,,,-"V;1----,-;:s-,,..?,.;..',..'-;7)--,-*$:--,`-,.,..4.'.
. t " '3, .,!. ... .' .. , i: .. . t....I . '.., ."...`-s.--.4,e- i-k,....- -",,,,, ,........ --,....01. .t.,,,L,, ',./1....! 0,: - ',1,,,,',... . r, 1,...,<.... /.....,...;-7.4, Ist,,!,,,,,..40 c! ,;,-0,,.p.,.1 ,,,
., r., ,,: ,1 ` :It ..71:;,.# .4. ,-.1.,,..A7.:s...,. \-;,..r. -,ss... ....,'-:1,4.7 ..7..., kg .i... ),
..., "lutiv......;?, ,- -L.'..0 '''''-..-. phrf. ,,....$ -A,r0;'...- 1;',1,1- / c it '',.' r.0.7 t*....".- ....
*4...N4', ,.,r4.4'. ,` 1 "-re' e'lr 't"v ,,,%krC o
Ar:
4e.* 'Zs;
;" ,
.t d'v'L ,
#4,.Vr T 1,'TAN,ns17z.,44t , .fx ,
, '130 ";,:',V;r1 Lk," re 1:4 r,. 4 2. h ked,' 00. r 1.r 1- r , 1 tf *1_ r s d' 4,
--.4_," 4.s 9tto4,
:71 (4)(217 11% .e'..1 7-, t T !t.:fiti'f -41 ZZ!..V CO,*' tiS ;AV1 -
r 1-,
, v.,) tvt7..5 4- 7 1*(t
*,. :*t, -t-1-_-,-;,?:-.1-;ht:-.;7,;.?,,,,?.4- ..,f, ',/ r:Ve3.-);47:.***:.'*..,r-pr...?.;,,-,,- \ 't. . f K..- -,, ,- , , ,. . , ....,,c,,:},-;-;.:
' -- .- , ., :, 1.- , ' ,4:, , . - ..t ,., ,. . , . .- , ' - 'fl ' : ',--...1.,:::::'- - _ -, ,. -. - tl , f ,,,4 I
'''' ^..' .... ' , `,. .." , S-4 - 444; .4..,,,, 4..7:: 7.4 ,-,;C . ,,,.. l 4/' 0 .."` '. L. 't t,.... . 0.-,.. 7 , ,,., k f ;
' ,: I : .-.' t. 1, ..iV'4';4,e. .,,,, .6 e .
4 1.4,1 s 24 ..1!,,,4 ,,Vi 4,
'''".7 ''''.'( .:1".1(+4.....
;..0"f Z,,1 13.tfis ,,)41.
1 .. ;...rAkz*,1, '''::::" ,;'`:r'.fr:i'.;.0,.... ...., ....;', ,,/,.,..?:
V. '1lr..#t ,fr417..'.7-1f7::.\4''',:';;-'4?.:11.::1/1'..4.2.:.;??.14t", ' . ,..; . ,.,Y,t4,
't /I' ;4 l, ,- \ :: L i- ''t..'.:''' 70.',0f.,,' ^.....L ..,.t..0
.A',7,,-,it,"t f,.,.,. ...; ,,,,,,i ,A;,1:-;..1
.) .1,...4 y ,.17.1 C.0
;"' ki 4 ' >,.tki i '1. ',V ei4 , -re, ii',., ; .'1 it a',..r'1.-"\c. -, -,
.- f -- .14' -''' 't '1'14', . c ,),-4:-
J ,AZ,J^, ,?., St\ et ,,,,,- 1;1,1.,1,1 1:,'" 4 is,4; ,,.a_,-. , ' t, ' i 1 $sit:',...f-,':tj ytet .1,,,,Y.0 . t
17-1" . ^ 1 ,., ..- : ,d, r- , _. ,
- 0 p,.,;,.. 1 . ; .,' ;', ,
"1" ' t", c : ,...;;;C ,
k : c"...'`..: ; '''" '- a ' '..r. - --- ,iw
-4:*.,,,,-,,,-. - r ...;,2"'-':=-14,-1......i..;0.
1 - ,-l'-'7::,.7} : ::14 ; :: :r: :ti;1; ..4:1:-f'r--: i: : .:::1''' 'iS1:1:1i:::1'.;:: : :,:i :::: 1..;1;1:
tip.
E;1).. 1:: : .:\51':::..;"::. i :::*:1'; \-:: :t.:::41:' ;7;1 ::. ,.:;;L.:E...,,'" ,1 0.,;e:,z1S-\:.711.,-,.-;:; s':...':' : "fr':::::.'::;:t...AT,;": '::.::::;:1 .1;1.::::'C:1; ''':.'::::;\:4.:::
,'-', 4-. : 21., A 0,-zy .X7 ik.10..,,x.*:: ,I.' ,, r 4, - , 1 ; v ,i14',,,1,- ',. , 1 t '..e, N i),-r...,1,,,. t*,;' - A 'y ` '''''' , k^. `,,,,' ,";,..'..:;!1..,1.4'e,, p';e,;!.....7 ''...N.ttr,:, .,.",'- ) .. f'. LI -/- ,',*.;,..,,, ..-4,A.,,,.
;-' ,-h.,. .. "4-',..",,, : ;J:vs'';:i7.7.r.'2,0z....' \,',',..,-,t. x.f,' e,,::4,. r*^.,.............: ".?
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
Bla
ckLJ
,Red
ar
- ST
7RId
4159
ItT
1--
1R
edB
lack
'1//4
,52
15.
85 11
019
0
I-4-
1=1
Bla
ck-
Red
71J2
6/(
,R
edI
(0/0
0B
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=
E5
qty
poi..
ptit-
roe
1
at,5
,00
is c
b.re
cer
Red
Bla
ckA
dditi
on S
ente
nce
20' ,
; 90
2.16
-/-9
0 =
//0
atyl
itv.,
ma,
- d
sso
t*A
s /
/se
6 ee
olec
.-1-
.
;1
Job
8L
AC
K14
.
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1101
1M11
1111
1111
1111
1111
1111
M11
1m
imm
umum
mum
mum
mi
mer
umm
alui
llmm
woo
mum
1111
1111
1011
1111
1111
1111
1111
1111
1111
1111
1114
1111
1111
1111
111
1111
1111
1111
1111
73M
1111
1111
1111
1011
1111
1111
1111
1111
0111
1
°111
1111
11M
IAM
M11
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1.11
1111
1111
11
IMU
MM
IIIE
ME
1111
1111
1.11
1111
1111
1111
111
" M
1111
1111
1111
11 IM
MO
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
0111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
M11
1111
1111
111I
IIMIN
NI
1011
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
1111
11.1
.111
1111
1111
1111
1111
1111
1111
1111
1111
111
1111
1111
1111
1111
1mm
usse
nmum
umm
mum
mem
mas
sani
me
mun
nusa
min
mni
amoo
min
num
mum
sim
mio
num
mm
moo
mm
ualis
min
mum
umal
mai
l5.
.1a
aM
a a
71
bo to
70
rrR
ED
o\
Use
a r
ed c
rayo
nD
olor
the
line
labs
.
10.
'
W O''''
-U
se a
bla
ckcr
ayon
for
the
blac
k lin
e.
.
IAA
wj d
RE
D.
r4_.
R.1
.0I
01
75-
'
CD
with
line
with
1@
ti
Ord
ered
pai
r Is
Sys
tem
is __.-
---
],-
,e\
(zt-
.
C'1
rila
nce-
d
\.tif-
M1-
21c
d),,
Circ
leyo
ur
Circ
leyo
ur
25 o
n th
e R
ED
line
red
cray
on.
75 o
n th
e B
LAC
Kbl
ack
cray
on.
88C
if jo
b 8
cont
inue
d
0 l'I
nd 2
5 on
the
RE
D li
ne.
Dra
w a
RE
D lu
re s
egm
ent
to th
e to
p of
the
grid
.
0 K
ind
75 o
n th
e B
LAC
K.It
ne.
Dra
w a
BLA
CK
line
seg
men
tto
the
right
sid
e of
the
grid
.
0 F
ind
the
Inte
rsec
tion
of th
etw
o lin
e se
gmen
ts y
ou d
rew
.La
bel t
he In
ters
ectio
n (2
5, 7
5).
r
1RE
.D
RIO
zs
!;LA
CK
dAC
K
0 Lo
ok a
t the
grid
on
the
prev
ious
page
.P
ut y
our
syst
em In
to s
tate
s A
. B,
Wan
d D
.
ST
AT
EO
RD
ER
ED
PA
IRB
ALA
NC
ED
or
NO
T B
ALA
NC
ED
BN
B
Alio
,bo
8B
415;
63-
C''
7o, 3
0
D.1
0,/0
89
0 0 A
/I
a.
IIIMINUM
Mum
mmEMMEM
IIMMUNEMMEM
MINNIMMENUMERMUNERWM
INORammINMENNEMMMEMEMMME
MMOMMEMMMMMUMMIME
MEMSMWOMMOMMIMMMME
Ausimummlimmommorno
osimmummummmummum
miumpmummmummammuma
ESIMIMEMEMN mamma
ARIMMEMOMMOINI mama
MMUMMENOMEMEMMMEM
MEMMUNNMMUMMOINMMEMAN
IIMMEMNLVAIIMEMEMEMEMEMS
N UMEMENAMMENIOMMMKROMMM
MIIIIMMEMEMEMERRIMIONEM
MUSEMEMEMMOIMMEMOMMENME
EMIUMMOZNIEMEMRNME
MMMMNEMMINIMMEM
MEENEMMINMENNUMMENNEN
NOMMUMIOMIREREMERMEM
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
'
.C.3
<p)
15 b <p)
,tr 1- <p)
,
% t7P
.
V
.6 NI
4C".
) f.'
a
0 T
orn
is 'r
otan
ding
on
the
com
er o
f10
Str
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
Xon
,he c
orne
r of
(5t
h S
tree
t, 12
th L
eone
l
.:.,1
11L
id ,s
0 T
he a
ddre
ss o
f poi
nt M
is (
116.
'Str
eet
/6 A
venu
e).
. ill
,I
-
i.
974
.
,.4
;.,
.,
..."
,'
,.
a.
-a
01
Job
15ra
t m y
our
own
cn.m
.1)
se-p
ur b
alun
ce.b
earn
to c
heck
you
Bita
nced
(qc)
,1.
0)0
..-_k
_o_-
_,A
,
, 35
)+
21 A
i.(
,6 (ZA
., 18
)ia
. -I-
147.
. wo
(ge
_12)
j44.
1.1
.= io
d,
(5S7
4.0
45_4
-1_,
L=
JA)
aWer
!pt
t 4n
ct,o
t t .r
ot
Not
Bai
,sne
rdl°
4 "
e'.r
isLt
t'
( 35
,+
.=..-
----
-...
-- 0
(43
.7+
=
(,
IL )
4^=
.
(726
)+
=.
......
...
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,
The
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
ec-c
.ohe
,.fi
ts$1
1,41
' elIp
14
Ct 1
0-
TE
ST
TE
ST
red
blac
kB
or
NB
red
blac
kB
or
NB
t
* * ,)
.
4.
I*
.?.
..
/
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
-:i-
---i
'--,
t,
.--tlli -,
---#
,
1
f
III
IIII
II
iI
711
H1
j ie
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
1111
L
i1
1i ,
4112
1111
1111
1111
1111
1111
11
j1
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