DOCUMENT RESUME

ED 084 082 SE 014 891

TITLE Introduction to Stretching Machines, Mathematics(Experimental): 5211.08.

INSTITUTION Dade County Public Schools, Miami, Fla.PUB DATE 72NOTE 24p.; An Authorized Course or Instruction for the

Quinmester Program

EDRS PRICE MF-$0.65 HC-$3.29DESCRIPTORS Behavioral Objectives; *Curriculum; Instruction; Low

Achievers; Mathematics Education; *NuLber Concepts;*Objectives; *Secondary School Mathematics; *TeachingGuides

IDENTIFIERS Equations; *Quinmester Program

ABSTRACTPerformance objectives are stated for this secondary

school instructional unit concerned with introduction to thestretcher and shrinker approach, solution of simple equations,factoring composite numbers into primes, definition of prime numbers,and communication skills with computational concepts. The course ofstudy is intended for students having competence in the basiccomputational skills with whole numbers. Comments are presentedconcerning teaching of the course. Included are a time schedule forinstruction of stretching machines; an outline of the topics.andobjectives included in the court tz content; suggestions foradministration of pre- and posttests; and lists of classroomsupplies, teaching aids, and state-adopted and other texts forenrichment and practice purposes. (CC)

5 DEPA TAr NTOG HF AL TMEUUCATtON It Ail FAWFNATIONAL INSTITUTE OF

EDUCATioN

AUTHORIZED COURSE OF INSTRUCTION FOR THE

INTRODUCTION TO STRETCHING MACHINES

5211.085212.08

MATHEMATICS

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DIVISION OF INSTRUCTION1971

QUINMESTER MATHEMATICS

COURSE OF STUDY

FOR

INTRODUCTION TO STRETCHING MACHINES

5211.085212.08

(EXPERIMEN'TAL)

DIVISION OF INSTPUCTIONDade County Public Schools

Miami, Florida 331321971-72

DADE COUNTY SCHOOL BOARD

Mr. William Lehman, ChairmanMr. G. Holmes Braddock, Vice-Chairman

Mrs. Ethel BeckhamMrs. Crutcher Harrison

Mrs. Anna Brenner MeyersDr. Ben Sheppard

Mr. William H. Turner

Dr. E. L. Whigham, Superintendent of SchoolsDade County Public Schools

Miami, Florida 33132

Published by the Dade County School Board

Copies of this publication may be obtained through

Textbook Services2210 S. W. Third StreetMiami, Florida 33135

PREFACE

The following course of study has been designed to set a minimum

standard for student performance after exposure to the material described

and to specify sources which can be the basis for the planning of daily

activities by the teacher.

The course sequence is suggested as a guide; an inCividual teacher

should feel free to rearrange the sequence whenever other alternatives

seem more desirable. Since the course content represents a minimum, a

teacher should feel free to add to the content specified.

Any ca meats and/or suggestions which will help to improve the

existing curriculum will be appreciated. Please direct your remarks to

the Consultant for Mathematics.

All courses of study have been edited by a subccruitteo of the

Mathematics Advisory acranittee.

CATALOGUE DESCRIPTION

The first of four quins designed to develop computational skillswith fractions, percents, and decimals using the "stretcher andshrinker" approach developed by the University of Illinnis Committeeon School Mathematics. Includes factoring, equations, primes, ex-ponents, and rearrangement (cormtativity).

Designed for the student who has competence in the basic computa-tional skills with whole numhers.

TABLE OF CONTENTS

Page

Goals 3

Performance Objectives 3

Strategies 4

Time Schedule 6

Outline

Stretching Machines 7

Hookups and Factoring 11

Factoring, Factors, and Equations 13

Punch Cards, Factoring, and Primes 15

Repeater Machines and Exponents 18

Posttest 20

Suggested Sources of Activities 20

GOALS

1. To introduce the student to the stretcher and shrinker approach

2. To give the student positive, success-oriented experiences inmath and increase his motivation

3. To develop and increase the student's understanding of, and skillwith, the concepts of multiplication, factoring primes, exponents,equations and rearrangement (cammutativity)

4. To increase the student's communication skills with computationalconcepts

PERRORMANCE OBJECTIVES

The student will be able to:

1. Solve simple equations involving stretching machines.

2. Given non-prime and prime numbers, state at least one other nameusing the process of multiplication for that number, where possible.

3. Solve simple word problems and puzzles utilizing multiplication andfactoring.

Factor a composite number less than 100 into primes and write thenumber in exponential notation.

5. Explain and give excuiples of the Rearrangement Principle(Commutativity).

6. Define prime nithers and list the first 20 primes.

7. Recall whole number multiplication facts with factors 1 10.

STRATEGIES

Before beginning this quin, the teacher should read carefullythe course description and comments contained in the Teacher'sEdition and the Activities Handbook. In adchtion to these, thefollowing general comments apply:

(1) Entering competencies required include: addition, sub-traction, multiplication of one and two-digit rumbers,and division of whole numbers by one and two-digit divi-sors. It is recommended that a pretest be given and item-analyzed. The Skills Mastery Test contained in the GOALSbook Dade County Public Schools Bulletin No. 7-H is recorrmended. If the answers are placed on computer cards, theservices of CDP (teacher-made text program) can be usedfor more complete test analysis. Retain these test re-sults to compare with the posttest for 5211.20/5212.20Decimals to assess student progress. This text can providethe pretest data for the four-quin series.

(2) In order to maintain student interest it is recommendedthat the teacher stick closely to the story line. Amachine either stretches or shrinks, not multiplies ordivides. The kids will realize the multiplier aJld dividerproperties of the machines and will verbalize this. Youshould acknowledge the correctness of this and then go onusing machine terminology. One of the major benefits ofthis course is its novelty, don't destroy it.

(3) The heart of the course is found in the hundreds of activ-ities found in the Activities Handbook, activities whichforeshadow, expand, drill and supplement the developmentfound in the text. The frequent and short quizzes foundin that handbook help students to gain confidence, whileat the same time reviewing small blocks of pages.

(4) It is practically impossible to use all the activities inthe handbook. The activities are divided into requiredand optional. Plan to use all the required activitiesand those optional activities that are appropriate to yourclass and the time available.

(5) It is recommended that the overhead projector be usedextensively. Many of the activities utilize transparencies,and transparencies of important workbook pages should bemade to assist with discussion. Transparencies of quizzesand tests are valuable so that students can either gradeeach other's paper or immediate feedback and discussionopportunities can be provided.

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(6) r.ne work text should be used primari:, in a class-discus-sion approach, 'lpefully utilizing the overhead projector.Tbo much can be lost by simply .,o. king pages. The impor-tant ideas should be expanc:ed and emphasized.

(7) Discretion should be used in deciding whether or not toallow books to go home. It has been found effective tohave row leaders distribute and collect books each day,leaving the books in the roam. A beginning activity onthe board can be used to expand on and review the material,and to settle the students while the raw leaders are dis-tributing the books.

(8) The course was designed to be used in a work text typeprogram. Much thought and planning should occur beforeutilizing the text as nonconsumable materials.

(9) Classroom supplies: Stretchers and Shrinkers

Assuming a class size of 30, the following supplies wouldbe consumed by the class in one year. (4 quips: 5211.08/

5212.08-5211.20/5212.20.

30 student sets (Book 1-4) of Stretchers and Shrinkers(if used consumably)

200 index cards (3x5)10 reams of duplicator paper

The materials which follow may be used with several classesand should be saved from year to year:

1 Teacher's Edition of Stretchers and Shrinkers1 - Activities Handbook for Stretchers and Shrinkers

1 meter stick1 yard stick1 box colored chalk2 boxes (100 sheets/box) thermal spirit masters2 boxes (100 sheets/box) transparency film for over-

head projector30 scissors30 rulers (marked in inches and centimeters)

200 file folders

You will also require the daily use of an overhead projec-

tor (with colored pens) and a file cabinet (at least three

drawers).

STRETCHERS AND SIEZDTKERS

Time Schedule

Introduction to Stretching Machines

Chapter 1

1-30 Stretching Machines 9 days2 Quizzes

Chapter 2

31-54 Hookups and Factoring 7 days2 Quizzes

Chapter 3

55-82 Factoring, Factors and Equations 8 days2 Quizzes Test

Chapter 4

83-126 Punch Cards, Factoring 11 daysPrimes

3 Quizzes

Chapter 5

127-144 Repeater Machines 5 daysExponents Test

40

If you are able to stick to this time schedule, you may use additionaldays for enridhment activities and computational review (see page

BOOK I, CHAPTER I: STRETCHING MACHINES

TEXT PAGES TOPIC OBJECTIVE

1-4 Labeling sticks ac-cording to length

Given a collection of sticks ofvarious lengths, the student isable to sort and label the stickshaving the same length.

Note: to fulfill this objectiveit is permissible for studentsto compare lengths with a card,a ruler, or any other measuringprocess they may devise.

5-7145-146

Comparing lengthsof sticks

a. Given the lengths of two sticks,expressed in the samethe student is abl: to tellwhich is longe-,shorter.

b. Given a wat of length (shownby a segment) and anotherlength (shown by another seg-ment), the student is able toexpress (tag) the secondlength in terms of the unitlength.

c. Given a unit of lengtn, andthe tag for a second length,the student is able to drawthe second length. (The stu-dent may be given a grid forthis.)

8-9 Using stretchingmachines on sticks

Introductory -- no performanceexpected.

10-16 Establishing the re-lationship betweeninput, machine, andoutput

a.

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Given an input length, astretching machine, and agrid, the student can drawand label the correspondingoutput length.

(BOOK I, CIAPTER I: STRETCHING MACHINES)

TEXT PAGES TOPIC OBJECTIVE

10-16 continued b. Given a machine, a grid, andan output length, the studentcan draw and label the cor-responding input length.

c. Given an input length and anoutput length, the studentcan tell the correspondingstretching machine.

17-20 Using tabular arrays Given:

147 for input, machine,and output problems a. an input length, a machine,

no grid,

b. a machine, an output length,no grid,

c. an input length, an outputlength,

the student will tell the cor-responding

a. output length

b. input length

c. stretching machine

Note: To meet this objectiveit is appropriate forstudents to transfer thegiven information to agrid in order tc con-struct the answer.

(BOOK I, CHAPTER I: SITEXVING MACHINES)

TEXT PAGES 'PDPIC OBJECTIVE

21-27 Establishirg nota-tional conv,:ntionsfor depicting input,machine, and outputrelationships onhistory cards andoutput tags

Given:

a. paired history cards and out-put tags for specified sticks,the student cal identify cor-rect and incorrect pairings.

b. pairings of incomplete historycards and output tags, thestudent can compute the mis-sing data.

28 Establishing nota-tional conventionsfor input, machine,output relationshipsin equation form

Given an equation in wh.:.e.th one

of the three critical variablesis missing, the student cancompute the missing data.

29 Establishing con- Given worded statements andventions for inter- their corresponding equations,preting worded state- students can make appropriatemeets about inputs, matchings.machines, and out-puts, and equationswhich describe therelationships

(BOOK I, CHAPTER I: STRETCHING MACHINES)

TEXT PAGES TOPIC OBJECTIVE

30 CHAPTER I SUMMARY a. By the completion of ChapterI, the student should beable to solve INPUT- MACHINE-OUTPUT exercises like thosedescribed in the Objectivesfor pp. 28-29, but he neednot be able to solve suchproblems by immediate re-call of a multiplicationfact. He may be permittedto construct solutions usinggrids, card measuring,skip counting, finger count-ing, etc.

b. Students should be able togive the output of a 1-machine, for any whole numberinput.

c. Students should be able torecognize the use of the 1-machine when the input andoutput lengths are the same.

BOOK I, CHAPTER 2: HOOKUPS AND FACTORING

TEXT PAGES TOPIC OBJECTIVE

31-34 Introducing hookupsand appropriateterminology forintermediate outputs

Given an input and a hookup ofstretchers on a grid, the stu-dent can draw the intermediateand final output sticks.

35-37 Using intermediatesticks

a. Given an input length, and ahookup of machines, the stu-dent can supply the inter-mediate output and the finaloutput lengths.

b. Given the input length, theintermediate output, and thefinal output, the studentcan supply the hookup ofmachines to do the job.

c. Given the input and outputlengths, and one machine ofa hookup, the student cansupply the intermediatelength and the other machineof the hookup.

38-41 Establishing nota-tional conventionsfor input, hookup,output relation-ships on historycards and outputtags

Similar objectives as for pp.35-37 except information isgiven in "history card" format.

42-43149-150

Establishing nota-tional conventionsfor input, hookup,output relation-ships in equationform

Similar objectives as for pp.35-37 except information isgiven in equation format.

(BOOK 1, CHAPTER 2: HOOKUPS AND FACTORING)

TEXT PAGES TOPIC OBJECTIVE

44-50 Finding differentways of doing ajob

a. Given nonprime and primemachines, the student cangive, when possible, atleast one hookup (no 1-machines) that will do thesame job.

b. Given a job, the student cangive either a single machineor at least one hookup thatdoes the same job.

51-52 Writing equationsto tell differentways of doing a

jab

Same objectives as for pp. 44-50 except information is givenin equation format.

53-54 Introducing theterminology offactoring anddifferentiatingmultiplying fromfactoring

Given a hookup of machines, thestudent can, by multiplying,find the equivalent singlemachine, and, given a singlemachine, can by factoring findan equivalent hookup.

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BOOK 1, CHAPTER 3: FACTORING, FACTORS, AND EQUATIONS

PAGES TOPIC OBJECTIVE

55-56 Doing jobs at theZabranchburgFactory

Introductory extending thedomain of machines to all non-zero whole numbers.

57-59 Establishing nota-tional conventionsfor factorizationson routing slips

Introductory -- Given a rout-ing slip, the student can in-dicate with a factoring diagraman alternative hookup to usewhen a desired machine isbroken.

60-63 Using factoringdiagras for canand cannot-do jobs

Practice Given a job, thestudent can use a factoringdiagram to indicate, whenpossible, alternative ways ofdoing the job.

64-67 Defining and find-ing factors

a. Given a machine or job,and one of its factors,the student can supply theother factor.

b. Given a machine or job,and a machine which is nota factor, the student recog-nizes that r hookup con-taining that machine willdo the job.

c. Given a number less than15, the student can giveall of its factors.

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(BOOK 1, CHAPTER 3: FACTORING, FACTORS, AND EQUATIONS)

68-74151-152

Introducing "equa-tion related"terminology andsolving equations

Given a hookup equation, thestudent can supply a missingfactor, or if there is nosolution, can recognize thisfact.

75-78 Extending thedomain of inputsand outputs tomagnitudes otherthan length

Given an equation (magnitudesother than stick lengths), thestudent can solve the equation.

79-82 Using equationsto solve problems

Given a problem, the studentcan use the stretcher machinemodel to find the solution,and use bar graphs to organizeand interpret data.

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BOOK 1, CHAPTER 4: PUNCH CARDS, FACTORING, AND PRIMES

TEXT PAGES TOPIC OBJECTIVE

83-84 Introducing and Given hookups of machines,.153-154 stating the re- the student can use the re-

arrangement arrangement principle to tellprinciple if they are equivalent re-

arrangements.

85-92 Using the rearrange-. Given specified jobs, the stu-ment principle to dent can (using punch card)establish notational show various ways of doing theconventions for jobs job.

or hookups on punchcards

93-94 Using punch cardsand the rearrange-ment principle

Given two hookups, one arearrangement of the other,the student recognizes thatthey do the same job.

95-101 Punching cards to a. Given a job, the studentshow factoriza- can give a hookup thattions will do the job and cor-

rectly complete a punchcard to show how the jobcan be done.

b. Given a punched card, thestudent can tell what jobis shown.

102-108 Deciding whether a. Given a hookup, the student155 jobs are even or can tell whether it does

odd odd or even jobs.

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BOOK 1, CHAPTER 4: PUNCH CARDS, FACTORING, AND PRIMES

TEXT PAGES TOPIC OBJECTIVES

102-108- continued

155b. Given a hookup, one of the

machines hidden, the stu-dent can tell whether thehookup does odd or evenjobs or it is impossible totell.

109-114156

115-116157

Getting rid ofmachines thatAnabru doesn'tneed

Defining primes

Exploratory and introductory.

117-120 Using primes inhookups and factor-ing diagrams

a.

b.

c.

Given a job, the studentcan construct a hookup ofprime machines.

Given a hookup, the studentcan tell if it's a hookup ofprimes and find the singlemachine equivalent to thehookup.

Given a machine, the studentcan factor it into primesand report the results in afactoring diagram.

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(BOOK 1, CHAPTER 4: PUNCH CARDS, FACTORING, AND PRIDES)

TEXT PAGES 10PIC OBJECTIVE

121-124 Using Anabru punchcards to depictprime factoriza-tions

Given a machine (a job) ,. the

student can show on the Anabrupunch card the prime factoriza-tion for the given machine and,conversely, given the primefactorization for a given jobshown on the Anabru punch card,the student can give the jobspecified (single machine).

125-126 Writing equationsfor prime factori-zations of jobs

Given a machine or job, thestudent can write an equationshowing the prime factorizationfor the given job.

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BOOK 1, CHAPTER 5: REPEATER MACHINES AND EXPONENTS

TEXT PAGES TOPIC OBJECTIVE

127-129 Introducing repeater Given a hookup of repeatedmachines machines, the student can ex-

press the hookup in repeatermachine form using a baseand exponent.

130-134 Using repeater a. Given a job, the studentmachines can use the appropriate

repeater machine to dothe job.

b. Given a repeater machine,the student can find thejob the machine can per-form.

c. Given a hookup of repeaters,the student can find thejob the hookup can do.

135-138 Introducing and Given a base and an exponent,using vocabulary the student can tell what job[base(s), the repeater will do.exponent(s),and power(s)]

139-142 Hooking up Given a hookup of repeaters160 repeater machines or non-repeaters, the student

can find the job the hookupwill do or a simpler hookupof repeaters to do the samejob.

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(BOOK 1, CHAPTER 5: REPEATER MACHINES AND EXPONENTS)

TEXT PAGES TOPIC OBJECTIVE

143-144 Using exponents in Given a job (compositeprime factoriza- number), the student cantions factor it into primes,

and then express the re-sult in exponent notation.

BOOK 1 Summary By the end of Book 1, it isdesirable that the studentacquire the following skills:

a. Recall whole numbermultiplication factswith factors 1-10.

b. Factor a compositenumber less than 100into primes.

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PRETEST

For the pretest use the Skills Mastery Test contained in theGoals book Dade County Public Schools Bulletin No. 7-H or a testthat is similar.

It is recommended that the students bubble their answers onIBM answer cards and the services of Central Data Processing beutilized to provide complete test analysis.

POSTTEST

Utilize Activity 47 (Test 1) and Activity 76 (Test 2) in theActivities Handbook as the posttest for this Quin.

Suggested Sources of Enrichment and Practice Activities:

A. State adopted

1. Crouch, William H. Coordinated Cross Number Puzzles A, B, C.New York: McCormiCkHMathers Publishing Co., 1970.

2. Denholm, R. A. and Blank, V. D. Mathematics Structure andSkills 1st Book. Chicago: Science Re,sarchAssociates, 1968.

3. Foley, Jack; Jacobs, Wayne and Basten, Elizabeth.Individualizing Mathematics. Menlo Park, California:Addison Wesley Publishing Co., 1970.

Skills and Patterns

Whole NumbersNumbers-Patterns-TheorySetsFractions--Addition and SubtractionFractionsMultiplication and DivisionDecimalsMeanings and Operations

4. Johnson, D. A., et al. Activities in Mathematics: FirstCourse: Number-Patterns. Glenview, Illinois: Scott,Foreman and OD., 1971.

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5. Sobel, Max A., et al. Essentials of Mathematics Series:Book 1. Boston: Ginn and Company, 1970.

6. Tucker and Wheeler. Mathematics Laboratory. New York:McCormick-Mathers Publishing Co., 1970.

7. Wirtz, Robert W., et al. Math Workshop Levels C, D, E.Chicago: Encyclopedia Britannica Educational Corp.,1964.

B. Non-state adopted

1. Brandes, Louis G. Yes, Math Can Be Fun. Portland, Maine:J. Weston Walch, 1960.

2. Dumas, Enoch. Arithmetic Games. Palo Alto, California:Fearon Publishers, Inc., 1960.

3. A Collection of Cross Number Puzzles.

4. Larsen, Harold D. Games to Play.

5.

6.

. Guzintas.

Ways to Multiply

7. . Brain Teasers

Evanston, Illinois: Harper and Row, Publishers, 1961.

8. Meyer, Jerome S. "Arithmetricks." Englewood Cliffs, N. J.:Scholastic Magazine, 1965.

9. Wagner, Guy, et al. Arithmetic Games and Activities.Darien, Connecticut, 1964.

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