DOCUMENT RESUME ED 051 200 SP 007 267 TITLE Elementary

Home , Associative property, Commutative property

DOCUMENT RESUME

ED 051 200 SP 007 267

TITLE Elementary Mathematics Guide, K-7. INSTITUTION Virginia State Dept. of Education, Richmond. PUB DATE 68 NOTE 189p.

EDRS PRICE ERRS Price MF-$0.65 BC-$6.58 DESCRIPTORS *Curriculum Guides, *Elementary School Curricv'um, *Elementary School Mathematics, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Kindergarten, *Mathematics

ABSTRACT GRADES OR AGES: Grades K-7. SUBJECT MATTER: Mathematics. ORGANIZATION AND PHYSICAL APPEARANCE: The main section of the guide entitled "Mathematical Strands with Teaching Suggestions" has the following subsections:1) sets and numbers, 2) numeration, 3) operations on whole numbers, 4) rational numbers, 5) geometry, and 6) measurement. Other chapters deal with problem solving and program objectives. The guide is printed and edition bound with a soft cover. OBJECTIVES AND ACTIVITIES: Objectives are listed for each grade under the six subsection headings. Activities are described in detail in the main section of the guide. INSTRUCTIONAL MATERTALS: None are listed. STUDENT ASSF1SMENT: No special provision is made for evaluation.(MBM) 0 rb4 Elementary Mathematics Guide ct0 T3, K - 7 s.DEPARTMENT OF HEAL EDUCATION & WELFARE OFFICE OF EDUCAPON THIS DOCUMENT HAS&FEN REPRO- RELIVED FROM DUCED EXACTLY AS ORIG- THE PERSON OR ORGANIZATION INATING IT POINTS OFVIEW OR °PIN. lOt S STATED DO NOTNECESSARILY fr-PRESENT OFFICIAL OFFICEOF EDU CATION POSITION OR POLICY

ELEMENTARY EDUCATION SERVICE STATE DEPARTMENT OF EDUCATION RICHMOND, VA. SEPTEMBER 1908

1 INTRODUCTION

The Elementary Mzthematics GuideK-7 was prepared to assist elementary teachers in planning and directing a logical, sequential program in modem mathematics,Attention is invited to the first section of the Guide which sets forth reasons for the development of a new guide in elementary mathematics. The Guide outlines basic ideas used in elementary mathematics and includes teaching suggestions for developing understanding of these ideas. The Statement of Objectives lists ideas and skills which most children might be expected to acquire at a given grade level. The form in which these objectives are presented was designed to give a picture of sequential development of mathematical ideas and skills throughout the elementary school and to indicate scope of emphasis at each grade level. Intuitive understanding of mathematical ideas which children akea.ly possess is used as the basis for developing greater competence in the use of numbers to communicate ideas and to solve problems. Physical model.) are used to introduce new ideas and to provide concrete means for solving onzsi er problems. Appreciation is expressed to a committee of classra.m teachers, elementary principals and supervisors, college persennel, and staff members of the Elementary Education Service of the State Department of Education, which worked over s period of three years to prepare the Guide. Appreciation is also expressed to other teachers, principals, and supervisors who joined members of the committee in summer workshops devoted to the preparation of material incorporated in the Guida.

WOODROW W. WILKERSON Superintendent of Public Instruction

2 ELEMENTARY MATHEMATICS COMMITTEE

MRS. JOSA C. BALDWIN DR. FRANCIS G. LANKFORD, JR. Teacher Professor of Education Buckingham University of Virginia

ELINORSue BIRNEY ROBERT G. LEONARD, JR. Teacher Director of Instruction Hopewell Warren-Rappahannock

MRS. ANN COLLIEK DR. JOSEPH N. PAYNE Teacher Frofessor of Education Waynesboro University of Michigan

DR. RICHARD W. COPELAND MRS. ELIZABETH REAVIS Associate Professor of Education Teacher Collegeof William and Mary Covington

MRS. BERTHA D. DETAMOR:' LEROY SALLEY Teacher Teacher Albemarle Mecklenburg

MR3, MAY C. DUNCAN CLARA E. SCOTT Elementary Principal Elementary Supervisor Roanoke County Pulaski

MRS. EDNA. L. FLAPAN MRS. ANNIE M. SEAWELL Elementary Supervisor Teacher FairfaxCounty Chesapeake

K. EDWARD GOODE Elementary Principal Franklin County

PARTICIPANTS IN SUMMER WORKSHOPS

BETTY GAIL ELLIOTT MRS. DOROTHY MUNDY Teacher Teacher Fairfax County Roanoke County

WILLIAM C. FLETCHER BOB Q. RAINES Elementary Coordinator Secondary Principal Quantico Dependents' Schools Washington

MRS. MARY G. GOEBEL BENJAMIN B. RUDISILL Teacher Teacher Fredericksburg Prince William

MRS. MARY GOODE MRS. ANNE WHITS Teacher Mathematics Supervisor Hopewell Portsmouth ii

3 STATE DEPARTMENT OF EDUCATION STAFF

S. P. JOHNSON, JR., Director HELEN S. LUKENS Elementary and Special Education Assistant SupervisorMathematics

BERNARD R. TAYLOR MRS. HATTIE H. RAGLAND Supervisor Assistant Supervisor

MRS, VIRGINIA S. CASHION ROBERT M. SANDIDGE Assistant Supervisor Assistant Supervisor

ELIZABETH ELLMORE MRS. CALLIE P. SHINGLETON Assistant Supervisor Assistant Supervisor

MRS. LOROTHY M. FAULCONER HOWELL L. GRUVER, Supervisor Assistant Supervisor Statistical Services

JOHN G. FOLEY HARRY L. SMITH, Director Assistant Supervisor Public Information and Publications

Iii 4 TABLE OF CONTENTS

POINT OF VIEW MATHEMATICAL STRANDS WIl A TEACHINGSUGGESTIONS Sets and Numbers The Meaning of a Set One-to-one Correspondence Number Variables Counting Set Terminology Set Operation Summary of Sets of Numbers Elementary Aspects of Number Theory Teaching Suggestions

NumerationHindu-Arabic Decimal Numeration ...... System Non-decimal Bases Ancient Numeration Systems Teaching Suggestions

Operations on Whole Numbers Addition Properties of Addition Algorithms for Addition Subtraction Properties of Subtraction Algorithms fir Subtraction Multiplication Properties of Maltiplication Algorithms for Multiplication Division Properties of Division Algorithms for Division Square Root of a Number Teaching Suggestions

Rational Numbers Fractional Numbers The Nature of Fractional Numbers Operations on Fractional Numbers The Integers The Nature of Integers The Order of Integers Operations on Integers Negatives of the Fractional Numbers The Nature of Negative Fractional Numbers Operations on Negative Fractional Numbers Decimal Fractions The Nature of Decimal Fractions Operations on Decimal Fractions Ratio and Proportion The Nature of Ratio and Proportion Properties and OperationsRatio andProportion TABLE OF CONTENTSContinued

Percent Nature of Percent Operations with Percent Properties For the Four Arithmetic Operations Teaching Suggestions

Geometry Basic Ideas of Geometry Relations Intersection Perpendicularity Parallelism Congruency Similarity Symmetry Plane Figures Space Figures Teaching Suggestions

Measurement Basic Ideas of Measurement The Metric System Estimation of Measurement Perimeter Area Polygons Closed Surfaces Volume Use of Similar Plane Figures in Indirect Measurement Scale Drawing Rectangular Coordinate System Introduction to Elementary Statistics Teaching Suggestions

PROBLEM SOLVING 153

IV. STATEMENT OF OBJECTIVES 157 Introduction Kindergarten *Int Grade Second Grade Third Grade Fourth Grade Fifth Grade Sixth Grade Seventh Grade V. SYMBOLS USED IN GUIDE 44` VI. INDEX ih Mathematics for the Elementary School POINT OF VIEW

A scientific and technological society demands more bers, the set of fractional numbers and the set of and better training in mathematics for its citizens. The rational numbers. A study of the unique properties needs in such a society and the changes in mathematics of these various sets of numbers and their applica- itself require changes in the teaching of mathematics at tions helps the child see the value of each. all levels. Curriculum changes in the elementary school Introduction and use of the properties of the various have not, however, been as drastic as popular opinion number systems give meaning to the learning of concerning the "new math" might lead one to expect. number facts and computational algorithms. The A sound curriculum in modern mathematics for the closure, commutative, associative, distributive, iden- elementary school is distinguished by two major char- tity and inverse properties are basic laws which pro- acteristics: vide the predominant structure for number systems The mathematical content of the program and used by elementary children. An approach to teaching that is consistent with the For example, to find the sum of 7 and 8, the child best that is known about learning. may think: "I need 3 of the 8 to put with 7 to make 10; 7 + 3 = 10; and 5 more is 15." The associati'e MATHEMATICAL CONTENT propertyisthebasicreasonforchanging the 7 + (3 + 5) to (7 + 3) + 5 to mlo the addition Traditionally the curriculum in the elementary school fact easier. was called arithmetic since it was centered around computation wIth whole riambers and fractional numbers. To find the product of 7 and 8 a child may think: Now we speak of mathematics in the elementary school "7 eights is (5 + 2) eights.This is because it has become apparent that the content may be (5 x 8) + (2 x 8) = 40+ 16 or 56." Naming made more meaningful and better continuity provided (5 + 2) x 8 as (5 x 8)+ (2 x 8) utilizes the by the introduction of new topics. distributive property.

The major new content for elementary school mathe- To find the product of 3and 4-1 the distributive matics consists of sets and set language, properties of 2 the various sets of numbers, number sentences, geom- property is used to name etry, and a variety of numeration systems. Much of this new content has as its main purpose the improve- x (4 +-21 ) as (3 x 4) + (x-2)1 ment of understanding of the usual content of arithmetic. 12+ 1or 13 2 2 The use of sets and set language helps to clarify and When we think of structure in work with numbers make more precise the concept of number. The in the elementary school, we think of properties of union of disjoint sets of objects provides a model the number systems (closure, commutative, associa- for the addition of numbers.Separating a set of tive, identity, inverse and distributive) and charac- physical objects provides a concrete model for sub- teristics of the numeration system (base, digits, place traction of numbers. value, and position). Together these provide the The ideas and language of sets provide the vehicle basic reasons or main ideas around which a learner for the study of geometric ideas.All geometric fig- organizes his thinking and from which he deduces ures may be viewed as sets of points. A line is a facts and computational algorithms. Heavy emphasis set of points; a sphere is a set of points; a circle is on the properties does not imply that elementary a set of points. Through the study of sets of points teachers begin with structure.Instead, by repeated and relations among them children can describe with use of the properties, the structure emerges in the precision and clarity the shape of objects found in child's mind. Many experiences, ranging from the their environment. intuitive and concrete to the abstract, in using these key ideas build for the learner a sense of their im- The use of sets and set language enables teachers portance and reel value. to focus attention on the various number systems that are studied in elementary schoolthe set of Use of number sentences, including equations and natural or counting numbers, the set of whole num- inequalities, helps in solving word problems. A use-

1 ful first step in solving a problem is translation of for whole numbers, fractional numbers and opera- the physical situation into the mathematici. lan- tions on them. The languageof guage of a number sentence. Through the study of a variety of numeration sys- mathematics contains symbols comparable to the tems, children are helped to broaden their view of nouns, verbs and phrases used in other communica- essential characteristics of numeration systems and tion. A situation which requires the equal appor- toappreciatetherichhistoricalbackgroundof tionment of three dozen cookies among eighteen mathematics. Naming the number of members in children, when translated into a number sentence, sn A as 12,,,one set of six may became: and 2 more gives contrast to (7 x 12) A (3 x 12) 18 = fj Or 04 the base ten in the numeration l8 system which we use, when X X X 12,. names the number of X X X Solving the problem then becomes a matter of com- members in set B. Naming the pleting familiar operations. number 4 in Roman numerals x as IV shows the use of sub- The use of both equations and inequalities also traction in a numeration sys- B helps in estimating computations. The cost of three tem which contrasts with the X X X X X and one-half dozen eggs at fifty cents per dozen may use of only additioninthe X X X X X be quickly estimated by the following inequalities: decimal place value system. The introduction of other num- X X 32 50>3 x 50 or $1.50 2 eration systems this broadens and deepens a fundamental mathematical idea and 31 X 50 < x 50 or $2.00 provides one avenue for building interests and appre- 2 ciations. Geometry in the elementary school is the study of The new content described above is belle...ed to make positions and locations in space. A position or loca- mathematics more meaningful for the learner. The new tion is viewed as a point.Space is the set of all content also enables he elementary school to provide points. Through the study of geometry greater un- for better continuity as students study more and more derstanding of shapes in the physical environment is mathematics, However, the new content does not con- obtained. For example: A soap bubble is a physical stitute the major portion of the elementary school model for a set of points called a sphere. The inter- mathematics curriculum. The dominant emphasis will section of a sphere and a plane continue, as in the past, to be the study of whole num- passing through the center of the bers and fractional numbers. A sphere is a great circle. The great 1 a+b=b+a1( circle is t., route for the shortest a xb=bxa ,4/ distance between any two points Spiral Development on the earth's surface. of Mathematical Content f (-3) + 2 = 2 + (1) \I %1-3) x 4 = 4 x (-3) sik. Geometry is necessary for precise and correct development of concepts of measurement. Length, With the goal to provide 91- for example, is a number of units obtained by com- ever-expandingviewsof + = + 3 4 paring a given line segment with a chosen unit. The central mathematical ideas, 4 3 length of AB is5 units, contentisspiraledfrom 5 4 4 5 /1 using the unit shown. Area grade to grade. Ideas are x 5 5 x e. A is a number ofunits ob- introduced at a given grade, tained by comparing a unit redeveloped and deepened 2 3 3 4 V region to a given region. unit in subsequent grades. For+3+2x4x3 ,e4, The area of parallelogram example, the commutative, 4 ; region ABCD is3, using associative and distributive 2 x 3 = 3 x 2 )1. the square region shown as properties for additicn and the unit. By stressing the multiplicationof whole Om ow mil 0.° association of numbers with numbers are developed in- {0) u {DO) geometric objects in mea- LI formally and experimentallyA {0 0) u (C1) surement and through the unit in the early grades. They ..0* use of a number line, a helpful model is provided are later named and used COMMUTATIVE PROPERTY

2 8 as basicreasons foralgorithms Then as theset Terminology is an integral on of the content of of fractional numbers is studied, properties for ad- elementary school mathematics, but if introduced be- dition and multiplication are viewed again for applica- fore understanding is developed, it may handicap or tion to fractional numbers. Likewise, when the set of inhibit development of ideas. By the age of nine or integers, the set of rational numbers and the set of ten many children arc interested and have readiness real numt-rs are studied, properties for addition and for new terminology.However, caution should be multiplication are again verified. With each successive exercised to insure that words and symbols do not extension of the number system, the goal is to ascertain become confused with mathematical content. which properties fit operatic- s on the new numbers and which properties are new for the extended number systems. APPROACH TO TEACHING Development of Understanding. One main goal of A numeration system with base of ten is introduced modern mathematics in schools is the development of in grade one by grouping objects into sets of ten. Place greater understanding on the part of pupils.Greater value is then introduced. The basic notation of base understanding means longer retention, better applica- and place value are spiraled throughout the curriculum tion of new ideas. and improved poser in reasoning with the goal of deepening the level of generality and and problem solving. 'this has been a goal for mathe- abstraction. The use of expanded notation, expanded matics education for many years ar l much of the notation with exponents, Roman numerals, other an- present movement is toward the further -calization and cient numeration systems, and other bases with place delineation of this goal. value are examples of the spiral natur; of numeration. As long as a person continues the study of mathe- Development of Competence in Computation. Con- matics, patterns will be generalized and abstracted at comitant with the goal of understanding is the develop- more sophisticatedlevels. mentofcompcten:eincomputation.Subsequent mathematical ideas are more difficult, if not impossible, to de clop without skill in computation. The skill to mbols The introduction of new topics into be expected by the teacher must vary with individual and the content of elementary school pupils. Knowledge of pupil maturity, and the mathe- Vocabulary mathematics has made necessary the matical or social need for the skill determine expecta- use of new symbols and new vv. tions set by the teacher. cabaciry. White vocabulary and symbols are useful in the communication of ideas in mathematics, the intro- Experiences milt., Concrete Materials.In developing duction of new terminology is second to the develop- understanding of mathematics and competence in com- ment of understanding of the designated idea.For putation itis important that a child's first experience example, the names of properties of number systems with any new topic be related to his previous back- closure, commutative, associative, identity, inverse and ground r..nd experience. For many new topics puoil:; distributiveshould be delayed until the student has a need to handle and manipulate physical objects to pro- reasonable degree of understanding of the ideas. There vide. xperience needed to make the transition to ab- should be opportunity to talk about tine idea, to use it, stract ideas. For example, experience which the child and then to learn its name for oral and written use. has in physically moving sets of objects together pro- This procedure is equally true fo: symbolsthe use of vides foundation for developing the concept of addi- braces to denote a set, and the symbols denoting union tion. Much ss'ch physical combining is needed for tb: and separation; for the inequalities symbol- denoting learner to make a smooth transition to imagined com- "is greater than" and ''is less than"; and for such new bining of objects and finally to the abstraction. In the vocabulary in geometi y as line segment, closed path. primary grades where formal work in making the tran- plane region and spec region. sition from concrete objects to abstract ide..s begins, it is important that each child have manipulative ma- While there is a slight increase in the total number terials at hand which he is free to use at any time to of technical words and symbols in general use, certain help in the discovery of a number fact or the verifica- words that have been in the program have been deleted. tion of a mathematical idea. Some words deleted, such as "minuend," "subtrahend," "multiplicand" and "multiplier," have doubtful value Transition from Concrete to Abstract.As pupils' for the learner and are not maintained in subsequent concepts become more mature, the need for experience mathematics courses.Other deletions such as "bor- with concrete materials decreases because they are able rowing" and "carrying" have been made because they to use previous ideas to learn new ones. However, the tend to convey misleading ideas. transition from concrete to abstract should not be too

3 9 abrupt. Many children may need to work from time sponses, the teacher leads pupils to see that the num- to time with sets of objects or physical models in order ber 3 shows the part that is shaded. The fractional to re-establish understanding or to visualize a new 5 concept. The teacher should encourage interplay be- 3 number is identified by using the whole numbers 3 tween concrete and abstract for purposes of application, 5 redevelopment and review. and 5 but is not a whole number itself. Involvement of Pupils in Instructional Goals.It is generally agreed that those goals which children ac- Variety in Algorithms.In a class where understand- cept as their own are most effective in motivating learning is valued, teachers encourage the use of longer ing. Help 4 children set worthwhile goals is difficult. algorithms. For example, the algorithms in A and in They more readily accept instructional goals when B are longer but may be more meaningful than the there is interest in the topic, when the topic is related shorter one. When pupils are introduced to the com- to .amiliar ideas, when application of knowledge and pact algorithm, transition from the longer to the shorter skill can be made to their own lives or when a chal- form must be smooth and understandable for the ienge is accepted. Having set a goal, children must learners. For some pupils transition to the more ab- have opportunity to gain a sense of achievement. The breviated algorithm C may be omitted completely teacher should guarantee this seme of achievement for without harm. each pupil who exerts a reasonable amount of effort. A 25 Involvement Through Discovery.It is the goal for B 25 C 25 every child in the class to become actively involved in 5 42)1050 42)1050 the learning process. An atmosphere of discovery helps 20 840 84 achieve this goal. In a discovery oriented class, teach- 42) /050 210 210 ers ask thought provoking questions, wait for responses 840 210 210 from children and build subsequent questions around ?l0 these responses. Such questions may lead pupils step- 210 by-step toward a desired conclusion or they may be open-ended to provide freedom and flexibility in the 42)1050 way learners approach problems. Encouraging children 84020 to talk about ideas gives teachers an inside view of 210 thinking and also gives a child insight into his own 210 5 thought processes. There is ego reinforcement when 25 someone listens.

Acceptance of Varied Responses.In a class where Understanding Followed by Practice. Questions and discovery by pupils is valued, teachers must be willing classroom discussion are vitalin developing under- to accept a variety of responses. For example, a teacher standing of an idea or procedure and are necessary may ask: "Can you use a whole number to tell what predecessors of practice.Practice that follows should part of the region is colored?" Some pupil may reply, provide for varied use and application of the topic. "Yes, the number 3." Another may reply, "Yes, the Work done by a child must be checked carefully by number 5."Both pupils show understanding of the the teacher as soon as possible to insure success for question. the learner and to provide the teacher with information for further planning. Systematic review work is necessary if the topic is to remain in the learner's repertoire. 11111ril Acceptance of Difference In Achleremcnt. In any classroom teachers should expect wide variationin pupil responses and achievement. From some students The teacher must recognize this degree of under- the teacher should expect more insightful answers to standingandproceedwithotherquestions.The vestions and more astute observations. From others teacher may then ask: "Does the number 3 tell what teachers sh..add :spect insight not so deep nor observa- part of the region is colored or does it tell the number tions so profound. But from all pupils teachers can of pieces?" "Does the number 5 tell the total number expect growth from year to year in mathematical ma- of pieces or the part that is shaded?" Through recog- turity, competence in computation and ability in prob- nition of the thinking reflected in the variety of re- lem solving.

4 0 10 Development of Problem Solving Attitude. In mathe- of computation, then evaluation must reveal degree of matics itis a goal for pupils to develop a problem understanding as well as degree of skill in computation. solving attitude. This attitude is encouraged and sup- If the development of active, or lifelong interest in the ported by a discovery oriented classroom where prob- study of mathematics is a goal for elementary school lems are posed by teachers and by pupils.If a prob- children, then means for determining extent and growth lem engages pupils, the ensuing activity points in the of interest must be devised and used. If understanding direction of solving it.This is problem solving in its and appreciation of the cot: tributions of mathematics broadest sense. to civilization--past, present and futureis a goal, then avenues must be opened to enable children to Solution of Word Problems. A proper subset of the communicate or demonstrate interest,understanding set of problems that engage pupils is the set of verbal and appreciation. orstoryproblems whichillustrateapplicationsof Standardized tests and other paper and pencil tests mathematics. Applications are drawn from everyday should continue to be used to assess both understand- experiences of pupils and of people they know. Where ing and skill in computation. Equally important is the feasible, applications from other subject aims, such as use of observations and oral questions as children science end social studies, are to be sought and ex- manipulate concrete materials, work independently on ploiter'. With each problem the goal is to select the practice material or explore ways to solve problems. mathematical model that fits the practical situation. Interviews or discussions with individuals or small The mathematical model, such as addition,is often groups may reveal understanding of basic ideas or stated as a number sentence.See Problem Solving, operational procedure, attitudes toward the study of page 153. The choice of a number sentence to express mathematics, or appreciation of the part played by the mathematical model forces pupils to give attention mathematics in the development of the modern world. to the ana'fsis of the problem rather thcm plungieg blindly into computation with the hope that the answer Evaluation ofthistype makes less demand on is correct. For many pupils, practical applications pro- teachers to correct papers after school.Instead, the vide the prime motivation frr studying mathematics. teacher assesses progress in the classroom by watch- This phase of problem solving then, must continue to ing children at work, checking papers as work goes receive emphasis in a sound elementary schoo; pro- on and analyzing questions and oral responses. Evalua- gram. tion which becomes an individual process related to the growth that a child has made and the help that is When enough time is spent on manipulation of con- needed as he develops understanding and skill cannot crete objects, discussion of ideas, utilization of varied be reduced to a sco e or letter grade that merely in- pupil responses, solution of practical problems and dicates the number of correct answers on a written appropriate practice, mathematical content makes sense. lesson. When content makes sense, the child becomes actively involved in the learning process and builds a firm Evaluation, a Basis for Planning Instruction. The foundation of competence on which to move ahead. real purpose of evaluation is the assessment of strengths and weaknesses to enable the teacher to plan sub- El aluation, a Rtflection of All Goals. Evaluation of sequent instruction more effectively.Machines and the outcomes of the teaching of mathematics must give testing devices are helpful in collecting data on strengths attention to all goals and must use techniques that focus and weaknesses, but no machine can make the kind of on appropriate pupil behavior. If teachers aim t(dye assessment which helps a child move ahead. Only a attention to goals of understan 'ins 33 well as to 16k. is knowledgeable, sensitive teacher can do this.

5 MATHEMATICAL. STRANDS WITH TEACHING SUGGESTIONS SETS AND NUMBERS

A set is a collection of things so described or defined that it is possibleto determine that a particular thing is a member or element of the set.Sets of objects have been used for many years to help children tocount and to answer questions involving the idea of "how many." Not all sets have thesame properties but all sets have the property of number. The idea of number is a central theme of the school mathematicsprogram. Ideas of sets and elementary notions of set theoryare used to develop and expand the concept of number in this guide. All sets which can be matched in ene-to-cne correspondenceshare a common property of number. Numerals are names and symbols assigned to a particular numberproperty. Numbers can be used in a cardinal sense to answer questions of howmany. They can also be used in an ordinal sense to answer questions of which one. When elements ofa set are arranged in a definite order, an ordinal number describes the position of an element in the givenset, as a student is third in line or he is reading page 20, the twentieth page. Although the study of number is extended in this guide to includethe set of rational numbers, emphasis in elementary school is concentrated on the set of whole numbers and theset of fractional or positive rational numbers.

Mathematical Ideas Illustrations and Explanations The Meaning of a Set A set is a well defined collection. The collection may consist of physical ob- 5 9 jects, alike or unlike, or abstract ideas, A alike or unlike. The things in a set to 7 .re usually called elements or mem- 41111 . bers. A set is well defined if a given 3 I object or idea may be identified as belonging to or not belonging to the set. Conventionally, set elements are en- Set A = or A A closed in braces and the set is named by a capital letter. BRA The elements of a set may be listed individually or identified by a de- C= (5, 9, 7, 3, I) All odd numbers 1 scriptive phrase. less than ten r

One 'o -one Correspondence Two sets A and B are in one-to-one correspondence if each element in A can be matched with a singleele- ment in B and each element in B can Set S Set P Set N SetC be matched withasingle element Set B in A. The members of Set S and Set Pare matched in one-to-one correspondence.

The members of Set N, Set C, and Set Barc in one-to-one correspondence with each other.

12 I: Mathematical Ideas Illustrations and Explanations

The fundamental basis for the con- One-to-one correspondence is the idea involved in set equivalence cept of number is one-to-one corre- and can be used in developing a sound concept of number. spondence or pairing of the elements of one set with the elements of another set.Before man devised a sys tern of counting, he used one-to-one correspondence tokeep records. Shepherds kept track of sheep matching a pebble with a sheep. record of time was kept by pairing each day with a mark on a stick or a stone. Set R Set Q Set Z Number The one property shared by all sets The number property of Set S above, for example, is 3. is the property of how many. This is The number property of Set R above is 5. called the number property ofthe sets. All other sets in one-to-one correspondence with Set S have the same number property or cardinal number, 3. MI other sets k one-to-one correspondence with Set R have the same number property or cardinal number, 5.

Equivalent sets:If two sets are in L (a, e,1, o, u} one-to-one correspondence, theyare said to be equivalent. Equivalent sets N{f, 14.1) have the same cardinal number.

Set A Set B

Sets A and B areinone-to- Sets L and N areinone-to- one correspondence. The cardi- one correspondence. The cardinal number of each is 3. nal number of each is 5. Non-equivalent sets:If two sets are notinone-to-onecorrespondence, Set Y has more members than they are non-equivalent sets and have Set Z. Set Z has fewer elements different cardinal numbers. The set than Set Y. The cardinal num- containing more members has a card- ber of Set Y is greater than the inal number greater than the cardinal cardinal number of Set Z. number of the other. The sct with fewer members has a cardinal number lean than the other. Set Y Set Z The comparison of the numbers may be shown by using the symbol < to mean is less than and the symbol > to mean is greater than.

4 > 3Four is greater than three. 3 < 4Three is less than four.

Also, N(Y) > N(Z) N(Z) < N(Y)

13' /11 Matl--natical Ideas Illustrations and Explanations

The N preceding the name of the set means "the numberproperty1 of the set." This use of N is net the same as theuse of n as a variable in sentence such as n + 5 = 7.

When sets are arranged in order of (*) (*, *) (*, *) (*, *, *, *) k, *, *, *} one more thane cardinal numbers 1 2 3 4 5 of the sets are ordered and become the counting numbers.

Variables Examples from the set of whole numbers:

A variable is a symbol that may be + 2 = 9This sentence is neither true nor false.If is replaced replaced by any element from a given by 7, the sentence reads 7 + 2 = 9 which isa true state- set. Symbols such as , P, 0 are ment. Thus 7 is a solution to the open sentence. used in early grades and letters such as n, x, or y in later grades.

The solution of a mathematical sen- y -. 7 = 4If y is replaced by 9, the sentence reads 9 7 = 4 tence with variables is the set of num- which is false. Thus 9 is not an element of thosolution bers that replaces the variables and set. However, if y is replaced by II, the sentence reads makes the sentence true. 11 7 = 4 and 11 is a solution.

When the variable in a sentence is re- 5 8 = n This sentence has no solution in the set of whole placed by an,element from the given numbers. There is no whole number thatcaa replace n such set, the sentence may be classified as that the sentence will be true.Hence, the solution is "true" or "false." the empty set.

Example from the set of integers: 5 8 = nThere is a number in the set of integers that can replace n such that the sentence is a true statement, namely -3. Thus the solution is -3 because 5 8 = -3. When a given variable occurs more Examples from the set of whole numbers: than once in a sentence, the same element is used for each occurrence of 0 x = 36 0+ 0 = 12 the variable. 6 x 6 = 36 6+ 6 { 6 } {6 } When different variables occur, they + =10 6 may represent different numbers or 8 2 the same numbers. Some pairs of numbers that make the 4 6 sentence true are shown. 0 10 S 5

Counting Counting isthe process of finding how many elements there are in a set. In counting, the elements of a set are matched with the counting numbers beginning with one. The last name The cardinal number of the set is "5."

8 saidinone-to-onecorrespondence of number names with elements of a set is the cartInal number of that set. This cardinal number names the number property of the set.

As elements of a set are counted, each January is the first month of the year. element is named in a definite order. February is the second month of the year. The order or position of each element is designated by an crdinal number. March is the third month of the year. First, second, and third are ordinal numbers which specify the order in which months are named in the year.

Set Terminology Equal sets: Two sets are equal if and A = (*) B = (*) A =B only if they contain exactly the same E= {1,3,5) F = {5,1, 3) E = F members. C = {2, 4, 6) D = {3, 5,7) C D

Set C and Set D are not equal. However, Set C is equivalent to Set D since the elements of Set C can be matched in one-to-one correspondence with the elements of Set D.

When the symbol "=" is used in mathematics, it means is identical to or is exactly the same as. For example, if A and B are sets, A = B means Set A is identical to Set B. For numbers, 5 + 3 = 8 means 5 + 3 represents the same number as 8 even though the numerals "5 + 3" and "8" are different. Likewise1=1means 2 4 that 1 names exactly the same number as1even though"I" and 2 4 2 "1"are different fractions (symbols). The symbol 7(--is used to 4 indicate is not identical to or is not exactly the same as.

Disjoint sets:Disjoint sets are sets which have no members in common. John Bill

Jane Nell

Alice Jo

Set A Set B Set C Set D

Empty set:The empty set is a set The Set M is containing no members or elements. { an empty set. Zero is the number associated with Set M the empty set andisitscardinal number. Another symbol for th r:. empty set is 0.

9 1,5 Mathematical Ideas Illustrations and Explanations Subset: A subset of A is a set B B JSet of all boys in the such that each element of B is an ele- All boys ischool. ment of A. Set B may contain every All boys A in school in element of Set A. By agreement, the the {Set of red-haired boys empty set is a subset of all sets. with red hair school in the school.

The set of red-haired boys is a subset of the set of all boys in school.

X Y Y

P Z Set B is a subset of Set A. Set A Set B

Finite set: A finite set is a set whose D = (All days in the week) members can be counted withthe counting coming to an end. C = {Mary, Jane, Jim) N = (All whole numbers less than 6)

Infinite set: An infinite set is a set The set of all counting numbers whose elements cannot be counted so that the counting comes to an end. Set C (I, 2, 3, 4, ...) There is no last number. The symbol... is placed after the last element of an infinite set to denote "continuation in this manner indefinitely."

Set Operation Y X U = I X Y Union of sets: The union of two sets Y X X Y is the set consisting of the elements contained in either set or in both setr. Set B Set C

Sets are joined, not added, and the symbol forset union is U.

{xxxx) U (yy) = xxxxyy I Set A U Set B= Set C

The operation of joining disjoint sets is the foundationfor addition. If an element is a member of both sets, the element is listed only once in the union a two sets.

Set A Set B A U B= (1, m, n, o, p) The union of a set with the empty set (A, B, C) U () = (A, B, C) is the set itself. The union of the empty set with another setmay be used to establish zero as the identity element in addition.

10 16 Nfathematical Ideas Illustrations and Fxptanatiors

Separationofsets:Setseparation involvespartitioning members of a set to form subsets.

A set of five boxes may be separated into several pairs of subsets such as: 3 boxes and 2 boxes 4 boxes and 1 box 5 boxes and 0 boxes

ISeparating a set into two subsets may be used to introduce the

1 idea of subtraction.

The symbol for inter- secron is fl. Intersection of sets: The intersection Set A A fl B = Jan of two sets is a set consisting of the Joe[Bob Nell Sallie 1 elements that ace common to the two sets.

Set B

A = {Joe, Bob, Jan, Nell, Sallie}

B = (Jane, Sue, Jan, Phil, Al) Set C Set D Cf1 D = (5, 6)

The intersection of two disjoint sets is the empty set. Set E Set F

The ideas involved in the intersection of se's areused in many topics in geometry.

Summary of Sets of Numbers Counting numbers: The set of count- Example: Natural Numbers N (1, 2, 3, 4, 5, .. .) ing or natural numbers begins with 1 2 one and continues without end. --ie Imsrmwimmtdi The set of whole numbers: The set Example: Whole Numbers of whole numbers con:sts of zero and W (0, 1, 2, 3, 4, 5, ...) 0 1 2 all of the counting or natural num- imurosimmummi rawilmmOi. bers. .4_, 1

11 7 Mathematical Ideas Illustrations and Explanations

The set of integers: The set of in- Example: Integers tegers consists of Or set of whole numbers and the set of negatives of = (. -2, -1, 0, +1, i-2, .) the natural numbers. -2 -1 0 1

The set of fractional numbers:The Example: Fractional Numbers set cf fractional numbers consists of numbers which 57 arewritteninthe F = {-0 , , 1. a 36 82 form, where a andbare whole num- b -2 -1 0 1 2 bers and b 0. .ammsmimanwriossauhamplmnomommaigissa 0 5i _ . 4 Ti 6 ' I.

The set of rational numbers:The set Example: Rational Numbers 8 of rational numbers consists ofnum- 1 7 betsbers which can be expressed in the R {3,1, 6, -1 ...} 2' 2'6'3' form of !- where a and b are integers -2 -1 0 1 2 and b 0. Asarammtlavamammloadmors=immsme ,rariassmoris The sets of counting -6 -5 -4 -3 -2 -1 1 2 3 4 numbers, the whole numbers, the frac- 3 3 3 3 3 3 2 2 tional numbers and the negatives of all of these numbers are subsets of the set of rational numbers.

The set of irrational numbers:The set Example: Irrational Numbers of irrational numbers consists ofnum- bers that match pointson a line and The symbol V representsthe square root of. which cannot be expressed inthe Since there is no rational number which when multiplied a by itself, form where a and b are integers, exactly equals 2, there is no rational representation of \if.However, b a rational approximation can be made to any desired degree of pre- b 0. cision.

There is a point on the number line that corresponds toevery rational number. However, there are points on the number line that do not correspond to any rational number. One of these points is demonstrated below.

-2 -1 2 otinimamimmomm..

By the Pythagorean Theorem the sum of the squares of the sides ofa right triangleis equal to the square of the hykotenuse.Applying theorem to the above triangle the hypotenuse is N,q. Usinga compass with rad;us equal to VI and describing an arc to intersect the number line, the point of intersection is \i/. This point of intersectioncorre- sponds to the number V.I. This same point does not correspondto a rational number.

12 18 Mathematical Ideas Illustrations and Explanations

The set of real numbers: The set of 2 -1 0 1 2 real numbers consists of the set of 41111114111111111111111%1MEIMINIMMIMMIONMINIMMINI.M... 8 rational numbers and the set of irra- -6 -3 -2 -1 1 2 3 4 N/2 Ti tional numbers. There is a one-to-one 3 i 3 3 -4 4 LT -4 correspondence between real numbers and all points on the number line. For each real number there is a point on the number line.For ach point on the number line there is a real number.

Elementary Aspects of Number T Even numbers: (0, 2, 4, 6, 8... .) Even numbers: A whole number that 0 = 2 x 0 is the product of 2 and a whole num- 2 = 2 x 1 ber is called an even number. 4 = 2 x 2 6 = 2 x 3 8 = 2 4

Odd numbers: Any whole number Odd numbers: (1, 3, 5, 7, 9,. .) that is not an even number is an odd 1 = (2 x 0) + I number.If a whole number isdi- 3 = (2 x 1) + 1 vided by 2 and yields a remainder of 5 = (2 x 2) + 1 1, the number is an odd number. 7 = (2 x 3) + 1 9 = (2 x 4) + I

Prime numbers: A prime number is The set of the first eight prime numbers = (2, 3, 5, 7, 11, 13,17, 19.) a counting number greater than 1 that has only itself and I as factors. With the exception of the number 2,all primenumbersareoddnumbers. There is no greatest prime number.

Compositenumbers: A composite number is a counting number greater The set of the first ten composite numbers = (4, 6, 8, 9, 10, 12,14, than 1 that is not a prime number. 15, 16, 18.)

Prime factorization: Every composite Pairs of factors of 36 are shown on the second line (from thetop) of number can be expressed as a product the factor tree. Each factor that is not prime is factored again.The of primes in exactly one way except prime factors of 36 are shown on the bottom row. for the order in which the prime fac- 36 tors appear in the product. This is the 36 /6\ fundamental theorem of arithmetic. /6\l\ 1 A A4 /\9 2 32 3 2 2 3 3 1\2 3i3/2 36\ A /2 36 -= 2 x 2 x 3 x 3 = 22 x 32

2 2 3 3

13 19 Mathematical Ideas Illustrations and Explanations

The greatest common factor of two There are two ways to find greatest common factors (g.c.f.). counting numbers: The greatest common factor of two counting numbers The g.c.f. of 12 and 18 is the greatest numberwhich is a factor isthegreatest number whichisa of both 12 and 18. factor of both given numbers. 1.List the factors of both numbers and note thegreatest factor common to them.

Factors of 12:I, 2,3,4,6,12 Factors of 18:1, 2,3,6,9,18 Common factors:1,2,3.6 g.c.f. is 6.

2. Factor each number into its prime factorization. 12 = 2 X 2 x 3 18 = 2 x 3 x 3

The prink factors commonto both numbers are 2 r.nd3. Then 2 x 3 = 6 is the g.c.f. of 12 and 18. The least common multiple of a set of countingnumbers: The least com- To find the least common multiple oftwo numbers: mon multiple(I.c.m.)of a set of counting numbers is the least number 1. List the counting number multiples ofeach number. Then note the that is a multiple of the numbers. least multiple common to the numbers. Example: Find the 1.c.m. of 8 and 12.

Counting number multiples of 8:8, 16 32, 40, 48...

Counting number nultiples of 12:12 36, 48, 60.. . The intersection of the two sets of multiples is theset of common multiples: 24, 48, 72, 96... The 1.c.m. is 24.

2. Factor each number into its prime factors.The I.c.m. is the product of the greatest common factor and the remainingfactors. Example: Find the I.c.m. of 8 and 12. 8= 2 X 2 X 2 11 = 2 X 2 x 3 Greatest common factor is 2 x 2 The remaining factors are 2 and 3 I.c.m. = (2 x 2) x 2 x 3 Lc m. is 24.

To find the 1.c.m. of three numbers:

I. Find the 1.c.m. of two of the numbers.Then find the 1.c.m. of these two numbers and the third number.In a similar way, the 1.c.m. of four or more numbers is found. Example: Find the I.c.m. of 8, 12, Lnd 15. 1.:.m. of 8 and 12 is 24 1.c.m. of 24 and 15 is 120 Wrice,l.c.m. of 8, 12, and 15 is 120.

14 Mathematical Ideas Illustrations and Explanations

2. Divide the numbers of the set by successive prime numbersuntil all the quotients are one. If a number is not divisible by theprime number, repeat the number on the next line. 28 12 15 24 6 15 22 3 15 3 1 3 15 5 1 1 5 1 1 1

Then the I.c.m. is the product of the prime divisors. l.c.m. =2X2X2X3X5 = 120

Although the Most attention is given in the elementary school to thepositive rational numbers and zero. theory behind imaginary numb( , is beyond the scope of aguide for elementary school mathematics, the setof imaginary numbers is included in the diagram below to indicate toteachers that later courses in mathematics may extend the study of numbers to encompass the entire system of complexnumbers.

Diagram of the Complex Number System

Imaginary Numbers

Complex I/umbers Irrational Numbers The Fractional Numbers and the Negatives Real Numbers of the Fractional Numbers

Negatives of the Rational Numbers Natural Numbers

Integers Zero

Whole Numbers

Natural Numbers

15 21 TEACHING SUGGESTIONS FOR SETS AND NUMBERS

The Meari;ng of Set Number

Children enter school with some underst.--ilding of Sound concepts of number develop slowly over long sets. From familiarity with a pair of shoes, a set of periods of time. Since concepts flow from individual dishes, a bunch of carrots, a flock of birds, or a collec- manipulation and observation of phys!. 1 objects, it is tion of rocks children recognize many synonyms for mandatory that provision be made for children to have the word set. Sets are recognized as crayons are placed many experiences with concrete and semi-concrete main boxes, books filed on shelves, pictures arranged on terials to develop understanding of the number property bulletin boards, dishes placed on trays or children of sets. grouped fcr games.Games may be developed to identify and describe many more sets in the environ- =A, A, , , i } ment. Familiar objects may be classifieci and grouped A=(t4,4,4 by size, shape, color or use. B if A} F= (X, X, X } Although a collection may consist of more than one cbject, the noun setissingular and refers to one group.It is use of the idea of set that is of real sig- (5, rit, Ili; it, It) = (4, 4 , 4 4 ) nificance in elementary mathematics. Over emphasis or over formalization of either set vocabulary or sym- H K=(4, A,X} bols may, in fact, be detrimental in helping children to , 0, ) develop understanding of the important uses of sets in number work and in geometry. Although ideas of equivalence and non-equivalence are developed almost simultaneously, it may be simpler to introduce non-equivalent sets first, particularly when One-to-ane Correspondence the two sets are obviously non-evivalentFor example, begin with sets such as A and B where com- As opportunity is provided for children to find their parison is obvious and move to other examples as own chairs, distribute books or materia,s, match pairs shown where careful matching is needed. of objects or objects to pictures, understanding of one- to-one correspondence develops. Matching fingers of glove to fingers on a hand, of mittens to hands, of hands to eyes, of eyes to ears, of feet to shoes arc examples of one-to-one correspondence with which a 0 child is personally concerned. Many of the folk tales with which the child is familiar, such as "The Three Kittens," "Goldie Locks and the Three Bears," "Snow White and the Seven Dwarfs," capitalize on the idea of correspondence. As children dramatize these stories, I they reinforce ideas of correspondence. 50

From manipulation of concrete objects teachers may Sets should be arranged in many patternsregular, move to semi-concrete representation on felt boards or irregular, compact and scattered--in order thatthe bulletin board to extend understanding. Care should be number property of the set may not be confused with taken to show that the one-to-one correspondence exists the size of elements or the amount of space occupied regardless of the position of elements within the sets. by objects in the set. Elements of sets arranged compactly may be scattered or rearranged to present many different patterns. Den- Counting sity and arrangement tend to influence a child's per- ception of one-to-one correspondence. Variety in kinds Care should be taken to see that number names are and shapes of elements, arrangements of the elements learned in relation to concrete objects and that the and ways in which sets are matched broadens the con- number names are ordered. Many children enter school cept of one-to-one correspondence. Equally important knowing number names and this knowledge should be is the need to provide opportunities for observation recognized and utilized. Songs such as "Ten Little In- and discussion as ideas of correspondence develop. dians," "One Potato," "This Old Man" and various

16 22 counting rhymes and finger plays are helpful in ordering the fact that the number of the set with no mem- ing the names.Counting :ischildren march utilizes bers is zero. rhythmic sense thatreinforces !mining of number To develop further the idea that zero is the number names in order. of the empty set, there may be a discussion of a series Although the terms cardinal and ordinal are not used of sets. Objects in a box may be counted to determine in elementary school, it is important that children be the number A elements in the set.Objects may be re- able to use number in the cardinal sense of "how many" moved one at a time, courting after each removal to and the ordinal sense of "which one." In using num- establish the number then in the set. When all elements ber in the ordinal sense care must be taken to insure have been removed, zero is established as the number that children know where counting has started in estab- name of tl'e set with no members, or the empty set. lishing the order: left, right, up, down, front and back. "Zero," not "none," isthe answer to the question, The cafeteria line may be used to show the importance "How many members in the empty set?" of a point of reference since it makes a difference when a child is fourth from the front or fourth from the Subsets:Such activities as the following may be back of the line. usi,c1 to develop the idea of subsets: The number name, four, may also be used in the 1. Using children in the room as members of a ordinal sense of "which one." For example, the posi- set then asking the subset of children wearing tion of the marked object, counting from the left can glasses to raise hands; the subset of children be named as "three" or "third." The context ha which with blond hair to stand; the subset of children names such as "three," "six," "ten," are used will indi- with orange eyes to come to the front of the cate whether the number is used in an ordinal or a room, etc. cardinal sense. 2. Separating sets of objects into a variety of subsets. to 3. Using yarn to encircle various subsets of felt objects on the flannel board. 4. Describing the members of subsets of a given set.For example, naming members of the Set Terminologs subset of months from June to December. Equivalent sets: See Number, page 7. Infinite sets: While it is not important to teach the Equal sets:The idea of equality may be most terms finite and infinite, these ideas are important in effectively developed by rearranging elementsina developing the concept that there is no greatest count- given set to demonstrate that change in pattern in no ing number.Even pfmary children can understand sense changes the set since it still contains the same that there is always a next greater number. members. Each new arrangement forms a set that is In upper elementary grades children may be asked equal to the original. to name the greatest number they know, then to determine whether there is a greater number. After dis- Empty sets: Meaning of the empty set may be de- cussion of this they discover that counting goes on and veloped through such activities as these: on. 1. Placing collections of materials :n 4 of 5 containers(bags, covered bo7.es, jars, pockets), Set Operation leasing one container with no members and pointing out that this set is called the empty Union of sets: The union of disjoint sets may be set. illustrated by: 2. Asking children to describe the sets in the room 1. Having four boys join three girls to form a new which have no members. For example: the set set of children. of boys with green hair, the set of elephants, 2. Moving two rows of desks together to demon- the set of girls two years old. strate joining of sets. After the meaning of empty set is understood, zero 3. Stringing sets of beaus. is introduced as the cardinal number of the empty set. Activities similar to those described above may be used 4. Joining sets of objects or pictures on the flannel by asking children to give the number of the set, stress- board or on the magnetic board.

17 23 Later children form the union of sets which are not 1. Giving each child a set of counters; asking that disjoint and may use sets whose members are words, these counters be separated into two sets. numbers, or geometric figures. For example: 2. Placing objects on the flannel board, and using A = {1, 3, 5, 7, 9} B {3, 6, 9} a ruler, pencil, or string to show separation into subsets. A U B {1, 3, 5, 6, 7, 9}

Intersection of sets:The intersection of sets may be Summary ofSetsofNumbers illustrated by: This materialis included to help teachers under- 1. Asking childrenin row one to raise hands, stand that there is relationship between sets of num- asking children in column one to stand, and dis- L is studied in elementary school and sets of numbers cussing the one who is both standing and raising which may he studied later.Intentionally. no teaching his hand to demonstrate that this child is in the suggestions are included in this section.Suggestions intersection. relative to teaching those sets of numbers pertinent to elementary mathematics will be found in other sections 2. Discussing activities in which pupils participate of the guide. outside of the classroom:band, gleeclub, science club, math club, or library club and listingchildren who belongtothevarious E' ntar) Aspects of Number Them} groups. Diagrams may be made to illustrate leand Com: osite numbers:An interesting set intersection. For example: sa of separating the prime numbers from the composite numbers is the Sieve of Eratosthenes. To M G find prime numbers less than 50, first list all counting Nora. numbers greater than 1 and less than 50: Jean John CBill Philip Draw a ring around each prime number and cross Bob Beth out each composite number.In this sense only the prime numbers will not pass through the sieve. To M = members of Math Club begin, draw a ring around 2 and cross out each multiple G = members of Glee Club of 2 thereafter.(If a number greater than 2 is a multiple of 2, it cannot bis primesinceit has 2 as a AnB = {Mary, Joe) factor ) Mary and Joe belong to the intersection of sets M and G. 0 3 / 5 if 7 9 1431 3. Using a set of books with a set of pencils to 11 / 13 ;( 15 / 17 3,8' 19 yl show that the intersection of two disjoint sets is an empty set. 21 joill 23 30( 25 X.27 3$' 29 X Books 31 ?if 33 ,e35 37 yd 39 VC 41 / 43 $.4' 45 106 47 $ig 49

Next, encircle 3 as a orime number and crossout 0 all multiples of 3 that remiiin. \, Pencils 5 G(111=( } 0 Set union and separation are introduced in early II 13 17 19 grades as foundation for the operations of addition and subtraction. Excessive time spent on set symbolization 23 25 is to be avoided. 31 35 37 Pc Separation of sets:Ideas relative to the separation of sets may be developed by: 41 43 47 49

18 Next, encircle 5 and cross out all multiples of 5 that remain. 00 0 0 11 13 17 19 7 00 CO 23 29 11 13 17 19 31 37 23 29 41 43 47 X 31 37 All the remaining numbers are prime numbers and 41 43 47 49 may be encircled. This can be verified by attempting to factor them.

Of the remaining numbers, encircle 7 and cross out In similar fashion, the set of prime numbers less its multiples. than any specified number can be determined.

19 i NUMERATION

A numeration system is a means of naming numbers.It involves a set of symbolsornumeralsand rules for combining these numerals to name numbers. An efficientsystem of numeration is a significant achievement for any people.It evolves as the result of ages of experimcmtal manipulationof concrete objects, tally marks, and symbols of various kinds. The ability to work with abstract symbolsas representations of real objects is a distinguishing characteristic of civilized man.

A section on numeration is included in this guide primarilyto can attention to five features of the Hindu- Arabic decimal system which make it an efficientsystem of numeration: 1, The Hindu-Arabic decimal systemuses a base of 'en. 2. The system employs a finite set of symbols represented by thenumerals or digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 3. A place value based on powers of ten, increasing in value fromright to left, is assigned to each position in the numeral to represent numbers larger than 9. 4. Each digit in the numeral represents the product of the numberit names and the place value assigned to its position.

5. Each number named is the sum of the productsmentioned above.

While other systems may have some of these characteristics,itis important to remember that the Hindu- Arabic system includes all five.It is also important to note that a place valuesystem makes it possible to extend the numeration system to the right of theones place so that any rational number may be represented.

The study of other systems of numeration is included in this guideas a means of providing deeper understanding and appreciation of the efficiency of the Hindu-Arabicsystem. These include systems that use a base other than ten, those that use different sets of symbols, andthose which do not employ place value. Use of bases other than ten is to be found in tables of measurements in dailyuse: Base sixty in sixty seconds make one minute, sixty minutes make one hour; base twelve in twelve inchesmake one foot; base sixteen in sixteenounces make one pound. Systems that use different sets of symbols are represented by Romannumeration and systems which do not employ place value by the Egyptian system.

Mathematical Ideas Illuctrations and FAplanations

Ifindu-Arabic Decimal Num:ration Sy stem Symbols: Basic sonbols for mmera- The finite set of symbols for tly Hindu-Arabicsystem is {0, 1, 2, 3, 4, tion are distinct, single characters as- 5, 6, 7, 8, 9). These unit symbolsare commonly calleddigits. signed to each whole numberzero through 9.

Base:The Hindu-Arabic system em- Example: ploys the use of a combination of two or more symbols when recording num- 1, 2, 3, 4, 5, 6, 7, 8. 9, base. Using the first digit and0, the bare bers above 9. The numeral for the becomes 10 which means 1 base andno units. In other numeration ",e number which is one greater than 9 is system might be: 1, 2, 3, 4, base, or 1, 2, 3, 4, 10 (1 baseano ..o called ten (10) and becomes the base units). of the system. The Hindu-Arabicsys- tem is called a decimal system of numeration because it uses a base of ten.

20 2 Mathematical Ideas Iliuslt.ations and Explanations

Place Value: The Hindu-Arabic system of numeration in addition to the idea of base also employs a positional pattern in naming any number greater than 9. The numeral for ten and all succeeding numerals of two or more digits are combinations made according to this pattern. In the numeral 10 the digit 1 occupies the position of I base and zero occupies the unite position.

The numerals from ten to nineteen Example: have been given unique names. 10 1 base or (1 x 10) + 0 Ten 11 (1 x 10) + 1 Eleven 12 (1 x 10) + 2 Twelve 13 (1 x 10) + 3 Thirteen 14 (1 x 10) + 4 Fourteen 15 (1 x 10) + 5 Fifteen 16(1 X 10) +6 Sixteen 17 (1 x 10) + 7 Seventeen 18 (1 x 10) + 8 Eighteen 19 (1 x 10) + 9 Nineteen

Unique names derived from names 202 bases or (2 x 10) + 0 Twenty for 2-9 have been assigned to each 21*(2 x 10) + 1 Twenty-one multiple of ten (the base) from ten to ninety. 30(3 x 10) + 0 Thirty 40(4 X 10) + 0 Forty 50 (5 x 10) + v Fifty 60(6 x 10) + 0 Sixty 70(7 x 10) + 0 Seventy 80 (8 x 10) + 0 Eighty 90(9 x 10) + 0 Ninety 100(10 x 10) Hundred

'The pattern used in counting from 1-9 continues with each multiple of 10 beyond 20.

21 27 Mathematical Ideas Illustrations and Explanations

Unique names have been assigned to Example: each group of ten tens or hundredsas each succeeding position to the left 1,000(10x 10x 10) Thousand become; 10 times the place value to 10,000(10 x 10< 10 x 10) Ten thousand its right. 100,000(10 x 10,< 10X 10X 10)Hundred thousand

Billions Millions Thousands Units

Two ideas are involved in the place value principle: I. There is a number assigned to each position in thenumeral. 2. Each digit represents the product of the rJmberit names and the place value assigned to its positior..

Zero:Zero is the numeral used to de- Zero is the number of the emptyset. note the idea of "not any." A = ( N(A) == 0

Zero matches the point on the number line whichseparates the set of positive real numbers from the set of negative realnumbers.

Zero is necessary in a numeration sys- Example: In the numeral 403, thezero indicates that there are no tem which uses place value. Where 0 groups of ten in the numeral. The numeral, therefore,names four appears in a numeral, it indicates not groups of one hundred, no groups of ten, and threeones or units. any groups in the place which is held by the zero.

Exponents:In the base ten decimal For example: 102 10 x 10 sys..m of numeration factors of ten play an important role.10 X 10 can 10s = 10 x 10 x 10 be written 102, where the 2, called the exponent, indicates how many times 10+ = r i< 10 x 10 x 10 10, the base, is used as a factor. 10J 10 x 10 x 10 x 10 x 10

For any number a, ar indicates thatahas been used as a factor n times.

10,000 1,000 100 10 X 10 x 10 x 1010 x 10 x 10 10 x 10 104 103 102

22 2 Mathematical Ideas Illustrations and E3 planations

Following this, pattern, 101 is 10 and 100 is 1.

Any number to the first power (having Example:101 = 10 151 = 15 an exponent of 1)is equal to the 51= 5 81= 8 number. For any number a, al = a

Any number (except zero) to the zero Example:100 = 1 150 = 1 power (having an c-ponent of e..ro) 50 = 1 80 = 1 is equal to one. For any number a, a 71- 0, a° =

In summary: 6,543 =6,000 + 500 + 40 + 3 =(6 x 1,000) + (5 x 100) + (4 x 10)+ (3x1) -=6 x (10 x 10x 10) + 5 x (10 x 10)+ (4 x10)+ 3 x 1 = (6 x 103) + (5 x 102) + (4 x 101) + (3 x 100)

Non-decimal Bases The important features of place value Example: The numeral 101, is read, "one zero one, base two," and systems of numeration are: indicates that instead of grouping by ones (units), tens, and ten tens, groups of ones, twos and two twos are used. 1. A base 2. The use of a finite set of symbols Example: 101,,.= 1 x (2 x 2) + (0 x 2) + (I x I) 3. The use of place value 4. The use of zero Note: Zero and one are the only symbols emplc yeti in base two.

Example: 231,0, = 2 x (4 x 4) + (3 x 4) -1-( I x 1) The HinduArabic system of numeration uses the number ten as a base. The INote: 0, 1, 2, 3 are symbolsed in base four. choke of ten as a base is arbitrary. Any other number, except zero and one, can be used a a base. Examples: 987i..1., = 9 x (12 x 12) + (8 x 12) + (7 x 1) 5e6,,,1,. = 5 x (12 x 12) (e x 12) + (6 x 1)

Note: 0, 1, 2, 3, ... 9, t, e are symbols used in base twelve. I

23 29 Mathematical Ideas Illustrations and Explanations

Work in non-decimal bases should be limited.Its value lies in helping students better understand the decimal system of notation by emphasiz- ing the ideas of place value irrespective of base.

Ancient Numeration Systems 21;?Egyptian numeration system:The Example: Egyptian numeration system used the number ten as a basis for grouping Names For (base ten). Place value was not em- Hindu-Arabic Egyptian Egyptian Symbols ployed since the numeration system had no symbol for zero. Each group I I stroke of ten and each group of ten tens, etc. 10 requiredthecreationofdifferent n heel bone symbols. 100 9 coil of rope 1,000 g lotus flower 10,000 ,-- bent reed 100,000 burbot fish 1,000,000 t astonished man Numerals: Hindu-Arabic Egyptian 121 onni 563 in In losa 9nnn The Roman numeration system:The Roman numeration system used a base ten but added symbols for five, fifty, Order makes no difference since place value is not involved. and five hundred so that numbers could be recorded with fewer repeti- tive symbols.Otherwise the Roman Hindu-Arabic Roman system was analogous to the Egyptian system. 1 5 V

10 X

50 1.

100 500 1,000 10,000

34,029 XXXIVXXIX

24 30 Mathematical Ideas Illustrations and Explanations

In order to simplifyfurther the re- Numerals: cording of numbers the Roman system did two things: Hindu-Arabic Roman

Employed the subtraction principle 463 CDLXIII by substituting IV for IIII and XL for ICXXX. 400 = CD

Placed a 'oar over a part or all of a 60 = LX Roman numeral to indicate multiplication by 1,000. 3

The Roman system has a practical significance since it is used for recording dates, in numerals on clocks, numbering chapters in books, and in outlining.

25 31 TEACHING SUGGESTIONS FOR NUMERATION

Hindu-Arabic Numeration Sstern d. The numeral between 16 and 26 which repre- Symbols: Many children are able to count sets of sents a number divisible by 7, or which has at 7 as a factor. least five objects when they enter school and somemay even recognize a few number symbols. The extent of 8. Giving time for children to invent game this knowledge is easily determined. The problem then becomes one of associating number symbols, both 9. Writing numerals: words and numerals, with the number property ofa set. a. In order from 1 through 9 Ability to associate symbol with number may be b. To make price tags extended through such activities as the following: c. To play games 1. Beginning with sets with a known number of d. To keep score. objects aid match'ng them with cards showing the numerals: Base: learned to associate one of the nu- Front: merals 0-9 with the number property of the set which it matches, children are ready to learn how numbers 5 larger than 9 are represented. The word ten has been Back: associated with the nor -,per of fingers on normal hands or the number of toes an normal feet, but numerals Ito which say one more than nine or two more than nixie 1111#/imay not be recognized. Children work with such concrete materials as sticks, crayons, blocks, acorns, paper clips, or other objects (Later ,k,cord names may be added to the backs. which they have collected.It may prove helpful to be- A set of cards should be available for each gin with objects that may be grouped as 1 ten and 1 child involved in the activity.) one, then as I ten and 2 ones, before introducing I ten and 0 ones. The next step, the use of bead frames, 2. Providing stencils for children to cut out sets counting boards and abacuses, should be followed by of numerals to use in games and activities like use of semi-concrete materials, pictures or cut outs on those which follow. flannel board, to illustrate grouping by ten or base ten, 3. Playing games where teacher holds upa set of fingers and children match with the correct numeral; where teacher holds up a numeral and children match with sets of fingers.

4. Giving a numeral card to each child to finda place in line. 5. Giving each child a group of numeral cards to arrangeinorder from leasttogreatest,or greatest to least, or top to bottom. 6. Giving older children numeral cards of fractional numbers with like, then unlike denominators, or decimal fractions to arrange in sequential order. 7. Using old calendar pages and drawinga ring 9 1 2 1 1 around: I 0 a. Different numerals as dictated It is agreed that one mark or bead on left hand h. A numeral between two numerals: J-5, 8-10, rod represents 10 marks or beads on the right 15-17 hand rod c. The numeral that represents the even num- Place value: Understanding of place value may be ber between 9 and 6, or 2A and 28 developed by establishing a place to the left for bundles

26 3 " of ten sticks each and to the right for any remaining The 5 names five thousand Of 5,000 sticks. One bundle of ten sticks and one single stick The 1 names one hundred Of 100 may be recorded as I ten and I one. By adding bundles to the tens side and single sticks to the ones side, The 9 names nine tens (ninety) Of 90 larger numbers may be represented. These numbers The 8 names eight ones (eight) or 8 may then be represented on the abacus or bead frame and notation recorded.It must be emphasized that 2. Using any five digits, such as 8, 3, 0. 9, 7, to numerals are read from left to right across the page. represent the largest possible number and tie least possible number not beginning with 0. Making place value charts can re-enforce the understanding of place value.It is most important that place 3. Playing such games as the following to re-en- value demonstrations on a bead frame be seen by the force and evaluate understanding of place value: crass in the proper left to right perspective. a. Circle the 5 which represents the largest num- When ten bundles of ten accumulate,theseare ber in each of the numerals: bound together to make one large bundle of ten tens. Procedure then is as with tens and ones; recording, 5345 demonstrating on the abacus and transferring notation 2545 to the number chart, 1 hundred, 6 tens, 2 ones, etc. 5555 b. Order the numbers from largest to smallest.

4. Writing numerals using any three digits, such as 1, 2, 3 repeating use of each in as many places lI as desired. For example: 2 123 1,123 13,231 10 tens 321 3,321 232,111 231 2,312 111,222.. This activity can lead to the realization that an infinite set of numbers can be represented by using a finite set of symbols.

Zero: Activities described for place value illu ,crate a 1 6 2 use of zero in a place value numeration system. Vihed Abacus recording numerals which represent the number of Number Chart objects grouped in ones, tens. hundreds, etc. or which represent a number illustrated on the abacus, attention When the idea of place value is established, numbers should be called to the use of 0. For example, just as may be written in expanded notation as: 7 is used to indicate the number of hundieds, zero is used to indicate that there are zero tens in seven hun- 6 tens, 2 ones z1 52 = 60 4. 2 dred three, 703, and to indicate zero hundreds in oi,e thousand eighty-nine,1,089.Understanding zero as 3 hundreds, 6 tens, 2 ones == 362 .= 300 + 60 + 2 the cardinal number of the empty set should be recalled I thousand, 3 hundreds, 6 tern, 2 ones =-= 1,362 = and utilized. 1000 + 300 + 60 + 2 Exponents: Exponents provide a convenient way to Children may work independently to establish un- record the powers of the base and the idea may be derstanding of place value using such activities as the deseloped by building on understanding of place value following: notation. 1. Indicating the value of each digit in a numeral: 100 is 10 tens or 10 x 10 which may be For example: In the numeral 365,198: written 102 The 3 names three hundred thousand or 300,000 1,000 is 10 x 10 x 10 or 103 The 6 names sixty thousand or 60,000 10,000 is 10 x 10x 10 x 10 or 104

27 I 105 = 100,000 Symbols for other bases may be introduced.For 104 = 10,000 example- Using base four and a single object, write 103 = 1,000 1 four.b., adding one object at a time, children see h 102 = 100 the base four symbol names each number of objects. What next? When four objects are bundled, the use of place value 101 = 10 in the notation for "one four and 0 ones (10?...) 100= 1 should be discussed. Continue adding objects a,adre- cording the symbol for each number of objects: Opportunity may be provided for children to write numerals in expanded notation using exponents: .6 429 = 400 + 20 + 9 = (4 x 100)+ (2 x 10) + (9 X 1) = 1 four 2ror (4 x 102)+ (2 x 101)+ (9 x 100) lOreUr

6,097 = 6,000 + 000 + 90 + 7 = 00 (6 x 1,000) + (0 X 100) -f II (9 x 10) (7 x 1) 1l four 12 ro4, 13/o, 20,0, = (6 x 103) + (0 x 102) + (9 x 101) + (7 x 100) Bundle objects whcn necessary. This processmay be continued until four fours are bundled into a sixteen. The numerals may be recorded in a place value chart similar to those above Non-Decimal Bases Non-decimal numeration systems are introduced to Place value charts my be built in other bases. re-enforce the understanding of the concepts of base Children may use what they know about place value and place value. to suggest how to write the numerals. As numerals arc recorded, the part played by base and place value Children may group objects and discuss the group- should be emphasized. ings. For example: Enrichment activities for Some pupils could include. 1. Making a calendar using different bases for A A. A A 1 five and 3 ones each month. 2. Writing age, weight, height, and birth date it other bases. 3 = 13,1 3. Completing the blanks in such exercises as th- following: 44i1,. = (4) fives (4) ones 2 threes and 2 ones 320, = (3) sixes (2) ones

201 rc,c = (2) nines (0) threes (1)on; s 2 2

Groupings may then be recorded in a place value chart. For example:

Twenty- Fives Th rees 1 Nines Tens fives Ones Hundreds Ones (Base) iv. (Base) (Base)

2 2 8

28 3 4- = 4 sevens 2 ones 3. Look in museums for samples of Egyptian (236.40 = 2 sixty-fours 3 eights 6 ones writings containing numerals. = 2 sixteens 3 ones The Roman Numeration System 4. Writing base ten numeral-, for: The Ro-ran numeration system may also help chil- 23, = (13) dren to avrcciate theefficiency of a place value ten decimal numeration system. After the introduction of 111 three = (13) the different symbols a discussion of the subtraction t en principle involved is necessary. Roman numerals can then be compared with Egyptian as well as with the 36,0, = (30) 104,, (40) Hindu-Arabic symbols. ten ten Discussion of ways Roman numerals are used today 101,.= (5) 321rnnr= (57) will include dates on buildings or movies; numerals ten ten on clocks; designations for outline or chapter headings or volumes of books. The Egyptian Numeration System Practice in writing Roman numerals and in converting Roman numerals to other systems may be used as By comparibg the Egyptian numeration system with enrichment activities or as teachers see practical need a place value system pupils can see advantages of a for this experience. place value system of numeration. Symbols employed by the Egyptians may be introduced with emphasis Pupils interested in demonstrating the efficiency of on the base ten idea in the system and on the fact that the Hindu-Arabic numeration system as contrasted with symbols were repeated because the system did not use another system may use multiplication examples such place value. as the following: Children with special inteiest in ancient numeration 237 CCXXXVII systems may: x 12 XII 474 CCXXXVII 1. Write base ten numerals for given Egyptian 237 CCXXXVII numerals. 2844 MMCCCLXX 2. Learn more about the Egyptian numeration MMCCCCCCCLXXXXXXXXVV1111= system from reference books. MMDCCCXLIV

29 35 OPERATIONS ON WHOLE NUMBERS

Understanding of the nature of number and the numeration system is necessary if children are to perform operations on numbers efficiently.Such understanding develops gradually as opportunities are provided for children to manipulate objects, discuss and test ideas, experiment with ways to solve problems, and apply principles or generalizations to operations on numbers. Operations studied in the elementary school are addition and its inverse operation subtraction, and multiplication and its inverse operation division.Helping childrentounderstandthese operations and developskills necessary to use them effectively is a special responsibility of the elementary school. As operations arc performed, certain properties become apparent. For example: The sum of two numbers is not affected by the order in which they are added (2 + 3 3 + 2); the product of any number and one is the number itself (8 x I = 8); the sum of any number anditsopposite,or additiveinverse,isalways zero, (3 + --3 = 0). The structure that these properties provide makes itunnecessary to memorize many unrelated facts. Use of the properties is a distinguishing characteristic of contemporary mathematicsprograms. Significant as properties are in providing structure for operation on numbers, it should be kept in mind that understanding is developed and the property used before effort is made to name or use vocabulary associated with the various properties. Algorithms are developed for use in operations on numbers with the idea that there are many ways of working and that a child may find the way that is best suited to him. As understanding of the process develops, most children will find and use efficient patterns.

Mathematical Ideas illustrations and Explanations

Addition Addition is the binary operation on Fro .n union of disjoint sets children are Ied to assign the number two numbers called addends to pro- property to each set and then to perform the operation of addition on duce a third number called the sum. the numbers. The addends are the numbers associated with two disjoint sets. The sum 0 0 X x x 0 x x isthe number of the set formed by U =F . 0 X x 0 x x joining or finding the union of the two sets. 3 5 = 8

Set A Set B Set C

In set notation this becomes: N (A) + N (B) = N (C) or a + b N (A) is read:The number property of set A." a and b are addends and c is the sum.

Capital letters are used to denote sets and lower case letters are used as variables that represent numbers. Letters have been used inthe above illustration, and willbe used subsequently throughout the guide, to express generalizations which arc true for operations on all numbers.

30 36 Mathematical Ideas Illustrations and Explanations

1. PROPERTIES OF ADDITION 3 + 5 = 83, 5 and 8 are whole numbers. From experience with many similar situations it becomes evident that, for any whole numbers a and Closure: When In operation on two b, a + b == c; that c is also a whole number and that the set of whole numbers produces a number within numbers is closed under addition. the same set of numbers, the set is said to be closed under that operation. The set of whole numbers is closed under addition, since the sum is an element of the set.

Commutative property: The sum of 3 + 5 = 8and5 + 3 =8 the: i 5 = 5 + 3 two numbers is not affected by reversing the order of the addends. In generalizing after much experience it becomes evident that for any This applies to addition in all sets two numbers, x and y: x+y-=--y+x of numbers.

Assoeiati,.eproperty: When three 4 -11 (3+ 2) 4+ 5= 9 or more numbers are addends, the sum is not affected by the way in and (4 + 3)+ 2= 7+ 2= 9 which the addends are paired. This then 4 + (3 + 2) = (4 + 3) + 2 applies to addition inallsets of numbers. Therefore, for any numbers, a, b, and c: a + (b + c) = (a + b) + c

identity element for additionzero 6+0=6 0+6=6 (0): When zero (0) is added to a Therefore, for any number a, a number, the sum is the number, and I- 0 -= a or 0 + a = a zero is called the additive identity. ADDEPDS 2.ALGORITHMS FOR ADDITION Any pattern of procedures used to s0in2 3 4 5 6 7 8 9 tame the result of a mathematical operation is called an algorithm or algorism.Algorithms for each of 1 i2 . the operations become more com- 2 2 3 4 .. plexas numbers become greater than the base of the numeratio 3 3 4 5 6 system. CI517 8 I6 DIU10 A:gorithms with two addends less than bare: This involves the addi- 6 7Ela101112 tion facts shown on the table to the 7 7 8 91011121314 right. 8 910111213141516

9 1011 12131415161718

When the commutative property is applied, only 55 facts need to be lea rned, and by applying the zero property, only 45.

31 ? 37 Mathematical Ideas Illustrations and Explanations

Many algorithms may be developed for any operation. A child should be permitted to use the one which has meaning for him. For example: 8 +9=8 -1-(2+ 7) (Renaming 9 as 2 + 7) =(8+ 2)+ 7 (Associative property) =10+ 7 (Fact previously learned) =17 (Place value notation)

35 3 tens + 5 ones 35 +2 + 2 ones +2 Algorithms with one addend less 3 tens + 7 ones 37 than the base and the other greater than base: 94 9 tens + 4 ones 94 +8 + 8 ones +8 9 tens I- 12 ones 102 Renamed as: 10 tens + 2 ones or 102

The expanded notation in these illustrations is used only to 42evelop understanding of the process involved in the algorithm.It should not be used in writing to the point that it becomes needless repeti- tion of an involved process. Some children may go quickly to a shorter algorithm.

15 27 10 10 Algorithms with addends equal to +10 +10 +36 +83 or greater thci base: 25 37 46 93

26 2 tens + 6 ones 20 + 6 + 42 4 tens + 2 ones 40 + 2 6 tens + 8 ones or 68 60 + 8 or 68

397 3 hundreds + 9 tens + 7 ones 1-486 4 hundreds + 8 tens + 6 ones 7 hundreds + 17 tens + 13 ones

Renamed as: 7 hundreds + 18 tens + 3 ones Renamed as: 8 hundreds + 8 tens + 3 ones or 883 300+ 90+ 7 397 400+ 80+ 6 + 486 700 + 170 + 13 883

= 700 + (100 + 70) + (10+ 3) = (700 + 100) + (70 + 10) + 3associative property = 800 + 80 + 3 =- 883

32 38 Illustrations and Explanations Mathematical Ideas corrunutativ,! and associative Algorithms with three or more ad- Since addition is a binary operation, both properties .,re used with emphasis always on tieideas involved rather dends: than on the terminology.

Tens Units 24 24 2 4 82 82 8 2 57 + 57 5 7 15 13Rename as: 13 1 hundred + (5 + 1) tens + 3 units or 150 1 hundred + 6 tens + 3 units or 163 163

Subtraction + 3 = 7 Subtractionisthe inverse operation 7 3 = because of addition.Itisthe operation of finding one of two addends when their 4 = n becauseL] + 4 =-- 7 sum and one addend are known.

1. PROPERTIES OF SUBTRACTION Closure: Since it is not always pos- 6 2 = 4, a whole number sible to subtract a whole number from a whole number and get a 4 6 does not name a whole number whole number, thesetof whole numbers is not closed under subtraction.

Commwative properly: Subtraction 5 3 = 2 5 14 2 is not commutative. Ifb

Associative property: Subtraction is (9 3) 2 = 4 9 (3 2) = 8 not associative.

Zero: Subtracting zero from a num- 8 0 = 8 ber dot's not change the number. For any whole number n n 0 n

33 39 Mathematical Ideas Illustrations and Explanations

2. ALGORITHMS FOR SUBTRACTION Withaddends less than base: Sub- traction facts may be derived from the addition table. ADDENDS

+ 04.--oi-1 214 5 6 7 8 9 0 1 ,

A = + = A

With known addendslessthan, 78 7 tens + 8 ones 70+ 8 78 equal to or greater than the base: 6 (0 tens I-6 ones) (0+ 6) 6 Algorithmsforsubtractionusing 7 tens +2 ones or 72 70+ 2 or 72 72 two or more digit numerals are developed first without renaming and then with renaming. 78 7 tens 48 ones 70+ 8 78 10 ( I ten + 0 ones) (10+ 0) 10 6 tens + 8 ones or 68 60+ 8 or 68 68

78 7 tens + 8 ones 70+ 8 78 64 (6 tens + 4 ones) (60+ 4) 64 1 ten + 4 ones or 14 10+ 4 or 14 14

74 6 tens + 14 ones 60+ 14 74 58 -- (5 tens + 8 ones) (50+ 8) 58 1 ten + 6 ones or 16 10+ 6 or 16 16

374 300 + 70 + 4 Rename the sum as 300 460 + 14 98 (000 + 90 + 8) then as 200 + 160 + 14 (000+ 90+ 8) 200 + 70 + 6 or 276

40 Mathematical Ideas Illustrations and Explanations

Multiplication Multiplication isa binary ope,,tion 6 x 3 = 18Factors, 6 and 3, are the numbers being multiplied. on numbers called factors to produce auniquethird number calledthe 18 is the product, the number obtained through multiplication. product.Multiplication of numbers greater than one can be performed * **** *** using repeated addition of the same * ***** 6 + 6 + 6 ***3 + 3 3 + 3 + 3+ 3 whole number. * *****(3 addends) *** (6 addends) *** The product is a multiple of each of *** the factors. ***

For any numbers, a x b = c, a andbare factors, c is the product.

1. PROPERTIES OF MULTIPLICATION Closure:The product of two whole 16 x 47 = 752 numbers is a whole number. For any whole numbers, a and b, a X b = c. and c is a unique whole number.

Commutative property:The product 6X3=18 3 x 6 = 18 6 x 3 = 3 x 6 of two numbers is not affected by For any numbers, a and b, a x b b x a reversing the order of the factors. This applies to multiplication in all sets of numbers.

Associative property:In finding the (6x 3) x 4 = 72 6x (3 x 4) = 72 product of three or more factors, 18 x 4 = 72 6 x 12 = 72 the way in which the factors are paired does not affect the product. (6 X 3) x 4 = 6 x (3 x 4) For any numbers a, b, and c, (a x b) Xc=ax tb X c)

Distributive property of multiplica- 10 x 7 70 tion over addition:This property 10 x (3 + 4) = (10 x 3) + (10 x 4) = 70 links addition and multiplication in finding the product. (6 + 4) x 7 = (6 X 7) 4 (4 X 7) =70 For any numbers a, b, and c a X (b + c) = (a X b) + (a X c) or (b + c) x a = (b X a) + (c X a)

Identity elementone:Multiplying 8 X 1 = 8 by one does not change the number. For any number a, ax 1= a and I Xa=a

Zero: Multiplying any number by 8X0 =0 zerogivesthenumber zeroas For any number a, a X 0 =. 0, 0 X a = 0 product.

35 41 Mathematical !duns Illustrations and Explanations

2. ALGORITHMS FOR MUL IIPLICATION Algorithms with twofactorsless Factors than base: x0 2 3 4 5 6 7 8 9 (This involves the multiplica- 0 tion facts shown othe table to the [VV..) 0 1 2 02 3B3 6 . 0 l' 4 4 8 16 )0 5R51015Etal LL 6 612182430 36 7 0 714212835 42 49 8 0 816243240 485664 9El918273645 546372 81

When the commutative property and the property of zero are applied, it can be observed that facts in the shaded portion of the chart are the same as facts in the unshaded ser-tion.

The distributive property may be If 5 X 8 = 40 is known, 7 x 8 may be found: used to arrive at facts with fac- 7 x 8 = (5 + 2) x 8 or 7x (4 + 4) tors 6 through 9. (5 x 8) + (2 x 8) = (7 x 4) + (7 x 4) = 40 + 16 = 28 + 28 =56 =56

Algorithms with one factor equal to I x 10 = 10 9 x 10 = 90 15 x 10= 150 the base: 2 X 10 = 20 10 x 10 = 100 3 x 10 = 30 11 X 10= 110 27 x 10 = 270 115 x 10= 1,150

Algorithms with one factor greater 3 x 23 = 69 23 23 3 Man base: x3_ x3 x 23 9(3x 3= 9) 69 9 without renaming 60 (3 X 20 = 60) 60 69 69

with re.taming 23= 2 tens + 3 ones x4 = x 4 ones 8 tens + 12 ones = 8 tens +I ten + 2 ones = 9 tens + 2 ones = 92

42 Mathematical Ideas Illustrations and Explanations

4 x 23 = 4 x (20 + 3) Renaming 23 as (20 + 3) = (4 x 20) + (4 x 3) Distributive property = 80 + 12 Multiplication = 80 + (10 + 2) Renaming 12 as (10 + 2) = (80 + 10) + 2 Associative property = 90 + 2 Addit:on = 92 Addition

23 = 20 + 3 23 x4 x 4 x4 12 80 + 12 or 92 92 80 92

Algorithms with two factors ,greater 23 x 46 =(20 + 3) x (40+ 6) Renaming 23 as (20 + 3) than base: and 46 as (40 + 6) = 20 X (40 + 6) + 3 x (40 + 6) Distributive property = (20 x 40) + (20 x 6) + (3 x 40) + (3 x 6) Distributive property = 800 + 120 + 120 + 18 Multiplication = 920 + 138 Addition = 1,058 Addition

46 46 x 23 x 23 18 = (3 x 6) 138 120 (3 x 40) 92 120 (20 x 6) 1,058 800(20 x 40) 1,058

Division Division isthe inverse operation of 138 + 6 = implies 6 x CI= 138 multiplication.Itis the operation of 138 + 23 = implies x 23 = 138 finding a factor when the product and one factor are known. The unknown For any natural numbers a and b, if axb=c with c a whole number, factor may be called the quotient, the thena=c+bandb-c+ a known factorthedivisor,andthe product may be called the dividend. 4)12 Division is the equivalent of repeated subtraction of the same whole number. 4 1 8 4 1 4 4+1 0 3

37 143 Mathematical Ideas Illustrations and Explanations

Zero in division: Division is considered 15 + 5 3 3 x 5 =15 as the inverse of multiplication. Zero 0 =6=0 0 x 6 =0 as a divisor is meaningless and division by zero is not possible in the set of 6 + 0 is meaningless becauseany number times zero is zero, never 6. whole numbers. Zero cannot be used as a divisor.

NEVER DIVIDE BY ZERO! Partitive application: Division can be Twenty-four girls are to be divided into fourteams with the same used to find the number of members number of girls on each team. How in each of n equivalent sets. many girls will be on each team? 24 + 4 = 6 Six girls on a team.

Measurement application: Division can Each day a child uses three pages froma notebook. If the book con- be used to find the number of equiva- tains twelve pages, how many days will it last? lent sets of n each in a given set. 12 3 =9 pages left after first day 9 3 =-6 pages left after second day 6 3 =3 pages left after third day 3 3 =0 pages left after fourth day or 12 + 3 = 4

The bookwill last for four days. 4-- 0 1 2 3 4 56 7 8 910 1112

4th day 3rd day 2nd dayI 1st day

PROPERTIES OF DIVISION Closure: Since it is riot alwayspos- 6 + 3 = 2 a whole number se}le to divide a whole number by a ttliole number and get a whole 3 + 6 51a whole number number, the set of whole numbers is not closed under division.

Commutative property:Division is 12 =4 =3 not commutative. 4 + 12 74 3

Associative property:Divisionis (18 + 2) + 3 = 3 not associative. 18 + (2 + 3) sA 3

Distributive property:Divisionis 369 + 3 = (300 + 60 + 9) + 3 distributive with respect to addition -= (300 + 3) + (60 -:3) + (9 + 3) except that the distribution must be = 100 + 20 + 3 over the dividend with each opera- = 123 tion producing a whole number. (a + b + c) d = (a + d) (b + d) + (c + d)

38 44 Mathematical Ideas Illustrations and Explanations

2.ALGORITHMS FOR DIVISION Algorithms with factors less than 45 9 = implies x 9 -= 45 base: Thisinvolvesfactslearned from the multiplication. 5 9)45

Factors

X 0 1 2 3 4 5 6 7

1 2

= 411 3 tti 4 x 0

The operation of division can be 37 ± 5= [7] implief X 5 = 37 carried out with zero remainders or non-zero remainders. 0 X 5= 37 7x 5= 35 37 35 = 2 remainder (7 x 5) + 2 = 37

7 5)37 b)a 35 2

a ---- (b x q) r

b = divisor, known factor a = dividend, product q = quotient, unknown factor r = remainder

39 Mathematical Ideas Illustrations and Explanations

Algorithms with one factor equal to IO÷ ICI = 1 150 =10 = 15 base: 20÷10=2 3,460 ÷ I0 = 346

Algorithms with factors greater than 137 base: With one or bothfactors 7 greater than base the algorithm may 30 be developed in several ways. 100 137 7)959 7)959 7)959 700 100 (7 x 100 = 700) 700 259 259 25 210 30 (7 X30= 210) 210 21 49 49 49 49 + 7 (7 x7 = 49) 49 49 0 137 959

959 7 = 1:1 implies x 7 = 959 959 ÷ 7 = 137 implies 137 x 7 = 959

53 31 3 SO = 153 50 100 153 7)374 24)3678 24)3678 350 2400 24 24 1278 127 21 1200 120 3 remainder 78 7F 72 72 6 remainder 6 remainder

Any problem with two known elemen's can be solved if itcan be cor- rectly stated in either of the following forms: addend + addend = sum factor X factor = product Subtraction is the operation of finding the unknown addend. Division is the operation of finding the unknown factor when the remainder is

40 Mathematical Ideas Illustrations and Explanations

3. SQUARE ROOT OF A NUMBER Example: The square root of a number is one = 2 because 2 x 2 = 4 of two equal factors whose product is the number. The symbol, V , Nig = 5 because 5 x 5 = 25 designates the square root of a number. /9- 3 because x-3- 2- 16 4 4 4 16 1.5 because 1.5 x 1.5 = 2.25

Since3 X 3 = 9 and (-3) x (-3) = 9, it is agreed that

=3 and N.5 --= 3.

Algorithm for finding the square Example: Find root of a number:Unike the other operations that have been encoun- 1. Since 4 x 4 = 16. try 4 as a first estimate ofN/10. tered, all of which are binary, the operation of finding the square root 2. 19 + 4 = 4.75 4.75 of a number isunaryin that it in- 4)19.00 volves only one number. 16 30 The algorithm for finding the square root of a number may be described 20 asa "divide and average" proce- 20 dure. The goal is to get the divisor and quotient to be the same or as 4 4.75 A close as you wish. 3. V19 7 lies between 4 and 4.75 ------J 2 1. Estimatethevalueofthe square root. 4. Repeat step 3, averaging 4.375 divisor and 4.342° (quotient) 2. Using the estimate as a divisor, 4.375 + 4.343 divide the number to find the 4.359 quotient. 2 3. Average the divisor and quo- 4.359 tient. The result is a closer approximation to the square root of the number. 4. Using this new approximation as a divisor, repeat steps 2 and 3 to obtain an approximation of the square root as correct as required. However, with practice, two to three repetitions should produce a square toot ascorrectasnecessaryfor most purposes.

41 7 TEACHING SUGGESTIONS FOR OPERATIONS ON WHOLE NUMBERS

Addition 2. Discuss a word problem and get from it mathematical ideas which are then transferred to a Readiness for introducing operation of additionis provided when disjoint sets are joined, i.e., by making mathematical sentence. the union of the two sets.After children have many 3. Solve the sentence and relate the result to the experiences with joining sets, th,: teacher may intro- problem situation. duce the operation of addition through the following 4. Make up other problems for classmates to solve. steps: 1. Demonstrating the joining of a set of four objects and a set of one object to form a set of I. PROPERTIES OF ADDITION five objects. Closure:Through discussion the idea of closure can 2. Explaining that joining sets of objects helps in be developed deductively. Number facts may be listed thinking about adding numbers. The addition on the chalkboard and children asked to supply the of numbers is stated as "four plus one equals sums.(This may also be done using addends of two five." and three digit numerals.) Such questions as the following may aid in developing the idea of closure: 3. Writing 4 + I = 5 on chalkboard and repeating "four plus one equals five." a. To what set of numbers do the addends belong? 4. Isolating + and = and explaining again their b. To what set of numbers do the sums belong? names and meanings. c. Is there an example when the sum of two whole 5. Using other examples as needed in group in- numbers is not a whole number? struction then repeating this development as d. What can always be said about the sum when children individually join sets of objects. the addends are wholenumbers? Children can discover answers to number facts or find other names for number<, (Example: 5 = 4 + 1, Understanding ofthe property provides a quick 5 = 3 + 2, or 5 = 0 + 5) by manipulating sets of ob- check for computation. When odd and even numbers jects, or using such manipulative devices as a bead are understood, a similar discussion can be carried frame, beads on a wire, colored plastic clothespins on to discover that the sum of even numbers is always attached to a coat hanger. The flannel board, magnetic an even number. When odd numbers are treated in the board, and dot cards can also be used for this same way, an additional question must be asked: What purpose. is true about the sum when even numbers are added that is not true for the sum of odd numbers? Charts Children may draw dominoes to illustrate number similar to the following to illustrate the idea may be sentences. prepared by children:

They may make cards showing numerals and also 0 E cards showing sets of dots. Use of these can vary in + many ways. 0 E 0 E 0 E 1111andIIIisIII While young children are able to gain understanding of closure, they are not expected to verbalize 111andIIIisIII property at an early age.

Commutative property:Joining sets of objects can Many opportunities for problem solving are avail- help children discover the commutative property since able following the introduction of the operation of itis easy to see that order is not important; that the addition. Primary children may: result is the same regardless of how the sets are killed 1. Dramatize an addition situation and then write together. As children understand that the ord:r in a mathematical sentence. which sets of objects are joined does not affect the sum,

42 48 pairs of adriends may be written on the chalkboard as When understanding of the property has been de- through manipulationof concreteobjects, follows: veloped 2 + 3 = 3 + 2 = D more abstract ideas may be introduced hv: a. Writing mathematical sentences onthe chalk- 2 3 board: 3+ 2 + 4 = 3 + 2 + 4 = +3 +2 b. Reviewing the idea that only two numbers may be added at one time. As sums are recorded and the processrepeated with c. Eliciting from children ways inwhich the sum other pairs of addends, children should beable to express in their own words thegeneralization that the might be found. order in which the operation of addition isperformed d. Putting parentheses around pairs of numbers as on two addends does notaffect the sum.Itisnot groupings are suggested: "commutative" be used with necessary that the term (3 + 2) + 4 =- 3 + (2 + 4) and calling young children since it is theidea and not the termi- attention to this use of parentheses. nology which is important. e. Completing the addition process andcomparing Associative property: Three sets of objects may be sums. usedtodevelop understandingoftheassociative property: f. Following this procedure with other examples as both group and individualactivity. g. Providing opportunity forchildren to generalize a. By joining the first andsecond sets, then by the property in their own words. joining to this union the third set. Identity element:Activitiessimilarto those described above may be used to develop understanding of zero as the identity element for addition. Byusing the empty set when concrete objects are manipulated and zero when writing mathematical sentences,chil- A dren can be led to see that with zero as one addend the sum is the number of the other addend. b. By using the same sets, joining secondand third sets, then to this union joining the first set. 3 + 0 = 3 0 + 3 = 3

2. ALGORITHMS FOR ADDITION Algorithms with two addends less than base: After the number facts are discovered, either through activities with concrete objects, through use of the number line, or through renaming and the applicationof the associative property, they should be learned. As motivation for learning, usefulness of the commutative Children can observe that the result is the same property in reducing the number of facts to belearned regardless of how the sets are grouped. They need many may be capitalized on. See tabiv oage31. experiences in joining sets in this way. Pictures of objects can also be used with children circling the order The number line may also be used in discovering in which grouping is done. sums for basic facts.

7 8 9 10 0 1 2 3 4 5 2 + 3 = 5 4 1ILmonamio.+1 Such games as the following may be used as drill in writes the sum of the sets on the chalkboard. Class- in learning number facts: mates take turns in guessing the number of objects each hand. The correct guess determines the next With a sr' of small objects in each hand, a child leader.

43 9 Algorithms with one addend less and one greater 4 tens and 5 ones than base: As when developing understanding of place 3 ones value children may manipulate bundles of ten sticks 4 tens and 8 ones. and single sticks to gain understanding of the process involved in the algorithm.For example, to find the Most children will advance quickly from the "con- sum of 45 and 3, they may join a set of four bundles crete stage" to the ability to do similar computations of ten and five single sticks to a set of three single using: sticks and record the results as

Word names Expanded notation Conventional algorithm 4 tens and 5 ones 40 + 5 45 3 ones 3 +3 4 tens and 8 ones 40+ 8 =48 48

Algorithms with addends equal to or greater than above. Through appropriate questions children are led base: This algorithm may also be introduccd through to see the need to regroup the thirteen single sticks the manipulation of bundles of sticks. For example, to into one bundle of ten and three ones. The procedure find the sum of 97 + 86, the procedure of joining sets may be recorded again as above using: of bundles and single sticksisfollowed as outlined

Word names Expended notation Conventional algorithm 9 tens and 7 ones 90+ 7 97 8 tens and 6 ones 80+ 6 + 86 17 tens and 13 ones = 17 tens + 1 ten + 3 ones 170+ 13 =170+ 10+ 3 183 = 18 tens + 3 ones =180+ 3 =183

Children may develop different algorithms as they 5 represents the sum or whole set and that1 and 4 gain understanding and skill that leads to the shorter name the parts or subsets is necessary. Children will more efficient algorithms. For example: be familiar with the equality sign from work with addition but minus () as the sign that means to sub- 94 94 94 +8 +8 +8 tract must be completely understood. 90 12 102 Using other examples children should manipulate 12 90 sets of counters at their desks with the teacher writing 102 102 mathematical sentence each time on the chalkboard. When this process is understood, a set of objects may Each child should be able to explain why numerals be displayed and separated in some such manner as belong in the various places. the following:

Subtraction

After children have had many experiences separating Children are asked to give the mathematical sentence sets of objects into subsets, the operation of subtrac- for the subtraction of two from five (5 2 = 3). tion may be introduced. Beginning with a set of five Demonstrating with objects the teacher can lead chil- objects, the teacher moves onc object slightly to the dren to scP that another way to write the sentence side, being sure that both subsets are in full view of might be 3 4 = 5 since 5 represents the sum or children; then asks for the number of the remaining whole set. This sentence then asks '3 and what make set.It should be clearly explained that separating sets 5?" helps with understanding of subtraction and that the subtraction is stated as "fi;e minus one equals four," Since they already know that 3 + 2 = 5, demonstra- It is then written on the chalkboard as 5 1 = 4 and tions with objects and questions should help them to read as "five minus one equals four." An illustration see that 5 3 = 2 and that + 3 = 5 is another with concrete objects and discussion of the fact that way of writing the same mathematical idea.

44 JO I he inverse relationship of addition and subtraction fact that subtraction is not commutative, they see, by may be further emphasized by: manipulating objects, that unlike answers are obtained when numbers to be subtracted are in a different order. 1. Displaying a set of five objects They understand that order does affect the result in 2. Separating them into sets of three and two subtraction.

ill Identity element:Children who discover that re- UI moving an empty set from a set of objects has no effect 3. Asking for mathematical sentences thatthis on the number of he set will be able to demonstrate, picture brings to mind as well as understand, that subtraction of zero from a number does not change the number. 4. Writing the sentences on the chalkboard:

3 2 =5 5 3--= 2 2.ALGORITHMS FOR SUBTRACTION 2 + 3 =5 5 2 3 With addends less than base:Children who are Much work with sets of objects may be necessary familiar with the table of additionfacts, and who before all children are able to give the four sentences understand the inverse relationship of addition and each time. Domino cards may be used to strengthen subtraction as involved in mathematical sentences such the understanding and the number line may be used as 4 + = 6 will quickly see how subtraction facts in the discovery of these and other subtraction facts. may be derived from the addition table. They should be led to see 'hat, if they know addition facts, they can subtract.

With known addends less than, equal to or greater than base: Activities during development of ideas of place value provide readiness for renaming needed in 1. PROPERTIES OP SUBTRACTION subtraction.Children can bundle and rebundle sticks to find different names for numbers.For example. Closure: The closure property for subtraction should use bundled sticks to show that: be developed very informally since children are not expected to verbalize the property. They can realize that 52 = 5 tens,2 ones in the set of whole numbers it is not always possible = 4 tens, 12 ones to subtract. By separating sets of objects and by look- ing at mathematical sentences such as: , 7 2 = 352 = 3 hundreds,5 tens,2 ones 2 4 = , 5 3 = , 5 7 = , 3 5 = , = 3 hundreds,4 tens, 12 ones 9 11 = , children will become aware that it is not 2 hundreds, 15 tens, 2 ones always possible to subtract using the set of whole numbers. 4 hundreds,6 tens,7 ones = 467 3 hundreds, 16 tens,7 ones = 467 Commutativity: Children may likewise manipulate 3 hundreds, 15 tens, 17 ones = 467 concrete objects to see that the order in which the sum and the known addend are arranged is important in Separating sets of sticks arranged in bundles of ones subtraction: and tensillustrates the meaning of the subtraction S 2 2 5 6 4 4 6. algorithm. As sticks are separated, the operation can be recorded on the chalkboard using word names. For Although children are not expected to verbalize the example:

Renamed Then expanded notation

5 tens, 2 ot.es 4 tens, 12 ones SO -1- 2 40 + 12 3 tens, 6 ones 3 tens,6 ones 30 +6 Renamed 30 + 6 1 ten ,6 ones or 16 10+ 6= 16

45 Because of individual differences, it is important to the conventional algorithm with understanding so each note that particularly in subtraction which requires child will get to this stage by following his own in- renaming, some children will need to stay on the con- dividual pattern. crete level longer while other children, building on prior experience, will readily move to word names and The abacus is also a useful device in developing the expanded notation forms. The goal is to be able to use subtraction algorithm.

52 -136 16

5 2 4 12 12 1 6 3 6

Multiplication Experience with such concrete and semi-concrete By manipulating equivalent sets of objects and pre- materials as those mentioned above leads to under- paring and using rectangular arrays children can dis- standing that the product for factors greater than one cover products. can be found by repeated addition of the same number. Also the different factorizations of one number may * * ** * ** ** ** ** * be illustrated with one set of blocks, squares, or other * * ** * ** ** ** ** * objects. For example: * * ** * ** ** ** ** * A ** ** ** ** * ** 1 12 3

Large demonstration arrays and peg boards on which 4 arrays of different sizes can be arranged may also 2 prove useful in developing understanding. 6

It may be helpful for children to cut squared paper 1, PROPERTIES OF MULTIPLICATION into strips representing arrays of increasing size to aid Closure:By discussing the sets of numbers to which in learning multiplication facts.For example: factors and products in multiplication of whole numbers belong children can be led to see that when factors are whole numbers, products are always whole numbers. 1 In this way understanding of closure develops although 6 5 the terminology may not be used at an early level.

41, 4 x 6 = 24 5 x 5 = 25 3 x 6 = la 2 6 6 Commutative property:By rotating a rectangular 6 array, thus reversing rows and columns, children can gain understanding of the commutative property of 3 multiplication. They can readily se:: that reversing the 6 factors does not change the product. 6 * * * * k ** ** * * * * * ** ** 3 x 5 = 6 x 3 J * * * * * ** ** 4 *** * * * 6 6 * * *

r46 9 Associative property:Much previous experience (2 X 3) x 4 = 2 x (3 x 4) with concrete and semi-concrete materials and with (5 x 8) x 6 = 5 x (8 x 6) properties of addition should enable children to understand readily that the way in which factors are grouped for the binary operation of multiplication does not Distributive property of multiplication over addi- affect the sum.Opportunity to experiment with the tion: By folding a paper showing an array, children can grouping of factors in multiplication should help all discover how the distributive property of multiplica- children to recognize that: tion over addition works.

.IK * .IK .IK .IK .IK .IK * 4( .IK .IK ***** * *** * ** *

.IK .IK .IK 4(* .IK -I( .IK .IK -I( .IK .IK .IK .IK .IK .IK 'IK 'IK * 'IK 'IK 'IK 'IK .1K

-I(-I( .IK -I(** .IK -I( -I( * * * .IK -I( .IK -I( * *** * ** *

* *-I(**** .IK -I( .IK .IK 4 -I( .IK .IK .IK * *** * ** * * *-I(***-I( .IK 4( .IK -I(-I(-I( .IK .IK .IK -I( 'IK 'IK 'IK 'IK -I(* 'IK

5 x 8 =(2 + 3) x 8 5 x 8 =5 x (4 +4) 5 x 8 =5 x (3 +5) =(2 X 8)+ (3 x 8) =(5 x 4)+ (5 x 4) =(5 x 3)+ (5 x 5) =16 + 24 =20 + 20 =15 + 25 =40 =40 =40

After understanding has been developed,the distribu- computation.Forexample:It may be used indis- tive property may be used in bothwritten and mental covering the more difficultmultiplication facts: 7 x 9=(5 +2)x9 7 x 9 = 7 x (4 +5) =(5 X 9)+ (2 x 9) =(7 x 4)+ (7 x 5) =45 + 18 =28 + 35 =50 + 13 =63 =63

It is also involved in multiplication algorithms: 765 x 32 renamed and multiplied as (2 + 30) X 765 = (2 x 765) + (30 x 765) = 1,530 1,530 + 22,950 22,950 added 24,480

Identity elementone: By working several examples 2. ALGORITHMS FOR MULTIPLICATION where one is a factor, children can readily set that the Algorithms with two factors less than base:See product is always the same as the other factor, (1 x 7 = 7; 9 X 1 = 9) and that one is a factor of Suggestions on page 46 for development of number facts. Once children understand the operation of multi- every whole number. plication as repeated addition of the same number and Zero: The union of empty sets, multiplication as know how to discover products for the basic facts, they repeatedaddition,demonstratesthatmultiplication should learn the facts. where one factor is zero gives a product of zero. Discovering patterns in the chart of basic multipli cationfactsmotivateslearning,providespractice needed for learning facts and stimulates observation. 0 B0c0D0 For example:By referring tomultiplication chart NtA) = 0;N(B) = 0;N(C) = 0; similar to the one on page 36 children may be stimu- N(D) = 0 lated to discover patterns by such questions as the A UBUCUD=0 Hence 4X0=0. following: This experience should be repeated if children persist a. What can you say about all products under the with errors in algorithms. column headed by two, by three, by four, etc.?

47 3 b. What patterns do you see formed by units digits facts. Children can invent games of their own. in products in the column under even factors? See also page 47 for suggestions regarding the use c. Is this true for products in columns under odd of distributive property to arrive at facts with factors factors? 6 through 9. This involves renaming one factor, then d. What pattern results when digits of products applying the distributive property and utilizing facts are added? previously learned.

Games can be used to motivate the learning of basic The number line may also be used to discover answers to basic facts.

9 x 5 = (2 + 7) x 3 9 x 3(2 x 3) + ( 7 x 3) 4 6 8 10 12 14 16 18 20 22 24 26 28 11/111i ti I A 2 x 3 7 X 3 6 21 27

Algorithms with one factor equal to base:Using 1 dime 1 x 10 = 10 I dime is 10 cents. dimes the pattern of multiplying by the base (10) becomes apparent: 2 dimes 2 x 10 = 20 2 dimes are 20 certs. 3 dimes 3 x 10 = 30 3 dimes are 30 cents.

Algorithms with one factor greater than base:The developed on page 47 produces the pattern for the distributive property of multiplication over additionas multiplication algorithm. For example:

23 x3

(20 + 3) x 3 = (20 x 3) + (3 x 3) 23 =60+9 x3 = 69 69

Algorithms with two factors greater thanbase: Ideas Division suggested for algorithms with one factor greater than base may be applied to develop understanding of al- Division may be presented as the inverse of multi- gorithms with two factors greater than base; although pliCation by asking questions to make the relation- most children should not need to rely heavily upon a ship clear. For example: physical model. Ideas presented on page 37 should be Five times what equals fifteen? helpful in developing and recording theprocess.Al- though some children may need to continue to work at What times seven equals twenty-one? the developmental level, most should be encouraged After answering many such questions children can see to use the conventional Algorithm. that when two factors are known, they multiply to

48 find the product and when the product and one factor Children will observe, as quotients are obtained, that are known they divide to find the missing factor. itisnot always possible to divide using the set of x 3 =- 15 15 + 3 =1;:] whole numbers.It should be kept in mind that understanding of the idea is more important than verbaliza- tion of the property. Zero indivision:Special attention should be given to the fact that division by zero is meaningless. The Commutative property: Mathematical sentences such idea that zero times any number is zero should be as the following may be written on the chalkboard. reviewed. The mathematical sentences 6 + 0 solved, and discussed: and x 0 = 6 may be written on the chalkboard, and followed by such questions as: 12+ 3 =4 3 + 12 4 18+ 6 =3 6+183 What times zero equals six? n x 0 = 6 Is there any number that answers the question? Children can see that division is not commutative. The (NO) way in which divisor and dividend are arranged does affect the quotient. Also 0 + 0 = implies x 0= 0 What number or numbers makes the statement Associative property: Mathematical sentences simi- true? (All numbers) lar to the following may be written on the chalkboard. solved, and discussed: If this procedure is followed by other examples and 18 + 6 + 3 = (18 6) -:- 3 = similar questions used, children will realize that division 18 + (6 + 3) = by zero is meaningless. 3= 3 = 1 18 + 2 =9 After experience witha number of similar mathe- Partative application: Children can gain understand- matical sentences, children can see that the associa- ing of the partative application of division by using tive property does not hold for division. The order in counters to find solutions to such problem situations which thedivisions are performed doesaffectthe as: quotient. a. Determining each person's share when 24 pieces Distributive property: An array may be folded to of candy are divided among 4 children illustrate the distributive property. For example: b. Determining how many children will ride in each car when 6 cars are used to take 24 children to the zoo. 45 + 9 = (27 + 18) + 9 Measurement application: Understanding ofthe (27 + 9) + (18 + 9) = measurement application of division may also be de- 3+2=5 veloped by the use of counters or pieces of string in the 45+9=5 solution of such problem situations as the following: a. Determining how many children may be served 2 hot dogs from a box of 36 * *** * *** 0- b. Determining how many 6 inch pieces may be * **31-2*7** * cut from 48 inches of ribbon. * *** * *** *

** **** * **1 8 * 1. PROPERTIES OF DIVISION * * * * Closure: Division examples similar to the following may be written on the chalkboard and discussed:

4)16 5)25 5)26 16)8 Care must be taken to see that children understand 5)S that the distribution car. be made only on the dividend and the renaming of the dividend unist make the division of each part exact.

49 55 2. ALGORITHMS FOR DIVISION Children can be led to see how the estimation using the largest whole number helps to find the quotient. Algorithms with factors less than base: When chil- As the work is recorded, attention should be called to dren understand the meaning of division, its inverse, the remainder. When the remainder isless than the multiplication, may be capitalized upon.Familiarity divisor, the operation is complete. with the chart of multiplication facts will point up this relationship and demonstrate that division facts are learned at the same time as the multiplication facts. Algorithms with known factor equal to base: By The more children use what they know about multipli- examining number sentences like the following, chil- cation the less time they will need to spend on division dren can see a pattern develop: facts. Children need many experiences telling simple word 10= 10= 90+ 10 =9 or story problems involving situations requiring divi- 20 + I0 =2 150+ 10 =15 sion; writing mathematical sentences for these prob- 30 + 10 = 230= 10 =23 lems; solving the problems; and testing solutions for reality. If number sentenceslikethe following arenext Before considering division with non-zero remain- introduced, children can see how a pattern involving ders,childrenneed many experiencesfindingthe remainders develops: greatest whole number that makes a sentence like the following true: 10+ 10 = 1 because 10 =1 x 10 ijj X 3 < 16 El X 5 < 26 11+ 10 =Irl because 11 =(1 X 10)+ 1 X 8 < 51 D x 9 < 82 12+ 10 = 1 r 2 because 12 =(1 x 10)+ 2 When understanding of this type of estimation has 26+ 10= 2 r 6 because 26 =(2 x 10) + 6 been developed, examples with non-zero remainders 153+ 10= 15 r 3 because 153 -=(15 x 10) + 3 may be introduced. For example: 928+ 10= 92 r 8 because 928 =(92 x 10) + 8

3 ) 155 Children may develop a table to show the remainders when using various divisors:

Number of Divisor Possible Remainders Possible Remainders

2 0, 1 2 3 0, 1,2 3 4 0, 1,2,3 4 5 0, 1,2,3,4 5

Algorithms with factor; greater than base: Many b. What is the greatest multiple of 10 that makes children find division using larger numbers difficult. the sentence n X 7 < 216 true? To insure success it is important to develop carefully c. What is the greatest multiple of 100 that makes and sequentially the ideas involved.Children need many experiences answering such questions as the fol- the sentence n x 3 < 602 true? lowing: Children need help in finding the number of places a. What is the greatest whole number that makes a quotient will have. For example, 234 + 7 will have the sentence n x 7 < 47 true? a two-place quotient because 10 x 7 < 234 and

50 56 100 x 7 > 234. Therefore, the answer will be a two- certain products. For example, since 49 is the product place numeral, more than 10, but less than 100. There of two like factors, 7, the square root of 49 is 7. should be many experiences of this type to help children determine the number of places in the quotient. As these experiences are examined and discussed. patterns will emerge: Square Roots --) _ quotient will have 1 place x0 1 2 3 4 5 6 7 8 9 0 quotient will have 1 or 2 places 1- 1 ) _quotient will have 2 or 3 places 2 PRODUCTS *Use the multiplication chart to see where the pattern 44, 3 changes for each one-place divisor. .17 4 Although children will be expected ultimately to use 5 the conventional algorithm for division, it is far better vai 7 . that they continue to use a longer, less efficient form 6 with understanding than to use the conventional al- 7 gorithm mechanically. 8 9 3. SQUARE ROOT OF A NUMBER The multiplication chart shows the square root of

51 RATIONAL NUMBERS

Many operations can be performed by using whole numbers. As work with problems increases and mathematical experience broadens, it becomes evident that there is need for other sets of numbers. For example, there are measuring situations for which the whole numbers are insufficient.In subtraction there are pairs of whole numbers for which there is no answer in the set of whole numbers, 3 5 = O. Similarly, in division, there are pairs of whole numbers which have no quotient in the, set of whole numbers, 5 2 = or 9 ± 13 = rj. These and other situations indicate that additional sets of numbers are needed in order to complete problems like those above. The set of numbers conventionally introduced after the set of whole numbers is the set of positive rational numbers and zero, which are the fractional numbers. Properties for operations on fractionalnun bers are developed from properties for operations on whole numbers. These two sets of numbers, the whole 'lumbers and the fractional numbers, are studied in detail in the elementary school and form the major portion of elementary mathematics.Included in the study of fractional numbers is decimal notation. A third set of numbers, the integers, is introduced in the elementary school. Integers and the real numbers are studied in detail in the junior and senior high school. Each succeeding number system represents an extension of the system previously studied in that definitions and properties of earlier systems are used to develop subsequent systems. Careful development of thesystems of whole numbers and rational numbers, particularly fractional numbers, is necessary ina program of elementary mathematics.

Mathematical Ideas Illustrations and Explanations

Fractional Numbers L NATURE OF FRACTIONAL NUMBERS Anordered pairof elementsis a (Dick, Joe) shows ni.1: first and Joe second. pair of elements with one designated as first and the other desig- (cart, car) shows cart first and car second. nated as second. (2, 3) shows 2 first and 3 second.

Ordered pairs of numbers with the Regions first number a whole number and the second number a counting num- 3 parts of the same size ber constitute the beginning idea of with 2 shaded fractional number. Ordered pairs of numbers can bz. related to regions, (2, 3) shows relationship segments, and sets of ob:ects. between shaded parts and unit

8 parts of the same size with 3 shaded (3, 8) shcws relationship between shaded parts and unit

(4,3)shows eachunit separated Into 3 parts of the same size with 4of these parts considered.

52 58 Illestrations and Explanations Mathematical Ideas

Segment.;

(I, 2) shows 1 or 2 parts of the same length.

The ordered pair (a, b) as appli:d to a unitregion or a unit segment shows that the unit is split into b pats with aof them being considered.

Se s of Objects Cyc :t (1, 3) shows 1 of 3 equivalent sets x x x x (2, 6) shows 2 of 6 equivalent sets Ot x x

A fraction is a symbol that names Ordered Pair Fraction ft also an ordered pair of numbers. R,ad two thirds or 2 divided nsraes the fractionalnumber. (2, 3) ? by 3 (3, 8) 3 Read three eighths or 3 di- 8 vided by 8 (4,3) 4 Read four thirds or 4 di- 3 vided by 3

The bar separating the two whole numbersis a symbol of division as in the division sign,"+."

'she quotient of any two whole numbei, 2, is called the numerator ,umbers may be expressed .s a In the ordered pair (2, 3) the first and the s,:ord number, 3, is calledthe denominator. fractional number. 2 4 2 3 4 ÷ 3 =; 8 ÷ 4=I 3 3 4

The ordered pair (a, b) is shown asthe fraction a where a, b are pair (a) is whole numbers, b 0. The first member of the n-lered called the numerator and thesecond nullifier (b) is called thede- nominator.

A fractional number can be thought .1111111111111111111 of as representing a purl,portion Of amount of the region, segment,or set. An infinite number ofordered pairsnames the same fractional 41 number. -1----11111111111111111111111

53 59 Mathematical Ideas Illustrations and Explanations _L=E=1111111MIME FT

Each siiaded bar is the same part or amount of the unit regionbut each part is named with a different fract::m. Each fractionnames the same fractional number.

Equivalent,fractions:Fractions that The fractions rssociated with each region aboveare different names name the same fractional number 1 2 3 4 are called equivalent fractions. for the same fractional number: 5. The fractions 2 4 6 8 10 are equivalent.

An equivalentfractionforany Example: fractional number can be obtained by: Fiactional number.: equivalent to1 : Multiplying the numerator and denominator by the same count- 1 x 2 2 x ,1 or 2 , ing number using the properties 2 2 X 2 4 2 of one, that one times a number is that number and that a num- 1 x 5 5 5 ber divided by itself is one. 2 x 5 10 =I Dividing the numerator and de- 5 5 + 5 1 5 nominator by the same counting + 1 or -= 1 15 IS + 5 3 number. 5

Two fractions with like denomina- 3 3 Example: Since denominators are alike and numeratorsare tors are equivalentif and only if 4,4 their numerators are the same. the same, 3 -=3. 4 4

5__10 5 10 8 16 8 16

5 x 2 10 x 1 10 104- 2 5 8 X 2 16x 1 16 16+ 2 8

10 10 5 5 16 16 8 8

5 15 8 24 5 x 24 :5x8 8 X 24 24x8 5 X 24 15 x 8 192 192

54 60 Mathematical Ideas IllustrationsandExplanations

120 120 _192 192

a b d axd cXb bxddxb

a Any fractional numbers and are equivalent if and only if a x d

Any one unknown term of two Examples: equivalent fractions may be found by using the idea that two frac- 4 n 4 x 35 = 7 x n a= 4 c n b = 7 d == 35 tional numbers are equivalent if 7 35 and only if a xd=b x c. 4 20 4 x = 7 x 20 7 n

n 20 n x 35 = 7 x 20 7 35

20 4 x 35 = n x 20 4= n 35

Along the number line:Each whole number can be matched with a single 2 3 4 point. almminimeimilmillmoimmilwirestomiso=ot 2 13 3 3 .5 3 Each fractional number can be matched Each unit segment is divided into 3 parts of the same size toindicate with a single point. thirds.

0 1 2 3 4 illowomi=siimilmmim.61111.1=6.11.rtrwilmuimhmtuNium6.118. 1 3. 4 4 4 4 4 Each unit segment is divided into 4 parts of the samc size toindicate fourths.

a The fraction is matched with a point on the number line by splitting the unit segment intobpieces of the same size and matching the point at the end of a pieces.

I t 61 Mathematical Ideas Illustrations and Explanations 0 Each fractional number can be named 2 by an infinite number of ordered pairs %ammo dram. of numbers. 0 2 1 ram2 0 4 1 2 4wissimmilismodmmowimoomirmaimaislimimmioNiq4 0 ftimsnmemiimiwomitwitemteimiscirimeiNeolmailW1 2

0 2

1 2 3 4 2 2 2 2 2

(1) 2 3 4 5 6 8 4 4 4 4 4 I I

0 1 2 3 4 5 6 78 9101112 14 1516 8 8 88 8 8 8 8 8 8 8 8 g 8 8 8 8

A fraction is said to be inits 8 4 X 2 4 is a common factor, therefore simplest form when the numera- 12 4 x 3 tor and denominator are relatively primehave no common fac- 8 8 ÷ 4 2 (the simplest form) tor greater than one. 12 12 ÷ 4 3

9 3 x 3 (no common factor 9 is thesimplest ((ma,) 10 5 x 2 10

9 and 10 arc relativelyprime.

23 34 Order of fractional numbers: The Since the length of the segment order of fractional numbers can be associated with 3 is longer than shown using the number line. A 4 given number is less than each number toitsrigut and greater than the segment associated with 2 1 3 each number to its left. 4 4 4 2 2 > 3 and 3 <4.3 2 3 3 3 3

Two fractional nunke.s can be Example: Compare 2and 3 compared by renaming one or 3 4 bothso that they have the same 2 8 denominators. Comparing the nu- 3 9 3 meratorswillshowwhich is 12 4 12 greater.

56 62 Mathematical Ideas Illustrations and FxplAnalions

2 Since 9 > 8,3> 4 3

C.a.n any two fractionsa and c one of the following is true:

a c a c a = >, or < b db d b d

Density of fractional numbers:An- other fractional number can elways be found between two given frac- 4- tional numbers. This propertyis 1 I known as density. Whole numbers I 16 are not dense because another whole 3 numbercannot befoundbetween 2 two consecutive whole numbers. 8

I Between1 and is the fractional number3. Between 1 and 3 4 2 8 4 8 5 is the fractional nu.nber. This process can becontinued indefinitely, 16 leading to the idea that there is an infinite number of fractional numbers between any two given fractional numbers.

2.OPERATIONS ON FRACTIONAL NUMBERS with the same de- Addition: Theadditionoffrac- To add fractional numbers named by fractions tional numbers, as with the addition nominators, the numerators are added and the samedenominator used of whole numbers, is the binary operation on two numbers called ad- 9 dends to produce a third number 9 called the sum.

5 911111111111.1=1-1

7 5 12 Therefore 9 9 9

57 33 Maths matical Ideas Illustrations and Explanations

Given two fractionsaand c with the same non-zero denominator, c a 4 c b b b

To add fractional numbers with unlike denominators the addendsare renamed so that all denominators are alike, and numeratorsare added as with fractions having like denominators.

Examples:

3 7 3 3 x 2 6 4+ 4 4 x 2 8

6 7 13 8 8 8

5 1 5 5 X 7 35 6+7 6 6 X 7 42

1 1 x 6 6 7 7 X 6 42

35 41 42 42 42

Another way to rename addends so that all denominatorsare alike, uses the idea of the least common multiple. (1.c.m.)

Examples:

1 4 To find the 1.c.m. of 6 and 9: 6 + 9 C3

Counting number multiples of 6 = 6, 12, 24, 30, 36, 42. ..

Counting number multiples of 9 = 9, 7, 36, 45, 54 I.c.m. of 6 and 9 = 18

Hence the least common multiple of the denominators is 18.

1 4 3 8 11 6 9 18 18 18

5 To find the 1.c.m. of 6, 9, and 12: 6 9 12

58 .'64 Mathematical Ideas illustrations and Explanations

Counting number multiples of 6 =6,12,18,24, 30,

Counting number multiples of 9 =9,18,27,6, .5, 54 ...

Counting number multiples of 12 = 12, 24,36,g, 60 ... c.m. of 6. 9. and 12 -= 36

1 4 5 6 16 15 37 6 9 12 36 36 4- 36

Subtraction:Thesubtractionof To subtract fractionai numbers named by fractions with the same de- fractional numbers, as with the sub- nominator, the numerators are subtracted and the like denominator traction of whole numbers, isthe used. inverse operation of addition and is the operation of finding an addend 12 5 7 when the sumand the other addend Since subtraction is the inverse of addition are known. 9 9 9 5=12 9 9 9

A picture of the separation of parts in subtraction can be shown by movement to the left on the number line.

0 1

7 :IF 5_ 9 9

12 5 7 9 9 9

Given a, b, c, whole numbers b 76 0 and a > c: a c ac b b

a > c assures that a c is an element of the set of whole numbers, so that the result of subtracting fractional numbers, under the stated restrictions, is another fractional number.

To subtract fractional numbers named by fractions with unlike denominators, the sum and the known addend are renamed so that denominators are aiike in the same manner as described for addition. Then the subtraction is perfoaned as with fractio.n having like denominators.

Multiplication:Multiplication of frac- To mulliply fractional numbers, the numerators are multiplied and tional numbers, as on the set of whole then the denominators are multiplied. numbers, isthe binary operation on 1 2 1x? 2 two numbers called factors to produce a third number called the product. 2 3 2 x 3 6

59 6 5 Mathe ;franca' Ideas Illustrations and Explanations

An algorithm for multiplication of fractionalnumbers can be developed by capitalizing on the student's knowledgeof the use of "of" in a mathematical sentence suchas:

3.1- of 4= 2; 1of 10e = Se 2

The uee of "of" in a mathematicalsentence may be interpreted through a pattern as follows: 8 x 6 = 48 1 1 of 8=4 4 x 6 = 24 of 48 =24 2 2

2 1 I of 4= 2 x 6 = 12 -7 of 24 =12 2 2

1 1- of 2=1 1 X 6 6 o f 12 =6 2 2

oft = 1 Ix=3 21 of 6=3 2 2 2

Where one factor is halved, the product is halved.As the pattern continues, it can be seen that 1 x 6 means thesame asI of 6. 2 2

1 2 To find of use the model: 3 5

5 2 2 5 3 15

Hence,1 x.= --2 3 5 15

It can be observed that the numerator of the productequals 1 x 2 and the denominator equals 3 x 5, thus verifying that 1 x? 1 x 2_ 2 3 5 3 x 5 15

Another way to illustrate the algorithm for themultiplication of fractional numbers is to use a ree.tangular region.

6 G Mathematical ideas Illustrations andEN14.111:ItionS

The area of a rectangular region with side measures that are whole numbers is found by multiplication.

Area = 3 rows of 4 sq. units each

Area = 3 x 4 = 12 sq. units

ft is reasonable to use multiplication to find the area if the side measures are fractional numbers.

Area of the region is1 square unit.Side measures areIlinear unit. The shaded region measures 2 by 4. The area of the rectangular 3 2 4 .8 region measuring by -- of the square unit. Hence. it is sensible 3 5 15 , 2 4 8 tor x tobe 3 5 15

The product of the denominators shows the number of equal-sized pieces in the unit square. The product of the numerators stows the number of pieces in the smaller region. 2 4 2 X 4 x = 8 3 5 3 x 5

a c a x c If a and are fractional numbers, the product is x b b d b x d

Divirion:The division of fractional Division of fractional numbers may be approached in more than one numbers, as in the division of whole way. Mathematical ideas used in these nproaches arc: numbers, is the inverse operation of 1. The quotient of any two numbers be expressed as a iractic.o: multiplication and is the operation of finding a factor when the product and a 4- h = b # 0 one factor are known.

61 .467 Mathematieci Ideas Illustrations and Explanation;

2. Any number divided by itself has I as a quotient:

a = 1, a 0 a

3. Any number multiplied by 1 has itself as product:

a b b

4. Any number divided by 1 has itself as quctient:

5Thereciprocalof a numberthat number by which the first number is multiplied to rive the product, 1:

a b a h ax x =- --- = 1, a S4 C, b 0 h a b > : a a x b

6. The product 1 two fractional :.umbersisthe product of the numerators divided by the product of the denominators:

a c a x c h d h d

7. The quotient of two fractional numbers isthe quotient of tl numerators divided by thi_ quotient of the denominators:

a c a 4- c b d b u

The quotient of ts,cf fractional numbers may be expressed as another fractional numbe.., frequently called a complex faction:

a a =

8. The inverse operation of division is multipliriticri: =cnndcxbr=aa b

62 "brr.7rtr-

Mathematical Ideas Illustrations and Explanations

The division algorithm may be developed by the use of a fraction where the terms are fractional numbers, often calleda coutplex fraction:

1 (1) Numbers refer to ideas on pages 61 -62. 1 1=2 2 ti 1 4

1 4 (3) (2) 2 1 X 1 4

4 1

1 4 (6) X 2 I 1 x4 4

1 x4 (5) 2 1

1 4 (6) 2 4=2 2

Division by a fractional number is the same as multiplication by its reciprocal. a (I) a = h ii c

a d (3) (2) b c = X c d d c a d (6) X b c c d X

a (5) X = b

d (6) X ba c

63 Mathematical Ideas Illustrations and Explanations

Division by - is the sameas nr)itiplice tion by its reciprocal,-d d c

Another approach uses the common denominatormethod. This methol involves renaming the two fractionsso that denominators are alike then dividing the whole numbernumerators and denominators. 1_ 2 2 (2) (3) x -- 2 4 2 2 4 2 1 2= 1 (7) 4 4 4 ÷ 4 2 ÷ 1 (4) =2

a c a ÷ c (7) b b b =b =a 4- c (4)

Inverse operation approachmay also be used in developing the division algorithm. 2 5 = {D means 5 2 3 7 7 3

In order to brake this mathematicalsentence tru,;, D must be a nurn7ler which has-2 as a factot. 3 0 must be (-2 x 3

The mathematical sentence becomes (- 2x A) x 5- = 2- 3 7 3

Applying the associative property; 2- x (A X-)5 =-2 3 7 3

2 If- X ( )= -2, then ( ) = I awl (A x5 3 3 -) = 1

7 7 Therefore, ti= - since - x - 1 5 5 7 2 7 (- x -) 5=2 3 5 7 3

2 7 Then (-x -) 0 Thererore, - -5 -2 x7 3 5 3 7 3 5 64 n U PROPERTIES OF ADDITION '.ND SUBTRACTION OF FRACTIONAL NUMBERS The same properties that hold true for add lion rind subtraction with respect to the set of whole numbers also bold true for addition and subtraction with respect to the set of fractional numbers. In brief these are:

Ariciltion Subtraction

1. Closure 1. Not closed 2. Cominutaiivi*.y 2. Not commutative 3. Associativity 3. Not associative 4. Identity clement (zero) 4. No identity (a 0 -= a, but 0 a a) 5. No inverse element 5. No inverse dement

PROPERTIES OF ME1.11Pt ICAT ION AND DIVISION OF FRAC.] IONAL NUMBERS The properties tirt hold true for multiplication and division with respect to the set of whole numbers also hold true for multiplicuiion and division with respect io the set of fractional numbers.

Aft4Wplicotion idon

L Closure I. Ciosure 2. Commutativity 2. Not commutative 3. Associativity 3. Not associative 4. Identity elenwnt (one) 4. Identityclement(one)on therightonly 5. Distribotivity over addition h b b b 6. Distributivity over stiotraction 5. Distributivity over additionfromtheright 7. Multiplication property of zero hand side only 8. Inverse 6. Distributivity over subtraction front the right hand side only

The reciprocal or inverse property is title undo: the multiplication of fractional numhe,:s but not for whole numbers.It states:If a and b are whole ta-nbers, a and b 0. a b =I ax _Itr.I and- x- a h a

1 is called the reciprocal ofa aid a is (ailed the reciprocal of1 a a

and-aare reciprocals ofcashother. a

The reciprocal of a number is the number by which the first number must be multiplied to give a product of 1the identity dement. Every fractional numbcr, except 0, has a reciprocal. Another name for the reciprocal property is the multiplicutiv., inverse property.

The closure property is added to the list of properties under division for fractional numbers. Since the division of one fractional number by another fractional number, excluding zero as a divisor, will always yield a fractional number as the 000tient, the set of fractional numbers is said to be dosed with respect to division.

65 71 Mathematical Ideas Illustrations and Explanations

The Integers

I. NATURE OF INTEGERS

The set of integers consists of the ,-3. , 1, 0, 1, 2, 3, 4, 5,... set of whole numbers and the set of negatives of the natural or counting numbers.

The positive i n t , gers arc the same 1 ,2, 3, 4, 5, ... or as the;et of %.eunting nmobers. Oftenfor emphasistheraised I. 42. +3, 4 4,I 5, ... plus sign is used for positive numbers. The number +.5isread "positive five."

The raised dash denotes a nega- number. The integer --5is read "negative five."

It should be kept in mind that these raised symbols, and -, are ;'art of the names of the numbers and not operational symbols. It is agreed that, if a numeral has no sign, it is a positive integer.

Positive and negative integers may be used to &rote change in direction from one position to another.

Positive may indicat.1 a change Growth of 2 inches; increasing change of 2; + 2 that is an increase. Deposit of $50; increasing change of $50; 450

95 feet above sea level; -195

Negative may indicate a ellinge Loss of $75; decreasing change of $75; -75 that is a decrease. Temperature of 5° below zero; decreasing change of5' from 0; -5

15 in the hole; decreasing change of 15 from 0;-15

Arrows may be used to represent (hang:. r 3 (gainof 3) +3 Onowin11.412ommuniolt. Arrows pointing to the right or up indicate positive integers.

669 INflithematical Ideas Illustrations and Explanations

Arrows pointing to theleftor down indicate negative integers. ..gmaiummanwimagmoms 3 3 (loss of 3)

Points on the number linethat match positive and negative integers +5 indicate change from 0. +4gain +3 +2 +1 1 --5 4 3 2 1 0 +1 +2 +3 +4 +5 0 4mimiimrilmhumiIimilmismkNi." 1 loss 4 wIffami4 gain 2 3 4loss 5

+10 +9 2.ORDER OF INTEGERS +g 6 5 4 3 2 1 0 +1 +"). +34-4+5 +6 On the number line any integer is+7 44ImkomiaNdikowissa4marowimmimosiorsisodaP +6 less than any other integer to the+5 right and is greater than any to its+4 left. Any integer is less than any+3 +2 Loss of 3 is less than a loss of I. 3 < 1. integer above it and greater than+1 any below it on the vertical num- 0 A temperature of +3' is higher than a temperatureof ber line. 12 7'.+3 >--7 43 0>2 2< 0 5 0 6 415>17 1 87 109

67 s'/3 Mathematical Ideas Illustrations and ExpNnations

Two integers the same distance from On number lines above the follow'.,..pposites may be noted: 0, one to the right, the other to the left on the numter line, are called opposites or inverses. Opposites 3 and +3 are opposit:s 5 and +5 are opposites --9 and +9 are opposites 10 and +10 are opposites

0 is neither positive nor negative; it is itsown opposite.

The Additive Inverse Property: The -6 is the additive inverse of +6 additive inverse of an integer is the same as the opposite of the integer. The sum of a number and its 0 +5 additive inverse is 0. +5Gain of 5 5 lawl 5 LOSS of 5

Gain of 5 then loss of 5 is 0 (+5) + (-5) = 0 (+7) (-7) = 0 (-10) + (+10) = 0

3. OPERAT'ONS ON INTEGERS Addition: The addition of integers, The sum of two integers may be found by using the numberline. as with the addition or whole numbers and fractional numbers, is the operation of finding the sum of two addends. 1 0 +3 +4 +5 +7 +g

The sum of two positive integers +3 is the same as the sum of the +4 equivalent counting numbers, a (+3) -I- (+4) = positive number. 8 7 4 3 2 1 0 +2 The sum of two negative integers Is a negal ye in:eget. This sum is 4 3 the negative of their combined distances from zero on the num- (-3) + (-4) = 7 ber line.

68 74 L4 Mathematical Ideas Illustrations and Explanations

The sum of a positive and a +5 1 0 +1 +2 +3 +4 negative integer when the positive 5 4 3 2 number is farther from 0 on the4."'""'""'"'mw"I''''' maw, number line is a positive number. (+5) + (-2) = +3 +5 The sum is the difference of their 2 distances from 0, 5 4 3 1 +2+3 +4 +5 The sum of positive and negative integers when the negative inte- (+3) + (-5) =-2 ger is farther from 0 of the number line is a negative integer. The sum is the negative of the dif- 5 ference of the distance from 0.

The additive inverse property is To find the sum (+9) + (-3), + 9 is renamed as ((+6) 4- (+3)] useful in finding the sum of a positive and negative integer. +9 + (-3) = ((+6) + (+3)) + (-3) = +6 + [(+3) + ( -3)] associative property + 6 + 0 additive inverse = +6

To find the sum 16 + +9,16 is renamed as [(-7) + (-9)) (-16) + (+9) = (( -7)(-9)1 + (+9) (-7) "r R-9) + (1-9)) associative property = 7 + 0 additive inverse = 7

Subtraction.. The subtraction of in- 4 (+3) is the number that added to +3 makes 4. tegers, as with the subtraction of 4 (+3) = 0 means 0 + (+3) = 4 whole numbers and fractional numbers is the clperation of finding the 6 (-5) is the number that added to 5 makes 6. missing addend when the sum and 6 (-5) = 0 means 0 + (-5) =6 one addend are known. If a, b, and c are integers and a + b = c, thm cb = a

The difference c b on a number 4 (+3) = D means 0 + (+3) = line is the change in moving from I 0 4 1 +2 +3 +4 b to c. The direction of movement4 3 2 determines whether the missing addend is positive or negative.

From +3 to 4 is 7 units. The direction is negative, hunce .= 7 4 (+3) = - -7

69 Mathematical Ideas Illustrations and Explanations

6 (-5) = means + (-5) =6

6 5 -4 3 2 1 1 2 3 4 5 6

From 5 to 46 is 11 units in a positive direction, hence + 11

Using the definition of subtraction, 5 (-7) =Li films +(-7) = 5 a + b = c and c b = a, it can be shown that to subtract b from c the + (-7)] + ( r7) = 5 1 (+7)(If equals are added to equals, additive inverse of b is added to c. the sums ar,; equal. 3 + 4 + 2 =- 7 + 2)

+ [(-7) + (+7)] = 5 + (+7) associative property

E] +(1, 5 4- (47) identity element = 5 + (+7)

= 2

To subtract 7, add its additive in-verse, + 7.

Multiplication. rhe multiplication of integers, as with the, multiplication of whole numbers and fractional numbers, is the operation of finding the product of two factors.

The product of two positive in- (+5) x (+6) ---- +30 tegers is a positive integer, the samt: as the product of the equiv- See Operations on Whole Numbers: Multiplication alent counting numbers.

The product of a positive number 13 -12--11-10-9 0 +1 +2 +3 and a negative number i.. a nega- 8-7 6 5 4 3 2 1 tive number.

The decreasing factor approach develops a patterr, which is useful in illustrating this product. +2 x +3 = +6 +1 x +3= +3 0 x +3 = 0 1 x3 =x The product decreases 3 each time.lt the pattern 2 X +3 =y continues, x must be 3 and y, 6.

70 76 Mathematical Ideas Illustrations and Explanations the product of two The product of two negative in- The decreasing factor approach is useful for finding tegers is a positive number. negative numbers. 5 x 4 = 20 4 x 4 = 16 3 x 4 = 12 2 x 4 = 8 1 x 4 = 4 0 x 4 = 0 --1X-4= x 2 x 4 = 3 x 4 = z Since each product increases by 4, it is sensible for x tobe4, y to be -I 8 and z be +12.

Division: The division of integers, as with the division of whole numbers and rational numbers, is the operation of finding the missing factor when the product and one factor are known. The missing factor is called the quotient.

The quotient of two positive in- +15 ÷ ..-3 --= means x +3 = +15 tegers is a positive oulnber, as in Since +5 x +3 = +15 the division of counting numbers. +15 ± +3 -= +5

The quotient of a positive integer +12= (-4) = r] mcans (--4) x = +12 and a negative integer is a nega- Since (--4) x (-3) = +12 tive integer. 12÷ (-4) --3

The quotient of a negative integer ( 12)(+3) = means (+3) x = --12 and a positive integer is a nega- Since +3 x (--4) = 12 tive integer. 12 ÷ +3 = 4

The quotient in the division of 12 ± (--3) = means 3 x = --12 two negative integers is a positive Since (-3) x (+4) = 12 integer (--12) (--3) = +4

The rules for operations on positive and negative nuribersshould be stated only after much work using concrete examplesand the number line as illustrated above. Theymay be delayed completelyuntil junior high school with no loss.Danger exists in the formulation of generalizations too early.

71 U;77 Mathematical Ideas Illustrations and Explanations

Negatives of the Fractional Numbers -1 0 +1 1. THE NATURE OF NEGATIVE FRAC-4111..iammlammaimmurolomid (tONAL NUMBERS -4 -3 -2 -1 0 +1 +2 +3 +4 4 Just as each whole aumliza- has its 4 4 4 4 4 4 4 a opposite, each fractional number has a corresponding number an equal Opposites distance to the left of 0 on the num- 3 +3 ber line, called the negative of that -ad 3+ 3=0 fractional number. The sum of any 4 4 4 4 pair of opposite fractional numbers =Sand +5 -5 +5 A is O. 4 4

2. OPERATIONS ON NEGATIVE FRAC- TIONAL NUMBERS Rules which apply to operations ln negative integers apply to operations on negative fractional numbers.

Addition: Examples: -1 +-1 -3+ = -- 4 3 12 12 12 +'_i -I +5 -2 +3 4 8 8 8

Subtroc (ion: ExampLa: -1-2 4 3 -1 2 = + additive inverse 4 3 -3 +8 + rename 12 12 +5 addition

+2 3

1 2 = additive inverse 2 3 -3-4 = + rename 6 6 -7 addition 6

72 '18 Nlathematical Ideas Illustrations and lApranations

Examples:

+1x 5 3 +1 x 2 5 X 3

=2 15

x 3 6 1 x 3 x 6

= 18

Division: E7.ample: +5 I 8 7

5x reciprocal 8 1. 5 x 7 8 x I 35 8

Decimals 1. NATURE of- DECIMALS Decimals are symbols for fractional

numbers having denominators which z.? e; are some power of ten. The decimal 44,4) 3 form of is .3. The denominator 10 7 =67 is not written but is indicated by the 6=- 6 7 10 to number of places to the rightofthe decimalpoint. The term decimal 2583 means that the place value notation 25 83 100 is used to express fractional numbers. 1298 2908 100

784 6784 6 = 6784 1000 1000

73 ,79 Mathematical Ideas Illustrations and Explanations

Decimals can be representedas a 5 0.5 sum of fractional numbers each of 10 whose denominators isa power of ten. 75 70 5 7 5 75 0.75 100 100 100 10 100 100 125 100 20 = 5 = 1 2 5 125 1000 1000 0.125 1000 WO 10 100 1000 1000

Every fractionat number, a (b 0), can be expressed usingpowers of ten as denominators eitheras terminating decimals or as repeating decimals:

Terminating decimals: Anyrr- 1 tional number whose denominator = 0.5 2 has only 2, 5, or 2 and 5as its prime factors may be writtenas a = 04 terminating decimal. 5

1= I 1 =0.25 4 2 x 2 4

1 1 1=0.125 8 2 x 2 x 2 8

I = 1 1=01 10 2 x 5 10

1 1 1 = 0.025 40 2 x 2 x 2 x 5 .10

1 1 Repeating decimals: Any rational 6 2 x 3 = 0.1666... (repeats 6) number whose denominatorhas 6

prime factors other than 2or S 1 1 may be written as a repeating 30 2 x 3 x 5 = 0.0333... (repeats 3) decimal.In repeating decimals 30 the sequence of digits which 3 re- = 02727 ... (repeats 2') peat in the quotient repeat peri- It odically in the same order. The number of digits in the repeating 3 = 0.428571428571... (repeats 428571) period will Leer begreater than 7 the divisor minus 1. A bar oNcr a set of digitsmeans that this set of digits is repeated. 3 = 0.2727 = 0.27 11

0.1666 = 0.16- 6

74 Mathematical Ideas Illustrations and Explanations

2. OPERATIONS ON DECIMALS 75 21 96 Addition: BL-cause a decimal and a .75 + .21 =- + = -= .96 fraction may be used to name the 100 100 100 same number, addition of decimals can be developed from whatis known about the addition of fractional numbers.

Because decimals use place value 0.75 + 0.21 = .96 or .75 notation, addition can be developed + .21 from what is known about the addi- .96 tion of whole numbers. 75 hundredths + 21 hundredths = 96 hundredths = .96

1.85 + .7 = 2.55 or 1.85 + .70 2.55

Subtraction: As with the addition, .75 ,6 =(.7 + .05) .60 .75 the subtraction of decimals may be - .60 developedfrom whatisknown =(.7 .6)+ .05 .15 about the subtraction of fractional numbers and whole numbers. =.1 + .05 .75 .6 =.15 .15

From knowledge of place value 6.45 6 ones + 4 tenths + 5 hundredths rotation and its use in addition 2,33 2 ones + 3 tenths + 3 hundredths and subtraction of decimalsit 4.12 4 ones I tenth + 2 hundredths or 4.12 may be generalizedthatones' place,tenths'place,etc.are aligned in vertical form by align- ing the decimal points.

3 2 6 .3 Multiplication:Multiplication .3x.2= 3 x 2- of - x x .2 decimals may be developed from 10 10 10 x 10 100 what is known about the multiplica- .06 tion of fractional numbers. 8 62 8 x 62 496 .62 8 x.62= X = 10 100 10 x 100 1000 x ,8 .496

The product equals the product of the %%hole numbers in the numerator divided by the product of the numbers in the denominator.

75 Mathematical Ideas Illustrations and Explanaiions

The development of a pattern showing divisionby powers of ten may prove useful for the understanding of the placement ofthe decimal point in a numeral:

Pattern:

1 = .1 10

= .01 100

1 = .001 1000

.0001 10000

805 3.5 x .023= 35 x 23 " x 23 .0805 10 1000 10 x 1000 10,000

3.5 x .023= .023 x 3.5 .023 3.5 x 3.5 x .023 .0115 .0105 .069 .070 .0805 .0805

The product equals 23 x 35 divided by 10,000.

6 3 Division: Division of decimalsmay .6 .3 = = 2 be developed from what is known 10 10 about the division of fractionalnumbers. The division of decimals is simplified by usingthe properties of1, n x I = n and = 1, n 0, to obtainan equivalent fraction. n

.96 100 .96 X 100 . 96 .12 = X .12 100 .12 x 100 12

.96 + .12 =.12).96 12)96

15 15 .125 15 x 1000 15.0;PO .125 .125 x 1000 125

.125)15. 125)15,000 ..8276 Mathematical ideas Illustrations and Explanations

Ratio and Proportion 1. THE NATURE OF RATIO AND PRO- PORTION Ratio: A ratio is a comparison of A ratio is a way to compare two numbers: two numbers by division. A ratio, therefore, is a rational number.It 3 shaded parts to may be exrressed in the following 5 congruent parts ways:

(3, 5); 3; 3 ÷ 5; or 3:5 The ordered pail (3, 5), meaning 3 out of 5, represents the ratio of the 5 3 shaded portion to the entire region. The shaded portion is-5- of the (read "3 is to 5" or "3 out of 5" or "3 per 5") region.

6 shaded parts to 10 congruent parts

The ordered pair (6, 10), meaning 6 out of 10, represents the ratio of the shaded portions to the entire regicn. The shaded portion is6 10 of the region.

Illustrated in another way:

1 piece of candy costs 3t

1 (I, 3) or-- 3 (I out of 3) or (I per 3)

Two pieces of candy cost 6E (2 out of 6) or 2 0 0 0 (2, 6) or or (2 per 6) 6 0 0

77 83 Mathematical Ideas Illustrations and Explanations

Three pieces of candy cost 90

(3 out of 9) Of (3, 9) or 3 or (3 per 9) 9

Proportio.t: A proportion is a state- As shown in the illustrations above: ment that two or more ratios are equal. 3 = 6 and 1 2 3 5 10 3 6 9

a C. = is read "a is to b as c is to d" b d

2. PROPERTIES AND OPERATIONS Since a ratio is a rational number, the operations and properties for rational numbers apply to ratios,

Percent

1. NATURE OF PERCENT 10 A percent is a special ratio which 10% means 10 per hundred, expressed as (10, 100)or always has 100 as its denominator. 100 100, therefore, is always the basis for comparison. 15% means 15 per hundred, expressed as (15, 100)or -?4T-:50-0

a a Ingeneral,rig rational number = isinterpreted;a is c percent of b or b 100 can be eklessed as percent by find- a per b is the same as c per 100, or a is to b as c is to 100. ing the number c in the statement, a c b 100

2, OPERATIONS WITH PERCENT Solving problems using percent. Example: 3 is what % of 8? 3 = c% of 8

78 84 Mathematical Ideas Illustrations and Explanations

Other ways of stating this are:

2 iswhat percent? 8 3 out of 8 is the same as c out of 100

3 c 8 100 3 X 100=8 xc definition of equivalent fractions

1 1 multiplication by the reciprocal of 8 x 8 xc=3 x 100x8 8

3 x 100 300 371 37.5 c 8 8 2

3 = 37'5 = 37.5% 8

This shows that 3 is 37.5% of 8.

Example: Bob sold 50% of his 12 rabbits. How many rabbits did he sell?

Using the fundamental statement:

a outofbis the same asc outof 100, = --c- b 100 aout of 12 is the same as 50 out of 100

Two ways of solving the proportion are: a a 50 or 12 100 12

a = 12 x50 a = 12 x .5 100

600 a= 6.0 I 100 a = 6

6 out of 12 is the same as SO out of 100.

Bob sold 6 rabbits.

79 85 Mathematical ideas Illustrations and Explanations

Example: Hal sold 6 rabbits. Th;s. was 50% of the rabbitshe owned. How many rabbits did he have before selling any?

6 out of b = 50 out of 100

6 50 6 or .5 b 100 b

6 x 100 = 50 x b 6 = .5 x b 600 = 50 x b .5 x b 6 6°° b b = 50 .5 12 = b b = 12

6 out of 12 is the same as 50 out of 100.

Hal had 12 rabbits before selling any.

Example: Leonard paid 6% interest for theyear on $400 he borrowed. How much interest did he pay?

a 6 400 100

He paid $6 for $100, therefore, 4 x 6 for $400or $24. Leonard paid $24 interest.

Find the number of hundreds (in thiscase 4), and multiply by the number per hundred (in this case 6), 4 x 6= 24.

In general, any problem involving percent can be solved by utilizing: 1. The definition of ratio as a rational number 2. Properties of rational numbers 3. Interpretation of the statement as the proportion:

a = or a per b is the same as c per 100. b 10c-0

Since the proportion given in 3 above may be used to solveany problem involving percent, the task becomes one of writingan appropriate proportion.

80 8(3 Properties for Four Aria: lions for Rational Numbers

To this point, the following sets of numbers havebeen introduced: 1. The counting numbers 2. The whole numbers 3. The fractional numbers 4. The integers 5. The negatives of the fractional numbers. nur.bers With each set of numbers, the four operations ofarithmetic and certain properties cf each set of with respect to these operations have been introduced.

The sets of numbers mentioned above include allnumbers studied in the elementary school.

These sets of numbers make up the set of rational numbers.The set of rational numb:rs is defined as the set of all numbers which can be expressed in the forma where a and b are integers, except b A 0. Since each of the above mentioned sets of numbers satisfies this definition,each is a subset of the set of rational numbers. rational num- Listed below are properties for each of the fourarithmetic operations with respect to the set of bersa, b, and c being rational numbers:

Addition 5:thiraction b is a rational number 1. Closurea + b is a rational number 1. Closure a a b b a 2. Commutativitya + b = b + a 2. Not Commutative (a b) c # a (b c) 3. Associativity(a + b) + c = a + (b + c) 3. Not Associative 4. Identity Element (Zero on the right) 4. Identity Element (Zero)a -1- 0 = 0 + a = a a 0 a, but 0 a A a 5. Additive Inversea + (a) = 0 (opposite)

Multiplication Division a -?- b is a rat cad number 1. Closurea X b is a rational number 1. Closure a b b .+ a 2. Commutativitya X b= b x a 2. Not Commutative (a + b) + c --A a + (b c) 3. Associativity(a X b) xc=aX (b x c) 3. Not Associative 4. Identity Element (One) ax1=1 xa=a 4, Identity Element (One on the right) a= 1 = a, but 1 a a 5. Multiplication property of zero ax0=Oxa=0 5. Distributivity over ad:ition (Distribution on 6. Reciprocal (Multiplicative Inverse) dividend ont'y): al : (c ± a) axl=j-xa= 1 (b + c) + a = (b 4 a a over subtraction: c) + a = (bat (c 4- a) 7. Distributivity over addition: (b a x (b + c) = (a x b) + (a xc) over subtraction: a x (bc) = (a X b) (a X c), b > c

81 87 TEACHING SUGGESTIONS FOR RATIONAL NUMBERS

Fractional Numbers 1. NATURE OF FRACTIONAL NUMBERS Young children beginto develop ideas of fractions as parts of wholes as they share halves of apples, cookies or candy bars. The idea that halves of objects may have many sizes and shapes grows as they divide different kinds of objects. This understanding may be applied to fractional numbecs in the primary grades D through paper folding.Each child may fold a sheet of paper into two parts of the same size and color one As parts are discussed, nambers may be recorded part blue.Such questions as the following should be on charts as tollows: used: A B CD a. How many parts are colored blue? Number of shaded parts 3 I 3 3 b. How many parts are there in all? Number of 4 2 parts in all 3 5 The fractional number I should be recorded as the 2 pint is made that the 1tells the number of blue parts and the 2 tells the number of parts in all. A B CD Number parts This procedure should be repeated with fourths: not shaded 1 0 2 folding the paper into two equal parts and again into Number parts 4 2 3 5 four equal parts, coloring one part red and two parts in all green and using such questions as the following: Ilmw a. Hew many parts al; colored red? Older children may writethe ordered pairsas b. How many parts are colored green? (3, 4).Considering fractional numbers as ordered c. How many parts are colored rcd and green? pairs is not difficult for childrenif the idea is con- nected with a physical model. For example: Using this d. How many parts are nol colored? model, asking questions similar to those above, record- e. How many parts are there in all? ing the ordered pair as (3, 4) and stressing that the first number of the ordered pair always tells the number of parts of the same sizethat are considered As the fractional number 1 is recorded, emphasis 4 (shaded) and the second number tells the total num- is again given to the idca that the 1tells the number ber of parts of the same size. of red parts and that 4 tells the number of parts in all. Using other models of fractional numbers children As 2 is recorded, children can see that 2 tells the num- may be asked to give the fractional number represented ber of green parts and 4 the number of parts in all. and to write its name as a fraction and as an ordered As understanding of the meaning of fractional number pair. grows, children will discover that1 and 2represent 2 4 When using physical models for developing under- the same fractional plc of the entire sheet of paper. standing of rational numbers, itis 'mportant touse different kines of models:Regions (circular, square, Models such as the following may be used: and rectangular), segment, and sets of objects. In this orr- way children will not get the notion that fractional numbers are just parts, but that they are numbers. A Each child should have his own kit of fractional modelsincludingvvholes,halves,fourths,eighths, gPF thirds, sixths, and a model of a number line to use at 82 '88 appropriate gradelevels.These may be made by Word problems using fractional numbers may be children. used. For example: Mother cut a cake into right equal pieces. She gave me 1 piece. What part of the cake A number line can be used in developing under- did she give me? What part of the cake is she keeping standing of the nature of fractional numbers. Starting for dinner? with a number line divided into unit segments, children may see how a unit segment may be divided into Exercises of the following types may aidinde- parts of the szme size and that fractions may be asso- veloping understanding of fractional numbers: ciated with points on the number line. Number lines showing halves, fourths, eighths, etc. may he made as a. Shade the cart of the figure that shows what needed. the fraction means:

3 2 4 2 2 4 3 b. color parts c. named by;

(1, 4) What does thenumber pair (2, 3) tell about (3, 3) the picture? What does thenumber pair (3, 5) tell about the picture? 1 1 1 (5, 8) LL.I1__LJ What does thenumber pair (5, 5) tell about the picture?

d. Matchthe ordered pair with the correctpictures: (1, 4); (1, 6); (2, 3); (6, 12); (3, 4)

e. Add labels where omitted on thefollowing number line:

0 2 3 L 1 1 I- 5 Ei 7 at Efl 4- T

Equivalent fractions:Congruent models should be used to help children discover that there are many names for the same fractionalnumber. For example: 111111E-TA (a) (b) (c) 83 89 Such questions as the following can aid in the dis- children to show 2 units divided into halves, fourths, covery: eighths, sixteenths ... As points are labeled and the chart is more complete, children discover that a frac- For model (a): tional number can be named by an infinite number of ordered pairs. How many parts are shaded? How many parts are there in all? In the beginning, children should use models to find er"iivalent fractions.Once the notionis developed, What fractional numberisillustrated bythis charts similar to those on page 53 may be placed on model?(At thispointthe fractional number bulletin boards for reference.Charts showing halves, should be r corded on the chalkboard as (1, 2) fourths, eighths, sixteenths ; charts of halves, thirds, 1, sixths, twelfths ... ; charts of halves, fifths, tenths or i). ... should be included.

Tae same questions should be repeated for model Preliminary work with factoring numbers, finding (b) and the fractional number recorded as (2, 4) or common factors, and finding equivalent fractionsis necessary before introducing the idea of fractions in 2, and for model (c) and the number recordedas simplestform.When introducing "simplest form," such 4 questions as the following may be used: (4, 8) or 4. 8 What common factors do the terms of 8 have? 16 When this has been done, the following questions What is the greatest common factor of 8 and 16? should be used: What is the result when you divide the numerator Does each fractionai number represent the same 8 amount of space in the model? and denominator of by the greatest common 16 I 4 factors of 8 and 16? Are names for the same fractionalnum- 2 ber? Are 8 and 1 equivalent fractions? 16 2 Are there other ntmes for this number? Do the terms of 1 have any common factors other Can you demonstrate or prove this with a model? than 1? This development should be repeated with other 3 1 2 3 It should be explained that 1 is a fraction in simplest fractional numbers, such as I ,2,, or, 2 369 5 1015 form because the numerator and denominator have etc. using different types of models. both concrete no common factors other than 1. This procedure should and semi-concrete. be followed with other examples including some fractions whose numerators will not be one in simplest A number line may be marked to show halves, form and some fractions already in simplest form. fourths, and eighths.Children may use this to show otl et names for segments of the line labeled 1 and 4. After many experiences with physical models, chil- 2 4 dren are able to discover that an equhalent fraction can be obtained by multiplying or dividing the nu- Attention may be directed, to the classroom where merator and denominator by the sa.-qt counting num- physicalmodelsrepresentingequivalentfractional ber. For example, by referring to r odels they can un-un- cambers may be seen: windo4anes, panels, floor tiles. 2=2 x 1 4 4 4 I t, derstandwhy: = ,etc. 4 2 x 28 8 -s- 4 2 After many experiences with physical models, children can be led to discover that an equivalent fraction Order o) fractional nmbers:The number line may be can be obtained by multiplying or dividing the numera- used to develop understanding of the order of frac- tor and denominator by the same cowing number. Ilona/ numbers. The idea may be introduced on a large For example: A chart similar to the one in column 2 demoi.stration number linesimilartothe one on pages 53.54 (Rational Numbers) may be built with page 56, with points located but unlabeled. When

84 90 points have been properly labeled, questions like the As the number of congruent parts of a whole following may be used to lead children to use the num- decreases, the size of each part increases. ber line to discover the order of fractional numbers: After comparing unit fractions, children are ready to Which segment on the number lineislonger, compare unlike fractions in which one fraction needs the one associated with 1 or the one associated renaming.

with --? Children may be asked such questions as: 4 Which is larger 2 or 5? ... the one associated withIor the one asso- 3 6 4 How can we find out? dated with. 1? 2 3 3 ... the one associated with2or the one asso- 3 elated with 3? 4

1 1 1 1 2 3 Which are true: > ; > ; > ? 2 4 4 3 3 4 T I -I 1 4 5 Children may use a strip of paper of a unit length 3 6 6 which rosy be folded to show halves. another strip of Models similar to those above may be aced and paper (same unit length) which may be folded to show comparisons made.As understan.ling develo:s that fourths, and another strip of paper (same unit length) this is a process of renaming, children may be led to which may be folded to show thirds. compare pairs of fractional numbers by renaming and then verifying answers with models. These may be compared to show order of fractional numbers. To compare fractions which require renaming of I both fractional numbers, children may make models.

2 2 For example: Models of and may be compared 3 i iychanging each to twelfths as follows: 2 3 Draw a unit rectangle.Diaw vertical searnents to show thirds. Color two sectio.zs, (step 1). CI I 1 1 2 4 4 4 After comparing many pairs cf fractions in this way, children should be able to gener.lize that a given fractional number on a number line is less than the one to its right and greater than the one to its left. Draw horizontal segments to show fourthsUse a contrasting color to shade three ot these new .sections. Physical models may also be used to compare unlike (step 2). fractions with numerators of I.Models similar to that below may be used with questions.

Which is larger, 1 or 1? After working with many 3 6 models of different unit fractions, children should be Answer these questions: able to generalize: As the number of congruent parts of a whole How many sections are blue? 8 increases, the size of each part decreases. How many small sections are led? 9

85 81 How many small sections are there? 12 2. OPERATIONS ON FRACTIONAL NUMBERS 8 8 2 Addition:Physical models should be used by chil- are blue-9are red. -= and -9- =- -3- 12 1-2 12 3 12 4 dren for concrete experience in adding fractional numbers with like denominators.Sets of cubes, strips of then -3 > 2. paper, or rectangular and circular regions to represent 4 3 fractional parts of a unit, may all be used to develop Transparencies of such models can be made by understanding of addition with like fractions. children. The larger fraction may be readily determined by comparing the number of twelfths.Other pairs of fractional numbers may thou be compared without using models and the answers verified by drawings if needed.

Density of fractional numbers:As the density of rational numbers is difficult for childree. to understand, :t is taught much later than are other fractional num- 5 3 ber concepts. The number line may be used by asking + ={,] appropriate questions.Children may discover that a 6 6 fractional number can always be found between two given fractional numbers. This may be demonstrated by associating fractions with points on the number line.

For example: between -1 and -11 found -3-, between 4 2 8 -1 and -3 is found -5-,etc. When the pllysical model 4 8 16 2 , I gets in the way and it becomes difficult to determine these points, the discussian may be carried on by continuing to name fractions between points.If a ques- + -3 = tion like. "Do you think itis possible to name all the 4 4 fractional numbers between two given fractional numbers?" brings the answer "yes," the child should try to prove it with examples. Someonx who answers "no" These experiences should be discussed as mathe- may attempt to justify his point of view. With experi- matical sentencesforthe addition of the fractional ences such as these , children will be able to make the numbers are written on chalkboard or overhead pro- generalization regarding the density of fractional num- jector. When enough experiences of this type have been bers; i.e.. between any two fractional numbers there is used to develop understanding of the idea, sums may be another fractional number. found for addition examples without wing manipulative materials. Generalization gout the addition with like fractions may then be expressed.

The number line may also beiused to find the sum of fractional numbers named with like denominators.

For example:

0 1 5 2

g 9`) Children who have had many experiences with If children experience difficulty in doing this,it may equivalent fractions should have little need for physical indicate that understanding of ideas involved in sub- models in adding fractions with unlike denominators. traction is not adequate. They should be able to generalize that it is necessary to rename these fractional numbers in order to find the sum. If there is need for use of physical models, it is Multiplication..In order to capitalize on children's clear that understanding of equivalent fractions is not understanding of "of" in such mathematical sentences adequate and this should be the point of emphasis as of 8 = 4, teachs., may introduce multiplication before practice of addition. 2 of fractional numbers by listing on the chalkboard Subtraction: Physical models similar to those used in such mathematical sentences as the following. using addition of fractional numbers may be used to develop accompanying questions to draw attention to the pattern as it develops: understanding of subtraction with like fractions. As -1 3 10 1 of 32 = 16 4 x 8 =32 is subtracted from -2 or -5 from usin" fractional 2 3 6 6 parts of unit models, children can be .ed to express of 16 = 8 2 x 8 =-16 the generalization.If understanding of equivalent frac- 2 tions and of the addition with umike denominators is complete, subtraction with unlike fractions by rewrit- of 8 = 4 1 x 8 =8 ing them with the least common denominators should 2 present few problems. of 4 = 2 -1 x 8=4 2 2 Cnildren should have opportunity to solve number 1 1 sentences written in several forms: of 2 = 1 x 8 =2 2 4 4 2 2 7 2 2 + 1 1 5 3 3 5 3 of 1 -1 x 8 = 2 2 8 They should have experience in writing number sen- 1 of1,= -I tences to solve number stories.For example: With -8 2 2 4 yards of ribbon on a spool, does Mary have enough to 1.of -41 = -81 2 make a bow if the pattern calls for -5 yards? 6 What happens to the product as one factor is Will there be any ribbon left over? halved? Does this also heppen when Iis the factor that _ - - here of a 8 6 24 24 24 2-4 is halved? sartf Does this happen when the factor halted is fess Pupils may etc ate number stories to fit mathematical than I? Is the same thing true when the multiplication sentences. For example: =2 sign is used instead of "of"? 16 16 2 Does the same pattern dock? for other sets of How much ribbonwas wed if Jane of 16 numbers? 9 a yard left froma roll whichcontained of 16 Theofrelationship may also be used with physi- a yard? calmotif:is to develop.liemultiplication algorithm. Bill lives 9 ofa mile from school.Joe lives 16 To develop understanding of the mathematical sen

of a mile fromschc Haw far apart are tencc of of -2= a rectargular region similar IC 16 3 5 their houses? following may be drawn on she chalkboard and oyes-

87 tions used to involve children in the process of finding region with area1 and with side measures of 1unit 1 of 2orIx3 ofthe region: may be displayed and discussed: 5 3 5 What are the dimensions of the region which is shaded? What is the area of the shaded region?

Is the sentence, 1 x 1 =1 true? Why? 2 2

Using other models, repeat above questions:

What are the di. Tensions of the region whichis shaded?

What is the area of the shaded region?

1 1 1 Is the sentence x = true? Why? Why isit necessary to split or separate there- 2 2 4 gion into five equri parts? 3 3 Is the sentence x = true? Why? How many fifths should be colored? 2 4 8 Why isit also necessary to split the region into three equal parts? 4 4 4 Hcw many thirds should be coloreci?

Into how many parts is the regionnow separated? How many of these fifteenths have been colored twice?

. 1 What is of 2 ? Comp let.tint mathematical 1 x1- 1 3xI 3 3 5 2 2 4 4 2 8 32 sentence: 1 of 2= x L" j 3 5 Shaded arca Shaded area I 3 iss of unit. is of unit. Understanding of multiplication of tractionalnum- 8 bers using a rectangular regionmay also be developed through the use of transparencies with overlays. Physi- After many experiences of this type children should cal models should be used until understanding is de- be able to generalize that to find the product of frac- veloped, Children may then make the'r own drawings tional numbers, the numerators are multiplied, the de- to illustrate other mathematical sentences, nominators are multiplied and the yoduct is written as a fraction. For example: 3x2 3 x 2 6 The area ofrectangular region may also be used to 5 7 5 x 7 35 illustrate the a torithm for the multiplication offrac- tional ,cumbers. A region whose side measures are Division:Division of fractional numbers may be whole numbers should be reviewed through theuse of physical models, approached in more than one way, three of whicharc outlined below. Teachers should selectone approach at a time to develop, discussing procedures s'tp by Questions may help to recall the fact that thearea step. When ideas are understood to the point where is found by mnItiplying the number of rows by the the operation can be performed, another approachmay number of square units in eachrow. Next a square be demonstrated in the same way to reinforce under-

88 ,94 standing of the principle that division by a fractional be urged to use the algorithm that they understand number is the same as multiplication by its reciprocal. best.Before teaching division of fractional numbers This is particularly important when either the complex by the complex fraction algorithm, children must under- fraction or inverse operation is the first approach used. stand the following:(See Rational Numbers, pages When the general principle is accepted cldldren should 61-62.

5 I. 6 4.3=-6; 2 4- 4 = 5 + I 1; a ÷b -= a 3 4 5 a 2 6 b 7 ± 7 =-- I; 7 a 2. 7)7 ' 7 1 1 5 2 6 b 4 4 3. 16 x I = 16; X 1 lx =--- ; a x 1 = 3 3' 3 3

8 8 8 ± 1 = 8; + 1 = a 4. 1)8' 8'

2 3 I 5 5. x = I; x = 1; 5x = 3 5 1 5

5 5 5 x 5 25. c _aXc 6 7 6 x 7 42-' b d b X d 11 a II 2 a c S 7. =I. ± 3 7 2 b b 7 d

It must be understood that the number one las many 2 I Let?= x and =y forms which are useful in working with complex frac- 3 8 3 8 tions. The numerator and denomiultor of a con plea 2 fraction can be multiplied by one in any of its forms then 2 1=-- x y x and 2 4- without changing its value in the same way that any 3 8 y 3 8 I For fractional number may be multiplied by one. 8 example: The following questions may be used to check for 5 5 3 15 5 5x 1 these understandings: 7 7 3 21 7 In how many ways can the operation of dividing 2 2 3 6 2 2 x 1 four by three be shown? 3 3 3 3 9 3 1] 4 ÷ 3 = =[7 5 a 3)4 5 3 15 1 7 3.21 7 b b What does the bar between the two whole numbers 2 c c 2x7 14 Y I in a fraction mean? d 3 7 21 3 d What do the following expressions mean:

I 11 It children have undelsranding the a fraction is one number, not rwq, or that division of fractional 2 4 2 12 numbers in%olves Iwo numbers, not four, the follov.ing 4'2'3 2 idea may be helpful: 4 2

89 95 By what number :nay any number be multiplied without changing its value?Give examples to 3x6 prove it. 4 5W!ly? (5) reciprocal 1 By what number must a number be multiplied to produce a product of 1?Give examples to prove it. 3x 6 Why? (2) z-- 1 4 5 N in tow many different ways may the following

complex fraction be expressed: = -3- Why? (6) = 20 b d h d 7

? =9 Why? (3), (2) factored 3 10 4 Then the process is understood, pupils should be 7 3 []x3 3x 7 able to generalize that division by a fractional number 8 4 4 4 8 is the same as multiplication by its reciprocal.

2 One algorithm for the division of fractional numbers 2 1 3 Does _ = ? may be developed through children's familiarity with 3 8 1 common denominators as used in addition and sub- 8 traction. This approach can provide a logical intro- 2 duction to division of fractional numbers. 2 I 2 4. Does : 3 8 3 4- 8 3 Mathematical sentences similar to the following may 8 be written on the chalkboard, and procedures discussed as each step of the solution is performed: 2 2

1 Does3 ..-,_ 17 = D 1 3 2 4 i 1 x2_2 1.c.m. of 2 and 4 is 4. 2 2 4 In how many ways may 1 be expressed? 2 1 2 4- 1 2 1 2 2 = 2 4 4 4 -4 1 1 1 416203 ; I416203 2 5 7 5 4 20 7 3 21 ÷ x = and x = 3 6 8 6 4 24 8 3 24 1.c.m. of 6 and 8 is 24. If it is clear that children understand basic ideas expressed above, the complex fraction method may be 20 21 20 4- 21 20 = 21 20 introddced, developed step by step and discussed as 24 24 24 4- 24 1 21 follows: If children experiencedifficultyin using this ap- 3 proach,itwill indicate that they do not understand 3 5 4Is this true? Why? one or more of the following: 4 6 5 6 Now tc. find the common denominator

3x 6 That a number divided by itself has 1 as the 4 5 Why may this operation be performed? quotient x6- 6 How to express the dumerator of the fraction 20 4- 21 (3) N x I = N as a quotient. (5) reciprocal 1 Attention sho.,1d be given to the area of difficul'y (2) I before practice on the algorithm.

90 9 Familiarity ivith division of whole numbers as the 5X-8 or 5- x -8 [-]Why? inverse of multiplication may lead some children to 6 3 6 3 ask if this relationship holds true for fractional numbers. As the procedure is developed and the relation- x _8_)x 3=5True? Why? SLibiitution ship understood, further evidence can be seen that 6 division by a fractional number is the same as multipli- ÷3=5x8 cation by its reciprocal. 6 8 6 y-

Before developing the algorithm, children must have The idea that division by a fractional number is the clear ,Inderstanding of statements given under the com- same as multiplication by its reciprocal should again plex fraction approach, page 63, especially statements be evident. Most children will accept this as a principle 5 and 8. They muss recognize dilision as the inverse to be trusted and be ready to use it as the reason for ofmultiplication they mustunderstandthat if asimple, direct approach todivisionof fractional 3 x 4 12, then 12 ± 3 = 4 and 12 ± 4 = 3; that numbers. 2 -2 -/means that -. x = that axb=c means 3 7 7 Estimation shoved be used to determine reasonableness of answers. This must be done through conver- = a, (c b a) and that- = b, (c ÷ a = b). a sation; it should not be done through seat work. Dis- cussion of such questions as the following may be Questions similar to those used to determine readi- helpful: nessfor the complex fraction approach, page 89 should be used to review understanding of these basic Which quotient is greater, 12 ÷ 2 or 12 ÷? 2 ideas. 1 2? Why? 18± - or 18 Why? Can physical 3 3 A problem may then be developed through steps as models be used to prove the above? outlined on page 63. The process must be discussed step by step with children supplying reasons for each Are the following true or false? operation. Questions should aid in making the process 3 1 1 1 clear. (T) -4 -2 For example: <2- I 3 71 5 . 3 and 0 x = 65_ Before practice for mastery there should be opportunity to discuss and so-.e such sentences as the fol- Is this true? Why? lowing: Islultiplication is the inverse of division 1 3=N 24-N= 1 N÷3_ -7 5 8 5 2 4 2 must NC-XA) 6 Opportunity should be provided for pupils to write, discuss and solve practical problems involving division (5- xtx =,5 Why? Substitution 6 8 6 of fractional numbers.

n x3 -5 Is thisalso true? Why? 6 8 6 PROPERTIES OF ADDITION AND SUBTRACrioN of FRAC- Assojatie properly TIO`:AL NUMBERS What children know about the properties of whole If x ( ) numbers may be used to determine whether these prop- 6 6 erties also hold for fractional numbers. Such activities what roust be in ( )? (1) Why? N 1 N as the following may be used: 3 Then Ax = 1True? Why? 1. Finding the sum of fractional numbers and dis- 8 cussing the set of numbers to which the sum belongs to enable children to see that 'he set of Then A = -8- Why? Kcciptocal 3 rational numbers is dosed for addition.

91 97 2. Finding the difference of several pairs of frac- system to include negative numbers. Activities like the tional numbers to discover that, using numbers following can be used to develop understanding of which are familiar at this point, it is not always integers: possible to subtract.It becomes apparent that subtraction isnot closed for the set of frac- a. Weather reported as 10° below zero, may be tional numbers. written as --10° after a thermometer has been examined. 3. Using the number line to see that addition is commutative and associative: b. Time before rocket blast off may be counted down, the significance of zero discussed and the 4 7 4 3 time written as 4, 3, 2, 1, 0. + =- 4 4 4 4 4 4 c. An event occurring in 55 BC may be discussed, +0 + )4. =3 "zero" identified and the time written as 55. %7 7 %7 7 / 7 d. Ocean depth recorded as 2000 feet below sea 4. Subtracting pairs of fractional numbers, then level may be discussed, comparison made with reversing the order cf the pairs to see that a mountain peak of 3000 feet elevation, the subtraction is not commutative. For example: "zero" identified and a diagram drawn. Dis- tances may be written as 2000 and +3000. 3_1=2 1_3 2 e. The number line may be used to illustrate the 8 8 8 8 8 8 route of a Scout troop that hikes 5 miles east, 5. Subtracting using three fractionalnumbers, 2 then 1 mile west, "zero" established as the start- numbers at a time; then changingthe order of ing point, and the distance recorded as +5 and subtraction to see that subtractionis not asso- 1. ciative. Fcr example: After discussion of many activities of this type, pupils should be able to generalize that positive integers may be used to denote change to the right or upward, and that negative integers may be used to denote change 8 r5 to the I,ft or downward. Care must be taken to insure 1)=9 that children understand that the symbols t- and 6. Using such examples as the following to dis- are part of the numerals for integers and not opera- cover the identity element of addition and sub- tional symbols for addition and subtraction. If symbols traction of fractional numbers. For example: for integers are written in the raised position and read as "positive" and "negative," much confusion can be 2 _ + 0=0 +2 2 avoided. 3 3 3

0= but 0 2.ORDER OP INTEGERS 5 5 5 As negative numbers are discussed, pupils may begin to use the number line extended to the left of zero. A thermometer represents the most familiar model and PROPERTIES OF MULTIPLICATION AND DIVISION OF can provide a starting point for discussion. FRACTIONAL NUMSERS Children usually understand that 0 is colder or less warm than Knowledge of properties of whole numbers may be a reading of 10° above zero, that 3' below zero or used to determine whether these also hold true for --3° is colder or lc,s warm than 2' above zero. From multiplication and division of fractional numbers. this beginning, the number line, both horizontal and vertical, can be used to answer such questions as the following: The Integers 1.NATURE OF INTEGERS Which are true? Which are false? Situations involving gain and loss, rise and fall, de- Is +6>+5? ) Is --7 <+3? (7 ) posit and withdrawal are familiar to most children. Is5<+9? (1) Is0 <-1? (F) There should be much discussionto associatethis Is -3<2? ( I ) Is4 =+4? (1r) practical experience with extension of the number Is -4<7? (F) Is0 <+ I? (I)

92 ,98 As relative values of positive and negative numbers When the relationship of opposites is understood, are understood, the idea of opposites may be introduced the generalization can be stated and the term additive through discussion of such problems as the following: inverse applied.

a. Starting from the same street corner, John walks Use should be made of the number lineitde- 3 blocks east and Charles walks 3 blocks west. veloping understanding of all operations on integers. Does the number line below represent the situa- If an integer is understood to be less than any integer tion? to the right or above it, and greater than any to its left or below it, movement of concrete objects along the line should enable children to understand and express basicprinciplesunderlying algorithmsinthe four Charles 4 John operations. w E F1.1. 3 Using the number line, opportunity should be provided to show many such operations as the following: Balance of 0.Deposit of $10. Withdrawal of $7. Balance of $5. Deposit of $3. Withdrawal of $8. Can 3 and +3 be thought of as opposites? Balance of $15. Deposit of $1. Withdrawal of $18. Why? Debt of $10. Deposit of $25. New 1- nce of $15. b. Mary earns 25¢ and loses it on the playground. How much money does she then have? A gain 3. OPERATIONS ON INTEGERS of 25 and a loss of 25 is equal to zero, +25 and 25 = 0. May +25 and 25 be thougi t Addition:By demonstrating on the number line, of as opposites? children may complete such sentences as the following:

a. +6 -r +3 = [11 c. 3 + 5 :---- e. + 3 = El g. 8 + +3 = ,171 b. +2 + 48= d +5 = f.+3 4- = h. 2 + +6 =

1 )411

Does the commutative property hold for addition of Sums of integers where both are negative may be integers? introduced thro.igh discovery of a pattern which will enable them to fill in blanks rem lining in each column Sums of integers where one is positive and one nega- below: tive may also be introduced by leading children to discover a pattern which will enable them tofillin 5+ 3 2 3 +4 : 1 3 +-5 = --2 blanks remaining in each column below: 4 + 3 I 2+ 4 =: 2 2+ 5 ---:-3.

4+1 =5 3+ 1= 4 2+ 1 = 3 3 +3 =0 I +4 - - 3 I + --5 --4

4+ 0 =_ _ 3+ 0= 2+ 0= _ _ 2 + --3 __ 0 + -4 0 + 5 ::--- _

4 + 1 = 3+ 1= 2+ 1= _ I + --3 _ --4 --1 + -S = 4 + 2 3+-2=_ 2+ 2= _ _ 0 + --3 = _ -2 +4 :: --2 + --5 r=

4 + 3 r--: 3+ 3= _ 2+ 3= _ + 3 _ 3 +4 3 + =

4 + 4 :---- 3+ 4= 2+ 4=- _ 2 + --3 =__ 4 +4 rr _ --4 + --5 4 + 5 = 3+ --5= 2+ 5 _ _ 3 + 3 =_ _ -t 4 :: _ _ --5 + --5

93 '89 After discussion of results obtained by use ofnum- of finding a missing addend when the sum andone ad- ber line and through observation of developing patterns, dend are known. children should be able to state and use basic principles that apply to addition of positive and negative integers. With the number sentence 2 5 = [:] rewritten as 5 + = 2 the number line may be used to find the missing addend. Starting at the pointon the line Subtraction: Before attempting subtraction involving named by the known adderd, 5, and movingto the negative numbers it is important that subtraction has point named by the sum, 2, three unitson the line been recognized Is the inverse of addition, the process are passed. The direction is to the right or positive so the missing addend is +3.

8 4 --3 --2 1 0 1 +2

Other number sentences similarto the following tion of integers. Examples likethe following may be may be solved in a like manner: used: 3 x 4 4 +2 = becomes +2 -1- = 4 4 + 4 + 4 = 12 +7 ---4 = becomes4 + = +7 3 x (-4) = (-4) + (-4)+ (-4) = 12 +3 +5 = becomes+5 -4- 0 = +3 (-3) X 4 = 4 x (--3) Commutative property. = (-3) + (-3) + (-3) + (-3) = 12 Number sentences involving negative numbersmay be solved in this way until ehiidren can reach generali- There should be much discussion and many examples zations stated on pages 69-70 of Rctional Numbers. like the one above before children are asked to state the generalization of the way a positive integer anda Patterns may also be used to develop this under- negative integer are multiplied. standing as children find numbers thatwill enable them to fill blanks in sequences like those below: The decreasing factor approach may provide the best introduction to the multiplication of two negative +7 +4 = +3 --7 +4 =-11 numbers.After using this approach and developing +6+4 = +2 7+3 = 10 several patterns, children should be able to generalize +5 +4 = +I +2 = 9 that the p:oduct of two negative integers isa positive 7 n unther. +4 +4 = 7 + 1 8 +3+4=-- 7 0 =7 Division: The idea that division is the inverse Gt multi- +2+4=-- --7 1 6 plication may be used to introduce division of integers. Examples like the following may be used: +1 +4 = 72 0+4=-- 2 73 = 12=3 = 4; =4 because 3 x 4 = 12 3 --1 +4 = __ --7 -4_ Inverse operation --2 +4 -= -7 --5 _ 12 --12 =3; =3 7- -6. -4 77 because 3 x (-4) 12 12 7 8= I 2 ÷ 3 4; = 4 3 because ( -3) x 4 -= --I2

2 Multiplication: Upderstand;ni: of multiplicationas re- I 2 4 4; = -44 peated addition may be used to introduce multiplica- because (- 3) x (- 4 ) = 12 94 109. When there has been careful discussion and practice 2. OPERATIONS ON POSITIVE AND NEGATIVE FRAC- with examples like the above, children should be able TIONAL NUMBERS to formulate generalizations for division based on the Since operations on positive and negative fractional generalizations for multiplication. numbers are performed in the same way as operations using integers, there should be little difficulty in developing algorithms. Negatives of Fractional Numbers I. THE NATURE OF NEGATIVE FRACTIONAL NUMBERS Decimals Ideas related to the nature of fractional numbers and integers also apply to negative fractional numbers. 1. THE NATURE OF DECIMALS Each fractional number has an opposite, a negative Decimals may be introduced through use of models fractional.-qber, an equal distance to the left of zero of regions separated into tenths. As parts of the region along the number line. The sum of pairs of opposite are colored, children may name the fractional number fractional numbers is always zero. identified with the shaded part.

The big region is one unit. (1.0) How many spaces are in the region? (10) What is another way to name the (1.0) unit? 10 What part of the region is gray? 1 10 What part of the region is lavender? 5 (.5) 10 What part of the region is lavender 7 (.7) and white? 10 What part of the region is lavender,10 (1 0) white, blue , and gray? 10 The fact that tenths can be written using decimal tenths, hundredths, and thousandths may also be used notation may be introduced, the questions above used to show relationships. Graph paper squared in tenths and answers written using decimals. may be used by children to construct individual models.

A model of the same size may be further sub- If responses to such questions as the above are re- divided to show ten rows of ten spaces each, similar corded on a chart, relationships may be observed and questions asked and deci-nal notation for hundredths understanding of a decimal as a fractional number with introduced.Transparencieswithoverlaystoshow some power of ten as a denominator may be further developed.

Tenths Hund edths I Thousandths Fractions Decimals Fractions Decimals Fractions Decimals

10 100 .1 .10 .100 10 100 1000

7 70 700 .7 .70 .700 10 100 1000

50 ,500 1 or 1.5 1.50 ia00 or 1.500 10 I"100 or 15 150 1500 10 100 1000

95 101 Decimals may also be linked to children's knowl- .875 edge of values of dollars, dimes, and pennies. An array means 7 ± 8 or 8)7.000 of pennies, ten rows of ten each, or a chart showing 8 this type of array, may be used to develop understand- If in performing the division a remainder of zero is ing of hundredths.Notation can be associated with obtained, the decimal is said to terminate. what is already known about use of the decimal point in writing money: 16 pennies = $.16; 8 pennies = $.08. Repeating Decimals: Afterdivisionofdecimalsis Such activities as the following may provide addi- familiar to children, it can be shown by division that tional help in understanding the nature of decimals and some fractional numbers can be expressed as decimals their relation to fractional numbers: which do not terminate and the digits of quotients re- Construct place value charts to use in reading and peat, such as in the following examples:

writing decimals. 1 = 3)1.000 = 0.333 ... 3 Arrange the following decimals in order, beginning with the least.36, .3, .03, 9.5, 4.16, .001, I= 7)1.000000 = .142857142857... 7 Use ruler, meter stick, or steel tape marked in tenths to measure heighths of children. Such quotients are known as repeating decimals. Use an odometer. "handy" grocery counter, ped- From the above it should beident that all frac- ometer or other devices for measuring which regis- tional numbers may be expressed in decimal notation. ter in tenths of units. Charts such as the following may help to bring out the patterns: Terminating Decimals: As understanding of the nature Repeating (R) Prime of decimals develops, it should be evident that itis Decimal Termi- Factors of 7 58 simple to express--cr as decimals. Such ques- Fraction Equivalent nating (T)Denominators 10 100 tions as the following may lead children to name fractional numbers as equivalent fractions having novters .5 T 2 of ten as denominators and to express the resulting 2 fractions as decimals:

I n . _2 .33 R 3 3 (a) (b) 5 10 (c) 4

7 1 (d) (c) _ .25 T 2 275 4 What is the value of n in examples (a) and (b) rbove? .2 T 5 5 Write the decimal equivalent of (a)and (b)

above. 1 .1666 R 2, 3 In (c), (d), and (e) above, what powers of ten 6 can be used as denominators to make equivalent fractions?

W'rite the dc imal notations for (c), (d), and (e) above. (.25. .28, .875)

Mien the decimal equivalent is not readily seen, it may be necessary to perform the indicated division:

96 10e, 2. OPERATIONS ON DECIMALS Multiplication:Multiplication of decimals may be related to what children know about multiplication of Addition:Additian of decimals may be introduced fractional numbers. by using what children know abnut addition of fractional numbers. Decimals may be written as fractions 3 2 3 x 2 6 and added. .3 x .2 = x = .05 10 10 10 x 10 100

.7 + .2 =-7+-2=-9 5 x .7 = -5 x -/- = 35 3-1i== 3.5 =.9 1 10 10 10 10 10 10 2i 7 x 21 21 96 .7 x .21 -= -70 .75 + .21 =-75+ .96 10 100 10x 100 100 100 100 147 = .147 16 2 16 20 1000 .16 + .2 + = T 100 10 WO 100 4 8 24 192 8 x 2.4 = x 2 = x 36 1 10 1 10 10 100 19-= = 19.2 10 9 .5 + .36+ .009=-5 + -364 10 100 1000 As many operations of this type are performed, chil- 500+ 360 9_ dren should be able to ge.teralize that ones multiplied 1000 1000 1000 by tenths Oyes tenths, tenths multiplied by tenths gives 869 hundredths, and tcnths multiplied by hundredths gives .869 thousandths. 1000 The operation can then be performed using decimal notation, products can be compared and generaliza- Addition of decimals may also be introduced by tion applied.As operations with decimals involving using knowledge of place value notation developed ten thousandths and hundred thousandths are per- throi.i3h use of place value charts. formed, generalizatio.,s may be extended to Include the idea that hundredths multiplied by hundredths gives ten thousandths. .7 .75 .16 .5 + .2 + .21 4- 2 .36 After solving many problems like the following using .9 .96 .36 + .009 fractions and changing products to decimals, a pattern .869 should emerge. Children should be able ti, see that to place the decimal point correctlyin the product, the total number of digits to the right of the decimal in Through practice in writing decimals for addition and numerals of factors any be counted subtraction using a Oz.:se value chart, children should be aware of the significance of the decimal point in Examples writing numerals for decimals. 3 6 .3 2= sit- =,6 10 10

Subtraction:Understanding of addition of decimals can 9 27 .9 .3= -3- be applied to subtraction, either by expressing the deci- 10 10 100 mal as a fraction or by usiniz, knowledge of place Naive _64 notation.Since decimals are base ten numerals and 4 x .16 = 4 16 use the same place value pattern as whole numbers, 100 1()0 children should expect the decimals to behave in the 29 5 145 operations of addition and subtraction in a way similar 145 to whole numbers. 1(10 10 1000

97 396 1584 Fractional number approach: .396 x 4 - x 4 = 1.584 1000 1000 3 9 3 10 30 3 1 .3 -2.-- .9 = - = - 9 135 - = - .15 x .09 =-15 x .0135 10 10 10 9 90 9 3 100 100 10,000 Inverse of multipli:ation approach: Pattern .21 ± .7 n n x .7 =.21 .3 x .7 = .21 1 x 1 =1 3 x 1 =3 n -= .3 x .1 =,1 3 x .1 =0.3 x .01 = .01 3 x .01 = 0.03 Place value approach: I x .001 = .001 3 x .001 = 0.003 Through multiplying by one the divisor is changed to a whole number and the algorithm is performed nd and using the place value notation.

. 1 . x 1 =.1 .4 X 1 =.4 76.5 IC, 765 9 8.5.)7675 x 85)765 x .1 -=.01 .4 x .1 =.04 8.5 10 85 x .01 =.001 .4 x .01 = .004 .85).765 .765 100 76.5 .9 .85 100 85 85)76.5 7.65 1000 7650 90 Multiplication of decimals may also be approached .085F7.67 through use of renaming, commutative and associative .085 1000 85 85)76.'7.0 properties as illustrated below: Patterns which can be seen by using t' .6 x .21 foqns of division may be helpful: = (6 x .1) x (21 x .01) renaming = (6 x 21) x (.1 X .01) commutative and 39- 1 = 39. 39 associative 1 properties 39 39-10= 3.9 7= 3.9 = 126 x .001 = .126 10

39 100 -=0.39 39-= 0.3') When theprincipleisunderstood and accepted, 100 many activities like the following may be introduced: 39=- 1000 0.039 Solve the following by using fractional notation and changing answer to a decimal: 48 ± 3 = 16 .7 x .8 = .5 x 8.6 --= 4.8 4- 3 = 1.6 4.25 x 1.2 = 3.02 x 5.8 = .48 =3= 0.16 Place the decimal point in products of the follow- .048 ± 3 0.016 ing: .9 x 3.2 = 2 8 8 After much practice children should be 7,1 o the generalization that: .4 x .36 = 1 4 4 When the divisor is a whole number.t .5x 1.82 = 9 1 0 point will have the same position in :b .05 x 1.82 == 0 0 9 I 0 as it has in the dividend. As a check, the quotient sbooIcl lk multij the division of decimals may be introduced divisor to determine if the product cquaic t: by recalling what children know about the division of fractional numbers, by using understandinj ol division The habit ol estimating answers before 5 as the inverse of multiplication, and by employing 'CMS isparticularly important in checki-q: L knowledge of place value notation. using decimals,Estimating quotients b.:

98 104 division examples makes the above generalization more .1f two candy bars cost 12 cents, what is the meaningful. Estimations should be recorded for every ratio of cost to candy bars? example. (12,2) -1-2 12 : 2 6:1

Ratio and Proportion 1. THE NATURE OF RATIO AND PROPORTION Proportion: Understanding of a porportion as a state- The idea that a ratio is a comparison of two num- ment that two or more ratios are equal maybe related bers by division may be developed through use of to knowledge of equivalent fractions: concrete objects which can be compared. Forexample: 1 4 red blocks and blue blocks, balls and bats, chairs and 2 tables, books and children.Illustrations such as the 2 4 8 followit.2, may be used and the anawers expressed first as ordered pairs, then as fractional numbers,indicated Such questions as the following may be used: division and ratios: Is thc, ratio of 2 candy bars for 121 the same as II Ili 5 bars for 30e? Is the rzttio of 3 adults to 20 scours the same as 9 adults to 60 scouts? a. What is the ratio of red blocks to blueblocks? Is the ratio of 30 cookies to 10 children the same ,t (4,3) 4 + 34:3 as 60 cookies for 15 children? 3 b. What is the ratio of blue blocks to red blocks? 2. PROPERTIES AND OPERATION.' (3. 4) -3 3 4 3:4 4 Since a ratio is a rational number, operationsand properties for rational numbers apply toproportions.

Per Cent 1. NATURE OF PER CENT raw-TRIFI? The fact that a per cent is a special kind ofratio and c. What is the ratio of balls to bats? that the term per cent means per centurn or per one hundred should be discussed.If children are familiar (5.2) 5 -:2 5:2 with ratios written in the form 3:5 and read as3 to 5 2 or 3 per 5. then 7% can be read as7 per 100 and d. What is the ratio of table to chairs? written in the following forms:

7 (1. 4 ) -I- 1 4 1:4 7%;7:100; .07;7 per 100 4 100' Story problems such as the following may be used in (Using the idea of sets means that for every 7 in developing understanding of ratio: one set there are 100 in thesecond set.) a. There are 30 cookies for tenchildren. What is the ratio of cookies to children? Problems invoking per cent may be solvedthrough 1--0 the use or 'roportions. (30. 10) 10 3:1 1(1 Ratios are usually expressed in lowest terms. 5 100

11.1 hree adults went on aeanipir±'tripwith n 60 twenty' scouts. What was the ratio of scouts to 5 100 adults? 20 3 60 (20, 3) 20 1 20:3 3 n 100 99 105 Properties for Four Arithmetic Operations for RationalNumbers

Children may test the properties-closure,commuta- (under two operations)-to see if they tivity, associativity, identity, inverse, and hvld true for distributivity operations on numbers from various sets. Forexample:

Addition-Using the set of even numbers:

I. 2 + 6 = 3, an even number Closure holds a + b = c 2. 2 + 4 = 4 + 2; 6 = 6 Commutativity holds a+ b=b+a 3. 4 + (8 + 6) = (4 + 8) + 6; 4 + 14 ---- 12 + 6; 18 = 18 Associativity holds a+ (b + c) = (a + b) +c 4. 6 + 0 = 0 + 6 = 6 Identity element (0) a + 0 ---- 0 + a = a 5. No inverse element Additive inverse a + -a = 0

Subtraction-Using the set of fractional numbers:

1.-5 = A, a fractional number 6 6 6 Closure does not hold -5is not a fractional number 6 6

2. 5 1 5 6 6 6 6 Commutativity does not hold /5 Associativity does not hold 7 7/ 7 7 -7 7/ 5 5 5 4.-0 = -5 0 - Identity element (0) dces not holdon 9 9 the left but does on the right

Multiplication-Using the setofdecimals:

I..3 x .5 = .15, a decimal Closure holcis 2. .05 x .6 = .6 X .05 .030 = .030 Commutativity holds 3.(.3 x .2) x .7 = .3 x (.2x .7) ,06 x .7 = .3 x .14 .042 = .042 Associativity holds 4. .16 x 1 = I x .16 = .16 Identity element (I) holds 5. .16 x 0 = 0 0 x .16 = 0 Multiplication property of 0 holds 6..7 x = 1 .7 .7 Multiplicative ins crse holds

Multiplication and addition 7. .6 x (.3 + .02) = (.6 x .3) 4- (.6x .02) Distributivity of multiplicationover addition holds

100 1 Division---Using the set of fractional numbers:

5 5 12 60 2 1.5 Closure holds 6 12 6 5 30 1

1 5 25- 1 8 9 769 8

5 x9 I X 8 Commutativity does not hols1 8 1 9 5 45 8 8 45

3. o(3_ 1 3 N4 %3 4/ 2 2/ Associativity doesnothold 2 2 8 1

4.3 : 1 = 3 5 5 Identity on right only , =5

101 1107 GEOMETRY

Geometry is the study of ides of points, lines, curves, planes, andspace.In the elementary school the approach is informal and intuitive.Relationships and properties are discovered and discussed beforeformal vocabulary is introduced. The language of geometry is the languageof sets; that is, sets of points.

In the natural world geometric formsare everywhere observable.Man employs geometric shapes inthe buildings which he designs, the utensils and instruments whichhe uses, the art with which he decorates his home, his clothes and his surroundings. It is desirable that children learnto see and identify some of these geometric designs, especially those that give a feeling for beauty andsymmetry.

The early study of geometry must not be entirely incidental. Thoughitis hoped that teachers will capitalize upon opportunities to emphasize geometric ideas, there must be sequentialdevelopment to reinforce concepts already discovered and to introduce new ones which followlogically. Thus, children will proceed at varyingrates depending upon previous experience and level of maturity.Teachers should use many activities with concrete models which lead to better understanding ofspace figures and their relationships.Pupils who have had experiences manipulating models of closed surfaces such zs balls and blocks willhave greater readiness for understanding the connection between concrete objects and geometricideas.

While geometry and measurement are treatedas separate ideas in this guide, they are interrelated in classroom teaching. A study of line segments provides the necessary geometric backgroundfor studying linear rneasums. Plane regions re studied and then area of plane regions. Space regions are studied thenvolume or capacity of space region.... Thus sin, the attachment of numbers to geometric objects, isinterrelated with shape and space.

Mathematical Ideas Illustrations and Explanations

Basic Ideas of Geonit try The basic ideas of geometry are concepts of point, line, plane, curve, and simple closed curve.Physical models from the world are used to develop ideas.

Point: A point can be thought of as Representation of a point an exact location in space which has no size or shape.UsJally, a point is .A represented by a dot whichinitself covers an infinite number of points. Capital letters are used for labeling points.

Line: A line may be thought ofas a Representation ofaline set of points which extends endLsdy in opposite directions. Unless other- Lines are named using two points of the line and thesymbol H over wise stated, a lineis thought to be the names of the points or by usinga lower case letter. straight butisnot thought to have f--1 either breadth or thickness. AR A rn lire nn

Line 3egmeni: A line segment isa I he cridpoints arc n inuJ by capital lettersand the Irne scgmcnt is portion of a line which consists of too designated by Al3 and read "line segment endpoints and all the points between them. p All

102 1 08 Mathematical Ideas Illustrations and 11,,,,planations

Ray: A ray consists of one ;endpoint Ray is written A13 and read "ray AB." The letter naming the endpoint and all of the points of a line on one of the ray is always written first and the arrow indicating endless ex- side of the endpoint. tension is placed over the letter naming the second point.

AB A B

CD C Plane: A plane may be thought of as a set of points forming a flat surface which extends inall directions without end.

Angie: An angle is a geometric figure The symbol (L) represents an angle. consisting of two rays having the same endpoint. Point A is the common endpoint of the two rays and is called the A vertex of the angle.

AB U AC forms [BAC. (SCAR) An a,,gle can be named with the three letters BAC but the vertex letter must be in the c,..nter. The angle can be named by the single capital letter that names the vertexA. The angle refers only to the set of points comprising the two rays and does not refer to any points off the rays.

A 7 Right nee: A right angle is the angle formed by two rays which form a square corner.

Z 4-1

II LZ is a right angle

Lines HG and 11 form four rid" anglesat JIG.z ano i IIKI. N

MP U MN forms rightang: PMN.

103 1109 Mathematical Ideas Illustrations and Explanations Path or curve: A path or curve is the set of allpoints passed throughin mo,ing from one point to another point in a plane. A B

Closed path or closed curve: A closed path or closed curve is a path which begins at one point and ends at the same point. AF

Simple closed curve: A simple closed curve is a closed curve that does not A cross itself.

Relationships 1. INTERSECTION The intersection of two sets of points is the set of points common to the two sets.

Intcrs:cting lines: When t)csetsof points which form two lines have only one point in common, the lines arc said Line n and line m ate intersectinglines. to intersect in this common pointthe Point P is the point of intersection. point of intersection. The linesare called intersecting lines. n f1m =P

104 i 11f} 111atheinatical Ideas Illustrations and Explanations

Thro, any point, A, an infinite number of lines may be drawn.

If 2 points of live n are also members n of line m, then line m and line n are Y the same line. Any two points may be contained in one and only one line.

Intersecting planes: When the sets of Plane m and plane 11 are intersecting points which form two pines have a planes.Line AB is the line of inte:- set of points in common, the planes section. are saidto intersectinthis setof points. The set of points is a line.It m rl n AB is called the line of intersection. The planes are called intersecting planes.

An infinite number of planes can intersect in the line XY.

If three points not in the same line are in plane n and also in plane in, plane n and plane m are the same.

Any three points not in the same line determine one and only one plane.

105 111 Mathematical Ideas illustrations and Explanations

Line intersecting a plane: When the set Line I intersects plane m in the point of points which forms a line h.t; only P. one point in common with the set of points which forms a plane, the line is , 1 nm = P said to intersect the plane in thi; point the point of intersection.

If two points of line 1 are in plane rn, if a line joins two points in a plane, then line I lies in plane m. the line lies wholly in the plane. The intersection of the plane and the line is the line.

If a line and a plane have no point in common, the intersection of the two sets of points is the empty set. The !ine is parallel to the plane.

4-0 4-) 2.PERPENDICULARITY AB 1_ CD

Perpendicular lines: If two lines inter- or sect to form square corners or right angles they arc called perpendicular lines. The symbol foris perpendicular CD 1 AB to" is " " A

14 4 PA j BC

I(16 142 Mathematical Ideas Illustrations and Explanations

Line perpendicular to a plane:A line :3 perpendicular to a plane if it is perpendicular to every line in the plane passing through its point of intersection with the plane.

P is the point of intersection of line and the plane. fikt] EP, AB L PD, ABI_ PC, AB_LPF,

AB ± plane m

Perpendicular planes:Two planes are perpendicular if a line in one plane is perpendicular to the second plane. Plane n ± plane m

3. PARALLELISM n 4 Parallel lines:Lines which lie in the same plane but have no pointsin common, do not intersect, are caller' parallel lines, n m line n is parallel to line m Rays and line segments are parallel Two lines in the same plane either 'intersect in one point or they are if they are subsets of parallel lines. pa calla

Two lines not in the same plane are calledskewlines.

/Parallelplanes:Two planes which neverintersectarecalledparallel planes. /Plane m li plane n

107 113 Mathematical Ideas Illustrations and Explanations

4. CONGRUENCY Congruency is the relationship of two geometric figures which have the same size and shape. The symbol for "is congruent to" is

Line segments: When two line seg- A B ments have the same measure they AB CD are congruent. C D

Angles: If all the points of one angle fit exactly over all the points of another angle, the angles are congruent.

F, iBAC IDEF A

Plane Figures:If all the points of a plane figurefitexactly over all the points of another, the figures are congruent. A ABC -z-e A DEF

ABCD JEFGH

The two rectangles are congruent.

5. SIMILARITY

Geometric shapes or forms are similar, All squares are similar. or similarity exists, when they have the All circles are similar. same shape but not necessarily the same size. The symbol for "is similar to" is

Triangles A and B are similar

Salt drawing is b3ccd on sruilArity

108

1.1. Mathematical Ideas Illustrations and Explanations

Triangles C and D are similar and congruent.

[ 13 I Rectangles A and B are similar, but not congruent.

C Rectangles C and D are similar and congruent.

6. SYMMETRY Line symmetry:When a line can be drawn which divides a plane figure so .4 that one side is an exact mirror image of the other, the figure is said to be A 13 symmetrical. The lineiscalled the 1 4 line of symmetryor the axis of symmetry.

Line 1is the !inn of symmetry. For ex cry point on one side of P there is a corresponding point on the other side the same distance from P.

If the figure is folded on the line of symmetry, the halves coincide exactly.

For each point on one side of the line of symmetry there is a corresponding point on the °the iside,

109 115 Mathematical Ideas Illustrations and Explanations

Plane symmetry:Some figdresare symmetrical. When this is true a plane This rectangular prismis --the plane of symmetryseparates symmetrical with respect to the figure so that one side is the exact plane m. It has three planes mirror image of the other. of symmetry.

Plane Figures--Closed Paths

A simple closed curveseparatesa interior plane into three sets of pointsthe set of points on the curve, the set of points curve n the interior and the t.,et of points -sn the exterior. exterior

exterior

CIRCLE: A circle is a simple closed curve in a plane every point of which isthe same distance from a given point in the interior called the center. The symbol for circle is 0 and is designated by the letter labeling the center.

A circle separates a plane into three AB is the diameter of 0 R. A diameter of a circle is a line segment sets of pointsinterior, the curve, and which passes through the center point with its end points on the circle. exterior sets of points. Every circle has an infinite number of diameters.

RB and RA are radii of 0 R.

A radius of a circle is a line segment with one endpoint on the center of the circle and the other endpoint on the curve. Every circle has an infinite number of radii. C An arc of a circle is that part of the circle travelledin moving from one designated mint ..._toanother designated point.AB, CD, 13D, AC

CD is a chord of OR.

A chord of a circle is the line segment with endpoints on the curve. The diameter is a special chord.

110 Alathematical Ideas Illustrations and Explanations

ELLIPSE: An ellipse is a simple closed curve having two fixed points on the interior called the foci. Each point of the curve has associated with it a number which is the same for every point. This num'er is the sum of the measures of the distances from the point to the two fixed points.

An ellipse has two lines of symmetry. Points A and B are foci.

PomoN: A polygon is a simple closed Examples: curve formed by the union of three or 'acre line segments. Two line segments will a common endpoint form an angle since line segments can be thought of as parts of rays. The common endpoints are called vertices of the polygon.

A regular polygon has line segments or sides of the same measure andits angles are congruent. Triangle.. A triangle is a polygon 0 with exactly three sides. The sides of a triangle may have one of three relationships: threesidesequalinmeasure (equilateral) two sides equal in measure (isosceles) all three sides of different measures (scalene).

The angles of a triangle may have certain relationships: equilateral scalene equiangular acute II ree angles that are congruent isosceles (equiangular) Iwo angles 0)21are congruent (isosceles) all three angles that are not congruent (acute)

I I I Mathematical Ideas Illustrations and Explanations

one angle a right angle (right) one angle greater than a right angle (obtuse) all abgles less than a right angle (acute). right obtuse

Quadrilateral: A quadrilateral is a polygon with exactly four sides.

PARALLELOGRAM: A parallelogram is a quadrilateral having opposite sides parallel and equal in measure.

RECTANGLE: A rectangleis quadrilateral having opposite sides parallel, equal in measure, and co gaining one right angle. A rectangle is a parallelogram.

SQUARE: A square is a quadrilateral which is a rectangle having all sides of the same measure. A square is a rectangle and a paral- lelogi am.

TRAPEZOID: A trapezoidisa quadrilateral having two and only two sides parallel

Pentagon: A pentagon is a polygon haring exactly five sides.

112 118 Mathematical Ideas Illustrations and Explanations

Hexagon: A hexagon is a polygon having exactly six sides.

Octagon: An octago is a polygon having exactly eight sides.

Plane Region A plane regionis the union of the curve and its interior set of points. 410 Space FiguresClosed Surfaces Any set of points is aspace figure. -IP Geometry includes the sets of points in all space as well as sets of points in a plane.

A closed surface separates space into three sets of pointsthose in the interior, those on the surface, and those on the exterior. A point in space ties 74-surface interior either on the surface. in the interior set of points, or on the exterior set of exterior points.

Sphere: A sphere is a closed surface formed by rotating a circle about any of its diameters. The center of the sphere is the center of the rotated circle. A sphere is the union 0 is the center of of all points in space Which are the the sphere. same distance from a fixed point or center.

OA, 013, and OC Pre radii of the sphere.

113

F 119 Mathematical Ideas Ilfustrations and Explanations

Cylinder: A right circular cylinder is a closed surface formed by rotating a rectangle around one ofits sides. The bases are circular regions in parallel planes. The surface other than the bases is called the lateral surface.i,rt element of the cylinder is a line segment perpendicular to the two bases with endpoints on the C B circles. ABCD is a rectangle E is an element of the cylinder.

Cone: A right circular cone isa closed surface formed by rotatinga right triangle about one of therays forming the right angle. Therays A is the vertex. forming the right angle are known as legs of Cie right triangle. The ABC is a right triangle, base of a right circular cone isa circular region. AB is one of its legs.

Prism: A prism is a closed surface with at least two congruent faces lying in parallel planes.All other faces are parallelograms.

Pyramid: A pyramid isa closed surface whose base may beany polygonal region and whose sidesare triangular regions having acommon vertex.

Pyramids are named according to triangular square the polygon which forms the base. pyramid pyramid

Space Region(Solid Region) A space region is the union of the closed surface and its interiorset of points.

114 120 TEACHING SUGGESTIONS FOR GEOMETRY

Bask Ideas of Georneir) Draw a picture of a point, label it M and discuss whether the point has thickness. The ideas of point, line, line segment, ray, plane, ?ngle, and space should be developed using models A from the physical world.

Point: The idea of a point may be introduced by discussing such questions as: What brings the idea of a point to your mind? V (Tip of a pencil, knife, scissors, thumb tacks, Draw pictures of lines through point M. pins, nails, corner of a book) How can the idea of a point be shown on paper? Discuss r;viestions such as: Which dot is the best representation of an exact How many lines can pass through point M? location in space? Could the paper be covered with pictures of lines through point M?

Line: String can be used to illustrate the idea of a Can the set of points be said to cover the top of line by beginning at each end and rolling the string the desk if pencil point is placed at M and the into two balls, attaching mid section to a board with paper turned around it? thumb tacks as illustrated, and unrolling the string in Does a line have thickness? both directions, pretending it has no ends. Is a plane flat? anti Does a plane have thickness? tack tack Place pencil point through the center hole of ruler, Line segment: The idea of a line segment is illus- (a number line), keep ruler flat on the desk, and spin trated above by the tacks and the string between them. it.Discuss whether this gives idea of a plane. The "lines" on the tablets of the students are pictures Hold pencil with point straight up, place ruler on of the idea, line segment, with edges of paper as the pencil tip and spin ruler.Does this action define a end points. plane? (Yes) Tilt pencil and spin ruler.Does this action define Ray: A seamstress' tape or a surveyor's tape may the same plane? (No) give the idea of a ray. The case is the end point with the tape extending from it in the one direction illustrating the ray. Angle: Understanding of angles can be developed The tight sent out from the bulb (end point) of a through use of a model prepared by children in the fol- flashlight can produce the idea of a ray. A ball of yarn lowing manner: and three children may also demonstrate the idea. The Cut two strips of tagboard first child holds the end of he yarn, the second holds the yarn at a point to draw istaut, as the third child about 1 inch wide and as long moves off unwinding the ball as he goespretending as desired.Join these strips that it has no second end at one end with a brass brad. Draw a ray dow n the center of each strip from the brad (13). Label apointsomewhere along eachstrip. The brad Plane: To produce the idea of a plane children may represents the endpoint of the do the following: rays and vertex of the angle.

115 The model can be used to show the special kinds of of cars and trucks, creek beds, spilled paint, and snail angles most commonly used today and to locate angles tracks. in the physical world. The idea of a path or curve may be demonstrated with finger paint as children dip a finger into the paint

and make a series of dots on the paper .... or make 1111111111141141MMENMI one dot and draw the finger from the dot One ray may be placed along one side of the desk across the page leaving a path behind. The path repre- and the other fitted around the corner.Corners of sents a set of points.Paths may be wiggly or straight rooms, doors, books may also be explored as children but all represent geometric curves. see other places that are models of angles. The position of Lhe rays which seems to be the most often used can be observed and discussed. A model such as this is Closed path or:: ^loved curve.Since a closed path exemplified in a pair of scissors. or curve is a path which begins at one point and ends at the same point, children may illustrate the idea by drawing many different closed paths.Interesting de- Right angle:A model for right angles can be pre- signs may be made and examples collected from maga- pared by: zines. 1. Cutting two strips of tag board of different measure. 2. Fastening them together by placing a brad through thecenterof the longer one and through the end of the shorter one. 3. Labeling the vertex B (the brad) and points A, C, D, as shown.

C41110111=1111111D

Ray BA can be moved so that point A is closer to point C than to point D. When ray BA is upright so that the angles CBA and DBA are the same, they form Simpleclosed curies:Simple closedcurves are right angles. Children may use this model to find where curves that do not cross themselves. Children may ex- right angles are used in construction, art, printing of hibit designs illustrating simple dosed curves. Some books, and other places. letters of the Ziphabet and some numerals may be drawn as simpio closed curves similar to the m. A model of a right angle may also be produced by: I. Folding and creasing a sheet of paper. 2. Folding the folded edge along itself.

I 40

The corner at the fold will represent a right angle.

Path or curve: Paths through the woods represent sets of points traveled in moving from one place to another. Children may see paths or curves in the track

116 122 Relationships (a set of points). When the line passes through the plane, the intersection is a point. When the line lies on I. INTERSECTION the plane, the intersection is the line itself. When the Ideas of intersection may be reviewed by recalling line lies off the plane and there is no intersection. the activities listed under the intersection of sets, page 18 line and the plane are parallel. of Sets and Numbers.

2. PERPENDICULARITY Intersecting lines:Stips of tagboard to represent lines may be used to illustrate the intersection of lines. Perpendicular lines: Theperpendicularity of lines Each child may experiment to find positions in which may be demonstrated by applying the folded paper the strips can be made to intersect. model of a right angle to the model described for right angles on page 116.It can be seen from the illustration at the right that the corners formed by the intersection of two lines are right angles. A A

Position A shows that all points intersect.Points have no thickness and the two lines are the same line. Position B shows that the two lines have just one point in common.

Intersecting planes:To make a physical model of Line perpendicular to a plane: A line perpendicular intersecting planes sheets of paper may be cut as shown to a plane may be demonstrated by : and interlocked. It can be seen that the points at which a. Using a sheet of paper to represent a plane the planes intersect become line XY. Children may discuss the possibility that a third or even more planes b. Drawing intersecting lines through point CI on may intersect at line XY. (See illustration page 105.) the plane c. Holding a pencil at point Co d. Placing folded paper model of right angle in angles formed by pencil and each line passing through point Q on the plane e. Discussing relationship of pencil to the lines and to the plane.

The idea that many planes may intersect at one line may also be illustrated by drawing a line on the chalk- Perpendicular planes: A model for pe:1,endiculat board, labeling points on the line, and holding a sheet planes may be found in classrooms. The folded paper of cardboard representing a plane against the line in model forright angles may be applied to corners many positions. formed by walls.If the model fits exactly, the angles of the walls are right angles and the planes of the walls If a third point not on the line is drawn, it becomes are perpendicular to each other. The model may be evident that the cardboard may contain all Om points applied in many corners. in only one position. One and only one plane may contain three points not in the sane fine. 3. PARALLELISM Line intersecting a plane.Children may show the Parallellines: Physical models of parillel lines may three relationships of a line and a plane by using a be observed in Venetian blinds,railto id tracks, tire double pointed knitting needle to represent a line (a set tracks, edges of tape or line segments on notebook of points) and a sheet of paper representing a plane paper. Other examples may be identified by children.

117 tangular boxes, shelves of book cases, or ceiling and floor.Additional models may be listed and discussed by children, applying the idea that parallel planesnever intersect no matter how far extended.

4.CONGRUENCY Models of congruent figures may be seen in sheets of paper from a notebook, floor tiles,stars in a flag, checkers, dominos, or playing cards.Others may be listed and discussed keeping in mind thatcongruent figures have the same size and shape.

A model for parallel lines may be constructed by: A a. Drawing line b and point G off the line b. Drawing many linesthrough G intecting line b c. Observing that lines through G almost cover the plane but leave a space, a missing linec which has no points on line b. Lines b andc are parallel lines and their intersection set is empty.

Line segment: Line segments with end points equally far apart may be constructed using acompass by: a. Drawing line segments AB and CD b. Placing compas.s point on A and openingcom- pass so that perm' point is on B c. Placing point of compass on point E on CD with setting unchanged d. Marking point G with pencil. Line segment EG is congruent to linesegment AB. EG AB

Parallel lines may also be Loustructedusing acompass and straight edge as follows: a. Draw line n. b. Label a point E. c. Construct a right angle at E.(See construcran of right angles below.) E d. Construct a line perpendicular toray EA at point C. e. Label line pi. Line rn and line it are parallel.

Parallel planes: Physical models of parallel planes Conetuent line segments may also be constructed may be seen in stacks of paper, opposite sides of rec- by b.cling, dividing into two congruent lineseg-

118 124 Y' fsYLCtaM1-

ments, a given line segment. The following procedure c. With A as center draw arcIintersecting rays may be usual: of angle A at J and M. a. Draw line segment XY. d. Without changing opening of compass and with E as center, draw arc 2 inter setting the ray at T. b. Open compass a distance greater than 1 the 2 e. With T as center and opening of compass the measure of XY. distance between J and M, draw arc intersecting arc 2 at N. c. With X and Y as centers, draw intersecting arcs above and below XY. f. Draw a ray from E through point N. d. Label points of intersection A and B. !_NET [JAM e. Draw line through points A and B. f. Label intersecting point of AB and XY, E. XE BEY

Since angles formed by the bisector of the line segment in the construction above are right angles, the same procedure may be used to construct: a. right angles b a line perpendicular to another line 1 c. a perpendicular bisector of a line. Congruent angles are also formed when an angle is bisected. This may be done by using the following pro- Angles: Models of congruent angles may be seen in cedure: the many right angles used in construction of buildings a. Draw angle B. or %Laving of materials, in the corners of hexagonal tiles or points of a star. b. With vertex, point B, as center, draw arc 1in- tersecting the rays of the angle. Models of different shaped figures may be cut from construction paper and angles checked for congruency c. Label points of intersection A and C. by applying one angle to another to match vertices and d. With A as center draw arc 2. rays. e. With compass opening unchanged and C as center, draw arc 3 intersecting arc 2. f. Label point of intersection D. g. Draw ray BD. Ray BD bisects the angle ABC. iABD c LDBC

Plane figures: Models of congruent figures may be seen in stacks of paper, stacks of index cards, pages E of a book or calendar. Duplicate models of many different figures may be cut from cardboard and shuffled in a box on bulletin Understanding of congruency may be further de- board.Different activities in which congruent figures veloped by the construction of congruent angles using are matched may be designed. the following procedure:

a. Draw angle A. 5. SILAILAR11 Y b. Draw ray and label endpoint E. Geometric figures that arc similar have the same

119 12i shape but notnecessarily the same size.Children mirror in such a way that the reflection is an exact may cut from construction paper geometric shapes image. such as triangles, squares, circles,rectangles, quadrilaterals, and others. They may observe that while all The lineof symmetry marks points between the squares have the same shape and all circles have the object and the reflected image.Differentkinds of same shape, all triangles and all rectangles do not. triangles and other plane figures may be examined for axes of symmetry as illustrated. Children may experiment with ideas of similarity by cutting two triangles exactly alike. They may discover that, by cutting a strip paralle' to any side of one of the triangles, the figure, although smaller,is of the same shape as the uncut triangle. The triangles, therefore, are similar but no longer the same size since the corresponding sides of one triangle have been shortened.

E Plane symmetry: A symmetrical space figure may b, divided by a plane so that one half is an exact image AB corresponds to DF of the other. Models of many different space figures BC corresponds to FE may be constructed or collected and examined for planes of symmetry. Through discussion it can be estab- AC corresponds to DE lished that while some figures may have no planes of symmetry, others have one or more. It can likewise be discovered that these corresponding sides are in proportion to each or.er: AB is to DF as BC is to FE as AC is to DE.

Children may also discover that the angles of the smaller triangle will fit over the angles of the larger triangle. Other figures may be tested for similarity in the same way. For use of similar plane figures in indirect measurement, see page 138.

6.SYMMETRY Plane Fit res--Closed Paths Line symmetry: A plane figure which is symmetrical A rubber band, a piece of string with ends tied to may be divided by a line so that one half is an exact form a ring, strings of beads may be used to illustrate image of the other.Small hand mirrors with one the idea that a simple closed cune separates the straight edge are useful in the study of symmetry. Ob- plane into three sets of pointsthose on the interior, jects of various kinds can be tested by placing the ose on the curves, and those on the exterior.

120 126 A game of marbles with a ring c. Inserting a pencil in the ring drawn on the ground uses the d. Pulling the string taut and drawing a closed ideas. The marbles on the interior are to be played. Those on the CUfVC. ring are neither on the interior nor the exterior set of points. The resulting figure should be an ellipse with tacks being focus points. See also page 111.

I.CIRCLE Places where the idea of a circle is used may be listed and discussed:cups, wheels of bus, end ofpencil,telephonedial, holesinthedial,clockface, screw head, basketball hoop.

Children may experiment with ways to draw circles without tracing circular objects. A compass may be improvised by: a. Cutting a hole in one end of a strip of tag board ss b. Fastening the other end to aflat surface with a Tack tack ie 3,POLYGON c. Placing a pencil through the hole and rotating the Many examples of polygons may be seen in the strip around the tack. physical world.Models of polygons may be constructed from strips of tag board, pipe cleaners. and The tack represents a point in drinking straws. the interior, the center, and the strip of tag board represents a Strips of tag board may be fastened together with line segment, the radius of the brads to experiment with the number of strips neces- circle.By extending the radius sary to form a simple closed path.Strips of the same through the center to a point on length and of different lengths may also be used in the curve, the new line segment the experiment. Such questions as the following may becomes the diameter. be used to guide discussion of figures: What kind of space figure is represented by one Other illustrations of the idea strip? of center, radius and diameter, as well as of chord and arc, may What plane figure is formed by two strips? be seen on page 110. What is the least number of strips necessary to represent a closed figure?

2. ELLIPSE What kind of figureis formed when allstrips have the same measure? From discussions and pictures or diagrams of satel- lites in orbit, children hase gained impressions of the What may be noted about angles when all sk'cs idea of an ellipse. are of the same measure?

A picture of an ellipse may be drawn by: Triangle.It has been noted that three line segments a. Placing two thumb tacks on a line fo-ni a triangle. Experiments may be performed using b. Forming a ring of string and placing it around brads and strips of tag board to construct triangles the tacks with all sides the same measure. with two sides the

121 127 same measure, and with all three sides having different measure.Children may learntoidentifytriangles named in terms of the sides. Se: page 111.

X A right triangle may be produced by using the procedure for constructing a right angle and continuing as follows: Draw line segment AY PAEY is a right triangle sir,ceYEA is a right Understanding of construction of congruent line seg- angle. ments may be used to construct equilateral triangles. The procedure folLws: Experiments may be extended to include identifica- tion by angles by: a. Draw line segment AB. a. Cutting models of equilateral triangle, tearing b. Witt compass opening equal to the measure off th! vertices tofit them along a line as of ,AB and with A as center, dray arc 1. illustrated

c. With B as center, draw arc intersecting arc 1 at C. d. Draw line segments AC and BC. PABC is an equilateral triangle since AB AC CB.

b. Repeating the experiment using isosceles and scalene triangles Isosceles triangles may also be constructed using ideas related to congruent line segments. The procedure c. Discussing the relationship of anglesin any follows: triangle to a line. a. Draw line segment CD. b. Open compass a distance greater than 1 the Quadrilateral:Four stripsof tag board fastened with brads may be used to develop ideas of quad- measure of CD. rilaterals by: c. With C and D as centers, draw intersecting arcs. Using all strips of different lengths d. Label point of intersection E. Using Iwo strips of the same length and two different lengths e. Draw line segments CE and DE. Fastening the strips described above in different PCDE is an isosceles triangle since CE c DE. combinations

122 Using two strips of one measure and two longer strips to experiment with combinations

Using all strips of the same measure

Pushing opposite cornersto skew figuresinto different shapes

Testing for right angles using folded paper model. 4 -41 Children may learn to ideality figures formed according to sides and angles. Pentagon: Five strip.; ctag board may be used to develop ideas of pentagons ii, the same manner as described above.

Hexagon: Six strips of tag board may be usedto develop ideas of hexagons inthe same manner as described above.

Octagon: Eight strips of tag board may be used to develop ideas of octagons in the same manner as described above,

Plane Region When the idea that a simple closed curve separates a plane into three sets of points is well established, children should have little difficulty understanding a plane region as the union of the set of points on the curve and the set of points on the interior. To illustrate the idea models of plane figures cut from paper may be contrasted with models constructed from strips Understanding of the relationship of parallel lines of tag board. may be used to construct a square with straight edge and compass as follows: a. Craw two parallel lines by constructing right angles.

b. Open compass the distance of the measure of CG. c. Mark off arcs along line m from point C and along line n from point G. Space Figures--Closed Surfaces d. Label points of intersections X and Y. Physical models of closed surfaces can be seen in inflatedballoons,footballs,andbasketballs; empty e. Draw line segment XY. boxes including rectangular prisms,cylinders, pyra- Closed t.urve GCXY is a square since all sides mids, cubes, and other shapes.ft can be seen that the are congruent, opposite sides are parallel and balloon separates the space into three sets of points; angles arc right angles. the interior, exterior. and the balloon itself, or the surface.

Relatiorship of angles of quadrilaterals to aline Children may collect boxes of interesting shapes segment may be determined in the same manner as for to be used in learning to describe space figures in pre- a triangle. cise geometric terms.

123 Sphere:Questions such as the following can be used Spinning a coin, describing the figure formed am to develop ideas of spheres: discussing relation of coin and sphere Is a football a closed surface?Isita sphere? Spinning a rectangle around one side, describim Why? the figure formed and discussing relation of rec Which cf the following might be considered ex- tangle to cylinder amples of closed surfaces: baseball, tennis ball, Spinning a right triangle around one of the ray ping pong ball, golf ball? Why? forming therightangle,describingthefigure Which oftheballslisted above most nearly formed and discussing relation of triangle to cone. represent spheres? Why? Is a globe representing the earth a sphere? Why? Prism:Collections of boxes of many sizes and shapes can be used in developing ideas of prisms. As each. Children may cut out the model of a circle and spin box is examined, such questions as the following may it about a diameter to see the relationship between a be used: circle and a sphere. Since the circle will spin on any of Is the figure a closed surface? its diameters, it can be seen that the diameters of the sphere and diameters of the circle are the same and that Horn many faces has it? radii of the sphere and radii of the circle are the same. How many pairs of faces lie in parallel planes? How many facesarecongruent? Cylinder:Physical models of cylinders can be found Is the figure a prism? easily: unopened cans, oatmeal boxes, mailing tubes, and cylinders from motors. A cardboard model of a rectangle may be rotated around one of its sides to Pyramid:Pictures of ancient pyramids may be dis- demonstrate the relationship of cylinder to rectangle. cussed to introduce the idea that the faces of all pyra- It can be seen that the height of the cylinder is the mids are triangular regions that meet at a common same as the length of the side of the rectangle, that the point or vertex.As children experiment with con- base is a circular region with a radius of the same structions of models it can be seen that any regular measure P.. the other side of the rectangle. This idea polygonal region may be used as t le base of 'a pyramid. may beillustratedalso by wrapping flexiblewire Such questions as the following may guide discussion of around a cylinder, then wrapping wire over the ends figures constructed: rq the cylinder and discussing: What is the shape of the base? The shape of the resulting figures What is the shape of each face? Relationship of figuresto height and diameter May any face of the pyramid be named as base? of cylinder. Why? May any corner of the pyramid be named as VCrteA? Why? What is the name of the pyramid constructed?

Space Region--(Solid Region) Use of the term "solid region" may be helpful in introducing the idea of a region that is the union of the dosed surface and the interior set of points.Discus- sions in relation to dosed surfaces may be reviewed to Vertex establish an understanding of the difference tetween ping pong and golf balls, tennis and baseballs. Glass Pattern for Constructing a Cone paper weights in many shapes in contrast with glass bottles or jars may be rsed to extend understanling Cone:One meydel, the ice cream cone, is familiar of the idea. A styrene foam ball and old tennis bail to most children. Models may also be constructed us- may be sliced in half and nature of the interior ex- ing the accompanying illustration.Discussions based amined and discussed to insure that children ha 0.e on activities similar to the following inay also serve clear understanding of thz differences between closed to extend ideas related to cones: surfaces and space regions.

126 113 MEASUREMENT

Measurement, or the process of measuring, associates a number with a physical object or a geometric figure to show quantity or amount. Measurement developed historically through needs of Man to compare and describe quantitative aspects of his environment. The continuing necd for measurement has led to the development of more precise measuring instruments and to applications in diver se situations.

One of the first ideas about measurement that a chili experiences is the comparison of physical objects that tells him which is longer or shorter, larger o: smaller, taller or shorter. Later the child learn; to choose a unit and associate numbers with the objects.

Important concepts related to all measurement are: The meaning of measurement

The arbitrary nature of units of measure r. The necessity for standard units of measure The relation of the property of the object to be neasured and the unit of measure used The approximate nature of measurement of physical objects.

Measurement is the most widely used area of applied mathematics inthe elementary school curriculum. Ideas of measurement are involved in the sequential development of setsof numbers, the number line,and geometry. Measurement provides teachers with many oppertunitiesforpracticalapplicationof mathematical ideas, for practice in operations on numbers, and for active participation and experimentation.

Mathematical Ideas Illustrations and Explanations

Basic Ideas of Nfeasurement

A measure is the number obtained by Objects to be Units of Measurement comparing an object with a unit of Measured Comparison measure which is of the same nature as the object. 1-- -4 4 t 4

Lite Segment Unit Segment Number of unit segments 7,-- 4

Plane Region Unit Plane Number of unit plane Region regions = 8

Space Region Unit Space Number of unit space Region regions = 8 Mathematical Ideas Illustrations :lad Explanations

Angle Unit Angle Number of unit angles = 2

Standard Units of Measure. Standard Three rains of Marks on a units of measure are selected arbi- Width of thumb barleycorn metal bar trarily by agreement. The historyc. measurement is the story of the need for and the development of standard units of measure. The history of meas- 1 1 1 urement provides one vehicle for developing understanding of the necessity for standard units of measure.

Each type of measurement may be Other Units thought of as having a standard unit arbitrarilyselected. Forexample: yard -=1 foot, 1760 yards = 1 mile meter or yard for length, gram or 3 pound for weight, second for time, and degree for angle. I- pound (lb.) = 1 ounce (oz.), 2000 = I ton 16

1 degree= Iminute, 360 degrees = 1 revolution 60

second = 1 millisecond. 60 seconds 1000 = I minute

Other units may be defined in terms Equisalents of the standard, as either a multipleor as a fractional part of the basic unit. 52 = in. 12 in. = 1 ft. 2

5!x 12 = 66 5 ft. =- 66 in. 2 2

80 oz. lbs. 16 oz. = 1 I.

80. 16 80 oz, = 5 lbs

Direct and Indirect Measure: Direct A measuring tape is used to determine length ofa classroom. measurements are made by comparing a unit of measurement with the object. A measuring cup is used to determine theamount of flsor for a cake recipe.

126 .132 NIathematical ideas Illustrations and Explanations

Indirect measurement is a measure ob- To measure rise or fall of temperature, the expansion or contraction of tainedbycalculationinsituations a liquid in a closed glass tube is measured. (Liquid thermometer) The where .l rect. comparison of a s'andard rise or fall of temperature is indicated by an appropriately calibrated unit of measure and the ob;ect is not scale attached to the tube. possible or practical. To measure the passage of time, a clock or watch, whiTh is a mechani- cal device of gear wheels driven by electricity or springs, is used.

To measure the distance a childlives from school, the number of revolutions of the wheel of an automobile may be counted. A series of appropriate gears enables the distance to be indicated on the automobile ockmeter.

To measure the area of a rectangular region, the lc ngrat of adjacent sides are measured and the areaisdetermined by an appropriate mathematical formula.

Approximate now., of measurement: Error All measurements determined by the use of a measuring irstrument are ap- 1 reI '-- 2t 13 proximate because of the c ,aracteris- tics of measuring instruments them- Length recorded as 2 inches to the nearest inch. selves and of human limitations in the use of the tools. Error is no more than one-half inch. 'rror

I 1 .1 . 2 Since measurements are made to the nearest given unit, an error is always involved. This error is no more than Length recorded as 1-1 inches to the nearest half-inch 2 one half the unit of measure being used. Error is no more than one-quarter inch.

1.1.'rror . 11. . r 1-1 .2 A I a

Length recorded as 1-3 inches to the nearest quarter inch. 4 Error is no more than one-eighth inch.

Precision The precision of a measure- Precision of mentisdetermined by theunit of Gil en Measurement Nfeasurtment measure used. 3. 8-- in. in. 8 8 3.6 miles 0.1 miles 2 lbs. 4 oz. 1 oz. 3 tons 1 ton

127 ,433 Mathematical Ideas Illustrations and Explanations

Ameasurement is only as precise as Precision of Appropriate the measuring instrument used:The Object Instrument Instrument smaller the unit of measure, the more 1 . precise the measurement. The degree Book in. Foot ruler of precision desired and the size of 2 the object to be measured determine Rug 1 ft. Yard stick the appropriate unit of measure.

Distance between mi. Odometer cities

of Types Measurement:Different types SYSTEMS of measurements are needed for de- scribingdifferenttypesofobjects, English Metric quanti'ies, or conditions. Each type Liquid teaspoon, tablespoon, milliliter, liter, kiloliter may include mole than one unit of cup, pint, quart, gallon measure. Dry teaspoon, tablespoon, milliliter, liter, kiloliter cup, pint,* quart,* peck, bushel

Weight ounce, pound, ton milligram, gram, kilogram

Time second, minute, hour, same day, week, month, year, decade, century

Temperature degree, Fahrenheit degree, Celsius

Counting unit, pair, dozen, gross, sa me ream

Angular second, minute, degree, same revolution

Length inch, foot, yard, rod, millimeter, centimeter, mile, light year meter, kilometer Area square inch, square foot, square millimeter, square square yard. acre centimeter, square meter, hectare Volume cubic inch, cubic foot, cubic millimeter, cubic cubic yard centimeter, cubic meter

Many industries, trades, and occupations have units ofmeasure particularly their own.

Not equivalent to the same unit of liquidmeasurement.

Pint Quart

3 Liquid measure 287- cubic inches 57-cubic inches 8 4

1 Dry measure 33-3 cubicinches 67cubic inchcr. 5 5

128 134 Mathematical Ideas Illustrations and Explanations

The Metric System

The Metric System of measure isa Relationships of Afettie Units decimal system since relationships between units are based on powers of Weight Linear Mass Volume ten. 1000 units kilometer kilogram kiloliter 100 units hectometer hectogram hectoliter

10 units decameter deco cram decaliter

Standard: Unit Meter Grain Liter

I unit decimeter decigram deciliter 10

1 unit centimeter centigram centiliter 100

1 unit millimeter milligram milliliter 1000

The English System In the English System units of mcas- Relationships of Units of English System ure a:e selected arbitrarily withn 1760 yds. = 1 mi. consistent pattern of relationships between units. 5-1yds. = 1 rd.2000 i),s. = 1 T.4 qts. = Igal. 2

Standard: Yard Pound Quart

1 yd.--=1 It. 1 lb. I oz . qt. = 1pt. 3 16 2

1 yd. 1in. _qt. = 1 cup 36 4

Relationship of metric units to English One inch is about 2.5 centimeters units: Since it is necessary at times to inch scale convert measurements from English to I It metric, or metric to English systems, it 1 2 3 I t t is desirable to develop understandingt-- ` of an approximate relationshipbe- 1 2 3 4 5 6 7 centimeter scale tween the more commonly used units.

129 'AIathematical Ideas Illustrations and Eyplariations

Excessive computation is to be avoided since the principle idea of relative re- One kilometer is a littlemore than one-half mile. lationships is often lost in thismanner. One kilogram is a little more thantwo pounds. One is a little more than one yard,

The metric system is used inmany countries and by scientists in all countries. Many industries in the UnitedStates now use the system since itis easier to convert fromone unit to another in a base ten system. Legal units of measure in the UnitedStates hale been determined in terms of the meter. gram, andliter since 1866,

Converting from one unit to onother: Two approaches for converting unitsare described oclow: A necessary phase of thestudy of measurement is learning how to con- The relationship between 2measures can be expressed as a ratio. This vert a given measurement to an equiva- fact can be used in converting from Iunit to another, For example: lent measurement expressed ina different unit of measure. The objective 1- 12 foot to inches is 1:12 or inches to foot is 12: Ior is to learn a systematic pattern based 12 on understanding which will enable the student to convert units successfully. ounces to pounds is 16 ior/ 6

inches to centimeters is 1:2.54or .00 2.54 In a proportion the numbersmust always be from the same denomi- nation. This fact can be used inconverting from one unit to another. If a given measurement of 60 inchesis to be convected to feet, apropor- tion may be used:

I- nor 1:12 = [1:60 12 60

Solving the equation results in fl 5 and 60 inches equals 5 feet.

If a given measurement of 51 pounds isto be converted to ounces. a proportion may again be used: 5.5 16 L

So !sing the equation results in [] ,8 and 5-1 pounds equals 88 2 ounces.

130 13G NlathentaticaI Ideus Illustrations and Explamitions

A second approach to converting units depends to some et:tent on experience and common sense.

To change from a large unit to a smaller unit, multiply by the number that tells how many smaller units are equivalent to one large unit.

Change 24 yards to feet 3 feet = i yard 24 x 3 = 72 24 yards = 72 feet

Change 47 kilometers to meters 1000 meters = 1 kilometer 47 x 1000 = 47,000 47 kilometers = 47,000 meters

To change from a small unit to a larger unit, divide by the number that tells how many smaller units are equivalent to one large unit. Change 488 ounces to pounds

1 pound = 16 ounces488 ÷ 16 = 301488 ounces = 301 pounds 2

Change .6 kilometers to miles I mile = 1.61 kilometers 36 ÷ 1.61 = 22.4 36 kilometers = 22.4 miles

Omation of Measurement Estimation is the mental process for arriving at a reasonable measurement by seeing an object in proper perspective to a unit of measure.Estimates are based on known facts or past experience. Experience and mental computation are essential in arriving at a satisfactory result when actual measurement is not feasible or not needed. Using parts of students' bodies as units of measure, tinge' joints, hand span, foot. Estimating measures Stepping or pacing to estimate distance. Estimating number of discrete objects P=3+3+3 Perimeter P=s+s+s The perimeter of .simple closed path isthedistancetraveledin moving from one point along the path back to the starting point.

131 137 Mathematical Ideas Illustrations and Explanations

Polygon:The perimeter of a polygon is the sum of the measures of the line segments which formits sides.

y = 4

P= 2 +4 +2 +4 s = 2 P = (2 x 2) + (2 x 4) = 2 + 2 + 2 + 2 P = 2y2x or 2(1 + w) P = 4s

Circle The perimeter of a circle, The constant relationship which exists bLtvkeen a diameter of :1 1( conventionally called thecircumfer- and its circumference may be demonstrated ehce,is the product ofpiand the in the: following 'kN,L) measure of a diameter of the circle. 1. Wrap string around model ofa .ircIe. 2. Stretch string out straigLt.

3. Cut strip of paper thesame length as a diameter.

4. Compare length of diameterwith length of string.Ret lit of comparison the circumference isa little more than 3 times the diameter. 5. More approximations of therelationship of the diameter to the circumference will give 31or 3.14. 7

6. This constant ratio, with valueapproximately 31 or 3.14 is known as D., the Greek letter pi. 7. C = 1rd C = 2wr

Area Theareaof a given plane region is the number of unit plane regions required Unit to cover the plane region with no over- 11E11 lapping. Plane regionSeepage 123, Geometry. Area is 4 Units

AdaUnit

Area is 12 Units

132 138 111athernatical Ideas Illustrations and Explanations

A Area is 9 Units Unit

Although any plane region can be used as the unit, a square plane region such as a square inch or a square centimeter is usually used as the standard unit of area measurement.

1. AREA OF POLYGONS Rectangular region:The area of a rectangular region is the product Count number of square units which of tne measure- of its length arid can be placed along one side of roc- width. tangle.

Count the number of rows.

The area is the product of the number of rows and the number of units in each row. A= 4 4- 6 = 24 sqvare units Rectangle

There areI squareunits along one w side, and w rows of the units. A I X v, Unit

Square

A=--sxsors2

Parallelogram regkn: The area of Cutting along a line perradicular to a parallelogram region is the prod- the bases, a parallelogram region can uct of the measuns ofitsbare be made into a rectangular region. (length) and height(altitude or width). A =bxh

133 139 Mathematical Ideas illustrations arl Explanation,:

Trapezoidal region:The area ofa trapezoidal region is one half of the product of the measures of the b2 heightk altitude) and the sum of b, the measures ofitsparallel sides (bases)

b2 b

By adding a congruent trapezoidal region.the above parallelogram region can be made with base (b, + b2).

Tea of the above parallelogram region = h x (b, + b2).

Thus, area of trapezoidal region h (b, + b2).

Triangular region: The area of a By using another congruent triangular triangular region is equal to one half the product of the meas,.re of its region, a parallelogram region may be formed. base and height.

r c b

The area of the parallelogram region= b x h tience, the area cr the triangular regionmay be expressed: 1 A = - b x h 2

Circular region:1 he area of a cir- An approximation for the area of a circularregion may be developed cular region is equal to the product by using the following model: of pi and the square of its radius, 2r

134 140 Nlatheniatical Ideas Illustrations and Explanations

The area of the square region is: 2r x 2r = (2r)2 = 4r2 Hence, the area of the circle is approximately 3r

A 3r2 means "approximately equal to")

The following model may also be used to develop the formula for finding the area of a circular region.

Cut a circular region into an even number of congruent wedge shaped sections. it these together to form an approximation of a parallelogram region; the smaller the wedges the better approximation.

If the region were a parallelogram, its base would be equal to 1 of the circumference of the circle and its height equal to the radius of the circle.

1 C 27rr b= C or wr h r A=bxh ArXr = trr2

2. AREA OF CLOSED Sukfitris;

Prism and pyramid:The surface h area of a prism or pyramid is found by adding the measures of the areas of the (aces or plane regions x.hich

make up the closed surfaces. 1

Total Area = 2 x (( x w) + 2 x (w x h) + 2 x(fxh)

Total Area = (2 x B) (h X p) where 13 is the area of base, h is the height of the prism and p is the perimeter of the base.

135 141 Mathematical Ideas Illustrations and Explanations Cylinder:The surface area of a cylinder is found by adding the area of its lateral surface and the areas of the two bases. c

Lateral Area: Think of a label ofa can cut in a line perpendicular to the bases. Stretched out the label makesa rectangular region with the same area as the lateral surface of the cylinder.

Lateral Area = c X h

c=-- 2wr or w d

Lateral Area = 2wr x h or wd x11

Area of 2 bases = 2wt2

Surface area of a cylinder = lateralarea + area of bases

A = 2wr x h + 2wr2or

A --= 2 B + (h x c) when 13is the area cf one base, his the height and c is the circumference of a base.

Volume 4 ,4 The volume of a space region is the /77.unit I --f I number of unit space regions required region L _ I .o fill the space region. 7 e Volume 's 4 units. u unit region LL Rectanplar prism: The volume of Vo7uine is 5 units. a rectangular prism is the product of the area of its base and the alti- Find number of cubic units of the tude or the product of its base layer. length, wicth, and altitude. 6 x 4 = 24 Hence, 24 cubic units will form the base layer.

The height tells the number of layers. Hence, volume = 3 x 24 = 72 cubic units.

136 142 Mathematical Ideas Illustrations and Explanations

B h If B is the area of the base, N

I V= (lxw)xh

Regular pyramid:The volume of a pyramid is one third the product of the area of its base and the altitude or height.

Given a prism and a pyramid congrucnt bases and altitudes, the pramid filled threc times with sand will fill the prism.

V= 1 B x h 3

Right Crcidar cylinder:Tht. volume of a right circular cylinder is the trait product of the area of its base and region the altitude.

The area of the base is 242. Hence, wr2 cubic units will cover the base completely Iunit deep. There are h layers. Hence: V=Bxh

V = 1,1.2 x h

137 143 Mathematical Ideas Illustrations and Explanations

Right circular cone:The volume of a right circular cone is one third the product of the area of its base and the altitude.

Given a cone and a cylinder with congruent basesand altitudes,tl cone filled three times with sand will fill the cylinder. Hence,

V= 1 B h or V = 1rr2h 3 3 Bis the area of the base or are.

Measures of some geometric figures, such as the surface area or volume of a sphere, have not been in:luded. Pi.easures of such figuresmay be more appropriitely introduced at the junior high or secondary levels.

Use of Similar Plane Figures in 16 5 Indirect Measv-ement -A 2 Use of similar polygons inindirect meas,.rernent depends uponunder- n standing that corresponding sides of similar polygons are in proportion. n 16 2 5 5 x 1=16 x 2

32 2 n = 6-5

12 a a 5 4

4 Sxa=4x12

48 3 3 n a 5

32 n 5 3 5 xn= 12 x 3 n==36 71 5 5 Mathematical Ideas Illustrations and Fxplanaticns

&ale Drawing Scale drawing is a representation of an A rectangular room w:iich measures 15 feetby 20 feetmay be repre- object equal in size, enlarged, or re- sented by a drawing, 1-1- inches h! 2 inches, where -1- inch represents duced to convenient or appropriate 2 2 size. The drawingissimilartothe 5 feet. origin s +,therefore,allmeasurements are in proportion to those of the original object. Tht scaleis the ratio of measurements im the object to measurements on the drawing.

0 5' 10'

Scale

Apeanut which measures 5 inch liay berepresented by a drawing 8

1 21inches whereinch represents inch. 2 4

11, 1" 0 3'' 4 2 4 1_ Scale

Rectangular Coordinate System

The rectangular coordinate system is Johnnylives2blockseast and 3 made by the perpendicular intersection blocks north of the cotter of town. of two number lines. In the rectangu- C is the center of town. His home is lar coordinate system, ordered pairs of shown by.I.Mary lives 1 block west E numbers are used to locate points in and 2 blocks south of the center of theplane.Thereisaone-to-one town. Her house is shown by M. correspondence between thesetof points in the plane and the set of or- Q:$ blocks west, 3 blccks north dered pairs of real numbers. S R: 3 blocks east, 2 blocks south

139 414 5 Mathematical Ideas Illustrations and Explanations

Locate the point that matches( +2, +3) A The point A (42,3) is located by moving to the right 2 and up 3. +J. 1 t I l I Locate the point (--1, 2) B Point B, ( -1, --2) is lccated by moving to the leftI and down 2,

N Longitude andlatitude:Longitude o° and latitudz are lines drawn on maps and globes to indicate distance and to 90 81170 60 5040 30 20 10 10 20 30 40 50 60 70 80 90 locate points.Lines of longitt:de are90 up mil 90 drawn north and south through the 80 80 North and South Poses. Latitude lines are drawn east and west parallel to70 . &la70 the equator.Longitude and latitude 60 represent a modification of the rec- III50 tangular coordinate system. 40 111.111.111111111130 20 m. I0 Equator W 0' 0° E 10 .i.10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90

90 80 70 60 50 40 30 20 10 1020 30 40 50 60 70 80 90

0' S Prime Meridian Richmond, Virginia is 77' west longitude, 37' north latitude. Johannes- burg, South Attica is 30' east longitude, 25' south latitude.

140 14 G Mathematical Ideas Illustrations and Explanations

Graphs:Most graphs are based on the rectangular coordinate system or Temperature at Noon Each School Day in January ore adaptation of it.Many graphs will utilize only the first quadrant of 44 the rectangular coordinate system since only positive numbers will be involved. 42wommummumormomum 40olimpuompirmnumminni 38=1presumunimummIgnimi 36N11111111101111MilMIMINNINIIII 34 32 MI 1111111111111M.IMI LINE GRAPH 30 28 26ImIonsimmi RupwAsmn 24IIIIIIMH11111111111orraM11111 22 20111111 2 3 4 5 69 10 11 12 13 16 17 18 19 20 23 24 25 26 27 Days in January

7B BAR GRAPH

10 15 20 25 30 35 40 45 Number cf Tickets Sold Other types of graphs may also be used in representingdata, Weights of Some Metals Compared to Water Natal PICTOGRAPH LeadMUMMA GoldLIALULUULLUSia3 Iron&MUM Sr/ Mathematical Ideas illustratio; and Explanations

CIRCLE GRAPH

Elements in the Earth's Crust

Graphs are particularly useful in statistics since relationships among items of data may be clearly and quickly seen and interpretation made simpler.

Introduction. to Elemental) Statistics To answer many questions it is neces- As part of the study of weather in a science class, students observe sary to collect, analyze, and Interpret and record the outside temperature in degrees Fahrenheit each day at facts. These facts are known as data. noon. During January the following temperatures were recorded: The study of numerical data is known as statistics. F

Collecting data: The first step in a First Week 38 40 39 38 41 statistical study is the collection of Second Week 39 35 34 36 38 information or data. Third Week 36 32 26 25 25 Fourth Week 28 30 33 35 34

Organizing data: For better under- Temperature in standing, data are arranged in some Degrees Fahrenheit Frequency systematic fashion. One way to do this is to write the data in order of 41 tt size from the highest to the lowest, 40 and then show how often each item 39 has occurred. Such an arrangement 38 is called a frequency distribution. 37 36 tf 35 34 fl 33 32 31 30 29 28 2' 26 25 ./t Total 20

142 .1'48 Lo4",43,00.,1.

Mathematical Ideas IllustrMions and Ik,:.inations

The data may consist of large numbers Temperature in Degrees Fahrenheit of items, or have items that differ so greatly in value that a frequency dis- Intervals Frequem.y tribution showing each item becomes unwieldy, In such cases, the Jata are 40-41 2 entered by intervals. This is called a 38-39 5 grouped frequency distribution. 36-37 2 34-35 4 32-33 2 30-31 1 28-29 1 26-27 1 24.25 2

>,4 .1 A c3 3

The informationinagrouped fre- 2 quency distribution, as above, may be z pictured in a vertical bar graph, called ahistogram. 1

It N ,- M sel Os ao 4 41, N N N M M re, Temperature in Degrees Fahrenheit

Instead of handling an entire table of data, it is often more convenient to use a number that is representative or typical of all Ito data in the table. Such numbers ark calledmeasuresof central tendencysmce they tend to be descriptive of all the data. There are three such measures and each has advantages.

143

4049 Mathematical Ide Es Illustrations and Explanations

the arithmetic mean (or average) The arithmetic mean for temperatures during of a set of data is equal to the five days of the first week sum of January (See table page 142) isthe sum of those temperatures of all the data values divided by the divided by the number of readings. number of values. 38 + 40 + 39 + 38 + 41 19639.2 5 5

39.2 is the mean or average temperature for fivedays in the first week of January.

The arithmetic mean for the time during 20days of 1?,,uary is the sum of temperatures for all 20 days (682) divhied bynumber of reddings (20).

682 =-- 34.1 20

34.1°F is themean or average temperature during twenty daysof January.

The median of a set of data is the There are 20 numbers in the frequencydistribution on page 142. middle number of the set when the Counting down from the top,or up from the bottom, the members are arranged in order of middle temperature is 35°F. Thus the mediantemperature for January is 35`F. size. To find the median, firstar- range the numbers in order of size. Given the numbers 21, 23, 24, 28, 34,the median is 24, The median is the middle number if the set contains an odd number of Given the numbers 62, 65, 70, 74, 78, 81, numbers.If the set contains an the median is 72. ever. uumberofnumbers, the median is a number halfway between the two numbers thatare nearest the middle.

The mode of a set of data is the The temperature 38°F occurs most frequentlyin the frequency distribu- number that occur- most frequently. tion on page 142. Thus 38'F is themode for temperature during To find the mode (ofa set of data) January. arrange the data 'n a frequency table and select the item having thegreatest frequency.

144 150 TEACHING SUGGESTIOYS FOR MEASUREMENT

Basic Ideas of Measureraent In addition to the classroom collection, each child should have availablefor individual use an appro- Chikiren should have experience measuring with priately marked ruler and, as the need develops, a arbitrary units of measure such as, pencils, pieces of protractor. In developing understanding of the need for chalk, or edges of books to measure length or distance; standard units the cubit referred to in Biblical litera- sheets of paper, cards, tiles, or flat sides of books or ture may be discussed. The length of a cubit was blocks to measure surface; blocks, books, or boxes to determined by the length of a man's arm from finger also be measure capacity or volume. Attempts may tip to elbow. The variation in cubit as represented by mark to measure a flat surface using circular or tri- forearm measurements obtainable in the classroom can angular regions, or to measure a circular region using help children to understand why this unit is no longer square or rectangular regions or smaller circular units. in use. If records are kept as children measure objects using arbitrary units, discussion of results ran bring out the Older children may study history of the developruent need for employing a unit which has tiro same nature as of instruments of measurement. This may range fr. rn the object being measured.It may also point out the research on primitive and archaic units, to explanation needforstandardunits which everyoneuses and of modern devices. understands.

Standard Units of Measure: Each child may 1,e pro- Direct and Indirect Measurement:First experiences vided with a strip of cardboard between 12 and 13 with measurement generally involve direct application inches long and a smaller strip measuring one inch of a unit to the object to be measured. Use of a ruler square.Starting at a point w!-ichis labeled 0 and to measure length or a cup to measure liquid is direct measurement. Although use of tile term is not stressed. the idea should be established. 1.377-1-? 11 1I I Discussion of other ways to measure can result in a classroom collection of such instruments as the following: thermometer, speedometer, clocks, scales (plat- using the 1 inch unit to mark unit segments, evh inch form, balance, and spring), barometer, gauges(air labeled. These rulers should be checked against is pressure, rain, wind. and steam), and meters. other rulers and yard sticks to show that an inch is the sante length no matter how long the measuringinstru- ment may be. The self-made rulers may then be used to Opportunities to use these instrurneuts to gain greater measure things at home and at school. understanding of indirect measurement should be provided.Large scale thermometers may be usedto Standard units of measure of many kinds should be measure temperatures of various substances. A weather availableinthe classroom and opportunity touse station may be visited to see gauges which measure them should be provided. Foot and metric rulers, yard and record temperature, barometric pressure, rainfall, and meter sticks,tape measures,teasroons,table- and wind velocity.Devices for measuring rise and fall spoons, balance scales and weights, units for liquid of the tide may be examined. and dry measure, clocks, and large scale thermometers should be accessible to children at all limes. Children As formulae arc developed and used in measuring should have experience filling half-pint,pint, quart, area and volume of geometric figures, ideas of indirect half-gallon, and gallon containers with sand or water as measurement arc extended. well as applying units of linear measure.Objects of different size but of the same weight Mould be available for comparison both with and without weighing. APProlirnate Nature of Measurement:Such activi- By applying standard units of measure to concrete ma- ties as the following may help children to understand terials, children discover relationships between cup, pint the approximate nature of measurement: and quart, ounces and pound, meter and centimeter. If relationships di.covered are recorded on ,harts and Recording and comparing results when several made accessible when needed, children learn equiva- children measure the same object using different lents in a meaningful way, instruments of measure; such as. 6 ini ruler. 12

145 '151 inch ruler, yard stick, or steel tape;tablespoon. cup, or pint measure. may be gained by keeping a record of types encountered in con, ersations, stories, newspaper articles,radio and Recording and comparing results when several television ads. or labels on packages. Typespeculiar children measure the same object using thesame to rpecial trades, industries, or occupathmsmay be measuring instrument. especially interesting. For example: Comparing the amount of liquid in a glass measur- Unit of Measure ing cup when the scale is readat eye level and Trade or Occupation below eye level. fathom maritime Discussing such possiblereasons for variations as: angle from which measurement is read,thickness knot maritime of pencil mark, thickness of markon instrument, dust or dampness on scales, humanmanipula- ream paper tion of the instrument. quire paper

hogshead Precision:Length and width of sheets ofpaper, desk tobacco, molasses tops, or any flat surface may be measuredusing rulers dram I,I, 1 drug marked in 1 inch, - and -1- inch units. If results 248 16 are recorded IA a chart sirm..to the one below, it cord wood can be seen that precision of a measurement is determined by the unit of measure usedand that as the size hand horses of the unit decreases themeasurement approaches closer and closer the a ualmeasure. carat gold, jewels chain Size of Unit Length of Object surveyors

1 inch 10 inches The Metric System It is important to develop understanding -I- inch 9-1inches that the 2 2 metric system of measurement is the most usedin the world today and is based onpowers of ten.Its basic I 3 units are meter, gram, and liter. Blank 9-inches charts similar 4i n- 4 to those used in developing understanding ofplace value and decimals may be made in the formof transparencies or wall charts. Units inch 9-sinches may be written in, and 8 8 appropriate prefixes and numerical valuesadded, as relationships are discussed. 1 9 IInches 16 16

Types of Measurement:The importance of a collection of measuring devices readily available ineach classroom cannot be over emphasized.It is through much contact with such devices that childrenacquire intuitive knowledge of the difference betweenpound and ounce, pint and quart for liquid anddry measure, quart and liter, square inch and cubic inch, and of the type of instrument appropriate formeasuring As a part of the study of the history of each given substance. measurement children may be interested in learning ofthe development of the metric system and the derivation ofp,e- Knowledge of less familiar types ofin trement fixes used.

146 152 The English System converting from a larger unit to a smaller.It makes Although the English system is the one most familiar sense that there will be a larger number of small pieces to children in the United StaLs, it may be necessary to (inches) than large pieces (feet)in a given quantity. emphasize the point thatitsunits of measure were Counters and egg cartons may be used to develop ....hosenarbitrarily over periods of time and are not related to a pAtern as in the metric system.Charts understanding of procedure for convcrtn g from small similar to the one given on page 129 may be developed units to larger units.Sixty counters may be dropped as units in the English system are used. into compartments of the cartons and ways to determine the number of dozen of counters discussed:

The full cartons may he counted, 5 Relationship of Metric Units to English Units: Since excessive computation in converting from English to Repeated subtraction may be used. metric system or from metric to English may tend to 60 12 =48 I carton obscure understanding ofthe approximate physical 48 12 36 2 cartons relationship, physical models should be usedinthe 36 12 = 24 3 cartons development of the ideas. 24 12 = 12 4 cartons 12 12 -= 0 5 cartons A yard stick may be measured against a meter stick Division may be used, 60 ± 12 = 5 to discover that a meter is a 1ittle more than a yard.

As thisrelationshipisunderstood, children may Liquid may be measured by using y,. iand liter generalizethat divisionisuscdin converting from measures to demonstrate that a literis a more small units to large units.It is sensible that the num- than a quart. ber of large units (dozens) should be less than the number of smaller units (ones). Two parallel line segments of the same length may be marked. one in inchcs. the other in centimeters to When manipulation of concrete objects has estab- show that ore inch is equal to about 21 or 2.5 centi- lished understanding of the process, relationships may be expressed as ratios, and a proportion used to find the meters. missing term in the same manner as for equivalent fractions. page 55. For example: Some packaged food products,particularlythose prepared abroad, are marked in metric units.Com- 1 foot = 12 inches; --I orI :12 parison may be made with similar products marked in 12 the English system. 4 4 feet = inches; or 4: El Converting Front One Unitto Another: Two ap- 4 proaches for converting from one unit to another are hence = 4 x 12 = 48 described on page 130. Since understanding of either 12 approach depends to a great degree on experience and common sense, many opportunities for the manipula- 12 units = 1 dozen; 12 or 12:1 tion of concrete materials should be provided.Four 1 one-foot ru;ers may be placed end to end and ways of detelmining the number of inches represeated in the 60 units = doun; 60 or 60: four feet discussed:

2 60 60 Inches may be counted, 48 hence = 7_7 .11 12 Repeeed addition may be used, 12 + 12 + 12 + 12 48 If ideas involved in c.onverting from one unit to an- othrr arc understood, there should be little need for Multiplication may be used, 12X A= 48 manipulation of concrete materials in developing algorithms for operations on numbers used in measuring. When the relationship is understood, children should Opportunity to experiment with examples like the fol- be able to generalize that multiplicationisused in lowing may he provided instead:

147 ADDITION:

1 /b. 12 oz. + 2 lb.5 oz. 3 lb. 17 oz. = (3 4 1) lb.I oz. Regrouping (17 oz. = Ilb.Ioz.)

= 4 lb.1 oz.

SUBTRACTION:

4 hr. 30 min. 3 hr. (60 + 30) min. = 3 hr. 90 min. (4 hr. 3 hr. 60 min.) 1 hr. 40 min. =- 1 hr. 40 min. = 1hr. 40 min. 2 hr. 50 min.

MULTIPLICATION:

8 yd.4 ft. 4

32 yd. 16 ft. --= (32 + 5) yd. 1 ft. Regrouping (16 ft. = 5 yd. 1 ft.) --= 37 yd. 1 ft.

DIVISION:

1yr. 7 mo. 4)6 yr. 4 mo. 4 yr. 2 yr. 4 mo. 28 mo. Regrowing (2 yr. = 24 mo.) 28 mo.

Estimation of Measurement opportunity for children to walk completely around many kinds of simple closed paths (painted lines on Intuitive understanding ofmeasures may be de- tennis,basketball,or shuffleboard courts) counting veloped through practice in estimating length, weight, steps as they go.Perimeter thus becomes associated height, or temperature, then measuring to check esti- with distance to be measured. mates. Children may step off distances, then measure length of each stride to form a basis for estimating The derivation of the word perimeter from Greek distance, They may use their own height and weight words for "around" and "measure" may be used to as basis for estimating height and weight of others. strengthen locos of perimeter. Objzets weighing one ounce, 8 ounces, and one pound may be handled to develop an intuitive idea for estimating weight. Sets of objects may be used to develop Polygon: As simple closed paths arcclassified by skillin estimating the number of discrete objects: shapes, polygons may be constructed from paper, number of seats in auditorium, number of students in strips of cardboard, pipe cleaners, string on pegboard, the cafeteria, number of windows in a building, number bent coat hangers, or other materials. As the perimeter of beans in a pound. of each type is measured, simple formulae may b: mitten and recorded in chart form. For example:

Perimeter regular hexagon Understanding of perimeter as distance along a P=s+s+s+s+s+s simple closed curve may be developed by providing P = 6 s

148 154 1,717ortgarre.,...

rectangle or parallelogram 2. Betsy's father is covering the tomato patch with mulchpaper. Must he know the areaor P=a+b+a+b perimeter of the patch? P 2a 21. P = 2 (a + b) 3. Jack is painting the dog house, Must he know the area or perimeter before buying paint? triangle s P =s+s+s or P=a+a+b or P=a+b+c 1. AREA OF POLYGONS To develop generalizations or formulae for finding areas of various polygons such activities as the follow- Circle: The perimeter of a circleis designated by a ing may be used: special name, circumference, which may be measured by using another measurement,thediameter. The relationship of these two measurements may be demonstrated by using string and a cardboard model of a Rectangular regions: circle. A detailed description of the manner in which a. Applying unit regions to various sized rectangu- this is done can be found in Measurement, page 132. lar regions and recording the process by listing the number of units per row, the number of This procedure may be repeated using many circles rows of units and the total number of units used of different size, demons rating that in each instance the circumference is approximately three times the length b. Counting in a given area of floor space the num- of the diameter. When children become familiar with ber of tiles in a row, the number of rows of operations on decimal fractions, a more precise quo- tiles and the total number of tiles used. tient, 3.14+ designated by the term pi or the Greek letter n, may be derived. c. Constructing a revealer as illustrated below to show row by row the measuring process. Since the circumference is approximately th-ee times the length of the diameter, a formula for finding the circumference of any circle may then be developed by using the diameter, C 3.14+ x D or C= n x D. Again, effort should not be made to have children memorize the formula but rather to be able to demonstrate a means of determining the circumference of any IF 1 circle if its diameter can be measured. Tab

Area 0 1 2 3 4 5 6 Understanding of area may be developed throuoi activities that apply unit regions of various sizes and shapes to flat surfaces. Desk tops or other rectangular regions may be covered with sheets of paper, rectangular unit regions, without overlapping. As the activity As the tab is pulled, a row of nnit regions appears is extended to include triangular regions to be meas- to illustrate1 row of 5 units.As the action is con- ured,itwill become evident that triangular shaped tinued, successive rows as marked along the horizontal units are more appropriate for covering these regions dimension of the region being measured are reached. without overlapping. 1 row of 5 units (1 x 5), 2 rows of 5 units (2 x 5)

. ..until the total region is covered, 6 rows of 5 units Discussion of situations like the following may help (6 x 5) or 30 units of measure. in development of understanding of the relationship of perimeter and area: d. Discussing the process as used in measuring 1. John's fatheris building a fence around the rectangufar spaces and stating the generaliza- yard. Must he know the area or perimeter of tion that the measure of the area of arec. the yard? tangle=tx worA=bxh.

149 155 Parallelogram region: circles with radii of the same measure, but a. Making paper models of parallelogram regions. cut into smaller and smaller sections b. Cutting parallelogram and reconstructing figure (4) Discussing that the smaller the size of the to form a rectangle as shown in illustration on section, the closer the approximation to page 133. the size of a parallelogram which has a c. Discussing the resulting figure and stating the bare approximately one half the circumference of thecircle generalization that the measure of the area of and height equal to the measurement of the radius of the parallelogram is length times width. A =1 x w or A = b x h. circle. c. Stating the generalizationthat the area of a

Trapezoidal region: circle is equal to the radius times 1 the circum- a. Making models of two congruent trapezoidal ference. The circumference is equal to 27,r. regions. A= 1 (2,rr) x r b. Suggesting that children arrange the two figures to produce a third figure with which they are A = x r familiar. (See illustration on page 134.) A = 742 c. Discussing the resulting figures and stating the generalization that the area of atrapezoidal region is equal to one half of the sum of the 2.AREA OF CLOSED SURFACES measure of the two bases times the measure of To develop formulae for finding the areas of closed the height. A = I (b1 b2) x h. surfaces, such activities as the following may be used:

Prism: Triangular region: a. Making models of two congruent triangular regions.

b. Suggesting that children arrange theseto form a familiar figure.(See illustration on page 134.) c. Discussing the resulting figures and stating the generalization that the area of a triangular'c- gion is equal to one half of the area ofa parallelogram region. A = b x h.

Circular region: a. Making models of a circular region and of a square with side measures twice the length of a. Open'ng corner scams of a large cardboard the radius of the circle; applying circle tosquare carton and spreading itflat as in illustration. as in diagram on page 134 to illustrate the relationship and to develop the generalization b. Measuring dimensions of each section of the box that the area of the circle is slightly less than c. Computing the area of each plane surface and the area of the square. developing a formula. b. Making model of a circular region: Area of A =Ix w A =B (1) Cutting it similar to illustration on page 135 Area of B=f x w C =D (2) Fitting sections together to form an ArcaofC=hXI E = F ap- Area of Dh x proximation of a parallelogram region AreaofE=hxw (3) Repeating the process in 2 above using AreaofF=hxw Total surface area =A+B+C+D+E+F = 2 (/ x w) + 2 (h x I) ;- 2 (h x w)OT = 2B + h x 2 (1 + w) or 2B + (h x p) where B = area of the bases and p = perimeter of the base.

Pyramid:Drawing flattened models of a regular pyramid with a square base as illustrated to the right and proceeding as for rectangular prism to develop formulae forfinding the area of the closed surface. AicaofA=sxs=s2

Area of B 1 (s x h) 2

Area of C = 1 (s x h) 2

Area of D =1 (s x h) 2

Area of E = 1 (s X h) 2

Area of closedsurface of regular pyramid B + 1 (h x p) where B = area of base and 2 p = perimeter of base.

Cylinder: Total surface area = R + A + B = (c X h) + 2 (r0) where c = circumference of base of cylinder

h Volume Understanding of volume may be developed through activities that fit unit space regions into space regions of various sizes and shapes. Blocks may be fitted into a box, or empty spools into a paper tube of the same 0 s diameter. a. Fitting paper around a can or circular box and If spools arc fittedinto arectangular box. it should cutting circles to match ends of cylinder become evident that space regions of appropriate na- turc must be used for accurate measurement. As under- b, Spreading paper and measuring lengthand standing of volume as measure of interior spaceis width of rectangle and radius of circles established. the ree.1 for use of standard units should c. Computing area of plane surfaces and develop- also be clear. ing formula. Discussion of such situations as the following nu) aid Area of R=c xh in further clarification of understanding of solurn : Area of A =x r2 A =B 1. Torn is ordering sand for his little sister's sand Area of B = rr X r2 box. Must he know volume. perimeter, or area?

151 157 2. in estimating water needed to fill the aquarium, illustrated on page 137 may be used to develop under- must the class know perimeter, volume, or area? standing of the relationship of prism and pyramid with 3. In estimating the amount of water in the water bases of the same measure. Models of prism andpyra- tank, must thecityengineer know volume, mids may be constructed.It can be demonstrated that perir,:cter, or surface area? the model of the pyramid filled three times willfill the prism.It should be evident that the volume ofa Rectangular prism:As the nature of units needed pyramid is equal to one third of the volume ofa prism to measure volume of rectangular prisms is discussed, with base and altitude of the same measuresor cubes of different sizes may be examined and dimen- Volume of a regular pyramid = 1 B h where sions for standard units discussed. 1 inch and 12 inch 3 cubes may be constructed from heavy cardboard. B represents the area of the base.

Right circular cylinder:By adapting ideas used in finding the volume of rectangularprisms,children should be able to visualize the volume ofa cylinder. Since the base is a circular region, the volume ofthe The bottom of a rectangular pasteboard boxmay be cylinder will be equal to the area ofa circle times the covered with unit regions to make a layer to cover the measurement of the height of the cylinder. V = 7,12 x h area of the base, and other layers added until the box is filled.If the box is slit along the edges of one side as shown in the illustration, the layers can be revealed. Right circular cone:When procedure for finding the It can be seen that the volume of the rectangular box volume of a cylinder is understood, the ideamay be or prism is equal to the area of the bottom layer times used to determine the volume of a cone by: the number of layers or a. Constructing a cone as illustrated on page 124 Volume of prism = (1 x w) x hor =BxhwhercB= b. Constructing a cylinder with the height and area of ',ase. diameter of the cone c. Using the cone to fill the cylinder. Regular pyramid: When procedure for finding the It will bt apparent that the volume ofa cone is one volume of a prism is understood, the ideamay be used to deters ine the volume of a pyramid. Construction third the volume of the cylinder. V = 1rr2 x PROBLEM SOLVING

As stated in the "Point of View," problem solving is D. The solutiontothemathematical modelis as much an approach to teaching application of mathe- 50,000.Interpreting this back into the physical matical content as itis the process of finding answers situation is shown by the sentence: 1 would travel to word or verbal problems. However, the solution of about 50,060 miles if I went around the earth word problems remains as a major goal at each grade twice along the equator. level of elementary mathematics.Solving word problems with applications to szience and other subject Illustrated helow are some practical suggestions that fieldsis of increasing importance. This makes itim- teachers have found helpful. Most of these suggestions perative that careful and systematic attention be given are designed to help with aspects of problem solving to problems in the mathematics program. described in A or B above.

Because problem solving is an important goal and 1. Children may make problems to fit a given num- because teachers often encounter difficulty in teaching ber sentence. This can help them to see the wide verbal problems. a special section of this guideis variety of situations that give riseto a single provided. mathematical model and it insures problems that relate to life experiences of the children involved. The essence of solving word problems may be Example: 5 + 3 = [1) may become a number summarized as follows: story about boys and girls at a party, apples and A. The words of verbal problems suggest some kind pears in a fruit bowl, or kittens and puppies at a pet show. ofaction--eitherphysicalmovementoran imagined operation. 2. Diagrams may be drawn to assist in formulation B. A mathematical model is set up to fit this action of the mathematical model. or imaginer; operation.Itis at this stage that

the decision is made on the mathematics needed x to fit the problem. x C. The mathematical problem is solved. x

D. The mathematical solutionisthen interpreted back into the problem situation described by the words. x x Example: At the equator, the distance around the Example: There are 28 pupils in the third grade. earth is about 25,000 miles. About how many miles In the room there are 3 tatles for 6 children each would you travelif ,ou went aroimd the earth twice and 2 tables for 4 children each. Are more tables along the equator? needed to give each pupil a space at a table? A. The action describedis"traveltwice around 6 + 6 + 6 + 4 + 4=26 the equator." (3 x 6) + (2 x 4)<28 Yes, 1 more table B, The leatner begins to search his repertoire of is needed to seat 2 children. mathematical models. He may think: "Around once; around twice; 25,000 miles each time" He 3. The problem may be dramatized to help pupils has chosen an addition model that fits the words. to visualize the action of the problem. The learner might think: "Around once; around 4. Children may dictate or write their own prob- twice; two times f went 25,000 miles." He may write: 2 x 15,000 = n. He has chosen a multi- lems, write appropriate number sentences, and il,en solve the sentences. This assures problems plication model that fits the word description. :hat are understood and within the difficulty level, C. Solving the mathrnatical problem: both language and mathematics, of each student. 25,000 + 25,000= n 2 X 25,000 = n 5. Teachers may make problems involving situa- 50,000 = n 50,000 = n tions of corent interest to the class.

153 ,159 6. Information giv.-n in the problem may be sum- 9. If problems dealing with large numbers, fractions, rnarized to emphasize the action suggested. or decimals present difficulties, they may be approached by: Example: Helen and Clara are stacking programs a. Substituting small whole numbers for numbers for the class play into sets of 20. If they have 180 in the problem programs, how many stacks should they have? b. Reading and solving the problem using the As possible summaries children may either whole numbers a. Draw pictures of the action c. Discussing the mathematical model used b. Put 180 counters into sets of 20 d. Using the same operation with the original c. Ask themselves questions: numbers. 180 programs make what number of stacks of 20? Example: The class had 15-2 dozen cookies for How many stacks of 20 are needed to make 3 I 180? a bake sale.If 1dozen were left at the end of 2 the sale, how many were sold? From the summary, a mathematical modelis easier to formulate into a number sentence: a. The class had 15 dozen cookies for a bake sale.If1 dozen were left at the end of the I3U = x 20 sale, how many were sold? El x 20 = 180 b. Think: 15 1 =nor 1 +n= 15 180 + 20 = c. Solve:

Finding the number that will make the equation 1 d. Think:15-2 1-1= n or 1+ 15? true will p'ovide solution to the problem. 3 2 2 3

7. The problem may be read, an estimation of the 10. Several number sentences related to a given prob- answtr made, and the problem reread to check lem may be written, children asked to select those the reasonableness of the answer. Such activity which may be used in the solution of the problem, helps inthe formulation of the mathematical and appropriateness of selections discussed. model to be used. Example: Jack earned 45each time he cleared Example:If Jack's father travels an average of snow from the front walk.If he shoveled snow 50 miles each hour on a trip, how many hours four times in January. how much did he earn that does it take him to drive 375 miles? Estimates month? might be made by thinking: a. 45 + 45 + 45 + 45 n 100 miles 2 hours; 300 miles-6 hours b. 45 x n = 4 375 miles almost 8 hours C. n = 45 x 4 When the problem is reread and the estimation d. 45 + 4 = n inserted, reasonableness may be discussed and checked by children's experiences with travel. 11. Number sentences may be written ter problems without attempting to solve them in order to 8. Problems containing information not necessary to focusattention on developing a mathematical the solution may be used to help children identify model that fits the problem. the true problem situation. Example: The grocer charges 700 a dozen for Example: 1 of the 150 pieces of mail delivered large eggs and 60t a dozen for small eggs. How 2 much does he get for 1 dozen large eggs and 3 by the postman in one city block were first class dozen small eggs? mail and I were second class. How many pieces 3 Physical models may be drawn, then mathematical of second class mail were delivered? models stated:

154 70 + (3 x 60) = n who contributed; 200 is the amount contrib- (3 x 60) + 70 = n uted by each pupil; n or is the amount of money for the party. 12. Problem situations described without using num- 25 x 20 = n bers may be used to focus attention on mathematical operations to be used. 500 = n There will be 500 cents or $5.(JJ to spend.) Example: If each member of c class contributes the same amount to a fund for a class party, how As a means of introducing new mathematical topics, much money will there be to spend? a practical problem may be presented for which students do not have an appropriate mathematical model Questions similar to the following may be asked or for which they may need a mare efficient model. to guide thinking: Such problems provide motivation for learning the topic and may give the learner a sense of realism in the study a. What do you know from reading the problem? of the topic. b. What are you asked to find? Po: example: As one way to introduce ratio and pro- c. Can you write a sentence that tells what can portion, the problem of increasing the number of cup the answer?(Number of be done to find cakes produced by a recipe may be presented.If the people multiplied by amount of money given recipe makes 2 dozen and 6 dozen are needed for a by each person is the amount of money to party, the relationship involved is 6 to 2 or a ratio of spend) . The recipe may be increasedinthe following d.If you insert numbers in the story, is the answer reasonable? (25 is the number of pupils manner:

Recipe for 2 dozen Proportion for 6 dozen cup cakes cup cakes

2 6 I c butter = 3 c butter I

2 6 2 csugar = 6 c sugar 2

2 6 4 eggs 12 eggs 4

2 6 I c milk = 3 c milk 1

2 6 3 cflour = 9 c flour 3

2 6 2 tbaking powder 6 I baking powder 2

2 6 1 1 tsalt = 1-tsalt 2 1 n 2 2

2 6 2 tflavoring = 6 t flavoring 2 fj Practical problems may be interwoven with mathe- cent inlife beyond the classroom. There should be matical topics being studied,particularly when the continuity' interplay between mathematics studied and mathematical top:c loses a sense of reality. verbal or practical problems.

For example: During the study of percent the fed- Certain actions o :cur often in practical problems. eral government announces an increase in the interest That actions relate to the four operationson numbers rate on Type F. savings bonds. This may be used not is not accidental. Summarized beloware actions that only to give reality to computation of interest but can lead to the formulation of given operationswith whole leadto greater understanding of application of per numbers.

Action Pro 'em Matheniatic,1 Model

Joiningor combining two sets with How many in the resulting set? Addition: numbers of each known 5 + 17 = n

Separatiaga set into two subsets How many in the other subset? Subtraction: with numbers of originalset and 17 5 n or one subset known 5 + n = 17

Comparing onesetwith another How many more in one set than in Subtraction: with numbers of each known the other? How many fewer in one 17 5 = n or set than the other? 5 + n = 17 Joining equivalentsetswiththe How many in the resulting set? Multiplication: number of sets and the members of 5 x 17 = n Or each set known Addition: 17 + 17 + 17 + 17 + 17 = n Separatinga set into equivalent sub- flow many sets? Division: sets with the number of the original 35 4 5 n zer3 set and of each subset known r = 35 remainder

How many sets and how many left 35 =in x 5or non -zero over? 37 = fn x 5) + rremainder Separating aset into equivalent sub- How many in each set? 35 4 5 = n zero sets with the number of the original 5 x n = 35 remainder setandthenumber ofsubsets known How many in each set and how 35 = 5 x n non-zero many left over? 38 = (5 x n) + rremainder

Pupiis must learn that to read word problemsin solving all sinrd problems, each must be carefully read. mathematics demands reading skill that differs from that used in other subject areas.Problems must be The assignment of pages of word problems fir pupils read carefully several times, each time fora different to do on their own is certain produce failure for reason. They must be read to: many. Alone they are unable to liormulate the mathematical model, the crucial step in problem 1. Get overview of the problem situation. solving. Many pupils proceed by trial anderror using each 2. Garner necessary facts. operation until they get a number that 'looksabout right." When this happens, little has been 3. Determine mathematical model. achiesed with the major goal of problem solving anda negative 4. Check the reasonableness of the solution ofthe attitude is often acquired.Pupils who are successful model. usually have experienced a carefully planned,sequentially develop( I program of instruction that 5. Interpret the solution in words of the problem. matched individual understanding with appropriateapplii:aticn. It is generally recognized that itis better for a pupil Children must realize that each problem is different to do fewer problems successfull!,than to attempt and that since no set pattern can be developedkir many and solve few.

156 162 STATEMENT OF OBJECTIVES

INTRODUCTION

The Statement of Objective.; outlines mathematical Organization of the Guide was determined by the knowledge and skills to be emphasized during the first beliefthatcertain broad baskideasunify mathe- eight years, grades K-7, of the elementary school. This matical content.Six major strandsSets and Num- Statement was designed as a guidein planning for bers,Numeration, Operations on Whole Numbers, sequential development of basic mathematical ideas. Rational Numbers, Geometry, and Measurementhave Objectives should assist teachers: been selected as having greatest significance in mathematics for the elementary school. Effort has been made In assessing competencies which each pupil brings to state objectives in a way that suggests interrelated- to a given grade ness of these ideas in a spiraling development of con- In planning experiences needed for further growth tent. in rnathcmatical understanding and skill Because mathematical ideas presentedinthesix Indevelopingunderstandingofmore advanced strands are closely interrelated,itis possible that an mathematics which children will encounter. idea may develop spirallythrough more than one strand.For example, manipulation of concrete and Effort has been made to express objectives in terms of semi-concrete materials basic to understanding opera- what children might be able to do after a year's ex- tions on whole numbers, may appear in objectives for perience in a given grade. The Statement is not in- Sets and Numbers and Numeration. This should be tended to provide a set of n:nimum essential! for a kept in mind as this section of the Guide is used. child at any Grade level. 'I he mr rarer in which mathematical ideas arc intro- Grade placement of contentasoutlinedissug- duce] and developed in the Guide should be carefully gestive. Objectives may need to be restated in terms of noted. Understanding which children may have gained iiditions existing inindividual schools.Special at- in pie-school years is used in expanding ideas of num- tention should be given to objectives for primary grades ber and space. Work with concrete and semi-concrete in schools where kindergartenis not a part of the materials develops these ideas in greater depth before system.Grade placement of content must be inter- symbols are used, generalizations stated, or operations preted and used by every teacher in light of what in- with numbers forn alized, dividual children know and are able to do at a given time. Mathematical content must be organized and pre- sentedina sequence which enables each childto A characteristic of the study of mathematics is the develop understanding of basic ideas.Each teacher spiraling of concepts from grade to grade over a period must be knowledgeable concerning content presented of years. For example, multiplication begins ir. the first at all levels in order to insure maintenance of skills and grade with the manipulation of equivilent sets of con- understanding and to plan next steps in learning. No crete objects, continues through the set of whole num- gaps may be allowed in gradual build up and expansion bers and expands toincludethesetof fractional ifthe school isto provide a complete program for numbers and the set of rational numbers. each

157 163 OBJECTIVES

KINDERGARTEN FIRST GRADE

Sets and Numbers

Identify two equivalent sets by placing the members Recognize thatthe equality symbol means thatthe of the sets in one-to-one correspondence. same number is named on either side of the symbol.

Recognize the common propertyofallsetsas Use 0 as the cardinal number of the empty set. number.

Name the number of members in sets with 1-9mem- Ord,.r sets of objccts with numbers to ten. bers by counting.

Order numbers using concept of "one more than." and one less than," and "equal to."

Recognize and use number words for ordinals through five. (number five and fifth)

Count to 100 by ones and tens, to 20 by twos, to 50 by fives.

Select 1-9 objects from a given !et using oral words Recognize and use number words through 10. and written numerals.

Compare sets of objects using such terms as more Compare two numbers usingis grate, than," is than, as many as, fewer than. less than," or "is equal toand use symbols ">," 44.<1111 4,...,1

Recognize and d:rronstrate one half andone fourth of a set of objects.

Join two sets to make a third set; separatea given set into two sets.

Numeration

Recognize and name the symbols 1 through 9. Group sets of objects using the base of ten andname the number ar, so many sets of ten andso niany ones.

Use place value notation for numbers through 99to show the number of tens and ones.

Read and ssrite numerals 0 through 99.

Read and write number words to ten,

Recognize that a number has mar.)names. (3 = 2 + 1 = 3 + 0 = 4-- 1)

158 i4 .E.SEIFI'Solnrk

SECOND GRADE THIRD GRADE

Sets and Numbers

Know odd and even numbers.

Use ordinal numbers through ten. (Number ten aml Use ordinal numbers beyond ten. tenth)

Count to 1000 by tens and hundreds. Use whole numbers to ten thousand.

Recognize and demonstrate one half, one fourth, one Recognize and demonstrate one half, one fourth, three third, using sets of objects. fourths, one third, and two thirds, using sets of objects.

Numeration

Read and write three-place numerals using place Read and write four-place numerals using place value, value notation to show the number of hundreds, tens. expanded notation, and ....crd names. and ones.

Count, read. and write numerals from 0 to 300. Read and write decimal notations for money.

Reau and ssrite many names for numbers. Example 5 + 3 = 8; 10 2 = 8; 4 + 4 = 8. Read and write Roman numerals through X.

159

18J KINDERGARTEN FIRST GRADE

Operations on Whole Numbers

Rearrange sets of objects to demonstrate the join- Know and use addition and subtraction facts forsums ing and separating of sets to develop a readiness for through 10. addition and subtraction. Explore the facts with sums through 18.

Recognize the commutative property of addition without naming it.

Add using three addends with sums through ten.

Use intuitively the associative property of addition.

Use vertical and horizontal forms for addition,

Use the symbols +, --, and = to form numbersen- tences. Example: 3 + 6 = ; 6 3 =

Explore addition of numbers named by two digits without renaming.

Demonstrate with sets of objects the inverse relationship between addition and subtraction. 4 + 2 = 6; 6 2 = 4; 6 4 = 2

Use multiplication and division intuitively without sytri bolization with products to ten.

11G1 SECOND GRADE THIRD GRADE

Operation on Whole Numbers

Know and use addition and subtraction factswith sums through 18.

Recognize that subtraction is not commutative.

Recognize and use the associative property of addition without naming it.Example: 74 (3 + 5) = (7 + 3) + 5.

Add and subtract numbers named by two digits Add and subtract numbers each named by two orthree without renaming. digits without renaming. Add numbers named by two digits. renaming as Add numbers named by two and three digitswith necessary. renaming.

Use the inverse relationship of subtraction and addi- Subtract numbers each named with two or three digits tion to derive subtraction facts from the addition with renaming. facts.

Check subtraction by addition.

Use odd and even numbers in developinz and check- Use odd and even numbers in developing ideas of ing addition and subtraction. multiplication and division and checking products and quotients.

Demonstrate m1.1tiptication through joining sets of objects, repeated addition, and rectangular arrays.

Know and use multiplication facts with products Know and use multiplication facts with one factor 0-6, Explore facts with both through ten. and the other factor 0-9. factors 7-9.

Use multiplication with one factor a number named by one digit and the other factor a numbernamed by either two digits or three digits.

161 167 KINDERGARTEN FIRST GRADE

Recognize the number of 2's, 3's, 4's, and 5's in multiples of these numbers to 10.

Use numbers with understam'ing in stories, games, and related activities.

Rational Numbers

Recognize that single unit objects can be split into Recognize and demonstrate 1arid of physical units two equal pieces, and that each piece is called one 2 4 and of sets of objects. half of the object.

Recognize informally that sets of 2, 4, 6, and 8 ob- Assign the number name 1to each of twoN equal sired jects can be separated .into two equivalent subsets 2 as readiness for understanding that each subset is pieces of a physical unit and to each of No equivalent one half of the original set. sublets of a set of objects.

162 168 SECOND GRADE TIIIIRD GRADE

Know and us,: the proprties of zero and one in mul i- plication.

Recognize theroleof thedistribt tive propertyof multiplication over addition and its relationship to the algorithm. Example:

3 x (2+ 4) = (3 >:2)+ (3 x 4) =6 + 12 = 13

4 x 23= 4 x (20 -f3)= (4 x 20)+ (4 x 3) = 80 + 12 =92

23 23 x4 x4 12 92 80 92

Recognize inverse relationship of multiplication ar d division through use of multiplication :acts.

Recognize that multiplication is commutative. Recognize that division is not commutative.

Demonstrate divisionthrough partitioning sets of Use long division algcrithm with divisor a number objects, repeated subtraction, and rectangular arrays. named by one digit (1-6) with and without remainders.

Apply addition and subtraction to problems involving Apply addition and subtraction to problems involvirg money using cents only. money, including dollar; and cents.

Rational 'Numbers

Recognize and demonstrate 1 , 1 , and of aphysi- Recognize the cater relationship (greater than,leis 3 4 than) for fractions of eighths, sixths, fifths, fourth,, calunit, a set of objects, a region, and a line seg- thirds, and halves. ment. Recognize that there is a point on the number line fcr every fractional number.

Recognize that there ate many fractions that name the same fractional number.

Recognize the inverse relationship of addition and sut tra'rt of fractional ninbcrs.

Find one half of each even number from 2 through Find one half of each (Nen number from 2 through 20. 10.

Recognize that a fracion names an ordered pair ig numbers, as well as a fractional number.

163 169 KINDERGARTEN FIRST GRADE

Geometry

Recognize and name squares, rectangles, circles,and Recognize that squares, rectangles, triangles, triangles. and circles are closed paths. Tell whether a given point isinside, outside, or on such a path.

Relate terms inside, outside, andon as they apply Recognize some distinguishing featuresof circles, rec- to squares, rectangles, circles, and triangles. tangles, triangles, and squares. Usesome related terms such as round corner,square corner, paths that are curved, and paths that are straight.

Recognize physical models and drawingsof points, Recognize and name a line with the understandingthat line segments, and portions ofa plane (flat surface). it means a straight path and that itextends indefinitely in both directions.

Identify objects shaped likea ball, box, or can. Identify objects shaped like a ball, a box,a can, an egg, a cone.

Measurement

Compare objects Orough useof such words as Use non-standard units of linear longer-shorter, hea der-lighter, higher measure and liquid - lower, larger- measure, such as a pencil for length and paper smaller. cup for liquid measure, to illustrate the necessity forstat.dard units.

164 17 0 SECOND GRADE THIRD GRADE

Interpret simple ratio situations, such as 3 pencils for 10 cents. as an ordered pair or fraction.

Geometry Distinguish between open and closed curves and inside and outside of simple closed curves.

Recognize and use properties of simple geometric Recognizebasicpropertiesoftriangles,rectangles, figures such as square corners, opposite sides, and squares, and circles without use of formal definitions. roundness.

Recognize that a rectangular sheet of paper can be folded into two or more parts of the same size and shape.

Recognize and use the words: point, line segment, Recognize a point as a location in space as distinct and line to describe geometric ideas. from its physical representation.

Recognize a tine as a set of points that continues on indefinitely.

Recognize a line segment as composed of the two endpoints and all the points between them.

Recognize that thereisonly one line through two points.

Recognize that two different lines can intersect at only one point.

Recognize and name rays and angles.

Recognize symmetry with respect to a line by using such activities as simple paper folding,

Recognize geometric figures and objects which have Ree.,gnize without formal terminology distinguishing the same size and shape. features of spheres, rectangular prisms (boxes), and cylinders,

Measurement Recognize proper instruments for measuring different Recognize the necessity for standard units of measure. things, such as ruler, clock, cup, calendar, and scales, as readiness for selecting units of measure. Recognize that the measure of a line segment is the number obtained by comparing the line segment to a selected unit line segment.

165 171 KiNDFFIGARILN MIST GRADE

Determine length of line segments to the nearest inch using a ruler marked only in inches.

Explore liquid measures using cops, pints. and quarts.

Read and use a calendar for days and weeks.

Tell time to the nearest half-hour.

Identify pennies, nickels. and dimes. Recognize the compatative values of pennies, nickels, and dimes and use them in making change.

Recognize that there is a point on a line for each whole number.

172

166 ,rttflte

SECOND GRADE T111RD GRADE

Use a ruler marked in half-inch, inch, and foot units Usc standard units: inches. tut, and yards in measur- to measure line segments. ing line segments; cups, pints, quarts, and gallons in measuring liquid,.

Measure liquids by cups, pints, quarts, and gallons.

Read and use a calcndarfordays, weeks, and Use year as mcasurc of time.

Tell time to the nearest quarter-hour. Tell time to the nearest minute and use notation such as 3:15 P.M.

Measure weight in pounds. Use ounces to measure weight.

Measure temperature on the Fahrenheit scale to nearest degree.

Find the perimeter of triangles and rectangles by adding the measures of the sides.

Recognize that a measurement of physical objects is always an approximate number.

Make change up to a quarter. Make change up to $1.00.

Read and interpret simple information given in tables.

Recognize that there is a point on a line for each fractional number.

167 OBJECTIVES

FOURTH GRADE EIFT11 GRADE

Sets and Numbers

Recognize and use sets of numbers, including count- Identify the set of mime numbers 2, 3, 5, 7, 11, ing numbers, whole numbers. even numbers, and odd numbers.

Find the prime factors of counting numbers through 100.

Identify the set of composite numbers 4, 6, 8, 9. 10, 12, ...

Determine the least common multiple of twocounting numbers by listing the maltiples of the two numbers and by prime factorization.

Determine the greatest common factor of the counting numbers by listing factors of both numbers and by prime factorization.

Recognize sets of fractional numbers and equivalent Identify the set of fractional numbers fractional numbers.

Construct sets of equivalent fraclionc.

Use set notation of union andintersection.

Numeration

Read and write seven place numerals as needed. Read and write decimal; utilizing extensio

value to two places to the right of the one's,

Use exponents2,3, or 4,Example: Ir 31 = 27; 4000 = 4 x 1000 = 4 x 10x 10 x 10 = 4 x 103

Use Roman numerals through one hundred with Read and write Roman numerals. emphasis on base of ten and the use of a subtractive Principle in a numeration system.Examples: Ten ones make one X, ten tens make one C, ten hun-

168

174 S/ XT11 GRADE SEVEN-ID GRADE

Sets and Numbers Use set of negative integers in practical situations Use the entire set of rational numbers. such as naming temperatures above and below zero.

Recognize that there are numbers called irrational numbers that ,re rot rational.

Use solution sets in equations and inequalities.

Use fractional numbers including decimal representation.

Jse set terminology in referring to sets of point.; in geometry.

Numeration Understand the role of place value in a numeration system through the study of non-positional systems such as the Egyptian,

Use expanded notation with exponents.Example: Express numbers by using scientificnotations.Ex- 4526 =: (4 x 103) + (5 x 103) + (2 X 101) + ample:The distance from the earthtothe sun, (6 X100) 93,000,000 miles, is 9.3 X 107 miles.

Represer,t fractional numbers by decimals and fractions.

169 FOURTH GRADE FIFTH GRADE

dreds make one M, IV = one less than V and XL = ten less than fifty.

Read and write numerals in a base other than 10.

Operations011Whole Numbers Add and subtract numbers named by four or more digit numerals without renaming.

Rename as necessary when using numbers named by four or more digits for addition and siibtraction

Add and subtract numbers applied to vari, us types Add and subtract numbers from measurement and of measures without renaming. money, renaming as needed.

Know and use multiplication facts with factors 0 through 10.

Multiply with one factor a number named by two Use multiplication with factors named by thicc and digits and tne other factor a number named by four digits. three digits.

Use the terms associative and commutative propzr- ties of addition and multiplication after understand. ing the ideas.

Use, the term distributivepropertyafterunder- standing the idea.

Use division algorithm with divisors named by two Use division with divisor named by two digits. digits ending in 1. 2, 3. or 4 with and without remainders.

Rational Numbers Determine order relationship for fractional numbers.

Find an equivalent fraction for agiven fraction through use of models.

1 23 Recognize that fractions such as , .. , are I 2 3 names for one

Know and uke decimals to thousand ,

170 176 SIXTH GRADE SEVENTH GRADE

Operations on Whole Numbers

Use simple addition and subtraction in bases other than ten.

Multiply and divide with numbers from measure- Use simple multiplication in bases other than ten. ment and money. renaming when needed.

Use division with divisors named by three digits. Use division with divisors named by four or more digits.

Find square roots of numbers represented by two and three-digit numerals.

Rational Numbers Use order relations >, < with integers and rat' .snal ',umbers.

Recognize that thereisafrictional number be- Recognize the idea that the integers are not dense, but tween every two fractional numbers on the number that rational numbers are dense. fine.

Demonstrate an undersianding of decimals as names Recognize theadditiv::inverse property that every for certain fractional numbers. fractional number has an opposite.

1 Know decimal equivalent krIra, lionssuchas . 2

1 3 1 2 1 1

4'43 3 5' 8

171 77 FOURTH GRADE FIFTH GRADE

Add and subtract fractional numbers with same Add and subtract fractional numbers with unlike de- denominator. nominators.

Add and subtract decimals to thousandths.

Recognize that subtraction is not commutative in the set of fractional numbers.

Know and use the commutative and associativeprop- erties for addition in the sot of fractional numbers.

Find a fractional part of a set of objects. Multiply fractional numbers.

Write mathematical sentences for finding a frac- I Know and use in the form 1, 2, 3 . . . ,as the tional part of a set of objects. I 2 3 identity element for multiplication of fractionalnumbers.

Use diagrams to show that multiplying or dividing Know and use the commutative and associativeprop- the numerator and denominator of a fraction by the erties for multiplication in the set of fractional numbers. same number results in an equivalent fraction.

Find the product when the first factorisrepre- Know and use the distributive property of multiplica- sented by a whole number and the second factor by tion over addition on the set of fractional numbers. a fractional number. Use repeated addition.

Express the quotient of whole numbers as a fraction or a mixed numeral.

Use ratios to represent situations such as 3 pencils Find the missing term in a proportion. for 10 cents as3 .10 Determine it two ratios are equivalent in same manner as determining if two fractions are equivalent. Find the missing term in two given equivalent fractions or ratios using common denominators.

172 178 SIXTH GRADE SEVENTH GRADE

Add and subtract fractional numbers named by decimals.

Recognize that subtraction is always possible in the set of integers.

Recognize that subtractionis always possible in the set of rational numbers.

Recognize thatallproperties of fractional numbers also hold true for iational numbers.

Recognize the inverse relationship of multiplication and division of fractional numbers.

Explore the four operations in the st t of integers and the set of rational numbers.

Recognize the reciprocal (multiplicative inverse) for every fractional number except zero and use it in division of fractional numbers.

Recognize that division is always possible in the :,et of fractional numbers.

Divide using set of fractional numbers.

Multiply and divide decimal fractional numbers.

Determine by division whether any fractional num- Demonstrate understanding of fractions and decimals ber has a repeating or terminating decimal equiva- that name the same fractional number. lent.

Interpret percent as a ratioin tvItichthe second Treat percentage problems as proportions inwhich number is always 100. one of the equivalent ratios always has 100 as the denominator. a b 100

173 .179 OURTH GRADE FIFTH GRADE

Geometry

Recognize that a quadrilateral may have pairs Recognize and name quadrilaterals. of parallel sides, one pair of parallel sides. or no pairs of parallel sides.

Recognize that the 3 sides of a triangle may have Construct an isosceles triangle and an equilateral tri- the same measure, 2 sides may have the same meas- angle. ure, or all three have different measures.

Construct the perpendicular bisector of a line segment.

Recognize without formal definitionthat a plane Recognize thata plane is determined by three points region is a closed path and the part of the plane not all 011 one line. which is inside the closed path.

Know and use the concept o; a circle as the set of Construct a circle given the radius. all points in a plane that are the same distance from a fixed point on the interior. Know relationship between the length of radius and diamPter.

Recognize the possibilities of the intersection of a line and a plane (one point, no pointsthe c,npty sct, line contained in Cie plane).

Recognize parallel planes as planes which do not intersect.

Recognize that when two planes intersect. the intersection is a line.

Recognize that the intersection of a plane and a sphere is a circle, in some cases a great circle, or a singie point.

Construct a line segment of the same length as a Know and use symbols for linc,line segment,ray, given line segment by use of compass and straight- angle,parallellines,perpendicularlines.and con- edge. gruence.

Recognize and name parallel lines as lines in a plane v, do not intersect.

Rugnizc a ray as starting at a point of origin and exn. riding in one direction indefinitely.

Recognize an angle as a figure formed by two rays having a common endpoint.

Know. use, and construct a rightangle,

Use model of a right angle to construct perpendicular lines, right triangle. square, and rectangle.

174 180 SEVENTH GRADE SIXTH GRADE

Geometry Recognize that two polygons havingthe same shape Recognize pentagons, hexagons, octagons. are similar and thattheir corresponding sidesarc proportional. Use informal methods to establishdeductively the Recognize an ellipse. properties of geometric figures.

Recognize that an arc of a circleis a part of the circle.

Recognize the subset relationshipbetween sets of points, such as point to line.line to plane, and plane to space. Recognize that there is a pointbetween any two points on a tine.

Know and use geometric symbolsin writing simple mathematical sentences.

Construct an angle of the same measureas a given Construct the bisector of an angle. angle.

Constt Jet parallel lines.

175 FOURTH GRADE FIFTH GRADE

Interpret, informally,space as the set of all points. Recognize the characteristics ofpyramids and cones. Recognize the characteristics ofspheres, rectangular prisms, and cylincers. Recognize without formal definitionthat a solid region is a closed surface and thepart of space which is inside the solid region.

Measurement

Use the metric system oflinear measure.

Determine distanci: to the nearest quarter-inch and Measure distance to nearest eighth-inch. recognize the mileas a unit of distance. Use odometer for measuringdistances.

Use measures of teaspoonand tablespoon. Recognize the drymeasure of pint, quart, peck, and bushel. Use dry measures of pint,quart, peck, and bushel.

Recognize the relationshipsof standard units for liquid measures.

Recognize the relat onships ofstandard units for dry measures.

Recognize freezing and boilingpoints.

Recognize the secondas a unit of time. Use decade Recognize standard timezones. and century as meat tres oftime.

Use ton to measure weight. Recognize that other standardunits of measure are used in special occupations.

Measure perimeter of quadrilaterals. Find the perimeter ofa polygon by addition.

Use a formula to find theperimeter of regular triangles and quadrilaterals.

Approximate therelatio- ship between thecircum- ference and diameter ofa circle as 3.

Recognize the degree as thestandard unit of measure for angles.

Recognize that a right anglehas the measure of 90°

Measure an angle by usinga protractor.

Recognize degree asa unit of measure for an arc of a circle.

176 182 SIXTH GRADE SEVENTH GRADE

measurement Recognize and use metric units of measure for weight and volume.

Recognize the light year as a unit of measure for space distances.

Estimate measurements of objects using appropriate units.

Measure temperature using the Celsius scale (centigrade).

Find perimeters of regular polygons by formula. Demonstrate the Pythagorean theorem by using models of 3-4-5 and 5 -12 -13 right triangles.

Find the circumference of a circle by formula.

Determine through paper folding or measurement that the sum of the measures of the angles of a triangle is 180% a straight line.

B3 FOURTH GRADE FIFTH GRADE

Measurement

Recognize that a plane region is measured by com- Develop and use a formula to find the area of a rec- paring the plane region to a selected unit plane tangular region. region to obtain a number. Find surface areas of closed space regions. Find area of plane region by covering the plane region w;th chosen unit plane region.

Recognize informally the concept of precision.

Recognize that a point in a plane (first quadrant) Use ordered pairs of numbers to locate points on a can be located by an ordered pair of numbers, plane (first quadrant).

Read maps. Understand scale drawings.

Recognize latitude and longitude as a means of locat- ing a point on the surface of a sphere, a globe

Perform the four arithmetic operations using numbers from measurements.

18(1

178 SIXTH GRADE . SEVENTH GRADE

Measurement

Find the area of a circular region. Compare areas of polygons having perimeters of the same measure.

Approximate areas of irregular and curved plane regions by use of a grid.

Recognize that a space region is measured by coin - paring the space region to a selectee unit space region to obtain a number.

Find volume of space region by filling the space Find the volume of a rectangular prism by experiment region. and by formula.

Recognize that measures may be determined by direct and indirect measurement.

Determine which of two given measuresis more Recognizethatthegreatestpossibleerrorof any

precise. 1 measure is -- the unit used. 2

Find average (mean). Find mean, median and mode of a set of Lumbers.

Recognize that thereisa point on a line for each integer.

Recognize that there is a point on a line for each rational number.

Locate points in a plane by an ordered pair of numbers (all four quadra..ts?.

Make scale drawings.

Use latitude and longitude to locate a point on the surface of sphere.

185

179 SYMBOLS USED IN GUIDE

+ plus, add x4 exponent

minus, subtract b2 subscript x, multiply (a, b) ordered pair

divide a fraction, indicated division b

square root 12,,,e base five

=--: ;s exactly the same as, is equal to % per cent

74,is not exactly the same as, is not equal to +5 positive five

> is greater than 5 negative five

AB line segment, chord < is less than 4-* AB line is similar to -4 AB ray R. is approximately the same as I angle is congruent to C) circle ... continued in this manner _L is perpendicular to O empty set, null set is parallel to U union AB arc nintersection 1rpi, 3.1415+

() parentheses 'degree

() braces " inch, inches, second

() brackets foot, feet, minute

.115 bar, repeated digits :is to

180

181; INDEX 31, 33, 35, Equator 140 Addend 30 Commutative property 38, 65, 81 Error 127 Addition 30 chart 31 itchle Estimate :14, 138 decimals 75 right circular measurement 131 154 fractional numbers 57, 72 Congruent 1 )8 problem solving 68 108 integers angle Expanded notation 32 repeated 35 line segment 108 whole numbers 30 plane figure 108 Exponent 22 31 110, 113 Algorism, a'gorithm Consecutive 57 Exterior Altitude 133 Co.iversion 130 Face 114, 135 Angle 103 139 Coordinate, rectangular Factor, greatest common 14 right 103 8 acute 111 Counting Factorization, prime 13 obtuse 112 numbers 11 Fae.tors 35 Approximate 127 Cube 136 111 104 Focus (foci) Arc 110 Curve closed 104, 110 Fahrenheit 128 Area 132 simple closed 104 circle 134 Fraction 53 62 closed surface 135 Cylinder complex 54 lateral 136 right circular 114, 136, 137 equivalent 133 parallelogram Fractional numbers 52 polygon 133 Data 142 negative 72 rectangle 133 trapezoid 134 Decimals 73 Frequency distribution 142-143 triangle. 134 74 repeating Fundamental theorem of terminating 74 Associative property 31, 33, 35, arithmetic 13 38, 65, 81 Denominatorom 53 Geometry 102 Average 143 Density 57 Gram 129 Diameter 10 Base centi- numeration system 20 20 deca- plane regions 133 deci- 126 space regions 135 Direct measure bed )- 30 Distributive property 35, 38, 65, 81 kilo- Binary milli- 118 Dividend 37 Bisect Gra& 141 37 141 Braces 6 Division bar decimals 76 circle 142 fractional numbers 61, 73 /43 Cardinal number 9 histogram integers 71 line 141 Celsius (centigrade) 128 measurement 33 picto- 141 partitive 38 CenterCen ' 110 37 whole numbers Hexagon 113 Central tendency 143 37 Divisor Histogram 143 Chord 110 31,35,65,81 Circle 110 Element Identity elenrnt 114 um 110 cylinder Indirect measure 126, 138 area 134 set 6 center 110 Inequality chord 110 111 (see Set non-equivalent) circumference 132 102 110 Endpoint Integers 12, 66 66 132 English sy stem of measure 128-129 negative Circumference positive 66 Equation 1 Closure property 11, 33, 35, 110 113 38, 65, 81 (set Mathematical sentence) Interior

181 1,87 Intersection 104 Multiplication 35 Pattern 31, 60 line 104 chart 36 Path, closed 104 plane 105 decimals 75 point of 104, 106 fractional numbers 59, 73 Pentagon 112 integers 70 68 Inverse whole numbers 35 Percent 78 additive 30, 68, 81 element 65 Perimeter 131 Negative 12 multiplicative 65 Perpendicular operation 30, 33, 37, 62 Number 7 line 106 cardinal 9 plane 107 complex 15 Lateral area composite 13 Pi 132 136 cylinder counting 11 Place value 21 prism 135 even 13 pyramid 135 fractional 12 Plane 103 Latitude 140 imaginary 15 figure 110 integer 12 region 113 Leg 114 irrational 12 Point 102 natural 11 Line 102 negative 12 intersection 104 Polygon 111 odd 13 symmetry 109 area 133 ordinal 9 perimeter 132 Line segment 102 prime 13 regular 111 property 7 Liter 129 rational 12,52 Power 23 centi- real 13 deca- Precision 127 theory 13 deci- whole 11 Prism 114,136 hecto- kilo- Number line 43,55.67 Problem solving 153 milli- Number sentence 1 Product 35 Longitude 140 (see Mathematical sentence) Properties Numerals 20 addition 31, 65, 81 Mathematical model 156 division 38, 65, 81 Numeration system 20 multiplication 35. 65, 81 (see Mathematical sentence) decimal 20 one 54 Egyptian Mathematical sentence 8 24 subtraction 33, 65, 81 Hindu-Arabic 20 Mean, arithmetic 143 non-decimal 23 Proportion 78 Roman 24 Measure 125 Pyramid 114, 137 Numerator 53 Measurement 125 Pythagorean theorem 12 systems 128 types 128 Octagon 113 Quadrant 141 One, properties 54 Median . 144 Quadrilateral 112 One-to-one correspondence 6 Men.ber of set 6 Quotient 37 Open sentence 8

Meridian, prime . 140 Operations 30 Meter 129 Radius 110 cent.- Opposites 68 Ratio 77 deca- Order 8 deci- fractional numbers 56 Ray 103 hecto- integers 67 kilo- Reciprocal 62.81 miili- Ordered pair 52, 77 Rectangle 112 Ordinal 9 Metric system of measure 128-129 Rectangular region 60-61

Mode 144 Parallel Region line 107 solid 114 Multiple 35 plane 107 space 114

Multiple, least common 14 Parallelogram 1 1 2 Relatively prime 56

182 188 Remainder 39 Statistics 142 Unary 41 Scale drawing 139 Subtraction 33 Union decimals 75 plane region 113 Set 6 fractional numbers 59, 72 sets 10 disjoint 9 integers 69 space region 114 element 6 repeated 37 125 empty 9 whole numbers 33 Unit of measure equal 9 angular 128 equivalent 7 Sum. 30 a va 128 10 counting 128 finite Su. face 113 infinite 10 dry 128 closed 113 length 128 intersection 11 lateral 114 member 6 liquid 128 non-equivalent 7 Symbols 20, 180 standard 126 separation 11 temperature 128 subset 10 Symmetry 109 time 128 union 10 axis 109 volume 128 line 109 weight 128 Similarity 108 plane 110 Variable 8 Skew 107 Trapezoid 112 Solution 8 Vertex 103, 114 Triangle 111 Space acute 112 Volume 136 figure 113 equiangular 111 rectangular prism 136 region 114 equilateral 111 regular pyramid 137 Sphere 113 isosceles 111 right circular cone 138 obtuse 112 right circular cylinder 137 Square 112 right 112 Square root 40 scalene 111 Zero 9,22,33,35,38,65,68,81

183 189