DOCUMEMT RESUME ED 051 106 SP 007 250 AUTHOR Henderson, George L.; And Others TITLE Guidelines to Mathematics, 6-8. Key Content Objectives, Student Behavioral Objectives, and Other Topics Related to Grade 6-8 Mathematics. INSTITUTION Wisconsin State Dept. of Public Instruction, Madison. NOTE 44p.: Bulletin No. 186 ECRS PRICE DESCRIPTORS EDRS Price MF-$0.65 HC-$3.29 *Curriculum Guides, *Grade 6, *Grade 7, *Grade 8, *Mathematics Curriculum ABSTRACT GRADES OR AGES: Grades 6-8. SUBJECT MATTER: Mathematics. ORGANIZATION AND PHYSICAL APPEARANCE: The guide is divided into three chapters. The central and longest chapter, which outlines course content, is further subdivided into 17 units, one for each of 17 content objectives.. This chapter is in lint form. The guide is offset printed and staple-bound with a paper cover. OBJECTIVES AND ACTIVITIES: The central chapter lists 17 mathematical concepts such as numeration systems, ratio and proportion, size and shape, measurement, and statistics and probability. A list of related behavioral objectives for each concept at each grade level is then presented. A "summary', list for grades K-5 is also included. No specific activities are mentioned, although one chapter gives general guidelines on developing problem-solving situations. INSTRUCTIONAL MATERIALS: No mention. STUDENT ASSESSMENT: Readers are referred to the guide for grades K-6 (SP 007 249) . (RT)
Transcript
DOCUMEMT RESUME
ED 051 106 SP 007 250
AUTHOR Henderson, George L.; And OthersTITLE Guidelines to Mathematics, 6-8. Key Content
Objectives, Student Behavioral Objectives, and OtherTopics Related to Grade 6-8 Mathematics.
INSTITUTION Wisconsin State Dept. of Public Instruction, Madison.NOTE 44p.: Bulletin No. 186
ABSTRACTGRADES OR AGES: Grades 6-8. SUBJECT MATTER:
Mathematics. ORGANIZATION AND PHYSICAL APPEARANCE: The guide isdivided into three chapters. The central and longest chapter, whichoutlines course content, is further subdivided into 17 units, one foreach of 17 content objectives.. This chapter is in lint form. Theguide is offset printed and staple-bound with a paper cover.OBJECTIVES AND ACTIVITIES: The central chapter lists 17 mathematicalconcepts such as numeration systems, ratio and proportion, size andshape, measurement, and statistics and probability. A list of relatedbehavioral objectives for each concept at each grade level is thenpresented. A "summary', list for grades K-5 is also included. Nospecific activities are mentioned, although one chapter gives generalguidelines on developing problem-solving situations. INSTRUCTIONALMATERIALS: No mention. STUDENT ASSESSMENT: Readers are referred tothe guide for grades K-6 (SP 007 249) . (RT)
Guidelines to Mathematics
6-8
key Conteat Objectives, Student Behavioral
Objectives, and Other Topics Related to
Grade 6-8 Mathematics
Issued by
William C. Kahl, State Superintendent
Wisconsin Department of Public Instruction
Bulletin No 186
700139
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U.S, DEPARTMENT OF HEALTH.EDUCATIONA WELFARE°MCC OF EDUCATION
THIS DOCUMENT HAS BEEN REPRO.
OUCH/ EXACTLY AS RECEIVED FROM
THE PERSON OR ORGANIZATIONMO-
MitiO IT -.DINTS OF Vl.W OR OPIN-
IONS STATED DO NOT NECESS:I,RILYREPRESENT OFFICIAL OFFICE Of SOU
CATION POS,NGN OR POUCY
Writing Committee
George L. Henderson, Supervisor-MathematicsWisconsin 1)epartment of Public InstructionChairman
Peter ChristiansenMathematics Coordinator'Madison
Lowell Olunn.\latheinaties CoordinatorMonona
David R. JohnsonChairman Mathematics DepartmentNieolet High School, Glendale
Walter LeffinEducation Department\Wisconsin State University, Oshkosh
4,004.1r,
Courtney LeonardChairman, 7-12 INIatheniaties Department:Manitowoc Public Schools
Ted K I,osbyMathematics Department.Tanies Madison .1elnoriai Jr. High SeJ!1P Madison
Henry 1). SnyderEducation DepartmentUniversity of Wisconsin, MilwaukeeDe Lloyd SteitzMathematics DepartmentWisconsin State University, Eau ClaireCarl J. Vander lin'Mathematics DepartmentUniversity of \Wisconsin Extension Division, .Madison
Advisory Committee
Walter LeffinWisconsin State University, Oshkosh
Courtney LeonardManitowoc Public Schools
Milton BleckeRipon Public Schools
George L. BulbsWisconsin State University, Platteville
Arnold M. ChandlerWisconsin Department of Piddle Instruction
Peter ChristiansenMadison Public Schools
Robert E. DaltonOshkosh Public Schools
John 0. DanielsonWisconsin State University, Superior
Kenneth \V. DowlingWisconsin Department of Public Instruction
Lowell GhinnMonona Public Schools
Elroy E. (]otter\Wisconsin State University, Eau Claire
John O. HarveyUniversity of Wisconsin, Madison
George I,. Henderson, ChairmanWisconsin Department of Public Instruction
David R. JohnsonMilwaukee Public Schools
Ted E. Losby.Madison Public Schools
Robert A. Iiistan\Wisconsin Departmeht of Public. Instruction
William RocrigKaukauna Vocational and Adult School
William C. SeiserMilwaukee Public Schools
Henry D. SnyderUniversity of Wisconsin-Milwaukee
Be Lloyd SteitzWisconsin State University, Ilan ClaireRoy J. StumpfUniversity of Wisconsin, Green Bay
Carl 3. Vander linUniversity of Wisconsin, Madison
Henry Van Engel'University of Wisconsin, Madison
Edwin F. Wilde13eloit College
Photos courtesy Racine Unified Schools.
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ForewordIn a dynamic, society such as ours, change is constant. People ac-
cept this fact more readily in many fields than in the area of mathe-matics. And yet, III recent years new pl'OjCCIS, new courses, and newmaterials remind us that a great potential exists for improved changesin the mathematics program.
The primary purpose of Guidelines to Mathematics, 6-8 is to helpthose who ha.e responsibility in local school districts for providinga well conceived mathematics program. Each of the sections of thisguide has been designed to meet specific objectives. Although nosection represents the final word on the topic being discussed, it isintended that all sections represent definitive statements which canserve as a sound foundation for further study and investigation.
The main section of this guide has been devoted to a careful de-velopnent of major concepts and associated behavioral objectives. Inaddition, other sections have been devoted to issues concerning theeffective teaching of mathematics.
The Wisconsin State Department of Public Instruction trusts thatthe efforts of all who helped make this 7mblication possible will resultin imprIved mathematical experiences for all Wisconsin youth.
WILLIAM C. K.% W.State Superintendent
I.M.,-ww..4--
Table of Contents
Mathematics Instruction: A Point of View
Key Mathematical Content Obja.:tives and
Related Student Behavioral Objectives 4
Problem Solving 40
Chapters on ...
Adjusting NlatInmatics Programs to Student Abilities
Evaluation and Testing
Inservice Education
Criteria for Program Development
...appear in Guidelines to Mathematics, K-6,Bulletin No. 141.
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Mcthematics instruction: A point of ViewThe variety of mathematics programs now
available for schools promises to benefit bothteachers and students alike. Having beenfreed from the fixed form that characterizedthe traditional program, teachers now have agreater opportunity than ever to select thematerials and teaching techniques best suitedto a student's needs. The basis used forselecting methods and malerials is importantand is dependent upon the viewpoint of theteacher in regard to his instructional task.Just as mathematics programs have changed,so might certain aspects of the traditionalview of instruction be changed.
It has been common practice heretofore toconsider mathematics skills and concepts asbeing "all-or-nothing" attainments. That is,masters' of a certain body of facts or skillshas been traditionally regarded as the properprovince of the teacher at a particular gradclevel. The failure of sonic students to achievethe expected level of proficiency has forcedthe teacher of the next grade to "reteach"these areas, a task that has long been a sourceof irritation to many teachers. This type ofcompartmentalization of a course of studyshould he minimized, if not eliminated. In-stead of expecting boys and girls at a givengrade level to master in a rather final sensea certain ortion of the mathematical train-ing, educators should develop an approachwhich recognizes the existence of individualgrowth rates and stresses the continuity ofthe instructional process.
Teaching for GrowthPerhaps the one factor most essential to
the success of the mathematics curriculum isan emphasis on understanding, that is, under-standing of mathematics. This emphasisrepresents a marked shift in the focus ofeducators' attention from overt behavioralskills and social applicati'llis to understand-ings developed as a person organizes andcodifies mathematical ideas. It is no longerdeemed sufficient for a teacher to be satisfiedwith competent computational perform/ niceby a student in spite of the obvious necessityfor such skill. The need to find a moreefficient, a more enjoyable, and a more illumi-
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nating method of instruction has led to theconsensus that clear and penetrating under-standing of certain essential mithematiesmust precede, but certainly not supplant thetraditional point of emphasis. computation.How is this to be achieved?Experiencing the Physical World
Certain principles of instruction deserverenewed emphasis. It is agreed that oneshould traverse from the known to the un-known; from the particular to the general;from the concrete to the abstract. Clearl,students should be encouraged to developmany concepts of mathematics from theirexperiences with physical objects.
This approach implies, in many eases, thatprior to the introduction of a concept, timeshould he provided for each student to ex-periment with physical objects appropriateto the objective. For instance, every stud(' itshould have the experience of compr,angvolumes of various three dimensional figuressuch as cones, cylinders, spheres and prismsby using sand or liquids as a prelude tovolume prolalcrw; he should work on makingand/or comparing thermometer scales be-fore using the formula to convert Celsiusto Ftihrenheit; he should have many experi-ences with locating cities on maps, tind play-ing various games on coordinate charts, be-fore graphing solution sets of linear equa-tions.
By such purposeful "playing" importantconcepts that have been left largely to chancedevelopment will Ins given appropriate atten-tion and will be established on a firm founda-tion.
Useful Unifying ConceptsBecause understanding is an ambiguous
term and because a teacher must make asclear as possible the relationship betweenformer objectives and present points of emphasis, it may he \I.11 to consider just a fewof the many concepts that are especially im-portant to clarify and develop from theearliest stages.
For the foreseeable future, the study ofnumber systems tvill continle to he the ssena. of the mathematics program. Involved
in this study are the collections of variouskinds of number ideas, such as natural 11 1.1 111-bers, together Kith the operation defined onthese number ideas and the properties ofthese operations (for example, commutative,Associative, distributive properties). Ob-viously, the development of an understandingof such ideas must take place over an ex-tended period of time and at considerablydifferent rates for different students. Thereis no such thing as complete understandingof imy particular number system by a stu-dent. Rather, teachers at each level shouldascertain that the instruction is directedtoward deepening and extending the broadmathematical streams cited in later sectionsof this report. ]low can this be done!
As a student's grasp of mathematicsgrows, lie must be guided toward the acquisi-tion of the lvoad and essential concepts whichtie together seemingly disconnected particu-lars into a coherent general structure. Thesame operation. are encounred in severalpeculiar and different contexts to succeedinggrades. Each particular use really representsa different operation.
For instance, the multiplication of twonatural numbers such as 3 x 5 may be prop-erly regarded as the process of starting withnothing and adding 5 objects to a set :1 times.But can A x Va be similarly. interpreted asthe process of repeatedly adding 1,44 objectsfor t/2 times! Or what meaning can beascribed to (-3) (-5)! Certainly not 5things repeatedl.y added for 3 times! Suchstatements require diffeient interpretationsbecause the operation called .mailipiicalionhas several meanings and definitions de-pending upon the kinds of numbers or setelements to which it is applied.
Natural numbers, integers, and rationalnumbers each combine somewhat differentlyunder a given operation. But rather than in-vent new symbols to represent the changingdefinitions of an operation as it applies tothe various number systems, one uses thesame symbol throughout. In this situation,as in many similar ones, students must hetaught to interpret symbols in context. Onlyin this way will mathematics shorthandserve the important purpose of clarify-
lag and improving communication. Further-more, the student should he led to under-stand that although the specific interpreta-tion of an operation differs from one systemto the next, a structural unity still remainswhich is founded on the commutative, as-sociative, and distributive properties as theypertain to the operations. Because of thisstructural unite, one can rely on the contextin which a symbol is used to determine itsmeaning, rather than create new symbols foreach particular variation of fi concept.
Another matter often treated too lightlyinvolves the freedoms or restrictions whichmay or may not exist for a given operation.When one matches the elements of a set withthe word names of the ordinal numbers, hemust know that the order in which the objectsare matched to too numerals will not affectthe count. On the other band, cases exist inwhich the order of presentation of irforma-tie,' is important. One cannot mix up nu-merator and denominator with impunity orname the coordinates of a point by Nvhim.The basis for free choice, where it exists,and the reasons for restrictions, as they oc-cur, must be made clear if understanding isto be achieved.
To sum up, students should come to realizethat mathematics is a logical system whichexisfs as a human inventimi, formulated, en-riched, extended, and revised in response tothe twin needs to perfect :t as a logical struc-ture and to use it as a convenient method ofdescribing certain aspects of nature as seenby man.
Undue Emphasis on Particulars
Educators should avoid giving undue out-phasis to any single aspect of the total mathe-matics program. The idea of sets, for in-stanco, should not be glorified beyond itsusefulness in contributing to the attainmentof the broad goals of the program. Similarly,it is um% ise to go to the extreme of down-gilding the importance of computationalproficiency to the point where long rangegoals are placed in jeopardy. Furthermore,big words or impressive terms and symbolsmust never interfere with a student 's under-standing of the concepts which the words or
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symbols represent. Clarity of communica-tion is the chief purpose of mathematicallanguage and symbolism. If a particularterm or symbol does not serve the cause ofclarity, then it should not be used. Students,of course, must eventually learn how to usethe language effectively and understrind it incontext. However, teachers should exercisegreat care in working toward such a goaland should avoid any proliferation of sym-bols or premature verbalizations that willhinner goal attainment.Problem Solving
One of the greatest challenges for theteacher is the development of appropriate,real-life problems which meaningfullyvolve learners. Classroom teachers recog-nize the limitations of textbooks in providingsuch verbal problems. Many problems fail tochallenge students; they fail to represent aworld which is real to young people and toooften they fail to contribute to the student'sskill in problem solving -4tuations outside thetextbook. Notwithstamling these limitations,textbook verbal problems will continue to beprominent in most classy ooms until individualteachers identify other ways of meeting theproblem solving uiiligatio».
There is increasing evidence that through(experimental programs and through. theefforts of individual classroom teach( rs im-proved was of meeting this obligation arebeing developed. Much is being heard aboutthe importance of relating mathematics toother areas, of the use of computers in mathe-matics instruction, of units of study inmathematics, of enrichment activities inmathematics, of the study of the history ofmathematics, and of students creating theirow n mathematics problems. Instead ostressing the social development of the stu-dent these efforts emphasize the structure ofmathematics through problem solving experi-ences. These activities are hopeful signspointing to a greater stress being plaeed, in(he near future, on the importance of mean-ingfully-structured problem solving experi-ences as a part of the mathematics program.Individual Differences
If mathematics instruction is viewed as aprocess of initiating understanding and of
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carefully nurturing this understanding as thestudent matures, it will then he necessary todiscover effective techniques for accommodat-ing the widely differing rates at which youngpeople. develop. implicit in this statement isthe need for schools and teachers to recog-nize that sonic students may not be able togrow substantially or may seem to terminatetheir potential for growth at some pointalong the way. In such cases, provisionshould be made for experiences most appro-priate to the individuals welfare.
To date, no satisfactory method has beendeveloped to cope with the vast range of in-dividual differences. Flexible grouping pro-cedures, nongraded classrooms, individual-ized instruction, team-teaching, computer-assisted instruction, and television teaching,are receiving extensive testing and evalua-tion. Perhaps the experiments currentlyunderway will yield effective techniques oncemore is learned about the problem. In toemeantime, each teacher must exercise pro-fessional judgment and common s'ense inadopting an optimal arrangement that is com-patible with his own abilities, with the charac-teristics of students hi his class, and withthe physical facilities and administrativepolicies of the school.
In summary, mathematics instructionshould he viewed as a continuous effort todevelop in the individual a knowledge ofmathematics that is characterized by itsdepth and connectedness. To the extent pos-sible, the student should be encouraged toexperiment with the objects of his en-vironment. Thus prepared, he may be ledto the invention or 'discovery of those ideaswhich provide both a broad basis for furtherexploration and a sense of delight in a well-founded mastery of the subject. The taskis a challenging one and deserves mucheffort. To this .end, the teacher should doeverything possible to see that instruction iswell-planned and is provided on a regularbasis. As an additional, but essential meas-ure, cooperative action such as inserviceeducation should be taken by school personnelto develop in the teaching staff a view ofinstruction appropriate to present day needsand opportunities.
Key Mathematical Content Topics
and
Related Student Behavioral Objectives
In the following outline, an attempt haslwen made to point out where key contenttopics might be introduced and developedin the school mathematics program, includ-ing a "summary" for grades K -5, andspecific student behavioral objectives forgrades 6, 7 and 8. The behavioral objectivessuggest a sequence of development of mathe-matical ideas which provides for their rein-forcement and continuity from t hoe of intro-duction through grade 8. By means of sucha sequence of development, key ideas can 1wextended to conform 'rith the maturity andbackground experiences of students.
The content topics have been organizedunder seventeen main topics: Sets and Num-bers; Numeration Systems; Order; NumberSystems; Ratio and Proportion; Computa-tion; Size and Shape; Sets of Poiats; Sym-metry; Congruence; Similarity; CoordinateSystems and Graphs; Constructions; Meas.uremcat ; Mathematical Sentences; OrderedPairs, Relations and Functions; and Statis-tics and Probability. The first fifteen topicsare indentical to those in Guidelines to Mathe-matics, published in 1967 by the Wis-consin Department of Public Instruction.
1I is not intended that the placement oftopics in this outline be considered as theonly correct arrangement or that all of thetopics necessarily be tauglIt at every gradelevel as presented here. The ideas listed foreach grade level should be regarded as asuggested guide for introducing varioustopics; the outline is not intended to be all-inclusive. Teachers will find it necessary toalter the order of topics to meet the needsof students or the needs of particular groupsof students.
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Furthermore, no fixed mnoant of time oremphasis has been suggested for any ohjee-tive in this outline. A disproportionateamount of space has been devoted to sometopics for purposes of clarity, and the amountof space devoted should not be considered anindication of the relative importance of atopic.
This outline extends for two more gradesthe outline originally presented in Guide-fines to Mathematics, K-c. Grade 6 objec-tives are veyem'ed in this outline withoutmodift-at ion.
School administrators ,,nd supervisors,mathematics curr;euluin committees, teach-ers, and university instructors should findthis outline useful for one or more of thefollowing purposes:
As an orientation to the key topics andbehavioral objectives of the schoolmathematics curriculum, grades 6-8.
As a source of knowledge of the relatedobjectives of introductory secondarymathenli,tics.
As a "yardstick" to compare againstpresent mathematics programs.
As a guide in determining the conceptsthat need to be highlighted in pre-service and in-service education pro-grams for teachers.As an aid to program evaluation.
Throughout this outline, an asterisk(*)appearing under any behavioral objective,or in lieu of objectives, indicates that theunderstanding and skills listed for previousgrades are to be expanded and reviewed.
Mathematical Topics
Sets and NumbersAny clearly defined collection of distin-
guishable objects is called a set. The obj-2ctsin this set are called members or elements. Aset may be identified by a description (pos-sibly containing a rule), a listing of the set'smembers, a Venn diagram, or a graph. Basicsot operations include union, intersection andcomplementation.
In grades 7 and 8 the emphasis is on theEtu,1,y of rational numbers, negative as rllas non-negative. A brief introduction is alsoincluded to the irrational numbers whichleads the students to the set of real numbers.A more complete development of the set of
irrational numbers and the set of real num-bers is instroduced in grades 9 through 1'?.
RemarksSet concepts eau assist in the understand-
ing of numerical and geometric ideas. Theuse of the Venn diagram might be employedto clarify definitions of such terms as primeand composite numbers, odd or even num-bers, universal sets, subsets, overlappiligsets, and disjoint sets. Special tttentionshould be given in the 7th and 8th grade,: tothe clarification of finite, infinite, equ,LI andequivalent sets.
Students should be able to: c)
Numeration SystemsThe understanding of base and place value
of a numeration system is extended in 7thand 8th grades. Expanded notation, scientificnotation using positive, zero and negativeintegral exponents, and the relationship be-tween fractional and decimal representationsof rational numbers are emphasized.
RemarksComparison of ha se ten with oilier nutncra-
fion and modern, strengthensunderstanding of base and place, but com-putational facility should be encouragedonly in base ten.
Students should be able to:
OrderThe concept of order is extended front
generalizations about cardinal numbers ofsets to order in the set of real numbers basedon the following two principles.
(1) If a and b are mny two real numbers, thcaone and only one of the following is true;
b, a < b, or a > b.(2) (liven three distinct real numbers, one is
between the other two.
In grades 7 and 8, emphasis is placed onthe order of rational numbers.
RemarksPractice in ordering numbers should in-
clude inspection of digits in decimals,(3.21089 < 3.2 1098), application of skillswith fractions anurdecimals, (3/8 < 5/9 and.375 < .5), and comparison of relative loca-tions on the number line.
Students should be able to:
Behavioral ObjectivesGrades K-5
Sets and Numbers
K Identify two equivalent sets by placingthe members of the set in one-to-one cor-respondence.
K Use such terms as more than, as many as,fewer than when comparing sets of ob-jects.
1 Count the members of a set containing onehundred or fewer members.
1 Use "0" as the symbol for the numberof elements in the empty set.
2 Identify 14, lia, 1/2 of a whole by usingphysical objects.
3 Determine the cardinality of a set to10,000 through appropriate experiences.
3 Use ordinal numbers beyond tenth.4 Determine the factors of a counting num-
ber (a whole number other than zero).5 Find the prime factou numbers
through 100.
5 Construct sets of equivalent fractiousthrough working with set...3 of ,bjects. Auexample of sue:t a set is 2/3, 4/6, 6/9,8/12 . .
Numeration Systems
K Identify the numerals 0 through 9.
1 Give different numerals for a given num-ber such as 6 + 2, 10 2, and 8 for eight.
1 Interpret the place-value concept forwriting whole numbers to one hundred;such as, 89 is the same as 8 tens, 9 ones.
2 Write three-digit numerals in expandednotation; for example, 765 700 + 60 5.
3 Recognize that numerals such as 57 canbe expressed as 40 + 17.
4 Interpret place value for large numbers.5 Write many nawea for the same rational
number.
5 Work with bases, such as 3, 4, 5, 6, and 7,to domonstrate an unders.anding of thebase of a numeration system.
Order*
K Determine whether two sets are equiva-lent (can be matched or placed hi a one-to-one correspondence).
1 Determine that 8 is greater than 5 andthat 5 is less than 8 by compering ap-propriate sets of objects and do this forany two numbers less than 10.
2 Use symbols >, <, and irr mathematicalsentences.
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3 Determine betweennesq, greater than, orless than for numbers through 999.
3 Recognize greater than or less than forthe fractions 1/2 with physical ob-jects.
5 Determine greater than, less than, andbetweenness for rational numbers.
Behavioral ObjectivesSixth Grade
Sets and Numbers
1. Use negative numbers in many differentsituations.
Numeration Systems
1. Represent rational numbers by decimalsand fractions.
2. Express large numbers by using s,2i(ntificnotation, such as the distance from earthto the sun as 9.3 x 10' miles.
3. Use exponential notation in representingnumbers; for example, 2345 = 2 X 10' +3 x 102+ 4 X 10 + 5.
4. Demonstrate an understanding of the re-lationship between decimals and commonfractions.
Order*
1. Determine greater than, less than, andbetweenness for (positive, negative, andzero) integers.
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Behavioral ObjectivesSeventh Grade
Sets and Numbers
1. Given n universal set describe or list themembers of a given set. For example, inthe universal set of counting numbersthe set of multiples of 3 is -13, 6, 9, 12
2. Determine whether an element is a mem-ber of a given set.
3. Determine whether a set, including theempty set, is a subset of a given set.
4. Determine whether two sets are equiva-lent.
5. Determine whether two sets are equal.6. Identify the elements in the intersection
of two sets.7. Identify the elements in the union of two
sets.8. Given a subset of a universal set identify
the members of its complement.
9. Express a counting number in primefactor form. For example, 24 = 2 x 2 x2 x 3 (2') (3).
10. Interpret a fraction as a part of a whole,as expressing a ratio, as part of a group,or as an indicated division.
11. Given a set of numerals, classify them asrepresenting whole numbers, and/or in-tegers, and/or rational numbers.Examples: 8/4 represents a whole num-
ber end an integer aid a rational num-ber .333 . . . . represents a rationalnumber; 4 represents an integer anda rational number.
Numeration Systems1.
2.
Use positive integral exponents to ex-press the power of a positive rationalnumber.Use expended notation in representingwhole numbers.(Examples: 314 = 3 X 7' + 1 X 7+ 4 X 1, and 314 ten = 3 X 10=+ 1 X 10+ 4 x 1.)
3. Use scientific notation to express numbersgreater than ten.
4. Recognize that rational numbers can be,expresse0 in the form where p and q
are integers and q -/' 0.
5. Write any positive rational number indecimal notation.
Order*
1. Recognize that one and only one of thefollowing statements is true when a andb are any two rational numbers:
< b, a b, a > b.
2. Determine, given two rational iminbers,NVICAll is greater.
:3. Determine, given three rational »umbers,which one is between the other two.
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Behavioral ObjectivesEighth Grade
Sets and Numbers
1. Show that a rational number can be ex-pressed as a repeating decimal, and thata repeating decimal can he expressed infractional form.
3. Determine the number of subsets in a setcontaining N elements. For example, aset containing 5 elements has 32 or 2'subsets.
Numeration Systems
1. Use positive, zero, and negative exponentscorrectly.
2. Use scientific notation to express positiverational minibus; for saniple, the lengthof a microwave is 2.2 x 10.2 meters.
3. Use the symbol to indicate one of twoequal factors whose product is a.
4. Use expanded notation to e.;press positiverational numbers.
(Example: 21.37 2 X 10' 1 X 10' +3 x 10-1 + 7 x 10.2)
Order*
1. Determine, given a rational nanther andall irrational number, which is greater.
For example: which is greater 2 or N/31
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Mathematical TopicsNumber Systems
A number system consists of a set of num-bers, t vo oue?-tions defined on the numbers,and properties of the operations.
The whole number system and the systemof positive rational numbers are studied inthe elementary grades and the system of in-tegers is introduced. The rational numbersystem is studied in more detail in grades7 and 8. Subsequent understanding of thereal number system depends on careful andcomplete development of the systems of in-tegers and rational numbers.
The important and useful properties of theoperations on the rational numbers include:
Addition MultiplicationClosure ClosureCOMMUtativity CommutativityAssociativity AssociativityIdentity Element Identity ElementInverse Elements Inverse Elements
DistributivityRemarks
Emphasis should be placed on how theproperties of number systems facilitate com-putation. Finite systems can be used asmodels to help students understand the con-cept of number system.
Students should be able to: ()
Ratio and Proportion
A ratio is a pair of numbers used to com-pare quantities or to express a rate.
Symbols commonly used for a ratio are(a,b), a tb, and a/b. Note that the symbol(a,b) is also used to denote the coordinatesof a pant in a plane and that the symbola/b is generally used to name a rational
number, when a and b are integers and b isnot zero.
A proportion is a statement that two ratiosare equivalent (that two pairs of numbersexpress the same rate). A proportion is writ-ten in the form a/b = c/d, when a x dc X b.
Students should be able to: [:).
Computation
Computation can be described as the ap-plication of systematic (algorithmic) pro-cedures to the process of renaming numbers.
Remarks
Understanding find appreciation of manymathematical concepts are facilitated by
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computational proficiency. It is expectedthat in grades 7 and 8 the computational skilldeveloped in earlier grades will be main-tained and extenied.
Students should be able to:
Behavioral ObjectivesGrades K-5
Number Systems
K Rearrange sets of objects to demonstratethe joining and separating of sets, andthereby develop a readiness for additionand subtraction.
1 Recognize examples of the commutativeproperty for addition in the set of wholenumbers. Demonstrate with sets of ob-jects the relationship between such sen-tences as 4 + 2 = 6, 6 4, and6 4 = 2.
2 Use the associative property of additionin the set of whole numbers; for exampl,,,(3 + 4) + 5 3 + (4 + 5) .
2 Recognize zero as the identity eleli (21t .for addition in the set of whole inm.Hersand its special role in subtract Hi.
3 Recognize the role of 1 as the identityelement for multiplication in the set ofwhole numbers. ..
3 lieu ;size the distributive property ofmultiplication over addition in tha set ofwhole numbers.
4 Recognize the inverse relation betweenaddition sentences and two subtractionsentences, such as 725 -I- 342 = 1067 acid1067 725 = 342 and 1067 342 = 725.
4 Use parentheses to show order of opera-tion; for example, 2 4- 4 x 3= 2 -i-
(4 X 3) = 11 and (2 4- 4) X 3 = 6 X 318.
5 Recognize that subtraction is not alwayspossible in the set of positive rationalnumbers and in the set of whole numbers.
Ratio and Proportion3 Interpret simple ratio situations, such as
2 apples for 15?, written 2 (apples)15 cents).
4 Determine if two ratios arc equivalent byusing the property of proportions com-monly called cross multiplication. Forexample, 3/4 9/12 because 3 X 124 X 9, whereas 6/7 0 7/8 because 6 X 8 07 X 7.
4 Find the missing whole number in tv oequivalent ratios like 2/3 = /9 or 5/
25/70.5 Use the ideas of ratio and equivalent ratio
with problems that include fractions asterms. For example, find the missingnumber in 2/3 = /20.
55 Use members of sets of equivalent ratios
with the same first term or the samesecond term to compare different ratios.
Computation
1 Use the addition facts through the sumof 10 and the corresponding subtractionfacts.
2 Use the multiplication facts through theproduct 18.
3 Use the vertical algorithm in additionand subtraction with two- and three- placenumerals when regrouping may be neces-sary.
iil
12
3 Estimate the sum of two numbers. Forexample, 287 + 520 is approximately 300Jr 500 or 800.
4 Do column addition with several four-place or five-place addends.
4 Use the subtractive division algorithmwith two-place divisors ending in 1, 2, 3, 4.
5 Add and subtract rational numbers.5 Express the quotient of integers as a
mixed numeral; for example, 24 ÷ 54 4/5.
Behavioral ObjectivesSixth Grade
Number Systems
1. Recognize that 1/1 or 1 is an identityelement for multiplication in the set ofrational numbers.
2. Recognize the multiplicative inverse(reciprocal) for every positive rationalnumber except zero and use it in thedivision of rational numbers. For ex-ample, 1/2 ---:--- 3/4 = 1/2 x 4/3.
3. Recognize that the operation of divisionis the inverse of multiplication in theset of positive rational numbers. Forexample, the sentences 34 X ')7.Y = 1A,1/2 :Y1 %, and IA havethis relationship.
4. Recognize that the;e is no smallest orlargest rational number between twopositive integers.
5. Recognize that the integers (positive andnegative whole numbers and zero) arean extension of the whole numbers.
6. Find the adaptive inverse (opposite) foreach integer by using the number line.
7. Recognize that the rational numbers(positive and negative whole numbers,positive and negative fractions, and zero)ar' an extension of the integers.
8. Recognize that finding an integral powerof a number involves repeated multipli-cation of the same number. For example,(2/' X X 2A3.
9. Use the commutative and associativeproperties of multiplication for rationalnumbers.
10. Use the distributive property of multi-plication with respect to addition ofrational numbers.
11. Use the commutative property of addi-tion for integers.
12. Recognize that the rational number sys-tem is dense; that is, between each twodifferent rational numbers, there is arational number.
Ratio and Proportion1. Interpret percent as a ratio in which the
second number is always 100.2. Solve all three cases of percentage prob-
lems as problems in which they find themissing term of two equivalent ratios.For example, 20% of 30 and: 20/100 --D/30 ; 30 is what percent of 55 and:0/100 = 30/55; 25 is 40% of what mini-her and: 40/100 = 25/9.
3. Use equivalent ratios to convert frac-tions to decimals and conversely; for ex-ample, to write 3/5 as hundredths, solvefor II in 3/5 = n/100; to write 44 hun-dredths as 25ths, solve for n in n/25 --44/100.
4. Solve ratio problems where some or all ofthe terms of the ratios are written asdecimals.
5. Use proportions in problems about thelengths of sides of similar triangles.
Computation
1. Multiply and divide non-negative rationalnumbers.
2. Use the conventional division algorithm.3. Add integers.
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13
Behavioral ObjectivesSeventh Grade
Number Systems
1. Recognize and apply the addition andmultiplication properties of the systemof positive rational numbers. (See descrip-tion of Number Systems.)
2. Recognize and apply the addition prop-erties of the system of integers.
3. Recognize and apply the distributiveproperty in the system of positive rationalnumbers.
Ratio and Proportion*
1. Recall that a ratio can be used to com-pare quantities as well as to express arate.
Computation1. Add and subtract integers by using the
number line.2. Express positive rational numbers in
decimal form.3. Add, subtract, multiply and divide non-
negative rational numbers in decimalform, recognizing that division by zero isundefined (impossible).
4. Average (compute the mean) sets of non-negative rational numbers.
6. Solve all types of percentage problems asproblems in which they find the missing
ton] in a proportion. (Including problem;involving percents less than one andgreater than onc. hundred.)
7. Given an; two rational numbers, find arational nutnbei that is between them.
8. Apply divisibility tests for 2, 3, 4, 5, 6,8, and 9. Example: 135 is divisible by :3because the sum of its digits is divisibleby 3, by 5 because the last digit is 5, andby 9 because the sum of its digits is di-visible by 9.
9. Using prime faAorization find the leastcommon multiple (LCNI) and greatestcommon divisor (0CD) of two whole num-
*bers and use tly,in in computation withnon-negative rational numbers.
14
17
Behavioral ObjectivesEighth Grade
Number Systems
1. Recognize and apply the addition andmultiplication properties of the rationalnumber system.
2. Recognize and apply the multiplicationproperties of the system of integers.
3. Define subtraction in terms of addition inthe systems of integers and rational mun-hers. (Example: a b means a + (b).)
Ratio and Proportion*
1. Use proportions to solve problems involv-ing similar triangles.
Computation
1. Add, subtract, multiply and divide in-tegers and rational numbers.Approximate square roots of positiveintegers.
3. K.press repeating decimals in fractionalform.
4. Compute the mean of a set of rationalnumbers.
2.
1815
5. Compute products and quotients of num-bers expressed in exponential notation(including scientific notation).Examples:
3'. 3' = 312; a' a' = a';(3 x 102) (2.3 x 10') = 6.9 X 10'.
6. Perform a series of operations in properolder when grouping symbols are omitted,i.e., multiplications and divisions are per-formed first, in left-to-right order, thenadditions and substraet ions are performedin left-to-right order.
Mathematical Topics
Size and ShapeThe classification of and disting.,ishing
characteristics of two- and three-dimensionalgeometric figures are determined by theirsize and shape. The work of earlier grades isextended to include a more careful and de-tailed examination of these figures.
Remarks
Sufficient time should be allowed for stu-dents to examine models, determine their ownclassification schemes, and to make and testconjectures about properties of plane andsolid figures.
Students should be able to: C>
Sets of PointsA point is represented by a location in
space; lines, planes pnd space are sets ofpoints with certain properties.
Remarks
These concepts should be developed in-tuitively using the real world as a model,emphasizing the subset relationships in-volved.
Students should be able to: C>
SymmetryMany geometric figures have a kind of
balance called symmetry. If a figure can befolded so that corresponding parts coincide,it is said to have a line of symmetry; in muchthe same way, a figure can be said to have aplane of symmetry.
RemarksIdeas developed in earlier grades should
be expanded to include the symmetry ofpositive and negative numbers illustrated bythe number line, the symmetry with respectto diagonals in addition and multiplicationtables, and symmetries with respect to linesin the coordinate plane.
Students should be able to:
Congruence
Intuitively, geometric figures are con-gruent if they "fit" each other exactlythat is, if they have the same size and shape.More precisely, two sets of points are con-gruent if there is a one-to-one correspondencebetween the two which preserve distancethat is, if two points are one inch apart, thecorresponding points of a congruent set areone 1,,ch apart.
RemarksStudents should see the necessity for the
more irreeise definition because of congruentsets of points (two- and three-dimeasional)which cannot be made to "fit."
Students should be able to: [:),
Behavioral ObjectivesGrades K-5
Size and Shape
K Recognize squares, rectangles, circles, andtriangles.
1 Observe distinguishing features of spher-es, rectangular prisms (boxes), cylinders,and other objects.
4 Recognize isosceles and equilateral tri-angles and parallelograms.
5 Recognize common polyhedra, such as atetrahedron, a cube, a rectangular prism.
5 Identify faces, edges, vertices, and diag-onals of common polyhedra.
Sets of Points
1 Recognize that squares, rectangles, tri-angles, and circles are closed curves andtell whether a point is inside, outside, oron such a curve.
2 Recognize a straight line as a set of pointswith no beginning and no end.
2 Recognize a simple curve (in a plane) asone that does not cross itself.
3 Recognize rays and angles.
3 Recognize that there is only one linethrough two points and that two lines canintersect at only one point.
4 Describe lines as intersections of planes.4 Interpret a circle as the set of all points
in a plane that are at the same distancefrom a fixed point.
5 Recognize parallel planes.5 Recognize perpendicular lines.5 Recognize that a plane is determined by
three points not all on one line.
Symmetry
3 Recognize symmetry with !expect to alino by folding paper containing sym-metrical figures.
4 Recognize that some figures have two orinure axes of symmetry through paperfolding.
5 Recognize symmetry with respect to apoint by folding a paper along a linethrough the center of such geometricfigures as a circle and a square.
Congruence
3 Recognize congrue.it angles.5 Recognize that triangles are congruent if
corresponding sides are congruent andcorresponding angles are congruent.
18
Behavioral ObjectivesSixth Grade
Size and Shape*
Sets of Points
1. Recognize the properties of isosceles tri-angles, equilateral triangles, and scalenetriangles, such as the fact that the longestside of a triangle is opposite the angle ofgreatest, measure.
0, Recognize that a line (on-dimensionalspace) is a subset of a plane (two-dimen-sional space) and that both ae. subsets ofspace (three-dimensional spa2e).
3. Recognize the relationship between thecircumference and the diameter of a circle.
Symmetry
1. Recognize the reflection of a plane figurein a mirror and draw diagrams such asthe figure at right.
Describe, in terms of set, subset, unionand/or intersection of sets of points, thefollowing:
Skew linesTangent, secantCentral angleArcDiagonal
2. Recognize the various plane figures formedwhen a plane intersects a given figuresuch as a square pyramid or a cube.
3. Use relationships such as those involvedin finding the solution of the following:
1 is parallel to12
Find the numberof degrees in <N,<y, <z and thestun of the meas-ues of <v and<w.
12
(Relationships involved: vertical angles, sup-plementary angles, alternate interior angles,exterior angle to remote interior angles of atriangle]
Symmetry
1. Identify symmetry with respect to aplane.
Congruence
1. '-?ccognize that two triangles are con-gruent if two sides and the included angleof one are congruent to the correspondingparts of the second (SAS).
2:3
21
2. Recognize that two triangles are co.)-gruent if two angles and the included sideof one are congruent to the correspondingpads of the second (ASA).
3. Recognize that two triangles arc con-gruent if the three sides of one arc con-gruent to the corresponding three sides ofthe second (SSS).
Mathematical Topics
SimilarityTwo geometric figures thot have the same
shape, though not necessarily the same size,are said to be similar.
Students should be able to: ri)
Co'rdinate Systems and Graphs
A line is called a wanber line if a one-to-one correspondence exists between a givenset of numbers and a subset of points onthe line and if, by means of this correspond-ence, the points are kept in the same orderas their coresponding numbers. The. nuni-
her corresponding to a point is the coordinateof that point. The idea of assigning numbersto points can he extended to points in a plane,that is, a one-to-one correspondence betweenordered pairs of numbers and points in aplane.
Students should be able to: ri)
Constructions
Remarks
In grades K-6 the emphasis was on usinginstruments to do "constructions." In grades7 and 8 the emphasis should be on usingidentified properties and relationships of
23
figures to do "constructions." For example,constructing a triangle given two sides andthe included angle utilizes the relationshipOf cw,gruence to insure the uniqueness of the"construction."
Students should be able to:
Behavioral ObjectivesGrades K-5
Similarity3 Recognize that figures are similar if they
have the same shape. For example, allsquares are similar.
4 Recognize that all congruent figures aresimilar, but not all similar figures arecongruent.
5 Recognize the similarity of maps madewith different scales.
Coordinate Systems and Graphs
1 Use the number line to illustrate additionand subtraction problems.
3 Recognize that a point On a line can bedescribed by a number (coordinate).
3 Use the number line to illustrate multipli-cation problems.
4 Recognize that points in a plane (thefirst quadrant) can he represented by(ordered) pairs of numbers (coordinates).
5 Construct simple picture, bar, and linegraphs.
5 Use the number line to represent negativeintegers.
Constructions
4 Demonstrate through paper folding anund?rstanding of a line as an intersectionof two planes.
4 Bisect a line segment by using a compassand straight edge.
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5 Demonstrate :in understanding of variouspolyhedra by making appropriate papermodels.
5 Bisect au angle. (Students may discoverseveral different MIA Met 1011S.)
5 Reconstruct an angle and a triangle byusing a compass and a straight edge.
Behavioral ObjectivesSixth Grade
Similarity*
Coordinate Systems and Graphs*
Constructions
I. Construct a line perpendicular to a givenline.
2. Construct parallel lines.3. Make models of various prising and find
their surface areas.
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25
Behavioral ObjectivesSeventh Grade
Similarity
1. Identify corresponding parts of similarplane figures.
2. Recognize that corresponding angles ofsimilar plane figures are congruent.
3. Recall that measures of correspondingsides of similar plane figures are pro-portional.
Coordinate Systems and Graphs*
(See Ordered Pairs, Relations and Functionsand Mathematical Sentences)
Constructions
1. Construct a rhombus and a square given 3. Construct the perpendicular bisector of aone side. line segment.Construct an equilateral triangle given 4. Make models of pyramids.one side.
26
Behavioral ObjectivcsEighth Grade
Similarity
1. Recall that the measures of radii, di-ameters, corresponding altitudes, and cor-responding diagonals of similar planefigures are proportional.
2. Recognize that triangles are similar ifcorresponding angles are congruent.
3. Recognize that triangles are similar ifthe measures of corresponding sides areproportional.
4. Recall that the ratio of the measures oftwo sides of a triangle is the same as theratio of the measures of the correspond-ing two 6.1,.s of a similar triangle.
5. Recall that sine, cosine and tangent ratiosare independent of the measures of thesides of the triangles involved.
Coordinate Systems and Graphs*
(See Ordered Pairs, Relations and Functionsand Mathematical Sentences)
Constructions
1, Given three sides, or two sides and theincluded angle, or two angles and the in-cluded side, construct tho triangle.
2,8.
97
2. Determine whether you can constructnone, one, two, or many triangles giventhree angles or given two sides and anangle not included betveen them.
3. Construct representations of \f-1,V5, etc.
Mathematical Topics
Measurement
The process of measuring associates anumber with a property of an object. Meas-uring an object is done either directly or in-directly. In direct measurement the numberassigned to nu object is determined by itsdirect comparison to a selected unit of meas-ure of the same nature as the object beingmeasured (a unit segment to measure seg-ments; a un't angle to measure angles; aunit closed region to measure closed regions;;nal a unit solid to measure solids). When ameasuring instrument cannot be applieddirectly to the object to he measured, indirectmeasurement is employed. In either case,measure of physical objects is approximate.
In grades 7 and 8, basic mensuration for-mulas for calculating areas and volumes :.redeveloped and applied to increasingly com-plex figures.
Remarks
The accuracy of the measure obtained isrestricted by the unevenness of the objectmeasured, by the limitations of the meast r-ing instrument us?d, mid by human
Stndents should be encouraged to de-vise their own units for measuring varioasobjects and then led to appreciate standardunits of measnremmt. Indirect measureme ntof things such as popularity, I.Q., humor,relative limnidity, etc., can be discussed.
StudEnts should be able to:
Mathematical Sentences
Mathematical sent ences and ordinarylinguistic sentences have the following com-mon characteristics:
1. Both types of sentences use symbolsto communicate ideas.
2. The const met ion of both types ofsentences follows a predetermined setof rifles.
3. Both types of sentences may expresstrue statements or fake statementsdepending upon the symbols used andthe contexts in which they are used
It is quite common in mathematics, how-ever, to use sentences that do not possesscharacteristic 3. Such sentences are called
0929
open sentences. For example, the sentence-+ 3 7 expresses neither n true state-
ment nor a false tatement until tt meaning-ful replacement for has been supplied.Tiie set of all replacements for wine Iproduce a trite statement is called the sohtion .set for the given sentence. Considersentence 12 > 5. The set of all allow-able replacements for is sometimes calledthe universe. Thus, if the universe is thset of positive integers, the solution set is11,2,3,4,15,0; but, if the universe is the setof rational number.4, the solution set con-sists of ;11 ration; 1 numbers less than 7, abinfinite s2t.
StudEnts should be able to: 0
Behavioral ObjectivesGrades K-5
Measurement
K Use appropriately such words as longer,shorter, heavier, lighter, higher, lower,larger, smaller.
1 Recognize the comparative value of coins(pennies, nickels, dimes) and use them inmaking change.
1 Identify various instruments of measure-ment of time, temperature, weight, andlength, such as eloeks, thermometers,scales, rulers.
2 Make a ruler with divisions showirg halfunits.
2 Tell time to the nearest quarter hour.3 Find the perimeter :A a rectangle or
parallelogram.4 Find areas of simple regions informally.
For example, a rectangular region withdimensions 2" by 3" elm be covered bysix one-inch squares (regions).
5 Measure an angle by using a protractor.5 Estimate distances to the nearest unit.5 Recognize that all measurement involves
approximat ion.
Mathematicul Sentences
1 Find solutions for sentences like + A7 in which many correct solutions are
possible.2 Use equivalent sentences like 3 7
and 7 3 to show subtraction as theinverse of addition.
3 Place the correct symbol (<, >, ) inthe place holder in sentences such as:3 x 5 0 7 + 8, 25 + 42 0 87 28, and65 :39 0 5 X 7.
BO
30
3 Find solutions for sentences like. + 2:392:39 + and 1987 + j 0 + 548) (1987
+ + 548 to generalize the idea of theeonnnutative and associative propertiesfor addition.
4 Use sentences like [I] X 5 45 and 455 El to show division as the inverse ofmultiplication.
4 Recognize that 3 X 7 has no wholenumber solution. Find solutions for math-ematical sentences involving more thanone operation such as (2 X 5) + 4 C]and (3 X 2) + 10.
5 Write sentences using fractions to repre-sent physical situations.
Behavioral ObjectivesSixth Grade
Measurement
1. Find the volume of a rectangular prism.Estimate and compare perimeters of poly-gons, such as rectangles, triangles, andparallelograms.
3. Estimate the area of an irregular planeregion by use of a grid where au approxi-mation to the area is the avenge of theinner and outer areas.
4. Use formulas for the areas of rectangles,parallelograms, and triangles.
5. Use the formula for the circumference ofa circle.
6. Use the metric system of measure forlength.
7. Use formulas of volume for commonsolids.
8. Work with approximate numbers. For ex-ample, know that the area m! a squarewhose sides measure 6.5 and 3.6 inches tothe nearest tenth of an inch has an areabetween 6.4 x 3.5 and 6.6 X 3.7 squareinches.
9. Solve problems involving the measure-ment of inaccessible heights and distancesindirectly by using the properties of simi-lar triangles.
Mathematical Sentences
1. Use all of the previously introducedsentence forms with decimal numerals.
2. Write sentences using decimal numeralsto represent physical situations.
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3. Use previously described sentence formsto generalize the commutative property ofmultiplication, the associative property ofmultiplication, and the distributive prop-erty of multiplication over addition forrational numbers in any form.
Behavioral ObjectivesSeventh Grade
Measurement
1. Use various measuring instruments(such as rulers having English and Met-ric scales, protraeters, calipers) andrecord results to the nearer smallest unitof the scales. Example: Use rulers cab-
bitted in tenths of aim inch, in sixteenthsof an inch, in centimeters and milli-meters, etc.
2. Determine the perimeters and areas ofregular polygons.
3. Use the fornmula for the area of a circle.4. Determine the areas of figures such as
shown below:
5. Determine the surface area and volumeof prisms and pyramids.
6. Recall that the sum of the measures ofthe angles of a triangle is one hundredeighty degrees.
7. Determine the sum of the measures ofthe angles of a quadrilateral.
8. Determine greatest possible error for
any calibration of a given measuring in-strument.
9. Recognize the relationship between pre-cision and greatest possible error.
10. Estimate linear measurement.; withoutusing measuring instruments. Example:Time length of a building, the width of astreet, the height of a flagpole.
Mathematical Sentences
1. Establish the truth value of simpre math-ematical sentences. (Examples: 3 H- 4 ----8 is false; 35 X 24 < 30 X 20 is false;13 7 5 is true.)
2. Find the solution sets of simple opensentences using as a universe the wholenumbers. (Example: 3N + 4 16.)
3. Graph using the number line solutionsets for equalities and inequalities whorethe replacement set is the set of integers,such as X >-2; X + 3<12; X 3; Xand X <5; X >5 and X <4; X >-2 orX <:i; X <5 or X <3; X >4 or X
ta232
Behavioral ObjectivesEighth Grade
Measurement
1. Use formulas h) calculate volumes andsurface areas of spheres, cylinders andcones.Determine the relative error given ameasuring instrument and an object tomeasure.Determine the sum of the measures of theangles of regular ptlygons.
4. Weal': the relationship between degrees,minutes and seconds as units of angularmeasurement.
5. Recall the relationships between units ofthe Metric System.
6. Do computation with measure:, (approxi-mate numbers) and determine the greatestpossible error and the relative error ofeach result. (Addition, subtraction,
.on, division)7. Use the Pythagorean relation:hip to de-
termine the length of :my side of a righttriangle given the lengths of the othertwo sides.
N. Solve, problems involving indirect meas-urement using the sine, cosine and tangentratios. (Including use of tables)Improve results obtained by estimatingthe area of plane figures through the useof grids by using grids containing smallerunits. Examples:
Mathematical Sentences
1. Find solution sets of simple open sen-tences using as a universe the integers.(Examples: 5X + 4 --16.)
2. Graph using the Cartesian plane I X I (in-tegers) solution sets of inequalities andequalities such as x + y V- 19; x v > 19;x y < 19; x y
Define oquivalmit open sem enees as s,11-tence:, having the same solution set.
4. Find solution sets of open sentences byfinding equivalent open sentences usingthe properties of equality.
3.3
33
Mathematical Topics
Ordered Pairs, Relations and Functions
An ordered pair of numbers is simply apair of numbers such as (4,7). The order hiwhieb the numbers arc named in the pair isimportant. Thus the ordered pair (4,7) isnot the same as the ordered pair ".,t).
Ordered pairs may be used to epreseutcates, comparisons, integers, rational 1/11M-hers, vectors, and coordinates of a point ina plane.
A relation is a set of ordered pairs. Afunction is a nlation where the first elementin each ordered pair is unique (not the sameas any other first element).
Remarks
Students should become familiar with thebasic properties of ordered pairs, relationsand functions and should know how to repre-sent relations and functions graphically.These ideas (ordered pair, relation, function)should be introduced using an intuitive ap-pcoach.
Students should be able to: (:).
Statistics and ProbabilityStatistics refers to the ways of collecting,
organizing, analyzing, interpreting and smn-marizing numerical data of all kinds. In-ferential statistics refers to a means of usingdata from a relatively 6inall representativesample to predict information about a totalpopulation. Since sample Mita generallydoes not give exact information about thepopulation, the art of decision-mal:ing or in-ference from incomplete data involves sonicknowledge about the theory of probability.The level of probability that one assigns toan inference about a certain population in-dicates the degree of certainty with whichone can expect the same result to occur ifrepresentative samples of the same size alerepeatedly drawn from the saint, parent popu-lation.
The study of probability includes the studyof experiments involving chance events, theoutcomes of such experiments, and the like-lihoods that particular outcomes will occur.
The probability of an outcome for a par-ticulay experiment is a numerical measure ofthe likelihood that the outcome will occur.Thus the theory of probability can be con-sidered to be the methods for assigningprobabilities to outcomes of experiments andthe study of the relationships among them.
RemarksIn grades 7 and 8 the study of statistics
should include drawing inferences from dataas well as the organization and representaticnof data. The study of probability should be-gin at the intuitive level, drawing upon thenotions of chance which the students havealready formulated based on their experi-ences. A substantial amount of experimenta-tion and work with concrete materials shouldbe included to "test" students' intuitivenotions. Abstract symbolism should only lieused to represent generalizations that are ob-tained through less formal laboratory ac-tivities.
q435
Students should be able to:
Behavioral ObjectivesGrades K-5
Ordered Pairs, Relations and Functions
Statistics and Probability
36
Behavioral ObjectivesSixth Grade
Ordered Pairs, Relations and Functions
Statistics and Probability
36
37
Behavioral ObjectivesSeventh Grade
Ordered Pairs, Relations and Functions
1. Determine equality of two given orderedpairs of numbers. Example: (2,3) 0 (3,2).Use ordered pairs of real numbers as co-ordinates of points in a plane. Example:(2,3), ( -5,7), (-1/2)-3)) (V2,S).
3. List the members of a Cartesian productsuch as A x B where Set A has elements1, 2 and 3, and Set B has elements 2, 4,6, and 0, and graph A X B. (A x B isread "A cross B").
4. Calculate the number of elements in aCartesian product given the number ofelements in each set.
5. Identifv relations such as parallelism,perpendicularity, equality, not-equal-to,congruence and similarity.
6. Identify relaticas such as is-a-factor-of,less than, greater than, is-a-subset-of, is-au- element -of, is-the-square-el, etc., andfoemulate a set of ordered pairs of ele-ments from each. Example: Using therelation is-a-factor-of in the following setof numbers, {1,23,0 formulate the set ofordered pairs identifying the relation.Answer: (1,1), (1,2), (1,3), (1,6), (2,2),(2,6), (3,3), (3,6), (6,6) }.
7. Graph relations (including functions) inthe planes I x 1 (integers), Q x Q (ra-tionals), and H X H (reels).
Statistics and Probability
1. Organize and present data using a fre-quency table and graphs.
2. Read and interpret the information abouta set of data presented in the form of afrequency table or a graph.
3, Distinguish between certain :Hid uncer-tain events.
4. Recognize when the outcomes for a givenexperiment are equally likely. For ex-ample, when flipping a fair coin the out-comes "head" and "tail" are equallylikely.
5. List and/or count all the possible out-comes for an experiment for which thesimple events are single elements. Forexample, spinning a spinner, tossing adie, thawing a lot, flipping a coin.
6. Assign probabilities to the outcomes ofan experiment for which 'he simpleevents are equally likely. For 1.xample,the probability of tossing a "2" withone toss of a die is I chance nut ot or1/6.
7. Recognize that the probabilit of anevent is a number p such that o p < 1,and recognize that the probabiiity of acertain event is 1 and the probability ofan impossible event is 0.
3138
8. Assign probabilities to the outcomes ofexperiments for which the simple eventsare not equallylikely. For ex-ample, the prob-ability of the ar-row stopping onred on the spin-ner at the rightis 1 Aimee outof 4 or 1.'1.
. Recognize that the probability of anevent does not guarantee how often theevent will occur. For example, the prob-ability of obtaining a "2" when a fairdie is tossed is 1/6; however, this doesnot mean that the "2" will occur onceout of every six tosses.
10. E5timate the probability of an outcomeof an experiment by empirical methods.For example, estimate the probability ofa bottle cap landing with the cork sideup when it is flipped (the student canflip a bottle cap 100 or more times, keep-ing a record of how the cap lands, andestimate the probability from his data).
Behavioral ObjectivesEighth Grade
Ordered Pairs, Relations and Functions
1. (liven a domain (sueb as a subset of theintegers) and given a description of arelation, determine the set of orderedpairs and graph the relation. Example:Domain is {-3,-2,--1,0,1,2,3 and thedescription of the relation is "double thenumber ;dui add 3." The determined setof ordered pairs should be { f
(-2,-1), (-1,1), (0,3), (1,5), (2,7),(3,11)
2. Determineor is not aExamples:
whether a given relation isfunction.
(a,b) 1)=-10, 11,1) r H } is afunction but
(a,b) a-1)2, a,b. r is nota function.
3. Given the graph of a linear function, de-termine a set of ordered pairs and de-scribe the function.
Statistics and Probability
1. List and/or count all the possible out-comes for an experiment for AIdell thesimple events are ordered pairs, simplecoinbinations, or simple pennittations. Forexample, flipping two coins, tossing a pairof dice, rearranging the order of threeletters of the alphabet, spinning a spinnermid simultaneously tossing a coin, etc.
2. Assign probabilities to the outcomes ofan experiment for which the shill& eventsare ordered pairs, combinations or per-mutations. For example, tin probabilityof getting 2 heads when 2 coins are nippedsimultaneously is 1 out of 4 or V4,
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3. Recognize how the probability of a specificoutcome will change from trial to trialwhen sampling is clone without replacc-ment. For example, from an urn contain-ing. 3 red marbles and 2 white marblesthe probability of drawing a red marbleis 3/5, but the probability of drawing a redmarble if the first is not replaced then iseither 2/4 or 1/.
4. Estimate the probability of an event fromprevious data. For example, a basketballplayer made 50 free throws out of 75 tries;what is the probability that he will makehis next free throw attempt!
5. Estimate information about a populationby random sampling. For example, outof 100 flash bulbs selected at random 5were defective.flow many out of (1000 are likely to bedefect ive t
Problem Solving
A situation becomes a problem for a stu-dent only if the following conditions exist:
I. A question is posed (either implicitlyby the materials being studied or ex-plicitly by the student) for which ananswer is not immediately available.
2. The student is sufficiently interested tofeel some intrinsic need to find a solu-tion.
3. The student has enough confidence inhimself to believe a solution is possible.
4. The student is required to use morethan immediate recall or previouslyestablished patterns of action to find amethod for arriving at a conclusion.
Not all pupils will consider a given situa-tion a problem. if a student can see a"method of solution" immediately, then thesituation is riot a problem for him, but simplymu exercise or an application of some processwhich lie has already inast.2red. For example,an exercise such as 3 X 14 is not a problemfor a stadent who has already mastered theidea of multiplication with two place nu-merals. However, this example could very\veil be a problem for a student who hasstudied only a few elementary multiplicationcombinations. In order for this student toobtain a correct solution, he would have tothink of sonic method of renaming 14 suchas 10 + 4 ;1/1(1 find sonic method of reducingthe problem to simpler stages with whichhe could work, such as 3 x 14 = 3 X (10 +4) = (3 X 10) + (3 X 4).
Heal problem solving is not recalling anumerical fact or fitting a situation into amemorized pattern, but is discovering one'sown method for extending previous learningsto new situations.
It is not the intent of this chapter to con-sider all of the general aspects of problemsolving. Only some of the problem situationsthat involve the use of known mathematicalideas by students will be discussed. Theverbal problems found in textbooks maize up
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a large portion of the problem situationsconsidered in mathematics programs. Whensuch problems are presented, the mathe-matical ideas necessary to solve them aregenerally known to the student, but thecontexts in which they are presented re-quire the student to find the methods ofsolution 01 to decide Whiell of the ideas healready knows will help hint find the solii-
fmnlTie solution of a typical textbook verbalproblem involves a series of four interrelatedstops. The complexity of each of the stepsVttrieS with the p;oblern, but the method ofsolution follows this general pattern.
Step 1 Recognizing the question beingasked.
Step 2 Traustiting the verbal probleminto a mathematical sentence.
Step 3 Finding a solution for the mathe-matical sentence.
Step 4 Analyzing the solution for thesentewe to see if it provides areasonable solution for the orig-inal verbal problem.
Each step of the solution can involve manyfacets. Although all steps iii the solution ofa verbal problem are important, Steps 1 and2, the abstraction or translation phase, can beconsidered the "heart" of the process.
The student may have to decide manythings before he is able to write the mathe-matical sentence which represents the prob-lem. If the question is not explicitly stated inthe verbal problem, the student 11111:1 formu-late his own question or questions.
Ile will have to decide if all of the neces-sary data for the solution is given in thestatement, and he will have to select thepertinent data from the given information.He will have to decide which mathematicaloperation is suggested by the "action" of theprEblem. Ile /mist determine the order inwhich the data 111( problem will appear inthe 1 he student
must also decide what relationships, if any,are involved that NVi II enable him to usepreviously solved problems as "models" andwhat approach is the most efficient or easiestfor him to use in obtaining a correct solution.Steps 1 mid 2 could also involve some trialand error activity in which different sen-tences are ,ested as to which one best fitsthe given situation. These steps includeproblem solving techniques conunoilly re-ferred to in many textbooks and teacherguides as "Reading and understanding theproblem" and "Restating the problem inyour own words."
Step 3 involves finding a solution to themathematical sentence. This procedure canbe very simple or very complex dependingon the given situation. In general this stepinvolves the npplieation of knowledge thatthe student bas already attained throughpractice work with similar sentence forms.
Step 4 is the familiar "check" in which thestudent determines if the solution is accept-able for the given problem situation. Thestudent must decide if the solution is reason-able or "make sense." If it is apparent thatthe solution is not correct, the student mustthen "rework" his problem, check his com-putations, examine his sentence to see if itactually symbolizes the "story" of the prob.'cm, and look for careless errors or possiblemisinterpretations or misrepresentations ofthe data.
Sonic exemples of typical verbal problemsare given below.
in the first grade, simple "picture" ororal problems are presented which can berepresented by sentences like 5 2where the sum does not exceed ten. The prob-lem might be presented to the students inthis form: "John has 5 toy cars. His mothergave him two more toy cars for his birth-day. How many toy cars does he have now !''
"action" of the story is the joining oftwo sets of toy cars, and the mathematicalsentence Nvhich represents this problem is5 -f- 2 Through such oresentat ions, thestudent will learn to associate the notion of"joining" with the operation of add;doil.
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lie will thus have a better understanding ofNV/1M addition means ;Ind will develop tech-niques for solving similar problems.
In the second and third grades, problemsituations are extended to include verbalproblems represented by sentences like 3
8, 4- 5 9, 7 2 -=-= , 85, and 2 6. A typical verbal prob-
lem at these grade levels is: "Mary had 15doll dresses. After she gave some of thedresses to her sister, she found that she had8 dresses left. How many doll dresses didshe give to her sister?" This problem isrepresented by the sentence 15 8.The "action" of the problem is "takingawa:."; therefore, the corresponding, opera-tion of the sentence must be subtraction.The 15 represents the number of dressesMary had in the beginning, the repre-sents the number of dresses she gave to hersister, and the 8 represents the number ofdresses she had left. Students should realizethat although l.i 8, ti 4- 15,and 15 can be represented by dif-ferent physical situations, the computationinvolved in solving each of the sentence.; isthe same.
In the later elementary grades, problemsare presented that involve larger numbersas in sentences such as 45 + 327 or 2:0-- - 76 and that use the same Inas do the problems first introduced in theprimary grades.
In the third and fourth grades, problemsituations represented by sentences such as3 x 7 El, 5 x 20, and ri X ti= 3tlareintroduced. A sample problem for this gradelevel is: "Tom found that he needed sixteensmall cartons to corer the bottom a a pack-ing If he needed tone layers of cartonsto fill the case, how ninny small cartons wirein the ease!" The "action" of this prob-lem is repeated addition or inottlidication.Thus the sentence used to represent the storycould he 16 -+- 16 16 -1- 16 or 4 X 16
The student should he allowed to us,.either sentence to represent the problem, butshould he led to realize that the sentence4 x 16 -- is the shorter way of writing- a
sentence for this type of problem. In thelatter sentence, the 4 represents the numberof sets, the 16 represents the number of ob-jects in each set, and the represents thetotal munber of objects in all of the sets.
In the sentence 5 X = 30, the 5 repre-sents the number of sets, the representsthe number of objects in each set, and the30, the number of objects in all. Likewisethe sentence x 6 = 30 represents a prob-lem situation in which the total number ofohects (30) is given, the lumber of objectsin each set (6) is given, and the number ofsets, represented by the , is the solution totlw problem. The sentences 5 X = 30 and
X 6 30 represent different physical situa-tions, but both types of sentences can besolved by the same computational method ofrepeated subtraction if the multiplicationfact that makes the sentence true is notknown.
Problem ,ithations introduced at the fourthand fifth grade levels are similar to the prob-lent: "If John separates 12 marbles into smallgroups with 4 marbles in each group, howmany groups of marbles will lie have!" Thisproblem can be represented by the sentence
X 4 12, for the question of the problemcan be restated: "How many sets of 4 arethere in 12!" The problem can also be in-terpreted as the division of a set of 12 ob-jects into sets of 4 objects. With the hitterinterpretation, the problem can be repre-sented by the division sentence 12 ÷ 4 = E.Both sentences, E X 4 = 12 and 12 ÷ 4 E,represent the saute physical situation andimplicitly ask the same question, "flow manysets of 4 are there hi 12!" Both sentencesare solved by repeated subtraction if thenumber which makes the scatences trne is notknown.
As fractions, decimal numerals, and in-tegers are introdneed and used in grades6-8, problem situations involving their usein sentence forms similar to those consideredpreviously are presented. It's important thatstudents be able to write the sentence repro-sentiug the "story" of a Verb,' problem.
The verbal "story" type problems are notthe only source of problem situations for usein the school mathematics program. Manyother nuithematical problems arise from phy-sical situations, from social or mathematicalapplications of mathematical ideas, or fromsituations made up by the teacher or thestudents. These varied situations can be real,imagined, or in the nature of a puzzle.
any of these problems fall into a categorywhich might be regarded as a "higher order"of problem solving than the typical textbook"story" problem. It is not possible to out-line a regular progression of steps to be fol-lowed in solving all such problem situationsas the process will vary with the nature oftho problem) presented and the ability andexperience of the student. The four stepsoutlined for the solution of verbal, problemsdo not necessarily apply in general problemsolving situations, for it is not always pos-sible to translate such a problem into amathematical sentence.
110NN-ever, it is possible to list a few of theactivities that are essential in time generalproblem solving process. These activities in-clude: translating the problem into a simplerform, critically examining the given data,forming hypotheses or conjectilteS, reasoning(Ma trial and error basis, analyzing or evalu-ating results on the basis of past experience,and forming generalizations from similarproblem situations. These activities are notthe only ones that may be involved, and notevery problem situation will require the stu-dent to become engaged in all of the activitieslisted above. It is important to note that theorder in the student pei-forms theseactivities may vary for different problemsituations.
Teachers can do much to help students im-prove their problem solving abilities for alltypes of nuithennitical problems. The teaell-(21''S 1111101011 :4110111(1 be to create a question-ing, challenging atmosphere; to introducelrrobletu situations; ;ILI to guide and (l-eum age students to devei their ir own prob-lem solving, t eliniques.
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A few suggestion:- that can help the teach-er promote good problem solving techniquesare listed below.
The teacher should present problemsituations that involve many basicmathematical priarAples. He should becertain that these situations are relatedto the kinds of experience the pupil hasalready had so that the pupil is able toapply the principles.
The teacher six uld let his students usetheir own methods. Many problems haveno single, best, method of solution. In-sistence on oiP "favorite" method ofsolution often lestroys enthusiasm andoriginal thinking.
Whenever the problem permits, theteacher should emphasize the writing ofa mathematical sentence that shows the"action" of the problem. If studentsare to acquire good problem solvinghabits, they must be able to describe theaction of the problem in terms of 'indite-
viatica] symbols Nvhenever applicable.Writing the mathematical sentence fora problem is as important mathemati-cally as finding the. solution for the sen-tence.
The teacher shonld encourage his stu-dents to use diagrams, estimates, dra-matizations, or other techniques thathelp them understand the problem.
The teacher should suggest that stu-dents try various approaches to a prob-lem when they are not certain of thecorrect method. Shutents can evaluatethe method used by ascertaining thatthe solution is reasonable. Studentsshould realize that the trial and errormethod can be an effective approach todifficult problems.
The teacher should confront studentswith some situations in which they lutistformulate their own questions. Thistype of presentation is similar to thekinds of problems t1u are apt to facein their future vocations.
